High-Fidelity Composite Pulse Techniques
for Quantum Computation and Simulation
Nikolay V. Vitanov
Quantum Optics and Quantum Information Group
Department of Physics, Sofia University, Bulgaria
Quantum Simulations with Trapped Ions Workshop 2013
Old Ship Hotel, Brighton
December 17, 2013
Quantum Optics & Quantum Info @ Sofia Uni
Andon
Rangelov
(SU)
Peter
Ivanov
(SU)
Svetoslav
Ivanov
(SU)
Genko
Vasilev
(SU)
Boyan
Torosov
(Milan)
Elica
Kyoseva
(MIT)
Iavor
Boradjiev
(BAS)
Tihomir
Tenev
(BAS)
Genko
Genov
(PhD)
Lachezar
Simeonov
(PhD)
Hristina
Hristova
(PhD)
Christo
Tonchev
(PhD)
ACTIVITIES: quantum optics & control, quantum computation,
quantum simulation, classical & nonlinear optics, femtosecond physics
EU: STREP iQIT
AvH: CLGrant
Outline
Two-state systems
Broadband, narrowband, passband, fractional-π pulses
Local addressing (subwavelength localization)
Composite adiabatic passage
Multistate systems
Composite STIRAP
Suppression of unwanted transitions
Conditional gates and entanglement
Dicke and NooN states
Toffoli gate and Cn -NOT gates
C-phase gate
Lineshapes
smooth pulses of finite duration
power narrowing
Outline
Two-state systems
Broadband, narrowband, passband, fractional-π pulses
Local addressing (subwavelength localization)
Composite adiabatic passage
Multistate systems
Composite STIRAP
Suppression of unwanted transitions
Conditional gates and entanglement
Dicke and NooN states
Toffoli gate and Cn -NOT gates
C-phase gate
Lineshapes
smooth pulses of finite duration
power narrowing
Coherent excitation
resonant
small area (fastest)
accurate
sensitive
Excitation Probability
1.0
adiabatic
0.8
composite
0.6
resonant
adiabatic
large area (slow)
inaccurate
robust
0.4
0.2
0
0
Nikolay Vitanov
1
2
3
Pulse Area (in units π)
(Uni Sofia)
4
CompPulses for QComp & QSim
composite
moderate area (fast)
robust, accurate
flexible
QSim, Brighton
1 / 26
Amplitudes
Phases
Composite pulses
PN
A sequence of phased pulses with an electric field E(t) = n=1 eiφn En (t)
PN
[Rabi frequencies Ω(t) = n=1 eiφn Ωn (t)] generates the propagator
U(φ1 , φ2 , . . . , φN ) = U(φN )U(φN −1 ) · · · U(φ2 )U(φ1 )
The phases φn are used as control parameters which are determined from the
condition to produce a transition probability profile of a desired shape.
composite pulses ≡ discrete optimal control theory
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
2 / 26
Composite Pulses
NMR
Levitt & Freeman, JMR 33, 473 (1979)
Freeman, Kempsell & Levitt, JJMR 38, 453 (1980)
Levitt, Prog. NMR Spectrosc. 18, 61 (1986)
Freeman, Spin Choreography (Spektrum, Oxford, 1997)
Polarization Optics
West & Makas, JOSA 39, 791 (1949)
Destriau & Prouteau, J. Phys. Radium 10, 53 (1949)
Pancharatnam, Proc. Indian Acad. Sci. A41, 130 (1955); A41, 137 (1955)
Harris, Ammann & Chang, JOSA 54, 1267 (1964)
McIntyre & Harris, JOSA 58, 1575 (1968)
Quantum Optics & Quantum Info
Blatt’s group: Nature 422, 408 (2003); Nature Phys. 4, 839 (2008); PRL 102, 040501 (2009)
Wunderlich’s group: PRL 98, 023003 (2007); PRA 77, 052334 (2008); PRL 110, 200501
(2013)
Composite pulses: SU(2) method
coherently driven two-state quantum system
i~∂t c(t) = H(t)c(t) H(t) = (~/2)Ω(t) e−iD(t) |ψ1 ihψ2 | + h.c.
