1.5 Open Quantum Systems
1.5.4 Phase-Space Methods: Wigner Representation
A further, convenient way to describe open quantum systems is via phase-space methods. They are appealing because of their close analogy to classical physics. There exist several standard methods:
P,Qand Wigner representation corresponding to the mapping of operator expectation values in normal, anti-normal and symmetric order to expectation values of quasi-probability distributions.
The Wigner distribution comes closest to a distribution over classical phase space. The standard reference is again Gardiner’s book [59] on quantum noise.
For certain special cases such as the open single-mode cavity, these distributions can be com-pletely positive functions that obey a proper Fokker-Planck equation. This is in general not true for interacting problems where, for instance, the Wigner representation maps a standard Hubbard-like quartic term to third-order derivatives in a “generalized” Fokker-Planck equation. Such an equation cannot be transformed into a stochastic differential equation by canonical methods. While there exists very interesting work on how such transformations may still be achieved [63–65], to the author’s knowledge these methods have not yet found widespread application.
More attention has been given to a strain of work that tries to avoid the appearance of generalized Fokker-Planck equations by increasing the dimension of the phase space to which the density matrix is mapped. This has led to the so-calledpositiveandgaugePrepresentations [66–68] which allow for proper Fokker-Planck equations and hence for standard transformations to stochastic differential equations even for interacting quantum systems. The drawback is that the interaction terms are in general mapped to multiplicative-noise contributions (cf.B.3.2) which are difficult to handle numerically for large noise (i.e. strong interactions). Thus there appears to be a “conservation of difficulty”, as expected. While all of these methods apply only to bosonic systems, work on fermionicQfunctions does exist [69,70].
To map the master equation of our open single-mode cavity (which follows from Eq. (1.82) for Lˆ =a) to a Wigner distributionW, we need the following correspondences
a†aˆρ←−
α∗−1
2
∂
∂α α+1 2
∂
∂α∗
W α, α∗
=
"
|α|2−1 2
∂
∂αα+1 2α∗ ∂
∂α∗ −1 4
∂2
∂α∂α∗
#
W α, α∗ ˆ
ρa†a←−
α−1
2
∂
∂α∗ α∗+1 2
∂
∂α
W α, α∗
=
"
|α|2−1 2
∂
∂α∗α∗+1 2α ∂
∂α −1 4
∂2
∂α∂α∗
#
W α, α∗ aˆρa†←−
α∗+1
2
∂
∂α α+1 2
∂
∂α∗
W α, α∗
=
"
|α|2+1 2
∂
∂αα+1 2α∗ ∂
∂α∗ +1 4
∂2
∂α∂α∗
#
W α, α∗ .
(1.91)
Applying these to the master equation, we find the corresponding proper Fokker-Planck equation (cf. App.B.1) for the Wigner distribution
W˙ α, α∗
=
"
−iω0
α∗ ∂
∂α∗ + 1 2−α ∂
∂α− 1 2
−λ
2 2|α|2−1−1 2
∂2
∂α∂α∗
!#
W α, α∗ +λ
"
|α|2+1 2
α ∂
∂α + 1
+1 2α∗ ∂
∂α∗ +1 4
∂2
∂α∂α∗
#
W α, α∗
=−iω0
α∗ ∂
∂α∗ −α ∂
∂α
W α, α∗ +λ
2
"
∂
∂αα+ ∂
∂α∗α∗+ ∂2
∂α∂α∗
#
W α, α∗
=
"
iω0+ λ 2
∂
∂αα−
iω0− λ 2
∂
∂α∗α∗+λ 2
∂2
∂α∂α∗
#
W α, α∗
=
"
ω0 ∂
∂pq− ∂
∂qp
+λ 2
∂
∂qq+ ∂
∂pp
+λ 4
1 2
∂2
∂q2 +1 2
∂2
∂p2
!#
W(q, p), (1.92) where in the last step, we have used α = q + ip and ∂α = ∂q−i∂p
/2 to emphasize the connection to a distribution on classical phase space. Observe how there is diffusion in both the spatial and the momentum coordinate. The corresponding stochastic differential equations are
dp=−ω0qdt−λpdt/2 +p
λ/4 dWp, dq= ω0pdt−λqdt/2 +p
λ/4 dWq. (1.93)
To bridge the gap to Sec.1.4, we note here that the correspondingclassicalpath integral is S[p, q,p,ˆ q] =ˆ
Z t t0
d¯th ˆ
p( ˙p+w0q+λp/2) + ˆq( ˙q−ω0p+λq/2) + iλ ˆ
p2+ ˆq2 /8i
≡ Z t
t0
d¯t h
φˆ∗(i∂¯t−ω0+ iλ/2)φ−φˆ(i∂¯t+ω0−iλ/2)φ∗+ iλφˆ∗φˆi ,
(1.94)
where the hatsdo notindicate operator-valuedness but label the response fields, and we have made the reverse transformation back to a complex representation according to
φ=√
2α=√
2 (q+ ip), φˆ= i
2 ˆ q+ iˆp
√2 . (1.95)
In the terminology of1.4.2, the path integral (1.94) has two response fieldsz1 = ˆpandz2 = ˆq.
