Long transient dynamics in the Anderson-Holstein model out of equilibrium

We calculate the time-dependent nonequilibrium current through a single-level quantum dot strongly coupled to a vibrational mode. The nonequilibrium real-time dynamics caused by an instantaneous coupling of the leads to the quantum dot is discussed using an approximate method. The approach, which is specially designed for the strong polaronic regime, is based on the so-called polaron tunneling approximation. Considering different initial dot occupations, we show that a common steady state is reached after times much larger than the typical electron tunneling times due to a polaron blocking effect in the dot charge. A direct comparison is made with numerically exact data, showing good agreement for the time scales accessible by the diagrammatic Monte Carlo simulation method.

Figure 1

First three Feynman diagrams of the Dyson series for the PTA. The decoupled dot's Green's function $D_{\text{dot}}(t,t^{\prime })$ is represented with a solid line, the decoupled leads' Green's function $g_{\alpha }(t,t^{\prime })$ with dashed lines. The two-point phonon correlator connecting two time integration variables $t,t^{\prime }$ is denoted by ${\Lambda }_2=\langle {\mathcal{T}}_{\mathcal{C}}X\left(t\right)X^{†}\left(t^{\prime }\right)\rangle$.

Figure 2

Steady-state current vs applied bias voltage calculated within the PTA with $n_{\text{D}}=0$ (brown dashed lines), $n_{\text{D}}=1$ (black dotted lines), and with the self-consistent charge (blue lines) determined by Eq. (10). The dot is off resonant with ${\tilde{\epsilon }}_{\text{D}}=-10\Gamma$, $\lambda =16\Gamma$, ${\omega }_0=8\Gamma$. Inset: The same plot but for ${\tilde{\epsilon }}_{\text{D}}=0$.

Figure 3

Spectral density $A_{\text{D}}\left(\omega \right)$ of the PTA with fully treating the dot occupation (blue lines) vs APTA (red lines) for ${\tilde{\epsilon }}_{\text{D}}=-10\Gamma$, $\lambda =16\Gamma$, ${\omega }_0=8\Gamma$, and $V=10\Gamma$. Inset: Zoom of the figure showing the difference between APTA and PTA spectral densities.

Figure 4

Comparison between the current from diagMC (symbols) and APTA (straight lines) for ${\tilde{\epsilon }}_{\text{D}}=-10\Gamma$, $\lambda =16\Gamma$, ${\omega }_0=8\Gamma$, $V=2\Gamma$. The current from the initially empty (occupied) dot are highlighted in red (green) color for the APTA and represented by dots (diamonds) for the diagMC. Inset: Same plot but for larger times.

Figure 5

The currents from diagMC and APTA are compared with the same color code and parameters as in Fig. 4 but with different voltages $V=5\Gamma$ (figure on the top) and $V=26\Gamma$ (bottom). The steady-state current cannot be reached even for times of the order of $10{\Gamma }^{-1}$.

Figure 6

The dot occupation for the diagMC and APTA are compared for $V=26\Gamma$. While for the dot occupation of the initially occupied dot a good agreement is observed, the APTA dot occupation of the initially empty dot is overestimating the blocking effect with respect to the exact diagMC results.

Figure 7

The current from diagMC and APTA are compared with the same color code and parameters as in Fig. 4 but with different voltages, $V=40\Gamma$ (top panel) and $V=80\Gamma$ (bottom panel).

Figure 8

Time-dependent current for long times with the same parameters as in Fig. 4 and with $V=5\Gamma$. A plateau value of the current for the initially empty dot is observed between $t\approx 20-100{\Gamma }^{-1}$. Inset: The same plot but for much longer times.

Figure 9

Time-dependent dot occupation for the same parameters as in Fig. 8. The charge is blocked for the initially empty dot, leading to a small change in the dot occupation up to $t\approx 100{\Gamma }^{-1}$. Inset: The same plot for a larger time scale.

Figure 10

Time-dependent current with the same parameters as in Fig. 8 but with $V=26\Gamma$. The length of the plateau is getting smaller, but still it is clearly visible between $t\approx 10-50{\Gamma }^{-1}$. Inset: The same plot for larger times.

Figure 11

Time-dependent current with the same parameters as in Fig. 10 but with $V=40\Gamma$. The current does not show a plateau for small times but the times until a joint steady state is reached is still large.