Rt
D(t) = 0 ∆(t0 )dt0 ; detuning ∆ = ω0 − ω; Rabi frequency Ω(t)
evolution described
by the
SU(2) propagator U: c(t) = U(t, ti )c(ti )
a
b
U=
Cayley-Klein parameters a and b
−b∗ a∗
transition probability p = |b|2 = 1 − |a|2
Rt
∆ = 0 =⇒ a = cos(A/2), b = −i sin(A/2)
A = tif Ω(t)dt pulse area
constant phase shift φ in the field: Ω(t) → Ω(t) eiφ
a
b eiφ
U(φ) =
−b∗ e−iφ a∗
composite sequence of N phased pulses =⇒ overall propagator
U(N ) = U(φN )U(φN −1 ) · · · U(φ2 )U(φ1 )
Torosov & NVV, PRA 83, 053420 (2011)
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
3 / 26
BB, NB and PB composite pulses
broadband (BB), narrowband (NB), passband (PB)
1.0
Excitation Probability
Excitation Probability
1.0
0.8
0.6
0.4
0.2
0.0
0.8
0.6
0.4
0.2
0.0
0
1
2
3
Pulse Area Hin units ΠL
4
0
1
2
3
Pulse Area Hin units ΠL
4
Excitation Probability
1.0
BB: flat top around A = π
k U (N) ]
[∂A
11 A=π = 0 (k = 0, 1, 2, . . .)
0.8
0.6
NB: flat bottom around A = 0 and 2π
k U (N) ]
[∂A
12 A=0 = 0 (k = 0, 1, 2, . . .)
0.4
0.2
0.0
0
Nikolay Vitanov
1
2
3
Pulse Area Hin units ΠL
(Uni Sofia)
4
PB: flat top around A = π and
flat bottom around A = 0
CompPulses for QComp & QSim
QSim, Brighton
4 / 26
BB, NB and PB composite pulses
broadband (BB), narrowband (NB), passband (PB), half-π
1.0
Excitation Probability
Excitation Probability
1.0
0.8
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0.0
0
1
2
3
Pulse Area Hin units ΠL
4
0
1
2
3
Pulse Area Hin units ΠL
4
0
1
2
3
Pulse Area Hin units ΠL
4
1.0
Excitation Probability
1.0
Excitation Probability
0.8
0.8
0.6
0.4
0.2
0.0
0.8
0.6
0.4
0.2
0.0
0
Nikolay Vitanov
1
2
3
Pulse Area Hin units ΠL
(Uni Sofia)
4
CompPulses for QComp & QSim
QSim, Brighton
4 / 26
Deviation
Transition Probability
Broadband composite pulses
1.0
0.8
3
a sequence of N = 2n + 1 identical pulses
with anagram phases
(φ1 , φ2 , · · · , φn , φn+1 , φn , · · · , φ2 , φ1 )
15 45
9 11 25
5 7
1
0.6
(N)
0.4
φk
0.2
0
100
j
k j k
k
= n + 1 − k+1
2
2
10-2
0,
10-4
45
10-6
0
25
15 11 9 7
0.5
5
3
1
1.0
Pulse Area (in units π)
1.5
range wherein error < 10−4
0.006π for 1 pulse
0.26π for 5 pulses
0.62π for 25 pulses
0.82π for 125 pulses
2π
N
0, 32 , 0 π
2 4
, , 0 π
5 5
8 4 6
, , ,0 π 7 7 7
8 12 6 8
, , , ,0 π
9 9 9 9
···
0, 45 ,
6 4
0, 7 , 7 ,
8 6 12
, , ,
9 9 9
p = 1 − cos2N (A/2)
2.0
transition probability
p → 1 for large N if A 6= 2mπ
optimize against variations in
pulse area to arbitrary order
A = π(1 + ) =⇒ p ∼ 1 − (π/2)2N
Torosov & NVV, PRA 83, 053420 (2011)
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
5 / 26
Deviation
Transition Probability
Broadband composite pulses
1.0
0.8
3
a sequence of N = 2n + 1 identical pulses
with anagram phases
(φ1 , φ2 , · · · , φn , φn+1 , φn , · · · , φ2 , φ1 )
15 45
9 11 25
5 7
1
0.6
(N)
0.4
φk
0.2
0
100
j
k j k
k
= n + 1 − k+1
2
2
10-2
0,
10-4
45
10-6
0
25
15 11 9 7
