We shall derive the master equation of an open quantum system. As a common assumption, we assume that the whole environment (called bath) and the system are quantum mechanical, in the sense that (1) the relevant degrees of freedom are completely characterized by state vectors (or density matrices), and (2) the time evolution of the whole system is unitary U(t) = exp( − iHt), where the full Hamiltonian H = HS + HB + V is assumed to be time-independent and consists of three parts, namely the system Hamiltonian HS, the bath Hamiltonian HB and the interaction term V between the system and the bath. The goal of the master equation is to find the dynamics of the system by tracing out the many degrees of freedom of the bath. This is not always possible and we shall assume that the system-bath interaction V is sufficiently weak, so that perturbation theory is applicable.
General equation of motion
In the interaction picture, the evolution of the total density matrix ρT is
- $i \hbar \frac{d}{dt}\tilde{\rho }_{T}=\left[ \tilde{V}\left( t \right),\tilde{\rho }_{T} \right]$,
where ρ̃T(t) ≡ U0 + ρTU0, Ṽ(t) ≡ U0 + VU0, and U0 = e − i(HS + HB)t/ℏ. This evolution is for the moment very general, and the solution can be formally written as $\tilde{\rho }_{T}\left( t \right)=\tilde{\rho }_{T}\left( 0 \right)+\frac{1}{i\hbar }\int_{0}^{t}{dt_{1}\left[ \tilde{V}\left( t_{1} \right),\tilde{\rho }_{T}\left( t_{1} \right) \right]}$. Iterate once more, we have
$$\tilde{\rho }_{T}\left( t \right)=\tilde{\rho }_{T}\left( 0 \right)+\frac{1}{i\hbar }\int_{0}^{t}{dt_{1}\left[ \tilde{V}\left( t_{1} \right),\tilde{\rho }_{T}\left( 0 \right) \right]}$$
$+\frac{1}{\left( i\hbar \right)^{2}}\int_{0}^{t}{\int_{0}^{t_{1}}{dt_{1}}dt_{2}\left[ \tilde{V}\left( t_{1} \right),\left[ \tilde{V}\left( t_{2} \right),\tilde{\rho }_{T}\left( t_{2} \right) \right] \right]}$.
In the following, we shall invoke several approximations to simplify calculations, namely Born approximation, product initial stat assumption and later Markov approximation.
Born approximation: weak coupling
Here we assume the interaction V is weak. So, suppose we continue to iterate, we expect that the series would converge and write the general solution as $\tilde{\rho }_{T}\left( t \right)=\tilde{\rho }_{T}\left( 0 \right)+\sum\limits_{n\ge 1}{\frac{1}{\left( i\hbar \right)^{n}}\int_{0}^{t}{dt_{1}...\int_{0}^{t_{n-1}}{dt_{n}}\left[ \tilde{V}\left( t_{1} \right),...,\left[ \tilde{V}\left( t_{n} \right),\tilde{\rho }_{T}\left( 0 \right) \right] \right]}}$. This way of terminating an iterative equation is often known as the Born approximation. However, we shall only consider the accuracy up to the second order in V. Taking the trace over the bath,
$$\tilde{\rho }\left( t \right)=\tilde{\rho }\left( 0 \right)+\frac{1}{i\hbar }\int_{0}^{t}{dt_{1}}Tr_{B}\left[ \tilde{V}\left( t_{1} \right),\tilde{\rho }_{T}\left( 0 \right) \right]+\frac{1}{\left( i\hbar \right)^{2}}\int_{0}^{t}{\int_{0}^{t_{1}}{dt_{1}dt_{2}}}Tr_{B}\left[ \tilde{V}\left( t_{1} \right),\left[ \tilde{V}\left( t_{2} \right),\tilde{\rho }_{T}\left( 0 \right) \right] \right]$$
, where ρ̃(t) ≡ TrB[ρ̃T(t)].
Product initial state assumption
Next, we need to invoke a rather important assumption, that the initial state between the system and the environment is uncorrelated, or mathematically represented by ρT(0) = ρ(0) ⊗ ρB(0). Another not essential but often valid assumption is that TrB[Ṽ(t1)ρB] = 0. This suggests that the first order term is zero. To second order accuracy, we write
ρ̃(t) = eM(t)ρ̃(0)
, where $M\left( t \right)\chi \equiv \frac{1}{\left( i\hbar \right)^{2}}\int_{0}^{t}{\int_{0}^{t_{1}}{dt_{1}dt_{2}}}Tr_{B}\left[ \tilde{V}\left( t_{1} \right),\left[ \tilde{V}\left( t_{2} \right),\chi \otimes \rho _{B} \right] \right]$ is a superoperator. Taking the time-derivative, we have $\frac{d}{dt}\tilde{\rho }\left( t \right)=\frac{d}{dt}M\left( t \right)\times \tilde{\rho }\left( t \right)$. Explicitly, we obtained the master equation,
$$\frac{d}{dt}\tilde{\rho }\left( t \right)=\frac{1}{\left( i\hbar \right)^{2}}\int_{0}^{t}{d\tau }Tr_{B}\left[ \tilde{V}\left( t \right),\left[ \tilde{V}\left( \tau \right),\tilde{\rho }\left( t \right)\otimes \rho _{B} \right] \right]$$
.
