Non-Equilibrium Master Equations
Vorlesung gehalten im Rahmen des GRK 1558
Wintersemester 2011/2012
Technische Universität Berlin
Dr. Gernot Schaller
April 29, 2014
2
Contents
1 An introduction to Master Equations
1.1 Master Equations – A Definition . . . . . . . . . . . . .
1.1.1 Example 1: Fluctuating two-level system . . . .
1.1.2 Example 2: Diffusion Equation . . . . . . . . .
1.2 Density Matrix Formalism . . . . . . . . . . . . . . . .
1.2.1 Density Matrix . . . . . . . . . . . . . . . . . .
1.2.2 Dynamical Evolution in a closed system . . . .
1.2.3 Most general dynamical Evolution . . . . . . . .
1.2.4 Master Equation for a cavity in a thermal bath
1.2.5 Master Equation for a driven cavity . . . . . . .
1.2.6 Tensor Product . . . . . . . . . . . . . . . . . .
1.2.7 The partial trace . . . . . . . . . . . . . . . . .
2 Obtaining a Master Equation
2.1 Phenomenologic Derivations (Educated Guesses) . . .
2.1.1 The single resonant level . . . . . . . . . . . .
2.1.2 Cell culture growth in constrained geometries
2.2 Derivations for Open Quantum Systems . . . . . . .
2.2.1 Weak Coupling Regime . . . . . . . . . . . . .
2.2.2 Strong-coupling regime . . . . . . . . . . . . .
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3 Solving Master Equations
3.1 Analytic Techniques . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Full analytic solution with the Matrix Exponential . .
3.1.2 Equation of Motion Technique . . . . . . . . . . . . . .
3.1.3 Quantum Regression Theorem . . . . . . . . . . . . . .
3.2 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Numerical Integration . . . . . . . . . . . . . . . . . .
3.2.2 Simulation as a Piecewise Deterministic Process (PDP)
4 Multi-Terminal Coupling : Non-Equilibrium Case I
4.1 Conditional Master equation from conservation laws .
4.2 Microscopic derivation with a detector model . . . . .
4.3 Full Counting Statistics . . . . . . . . . . . . . . . . .
4.4 Fluctuation Theorems . . . . . . . . . . . . . . . . .
4.5 The double quantum dot . . . . . . . . . . . . . . . .
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4
CONTENTS
5 Direct bath coupling :
Non-Equilibrium Case II
65
5.1 Monitored SET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Monitored charge qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6 Open-loop control:
Non-Equilibrium Case III
6.1 Single junction . . . . . . . . . . . . .
6.2 Electronic pump . . . . . . . . . . . . .
6.2.1 Time-dependent tunneling rates
6.2.2 With performing work . . . . .
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7 Closed-loop control:
Non-Equilibrium Case IV
7.1 Single junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Maxwell’s demon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Qubit stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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CONTENTS
5
This lecture aims at providing graduate students in physics and neighboring sciences with a
heuristic approach to master equations. Although the focus of the lecture is on rate equations,
their derivation will be based on quantum-mechanical principles, such that basic knowledge of
quantum theory is mandatory. The lecture will try to be as self-contained as possible and aims at
providing rather recipes than proofs.
As successful learning requires practice, a number of exercises will be given during the lecture,
the solution to these exercises may be turned in in the next lecture (computer algebra may be used
if applicable), for which students may earn points. Students having earned 50 % of the points at
the end of the lecture are entitled to three ECTS credit points.
This script will be made available online at
http://wwwitp.physik.tu-berlin.de/~ schaller/.
In any first draft errors are quite likely, such that corrections and suggestions should be addressed to
[email protected].
I thank the many colleagues that have given valuable feedback to improve the notes, in particular Dr. Malte Vogl, Dr. Christian Nietner, and Prof. Dr. Enrico Arrigoni.
6
CONTENTS
literature:
[1] Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information,
Cambridge University Press, Cambridge (2000).
[2] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University
Press, Oxford (2002).
[3] H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University
Press, Cambridge (2010).
[4] G. Lindblad, Communications in Mathematical Physics 48, 119 (1976).
[5] G. Schaller, C. Emary, G. Kiesslich, and T. Brandes, Physical Review B 84, 085418 (2011).
[6] G. Schaller and T. Brandes, Physical Review A 78, 022106, (2008); G. Schaller, P. Zedler,
and T. Brandes, ibid. A 79, 032110, (2009).
7
8
LITERATURE:
Chapter 1
An introduction to Master Equations
1.1
Master Equations – A Definition
Many processes in nature are stochastic. In classical physics, this may be due to our incomplete
knowledge of the system. Due to the unknown microstate of e.g. a gas in a box, the collisions of gas
particles with the domain wall will appear random. In quantum theory, the evolution equations
themselves involve amplitudes rather than observables in the lowest level, such that a stochastic
evolution is intrinsic. In order to understand such processes in great detail, a description should
include such random events via a probabilistic description. For dynamical systems, probabilities
associated with events may evolve in time, and the determining equation for such a process is
called master equation.
Box 1 (Master Equation) A master equation is a first order differential equation describing the
time evolution of probabilities, e.g. for discrete events
dPk X
=
[Tkℓ Pℓ − Tℓk Pk ] ,
dt
ℓ
(1.1)
where the Tkℓ are transition rates between events ℓ and k. The master equation is said to satisfy
detailed balance, when for the stationary state P̄i the equality Tkℓ P̄ℓ = Tℓk P̄k holds for all terms
separately.
When the transition matrix Tkℓ is symmetric, all processes are reversible at the level of the
master equation description. Mostly, master equations are phenomenologically motivated and
not derived from first principles, but in the case of open quantum systems we will also discuss
microscopic derivations. Already the Markovian quantum master equation does not only involve
probabilities (diagonals of the density matrix) but also coherences (off-diagonals) in the master
equation. Here, therefore the term master equation is used in a weaker condition as simply the
evolution equation for probabilities.
It is straightforward to show that the master equation conserves the total probability
X dPk
k
dt
=
X
kℓ
(Tkℓ Pℓ − Tℓk Pk ) =
9
X
kℓ
(Tℓk Pk − Tℓk Pk ) = 0 .
(1.2)
10
CHAPTER 1. AN INTRODUCTION TO MASTER EQUATIONS
Beyond this, all probabilities must remain positive, which is also respected by a normal master
equation. Evidently, the solution of the master equation is continuous, such that when initialized
with valid probabilities 0 ≤ Pi (0) ≤ 1 all probabilities are non-negative initially. After some time,
the probability Pk may approach zero (but all others are non-negative). Its time-derivative is then
given by
X
dPk =
+
Tkℓ Pℓ ≥ 0 ,
(1.3)
dt Pk =0
ℓ
such that Pk < 0 is impossible. Finally, the probabilities
Pmust remain smaller than one throughout
the evolution. This however follows immediately from k Pk = 1 and Pk ≥ 0 by contradiction. In
conclusion, a master equation of the form (1.1) automatically preserves the sum of probabilities
and also keeps 0 ≤ Pi (t) ≤ 1 – a valid initialization provided. That is, under the evolution of a
master equation, probabilities remain probabilities.
1.1.1
Example 1: Fluctuating two-level system
Let us consider a system of two possible events, to which we associate the time-dependent probabilities P0 (t) and P1 (t). These events could for example be the two conformations of a molecule,
the configurations of a spin, the two states of an excitable atom, etc. To introduce some dynamics,
let the transition rate from 0 → 1 be denoted by T10 > 0 and the inverse transition rate 1 → 0 be
denoted by T01 > 0. The associated master equation is then given by
d
P0
−T10 +T01
P0
(1.4)
=
P1
+T10 −T01
dt P1
Exercise 1 (Temporal Dynamics of a two-level system) (1 point)
Calculate the solution of Eq. (1.4). What is the stationary state?
Exercise 2 (Tunneling mice) (1 point)
Two identical mice are put in a box consisting of two (left and right) compartments. Afterwards
they randomly change compartments with rate T . Set up the master equation for the probabilities
to find both mice in the left (PL ) or right (PR ) compartments or in a balanced configuration (PB ).
1.1.2
Example 2: Diffusion Equation
Consider an infinite chain of coupled compartments as displayed in Fig. 1.1. Now suppose that
along the chain, molecules may move from one compartment to another with a transition rate
T that is unbiased, i.e., symmetric in all directions. The evolution of probabilities obeys the
infinite-size master equation
Ṗi (t) = T Pi−1 (t) + T Pi+1 (t) − 2T Pi (t)
Pi−1 (t) + Pi+1 (t) − 2Pi (t)
,
= T ∆x2
∆x2
(1.5)
11
1.1. MASTER EQUATIONS – A DEFINITION
Figure 1.1: Linear chain of compartments coupled with a transition rate T , where only next
neighbors are coupled to each other symmetrically.
which converges as ∆x → 0 and T → ∞ such that D = T ∆x2 remains constant to the partial
differential equation
∂P (x, t)
∂ 2 P (x, t)
= D
∂t
∂x2
with
D = T ∆x2 .
(1.6)
We note here that while the Pi (t) describe (dimensionless) probabilities, P (x, t) describes a timedependent probability density (with dimension of inverse length).
Such diffusion equations are used to describe the distribution of chemicals in a soluble in the
highly diluted limit, the kinetic dynamics of bacteria and further undirected transport processes.
From our analysis of master equations, we can immediately conclude that the diffusion equation
+∞
R
P (x, t)dx = 1. Note that it is
preserves positivity and total norm, i.e., P (x, t) ≥ 0 and
−∞
straightforward to generalize to the higher-dimensional case.
One can now think of microscopic models where the hopping rates in different directions are
not equal (drift) and may also depend on the position (spatially-dependent diffusion coefficient).
A corresponding model (in next-neighbor approximation) would be given by
Ṗi = Ti−1,i Pi−1 (t) + Ti+1,i Pi+1 (t) − (Ti,i−1 + Ti,i+1 )Pi (t) ,
(1.7)
where Ta,b denotes the rate of jumping from a to b. An educated guess is given by the ansatz
∂P
∂t
∂2
∂
[A(x)P (x, t)] +
[B(x)P (x, t)]
2
∂x
∂x
Ai−1 Pi−1 − 2Ai Pi + Ai+1 Pi+1 Bi+1 Pi+1 − Bi−1 Pi−1
≡
+
2
∆x
2∆x 2Ai
Ai−1 Bi−1
Ai+1 Bi+1
=
Pi−1 −
Pi+1 ,
−
Pi +
+
∆x2
2∆x
∆x2
∆x2
2∆x
=
(1.8)
which is equivalent to our master equation when
∆x2
[Ti,i−1 + Ti,i+1 ] ,
Ai =
2
Bi = ∆x [Ti,i−1 − Ti,i+1 ] .
(1.9)
We conclude that the Fokker-Planck equation
∂P
∂t
=
∂2
∂
[A(x)P (x, t)] +
[B(x)P (x, t)]
2
∂x
∂x
with A(x) ≥ 0 preserves norm and positivity of the probability distribution P (x, t).
(1.10)
12
CHAPTER 1. AN INTRODUCTION TO MASTER EQUATIONS
Exercise 3 (Reaction-Diffusion Equation) (1 point)
Along a linear chain of compartments consider the master equation for two species
Ṗi = T [Pi−1 (t) + Pi+1 (t) − 2Pi (t)] − γPi (t) ,
ṗi = τ [pi−1 (t) + pi+1 (t) − 2pi (t)] + γPi (t),
where Pi (t) may denote the concentration of a molecule that irreversibly reacts with chemicals in
the soluble to an inert form characterized by pi (t). To which partial differential equation does the
master equation map?
In some cases, the probabilities may not only depend on the probabilities themselves, but
also on external parameters, which appear then in the master equation. Here, we will use the
term master equation for any equation describing the time evolution of probabilities, i.e., auxiliary
variables may appear in the master equation.
1.2
1.2.1
Density Matrix Formalism
Density Matrix
Suppose one wants to describe a quantum system, where the system state is not exactly known.
That is, there is an ensemble of known normalized states {|Φi i}, but there is uncertainty in which
of these states the system is. Such systems can be conveniently described by the density matrix.
Box 2 (Density Matrix) The density matrix can be written as
X
ρ=
pi |Φi i hΦi | ,
(1.11)
i
P
where 0 ≤ pi ≤ 1 denote the probabilities to be in the state |Φi i with i pi = 1. Note that we
require the states to be normalized (hΦi |Φi i = 1) but not generally orthogonal (hΦi |Φj i 6= δij is
allowed).
Formally, any matrix fulfilling the properties
• self-adjointness: ρ† = ρ
• normalization: Tr {ρ} = 1
• positivity: hΨ| ρ |Ψi ≥ 0 for all vectors Ψ
can be interpreted as a valid density matrix.
For a pure state one has pī = 1 and thereby ρ = |Φī i hΦī |. Evidently, a density matrix is pure
if and only if ρ = ρ2 .
13
1.2. DENSITY MATRIX FORMALISM
The expectation value of an operator for a known state |Ψi
hAi = hΨ| A |Ψi
(1.12)
can be obtained conveniently from the corresponding pure density matrix ρ = |Ψi hΨ| by simply
computing the trace
hAi ≡ Tr {Aρ} = Tr {ρA} = Tr {A |Ψi hΨ|}
!
X
X
=
hn| A |Ψi hΨ|ni = hΨ|
|ni hn| A |Ψi
n
n
= hΨ| A |Ψi .
(1.13)
When the state is not exactly known but its probability distribution, the expectation value is
obtained by computing the weighted average
X
Pi hΦi | A |Φi i ,
(1.14)
hAi =
i
where Pi denotes the probability to be in state |Φi i. The definition of obtaining expectation values
by calculating traces of operators with the density matrix is also consistent with mixed states
(
)
X
X
hAi ≡ Tr {Aρ} = Tr A
pi |Φi i hΦi | =
pi Tr {A |Φi i hΦi |}
i
=
X
i
=
X
i
pi
X
n
i
hn| A |Φi i hΦi |ni =
pi hΦi | A |Φi i .
X
i
pi hΦi |
X
n
!
|ni hn| A |Φi i
(1.15)
Exercise 4 (Superposition vs Localized States) (1 point)
Calculate the density matrix for a statistical mixture in the states |0i and |1i with probability
p0 = 3/4 and p
p1 = 1/4. p
What is the density p
matrix for p
a statistical mixture of the superposition
states |Ψa i = 3/4 |0i+ 1/4 |1i and |Ψb i = 3/4 |0i− 1/4 |1i with probabilities pa = pb = 1/2.
1.2.2
Dynamical Evolution in a closed system
The evolution of a pure state vector in a closed quantum system is described by the evolution
operator U (t), as e.g. for the Schrödinger equation
E
(1.16)
Ψ̇(t) = −iH(t) |Ψ(t)i
the time evolution operator
U (t) = τ̂ exp



−i
Zt
0
H(t′ )dt′



(1.17)
14
CHAPTER 1. AN INTRODUCTION TO MASTER EQUATIONS
may be defined as the solution to the operator equation
U̇ (t) = −iH(t)U (t) .
(1.18)
For constant H(0) = H, we simply have the solution U (t) = e−iHt . Similarly, a pure-state density
matrix ρ = |Ψi hΨ| would evolve according to the von-Neumann equation
ρ̇ = −i [H(t), ρ(t)]
(1.19)
with the formal solution ρ(t) = U (t)ρ(0)U † (t), compare Eq. (1.17).
When we simply apply this evolution equation to a density matrix that is not pure, we obtain
X
ρ(t) =
pi U (t) |Φi i hΦi | U † (t) ,
(1.20)
i
i.e., transitions between the (now time-dependent) state vectors |Φi (t)i = U (t) |Φi i are impossible
with unitary evolution. This means that the von-Neumann evolution equation does yield the same
dynamics as the Schrödinger equation if it is restarted on different initial states.
Exercise 5 (Preservation of density matrix properties by unitary evolution) (1 point)
Show that the von-Neumann (1.19) equation preserves self-adjointness, trace, and positivity of the
density matrix.
Also the Measurement process can be generalized similarly. For a quantum state |Ψi, measurements are described by a set of measurement operators {MP
m }, each corresponding to a certain
†
measurement outcome, and with the completeness relation m Mm
Mm = 1. The probability of
obtaining result m is given by
†
pm = hΨ| Mm
Mm |Ψi
(1.21)
and after the measurement with outcome m, the quantum state is collapsed
m
|Ψi → q
Mm |Ψi
†
hΨ| Mm
Mm
.
(1.22)
|Ψi
The projective measurement is just a special case of that with Mm = |mi hm|.
Box 3 (Measurements with density matrix) For a set of measurementP
operators {Mm } cor†
responding to different outcomes m and obeying the completeness relation m Mm
Mm = 1, the
probability to obtain result m is given by
†
pm = Tr Mm
Mm ρ
(1.23)
and action of measurement on the density matrix – provided that result m was obtained – can be
summarized as
m
ρ → ρ′ =
M ρM †
nm m o
†
Tr Mm
Mm ρ
(1.24)
15
1.2. DENSITY MATRIX FORMALISM
It is therefore straightforward to see that description by Schrödinger equation or von-Neumann
equation with the respective measurement postulates are equivalent. The density matrix formalism conveniently includes statistical mixtures in the description but at the cost of quadratically
increasing the number of state variables.
Exercise 6 (Preservation of density matrix properties by measurement) (1 point)
Show that the measurement postulate preserves self-adjointness, trace, and positivity of the density
matrix.
1.2.3
Most general dynamical Evolution
Any dynamical evolution equation for the density matrix should (at least in some approximate
sense) preserve its interpretation as density matrix, i.e., trace, hermiticity, and positivity must
be preserved. By construction, the mesurement postulate and unitary evolution preserve these
properties. However, more general evolutions are conceivable. If we constrain ourselves to master
equations that are local in time and have constant coefficients, the most general evolution that
preserves trace, self-adjointness, and positivity of the density matrix is given by a Lindblad form [4].
Box 4 (Lindblad form) A master equation of Lindblad form has the structure
ρ̇ = Lρ = −i [H, ρ] +
2 −1
N
X
α,β=1
γαβ
Aα ρA†β
o
1n †
Aβ Aα , ρ
−
2
,
(1.25)
∗
where the hermitian operator H = H † can be interpreted
P ∗ as an effective Hamiltonian and γαβ = γβα
is a positive semidefinite matrix, i.e., it fulfills
xα γαβ xβ ≥ 0 for all vectors x (or, equivalently
αβ
that all eigenvalues of (γαβ ) are non-negative λi ≥ 0).
Exercise 7 (Trace and Hermiticity preservation by Lindblad forms) (1 points)
Show that the Lindblad form master equation preserves trace and hermiticity of the density matrix.
The Lindblad type master equation can be written in simpler form: As the dampening matrix
it can be diagonalized by a suitable unitary transformation U , such that
P γ is hermitian,
†
αβ Uα′ α γαβ (U )ββ ′ = δα′ β ′ γα′ with γα ≥ 0 representing its non-negative
P eigenvalues. Using this
unitary operation, a new set of operators can be defined via Aα = α′ Uα′ α Lα′ . Inserting this
16
CHAPTER 1. AN INTRODUCTION TO MASTER EQUATIONS
decomposition in the master equation, we obtain
2 −1
N
X
o
1n †
ρ̇ = −i [H, ρ] +
γαβ
−
Aβ Aα , ρ
2
α,β=1
"
#
o
X X
1n †
†
∗
L β ′ L α′ , ρ
Lα′ ρLβ ′ −
γαβ Uα′ α Uβ ′ β
= −i [H, ρ] +
2
′
′
αβ
α ,β
X 1 †
†
,
Lα Lα , ρ
= −i [H, ρ] +
γα Lα ρLα −
2
α
Aα ρA†β
(1.26)
where γα denote the N 2 − 1 non-negative eigenvalues of the dampening matrix. Evidently, the
representation of a master equation is not unique. Any other unitary operation would lead to a
different non-diagonal form which however describes the same master equation. In addition, we
note here that the master equation is not only invariant to unitary transformations of the operators
Aα , but in the diagonal representation also to inhomogeneous transformations of the form
Lα → L′α = Lα + aα
1 X
H → H′ = H +
γα a∗α Lα − aα L†α + b ,
2i α
(1.27)
with complex numbers aα and a real number b. The first of the above equations can be exploited
to choose the Lindblad operators Lα traceless.
Exercise 8 (Shift invariance) (1 points)
Show the invariance of the diagonal representation of a Lindblad form master equation (1.26) with
respect to the transformation (1.27).
We would like to demonstrate the preservation of positivity here. Let us get rid of the unitary
evolution term by transforming to a comoving frame ρ = e−iHt ρ̃e+iHt , where the master equation
assumes the form
X 1 †
†
ρ̇ =
γα Lα (t)ρLα (t) −
(1.28)
Aα (t)Aα (t), ρ
2
α
with the transformed time-dependent operators Lα (t) = e+iHt Lα e−iHt . It is also clear that if
the differential equation preserves positivity of the density matrix, then it would also do this for
time-dependent rates γα . Define the operators with K = N 2 − 1
W1 (t) = 1 ,
1X
γα (t)L†α (t)Lα (t) ,
W2 (t) =
2 α
W3 (t) = L1 (t) ,
..
.
WK+2 (t) = LK (t) .
(1.29)
17
1.2. DENSITY MATRIX FORMALISM
Discretizing the time-derivative in (1.28) one transforms the differential equation for the density
matrix into an iteration equation
X 1 †
†
ρ(t + ∆t) = ρ(t) + ∆t
γα Lα (t)ρ(t)Lα (t) −
Lα (t)Lα (t), ρ(t)
2
α
X
=
wαβ (t)Wα (t)ρ(t)Wβ† (t) ,
(1.30)
αβ
where the wαβ matrix assumes block form

0
···
0
1
−∆t
 −∆t
0
0
···
0

 0
0
∆tγ
(t)
1
w(t) = 
 ..
..
...
 .
.
0
0
∆tγK (t)




