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Physical review. B, Condensed matter

Sudden theory for tunneling in dissipative systems

Alfred M. Levine, Waldemar Hontscha, Eli Pollak

Published Aug 1, 1989• 25 citations
Publisher

Abstract

The effect of a low-frequency dissipative medium on the tunneling rate of a particle trapped in a metastable potential is investigated with the aid of a frozen-bath sudden approximation. The sudden theory is formulated for arbitrary time-dependent friction and for all temperatures at which the escape process is dominated by tunneling. The validity criterion of the theory is only that the bath frequency spectrum be substantially lower than the system frequency. It is applicable both in the weak- and strong-damping limits. The sudden theory is applied to tunneling in a cubic potential and a piecewise harmonic potential where the barrier frequency differs from the well frequency. In contrast to the ImF method, the sudden theory, when valid, can provide estimates of tunneling rates from well-defined excited resonance states and is not limited to estimates of only thermal rates. We find that, if the barrier is thin relative to the well, dissipation can serve to enhance tunneling rates from excited states.

DOI
10.1103/physrevb.40.2138

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Physica Scripta•Apr 1, 1988•1 citation
The effect of a low-frequency dissipative medium on the tunneling rate of a particle trapped in a metastable potential is investigated with the aid of a frozen-bath sudden approximation. The sudden theory is formulated for arbitrary time-dependent friction and for all temperatures at which the escape process is dominated by tunneling. The validity criterion of the theory is only that the bath frequency spectrum be substantially lower than the system frequency. It is applicable both in the weak- and strong-damping limits. The sudden theory is applied to tunneling in a cubic potential and a piecewise harmonic potential where the barrier frequency differs from the well frequency. In contrast to the ImF method, the sudden theory, when valid, can provide estimates of tunneling rates from well-defined excited resonance states and is not limited to estimates of only thermal rates. We find that, if the barrier is thin relative to the well, dissipation can serve to enhance tunneling rates from excited states.
The problem of metastability due to quantum and thermal fluctuations in a weakly damped system is treated within Feynman's influence-functional theory. It is shown that quite generally the nonequilibrium density matrix may be written as a single path integral. For a cubic metastable potential it has the form of a Feynman propagator with a complex Lagrangian containing second-order derivatives in time and terms noninvariant under time reversal. Dissipative tunneling and thermal activation emerge together as different solutions of the associated Lagrange equations. The low-temperature correction to the exponent of the tunneling rate behaves as Tn+1{T}^{n+1}Tn+1 for a spectrum of bath excitations vanishing as ${\ensuremath{\omega}}^{n}$. It is shown, moreover, how this result may be understood in terms of environment-assisted tunneling. Thus the low-temperature behavior of the tunneling rate provides a method to measure the dissipation mechanism in the quantum regime.
A phenomenological approach to dissipative tunneling, originally put forth by Pollak [Phys. Rev. A 33, 4244 (1986)], is discussed and contrasted with the bounce formalism. This phenomenological approach is based on the WKB theory for parabolic barrier tunneling. It is quite simple and readily evaluated. We study the results for the leading exponential dissipative decay rate in metastable potentials and contrast them with previous instanton-type (bounce-technique) calculations. In doing so, we compare the zero-temperature decay rate in various metastable potentials, different dissipation mechanisms, and also consider the rate enhancement at very low temperatures. One finds between the two methods unexpected good agreement at weak to moderate dissipation strength; the accuracy decreases with increasing dissipation strength and increasing deviation from a harmonic barrier shape.
The recently developed theory for tunneling in dissipative systems is rederived using quantal transition-state theory. As predicted by Caldeira and Leggett, we find an exponential damping of the tunneling rate at 0 K. The exponential rate enhancement at low temperatures as well as the crossover temperature are also obtained with this approach. Moreover, the rate enhancement is given explicitly in terms of energy transfer from the bath to the dissociative mode. The present derivation includes memory effects.
Quantum tunneling under the influence of an environment with nearly degenerate spectrum is considered. We first give a general derivation of the zero point motion formula in the sudden tunneling limit. The effects of a finite excitation energy of the environment are then considered by a perturbation method. Examples of linear oscillator coupling and rotational coupling show that the finite excitation energy can be represented by a dissipation factor and reduces the tunneling probability estimated in the limit of sudden tunneling. These examples clearly show that the applicability of the sudden tunneling approximation is governed by the details of the coupling as well as the relative time scales of the tunneling and the environmental degrees of freedom. We discuss some applications to heavy-ion fusion reactions.
A real-time functional-integral method is used to derive an effective action that gives the transmission spectrum of a tunneling particle interacting with a bath of harmonic oscillators. The transmission spectum is expressed in terms of double functional integrals with respect to the coordinate of the particle which are evaluated by means of the stationary-phase approximation. The equations of motion for the stationary-phase trajectories are solved exactly for an arbitrary spectral density function of the bath, and the obtained solutions are used to find the transmission spectra for specific examples. For a bath with single frequency \ensuremath{\omega}, an analytic expression of the transmission spectrum is obtained which covers from the sudden tunneling (\ensuremath{\omega}T0{\mathit{T}}_{0}T0​\ensuremath{\ll}1) to the adiabatic one (\ensuremath{\omega}T0{\mathit{T}}_{0}T0​\ensuremath{\gg}1), where T0{\mathit{T}}_{0}T0​ is the time it would take a classical particle to traverse the inverted bare potential barrier. For a bath with an Ohmic spectrum, the differential tunneling conductance at low bias voltage V and for \ensuremath{\eta}T0{\mathit{T}}_{0}T0​\ensuremath{\ll}1 is found to obey a power law \ensuremath{\sim}(eVT0{\mathit{eVT}}_{0}eVT0​/\ensuremath{\Elzxh}${)}^{\mathrm{\ensuremath{\eta}}{\mathit{T}}_{0}{\mathit{S}}_{0}/2\mathrm{\ensuremath{\pi}}\mathrm{\ensuremath{\Elzxh}}}$, where \ensuremath{\eta} is the friction coefficient and S0{\mathit{S}}_{0}S0​ is the tunneling exponent in the absence of interaction. \textcopyright{} 1996 The American Physical Society.
The role of memory friction on dissipative quantum reaction rates is emphasized. The domain of low temperatures, bearing a variety of important tunnelling phenomena, is difficult to handle analytically. In order to fill this gap, we present new scaling relations for the bounce length τ B as a function of the dissipation renormalized barrier frequency μ. In addition, we propose a simple estimate for τ B , which in turn yields directly the thermal low temperature enhancement of the quantum decay rate for Ohmic friction, as well as for memory friction. Finally the range of validity of the theory is discussed and pictorally represented in a rate phase diagram.