Exponential speedup of incoherent tunneling via dissipation

We study the escape rate of a particle in a metastable potential in the presence of a dissipative bath coupled to the momentum of the particle. Using the semiclassical bounce technique, we find that this rate is exponentially enhanced. In particular, the influence of momentum dissipation depends on the slope of the barrier that the particle is tunneling through. We investigate also the influence of dissipative baths coupled to the position, and to the momentum of the particle, respectively. In this case the rate exhibits a nonmonotonic behavior as a function of the dissipative coupling strengths. Remarkably, even in the presence of position dissipation, momentum dissipation can enhance exponentially the escape rate in a large range of the parameter space. The influence of the momentum dissipation is also witnessed by the substantial increase of the average energy loss during inelastic (environment-assisted) tunneling.

Figure 1

(a) The considered model of a particle in a (semidouble parabolic) potential and coupled to an external bath via the momentum operator with coupling constant ${\tau }_p$. The slope of the energy barrier on the right can be changed by varying the minimum $\mathrm{\Sigma }$, the different colored lines correspond to different values of $\mathrm{\Sigma }$. (b) Exponential enhancement $\mathcal{E}$ of the escape rate $\mathrm{\Gamma }$ as a function of ${\tau }_p$. Different lines correspond to different regimes: $\mathrm{\Sigma }=0.01V_0$, $\mathrm{\Sigma }={10}^4{V}_0$, $\mathrm{\Sigma }=\infty$; the dotted lines correspond to analytic expansions (see text). The frequency of the harmonic well is ${\omega }_0$, whereas ${\omega }_c$ is the high-frequency cutoff of the bath spectral density. In all cases the escape rate increases exponentially. Parameters $V_0/\hslash {\omega }_0=12.5$ and ${\omega }_c=8000{\omega }_0$.

Figure 2

(a) Example of the inverted potential $-V\left(x\right)$ for the motion of the minimizing (classical) path $x_{cl}\left(\tau \right)$. (b) Different paths $x_{cl}\left(\tau \right)$ for different values of $\mathrm{\Sigma }$ in the nondissipative case $\gamma ={\tau }_p=0$. The path shrinks with increasing $\mathrm{\Sigma }$. For $\mathrm{\Sigma }=\infty$ the particle is instantly reflected at the turning point $x_{\mathrm{esc}}$.

Figure 3

(a) Particle in a metastable well in the presence of both dissipative couplings. (b) Logarithmic plot of $\mathcal{E}$ as a function of $\gamma$, at fixed ratio ${\tau }_p{\omega }_0^2/\gamma =0.5$ and for different values of $\mathrm{\Sigma }$. The gray area and the violet and black lines correspond to the shape in panel (a). The dashed line is the analytical result shown in Appendix pp2. (c) Logarithmic plot of $\mathcal{E}$ as a function of $\gamma$, for $\mathrm{\Sigma }=2V_0$ and different values of the ratio ${\tau }_p{\omega }_0^2/\gamma$. (d) $\mathcal{E}$ as a function of $\mathrm{\Sigma }$ and $\gamma$, at fixed ratio ${\tau }_p{\omega }_0^2/\gamma =0.5$. For (b), (c), and (d) we used $V_0/\hslash {\omega }_0=12.5$.

Figure 4

(a) Average energy loss during the tunneling process $\langle \mathrm{\Delta }E\rangle =V\left(0\right)-V\left(x_{\mathrm{esc}}\right)$ for a particle coupled to two baths via the momentum and the position operators. (b) The quantity $\langle \mathrm{\Delta }E\rangle$ as a function of $\mathrm{\Sigma }$ for different dissipative cases. The solid (purple) line is for both dissipative couplings, whereas the dashed (red) line and the dotted (blue) line are in the presence of a single bath with coupling through the momentum and the position, respectively.

Figure 5

(a) Metastable well potential discussed in the present article. (b) Example of a single bounce path which minimizes the action without dissipation. The bounce time ${\xi }_B$ denotes the imaginary time interval in which the path is in the region $x>a$.

Figure 6

Example of bounce paths with the two dissipative couplings for (a) $\mathrm{\Sigma }=V_0$ and (b) $\mathrm{\Sigma }=20V_0$. The returning point $x_{\mathrm{esc}}$ shifts to higher values in both dissipative cases.

Figure 7

Change in the quantities contained in the prefactor $K$ due to dissipation for ${\tau }_p{\omega }_0=0.5\gamma /{\omega }_0$ and ${\omega }_c/{\omega }_0=8000$. (a) Ratio of the determinants scaled with its value without dissipation, (b) prefactor of the Jacobian transformation scaled with its value without dissipation, (c) negative eigenvalue scaled with its value without dissipation, (d) the prefactor $K$ of the escape rate scaled with its value $K_0$ without dissipation.

Figure 8

(a) The average energy loss $\langle \mathrm{\Delta }E\rangle$ as a function of the dissipative coupling strength, $c_s=\gamma /{\omega }_0$ for position dissipation and $c_s={\omega }_0{\tau }_p$, for two values of $\mathrm{\Sigma }$, for pure position dissipation (solid blue line) and for pure momentum dissipation (dashed red line) In the overdamped limit $\langle \mathrm{\Delta }E\rangle$ saturate to the same value. (b) Saturation of the energy loss in the presence with pure momentum dissipation as a function of $\mathrm{\Sigma }$, for different ${\omega }_c$ at ${\tau }_p{\omega }_0=0.5$.

Figure 9

(a) Smooth potential studied in this section. By increasing ${\omega }_B/{\omega }_0$ the potential becomes comparable to the semidouble parabolic potential. (b) Perturbative results for $\mathcal{E}$ for different ratios ${\omega }_B/{\omega }_0$ as a function of ${\tau }_p{\omega }_0$. (c) Comparison between the results for the semidouble parabolic and for the smooth potential as discussed in the text. Dotted lines correspond to results for the semidouble parabolic potential, while colored lines are for the smooth potential. (d) Logarithmic plot of the analytical formula (8) and the perturbative results for the smooth potential. Both potentials have the same slope at the turning point ${dV\left(x\right)/dx|}_{x_{\mathrm{esc}}}=-700m{\omega }_0^2$.