,


(1.31)
which makes it particularly easy to diagonalize it: The lower right block is already diagonal and
the eigenvalues of the upper two by two block may be directly obtained from solving for the roots
of the characteristic polynomial
λ2 − λ − ∆t2 = 0. Again, we introduce the corresponding unitary
P
transformation W̃α (t) =
uα′ α (t)Wα′ (t) to find that
α′
ρ(t + ∆t) =
X
wα (t)W̃α (t)ρ(t)W̃α† (t)
(1.32)
α
with wα (t) denoting the eigenvalues of
√the matrix
(1.31) and in particular the only negative eigen1
2
value being given by w1 (t) = 2 1P
− 1 + 4∆t . Now, we use the spectral decomposition of the
density matrix at time t – ρ(t) = a Pa (t) |Ψa (t)i hΨa (t)| – to demonstrate approximate positivity
of the density matrix at time t + ∆t
2
X
hΦ| ρ(t + ∆t) |Φi =
wα (t)Pa (t)hΦ| W̃α (t) |Ψa (t)i
α,a
2
X
√
1
2
≥
Pa (t)hΦ| W̃1 (t) |Ψa (t)i
1 − 1 + 4∆t
2
a
2
X
∆t→0
≥ −∆t2
Pa (t)hΦ| W̃1 (t) |Ψa (t)i → 0 ,
(1.33)
a
such that the violation of positivity vanishes faster than the discretization width as ∆t goes to
zero.
1.2.4
Master Equation for a cavity in a thermal bath
Consider the Lindblad form master equation
ρ˙S
1
1
= −i Ωa a, ρS + γ(1 + nB ) aρS a − a† aρS − ρS a† a
2
2
1
1
+γnB a† ρS a − aa† ρS − ρS aa† ,
2
2
†
†
(1.34)
18
CHAPTER 1. AN INTRODUCTION TO MASTER EQUATIONS
−1
with bosonic operators a, a† = 1 and Bose-Einstein bath occupation nB = eβΩ − 1
and cavity
√
†
frequency Ω. In Fock-space representation, these operators act as a |ni = n + 1 |n + 1i (where
0 ≤ n < ∞), such that the above master equation couples only the diagonals of the density matrix
ρn = hn| ρS |ni to each other
ρ˙n = γ(1 + nB ) [(n + 1)ρn+1 − nρn ] + γnB [nρn−1 − (n + 1)ρn ]
= γnB nρn−1 − γ [n + (2n + 1)nB ] ρn + γ(1 + nB )(n + 1)ρn+1
(1.35)
in a tri-diagonal form. That makes it particularly easy to calculate its stationary state recursively,
since the boundary solution nB ρ̄0 = (1 + nB )ρ̄1 implies for all n the relation
ρ̄n+1
nB
=
= e−βΩ ,
ρ̄n
1 + nB
(1.36)
i.e., the stationary state is a thermalized Gibbs state with the reservoir temperature.
Exercise 9 (Moments) (1 points)
Calculate the expectation value of the number operator n = a† a and its square n2 = a† aa† a in the
stationary state of the master equation (1.34).
1.2.5
Master Equation for a driven cavity
When the cavity is driven with a laser and simultaneously coupled to a vaccuum bath nB = 0, we
obtain the master equation
P +iωt
P ∗ −iωt †
1
1 †
†
†
ρ˙S = −i Ωa a + e
a+
e
a , ρS + γ aρS a − a aρS − ρS a a
2
2
2
2
†
†
with the Laser frequency ω and amplitude P . The transformation ρ = e+iωa at ρS e−iωa
a time-independent master equation
P∗ †
1 †
1 †
P
†
†
a , ρ + γ aρa − a aρ − ρa a .
ρ̇ = −i (Ω − ω)a a + a +
2
2
2
2
(1.37)
† at
maps to
(1.38)
This equation obviously couples coherences and populations in the Fock space representation.
Exercise 10 (Coherent state) (1 points)
Using the driven cavity master equation, show that the stationary expectation value of the cavity
occupation fulfils
lim a† a =
t→∞
|P |2
γ 2 + 4(Ω − ω)2
19
1.2. DENSITY MATRIX FORMALISM
1.2.6
Tensor Product
The greatest advantage of the density matrix formalism is visible when quantum systems composed
of several subsystems are considered. Roughly speaking, the tensor product represents a way to
construct a larger vector space from two (or more) smaller vector spaces.
Box 5 (Tensor Product) Let V and W be Hilbert spaces (vector spaces with scalar product) of
dimension m and n with basis vectors {|vi} and {|wi}, respectively. Then V ⊗ W is a Hilbert
space of dimension m · n, and a basis is spanned by {|vi ⊗ |wi}, which is a set combining every
basis vector of V with every basis vector of W .
Mathematical properties
• Bilinearity (z1 |v1 i + z2 |v2 i) ⊗ |wi = z1 |v1 i ⊗ |wi + z2 |v2 i ⊗ |wi
• operators acting on the combined Hilbert space A ⊗ B act on the basis states as (A ⊗ B)(|vi ⊗
|wi) = (A |vi) ⊗ (B |wi)
P
• any linear operator on V ⊗ W can be decomposed as C = i ci Ai ⊗ Bi
P
• the scalar P
product is inherited in the natural way, i.e.,Pone has for |ai = ij aij |vP
i i ⊗ |wj i
and |bi = kℓ bkℓ |vk i⊗|wℓ i the scalar product ha|bi = ijkℓ a∗ij bkℓ hvi |vk i hwj |wℓ i = ij a∗ij bij
If more than just two vector spaces are combined to form a larger vector space, the dimension of
the joint vector space grows rapidly, as e.g. exemplified by the case of a qubit: Its Hilbert space is
just spanned by two vectors |0i and |1i. The joint Hilbert space of two qubits is four-dimensional, of
three qubits 8-dimensional, and of n qubits 2n -dimensional. Eventually, this exponential growth of
the Hilbert space dimension for composite quantum systems is at the heart of quantum computing.
Exercise 11 (Tensor Products of Operators) (1 points)
Let σ denote the Pauli matrices, i.e.,
+1 0
0 −i
0 +1
3
2
1
σ =
σ =
σ =
0 −1
+i 0
+1 0
Compute the trace of the operator
Σ = a1 ⊗ 1 +
3
X
i=1
i
αi σ ⊗ 1 +
3
X
j=1
j
βj 1 ⊗ σ +
3
X
i,j=1
aij σ i ⊗ σ j .
Since the scalar product is inherited, this typically enables a convenient calculation of the trace
20
CHAPTER 1. AN INTRODUCTION TO MASTER EQUATIONS
in case of a few operator decomposition, e.g., for just two operators
X
Tr {A ⊗ B} =
hnA , nB | A ⊗ B |nA , nB i
nA ,nB
=
"
X
nA
hnA | A |nA i
#"
= TrA {A}TrB {B} ,
X
nB
hnB | B |nB i
#
(1.39)
where TrA/B denote the trace in the Hilbert space of A and B, respectively.
1.2.7
The partial trace
For composed systems, it is usually not necessary to keep all information of the complete system
in the density matrix. Rather, one would like to have a density matrix that encodes all the
information on a particular subsystem only. Obviously, the map ρ → TrB {ρ} to such a reduced
density matrix should leave all expectation values of observables acting on the considered subsystem
only invariant, i.e.,
Tr {A ⊗ 1ρ} = Tr {ATrB {ρ}} .
(1.40)
If this basic condition was not fulfilled, there would be no point in defining such a thing as a
reduced density matrix: Measurement would yield different results depending on the Hilbert space
of the experimenters feeling.
Box 6 (Partial Trace) Let |a1 i and |a2 i be vectors of state space A and |b1 i and |b2 i vectors of
state space B. Then, the partial trace over state space B is defined via
TrB {|a1 i ha2 | ⊗ |b1 i hb2 |} = |a1 i ha2 | Tr {|b1 i hb2 |} .
(1.41)
The partial trace is linear, such that the partial trace of arbitrary operators is calculated
similarly. By choosing the |aα i and |bγ i as an orthonormal basis in the respective Hilbert space,
one may therefore calculate the most general partial trace via
(
)
X
cαβγδ |aα i haβ | ⊗ |bγ i hbδ |
TrB {C} = TrB
αβγδ
=
X
αβγδ
=
X
αβγδ
=
X
αβγδ
=
cαβγδ TrB {|aα i haβ | ⊗ |bγ i hbδ |}
cαβγδ |aα i haβ | Tr {|bγ i hbδ |}
cαβγδ |aα i haβ |
"
X X
αβ
γ
#
X
ǫ
hbǫ |bγ i hbδ |bǫ i
cαβγγ |aα i haβ | .
The definition 6 is the only linear map that respects the invariance of expectation values.
(1.42)
1.2. DENSITY MATRIX FORMALISM
Exercise 12 (Partial Trace) (1 points)
Compute the partial trace of a pure density matrix ρ = |Ψi hΨ| in the bipartite state
1
1
|Ψi = √ (|01i + |10i) ≡ √ (|0i ⊗ |1i + |1i ⊗ |0i)
2
2
21
22
CHAPTER 1. AN INTRODUCTION TO MASTER EQUATIONS
Chapter 2
Obtaining a Master Equation
2.1
Phenomenologic Derivations (Educated Guesses)
Before discussing rigorous derivations, we give some more examples of phenomenologically motivated master equations.
2.1.1
The single resonant level
Consider a nanostructure (quantum dot) capable of hosting a single electron with on-site energy
ǫ. Let P0 (t) denote the probability of finding the dot empty and P1 (t) the opposite probability
of finding an electron in the dot. When we do now couple the dot to a junction with tunneling
rate Γ, its occupation will fluctuate depending on the Fermi level of the junction, see Fig. 2.1. In
Figure 2.1: A single quantum dot coupled to a single junction, where the electronic occupation of
energy levels is well approximated by a Fermi distribution.
particular, if at time t the dot was empty, the probability to find an electron in the dot at time
t + ∆t is roughly given by Γ∆tf (ǫ) with the Fermi function defined as
f (ω) =
1
eβ(ω−µ) + 1
,
(2.1)
where β denotes the inverse temperature and µ the chemical potential of the junction. The
transition rate is thus given by the tunneling rate Γ multiplied by the probability to have an
electron in the junction at the required energy ǫ ready to jump into the system. The inverse
23
24
CHAPTER 2. OBTAINING A MASTER EQUATION
probability to find an initially filled dot empty reads Γ∆t [1 − f (ǫ)], i.e., here one has to muliply
the tunneling probability with the probability to have a free slot in the junction, such that we have
d
P0
−Γf (ǫ) +Γ(1 − f (ǫ))
P0
,
(2.2)
=
P1
+Γf (ǫ) −Γ(1 − f (ǫ))
dt P1
where the diagonal elements follow directly from trace conservation. The stationary state fulfils
P1
f (ǫ)
= e−β(ǫ−µ) ,
=
P0
1 − f (ǫ)
(2.3)
i.e., the quantum dot equilibrates with the reservoir. Here we have been able to describe the
system only by occupation probabilities (no coherences), which is possible since due to charge
conservation one cannot create superpositions of differently charged states. In the corresponding
density matrix formalism, the coherences would simply vanish.
2.1.2
Cell culture growth in constrained geometries
Consider a population of cells that divide (proliferate) with a certain rate α. These cells live
in a constrained geometry (e.g., a petri dish) that admits at most K cells. Let Pi (t) denote the
probability to have i cells in the compartment. Assuming that the proliferation rate α is sufficiently
small, we can easily set up a master equation
Ṗ0 = 0 ,
Ṗ1 = −1 · α · P1 ,
Ṗ2 = −2 · α · P2 + 1 · α · P1 ,
..
.
Ṗℓ = −ℓ · α · Pℓ + (ℓ − 1) · α · Pℓ−1 ,
..
.
ṖK−1 = −(K − 1) · α · PK−1 + (K − 2) · α · PK−2 ,
ṖK = +(K − 1) · αPK−1 .
(2.4)
The prefactors arise since any of the ℓ cells may proliferate. Arranging the probabilities in a single
vector, this may also be written as Ṗ = LP , where the matrix L contains the rates. When we
have a single cell as initial condition (full knowledge), i.e., P1 (0) = 1, one can change the carrying
capacity K = {1, 2, 3, 4, . . .} and
for each K the resulting system of differential equations for
Psolve
K
the expectation value of hℓi = ℓ=1 ℓPℓ (t). These solutions may then be generalized to
h
K i
.
(2.5)
hℓi = e+αt 1 − 1 − e−αt
Similarly, one can compute the expectation value of hℓ2 i.
However, one may not be only interested in the evolution by mean values (especially when one
wants e.g. to model tumour growth) but also would like to have some idea about the evolution of
a single trajectory. In case of a rate equation (where coherences are not included), it is possible
to generate single trajectories also from the master equation by Monte-Carlo simulation. Suppose
at time t, the system is in the state ℓ, i.e., Pα (t) = δℓα . After a sufficiently short time ∆t, the
probabilities to be in a different state read
P (t + ∆t) ≈ [1 + ∆tL] P (t) ,
(2.6)
25
2.2. DERIVATIONS FOR OPEN QUANTUM SYSTEMS
which for our simple example boils down to
Pℓ (t + ∆t) ≈ (1 − ℓα∆t)Pℓ (t) ,
Pℓ+1 (t + ∆t) ≈ +ℓα∆tPℓ (t) .
(2.7)
To simulate a single trajectory, one may simply draw a random number and determine whether
the transition has occurred (Pjump = ℓα∆t) or not. If the transition occurs, we do again have
complete knowledge and may set Pα (t + ∆t) = δℓ+1,α . If the transition does not occur, we also
have complete knowledge and remain at Pα (t + ∆t) = δℓα . In any case, this looks effectively like a
measurement of the cell number that is performed at intervals of ∆t. The simple average of many
such trajectories will yield the mean evolution predicted by the master equation, see Fig. 2.2.
120
specific trajectories
mean and error estimate
logistic growth
logistic growth with l0=1.98
100
cell number
80
60
40
20
0
0
2
4
αt
6
8
10
Figure 2.2: Population
dynamics for the linear master equation (black curve hℓi and shaded area
q
determined from hℓ2 i − hℓi2 ) and for the logistic growth equation (dashed red curve) for carrying
capacity K = 100. For identical initial and final states, the master equation solution overshoots
the logistic growth curve. A slight modification (dotted green curve) of the initial condition in
the logistic growth curve yields the same long-term asyptotics. An average of lots of specific
trajectories would converge towards the black curve.
2.2
Derivations for Open Quantum Systems
In some cases, it is possible to derive a master equation rigorously based only on a few assumptions.
Open quantum systems for example are mostly treated as part of a much larger closed quantum
26
CHAPTER 2. OBTAINING A MASTER EQUATION
system (the union of system and bath), where the partial trace is used to eliminate the unwanted
(typically many) degrees of freedom of the bath, see Fig. 2.3. Technically speaking, we will consider
Figure 2.3: An open quantum system can be conceived as being part of a larger closed quantum
system, where the system part (HS ) is coupled to the bath (HB ) via the interaction Hamiltonian
HI .
Hamiltonians of the form
H = HS ⊗ 1 + 1 ⊗ HB + HI ,
(2.8)
where the system and bath Hamiltonians act only on the system and bath Hilbert space, respectively. In contrast, the interaction Hamiltonian acts on both Hilbert spaces
X
HI =
Aα ⊗ Bα ,
(2.9)
α
where the summation boundaries are in the worst case limited by the dimension of the system
Hilbert space. As we consider physical observables here, it is required that all Hamiltonians are
self-adjoint.
Exercise 13 (Hermiticity of Couplings) (1 points)
Show that it is always possible to choose hermitian coupling operators Aα = A†α and Bα = Bα† using
that HI = HI† .
2.2.1
Weak Coupling Regime
When the interaction HI is small, it is justified to apply perturbation theory. The von-Neumann
equation in the joint total quantum system
ρ̇ = −i [HS ⊗ 1 + 1 ⊗ HB + HI , ρ]
(2.10)
describes the full evolution of the combined density matrix. This equation can be formally solved
by the unitary evolution ρ(t) = e−iHt ρ0 e+iHt , which however is impractical to compute as H
involves too many degrees of freedom.
Transforming to the interaction picture
ρ(t) = e+i(HS +HB )t ρ(t)e−i(HS +HB )t ,
(2.11)
27
2.2. DERIVATIONS FOR OPEN QUANTUM SYSTEMS
which will be denoted by bold symbols throughout, the von-Neumann equation transforms into
ρ̇ = −i [HI (t), ρ] ,
(2.12)
where the in general time-dependent interaction Hamiltonian
X
HI (t) = e+i(HS +HB )t HI e−i(HS +HB )t =
e+iHS t Aα e−iHS t ⊗ e+iHB t Bα e−iHB t
α
=
X
α
Aα (t) ⊗ Bα (t)
(2.13)
allows to perform perturbation theory.
Born-Markov-Secular Approximations
Without loss of generality we will assume here the case of hermitian coupling operators Aα = A†α
and Bα = Bα† . One heuristic way to perform perturbation theory is to formally integrate Eq. (2.13)
and to re-insert the result in the r.h.s. of Eq. (2.13). The time-derivative of the system density
matrix is obtained by performing the partial trace
ρ˙S = −iTrB {[HI (t), ρ0 ]} −
Zt
TrB {[HI (t), [HI (t′ ), ρ(t′ )]] dt′ } .
(2.14)
0
This integro-differential equation is still exact but unfortunately not closed as the r.h.s. does not
depend on ρS but the full density matrix at all previous times.
To close the above equation, we employ factorization of the initial density matrix
ρ0 = ρ0S ⊗ ρ̄B
(2.15)
together with perturbative considerations: Assuming that HI (t) = O{λ} with λ beeing a small
dimensionless perturbation parameter (solely used for bookkeeping purposes here) and that the
environment is so large such that it is hardly affected by the presence of the system, we may
formally expand the full density matrix
ρ(t) = ρS (t) ⊗ ρ̄B + O{λ} ,
(2.16)
where the neglect of all higher orders is known as Born approximation. Eq. (2.14) demonstrates
that the Born approximation is equivalent to a perturbation theory in the interaction Hamiltonian
ρ˙S = −iTrB {[HI (t), ρ0 ]} −
Zt
TrB {[HI (t), [HI (t′ ), ρS (t′ ) ⊗ ρ̄B ]] dt′ } + O{λ3 } .
(2.17)
0
Using the decomposition of the interaction Hamiltonian (2.9), this obviously yields a closed equation for the system density matrix
ρ˙S = −i
X
α
X
Aα (t)ρ0S Tr {Bα (t)ρ̄B } − ρ0S Aα (t)Tr {ρ̄B Bα (t)} −
+Aα (t)Aβ (t′ )ρS (t′ )Tr {Bα (t)Bβ (t′ )ρ̄B }
−Aα (t)ρS (t′ )Aβ (t′ )Tr {Bα (t)ρ̄B Bβ (t′ )}
−Aβ (t′ )ρS (t′ )Aα (t)Tr {Bβ (t′ )ρ̄B Bα (t)}
i
+ρS (t′ )Aβ (t′ )Aα (t)Tr {ρ̄B Bβ (t′ )Bα (t)} dt′ .
Zt h
αβ 0
(2.18)
28
CHAPTER 2. OBTAINING A MASTER EQUATION
Without loss of generality, we proceed by assuming that the single coupling operator expectation
value vanishes
Tr {Bα (t)ρ̄B } = 0 .
(2.19)
This situation can always be constructed by simultaneously modifying system Hamiltonian HS
and coupling operators Aα , see exercise 14.
Exercise 14 (Vanishing single-operator expectation values) (1 points)
Show that by modifying system and interaction Hamiltonian
X
HS → HS +
gα Aα ,
Bα → Bα − g α 1
(2.20)
α
one can construct a situation where Tr {Bα (t)ρ̄B } = 0. Determine gα .
Using the cyclic property of the trace, we obtain
ρ˙S = −
XZ
αβ 0
t
h
dt′ Cαβ (t, t′ ) [Aα (t), Aβ (t′ )ρS (t′ )]
i
(2.21)
Cαβ (t1 , t2 ) = Tr {Bα (t1 )Bβ (t2 )ρ̄B } .
(2.22)
+Cβα (t′ , t) [ρS (t′ )Aβ (t′ ), Aα (t)]
with the bath correlation function
The integro-differential equation (2.21) is a non-Markovian master equation, as the r.h.s.
depends on the value of the dynamical variable (the density matrix) at all previous times – weighted
by the bath correlation functions. It does preserve trace and hermiticity of the system density
matrix, but not necessarily its positivity. Such integro-differential equations can only be solved in
very specific cases. Therefore, we motivate further approximations, for which we need to discuss
the analytic properties of the bath correlation functions.
It is quite straightforward to see that when the bath Hamiltonian commutes with the bath
density matrix [HB , ρ̄B ] = 0, the bath correlation functions actually only depend on the difference
of their time arguments Cαβ (t1 , t2 ) = Cαβ (t1 − t2 ) with
Cαβ (t1 − t2 ) = Tr e+iHB (t1 −t2 ) Bα e−iHB (t1 −t2 ) Bβ ρ̄B .
(2.23)
Since we chose our coupling operators hermitian, we have the additional symmetry that Cαβ (τ ) =
∗
Cβα
(−τ ). One can now evaluate several system-bath models and when the bath has a dense
spectrum, the bath correlation functions are typically found to be strongly peaked around zero,
see exercise 15.
29
2.2. DERIVATIONS FOR OPEN QUANTUM SYSTEMS
Exercise 15 (Bath Correlation Function)
R (1 points)
Evaluate the Fourier transform γαβ (ω) = Cαβ (τ )e+iωτ dτ of the bath correlation functions for
P
P
P
the coupling operators B1 = k hk bk and B2 = k h∗k b†k for a bosonic bath HB = k ωk b†k bk in
−βH
the thermal equilibrium state ρ̄0B = Tr e e−βHBB . You may use the continous representation Γ(ω) =
{
}
P
2π k |hk |2 δ(ω − ωk ) for the tunneling rates.
This implies that the integro-differential equation (2.21) can also be written as ρ˙S =
Rt
0
W(t − t′ )ρS (t′ )dt′ ,
where the kernel W(τ ) assigns a much smaller weight to density matrices far in the past than
to the density matrix just an instant ago. In the most extreme case, we would approximate
Cαβ (t1 , t2 ) ≈ Γαβ δ(t1 − t2 ), but we will be cautious here and assume that only the density matrix
varies slower than the decay time of the bath correlation functions. Therefore, we replace in the
r.h.s. ρS (t′ ) → ρS (t) (first Markov approximation), which yields in Eq. (2.17)
ρ˙S = −
Zt
TrB {[HI (t), [HI (t′ ), ρS (t) ⊗ ρ̄B ]]} dt′
(2.24)
0
This equation is often called Born-Redfield equation. It is time-local and preserves trace and
hermiticity, but still has time-dependent coefficients (also when transforming back from the interaction picture). We substitute τ = t − t′
ρ˙S = −
Zt
TrB {[HI (t), [HI (t − τ ), ρS (t) ⊗ ρ̄B ]]} dτ
(2.25)
0
= −
XZ
αβ 0
t
{Cαβ (τ ) [Aα (t), Aβ (t − τ )ρS (t)] + Cβα (−τ ) [ρS (t)Aβ (t − τ ), Aα (t)]} dτ
The problem that the r.h.s. still depends on time is removed by extending the integration bounds
to infinity (second Markov approximation) – by the same reasoning that the bath correlation
functions decay rapidly
ρ˙S = −
Z∞
TrB {[HI (t), [HI (t − τ ), ρS (t) ⊗ ρ̄B ]]} dτ .
(2.26)
0
This equation is called the Markovian master equation, which in the original Schrödinger
picture
∞
ρ˙S = −i [HS , ρS (t)] −
−
X
Z∞
αβ 0
XZ
αβ 0
Cαβ (τ ) Aα , e−iHS τ Aβ e+iHS τ ρS (t) dτ
Cβα (−τ ) ρS (t)e−iHS τ Aβ e+iHS τ , Aα dτ
(2.27)
is time-local, preserves trace and hermiticity, and has constant coefficients – best prerequisites for
treatment with established solution methods.
30
CHAPTER 2. OBTAINING A MASTER EQUATION
Exercise 16 (Properties of the Markovian Master Equation) (1 points)
Show that the Markovian Master equation (2.27) preserves trace and hermiticity of the density
matrix.
In addition, it can be obtained easily from the coupling Hamiltonian, since we have not even
used that the coupling operators should be hermitian.
There is just one problem left: In the general case, it is not of Lindblad form. Note that
there are specific cases where the Markovian master equation is of Lindblad form, but these rather
include simple limits. Though this is sometimes considered a rather cosmetic drawback, it may
lead to unphysical results such as negative probabilities.
To obtain a Lindblad type master equation, a further approximation is required. The secular
approximation involves an averaging over fast oscillating terms, but in order to identify the
oscillating terms, it is necessary to at least formally calculate the interaction picture dynamics of
the system coupling operators. We begin by writing Eq. (2.27) in the interaction picture again
explicitly – now using the hermiticity of the coupling operators
ρ˙S = −
Z∞ X
0
= +
αβ
Z∞ X
0
{Cαβ (τ ) [Aα (t), Aβ (t − τ )ρS (t)] + h.c.} dτ
Cαβ (τ )
αβ
X n
a,b,c,d
|ai ha| Aβ (t − τ ) |bi hb| ρS (t) |di hd| Aα (t) |ci hc|
o
− |di hd| Aα (t) |ci hc| |ai ha| Aβ (t − τ ) |bi hb| ρS (t) dτ + h.c. ,
(2.28)
where we have introduced the system energy eigenbasis
HS |ai = Ea |ai .
(2.29)
We can use this eigenbasis to make the time-dependence of the coupling operators in the interaction
picture explicit
ρ˙S = +
Z∞ X
0
Cαβ (τ )
αβ
X n
a,b,c,d
e+i(Ea −Eb )(t−τ ) e+i(Ed −Ec )t ha| Aβ |bi hd| Aα |ci |ai hb| ρS (t) |di hc|
o
−e+i(Ea −Eb )(t−τ ) e+i(Ed −Ec )t ha| Aβ |bi hd| Aα |ci |di hc| |ai hb| ρS (t) dτ + h.c. ,
=
∞
X X Z
αβ a,b,c,d 0
Cαβ (τ )e
+i(Eb −Ea )τ
dτ e
−i(Eb −Ea −(Ed −Ec ))t
ha| Aβ |bi hc| Aα |di
o
+ |ai hb| ρS (t) (|ci hd|)† − (|ci hd|)† |ai hb| ρS (t) + h.c.
∗
n
(2.30)
The secular approximation now involves neglecting all terms that are oscillatory in time t
31
2.2. DERIVATIONS FOR OPEN QUANTUM SYSTEMS
(long-time average), i.e., we have
X X
ρ˙S =
Γαβ (Eb − Ea )δEb −Ea ,Ed −Ec ha| Aβ |bi hc| Aα |di∗ ×
αβ a,b,c,d
n
†
†
o
× + |ai hb| ρS (t) (|ci hd|) − (|ci hd|) |ai hb| ρS (t)
X X
+
Γ∗αβ (Eb − Ea )δEb −Ea ,Ed −Ec ha| Aβ |bi∗ hc| Aα |di ×
αβ a,b,c,d
n
× + |ci hd| ρS (t) (|ai hb|)† − ρS (t) (|ai hb|)† |ci hd|
o
,
(2.31)
where we have introduced the half-sided Fourier transform of the bath correlation functions
Z∞
(2.32)
Γαβ (ω) = Cαβ (τ )e+iωτ dτ .
0
This equation preserves trace, hermiticity, and positivity of the density matrix and hence all desired
properties, since it is of Lindblad form (which will be shown later). Unfortunately, it is typically
not so easy to obtain as it requires diagonalization of the system Hamiltonian first. By using the
transformations α ↔ β, a ↔ c, and b ↔ d in the second line and also using that the δ-function is
symmetric, we may rewrite the master equation as
X X Γαβ (Eb − Ea ) + Γ∗βα (Eb − Ea ) δEb −Ea ,Ed −Ec ×
ρ˙S =
αβ a,b,c,d
× ha| Aβ |bi hc| Aα |di∗ |ai hb| ρS (t) (|ci hd|)†
X X
−
Γαβ (Eb − Ea )δEb −Ea ,Ed −Ec ×
αβ a,b,c,d
× ha| Aβ |bi hc| Aα |di∗ (|ci hd|)† |ai hb| ρS (t)
X X
−
Γ∗βα (Eb − Ea )δEb −Ea ,Ed −Ec ×
αβ a,b,c,d
× ha| Aβ |bi hc| Aα |di∗ ρS (t) (|ci hd|)† |ai hb| .
(2.33)
We split the matrix-valued function Γαβ (ω) into hermitian and anti-hermitian parts
1
γαβ (ω) +
2
1
γαβ (ω) −
Γ∗βα (ω) =
2
Γαβ (ω) =
1
σαβ (ω) ,
2
1
σαβ (ω) ,
2
(2.34)
∗
∗
with hermitian γαβ (ω) = γβα
(ω) and anti-hermitian σαβ (ω) = −σβα
(ω). These new functions can
be interpreted as full even and odd Fourier transforms of the bath correlation functions
Z+∞
Cαβ (τ )e+iωτ dτ ,
γαβ (ω) = Γαβ (ω) + Γ∗βα (ω) =
−∞
Z+∞
Cαβ (τ )sgn(τ )e+iωτ dτ .
σαβ (ω) = Γαβ (ω) − Γ∗βα (ω) =
−∞
(2.35)
32
CHAPTER 2. OBTAINING A MASTER EQUATION
Exercise 17 (Odd Fourier Transform) (1 points)
Show that the odd Fourier transform σαβ (ω) may be obtained from the even Fourier transform
γαβ (ω) by a Cauchy principal value integral
i
σαβ (ω) = P
π
Z+∞
−∞
γαβ (Ω)
dΩ .
ω−Ω
In the master equation, these replacements lead to
X X
ρ˙S =
γαβ (Eb − Ea )δEb −Ea ,Ed −Ec ha| Aβ |bi hc| Aα |di∗ ×
αβ a,b,c,d
o
1n
†
× |ai hb| ρS (t) (|ci hd|) −
(|ci hd|) |ai hb| , ρS (t)
2
X X 1
−i
σαβ (Eb − Ea )δEb −Ea ,Ed −Ec ha| Aβ |bi hc| Aα |di∗ ×
2i
αβ a,b,c,d
h
i
†
× (|ci hd|) |ai hb| , ρS (t)
X X
=
γαβ (Eb − Ea )δEb −Ea ,Ed −Ec ha| Aβ |bi hc| Aα |di∗ ×
†
αβ a,b,c,d
o
1n
†
(|ci hd|) |ai hb| , ρS (t)
× |ai hb| ρS (t) (|ci hd|) −
2
"
#
XX 1
−i
σαβ (Eb − Ec )δEb ,Ea hc| Aβ |bi hc| Aα |ai∗ |ai hb| , ρS (t) .
2i
αβ a,b,c
†
(2.36)
To prove that we have a Lindblad form, it is easy to see first that the term in the commutator
XX 1
σαβ (Eb − Ec )δEb ,Ea hc| Aβ |bi hc| Aα |ai∗ |ai hb|
(2.37)
HLS =
2i
αβ a,b,c
is an effective Hamiltonian. This Hamiltonian is often called Lamb-shift Hamiltonian, since it
renormalizes the system Hamiltonian due to the interaction with the reservoir. Note that we have
[HS , HLS ] = 0.
Exercise 18 (Lamb-shift) (1 points)
†
Show that HLS = HLS
and [HLS , HS ] = 0.
To show the Lindblad-form of the non-unitary evolution, we identify the Lindblad jump operator Lα = |ai hb| = L(a,b) . For an N -dimensional system Hilbert space with N eigenvectors of
2
HS we would
Phave N such jump operators, but in a rotated basis, we may leave out the identity
matrix 1 = a |ai ha| (which has trivial action). It remains to be shown that the matrix
X
γ(ab),(cd) =
γαβ (Eb − Ea )δEb −Ea ,Ed −Ec ha| Aβ |bi hc| Aα |di∗
(2.38)
αβ
33
2.2. DERIVATIONS FOR OPEN QUANTUM SYSTEMS
P
is non-negative, i.e., a,b,c,d x∗ab γ(ab),(cd) xcd ≥ 0 for all xab . We first note that for hermitian coupling
operators the Fourier transform matrix at fixed ω is positive (recall that Bα = Bα† and [ρ̄B , HB ] = 0)
X
Γ =
x∗α γαβ (ω)xβ
αβ
Z+∞
(
dτ e+iωτ Tr eiHS τ
=
−∞
"
X
#
x∗α Bα e−iHS τ
α
"
X
#
xβ Bβ ρ̄B
β
)
Z+∞
X
e+i(En −Em )τ hn| B † |mi hm| B ρ̄B |ni
dτ e+iωτ
=
nm
−∞
X
=
nm
2πδ(ω + En − Em )|hm| B |ni|2 ρn
≥ 0.
(2.39)
Now, we replace the Kronecker symbol in the dampening coefficients by two via the introduction
of an auxiliary summation
X
x∗ab γ(ab),(cd) xcd
Γ̃ =
abcd
=
XXX
ω
=
"
XX X
ω
=
αβ abcd
γαβ (ω)δEb −Ea ,ω δEd −Ec ,ω x∗ab ha| Aβ |bi xcd hc| Aα |di∗
cd
αβ
XX
ω
αβ
#
xcd hc| Aα |di∗ δEd −Ec ,ω γαβ (ω)
yα∗ (ω)γαβ (ω)yβ (ω) ≥ 0 .
"
X
ab
x∗ab ha| Aβ |bi δEb −Ea ,ω
#
(2.40)
Transforming Eq. (2.36) back to the Schrödinger picture (note that the δ-functions prohibit
the occurrence of oscillatory factors), we finally obtain the Born-Markov-secular master equation.
Box 7 (BMS
of the
P master equation) In the weak coupling limit, an †interaction Hamiltonian
†
form HI = α Aα ⊗Bα with hermitian coupling operators (Aα = Aα and Bα = Bα ) and [HB , ρ̄B ] =
0 and Tr {Bα ρ̄B } = 0 leads in the system energy eigenbasis HS |ai = Ea |ai to the Lindblad-form
master equation
"
#
X
ρ˙S = −i HS +
σab |ai hb| , ρS (t)
ab
+
X
a,b,c,d
γab,cd =
X
αβ
γab,cd
o
1n
†
,
(|ci hd|) |ai hb| , ρS (t)
|ai hb| ρS (t) (|ci hd|) −
2
†
γαβ (Eb − Ea )δEb −Ea ,Ed −Ec ha| Aβ |bi hc| Aα |di∗ ,
P
where the Lamb-shift Hamiltonian HLS = ab σab |ai hb| matrix elements read
XX 1
σαβ (Eb − Ec )δEb ,Ea hc| Aβ |bi hc| Aα |ai∗
σab =
2i
c
αβ
(2.41)
(2.42)
34
CHAPTER 2. OBTAINING A MASTER EQUATION
and the constants are given by even and odd Fourier transforms
Z+∞
γαβ (ω) =
Cαβ (τ )e+iωτ dτ ,
−∞
Z+∞
Z+∞
γαβ (ω ′ ) ′
i
dω
σαβ (ω) =
Cαβ (τ )sgn(τ )e+iωτ dτ = P
π
ω − ω′
(2.43)
−∞
−∞
of the bath correlation functions
Cαβ (τ ) = Tr e+iHB τ Bα e−iHB τ Bβ ρ̄B .
(2.44)
The above definition may serve as a recipe to derive a Lindblad type master equation in the
weak-coupling limit. It is expected to yield good results in the weak coupling and Markovian limit
(continuous and nearly flat bath spectral density) and when [ρ̄B , HB ] = 0. It requires to rewrite
the coupling operators in hermitian form, the calculation of the bath correlation function Fourier
transforms, and the diagonalization of the system Hamiltonian.
In the case that the spectrum of the system Hamiltonian is non-degenerate, we have a further
simplification, since the δ-functions simplify further, e.g. δEb ,Ea → δab . By taking matrix elements
of Eq. (2.41) in the energy eigenbasis ρaa = ha| ρS |ai, we obtain an effective rate equation for the
populations only
#
"
X
X
γba,ba ρaa ,
(2.45)
ρ̇aa = +
γab,ab ρbb −
b
b
i.e., the coherences decouple from the evolution of the populations. The transition rates from state
b to state a reduce in this case to
X
γab,ab =
γαβ (Eb − Ea ) ha| Aβ |bi ha| Aα |bi∗ ,
(2.46)
αβ
which – after inserting all definitions condenses basically to Fermis Golden Rule. A description of
open quantum systems is in this limit possible with the same complexity as for closed quantum
systems, since only N dynamical variables have to be evolved. Also, this may serve to demonstrate
the correspondence between quantum and classical mechanics: If only diagonals of the density
matrix remain in the (orthogonal) energy system eigenbasis, this implies that the quantum master
equation boils down to a classical rate equation. Even more, if coupled to a single thermal bath,
the quantum system simply relaxes to the Gibbs equilibrium, i.e., we obtain simply equilibration
of the system temperature with the temperature of the bath.
Coarse-Graining
Although the BMS approximation respects of course the exact initial condition, we have in the
derivation made several long-term approximations. For example, the Markov approximation implied that we consider timescales much larger than the decay time of the bath correlation functions.
Similarly, the secular approximation implied timescales larger than the inverse minimal splitting
35
2.2. DERIVATIONS FOR OPEN QUANTUM SYSTEMS
of the system energy eigenvalues. Therefore, we can only expect the solution originating from the
BMS master equation to be an asymptotically valid long-term approximation.
Coarse-graining in contrast provides a possibility to obtain valid short-time approximations
of the density matrix with a generator that is of Lindblad form, see Fig. 2.4. We start with the
Figure 2.4: Sketch of the
perturbation theory. Calculating the exact time evolution
coarse-graining
Rt
operator U (t) = τ exp −i HI (t′ )dt′ in a closed form is usually prohibitive, which renders the
0
calculation of the exact solution an impossible task (black curve). It is however possible to expand
Rt
Rt
the evolution operator U2 (t) = 1−i HI (t′ )dt′ − dt1 dt2 HI (t1 )HI (t2 )Θ(t1 −t2 ) to second order in
0
0
HI and to obtain the corresponding reduced approximate density matrix. Calculating the matrix
exponential of a constant Lindblad-type generator LCG
is also usually prohibitive, but the first
τ
order approximation may be matched with U2 (t) at time t = τ to obtain a defining equation for
LCG
τ .
von-Neumann equation in the interaction picture (2.12). For factorizing initial density matrices,
it is formally solved by U (t)ρ0S ⊗ ρ̄B U † (t), where the time evolution operator
U (t) = τ̂ exp



