Technische
Universität
München
Fakultät
für
Physik
Walther-MeißnerInstitut für
Tieftemperaturforschung
Time-domain control
of
light-matter interaction
with
superconducting circuits
Diploma Thesis
Thomas Losinger
Advisor: Prof. Dr. Rudolf Gross
Garching, 2012-11-07
Bayerische
Akademie der
Wissenschaften
Time-domain control of light-matter interaction with superconducting circuits
Contents
List of Figures
iii
List of Tables
v
List of Symbols and Abbreviations
vii
1 Introduction and Motivation
2 Theoretical background
2.1 Josephson physics . . . . . . . . . . . .
2.1.1 Josephson junction . . . . . . .
2.1.2 RCSJ model . . . . . . . . . . .
2.1.3 3 Josephson junction flux qubit
2.2 Quantum harmonic oscillator . . . . .
2.3 Jaynes-Cummings model . . . . . . . .
2.4 Dynamic and decoherence . . . . . . .
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5
5
6
7
9
13
18
21
3 Experimental setup
3.1 Cryogenic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1.1 Superconducting niobium resonator and antenna . . .
3.1.1.2 3 Josephson junction flux qubit and coupling junction
3.1.2 Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2.1 Microwave input lines . . . . . . . . . . . . . . . . . .
3.1.2.2 Microwave output line . . . . . . . . . . . . . . . . . .
3.1.2.3 DC lines . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Room temperature setup . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Continuous wave spectroscopy . . . . . . . . . . . . . . . . . . .
3.2.1.1 Single tone continuous wave spectroscopy . . . . . . .
3.2.1.2 Two-tone continuous wave spectroscopy . . . . . . . .
3.2.2 Pulsed wave spectroscopy and time-domain measurements . . .
3.2.2.1 Pulse generation . . . . . . . . . . . . . . . . . . . . .
3.2.2.2 Pulse detection . . . . . . . . . . . . . . . . . . . . . .
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23
23
23
23
25
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29
29
31
31
31
32
33
34
35
4 Measurement results
4.1 Continuous wave spectroscopy . . . . . . . . . . .
4.1.1 Flux calibration . . . . . . . . . . . . . . .
4.1.2 High power continuous wave spectroscopy
4.1.3 Photon number calibration . . . . . . . .
4.1.4 Low power continuous wave spectroscopy
4.2 Time-domain measurements . . . . . . . . . . . .
4.2.1 ACQIRIS card measurements . . . . . . .
4.2.1.1 Pulsed two-tone spectroscopy . .
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47
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62
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i
Time-domain control of light-matter interaction with superconducting circuits
4.2.2
4.2.1.2 Rabi oscillation measurements . . . . . . . . . . . . . . . . . . .
FPGA board measurements . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2.1 Rabi oscillation measurements . . . . . . . . . . . . . . . . . . .
64
68
69
5 Conclusion and Outlook
77
6 Acknowledgments
79
Bibliography
81
A Digital heterodyne IQ mixer calibration
A.1 Mathematical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 MATLAB code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
85
94
B Photon number calibration
101
C Persönliche Erklärung
105
ii
Time-domain control of light-matter interaction with superconducting circuits
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
Josephson junction . . . . . . . . . . . . . . . . . . . . .
RCSJ model of a Josephson junction . . . . . . . . . . .
Tilted washboard potential . . . . . . . . . . . . . . . .
3 Josephson junction flux qubit . . . . . . . . . . . . . .
Potential of a 3 Josephson junction flux qubit . . . . . .
Qubit hyperbola . . . . . . . . . . . . . . . . . . . . . .
Energy levels of a multi-mode resonator . . . . . . . . .
Lumped element circuit for a superconducting resonator
Energy levels of the Jaynes-Cummings Hamiltonian . . .
The Bloch sphere . . . . . . . . . . . . . . . . . . . . . .
The π-pulse pattern . . . . . . . . . . . . . . . . . . . .
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6
7
8
9
11
12
14
16
20
21
22
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
The chip design . . . . . . . . . . . . . . . . . . . . . . . .
Shadow evaporation techique . . . . . . . . . . . . . . . .
Schematic of the experimental setup . . . . . . . . . . . .
Photograph of the cryogenic stage . . . . . . . . . . . . . .
Continuous wave spectroscopy setup . . . . . . . . . . . .
Functional principle of two-tone spectroscopy . . . . . . .
Pulse generation setup . . . . . . . . . . . . . . . . . . . .
IQ detector setup . . . . . . . . . . . . . . . . . . . . . . .
Functional principle of an IQ mixer . . . . . . . . . . . . .
Effects of the digital filters on ACQIRIS card data . . . .
Origin of the ACQIRIS card artifacts . . . . . . . . . . . .
1 dB compression point of amplifiers in the IQ detector . .
IQ mixer calibration . . . . . . . . . . . . . . . . . . . . .
Recorded raw data in comparison to calibrated data . . .
Reconstructed amplitude and phase for a calibration pulse
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24
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30
32
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34
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36
37
39
40
42
44
45
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
Second harmonic resonator mode . . . . . . . . . . . . . . . . . . . . . . . . .
Flux calibration: Full range scan . . . . . . . . . . . . . . . . . . . . . . . . .
Flux calibration: Detailed scan of the anticrossings . . . . . . . . . . . . . . .
Single tone continuous wave spectroscopy at high power . . . . . . . . . . . .
Numerical fit of the Jaynes-Cummings Hamiltonian to high power data . . . .
Photon number calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Low power continuous wave spectroscopy with fits . . . . . . . . . . . . . . .
Low power two-tone spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . .
Fittet energy spectrum of the sample . . . . . . . . . . . . . . . . . . . . . . .
Pulsed wave two-tone spectroscopy cabling and pulse patterns . . . . . . . . .
Pulsed wave two-tone spectroscopy measurement result . . . . . . . . . . . . .
Time-domain measurements cabling and pulse patterns . . . . . . . . . . . . .
Recorded quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Amplitude and phase of a time-domain measurement with the ACQIRIS card
Energy relaxation times for an ACQIRIS card measurement . . . . . . . . . .
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48
49
50
51
52
54
57
59
61
62
63
64
65
66
67
iii
Time-domain control of light-matter interaction with superconducting circuits
iv
4.16
4.17
4.18
4.19
4.20
4.21
ACQIRIS and FPGA measurement in comparison . . . . . . .
Energy relaxation times for a FPGA board measurement . . .
Heaviside window function . . . . . . . . . . . . . . . . . . . .
Rabi osclillations for different drive power values . . . . . . .
Fourier spectrum and power dependence of the oscillations . .
Rabi oscillation measurement detuned form degeneracy point
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69
70
72
73
74
75
A.1
A.2
A.3
A.4
Points in the real projective plain RP2 . . . . . . . . .
IQ mixer calibration: The transformations step by step
IQ mixer calibration: Amplitude extraction . . . . . .
IQ mixer calibration: Signal vs. Time . . . . . . . . .
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86
88
89
93
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.
Time-domain control of light-matter interaction with superconducting circuits
List of Tables
3.1
3.2
3.3
Maximum power dissipation per temperature stage . . . . . . . . . . . . . . . . .
Resonator’s thermal noise and dissipated power . . . . . . . . . . . . . . . . . . .
Antenna’s thermal noise and dissipated power . . . . . . . . . . . . . . . . . . . .
27
28
28
4.1
4.2
4.3
4.4
4.5
Resonator modes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flux calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Photon numbers on average and corresponding probe powers . . .
Fit parameters for high and low power spectroscopy in comparison
Vacuum Rabi levels . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
49
56
58
61
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v
Time-domain control of light-matter interaction with superconducting circuits
vi
Time-domain control of light-matter interaction with superconducting circuits
List of Symbols and Abbreviations
a(~r,t)
b
a† , b
a
A(t)
~
A
α
AC
β
c
C
Ck
Cin
Cout
CPW
δΦ
δωi
∆
d
DC
DTG
e
EJD
EJS
EJ0
FWHM
Γ1
Γ2
ΓDS
En
EQ
EQE
gi
|gi , |ei
Hi
disp
b
HJC
b JC
H
b MM
H
HMMJC
amplitude of a wave function
photon creation and annihilation operator
time dependent amplitude extracted from an IQ measurement
vector potential
relative form factor of the third Josephson junction of the qubit
alternating current
relative form factor of the coupling junction
velocity of light in vacuum c = 2.99792 · 108 m/s
capacity in the RCSJ model
capacity per unit length in the RCSJ model
capacity of the input port of the resonator
capacity of the output port of the resonator
coplanar waveguide
detuning of the flux from half a flux quanta
detuning of the mode i form the qubit excitation frequency ωQ
energy gap of a qubit
thickness of the insulating layer in a Josephson junction
direct current
data timing generator
elementary charge e = 1.60218 · 10−19 C
flux dependent energy bias of a qubit
potential energy of a driven Josephson junction
potential energy of an undriven Josephson junction
Josephson energy
f ull width half maximum
energy relaxation rate of the qubit
total dephasing rate of the qubit
energy relaxation rate of the dressed state
excitation energy of n photons in a resonator
qubit potential
qubit excitation energy
coupling strength of the qubit to the i-th resonator mode
ground and excited state of the qubit
Hilbert space of a quantum harmonic oscillator with mode i
Jaynes-Cummings Hamiltonian in the dispersive limit
Jaynes-Cummings Hamiltonian
multi-mode harmonic oscillator Hamiltonian
Hilbert space of a multi-mode Jaynes-Cummings Hamiltonian
vii
Time-domain control of light-matter interaction with superconducting circuits
b NLT
H
b SM
H
HSMJC
bQ
H
HQ
b ZPE
H
~
HEMT
I
IC
ID
IP
IR
IS
kB
κi
l
Lk
LJ
LS
M
me
m∗
nCP
n
bi
nPh
|n, ±ii
ωi
ωIF
ωJ
ωLO
ωP
ωQ
ω
eQ
ωS
ωSa
φ
ϕ
Φ
Φ0
Φ
Ψ
|Ψi
PSG
viii
transmission line Hamiltonian with a nonlinearity
single mode harmonic oscillator Hamiltonian
Hilbert space of a single mode Jaynes-Cummings Hamiltonian
qubit Hamiltonian
Hilbert space of a qubit
zero point energy of a multi-mode harmonic oscillator Hamiltonian
reduced Planck constant ~ = h/2π = 1.05457 · 10−34 Js
high electron mobility transistor
external driving current
critical supercurrent of a Josephson junction
displacement current trough the capacitively part of the RCSJ model
persistent current of a qubit
current trough the resistively part of the RCSJ model
supercurrent of a Josephson junction
Boltzmann constant kB = 1.38065 · 10−23 J/K
loss rate of the i-th resonator mode
geometrical length of the resonator
inductance per unit length in the RCSJ model
Josephson inductance
inductance of a Josephson junction
mutual inductance
electron mass me = 9.10938 · 10−31 kg
effective mass of a particle
density of Cooper pairs
photon number operator of mode i
number of thermal photons in the resonator
dressed state of mode i with n photons and the qubit
angular frequency of the cavity mode i
intermediate frequency of an IQ mixer
Josephson frequency
local oscillator frequency of an IQ mixer
probe tone frequency
qubit excitation frequency
dispersively shifted qubit excitation frequency
spectroscopy tone frequency
sampling frequency of a data acquisition card
gauge invariant phase difference over a Josephson junction
Bloch azimuthal angle
magnetic flux through the qubit loop
magnetic flux quantum Φ0 = h/2e = 2.06783 · 10−15 Wb
scalar potential in the Schrödinger equation
wave function of a superconductor
Bloch vector
Agilent E8267D PSG
Time-domain control of light-matter interaction with superconducting circuits
poa
q
QED
QIP
ρ(t) )
RCSJ
RF
σ
bx , σ
bz
SMF
SNR
T1
TDS
τ
Θ
ϑ
R
U
UV
VNA
WMI
ξ(~r,t)
Z0
photons on average
charge
quantum electrodynamics
quantum information processing
time dependent phase extracted from an IQ measurement
resistively capacitively shunted junction
radio frequency
Pauli operators
Rohde & Schwarz SMF100A
signal to noise ratio
energy relaxation time of the qubit
energy relaxation time of the dressed state
pulse width in a time-domain measurement
Bloch polar angle
mixing angle
resistor in the RCSJ model
voltage drop over a Josephson junction
ultraviolet
vector network analyzer
Walther-Meissner-Institut
phase of a superconductor’s wave function
characteristic impedance of an element in the RCSJ model
ix
Time-domain control of light-matter interaction with superconducting circuits
x
Time-domain control of light-matter interaction with superconducting circuits
1
Introduction and Motivation
In classical computation the information is stored in the classical states 0 and 1. Considering a quantum computer where the information is processed in quantum mechanical two level
systems with eigenstates |gi and |ei the situation is more complex, as the system is no longer
described by classical electrodynamics but by quantum mechanics. Already in the 1980s it was
proposed by Feynman and Deutsch to use quantum mechanical phenomena for simulation [1]
and computation [2]. These two proposals have given rise to the field of quantum information
processing (QIP). In quantum computation one solves problems that would not be computable
with classical computers in a suitable time. A famous example is the prime factorization of large
numbers via Shor’s algorithm [3]. The fact that the time it takes to factorize a large number
grows exponentially with the length of the key on a classical computer is a key element of cryptography where the key is produced from the product of two prime numbers. To decrypt the
information without knowledge of the two prime numbers one has to factorize the key. If the key
is long enough this is not computable on a classical computer within a suitable amount of time.
But if a quantum computer and Shor’s algorithm is used the time increases only polynomially
with the length of the key. It was recently demonstrated that 143 could be factorized with a
NMR system [4] and 15 was successfully factorized in a superconducting phase qubit system [5].
A second application for QIP is quantum simulation [6, 7], where one quantum system which
is hard to investigate is mapped onto a different quantum system that could be well controlled
and realized more straight forwardly. For example one can simulate a relativistic quantum system with a non-relativistic quantum system [8, 9]. For a functional quantum computer we need
quantum bits (qubits) which fulfill the DiVincenzo criteria [10] which were proposed in 2000.
These criteria demand the existence of a quantum two level system which can be used as a qubit
and that it can be prepared in an initial state. Further the decoherence times have to be large
enough such that a sufficient number of operations can be performed within that time. These
operations are realized by quantum gates. After the operations have been applied to the system
the readout has to be performed with high accuracy. Finally the system should be extendable
to a larger number of qubits easily. There are several approaches to implement qubits which try
to fulfill all these criteria.
Promising realizations can be found in the field of cavity quantum electrodynamics (QED). Here
one uses highly reflective mirrors to confine an optical light field which interacts with trapped
atoms or ions. These systems allow to investigate the interaction of the atom/ion with the light
field. This interaction is characterized by the coupling strength which should be large in order to
transfer photons efficiently from the light field to the atom such that the atom is excited. In this
field it has been shown that one can store information in a single atom [11] and that it is possible
to entangle up to 14 qubits [12]. The main benefit of these approaches is the relatively long
coherence time of several milliseconds [13] while the increase of the coupling strength is still an
issue. With this approach it is also possible to perform a quantum non demolition measurement
so it is possible to read out the quantum state without destroying the quantum state. For the
development of such a measurement technique where the wave function does not collapse through
the readout Serge Haroche and David J. Wineland were awarded the Nobel Prize in 2012.
1
Time-domain control of light-matter interaction with superconducting circuits
Another promising approach to study light-matter interaction is circuit QED. Here artificial
atoms (qubits) such as flux qubits [14, 15, 16, 17] or transmons [18, 19, 20] are realized with
a superconducting circuit. A superconducting resonator forms the equivalent to the highly reflective mirrors in cavity QED. Here the confined photon field is shifted from the optical to the
microwave regime. The transition frequency of the artificial atom and the resonance frequency of
the resonator are designed via electronic engineering which yields a large experimental ground.
This approach is very favorable since the needed micro and nano fabrication technology is already available. A flux qubit can be realized with a superconducting loop which is interrupted
by 3 Josephson junctions [14, 15]. A flux can be trapped in the loop. This flux is used to vary
the transition frequency of the artificial atom. In circuit QED the coherent transfer of photons
in a three-resonator circuit was recently demonstrated [21]. The coherence times still are not
sufficiently long since they are in the order of tens of microseconds [22, 23]. The coherence times
are very important because only within this time one can perform algorithms. If one wants to
read out the information after a certain time t longer than the coherence time the information
is lost.
Two important components of a quantum computer are, first quantum bits (qubits) where the
information is processed and second a bus system such that information can be transferred from
one qubit to another. To perform operations it is essential to have gates with a very high fidelity. A fundamental gate in QIP is the so called controlled-NOT (CNOT) gate [24]. Another
promising gate where a high fidelity could be achieved is the Mølmer-Sørenson gate [25] since
it is based on second order transitions. Gates are realized by applying pulse sequences on the
qubits. These pulses have to be performed within the coherence time of the system. Therefore,
one can estimate the number of possible operations by the quotient of the coherence time and
the time needed to perform one gate operation. Since the time for the gate operation is more or
less fixed it is essential to increase the coherence time to compute complex quantum algorithms.
At the WMI Josephson junction flux qubits have been successfully investigated spectroscopically
for several years [17, 26, 27]. Up to now the coherence times of these qubits have not been
investigated in details.
At the WMI three Josephson junction flux qubits are used since they yield larger coupling
strengths than for example transmons. It was demonstrated [17] that the ultra strong coupling
regime can be realized by using a coupling junction in a shared edge of the qubit and the resonator. The size of the coupling junction influences the coupling strength. A current field of
research at the WMI is the determination of the maximal coupling strength of a flux qubit
to a superconducting resonator. So the first issue we address in this thesis is the determination of the coupling strength for an area of the coupling junction different than in ref. [17] in
frequency-domain measurements. The second issue we would like to address in this thesis is the
measurement of the coherence times of a coupled qubit resonator system. Therefore, we have to
build up, characterize and calibrate a detector for time-domain measurements.
2
Time-domain control of light-matter interaction with superconducting circuits
In this thesis we successfully measure the energy relaxation time of qubits in a coplanar transmission line resonator fabricated at the WMI for the first time. Therefore, we introduce the
theory necessary to describe the artificial atom and the light field in the resonator. We use the
Jaynes-Cummings model [28] which was derived in the field of quantum optics but also holds
for our artificial system. Furthermore, we introduce the physics of the measurement technique
used in two-tone spectroscopy and the time-domain measurements. For the latter one we give a
short introduction into the loss mechanisms of the system since these determine the coherence
times. Afterwards we proceed with a detailed description of the used measurement setup. We
introduce the production steps of our sample, depict the configuration of the measurement setup
inside the dilution refrigerator and the room temperature setup. Furthermore, we concentrate on
the detector which we built up for the time-domain measurements, here we perform a detailed
characterization and introduce a digital heterodyne IQ mixer calibration algorithm which we
developed during the course of this thesis. Finally we present the experimental results of the
performed measurements. We begin with detailed continuous wave single and two-tone spectroscopy such that we can calibrate the applied flux to the qubit and the photon number in the
resonator. Afterwards we perform spectroscopy measurements with negligible photon number
in the readout resonator mode. From a fit of the Jaynes-Cummings Hamiltonian to the data
acquired we are able to determine the coupling strengths gi of the qubit to the resonator mode
i. Furthermore, we gain knowledge of the characteristic frequencies of our system such that we
can apply a signal of the correct frequency in the time-domain measurements. During these
measurements we observe an oscillatory behavior of the response of the sample with the length
τ of the applied pulse which strongly indicates the observation of Rabi oscillations.
3
Time-domain control of light-matter interaction with superconducting circuits
4
Time-domain control of light-matter interaction with superconducting circuits
2
Theoretical background
In this thesis we investigate a coupled qubit resonator system. This quantum mechanical system
is theoretically described by the Hamiltonian of the Jaynes-Cummings model [28]. In this chapter
we introduce the basic ingredients needed to develop the Jaynes-Cummings Hamiltonian i.e. the
3 Josephson junction flux qubit and the quantum mechanical oscillator with multiple mode
excitations. Therefore, we start with an introduction to Josephson physics of one Josephson
junction and develop a model for a 3 Josephson junction flux qubit. We proceed with a description
of the single mode quantum harmonic oscillator and expand it to a multi-mode picture. In the
next step we use a fourth Josephson junction located at the shared edge of the qubit and the
resonator. We treat this coupling junction as a nonlinearity in the resonator which should
enhance the coupling. In the last part of this chapter we provide the basic theory necessary to
interpret the time-resolved measurement results of section 3.2.2.
2.1
Josephson physics
The history of superconductivity began in 1911 when Heike Kamerlingh Onnes discovered the
disappearance of the resistance of mercury at liquid helium temperature [29, 30]. In 1935,
the London brothers developed a phenomenological theory of superconductivity [31] where they
introduced a macroscopic wave function of the form
Ψ(~r,t) = a(~r,t) exp iξ(~r,t) .
(2.1)
With a time and space dependent amplitude a(~r,t) and phase ξ(~r,t) . For better readability we
drop the dependence on (~r,t) in the rest of this thesis. This ansatz is used to solve the time
dependent Schrödinger equation
2
~ − qA
~
−i~
∇
c
∂
i~ − qΦ Ψ =
Ψ
(2.2)
∗
∂t
2m
with the reduced Planck constant ~ = h/2π = 1.05457 · 10−34 Js [32], the charge q, the scalar
~
potential Φ, the velocity of light in vacuum c = 2.99792 · 108 m/s [32], the vector potential A
∗
and the effective mass m . It took more than 20 years before Bardeen, Cooper, and Schrieffer
were able to developed a microscopic theory [33, 34] to solve the time dependent Schrödinger
equation (2.2) in 1957. For their BCS theory they received the Nobel Prize in 1972. They found
that the amplitude of the wave function in equation (2.1) is given as
√
(2.3)
a = nCP .
Where nCP is the density of the Cooper pairs in the superconductor. Cooper pairs are quasiparticles formed by two electrons in momentum space. As a result the charge is given as q = 2e and
the effective mass m∗ = 2me in equation (2.2) with the elementary charge e = 1.60218 · 10−19 C
[32] and the mass of the electron me = 9.10938 · 10−31 kg [32]. So the wave function
√
Ψ = nCP exp (iξ)
(2.4)
describes the condensate of all Cooper pairs and not the single particles. Thus we have a beautiful
example for quantum mechanics on a macroscopic scale.
5
Time-domain control of light-matter interaction with superconducting circuits
2.1.1
Josephson junction
Already in 1962 Brian Josephson considered two superconducting materials (SC) with wave
√
functions Ψ1,2 = n1,2 exp (iξ1,2 ) separated by a thin insulating layer I with thickness d [35] as
depicted in Figure 2.1. If we observe coherent Cooper pair tunneling we call such a configuration
a Josephson junction.
Figure 2.1: A sketch of a Josephson junction. The two superconductors with the macroscopic
wave functions Ψ1 and Ψ2 are separated by a thin insulating layer I. For a given insulating
material, the thickness d determines the tunneling rate of the Cooper pairs because it determines
the overlap of the two wave functions.
He realized that due to the macroscopic wave function and the formation of Cooper pairs in the
momentum space tunneling is a coherent process over a length of several thousands of atoms
(due to Heisenberg’s uncertainty relation [36]) and as a result the Cooper pairs are able to tunnel
through the thin barrier if the thickness d is not too large such that the wave functions on both
sides overlap. For this work and the prediction of the Josephson effects via equations (2.5) and
(2.6) he was awarded the Nobel Prize in 1973.
IS = IC sin (φ)
∂φ
2π
=
U
∂t
Φ0
(2.5)
(2.6)
So the supercurrent IS over the junction is determined by a critical supercurrent IC and the
gauge invariant [37] phase difference φ of the two wave functions of the superconductors
Z
2π 2 ~ ~
Adl.
(2.7)
φ := ξ2 − ξ1 −
Φ0 1
The relation (2.5) is known as the first Josephson equation or current phase relation. This
phenomenon is also known as DC Josephson effect because a DC current can flow without any
external applied voltage. The second Josephson equation (2.6) or voltage phase relation combines
the partial time derivative of the phase difference φ with the voltage U across the junction via the
flux quantum Φ0 = 2.06783 · 10−15 Wb. If the voltage V is time-independent, we can solve the
partial differential equation (2.6) via separation of variables. With the result we enter equation
(2.5) and end up with the AC Josephson effect (2.8).
2πU
IS = IC sin φ0 +
t
(2.8)
Φ0
Equation (2.8) is a remarkable example for a nonlinear effect in electronic circuits. Here a DC
voltage induces an alternating current with Josephson frequency ωJ = 2πU/Φ0 . Since we are
now familiar with the Josephson effects we can proceed with a derivation of the potential energy
EJS stored in the junction. This is done by integrating the power with respect to time from time
t0 = 0 where the phase difference φ0 = 0 to a certain time t where the phase difference is φ [38].
Z t
Z t
Z t
Z
0 0 IC Φ0 φ
0
IC Φ0
0
0
0 ∂φ
0
sin φ
dt
=
sin
φ
dφ
EJS =
P dt =
IS U dt =
2π
∂t0
2π 0
(2.9)
0
0
0
= EJ0 (1 − cos (φ))
6
Time-domain control of light-matter interaction with superconducting circuits
Where we introduced the Josephson energy EJ0 = IC Φ0 /2π. Equation (2.9) represents an energy
which is 2π periodic in φ with minima for 2πn and n ∈ Z. The fact that energy can be stored
in the junction also indicates that a Josephson junction is a nonlinear device. We use relation
(2.9) in sections 2.1.3 and 2.2 again. Another useful quantity we can derive from the Josephson
equations (2.5) and (2.6) is the so called Josephson inductance LJ . Therefore, we have a look at
the time derivative of the first Josephson equation (2.5)
dIS
dφ
2π
= IC cos (φ)
= IC cos (φ) . U
dt
dt
Φ0
When we rearrange this equation we end up with
Φ0
dIS −1
1
=
LS = U
= LJ
,
dt
2πIC cos (φ)
cos (φ)
(2.10)
(2.11)
where LJ = Φ0 /(2πIC ) denotes the Josephson inductance. So the ideal Josephson junction acts
as a nonlinear inductance where the energy EJS is stored in the inductance LS [38].
2.1.2
RCSJ model
To model a Josephson junction such that it is usable for electronic engineering it is necessary
to develop an equivalent circuit diagram (see Figure 2.2). This was achieved with the so called
resistively and capacitively shunted junction model (RCSJ model) which was developed by
Stewart and McCumber in 1968 [39, 40].
Figure 2.2: The Josephson junction in the RCSJ model. The current source applies a current I
to the junction (dashed box) which is modeled as a parallel circuit of a normal resistance R, an
ideal Josephson junction with critical current IC and a capacitor with capacity C. The voltage
U drops on that equivalent circuit.
The current I generated by the source splits up into three parts, one enters the resistor such that
IR = U/R, the second flows through the junction IS = IC sin (φ) and the third is a displacement
current in the capacitor ID = CdU/dt. Where R denotes the normal resistance and C the
capacity of the junction. By applying Kirchhoff’s current law [41], we obtain the equation
X
dU
U
I=
Ii = IC sin (φ) + + C
.
(2.12)
R
dt
i=S,R,D
In the small junction limit the spatial derivative of the phase difference φ over the junction is
negligible [42] as a result the total and the partial time derivatives are identical such that we
can use the second Josephson equation (2.6) to replace the voltage U by the phase difference φ
or vice versa [37] and end up with
I = IC sin (φ) +
Φ0 dφ CΦ0 d2 φ
+
.
2πR dt
2π dt2
(2.13)
7
Time-domain control of light-matter interaction with superconducting circuits
Equation (2.13) is a nonlinear differential equation of second order without an analytical solution
except for the case when the capacity vanishes such that it is reduced to first order [37]. However,
by multiplying equation (2.13) with the constant term Φ0 /2π and neglecting the dissipation term
∝ dφ/dt, we end up with the equation
C
Φ0
2π
2
d2 φ
Φ0
=
I − EJ0 sin (φ) .
2
dt
2π
(2.14)
~ JD and we can treat the phase difference
Equation (2.14) is of the same type as m~a = F~ = −∇E
φ as a phase particle. It can be shown [38] that the potential EJD when the junction is driven
by a current is given as
I
(2.15)
EJD = EJ0 1 − cos (φ) − φ .
IC
The potential described by equation (2.15) is called tilted washboard potential. In Figure 2.3 we
depict the potential for various values of the driving current I.
Figure 2.3: The so called tilted washboard potential for a driven Josephson junction in the
RCSJ model. When the driving current I is turned off, the potential is identical to the undriven
potential EJS given in equation (2.9) and tunneling of a phase particle in both directions is
possible with the same probability (blue line). If we turn on the current and are below the
critical current IC of the junction tunneling into one direction is more probable than in the other
direction (red line). If the driving current is higher than the critical current the potential does
no longer exhibit stable minima (magenta line).
8
Time-domain control of light-matter interaction with superconducting circuits
In the case that the driving current I is switched off (see blue line of Figure 2.3), we end up
with the potential of the undriven junction EJS given in equation (2.9) where the equilibrium
is stable and tunneling of a phase particle in both directions appears with the same probability
due to the height of the potential barrier and its width. If we slowly increase the driving current
tunneling into one direction becomes more favorable than in the other (red line) because the
potential barriers have different heights. When the driving current I is equal to the critical
current IC of the junction (green line) the system is in an unstable equilibrium because an
infinitesimal distortion results in an acceleration of the phase particle. If the current is further
increased the system is no longer stable (magenta line). A great advantage of the RCSJ model
is that it describes the dynamics of the junction quite well without a detailed description of the
microscopic current transport [37].
2.1.3
3 Josephson junction flux qubit
In this section we introduce the 3 Josephson junction flux qubit [14, 15], which serves as an
artificial atom and is one of the main components of our sample of a coupled qubit resonator
system. Therefore, we have a look at the qubit potential and its dependencies, further on we
identify the ground |gi and excited state |ei with the direction in which the persistent current IP
is flowing. If we place 3 Josephson junctions in series and connect the two open ends, we end up
with a superconducting ring interrupted by 3 Josephson junctions. Such a device can be used as
a flux qubit. We assume that two of the junctions are equal and the third is smaller by a design
factor α with respect to the area of the junction [14, 15] as depicted in Figure 2.4.
Figure 2.4: Sketch of a three Josephson junction flux qubit. A superconducting loop is intersected
by three Josephson junctions (blue blocks). Two of the junctions are designed to be equal such
that they have the same potential energy EJ0 , capacity C and inductance LJ . The third differs
by a parameter α ≈ 0.7 with respect to the area of the junction. The ground |gi and exited
state |ei are coded in clockwise (green arrow) or counter clockwise (magenta arrow) rotation of
the persistent current IP .
The ground |gi and excited state |ei are coded in clockwise (green arrow) or counter clockwise
(magenta arrow) rotation of the persistent current IP which is caused by the flux Φ trapped in
the qubit loop. If the inductance of the superconducting loop is small compared to the Josephson
inductance LJ of the junctions we can express the potential energy of the loop as the sum of the
potential energies of each junction derived in equation (2.9)
EQ = EJ0 (2 + α − cos (φ1 ) − cos (φ2 ) − α cos (φ3 )) ,
(2.16)
where i = 1, 2 indicates the two equal junctions and i = 3 is the so called α-junction.
9
Time-domain control of light-matter interaction with superconducting circuits
Further it is justified to use fluxoid quantization which was discovered by R. Doll and M. Näbauer
at the Walther-Meissner-Institut (WMI) [43] and by B.S. Deaver and W.M. Fairbank [44] in 1961.
Hence we find the relation [15]
φ1 − φ2 + φ3 = −2π
Φ
.
Φ0
We use equation (2.17) to eliminate φ3 from equation (2.16)
Φ
.
EQ = EJ0 2 + α − cos (φ1 ) − cos (φ2 ) − α cos 2π
+ φ1 − φ2
Φ0
(2.17)
(2.18)
Since the significant variations of the persistent current IP occur in a flux range of several mΦ0
around the degeneracy point of the qubit at (n + 1/2) Φ0 with n ∈ Z [14, 15] it is reasonable to
introduce the flux detuning δΦ as
2n + 1
δΦ := min Φ −
(2.19)
Φ0 .
n∈Z
2
So δΦ measures the minimal distance from the flux Φ to the nearest half flux quantum. Therefore,
the lower and upper bounds are given as −0.5Φ0 ≤ δΦ < 0.5Φ0 . For simplicity we assume for
the rest of this thesis that Φ is located around 1/2 Φ0 such that the minimum is given at n = 0,
so finally the qubit potential is of the form
δΦ
EQ = EJ0 2 + α − cos (φ1 ) − cos (φ2 ) − α cos π + 2π
+ φ1 − φ2
.
(2.20)
Φ0
A contour plot of the qubit potential EQ for α = 0.72 and two different values of δΦ is presented
in Figure 2.5 (a) and (b). In Figure 2.5 (a) the flux detuning is zero and the potential is periodic
with period 2π in each degree of freedom. There are two identical minima around (0, 0)T , the
minima of the double well potential are located at [45]
1
φ1 = −φ2 = ± arccos
.
(2.21)
2α
where we used that the flux detuning δΦ = 0. We would like to mention that we assume that the
so called intra cell minima [14] remain at the constant value given in equation (2.21) for small
perturbations of the flux. The potential energy EQ which belong to these minima are no longer
