Quantum Kramers model: Corrections to the linear response theory for continuous bath spectrum

Decay of the metastable state is analyzed within the quantum Kramers model in the weak-to-intermediate dissipation regime. The decay kinetics in this regime is determined by energy exchange between the unstable mode and the stable modes of thermal bath. In our previous paper [Phys. Rev. A 42, 4427 (1990)], Grabert's perturbative approach to well dynamics in the case of the discrete bath [Phys. Rev. Lett. 61, 1683 (1988)] has been extended to account for the second order terms in the classical equations of motion (EOM) for the stable modes. Account of the secular terms reduces EOM for the stable modes to those of the forced oscillator with the time-dependent frequency (TDF oscillator). Analytic expression for the characteristic function of energy loss of the unstable mode has been derived in terms of the generating function of the transition probabilities for the quantum forced TDF oscillator. In this paper, the approach is further developed and applied to the case of the continuous frequency spectrum of the bath. The spectral density functions of the bath of stable modes are expressed in terms of the dissipative properties (the friction function) of the original bath. They simplify considerably for the one-dimensional systems, when the density of phonon states is constant. Explicit expressions for the fourth order corrections to the linear response theory result for the characteristic function of the energy loss and its cumulants are obtained for the particular case of the cubic potential with Ohmic (Markovian) dissipation. The range of validity of the perturbative approach in this case is determined ($\gamma /{\omega }_b<0.26$), which includes the turnover region. The dominant correction to the linear response theory result is associated with the “work function” and leads to reduction of the average energy loss and its dispersion. This reduction increases with the increasing dissipation strength (up to $\sim 10\%$) within the range of validity of the approach. We have also calculated corrections to the depopulation factor and the escape rate for the quantum and for the classical Kramers models. Results for the classical escape rate are in very good agreement with the numerical simulations for high barriers. The results can serve as an additional proof of the robustness and accuracy of the linear response theory.

Figure 1

Frequency dependence of the coupling constants $g^2\left(\nu \right)$ to the $s$ modes. The curves correspond to the different values of the dissipation strength ($\mu$ parameter). Solid line, $\mu =1.1$; dashed line, $\mu =1.2$; dashed-dotted line, $\mu =1.3$.

Figure 2

Dependence of the average energy loss of the $u$ mode on the dissipation strength ($\mu$ parameter). Solid line, total result $-〈\delta \varepsilon 〉$; dotted line, linear response theory result $-{〈\delta \varepsilon 〉}_0$ [Eq. (4.33)]; dashed line, ${〈\delta \varepsilon 〉}_1$ [Eq. (4.35)]; dashed-dotted line, $-{〈\delta \varepsilon 〉}_2$ [Eq. (4.36)]. The barrier height has been set equal to $\beta V_0=10$ and the barrier frequency to $\beta \hslash {\omega }_b=2$.

Figure 3

The “quantum” contribution to the average energy loss of the $u$ mode, $-{〈\delta \varepsilon 〉}_2$, as a function of the dissipation strength ($\mu$ parameter). The curves correspond to the different values of the barrier frequency: solid line, $\beta \hslash {\omega }_b=4$; dashed line, $\beta \hslash {\omega }_b=2$; dashed-dotted line, $\beta \hslash {\omega }_b=0$ (classical limit).

Figure 4

The dispersion of the average energy loss of the $u$ mode. The curves in the figure correspond to the different values of the barrier frequency: solid line, $\beta \hslash {\omega }_b=4$; dashed line, $\beta \hslash {\omega }_b=2$; dashed-dotted line, $\beta \hslash {\omega }_b=0$ (classical limit). The corresponding linear response theory results, ${〈{\delta }^2\varepsilon 〉}_0$, are also shown for comparison (dotted lines). The barrier height has been set equal to $\beta V_0=10$.

Figure 5

The ${〈{\delta }^2\varepsilon 〉}_2$ contribution to the dispersion of the average energy loss. Solid line, $\beta \hslash {\omega }_b=4$; dashed line, $\beta \hslash {\omega }_b=2$; dashed-dotted line, $\beta \hslash {\omega }_b=0$ (classical limit).

Figure 6

The characteristic function of the energy loss, $\tilde{P}\left(\tau -i/2\right)$. The curves in the figure correspond to the different values of the dissipative $\mu$ parameter: solid line, $\mu =1.1$; dashed line, $\mu =1.2$; dashed-dotted line, $\mu =1.3$. The barrier height has been taken equal to $\beta V_0=10$ and the barrier frequency equal to $\beta \hslash {\omega }_b=2$. The linear response theory results, ${\tilde{P}}_0\left(\tau -i/2\right)$, are also shown for comparison by dotted lines.

Figure 7

The characteristic function of the energy loss, $\tilde{P}\left(\tau -i/2\right)$. The curves in the figure correspond to the different values of the barrier frequency: solid line, $\beta \hslash {\omega }_b=4$; dashed line, $\beta \hslash {\omega }_b=2$; dashed-dotted line, $\beta \hslash {\omega }_b=1$; and dotted line, $\beta \hslash {\omega }_b=0$ (classical Kramers model). The barrier height has been set equal to $\beta V_0=10$ and the dissipative $\mu$ parameter set equal to $\mu =1.2$.