−i
Zt
HI (t′ )dt′
0


(2.47)

obeys the evolution equation
U̇ = −iHI (t)U (t) ,
(2.48)
which defines the time-ordering operator τ̂ . Formally integrating this equation with the evident
36
CHAPTER 2. OBTAINING A MASTER EQUATION
initial condition U (0) = 1 yields
U (t) = 1 − i
Zt
HI (t′ )U (t′ )dt′
Zt
0
Zt
Zt
Zt
0
= 1−i
≈ 1−i
HI (t′ )dt′ −
0
HI (t′ )dt′ −
0

dt′ HI (t′ ) 
Zt′
0

dt′′ HI (t′′ )U (t′′ )
dt1 dt2 HI (t1 )HI (t2 )Θ(t1 − t2 ) ,
(2.49)
0
where the occurrence of the Heaviside function is a consequence of time-ordering. For the hermitian
conjugate operator we obtain
†
U (t) ≈ 1 + i
Zt
′
′
HI (t )dt −
0
Zt
dt1 dt2 HI (t1 )HI (t2 )Θ(t2 − t1 ) .
(2.50)
0
To keep the discussion at a moderate level, we assume Tr {Bα ρ̄B } = 0 from the beginning. The
exact solution is approximated by


Zt
Zt
n
(2)
ρS (t) ≈ TrB 1 − i HI (t1 )dt1 − dt1 dt2 HI (t1 )HI (t2 )Θ(t1 − t2 ) ρ0S ⊗ ρ̄B ×
0

× 1 + i
Zt
0
HI (t1 )dt1 −
0
Zt

dt1 dt2 HI (t1 )HI (t2 )Θ(t2 − t1 )
0
 t


 Z
0
0


HI (t1 )dt1 , ρS ⊗ ρ̄B
= ρS − iTrB


0
 t

Zt
Z

0
+TrB
dt1 dt2 HI (t1 )ρS ⊗ ρ̄B HI (t2 )


0
−
Zt
0
o
0
dt1 dt2 Θ(t1 − t2 )HI (t1 )HI (t2 )ρ0S ⊗ ρ̄B + Θ(t2 − t1 )ρ0S ⊗ ρ̄B HI (t1 )HI (t2 ) .
(2.51)
Again, we introduce the bath correlation functions with two time arguments as in Eq. (2.22)
Cαβ (t1 , t2 ) = Tr {Bα (t1 )Bβ (t2 )ρ̄B } ,
(2.52)
such that we have
(2)
ρS (t)
=
ρ0S
+
XZ
αβ 0
t
dt1
Zt
0
h
dt2 Cαβ (t1 , t2 ) Aβ (t2 )ρ0S Aα (t1 )
i
−Θ(t1 − t2 )Aα (t1 )Aβ (t2 )ρ0S − Θ(t2 − t1 )ρ0S Aα (t1 )Aβ (t2 ) .
(2.53)
2.2. DERIVATIONS FOR OPEN QUANTUM SYSTEMS
At time t, this should for weak-coupling match the evolution by a Markovian generator
CG
ρS CG (τ ) = eLτ ·τ ρ0S ≈ 1 + LCG
· τ ρ0S ,
τ
37
(2.54)
such that we can infer the action of the generator on an arbitrary density matrix
LCG
τ ρS
1X
=
τ αβ
Zτ
dt1
Zτ
0
0
h
dt2 Cαβ (t1 , t2 ) Aβ (t2 )ρS Aα (t1 )
i
−Θ(t1 − t2 )Aα (t1 )Aβ (t2 )ρS − Θ(t2 − t1 )ρS Aα (t1 )Aβ (t2 )


Zτ
Zτ
X
1
= −i 
Cαβ (t1 , t2 )sgn(t1 − t2 )Aα (t1 )Aβ (t2 ), ρS 
dt1 dt2
2iτ
αβ
0
0
1
+
τ
Zτ
dt1
0
Zτ
dt2
X
αβ
0
1
Cαβ (t1 , t2 ) Aβ (t2 )ρS Aα (t1 ) − {Aα (t1 )Aβ (t2 ), ρS } (,2.55)
2
where we have inserted Θ(x) = 12 [1 + sgn(x)].
Box 8 (CG
PMaster Equation) In the weak coupling limit, an interaction Hamiltonian of the
form HI = α Aα ⊗ Bα leads to the Lindblad-form master equation in the interaction picture


Zτ
Zτ
X
1
ρ˙S = −i 
dt1 dt2
Cαβ (t1 , t2 )sgn(t1 − t2 )Aα (t1 )Aβ (t2 ), ρS 
2iτ
αβ
0
1
+
τ
Zτ
0
dt1
0
Zτ
0
dt2
X
αβ
1
Cαβ (t1 , t2 ) Aβ (t2 )ρS Aα (t1 ) − {Aα (t1 )Aβ (t2 ), ρS } ,
2
where the bath correlation functions are given by
Cαβ (tt , t2 ) = Tr e+iHB t1 Bα e−iHB t1 e+iHB t2 Bβ e−iHB t2 ρ̄B .
(2.56)
We have not used hermiticity of the coupling operators nor that the bath correlation functions
do typically only depend on a single argument. However, if the coupling operators were chosen
hermitian, it is easy to show the Lindblad form. Obtaining the master equation requires the calculation of bath correlation functions and the evolution of the coupling operators in the interaction
picture.
Exercise 19 (Lindblad form) (1 point)
By assuming hermitian coupling operators Aα = A†α , show that the CG master equation is of
Lindblad form for all coarse-graining times τ .
38
CHAPTER 2. OBTAINING A MASTER EQUATION
Let us make once more the time-dependence of the coupling operators explicit, which is most
conveniently done in the system energy eigenbasis. Now, we also assume that the bath correlation
functions only depend on the difference of their time arguments Cαβ (t1 , t2 ) = Cαβ (t1 − t2 ), such
that we may use the Fourier transform definitions in Eq. (2.35) to obtain


τ
τ
Z
Z
X
1 X
ρ˙S = −i 
dt1 dt2
Cαβ (t1 − t2 )sgn(t1 − t2 ) |ai ha| Aα (t1 ) |ci hc| Aβ (t2 ) |bi hb| , ρS 
2iτ abc
αβ
0
+
1
τ
Zτ
0
dt1
Zτ
0
0
dt2
XX
αβ abcd
h
Cαβ (t1 − t2 ) |ai ha| Aβ (t2 ) |bi hb| ρS |di hd| Aα (t1 ) |ci hc|
i
1
− {|di hd| Aα (t1 ) |ci hc| · |ai ha| Aβ (t2 ) |bi hb| , ρS }
2
τ
Zτ
Z
X
XZ
1
dt1 dt2
σαβ (ω)e−iω(t1 −t2 ) e+i(Ea −Ec )t1 e+i(Ec −Eb )t2 ×
dω
= −i
4iπτ
αβ
abc
0
0
× hc| Aβ |bi hc| A†α
Z
Zτ
∗
|ai [|ai hb| , ρS ]
Zτ
XX
dt1 dt2
γαβ (ω)e−iω(t1 −t2 ) e+i(Ea −Eb )t2 e+i(Ed −Ec )t1 ha| Aβ |bi hc| A†α |di∗ ×
1
dω
2πτ
αβ abcd
0
0
o
1n
†
†
.
(|ci hd|) |ai hb| , ρS
× |ai hb| ρS (|ci hd|) −
2
+
(2.57)
We perform the temporal integrations by invoking
Zτ
e
iαk tk
0
dtk = τ e
iαk τ /2
hα τ i
k
sinc
2
(2.58)
with sinc(x) = sin(x)/x to obtain
Z
i
hτ
i
hτ
XX
τ
ρ˙S = −i
dω
σαβ (ω)eiτ (Ea −Eb )/2 sinc (Ea − Ec − ω) sinc (Ec − Eb + ω) ×
4iπ
2
2
abc αβ
× hc| Aβ |bi hc| A†α |ai∗ [|ai hb| , ρS ]
Z
i
hτ
i
hτ
XX
τ
+
dω
γαβ (ω)eiτ (Ea −Eb +Ed −Ec )/2 sinc (Ed − Ec − ω) sinc (ω + Ea − Eb ) ×
2π
2
2
αβ abcd
o
1n
†
†
∗
†
(|ci hd|) |ai hb| , ρS
.
(2.59)
× ha| Aβ |bi hc| Aα |di |ai hb| ρS (|ci hd|) −
2
Therefore, we have the same structure as before, but now with coarse-graining time dependent
dampening coefficients
#
"
X
τ
σab
|ai hb| , ρS
ρ˙S = −i
ab
+
X
abcd
τ
γab,cd
o
1n
|ai hb| ρS (|ci hd|) −
(|ci hd|)† |ai hb| , ρS
2
†
(2.60)
39
2.2. DERIVATIONS FOR OPEN QUANTUM SYSTEMS
with the coefficients
Z
hτ
i
hτ
i
X
1
τ
iτ (Ea −Eb )/2 τ
sinc (Ea − Ec − ω) sinc (Eb − Ec − ω) ×
σab =
dω
e
2i
2π
2
2
c
"
#
X
×
σαβ (ω) hc| Aβ |bi hc| A†α |ai∗ ,
αβ
τ
γab,cd
=
Z
hτ
i
hτ
i
τ
dωeiτ (Ea −Eb +Ed −Ec )/2 sinc (Ed − Ec − ω) sinc (Eb − Ea − ω) ×
2π
2
2
#
"
X
(2.61)
×
γαβ (ω) ha| Aβ |bi hc| A†α |di∗ .
αβ
Finally, we note that in the limit of large coarse-graining times τ → ∞, these dampening coefficients
converge to the ones in definition 7, i.e.,
τ
lim σab
τ →∞
τ
lim γab,cd
τ →∞
= σab ,
= γab,cd .
(2.62)
Exercise 20 (CG-BMS correspondence) (1 points)
Show for hermitian coupling operators that when τ → ∞, CG and BMS approximation are equivalent. You may use the identity
i
hτ
i
hτ
lim τ sinc (Ωa − ω) sinc (Ωb − ω) = 2πδΩa ,Ωb δ(Ωa − ω) .
τ →∞
2
2
Thermalization
The BMS limit has beyond its relatively compact Lindblad form further appealing properties in
the case of a bath that is in thermal equilibrium
ρ̄B =
e−βHB
Tr {e−βHB }
(2.63)
with inverse temperature β. These root in further analytic properties of the bath correlation
functions such as the Kubo-Martin-Schwinger (KMS) condition
Cαβ (τ ) = Cβα (−τ − iβ) .
Exercise 21 (KMS condition) (1 points)
Show the validity of the KMS condition for a thermal bath with ρ̄B =
(2.64)
e−βHB
.
Tr{e−βHB }
40
CHAPTER 2. OBTAINING A MASTER EQUATION
For the Fourier transform, this shift property implies
Z+∞
Z+∞
−iωτ
Cβα (−τ − iβ)e−iωτ dτ
Cαβ (τ )e
dτ =
γαβ (−ω) =
−∞
−∞
=
−∞−iβ
Z
Cβα (τ ′ )e
+iω(τ ′ +iβ)
′
Cβα (τ ′ )e+iωτ dτ ′ e−βω
(−)dτ ′ =
−∞−iβ
+∞−iβ
=
+∞−iβ
Z
Z+∞
′
Cβα (τ ′ )e+iωτ dτ ′ e−βω = γβα (+ω)e−βω ,
(2.65)
−∞
where in the last line we have used that the bath correlation functions are analytic in τ in the complex plane and vanish at infinity, such that we may safely deform the integration contour. Finally,
the KMS condition can thereby be used to prove that for a reservoir with inverse temperature β,
the density matrix
ρ¯S =
e−βHS
Tr {e−βHS }
(2.66)
is one stationary state of the BMS master equation (and the τ → ∞ limit of the CG appraoch).
Exercise 22 (Thermalization) (1 points)
−βH
Show that ρ¯S = Tr e e−βHSS is a stationary state of the BMS master equation, when γαβ (−ω) =
{
}
−βω
γβα (+ω)e .
When the reservoir is in the grand-canonical equilibrium state
ρ̄B =
e−β(HB −µNB )
,
Tr {e−β(HB −µNB ) }
(2.67)
where N = NS + NB is a conserved quantity [HS + HB + HI , NS + NB ] = 0, the KMS condition is
not fulfilled anymore. However, even in this case one can show that a stationary state of the BMS
master equation is given by
ρ¯S =
e−β(HS −µNS )
,
Tr {e−β(HS −µNS ) }
(2.68)
i.e., both temperature β and chemical potential µ equilibrate.
2.2.2
Strong-coupling regime
So far we have considered the situation where the interaction Hamiltonian was the weakest part.
One may easily generalize to a situation where the system Hamiltonian is weakest, i.e., formally
we are interested in the limit ǫ → 0 where H = HS + ǫ−1 HI + ǫ−2 HB . The derivation based on
the conventional Born, and Markov approximations is performed in complete analogy, except that
41
2.2. DERIVATIONS FOR OPEN QUANTUM SYSTEMS
the secular approximation is not necessary, since in the considered limit, all system energies are
effectively zero.
We start in the same interaction picture and perform Born- and Markov approximations, see
Eq. (2.27). Now however, the system density matrix changes much faster even than the system
coupling operators, such that we may effectively replace e±HS τ → 1
∞
ρ˙S = −i [HS , ρS ] −
XZ
αβ 0
∞
Cαβ (τ ) [Aα , Aβ ρS (t)] dτ −
XZ
Cβα (τ ) [Aα , ρS (t)Aβ ] dτ . (2.69)
αβ 0
It is straightforward (for hermitian coupling operators) to show that this expression is of Lindblad
form.
Box 9 (Singular P
Coupling Limit) In the singular coupling limit, an interaction Hamiltonian
of the form HI = α Aα ⊗ Bα with hermitian coupling operators (Aα = A†α and Bα = Bα† ) and
[HB , ρ̄B ] = 0 and Tr {Bα ρ̄B } = 0 leads to the Lindblad-form master equation
#
"
X
X 1
1
σαβ (0)Aα Aβ , ρS (t) +
γαβ (0) Aα ρS (t)Aβ − {Aβ Aα , ρS (t)} ,(2.70)
ρ˙S = −i HS +
2i
2
αβ
αβ
where the constants are given by
Z+∞
γαβ (0) =
Tr e+iHB τ Bα e−iHB τ Bβ ρ̄B dτ ,
−∞
Z+∞
Tr e+iHB τ Bα e−iHB τ Bβ ρ̄B sgn(τ )dτ .
σαβ (0) =
(2.71)
−∞
Formally, the singular coupling limit may also be obtained from the BMS limit by setting all
system energies to zero.
42
CHAPTER 2. OBTAINING A MASTER EQUATION
Chapter 3
Solving Master Equations
The wide range of applications of master equations have led to multiple solution methods, many
specialized to a specific form of the master equation. We consider the special case of Markovian
master equations here, i.e., time-local master equations with constant generators. Lindblad form
is not explicitly required, just linearity in the density matrix and constant coefficients. We can
thus always map to a matrix vector notation, i.e.,
ρ˙S = LρS ,
(3.1)
where ρS is a vector (of at most dimension d2 where d is the dimension of the system Hilbert space)
and L is a square matrix (of at most dimension d2 × d2 ). For example, when only rate equations
are considered (as in the BMS approximation for non-degenerate levels), the vector ρS may only
contain the populations and the mapping to the superoperator description becomes trivial. In
the general case, we map the density matrix to a density vector, where we have conventionally d
populations in the first place, and (at most) d(d − 1) coherences afterwards. The Liouvillian acting
on the density matrix is then mapped to a Liouville superoperator acting on the density vector by
identifying the coefficients in the linear system in an arbitrary basis, e.g.
o X
1 n †
†
ρ̇ij = −i hi| [H, ρ] |ji +
γαβ hi| Aα ρAβ |ji − hi| Aβ Aα , ρ |ji .
(3.2)
2
αβ
As an example, we consider the Liouvillian
1 + − σ σ ,ρ
L[ρ] = −i [Ωσ , ρ] + γ σ ρσ −
2
z
−
+
(3.3)
with σ ± = 21 (σ x ± iσ y ) in the eigenbasis of σ z |ei = |ei and σ z |gi = − |gi, where we have σ + |gi =
|ei and σ − |ei = |gi. From the master equation, we obtain
ρ̇ee = −γρee ,
ρ̇gg = +γρee ,
γ
γ
ρ̇eg = − − 2iΩ ρeg ,
ρ̇ge = − + 2iΩ ρge ,
2
2
such that when we arrange the
equation reads

 
ρ̇gg
 ρ̇ee  

 
 ρ̇ge  = 
ρ̇eg
(3.4)
matrix elements in a vector ρ = (ρgg , ρee , ρge , ρeg )T , the master
0 +γ
0 −γ
0 0 − γ2
0 0
0
0
+ 2iΩ
0
− γ2
43