degenerated for a small detuning of δΦ = −20 mΦ0 in Figure 2.5 (b). A one dimensional cut along
the line φ1 +φ2 = 0 where the intra cell minima are located is visualized in Figure 2.5 (c) and (d).
It is worth mentioning that detuning the flux through the qubit loop causes a manipulation of
the qubit potential. The two level system needed to encode the qubit is represented by the lowest
excitation in each minimum of the double well potential (d). We would like to point out that
we can change the double well by tuning the flux Φ as a result we lower and higher the ground
state |gi and exited state |ei of the qubit. This also causes a change of the qubits excitation
energy EQE which is the energy difference of the ground and excited state. The amount of flux
detuning is characterized by the value of δΦ.
10
Time-domain control of light-matter interaction with superconducting circuits
Figure 2.5: Equipotential lines of the qubit potential for two different detuning values. The qubit
potential (2.20) for phases φ1 , φ2 from −2π to 2π, δΦ = 0 and α = 0.72 is plotted in (a). It turns
out that the potential is 2π-periodic in each direction and it results in a double well potential
along the blue line. In (b) we detuned the flux by δΦ = −20 mΦ0 , the periodicity remains but
the double well potential is no longer symmetric. In (c) we present a one dimensional cut along
the line φ1 + φ2 = 0 [blue and red lines in (a) and (b)] through the intra cell minima (2.21). The
inset of the black box is visualized in (d).
b Q for values around δΦ = 0 can be written
According to ref. [45] the Hamiltonian of the qubit H
as
∆
1 ∆
b
HQ = σ
bz + σ
bx =
.
(2.22)
2
2
2 ∆ −
With σ
bx and σ
bz as the Pauli operators, ∆ as the energy gap of the qubit and a flux dependent
energy bias
∂EQ := 2δΦ
= 2δΦIP ,
∂δΦ (φ1 ,−φ1 )T =min
(2.23)
where the minimum is given in equation (2.21). According to ref. [45] we quantify the persistent
current IP of the qubit to
s
IP = IC
1−
1
2α
2
.
(2.24)
11
Time-domain control of light-matter interaction with superconducting circuits
Therefore, we express the qubit exitation energy as
p
EQE = ~ωQ = ∆2 + 2 ,
(2.25)
where ωQ denotes the qubit excitation frequency. The minimum of the qubit excitation energy
EQE is given by the flux independent energy gap ∆ for a flux detuning of δΦ = 0. The resulting
energy diagram depending on the flux detuning δΦ of the qubit hyperbola is depicted in Figure
2.6 (a).
Figure 2.6: The qubit hyperbola and the persistent current dependencing on the flux detuning
δΦ. In (a) we depict the qubit energy diagram according to equation (2.25). It has a hyperbolic
dependence on the flux detuning δΦ. With vanishing detuning the levels are separated by the
qubit energy gap ∆. In (b) the slope of the hyperbola is plotted. For large detunings the value
of the slope converges to the persistent current of the qubit IP .
At detuning δΦ = 0 we need the energy gap ∆ to excite the qubit from state |gi to state |ei. The
red lines indicate the case for vanishing energy gap. For large detunings the slope of the qubit
hyperbola converges to
∂EQE
= ±IP
δΦ→±Φ0 /2 ∂δΦ
lim
(2.26)
which is visualized in Figure 2.6 (b). Here the current circulating in the qubit loop is shown
dependencing on the flux detuning. For large detunings we can distinguish between clockwise
and counter clockwise rotating currents, but when we are close to zero detuning the two currents
overlap. This can be understood by a degeneracy of the qubit energy levels in the double well
potential of equation (2.20) which is visualized in Figure 2.5.
12
Time-domain control of light-matter interaction with superconducting circuits
2.2
Quantum harmonic oscillator
The second important component of our sample is the superconducting niobium resonator, which
in good approximation is a one dimensional structure with negligible width and height terminated
by a capacitor at each end. Each mode of the resonator is modeled as a harmonic oscillator.
We present different scenarios of such models depending on the modes which are present in the
resonator. In an analogous way to its mechanical equivalent, a mass attached to a spring [32],
the Hamiltonian of the harmonic oscillator is given by
1
†
b SM = ~ω b
,
(2.27)
H
ab
a+
2
where ω is the eigenfrequency of the oscillator, b
a annihilates a photon at frequency ω and b
a†
creates a photon at frequency ω. In the Fock space where |ni denotes the state with n photons
in the resonator we find the expressions [32]
√
b
a |ni =
n |n − 1i
(2.28)
√
†
b
a |ni =
n + 1 |n + 1i
(2.29)
b
a |0i = 0,
(2.30)
since |ni is an eigenstate of the photon number operator n
b := b
a† b
a. These three relations hold
for any bosonic particle. Since this is an orthonormal eigenbasis we can define the Hilbert space
of harmonic oscillator with a single mode as
HSM = {|ni , n ∈ N0 } .
According to ref. [32] the eigenenergies of the time-independent Schrödinger equation
1
†
b
Ψ = EΨ
HSM Ψ = ~ω b
ab
a+
2
(2.31)
(2.32)
are given as
1
En = n +
~ω.
2
(2.33)
This energy spectrum is visualized in Figure 2.7 (a).
13
Time-domain control of light-matter interaction with superconducting circuits
Figure 2.7: The energy levels for different scenarios of a resonator. In (a) only a single mode is
present, the Fock state |0i denotes the vacuum level. The higher excitations are separated by
the energy ~ω. The energy spectrum of a multi-mode resonator with equidistant mode spacing
is illustrated in (b) where |nii denotes the state of n photons of mode i. The equidistant mode
spacing can be seen as a resonance condition which can be written as ωi = (i + 1)ω0 with i ∈ N0 .
It turns out, that for example the states |2i0 which corresponds to the lowest mode with respect
to frequency has the same excitation energy as the state |1i1 . In (c) we illustrate the most
general case of a multi-mode resonator. Here the resonance condition or the equidistant mode
spacing is no longer fulfilled and the energy spectrum is rich. Such a situation can be caused
by a nonlinearity such as a Josephson junction embedded in the resonator such that we create
a mode dependent phase drop [46]. This situation causes more than one parabola in the energy
spectrum for clarity we decided to show only one. A sketch of the current density of the first
three modes of a multi-mode resonator without a nonlinearity (d). The electrical length of the
resonator is proportional to the geometric length independent of the mode and we end up with
the energy spectrum depicted in (b). In (e) a nonlinearity is placed at 1/4 of the resonators
length. The mode dependent phase drop [46] modifies the electrical length such that modes with
other wavelengths than in the undistorted case fit into the resonator. Since the nonlinearity is
placed at 1/4 of the resonators length the third harmonic ω3 would not be affected by the phase
drop since at this position the current distribution i(z) is zero. This scenario causes the energy
spectrum depicted in (c).
14
Time-domain control of light-matter interaction with superconducting circuits
Up to now we only talked about a resonator with a single mode, but for a sufficiently large amount
of energy we can also excite higher harmonics. This can be expressed as the superposition of the
single mode Hamiltonians and the Hamiltonian of the new system can be written as
X
1
†
b
ai b
ai +
with i ∈ N0 .
(2.34)
H=
~ωi b
2
i
This equation is a functional in an infinite dimensional Hilbert space formed from a tensor
product of the Hilbert spaces Hi for every single mode i. However to make it accessible for a
numerical simulation in a classical computer one has to assign a maximum number of Fock states
N which leads to a square matrix of finite dimension. Under this assumption we can subtract
b ZPE which is given as
the zero point energy H
b ZPE =
H
N
X
~ωi
i=0
2
,
such that we end up with the new multi-mode Hamiltonian
X
b MM = H
b −H
b ZPE =
H
~ωi b
a†i b
ai .
(2.35)
(2.36)
i
We end up with a single vacuum mode and count the number of excitations. We would like to
mention that we are still able to normalize the energy level with zero excitations to a certain
value. For example in Figure 2.7 (b) and (c) we set the constant to ~ω0 /2 to emphase the analogy
to the situation depicted in (a). While in the energy level diagram of our qubit resonator system
in Figure 4.9 the vacuum level is set to zero. However in a multi-mode resonator we are able to
distinguish two cases. The first case is when the different modes are resonant which means that
ωi = (i + 1)ω0
with i ∈ N0 .
(2.37)
Here the mode spacing of each mode in the resonator is equidistant such that the first harmonic
ω1 has twice the frequency/energy of the fundamental mode ω0 . An example of an energy
spectrum in the resonant case with renormalized vacuum mode is visualized in Figure 2.7 (b).
The most general and therefore the most interesting case is a scenario where the condition (2.37)
is violated. As proposed in ref. [46] we can create such a system by adding a Josephson junction
to the resonator. This nonlinearity in the system causes a mode dependent phase drop [46].
Therefore, we shift the modes such that they are no longer on resonance, which means that
ωi 6= (i + 1)ω0
with i ∈ N.
(2.38)
We would like to mention that equation (2.36) does not contain the nonlinearity explicitly.
Furthermore, the mode ω1 is still called the first harmonic and so forth to keep a commonly used
term. The interaction between the oscillator and the nonlinearity shifts the modes in a nontrivial
correlation [46]. In the sample used in this thesis the coupling junction is placed at 1/4 of the
resonators length [see Figure 3.1 (a)]. So the third harmonic ω3 is not affected by a phase drop
since the current distribution i(z) at this position is zero. Up to now we modeled the multi-mode
resonator as sum of quantum harmonic oscillators to investigate the energy spectrum, but we
did not describe the resonator from an electrical engineering point of view. In network theory
the harmonic oscillator is modeled as a transmission line with lumped elements because the
physical dimensions of the circuit are comparable to the electrical wavelength of the microwave
[47, 48]. As we use a superconducting niobium resonator the appropriate lumped element circuit
is depicted in Figure 2.8.
15
Time-domain control of light-matter interaction with superconducting circuits
Figure 2.8: The superconducting niobium resonator in a coplanar waveguide (CPW) architecture.
In (a) the niobium (blue layer) is fabricated on the silicon substrate (gray layer) as described in
section 3.1.1.1. It is capacitively coupled to the center conductor of the measurement cables via
the capacitors Cin and Cout . The length of the resonator l is defined by the distance between the
coupling capacitors Cin and Cout . The equivalent circuit diagram consisting of discrete capacitors
and inductors is shown in (b).
16
Time-domain control of light-matter interaction with superconducting circuits
In Figure 2.8 (a) the fabricated superconducting niobium resonator is characterized by its length
l and its input and output capacitors Cin and Cout . We use a coplanar waveguide architecture
(CPW) which is the two dimensional equivalent to a coaxial cable. In (b) we show the equivalent circuit diagram for the modeled lossless transmission line. We divide the total length l
of the resonator into n discrete elements with a length ∆zk where the transmission line has an
inductance Lk and capacity Ck per unit length such that the equations
l =
L =
C =
n
X
k=1
n
X
k=1
n
X
∆zk
(2.39)
Lk
(2.40)
Ck
(2.41)
k=1
are fulfilled. We would like to mention that the distances ∆zk do not have to be equal, only
the inductances Lk and the capacitances Ck have the same value. If we assume a homogeneous
waveguide and substrate the lengths ∆zk will become equal, too. Under that conditions we can
quantify the characteristic impedance to [47, 48]
r
Lk
.
(2.42)
Z0 =
Ck
The microwave signal of angular frequency ω in the transmission line is then given as the solution
of the so called telegrapher equations. The solutions can be found in refs. [47, 48]. According
to that solutions the voltage and current in the transmission line is given as propagating waves
traveling in both directions. These propagating waves reduce to an infinite discrete set of standing
waves when we introduce the boundary condition given by the electrical length of the resonator
fixed by the capacitors Cin and Cout . As a result the angular frequencies ω are also reduced to
an infinite set of the resonator modes ωi . In the case of a multi-mode resonator with equidistant
mode spacing we end up with the situation depicted in Figure 2.7 (b) and (d). If we take into
account a nonlinearity such as a Josephson junction which causes a mode dependent phase drop
[46] the situation is of more interest. According to ref. [46] that such a system can be modeled
by the Hamiltonian in a lumped element formalism
q2
δΦ2
2πδΦ
b NLT =
H
+
− EJ0 cos
.
(2.43)
2 (C + CJ )
2L
Φ0
Where q is the charge in the resonator and δΦ is the flux detuning introduced in equation (2.23).
The first term takes into account the capacity of the resonator and the junction, while the second
and the third term represent the inductance of the resonator and the junction respectively.
Depending on which of the two conjugate variables (flux or charge) is used one has to verify
which terms in equation (2.43) can be neglected. As we are using flux qubits we have to estimate
if we can neglect the charge term.
17
Time-domain control of light-matter interaction with superconducting circuits
2.3
Jaynes-Cummings model
So far we discussed the quantum behavior of a superconducting flux qubit and a superconducting
resonator as two independent systems. In this thesis we couple these two systems, so the Hamiltonian of the new system is the superposition of the two independent systems plus an additional
interaction term. In the most general case this results in the Hamiltonian
X
b = ~ ωQ σ
bz +
~ωi b
a†i b
ai + ~gi b
a†i + b
ai (b
σ+ + σ
b− ) .
(2.44)
H
2
i
The first term corresponds to the qubit eigenstates |gi and |ei the series represents the multimode resonator modeled as a sum of quantum harmonic oscillators and the interaction of each
mode i with the qubit. The mode dependent coupling strength is denoted by gi . According to
ref. [45] this dependence arises from
hgi = M IP Ii ,
(2.45)
where M is the so called mutual inductance between the qubit and the resonator, IP is the
persistent current of the qubit and Ii is the vacuum current of the i-th cavity mode [45] given as
r
~ωi
Ii =
.
(2.46)
L
So the current in the resonator Ii depends on the mode and the total inductance of the resonator
given in equation (2.41). If the condition ωQ + ωi > |ωQ − ωi | > gi is fulfilled, where the latter
condition is the more important one, we can neglect the fast rotating terms ωQ + ωi . This is the
so called rotating wave approximation (RWA) which simplifys the Hamiltonian to [49]
X
b JC = ~ ωQ σ
H
bz +
~ωi b
a†i b
ai + ~gi b
a†i σ
b− + b
ai σ
b+ ,
(2.47)
2
i
where we neglected the so called counter rotating terms b
a†i σ
b+ and b
ai σ
b− . In 1963 this ansatz
was derived for the first time by Jaynes and Cummings [28]. Therefore, equation (2.47) is called
the Jaynes-Cummings Hamiltonian. The operator combination b
a†i σ
b− and b
ai σ
b+ represent the
process that a photon is absorbed by the qubit from the photon field of the resonator mode i
or vice versa. By considering these processes the Jaynes-Cummings Hamiltonian preserves the
number of excitations in the system. Due to the second diagonal entries of the σ
bz operator the
Hamiltonian (2.47) has to be diagonalized in a suitable Hilbert space. In our case this Hilbert
space is given by
HMMJC = Hq ⊗i Hi
(2.48)
where ⊗i denotes the tensor product. The subscript i should indicate that for every resonator
mode this multiplication has to be executed and HQ is the Hilbert space of the qubit represented
as
1
0
HQ =
,
.
(2.49)
0
1
In this thesis we choose that the ground state of the qubit is be represented by |gi = (1, 0)T and
exited state by |ei = (0, 1)T . The Hi correspond to the Fock states of the resonator mode ωi .
So in analogy to the Hilbert space (2.31) it is given by
Hi = {|nii , n ∈ N0 } .
18
(2.50)
Time-domain control of light-matter interaction with superconducting circuits
In a multi-mode picture the vacuum mode still has to be renormalized, this is very important for
the numerical fits of the theory presented here with the experimental data discussed in Chapter
4. However we now concentrate on one mode i as the Jaynes-Cummings Hamiltonian (2.47) does
not contain a coupling of different resonator modes. We would like to mention that we do not
drop the index i. In analogy to ref. [49] we reduce the Hilbert space (2.48) to
0
|nii
(2.51)
HSMJC = HQ ⊗ Hi = |n, eii =
, |n, gii =
, n ∈ N0 .
|nii
0
In this Hilbert space the eigenstates can be calculated analytically by introducing the mixing
angle ϑ [45, 49, 50]
|n, −ii
cos (ϑ) − sin (ϑ)
|nii |gi
=
.
(2.52)
|n, +ii
sin (ϑ) cos (ϑ)
|n − 1ii |ei
For a detailed derivation on the mixing angle ϑ we would like to refer to ref. [49]. According to
ref. [45] the mixing angle is given by the expression
√ 1
2gi n
ϑ = arctan
(2.53)
2
δωi
where we introduced the frequency detuning δωi := ωQ − ωi of the resonator mode i and the
qubit excitation frequency ωQ given in equation (2.25). With this new parameter we are able to
distinguish two different cases. In the first case we find that the qubit frequency ωQ ≈ ωi such
that the frequency detuning δωi is closed to zero. In this case we receive [45, 49]
1
sin (ϑ) = lim cos (ϑ) = √
δωi →0,gi >0
δωi →0
2
lim
As a result the dressed eigenstates are of the form
1 |nii |gi − |n − 1ii |ei
|n, −ii
=√
|n, +ii
2 |nii |gi + |n − 1ii |ei
(2.54)
(2.55)
and with energy levels separated by a factor
√
2gi n
(2.56)
as visualized in Figure 2.9 (a). In the uncoupled system (dashed lines) the energy states where
the total number of excitations is equal are degenerate, while by introducing the mixing angle ϑ
the eigenstates are separated with a photon number dependent energy splitting (solid lines). In
the case when the frequency detuning |δωi | > gi which is the so called dispersive limit [45, 50]
the interaction changes and the Hamiltonian can be approximated as [51]
~
gi2
gi2
disp
†
b
HJC ≈ ~ ωi +
σ
bz b
ai b
ai +
ωQ +
σ
bz ,
(2.57)
δωi
2
δωi
such that we end up with an energy spectrum as depicted in Figure 2.9 (b). Where the first
term represents the resonator with a shifted resonance frequency and the second corresponds to
the qubit whose excitation frequency is also shifted i.e. the resonator frequency of the coupled
system (solid lines in 2.9 (b)) is shifted by the value ±gi2 /δωi depending on the qubit state.
In both cases of the resonant and the dispersive coupled qubit resonator system the resonator
is affected by a frequency shift such that this mechanism forms the physical basis of two-tone
spectroscopy introduced in section 3.2.1.2 and used in Chapter 4.
19
Time-domain control of light-matter interaction with superconducting circuits
Figure 2.9: The energy spectrum of the Jaynes-Cummings Hamiltonian for two different cases.
In (a) the excitation frequency of the qubit ωQ and the resonator mode ωi are equal. The dashed
green lines represent the uncoupled system when the qubit is in state |gi while the magenta lines
represent the excited state |ei. For the coupled system the eigenstates split up and the level
spacing is depends on the photon number. In the dispersive case the frequency detuning δωi
is large in comparison to the coupling gi of the qubit to the resonator mode ωi is depicted in
(b). The resonator mode is shifted by ±gi2 /δωi depending on the qubit state, while the qubit
excitation frequency is changed by (2n + 1)gi2 /δωi [50].
However the most interesting term in the Hamiltonian (2.57) is
~gi2 † ~g 2
b
ai b
ai σ
bz = i n
bi σ
bz
δωi
δωi
(2.58)
which from this point of view causes a photon number dependent shift of the qubit frequency.
Where n
bi is the photon number operator of mode i which counts the number of photons with
mode i in the resonator. This is more obvious if we rearrange the Hamiltonian (2.57) to
~
gi2
2gi2 †
~
gi2
†
disp
†
b
HJC ≈ ~ωi b
ai b
ai +
ωQ +
+
b
ab
ai σ
bz = ~ωi b
ai b
ai +
ωQ + (2b
ni + 1)
σ
bz .(2.59)
2
δωi
δωi i
2
δωi
In the dispersive limit of the coupled system the qubit frequency ωQ is shifted by the value
2ngi2 /δωi for a fixed number of photons and a vacuum Lamb shift gi2 /δωi [50]. This photon
number dependence of the AC-Zeeman shift helps us to calibrate the photon number in section
4.1.3. However, up to now we only talked about steady states which is experimentally accessible
trough spectroscopy. When we are interested in dynamics and the characteristic time scales we
have to consider decoherence effects. Therefore, we discuss the corresponding theory in the next
part.
20
Time-domain control of light-matter interaction with superconducting circuits
2.4
Dynamic and decoherence
In classical computation, information is stored in the classical states 0 and 1. When information
is stored in a quantum system with eigenstates |gi and |ei the situation is more complex since
one has to care about superposition of states, which can be expressed as [38]
Θ
Θ
|gi + sin
exp (iϕ) |ei .
(2.60)
|Ψi = cos
2
2
The state |Ψi is a superposition of the two pure eigenstates on the so-called Bloch sphere with
the polar angle Θ and the azimuthal angle ϕ as depicted in Figure 2.10.
Figure 2.10: The Bloch sphere representation of a qubit. The Bloch vector |Ψi at a certain time
is a superposition of the ground |gi and excited state |ei characterized by the angles ϕ and Θ.
It is important to keep this picture in mind to understand the physics and experimental accessibility of decoherence and the characteristic time scales linked to it. The qubit or in general the
two level system can be driven from the ground state |gi to the excited state |ei by applying
a π-pulse (see Figure 2.11) at its excitation frequency ωQ along the z-axis, so via a coupling
to σ
bz and therefore changing the value of the polar angle Θ. However, due to coupling of the
two level system to the environment various noise sources [26, 52] perturb the system and cause
decoherence. An important parameter for decoherence processes is the energy relaxation rate
Γ1 . We can calculate this rate via Bloch-Redfield-Theory [53, 54], according to ref. [26] in the
case of a flux qubit as the two level system the relaxation rate is given as
Γ1
Γ1 = πSΦ
,
(ωQ )
(2.61)
Γ1
where SΦ
is the symmetrized noise spectral density [26] depending on the qubit excitation
(ωQ )
frequency ωQ .
21
Time-domain control of light-matter interaction with superconducting circuits
Figure 2.11: An example of a Rabi or π-pulse which excites the two level system from the ground
state |gi to the exited state |ei. In time-domain measurements the length τ of a π-pulse at the
qubit excitation frequency ωQ is the first value which has to be specified.
The inverse of the energy relaxation rate is called the energy relaxation time T1 := (Γ1 )−1
and it can be interpreted as the time which the qubit needs to relax from the exited state |ei
to the ground state |gi. This time is experimentally accessible via so called Rabi oscillation
measurements which are presented in Chapter 4. Despite time dependent motion of the polar
angle Θ the azimuthal angel ϕ can also vary in time. This process is often called dephasing
[26, 50]. According to refs. [26, 55] there are several mechanisms which cause dephasing. The
total dephasing is a sum of all these mechanisms and we find the expression [50]
Γ2 =
Γ1 X
+
Γ2,m .
2
m
(2.62)
For a detailed analysis of the different dephasing mechanisms we would like to refer to refs.
[26, 50, 52, 55]. We would like to mention that the measured dephasing time T2 := (Γ2 )−1
depends on the experimental technique one applies [26]. For all these experimental techniques
like Spin-Echo and Ramsey fringes [26, 50] the pulse length τ of a π-pulse has to be known.
The determination of that pulse width is inevitable and should be one of the first measurements
in the time-domain. All rates introduces so far influence the timescale for which a functional
quantum computer can store the information and perform algorithms accurately. In summary
since the two level system is not perfectly isolated from the environment1 the system is affected
by decoherence. To determine the decoherence rates one has to perform time-domain measurements. In a first time-domain measurement one has to determine the length of a π-pulse in order
to gain access to the explicit form of the pulse sequences which one has to apply in subsequent
measurements.
In this chapter we presented the necessary theory to describe the coupled qubit resonator system.
The flux qubit is described in an RCSJ model and its energy spectrum is determined by the flux
trough the qubit loop which causes a persistent current in the loop. The resonator is modeled
as a sum of harmonic oscillators where we described two scenarios. In the first scenario the
mode spacing is equidistant and in the second case the mode spacing is non-equidistant due to a
Josephson junction which causes a mode dependent phase drop [46]. The energy spectrum of the
coupled qubit resonator system can be understood by concerning the Jaynes-Cummings Hamiltonian [28]. In the last section we presented a short introduction to decoherence in our system
and depicted the idea how to determine the relaxation and dephasing rates in an experimental
setup.
1
On the other side if the system would be perfectly isolated, we would not be able to determine the state of
the system by measurement.
22
Time-domain control of light-matter interaction with superconducting circuits
3
Experimental setup
As mentioned in the previous parts the aim of this work is to couple a superconducting resonator
to a flux qubit and observe the response of it in continuous wave spectroscopy, pulsed wave
spectroscopy and time-domain measurements. As we intend to observe quantum mechanical
phenomena the sample has to be cooled down to very low temperatures (in our case approximately
50 mK) to reduce the number of thermal photons in the resonator to a negligible value and as
a result the quantum properties become dominant. We first introduce the experimental setup
used in the dilution refrigerator including the sample and afterwards treat the room temperature
components.
3.1
Cryogenic setup
In this section we introduce the sample used for our experiments and explain the different production steps. The sample is mounted into a sample holder which is then built into an in-house
fabricated cryostat with input and output chains. These chains consist of microwave cables,
attenuators, circulators and amplifiers and have an optimized signal to noise ratio under the
boundary condition of a limited cooling power at the different temperature stages. To couple
magnetic flux into the qubit loop we use DC lines connected to a superconducting coil with a
persistent current switch. These components are also introduced.
3.1.1
Sample
The sample we investigate is produced with the help of optical and electron beam lithography
as well as reactive-ion etching at the WMI. Here we give a short overview of several production
steps which are necessary to fabricate a superconducting resonator and a flux qubit. At this
point the author wants to thank M. Haeberlein for the production of the sample, because the
author was not involved in the production of the sample and without the contribution of Mr.
Haeberlein this thesis would not have been possible.
3.1.1.1
Superconducting niobium resonator and antenna
The resonator is produced via optical lithography and reactive-ion etching. The main production
steps are as follows. At the beginning one sputters approximately 100 to 200 nm niobium onto
a Si-substrate. Afterwards one spins a resist on top of the niobium. In the next step one covers
the resist with a chromium-plated mask which contains a negative of the pattern one wants to
produce [see Figure 3.1 (a)] and applies a flood exposure of UV light on the resist which is not
covered by the mask in a mask aligner. Afterwards one applies chemical development to remove
the resist which was exposed to UV light followed by a reactive-ion etching process to remove
the niobium which is not covered by the resist. Finally, one does a resist stripping. Now one
has produced a chip with a niobium resonator and an antenna on it as well as the connection
pads needed to contact the sample. If one has produced more than one sample it might be of
interest to check one of the samples in a 4 K cryostat (as the critical temperature of niobium is
TCNb = 9.2 K) to gain information about the superconducting resonator for example its quality
factor.
23
Time-domain control of light-matter interaction with superconducting circuits
Figure 3.1: The Chip design (5 by 10 mm) used in the course of this thesis with a transmission
line resonator (blue). The resonators coupling capacitors Cin and Cout located at the green
boxes fix the boundary conditions for the resonator modes since they fix the geometrical length
to 19 mm. The shape of the resonator takes into account that the so called box modes, which
arise from the geometrical extensions are shifted to higher frequencies such that they are not in
the bandwidth of our experiment. One of the coupling capacitors is depicted in (b). During the
reactive-ion etching the inner conductor of the CPW has been interrupted in order to build a
coupling capacitor. The qubit is located at 1/4 of the resonator length [magenta box in (a)] such
that it can be excited by a signal in the antenna (red). An SEM picture of the qubit is depicted
in (c). The qubit is colored for clarity. The Josephson junctions of the qubit [blue boxes in (c)]
and the coupling junction [red box in (c)] are clearly visible. The right edge of the loop also
forms the shared edge of the qubit and the resonator and determines the mutual inductance M .
During the course of this thesis a chip with the same fabrication parameters is used.
24
Time-domain control of light-matter interaction with superconducting circuits
3.1.1.2
3 Josephson junction flux qubit and coupling junction
In order to fabricate the 3 Josephson junction flux qubit on a chip we have to produce Josephson
junctions out of aluminum and aluminum oxide. There are two identical junctions and the third
so called α-junction with a relative size α ≈ 0.7 compared to the others. This qubit is located
at 1/4 of the length of the resonator. Via a shared edge the qubit is coupled galvanically and
inductively with an geometric inductance Lgeo to the resonator [45]. We try to increase the
inductive coupling with the help of a fourth junction with a relative size factor β ≈ 1 compared
to the other junctions of the qubit [see Figure 3.1 (c)]. The inductive coupling is determined by
the mutual inductance M as could be seen from equation (2.45). In our case we assume that the
mutual inductance is given as M = Lgeo +LJ . Since the geometric inductance Lgeo is fixed by the
design of the qubit the inductance the coupling junction LJ could be changed easier1 . However,
we can tune the inductance of the coupling junction and therefore enhance the coupling. It was
reasonable to reduce the size of the fourth junction because according to ref. [45] the mutual
inductance M is proportional to the Josephson inductance LJ ∝ 1/A of the coupling junction,
where A denotes the area of the junction. We can enhance the coupling gi by reducing the size
of the junction as could be seen from equation (2.45). To produce Josephson junctions we have
to produce a pattern for the shadow evaporation process via electron beam lithography [56, 57]
at first. A sketch of the shadow evaporation process is depicted in Figure 3.2.
Figure 3.2: A sketch of the shadow evaporation technique. By depositing material under two
different angles on a substrate with the help of a mask we would deposit it on two different
locations and we would end up with a nonfunctional sample (a). This can be solved by using
a mask structure as depicted in (b). After the evaporation and oxidation process we produced
tunnel junctions but also ghost structures [56, 57].
1
Both values are fixed during the fabrication process. But the production of a functional flux qubit with
different sizes of the coupling junction is more straight forward than a in the case of different areas.
25
Time-domain control of light-matter interaction with superconducting circuits
In the shadow evaporation process we deposit aluminum under an angle ζ on the chip [red trace
in Figure 3.2 (a)] in a first step. To produce the thin insulating tunnel barrier of the Josephson
junction we oxidize the aluminum we evaporated on the chip in the first step. The oxidation time
determines the thickness of the insulating barrier. With a second evaporation under an angle
−ζ (blue trace) we deposit another layer of aluminum afterwards. After a lift-off we produced
the desired sample [see Figure 3.1 (c)] for this thesis. For a more detailed description of these
processes we would like to refer to refs. [45, 58].
3.1.2
Cryostat
After the sample is mounted in a suitable sample holder it is cooled down to a base temperature of 43 mK of the 3 He/4 He dilution refrigerator. For a detailed introduction to the different
components of a dilution refrigerator we would like to refer to refs. [45, 59]. The sample stage
temperature is stabilized at approximately 50 mK via a resistance measurement bridge and a temperature controller. To protect the experiment from fields that might perturb the measurement
the cryostat is covered by a cryoperm and mu-metal shield which are surrounded by a shielding
room finally. The basic experimental setup which is used during this thesis and mounted inside
this shielding room is depicted in Figure 3.3. Other measurement equipment located outside of
the shielding room is introduced in section 3.2.
Figure 3.3: A schematic of the experimental setup inside the shielding room. The different temperature stages are visualized including the devices at the specific temperature. The attenuators
thermalize inner and outer conductor of the feed lines and are chosen such that the signal to
noise ratio (SNR) arriving at the sample is optimized, the numbers can be found in the text. The
circulators in the output guarantee that no signal is entering the sample from higher temperature
stages but the signal from the sample can pass trough.
26
Time-domain control of light-matter interaction with superconducting circuits
3.1.2.1
Microwave input lines
The input line of the resonator is connected to a Rohde & Schwarz ZVA24 [60] vector network
analyzer (VNA) during continuous wave spectroscopy and thermalized at each temperature stage
via an attenuator to reduce thermal noise by thermalizing inner and outer conductor at the
same time. According to ~ω = kB T where kB = 1.38065 · 10−23 J/K represents the Boltzmann
constant a thermal photon with a frequency ω/2π of 5 GHz would correspond to a temperature
T of approximately 240 mK. If we would connect the inner and outer conductor which are at
room temperature to the resonator without thermalization we would populate the resonator with
more than 1200 thermal photons. By thermalizing the lines at different temperature stages we
attenuate not only the signal but also the noise of the higher temperature stage but we add an
amount of thermal photons corresponding to the actual temperature stage. The added noise
quanta can be calculated with [59]
nPh =
1
1
+
.
2 exp ~ω − 1
kB T
(3.1)
The term 1/2 is due to the vacuum fluctuations. The task is to optimize the signal to noise
ratio (SNR) entering the sample. So obviously it would be the best to attenuate only at the last
temperature stage. Because there one attenuates the signal and the high temperature noise by
a desired value of the attenuator and adds only a small number of thermal photons. However,
due to the limited cooling power this is not possible and one has to dissipate power at higher
temperature stages where the cooling power is sufficient. Therefore, a boundary condition of our
optimization problem is the limited cooling power at each temperature stage. According to ref.
[61] the maximum allowed power dissipation values for each stage of our cryostat are as given in
Table 3.1.
stage
1K-pot
still
step exchanger
sample stage
temperature (K)
1.2
0.700
0.100
0.050
max. dissipation (µW)
20000
500
200
50
