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Physical review. B, Condensed matter

Dissipative quantum tunneling: A variational principle for phase shifts

Chang Han, Peter S. Riseborough

Published Aug 1, 1989• 1 citation
Publisher

Abstract

The tunneling process in dissipative quantum systems can be described in terms of a Feynman functional integral. The quasiclassical approximation entails the evaluation of solutions of the Euler-Lagrange equations in imaginary time and diagonalizing the small-amplitude excursions about the extremal trajectories. The latter process is equivalent to evaluating all the eigenvalues of integro-differential eigenvalue equations. We show that the exact eigenvalues are related to the phase shifts and derive a Hulth\'en-type of variational principle for the phase shifts appropriate for the above-described eigenvalue equation. We illustrate the method by applying it to the problem of the decay of a metastable state coupled to an Ohmic heat bath at T=0. We compare the accuracy with the Born approximation, rigorous results, and numerical evaluations.

DOI
10.1103/physrevb.40.2120

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The tunneling process in dissipative quantum systems can be described in terms of a Feynman functional integral. The quasiclassical approximation entails the evaluation of solutions of the Euler-Lagrange equations in imaginary time and diagonalizing the small-amplitude excursions about the extremal trajectories. The latter process is equivalent to evaluating all the eigenvalues of integro-differential eigenvalue equations. We show that the exact eigenvalues are related to the phase shifts and derive a Hulth'en-type of variational principle for the phase shifts appropriate for the above-described eigenvalue equation. We illustrate the method by applying it to the problem of the decay of a metastable state coupled to an Ohmic heat bath at T=0. We compare the accuracy with the Born approximation, rigorous results, and numerical evaluations.
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.
We consider the decay of a metastable state, coupled to a thermal reservoir, by quantum tunnelling. We calculate the leading exponential term in the W.K.B. expansion of the decay rate through a variational approximation. The approach is simple, quick and versatile, when compared with the alternative method which involves the numerical solution of non-linear, non-local, integro-differential equations. We have demonstrated the versatility of the variational approach by applying it to a few types of potentials, most of which have not been previously studied. In the few cases where numerical results are available, the variational approximation gives excellent agreement. In the cases where no numerical results are available for comparison, the variational approximation reproduces trends that are either physically reasonable or that may be inferred by interpolation between exactly soluble limits. We also address the question: What are the effects of frequency dependent friction, or a finite noise correlation time, on the quantum tunnellling rate? This question is one that has been recently broached in the discussion of experiments on highly damped r.f. SQUIDs and Josephson junctions. We find that the effect of the memory time is to reduce the lifetime of the metastable state, towards its undamped value. Again, we find that the variational approximation reproduces the exact variation of the action to a surprisingly good degree of accuracy.
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The canonical treatment of the quantum dynamics of systems with dissipation affected degrees of freedom proceeds within the sum over histories picture of quantum theory. This paper proposes to supplement that approach with a Schrodinger equation approach to the same physics. Starting from a Lagrangian that is of the same form as those obtained by path integration, one can arrive at a quantized Hamiltonian analogue using the standard rules applied to non-dissipative systems. The resulting Hamiltonian is non-diagonal in position representation, so that the corresponding Schrodinger equation is a nonlinear integro-differential equation reminiscent of the Hartree equation. In the case of the normal tunnel junction, with a quasiparticle heat bath, this equation has an exact solution.
We consider the motion of a quantum particle placed in a metastable minima of a potential well and examine the effect of interactions with the thermal reservoir at temperature T. We use the dilute, bounce approximation for the Green's function. We find a particular value of the dissipative coupling strength for which we can solve the Euler-Lagrange equations exactly, in the range of temperatures between zero and the crossover to the regime where thermal hopping dominates. We obtain analytic expressions for the tunneling rate in this temperature regime.
The quantum-mechanical problem of tunneling of a particle coupled to a heat bath that consists of a large number of harmonic oscillators is studied in this paper. Here the equation of motion for the particle which is a nonlinear integro-differential equation is found by eliminating the degrees of freedom of the oscillators. The nonlinear operator equation may be solved by noting that the position operator at the time t can be expanded as a power series in terms of the initial position and momentum with time-dependent c-number coefficients. These coefficients also satisfy nonlinear integro-differential equations. For the problem of dissipative tunneling of a particle in a double-well potential, the first few terms of expansion have been calculated numerically. The expansion parameter in this formulation is a small dimensionless parameter which is inversely proportional to the product of the width and the square root of the height of the barrier.