0
ρgg


0
  ρee
  ρge
0
− 2iΩ
ρeg


,

(3.5)
44
CHAPTER 3. SOLVING MASTER EQUATIONS
such that populations and coherences evolve apparently independently. Note however, that the
Lindblad form nevertheless ensures for a positive density matrix – valid initial conditions provided.
Exercise 23 (Preservation of Positivity) (1 points)
Show that the superoperator
preserves positivity of the density matrix provided that
2 Eq.0 (3.5)
0 in
0
initial positivity (−1/4 ≤ ρge − ρgg ρee ≤ 0) is given.
The expectation value of an operator A is then mapped to a product with the trace vector.
Exercise 24 (Expectation values from superoperators) (1 points)
Show that for a Liouvillian superoperator connecting N populations (diagonal entries) with M
coherences (off-diagonal entries) acting on the density matrix ρ(t) = (P1 , . . . , PN , C1 , . . . , CM )T ,
the trace in the expectation value of an operator can be mapped to the matrix element
hA(t)i = (1, . . . , 1, 0, . . . , 0) · A · ρ(t) ,
| {z } | {z }
N×
M×
where the matrix A is the superoperator corresponding to A.
3.1
3.1.1
Analytic Techniques
Full analytic solution with the Matrix Exponential
Trivially, as we have a linear system with constant coefficients, we may obtain the solution of the
master equation via
ρ(t) = eLt ρ0 ,
(3.6)
such that one would have to calculate the matrix exponential. This is usually quite difficult and
constrained to very small dimensions of L, escpecially since the Liouville superoperator L is not
hermitian such that it is no spectral decomposition.
Exercise 25 (Single Resonant Level) (1 points)
Calculate the matrix exponential of the Liouvillian superoperator for a single resonant level tunnelcoupled to a single junction
−Γf +Γ(1 − f )
L=
+Γf −Γ(1 − f )
when the dot level is much lower than the Fermi edge (f → 1) and when it is much larger than the
Fermi edge f → 0.
45
3.1. ANALYTIC TECHNIQUES
3.1.2
Equation of Motion Technique
Instead of solving the master equation for the density matrix, it may be more favorable to derive
a related linear set of first oder differential equations for observables hBi i (t) of interest instead. In
fact, for infinitely large system Hilbert space dimensions such a procedure might even be necessary
D
E
Ḃi (t) = Tr {Bi Lρ(t)}
X
1 †
†
= −iTr {Bi [H, ρ(t)]} +
Lα Lα , ρ(t)
γα Tr Bi Lα ρ(t)Lα −
2
α
(
)
!
X 1
= Tr
+i [H, Bi ] +
γα L†α Bi Lα −
L†α Lα , Bi
ρ(t)
2
α
#
)
("
X
X
Gij Bj ρ(t) =
Gij hBj (t)i ,
(3.7)
= Tr
j
j
where in the last line we have used that there is for a finite dimensional system Hilbert space only
a finite set of linearly independent operators. In practise, one can often hope to end up with a
much smaller set of equations.
Exercise 26 (EOM for the harmonic oscillator) (1 points)
Calculate the expectation value of a + a† for a cavity in a vacuum bath i.e., for nB = 0 in Eq.
(1.34).
3.1.3
Quantum Regression Theorem
As with the Heisenberg picture for closed quantum systems, it may be favorable to keep the density
matrix as constant and to shift the complete time-dependence to the operators. From the previous
Equation we can conclude for the operators
X 1 †
†
†
Lα Lα , Bi (t)
Ḃi (t) = L Bi (t) = +i [H, Bi (t)] +
γα Lα Bi (t)Lα −
2
α
X
=
Gij Bj (t) ,
(3.8)
j
where we have introduced adjoint Liouvillian L† . For open quantum systems, it is however often
important to calculate the expectation values of operators at different times, which may be facilitated with the help of the quantum regression theorem. We find directly from properties of the
matrix exponential
X
d
Gij Bj (t + τ ) .
Bi (t + τ ) = L† Bi (t + τ ) =
dτ
j
(3.9)
Using this relation, we find the quantum regression theorem for two-point correlation functions.
46
CHAPTER 3. SOLVING MASTER EQUATIONS
Box
(Quantum Regression) Let single observables follow the closed
D E 10 P
Ḃi = j Gij hBj i. Then, the two-point correlation functions obey the equations
X
d
hBi (t + τ )Bℓ (t)i =
Gij hBj (t + τ )Bℓ (t)i
dτ
j
equation
(3.10)
with exactly the same coefficient matrix Gij .
The advantage of the Quantum regression theorem is that it enables the calculation of expressions for two-point correlation functions just from the evolution of single-operator correlation
functions.
Let us consider the example of an SET at infinite bias (fL → 1 and fR → 0). The singleoperator expectation values obey
d
dt
† −Γ
+Γ
dd
(t)
L
R
†
=
+ΓL −ΓR
d d(t)
such that the quantum regression theorem tells us
d
dτ
3.2
3.2.1
†
†
−Γ
+Γ
dd
(t
+
τ
)d
d(t)
L
R
†
=
+ΓL −ΓR
d d(t + τ )d† d(t)
† (t)
dd
,
†
d d(t)
†
†
dd
(t
+
τ
)d
d(t)
†
.
d d(t + τ )d† d(t)
(3.11)
(3.12)
Numerical Techniques
Numerical Integration
Numerical integration is generally performed by discretizing time into sufficiently small steps. Note
that there are different discretization schemes, e.g. explicit ones
ρ(t + ∆t) − ρ(t)
= Lρ(t) ,
∆t
(3.13)
where the right hand side depends on time t and implicit ones as e.g.
1
ρ(t + ∆t) − ρ(t)
= L [ρ(t) + ρ(t + ∆t)] ,
∆t
2
(3.14)
which first have to be solved (typically requiring matrix inversion) for ρ(t + ∆t) to propagate the
solution from time t to time t + ∆t. As a rule of thumb, explicit schemes are easy to implement but
may be numerically unstable (i.e., an adaptive stepsize may be required to bound the numerical
error). In contrast, implicit schemes are usually more stable but require a lot of effort to propagate
the solution. Here, we will just discuss a fourth-order Runge-Kutta solver.
In order to propagate a density matrix ρn at time t, to the density matrix ρn+1 at time t + ∆t,
47
3.2. NUMERICAL TECHNIQUES
the fourth-order Runge-Kutta scheme proceeds requires the evaluation of some intermediate values
σn,1 = ∆tLρn ,
σn,2 = ∆tL ρn +
σn,3 = ∆tL ρn +
σn,4
ρn+1
1
σn,1 ,
2
1
σn,2 ,
2
= ∆tL (ρn + σn,3 ) ,
1
1
1
1
= ρn + σn,1 + σn,2 + σn,3 + σn,4 + O{∆t5 } .
6
3
3
6
(3.15)
This explicit scheme requires four matrix-vector multiplications per timestep. It should always be
used with an adaptive stepsize, which can be controlled by comparing (e.g. by computing the norm
of the difference) the result from two successive propagations with stepsize ∆t with the result of a
single propagation with stepsize 2∆t. If the difference between solutions is too large, the stepsize
must be reduced (and the intermediate result should be discarded). If it is not too large, the
result can always be accepted. In case the error estimate is too small, the stepsize can later-on be
cautiously increased.
Exercise 27 (Order of the RK scheme) (1 points)
Acting with the Liouville superoperator performs the time-derivative of the density matrix. Show
that the presented scheme (3.15) is of fourth order in ∆t, i.e., that
2
3
4
2 ∆t
3 ∆t
4 ∆t
ρn+1 = 1 + L∆t + L
ρn .
+L
+L
2!
3!
4!
3.2.2
Simulation as a Piecewise Deterministic Process (PDP)
Suppose we would like to solve the Lindblad form master equation (in diagonal representation)
X 1
†
†
ρ̇ = −i [H, ρ] +
γα Lα ρLα −
(3.16)
ρ, Lα , Lα
2
α
numerically but do not have the possibility to store the N 2 matrix elements of the density matrix.
Instead, the master equation can be unravelled to a piecewise deterministic process for a pure
quantum state. The advantage here lies in the fact that a pure state requires only N complex
observables to be evolved.
Consider the nonlinear but deterministic equation
"
#
#
"
X
X
1
i
˙ = −i H −
γα hΨ| L†α Lα |Ψi |Ψi .
(3.17)
γα L†α Lα |Ψi +
|Ψi
2 α
2 α
Although this is nonlinear in |Ψ(t)i, one can show that the solution is given by
|Ψi =
e−iM t |Ψ0 i
hΨ0 | e+iM † t e−iM t |Ψ0 i
1/2
,
(3.18)
48
CHAPTER 3. SOLVING MASTER EQUATIONS
where we have used the operator M = H −
non-hermitian Hamiltonian.
i
2
P
α
γα L†α Lα , which is also often interpreted as a
Exercise 28 (Norm for continuous evolution) (1 points)
Calculate the norm of the state vector hΨ(t)|Ψ(t)i from Eq. (3.18).
We show the validity of the solution by differentiation
E
h
i
1
e−iM t |Ψ0 i
+iM † t
† −iM t
+iM † t
−iM t
i
hΨ
|
e
M
e
|Ψ
i
−
hΨ
|
e
M
e
|Ψ
i
Ψ̇ = −iM |Ψi −
0
0
0
0
2 hΨ0 | e+iM † t e−iM t |Ψ0 i3/2
†
−iM t
1
e−iM t |Ψ0 i
+iM † t
i
hΨ
|
e
M
−
M
e
|Ψ0 i
0
3/2
2 hΨ0 | e+iM † t e−iM t |Ψ0 i
X
1
e−iM t |Ψ0 i
+iM † t †
= −iM |Ψi +
γ
hΨ
|
e
Lα Lα e−iM t |Ψ0 i
α
0
3/2
2 hΨ0 | e+iM † t e−iM t |Ψ0 i
α
X
1
= −iM |Ψi + |Ψi
γα hΨ| L†α Lα |Ψi .
2
α
= −iM |Ψi −
(3.19)
However, the name PDP already suggests that the process is only piecewise deterministic.
Hence, it will be interruppted by stochastic events. The total probability that a jump of the wave
function will occur in the infinitesimal interval [t, t + ∆t] is given by given by
Pjump = ∆t
X
α
γα hΨ| L†α Lα |Ψi .
(3.20)
That is, if a jump has occurred, one still has to decide which jump. Choosing a particular jump
Lα |Ψi
|Ψi → q
hΨ| L†α Lα |Ψi
(3.21)
is performed randomly with conditional probability
γα hΨ| L†α Lα |Ψi
,
Pα = P
†
α γα hΨ| Lα Lα |Ψi
(3.22)
where the normalization is obvious. This recipe for deterministic (continuous) and jump evolutions
may also be written as a single stochastic differential equation, which is often called stochastic
Schrödinger equation.
Box 11 (Stochastic Schrödinger Equation) A Lindblad type master equation of the form
X 1
†
†
ρ̇ = −i [H, ρ] +
(3.23)
ρ, Lα , Lα
γα Lα ρLα −
2
α
49
3.2. NUMERICAL TECHNIQUES
can be effectively modeled by the stochastic differential equation
#
"
X
1
1X
γα L†α Lα +
γα hΨ| L†α Lα |Ψi |Ψi dt
|dΨi = −iH −
2 α
2 α


X
 q Lα |Ψi
+
− |Ψi dNα ,
†
α
hΨ| Lα Lα |Ψi
(3.24)
where the Poisson increments dNα satisfy
dNα dNβ = δαβ dNα ,
E (dNα ) = γα hΨ| L†α Lα |Ψi dt
(3.25)
and E (x) denotes the classical expectation value (ensemble average).
The last two equations simply mean that at most a single jump can occur at once (practically we have dNα ∈ {0, 1}) and that the probability for a jump at time t is given by Pα =
γα hΨ| L†α Lα |Ψi dt. Numerically, it constitutes a simple recipe, see Fig. 3.1.
jump
no jump
which jump
Figure 3.1: Recipe for propagating the stochastic Schrödinger equation in Box 11. At each timestep
one can calculate the total probability of a jump occuring during the next timestep. Drawing a
random number yields whether a jump should occur or not. Given that a jump occurs, we determine
which jump by drawing another random number: Setting the particular dNα = 1 and dt = 0 we
solve for |Ψ(t + ∆t)i = |Ψ(t)i + |dΨ(t)i and proceed with the next timestep. Given that no jump
occurs, we simply set dNα = 0 for all α, solve for |Ψ(t + ∆t)i = |Ψ(t)i + |dΨ(t)i and proceed with
the next timestep.
It remains to be shown that this PDP is actually an unravelling of the master equation, i.e.,
that the covariance matrix
ρ = E (π̂) = E (|Ψi hΨ|)
(3.26)
50
CHAPTER 3. SOLVING MASTER EQUATIONS
fulfils the Lindblad type master equation. To do this we first note that hΨ| L†α Lα |Ψi = Tr L†α Lα π̂ .
The change of the covariance matrix is given by
dπ̂ = |dΨi hΨ| + |Ψi hdΨ| + |dΨi hdΨ| .
(3.27)
Note that the last term cannot be neglected completely since the term E (dNα dNβ ) does not vanish.
Making everything explicit we obtain
(
)
X
†
1X †
dπ̂ = +dt −i [H, π̂] −
γα Lα Lα , π̂ +
γα π̂Tr Lα Lα π̂
2 α
α


X
Lα π̂L†α
o − π̂  + O{dt2 , dtdNα } .
+
dNα  n
(3.28)
†
Tr
L
L
π̂
α α
α
We now use the general relation
E (dNα g(π̂)) = γα dtE Tr L†α Lα π̂ g(π̂)
(3.29)
for arbitrary functions g(π̂) of the projector. This relation can be understood by binning all K
values of the actual state π̂ (k) (t) in the expectation value into L equal-sized compartments where
π̂ (k) ≈ π̂ (ℓ) . In each compartment, we have Nℓ realizations of dNαℓm with 1 ≤ m ≤ Nℓ and
P
ℓ Nℓ = K, of which we can compute the average first
1 X
dNα(k) (t)g(π̂ (k) (t))
E (dNα g(π̂)) = lim
K→∞ K
k
P
P
(ℓm)
Nℓ N1ℓ
dNα g(π̂ (ℓ) (t))
m
P
=
lim ℓ
L,Nℓ →∞
Nℓ
ℓ
P
Nℓ γα dtTr L†α Lα π̂ (ℓ) g(π̂ (ℓ) (t))
P
=
lim ℓ
L,Nℓ →∞
Nℓ
ℓ
P
Nℓ γα dtE Tr L†α Lα π̂ (ℓ) g(π̂ (ℓ) (t))
P
=
lim ℓ
L,Nℓ →∞
Nℓ
ℓ
= γα dtE Tr L†α Lα π̂ (ℓ) g(π̂ (ℓ) (t)) ,
(3.30)
P
N x̄
Pi i i
i Ni
when x̄i represent averages of disjoint subsets of
where we have used the relation that x̄ =
the complete set. Specifically, we apply it on the expressions




π̂
π̂
o  = γα dtE Tr L†α Lα π̂
n
o  = γα dtρ ,
E dNα n
†
†
Tr Lα Lα π̂
Tr Lα Lα π̂
†
E (dNα π̂) = γα dtE Tr Lα Lα π̂ π̂
(3.31)
This implies
(
dρ = dt −i [H, ρ] +
X
α
γα
1 †
Lα ρL†α −
Lα Lα , ρ
2
)
,
(3.32)
51
3.2. NUMERICAL TECHNIQUES
i.e., the average of trajectories from the stochastic Schrödinger equation yields the same solution
as the master equation.
This may be of great numerical use: Simulating the full master equation for an N -dimensional
system Hilbert space may involve the storage of O{N 4 } real variables in the Liouvillian, whereas
for the generator of the stochastic Schrödinger equation one requires only O{N 2 } real variables.
This is of course weakened since in order to get a realistic estimate on the expectation value, one
has to compute K different trajectories, but since typically K ≪ N 2 , the stochastic Schrödinger
equation is a useful tool in the numeric modelling of a master equation.
As an example, we study the cavity in a thermal bath. We had the Lindblad type master
equation describing the interaction of a cavity mode with a thermal bath
†
1 †
1
†
†
ρ˙S = −i Ωa a, ρS + γ(1 + nB ) aρS a − a aρS − ρS a a
2
2
1
1
(3.33)
+γnB a† ρS a − aa† ρS − ρS aa† .
2
2
We can immediately identify the jump operators
L1 = a
and
L 2 = a†
(3.34)
and the corresponding rates
γ1 = γ(1 + nB )
and
γ2 = γnB .
(3.35)
From the master equation, we obtain for the occupation number n = a† a the evolution equation
d
n = −γn + γnB , which is solved by
dt
(3.36)
n(t) = n0 e−γt + nB 1 − e−γt .
The corresponding stochastic differential equation reads
1
1
†
†
†
|dΨi =
−iΩa a − γ(1 + 2nB )a a + γnB + γ(1 + 2nB ) hΨ| a a |Ψi + γnB |Ψi dt
2
2
!
!
a† |Ψi
a |Ψi
− |Ψi dN1 + p
− |Ψi dN2 ,
(3.37)
+ p
hΨ| a† a |Ψi
hΨ| aa† |Ψi
which becomes particularly simple when we do not consider superpositions of Fock basis states
from the beginning. E.g., for |Ψi = |ni we obtain
1
1
|dni =
−iΩn − [γ(1 + 2nB )n + γnB ] + [γ(1 + 2nB )n + γnB ] |ni dt
2
2
+ (|n − 1i − |ni) dN1 + (|n + 1i − |ni) dN2
= −iΩndt |ni + (|n − 1i − |ni) dN1 + (|n + 1i − |ni) dN2
(3.38)
such that superpositions are never created during the evolution. The total probability to have a
jump in the system during the interval dt is given by
Pjump = γdt (1 + nB ) hn| a† a |ni + nB hn| aa† |ni = γ [(1 + nB )n + nB (n + 1)] dt .
(3.39)
52
CHAPTER 3. SOLVING MASTER EQUATIONS
If no jump occurs, the system evolves only oscillatory, which has no effect on the expectation value
of a† a. However, if a jump occurs, the respective conditional probability to jump out of the system
reads
P1 =
(nB + 1)n
(nB + 1)n + nB (n + 1)
(3.40)
and to jump into the system consequently (these must add up to one)
P2 =
nB (n + 1)
.
(nB + 1)n + nB (n + 1)
(3.41)
Computing trajectories with a suitable random number generator and averaging the trajectories,
we find convergence to the master equation result as expected, see Fig. 3.2.
expectation value <n(t)>
10
master equation solution
average of 10 trajectories
average of 100 trajectories
average of 1000 trajectories
8
6
4
2
0
0
2
4
time γ t
6
8
10
Figure 3.2: Single trajectories of the stochastic Schrödinger equation (curves with integer jumps).
The averages of 10, 100, and 1000 trajectories converge to the prediction from the associated
master equation. Parameters: γdt = 0.01, nB = 1.5.
Chapter 4
Multi-Terminal Coupling :
Non-Equilibrium Case I
The most obvious way to achieve non-equilibrium dynamics is to use reservoir states that are nonthermalized, i.e., states that cannot simply be characterized by just temperature and chemical
potential. Since the derivation of the master equation only requires [ρ̄B , HB ] = 0, this would still
allow for many nontrivial models, hn| ρ̄B |ni could e.g. follow multi-modal distributions. Alternatively, a non-equilibrium situation may be established when a system is coupled to different thermal
equilibrium baths or of course when the system itself is externally driven – either unconditionally
(open-loop feedback) or conditioned on the actual state of the system (closed-loop feedback).
First, we will consider the case of multiple reservoirs at different thermal equilibria that are only
indirectly coupled via the system: Without the system, they would be completely independent.
Since these are chosen at different equilibria, they drag the system towards different thermal states,
and the resulting stationary state is in general a non-thermal one. Since the different compartments
interact only indirectly via the system, we have the case of a multi-terminal system, where one can
most easily derive the corresponding master equation, since each contact may be treated separately.
Therefore, we do now consider multiple (K) reservoirs held at different chemical potentials and
different temperatures
(1)
(1)
(K)
(K)
e−β(HB −µNB )
e−β(HB −µNB )
o
o.
n
n
ρ̄B =
⊗
.
.
.
⊗
(1)
(1)
(K)
(K)
−β(HB −µNB )
−β(HB −µNB )
Tr e
Tr e
(4.1)
To each of the reservoirs, the system is coupled via different coupling operators
HI =
X
α
k
X
Aα ⊗
Bα(ℓ) .
(4.2)
ℓ=1
Since we assume that the first order bath correlation functions vanish Bαℓ ρ̄B = 0, the tensor
product implies that the second-order bath correlation functions may be computed additively
Cαβ (τ ) =
K
X
(ℓ)
Cαβ (τ ) ,
(4.3)
ℓ=1
which transfers to their Fourier transforms and thus, also to the final Liouvillian (to second order
53
54
CHAPTER 4. MULTI-TERMINAL COUPLING : NON-EQUILIBRIUM CASE I
in the coupling)
L=
K
X
ℓ=1
L(ℓ) .
(4.4)
The resulting stationary state is in general a non-equilibrium one. Let us however first identify a
trivial case where the coupling structure of all Liouvillians is identical
h
i
L(ℓ) = Γ(ℓ) L0 + β (ℓ) L1 ,
(4.5)
where β (ℓ) is a parameter encoding the thermal properties of the respective bath (e.g. a Fermifunction evaluated at one of the systems transition frequencies) and L0/1 are superoperators independent of the respective bath and Γ(ℓ) represent different coupling constants. For coupling to a
single reservoir, the stationary state is defined via the equation
h
i
L(ℓ) ρ̄(ℓ) = Γ(ℓ) L0 + β (ℓ) L1 ρ̄(ℓ) = 0
(4.6)
and thus implicitly depends on the thermal parameter ρ̄(ℓ) = ρ̄(β (ℓ) ). For the total Liouvillian, it
follows that the dependence of the full stationary state on all thermal parameters simply given by
the same dependence on an average thermal parameter
"
#"
P (ℓ) (ℓ) #
h
i
X
X
X
(ℓ)
(ℓ)
(ℓ)
(ℓ)
ℓΓ β
Γ
L0 + β L1 ρ̄ =
Γ
L ρ̄ =
L0 + P
L1 ρ̄ ,
(4.7)
(ℓ′ )
ℓ′ Γ
ℓ
ℓ
ℓ
P
(ℓ)
(ℓ) β
ℓΓ
P
such that the dependence is simply ρ̄ = ρ̄
Γ(ℓ)
ℓ
This can be illustrated by upgrading the Liouvillian for a single resonant level coupled to a
single junction
−Γf +Γ(1 − f )
,
(4.8)
L=
+Γf −Γ(1 − f )
−1
where the Fermi function f = eβ(ǫ−µ) + 1
of the contact is evaluated at the dot level ǫ, to the
Liouvillian for a single-electron transistor (SET) coupled to two (left and right) junctions
−ΓL fL − ΓR fR +ΓL (1 − fL ) + ΓR (1 − fR )
.
(4.9)
L=
+ΓL fL + ΓR fR −ΓL (1 − fL ) − ΓR (1 − fR )
Now, the system is coupled to two fermionic reservoirs, and in order to support a current, the dot
level ǫ must be within the transport window, see Fig. 4.1. This also explains the name singleelectron transistor, since the dot level ǫ may be tuned by a third gate, which thereby controls the
current.
Exercise 29 (Pseudo-Nonequilibrium) (1 points)
Show that the stationary state of Eq.(4.9) is a thermal one, i.e., that
ρ̄11
f¯
.
=
ρ̄00
1 − f¯
Determine f¯ in dependence of Γα and fα .
4.1. CONDITIONAL MASTER EQUATION FROM CONSERVATION LAWS
55
Figure 4.1: Sketch of a single resonant level (QD at energy level ǫ) coupled to two junctions with
different Fermi distributions (e.g. with different chemical potentials or different temperatures. If
the dot level ǫ is changed with a third gate, the device functions as a transistor, since the current
through the system is exponentially suppressed when the the dot level ǫ is not within the transport
window.
4.1
Conditional Master equation from conservation laws
Suppose we have derived a QME for an open system, where the combined system obeys the
conservation of some conserved quantity (e.g., the total number of particles). Then, it can be
directly concluded that a change in the system particle number by e.g. minus one must be directly
accompagnied by the corresponding change of the reservoir particle number by plus one. For
couplings to multiple reservoirs these terme can also be uniquely identified, since the Liouvillians
are additive. Whereas its actual density matrix says little about the number of quanta that have
already passed the quantum system, a full trajectory would reveal this information. In such cases,
it is reasonable to discretize the master equation in time, where it can most easily be upgraded to
an n-resolved (conditional) master equation. In what follows, we just track the particle number in
a single attached reservoir and denote by ρ(n) (t) the system density matrix under the condition that
n net particles have left into the monitored reservoirs, but the method may easily be generalized.
Assuming that at time t, we have n particles transferred to the reservoir, we may discretize the
conventional master equation ρ̇ = Lρ in time and identify terms that increase or decrease the
particle number (often called jumpers)
˙ = L ρ(n) + L+ ρ(n−1) + L− ρ(n+1) .
ρ(n)
0
(4.10)
Since the total number of transferred particles is normally not constrained, this immediately yields
an infinite set of equations for the system density matrix. Exploiting the translational invariance
suggests to use the Fourier transform
ρ(χ, t) =
X
ρ(n) (t)e+inχ ,
(4.11)
n
which again reduces the dimension of the master equation
ρ̇(χ, t) = L0 + e+iχ L+ + e−iχ L− ρ(χ, t) = L(χ)ρ(χ, t) .
(4.12)
As an example, we convert the SET master equation (4.9) into an n-resolved version, where n
56
CHAPTER 4. MULTI-TERMINAL COUPLING : NON-EQUILIBRIUM CASE I
should denote the number of particles that have tunneled into the right reservoir
−ΓL fL − ΓR fR
+ΓL (1 − fL )
(n)
ρ(n)
ρ̇
=
+ΓL fL
−ΓL (1 − fL ) − ΓR (1 − fR )
0
0
0 ΓR (1 − fR )
(n−1)
ρ(n+1) .
ρ
+
+
+ΓR fR 0
0
0
(4.13)
Performing the Fourier transformation reduces the dimension (exploiting the shift invariance) at
the price of introducing the counting field
−ΓL fL − ΓR fR
+ΓL (1 − fL ) + ΓR (1 − fR )e+iχ
ρ(χ, t) .
(4.14)
ρ̇(χ, t) =
+ΓL fL + ΓR fR e−iχ
−ΓL (1 − fL ) − ΓR (1 − fR )
Formally, this just corresponds to the replacement (1 − fR ) → (1 − fR )e+iχ and fR → fR e−iχ in
the off-diagonal matrix elements of the Liouvillian (which correspond to particle jumps).
4.2
Microscopic derivation with a detector model
Unfortunately, we do not always have a conserved quantity that may only change when tunneling
across the system-reservoir junction takes place. A counter-example may be the tunneling between
two reservoirs, that is merely modified by the presence of the quantum system (e.g. quantum point
contact monitoring a charge qubit) and is not connected with a particle change in the system. In
such cases, we may not identify a change in the system state with a microcanonical change of the
reservoir state. However, such problems can still be handled with a quantum master equation by
introducing a virtual detector at the level of the interaction Hamiltonian. Suppose that in the
interaction Hamiltonian we can identify terms associated with a change of the tracked obervable
in the reservoir
X
HI = A+ ⊗ B+ + A− ⊗ B− +
Aα ⊗ Bα ,
(4.15)
α6={+,−}
where B+ increases and B− decreases the reservoir particle number. We extend the system Hilbert
space by adding a virtual detector
HS → HS ⊗ 1 ,
HB → HB
HI → + A+ ⊗ D† ⊗ B+ + [A− ⊗ D] ⊗ B− +
X
α6={+,−}
[Aα ⊗ 1] ⊗ Bα ,
(4.16)
P
P
where D = n |ni hn + 1| and D† = n |n + 1i hn|. Considering the detector as part of the system
and decomposing the system density matrix as
X
ρ(t) =
ρ(n) (t) ⊗ |ni hn| ,
(4.17)
n
we can use the method of our choice to obtain the conditional master equation by virtue of
DD† = D† D = 1 ,
hn| DA− ρA+ D† |ni = A− ρ(n+1) A+ ,
hn| D† A+ ρA− D |ni = A+ ρ(n−1) A− .
(4.18)
57
4.3. FULL COUNTING STATISTICS
4.3
Full Counting Statistics
To obtain the probability of having n tunneled particles into a respective reservoir, we have to sum
over the different system configurations of the corresponding conditional density matrix
Pn (t) = Tr ρ(n) (t) .
(4.19)
In view of the identities
X
Pn (t)e+inχ ≡ M(χ, t) ,
Tr {ρ(χ, t)} = Tr eL(χ)t ρ0 =
(4.20)
n
k
we see that in order to evaluate moments n , we simply have to take derivatives with respect to
the counting field
X
k X k
n (t) =
n Pn (t) = (−i∂χ )k
Pn (t)e+inχ = (−i∂χ )k M(χ, t)|χ=0 .
(4.21)
n
n
χ=0
Based on a conditional master equation, this enables a very convenient calculation of the stationary
current. Taking the stationary density matrix L(0)ρ̄ = 0 as initial condition (the stationary current
is independent on the initial occupation of the density matrix), we obtain for the current as the
time derivative of the first moment
I = hṅ(t)i
d
= −i∂χ Tr eL(χ)t ρ̄ χ=0
dt = −i∂χ Tr L(χ)eL(χ)t ρ̄ χ=0
n
= −iTr L′ (0)ρ̄ + [L′ (0)L(0) + L(0)L′ (0)] tρ̄
o
t2
ρ̄ + . . .
+ L (0)L (0) + L(0)L (0)L(0) + L (0)L (0)
2
= −iTr {L′ (0)ρ̄} ,
′
2
′
2
′
(4.22)
where we have used L(0)ρ̄ = 0 and also Tr {L(0)σ} = 0 for all operators σ (trace conservation).
Exercise 30 (Stationary SET current) (1 points)
Calculate the stationary current through the SET (4.14).
Sometimes a description in terms of cumulants is more convenient. The cumulant-generating
function is defined as the logarithm of the moment-generating function
C(χ, t) = ln M(χ, t) ,
such that all cumulants may be obtained via simple differentiation
k = (−i∂χ )k C(χ, t)|χ=0 .
n
(4.23)
(4.24)
58
CHAPTER 4. MULTI-TERMINAL COUPLING : NON-EQUILIBRIUM CASE I
Cumulants and Moments are of course related, we just summarize relations for the lowest few
cumulants
2 2 hhnii = hni ,
n
= n − hni2 ,
3 n
= n3 − 3 hni n2 + 2 hni3 ,
4 2
(4.25)
n
= n4 − 4 hni n3 − 3 n2 + 12 hni2 n2 − 6 hni4 .
The advantage of this description lies in the fact that the long-term evolution of the cumulantgenerating function is usually given by the dominant eigenvalue of the Liouvillian
C(χ, t) ≈ λ(χ)t ,
(4.26)
where λ(χ) is the (uniqueness assumed) eigenvalue of the Liouvillian that vanishes at zero counting
field λ(0) = 0. For this reason, the dominant eigenvalue is also interpreted as the cumulantgenerating function of the stationary current.
Exercise 31 (current CGF for the SET) (1 points)
Calculate the dominant eigenvalue of the SET Liouvillian (4.14) in the infinite bias limit fL → 1
and fR → 0. What is the value of the current?
We show this by using the decomposition of the Liouvillian in Jordan Block form
L(χ) = Q(χ)LJ (χ)Q−1 (χ) ,
(4.27)
where Q(χ) is a (non-unitary) similarity matrix and LJ (χ) contains the eigenvalues of the Liouvillian on its diagonal – distributed in blocks with a size corresponding to the eigenvalue multiplicity.
We assume that there exists one stationary state ρ̄, i.e., one eigenvalue λ(χ) with λ(0) = 0 and that
all other eigenvalues have a larger negative real part near χ = 0. Then, we use this decomposition
in the matrix exponential to estimate its long-term evolution
n
o
−1
M(χ, t) = Tr eL(χ)t ρ0 = Tr eQ(χ)LJ (χ)Q (χ)t ρ0 = Tr Q(χ)eLJ (χ)t Q−1 (χ)ρ0




λ(χ)·t


e








0

 −1
→ Tr Q(χ) 
Q
(χ)ρ

0
...










