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Journal of Physics A Mathematical and General

Microwave-assisted tunnelling in the presence of dissipation

G J Papadopoulos

Published Aug 7, 1997• 6 citations
Publisher

Abstract

The case of an electron tunnelling through a parabolic repeller and acted upon by a microwave electric field is presently treated taking account of dissipation. Dissipation is introduced using a frictional force and the relevant quantum mechanics stems from an appropriate Lagrangian capable of generating the dissipative force. Considerations are presented for deriving a suitable continuity equation adapted to the dissipative processes involved. Starting with a wavepacket as the particle's initial state, expressions for the probability and current densities, as well as the transmission coefficient, are derived. These are used to see the influence of friction, frequency, amplitude and initial phase of the applied field on the tunnelling effect. Furthermore, the localizing effect of friction is made visible on the reflected portion of the wavepacket, where the main body of the probability resides.

DOI
10.1088/0305-4470/30/15/031

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The case of an electron tunnelling through a parabolic repeller and acted upon by a microwave electric field is presently treated taking account of dissipation. Dissipation is introduced using a frictional force and the relevant quantum mechanics stems from an appropriate Lagrangian capable of generating the dissipative force. Considerations are presented for deriving a suitable continuity equation adapted to the dissipative processes involved. Starting with a wavepacket as the particle's initial state, expressions for the probability and current densities, as well as the transmission coefficient, are derived. These are used to see the influence of friction, frequency, amplitude and initial phase of the applied field on the tunnelling effect. Furthermore, the localizing effect of friction is made visible on the reflected portion of the wavepacket, where the main body of the probability resides.
A wave equation formulation of the problem of quantum tunneling in a dissipative medium is developed by considering a many-body system in which the central particle is subject to an arbitrary force law, and at the same time is coupled to a bath of noninteracting harmonic oscillators. For the motion of the central particle it is possible to obtain an effective Lagrangian and Hamiltonian by eliminating the degrees of freedom of the oscillators. However both of these operators are nonlocal, and it is difficult to derive a wave equation for this motion. As an alternative method one can write a many-body Schrödinger equation for the whole system, and then eliminate the wave functions of all of the oscillators. This result is a many-channel Schrödinger equation for the motion of the central particle. By truncating this set of coupled equations, one can solve the problem for different force laws. In particular, in this work, the cases of dissipative tunneling, hopping, and quantum coherence are studied in detail. It is also shown how this approach can be generalized to multidimensional dissipative systems.
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 bi-linearly 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.
Ignoring thermal fluctuations of the environment, taking into account only its dissipative effects, free propagation of a Gaussian wavepacket is studied in the framework of the linear Caldirola-Kanai (CK) equation and non-linear equations, the Schrodinger-Langevin (SL) equation, known as Kostin equation, and the Schuch-Chung-Hartmann (SCH) equation. By a Gaussian ansatz for the probability density one obtains two equations, one for the evolution of the center of the wave packet which is just the classical Langeving equation and one for the evolution of the width of the wavepacket. This last equation has different forms in different approaches having only an analytic solution in the CK and SCH frameworks. Computations show that for a given friction, the width of the wavepacket increases with time in all approaches. In a given time, it reduces with friction in both CK and SL approaches revealing localization effects of dissipation, while has opposite behavior in the SCH approach. Furthermore, energy expectation value and its time-derivative is computed and compared in all approaches. It is shown that the rate of energy is given by the expectation value of momentum field in the SL framework. Finally, transmission of a Gaussian wavepacket from a rectangular barrier is numerically studied in the context of CK and compared to the non-dissipative case. The transmission decreases considerably with the dissipation.
It is shown that the propagator K(xt mod x'0) for a particle in a potential field is derivable from the classical path starting from x' and reaching x at time t. Considering a wavepacket as the particle initial state lying mainly on one side of a barrier the propagator is used for obtaining the wavefunction on the other side. Tunnelling is then discussed in terms of the evolving wavefunction. It is argued that while the particle's expectation energy is taken to be lower than the barrier's height the transformation by which the wavefunction is produced entails energetically crossover flights as required by the dynamics for the classical paths involved in the propagation process. The parabolic repeller exemplifies the formalism and exact results for the probability and current densities are given. A computation involving ballistic tunnelling shows that the current density rises to a saturation value proportional to the particles' injection rate.