Sub-Ohmic to super-Ohmic crossover behavior in nonequilibrium quantum systems with electron-phonon interactions

The transition from weakly damped coherent motion to localization in the context of the spin-boson model has been the subject of numerous studies with distinct behavior depending on the form of the phonon-bath spectral density $J\left(\omega \right)\propto {\omega }^s$. Sub-Ohmic ($s<1$) and Ohmic ($s=1$) spectral densities show a clear localization transition at zero temperature and zero bias, while for super-Ohmic ($s>1$) spectral densities this transition disappears. In this paper, we consider the influence of the phonon-bath spectral density on the nonequilibrium dynamics of a quantum dot with electron-phonon interactions described by the extended Holstein model. Using the reduced density matrix formalism combined with the multilayer multiconfiguration time-dependent Hartree approach, we investigate the dynamic response, the time scales for relaxation, as well as the existence of multiple long-lived solutions as the system-bath coupling changes from the sub- to the super-Ohmic cases. Bistability is shown to diminish for increasing powers of $s$ similar to the spin-boson case. However, the physical mechanism and the dependence on the model parameters such as the typical bath frequency ${\omega }_c$ and the polaron shift $\lambda$ are rather distinct.

Figure 1

The phonon spectral density $J\left(\omega \right)=\frac{\lambda \pi {\omega }^s}{\mathrm{\Gamma }\left(s\right){\omega }_c^s}{e}^{-\frac{\omega }{{\omega }_c}}$ for various values of $s$ for ${\omega }_c=100{\mathrm{cm}}^{-1}$ and $\lambda =1000{\mathrm{cm}}^{-1}.$

Figure 2

Transient dynamics of the average quantum dot population for the Ohmic (left set of panels, $s=1$) and sub-Ohmic case (right set of panels, $s=1/2$). The results are shown for all four initial conditions: Black and red curves correspond to the unoccupied/occupied dot at phonon initial condition ${\delta }_j=0$, whereas blue and green curves correspond to the unoccupied/occupied dot at ${\delta }_j=1$, respectively. In all results the cutoff time used to generate the memory kernel is $t_c=100\text{fs}.$

Figure 3

Transient dynamics of the average quantum dot population for the super-Ohmic case ($s=2$ and $s=3$ for the left and right set of panels, respectively). The results are shown for all four initial conditions: Black and red curves correspond to the unoccupied/occupied dot at phonon initial condition ${\delta }_j=0$, whereas blue and green curves correspond to the unoccupied/occupied dot at ${\delta }_j=1$, respectively. In all results the cutoff time used to generate the memory kernel is $t_c=100\text{fs}.$ Converging the results ${\omega }_c=500{\text{cm}}^{-1}, \lambda =2000{\text{cm}}^{-1}$ for $s=2$ and ${\omega }_c=100{\text{cm}}^{-1}, \lambda =2000{\text{cm}}^{-1}$ for $s=3$ is difficult, and thus we only show the direct ML-MCTDH-SQR to $t=t_c$.

Figure 4

Left: Dot population in the super-Ohmic case for ${\varepsilon }_d=0.25\text{eV}, \lambda =2000{\text{cm}}^{-1}$, and ${\omega }_c=500{\text{cm}}^{-1}$. The results are shown for all four initial conditions: Black and red curves correspond to the unoccupied/occupied dot at phonon initial condition ${\delta }_j=0$, whereas blue and green curves correspond to the unoccupied/occupied dot at ${\delta }_j=1$, respectively. Right: $\Im \left\{\alpha \left(t\right)\right\}$ for different values of $s$ for ${\omega }_c=100{\text{cm}}^{-1}$ (upper panel) and ${\omega }_c=500{\text{cm}}^{-1}$(lower panel).

Figure 5

Coherent to incoherent transition in the super-Ohmic case $s=2$. Shown are results of the dot population for different characteristic frequencies ${\omega }_c$ of the phonon bath as indicated in the plot and for four different initial conditions as indicated by the different colors. The color code is the same as in Fig. 4.

Figure 6

The reaction coordinate $〈Q\left(t\right)〉$ for several values of $s$ and for ${\varepsilon }_d=0.25\text{eV}, \lambda =2000{\text{cm}}^{-1}, {\omega }_c=500{\text{cm}}^{-1}$, and temperature $T=0$. Black—unoccupied $n_d\left(0\right)=0$ and ${\delta }_j=0$; red—occupied $n_d\left(0\right)=1$ and ${\delta }_j=0$; blue—unoccupied $n_d\left(0\right)=0$ and ${\delta }_j=1$; green—occupied $n_d\left(0\right)=1$ and ${\delta }_j=1$. In addition, the dashed lines depict $Q_1\left(t\right)$, the part of $〈Q\left(t\right)〉$ which describes the relaxation of the reaction mode due to the coupling to the phonon bath in the absence of coupling to the leads [see Eq. (21)].