Dissipation can enhance quantum effects

Usually one finds that dissipation tends to make a quantum system more classical in nature. We study the effect of momentum dissipation on a quantum system. The momentum of the particle is coupled bilinearly to the momenta of a harmonic oscillator heat bath. For a harmonic oscillator system we find that the position and momentum variances for momentum coupling are, respectively, identical to momentum and position variances for spatial friction. This implies that momentum coupling leads to an increase in the fluctuations in position as the temperature is lowered, exactly the opposite of the classical-like localization of the oscillator, found with spatial friction. For a parabolic barrier, momentum coupling causes an increase in the unstable normal mode barrier frequency, as compared to the lowering of the barrier frequency in the presence of purely spatial coupling. This increase in the frequency leads to an enhancement of the thermal tunneling flux, which below the crossover temperature becomes exponentially large. The crossover temperature between tunneling and thermal activation increases with momentum friction so that quantum effects in the escape are relevant at higher temperatures.

Figure 1

Position variances (left, scaled with $\hslash ∕\Omega$) and momentum variances (right, scaled with $\hslash \Omega$) for momentum friction (solid lines) and for the corresponding spatial case (dotted lines) for various values of the friction strength $\gamma \Omega =0.5,2,7$: left, solid from bottom to top; left dotted from top to bottom; right, solid from top to bottom; right, dotted from bottom to top. The cutoff frequency is ${\omega }_c∕\Omega =10$. For the spatial case a model with Drude damping (cutoff ${\omega }_c$) is used with equivalent friction strength ${\gamma }_s=\gamma {\Omega }^2\pi ∕2$.

Figure 2

Classical enhancement of the reactive flux as a result of momentum coupling to a harmonic bath: The parabolic barrier frequency is plotted as a function of the coupling strength for three different values of the cutoff frequency. The reduced barrier frequency is also the ratio of the exact thermal unidirectional flux to the flux in the absence of coupling to the bath. The solid line is for $w_c=10$, the dashed line is for $w_c=4$, and the dotted line is for $w_c=0.5$.

Figure 3

VTST solution for the classical transmission factor through a parabolic barrier potential as compared to choosing the system coordinate as the reaction coordinate. The three lines correspond to the same values of the cutoff frequency as in Fig. 2.

Figure 4

Macroscopic enhancement of the quantum tunneling rate above crossover due to momentum coupling to a harmonic bath. Panels (a), (b), and (c) correspond to the cutoff frequencies 10, 4, and 0.5, respectively. In each panel we plot the dependence of the Wolynes enhancement factor as a function of the coupling strength for three different temperature values. Defining $\theta =\hslash \beta {\Omega }^{‡}$, the dotted, dashed, and solid lines correspond to $\theta =0.5$, 1, and 3, respectively. Note the change in scale for the coupling strength as one goes from panel (a) to panel (c). For low cutoff frequencies, the enhancement is weakened.