0.5
5
3
1
1.0
Pulse Area (in units π)
1.5
range wherein error < 10−4
0.006π for 1 pulse
0.26π for 5 pulses
0.62π for 25 pulses
0.82π for 125 pulses
AG Halfmann (Darmstadt); Pr3+ : Y2 Si O5
Nikolay Vitanov
(Uni Sofia)
2π
N
0, 32 , 0 π
2 4
, , 0 π
5 5
8 4 6
, , ,0 π 7 7 7
8 12 6 8
, , , ,0 π
9 9 9 9
···
0, 45 ,
6 4
0, 7 , 7 ,
8 6 12
, , ,
9 9 9
p = 1 − cos2N (A/2)
2.0
transition probability
p → 1 for large N if A 6= 2mπ
optimize against variations in
pulse area to arbitrary order
A = π(1 + ) =⇒ p ∼ 1 − (π/2)2N
Torosov & NVV, PRA 83, 053420 (2011)
CompPulses for QComp & QSim
QSim, Brighton
5 / 26
Narrowband composite pulses
a sequence of N = 2n + 1 identical pulses
with sign-alternating phases
(φ1 , φ2 , −φ2 , φ3 , −φ3 , · · · , φn+1 , −φn+1 )
(N )
φk
= (−)k
k 2π
2
N
(k = 1, 2, . . . , N )
Transition Probability
1.0
0, 32 , − 23 π
2
0, 5 , − 25 , 54 , − 45 π
2
2 4
4 6
0, 7 , − 7 , 7 , − 7 , 7 , − 67 π
2 4
4 6
6 8
2
, − 9 , 9 , − 9 , 9 , − 9 , 9 , − 98 π
9
0.8
0.6
0,
···
0.4
p = sin2N (A/2) = pN
0.2
0
0
0.5
1.0
1.5
Pulse Area (in units π)
2.0
transition probability
NVV, PRA 84, 065404 (2011)
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
6 / 26
Local addressing
Electric Field
(arb. units)
Blatt, PRL 102, 040501 (2009): Toffoli gate
1.0
0.5
Excitation can be localized in any
desired spatial range around the center.
⇓
We can “beat the diffraction limit”!!!
0
1.0
Excitation
Bloch, Nature 471, 319 (2011)
0.8
0.6
The physical reason is the destructive
interference of the ingredient pulses
in the composite sequence.
3
0.4
9
25
0.2
125
0
-2
-1
0
1
Radial Distance (arb. units)
2
SS Ivanov & NVV, Opt. Lett. 36, 1275 (2011)
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
7 / 26
Passband composite pulses
nested NB (ν1 , ν2 , . . . , ν2m+1 )
and BB (β1 , β2 , . . . , β2n+1 )
(N )
νk
PN(B) = [1 − (1 − p)Nb ]Nn
PB(N) = 1 − (1 − pNn )Nb
(N )
φk
=
2π
N
Population Inversion P
1.0
0.8
1
0.6
0.4
0.2
0
1.0
0.8
0.6
0.4
0.2
0
(Uni Sofia)
2π
2m+1
k j
k
n + 1 − k+1
2
2
0.2
0.4
E. Kyoseva & NVV, PRA, Dec 2013
Nikolay Vitanov
2
P = 1 − cos2N (A/2) = 1 − (1 − p)N
Population Inversion P
nested BB into NB
N3 (B3 ) = (β1 + ν1 , β2 + ν1 , β3 + ν1 ,
β1 + ν2 , β2 + ν2 , β3 + ν2 ,
β1 + ν3 , β2 + ν3 , β3 + ν3 )
N3 (B3 ) = (0, 23 , 0, 32 , 34 , 23 , 43 , 0, 34 )π
k
P = sin2N (A/2) = pN
N3 = (ν1 , ν2 , ν3 ) = 0, 23 , − 23 π
2
B3 = (β1 , β2 , β3 ) = 0, 3 , 0 π
nested NB into BB
B3 (N3 ) = (β1 + ν1 , β1 + ν2 , β1 + ν3 ,
β2 + ν3 , β2 + ν2 , β2 + ν1 ,
β3 + ν1 , β3 + ν2 , β3 + ν3 )
B3 (N3 ) = (0, 23 , 34 , 0, 34 , 23 , 0, 32 , 43 )π
= (−)k
CompPulses for QComp & QSim
0.6 0.8 1.0 1.2 1.4
Pulse Area (in units of π)
1.6
1.8
2.0
QSim, Brighton
8 / 26
Multi-point composite pulses
√
√
P = 1 at areas π, π 2, π 3
√
π, π 2
√P = 1 at areas √
[(A/ 8)0 , Aπ/2 , (A/ 8)0 ]
[(0.83A)0 , (0.83A)0.65π , (0.83A)0 ]
