2.2 Derivation of Master Equation by Perturbation Theory
2.2.1 Master Equation in Integral Form
In this section, we will derive the master equation using time-dependent perturbation theory.
To retain the generality and applicability of the master equation, the equation of motion will be derived with minimum amount of approximations, in which they will be stated explicitly in the section.
First, we consider the system and the bath to be given by the Hamiltonian
Htot =HS(t) +HB(t) + V(t) (2.2) where HS(t) is the Hamiltonian of the system of interest, while HB(t) is the Hamiltonian of the environment, and V(t) is the system-environment interaction Hamiltonian. is a constant that is used to track the order of the perturbation that will be set to unity at the end of the derivation.
From standard quantum mechanics, the evolution of the density matrix at later time will be given by
2.2. DERIVATION OF MASTER EQUATION BY PERTURBATION THEORY
⇢tot(t) = Utot(t)⇢tot(t0)Utot† (t)
=US(t)UB(t)UI(t)⇢tot(t0)UI†(t)UB†(t)US†(t)
(2.3)
where US(t), UB(t) and UI(t) are the unitary operators associated with the Hamiltonian of the system, environment and the interaction term respectively. In particular, we note that the interaction term in the interaction picture is given by ˜V(t) = U0†(t)V(t)U0(t), with ˜V(t) the interaction term in the interaction picture and U0(t) = US(t)UB(t), therefore it always contains explicit time dependence. As a result, UI(t) can be formally be written as follows4:
UI(t) =I Z t
t0
V˜(t0)UI(t0)dt0 (2.4) As a standard approximation, we will assume that the system and the environment are weakly coupled such that the initial state can be written in factorised form ⇢S(t0)⌦⇢B, i.e. the system and the environment are initially uncorrelated. Such form of approximation is common used as in many systems, the system and environment is sufficiently weakly coupled such that the system would have little influence on the environment statistics initially [88]. Nonetheless, systems with initial system-environment correlation has also been studied recently [89, 90]. We also assume that the interaction Hamiltonian can be written in a factorised form V(t) = S(t)⌦B(t), where S(t) and B(t) are the system and environment operators respectively. With such approximations, the system reduced density matrix can now be written as follows:
⇢S(t) = TrB[US(t)UB(t)UI(t)(⇢S(t0)⌦⇢B)UI†(t)UB†(t)US†(t)] (2.5)
Furthermore, because of the weak coupling between system and environment, we can also perform a Dyson expansion on Eq. (2.4), such that
4See the appendix for the details of derivation.
UI(t) = I i Z t
t0
V˜(t1)dt1 2
Z t t0
Z t1
t0
V˜(t1) ˜V(t2)dt1dt2 +O( 3) (2.6) where I is the identity matrix and any operatorX given by ˜X(t) =U0†(t)X(t)U0(t).
By substituting Eq. (2.6) into the Eq. (2.5), we obtain the following with respect to : For the term independent of :
TrB[U0(t)(⇢S(t0)⌦⇢B)U0†(t)] = US(t)⇢S(t0)US†(t) = ¯⇢S(t) (2.7) For the term proportional to :
iTrB[U0(t) Z t
t0
V˜(t1)dt1(⇢S(t0)⌦⇢B)U0†(t)]
= i Z t
t0
dt1US(t) ˜S(t1)⇢S(t0)US†(t)TrB[UB(t) ˜B(t1)⇢BUB†(t)]
= i Z t
t0
dt1US(t) ˜S(t1)⇢S(t0)US†(t)D
B(t˜ 1)E
B
= 0
(2.8)
Here, we assume that D
B(t˜ 1)E
B yields zero, and in an event that it does not, we can always redefine the environment operator B(t)!B(t) D
B˜(t)E
B such that the expectation value will once again be zero. In this case, the other term that is proportional to will also yield zero, hence the first order terms have null contribution to the ⇢S(t).
For the term proportional to 2, there will be two similar terms that will arise from the derivation, for conciseness, we will demonstrate the derivation for one of the terms below:
TrB
U0(t)
Z t t0
V˜(t1)dt1(⇢S(t0)⌦⇢B) Z t
t0
V˜(t2)dt2U0†(t)
= Z t
t0
dt1
Z t t0
dt2US(t) ˜S(t1)⇢S(t0) ˜S(t2)US†(t)TrB[UB(t) ˜B(t1)⇢BB(t˜ 2)UB(t)]
= Z t
t0
dt1 Z t1
t0
dt2
US(t) ˜S(t1)⇢S(t0) ˜S(t2)US†(t)D
B(t˜ 1) ˜B(t2)E
B
US(t) ˜S(t2)⇢S(t0) ˜S(t1)US†(t)D
B˜(t2) ˜B(t1)E
B
(2.9)
2.2. DERIVATION OF MASTER EQUATION BY PERTURBATION THEORY
As a result, by setting to unity, the system density matrix ⇢S(t) is given by:
⇢S(t) = ¯⇢S(t) + Z t
t0
dt1
Z t1
t0
dt2 US(t)[ ˜S(t2)⇢S(t0),S(t˜ 1)]US†(t)D
B˜(t1) ˜B(t2)E
B+h.c. (2.10) Hereh.c. denotes the hermitian conjugate.
At this stage, such form of master equation has already encapsulated all the essential properties of the e↵ects of the environment onto the system through the bath correlation.
In principle, so long one specifies the type of system operator and the details of the bath, this equation can be solved directly to reveal the dynamics of the system. By casting the master equation into an integral form, one realises that the system unitary operator is now an expression in terms of t, and hence this term needs to be computed once for every time step during calculations. Such an expression may be useful for driven systems, where the evaluation of the next time step depends on the values of the current time step (such a property is known in literature as the back-propagation [91, 92]). Hence, any reduction in the amount of integrals that are required to be computed at each time step will potentially significantly reduce the computational time. Nonetheless, in practice, this form of master equation still involves a double integral and may be computationally challenging to evaluate for large Hilbert spaces.
In chapter 3, the system that we are dealing with are time-independent of nature, hence the integral form in Eq. (2.10) does not serve any computational advantage during the evaluation due to the double integrals involved. Hence, in this case we will attempt to remove the double integrals by recasting the master equation in a di↵erential form.