0






1








0
 −1

λ(χ)·t
Q
(χ)ρ
= eλ(χ)t c(χ)
(4.28)
= e
Tr Q(χ) 

0
...










0
with some polynomial c(χ) depending on the matrix Q(χ). This implies that the cumulantgenerating function
C(χ, t) = ln M(χ, t) = λ(χ)t + ln c(χ) ≈ λ(χ)t
becomes linear in λ(χ) for large times.
(4.29)
59
4.4. FLUCTUATION THEOREMS
4.4
Fluctuation Theorems
The probability distribution Pn (t) is given by the inverse Fourier transform of the momentgenerating function
1
Pn (t) =
2π
Z+π
Z+π
1
M(χ, t)e−inχ dχ =
eC(χ,t)−inχ dχ .
2π
−π
(4.30)
−π
Accordingly, a symmetry in the cumulant-generating function (or moment-generating function) of
the form
C(−χ, t) = C(+χ + i ln(α), t)
leads to a symmetry of the probabilities
R +π C(χ,t)−inχ
R +π C(χ,t)−inχ
1
e
dχ
e
dχ
P+n (t)
2π −π
−π
= 1 R +π
= R +π
P−n (t)
eC(−χ,t)−inχ dχ
eC(χ,t)+inχ dχ
2π −π
−π
R +π C(χ,t)−inχ
R +π C(χ,t)−inχ
e
dχ
e
dχ
−π
−π
= R +π
= R +π+i ln(α)
eC(χ+i ln(α),t)−inχ dχ
eC(χ,t)−in[χ−i ln(α)] dχ
−π
−π+i ln(α)
= e+n ln(α) ,
(4.31)
(4.32)
where we have used in the last step that C(−π + i ln(α), t) = C(+π + i ln(α), t), such that we
can add two further integration paths from −π to −π + i ln(α) and from +π + i ln(α) to +π.
The value of the cumulant-generating function along these paths is the same, such that due to
the different integral orientation there is not net change. Note that the system may be very far
from thermodynamic equilibrium but still obey a symmetry of the form (4.31), which leads to a
fluctuation theorem of the form (4.32) being valid far from equilibrium.
As example, we consider the SET (which is always in thermal equilibrium). The characteristic
polynomial D(χ) = |L(χ) − λ1| of the Liouvillian (4.14) and therefore also all eigenvalues obeys
the symmetry
fL (1 − fR )
D(−χ) = D +χ + i ln
= D (χ + i [(βR − βL ) ǫ + βL µL − βR µR ]) ,
(4.33)
(1 − fL )fR
which leads to the fluctuation theorem
P+n (t)
= en[(βR −βL )ǫ+βL µL −βR µR ] .
t→∞ P−n (t)
lim
(4.34)
For equal temperatures, this becomes
P+n (t)
= enβV ,
t→∞ P−n (t)
lim
(4.35)
which directly demonstrates that the current
+∞
∞
∞
X
X
d X
d
nPn (t) =
n [P+n (t) − P−n (t)] =
hn(t)i =
nPn (t) 1 − e−nα
I=
dt
dt n=−∞
n=1
n=1
always follows the voltage.
(4.36)
60
4.5
CHAPTER 4. MULTI-TERMINAL COUPLING : NON-EQUILIBRIUM CASE I
The double quantum dot
We consider a double quantum dot with internal tunnel coupling T and Coulomb interaction U
that is weakly coupled to two fermionic contacts via the rates ΓL and ΓR , see Fig. 4.2. The
Figure 4.2: A double quantum dot (system) with on-site energies ǫA/B and internal tunneling
amplitude T and Coulomb interaction U may host at most two electrons. It is weakly tunnelcoupled to two fermionic contacts via the rates ΓL/R at different thermal equilibria described by
the Fermi distributions fL/R (ω).
corresponding Hamiltonian reads
HS =
HB
HI
ǫA d†A dA
+
ǫB d†B dB
dA d†B
+T
+
X
X
=
ǫkL c†kL ckL +
ǫkR c†kR ckR ,
k
dB d†A
+ U d†A dA d†B dB ,
k
X
X
=
tkL dA c†kL + t∗kL ckL d†A +
tkR dB c†kR + t∗kR ckR d†B .
k
(4.37)
k
Note that we do not have a tensor product decomposition in the interaction Hamiltonian, as the
coupling operators anti-commute, e.g.,
{d, ckR } = 0 .
(4.38)
We may however use the Jordan Wigner transform, which decomposes the Fermionic operators in
terms of Pauli matrices acting on different spins
dA = σ − ⊗ 1 ⊗ 1 ⊗ . . . ⊗ 1 ,
dB = σ z ⊗ σ − ⊗ 1 ⊗ . . . ⊗ 1 ,
z
ckL = σ z ⊗ σ z ⊗ σ
. . ⊗ σ}z ⊗σ − ⊗ 1 ⊗ . . . ⊗ 1 ,
| ⊗ .{z
k−1
z
z
z
z
ckR = σ ⊗ σ ⊗ σ
. . ⊗ σ}z ⊗ σ
. . ⊗ σ}z ⊗σ − ⊗ 1 ⊗ . . . ⊗ 1
| ⊗ .{z
| ⊗ .{z
KL
(4.39)
k−1
to map to a tensor-product decomposition of the interaction Hamiltonian, where σ ± = 21 [σ x ± iσ y ].
†
The remaining operators follow from (σ + ) = σ − and vice versa. This
automat decomposition
†
ically obeys the fermionic anti-commutation relations such as e.g. ck , d = 0 and may therefore also be used to create a fermionic operator basis with computer algebra programs (e.g. use
KroneckerProduct in Mathematica).
61
4.5. THE DOUBLE QUANTUM DOT
Exercise 32 (Jordan-Wigner transform) (1 points)
Show that for fermions distributed on N sites, the decomposition
z
. . ⊗ 1}
ci = σ
. . ⊗ σ}z ⊗σ − ⊗ |1 ⊗ .{z
| ⊗ .{z
i−1
N −i
preserves the fermionic anti-commutation relations
o
n
† †
{ci , cj } = 0 = ci , cj ,
n
ci , c†j
o
= δij 1 .
Show also that the fermionic Fock space basis c†i ci |n1 , . . . , nN i
σiz |n1 , . . . , nN i = (−1)ni +1 |n1 , . . . , nN i.
=
ni |n1 , . . . , nN i obeys
Inserting the decomposition (4.39) in the Hamiltonian, we may simply use the relations
(σ x )2 = (σ y )2 = (σ z )2 = 1 ,
σ z σ − = −σ − ,
1
[1 + σ z ] ,
2
σ z σ + = +σ + ,
1
[1 − σ z ] ,
2
σ + σ z = −σ +
σ+σ− =
σ − σ z = +σ − ,
σ−σ+ =
(4.40)
to obtain a system of interacting spins
1
1
1
1
HS = ǫA [1 + σAz ] + ǫB [1 + σBz ] + T σA− σB+ + σA+ σB− + U [1 + σAz ] [1 + σBz ]
2
2
2
2
X
X
1
1
z
z
HB =
ǫkL [1 + σkL
]+
ǫkR [1 + σkR
]
2
2
k
k
"
#
"
#
Y
X
X
Y
+
−
σkz′ L σkL
HI = σA− σBz ⊗
tkL
+ σA+ σBz ⊗
t∗kL
σkz′ L σkL
k′ <k
k
+σB− ⊗
.
X
k
tkR
"
Y
k′
k
σkz′ L
#"
Y
k′′ <k
#
+
+ σB+ ⊗
σkz′′ R σkR
k′ <k
X
k
t∗kR
"
Y
k′
σkz′ L
#"
Y
k′′ <k
#
−
σkz′′ R σkR
(4.41)
With this, we could proceed by simply viewing the Hamiltonian as a complicated total system
of non-locally interacting spins. However, the order of operators in the nonlocal Jordan-Wigner
transformation may be chosen as convenient without destroying the fermionic anticommutation
relations. We may therefore also define new fermionic operators on the subspace of the system (first
two sites, with reversed order) and the baths (all remaining sites with original order), respectively
d˜A = σ − ⊗ σ z ,
d˜B = 1 ⊗ σ − ,
c̃kL = |σ z ⊗ .{z
. . ⊗ σ}z ⊗σ − ⊗ 1 ⊗ . . . ⊗ 1 ,
k−1
z
z
c̃kR = |σ ⊗ .{z
. . ⊗ σ}z ⊗ σ
. . ⊗ σ}z ⊗σ − ⊗ 1 ⊗ . . . ⊗ 1 .
| ⊗ .{z
KL
(4.42)
k−1
These new operators obey fermionic anti-commutation relations in system and bath separately
(e.g. {d˜A , d˜B } = 0 and {c̃kL , c̃k′ L } = 0), but act on different Hilbert spaces, such that system and
62
CHAPTER 4. MULTI-TERMINAL COUPLING : NON-EQUILIBRIUM CASE I
bath operators do commute by construction (e.g. [d˜A , c̃kL ] = 0). In the new operator basis, the
Hamiltonian appears as
h
i
† ˜ ˜† ˜
† ˜
† ˜
†
†
˜
˜
˜
˜
˜
˜
˜
H S = ǫA d A d A + ǫB d B d B + T d A d B + d B d A + U d A d A d B d B ⊗ 1 ,
#
"
X
X
†
†
ǫkL c̃kL c̃kL +
ǫkR c̃kR c̃kR ,
HB = 1 ⊗
k
HI = d˜A ⊗
k
X
tkL c̃†kL
+
d˜†A
k
⊗
X
k
t∗kL c̃kL + d˜B ⊗
X
k
tkR c̃†kR + d˜†B ⊗
X
t∗kR c̃kR ,
(4.43)
k
which is the same (for this and some more special cases) as if we had ignored the fermionic nature
of the annihilation operators from the beginning. We do now proceed by calculating the Fourier
transforms of the bath correlation functions
γ12 (ω) = ΓL (−ω)fL (−ω) ,
γ34 (ω) = ΓR (−ω)fR (−ω) ,
with
theEcontinuum tunneling rates Γα (ω) = 2π
D
c†kα ckα .
γ21 (ω) = ΓL (+ω)[1 − fL (+ω)] ,
γ43 (ω) = ΓR (+ω)[1 − fR (+ω)]
P
k
(4.44)
|tkα |2 δ(ω − ǫkα ) and Fermi functions fα (ǫkα ) =
Exercise 33 (DQD bath correlation functions) (1 points)
Calculate the Fourier transforms (4.44) of the bath correlation functions for the double quantum
dot.
Next, we diagonalize the system Hamiltonian (in the Fock space basis)
E0 = 0 ,
E− = ǫ −
E+ = ǫ +
√
√
|v0 i = |00i ,
∆2
+
T2
,
∆2 + T 2 ,
E2 = 2ǫ + U ,
|v− i ∝
|v+ i ∝
|v2 i = |11i ,
h
h
∆+
∆−
√
√
∆2
+
T2
∆2 + T 2
i
|10i + T |01i ,
i
|10i + T |01i ,
(4.45)
where ∆ = (ǫB − ǫA )/2 and ǫ = (ǫA + ǫB )/2 and |01i = −d˜†B |00i, |10i = d˜†A |00i, and |11i =
d˜†B d˜†A |00i. We have not symmetrized the coupling operators but to obtain the BMS limit, we may
alternatively use Eqns. (2.60) and (2.61) when τ → ∞ . Specifically, when we have no degeneracies
in the system Hamiltonian (∆2 + T 2 > 0), the masterP
equation in the
Penergy eigenbasis (where
a, b ∈ {0, −, +, 2}) becomes a rate equation ρ̇aa = + b γab,ab ρbb − [ b γba,ba ] ρaa with the rate
coefficients
X
γab,ab =
γαβ (Eb − Ea ) ha| Aβ |bi ha| A†α |bi∗ .
(4.46)
αβ
We may calculate the Liouvillians for the interaction with the left and right contact separately
L
R
+ γab,ab
,
γab,ab = γab,ab
(4.47)
63
4.5. THE DOUBLE QUANTUM DOT
since we are constrained to second order perturbation theory in the tunneling amplitudes. Since
we have d˜A = A†2 = A1 = d˜A and d˜B = A†4 = A2 = d˜B , we obtain for the left-associated dampening
coefficients
L
γab,ab
= γ12 (Eb − Ea )|ha| A2 |bi|2 + γ21 (Eb − Ea )|ha| A1 |bi|2 ,
R
γab,ab
= γ34 (Eb − Ea )|ha| A4 |bi|2 + γ43 (Eb − Ea )|ha| A3 |bi|2 .
(4.48)
In the wideband (flatband) limit ΓL/R (ω) = ΓL/R , we obtain for the nonvanishing transition rates
in the energy eigenbasis
√
√
L
R
γ0−,0−
= ΓL γ+ [1 − fL (ǫ − ∆2 + T 2 )] ,
γ0−,0−
= ΓR γ− [1 − fR (ǫ − ∆2 + T 2 )] ,
√
√
L
R
γ0+,0+
= ΓL γ− [1 − fL (ǫ + ∆2 + T 2 )] ,
γ0+,0+
= ΓR γ+ [1 − fR (ǫ + ∆2 + T 2 )] ,
√
√
R
L
γ−2,−2
= ΓR γ+ [1 − fR (ǫ + U + ∆2 + T 2 )] ,
γ−2,−2
= ΓL γ− [1 − fL (ǫ + U + ∆2 + T 2 )] ,
√
√
L
R
γ+2,+2
= ΓL γ+ [1 − fL (ǫ + U − ∆2 + T 2 )] ,
γ+2,+2
= ΓR γ− [1 − fR (ǫ + U − ∆2 + T 2 )] ,
√
√
R
L
γ−0,−0
= ΓR γ− fR (ǫ − ∆2 + T 2 ) ,
γ−0,−0
= ΓL γ+ fL (ǫ − ∆2 + T 2 ) ,
√
√
L
R
γ+0,+0
= ΓL γ− fL (ǫ + ∆2 + T 2 ) ,
γ+0,+0
= ΓR γ+ fR (ǫ + ∆2 + T 2 ) ,
√
√
R
L
γ2−,2−
= ΓR γ+ fR (ǫ + U + ∆2 + T 2 ) ,
γ2−,2−
= ΓL γ− fL (ǫ + U + ∆2 + T 2 ) ,
√
√
L
R
= ΓL γ+ fL (ǫ + U − ∆2 + T 2 ) ,
γ2+,2+
γ2+,2+
= ΓR γ− fR (ǫ + U − ∆2 + T 2 ) ,
(4.49)
with the dimensionless coefficients
∆
1
1± √
γ± =
2
∆2 + T 2
(4.50)
arising from the matrix elements of the system coupling operators. This rate equation can also be
visualized with a network, see Fig. 4.3.
Figure 4.3: Configuration space of a serial
double quantum dot coupled to two leads.
Due to the hybridization of the two levels,
electrons may jump directly from the left
contact to right-localized modes and vice
versa, such that in principle all transitios are
driven by both contacts. However, the relative strength of the couplings is different,
such that the two Liouillians have a different
structure. In the Coulomb-blockade limit,
transitions to the doubly occupied state are
forbidden (dotted lines), such that the system dimension can be reduced.
As the simplest example√of the resulting rate equation,
√ we study the high-bias√and Coulomb2
2
blockade limit fL/R (ǫ+U ± ∆ + T ) → 0 and fL (ǫ± ∆2 + T 2 ) → 1 and fR (ǫ± ∆2 + T 2 ) → 0
when ∆ → 0 (such that γ± → 1/2). This removes any dependence on T and therefore, a current
will be predicted even when T → 0 (where we have a disconnected structure). However, precisely in
64
CHAPTER 4. MULTI-TERMINAL COUPLING : NON-EQUILIBRIUM CASE I
this limit, the two levels E− and E+ become energetically degenerate, such that we cannot neglect
the couplings to the coherences anymore and the BMS limit is not applicable. The resulting
Liouvillian reads


−2ΓL ΓR
ΓR
0
1
−ΓR
0
ΓL + ΓR 
 ΓL
,
L =
(4.51)

ΓL
0
−ΓR
ΓL + ΓR 
2
0
0
0
−2(ΓL + ΓR )
where it becomes visible that the doubly occupied state will simply decay and may therefore –
since we are interested in the long-term dynamics – be eliminated completely


−2ΓL ΓR
ΓR
1
ΓL
−ΓR
0 .
LCBHB =
(4.52)
2
ΓL
0
−ΓR
Switching to a conditional Master equation by counting all electrons in the right junction and
performing the Fourier summation corresponds formally to the replacement ΓR → ΓR e+iχ in all
off-diagonal matrix elements (corresponding to single-electron jumps), such that we have


