Quantum Smoluchowski equation: A systematic study

The strong-friction regime at low temperatures is analyzed systematically starting from the formally exact path-integral expression for the reduced dynamics. This quantum Smoluchowski regime allows for a type of semiclassical treatment in the inverse friction strength so that higher-order quantum corrections to the original quantum Smoluchowski equation [J. Ankerhold, P. Pechukas, and H. Grabert, Phys. Rev. Lett. 87, 086802 (2001); J. Ankerhold and H. Grabert, Phys. Rev. Lett. 101, 119903 (2008)] can be derived. Drift and diffusion coefficients are determined by the equilibrium distribution in position and are directly related to the corresponding action of extremal paths and fluctuations around them. It is shown that the inclusion of higher-order corrections reproduces the quantum enhancement above crossover for the decay rate out of a metastable well exactly.

Figure 1

(Color online) Escape rates according to the QSE compared to the semiclassical exact rate vs friction strength for a metastable potential with ${\omega }_0={\omega }_b$ in the high-temperature range. Solid line depicts the leading-order result [${\lambda }_0$ term in Eq. (67)] for temperatures ${\omega }_0\hslash \beta =0.2,0.4,0.6$; differences between the three data sets are indistinguishable on this scale. Also shown are results obtained according to [23] with a friction-independent quantum enhancement factor for ${\omega }_0\hslash \beta =0.2$ (dotted), 0.4 (short-dashed), and 0.6 (long-dashed); see text for details.

Figure 2

(Color online) Same as in Fig. 1 but in the low-temperature range ${\omega }_0\hslash \beta =5$ and only for the QSE. The leading-order approximation (dashed line) and the next-order approximation including terms of order ${\lambda }_2$ (solid) are shown.

Figure 3

Equilibrium distribution scaled with $P_{\beta }^{\left(\text{cl}\right)}\left(x=0\right)$ in a double-well potential vs the scaled position $x=q/\sqrt{\hslash /m{\omega }_0}$ for various orders of perturbation theory. The classical result (dotted) is depicted together with the leading-order expression (short-dashed) according to Eq. (47), the expression in local harmonic approximation (dashed) according to Eq. (54), and the expression beyond (solid) according to Eq. (81). Parameters are $\gamma /{\omega }_0=3$, ${\omega }_0\hslash \beta =1$ for the bath and $\alpha \hslash /m^2{\omega }_0^3=0.5$ for the potential so that the well minimum is located at $x=\sqrt{2}$.

Figure 4

Quantum corrections $\delta D_2^{\left(n\right)}$ (87) in the diffusion coefficient for a harmonic system for $n=1$ (thin) and $n=2$ (thick). The systematic strong-friction expansion according to Eq. (28) (solid) is shown together with the without-$\gamma$ expansion [23] (dotted). The parameter $\gamma \hslash \beta =0.4$ is kept fixed, while $\gamma /{\omega }_0$ is varied. For $n=1$, both expansions coincide (see text).