Model

$$\begin{align} H = H_S + H_B + H_I \end{align}$$

von Neumann equation in the INTERACTION PICTURE

$$\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\rho(t) = - \mathrm{i} [H_I(t), \rho(t)] \end{align}$$

A formal solution

$$\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\rho(t) = - \mathrm{i} [H_I(t), \rho(0)] -\int_0^t \mathrm{d} s[H_I(t), [H_I(s), \rho(s)]] \end{align}$$

Four Approximation

1. Separability

At $t=0$ , there are no correlations between the system

$$\begin{align} \rho(0) \approx \rho_S(0) \otimes \rho_B(0) \end{align}$$

2. Born approximation

Large environment means:

  • that the state of the environment does not significantly change as a result of the interaction with the system, $\rho(t) \approx \rho_S(t) \otimes \rho_B(t)$

Weak coupling means:

  • the system and the environment are noncorrelated during all the time evolution, $\rho_B(t) = \rho_B(0) \equiv \rho_B$

In summary

$$\begin{align} \rho(t) \approx \rho_S(t) \otimes \rho_B \end{align}$$

3. Markov approximation

Short-memory environment

The reservior correlation functions decay faster compared to the system.

4. Secular approximation(Not necessary for all systems)

Only consider the resonance transition.

Not necessary for all master equation.

Derivations

In SCHRODINGER PICTURE

$$\begin{align} H_I^{SP} = \sum_{\alpha} A_{\alpha} \otimes B_{\alpha} = \sum_{\alpha, \omega}A_{\alpha}(\omega) \otimes B_{\alpha} \end{align}$$

where

$$\begin{align} A_{\alpha}(\omega) = \sum_{\varepsilon } |\varepsilon\rangle\langle\varepsilon| A_{\alpha} |\varepsilon + \omega \rangle\langle\varepsilon + \omega| \end{align}$$

So, in the interaction picture

$$\begin{align} H_I(t) = \sum_{\alpha,\omega} e^{- \mathrm{i}\omega t}A_{\alpha}(\omega)\otimes B_{\alpha}(t) = \sum_{\alpha,\omega} e^{ \mathrm{i}\omega t}A^{\dagger}_{\alpha}(\omega)\otimes B^{\dagger}_{\alpha}(t) \end{align}$$

use SEPARABILITY, we get

$$\begin{align} [H_I(t), \rho(0)] =& \sum_{\alpha,\omega}e^{\mathrm{i}\omega t}[A^{\dagger}_{\alpha}(\omega)\otimes B^{\dagger}_{\alpha}(t), \rho_S(0)\otimes \rho_B(0)] \\ =& \sum_{\alpha,\omega}e^{\mathrm{i}\omega t} [A^{\dagger}_{\alpha}(\omega) \rho_S(0) \otimes B^{\dagger}_{\alpha}(t) \rho_B(0) - \rho_S(0) A^{\dagger}_{\alpha}(\omega) \otimes\rho_B(0) B^{\dagger}_{\alpha}(t) ] \\ =& \sum_{\alpha,\omega}e^{\mathrm{i}\omega t} [A^{\dagger}_{\alpha}(\omega), \rho_S(0)] \langle B_\alpha(t) \rangle \\ \end{align}$$

where

$$\begin{align} \langle B_\alpha(t) \rangle \equiv \mathrm{Tr}_B[B^{\dagger}_{\alpha}(t) \rho_B] = 0 \end{align}$$

use $[H_I(t), \rho(0)] = 0$ and BORN APPROXIMATION, we get

$$\begin{align} \mathrm{Tr}_B \left[\frac{\mathrm{d}}{\mathrm{d}t}\rho(t) \right] = \frac{\mathrm{d}}{\mathrm{d}t}\rho_S(t) = -\int_0^t \mathrm{d} s \cdot \mathrm{Tr}_B[H_I(t), [H_I(s), \rho_S(s)\otimes \rho_B]] \end{align}$$