−2ΓL ΓR e+iχ ΓR e+iχ
1
.
ΓL
−ΓR
0
(4.53)
LCBHB (χ) =
2
ΓL
0
−ΓR
Exercise 34 (Stationary Current) (1 points)
Calculate the stationary current of Eq.(4.53).
Exercise 35 (Nonequilibrium Stationary State) (1 points)
Show that the stationary state of Eq. (4.52) cannot be written as a grand-canonical equilibrium
state by disproving the equations ρ̄−− /ρ̄00 = e−β(E− −E0 −µ) , ρ̄++ /ρ̄00 = e−β(E+ −E0 −µ) and ρ̄++ /ρ̄−− =
e−β(E+ −E− )
Chapter 5
Direct bath coupling :
Non-Equilibrium Case II
Another nonequilibrium situation may be generated by a multi-component bath with components
interacting directly (i.e., even without the presence of the system) via a small interface. However,
the interaction may be modified by the presence of the quantum system. A prototypical example
for such a bath is a quantum point contact: The two leads are held at different potentials and
through a tiny contact charges may tunnel. The tunneling process is however highly sensitive to
the presence of nearby charges, in the Hamiltonian this is modeled by a capacitive change of the
tunneling amplitudes. When already the baseline tunneling amplitudes (in a low transparency
QPC) are small, we may apply the master equation formalism without great efforts, as will be
demonstrated at two examples.
5.1
Monitored SET
High-precision tests of counting statistics have been performed with a quantum point contact that
is capacitively coupled to a single-electron transistor. The Hamiltonian of the system depicted in
Fig. 5.1 reads
Figure 5.1: Sketch of a quantum point contact (in fact, a two component bath with the components
held at different chemical potential) monitoring a single electron transistor. The tunneling through
the quantum point contact is modified when the SET is occupied.
65
66
CHAPTER 5. DIRECT BATH COUPLING :
NON-EQUILIBRIUM CASE II
HS = ǫd† d ,
X
X
X
X
†
†
HB =
ǫkL c†kL ckL +
ǫkL c†kR ckR +
εkL γkL
γkL +
εkL γkR
γkR ,
HI =
"k
X
k
tkL dc†kL +
k
X
#k
tkR dc†kR + h.c. +
k
"
k
X
#
tkk′ + d† dτkk′ γkL γk†′ R + h.c. ,
kk′
(5.1)
where ǫ denotes the dot level, ckα annihilate electrons on SET lead α and γkα are the annihilation
operators for the QPC lead α. The QPC baseline tunneling amplitude is given by tkk′ and describes
the scattering of and electron from mode k in the left lead to mode k ′ in the right QPC contact.
When the nearby SET is occupied it is modified to tkk′ + τkk′ , where τkk′ represents the change of
the tunneling amplitude.
We will derive a master equation for the dynamics of the SET due to the interaction with the
QPC and the two SET contacts. In addition, we are interested not only in the charge counting
statistics of the SET but also the QPC. The Liouvillian for the SET-contact interaction is well
known and has been stated previously (we insert counting fields at the right lead)
LSET (χ) =
−ΓL fL − ΓR fR
+ΓL (1 − fL ) + ΓR (1 − fR )e+iχ
+ΓL fL + ΓR fR e−iχ
−ΓL (1 − fL ) − ΓR (1 − fR )
.
(5.2)
We will therefore derive the dissipator for the SET-QPC interaction separately. To keep track
of the tunneled QPC electrons, we insert a virtual detector operator in the respective tunneling
Hamiltonian
HIQPC =
X
kk′
X
†
∗
t∗kk′ 1 + d† dτkk
tkk′ 1 + d† dτkk′ B † γkL γk†′ R +
′ Bγk ′ R γkL
kk′
†
= 1⊗B ⊗
†
X
tkk′ γkL γk†′ R
kk′
†
+d d ⊗ B ⊗
X
+1⊗B⊗
τkk′ γkL γk†′ R
kk′
†
X
†
t∗kk′ γk′ R γkL
kk′
+d d⊗B⊗
X
†
∗
τkk
′ γk ′ R γkL .
(5.3)
kk′
Note that we have implicitly performed the mapping to a tensor product representation of the
fermionic operators, which is unproblematic here as between SET and QPC no particle exchange
takes place and the electrons in the QPC and the SET may be treated as different particle types.
To simplify the system, we assume that the change of tunneling amplitudes affects all modes in
the same manner, i.e., τkk′ = τ̃ tkk′ , which enables us to combine some coupling operators
HIQPC =
X
X
†
t∗kk′ γk′ R γkL
.
tkk′ γkL γk†′ R + 1 + τ̃ ∗ d† d ⊗ B ⊗
1 + τ̃ d† d ⊗ B † ⊗
(5.4)
kk′
kk′
The evident advantage of this approximation is that only two correlation functions have to be
computed. We can now straightforwardly (since the baseline tunneling term is not included in the
bath Hamiltonian) map to the interaction picture
B1 (τ ) =
X
kk′
tkk′ γkL γk†′ R e−i(εkL −εk′ R )τ ,
B2 (τ ) =
X
kk′
† +i(εkL −εk′ R )τ
e
.
t∗kk′ γk′ R γkL
(5.5)
67
5.1. MONITORED SET
For the first bath correlation function we obtain
E
D
XX
†
tkk′ t∗ℓℓ′ e−i(εkL −εk′ R )τ γkL γk†′ R γℓ′ R γℓL
C12 (τ ) =
kk′
=
X
kk′
1
=
2π
ℓℓ′
|tkk′ |2 e−i(εkL −εk′ R )τ [1 − fL (εkL )] fR (εk′ R )
Z Z
′
T (ω, ω ′ ) [1 − fL (ω)] fR (ω ′ )e−i(ω−ω )τ dωdω ′ ,
(5.6)
P
where we have introduced T (ω, ω ′ ) = 2π kk′ |tkk′ |2 δ(ω − εkL )δ(ω − εk′ R ). Note that in contrast
to previous tunneling rates, this quantity is dimensionless. The integral factorizes when T (ω, ω ′ )
factorizes (or when it is flat T (ω, ω ′ ) = t). In this case, the correlation function C12 (τ ) is expressed
as a product in the time domain, such that its Fourier transform will be given by a convolution
integral
Z
γ12 (Ω) =
C12 (τ )e+iΩτ dτ
Z
= t dωdω ′ [1 − fL (ω)] fR (ω ′ )δ(ω − ω ′ − Ω)
Z
= t [1 − fL (ω)] fR (ω − Ω)dω .
(5.7)
For the other correlation function, we have
Z
γ21 (Ω) = t fL (ω) [1 − fR (ω + Ω)] dω .
(5.8)
Exercise 36 (Correlation functions for the QPC) (1 points)
Show the validity of Eqns. (5.8).
The structure of the Fermi functions demonstrates that the shift Ω can be included in the
chemical potentials. Therefore, we consider integrals of the type
Z
I = f1 (ω) [1 − f2 (ω)] dω .
(5.9)
At zero temperature, these should behave as I ≈ (µ1 − µ2 )Θ(µ1 − µ2 ), where Θ(x) denotes the
Heaviside-Θ function, which follows from the structure of the integrand, see Fig. 5.2. For finite
temperatures, the value of the integral can also be calculated, for simplicity we constrain ourselves
to the (experimentally relevant) case of equal temperatures (β1 = β2 = β), for which we obtain
Z
1
dω
+ 1) (e−β(µ1 −ω) + 1)
Z
1
δ2
dω ,
= lim
δ→∞
(eβ(µ2 −ω) + 1) (e−β(µ1 −ω) + 1) δ 2 + ω 2
I =
(eβ(µ2 −ω)
(5.10)
68
CHAPTER 5. DIRECT BATH COUPLING :
NON-EQUILIBRIUM CASE II
1
f1(ω)[1-f2(ω)]
f1(ω)
1-f2(ω)
0,5
0
kB T 2
Figure 5.2: Integrand in Eq. (5.9). At zero
temperature at both contacts, we obtain a
product of two step functions and the area
under the curve is given by the difference
µ1 − µ2 as soon as µ1 > µ2 (and zero otherwise).
kB T 1
µ2
µ1
where we have introduced the Lorentzian-shaped regulator to enforce convergence. By identifying
the poles of the integrand
∗
ω±
= ±iδ ,
π
(2n + 1)
β
π
= µ2 + (2n + 1)
β
∗
ω1,n
= µ1 +
∗
ω2,n
(5.11)
where n ∈ {0, ±1, ±2, ±3, . . . we can solve the integral by using the residue theorem, see also
Fig. 5.3 for the integration contour. Finally, we obtain for the integral
Figure 5.3: Poles and integration contour for
Eq. (5.9) in the complex plane. The integral
along the real axis (blue line) closed by an
arc (red curve) in the upper complex plane,
along which (due to the regulator) the integrand vanishes sufficiently fast.
∞
X
δ 2 δ 2 +
Res f1 (ω) [1 − f2 (ω)] 2
I = 2πi lim Res f1 (ω) [1 − f2 (ω)] 2
δ→∞
δ + ω 2 ω=+iδ n=0
δ + ω 2 ω=µ1 + π (2n+1)
β
∞
o
2
X
δ
+
Res f1 (ω) [1 − f2 (ω)] 2
2
δ
+
ω
ω=µ2 + π (2n+1)
n=0
n
β
µ1 − µ2
,
=
1 − e−β(µ1 −µ2 )
(5.12)
which automatically obeys the simple zero-temperature (β → ∞) limit. With the replacements
69
5.1. MONITORED SET
µ1 → µR + Ω and µ2 → µL , we obtain for the first bath correlation function
γ12 (Ω) = t
Ω−V
,
1 − e−β(Ω−V )
(5.13)
where V = µL − µR is the QPC bias voltage. Likewise, with the replacements µ1 → µL and
µ2 → µR − Ω, the second bath correlation function becomes
γ21 (Ω) = t
Ω+V
.
1 − e−β(Ω+V )
(5.14)
Now we can calculate the transition rates in our system (containing the virtual detector and the
quantum dot) for a non-degenerate system spectrum. However, now the detector is part of our
system. Therefore, the system state is not only characterized by the number of charges on the
SET dot a ∈ {0, 1} but also by the number of charges n that have tunneled through the QPC and
have thereby changed the detector state
#
"
X
X
(5.15)
γ(b,m)(a,n),(b,m)(a,n) ρ(a,n)(a,n) .
ρ̇(a,n)(a,n) =
γ(a,n)(b,m),(a,n)(b,m) ρ(b,m)(b,m) −
b,m
b,m
Shortening the notation by omitting the double-indices we may also write
#
"
X
X
(m)
γ(b,m),(a,n) ρ(n)
ρ̇(n)
γ(a,n),(b,m) ρbb −
aa ,
aa =
b,m
(5.16)
b,m
(n)
where ρaa = ρ(a,n),(a,n) and γ(a,n),(b,m) = γ(a,n)(a,n),(b,m)(b,m) . It is evident that the coupling operators
A1 = (1 + τ̃ d† d) ⊗ B † and A2 = (1 + τ̃ ∗ d† d) ⊗ B only allow for sequential tunneling through the
QPC at lowest order (i.e., m = n ± 1) and do not induce transitions between different dot states
(i.e., a = b), such that the only non-vanishing contributions may arise for
γ(0,n)(0,n+1) = γ12 (0) h0, n| A2 |0, n + 1i h0, n| A†1 |0, n + 1i∗ = γ12 (0) ,
γ(0,n)(0,n−1) = γ21 (0) h0, n| A1 |0, n − 1i h0, n| A†2 |0, n − 1i∗ = γ21 (0) ,
γ(1,n)(1,n+1) = γ12 (0) h1, n| A2 |1, n + 1i h1, n| A†1 |1, n + 1i∗ = γ12 (0)|1 + τ̃ |2 ,
γ(1,n)(1,n−1) = γ21 (0) h1, n| A1 |1, n − 1i h1, n| A†2 |1, n − 1i∗ = γ21 (0)|1 + τ̃ |2 .
The remaining terms just account for the normalization.
Exercise 37 (Normalization terms) (1 points)
Compute the remaining rates
X
γ(0,m)(0,m),(0,n)(0,n) ,
and
m
explicitly.
X
m
γ(1,m)(1,m),(1,n)(1,n)
(5.17)
70
CHAPTER 5. DIRECT BATH COUPLING :
NON-EQUILIBRIUM CASE II
Adopting the notation of conditional master equations, this leads to the connected system
(n)
= γ12 (0)ρ00
(n)
= |1 + τ̃ |2 γ12 (0)ρ11
ρ̇00
ρ̇11
(n+1)
(n−1)
+ γ21 (0)ρ00
(n+1)
(n)
− [γ12 (0) + γ21 (0)] ρ00
(n−1)
+ |1 + τ̃ |2 γ21 (0)ρ11
(n)
− |1 + τ̃ |2 [γ12 (0) + γ21 (0)] ρ11 , (5.18)
such that after Fourier transformation with the counting field ξ for the QPC, we obtain the
following dissipator
γ21 e+iχ − 1 + γ12 e−iχ − 1
0
,
LQPC (ξ) =
0
|1 + τ̃ |2 γ21 e+iχ − 1 + γ12 e−iχ − 1
(5.19)
which could not have been deduced directly from a Liouvillian for the SET alone. More closely
analyzing the Fourier transforms of the bath correlation functions
V
,
1 − e−βV
V
= γ12 (0) = t +βV
e
−1
γ21 = γ21 (0) = t
γ12
(5.20)
we see that for sufficiently large QPC bias voltages, transport becomes unidirectional and only one
contribution remains and the other one is exponentially suppressed. The sum of both Liouvillians
(5.2) and (5.19) constitutes the total dissipator
L(χ, ξ) = LSET (χ) + LQPC (ξ) ,
(5.21)
which can be used to calculate the probability distributions for tunneling through both transport
channels (QPC and SET).
Exercise 38 (QPC current) (1 points)
Show that the stationary state of the SET is unaffected by the additional QPC dissipator and
calculate the stationary current through the QPC for Liouvillian (5.21).
When we consider the case {ΓL , ΓR } ≪ {tV, |1 + τ̃ |tV }, we approach a bistable system, and
the counting statistics approaches the case of telegraph noise. When the dot is empty or filled
throughout respectively, the current can easily be determined as
I0 = [γ21 (0) − γ12 (0)] ,
I1 = |1 + τ̃ |2 [γ21 (0) − γ12 (0)] .
(5.22)
For finite time intervals ∆t, the number of electrons tunneling through the QPC ∆n is determined
by the probability distribution
1
P∆n (∆t) =
2π
Z+π
Tr eL(0,ξ)∆t−i∆nξ ρ(t) dξ ,
(5.23)
−π
where ρ(t) represents the initial density matrix. This quantity can e.g. be evaluated numerically.
When ∆t is not too large (such that the stationary state is not really reached) and not too small
71
5.1. MONITORED SET
(such that there are sufficiently many particles tunneling through the QPC to meaningfully define a
current), a continuous measurement of the QPC current maps to a fixed-point iteration as follows:
Measuring a certain particle number corresponds to a projection, i.e., the system-detector density
matrix is projected to a certain measurement outcome which occurs with the probability P∆n (∆t)
ρ=
X
ρ
n
(n)
ρ(m)
⊗ |ni hn| →
.
Tr {ρ(m) }
m
(5.24)
The density matrix after the measurement is then used as initial state for the next iteration, and
the ratio of measured particles divided by measurement time gives a current estimate I(t) ≈ ∆n
.
∆t
Such current trajectories are used to track the full counting statistics through quantum point
contacts, see Fig. 5.4
150
QPC current
finite sampling
infinite sampling
100
50
0
0
1000
2000
3000
time [∆t]
4000
Pn(∆t)
Figure 5.4: Numerical simulation of the time-resolved QPC current for a fluctuating dot occupation. At infinite SET bias, the QPC current allows to reconstruct the full counting statistics of the SET, since each current blip from low (red line) to high (green line) current corresponds to an electron leaving the SET to its right junction. Parameters: ΓL ∆t = ΓR ∆t = 0.01,
γ12 (0) = |1 + τ̃ |2 γ12 (0) = 0, γ21 (0) = 100.0, |1 + τ̃ |2 γ21 (0) = 50.0, fL = 1.0, fR = 0.0. The right
panel shows the corresponding probability distribution Pn (∆t) versus n = I∆t, where the blue
curve is sampled from the left panel and the black curve is the theoretical limit for infinitely long
times.
72
5.2
CHAPTER 5. DIRECT BATH COUPLING :
NON-EQUILIBRIUM CASE II
Monitored charge qubit
A quantum point contact may also be used to monitor a nearby charge qubit, see Fig. 5.5. The
A
B
Figure 5.5: Sketch of a quantum point contact monitoring a double quantum dot with a single
electron loaded (charge qubit). The current through the quantum point contact is modified by the
position of the charge qubit electron, i.e., a measurement in the σ z basis is performed.
QPC performs a measurement of the electronic position, since its current is highly sensitive on
it. This corresponds to a σ x measurement performed on the qubit. However, the presence of a
detector does of course also lead to a back-action on the probed system. Here, we will derive a
master equation for the system to quantify this back-action.
The Hamiltonian of the charge qubit is given by
HCQB = ǫA d†A dA + ǫB d†B dB + T dA d†B + dB d†A ,
(5.25)
where we can safely neglect Coulomb interaction, since to form a charge qubit, the number of
electrons on this double quantum dot is fixed to one. The matrix representation is therefore just
two-dimensional in the |nA , nB i ∈ {|10i , |01i} basis and can be expressed by Pauli matrices
HCQB =
ǫA T
T ǫB
i.e., we may identify d†A dA =
Hamiltonian reads
1
2
=
(1 + σ z ) and d†B dB =
HI = d†A dA ⊗
A/B
ǫA + ǫB
ǫA − ǫB z
1+
σ + T σ x ≡ ǫ1 + ∆σ z + T σ x ,
2
2
X
kk′
1
2
(5.26)
(1 − σ z ). The tunneling part of the QPC
†
†
tA
kk′ γkL γk′ R + dB dB ⊗
X
†
tB
kk′ γkL γk′ R + h.c. ,
(5.27)
kk′
where tkk′ represents the tunneling amplitudes when the electron is localized on dots A and B,
respectively. After representing the charge qubit in terms of Pauli matrices, the full Hamiltonian
73
5.2. MONITORED CHARGE QUBIT
reads
H = ǫ1 + ∆σ z + T σ"x
#
X
X
1
†
†
tA∗
tA
+ [1 + σ z ] ⊗
kk′ γk′ R γkL
kk′ γkL γk′ R +
2
′
kk′
#
" kk
X
X
1
†
†
tB∗
tB
+ [1 − σ z ] ⊗
kk′ γk′ R γkL
kk′ γkL γk′ R +
2
kk′
kk′
X
X
†
†
+
εkL γkL
γkL +
εkR γkR
γkR .
k
(5.28)
k
To reduce the number of correlation functions we again assume that all tunneling amplitudes are
B
modified equally tA
kk′ = τ̃A tkk′ and tkk′ = τ̃B tkk′ with baseline tunneling amplitudes tkk′ and real
constants τA and τB . Then, only a single correlation function needs to be calculated
#
"X
X
τ̃A
τ̃
B
†
HI =
,
(5.29)
t∗kk′ γk′ R γkL
tkk′ γkL γk†′ R +
(1 + σ z ) +
(1 − σ z ) ⊗
2
2
′
′
kk
kk
which becomes explicitly
+
*
i
ih
Xh
†
† +i(εkL −εk′ R )τ
tℓℓ′ γℓL γℓ†′ R + t∗ℓℓ′ γℓ′ R γℓL
tkk′ γkL γk†′ R e−i(εkL −εk′ R )τ + t∗kk′ γk′ R γkL
e
C(τ ) =
kk′ ℓℓ′
=
X
kk′
|t
kk′
|
2
h
e
−i(εkL −εk′ R )τ
D
†
γkL γk†′ R γk′ R γkL
E
+e
+i(εkL −εk′ R )τ
D
γ
k′ R
†
γkL
γkL γk†′ R
Ei
Z
h
i
1
′
′
=
dωdω ′ T (ω, ω ′ ) e−i(ω−ω )τ [1 − fL (ω)]fR (ω ′ ) + e+i(ω−ω )τ fL (ω)[1 − fR (ω ′ )] (5.30)
2π
where
in the last step replaced the summation by a continuous integration with T (ω, ω ′ ) =
P we have
2
2π kk′ |tkk′ | δ(ω − εkL )δ(ω ′ − εk′ R ). We directly conclude for the Fourier transform of the bath
correlation function
Z
γ(Ω) =
dωdω ′ T (ω, ω ′ ) [δ(Ω − ω + ω ′ )[1 − fL (ω)]fR (ω ′ ) + δ(Ω + ω − ω ′ )fL (ω)[1 − fR (ω ′ )]]
Z
=
dω [T (ω, ω − Ω)[1 − fL (ω)]fR (ω − Ω) + T (ω, ω + Ω)fL (ω)[1 − fR (ω + Ω)]] . (5.31)
In what follows, we will consider the wideband limit T (ω, ω ′ ) = 1 (the weak-coupling limit enters
the τ̃A/B parameters), such that we may directly use the result from the previous section
Ω+V
Ω−V
+
,
(5.32)
−β(Ω+V
)
1−e
1 − e−β(Ω−V )
where V = µL − µR denotes the bias voltage of the quantum point contact. Since we are not
interested in its counting statistics here, we need not introduce any counting fields. The derivation
of the master equation in the system energy eigenbasis requires diagonalization of the system
Hamiltonian first
√
√
∆ − ∆2 + T 2 |0i + T |1i
2
2
E− = ǫ − ∆ + T ,
|−i = q
√
2
T 2 + ∆ − ∆2 + T 2
√
√
∆ + ∆2 + T 2 |0i + T |1i
2
2
E+ = ǫ + ∆ + T ,
(5.33)
|+i = q
√
2 .
2
2
2
T + ∆+ ∆ +T
γ(Ω) =
74
CHAPTER 5. DIRECT BATH COUPLING :
NON-EQUILIBRIUM CASE II
Exercise 39 (Diagonalization of a single-qubit Hamiltonian) (1 points)
Calculate eigenvalues and eigenvectors of the system Hamiltonian.
Following the Born-, Markov-, and secular approximations – compare definition 7 – we obtain
a Lindblad Master equation for the qubit
o
X
1n
†
†
ρ̇ = −i [HS + HLS , ρ] +
γab,cd |ai hb| ρ (|ci hd|) −
, (5.34)
(|ci hd|) |ai hb| , ρ
2
abcd
where the summation only goes over the two energy eigenstates and HLS denotes the frequency
renormalization. Since the two eigenvalues of our system are non-degenerate, the Lamb-shift
Hamiltonian is diagonal in the system energy eigenbasis and does not affect the dynamics of the
populations, which decouples according to the rate equation
#
"
X
X
γba,ba ρaa
(5.35)
ρ̇aa =
γab,ab ρbb −
b
b
completely from the coherences. In particular, we have
ρ̇−− = γ−−,−− ρ−− + γ−+,−+ ρ++ − γ−−,−− ρ−− − γ+−,+− ρ−−
= γ−+,−+ ρ++ − γ+−,+− ρ−− ,
ρ̇++ = γ+−,+− ρ−− − γ−+.−+ ρ++ ,
which when written as a matrix becomes
ρ̇−−
−γ+−,+− +γ−+,−+
ρ−−
=
.
+γ+−,+− −γ−+,−+
ρ++
ρ̇++
(5.36)
(5.37)
The required dampening coefficients read
√
γ−+,−+ = γ(E+ − E− )|h−| A |+i|2 = γ(+2 ∆2 + T 2 )
γ+−,+−
T2
(τ̃A − τ̃B )2 ,
4(∆2 + T 2 )
√
T2
(τ̃A − τ̃B )2 .
= γ(E− − E+ )|h+| A |−i|2 = γ(−2 ∆2 + T 2 )
2
2
4(∆ + T )
(5.38)
Exercise 40 (Qubit Dissipation) (1 points)
Show the validity of Eqns. (5.38).
This shows that when the QPC current is not dependent on the qubit state τ̃A = τ̃B , the dissipation on the qubit vanishes completely, which is consistent with our initial interaction Hamiltonian.
In addition, in the pure dephasing limit T → 0, we do not have any dissipative back-action of the
measurement device on the qubit. Equation (5.37) obviously also preserves the trace of the density
matrix. The stationary density matrix is therefore defined by
√
γ+−,+−
γ(−2 ∆2 + T 2 )
ρ̄++
√
=
=
.
(5.39)
ρ̄−−
γ−+,−+
γ(+2 ∆2 + T 2 )
5.2. MONITORED CHARGE QUBIT
75
When the QPC bias voltage vanishes (at equilibrium), we have
√
√
γ(−2 ∆2 + T 2 )
2
2
√
(5.40)
→ e−β2 ∆ +T = e−β(E+ −E− ) ,
2
2
γ(+2 ∆ + T )
i.e., the qubit thermalizes with the temperature of the QPC. When the QPC bias voltage is large,
the qubit is driven away from this thermal state.
Exercise 41 (Strongly monitored qubit) (1 points)
Calculate the stationary qubit state in the infinite bias regimes V → ±∞.
The evolution of coherences decouples from the diagonal elements of the density matrix. The
hermiticity of the density matrix allows to consider only one coherence
ρ̇−+ = −i (E− + ∆E− − E+ − ∆E+ ) ρ−+
1
+ γ−−,++ − (γ−−,−− + γ++,++ + γ−+,−+ + γ+−,+− ) ρ−+ ,
(5.41)
2
where ∆E± corresponds to the energy renormalization due to the Lamb-shift, which induces a
frequency renormalization of the qubit. The real part of the above equation is responsible for the
dampening of the coherence, its calculation requires the evaluation of all remaining nonvanishing
dampening coefficients
h
i
√
√
T2
(τ̃A − τ̃B )2 ,
γ−+,−+ + γ+−,+− = γ(+2 ∆2 + T 2 ) + γ(−2 ∆2 + T 2 )
4(T 2 + ∆2 )
∆2
(τ̃A − τ̃B )2 .
(5.42)
γ−−,−− + γ++,++ − 2γ−−,++ = γ(0) 2
T + ∆2
Using the decomposition of the dampening, we may now calculate the decoherence rate
1
1
γ =
(γ−+,−+ + γ+−,+− ) + (γ−−,−− + γ++,++ − 2γ−−,++ )
2
2
2
n
2
√
√
T
(τ̃A − τ̃B )
β
2
2
2
2
V + 2 ∆ + T coth
V +2 ∆ +T
=
8
∆2 + T 2
2
o
√
√
β
2
2
2
2
V −2 ∆ +T
+ V − 2 ∆ + T coth
2
(τ̃A − τ̃B )2 ∆2
βV
+
V coth
,
(5.43)
2
∆2 + T 2
2
which vanishes as reasonably expected when we set τ̃A = τ̃B . Noting that x coth(x) ≥ 1 does not
only prove its positivity (i.e., the coherences always decay) but also enables one to obtain a rough
lower bound
(τ̃A − τ̃B )2 T 2 + 2∆2
(5.44)
γ≥
2β
T 2 + ∆2
on the dephasing
√ rate. This lower bound is valid when the QPC voltage is rather small. For large
voltages |V | ≫ ∆2 + T 2 , the dephasing rate is given by
(τ̃A − τ̃B )2 T 2 + 2∆2
|V |
γ≈
2
T 2 + ∆2
and thus is limited by the voltage rather than the temperature.
(5.45)
76
CHAPTER 5. DIRECT BATH COUPLING :
NON-EQUILIBRIUM CASE II
Chapter 6
Open-loop control:
Non-Equilibrium Case III
Time-dependent equations of the form
ρ̇ = L(t)ρ(t)
(6.1)
are notoriously difficult to solve unless L(t) fulfills special properties. One such special case is e.g.
the case of commuting superoperators, where the solution can be obtained from the exponential
of an integral
Z t
′
′
′
(6.2)
[L(t), L(t )] = 0
=⇒
ρ(t) = exp −i
L(t )dt ρ0 .
0
Another special case is one with a very slow time-dependence and a unique stationary state L(t)ρ̄t =
0, where the time-dependent density matrix can be assumed to adiabatically follow the stationary
state ρ(t) ≈ ρ̄t . For fast time-dependencies, there exists the analytically solvable case of a train of
δ-kicks
∞
X
L(t) = L0 +
ℓi δ(t − ti ) ,
(6.3)
i=1
where the solution reads (tn < t < tn+1 )
ρ(t) = eL0 (t−tn ) eℓn eL0 (tn −tn−1 ) × . . . × eℓ2 eL0 (t2 −t1 ) eℓ1 eL0 (t1 ) ρ0 ,
(6.4)
which can be considerably simplified using e.g. Baker-Campbell-Haussdorff relations when the
kicks are identical ℓi = ℓ.
Here, we will be concerned with piecewise-constant time-dependencies, such that the problem
can be mapped to ordinary differential equations with constant coefficients, which are evolved a
certain amount of time. After an instantaneous switch of the coefficients, one simply has initial
conditions taken from the final state of the last evolution period, which can be evolved further and
so on.
6.1
Single junction
The simplest model to study counting statistics is that of a single junction. Not making the
microscopic model explicit, it will in the small tunneling limit obey the equations
Ṗn = +γPn−1 (t) + γ̄Pn+1 − [γ + γ̄] Pn (t) ,
77
(6.5)
78
CHAPTER 6. OPEN-LOOP CONTROL:
NON-EQUILIBRIUM CASE III
where Pn (t) denotes the probability to have n particles passed the junction after time t. Accordingly, γ represents the tunneling rate from left to right and γ̄ the opposite tunneling rate, see
Fig. 6.1. Depending on the microscopic underlying model, these parameters may depend on parti-
Figure 6.1: Sketch of a single junction with
left-to-right and right-to-left tunneling rates γ
and γ̄, respectively. These parameters can be
changed in a time-dependent manner by modifying the microscopic model (double-headed
arrows).
cle concentrations on left and right side of the tunneling barrier and on the height of the barrier etc
and may thus be modifiedPin time. First let us consider the time-independent case. After Fourier
transformation P (χ, t) = n Pn (t)e+inχ , the n-resolved equation becomes
Ṗ (χ, t) = γ(e+iχ − 1) + γ̄(e−iχ − 1) P (χ, t) .
(6.6)
This is thus in perfect agreement with what we had for the quantum point contact statistics in
Eq. (5.19). With the initial condition P (χ, 0) = 1 it is solved by
P (χ, t) = exp
γ(e+iχ − 1) + γ̄(e−iχ − 1) t .
(6.7)
Exercise 42 (Cumulants) (1 points)
Show that the cumulants of the probability distribution Pn (t) are given by
k n
= γ + (−1)k γ̄ t .
This initial condition is chosen because we assume that at time t = 0, no particle has crossed
the junction Pn (0) = δn,0 . The probability to count n particles after time t can be obtained from
the inverse Fourier transform
1
Pn (t) =
2π
Z+π
exp γ(e+iχ − 1) + γ̄(e−iχ − 1) t e−inχ dχ .
(6.8)
−π
This probability can for this onedimensional model be calculated analytically even in the case of
79
6.1. SINGLE JUNCTION
bidirectional transport
Z+π
∞
a
b
X
(γt)
(γ̄t)
1
Pn (t) = e−(γ+γ̄)t
e+i(a−b−n)χ dχ
a!
b!
2π
a,b=0
−π
∞
X
(γt)a (γ̄t)b
δa−b,n
a!
b!
a,b=0
 P
∞
(γt)a (γ̄t)a−n