[(0.72A)0 , (0.72A)0.63π , (0.72A)0 ]
Excitation Probability
0.8
0.6
0.4
0.2
0.0
0.0
Excitation Probability
1.0
0.5
1.0
1.5
Pulse Area Hin units ΠL
0.8
0.6
0.4
0.2
0.0
0.0
2.0
1
1
0.1
0.1
Excitation Probability
Excitation Probability
1.0
0.01
0.001
10-4
10-5
10-6
0.0
Nikolay Vitanov
0.5
1.0
1.5
Pulse Area Hin units ΠL
(Uni Sofia)
2.0
0.5
1.0
1.5
Pulse Area Hin units ΠL
2.0
0.5
1.0
1.5
Pulse Area Hin units ΠL
2.0
0.01
0.001
10-4
10-5
10-6
0.0
CompPulses for QComp & QSim
QSim, Brighton
9 / 26
Multi-point composite pulses
√
√
√
√
P = 1 at areas π, π 2, π 3, π 4, π 5
√
√
√
P = 1 at areas π, π 2, π 3, π 4
1.0
Excitation Probability
Excitation Probability
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.5
1.0
1.5
2.0
Pulse Area Hin units ΠL
2.5
Excitation Probability
Excitation Probability
0.1
0.001
10
-5
10-6
0.0
0.4
0.2
0.5
1.0
1.5
2.0
Pulse Area Hin units ΠL
2.5
3.0
0.5
1.0
1.5
2.0
Pulse Area Hin units ΠL
2.5
3.0
1
0.01
10
0.6
0.0
0.0
3.0
1
-4
0.8
0.5
1.0
1.5
2.0
Pulse Area Hin units ΠL
2.5
3.0
0.1
0.01
0.001
10-4
10-5
10-6
0.0
may be useful for sideband cooling
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
10 / 26
Composite adiabatic passage (level crossing)
Transition Probability p
Torosov, Guerin & NVV, PRL 106, 233001 (2011)
1.0
9
5
0.8
3
0.6
1
1
0.2
(a)
0
0
10
1
3
(c)
1
(b)
transition probability
=⇒ p → 1 for large N (a 6= ±1)
5
9
-4
10
-6
10
0
2
4
6 0
2
4
6
8
Peak Rabi Frequency Ω0 (in units 1/T)
Nikolay Vitanov
(Uni Sofia)
2π
N
p = 1 − a2N
(d)
5
10
k j k
j
k
= n + 1 − k+1
2
2
these “magic phases” optimize AP
against variations in the pulse area
and the chirp rate to arbitrary order
0.4
-2
1−p
(N)
φk
5
CompPulses for QComp & QSim
(0, 2, 0) π/3
(0, 4, 2, 4, 0) π/5
(0, 6, 4, 8, 4, 6, 0) π/7
(0, 8, 6, 12, 8, 12, 6, 8, 0) π/9
···
QSim, Brighton
11 / 26
Composite adiabatic passage (level crossing)
Transition Probability p
Torosov, Guerin & NVV, PRL 106, 233001 (2011)
1.0
9
5
0.8
3
0.6
1
1
0.2
(a)
0
0
10
1
3
(c)
1
(b)
transition probability
=⇒ p → 1 for large N (a 6= ±1)
5
9
-4
10
-6
10
0
2
4
6 0
2π
N
p = 1 − a2N
(d)
5
10
k j k
j
k
= n + 1 − k+1
2
2
these “magic phases” optimize AP
against variations in the pulse area
and the chirp rate to arbitrary order
0.4
-2
1−p
(N)
φk
5
2
4
6
8
Peak Rabi Frequency Ω0 (in units 1/T)
(0, 2, 0) π/3
(0, 4, 2, 4, 0) π/5
(0, 6, 4, 8, 4, 6, 0) π/7
(0, 8, 6, 12, 8, 12, 6, 8, 0) π/9
···
experiment in Pr3+ : Y2 Si O5
Schraft, Halfmann, Genov & NVV, PRA 88, 063406 (2013)
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
11 / 26
Universal composite pulses
U(φ) =
a
−b∗ e−iφ
b eiφ
a∗
U(N) = U(φN )U(φN−1 ) · · · U(φ2 )U(φ1 )
eliminate the coefficients of the lowest powers of a in U (N)
=⇒ a set of equations for the phases
the phases do not depend on the interaction parameters!
p = 1 − |a|m → 1
for sufficiently large m(N ) for any field!