Later on, we will see that it is both identical to the Schwinger-Keldysh path integral of the open single-mode cavity and the Martin-Siggia-Rose functional integral belonging to Eqs. (1.42).16
16This is actually no surprise, but still instructive.
Non-Equilibrium Quantum Field Theory
The upcoming Chp. begins with an introduction to the Schwinger-Keldysh formalism and its relation to classical stochastic processes in Sec.2.1. This is extended by a detailed illustration of thetwo-particle-irreducible effective action(2PIEA) as an approach to self-consistent perturbation theory in Sec.2.2. Finally, we discuss the numerical solution of the so-calledKadanoff-Baym equationsin Sec.2.3.
2.1 Schwinger-Keldysh Formalism
When dealing with non-equilibrium quantum field theory (QFT), one necessarily encounters the question “What is the Schwinger-Keldysh formalism?”, which we shall try to answer in the following. In recent years, with the growing interest in non-equilibrium phenomena, the technique has become more relevant. To properly put it into perspective, one should contrast it with QFT in particle physics, and with QFT in equilibrium.
The two classic authors on the subject are Schwinger [71] and Keldysh [72].1 Schwinger describes the method rather sharply in terms of operators. Keldysh introduces the rotation to the more physical “RAK” representation which is named after him. There exist a number of review articles that are decent starting points for getting familiar with the technique. The first is the classic by Kubo [74] on the fluctuation-dissipation theorem in equilibrium, which can give a good motivation for the more general non-equilibrium problem. Ref. [11] is a good place to start: it derives the Keldysh path integral from a master equation and introduces the one-particle irreducible (1PI) effective action; it also treats the functional renormalization group. The two reviews by Berges [75,76] are highly recommendable, although perhaps more advanced, as they are based on thetwo-particleirreducible effective action (2PIEA). They have detailed discussions of initial conditions and non-perturbative approximation techniques. Ref. [77] is an older piece that is possibly good for comparison and further reading, but mostly here for completeness.
Kamenev’s book [56] is sort of the standard for the path-integral based approach. It gives another derivation of the path integral from a discrete-time expansion in terms of coherent states.
1A precursor to Ref. [71] is [73].
Among the applications are interacting Bose gases. It uses effective-action concepts without ever stating this clearly, which is a drawback, but has a relevant chapter on classical stochastics. The latest version of Altland & Simons’s book [78] has a lot of material on field theory methods both for classical and quantum non-equilibrium problems, presented in two separate chapters. Their derivation of the Keldysh path integral is practically identical to Kamenev’s, but there is further material that may provide a complementary perspective. Calzetta’s book [79] is slightly cryptic when it comes to notation, but it has a very good chapter on Bose gases with a discussion of existing approaches (Φ-derivable, conserving and gapless approximations from 2PIEA). Rammer [61] is a reference work based solely on operators, for those who prefer that. It is very abstract, yet good for looking up identities and reading about general diagrammatics concepts. Stefanucci’s book [80] is based on operators as well and derives everything in terms of the so-calledLangereth rules. It is also strong on advanced diagrammatics.
The following two articles are about applications of the method to cold Bose gases. Ref. [81] is a good resource for an example of how the 2PIEA technique can be brought to bear on an actual problem. It also discusses the famous Hartree-Fock-Bogoliubov equations and their derivation.
Ref. [82] is interesting because it compares classical and quantum dynamics from the perspective of the Schwinger-Keldysh path integral, showing how the cubic response-field vertex gives rise to the non-classical dynamics (this will be explained below).
Concerning the relation of master equation and Keldysh functional integral, Ref. [83] can be recommended. It contains a relatively detailed derivation of the “Markovian dissipative action”
that is the functional equivalent of the Lindblad master equation in the usual Born-Markov approximation and is well suited for comparison to Schwinger’s original treatment. Ref. [84] is an application where the method is used in a way close to the original style given by Schwinger, which puts the emphasis on the fact that at least for interacting systems the method is about the response of the system to a “sudden modulation”.
The Schwinger-Keldysh path integral should not be discussed without relating it to classical non-equilibrium systems. Ref. [85] is the classic resource for the Martin-Siggia-Rose (MSR) functional integral, although [85] is based entirely on operators and the functional integral was introduced only later in [86] and [87]. The “response field” also occurring in the Keldysh formalism is introduced for classical systems, which provides an alternative approach to the Onsager-Machlup path integral (s.1.4.2).