a! (a−n)!
a=n
= e−(γ+γ̄)t
∞
P (γt)a (γ̄t)a−n


a! (a−n)!
= e−(γ+γ̄)t
= e
−(γ+γ̄)t
γ
γ̄
a=0
n/2
:
n≥0
:
n<0
√
Jn (2 γγ̄t) ,
(6.9)
where Jn (x) denotes a modified Bessel function of the first kind – defined as the solution of
z 2 Jn′′ (z) + zJn′ (z) − (z 2 + n2 )Jn (z) = 0 .
In the unidirectional transport limit, this reduces to a normal Poissonian distribution
−γt (γt)n
:
n≥0
e
n!
.
lim Pn (t) =
γ̄→0
0
:
n<0
(6.10)
(6.11)
Exercise 43 (Poissonian limit) (1 points)
Show the Poissonian distribution in the unidirectional transport limit.
Now we consider the case of a time-dependent rate γ → γ(t) with a piecewise-constant time
dependence and for simplicity constrain ourselves to unidirectional transport as shown in Fig. 6.2.
The fact that the model is scalar (has no internal structure) implies that the probability distribution
Figure 6.2: Time-dependent tunneling rate
which is (nearly) piecewise constant during the
intervals ∆t. In the model, we neglect the
switching time τswitch completely.
of measuring particles in the αth time interval is completely independent from the outcome
the
of
interval α − 1. If we denote the cumulant during the interval ∆t in the α-th interval by nk α ,
we find for the average
N
N
1 X k 1 X
hhn¯k ii =
γα ∆t = hγi ∆t ,
n α=
N α=1
N α=1
(6.12)
i.e., the average cumulant is simply described by an average tunneling rate – regardless of the
actual form of the protocol.
80
6.2
CHAPTER 6. OPEN-LOOP CONTROL:
NON-EQUILIBRIUM CASE III
Electronic pump
We consider a system with the simplest possible internal structure, which has just the two states
empty and filled. A representative of such a system is the single-electron transistor. The tunneling
of electrons through such a transistor is stochastic and thereby in some sense uncontrolled. This
implies that e.g. the current fluctuations through such a device (noise) can not be suppressed
completely.
6.2.1
Time-dependent tunneling rates
We consider piecewise-constant time-dependencies of the tunneling rates ΓL (t) and ΓR (t) with an
otherwise arbitrary protocol
−ΓL (t)fL − ΓR (t)fR
+ΓL (t)(1 − fL ) + ΓR (t)(1 − fR )e+iχ
.
(6.13)
L(χ, t) =
+ΓL (t)fL + ΓR (t)fR e−iχ
−ΓL (t)(1 − fL ) − ΓR (t)(1 − fR )
The situation is also depicted in Fig. 6.3.
Figure 6.3: Time-dependent tunneling rates
which follow a piecewise constant protocol.
Dot level and left and right Fermi functions
are assumed constant.
Let the tunneling rates during the i-th time interval be denoted by ΓiL and ΓiR and the corresponding constant Liouvillian during this time interval by Li (χ). The density matrix after the
time interval ∆t is now given by
ρi+1 = eLi (0)∆t ρi ,
(6.14)
such that it has no longer the form of a master equation but becomes a fixed-point iteration. In
what follows, we will without loss of generality assume fL ≤ fR , such that the source lead is right
and the drain lead is left. Utilizing previous results, the stationary state of the Liouvillian Li is
given by
Γi fL + ΓiR fR
f¯i = L i
ΓL + ΓiR
(6.15)
and thereby (ΓiL/R ≥ 0) obeys fL ≤ f¯i ≤ fR . This implies that after a few transient iterations, the
density vector will hover around within the transport window
∀
i ≥ i∗ .
(6.16)
f L ≤ ni = d † d i ≤ f R
The occupations will therefore follow the iteration equation
1 − ni
1 − ni+1
Li (0)∆t
.
=e
ni
ni+1
(6.17)
81
6.2. ELECTRONIC PUMP
To calculate the number of particles tunneling into the source reservoir, we consider the momentgenerating function
1 − ni
Li (χ)∆t
(6.18)
Mi (χ, ∆t) = Tr e
ni
and calculate the first moment
hnii = (−i)∂χ Mi (χ, ∆t)|χ=0
h
1
2
−(ΓiL +ΓiR )∆t
=
−
Γ
(f
−
n
)
1
−
e
i
R R
(ΓiL + ΓiR )2
i
i
i
−(ΓiL +ΓiR )∆t
i i
− (fR − fL )(ΓL + ΓR )∆t
≤ 0.
+ΓL ΓR (ni − fL ) 1 − e
(6.19)
The first term is evidently negative when ni is within the transport window fL ≤ ni ≤ fR , and
that the second term is also negative follows from its upper bound ni → fR , where we can use that
1 − e−x ≤ x for all x ≥ 0 which is the case for x = (ΓiL + ΓiR )∆t.
It follows that the average current always points from right (source for constant rates) to left
(drain for constant rates) regardless of the actual protocol ΓαL/R chosen. In other words: With
simply modifying the tunneling rates (not taking into account whether the dot is occupied or not)
it is not possible to revert the direction of the current.
6.2.2
With performing work
To obtain transport against the bias, it is therefore necessary to modify the Fermi functions e.g.
by changing the chemical potentials or temperatures in the contacts. However, this first possibility
may for fast driving destroy the thermal equilibrium in the contacts, such that we consider changing
the dot level ǫ(t). Intuitively, it is quite straightforward to arrive at a protocol that should lift
electrons from left to right and thereby pumps electrons from low to high chemical potentials, see
Fig. 6.4. The propagators for the first and second half cycles read
min
min
min
min
−Γmax
) − Γmin
)
+Γmax
)] + Γmin
)]e+iχ
L fL (ǫ
R fR (ǫ
L [1 − fL (ǫ
R [1 − fR (ǫ
L1 (χ) =
,
min
min −iχ
min
min
+Γmax
) + Γmin
)e
−Γmax
)] − Γmin
)]
L fL (ǫ
R fR (ǫ
L [1 − fL (ǫ
R [1 − fR (ǫ
max
max
max
max
−Γmin
) − Γmax
)
+Γmin
)] + Γmax
)]e+iχ
L fL (ǫ
R fR (ǫ
L [1 − fL (ǫ
R [1 − fR (ǫ
L2 (χ) =
,
max
max −iχ
max
max
+Γmin
) + Γmax
)e
−Γmin
)] − Γmax
)]
L fL (ǫ
R fR (ǫ
L [1 − fL (ǫ
R [1 − fR (ǫ
(6.20)
respectively. The evolution of the density vector now follows the fixed point iteration scheme
ρ(t + T ) = P2 P1 ρ(t) = eL2 (0)T /2 eL1 (0)T /2 ρ(t) .
(6.21)
After a period of transient evolution, the density vector at the end of the pump cycle will approach
a value where ρ(t + T ) ≈ ρ(t) = ρ̄. This value is the stationary density matrix in a stroboscopic
sense and is defined by the eigenvalue equation
eL2 (0)T /2 eL1 (0)T /2 ρ̄ = ρ̄
(6.22)
with Tr {ρ̄} = 1. Once this stationary state has been determined, we can easily define the momentgenerating functions for the pumping period in the (stroboscopically) stationary regime
M1 (χ, T /2) = Tr eL1 (χ)T /2 ρ̄ ,
M2 (χ, T /2) = Tr eL2 (χ)T /2 eL1 (0)T /2 ρ̄ ,
(6.23)
82
CHAPTER 6. OPEN-LOOP CONTROL:
NON-EQUILIBRIUM CASE III
Figure 6.4: Here, tunneling rates and the dot
level also follow a piecewise constant protocol,
which admits only two possible values (top left
and right) and is periodic (see left panel). The
state of the contacts is assumed as stationary.
For small pumping cycle times T , the protocol
is expected to pump electrons from the left towards the right contact and thereby lifts electrons to a higher chemical potential.
where M1 describes the statistics in the first half cycle and M2 in the second half cycle and the
moments can be extracted in the usual way. The distribution for the total number of particles
tunneling during the complete pumping cycle n = n1 + n2 is given by
X
X
Pn (T ) =
Pn1 (T /2)Pn2 (T /2) =
δn,n1 +n2 Pn1 (T /2)Pn2 (T /2) .
(6.24)
n1 ,n2 : n1 +n2 =n
n1 ,n2
Then, the first moment may be calculated by simply adding the first moments of the particles
tunneling through both half-cycles
X
X X
X
hni =
nPn =
n
δn,n1 +n2 Pn1 (T /2)Pn2 (T /2) =
(n1 + n2 )Pn1 (T /2)Pn2 (T /2)
n
n
= hn1 i + hn2 i .
n1 ,n2
n1 ,n2
(6.25)
In the following, we will constrain ourselves for simplicity to symmetric tunneling rates Γmin
=
L
max
min
min
max
max
ΓR = Γ
and ΓL = ΓR = Γ . In this case, we obtain the simple expression for the average
number of particles during one pumping cycle
hni =
Γmax Γmin T fL (ǫmax ) − fR (ǫmax ) + fL (ǫmin ) − fR (ǫmin )
max
min
Γ
+Γ
2
max
min Γ
−Γ
max
min
max
min
min
max
+
Γ
f
(ǫ
)
−
f
(ǫ
)
+
Γ
f
(ǫ
)
−
f
(ǫ
)
×
L
R
R
L
(Γmax + Γmin )2
T min
max
.
(6.26)
Γ +Γ
× tanh
4
The first term (which dominates for slow pumping, i.e., large T ) is negative when fL (ω) < fR (ω).
It is simply given by the average of the two SET currents one would obtain for the two half cycles.
83
6.2. ELECTRONIC PUMP
The second term however is present when Γmax > Γmin and may be positive when the dot level
is changed strongly enough souch that fL (ǫmin ) > fR (ǫmax ) and fR (ǫmin ) > fL (ǫmax ). For large
differences in the tunneling rates and small pumping times T it may even dominate the first term,
such that the net particle number may be positive even though the bias would favor the opposite
current direction, see Fig. 6.5. The second cumulant can be similarly calculated
2
n − hni2 = n21 − hn1 i2 + n22 − hn2 i2 .
(6.27)
Exercise 44 (Second Cumulant for joint distributions) (1 points)
Show the validity of the above equation.
particle number <n(T)> / Fano factor F
2
2
2
<n> (error sqrt[<n >-<n> ])
1,5
2
2
Fano factor FT = [<n >-<n> ]/|<n>|
1
0,5
0
50
100
150
200
-0,5
-1
pump cycle time T
Figure 6.5: Average number of tunneled particles during one pump cycle. Parameters have been
chosen as Γmax = 1, Γmin = 0.1, fL (ǫmin ) = 0.9, fR (ǫmin ) = 0.95, fL (ǫmax ) = 0, fR (ǫmax ) = 0.05,
such that for slow pumping (large T ), the current I = hni /T must become negative. The shaded
region characterizes the width of the distribution, and the Fano factor demonstrates that around
the maximum pump current, the signal-to-noise ratio is most favorable.
When the dot level is shifted to very low and very large values (large pumping power), the bias
is negligible (alternatively, we may also consider an unbiased situation from the beginning). In
this case, we can assume fL (ǫmin ) = fR (ǫmin ) = f (ǫmin ) and fL (ǫmax ) = fR (ǫmax ) = f (ǫmax ). Then,
the second term dominates always and the particle number simplifies to
min
Γmax − Γmin
T min
max
max
hni = f (ǫ ) − f (ǫ ) max
.
(6.28)
tanh
Γ +Γ
Γ
+ Γmin
4
84
CHAPTER 6. OPEN-LOOP CONTROL:
NON-EQUILIBRIUM CASE III
This becomes maximal for large pumping time and large power consumption (f (ǫmin ) − f (ǫmax ) =
1). In this idealized limit, we easily obtain for the Fano factor for large cycle times T
FT =
hn2 i − hni2 T →∞ 2Γmax Γmin
→
,
|hni|
(Γmax + Γmin )2
(6.29)
which demonstrates that the pump works efficient and noiseless when Γmax ≫ Γmin .
At infinite bias fL (ω) = and fR (ω) = 0, the pump does not transport electrons against any
potential or thermal gradient but can still be used to control the statistics of the tunneled electrons.
For example, it is possible to reduce the noise to zero also in this limit when Γmin → 0.
Chapter 7
Closed-loop control:
Non-Equilibrium Case IV
Closed-loop (or feedback) control means that the system is monitored (either continuously or at
certain times) and that the result of these measurements is fed back by changing some parameter of
the system. Under measurement with outcome m (an index characterizing the possible outcomes),
the density matrix transforms as
m
ρ→
M ρM †
n m m o,
†
Tr Mm
Mm ρ
(7.1)
†
and the probability at which this outcome occurs is given by Tr Mm
Mm . This can also be
written in superoperator notation
Pm ρ
m
ρ→
.
(7.2)
Tr {Pm ρ}
Let us assume that conditioned on the measurement result m at time t, we apply a propagator
for the time interval ∆t. Then, a measurement result m at time t provided, the density matrix at
time t + ∆t will be given by
Pm ρ
(m)
.
(7.3)
ρ(m) (t + ∆t) = eL ∆t
Tr {Pm ρ}
However, to obtain an effective description of the density matrix evolution, we have to average
over all measurement outcomes – where we have to weight each outcome by the corresponding
probability
X
X (m)
Pm ρ
(m)
ρ(t + ∆t) =
Tr {Pm ρ(t)} eL ∆t
=
eL ∆t Pm ρ(t) .
(7.4)
Tr
{P
ρ}
m
m
m
Note that this is an iteration scheme, the conditioned Liouvillian L(m) may well depend on the time
t (at which the measurement is performed) as long as it is constant during the interval [t, t + ∆t].
7.1
Single junction
To start with the simplest possible system, we reconsider here the case of a single junction in the
unidirectional transport limit, which is described by
ρ̇n = γρn−1 − γρn ,
85
(7.5)
86
CHAPTER 7. CLOSED-LOOP CONTROL:
NON-EQUILIBRIUM CASE IV
where the parameter γ describes the speed at which the resulting Poissonian distribution
−γ∆t (γ∆t)n
e
: n≥0
n!
ρn (∆t) =
0
: n<0
(7.6)
moves towards larger n. When we arrange the probabilities in an infinite-dimensional vector, the
master equation becomes

 .   .
.. 
.
..
.
 . 
  ..
  ρn−1 
d 
 ρn−1  
. −γ

.
(7.7)

=
  ρn 
dt  ρn  

+γ
−γ


..
..
... ...
.
.
For the initial state ρn (0) = δn,0 we have written the solution explicitly. Using the translational
invariance in n and the superposition principle, we can therefore write the general solution explicitly
as





1
ρ0 (t)
ρ0 (t + ∆t)
  ρ (t) 
 γ∆t
1
 ρ1 (t + ∆t) 
 1
 (γ∆t)2


  ρ (t) 


 ρ2 (t + ∆t) 
γ∆t
1
 2

2



−γ∆t 

.
.
.
.
.
.
(7.8)
=
e


.. ..  
..
.. 
..
 ..









 (γ∆t)n (γ∆t)n−1 . . . . . .  
 ρn (t + ∆t) 
  ρn (t) 
 n!
(n−1)!
..
..
..
..
.
.
.
.
which takes the form ρ(t + ∆t) = W(∆t)ρ(t) with the infinite-dimensional propagation matrix
W(∆t).
Exercise 45 (Probability conservation) (1 points)
Show
that the above
P
P introduced propagator W(∆t)preserves the sum of all probabilities, i.e., that
ρ
(t
+
∆t)
=
n n
n ρn (t).
We have found previously that an open-loop control scheme does not drastically modify the
picture. We do now consider regular measurements of the number of tunneled particles being
performed at time intervals ∆t. The major difference to our previous considerations is now that
we modify the tunneling rate γ in dependence on the initial state ρ(t) and on the time t, i.e., we
have γ → γm (t), where m denotes the total number of particles measured at time t. Then, the
propagation matrix becomes (omitting for brevity the dependence on t in the rates)


e−γ0 ∆t
 e−γ0 ∆t (γ0 ∆t)

e−γ1 ∆t


2
−γ1 ∆t
−γ2 ∆t
 e−γ0 ∆t (γ0 ∆t)

e
(γ1 ∆t)
e


2

.
.
.
.
W(t, ∆t) = 
(7.9)
.
.
.
.

. .
.
.
.


n
n−1
n−2
2 ∆t)
 e−γ0 ∆t (γ0 ∆t) e−γ1 ∆t (γ1 ∆t)
e−γ2 ∆t (γ(n−2)!
... 


