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Modern Physics Letters B

DISSIPATIVE QUANTUM TUNNELLING AND THERMAL ACTIVATION: AN EXACTLY SOLVABLE MODEL

H. Dekker

Published Aug 1, 1988• 5 citations
Publisher

Abstract

By balancing energy diffusion (inside a local potential hole) with ohmic quantum tunnelling (through a parabolic barrier), an exact quasi-equilibrium metastable state is found with a decay rate which includes Bell’s result (for moderate damping), Melnikov’s formula (for weak damping), the Büttiker, Harris and Landauer expression (in the classical limit), both Kramers’ weak and strong damping results, and the correct quantum limit as special cases.

DOI
10.1142/s0217984988000588

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By balancing energy diffusion (inside a local potential hole) with ohmic quantum tunnelling (through a parabolic barrier), an exact quasi-equilibrium metastable state is found with a decay rate which includes Bell’s result (for moderate damping), Melnikov’s formula (for weak damping), the Büttiker, Harris and Landauer expression (in the classical limit), both Kramers’ weak and strong damping results, and the correct quantum limit as special cases.
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We study the real time dynamics of a dissipative quantum system in a metastable state which may decay by crossing a potential barrier. Starting from an initial state where the system is in thermal equilibrium on one side of the barrier, the time evolution of the density matrix is evaluated analytically in the semiclassical approximation for coordinates near the barrier top. In a region about a critical temperature Tc{\mathrm{T}}_{\mathrm{c}}Tc​ large quantum fluctuations render the harmonic approximation of the potential insufficient and anharmonicities become essential. Accounting for non-Gaussian fluctuation modes, we show that the density matrix approaches a quasistationary state with a constant flux across the barrier. This extends our earlier results [Phys. Rev. E 51, 4267 (1995)] on the quantum generalization of the Kramers flux state to the region about Tc{\mathrm{T}}_{\mathrm{c}}Tc​. By matching the flux state onto the equilibrium state on one side of the barrier, we determine the decay rate out of the metastable state. The rate constant shows a changeover from thermally activated decay to quantum tunneling for temperatures below Tc{\mathrm{T}}_{\mathrm{c}}Tc​.
The quantum-mechanical version of the Kramers turnover problem is considered. The multidimensional character of the problem is taken into account via transformation to normal modes. This eliminates the coupling to the bath near the barrier top allowing the use of a simple harmonic transmission coefficient for the barrier dynamics. The well dynamics is described by a continuum form of a master equation for the energy in the unstable normal mode. Within first-order perturbation theory, the equations of motion for the stable normal modes have the form of a forced oscillator. The transition probability kernel is found using the known solution for the quantum forced oscillator problem. An expression for the quantum escape rate is derived. It encompasses all previously known limiting results in the thermally activated tunneling regime. The depopulation factor, which accounts for the nonequilibrium energy distribution is evaluated. The quantum transition probability kernel is broader than the classical and is skewed towards lower energies. Interplay between these two effects, together with a positive tunneling contribution, determines the relative magnitude of the quantum rate compared to the classical one. The theory is valid for arbitrary dissipation. Its use is illustrated for the case of a cubic potential with Ohmic (Markovian) dissipation.
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As a prelude to a discussion of tunneling in the presence of dissipation, we discuss the damped harmonic oscillator. We show that for a harmonic-oscillator potential, the Caldeira-Leggett Hamiltonian can be reduced to a sum of independent harmonic-oscillator Hamiltonians by a normal-mode transformation. We then discuss tunneling through a parabolic barrier and explain the apparently contradictory results in the literature. The decay of a metastable state is then discussed. We argue that much of the published work on tunneling in dissipative systems has assumed unphysical initial conditions. We calculate the tunneling out of a metastable state, on the assumption that the environment is initially at zero temperature, and find that quantum tunneling is enhanced by dissipation.
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.