derivation 1. time-dependent

$$H = \sum \epsilon _\alpha ^0 c_\alpha ^\dag c_a + \sum \epsi... ...i\} ) + \sum ( t_{i\alpha } d_i^\dag c_\alpha + {\rm h.c.} )$$ (107)

current from L region is

$$J_L = - (-e) \langle \dot{N}_L \rangle = i e \langle [ H, N_L ] \rangle$$ (108)

$$[H,N_L] = [ H, \sum d_k^\dag d_k]$$ (109)

$$= [ t_{i\alpha } d_i^\dag c_\alpha + t_{\alpha i} c_\alpha ^\dag d_i ,\sum d_k^\dag d_k]$$ (110)

$$= -t_{i\alpha } d_i^\dag c_\alpha + t_{\alpha i} c_\alpha ^\dag d_i$$ (111)

$$J_L = i e \sum ( -t_{i\alpha } \langle d_i^\dag c_\alpha \rangle + t_{\alpha i} \langle c_\alpha ^\dag d_i \rangle )$$ (112)

interesting that the prefactor of the current is imaginary!

Define Keldysh Green functions

$$G^<_{j\alpha }(t,t') = i \langle c_\alpha ^\dag (t') d_j(t) \rangle$$ (113)

$$G^<_{ji}(t,t') = i \langle d_i^\dag (t') d_j(t) \rangle$$ (114)

$$g^<_{\alpha }(t,t') = i \langle c_\alpha ^\dag (t') c_\alpha (t) \rangle$$ (115)

Dyson equation, , reads

$$\left( \begin{array}{cc} i\Longleftarrow i & i\Longleftarrow \alpha \\ \alpha \Longleftarrow i & \alpha \Longleftarrow \alpha \end{array} \right) = \left( \begin{array}{cc} i\longleftarrow i & 0 \\ 0 & \alpha \longleftarrow \alpha \end{array} \right)$$ (116)

$$+ \left( \begin{array}{cc} i\longleftarrow i & 0 \\ 0 & \alpha \... ... \\ \alpha \Longleftarrow i & \alpha \Longleftarrow \alpha \end{array} \right)$$ (117)

$\longleftarrow$ means non-interactiong Green function and is diagonal. $\Longleftarrow$ means interactiong Green function which includes self-energies. $i$ is indexes in the region C and $\alpha$ is in the region L, e.g.,

$$i\longleftarrow i = \left( \begin{array}{cccc} 1\longleftarrow 1 &0&0& \\ 0& 2\longleftarrow 2 &0 & \\ 0& 0& 3\longleftarrow 3 & \\ & & & \ddots \end{array} \right)$$ (118)

and so on.

The second term reads

$$\left( \begin{array}{cc} (i\longleftarrow i, \Sigma, \alpha ... ...a , t_{\alpha i}, i \Longleftarrow \alpha \end{array} \right)$$ (119)

The (2,1) component of eq.(118) gives

$$(\alpha \Longleftarrow i) = (\alpha \longleftarrow \alpha , t_{\alpha i}, i \Longleftarrow i)$$ (120)

Thus for equilibrium Green functions,

$$G_{i\alpha }(t,t') = \int d t_1 \sum_j G_{ij}(t,t_1) t_{j\alpha }(t_1) g_\alpha (t_1,t)$$ (121)

For non equilibrium Green functions,

$$G_{i\alpha }^<(t,t') = \int d t_1 \left\{ \sum_j G_{ij}^R(t... ...{ij}^<(t,t_1) t_{j\alpha }(t_1) g_\alpha ^A(t_1,t) \right\}$$ (122)

time dependence of $\epsilon _\alpha (t)$

$$\epsilon _\alpha (t) = \epsilon _a^0 + \Delta_\alpha (t)$$ (123)

In the stochastic approximation,

$$g_\alpha ^<(t,t') = i f(\epsilon _\alpha ^0) \exp( -i \int_{t'}^{t} d t_2 \epsilon _\alpha (t_2) )$$ (124)

$$g_\alpha ^A(t,t') = i \theta(t'-t) \exp( -i \int_{t'}^{t} d t_2 \epsilon _\alpha (t_2) )$$ (125)

return to eq.(113), ⁵

$$J_L(t) = e \sum\left[ t_{\alpha i} G_{i \alpha }^<(t,t) + {\rm h.c.} \right]$$ (126)

$$= 2e \; {\rm Re} \sum_{\alpha i} t_{\alpha i} G_{i \alpha }^<(t,t) = 2e \; {\rm Re} \; {\rm tr} \left[ t G^< \right]$$ (127)

See also the Appendix.

Use eq.(123), eq.(125) and eq.(126)

$$\sum_{\alpha i} t_{\alpha i} G_{i \alpha }^<(t,t) = \sum_{\alpha i} t_{\alpha i} \sum_j \int d t_1 \left[ G_{ij}^R(t,... ..._\alpha ^<(t_1,t) + G_{ij}^<(t,t_1) t_{\alpha }(t_1) g_\alpha ^A(t_1,t) \right]$$ (128)

$$= i \sum_{\alpha ij} \int d t_1 t_{\alpha i}(t_1) t_{j \alpha }(t_1... ...^R(t,t_1) f(\epsilon _\alpha ^0) + G_{ij}^<(t,t_1) \theta(t-t_1) \right] \times$$

$$\exp\left( -i \int_t^{t_1} d t_2 \epsilon _\alpha (t_2) \right)$$ (129)

insert

$$\sum_{\alpha i} t_{\alpha i} G_{i \alpha }^<(t,t) = i \sum_{\alpha ij} \int d t_1 \int d\epsilon \sum_\alpha \delta (... ...0) t_{\alpha i}(t) t_{j \alpha }(t) \exp( -i\epsilon _\alpha ^0(t_1-t) ) \times$$

$$\exp \left( -i \int_{t_1}^{t} d t_2 \Delta_\alpha (t_2) \right) \... ... G_{ij}^R(t,t_1) f(\epsilon _\alpha ^0) + G_{ij}^<(t,t_1) \theta(t-t_1) \right]$$ (130)

Define

$$\Gamma_{ji}(\epsilon ,t_1,t) = 2\pi \sum_\alpha \delta (\eps... ... \exp\left( -i \int_t^{t_1} d t_2 \Delta_\alpha (t_2) \right)$$ (131)

Then

$$\sum_{\alpha i} t_{\alpha i} G_{i \alpha }^<(t,t) = i \sum_... ...f(\epsilon _\alpha ^0) + G_{ij}^<(t,t_1) \theta(t-t_1) \right]$$ (132)

Taking into account of , upper limit of $\int d t_1$ is $t$.

$$\sum_{\alpha i} t_{\alpha i} G_{i \alpha }^<(t,t) = i \sum_... ... \left[ G_{ij}^R(t,t_1) f(\epsilon ) + G_{ij}^<(t,t_1) \right]$$ (133)

Therefore

$$J_L(t) = 2 e \int_{-\infty}^{t} d t_1 \int \frac{d\epsilon }... ...amma(e,t_1,t) ( G^R(t,t_1) f(\epsilon ) + G^<(t,t_1)) \right]$$ (134)