5 pulses:
(0, 5, 2, 5, 0)π/6
(0, −1, 2, −1, 0)π/6
13 pulses:
(0, 9, 42, 11, 8, 37, 2, 37, 8, 11, 42, 9, 0)π/24
(0, 15, 6, 13, 40, 35, 46, 35, 40, 13, 6, 15, 0)π/24
Genov, Schraft, NVV & Halfmann, in preparation
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
12 / 26
Universal composite pulses
U(φ) =
a
−b∗ e−iφ
b eiφ
a∗
U(N) = U(φN )U(φN−1 ) · · · U(φ2 )U(φ1 )
eliminate the coefficients of the lowest powers of a in U (N)
=⇒ a set of equations for the phases
the phases do not depend on the interaction parameters!
p = 1 − |a|m → 1
for sufficiently large m(N ) for any field!
5 pulses:
(0, 5, 2, 5, 0)π/6
(0, −1, 2, −1, 0)π/6
13 pulses:
(0, 9, 42, 11, 8, 37, 2, 37, 8, 11, 42, 9, 0)π/24
(0, 15, 6, 13, 40, 35, 46, 35, 40, 13, 6, 15, 0)π/24
Genov, Schraft, NVV & Halfmann, in preparation
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
12 / 26
Outline
Two-state systems
Broadband, narrowband, passband, fractional-π pulses
Local addressing (subwavelength localization)
Composite adiabatic passage
Multistate systems
Composite STIRAP
Suppression of unwanted transitions
Conditional gates and entanglement
Dicke and NooN states
Toffoli gate and Cn -NOT gates
C-phase gate
Lineshapes
smooth pulses of finite duration
power narrowing
Composite STIRAP
2
2
S
Wp
P
P
S
S
P
Ws
Wp
Ws
3
3
1
1
2
D
Wp
Ws
S
P
S
P
S
P
3
1
Nikolay Vitanov
2
D
Wp
Ws
3
1
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
13 / 26
Composite STIRAP
2
2
S
P
P
S
S
Wp
(0, 1; 3, 3; 1, 0)π/3
P
Ws
(0, 4; 5, 8; 3, 3; 8, 5; 4, 0)π/5
Ws
Wp
3
3
1
D
Wp
Ws
S
P
S
P
S
P
20
2
D
Wp
Ws
3
3
Infidelity 1-P3
Infidelity 1-P3
1
1
0.1
0.01
0.001
10-4
10-5
10-6
Peak Rabi Frequency W0 H1TL
2
1
1
D=0
Gaussian
3
5
0
5
10
1
0.1
3
0.01 D=30T
Gaussian
0.001
5
10-4
10-5
10-6
0
5 10 15
15
20
1
20
25
30
Peak Rabi Frequency W0 H1TL
Nikolay Vitanov
(Uni Sofia)
1
0.1
0.01
0.001
10-4
10-5
10-6
1
0.1
0.01
0.001
10-4
10-5
10-6
D=0
sine sq.
1
3
5
10
15
3
D=100T
sine sq.