Table 3.1: The dissipated power of each temperature stage in the cryostat we use is limited by
the values listed here.
So we calculate the power Pdiss dissipated by the attenuator Att at the different temperature
stages, via the formula [61]
Pin (dBm)
Pin (dBm)−Att(dB)
10
Pdiss (µW) = 1000 10 10
− 10
.
(3.2)
Here we only consider the input power Pin of the signal entering from the next higher temperature
stage via the resonator and the antenna chain because the power of the thermal photons is several
orders of magnitude lower. By using equations (3.1) and (3.2) consistently for the two feed lines
in Figure 3.3 we end up with the numbers presented in Table 3.2 for the resonator feed line at
a frequency of ω/2π = 7.106 GHz which corresponds to the 3λ/2 mode of the resonator. This
mode is used as probe frequency in two-tone spectroscopy measurements in section 4.1. The
dissipated power is calculated for an output of the VNA of −45 dBm2 which is one of the highest
input powers used during continuous wave spectroscopy.
2
The output power of the VNA is −5 dBm which is attenuated by 50 dB at roomtemperature (see Figure 3.5)
such that −45 dBm enter the resonator input line.
27
Time-domain control of light-matter interaction with superconducting circuits
stage
room temperature
He bath
still
step exchanger
sample stage
T (K)
300
4.2
0.700
0.100
0.050
Att (dB)
—
10
10
10
20
nPh
880
100
12.1
1.75
0.519
Pdiss (µW)
0.028
< 0.005
< 0.005
< 0.005
< 0.005
Table 3.2: The dissipated power the resonator feed line for the used attenuator configuration
introduced in Figure 3.3. The numbers in the fourth column are the sums of the attenuated
photons from the next higher temperature stage and the thermal photons added at the specific
temperature stage. Here we used a resonator frequency of ω/2π = 7.106 GHz. It turns out
that we are close to the limit given by vacuum noise. The dissipated power is calculated for an
input power of −45 dBm at the resonator feed line. The output power of 5 dBm of the VNA
was attenuated by 50 dB and applied to the resonator during the photon number calibration
measurements.
During continuous wave spectroscopy the on-chip-antenna is driven by a microwave source SMF
100A from Rohde & Schwarz [62]. While in the time-domain measurements we use a vector signal
generator E8267D PSG from Agilent [63] to drive the antenna. The inner and outer conductor of
this feed line is also thermalized by some attenuators at different temperatures. In Table 3.3 we
calculate the dissipated power in the antenna feed line for a frequency of ω/2π = 4.96 GHz which
is proportional to the energy gap ∆ of the qubit. According to equation (2.25) it is reasonable to
use this frequency as we want to excite the qubit via the antenna. Therefore, we have at least to
spend the energy of the gap at the degeneracy point. The relative high input power of −15 dBm
which is used to calculate the dissipated power is applied by the SMF to the antenna input port
at the shielding room during the first two-tone spectroscopy experiments in this thesis.
stage
room temperature
He bath
still
step exchanger
sample stage
T (K)
300
4.2
0.700
0.100
0.050
Att (dB)
—
20
3
5
20
nPh
1260
30.3
18.1
6.34
0.572
Pdiss (µW)
—
31.3
0.158
0.108
0.050
Table 3.3: The dissipated power the antenna feed line for the used attenuator configuration
introduced in Figure 3.3. The numbers in the fourth column are the sums of the attenuated
photons from the next higher temperature stage and the thermal photons added at the specific
temperature stage. Here we used a frequency of ω/2π = 4.96 GHz, which is equivalent to the
qubits energy gap ∆. The dissipated power is calculated for an input power of −15 dBm at the
antenna feed line. So this input power can be provided from the SMF in the case of continuous
wave spectroscopy or the PSG during the pulsed wave spectroscopy.
So the dominant part of the total dissipated power which is the sum of all power which are
dissipated is that from the antenna feed line. However, these values are still smaller than the
maximum values in Table 3.1 therefore, the cryostat is able to cool our experiment sufficiently.
28
Time-domain control of light-matter interaction with superconducting circuits
3.1.2.2
Microwave output line
The output of the resonator is connected to two circulators in series to protect it against noise
from higher temperature stages, the first one (7) in Figure 3.4 located on the sample stage at a
temperature of approximately 50 mK (see Figure 3.3 for wiring and Figure 3.4 for photos) and
the second (8) in Figure 3.4 is placed at the still stage (3) and therefore at a temperature of
700 mK. A circulator is a three port device, in our case lets call the ports input, output and
terminated. As depicted in Figure 3.3 the input of the first circulator is connected to the output
of the sample holder (5) and the output of the circulator is linked to a higher temperature stage.
The ferrite attached to the wave guides inside the circulator causes the scattering matrix of the
circulator to become antisymmetric, which means that a signal entering from the input port
is weakly damped to the output port but highly damped to the terminated port. The physics
behind this process is as follows in the case of a phase shift circulator. The signal entering the
circulator is split up in two equal parts where the ferrite causes a decrease of the group velocity
in one of the arms. So as a result on one port constructive and on the other port destructive
interference is realized [47, 64]. In our case this means that a signal entering the input interferes
constructively at the output and destructively at the terminated port. If now a signal or noise
is entering the output port it interferes destructively at the input port and so it is not able to
pass trough to the sample. At the terminated port it interferes constructively but that is not
relevant because this port is terminated by 50 Ω. A circulator where one of the three ports is
terminated by 50 Ω is also called an isolator. However, due to the ferrite a circulator used in
combination with a flux qubit has to be shielded such that the magnetic field of the ferrite does
not perturb our qubit. A cold high electron mobility transistor (HEMT) amplifier (9) in Figure
3.4 from Low Noise Factory Model LNF-LNC4_8A is mounted at 4 K to amplify the measured
signal with a gain of 40 dB. According to the data sheet [65] the bandwidth is limited to a range
of 4 to 8 GHz, here the gain and the added noise are almost constant. At room temperature
there is a third circulator and a second amplifier JS2 from Miteq with a gain of approximately
27 dB and a bandwidth from 2 to 8 GHz. Finally the signal enters the detection port of the VNA
in the case of continuous wave spectroscopy. In summary the bandwidth of our experiment is of
a limited range from 4 to 8 GHz.
3.1.2.3
DC lines
Up to now we introduced the microwave cables built into the cryostat which are necessary to
provide the RF signals to our experiment. However, as we intend to investigate a coupled
flux qubit resonator system we have to provide external flux to the qubit loop. So we have
to apply a magnetic field with the help of a superconducting magnet coil [(6) in Figure 3.4]
which is connected to DC lines. The coil is fabricated out of NbTi wire in a Cu matrix and
has an inductance of approximately 1 mH [45]. Driving a current If through the coil generates
a magnetic field and as a result a flux which can penetrate the qubit loop. When the desired
flux is trapped in the qubit loop we use a persistent current switch to disconnect the current
source from the magnet coil by keeping the flux in the coil as well as in the qubit loop constant.
This is useful for long time measurements at a constant flux value because we are not sensitive
to current fluctuations in the current source.
29
Time-domain control of light-matter interaction with superconducting circuits
Figure 3.4: A photograph of the cryogenic stage. In (a) the temperature decreases from 4.2 K
at the 4K-flange (1) over the 1K-pot (2), the still (3) at approximately 700 mK and the mixing
chamber (4) at a base temperature of approximately 43 mK. A view from the left side of the
upper part of the sample stage (green box) is depicted in (b). The sample mounted in the sample
holder (5) is biased with an external flux via the superconducting magnet coil (6). The backside
of the lower part of the sample stage (red box) is illustrated in (c). After the signal passed the
sample it enters a circulator (7), which is thermalized at approximately 50 mK. Then the signal
goes all the way up and enters a second circulator (8) at the still stage and is amplified by a cold
HEMT (9) at 4.2 K before it is sent to the room temperature components. More details can be
found in the text.
30
Time-domain control of light-matter interaction with superconducting circuits
3.2
Room temperature setup
In addition to continuous wave spectroscopy we also want to perform pulsed wave spectroscopy
and time-domain measurements. While continuous wave spectroscopy is a well established technique at the WMI pulsed measurements in the context of quantum information processing are
not. In this section we explain the continuous wave setup which is in use and then introduce in
detail how we generate the pulses sent to the sample inside the cryostat and how we detect the
response of our sample via an IQ mixer and a data acquisition card.
3.2.1
Continuous wave spectroscopy
A well established technique to investigate the interplay of a qubit and a resonator coupled
together is continuous wave spectroscopy [17, 18, 20]. This can be done by using a single probe
tone or by using two different tones, a so called probe tone with frequency ωP and a spectroscopy
tone ωS . The latter method is also a part of the basic functional principle of a Rabi oscillation
measurement where the probe tone is no longer continuous but pulsed. However, in this part
we introduce the devices used to perform single and two-tone continuous wave spectroscopy and
explain the basic functional principle of these measurements.
3.2.1.1
Single tone continuous wave spectroscopy
Single tone continuous wave spectroscopy is a fundamental technique. We need a signal source
which continuously generates a signal at a desired frequency. This signal is then sent to the sample
we investigate and the response of this signal is recorded by a spectrometer which detects the
same frequency at which the signal source is emitting the signal. If one hits a resonance frequency
of the sample it can transmit the signal to the detector, hence the measured transmission is large.
If one is now off resonance all the power is reflected and the detected transmission decreases.
In our case we send a continuous microwave signal and observe the response of the sample
with a vector network analyzer VNA. Therefore, we connect the two ports of our VNA ZVA24
from Rohde & Schwarz to the input and output of our superconducting resonator as labeled
in Figure 3.3. A VNA is able to generate a signal of desired amplitude and frequency within
its specifications with a built-in microwave generator and detect the response in transmission
or if desired even in reflection at the same time. In this thesis we concentrate on transmission
measurements during continuous wave spectroscopy. A VNA is even more powerful, it can detect
not only the amplitude but also the phase of the response. Therefore, the generated signal is
split up, the first part is sent to the sample and the second to a internal receiver, where it is
used as a phase reference. That is why we are able to measure a complex signal (amplitude and
phase) which we can interpret as a vector in R2 [66]. As mentioned, a VNA is able to measure
transmission and reflection so it contains an S-parameter test set.
31
Time-domain control of light-matter interaction with superconducting circuits
3.2.1.2
Two-tone continuous wave spectroscopy
Up to now we only talked about single tone continuous wave spectroscopy, where we could investigate the behavior of the coupled qubit resonator system dependent on the applied frequency
and power. The next step and therefore a more sophisticated technique is the so called two-tone
continuous wave spectroscopy, where one applies two signals which can differ in frequency, power
or both to the experiment. If one would only have one input line one has to use a power combiner to send the signals to the sample as it was done in ref. [45]. However, one can also use an
on-chip antenna to apply the second tone to the qubit. In this thesis this experimental approach
is realized at the WMI for the first time. We use once again the ZVA24 to apply the so called
probe tone ωP to the resonator and the second spectroscopy tone ωS is generated by the SMF
100A and sent to the experiment via the antenna input line in the cryostat as depicted in Figure
3.5.
Figure 3.5: The continuous wave setup used in single and two-tone experiments in this thesis. For
single tone continuous wave spectroscopy one does not need the SMF 100A so one can disconnect
it and terminate the antenna feed line with a 50 Ω resistor. The input signal to the resonator is
attenuated by 50 dB due to the limited dynamical range of the ZVA24.
All signal generators are connected to a 10 MHz Rubidium reference source from Stanford Research Systems Model FS 725 [67]. In our setup the measurement principle is as follows. Imagine
that your probe tone ωP is on resonance with the sample and the spectroscopy tone is off resonance or switched off as in Figure 3.6. So the VNA detects a high transmission of the total signal
which enters the sample. If the spectroscopy tone remains off resonant with the sample, the latter
can not be excited and the transmission remains almost constant. If now the spectroscopy tone
hits a resonance of the sample it can excite the system with microwave photons of frequency
ωS . According to the mechanism described in section 2.3 this causes an AC-Zeeman shift [blue
line in Figure 3.6 (a)]. Due to the linewidth of the signals the detected transmission measured
at the fixed probe frequency ωP decreases by ∆T r. We analyze the measured transmission of
the probe signal as a function of the spectroscopy tone ωS to detect the resonance of the qubit
i.e. we are able to measure the qubit hyperbola if we would apply an additional flux sweep. In
the time-domain measurement we use the phase change ∆P h depicted in Figure 3.6 (b) to gain
information about the sample. The phase shift is also caused by the AC-Zeeman shift of the
probe tone frequency.
32
Time-domain control of light-matter interaction with superconducting circuits
Figure 3.6: The functional principle of a two-tone spectroscopy for the measured transmission
(a) and phase (b). When signal generator SMF attached to the antenna is off (red line) we look
for a transmission peak of the resonator to assign the probe tone frequency ωP . By operating the
signal generator (blue line) on a resonance frequency of the system it affected by an AC-Zeeman
shift as described in section 2.3, due to the linewidth of the signals this causes also a reduced
transmission on the probe tone frequency ωP as pointed out in detail on the text. For this data
we used a driving amplitude of the antenna of −30 dBm and a frequency of ωS = 5.04 GHz at the
output of the SMF. We measured at the degeneracy point of the qubit such that ωS = ωQ = ∆/h.
3.2.2
Pulsed wave spectroscopy and time-domain measurements
With the experimental techniques introduced so far we are able to characterize our sample very
well in the frequency-domain, but we still suffer a lack of information regarding the timescales.
First of all we are interested in the relaxation time of our qubit T1 and second in the dephasing
time T2 . A naive approach to access these times is that one performs a Fourier analysis to the
data received from spectroscopy experiments. In the theory of Fourier analysis one has access to
the full spectrum i.e. to all frequencies but due to the limited bandwidth of the experiment and
the fact that one would have to perform a discrete and not an analytical Fourier transformation
one would end up with an approximation of the time response which is not satisfying. So one has
to perform a time based measurement. Therefore, we build up an IQ detector. Furthermore, we
implement protocols that generate and detect pulses with a length of a few ns up to µs efficiently.
In this section we introduce our approach to this task.
33
Time-domain control of light-matter interaction with superconducting circuits
3.2.2.1
Pulse generation
Pulsed measurements with more than one signal source are very sensitive to the phase position
of the different tones, so it is important to provide a reference signal which guarantees a stable
phase difference between each signal source. This is very important since we average over more
than 106 traces. If the phase difference would not be stable it might happen that the information
of the experiment one is interested in cancels out. The basic hardware and its wiring is depicted
in Figure 3.7.
Figure 3.7: The 10 MHz reference signal is applied to the two microwave sources (SMF and
PSG) and to the data timing generator (DTG) from Tektronix. The latter is used to apply a
trigger signal to the two sources as well as to the data acquisition card. With the SMF we drive
our resonator. We use the PGS to apply a pulse to the qubit via the on chip antenna to excite
the qubit. The two channels of the data acquisition card are recording a signal from an IQ mixer.
We use a 10 MHz Rubidium reference source from Stanford Research Systems Model FS 725 [67].
This reference signal is sent to a microwave source SMF 100 A, a vector signal generator E8267D
PSG from Agilent [63] and a data timing generator from Tektronix Model DTG 5334 [68]. From
the latter device a trigger pulse of desired shape with respect to delay and width is applied to
the microwave sources and the data acquisition card. In this thesis we use rectangular pulses
with a width from 5 ns up to 3 µs. The SMF applies a microwave signal of desired amplitude
and frequency to the input port of the resonator while the qubit is driven by the PSG. We use
the internal mixer of the PSG to generate pulses similar to that depicted in Figure 2.11 during
the time-domain measurements in section 4.2. The output of the resonator is connected to an
IQ mixer which splits up the signal into two parts. These two chains are connected to the data
acquisition card which records the amplitude and phase of the response at a certain time, the
details on this procedure are explained in the next part.
34
Time-domain control of light-matter interaction with superconducting circuits
3.2.2.2
Pulse detection
The pulses generated as described in the previous part are sent into the cryostat. After they
passed the sample and the amplification chain depicted in Figure 3.3 the pulses are split up with
an IQ mixer into two channels. Finally these two channels are detected with the help of an
ACQIRIS card from Agilent Model DC440 [69] with a sampling rate of 400 MSa/s3 . A sampling
rate of 150 MSa/s4 is used in the FPGA (f ield programmable gate array) board from Innovative
Integration [70] to speed up the measurements. We can speed up the measurements due to the
fact that the FPGA is able to average the acquired data very efficient such that we are able to
use every second trigger pulse for data acquisition. This is not possible with the ACQIRIS card,
where on average from 16 pulses only the first one is recorded while the other 15 pulses are lost
because the average calculations are still taking place. As a result the measurement time for the
same experiment carried out with the FPGA card is reduced by a factor of 8 in comparison to
the ACQIRIS card measurement but with a reduced time resolution of 150 points per µs instead
of 400 points per µs. However, we first concentrate on the ACQIRIS card. Due to the fact that
the data acquisition card is not able to work at GHz frequencies the signal has to be converted
down to several MHz by an IQ mixer with additional components such as amplifiers and filters.
The used hardware is depicted in Figure 3.8.
Figure 3.8: The signal from the experiment is first filtered using a band pass and then converted
down in frequency and split up into an in phase signal (I) and a quadrature component (Q).
To avoid compression in the amplifiers of the I- and Q-chain we attenuate the signal by 20 dB.
Before entering the ACQIRIS card we apply a low pass filter with a bandwidth from DC up to
32 MHz.
The signal from the output chain of the experiment inside the shielding room is band pass filtered
by a Mini-Circuit VHF-4600+ [71] before it enters the RF port of the IQ mixer from MITEQ
Model IR0408LC2Q , which is connected to a second microwave source. In our case a second
SMF which provides the local oscillator frequency at an output power of the SMF of 12.5 dBm.
The first SMF is used to drive the resonator. The bandwidth of the used IQ mixer is from 4 to
8 GHz. The basic functional principle of an ideal IQ mixer is shown in Figure 3.9.
3
4
the maximum sampling rate is 420 MSa/s, during the experiments we use a sampling rate of 400 MSa/s
the maximum sampling rate is 200 MSa/s but the used logic supports a sampling rate of 150 MSa/s
35
Time-domain control of light-matter interaction with superconducting circuits
Figure 3.9: The basic idea of an ideal IQ mixer is to convert the frequency down from the RF
input ωP with help of a local oscillator ωLO to the intermediate frequency ωIF := ωP − ωLO and
to split the signal into an in phase signal I and a quadrature component Q
One splits the measured RF signal into an in phase channel I and an a quadrature component
channel Q and converts it down to an intermediate frequency ωIF := ωP − ωLO such that one can
extract the amplitude A(t) and phase ρ(t) at any time of the experiment [50]. The intermediate
frequency ωIF is obtained from the local oscillator frequency ωLO and the probe tone ωP at the
RF input port by using an addition theorem. In this theorem we end up with the sum and
the difference of the two frequencies. The sum is filtered out from a band pass inside the IQ
mixer while the difference is kept as intermediate frequency ωIF . However, due to imperfections
in both output arms e.g. the two amplifiers (both from MITEQ model AU15-25 [72], with
a typical gain of 65 dB within their bandwidth from 1 to 300 MHz) do not have exactly the
same characteristics and the phase shift of the IQ mixer is not exactly π/2 this setup has to be
calibrated as described in Appendix A. Nevertheless, it is also worth to have a look at the raw
data before starting calibration. The raw data is low pass filtered with a Mini-Circuit SLP-30+
[73] and recorded by the ACQIRIS card. To record the calibration pulse we operate the PSG at
an output power of −100 dBm with a frequency ωP /2π = 7.109 GHz and connect it to the RF
input of the IQ mixer setup depicted in Figure 3.8 via a microwave cable directly without the
sample between the PSG and the IQ mixer. The local oscillator microwave source was operating
at ωLO = 7.099 Ghz and a power of 12.5 dBm such that the intermediate frequency is ωIF =
10 MHz. This intermediate frequency is identical to the one used in section 4.2. We set ωIF > 0
to avoid signal spectrum inversion [64]. The internal mixer of the PSG was used to fold the
continuous microwave generated by the PSG itself with a rectangular pulse with a period of 3 µs
and a delay and width of 1 µs generated by the DTG (see Figure 3.7 for wiring). So if the IQ
mixer would work perfectly we expect a recorded signal which is zero for the first microsecond,
a sine with a fixed amplitude A(t) = const. and a time independent phase ρ(t) == const. in
the next microsecond and once again zero for the rest of the pulse. The data of the recorded
calibration pulse is depicted in Figure 3.10 (a).
36
Time-domain control of light-matter interaction with superconducting circuits
Figure 3.10: A calibration pulse recorded by the ACQIRIS card for 3 µs and averaged 20 million
times for a output power of −100 dBm at the PSG. The raw data in (a) still seems to be noisy
and it is clearly visible that channel 2 has a DC offset. In (d) we show a zoom in of the green
box in (a) detect a spurious signal with a frequency of 200 Mhz. By applying a digital sliding
window filter of that frequency on both channels we end up with the figure depicted in (b).
Channel 2 is still perturbed by the pattern shown in (e) as a zoom in of the magenta box in (b).
We would like to stress that (e) shows the pattern only once and that it has a width of 32 data
points corresponding to a frequency of 12.5 MHz. If we compensate this pattern with a hopping
window filter we end up with the figure in (c).
37
Time-domain control of light-matter interaction with superconducting circuits
As depicted in Figure 3.10 (a) the data is of desired shape, the fact that the sine does not start at
exactly 1 µs is due to delays caused by the internal mixer of the PSG. The intermediate frequency
seems to be 10 MHz because we observe 10 periods in one microsecond. Although the data are
averaged 20 million times they still seem to be noisy, which is not expected because for a large
number of averages and due to the law of large numbers the signal trace should be smooth and
noise should cancel out. By having a closer look at the data it turns out that these are artifacts
added by the ACQIRIS card, which could be filtered digitally. The first feature we discover is
that there is a high frequency sinusoidal structure with a frequency of 200 MHz on the signal s
as shown in Figure 3.10 (d). This is exactly half the sampling frequency of the AQCIRIS card
and it is hard to imagine that is caused by the signal because of the mounted low pass filter
from DC to 32 MHz. This pattern could be removed easily by applying a so called digital sliding
window filter. The idea of such a filter is that one averages over one period τ from a time t
to a time t + τ and takes this new value as the value for time t. We have to choose the size
of the window w such that the condition 2πw/ωSa = τ is fulfilled, where ωSa /2π = 400 MHz
is the sampling frequency of the ACQIRIS card. In this case if we want to filter a 200 MHz
artifact from the signal s we have to choose a window which consists of two data points such
that the width w = 2. By integrating a sine or cosine function over a whole period the result
is zero because the positive and negative areas are equal. So we use numerical quadrature and
implement a modified trapezoidal rule for this task in the form of
si =
w−1
1 X
s(i+j)
w
with i = 1, 2, . . . S − w,
(3.3)
j=0
where S denotes the length of the signal (product of the sampling rate and the readout time).
Due to the fact that the filter acts from the first data point to the S − w-th data point, we loose
one period by applying this filter. As a result, by applying the sliding window filter one filters
out the frequency 1/τ and also higher harmonics and ends up with the average value of all other
frequencies. A MATLAB function of this filter can be found in Appendix A.2. The result of this
operation on every channel is visualized in Figure 3.10 (b). Channel 1 now appears smooth and
as expected for 20 million averages, none the less channel 2 still has a pattern with a period of
80 ns on it. According to the sampling rate ωSa of the ACQIRIS card the width of the pattern
is w = 32 data points. The shape of the pattern is depicted in Figure 3.10 (e). To obtain the
shape of the pattern and filter it out one could do something similar to a sliding window filter.
So we use what we call hopping window digital filter. The respective MATLAB function can
be found in Appendix A.2. To extract the shape of the pattern we average not over a period
as we do in the sliding window filter but now we average over the value of the signal s at a
certain position of the pattern pi over a number of periods K, this operation is given by the
mathematical expression (3.4).
pi =
K−1
1 X
s(i+jw)
K
with i = 1, 2, . . . w
(3.4)
j=0
Where pi is the value of the pattern at position i and w is the width of the pattern/window as
defined in the case of a sliding window filter. The next step is to subtract this pattern from the
signal trace so one has to shift the window by a fixed steps of with w. In detail we subtract that
pattern p from the first 32 points of the signal s than the pattern is hopping to the next 32 data
points where it is subtracted and so forth. There exists a pitfall if the length of the recorded
signal is not a multiple of the width w of the window. There are two possible solutions either
one disregards the tail of the signal or one subtracts only the first part of the pattern which has
the same length as the tail. We decided to disregard the tail of the data presented in this thesis
because the readout times where chosen long enough such that we do not loose any information.
38
Time-domain control of light-matter interaction with superconducting circuits
Another advantage of the hopping window filter is that it also acts as a DC filter, because one
subtracts a fixed structure from the signal and as one could see from equation (3.4) it takes into
account constant terms such as DC components, as could be seen in in Figure 3.10 (c). Here
we end up with a smooth signal which could now be used to calibrate the IQ mixer. However,
we had a closer look at the origin of the artifacts and it turns out that they are added by the
ACQIRIS card itself and are linked to the usage of the external reference input port where the
SRS FS 725 is linked to, so a possible reason for the presented features is an internal problem of
the ACQIRIS card as could be seen from the three different scenarios in Figure 3.11.
Figure 3.11: Three calibration pulses recorded by the ACQIRIS card for 3 µs and averaged 20
million times for a power output of −100 dBm at the PSG for different reference sources. In
(a) we connected the SRS to the external reference port of the ACQIRIS and the 10 MHz signal
from the SRS is used as reference frequency. In (b) we disconnected the SRS and measured with
the internal reference frequency. Compared to (a) we got rid of the artifacts but received bigger
DC offsets and the phase of the sinusoidal signal changed. The external reference was connected
but the internal was used for the measurement in (c). Despite different DC offsets there is no
difference in comparison to (b).
39
Time-domain control of light-matter interaction with superconducting circuits
In Figure 3.11 (a) the raw data first time presented in Figure 3.10 (a) is shown again for comparison, here we connected the SRS to the external reference port of the ACQIRIS and the 10 MHz
signal from the SRS is used as reference frequency, while in (b) the SRS is disconnected, the
input port is terminated and the internal reference is used during the measurement. It turned
out that we get rid of the artifacts we have in (a) but we observe larger DC offsets. The phase
of the sinusoidal signals at the beginning of the pulse also changed, which is not surprising if an
other phase reference is in use. In the last case depicted in (c) we connect the SRS again but
still use the internal reference source, despite different DC offsets the recorded data is the same
as in (b). So there is strong indication that the circuit of the external reference of the ACQIRIS
card is damaged. However, this does not perturb our measurements because as we showed here
we implement a powerful digital filter which cancels out the artifacts. The next thing one has
to guarantee is that one does not operate the amplifiers above their compression points. So we
measured the 1 dB compression point of the devices mounted as visualized in Figure 3.8 at room
temperature. Therefore, the PSG was directly connected to the RF input of the IQ mixer. The
compression point denotes the lowest input power at which the gain is reduced by more than
1 dB. The results of this measurement for an uncalibrated but digitally filtered IQ mixer is shown
in Figure 3.12.
Figure 3.12: The 1 dB compression point of the used amplifiers in the IQ detector depicted
in Figure 3.8. We plotted the measured gain vs. the RF-input power of the IQ mixer. The
dashed horizontal lines are a fit to the part where the gain is almost constant (input power from
−65 dBm to −30 dBm.) and provide a gain of 44.7 dB for Q and 44.2 dB for I. The small steps
at low powers arise from internal step attenuator from the input source. The 1 dB compression
point is then given by −23.1 dBm for the I-channel and −23.6 dBm for channel Q, which is more
clearly visible in the inset.
40
Time-domain control of light-matter interaction with superconducting circuits
As we now want to concentrate on the IQ mixer itself, channel 1 of the ACQIRIS card will be
denoted by I and channel 2 with Q for the same reason. From the time trace recorded by the
ACQIRIS card we fitted a sine for each power value to receive a fitted amplitude u
b. Because the
circuit is matched to R = 50 Ω the gain could be calculated via:
Pout =
2
Ueff
u
b2
=
R 2R
gain(dB) = 10 1 + 2 log10
(3.5)
u
b(V)
1V
− Pin (dBm)
(3.6)
The steps at low power in both channels arise from the internal step attenuators of the signal
source. If there were no other loss mechanisms in the two chains one would expect a theoretical
gain of 45 dB. Here the measured values of 44.7 dB for Q and 44.2 dB for I are in the expected
range. It turns out that the 1 dB compression point is at −23.1 dBm for the I-channel and
−23.6 dBm for channel Q. The difference in power shows that a calibration of an IQ mixer is
inevitable. The measurements presented in section 3.2.2 is driven at powers such that the RF
input power of the IQ mixer is approximately −100 dBm, so that we do not operate the amplifiers
beyond their compression point. After we got rid of the artifacts added by the ACQIRIS card
and we showed that we do not operate the amplifiers beyond their compression point, we can
now concentrate on calibrating the IQ mixer i.e. we apply a correction matrix which removes
DC offsets, equals the amplitudes of the two channels and sets the phase shift between them
to π/2. Despite the fact that there already exists a digital homodyning calibration method [50]
we decided to implement our own digital heterodyning calibration method. In a homodyning
calibration method the intermediate frequency ωIF is set to zero while in a heterodyning method
it is of the order of several MHz [50]. Our heterodyning method is based on fitting an ellipse
[74] in homogeneous coordinates in the real projective plain RP2 [75, 76] to the recorded data
in a numerical stable way [77] and afterwards transforming the ellipse into a circle. By doing
so we are able to remove DC offsets of both channels, let them be phase separated by π/2 and
the amplitudes of the two channels will become equal to the radius of the circle. A detailed
derivation of the correction values from the fitted ellipse parameters can be found in Appendix
A. Here we introduce the main steps of the correction algorithm and show the effects to the data
presented in Figure 3.10. We take the calibration pulse shown in Figure 3.10 (c), remember that
there have been applied digital filters on it, and plot I vs. Q for a time from 1.2 to 2.1 µs, this
is visualized in Figure 3.13 (b).
41
Time-domain control of light-matter interaction with superconducting circuits
Figure 3.13: A calibration pulse recorded by the ACQIRIS card for 3 µs and averaged 20 million
times for a output power of −100 dBm at the PSG. The raw data in (a) is already digitally
filtered to remove the artifacts generated by the ACQIRIS card. For the calibration of the IQ
mixer only the region between 1.2 µs and 2.1 µs is used. Therefore, we plot I vs. Q in (b) and
fit an ellipse to the data points. After the calibration algorithm was applied to the data and the
fitted ellipse we end up with a circle depicted in (c). In (d) channel I and Q are plotted versus
time here it is visible that the two signals have the same amplitude and are phase separated by
π/2.
It turns out, that as expected the signal is shifted by a DC offset in both channels. The fitted
values are DCI = −34 µV for I and DCQ = −52 µV for Q. The amplitude of the respective
channels is AMI = 510 µV and AMQ = 543 µV. The phase difference of the signal is fitted to
be δ = 4.5◦ . We apply a correction matrix C of the form written down in equation (3.7) to the
recorded data given in homogeneous coordinates.