Markov approximation: short memory
Here we need to evaluate the terms involving taking the average with respect to the thermal bath, which is assumed to have short memory in the sense that the correlation time is very short. Mathematically,
∫0tdτTrB[Ṽ(t)Ṽ(τ)ρB] = ∫0tdτTrB[Ṽ(t − τ)Ṽ(0)ρB] ≈ ∫0∞dτTrB[Ṽ(t − τ)Ṽ(0)ρB]
.
In other words, the two-point correlation function is significant only when t ≈ τ, and it is valid to extend the upper limit to infinity. This is the Markov approximation.
Example: damped quantum harmonic oscillator
As an application, let us consider an example of Brownian motion of a quantum harmonic oscillator. The system-bath coupling is assumed to be of the form
V = ℏ(a † Γ(t)eiΩt + aΓ † (t)e − iΩt)
, where $\Gamma \left( t \right)=\sum\limits_{k}{g_{k}b_{k}e^{-i\omega _{k}t}}$, the bosonic operators a and bk act on respectively the system (with frequency Ω) and bath (with frequency ωk). Here gk characterizes the strength of the coupling between the system and the bath oscillators. We shall need to evaluate quantities like
TrB[V(t)V(s)ρB] = ℏ2a † a⟨Γ(t)Γ † (s)⟩BeiΩ(t − s) + ℏ2aa † ⟨Γ † (t)Γ(s)⟩Be − iΩ(t − s)
, where ⟨Γ(t)Γ † (s)⟩B ≡ TrB[Γ(t)Γ † (s)ρB], and for a thermal bath, ⟨bj † bk⟩ = δjknk and nk = (eβℏωk − 1) − 1. Next, we will need to use the relation
$$\int_{0}^{\infty }{d\tau }e^{\pm i\varepsilon \tau }=\pi \delta \left( \varepsilon \right)\pm iPV\left( \frac{1}{\varepsilon } \right)$$
, where PV denotes the Cauchy principal value part. These correspond to the Lamb shift and Stark shift in the frequency, and is assumed to be small compared with Ω and shall be neglected here.
Define $\frac{\gamma }{2}\equiv \sum\limits_{k}{g_{k}^{2}\delta \left( \omega _{k}-\Omega \right)}$, we then have the master equation for a damped harmonic oscillator,
$$\frac{d}{dt}\tilde{\rho }=\frac{\gamma }{2}\left( N+1 \right)\left( 2a\tilde{\rho }a^{\dagger}-a^{\dagger}a\tilde{\rho }-\tilde{\rho }a^{\dagger}a \right)+\frac{\gamma }{2}N\left( 2a^{\dagger}\tilde{\rho }a-aa^{\dagger}\tilde{\rho }-\tilde{\rho }aa^{\dagger} \right)$$
,
where N ≡ (eβℏΩ − 1) − 1.
Thermalization
To complete the discussion, let us consider the time development of the mean photon number ⟨a † a⟩. Note that Tr(a † aρ̃) = Tr(a † aρ), and it may be useful to use (with n̂ ≡ a † a) n̂a = an̂ − a and n̂a † = a † n̂ + a † to simplify the right hand side of the master equation. We have
$$\frac{d}{dt}\left\langle a^{+}a \right\rangle =-\gamma \left\langle a^{+}a \right\rangle +\gamma N$$
, and the solution to this equation is
⟨n(t)⟩ = ⟨n(0)⟩e − γt + N(1 − e − γt)
, which suggests that ⟨n(t → ∞)⟩ → N = (eβℏΩ − 1) − 1 , as expected for the reason of thermalization.
References
Ravinder R. Puri, Mathematical Methods of Quantum Optics (Springer Series in Optical Sciences).
For a discussion about the validity about the Born-Markov approximation, see G. M. Moy, J. J. Hope, and C. M. Savage, Born and Markov approximations for atom lasers, Phys. Rev. A 59, 667 - 675 (1999).
Relationship between the Markov approximation and Fermi Golden Rule, see Robert Alicki, The Markov master equations and the Fermi golden rule, International Journal of Theoretical Physics (1977).