0.9999
0.9
0.7
10
0.5
0.3
5
0.1
0.5
1.0
1.5
2.0
20
5
0
15
0
0.0
20
1
5
Peak Rabi Frequency W0 H1TL
1
15
P > 1-10-4
0.9999
0.9
0.7
10
0.5
0.3
0.1
5
0.0
0.5
1.0
1.5
2.0
Delay Τ HTL
0
10
20
30
40
50
Peak Rabi Frequency W0 H1TL
Torosov& NVV, PRA 87, 043418 (2013)
CompPulses for QComp & QSim
QSim, Brighton
13 / 26
Suppression of Unwanted Transitions
3
3
2
2
Ω23
Ω13
Ω12
Δ
2
Ω13
Δ
1
1
0
σ+
σ−
1
Ω01
Ω12
Δ
1.5
Ω12
σ−
0.9
1
0.8
0.6
0.99 0.9
0.7 0.5 0.3
0.1
0.3
0.5
0.7
0.99
0.5
2
Detuning
(in units of 1/T)
1.0
0.1
RMS Pulse Area
(in units of π)
3
0.1
0.3
0.7
0
0.99 0.9
0.7 0.5 0.3
0.1
0.99
0.5
0.9
0.4
-2
0
0.2
0
-0.50
-0.25
0
Ellipticity θ
0.25
0.25
0.5 0
0.25
Mixing Angle θ (in units of π)
0.5
0.50
GT Genov & NVV, PRL 110, 133002 (2013)
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
14 / 26
Outline
Two-state systems
Broadband, narrowband, passband, fractional-π pulses
Local addressing (subwavelength localization)
Composite adiabatic passage
Multistate systems
Composite STIRAP
Suppression of unwanted transitions
Conditional gates and entanglement
Dicke and NooN states
Toffoli gate and Cn -NOT gates
C-phase gate
Lineshapes
smooth pulses of finite duration
power narrowing
Entangled Dicke and NooN states
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
15 / 26
Entangled Dicke and NooN states
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
15 / 26
Toffoli gate (C2 -NOT)
|ccti → |ccti
|000i → |000i
|001i → |001i
|010i → |010i
|011i → |011i
|100i → |100i
|101i → |101i
|110i → |111i
|111i → |110i











1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0











6 CNOT gates by the circuit model
5 CNOT gates with an ancilla state
Lanyon et al., Nature Phys. 5, 134 (2009)
Beyond the circuit model
Blatt’s group: trapped ions, 15 laser pulses, 71% fidelity, LD regime
T. Monz et al., Phys. Rev. Lett. 102, 040501 (2009)
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
16 / 26
Toffoli gate (C2 -NOT)
|110i → |111i
|111i → |110i
ν=0
111
Ω02
SS Ivanov & NVV, Phys. Rev. A 84, 022319 (2011)
for the second-sideband transition
2
Ων,ν+2 = Ωe−η /2 η 2 χν,ν+2
L2 (η 2 )
χν,ν+2 = √ ν
(ν+1)(ν+2)
ν=2
110
101
Ω24
χ02 =
011
Ω24
χ24 =
ν=4
100
010
001
Ω46
ν=6
000
χ46 =
1
√
2
12−8η 2 +η 4
√
4 3
4
360−480η 2 +180η
−24η 6 +η 8
√
24 30
Toffoli gate: Ω02 is a π-pulse
Ω24 and Ω46 are even-π-pulses
C. Monroe et al., PRA 55, R2489 (1997); B. DeMarco et al., PRL 89, 267901 (2002)
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
17 / 26
Toffoli gate (C2 -NOT) and Cn -NOT gates
SS Ivanov & NVV, Phys. Rev. A 84, 022319 (2011)
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
18 / 26
C-phase gate with overlapping pulses
MSJ gate
bichromatic field (first red and blue sidebands)
−
−
P
+
† −iφk + aeiφk )
H(t) = N
k=1 gk (t)σ(φk )(a e
control-phase gate
U = D(α)G(Jk,l )
U = exp i π σ1x σ2x
Pulse Amplitude
g0
4
−iφk σ x
k=1 Ak (t, ti )e
k
Ak (t, ti ) =
0
0
Rt
ti
h P
i
x x
G(Jk,l ) = exp i N
k<l Jk,l σk σl
R
R
Jk,l (t, ti ) = sin (φk − φl ) tt dt1 tt1 dt2 [gk (t2 )gl (t1 ) − gk (t1 )gl (t2 )]
i
T
2T
Time
dt1 gk (t1 )
i
3T
4T
present gate, type I
g0