∗ ∗ ∗
C = Rη−δ AM R−η DC = ∗ ∗ ∗
(3.7)
0 0 1
Here the matrix DC takes into account DC

0

DC = 0
0
offsets in both channels so its explicit shape is

0 −DCI
0 −DCQ  .
(3.8)
0
1
With DCI and DCQ as the fitted values for a DC offset.
42
Time-domain control of light-matter interaction with superconducting circuits
The next three transformation matrices Rη−δ AM R−η correct the amplitude and phase difference
of both channels at the same time in the eigenbasis of the ellipse. Therefore, we lost the phase of
channel I at the beginning of the pulse, as we want to recover the phase of channel I we correct
the angle error received in the dilation AM by an additional rotation with the angle −δ. The
origin of this angle error is due to the fact that the dilution factor in the first direction of the
eigenbasis of the ellipse differs from the factor according to the second eigenvector.
Rη−δ AM R−η =


  AMI

cos(η − δ) − sin(η − δ) 0
0
0
cos(η) sin(η) 0
SMA
AMI
 sin(η − δ) cos(η − δ) 0  0
0 − sin(η) cos(η) 0
SMI
0
0
1
0
0
1
0
0
1
(3.9)
Where η denotes the angle between the semi-mayor axis SMA and the x-axis. We decided to
set the amplitude of the calibrated signal for both channels to the analytically calculated value
for the amplitude of channel I AMI . SMI corresponds to the length of the semi-minor axis of
the ellipse. All the correction values presented here can be calculated from the fitted parameters
a, b, . . . , f of the ellipse, which we extracted from the recorded data by using Fitzgibbon’s method
[74]. The main differences to a homodyning calibration method [50] are that we do not have to
transform the data in a rotating frame which results in a multiplication of a universal matrix
(due to time dependency) to each signal point. Further we are in a more general case where
no small angle approximation is used to assign the correction matrix and finally we are able
to remove DC offsets. These would be converted up to the intermediate frequency ωIF in the
rotating frame due to the matrix multiplication in the homodyne method. After we applied the
correction matrix C we end up with Figure 3.13 (c). Now I and Q are calibrated which means
that they have the same amplitude and are phase separated by π/2. This is more clearly visible
in a parametric representation with respect to the time in Figure 3.13 (d). All four graphs have
in common that they are affected by a drift which arises from the nearly linear increase of the
measured amplitude in I and Q over the calibration region [see Figure 3.13 (a)]. The origin of this
increase might be a ground loop or a periodic signal with a frequency of a few kHz. This results
in an oscillation of the amplitude A(t) and phase ρ(t) at the intermediate frequency ωIF . If we
would apply a digital sliding window filter at that frequency on the channels I and Q we would
also cancel out the information we want to get through the measurement. Therefore, we apply
a digital sliding window filter after we extracted the time dependent amplitude A(t) and phase
ρ(t) . This has to be clarified in future works. However, before we proceed with a reconstruction
of the time dependent amplitude A(t) and phase ρ(t) which carries the information about our
experiment we would like to compare the recorded raw data with the filtered and calibrated data,
this is visualized in Figure 3.14.
43
Time-domain control of light-matter interaction with superconducting circuits
Figure 3.14: In (a) the recorded raw data of the ACQIRIS card is shown, while in (b) we present
data filtered by using a sliding window filter with a frequency of 200 MHz, a hopping window
filter at a frequency of 12.5 MHz and finally calibrating the two channels of the IQ mixer by
using the digital heterodyning method introduced in this thesis.
The applied digital filters and the heterodyne calibration method improves the data quality a lot,
when we compare the raw data in Figure 3.14 (a) to the calibrated and filtered data in (b). The
sliding window filter smoothed the two curves while the hopping window filter removed the DC
offset of channel Q and the periodic structure presented in Figure 3.10 (e). Finally the heterodyne
calibration method causes the amplitudes of channel I and Q to become equal and let the two
signal channels be phase shifted by π/2. The so recorded and calibrated pulses can now be used
to extract the time dependent amplitude A(t) and phase ρ(t) of the experiment. In the case of a
calibration pulse (which in our case is a pure sine signal with a time independent amplitude and
phase) the amplitude and phase deviation should be almost constant since there is no device in
the measurement chain that disturbs the signal generated by the microwave source. However,
with a sample mounted between the signal generator and the detector we get a response from
which we can extract for example the characteristic time scales of our coupled qubit resonator
system. It can be shown [50] that one can calculate the amplitude and phase out of the recorded
quadratures I and Q via
q
2 + Q2
A(t) =
I(t)
(3.10)
(t)
Q(t)
ρ(t) = arctan
(3.11)
I(t)
By applying these formulas to the data presented 3.13 (d) we end up with an amplitude and
phase for every recorded time increment. The result is visualized in Figure 3.15.
44
Time-domain control of light-matter interaction with superconducting circuits
Figure 3.15: The reconstructed amplitude and phase for a calibration pulse. In (a) the reconstructed amplitude A(t) for the data presented in Figure 3.13 (d) is plotted for each time
increment. The blue circles depict the calibrated data before a digital sliding window filter at
the intermediate frequency ωIF was applied while the green circles show the recorded data after
that filter was used. The red line indicates the case of perfect calibration without any source of
noise. The same as in (a) but for the extracted phase ρ(t) is shown in (b).
In Figure 3.15 (a) the amplitude at every recorded time increment is centered around 510 µV
which is the fitted amplitude of channel I and so the desired value as we set it equation (3.9). In
(b) the phase is centered around zero as expected for a time independent phase of the calibration
pulse. The blue circles visualize the data which was filtered by a digital sliding window filter
of 200 MHz, a hopping window filter of 12.5 MHz and calibrated via the digital heterodyning
calibration method introduced in this thesis on detail in Appendix A. Due to the linear increase
of the test pulse in the calibration region [see Figure 3.13 (a)] we apply a digital sliding window
filter at the intermediate frequency ωIF = 10 MHz to compensate this error such that we end up
with the green circles. We would like to mention that we lost one period in comparison to the
unfiltered data (blue circles) as a result of the definition of the digital sliding window filter (3.3).
By using all these filter and calibration methods presented in this chapter we improve the data
quality a lot such that we can extract the physical information of time-domain measurements
presented in section 4.2 with high accuracy. In summary we showed that we can handle the
hardware required to perform pulsed measurements, split up the recorded signal with the help
of an IQ mixer and record both channels with an ACQIRIS card. Furthermore, we understand
the shape of the raw data, can identify the source of the artifacts as the reference port of the
ACQIRIS card and filter them out digitally. We demonstrated how to calibrate the IQ mixer
used in the setup with a powerful heterodyne calibration algorithm.
45
Time-domain control of light-matter interaction with superconducting circuits
46
Time-domain control of light-matter interaction with superconducting circuits
4
Measurement results
As mentioned in the previous parts we study a superconducting resonator coupled to a flux qubit
and observe the response for continuous wave spectroscopy and time-domain measurements. As
we intend to observe quantum mechanical phenomena the sample is cooled down to very low
temperatures to neglect thermal population in the resonator as described in detail in section
3.1.2.1. Here we present the necessary measurements to characterize our system in the frequencyand time-domain. We begin with a determination of the resonator modes followed by a flux
calibration. Afterwards we perform continuous wave single- and two-tone spectroscopy including
a photon number calibration. Finally we present pulsed wave spectroscopy and time-domain
measurements of the coupled qubit resonator system which were successfully performed in the
qubit group of the WMI for the first time.
4.1
Continuous wave spectroscopy
In this section we present all important measurements to characterize the coupled qubit-resonator
system in the frequency-domain. First we are interested in the modes of the resonator when the
sample is cooled down to approximately 50 mK. Therefore, we perform a wide range frequency
scan with the vector network analyzer (VNA) and search for peaks in the transmitted amplitude
in a single tone continuous wave spectroscopy (data not shown). Afterwards we perform a more
detailed scan on the peaks and fit a Lorentzian to the recorded data. In Figure 4.1 this is
visualized for the second harmonic and yields a resonance frequency of ω2 /2π = 7.1057 GHz for
the resonator far detuned from the degeneracy point of the qubit. Taking into account the f ull
width half maximum (FWHM) κ2 = 831 kHz from the fit data we can estimate the quality factor
of our superconducting resonator Q2 = ω2 /(2πκ2 ) ≈ 8500. By repeating this procedure for the
fundamental mode and the first harmonic we enumerate that the three experimentally accessible
modes have resonance frequencies given in Table 4.1. It is worth mentioning that the limited
amplifier bandwidth ranging from 4 to 8 GHz causes a low signal to noise ratio (SNR) of the
fundamental mode, making it challenging to record a clear spectrum such that the uncertainties
prevail the accuracy of the fit. We would like to mention that all three experimentally accessible
modes are affected by a mode-dependent phase drop as discussed on detail in section 2.2. Due
to the limited bandwidth we are not able to determine the mode ω3 which would not be affected
by a phase drop.
mode i
0 (λ/2)
1 (λ)
2 (3λ/2)
frequency ωi /2π (GHz)
2.642
5.067
7.106
κi (MHz)
—
1.25
0.831
Qi
—
4054
8551
(i + 1) · ω0 /2π (GHz)
2.642
5.284
7.926
Table 4.1: The three experimentally accessible resonator modes with their center frequencies.
If we compare the second and fifth column, it turns out that the coupling junctions causes a
non-equidistant spacing of the modes.
47
Time-domain control of light-matter interaction with superconducting circuits
Figure 4.1: A frequency scan of the second harmonic far detuned from the degeneracy point of
the qubit. The Lorentzian fit yields a resonance frequency of 7.1057 GHz. The deviations of the
fit from the measured data for low transmission values arise from noise.
4.1.1
Flux calibration
Now that we know the resonance frequencies of our resonator modes we can continue with a flux
sweep to determine which amount of current through the magnet coil corresponds to a flux of
1 Φ0 in the qubit loop. We vary the current from −10 mA to +10 mA and use a frequency window
from 7.105 ± 0.015 GHz to record the measured transmission with the VNA. We observe periodic
anticrossings with a period of 1 Φ0 [14, 15]. The result of this measurement for an input power
of approximately 2 photons on average (poa) in the resonator is visualized in Figure 4.2 and it
turns out that we observe three anticrossings. We express the power used for the measurement
in number of photons in the resonator. These numbers are calculated in section 4.1.3. To carry
out the photon number calibration we have to do a flux calibration and a determination of the
coupling strengths gi first. When this is done we can determine the average photon number
in the measurements. However, the measurement presented in Figure 4.2 gives us a rough
estimation of the current to flux conversion factor. Therefore, we performed measurements with
a higher resolution of each anticrossing to determine our conversion factor with high precision.
By identifying distinctive points pj,i (see Figure 4.3) and taking the average of the degeneracy
point qj for each current range we are able to define a conversion factor from current to flux.
The numbers for the points pj,i presented in Table 4.2 represent the current value at frequencies
of 7.1 GHz and 7.1136 GHz where the recorded transmission through the coupled qubit resonator
system reaches a maximum (see Figure 4.3). As a result we are able to label the axis in units of
flux detuning δΦ instead of the coil current If which was applied for the measurement. Therefore,
the mean value of the current q corresponding to 1 Φ0 is given by:
3
1X
q3 − q1
q=
(qj − qj−1 ) =
= 6.444 mA
2
2
j=2
48
(4.1)
Time-domain control of light-matter interaction with superconducting circuits
Figure 4.2: A scan over the full range of the coil current If from −10 mA to +10 mA for an input
power of 2 poa in the resonator mode ω2 . Within this range we observe three anticrossings (blue
windows) which we use for flux calibration. The discontinuity at approximately 7 mA is caused
by a temperature instability which occurred at that time as could be seen from our temperature
monitoring (data not shown).
Measurement j
1
2
3
pj,1 (mA)
−8.257
−1.821
+4.630
pj,2 (mA)
−8.249
−1.813
+4.678
pj,3 (mA)
−8.207
−1.774
+4.678
pj,4 (mA)
−8.199
−1.766
+4.688
P
qj = 1/4 4i=1 pj,i (mA)
−8.228
−1.794
+4.659
Table 4.2: The flux current values of the four distinctive points taken from Figure 4.3 for each
anticrossing available in the range of the current source. In the last column we present the
average value of the current where the corresponding flux is n/2 · Φ0
Due to the fact that this mean value q is a telescoping series we calculate the variance σq with
the formula:
σq =
1
(|q2 − q1 − q| + |q3 − q2 − q|) = 9.5 µA
2
(4.2)
so we define the variance as the average of the absolute deviations in every single period. Finally
the applied coil current If can be converted into flux in units of Φ0
If (mA)
= 6.444 ± 0.009.
Φ(Φ0 )
(4.3)
According to ref. [45] the systematic error of the used 16 bit analog digital converter which is
used to set the current If to a certain value is given by 0.3 nA such that the statistical errors
prevail.
49
Time-domain control of light-matter interaction with superconducting circuits
Figure 4.3: A more detailed scan over the three anticrossings first observed in Figure 4.2 in a
frequency window of 7.106±0.010 GHz over a coil current range of 0.2 mA. The distinctive points
pj,i (black circles) are taken at a frequency of 7.1 GHz for pj,{1,4} and 7.1136 GHz for pj,{2,3} . In
all three measurements the resonator mode is populated with approximately 2 poa.
50
Time-domain control of light-matter interaction with superconducting circuits
4.1.2
High power continuous wave spectroscopy
In this section we provide first results of high power continuous wave spectroscopy experiments.
Furthermore, we fit the Jaynes-Cummings Hamiltonian (2.47) to the data to estimate the coupling strengths gi , the energy gap of the qubit ∆ and its persistent current IP . This enables us
to determine the photon number population in the resonator in section 4.1.3. We do this for all
three experimentally accessible modes, the fundamental mode, the first harmonic and the second
harmonic in a single tone continuous wave experiment. In the data presented in Figure 4.4 the
output power of the VNA is set to a value such that according to the photon number calibration
in section 4.1.3 the second harmonic of the resonator is populated with 2 poa.
Figure 4.4: The three experimentally accessible modes recorded in a continuous wave spectroscopy experiment with the setup depicted in Figure 3.5. The second harmonic is depicted
in (a). We observe a clearly visible anticrossing symmetric around the degeneracy point with
detuning δΦ = 0. The discontinuities (black arrows) arise from higher order transitions [45]. In
(b) and (c) we show spectroscopy data for the first harmonic and the fundamental mode. Due
to the limited bandwidth of the cold HEMT amplifier the signal to noise in the (c) is lower.
51
Time-domain control of light-matter interaction with superconducting circuits
For the second harmonic in Figure 4.4 (a) we observe an anticrossing where the qubit excitation
frequency ωQ is on resonance with the second harmonic of the resonator ω2 at δΦ ≈ ±4 mΦ0 .
In this region the recorded resonator transmission is reduced due to the new eigenstates of the
coupled qubit resonator system as described in section 2.3. The discontinuities (black arrows)
are caused by higher order transitions which can occur at large drive powers [45]. The recorded
transmission spectrum for the first harmonic in Figure 4.4 (b) is reduced for the same reason
around the degeneracy point of the qubit. Finally the fundamental mode (c) is dispersively
shifted around the degeneracy point of the qubit. We would like to mention that for a photon
number calibration the coupling strengths gi have to be known [45]. These can be estimated
from a numerical fit of the multi-mode Jaynes-Cummings Hamiltonian (2.47) to the recorded
data. The result of such a fit is presented in Figure 4.5.
Figure 4.5: A numerical fit of the recorded spectra to the theoretical Hamiltonian (2.47). The
data of the second (a) and first harmonic (b) agree well with the fit. In (c) the qubit excitation
frequency was recorded in a two-tone spectroscopy experiment. In (d) we present the fit to
the fundamental mode. The used fitting parameters are the coupling strengths to each mode
g0 , g1 , g2 , the energy gap of the qubit ∆ and the persistent current of our qubit IP . Some of
these parameters are used to calibrate the photon number n in our system.
52
Time-domain control of light-matter interaction with superconducting circuits
In Figure 4.5 we use the three single tone continuous wave spectroscopy measurements presented
in Figure 4.4 and a two-tone continuous wave spectrum [see Figure 4.5 (c)] for the fit. At this
point the author wants to thank Juan José García-Ripoll from Universidad Complutense de
Madrid for coding the main parts of the fitting program. The numerical fit of the multi-mode
Jaynes-Cummings Hamiltonian (2.47) shows that the recorded experimental data agrees very
well with the theory. The fitted coupling of the qubit to the second harmonic in Figure 4.5 (a)
is approximately 76.8 MHz which results in a relative coupling strength g2 /ω2 ≈ 1 %. In (b)
the fit of the theory to the experimental data yields a relative coupling rate of approximately
1.5 %. A two-tone spectroscopy was performed in (c). We are able to detect the qubit hyperbola
yielding a qubit delta ∆/h ≈ 4.94 GHz and a persistent current IP ≈ 205 nA. The fit also shows
that the mode spacing in the resonator is non-equidistant as the first exited state of the first
harmonic |1i1 has an eigenenergy of ω1 /~ = 5.067 GHz [lowest horizontal line on the right hand
side of (c)] while the eigenenergy of the eigenstate |2i0 is 2ω0 /~ = 5.284 GHz (middle horizontal
line). According to ref. [46] this is caused by a mode dependent phase drop over the coupling
junction. We would like to point out that this eigenstate is present in the theory as well as in
the numerical fit. But we do not observe it in the experimental recorded frequency spectrum
because it refers to a two photon process with an energy of 2~ω0 in the fundamental mode. This
two-tone spectroscopy data also shows that the second harmonic ω2 (upper horizontal line) is
almost constant in the recorded region, this is important as we can use the second haromic as the
read out mode in the photon number calibration. In Figure 4.5 (d) we fit the energy spectrum to
the fundamental mode ending up with a relative coupling of approximately 0.5 %. We would like
to point out that all three coupling rates are on the order of 1 % of the respective resonator mode
and there is no evidence for ultra strong or even deep ultrastrong coupling as expected from a
decrease in the size of the coupling junction. This relation was proposed in ref. [45]. However, we
discuss this in more detail in section 4.1.4. The next step is a calibration of the photon numbers
in the resonator. Therefore, we present a two-tone spetroscopy in the next section.
4.1.3
Photon number calibration
In this section we calibrate the photon number in the resonator via a two-tone spectroscopy.
We perform a frequency sweep for different probe tone powers PP from −25 to +5 dBm in steps
of 1 dBm generated by the VNA. During the measurement the flux detuning is constant at a
constant flux value close to δΦ = 0. The probe tone frequency is fixed to the resonance frequency
of the second harmonic ωP /2π = ω2 /2π = 7.109 GHz. The spectroscopy tone generated by the
SMF at a power of PS = −30 dBm is swept from ωS /2π = 4.5 GHz to 5.3 GHz and applied to
the qubit via the antenna. The condition PS > PP is fulfilled due to the 50 dB attenuator at
room temperature at the output port of the VNA as depicted in Figure 3.5. The experimental
result for the power range from PP = −25 dBm to 5 dBm is depicted in Figure 4.6. The blue
region at PP = −22 dBm for a spectroscopy frequency from 5.2 GHz to 5.3 GHz is caused by a
communication error between the VNA and the measurement computer such that we could not
record the acquire data.
53
Time-domain control of light-matter interaction with superconducting circuits
Figure 4.6: Photon number calibration in a two-tone spectroscopy. In (a) we depict the recorded
spectra for each probe power PP and the spectroscopy tone ωS . For 0 to 5 dBm we observe a
single dip which represents the dispersively shifted qubit excitation energy. For probe power
values below −1 dBm we observe two dips. The fitted center frequencies of a Lorentzian fit
to each dip are visualized in (b). We simulate the interaction of the first harmonic ω1 with
the photon number dependent dispersively shifted qubit excitation energy ω
eQ and diagonalized
the Hamiltonian such that we end up with the eigenstates depicted in (c). We use numerical
optimization to find the linear relation of the average photon number n and the applied probe
power PP visualized in (d). To verify that theory and the measured spectra agree we use the
fitted line (red line in (d)) to plot the experimental data (circles) and the theoretical model (solid
lines) in (e).
54
Time-domain control of light-matter interaction with superconducting circuits
According to equation (2.59) the dispersively shifted qubit excitation frequency is given by
ω
e Q = ωQ +
(g2 )2
(2n + 1) .
δω2
(4.4)
Since there is a linear dependence between the dispersively shifted qubit excitation frequency ω
eQ
and the photon number n in the second harmonic resonator mode ω2 we expect to measure a
single transmission dip in a two-tone spectroscopy measurement which represents this dispersively
shifted excitation frequency. If we consider the used probe tone frequency ωP /2π = ω2 /2π =
7.109 GHz and a qubit excitation frequency ωQ ≈ ∆/h = 4.94 GHz we determine the frequency
detuning to δω2 /2π = −2.166 GHz for a measurement closed to vanishing flux detuning. For
a coupling strengt g2 /2π = 76.8 MHz the shift per photon is expected to be 2 (g2 )2 /δω2 =
2π (−5.45 MHz), so the measured qubit excitation energy should decrease with increasing power.
If we compare these estimations to the acquired data in Figure 4.6 (a) we find that this holds
only for probe powers PP larger than 0 dBm. Below this power value we observe two dips. By
fitting a Lorentz distribution to each dip, we end up with the frequencies depicted in Figure 4.6
(b). For the rest of this section we denote the dip at higher frequency (blue circles) with ν+ and
the one at lower frequency (green circles) is denoted by ν− . For low powers the two dips seem
to remain at a constant value within the frequency window ω1 ± g1 around the first harmonic
resonator mode. This gives rise to the assumption that the condition δω1 < g1 is fulfilled and
that the observed two dips are the dressed states |m, +i1 and |m, −i1 where m denotes the
photon number in the first harmonic of the resonator as described in section 2.3. So we describe
this system with the Hamiltonian
b
H
ω1 g1
=
,
g1 ω
eQ
~
(4.5)
where the first harmonic of the resonator ω1 interacts with the dispersively shifted qubit excitation frequency ω
eQ (4.4) via the coupling g1 . According to the calculations carried out in
Appendix B the eigenstates of the diagonalized Hamiltonian (4.5) are given as
r
2
λ± =
ω1 + ωQ + (2n + 1) (gδω2 )2
±
2
ω1 − ωQ + (2n + 1) (gδω2 )2
2
2
+ 4 (g1 )2
.
(4.6)
The eigenstates λ± for a qubit excitation frequency of ωQ /2π = 5.04 GHz, coupling strengths
of g1 /2π = 86.5 MHz, and g2 /2π = 88.2 MHz are visualized in Figure 4.6 (c). The coupling g1
is chosen such that the level splitting for the lowest probe tone power PP is given by 2g1 . The
qubit excitation energy ωQ is close to the fitted value for the qubit energy gap ∆/h = 5.02 GHz
in the low power two-tone spectroscopy of Figure 4.7 (f). The small deviation of 20 MHz of the
used ωQ /2π = 5.04 GHz and the expected qubit excitation frequency at the degeneracy point of
the fit ∆/h = 5.02 GHz might be due to the accuracy of the fitting algorithm from which we
obtained ∆. The used coupling strength g2 is also obtained from the low power spectroscopy
fit. If we compare the measured eigenfrequencies ν± with the simulated eigenfrequencies λ± we
find that they agree very well. This gives rise to the assumption that the Hamiltonian (4.5)
describes the measured data properly. In the next step we have to correlate the average photon
number n from which we simulate the eigenstates λ± with the applied probe power PP for the
two measured frequencies ν± . Therefore, we use numerical optimization via the ansatz
λ+
ν+ .
min −
(4.7)
n λ−
ν− 2
55
Time-domain control of light-matter interaction with superconducting circuits
So for each pair of measured frequencies ν± we determine the average photon number n for
which the distance between the theoretical eigenfrequencies λ± and the measured frequencies ν±
is minimal in the Euclidean norm k k2 . After these calculations we end up with pairs of probe
powers PP and average photon numbers n for which the condition (4.7) is fulfilled. These pairs
are visualized in Figure 4.6 (d). To complete the photon number calibration we fit a line trough
the origin (red line in (d)). By converting the applied probe powers PP to photon numbers on
average n we find that the measured frequency pairs ν± agree very well (see Figure 4.6 (e))
with the theoretical eigenfrequencies described by the Hamiltonian (4.5). Finally we are able
to calibrate the average photon number n of the second harmonic mode ω2 by simulating the
interaction of the dispersively shifted qubit excitation energy ω
eQ and the first harmonic mode
ω1 . An excerpt of average photon numbers n and corresponding probe power PP are given in
Table 4.3.
average photon number n
0.25
1
2
probe power PP (dBm)
-20
-14
-10.5
Table 4.3: The probe power PP in dependence of the average photon number n in the second
harmonic ω2 according to the calibration.
Here we use the dressed states of a qubit and the resonator mode ω1 to calibrate the photon
number in the resonator mode ω2 which dispersively shifts the qubit excitation energy. With
the performed photon number calibration, we are now able to gauge the additional losses in the
resonator input line which are caused for example by cable loss and insertion loss. For n = 1 poa
we send in −14 dBm from the output port of the VNA. Since we know our resonator frequency
ω2 /2π = 7.106 GHz and the FHWM κ2 = 831 kHz of the second harmonic from Figure 4.1 the
power in the resonator is
n~ω2 κ2
Prf = 10 log10
= −144 dBm
(4.8)
1 mW
The theoretical power which arrives in the resonator is Pth = −114 dBm. This value is calculated
from the output power of the VNA and the attenuator configuration of 50 dB at roomtemperature
(see Figure 3.5) and 50 dB inside the shielding room (see Figure 3.3). So the difference of 30 dB
is caused by the additional losses mentioned above. However, since we calibrated the photon
number in the second harmonic of the resonator we are now able to perform continuous wave
experiments with negligible photon number on average in the resonator in the next section.
56
Time-domain control of light-matter interaction with superconducting circuits
4.1.4
Low power continuous wave spectroscopy
Based on the performed photon number calibration in the last section we are now able to repeat
the continuous wave spectroscopy experiments of section 4.1.2 with a negligible number of photons in the resonator. As a result we are able to determine the characteristic frequencies in our