n!
(n−1)!
..
..
..
...
.
.
.
87
7.1. SINGLE JUNCTION
When we would denote the normal propagator in Eq. (7.8) with an index i as W (i) = W|γ=γi , the
full effective feedback propagator can also be written as
X
W(t, ∆t) =
W (n) (t, ∆t)P (n) ,
(7.10)
n
(n)
where the matrix elements of the projection superoperator Pij = δin δjn vanish except at position
n on the diagonal.
Exercise 46 (Effective Feedback Propagator) (1 points)
Show the validity of Eq. (7.10).
This naturally includes the measurement process into the description of feedback: Measuring
n tunneled particles at time t projects the density vector as
P (n) ρ(t)
,
(7.11)
Tr {P (n) ρ(t)}
which occurs with probability Tr P (n) ρ(t) . When we compute the average outcome of such
a measurement but do not perform any feedback, we have to multiply with the corresponding
probability to obtain the average density matrix at time t + ∆t
X
ρ̄(t + ∆t) =
W(t, ∆t)P (n) ρ(t) = W(t, ∆t)ρ(t) .
(7.12)
ρ(t) →
n
With feedback however, a propagator that is conditioned on the measurement result has to be
applied
X
ρ̄(t + ∆t) =
W (n) (t, ∆t)P (n) ρ(t) ,
(7.13)
n
which corresponds to Eq. (7.10).
The matrix elements of the effective feedback propagatoor read
(
∆t)(n−m)
: n≥m
e−γm ∆t (γm(n−m)!
,
Wnm (t, ∆t) =
0
: n<m
where e.g. one simple feedback protocol would be given by
γ + α γ − mt
: m ≤ γt 1 + α1
γm (t) =
.
0
:
else
(7.14)
(7.15)
Such a feedback protocol would aim to stabilize a mean of m = γt and should also keep the width
of the distribution at a finite value. For general feedback protocols, we obtain for the first moment
at time t + ∆t
"
#
X
X X
X X
hnit+∆t =
nρn (t + ∆t) =
n
Wnm (t, ∆t)ρm (t) =
nWnm (t, ∆t) ρm (t)
=
n
" ∞
X X
m
=
X
m
n=m
n
ne−γm ∆t
m
(n−m)
(γm ∆t)
(n − m)!
#
n
#
" ∞m
n
X X
(γ
∆t)
m
ρm (t) =
ρm (t)
(n + m)e−γm ∆t
n!
m
n=0
(γm ∆t + m) ρm (t) = hγn it ∆t + hnit ,
(7.16)
88
CHAPTER 7. CLOSED-LOOP CONTROL:
NON-EQUILIBRIUM CASE IV
such that the change of the first moment during time interval ∆t is determined by the average
tunneling rate. Similarly, we obtain for the second moment
#
"∞
n
X
X X
2
(γm ∆t)
ρm (t)
n t+∆t =
n2 ρn (t + ∆t) =
(n + m)2 e−γm ∆t
n!
n
m
n=0
X
γm ∆t (1 + γm ∆t) + 2γm ∆tm + m2 ρm (t)
=
m2 = γn t ∆t2 + hγn (1 + 2n)it ∆t + n2 t .
This implies for the variance under feedback
2
n t+∆t − hni2t+∆t = ∆t2 γn2 t − hγn i2t + ∆t [hγn it + 2 hγn nit − 2 hγn it hnit ]
+ n2 t − hni2t .
(7.17)
(7.18)
For
the first term on the r.h.s. will dominate, such that due to hγn2 it − hγn i2t =
large ∆t,
(γn − hγn i)2 ≥ 0 the variance will always increase. For small ∆t however, the second term
may dominate, such that for adapted feedback protocols, the variance of the distribution may
decrease during ∆t.
Exercise 47 (Variance evolution without feedback) (1 points)
Show that without feedback γm (t) = γ, the variance
during the iteration will for arbitrary distribu
tions always increase as hn2 it+∆t − hni2t+∆t − hn2 it − hni2t = γ∆t.
It is however clear that the resulting distribution cannot have a completely vanishing width,
as for this case one would also obtain a growing variance in time.
Exercise 48 (Variance evolution of a localized distribution) (1 points)
Show that for arbitrary feedback, the variance of a localized distribution ρm (t) = δmm̄ the variance
will always increase unless γm̄ = 0.
Therefore, the resulting time-dependent distribution may be expected to have a finite width if
the iteration scheme is performed in a range where the feedback is negative. This can be illustrated
at the example of a continuous (n → x) Gaussian distribution
ρx (t) = √
(x−γt)2
1
e− 2σ2
2πσ
(7.19)
with mean µ = γt = hxit and width σ. The control scheme
αx
γx (t) = γeα e− γt
(7.20)
leads to the following expectation values (replacing sums by integrals and extending the integration
range over the complete real axis)
α2 σ 2
α2 σ 2
α2 σ 2
2
ασ 2
2
2 2 γ 2 t2
2 t2
2 t2
2γ
2γ
,
(7.21)
,
γx = γ e
,
hxγx i = e
γ t−
hγx i = γe
t
89
7.1. SINGLE JUNCTION
which leads to a change of the variance of
2 γx t − hγx i2t + ∆t [hγx it + 2 hxγx it − 2 hxit hγx it ]
2 2
α2 σ 2
α σ
α2 σ 2 ∆t 2γ
2 2 γ 2 t2
2
2
= ∆t γ e
eγ t −1 +
e 2 t2 γt − 2ασ 2 .
t
∆σ 2 = ∆t2
(7.22)
Neglecting the quadratic contribution (small ∆t) the variance is stabilized when the feedback
γt
strength is adapted with time and width of the distribution α ≥ 2σ
This is also confirmed
2.
by the numerical propagation of an initially localized probability distribution with the iteration
scheme (7.12) in Fig. 7.1. Without feedback (thin lines), the distribution propagates, but also
1
-1
feedback distribution Pn(t)
10
0,8
-2
10
-3
10
0,6
t=100
t=200
t=300
t=400
t=500
Gauss fit: σ=7.1
-4
10
t=100
t=80
t=60
t=40
t=20
t=0
0,4
-5
10
4
sampling N=10
-6
10
-50
-25
0
n-γt
60
80
25
50
0,2
0
0
20
40
100
120
particle number n
Figure 7.1: Plot of probability distributions Pn (t) without feedback (Poissonian evolution, thin
lines), with constant feedback strength (medium thickness) and with time-proportional feedback
strength (thick curves). The inset shows the long-time evolution for constant feedback strength
(increasing width) and time-proportional feedback strength (stationary width). The fit (dashed
curve) demonstrates that the stationary distribution is Gaussian. In addition, a direct sampling of
the feedback protocol yields the same distribution. Parameters: γ = 1, ∆t = 1, α = 1 for constant
feedback and α = 0.01t for time-proportional feedback.
increases its width in time. With constant feedback strength, the distribution propagates at the
same speed and its width is reduced but still increases with time for sufficiently large times. Only
when the feedback strength is chosen to scale linarly in time, the shape of the distribution becomes
stabilized. This in turn allows to determine the stationary width in dependence of the feedback
strength α = α0 t as follows: Inserting this dependence in Eq. (7.22) leads to a condition under
90
CHAPTER 7. CLOSED-LOOP CONTROL:
NON-EQUILIBRIUM CASE IV
which the variance of a Gaussian distribution remains constant
0 = (γ∆t)ey/2 [ey − 1] + 1 − 2
γ
y
α0
(7.23)
with the variable
y=
α02 σ 2
.
γ2
(7.24)
This transcendental equation can for given ∆t, γ and α0 be solved numerically for y, which in turn
enables one to determine the stationary width σ. For example, for γ∆t = 1 and γ/α0 = 100 we
find a numerical solution y ≈ 0.005025, which corresponds to a width of σ ≈ 7.1. This width is
exactly found in the Gaussian fit of the numerical solution (compare the inset of Fig. 7.1). The
validity of this effective description under feedback is further underlined by a direct sampling of
the feedback protocol (also compare the inset): Here, the feedback parameter is adjusted at finite
time intervals ∆t, and dependent on the previous measurement result. Therefore, under the chosen
feedback protocol
γn (t) = γeα0 t e−
α0 n
γ
(7.25)
the long-term limit leads to a frozen distribution which propagates with a constant shape.
These findings can be made more explicit for linear feedback of the form
γn (t) = γ [1 − g(n − γt)]
(7.26)
with the dimensionless feedback parameter g > 0. It can be thought of an approximation of scheme
(7.25) for small g = α0 /γ ≪ 1. Of course, the above scheme formally allows for negative rates
when n ≫ γ0 t. In reality however, the probability for such a process is exponentially suppressed
for sufficiently large times, since for large
√ times the width of a Poissonian process is sufficiently
smaller than its mean value σ/µ = 1/ γt. The linear feedback scheme has the advantage that
Eqns. (7.16) and (7.18) only couple to themselves and can thus be solved
C1 (t + ∆t) − C1 (t) = γ∆t [1 − gC1 (t) + gγt] ,
C2 (t + ∆t) − C2 (t) = γ 2 ∆t2 g 2 C2 (t) + γ∆t [1 + γgt − gC1 (t) − 2gC2 (t)] .
(7.27)
In the limit of small ∆t, the differential version of these equations read
Ċ1 = γ [1 − gC1 (t) + gγt] ,
Ċ2 = γ [1 + γgt − gC1 (t) − 2gC2 (t)]
(7.28)
and admit for the initial conditions C1 (0) = 0 and C2 (0) = 0 the simple solution
C1 (t) = γt ,
C2 (t) =
1 − e−2gγt
,
2g
which shows a continuous evolution towards a constant width of σ̄ =
(7.29)
q
lim C2 (t) =
t→∞
√1 .
2g
91
7.2. MAXWELL’S DEMON
7.2
Maxwell’s demon
measured QPC current
Maxwell invented his famous demon as a thought experiment to demonstrate that thermodynamics
is a macroscopic effective theory: An intelligent being (the demon) living in a box is measuring
the speed of molecules of some gas in the box. An initial thermal distribution of molecules implies
that the molecules have different velocities. The demon measures the velocities and inserts an
impermeable wall whenever the the molecule is too fast or lets it pass into another part of the box
when it is slow. As time progresses, this would lead to a sorting of hot and cold molecules, and
the temperature difference could be exploited to perform work.
This is nothing but a feedback (closed-loop) control scheme: The demon performs a measurement (is the molecule slow or fast) and then uses the information to perform an appropriate
control action on the system (inserting a wall or not). Classically, the insertion of a wall requires
in the idealized case no work, such that only information is used to create a temperature gradient.
However, the Landauer principle states that with each bit of information erased, heat of at least
kB T ln(2) is dissipated into the environment. To remain functionable, the demon must at some
point start to delete the information, which leads to the dissipation of heat. The dissipated heat
will exceed the energy obtainable from the thermal gradient.
An analog of a Maxwell demon may be implemented in an electronic context: There, an
experimentalist takes the role of the demon. The box is replaced by the SET (including the
contacts), on which by a nearby QPC a measurement of the dot state (simply empty or filled) is
performed. Depending on the measurement outcome, the tunneling rates are modified in time in
a piecewise constant manner: When there is no electron on the dot, the left tunneling rate ΓL is
increased (low barrier) and the right tunneling rate ΓR is decreased (high barrier). The opposite
is done when there is an electron on the dot, see Fig. 7.2. Thus, the only difference in comparison
SET
0,3
0,2
0,1
0
SET
200
400
600
800
1000
Figure 7.2: Sketch of the feedback scheme: For a filled dot (low QPC current), the left tunneling
rate is minimal and the right tunneling rate is maximal and vice-versa for an empty dot. The dot
level itself is not changed.
to Sec. 6.2.1 is that now information of the system is used to modify the tunneling rates. Very
simple considerations already demonstrate that with this scheme, it will be possible to transport
electrons against an existing bias only with time-dependent tunneling rates. When one junction is
92
CHAPTER 7. CLOSED-LOOP CONTROL:
NON-EQUILIBRIUM CASE IV
completely decoupled Γmin
L/R → 0, this will completely rectify the transport from left to right also
against the bias (if the bias is finite). In the following, we will address the statistics of this device.
The first step is to identify an effective evolution equation for the density matrix accounting
for measurement and control. In this case, both processes occur continuously, which enables
to
us 1
,
consider infinitesimally small time intervals ∆t. For an empty dot at time t, i.e., for ρE (t) =
0
the density vector at time t + ∆t will be given by
ρE (t + ∆t) = 1 + ∆tL(E) ρE (t) .
(7.30)
0
Similarly, we have for a filled dot ρE (t) =
1
(7.31)
ρF (t + ∆t) = 1 + ∆tL(F ) ρF (t) .
Both equations can be combined using a projection superoperator
ρ(t + ∆t) = 1 + ∆tL(E) PE + 1 + ∆tL(F ) PF ρ(t)
(7.32)
with the operators
PE =
1 0
0 0
,
PF =
0 0
0 1
.
(7.33)
Since we have PE + PF = 1, this defines when ∆t → 0 (continous control limit) an effective
Liouvillian
Leff = L(E) PE + L(F ) PF ,
(7.34)
which is in complete equivalence with Eq. (7.10) when ∆t → 0. Note that this can be performed
with and without counting fields. Taking into account the diagonal structure of the projection
superoperators, this simply implies that the effective Liouvillian under feedback has the first column
from the Liouvillian conditioned on an empty dot and the second column from the Liouvillian
conditioned on the filled dot
E
F
+iχL
F
+iχR
−ΓE
f
−
Γ
f
+Γ
(1
−
f
)e
+
Γ
(1
−
f
)e
L
R
L
R
R
L
L
R
. (7.35)
Leff (χL , χR ) =
−iχL
−iχR
+ΓE
+ ΓE
−ΓFL (1 − fL ) − ΓFR (1 − fR )
L fL e
R fR e
Evidently, it still obeys trace conservation but now the tunneling rates in the two columns are
different ones.
Exercise 49 (Current at zero bias) (1 points)
Calculate the feedback-current at zero bias fL = fR = f in dependence on f . What happens at zero
temperatures, where f → {0, 1}?
The effective Liouvillian describes the average evolution of trajectories under continuous monitoring and feedback. The validity of the effective description can be easily checked by calculating
Monte-Carlo solutions as follows:
93
7.2. MAXWELL’S DEMON
Starting e.g. with a filled dot, the probability to jump out e.g. to the right lead during the
(F )
small time interval ∆t reads Pout,R = ΓFR (1 − fR )∆t. Similarly, we can write down the probabilities
to jump out to the left lead and also the probabilities to jump onto an empty dot from either the
left or right contact
(F )
Pout,R = ΓFR (1 − fR )∆t ,
(E)
Pin,R = ΓE
R fR ∆t ,
(F )
(E)
Pout,L = ΓFL (1 − fL )∆t ,
Pin,L = ΓE
L fL ∆t .
(7.36)
Naturally, these jump probabilities also uniquely determine the change of the particle number on
either contact. The remaining probability is simply the one that no jump occurs during ∆t. A
Monte-Carlo simulation is obtained by drawing a random number and choosing one out of three
possible outcomes for empty (jumping in from left contact, from right contact, or remaining empty)
and for a filled (jumping out to left contact, to right contact, or remaining filled) dot. Repeating
the procedure several times yields a single trajectory for n(t), nL (t), and nR (t). The ensemble
average of many such trajectories agree perfectly with the solution
hnit = Tr d† deLeff (0,0)t ρ0 ,
hnL it = (−i∂χ ) Tr eLeff (χ,0)t ρ0 χ=0 ,
hnR it = (−i∂χ ) Tr eLeff (0,χ)t ρ0 χ=0
(7.37)
of the effective feedback master equation, see Fig. 7.3. To compare with the case without feedback,
we parametrize the change of tunneling rates by dimensionless constants
E
δL
ΓE
L = e ΓL ,
E
δR
ΓE
R = e ΓR ,
F
ΓFL = eδL ΓL ,
F
ΓFR = eδR ΓR ,
(7.38)
where δαβ → 0 reproduces the case without feedback and δαβ > 0(< 0) increases (decreases) the
tunneling rate to contact α conditioned on dot state β. The general current can directly be
calculated as
I=
F
F E
fL (1 − fR )ΓE
L ΓR − (1 − fL )fR ΓL ΓR
,
F
E
F
ΓE
L fL + ΓL (1 − fL ) + ΓR fR + ΓR (1 − fR )
(7.39)
which reduces to the conventional current without feedback when Γβα → Γα . For finite feedback
strength however, this will generally induce a non-vanishing current at zero bias, see Fig. 7.4.
In our idealized setup, this current is only generated by the information on whether the dot is
occupied or empty – hence the interpretation as a Maxwell demon. When the contacts are held at
equal temperatures βL = βR = β, this raises the question for the maximum power
P = −IV
(7.40)
generated by the device.
In what follows, we will consider symmetric feedback characterized by a single parameter
F
F
E
∆E
L = ∆R = −∆L = −∆R = +δ ,
(7.41)
where δ > 0 favors transport from left to right and δ < 0 transport from right to left and also
symmetric tunneling rates Γ = ΓL = ΓR . With these assumptions, it is easy to see that for large
feedback strengths δ ≫ 1, the current simply becomes
I → Γeδ
fL (1 − fR )
.
fL + (1 − fR )
(7.42)
94
CHAPTER 7. CLOSED-LOOP CONTROL:
NON-EQUILIBRIUM CASE IV
nSET(t)
1
0,8
0,6
0,4
0,2
0
no feedback reference
Ntraj=1
nL(t)/t
0,5
Ntraj=100
0
Ntraj=10000
feedback ME
-0,5
nR(t)/t
0,5
0
5
10
15
-0,5
dimensionless time Γ t
Figure 7.3: Comparison of a single (thin red curve with jumps, same realization in all panels) and
the average of 100 (medium thickness, green) and 10000 (bold smooth curve, turquoise) trajectories
with the solution from the effective feedback master equation (thin black) for the dot occupation
(top), the number of particles on the left (middle), and the number of particles on the right
(bottom). The average of the trajectories converges to the effective feedback master equation
result. The reference curve without feedback (dashed orange) may be obtained by using vanishing
feedback parameters and demonstrates that the direction of the current may actually be reversed
via sufficiently strong feedback. Parameters: ΓL = ΓR ≡ Γ, fL = 0.45, fR = 0.55, δLE = δRF = 1.0,
δRE = δLF = −10.0, and Γ∆t = 0.01.
To determine the maximum power, we would have to maximize with respect to left and right
chemical potentials µL and µR , the lead temperature β and the dot level ǫ. However, as these
parameters only enter implicitly in the Fermi functions, it is more favorable to use that for equal
temperatures
fL (1 − fR )
,
(7.43)
β(µL − µR ) = βV = ln
(1 − fL )fR
such that we can equally maximize
1
fL (1 − fR )
Γeδ
fL (1 − fR )
P = −IV = (−IβV ) →
−
.
ln
β
β
fL + (1 − fR )
(1 − fL )fR
(7.44)
The term in square brackets can now be maximized numerically with respect to the parameters
fL and fR in the range 0 ≤ fL/R ≤ 1, such that one obtains for the maximum power at strong
feedback
P ≤ kB T Γeδ 0.2785
at
fL = 0.2178
fR = 0.7822 .
(7.45)
The average work performed between two QPC measurement points at t and t + ∆t is therefore
given by
W ≤ kB T Γeδ ∆t0.2785 .
(7.46)
95
7.2. MAXWELL’S DEMON
current I [Γ]
no feedback
piecewise constant feedback
-4
-3
-2
-1
0
1
2
3
4
Figure 7.4: Current voltage characteristics
for finite feedback strength δ = 1 (red curve)
and without feedback δ = 0 (black curve).
For finite feedback, the current may point
in the other direction than the voltage leading to a positive power P = −IV (shaded
region) generated by the device.
dimensionless bias voltage β V
However, to perform feedback efficiently, it is required that the QPC sampling rate is fast enough
that all state changes of the SET are faithfully detected (no tunneling charges are missed). This
requires that Γeδ ∆t < 1. Therefore, we can refine the upper bound for the average work
W ≤ kB T 0.2785 .
(7.47)
This has to be contrasted with the Landauer principle, which states that for each deleted bit in
the demons brain (each QPC data point enconding high current or low current) heat of
Q ≥ kB T ln(2) ≈ kB T 0.6931
(7.48)
is dissipated. The second law is therefore not violated.
The conventional fluctuation theorem for the SET at equal temperatures
P+n (t)
= enβV
P−n (t)
(7.49)
is modified in presence of feedback. However, since the Liouvillian still contains the counting
fields in the conventional way, simply the factor in the exponential, not the dependence on the
number of tunneled electrons n is changed. To evaluate the FT, we identify symmetries in the
moment-generating function (or alternatively the eigenvalues of the Liouvillian)
+4δ fL (1
− fR )
λ(−χ) = λ +χ + i ln e
(1 − fL )fR
= λ(+χ + i(4δ + βV )) .
= λ +χ + i ln e+4δ eβV
(7.50)
From this symmetry of the cumulant-generating function we obtain for the fluctuation theorem
under feedback
P+n (t)
∗
= en(βV +4δ) = enβ(V −V ) ,
t→∞ P−n (t)
lim
where V ∗ = −4δ/β denotes the voltage at which the current (under feedback) vanishes.
(7.51)
96
CHAPTER 7. CLOSED-LOOP CONTROL:
NON-EQUILIBRIUM CASE IV
Exercise 50 (Vanishing feedback current) (1 points)
Show for equal temperatures that the feedback current vanishes when V = V ∗ = −4δ/β.
measured QPC current
However, piecewise constant feedback is not the only relevant scheme. As a further example,
let us consider instantaneously applying a fast and short pulse on the tunneling rates whenever
the QPC detects a change in the dots occupation, see Fig. 7.5. To change the counting statistics,
0,3
0,2
0,1
0
200
SET
400
600
800
1000
SET
Figure 7.5: Sketch of the modified feedback
scheme: This time, feedback control operations are not conditioned on the state of
the dot, but on its change. Whenever an
electron has jumped into the dot (red arrows), the left tunneling rate is decreased
and the right tunneling rate is increases with
an instantaneous pulse. The opposite is done
when an electron has jumped out of the dot
(green arrows). The dot level itself is not
changed. To obtain a significant change of
the statistics, it is necessary that particles
may also tunnel during the pulses. This
opens the possibility of setting up feedback
schemes with a finite recursion depth: Electrons tunneling during the control operations may trigger further control operations
and so on – eventually limited by the finite
processing speed of the apparatus and other
factors.
the infinitely short pulse of this feedback scheme must be infinitely large and can thus be well
represented by a δ function. Since for an SET the QPC does not resolve the direction of the
electronic jumps, we can only perform control operations conditioned on particles jumping in and
out
X
X
L(t) = L0 +
L(I) +
L(O) ,
(7.52)
tin
tout
where tin and tout denote the times at which an electron jumps in or out of the dot, respectively.
The control Liouvillians have the form
+δLI (1 − fL )e+iχL + δRI (1 − fR )e+iχR
−δLI fL − δRI fR
(I)
L
= δ(t − tin )
,
+δLI fL e−iχL + δRI fR e−iχR
−δLI (1 − fL ) − δRI (1 − fR )
−δLO fL − δRO fR
+δLO (1 − fL )e+iχL + δRO (1 − fR )e+iχR
(O)
L
= δ(t − tout )
(7.53)
,
+δLO fL e−iχL + δRO fR e−iχR
−δLO (1 − fL ) − δRO (1 − fR )
where δαβ > 0 denote dimensionless feedback parameters and δαβ = 0 reproduces the case without
97
7.2. MAXWELL’S DEMON
feedback. This implies a simple form of the corresponding (dimensionless) propagators
−δLI fL − δRI fR
+δLI (1 − fL )e+iχL + δRI (1 − fR )e+iχR
κI
e
= exp
,
+δLI fL e−iχL + δRI fR e−iχR
−δLI (1 − fL ) − δRI (1 − fR )
−δLO fL − δRO fR
+δLO (1 − fL )e+iχL + δRO (1 − fR )e+iχR
κO
e
= exp
(7.54)
+δLO fL e−iχL + δRO fR e−iχR
−δLO (1 − fL ) − δRO (1 − fR )
during such an infinitesimally short pulse.
Exercise 51 (δ-kick propagator) (1 points)
Calculate the general propagator for a time-dependent Liouvillian of the piecewise-constant form
L = L0 + Lc Θ(t − tc )Θ(tc + τc − t) ,
where tc denotes the control time and τc the control duration. What is the propagator when Lc =
κ/τc in the limit τc → 0?
With these Liouvillians it is still necessary to derive an effective Liouvillian under feedback.
For tunneling events during the baseline propagation, we have for the conditional density matrix
at time t + ∆t
ρ(nL ,nR ) (t + ∆t) =
1
2π
2 Z+π
exp [L0 (χL , χR )∆t − inL χL − inR χR ] dχL dχR ρ(t) ,
(7.55)
−π
where nL and nR denote the number of particles that have tunneled to the left and right junction
during ∆t, respectively. Since we are interested in continous monitoring, it suffices to expand to
first order in ∆t. With the splitting of the Liouvillian
L0 (χL , χR ) = L(0,0) + L(1,0) e+iχL + L(−1,0) e−iχL + L(0,1) e+iχR + L(0,−1) e−iχR
(7.56)
it becomes obvious that for small ∆t only single-tunneling events need to be considered. The
superoperators
J 00 (∆t) =
J ±1,0 (∆t) =
J 0,±1 (∆t) =
1
2π
1
2π
1
2π
2 Z+π
exp [L0 (χL , χR )∆t] dχL dχR ≈ 1 + L(0,0) ∆t ,
−π
2 Z+π
exp [L0 (χL , χR )∆t ∓ iχL ] dχL dχR ≈ L±1,0 ∆t ,
−π
2 Z+π
exp [L0 (χL , χR )∆t ∓ iχR ] dχL dχR ≈ L0,±1 ∆t
(7.57)
−π
now take the role of projection superoperators for a measurement of the particle numbers nL and
nR during ∆t. The effective propagator under feedback from t to t + ∆t can be written as
X
ρ(t + ∆t) =
Wα (∆t)Pα ρ(t) ,
(7.58)
α
98
CHAPTER 7. CLOSED-LOOP CONTROL:
NON-EQUILIBRIUM CASE IV
where Wα (∆t) denotes the propagator conditioned on measurement result α and Pα the corresponding projection superoperator. In the continous measurement limit, this defines an effective
Liouvillian via solving
h
i
κI
(0,−1)
(−1,0)
0,0
κO
(0,1)
(1,0)
ρ(t) (7.59)
+ e ∆t L
+L
ρ(t + ∆t) = 1 + ∆tL + e ∆t L
+L
for the difference quotient (ρ(t + ∆t) − ∆t)/∆t and letting ∆t → 0. With counting fields, it reads
I
O
Leff = L(0,0) + eκ (χL ,χR ) e+iχR L(0,1) + e+iχL L(1,0) + eκ (χL ,χR ) e−iχR L(0,−1) + e−iχL L(−1,0) (, 7.60)
where it becomes quite obvious that the dependence on the counting fields becomes nontrivial.
This is valid for a recursion depth zero, i.e., particles tunneling during the control operations do
not trigger further control operations. As with the piecewise constant scheme, this may also be
compared to a Monte-Carlo simulation of the feedback scheme, see Fig. 7.6 . For some analytic
nSET(t)
1
0,8
0,6
0,4
0,2
0
nL(t)/t
0,5
0
no-feedback reference
Ntraj=1
-0,5
Ntraj=100
Ntraj=10000
nR(t)/t
0,5
feedback ME
0
5
10
-0,5
dimensionless time Γ t
15
Figure 7.6: The average of the Monte-Carlo
trajectories converges to the effective feedback master equation result. Control operations (bold kinks in top graph) are not
always correlated with a change in the occupation or the particle number left and right,
since only the net particle number after control is displayed. Parameters: ΓL = ΓR ≡ Γ,
fL = 0.45, fR = 0.55, δLI = δRO = 0,
δRI = δLO = 1.0, and Γ∆t = 0.01.
understanding, let us assume that the control pulses only affect a single junction at a time with a
similar feedback strength
−δfR
+δ(1 − fR )e+iχR
1 − fR e+iχR (1 − fR )
δ→∞
κI (χR )
e
= exp
,
→
+δfR e−iχR
e−iχR fR
−δ(1 − fR )
fR
−δfL
+δ(1 − fL )e+iχL
1 − fL e+iχL (1 − fL )
δ→∞
κO (χL )
e
= exp
, (7.61)
→
+δfL e−iχL
−δ(1 − fL )
e−iχL fL
fL
where the propagators have a finite limit even in the limit of infinite feedback strength.
Exercise 52 (Infinite Feedback Strength) (1 points)
Show that for infinite feedback strength coupling to a single junction leads to an immediate equilibration of the density matrix with the contact to which it is coupled during control.
Therefore, the effective Liouvillian for infinite feedback strength is given by
−ΓL fL − ΓR fR
0
Leff (χL , χR ) =
0
−ΓL (1 − fL ) − ΓR (1 − fR )
1 − fL e+iχL (1 − fL )
0 ΓL (1 − fL )e+iχL + ΓR (1 − fR )e+iχR
+
e−iχL fL
fL
0
0
+iχF
1 − fR e
0
0
(1 − fR )
.
(7.62)
+
ΓL fL e+iχL + ΓR fR e+iχR 0
e−iχR fR
fR
99
7.3. QUBIT STABILIZATION
The non-standard dependence on the counting fields (there are for example counting fields on the
diagonal and products of left- and right-associated counting fields) severly disturbs the symmetries,
such that the fluctuation theorem is broken
P+n
6= enα .
P−n
7.3
(7.63)
Qubit stabilization
Qubits – any quantum-mechanical two-level system that can be prepared in a superposition of its
two states |0i and |1i – are at the heart of quantum computers with great technological promises.
The major obstacle to be overcome to build a quantum computer is decoherence: Qubits prepared
in pure superposition states (as required for performing quantum computation) tend to decay into
a statistical mixture when coupled to a destabilizing reservoir (of which there is an abundance in
the real world). Here, we will approach the decoherence with a quantum master equation and use
feedback control to act against the decay of coherences.
The system is described by
X
Ω z
σ ,
HB1 =
ωk1 (b1k )† b1k ,
2
k
X
1 1
z
hk bk + (h1k )∗ (b1k )† ,
= σ ⊗
HS =
HI1
k
HB2 =
HI2
X
ωk2 (b2k )† b2k
k
x
=σ ⊗
X
k
h2k b2k + (h2k )∗ (b2k )† ,
(7.64)
where σ α represent the Pauli matrices and bk bosonic annihilation operators. We assume that
the two bosonic baths are independent and at the same temperature, such that we can calculate
the dissipators separately. Then, the bath correlation function is also independent on the chosen
interaction Hamiltonian (therefore omitting the indices)
C(τ ) =
X D
kk′
=
X
k
1
=
2π
hk bk e−iωk τ + h∗k b†k e+iωk τ
E
hk′ bk′ + h∗k′ b†k′
|hk |2 e−iωk τ (1 + n(ωk )) + e+iωk τ n(ωk )
Z∞
0
J(ω) e−iωτ [1 + n(ω)] + e+iωτ n(ω) dω ,
(7.65)
−1
where we have introduced the bosonic occupation n(ω) = eβω − 1
with the inverse bath temP
2
perature β and the spectral density J(ω) = 2π k |hk | δ(ω − ωk ). We would like to identify the
Fourier transform, such that we have to transform the integral to one involving the full real axis.
Since all ωk > 0, the spectral density is defined for positive frequencies but can be analytically
continued by defining J(−ω) = −J(+ω). Making use of the relation n(−ω) = − [1 + n(ω)], we
100
CHAPTER 7. CLOSED-LOOP CONTROL:
NON-EQUILIBRIUM CASE IV
rewrite the correlation function as
1
C(τ ) =
2π
Z∞
J(ω) [1 + n(ω)] e
−iωτ
1
dω +
2π
0
1
=
2π
Z∞
1
2π
J(−ω)n(−ω)e−iωτ dω
Z0
J(ω) [1 + n(ω)] e−iωτ dω
−∞
J(ω) [1 + n(ω)] e
−iωτ
1
dω +
2π
0
=
Z0
Z∞
−∞
J(ω) [1 + n(ω)] e−iωτ dω .
(7.66)
−∞
We conclude for the Fourier-transform of the bath correlation function
γ(ω) = J(ω) [1 + n(ω)] .
(7.67)
It is easy to show that it fulfils the KMS condition
γ(−ω) = −J(ω) [−n(ω)] = J(ω)n(ω) = e−βω γ(+ω) ,
(7.68)
such that we may expect thermalization of the qubits density matrix with the bath temperature.
Note that due to the divergence of n(ω) at ω → 0, it is favorable to use an Ohmic spectral density
such as e.g.
J(ω) = J0 ωe−ω/ωc ,
(7.69)
which grants an existing limit γ(0). For a single system coupling operator A, the dampening
coefficients in the BMS approximation in Box 7 simplify a bit
γab,cd = δ(Eb − Ea , Ed − Ec )γ(Eb − Ea ) ha| A |bi hc| A |di∗
1 X
σab = δEa ,Eb
σ(Eb − Ec ) hc| A |bi hc| A |ai∗ ,
2i c
(7.70)
where the odd Fourier transform σ(ω) of the bath correlation function can be obtained from γ(ω)
by a Cauchy-principal value integral and the vectors denote the energy eigenbasis σ z |0i = |0i and
σ z |1i = − |1i. The Kronecker symbols automatically imply that many dampening coefficients
vanish, such that the action of the full Liouvillian can be written as
Ω
Ω
+ σ00 |0i h0| + − + σ11 |1i h1| , ρ
ρ̇ = −i
2
2
1
1
+γ00,00 |0i h0| ρ |0i h0| − |0i h0| ρ − ρ |0i h0|
2
2
1
1
+γ11,11 |1i h1| ρ |1i h1| − |1i h1| ρ − ρ |1i h1|
2
2
+γ00,11 [|0i h0| ρ |1i h1|] + γ11,00 [|1i h1| ρ |0i h0|]
1
1
+γ01,01 |0i h1| ρ |1i h0| − |1i h1| ρ − ρ |1i h1|
2
2
1
1
+γ10,10 |1i h0| ρ |0i h1| − |0i h0| ρ − ρ |0i h0| .
(7.71)
2
2
101
7.3. QUBIT STABILIZATION
It is a general feature of the BMS approximation that for a nondegenerate system Hamiltonian
(here Ω 6= 0) the populations and coherences in the energy eigenbasis decouple
ρ̇00 = −γ10,10 ρ00 + γ01,01 ρ11 ,
ρ̇11 = +γ10,10 ρ00 − γ01,01 ρ11 ,
1
ρ̇01 = − (γ00,00 + γ11,11 − 2γ00,11 + γ01,01 + γ10,10 ) − i (Ω + σ00 − σ11 ) ρ01 ,
2
1
ρ̇10 = − (γ00,00 + γ11,11 − 2γ11,00 + γ01,01 + γ10,10 ) + i (Ω + σ00 − σ11 ) ρ10 .
2
(7.72)
For the two interaction Hamiltonians chosen, we can make the corresponding coefficients explicit
coefficient
γ00,00
γ00,11
γ11,00
γ11,11
γ01,01
γ10,10
σ00
σ11
A: pure dephasing A = σ z
+γ(0)
−γ(0)
−γ(0)
+γ(0)
0
0
B: dissipation A = σ x
0
0
0
0
γ(+Ω)
γ(−Ω)
σ(0)
2i
σ(0)
2i
σ(−Ω)
2i
σ(+Ω)
2i
and rewrite the corresponding Liouvillian in the ordering ρ00 , ρ11 , ρ01 , ρ10 as a superoperator (further abbreviating γ0/± = γ(0/ ± Ω), Σ = σ00 − σ11 )


0 0
0
0
 0 0

0
0

LA = 
 0 0 −2γ0 − iΩ

0
0 0
0
−2γ0 + iΩ


−γ− +γ+
0
0

 +γ− −γ+
0
0
.
(7.73)
LB = 
γ
+γ

 0
0
0
− − 2 + − i(Ω + Σ)
+
+ i(Ω + Σ)
0
0
0
− γ− +γ
2
Both Liouvillians lead to a decay of coherences with a rate (we assume Ω > 0)
γA = 2γ0 = 2 lim J(ω) [1 + n(ω)] = 2
ω→0
γB
J0
= 2J0 kB T ,
β
1
1
γ− + γ+
= [J(Ω)[1 + n(Ω)] + J(−Ω)[1 + n(−Ω)]] = [J(Ω)[1 + n(Ω)] + J(Ω)n(Ω)]
=
2
2
2
1
Ω
=
,
(7.74)
J(Ω) coth
2
2kB T
which both scale proportional to T for large bath temperatures. Therefore, the application of
either Liouvillian or a superposition of both will in the high-temperature limit simply lead to
rapid decoherence. The same can be expected from a turnstyle (open-loop control), where the
Liouvillians act one at a time following a pre-defined protocol.
The situation changes however, when measurement results are used to determine which Liouvillian is acting. We choose to act with Liouvillian LA throughout and to turn on Liouvillian LB in
102
CHAPTER 7. CLOSED-LOOP CONTROL:
NON-EQUILIBRIUM CASE IV
addition – multiplied by a dimensionless feedback parameter α ≥ 0 – when a certain measurement
result is obtained. Given a measurement with just two outcomes, the effective propagator is then
given by
W(∆t) = eLA ∆t P1 + e(LA +αLB )∆t P2 ,
(7.75)
where Pi are the superoperators corresponding to the action of the measurement operators Mi ρMi†
on the density matrix. First, to obtain any nontrivial effect (coupling between coherences and
populations), the measurement superoperators should not have the same block structure as the
Liouvillians. Therefore, we consider a projective measurement of the σ x expectation value
M1 =
1
[1 + σ x ] ,
2
M2 =
1
[1 − σ x ] .
2
(7.76)
These projection operators obviously fulfil the completeness relation M1† M1 + M2† M2 = 1. The
superoperators corresponding to Mi ρMi† are also orthogonal projectors




1 1 1 1
1
1 −1 −1
1 1 1 1 1 
1 1
1 −1 −1 
.
,
P1 = 
(7.77)
P2 = 



−1 −1 1
1 
1 1 1 1
4
4
−1 −1 1
1
1 1 1 1
Exercise 53 (Measurement superoperators) (0 points)
Show the correspondence between Mi and Pi in the above equations.
In contrast to the previous cases however, they are not complete in this higher-dimensional
space P1 + P2 6= 1. Whereas previously without feedback the average of many measurements led
to no effect in the rate equations, this is now different.
Without feedback (α = 0), it is easy to see that the measurements still have an effect in contrast
to an evolution without measurements


1 1
0
0

1
 1 1 −(2γ 0+iΩ)∆t −(2γ 0+iΩ)∆t  = eLA ∆t (P1 + P2 ) 6=
0
0

e
2 0 0 e
−(2γ0 −iΩ)∆t
−(2γ0 −iΩ)∆t
0 0 e
e


1 0
0
0
 0 1

0
0
.
eLA ∆t = 
(7.78)
−(2γ
+iΩ)∆t
0
 0 0 e

0
0 0
0
e−(2γ0 −iΩ)∆t
This may have significant consequences – even without dissipation γ0 = 0: The repeated application
of the propagator for measurement without feedback (γ0 = 0 and α = 0) yields


1 1
0
0

L ∆t
n 1  1 1
0
0
.
(7.79)
e A (P1 + P2 ) = 
−iΩ∆t
n−1
−iΩ∆t
n−1

0 0 e
cos (Ω∆t) e
cos (Ω∆t) 
2
0 0 e+iΩ∆t cosn−1 (Ω∆t) e+iΩ∆t cosn−1 (Ω∆t)
103
7.3. QUBIT STABILIZATION
Exercise 54 (Repeated measurements) (0 points)
Show the validity of the above equation.
In contrast, without the measurements we have for repeated application of the propagator
simply
L ∆t n
= eLA n∆t .
(7.80)
e A
When we now consider the limit n → ∞ and ∆t → 0 but n∆t = t remaining finite, it becomes
obvious that the no-measurement propagator for γ0 = 0 simply describes coherent evolution. In
contrast, when the measurement frequency becomes large enough, the measurement propagator in
Eq. (7.78) approaches


1 1 0 0
L ∆t
n 1  1 1 0 0 

(7.81)
e A (P1 + P2 ) = 
2 0 0 1 1 
0 0 1 1
and thereby freezes certain states such as ρ̄ = 21 [|0i + |1i] [h0| + h1|]. This effect is known as
Quantum-Zeno effect (a watched pot never boils) and occurs when measurement operators and
system Hamiltonian do not commute and the evolution between measurements is unitary (here
γ0 = 0). When the evolution between measurements is an open one (γ0 > 0), the Quantum-Zeno
effect cannot be used to stabilize the coherences, which becomes evident from the propagator in
Eq. (7.78).
With feedback (α > 0) however, the effective propagator W(∆t) does not have the Block
structure anymore. It can be used to obtain a fixed-point iteration for the density matrix
ρ(t + ∆t) = W(∆t)ρ(t) .
(7.82)
Here, we cannot even for small ∆t approximate the evolution by another effective Liouvillian, since
lim W(∆t) 6= 1. Instead, one can analyze the eigenvector of W(∆t) with eigenvalue 1 as the (in
∆t→0
a stroboscopic sense) stationary state. It is more convenient however to consider the expectation
values of hσ i it that fully characterize the density matrix via
ρ00
1 + hσ z i
,
=
2
ρ11
1 − hσ z i
=
,
2
ρ01
hσ x i − i hσ y i
=
,
2
ρ10
hσ x i + i hσ y i
=
. (7.83)
2
Note that decoherence therefore implies vanishing expectation values of hσ x i → 0 and hσ y i → 0 in
our setup. Converting the iteration equation for the density matrix into an iteration equation for
the expectation values of Pauli matrices we obtain
e−2γ0 ∆t (1 + hσ x it ) cos (Ω∆t) − (1 − hσ x it ) e−(γ− +γ+ )α∆t/2 cos [(Ω + α(Ω + Σ)) ∆t]
2
−2γ0 ∆t e
(1 + hσ x it ) sin (Ω∆t) − (1 − hσ x it ) e−(γ− +γ+ )α∆t/2 sin [(Ω + α(Ω + Σ)) ∆t]
=
2
(γ+ − γ− ) (1 − hσ x it )
(7.84)
1 − e−(γ− +γ+ )α∆t ,
=
2(γ− + γ+ )
hσ x it+∆t =
hσ y it+∆t
hσ z it+∆t
104
CHAPTER 7. CLOSED-LOOP CONTROL:
NON-EQUILIBRIUM CASE IV
which (surprisingly) follow just the expectation values hσ x it on the r.h.s. The first of the above
equations can be expanded for small ∆t to yield
hσ x it+∆t − hσ x it
1
1
= − [8γ0 + α (γ− + γ+ )] hσ x it + α (γ− + γ+ ) + O{∆t} .
∆t
4
4
(7.85)
When ∆t → 0, this becomes a differential equation with the stationary state
hσ̄ x i =
α(γ− + γ+ )
,
8γ0 + α(γ− + γ+ )
(7.86)
which approaches 1 for large values of α. Taking into account the large-temperature expansions
for the dampening coefficients
γ0 = J0 kB T ,
γ− + γ+ ≈ 2J0 e−Ω/ωc kB T ,
(7.87)
we see that this stabilization effect also holds at large temperatures – a sufficiently strong (and
perfect) feedback provided. An initially coherent superposition is thus not only stabilized, but
also emerges when the scheme is initialized in a completely mixed state. Also for finite ∆t, the
fixed-point iteration yields sensible evolution for the expectation values of the Pauli matrices, see
Fig. 7.7.
0,8
expectation value
0,6
0,4
differential equation limit
0,2
0
-0,2
0
2
4
6
8
10
dimensionless time
Figure 7.7: Expectation values of the Pauli matrices for finite feedback strength α = 10 and finite
stepsize ∆t (spacing given by symbols). For large ∆t, the fixed point is nearly completely mixed.
For small ∆t, the curve for hσ x it approaches the differential equation limit (solid line), but the
curve for hσ y it approaches 0. For γ− = γ+ , the iteration for hσ z it vanishes throughout. Parameters:
γ− = γ+ = γ0 = Γ, Ω∆t =∈ {1, 0.1}, and Σ∆t ∈ {0.5, 0.05}.