Pulse Amplitude
α(t, ti ) = −i
PN
Sorensen-Molmer: off-resonant, sequential pulses
0
0
Τ
T T+Τ
2T+2Τ
Time
Milburn-Schneider-James: resonant, sequential pulses
present gate, type II
Sorensen & Molmer, PRA 62, 022311 (2000)
Milburn, Schneider & James, FP 48, 801 (2000)
Simeonov, Ivanov & NVV, PRA (2014)
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
g0
Pulse Amplitude
Simeonov et al: resonant, overlapping pulses
equal areas; phases 0, π2 , π, − π2
0
0
Τ
T
T+Τ 2T
Time
QSim, Brighton
2T+Τ
19 / 26
C-phase gate with overlapping pulses
p
MSJ gate
g0
Pulse Amplitude
HaL
- А2
0
x
4
3
1
2
0
0
T
2T
Time
3T
4T
- А2
p
present gate, type I
g0
Pulse Amplitude
HbL
-2 Π  6
- А6
0
x
- А6
-2 Π  6
0
0
Τ
T T+Τ
2T+2Τ
Time
p
present gate, type II
HcL
Pulse Amplitude
g0
-2 Π  5
- А5
0
- А5
x
pulse
shape
rect
sin
sin2
sin3
sin4
sin5
sin6
MSJ
T
0.627
0.984
1.253
1.477
1.671
1.846
2.005
pulse
shape
rect
sin
sin2
sin3
sin4
sin5
sin6
MSJ
T
0.627
0.984
1.253
1.477
1.671
1.846
2.005
present, type I
τ /T
T
speed-up
0.500
0.724
15.5%
0.413
1.094
27.3%
0.362
1.370
34.4%
0.327
1.597
39.3%
0.303
1.794
43.0%
0.284
1.971
45.9%
0.268
2.131
48.4%
present, type II
τ /T
T
speed-up
0.500
0.793
26.5%
0.454
1.091
47.1%
0.424
1.327
55.8%
0.393
1.539
60.4%
0.365
1.731
63.3%
0.341
1.905
65.5%
0.321
2.065
67.3%
gate
duration
4T
for MSJ
2T + 2τ
for type I
2T + τ
for type II
-2 Π  5
0
0
Τ
T
T+Τ 2T
Time
2T+Τ
L Simeonov, P Ivanov & NVV, PRA 2014
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
20 / 26
C-phase gate with overlapping pulses
p
MSJ gate
g0
Pulse Amplitude
HaL
- А2
0
x
4
3
1
2
0
0
T
2T
Time
3T
4T
- А2
p
present gate, type I
g0
Pulse Amplitude
HbL
-2 Π  6
- А6
0
x
- А6
-2 Π  6
0
0
Τ
T T+Τ
2T+2Τ
Time
p
present gate, type II
HcL
Pulse Amplitude
g0
-2 Π  5
- А5
0
- А5
x
pulse
shape
rect
sin
sin2
sin3
sin4
sin5
sin6
MSJ
T
0.627
0.984
1.253
1.477
1.671
1.846
2.005
pulse
shape
rect
sin
sin2
sin3
sin4
sin5
sin6
MSJ
T
0.627
0.984
1.253
1.477
1.671
1.846
2.005
present, type I
τ /T
T
speed-up
0.500
0.724
15.5%
0.413
1.094
27.3%
0.362
1.370
34.4%
0.327
1.597
39.3%
0.303
1.794
43.0%
0.284
1.971
45.9%
0.268
2.131
48.4%
present, type II
τ /T
T
speed-up
0.500
0.793
26.5%
0.454
1.091
47.1%
0.424
1.327
55.8%
0.393
1.539
60.4%
0.365
1.731
63.3%
0.341
1.905
65.5%
0.321
2.065
67.3%
gate
duration
4T
for MSJ
2T + 2τ
for type I
2T + τ
for type II
-2 Π  5
0
0
Τ
T
T+Τ 2T
Time
2T+Τ
L Simeonov, P Ivanov & NVV, PRA 2014
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
20 / 26
Outline
Two-state systems
Broadband, narrowband, passband, fractional-π pulses
Local addressing (subwavelength localization)
Composite adiabatic passage
Multistate systems
Composite STIRAP
Suppression of unwanted transitions
Conditional gates and entanglement
Dicke and NooN states
Toffoli gate and Cn -NOT gates
C-phase gate
Lineshapes
smooth pulses of finite duration
power narrowing
Lineshapes in Coherent Excitation
1.0
Ω = 4Γ
Ω = 3Γ
Ω = 2Γ
Ω=Γ
Ω = 0.1Γ
Excitation Probability
0.8
CW laser excitation:
power broadening is a paradigm!
Pulsed excitation:
extent of power broadening depends on
how the signal is collected!