system with more precision since they are no longer perturbed by the power dependent shifts discussed in section 2.3. This step is very important since we want to perform pulsed measurements
with fixed frequencies in the last part of the thesis. The result of a single tone continuous wave
spectroscopy measurement of the first and second harmonic as well as a two-tone continuous
wave spectroscopy experiment is visualized in Figure 4.7. In the two-tone spectroscopy we use a
probe tone frequency of ωP = 7.109 GHz and a spectroscopy power of PS = −30 dBm.
Figure 4.7: Measurement results for low power spectroscopy. In (a) we observe an anticrossing
when the second harmonic ω2 is resonant with the qubit excitation energy ωQ at a negligible
number of photons in the resonator in a single tone continuous wave spectroscopy. In (b) the
spectroscopy data for the first harmonic is visualized. A two-tone spectroscopy experiment with
a higher resolution than in Figure 4.5 was performed in (c). The pictures (d) to (f) show a fit
of the Jaynes-Cummings Hamiltonian (2.47) to the experimental data. As a result we end up
with coupling strengths g1 /2π ≈ 71.9 MHz and g2 /2π ≈ 88.2 MHz. The fit yields a qubit delta
of ∆/h ≈ 5.02 GHz and a persistent current of IP ≈ 208 nA. Futhermore we find |δω1 | < g1
around vanishing flux detuning δΦ.
57
Time-domain control of light-matter interaction with superconducting circuits
In Figure 4.7 (a) we observe an anticrossing of the second harmonic with the qubit hyperbola.
Since we measure with a negligible number of photons the higher order transitions we observe
in Figure 4.4 (a) are no longer present in the spectrum. The first harmonic is shifted to lower
frequencies around the degeneracy point of the qubit at δΦ = 0. In (c) we depict the experimental
results of a two-tone spectroscopy. This picture shows a very nice coincidence since for vanishing
flux detuning δΦ the resonance detuning |δω1 | < g1 such that we can investigate the dressed
states as depicted in Figure 2.9 (a). As a result we are able to observe the transition from the
dispersive to the resonant interaction regime of the Jaynes-Cummings Hamiltonian described
in section 2.3 by sweeping the flux detuning δΦ from finite values to zero. In analogy to the
theoretical results described in ref. [45] we observe a reduction of the measured transmission
when we undergo the transition from the dispersive to the resonant interaction in the JaynesCummings Hamiltonian. The pictures (d) to (f) show fits to the Jaynes-Cummings Hamiltonian.
We find relative coupling rates of approximately 1 % such that the qubit is strongly coupled to
the resonator. In agreement with the fact that we are at zero frequency detuning δω1 when the
flux detuning δΦ is also closed to zero the fit shows that the qubits energy gap ∆/h ≈ 5.02 GHz
is close to the excitation energy of the first harmonic of the resonator ω1 /2π = 5.067 GHz so the
condition
|δω1 /2π| = |ωQ − ω1 | ≈ |∆/h − ω1 | = 47 MHz < g1 /2π ≈ 71.9 MHz
(4.9)
is fulfilled. For flux detuning close to zero the dressed states |1, −i1 and |1, +i1 are formed from
the energy levels |0i1 ,|1i1 ,|gi and |ei as described by equation (2.55). We do not observe an
anticrossing in which the state |2i0 with two photons is involved since this would not preserve
the number of excitations which is one fundamental assumption of the Jaynes-Cummings model
introduced in section 2.3. If we compare the results of the fit in section 4.1.2 to that in this
section we find the numbers presented in Table 4.4.
g1 /2π (MHz)
g2 /2π (MHz)
∆/h (GHz)
IP (nA)
high power spectroscopy
72.4
76.8
4.94
205
low power spectroscopy
71.9
88.2
5.02
208
Table 4.4: Fit parameters of the Jaynes-Cummings Hamiltonian (2.47) for the performed high
power and low power continuous wave spectroscopy measurement.
The values for the coupling g1 of the qubit to the first harmonic of the resonator and the persistent
current IP agree very well. The deviations of the values for the qubit energy gap ∆ and the
coupling g2 also agree within the accuracy of the fit. With the higher resolution of the performed
two-tone spectroscopy presented in Figure 4.7 (c) in comparison to that in Figure 4.5 (c) we are
able to detect a rich stucture around the degeneracy point of the qubit. Since the probe tone of
the VNA is set to a value such that the second harmonic of the resonator is populated with a
negligible number of photons but the spectroscopy tone power PS = −30 dBm is still two orders
of magnitude larger in the two-tone measurement we observe some less pronounced side arms
[blue arrows in Figure 4.7 (c)]. We have a closer look on that before we proceed with timedomain measurements. A two-tone spectroscopy experiment with a higher frequency resolution
than in Figure 4.7 (c) for a probe tone frequency of ωP = 7.109 GHz and a spectroscopy power
of PS = −30 dBm is presented in Figure 4.8 (a).
58
Time-domain control of light-matter interaction with superconducting circuits
Figure 4.8: Zoom in around the degeneracy point of the qubit in two-tone spectroscopy is shown
in (a). We present a cut for vanishing detuning δΦ = 0 in (b). It turns out that there is an
additional pattern which we analyzed for a flux detuning of δΦ = 1.37 mΦ0 in (c). We multiplied
the data in (c) by −1 and plot it on a linear scale.
In Figure 4.8 (a) we observe two side arms (black arrows). As already mentioned the condition
|δω1 | < g1 is fulfilled such that we observe the dressed states |1, −i1 and |1, +i1 for vanishing flux
detuning δΦ. Therefore, we present a one dimensional cut through the experimental data taken
at δΦ = 0 in (b). It turns out that the dip at higher frequency is sharp and the other is flattened.
Since we are interested in spectroscopy frequencies at which they reach their minimum, we fitted
a Lorentzian to them [red and green lines in Figure 4.8 (b)]. The two center frequencies are fitted
to
ν+ = 5.132 GHz,
(4.10)
ν− = 4.968 GHz.
(4.11)
(4.12)
As a result they are split by the value
δν = ν+ − ν− = 164 MHz.
(4.13)
59
Time-domain control of light-matter interaction with superconducting circuits
According to equation (2.56) the photon number dependent energy level splitting in the resonant
case is given by
√
√
√
(4.14)
2g1 m = 2 · 71.9 MHz m = 143.8 MHz m
Where m denotes the photon number corresponding to the first harmonic ω1 . If we assume
√
δν = 2g1 m we find that m ≈ 1.3. The deviation from m = 1 might be due to the fact that the
Lorentzian from which we extracted ν− is a superposition of more than one vacuum Rabi level or
due to the accuracy of the fitting program which determines the coupling strengths. However, the
fitted and calculated energy level splitting agree well within the scope of measurement accuracy.
This indicates that we are able to observe the dressed states of a resonantly coupled qubit
resonator system with a dispersive readout via a second resonator mode. If we apply a larger
flux detuning δΦ ≈ 1.37 mΦ0 we change the qubit excitation energy via the flux dependent
energy bias [see blue line in Figure 4.8 (a)]. Therefore, the measured signal and linewidth of
the dip which corresponds to the dressed state |1, −i1 decreases. Furthermore, the dip is shifted
to slightly higher frequencies. Its limit is given by the resonator mode ω1 /2π = 5.067 GHz
which occurs for large detuning such that we end up in the dispersive limit. This is caused
by the increase of the frequency detuning δω1 since by an increase of the flux detuning δΦ
we increase the flux dependent energy bias given in equation (2.23) and therefore the qubit
excitation frequency ωQ [see equation (2.25)]. As a result the mixing angle ϑ form equation (2.53)
converges to zero for large flux detuning δΦ. So for increasing flux detuning the contribution of
the qubit to the state |1, −i1 decreases [see equation (2.55)] and at a certain flux detuning the
dressed states are no longer present since the condition g1 > δω1 is violated. Furthermore, the
peak of the dressed state |1, +i1 is also shifted to higher frequencies and exhibits side arms. This
is more clearly visible in Figure 4.8 (c). Here we are able to determine the center frequencies of
the peaks to
ν|1,+i1
= 5.316 GHz
(4.15)
ν|3,+i1
= 5.211 GHz
(4.16)
ν|4,+i1
= 5.194 GHz
(4.17)
ν|1,−i1
= 5.054 GHz.
(4.18)
The frequencies are given as the center frequencies of a Lorentzian fit. The choice of the indices
for this frequencies will become clear below. These side arms look very similar to the vacuum
Rabi resonances observed in ref. [78]. We are able to identify two side arms |3, +i1 (magenta)
and |4, +i1 (cyan) in Figure 4.8 (c). According to ref. [78] the energy levels of the n-photon
subspaces are given as
g
2πEn = ω1 ± √ ,
n
(4.19)
where g denotes the mode spacing. If we assume that the center frequency ν|1,+i1 = 5.316 GHz
correspondes to the state with n = 1 we are able to calculate the mode spacing to
g/2π = ν|1,+i1 − ω1 /2π = 5.316 GHz − 5.067 GHz = 240 MHz.
(4.20)
With this result we enter equation (4.19) again to end up with the numbers presented in Table
4.5.
60
Time-domain control of light-matter interaction with superconducting circuits
photon number n
1
2
3
4
state
|1, +i1
|2, +i1
|3, +i1
|4, +i1
energy En (GHz)
5.136
5.268
5.206
5.187
Table 4.5: The excitation energy of the four lowest vacuum Rabi levels according to equation
(4.19).
The black horizontal lines in Figure 4.8 represent these numbers. It is worth mentioning that the
fitted center frequencies ν|3,+i1 and ν|4,+i1 agree well with the excitation energies of the vacuum
Rabi levels. The fact that we could not observe a peak where the photon number should be
equal to two could be caused by the line width of the resonance peak located around 5.3 GHz.
We are not able to observe the symmetric behavior of the energy level splitting in contrast to the
vacuum Rabi levels observed in ref. [78] since for the minus in equation (4.19) we are perturbed
by the levels which would arise from the vacuum Rabi levels for negative flux detuning. However,
the results presented here seem to be consistent with vacuum Rabi transition.
Up to now we characterized the coupled qubit resonator system in the frequency-domain. We
used single tone and two-tone continuous wave spectroscopy methods and fitted the theory of the
Jaynes-Cummings Hamiltonian (2.47) to the data acquired with a negligible number of photons
in the readout mode of the resonator. As a result we present the theoretical energy spectrum of
our sample in Figure 4.9.
Figure 4.9: The fitted energy spectrum of the Jaynes-Cummings Hamiltonian (2.47) to the data
presented in Figure 4.7. The vacuum mode is denoted by |0i For a large flux detuning δΦ we
are able to identify the pure resonator eigenstates |nii where n denotes the number of photons
in the resonator mode i.
61
Time-domain control of light-matter interaction with superconducting circuits
Figure 4.9 visualizes the energy levels of our sample for various flux detunings from −15 mΦ0 to
+15 mΦ0 . The spectrum is symmetric around zero detuning. We find that the Jaynes-Cummings
Hamiltonian (2.47) describes our sample very well. For zero flux detuning the qubit’s excitation
energy ωQ is equal to the energy gap ∆/h ≈ 5.02 GHz such that for the first harmonic the
condition δω1 < g1 is fulfilled which leads to a rich structure around zero flux detuning.
4.2
Time-domain measurements
In this part we present time-domain measurements on the coupled qubit resonator system. To
perform these experiments we use pulses with a pulse width τ at the resonance frequencies of
our sample for various flux values. To find the correct frequency we use the energy spectrum
of the Jaynes-Cummings Hamiltonian (2.47) presented in Figure 4.9. Therefore, it was essential
to perform the spectroscopy experiments in the first part of this thesis. In a first approach
to this task we perform a pulsed wave two-tone spectroscopy to check if the used hardware
introduced and calibrated in section 3.2.2 works well in the experimental setup. Since the check
of the calibration and filtering methods introduced in section 3.2.2.2 was performed under testing
conditions without a mounted sample between the signal generator and the detector. Afterwards
we successfully perform a time-domain measurement of a coupled qubit resonator system with
the ACQIRIS card in the qubit group of the WMI for the first time. In the last part we repeat
this measurement with an FPGA board to speed up the measurement.
4.2.1
ACQIRIS card measurements
In this section we present the measurement results recorded with the ACQIRIS card. First we
perform a pulsed wave two-tone spectroscopy experiment to check the hardware and the software
algorithms developed during this thesis. Afterwards we perform a time-domain measurement of
the coupled qubit resonator system to investigate the response of the sample on Rabi pulses.
4.2.1.1
Pulsed two-tone spectroscopy
To test the IQ mixer, the digital filter methods and the heterodyne calibration algorithm we
perform a pulsed wave two-tone spectroscopy. A simplified circuit diagram and the used pulse
pattern are depicted in Figure 4.10.
Figure 4.10: The setup of the pulsed wave two-tone spectroscopy experiment. On the left hand
side we depicted a simplified circuit diagram which indicates which signal source generates the
signals used in the setup. The used pulse pattern for the readout time t of the ACQIRIS card
and the pulse width τ are depicted on the right hand side.
62
Time-domain control of light-matter interaction with superconducting circuits
We use one SMF to apply the spectroscopy tone ωS to the qubit via the antenna and a second
one is used as local oscillator ωLO for the IQ mixer, both are driven continuously. The PSG is
driven at ωP /2π = 7.109 GHz and at an output power such that the resonator is populated with
1 poa (see Table 4.3). The output power of the SMF is PS = −30 dBm. The DTG is used with a
period of 4 µs to send a trigger of 3 µs to the ACQIRIS card to define the length of the readout
time t. We use a second channel of the DTG to set the pulse width τ of the probe tone signal
to 1 µs. In this experiment we vary the spectroscopy tone ωS from 4.8 to 5.6 GHz in steps of
25 MHz and the flux is swept from δΦ ≈ −1.5 mΦ0 to approximately 2 mΦ0 . For each data point
we record over 2.3 · 106 traces for averaging. From the recorded time traces which look similar
to the calibration pulses presented in section 3.2.2.2 we extract the amplitude for a certain flux
value and a specific spectroscopy tone from the two fitted quadrature amplitudes AMI and AMQ
via the formula
q
B = (AMI )2 + (AMQ )2 .
(4.21)
We are interested in the average amplitude B while the probe tone ωP was applied to the
experiment and not in the amplitude A(t) at every time increment of the readout time t, since we
want to perform a spectroscopy experiment. The result of this experiment is depicted in Figure
4.11.
Figure 4.11: The recorded transmission spectra for a pulsed wave two-tone spectroscopy experiment. We are able to reproduce the results of the continuous wave two-tone spectroscopy in
Figure 4.7 (c). However, this measurement technique is very inefficient, due to the large amount
of time which is needed for averaging the time traces in the ACQIRIS card. We are able to
resolve the side arms (black arrows) first presented in Figure 4.7 (c).
63
Time-domain control of light-matter interaction with superconducting circuits
In general we can reproduce the experimental result of a continuous wave two-tone experiment
with the pulsed measurement technique presented in Figure 4.11. The mechanism for a decrease
of the recorded amplitude while the spectoscopy tone ωS is on resonance with the system is the
same as for continuous wave two-tone spectroscopy and described in section 2.3 and depicted in
Figure 3.6. Due to the fact that the averaging methods in a VNA are much more sophisticated
than the simple method used here the measurement time is a factor of two larger and the
resolution is not even a fourth in comparision to the measurement presented in Figure 4.7 (c).
However, this measurement where the sample is mounted beteen the signal generators and the
built up and calibrated IQ detector for the first time confirms that the detector operates as
expected and is able to deliver information about the experiment. With this measurement
technique we are also able to detect the side arms which were depicted in Figure 4.7 (c). However,
in the next step we reduce the pulse width τ by two to three orders of magnitude to perform
time-domain measurements.
4.2.1.2
Rabi oscillation measurements
In this section we present experimental results of time-domain measurements on a coupled qubit
resonator system. A simplified picture of the experimental setup and the pulse pattern used
in the experiment is presented in Figure 4.12. For a detailed introduction of the measurement
setup used in the experiments see section 3.2.2, here we provide an overview of the experimental
parameters.
Figure 4.12: A simplified circuit diagram for the time-domain measurements on the left hand
side. The used pulse pattern for the readout time t of the ACQIRIS card and the pulse width τ
are depicted on the right hand side. The readout time t and the pulse width τ differ by two to
three orders of magnitude and that the pulse always starts at t = 1 µs and ends at t = 1 µs + τ .
As we intend to measure Rabi oscillations we have to excite the qubit from the ground state
|gi to the excited state |ei at its excitation frequency ωQ such that we can extract the energy
relaxation time T1 of the qubit. To reduce the effect of flux fluctuations shifting the qubit’s
excitation frequency we perform this measurement around the degeneracy point of the qubit
where the qubit hyperbola is very flat. From the spectroscopy experiments in section 4.1.4 we
know that for a flux detuning of δΦ ≈ 0 the excitation frequency of the dressed state |m, −i
where the qubit is in the ground state has a frequency of ω|m,−i /2π = 4.96 Ghz.
64
Time-domain control of light-matter interaction with superconducting circuits
Therefore, we operate the PSG at ωS /2π = 4.96 GHz and an output power of −20 dBm. We
keep the indices P and S which indicated the probe tone and spectroscopy tone in a continuous
wave two-tone spectroscopy since the spectroscopy is the long time limit for the time-domain
measurements presented here. The SMF generates a weak continuous microwave signal such that
the second harmonic of the resonator is populated with approximately 1 poa at a fixed frequency
ωP /2π = 7.109 GHz, since this is the dispersively shifted resonator frequency according to the
spectroscopy experiments in section 4.1.4. The second SMF provides the local oscillator frequency
ωLO /2π = 7.099 GHz at a power of 12.5 dBm. The response of the sample is then converted down
to an intermediate frequency ωIF /2π = 10 MHz with the IQ detector depicted in Figure 3.8. The
signal of the channels I and Q is recorded with the ACQIRIS card for a readout time t of 3 µs.
The pulse width τ is swept from 5 to 38 ns and applied to the antenna with a delay of 1 µs after
the ACQIRIS card has begun to record the data. Each time trace for a certain pulse width is
recorded and averaged in the ACQIRIS card over 23·106 times. Two examples for these averaged
time traces for two different pulse widths τ are presented in Figure 4.13.
Figure 4.13: Recorded raw data traces for I and Q and two different pulse widths τ . In (a) we
depicted the recorded I and Q channel for a pulse width τ = 15 ns. While the delay is set to 1 µs
the response is seen at 1.2 µs due to the signal processing in the internal mixer of the PSG. In
(b) the pulse width is 22 ns, in comparrison to (a) the decrease in the measured amplitude from
approximately 1.3 to 1.7 µs is larger. The raw data presented here has already been filtered by
a sliding window and a hoppping window filter as described in section 3.2.2.2. The heterodyne
calibration algorithm uses the region from 0.1 to 1.0 µs to extract the correction matrix which is
applied to the complete trace.
65
Time-domain control of light-matter interaction with superconducting circuits
In Figure 4.13 (a) we depict the recorded and filtered data for a pulse width τ = 15 ns. A decrease
of the measured amplitude is clearly visible for both channels after approximately 1.3 µs. For a
pulse width τ = 22 ns [see Figure 4.13 (b)] the decrease is even more prominent. In both cases
the decrease of the measured amplitude can be explained with the same mechanism as for the
two-tone spectroscopy, namely an AC-Zeeman shift. After t ≈ 2.5 µs the signal has the same
amplitude as within the first microsecond of the recorded trace. This is a first hint that we
can excite the qubit with the applied pulse and that the system relaxes after a certain time.
To determine the so called energy relaxation time T1 we have to extract the time dependent
amplitude A(t) and phase ρ(t) from the recorded I and Q channel as described in section 3.2.2.2.
Therefore, we use the recorded data for a readout time t from 0.1 to 1.0 µs to apply the heterodyne
calibration algorithm from Appendix A to obtain a correction matrix C for each pulse width τ .
For every readout time increment we multiply the data of channel I and Q with the correction
matrix of the respective pulse width. Afterwards we calculate the amplitude A(t) and phase ρ(t)
via the equations (3.10) and (3.11). Finally we apply a sliding window filter at the intermediate
frequency ωIF for the reasons explained in section 3.2.2.2. The result of these calculations is
visualized in Figure 4.14.
Figure 4.14: The extracted amplitude (a) and phase (b) are depicted as a function of the readout
time t for pulse widths τ = 5, 6, . . . 38 ns. We would like to mention that all filter and calibration
algorithms as described in section 3.2.2.2 in detail have already been applied to the depicted
data. The MATLAB code to calculate the data presented here can be found in the Appendix
A.2.
The amplitude A(t) in Figure 4.14 (a) for a fixed pulse width τ shows a clearly visible response.
After the pulse was applied we detect a decrease in the amplitude and after a certain time it
relaxes to the initial value. The observed response varies for different pulse widths τ . In (b)
we depict the phase response of the system to the performed experiment. As expected for a
successful calibration the phase difference in the first third of the readout time t is almost zero
for every pulse width τ but for different pulse widths we observe an oscillatory behavior of the
phase. For all applied pulse widths τ we find that we can excite the system. The obtained
result of Figure 4.14 (b) looks very similar to that in ref. [19]. For the rest of this thesis we
concentrate on the phase ρ(t) to analyze the experimental results, calculate energy relaxation
times and investigate the oscillatory behavior of the response on the pulse width τ . The result
of this analysis is depicted in Figure 4.15.
66
Time-domain control of light-matter interaction with superconducting circuits
Figure 4.15: The extracted phase ρ(t) for different pulse widths τ is depicted in (a). The fitted
energy relaxation rate on a minimum of the observed response [magenta line in (a)] is given by
TDS ≈ 108 ns which is visualized in (b). In (c) we depict the same as in (b) but on a maximum
[cyan line in (a)], here the energy relaxation time is TDS ≈ 110 ns. We extract the maximum of
the phase shift (d) by applying the Chebyshev norm for readout times from 1.17 to 1.36 µs at
each pulse width τ . We find that the phase shift shows an oscillatory behavior. The fitted sine
yields a Rabi frequency of ωRabi /2π ≈ 78 MHz [red line in (b)].
67
Time-domain control of light-matter interaction with superconducting circuits
First we concentrate on the energy relaxation time. In Figure 4.15 (b) we present the recorded
response for a pulse width τ = 15 ns [magenta line in (a)]. We use the measured data marked
as green dots to fit an exponential decay to the data. Here we end up with a relaxation time
TDS ≈ 108 ns. In the spectroscopy experiments we found that at the degeneracy point of the
qubit at which we perform the time-domain measurements presented here we also drive the
dressed states, since we fulfill the condition |ωQ − ω1 | < g1 . Therefore, the energy relaxation
time of the qubit T1 can be calculated via a Matthiessen’s rule [79]
(TDS )−1 = ΓDS = Γ1 + κ1 ,
Γ1 = (TDS )−1 − κ1 =
(4.22)
1
− 1.25 MHz = 8.01 MHz,
108 ns
T1 = (Γ1 )−1 = 125 ns.
(4.23)
(4.24)
So the fitted relaxation rate of the dressed states ΓDS is a superposition of the resonator loss
rate κ1 and the qubit energy relaxation rate Γ1 . We have to regard the relaxation time TDS
with suspicion since the relaxation rate ΓDS = 9.25 MHz is close to the intermediate frequency
ωIF which was filtered with a sliding window filter. So the numbers presented here only are able
to quantify the order of magnitude of the qubit relaxation time T1 . If we apply the equation
(4.23) to the fitted relaxation rate TDS = 110 ns for the time trace presented in Figure 4.15 (c)
we can determine the qubit energy relaxation rate to T1 = 127 ns. We would like to mention
that the calculated T1 time for a pulse width of τ = 15 ns agrees well with the calculated T1 time
for a pulse width τ = 22 ns, which indicates that the qubit is excited for every pulse width we
applied in this experiment. In the next step we investigate the oscillatory behavior of the phase
ρ(t) with the pulse width τ . Therefore, we apply the Chebyshev norm1 to the data presented
in Figure 4.15 (a) for readout times from 1.17 to 1.36 µs at each pulse width τ . The maximum
phase shift for the different pulse widths τ is depicted in 4.15 (d). It might be possible that
the reduced visibility for pulse widths between 6 and 14 ns is caused by the response time of
the resonator. We use the measured data marked with green dots to fit a sine (red curve) to
the data. This fit yields a frequency of ωRabi /2π ≈ 78 MHz. The white curve in (a) is a sine
of this frequency. If we could proof that this frequency scales linear with the amplitude of the
spectroscopy power PS then it would be another indication that we observed Rabi oscillations.
With a pulse width sweep increment of 1 ns we can not observe oscillations with a frequency
greater than (1 ns/2)−1 = 500 MHz. However, such a time-domain measurement is successfully
performed in the qubit group of the WMI for the first time. To speed up the measurements and
to confirm the results received up to now we replace the ACQIRIS card by an FPGA board from
Innovative Integration (Model: X5-RX).
4.2.2
FPGA board measurements
To speed up the measurements and to avoid the digital filtering of the raw data we replace the
ACQIRIS card by an FPGA board. The rest of the measurement setup depicted in Figures
3.7 and the pulse sequences in 4.12 remain unchanged. In a first experiment we reproduce the
measurement results of the ACQIRIS card, afterwards we investigate the oscillatory behavior of
the sample with the pulse width τ and its power dependence.
1
68
The Chebyshev norm in a compact space like Rn is equivalent to the maximum norm.
Time-domain control of light-matter interaction with superconducting circuits
4.2.2.1
Rabi oscillation measurements
Since we replace the ACQIRIS card by the FPGA board we no longer have to apply the sliding
window filter at 200 MHz and the hopping window filter at 12.5 MHz, since the artifacts which
made it necessary to apply these filters are caused by internal errors of the ACQIRIS card.
However, the sliding window filter at the intermediate frequency ωIF still has to be used after we
calibrated the IQ mixer since the ground loop is still not compensated (see section 3.2.2.2 for a
detailed discussion). For a first check we perform a measurement with the same parameters as
in ACQIRIS card measurement in section 4.2.1.2. So the SMF operates at ωP /2π = 7.109 GHz
and an output power such that the second harmonic resonator mode is populated with 1 poa
(see Table 4.3). The pulse at a frequency of ωS /2π = 4.96 GHz at an output power of PS =
−20 dBm is generated by the PSG. The local oscillator is driven continuously at a frequency
ωLO /2π = 7.099 GHz and a power of 12.5 dBm. The results of this measurement in comparison
to the ACQIRIS card measurement are presented in Figure 4.16.
Figure 4.16: We compare the measurement results where we use the ACQIRIS card for readout
(a) to the measurement with the FPGA board (b) for the same experimental parameters. In
(b) the signal is slightly weaker despite the fact that less digital filters are used. In general the
measurement result is reproducible.