According MARKOV APPROXIMATION, we can approximate $\rho_S(s) \approx \rho_S(t)$ , and extend $t$ to $\infty$ . Substitute $s \to t - s$ ,

$$\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\rho_S(t)= \frac{\mathrm{d}}{\mathrm{d}t}\rho_S(t) = -\int_0^{\infty} \mathrm{d} s \cdot \mathrm{Tr}_B[H_I(t), [H_I(t - s), \rho_S(t)\otimes \rho_B]] \end{align}$$

decompose $H_{I}(t)$ , we get (convolution theorem)

$$\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\rho_S(t) = \sum_{\omega, \omega'} \sum_{\alpha, \beta} e^{\mathrm{i}(\omega' - \omega)t} \Gamma_{\alpha\beta}(\omega)\left[A_{\beta}(\omega)\rho_S(t)A^{\dagger}_{\alpha}(\omega') - A_{\alpha}^{\dagger}(\omega') A_{\beta}(\omega)\rho_s(t)\right] + \mathrm{h.c.} \end{align}$$

where

$$\begin{align} \Gamma_{\alpha\beta}(\omega) = \int_0^{\infty} \mathrm{d}s\cdot e^{\mathrm{i}\omega s} \langle B_{\alpha}^{\dagger}(t) B_{\beta}(t - s)\rangle \end{align}$$

Then SECULAR APPROXIMATION, only keep the resonance term, that is $\omega' = \omega$

$$\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\rho_S(t) = \sum_{\omega} \sum_{\alpha, \beta} \Gamma_{\alpha\beta}(\omega)\left[A_{\beta}(\omega)\rho_S(t)A^{\dagger}_{\alpha}(\omega) - A_{\alpha}^{\dagger}(\omega) A_{\beta}(\omega)\rho_s(t)\right] + \mathrm{h.c.} \end{align}$$

if we define

$$\begin{align} \left\{ \begin{matrix} S_{\alpha\beta}(\omega) \equiv& \frac{1}{2\mathrm{i}} \left[ \Gamma_{\alpha\beta}(\omega) - \Gamma_{\beta\alpha}^{ * }(\omega) \right]& \\ \gamma_{\alpha\beta}(\omega) \equiv& \Gamma_{\alpha\beta}(\omega) - \Gamma_{\beta\alpha}^{ * }(\omega) =& \int_{-\infty}^{\infty} \mathrm{d}s e^{\mathrm{i}\omega s} \langle B^{\dagger}_{\alpha}(s) B_{\beta}(0)\rangle\\ \Gamma_{\alpha\beta}(\omega) =& \frac{1}{2}\gamma_{\alpha\beta}(\omega) + \mathrm{i}S_{\alpha\beta}(\omega)& \end{matrix} \right. \end{align}$$

and (Lamb shift Hamiltonian. It seems like the interaction between vacuum energy fluctuations and the hydrogen electron in different orbitals, ref Atomic and Optical Physics I, 07 Atoms III: Fine Structure)

$$\begin{align} H_{LS} = \sum_{\omega} \sum_{\alpha,\beta} S_{\alpha,\beta}(\omega)A_{\alpha}^{\dagger}(\omega) A_{\beta}(\omega) \end{align}$$

then we get our master equation in the interaction picture

$$\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\rho_{S}(t) = -\mathrm{i}[H_{LS}, \rho_S(t)] + \mathcal{D}[\rho_S(t)] \end{align}$$

where

$$\begin{align} D(\rho_S) = \sum_{\omega}\sum_{\alpha,\beta} \gamma_{\alpha\beta}(\omega) \left[ A_{\beta}(\omega) \rho_SA_{\alpha}^{\dagger}(\omega) - \frac{1}{2}{A^{\dagger}_{\alpha}(\omega)A_{\beta}(\omega), \rho_S} \right] \end{align}$$

For Schrodinger picture, just shift $H_{LS}\to H_{LS} + H_S$ .

Reference