0.6
0.4
0.2
0
-8
-6
P ∝
-4 -2
0
2
4
Detuning (in units Γ)
6
If collected during excitation:
→ typical power broadening
8
If collected after excitation:
strong dependence on the pulse shape
due to coherent adiabatic population return!
1
∆2 + Γ2 /4 + Ω2 /4
Shape
rectangular
sinn
gaussian
sech
lorentziann
Nikolay Vitanov
Duration
finite
finite
infinite
infinite
infinite
(Uni Sofia)
Smooth
no
partly
yes
yes
yes
Linewidth
∆∝Ω
∆ ∝ Ω1/(n+1)
∆ ∝ ln Ω
∆ ∝ const
∆ ∝ Ω−1/(2n−1)
CompPulses for QComp & QSim
Broadening
linear
root
logarithmic
no
NARROWING!
QSim, Brighton
21 / 26
Power non-broadening in pulsed excitation
Fluorescence
7.5 GW/cm2
2.5 GW/cm2
0.75 GW/cm2
Ionization
Halfmann, Rickes, NVV & Bergmann, Opt. Commun. 220, 353 (2003)
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
QSim, Brighton
22 / 26
Smooth pulses of finite duration
1
Transition Probability
Rabi Frequency Hin units 1TL
3Π
2Π
Π
0.1
0.01
0.001
10-4
10-5
10-6
0
10
20
30
Detuning Hin units 1TL
40
50
0
10
0
1
T
2
0
T
n=1
-2
10
f (t) = sinn (πt/T ) (0 5 t 5 T )
non-analytic!
(n!π n Ω)2
P =
sin2 ηn+1
n
(n!π Ω)2 + (∆n+1 T n )2
∆1 =
2
n!π n Ω
Tn
Excited State Population
Time Hin units TL
-4
10
-6
10
0
10
n=2
-2
10
1
n+1
-4
10
-6
10
0
Boradjiev & NVV, PRA 88, 013402 (2013)
Nikolay Vitanov
(Uni Sofia)
CompPulses for QComp & QSim
10
20
30
40
50
Detuning (in units 1/T)
QSim, Brighton
23 / 26
2
1
Deviation from Complete Inversion
Smooth pulses of finite duration
∆
Ω
0
0.1
0.01
0.001
1
2
3
0.0001
0
4
0.00001
0
10
20
30
40
50
40
50
Detuning (in units 1/T)
È0; 0
È1; 1
Inversion Infidelity
È1; 0
È0; 1
È0; 2
(Uni Sofia)
-2
10
1
-3
10
2
0
-4
10
-5
10
-6
10
0
Boradjiev & NVV, PRA 88, 013402 (2013)
Nikolay Vitanov
-1
10
È1; 2
10
20
30
Trap Frequency ν (in units 1/T)
CompPulses for QComp & QSim
QSim, Brighton
24 / 26
Lineshapes: Power narrowing
f (t) =
∆1 T =
2
"
Raman-driven fine-structure doublet
in Na Rydberg atoms by microwave pulses
lz, lz2 , sech, sech2 , gs
1
[1 + (t/T )2 ]n
1
1
(2n + 1)n+ 2 (2n − 1)2n− 2
(4n)n ΩT
#
1
2n−1
from adiabatic condition
Boradjiev & NVV, Opt. Commun. 288, 91 (2013)
Nikolay Vitanov
(Uni Sofia)
Conover, PRA 84, 063416 (2011)
CompPulses for QComp & QSim
QSim, Brighton
25 / 26
CAMEL 10
Control of Atoms, Molecules and Ensembles by Light
Nessebar, Black Sea coast, 23-27 June 2014
camel10.quantum-bg.org
CAMEL 10
Control of Atoms, Molecules and Ensembles by Light
Nessebar, Black Sea coast, 23-27 June 2014
camel10.quantum-bg.org
Outline
Two-state systems
Broadband, narrowband, passband, fractional-π pulses
Local addressing (subwavelength localization)
Composite adiabatic passage
Multistate systems
Conditional gates and entanglement
smooth pulses of finite duration
power narrowing
TH
AN
K
Lineshapes
YO
Dicke and NooN states
Toffoli gate and Cn -NOT gates
C-phase gate
U!
Composite STIRAP
Suppression of unwanted transitions