When we compare the two subfigures in Figure 4.16 it turns out that as expected they look very
similar. We observe a low phase response for pulse widths between 7 and 18 ns in (b) as it is also
the case in (a) where the ACQIRIS card is used. This indicates that the reduced visibility is either
caused by the response time of the resonator or by the used IQ mixer. However, there are two
main differences of the ACQIRIS card measurement in (a) and the FPGA board measurement
in (b). First the signal to noise ratio is higher in (b), this may indicate that the digital filters
applied to the ACQIRIS card can be optimized. The second and more important difference is the
time needed to perform the measurement. Each measurement trace is averaged 23 · 106 times but
it took approximately 17 h to acquire the data with the ACQIRIS card but less than one hour
to record the data with the FPGA board. The better signal to noise ratio and the reduction of
measurement time for achieving similar measurement results suggests to use the FPGA board in
future experiments. We now proceed with a detailed analysis of the FPGA board measurement
presented in Figure 4.17 to confirm the measurement results recorded with the ACQIRIS card.
69
Time-domain control of light-matter interaction with superconducting circuits
Figure 4.17: In agreement with the ACQIRIS measurement of section 4.2.1.2 we find that the
phase shift shows an oscillatory behavior when we use the FPGA board (a). The fitted energy
relaxation rate on a maximum [magenta line in (a)] is given as TDS ≈ 111 ns which is visualized
in (b). In (c) we depicted the same as in (b) but on a minimum [cyan line in (a)], here the
energy relaxation time is found to be TDS ≈ 138 ns. The fitted sine yields a Rabi frequency of
ωRabi /2π ≈ 70 MHz [red curve in (d)].
70
Time-domain control of light-matter interaction with superconducting circuits
Figure 4.17 (a) visualizes the measurement result of an experiment which is performed with the
same parameters as the measurement discussed in section 4.2.1.2 but now with the FPGA board
as recording device instead of the ACQIRIS card. Despite the problem of the applied sliding
window filter at the intermediate frequency ωIF it is worth to analyze the energy relaxation
time in analogy to the results of section 4.2.1.2. The time trace for a pulse width τ = 25 ns
is visualized in Figure 4.17 (b). We use equation (4.23) to calculate a qubit energy relaxation
time of T1 ≈ 129 ns from the fitted dressed state relaxation rate TDS ≈ 111 ns. In Figure 4.17
(c) we visualize the recorded time trace for a minimum in the oscillation. Here we find a qubit
relaxation time of T1 ≈ 167 ns. This value is approximately one third larger than the value
at the maximum. However, since all qubit relaxation times presented so far are between 100
and 200 ns we can assume that the real relaxation time value is of this order of magnitude.
Similar to the measurements with the ACQIRIS card we observe an oscillatory behavior of the
recorded time traces with the pulse width τ in Figure 4.17 (a). This is more clearly visualized
in Figure 4.17 (d) where we plot the phase shift extracted form the Chebyshev norm for readout
times from 1.17 to 1.36 µs at each pulse width τ . The fitted sine (red curve) yields a Rabi
frequency of ωRabi /2π ≈ 70 MHz. We are affected by a reduced visibility for pulse width form
7 to 18 ns. However, the phase shift for pulse widths of 5 and 6 ns agree with the fitted sine.
This indicates that the optimization of the detector might be an issue in our setup. In the next
step we concentrate on the oscillatory behavior of the recorded traces with the pulse width τ .
It was shown in refs. [16, 19] that the Rabi oscillation frequency ωRabi depends linearly on the
driving amplitude. Therefore, we vary the spectroscopy tone power PS from −26 to −10 dBm
in steps of 2 dBm. Since we measured at zero flux detuning the spectroscopy tone frequency is
ωS /2π = 4.96 GHz. The resonator readout mode ω2 /2π = 7.109GHz is populated with 1 poa
and driven continuously by a SMF. The local oscillator is driven continuously at a frequency
ωLO /2π = 7.099 GHz and a power of 12.5 dBm. The pulse width is varied from 5 to 55 ns in
steps of 1 ns. Each recorded time trace for the respective pulse width is averaged 20 · 106 times.
For all power values PS we receive data similar to that visualized in Figure 4.17 (a). To end up
with comparable data we determine the phase shift no longer via the Chebyshev norm since it
is not robust against noise. To determine the phase shift ρ in an automated data analysis script
we have to determine the readout time increment in which the pulse of width τ ends. Therefore,
we analyzed the readout times for a pulse width of τ = 5 ns for all probe powers PP in the
performed measurement. In a histogram it turned out that the maximum of the observed phase
shift is located at the readout time t0 = 1.2533 µs (data not shown). We assume that the pulse
with width τ = 5 ns ends in this time increment. To weaken this assumptions we average over
7 time increments centered around t0 . As the pulses are defined to start at t = 1 µs and end
at t = 1 µs + τ we expect the maximal response of the sample is also shifted to larger readout
times t. The sampling rate of the FPGA board is ωSa /2π = 150 MHz which yields that two time
increments are separated by (150 MHz)−1 = 6.667 ns ≈ 7 ns. Therefore, we assume that for pulse
widths between 5 and 11 ns the respective pulse also ends at the time increment t0 . But for a
pulse width τ = 12 ns the pulse will end at the time increment t0 + 6.6667 ns and so forth. So
for longer pulse widths the maximum response of the observed phase shift is recorded in a time
increment after t0 . The average window has to be shifted too for larger pulse widths τ . So we
define a Heaviside window function which averages the recorded data as
(
1
0 )ωSa
−3 ≤ (t−t2π
− τ −5
≤3
7
7
H(t,τ ) =
(4.25)
0 else
where the center time is t0 = 1.2533 µs as introduced above. The b c denotes the floor operation.
71
Time-domain control of light-matter interaction with superconducting circuits
We would like to mention that the argument of the condition in equation (4.25)
τ −5
(t − t0 ) ωSa
−
2π
7
(4.26)
is always an integer, such that the statement is true for exactly 7 time increments. So the
Heaviside function (4.25) is built in a way such that we can compensate the shift of the response
for larger pulse widths τ . Since for every seventh pulse width the center time t0 is shifted to the
next readout time increment (see Figure 4.18).
Figure 4.18: A visualization of the Heaviside window function (4.25) which we used for an
automated determination of the phase ρ during the data analysis. Every seven pulse width
increments, which is in a good approximation the inverse of the sampling rate of the FPGA
board, the Heaviside window is shifted one readout time increment higher.
The weighting is set to 1/7 since we want to average the phase ρ(t) over seven readout time
increments by using the scalar product
X
ρ=
ρ(t) H(t,τ ) .
(4.27)
t
The phases ρ calculated in this way for every pulse width τ are depicted in Figure 4.19 for four
different powers PS .
72
Time-domain control of light-matter interaction with superconducting circuits
Figure 4.19: Typical data for the mean phase deviation ρ around the maximum as a function of
the pulse width τ for different drive power values PS . We observe a linear increase of the phase
as well as an oscillation. For large power (a) and (b), the frequency seems to be nearly constant.
while for low power (c) and (d) the frequency is decreasing with decreasing power.
In Figure 4.19 it turns out that for every power we observe a linear increasing trend in the phase
ρ for increasing pulse widths τ . This trend is superposed by an oscillatory signal. Therefore, we
use a fit function of the form
ρ = c0 cos (ωRabi τ ) + c1 τ + c2
(4.28)
with fitting parameter c0 , c1 , c2 and ωRabi . The red lines show the result of this fitting function.
For large power PS the frequency of the sinusoidal signal seems to saturate [see (a) and (b) in
Figure 4.19. However, for lower powers (c) and (d) the frequency decreases for decreasing power
PS . In addition to the fit of equation (4.28) we perform a Fourier analysis of the data. To
suppress low frequencies in the spectra we subtract the linear term c1 τ + c2 from the data such
that we end up with the spectra presented in Figure 4.20. For the five lowest power values in
Figure 4.20 (a) we are able to identify a single peak for each power PS while for higher power we
observe more than one peak. This may indicate that we would have to modify our fit function
for higher driving power since equation (4.28) is only able to model one frequency.
73
Time-domain control of light-matter interaction with superconducting circuits
Figure 4.20: The Fourier spectrum of the phase deviation for different drive powers PS is depicted
in (a). We subtracted the linear term of the fit function (4.28) to end up with the spectra
presented here (green lines). We would like to mention that they are shifted for clarity. The
blue circles indicate the Rabi frequency from the fit function (4.28). They agree well with the
respective maximum in the spectrum. The frequencies from the fit function and the prominent
frequencies of the Fourier spectrum for the respective drive amplitude are depicted in (b). The
red line indicates a linear increase of the frequency. We performed a power sweep and analyzed
the Rabi frequencies as a function of the drive amplitude. For low powers the Rabi frequency
ωRabi increases linearly with the amplitude. For large drive powers they seem to converge to a
constant value.
The blue circles in Figure 4.20 (a) indicate the fitted frequencies of that fit function. They agree
well with the peaks in the spectra. To clarify if the observed oscillations are in agreement with
Rabi oscillations we have to analyze the linear dependence of the frequencies on the driving
amplitude. The result of this analysis is visualized in Figure 4.20 (b). It turns out that for low
driving amplitudes the frequencies obtained from the fit seem to depend linearly but for higher
amplitudes (blue circles) they converge to a constant value which coincides with the coupling
strengths we determined when we analyzed the level splitting of the dressed state in section
4.1.3 and 4.1.4. By taking into account the prominent frequencies of the Fourier spectrum [green
triangles in Figure 4.20 (b)] we find that these frequencies agree well with that from the fit
function (4.28). However, if we also take into account the peaks labeled with the arrows in the
spectra they seem to depend linear on the driving amplitude. Such a phenomenon where the
Rabi frequency converges towards resonant frequencies present in the system under study was
also observed in ref. [16] but in a frequency range which was two orders of magnitude larger then
in our case. However, the data presented so far do not disagree with Rabi oscillations. With
an improved experimental setup the origin of this oscillatory behavior should be investigated in
future experiments.
74
Time-domain control of light-matter interaction with superconducting circuits
In the next measurement we apply a flux detuning δΦ ≈ 1 mΦ0 such that the qubit excitation
frequency ωQ /2π = 5.34 GHz. As a result the dressed states of the qubit and the first harmonic
ω1 of the resonator should no longer be present and we expect to observe driven Rabi oscillations
of the qubit and the second harmonic ω2 of the resonator. Therefore, we set the PSG to ωS /2π =
5.34 GHz and a drive power of PS = −10 dBm. The SMF which drives the resonator continuously
at a frequency ωP = 7.109 GHz at an output power corresponding to 1 poa in the resonator (see
Table 4.3). We record time traces from 0 to 3 µs with the FPGA board for pulse widths τ from
5 to 55 ns. Each readout time trace is averaged 20 · 106 times. The result of this measurement
is visualized in Figure 4.21 (a).
Figure 4.21: Typical data for a Rabi oscillation measurement in the case where the qubit is
detuned from its degeneracy point is depicted in (a). We observe a response of the sample
independent of the applied pulse widths. In (b) we visualize the extracted phase shift around
the maximum of the observed response. We still observe a slight linear trend of the phase ρ.
The fit (red line) yields a Rabi frequency of ωRabi /2π = 144.2 MHz. The Fourier spectrum of the
points in (b) corrected by the linear trend is depicted in (c). The peak is found at a frequency
of 143.9 MHz.
75
Time-domain control of light-matter interaction with superconducting circuits
If we compare Figure 4.21 (a) to Figure 4.17 (a) we find that the oscillatory behavior is clearly
visible even for short pulse lengths τ and that the periodicity is shorter. If we use the Heaviside
window function (4.25) for a center readout time t0 = 1.2533 µs we end up with phase shifts ρ
for every pulse width τ as depicted in (b). The red curve is a fit to the function
−τ
ρ = c0 exp
cos (ωRabi τ ) + c1 τ + c2
(4.29)
c4
So we extended the fit function (4.28) by an exponential decay of the amplitude of the oscillatory term. This fit yields a Rabi frequency ωRabi /2π = 144.2 MHz. To confirm this frequency
we perform a Fourier analysis in Figure 4.21 (c). To do this, we subtract the linear trend of the
fit function (4.29) from the phase ρ presented in Figure 4.21 (b). The spectrum is dominated by
a single peak at a center frequency of 143.9 MHz. If we compare the fitted Rabi frequency and
the center frequency of the Fourier spectrum we find that they agree very well. If we take into
account the fitted coupling strength to the low power spectroscopy data presented in Figure 4.7
(e) we find 2g2 /2π = g2 /π = 143.8 MHz. According to ref. [50] the frequency of driven Rabi
√
oscillations is given by νRabi = nS g2 /π. If we assume nS = 1 we have evidence that the data
presented in Figure 4.21 shows driven Rabi oscillations. To clarify this we have to perform a
power sweep at this flux detuning value. Since here the qubit hyperbola presented in Figure
4.7 (f) is quite steep we are very sensitive to flux fluctuations which change the qubit excitation
frequency ωQ via the flux dependent energy bias given in equation (2.23). This difficulty is
still an issue in the experimental setup. However, for the measurement presented here we found
a Rabi frequency of 143.9 MHz this corresponds to a time of approximately 7 ns which it takes
to apply a full rotation on the Bloch sphere. If we now want to excite the qubit from the ground
state |gi to the excited state |ei we have to use a pulse width τ ≈ 3.5 ns.
In this Chapter we performed continuous wave spectroscopy experiments to investigate the characteristic frequencies of our sample and its dependence on the flux detuning δΦ. We determined
the coupling strengths of the qubit resonator system at a negligible photon number population
in the resonator. Further we found indications for vacuum Rabi levels in our system. We successfully performed pulsed two-tone spectroscopy and time-domain measurements in the qubit
group of the WMI for the first time. We are able to detect a reproducible oscillatory behavior of
the phase response of the sample with two different data acquisition cards. The acquired data
indicates that these oscillations are similar to Rabi oscillations but this has to be clarified in
future experiments with an improved setup.
76
Time-domain control of light-matter interaction with superconducting circuits
5
Conclusion and Outlook
During the course of this thesis we have extended the measurement tool box available at the
WMI to time-domain measurements. Therefore, we have built up a functional IQ detector and a
powerful heterodyne calibration algorithm. After a detailed characterization of the detector we
investigated a coupled qubit resonator system. In preliminary measurements we have performed
spectroscopy measurements to determine the coupling strengths.
The study of a coupled qubit-resonator system is an interesting field. The usage of an artificial atom which is realized in a superconducting circuit yields larger coupling strengths than
in cavity QED where natural atoms are used. These artificial atoms are coupled to a superconducting resonator. At the WMI three Josephson junction flux qubits are used to realize artificial
atoms. Flux qubits are favorable since they yield larger coupling strengths than for example
transmons. At the WMI it was demonstrated that the ultra strong coupling regime can be realized by using a coupling junction in the shared edge of the qubit and the resonator. A current
field of research at the WMI is the determination of the maximal coupling strength of a flux qubit
to a superconducting resonator. It was proposed in ref. [45] that the coupling can be enhanced
by a decrease of the area of the coupling junction. In the course of this thesis we used a coupled
qubit resonator system with a relative size β = 1 of the coupling junction for two reasons. First
we investigate this sample in the frequency-domain to verify the relation M ∝ 1/A of the mutual
inductance and the area of the coupling junction and second to determine the energy relaxation
rate of the system in the time-domain.
In the frequency-domain measurements we have found coupling strengths of g1 /2π ≈ 72 MHz
and g2 /2π ≈ 88 MHz of the qubit to the first and second harmonic of the resonator. This shows
that the relation of the mutual induction and the size of the coupling junction is more complex than M ∝ 1/A. For experimentalists it would be useful to have a theory which predicts
the coupling strength as a function of the size of the coupling junction, here work is already in
progress. Furthermore, we have determined the qubit energy gap to ∆/h ≈ 5.02 GHz and its
persistent current to IP ≈ 208 nA. We have calibrated the photon number of the readout mode
of the resonator. With the on-chip antenna we have been able to perform two-tone spectroscopy
measurements. This has demonstrated that on-chip antennas can be implemented successfully
in the qubit group at the WMI. In the recorded two-tone spectroscopy we have found that at the
degeneracy point δΦ = 0 of the qubit the frequency detuning δω1 of the first harmonic ω1 with
the qubit excitation frequency ωQ is less than the coupling strength g1 to this mode such that
the spectrum exhibits the dressed states of our qubit and the first harmonic ω1 of the resonator.
As a result we have found that the excitation levels of the observed side arms in the recorded
spectra agree well with vacuum Rabi levels.
77
Time-domain control of light-matter interaction with superconducting circuits
We have been able to build up a functional detector for time-domain measurements whose main
functional device is an IQ mixer. Furthermore, we have demonstrated that the developed digital
heterodyne calibration method of the IQ mixer is able to compensate the imperfections caused
by the manufacturing tolerances. In the next step of this thesis we successfully have performed
time-domain measurements on a coupled qubit resonator system at the WMI for the first time.
To speed up these measurements by a factor of 10 and to confirm the results recorded with the
ACQIRIS card we have repeated the measurement with an FPGA board. It turned out that with
both data acquisition cards we get highly reproducible results. The decrease of the measurement
time while the signal to noise ratio remains constant shows that the FPGA board should be the
most favorable data acquisition card in future works for these kind of measurements. We have
been able to determine the energy relaxation time T1 of our flux qubit of approximately 120 ns.
The observed oscillatory behavior of the response as a function of the pulse width is a strong
indication that we observed Rabi oscillations. In the performed power sweep we have found a
correlation of the observed frequencies and the drive amplitude which supports this statement.
We have found that the dependence of the observed Rabi oscillations on the driving amplitude
at the degeneracy point of our qubit where we have been affected by the dressed states yields
more interesting data as in the case of driven Rabi oscillations when we detune our flux qubit
from its degeneracy point. Nonetheless this has to be confirmed in future experiments with an
optimized pulse generation and detection setup. For first measurements without an optimized
detector and pulse generation setup the signal to noise ratio is very good.
To improve future time-domain measurements we have to optimize the pulse generation which
means that the envelope of the pulse has to be modified. Furthermore, ghost pulses and the slew
rate of the signal might also be an issue towards an optimization of the experimental setup. The
required hardware for this task is an arbitrary waveform generator (AWG). Further we should
improve the detector by identifying the source of the ground loop and compensate it such that
we do no longer have to use digital filters. However, since we are now able to determine the
length of a π-pulse we would be able to perform Ramsey and Spin echo experiments to access
the dephasing time of the qubit with an improved detector. With this π-pulse we are in principle
able to create a single photon source, since we excite the qubit with one photon by applying such
a pulse. A single photon source is one of the main requirements to test a CNOT gate operation
which may pose a longterm goal in the qubit group at the WMI.
The study of a coupled qubit-resonator system is an interesting field. The usage of an artificial atom which is realized in a superconducting circuit provides a larger experimental ground
than in the case of natural atoms. The performed experiments in the frequency- and time-domain
provided an inside towards a functional quantum computer which can be used for quantum computation and simulation.
78
Time-domain control of light-matter interaction with superconducting circuits
6
Acknowledgments
First of all I would like to thank Prof. Rudolf Gross for providing me the opportunity to do my
Diploma Thesis in the qubit group at the Walther Meissner Institut. His advice and inspiring
didactic style during the interesting discussions was inestimably to solve the tasks during this
work.
I thank Dr. Achim Marx for his advice and great experience concerning any question about the
dilution refrigerator I was using during the experimental term of this thesis. His contributions
helped a lot to bring the theoretical knowledge from lecture courses and the practice of using a
cryostat together.
I am especially thankful to Alexander Baust for the great collaboration during the last year,
the introduction to the cryostat and the measurement instruments. He always was available for
questions and his experience was essential for the progress concerning measurement technique
development at the WMI. I felt great pleasure to explore the sample together with him. This
symbiosis was a big advantage to solve several issues.
Dr. Frank Deppe is to be thanked for the fruitful discussions and for his introduction to the
physics of superconductors and low temperature physics. He was able to awaken my demand to
investigate coupled qubit resonator systems in the qubit group of the WMI during my advanced
study period.
Dr. Hans Huebl’s knowledge on measurement techniques especially the technical issues combined
with the physics behind it was very helpful to solve several tasks. His abundance of patience is
really impressive. Therefore, I want to thank him.
Special thanks go to Max Haeberlein for the production of the sample which I investigated during
the last 12 months. Without his contribution this thesis and the results presented here would
never had been possible. His sophisticated knowledge on theoretical topics and production technique was very helpful in several discussions.
I also would like to thank Elisabeth Hoffmann for the kind introduction into the clean room
during the very beginning of my thesis.
Furthermore, I would like to thank Edwin Menzel and Peter Eder. Their detailed knowledge on
measurement techniques and instruments helped a lot to implement the measurement protocols
and filtering techniques in the time based measurements in this work.
79
Time-domain control of light-matter interaction with superconducting circuits
I’m also thankfully to all members of the qubit group for the fruitful discussions and the great
atmosphere not only concerning the work at the institute but also some group activities during
free time. This provided me a picture of the WMI not only as a place to work but also as a place
to meet friends.
The WMI workshop is to be thanked for the accurate fabrication of various parts which improved
the measurement setup in the laboratory. Robert Müller, head of the diploma student’s workshop, was always willing to provide a workbench when mechanical works have been necessary.
For this, I would like to thank him very much.
Dr. Juan José García-Ripoll from Universidad Complutense de Madrid has to be thanked for
his theoretical work on coupled qubit resonator systems and the fitting algorithms he provided
to proof the correlation between the measurement carried out at the WMI and the developed
theory. This collaboration and his will to provide a theoretical basis for the experiments is very
fruitful.
Further I would like to thank my office partners, especially Michael Schreier and Nikolaj Bittner.
The interesting discussions and their ideas from other qualified fields provided a great working
atmosphere in our room and from time to time a nice chit-chat.
Finally I would like to thank my parents Brigitte and Johann Losinger for their never ending
love. Their support and encouragement was priceless during my studies at the TU Munich.
80
Time-domain control of light-matter interaction with superconducting circuits
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Time-domain control of light-matter interaction with superconducting circuits
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Time-domain control of light-matter interaction with superconducting circuits
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Properties of ellipses,
http://mathworld.wolfram.com/
Time-domain control of light-matter interaction with superconducting circuits
A Digital heterodyne IQ mixer calibration
Here we introduce a calibration method for an IQ mixer where the two channels are represented
by points in the real projective plain. Therefore, we first give a short introduction to the real
projective plain RP2 . The central element of the calibration algorithm is fitting a ellipse with the
help of Fitzgibbon’s method [74, 77]. The basic idea of any projective space is that we embed the
Euclidean space which we are interested in in a space of higher dimension than the dimension of
Euclidean space itself. Furthermore we identify scalar multiples as the same object. Therefore,
a point in the Euclidean plain which is equivalent to the vector space R2 is represented by an
equivalence class in the real projective plain RP2 [75, 76].´
A.1
Mathematical calculations
Definition 1 (Set of points P in the real projective plain RP2 ) The set of all points in
the real projective plain RP2 is given by the quotient structure
n o
R3 \ ~0
P=
.
R \ {0}
An illustration of one object in P is shown in Figure A.1. We introduce a disjoint copy of P and
identify it as lines.
Definition 2 (Set of lines L in the real projective plain RP2 ) The set of all lines in the
real projective plain RP2 is given by the quotient structure
n o
R3 \ ~0
L=
.
R \ {0}
The last definition we have to assign before we are able to introduce the real projective plain
RP2 is a relation which indicates the condition that a point is an element of a line, if we think
of a line as an infinite set of points.
Definition 3 (Incidence relation I) Let [P ] ∈ P and [l] ∈ L be homogenous coordinates,
then the Incidence relation I is defined by:
[P ] I [l] ⇔ hP, li =
3
X
Pi li = 0
i=1
and we say P lies on l.
So here the Incidence relation I is defined as a scalar product. Finally we define the real projective
plain RP2 .
Definition 4 (real projective plain RP2 ) The triple (P, L, I) is called real projective plain
RP2 .
85
Time-domain control of light-matter interaction with superconducting circuits
Figure A.1: Points in the real projective plain RP2 [76]. The blue rectangle should represent
RP2 , while the green line is the equivalence class of a point [P ]. The red dot is one representative
P of the point [P ] here with a z-coordinate z = 1.
Since we defined points in the real projective plain we want to link them with points in the
Euclidean plain, therefore, we define two mappings.
Definition 5 (Homogenisation) The mapping
H : R2 → P
 
