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

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.



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


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

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

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


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

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)

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


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

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

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


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

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