x
x


y  .
7→
y
1
is called Homogenisation.
So we are able to transform points from the Euclidean plain to the real projective plain RP2 .
P and L are powerful objects, as we are now allowed to talk about infinite far points and one
infinite far away line. Therefore, we have to take care when we define the inverse mapping of H,
since infinite far away objects do not have a representative in R2 .
Definition 6 (Dehomogenisation) The mapping
n
o
D : R3 \ (x, y, 0)T | x, y ∈ R
→ R2
 
x
1
x
 y  7→
.
y
z
z
is called Dehomogenisation.
Now we are able to talk about points and lines in the real projective plain RP2 and we are able to
identify points in the Euclidean plain with points in the real projective plain RP2 , but we would
also like to talk about the line which connects two points and a way to calculate this specific
line.
86
Time-domain control of light-matter interaction with superconducting circuits
Definition 7 (Join ∨) For two different points [P1 ] , [P2 ] ∈ P we define the operation Join ∨
to calculate the line [l] ∈ L which connects the two points via:
l := P1 ∨ P2 = P1 × P2 .
As we would like to deal with ellipses in a proper way we define an ellipse as follows with the
help of homogeneous coordinates.
Definition 8 (Ellipse) Let p1 , ..., pn be a number of points in RP2 with n ≥ 5, then these points
are on an ellipse, if and only if there exists a matrix E of the form:


a b d
a b


E= b c e
with det(E) 6= 0, det
> 0 and det(E)/(a + c) < 0
b c
d e f
such that the equation pT
i Epi = 0 is fulfilled ∀ i = 0, . . . , n. We call E the descriptive matrix of
the ellipse or just ellipse.
For further details on the fascinating world of projective geometry we refer to ref. [75]. If we
drop the conditions to the determinants in Definition 8 we get a more general form i.e. a conic.
However, to extract the parameters a, b, . . . , f representing the ellipse we use the well established
Fitzgibbon method [74] which includes a constraint such that always an ellipse is fitted to the
points which gained from the experiment. We use Fitzgibbon’s method because it was shown,
that it’s robust against noise [77]. Now that we know all coefficients of the Matrix E, we extract
the correction values out of the matrix. It is worth mentioning that all the calculated values
only use the parameters a, b, . . . , f of the ellipse and they do not take into account in which
mathematical structure the ellipse is embedded. For example the algorithm of Fitzgibbon fits
the parameter in RP5 to a given set of points, the calculations we present here work in the real
projective plain RP2 and the figures depicted in this section are plotted in the Euclidean plain.
We begin with the DC offset of both channels, this manifests itself in an ellipse not centered at
O := (0, 0, 1)T as it was shown in ref. [80] the offset’s are given by
DC1 =
DC2 =
cd − be
,
b2 − ac
ae − bd
.
b2 − ac
(A.1)
(A.2)
and therefore, the first projective transformation one wants to apply is given by a translation of
the form


0 0 −DC1
DC = 0 0 −DC2  .
(A.3)
0 0
1
Here the power of using homogeneous coordinates becomes clearly visible for the first time, we
are able to express a translation as a matrix multiplication instead of a vector addition as a
result the correction matrix presented here is able to consider and correct DC offsets as well as
amplitude offsets and phase deviation errors of the channels I and Q. In comparison to the method
used in ref. [50] were one would need a 2 by 2 correction matrix and a 2 dimensional translation
vector to be as powerful as the method presented here. The result of this transformation can be
seen in Figure A.2 (b). The next step consists of correcting the amplitude and phase difference
of channels at the same time, therefore, the angle between the x-axis and the semi-major axis
has to be known. It can be shown [80] that it is given by
(
1
arctan a−c
for b 6= 0 and a < c
2
2b
.
(A.4)
η= π 1
a−c
for b 6= 0 and a > c
2 + 2 arctan 2b
87
Time-domain control of light-matter interaction with superconducting circuits
Figure A.2: The transformations for the IQ mixer calibration for each step on simulated data
without noise. The raw data with a DC offset, a mismatch in the amplitude of both channels
and a difference in phase unequal to π/2 is shown in (a). In (b) the DC offset is removed by
applying a translation matrix DC. We rotate the ellipse by the angle −η to go into the eigenbasis
where we can easily transform the ellipse into a circle by the dilation AM and finally rotating
back by η (c) to (e). In the last step (f) we restore the phase of channel 1 to its original value
from the experiment by rotating with the angle −δ.
88
Time-domain control of light-matter interaction with superconducting circuits
Figure A.3: Idea how to extract amplitudes of both channels from the fitted ellipse. The amplitude of both channels is given by the projection of the ellipse on the respective axis. So one
starts with solving a quadratic equation (a) and looks up for the case where the discriminant in
Vieta’s Theorem vanishes (b).
Furthermore we need the length of the semi-axes and the amplitudes of both channels, first the
two semi-axes lengths can be calculated via [80]
v
u 2(ae2 + cd2 + f b2 − 2bde − acf )
u
p
,
sa1 = t
(A.5)
(b2 − ac)
(a − c)2 + 4b2 − (a − c)
v
u
2(ae2 + cd2 + f b2 − 2bde − acf )
u
p
,
(A.6)
sa2 = t
(b2 − ac) − (a − c)2 + 4b2 − (a − c)
SMA = max {sa1 , sa2 } ,
(A.7)
= min {sa1 , sa2 } .
(A.8)
SMI
With SM A as the semi-mayor axis and SMI as the semi-minor axis. Now we can extract the
amplitudes of channel 1 and 2 form matrix E, this can be done analytically by considering the
idea depicted in Figure A.3.
Theorem 1 Let E be an ellipse as defined in Definition 8 and centered at O then the horizontal
and vertical projection to the axes are given by
r
(e2 −cf )(b2 −ac)
(cd − be) 1 ± 1 −
(dc−be)2
AM1 =
,
(A.9)
b2 − ac
r
2
)(b2 −ac)
(ae − bd) 1 ± 1 − (d −af
(ae−bd)2
AM2 =
.
(A.10)
b2 − ac
The condition that the ellipse is centered at O guarantees that |AMi+ | = |AMi− | for i = 1, 2,
where i corresponds to the channel numbers, otherwise the solution is shifted by the DC offsets
mentioned above.
89
Time-domain control of light-matter interaction with superconducting circuits
Proof 1 For every point p = (x, y, 1)T the condition of Definition 8 is fulfilled. This condition
can be rewritten by
ax2 + 2bxy + cy 2 + 2dx + 2ey + f = 0.
(A.11)
Now we set x = const. and rearrange
cy 2 + 2(e + bx)y + ax2 + 2dx + f = 0.
(A.12)
Vieta’s Theorem yields
−2(e + bx) ±
q
4 (e + bx)2 − 4c (ax2 + 2dx + f )
y1/2 =
.
2c
(A.13)
Thus we get the two y-values for which the line x = const. intersects the ellipse E [see Figure
A.3 (a)], at the maximum value and therefore at the projection to the x-axis, the two points are
identical and therefore the discriminant vanishes [see Figure A.3 (b)], this implies:
e2 + 2bex + b2 x2 − acx2 − 2cdx − cf = 0
2
2
2
(b − ac)x + 2(be − cd)x + e − cf = 0
(A.14)
(A.15)
Now we use Vieta’s Theorem again:
−2 (be − cd) ±
q
4 (be − cd)2 − 4 (e2 − cf ) (b2 − ac)
AM1 =
2 (b2 − ac)
(cd − be) ±
q
(cd − be)2 − (e2 − cf ) (b2 − ac)
=
b2 − ac
r
(cd − be) 1 ± 1 −
=
(e2 −cf )(b2 −ac)
(dc−be)2
(A.16)
b2 − ac
Repeating this with y = const. to receive AM2 :
ax2 + 2(d + by)x + cy 2 + 2ey + f = 0
q
−2(d + by) ± 4 (d + by)2 − 4a (cy 2 + 2ey + f )
x1/2 =
2a
d2 + 2bdy + b2 y 2 − acy 2 − 2cey − af = 0
2
2
2
(b − ac)x + 2(bd − ae)x + d − af = 0
−2 (bd − ae) ±
AM2 =
(A.19)
(A.20)
4 (bd − ae)2 − 4 (d2 − af ) (b2 − ac)
q
(ae − bd)2 − (d2 − af ) (b2 − ac)
b2 − ac
r
(ae − bd) 1 ± 1 −
=
(A.18)
2 (b2 − ac)
(ae − bd) ±
=
q
(A.17)
(d2 −af )(b2 −ac)
(ae−bd)2
(A.21)
b2 − ac
90
Time-domain control of light-matter interaction with superconducting circuits
Now we have all ingredients to transform the ellipse to a circle with desired radius r, here we
choose r = AM1 . One can also calibrate to the amplitude of channel 2 AM2 or to any other
value of interest. This is now applied by a series of projective transformation of the form.

  AM1


cos(η) − sin(η) 0
0
0
cos(η)
sin(η)
0
SMA
AM1
Rη AM R−η =  sin(η) cos(η) 0  0
0 − sin(η) cos(η) 0 . (A.22)
SMI
0
0
1
0
0
1
0
0
1
We rotate the ellipse by the angle −η such that the semi-mayor axis is equivalent with the x-axis,
and the semi-minor axis with the y-axis and as a result we are in an orthonormal eigenbasis of
the ellipse, then by applying a dilation to the ellipse such that it is now a circle and finally
rotating back by the angle η, see Figure A.2 (c) - (e) for a visualization of these steps. We now
already have achieved that the two channels have the same amplitude and are phase shifted by
π/2, but due to the fact that we correct this in the eigenbasis of the ellipse we lost the phase of
channel 1 and 2 at the beginning of the applied pulse. The phase of channel 2 at the beginning
is lost because we want it to be phase shifted from channel 1 by π/2. However, there is a way
to recover the phase of channel 1 by correcting the angle in an orthogonal basis. We do this, so
that we can easily compare this result to an other method used by another group [50].
Theorem 2 Let p be a point on an ellipse E, then the angle error caused by the dilation AM is
given by
SMA − SMI
δ := arctan
.
(A.23)
SMA + SMI
Proof 2 As p ∈ RP2 its coordinates are given by p = (x, y, 1)T and
    AM1 
 AM1
x
0
0
SMA
SMA x
AM1
 · y  =  AM1 y  .
AM · p =  0
0
SMI
SMI
1
0
0
1
1
We show that the point O is a fixpoint of the
 AM1
0
SMA
AM1
AM · O =  0
SMI
0
0
transformation AM
    
0
0
0
0 · 0 = 0 = O.
1
1
1
We are now interested in the angle between the two lines h1 , h2 defined as
     
x
0
y





h1 := p ∨ O = y × 0 = −x ,
1
1
0
 AM1     AM1 
0
SMA x
SMI y
AM1 
1  × 0 = 
.
h2 := AM p ∨ O =  AM
y
−
SMI
SMA x
1
1
0
(A.24)
(A.25)
(A.26)
(A.27)
Since we are only interested in measuring angles and not distances, it is sufficient to switch from
the orthonormal basis {(1, 0, 0)T , (0, 1, 0)T , (0, 0, 1)T } to the orthogonal basis
{(y, 0, 0)T , (0, x, 0)T , (0, 0, 1)T } and receive new coordinates
 
1

−1 ,
h1 =
(A.28)
0
 AM1 
SMI
AM1 
h2 = − SMA
.
0
(A.29)
91
Time-domain control of light-matter interaction with superconducting circuits
We now use Laguerre’s Formula [75], therefore we define the two complex numbers
q1 := 1 − i,
AM1
AM1
−i
= u − iv.
q2 :=
SMI
SMA
(A.30)
(A.31)
so the angle can be calculated:
∗ 1 (1 + i)(u − iv)
q1 q2
1
u + v + i(u − v)
1
= ln
δ = ln
=
2i
q1 q2∗
2i (1 − i)(u + iv)
2i
u + v − i(u − v)
 



 q3 
1 
1
q3
  + i arg q3 
= ln ∗ = 
ln
∗
∗



2i
q3
2i
q3
q3 
|{z}
=1





q3
q3
Im
Im
∗
q3
q3∗
1 



.

= arctan
=
2i arctan
q3 q3
q3
2i
Re ∗ + ∗ Re ∗ + 1
q3
q3
q3
We have a closer look at the argument of arctan:
q3 q3
Im |q
Im qq3∗
2
Im (q3 q3 )
|
3
3
=
.
2 =
|q
|
q3
q3 q3
3
Re (q3 q3 ) + |q3 |2
Re q∗ + 1
+
Re |q
2
2
|
|q |
3
3
(A.32)
(A.33)
3
So we calculate q3 q3
q3 q3 = (Re (q3 ) + i Im (q3 )) (Re (q3 ) + i Im (q3 ))
= (Re (q3 ))2 − (Im (q3 ))2 + 2i Re (q3 ) Im (q3 ) .
With these results we enter equation (A.32) again
Im (q3 q3 )
δ = arctan
Re (q3 q3 ) + |q3 |2
2 Re (q3 ) Im (q3 )
= arctan
(Re (q3 ))2 − (Im (q3 ))2 + (Re (q3 ))2 + (Im (q3 ))2
!
AM1
AM1
−
Im (q3 )
u−v
SMI
SMA
= arctan
= arctan AM
= arctan
AM1
1
Re (q3 )
u+v
+
SMI
SMA
SMA − SMI
= arctan
.
SMA + SMI
(A.34)
(A.35)
It is worth mentioning that δ only depends on the value of the semiaxes so it is a global property
of the ellipse and therefore to the imperfections off the signal and not of each individual point.
A lower and upper bound of δ is given by 0 ≤ δ ≤ π/4.
92
Time-domain control of light-matter interaction with superconducting circuits
Figure A.4: The signal of the two channels plotted versus time. The dashed lines represents the
uncalibrated data from Figure A.2 (a). If one would stop calibration with the situation in Figure
A.2 (e) one would end up with the dot-dashed lines. It is clearly visible that the two signals have
the same amplitude and are phase shifted by π/2. However, we can even reconstruct the phase
of channel 1 (solid lines) by rotating by −δ, as a result the sinusoidal movement of the blue solid
line follows perfectly the movement of the uncalibrated channel 1 data. If channel 1 would not
have been perturbed by a DC offset the dashed and the solid blue line would be identical.
The lower bound due to SMA = SMI which is the case if the data is already calibrated and
we have a perfect circle. The upper bound can be found with the condition SMI = 0. In that
case the ellipse is a degenerated double line which occurs if and only if the two signals are in
phase or phase shifted by π. The exact mathematical condition on here is (2n − 1)π with n ∈ N.
Only under this condition the method presented here does not work accurately because of the
breakdown of Fitzgibbon method. To correct this error δ we define a new rotation matrix


cos(δ) sin(δ) 0
R−δ = − sin(δ) cos(δ) 0 .
(A.36)
0
0
1
The effect of this rotation is depicted in Figure A.2 (f). As it is difficult to extract the phase
information from Figure A.2 we present the two signals in parameter form i.e. plotted against
time in Figure A.4. Without the rotation R−δ we would receive a calibrated signal with identical
amplitudes and a phase shift of π/2 of the two arms (dot-dashed lines in Figure A.4), but with
the extra rotation we are able to recover the phase of channel 1 (solid lines). One would expect
to apply R−δ directly after AM . Since rotation matrices of the same dimension form an Abelean
group and we do not have to care about the order how we apply it to the points.
93
Time-domain control of light-matter interaction with superconducting circuits
Finally the correction matrix C is given by


∗ ∗ ∗
C = R−δ Rη AM R−η DC = ∗ ∗ ∗ .
0 0 1
(A.37)
By multiplying this matrix C from the left to any set of points given in homogeneous coordinates
with the measured response of channel 1 on the x-coordinate and channel 2 on the y-coordinate
one is able to calibrate your measurement a posteriori. The result of this for ideal data is shown
in Figure A.4. If one would like to implement this correction matrix C in a numeric program,
one would use the group ability of rotation matrices and replace R−δ Rη by Rη−δ to reduce the
number of operations applied and therefore speed up the program. However, here we decided
to do it step by step to show the influence of every single matrix operation. Finally we would
like to present the MATLAB code in the case of a time-domain measurement with the AQCIRIS
card as data acquisition card.
A.2
MATLAB code
Here we present the MATLAB code to analyze a time-domain measurement, therefore we call
the script CallAuswPulsedTwotone.m.
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%This script calculates the result of a time−domain measurement recorded
%with the ACQIRIS card. If we use the FPGA board the applied filters on the
%I and Q raw data do not have to be applyed and the function
%"readPulsedTwoToneAQCIRIS.m" has to be replaced by
%"readPulsedTwoToneFPGA.m"
%
% written by Thomas Losinger
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%clear workspace
clear
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%input arguments:
%strings
path = ''; % string
%scalars
mixfreq = 10;
samplingRate = 400;
NumSamples = 1201;
%vectors
power = [−20];
times = [5:38];
with path to raw data
% IF frequency of the IQ−Mixer in MHz
% sampling rate of the ACQIRIS card in MHz
% total number of readout time increments
% probe powers in dBm
% pulse width in ns
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%begin execution
[Iraw,Qraw]=readPulsedTwotoneACQIRIS(path,NumSamples,times,power);
% The function "readPulseTwoToneACQIRIS" reads the rawdata from a WMI intern
% data format and writes it into the two MATLAB variables Iraw and Qraw.
% Therefore we will not publish the code of this function. The output is of
% the form
%
%
dim(Iraw) = lenght(NumSamples)+1, lenght(times)+1, lenght(power)
%
Iraw(1,1,jj) = power(jj)
%
Iraw(1,2:end,:) = times
%
Iraw(2:end,1,:) = recorded readout times in us
%
Iraw(2:end,2:end,:) = recorded data in uV
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Time-domain control of light-matter interaction with superconducting circuits
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%apply ditital filters (these can be neglected if the FPGA board is used)
% 200 MHz sliding window filter on both cannels
Ifil=slidingWindow(Iraw,200,samplingRate);
Qfil=slidingWindow(Qraw,200,samplingRate);
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% 12.5 MHz hopping window filter only on channel Q
Qfilpat=hoppingWindow(Qfil,12.5,samplingRate);
Ifilpat=Ifil(1:length(Qfilpat),:,:);
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%calibrate IQ mixer
[Amplitude, Phase, CorrValues]=calibrateIQ(Ifilpat,Qfilpat, mixfreq);
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%apply IF sliding window filter (also with the FPGA board)
AmplitudeFil=slidingWindow(Amplitude,mixfreq,samplingRate);
PhaseFil=slidingWindow(Phase,mixfreq,samplingRate);
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%draw a nice picture
plotresult(AmplitudeFil,path,1);
plotresult(PhaseFil,path,0);
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%save workspace in subfolder Analysis (this folder must exist)
save([path,'\Analysis\',datestr(now,'yyyymmdd'),'data']);
This script uses the functions slidingWindow.m
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function [ filtered ] = slidingWindow( unfiltered, filterFreq, SamplingRate)
%slidingWindow applies a digital sliding window filter to the rawdata to
%filter the filterFreq and higher harmonics
%
input arguments
%
unfiltered contains the raw data in the form
%
dim(unfiltered) = n,m,p
%
unfiltered(1,1,jj) = probe power in dBm
%
unfiltered(1,2:end,:) = pulse widths in ns
%
unfiltered(2:end,1,:) = recorded readouttimes in us
%
unfiltered(2:end,2:end,:) = recorded data which should be filtered
%
filterFreq contains the frequency at which the filter should operate
%
SamplingRate contains the sampling rate of the data acquisition card
%
%
output arguments
%
filtered contains the filtered data of the same format as unfiltered
%
but with fewer readout times
%
% written by Thomas Losinger
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[n,m,p]=size(unfiltered);
fillen=floor(SamplingRate./filterFreq)−1; % floor to receive an integer
% initiate memory
filtered=unfiltered(1:n−fillen,:,:);
%apply filter
for jj=1:p
for ii=2:m
for kk=2:n−fillen
filtered(kk,ii,jj)=mean(unfiltered(kk:(kk+fillen),ii,jj));
end
end
end
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end
and hoppingWindow.m
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Time-domain control of light-matter interaction with superconducting circuits
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function [ filtered ] = hoppingWindow( unfiltered, filterFreq, SamplingRate)
%hoppingWindow applies a digital hopping window filter to the rawdata to
%filter a reqular stucture with frequency filterFreq
%
input arguments
%
unfiltered contains the raw data in the form
%
dim(unfiltered) = n,m,p
%
unfiltered(1,1,jj) = probe power in dBm
%
unfiltered(1,2:end,:) = pulse widths in ns
%
unfiltered(2:end,1,:) = recorded readouttimes in us
%
unfiltered(2:end,2:end,:) = recorded data which should be filtered
%
filterFreq contains the frequency at which the filter should operate
%
SamplingRate contains the sampling rate of the data acquisition card
%
%
output arguments
%
filtered contains the filtered data of the same format as unfiltered
%
but with fewer readoutimes
%
% written by Thomas Losinger
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[n,m,p]=size(unfiltered);
fillen=floor(SamplingRate./filterFreq); % floor to receive an integer
% initiate memory
filtered=unfiltered(1:(floor((n−1)./fillen)*fillen+1),:,:);
%apply filter
for jj=1:p
for ii=2:m
%determine the pattern which should be filtered over 5 periods
pattern=zeros(fillen,1);
for k=1:5
pattern=pattern+unfiltered(1+((fillen*k−(fillen−1)):(k*fillen)),ii,jj);
end
pattern=pattern/5;
% filter the pattern
for kk=1:floor((n−1)/fillen)
filtered(1+((fillen*kk−(fillen−1)):(32*kk)),ii,jj)= ...
unfiltered(1+((fillen*kk−(fillen−1)):(fillen*kk)),ii,jj)−pattern;
end
end
end
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end
to filter the digital artifact added by the ACQIRIS card presented in Figure 3.10 (d) and (e). In
the next step we calibrate the IQ mixer via the function calibrateIQ.m.
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function [ Amp, Phase, CorrVal] = calibrateIQ( Iraw, Qraw, mixfreq)
%calibrateIQ calibrates an IQ mixer
%
input parameters
%
Iraw and Qraw contains the uncalibrated data in the form
%
dim(Iraw) = n, m, p
%
Iraw(1,1,jj) = probe power in dBm
%
Iraw(1,2:end,:) = pulse widths in ns
%
Iraw(2:end,1,:) = recorded readout times in us
%
Iraw(2:end,2:end,:) = recorded data in uV
%
mixfreq is the IF frequency of the IQ mixer in MHz
%
%
output parameters
%
Amp and Phase contain the extraced amplitude in uV
%
and phase in deg in the same form as the input parameters Iraw and Qraw
%
CorrVal contains the important correction values (DC offset of the
%
channels and their amplitudes in uV such as the phase deviation in deg
%
for each pulse width in ns
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Time-domain control of light-matter interaction with superconducting circuits
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%
% written by Thomas Losinger
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[n,m,p]=size(Iraw);
%initialize memory for Amp, Phase and CorrVal
Amp=Iraw;
Phase=Iraw;
CorrVal=zeros(6,m−1,p);
%write pulsewidth into first row
CorrVal(1,:,:)=Iraw(1,2:end,:);
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%enable user to select the calibration region therefore we plot the first
%recorded trace
rawdataplot=figure();
hold on
plot(Iraw(2:end,1,1),Iraw(2:end,2,1),'−',Qraw(2:end,1,1),Qraw(2:end,2,1),'−r');
legend('Iraw','Qraw');
xlabel('time (\mus)');
ylabel('Iraw, Qraw (V)');
title(['raw data for pulse width = ',num2str(Iraw(1,2,1)),' ns and probe power = ...
',num2str(Iraw(1,1,1)),' dBm']);
hold off
%enable user selected start stop for calibration:
startpoint=input('Signaltime start in \mus: ','s');
startpoint=str2double(startpoint);
stoppoint=input('Signaltime stop in \mus: ','s');
stoppoint=str2double(stoppoint);
time=Iraw(startpoint≤Iraw(2:end,1) & Iraw(2:end,1)≤stoppoint,1);
omegat=2.*pi.*mixfreq.*Iraw(2:end,1,1);
for jj=1:p
for ii=2:m
y1=Iraw(2:length(time)+1,ii,jj);
y2=Qraw(2:length(time)+1,ii,jj);
Y=[y1,y2,ones(size(time))]';
%start calibtation by using Fitzgibbon method
ellipsePara=EllipseDirectFit([Y(1:2,:)]');
%assign Matrix from the extracted fitparameters
E=[ellipsePara(1),ellipsePara(2)/2,ellipsePara(4)/2;...
ellipsePara(2)/2,ellipsePara(3),ellipsePara(5)/2;...
ellipsePara(4)/2,ellipsePara(5)/2,ellipsePara(6)];
%calculate correction matrix
corrvalues=extractCorrectionValues(E);
%write corrvalues into Corval
CorrVal(2:end,ii−1,jj)=[corrvalues(1:2);corrvalues(6:8)];
%build correction matrix
RotMat1=[cos(corrvalues(5)),sin(corrvalues(5)),0;...
−sin(corrvalues(5)),cos(corrvalues(5)),0;...
0,0,1];
AmpMat=[corrvalues(6)./corrvalues(3),0,0;...
0,corrvalues(6)./corrvalues(4),0;...
0,0,1];
etamindel=corrvalues(5)−corrvalues(8);
RotMat2=[cos(etamindel),−sin(etamindel),0;...
sin(etamindel),cos(etamindel),0;...
0,0,1];
CorrMat=RotMat2*AmpMat*RotMat1*[1,0,−corrvalues(1);0,1,−corrvalues(2);0,0,1];
%multiply correction matrix to all data points of current trace
temp=CorrMat*[Iraw(2:end,ii,jj),Qraw(2:end,ii,jj),ones(n−1,1,1)]';
%calculate Amplitude and Phase
Amp(2:end,ii,jj)=sqrt(temp(1,:).^2+temp(2,:).^2);
Phase(2:end,ii,jj)=unwrap(atan2(temp(2,:),temp(1,:)));
%now subtract ideal phase from unwrapped phase
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Time-domain control of light-matter interaction with superconducting circuits
Phase(2:end,ii,jj)=(Phase(2:end,ii,jj)−(Phase(2,ii,jj)+omegat)).*180./pi;
%and correct it by mean phase during the calibration
meanPhase=mean(Phase(startpoint≤Iraw(2:end,1) & Iraw(2:end,1)≤stoppoint,ii,jj));
Phase(2:end,ii,jj)=Phase(2:end,ii,jj)−meanPhase;
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end
end
% convert phase deviation from rad in deg
CorrVal(6,:)=CorrVal(6,:)./pi.*180;
end
In the first step of the calibration we use Fitzgibbon [74, 77] method to fit an ellipse with the
function EllipseDirectFit.m.
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function A = EllipseDirectFit(XY);
%
% Direct ellipse fit, proposed in article
%
A. W. Fitzgibbon, M. Pilu, R. B. Fisher
%
"Direct Least Squares Fitting of Ellipses"
%
IEEE Trans. PAMI, Vol. 21, pages 476−480 (1999)
%
% Our code is based on a numerically stable version
% of this fit published by R. Halir and J. Flusser
%
%
Input: XY(n,2) is the array of coordinates of n points x(i)=XY(i,1), ...
y(i)=XY(i,2)
%
%
Output: A = [a b c d e f]' is the vector of algebraic
%
parameters of the fitting ellipse:
%
ax^2 + bxy + cy^2 +dx + ey + f = 0
%
the vector A is normed, so that ||A||=1
%
% This is a fast non−iterative ellipse fit.
%
% It returns ellipses only, even if points are
% better approximated by a hyperbola.
% It is somewhat biased toward smaller ellipses.
%
centroid = mean(XY);
% the centroid of the data set
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D1 = [(XY(:,1)−centroid(1)).^2, (XY(:,1)−centroid(1)).*(XY(:,2)−centroid(2)),...
(XY(:,2)−centroid(2)).^2];
D2 = [XY(:,1)−centroid(1), XY(:,2)−centroid(2), ones(size(XY,1),1)];
S1 = D1'*D1;
S2 = D1'*D2;
S3 = D2'*D2;
T = −inv(S3)*S2';
M = S1 + S2*T;
M = [M(3,:)./2; −M(2,:); M(1,:)./2];
[evec,eval] = eig(M);
cond = 4*evec(1,:).*evec(3,:)−evec(2,:).^2;
A1 = evec(:,find(cond>0));
A = [A1; T*A1];
A4 = A(4)−2*A(1)*centroid(1)−A(2)*centroid(2);
A5 = A(5)−2*A(3)*centroid(2)−A(2)*centroid(1);
A6 = A(6)+A(1)*centroid(1)^2+A(3)*centroid(2)^2+...
A(2)*centroid(1)*centroid(2)−A(4)*centroid(1)−A(5)*centroid(2);
A(4) = A4; A(5) = A5; A(6) = A6;
A = A/norm(A);
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end
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%
EllipseDirectFit
Time-domain control of light-matter interaction with superconducting circuits
Afterwards we extract the correction values from the fit parameters based on the calculations in
section A.1. This is achieved by using the function extractCorrectionValues.m.
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function [ erg ] = extractCorrectionValues( E )
%extractCorrectionValues takes a fitted Ellipse E and extracts correction
%values such as the DCoffset of Channel 1 and 2, the semiaxes, the angle
%eta from the majoraxis with respect to the x−axis, the amplidute of
% Channel 1 and 2 as well as the phase deviation ∆
%
%
input argument
%
A 3x3 Matrix E of the form (a,b,d;b,c,e;d,e,f)
%
%
output argument erg with values
%
DC offset Channel 1
%
DC offset Channel 2,
%
semimajor axis SMA,
%
semiminor axis SMI,
%
angle eta between semimajor axis and x−axis,
%
amplitude Channel 1,
%
amplitude Channel 2,
%
phase deviation ∆
%
% written by Thomas Losinger
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%initialize memory
erg=zeros(8,1);
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%start calculation
alpha= E(1,2).^2−E(1,1).*E(2,2);
%calculate center of ellipse DC_1 and DC_2
erg(1)=(E(2,2).*E(1,3)−E(1,2).*E(2,3))./alpha;
erg(2)=(E(1,1).*E(2,3)−E(1,2).*E(1,3))./alpha;
%calculate angle eta between semimayor axis and −axis
erg(5)=−0.5.*acot((E(2,2)−E(1,1))./(2.*E(1,2)));
%calculate semiaxes SMA and SMI
beta=2.*(E(1,1).*E(2,3).^2+E(2,2).*E(1,3).^2+E(3,3).*E(1,2).^2 ...
−2.*E(1,2).*E(1,3).*E(2,3)−E(1,1).*E(2,2).*E(3,3));
gamma=sqrt(1+(4.*E(1,2).^2)./(E(1,1)−E(2,2)).^2);
a=sqrt(beta./(alpha.*((E(2,2)−E(1,1)).*gamma−(E(2,2)+E(1,1)))));
b=sqrt(beta./(alpha.*((E(1,1)−E(2,2)).*gamma−(E(2,2)+E(1,1)))));
erg(3)=max(a,b);
erg(4)=min(a,b);
if(a<b)
erg(5)=pi./2+erg(5);
end
%calculate the amplitudes of the channels AM_1 and AM_2
beta=E(2,2).*E(1,3)−E(1,2).*E(2,3);
gamma=E(1,1).*E(2,3)−E(1,2).*E(1,3);
erg(6)=((beta+sqrt(beta.^2+alpha.*(E(2,2).*E(3,3)−E(2,3).^2)))./alpha)−erg(1);
erg(7)=((gamma+sqrt(gamma.^2+alpha.*(E(1,1).*E(3,3)−E(1,3).^2)))./alpha)−erg(2);
if erg(6)<0
erg(6)=((beta−sqrt(beta.^2+alpha.*(E(2,2).*E(3,3)−E(2,3).^2)))./alpha)−erg(1);
end
if erg(7)<0
erg(7)=((gamma−sqrt(gamma.^2+alpha.*(E(1,1).*E(3,3)−E(1,3).^2)))./alpha)−erg(2);
end
%calculate phase deviation ∆
erg(8)=atan((erg(3)−erg(4))./(erg(3)+erg(4)));
end
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Time-domain control of light-matter interaction with superconducting circuits
Finally we visualize the results via plotresult.m
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function [] = plotresult( data, path, caseselect )
%plotresult plots the data in an imagesc plot and saves it to path
%
data contains the information to plot in the form
%
dim(data) = n, m, p
%
data(1,1,jj) = prob power in dBm
%
data(1,2:end,:) = pulse widths in ns
%
data(2:end,1,:) = recorded readouttimes in us
%
data(2:end,2:end,:) = recorded data
%
path is a string it contains the path to save the picture
%
caseselect is 0 to plot the phase
%
is 1 to plot the amplitude
%
% written by Thomas Losinger
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p=size(data,3);
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for jj=1:p
%determine if phase or amplitude should be ploted
if caseselect == 0
zaxtit = 'phase (deg)';
titlename = ['PhaseProbePower',num2str(data(1,1,jj)),'dbM'];
else
zaxtit = 'amplitude (\muV)';
titlename = ['AmplitudeProbePower',num2str(data(1,1,jj)),'dbM'];
end
%draw the picture
PlotRes=figure();
imagesc( data(1,2:end,jj),data(2:end,1,jj), data(2:end,2:end,jj));
colorbar;
xlabel('pulse width (ns)');
ylabel('readout time (\mus)');
zaxis = zaxtit;
cb = colorbar('vert');
zlab = get(cb,'ylabel');
set(zlab,'String',zaxis);
set(gca,'YDir','normal');
title(titlename);
%save image to the subfolder Analysis (this folder must exist)
file=[path,'\Analysis\',titlename,datestr(now,'yyyymmdd')];
saveas(PlotRes,file,'jpg');
saveas(PlotRes,file,'fig');
end
end
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Time-domain control of light-matter interaction with superconducting circuits
B
Photon number calibration
In this chapter we carry out the necessary calculations on the photon number calibration of
section 4.1.3. We assume that the Hamiltonian of the observed dressed states is of the form
b
H
ω1 g 1
=
(B.1)
g1 ω
eQ
~
where ω1 denotes the eigenfrequency of the first harmonic of the resonator, g1 is the coupling of
the qubit and the first harmonic of the resonator. Finally ω
eQ denotes the dispersively shifted
qubit excitation frequency from the Hamiltonian (2.57). Since we use the second harmonic for
the photon number calibration in the experiment we find the explicit form.
ω
eQ = ωQ + (2n + 1)
(g2 )2
.
δω2
(B.2)
Here ωQ denotes the pure qubit excitation frequency as defined in equation (2.25), g2 is the
coupling of the qubit to the second harmonic and δω2 = ωQ − ω2 is the frequency detuning of
the qubit and the second harmonic. In the next step we diagonalize the Hamiltonian (B.1) to
find the eigenfrequencies λ± of the system. We use the ansatz
b − ~λ · Id2 = 0
det H
(B.3)
ω1 − λ
g1
det
= (ω1 − λ) (e
ωQ − λ) − (g1 )2 = 0
(B.4)
g1
ω
eQ − λ
λ2 − (ω1 + ω
eQ ) λ + ω1 ω
eQ − (g1 )2 = 0
(B.5)
According to Vieta’s Theorem the solutions of the characteristic polynom (B.5) are given as
r
e Q ) 2 − 4 ω1 ω
eQ − (g1 )2
(ω1 + ω
eQ ) ± (ω1 + ω
λ± =
2
q
eQ + (e
ωQ )2 − 4ω1 ω
eQ + 4 (g1 )2
(ω1 + ω
eQ ) ± (ω1 )2 + 2ω1 ω
(B.6)
=
2
q
(ω1 + ω
eQ ) ± (ω1 − ω
eQ )2 + 4 (g1 )2
=
=: η ± µ.
2
If we use the explicit form of ω
eQ given in equation (B.2) we end up with
r
2 2
(g2 )2
ω1 + ωQ + (2n + 1) δω2 ±
ω1 − ωQ + (2n + 1) (gδω2 )2
+ 4 (g1 )2
λ± =
.
2
(B.7)
If ω
eQ = ω1 the level splitting is 2g1 which agrees well with the data depicted in Figure 2.9 (a).
We would like to mention that the photon number dependency of g1 is assumed to be constant
since the spectroscopy power PS is constant.
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Time-domain control of light-matter interaction with superconducting circuits
However, since we now know the eigenfrequencies of the Hamiltonian (B.1) for a given photon
number n in the second harmonic mode of the resonator and since we have some fitted eigenfrequencies of the performed power sweep in section 4.1.3 at a given output power of the VNA, we
have to link the photon number scale and the power scale in the next step. Therefore, we use
nonlinear optimization i.e. we minimize the distance between the measured1 eigenfrequencies ν±
and the theoretical eigenfrequencies λ± . First we define the function
λ+
f (n) : R → R2 n 7→
.
(B.8)
λ−
Further we treat the measured frequencies ν± as a vector in R2 of the form (ν+ , ν− )T with
ν+ > ν− such that we can formulate the optimization problem
ν .
(B.9)
min f (n) − + n
ν− 2
For each measured frequency pair ν± from which we know the output power of the VNA we
determine the photon number n in the eucledian norm k k2 for which the deviation of the
measured frequencies ν± to the theoretical eigenfrequencies λ± is minimal. As a result we end
up with pairs of values for the output power of the VNA and a photon number. To these set
of points we fit a line through the origin [see Figure 4.6 (d)] to calibrate the photon number.
However, here we present a more detailed expression on the objective function of the optimization
problem (B.9).
f (n) − ν+ = λ+ − ν+ ν− 2
λ−
ν− 2
q
(B.10)
= (λ+ − ν+ )2 + (λ− − ν− )2
q
= (λ+ )2 + (λ− )2 − (2λ+ ν+ + 2λ− ν− ) + (ν+ )2 + (ν− )2
With the two auxiliary calculations based on the notation in equation (B.6).
Auxiliary calculation 1
(λ+ )2 + (λ− )2 = (η + µ)2 + (η − µ)2 = 2η 2 + 2µ2
i
1h
1
= (ω1 + ω
e Q )2 +
(ω1 − ω
eQ )2 + 4 (g1 )2
2
2
= (ω1 )2 + (e
ωQ )2 + 2 (g1 )2
(B.11)
Auxiliary calculation 2
2λ+ ν+ + 2λ− ν− = 2 (η + µ) ν+ + 2 (η − µ) ν− = 2η (ν+ + ν− ) + 2µ (ν+ − ν− )
q
= (ω1 + ω
eQ ) (ν+ + ν− ) + (ω1 − ω
eQ )2 + 4 (g1 )2 (ν+ − ν− )
(B.12)
At this point we stop our analytic calculations and determine the photon number n via numerical
calculations in MATLAB. Therefore, we call the script PhotonCalScript.m.
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From the measured spectra we extract center frequencies of a Lorentzian fit, these frequencies are denoted
with ν± .
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This script calculates a photon number calibration in the case that the
dressed states were observed in a two−tone spectroscopy power sweep of
the probe tone with k power values. It needs a vector v of the form:
dim(v) = 3,k
v(1,:) = probe power values in dBm
v(2,:) = center frequency of the state |m,+>
v(3,:) = center frequency of the state |m,−>
written by Thomas Losinger
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%Set parameters
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omegaResOne = 5.067;
omegaResTwo = 7.106;
omegaQubit = 5.04;
gOne = 0.0865;
gTwo = 0.0882;
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first harmonic resonator mode in GHz
second harmonic resonator mode in GHz
qubit excitation frequency in GHz
coupling of qubit and first harmonic
coupling of qubit and second harmonic
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% initiate variable
photonnumbers=[v(1,:);zeros(1,length(v))];
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%
a
b
c
d
e
f
start calculation
= omegaQubit−omegaResTwo; % frequency detuning
= gTwo.^2./a;
% dispersive frequency shift
= omegaResOne.^2;
% first harmonic squared
= 2.*gOne.^2;
% twice g_1 squared
= omegaResOne+omegaQubit; % sum of first harmonic and qubit excitation frequency
= omegaResOne−omegaQubit; % difference of first harmonic and qubit excitation ...
frequency
for jj = 1:length(v)
g = v(2,jj)+v(3,jj);
% sum of the center frequencies
h = v(2,jj)−v(3,jj);
% difference of the center frequencies
l = v(2,jj).^2;
% squared center frequency of the state |m,+>
m = v(3,jj).^2;
% squared center frequency of the state |m,−>
%build objective function
objfunc = @(n) sqrt(c+(omegaQubit+(2*n+1)*b).^2+d−((e+(2*n+1)*b)*g+ ...
h*sqrt((f−(2*n+1)*b).^2+2*d))+l+m);
photonnumbers(2,jj)=fminsearch(objfunc,1);
end
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% determine output power to photon number conversion factor by fitting a
% line trough the origin on a linear−linear−scale
y = inline('a.*x','a','x');
slope = nlinfit(10.^(photonnumbers(1,:)./10),photonnumbers(2,:),y,1);
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Time-domain control of light-matter interaction with superconducting circuits
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Persönliche Erklärung
Mit der Abgabe der Diplomarbeit versichere ich, dass ich die Arbeit selbständig verfasst und
keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe.
Ort, Datum, Unterschrift
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