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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Dissipative tunnelling of coherent states

Sotirios Baskoutas

Published Oct 1, 1996• 5 citations
Publisher

Abstract

In the present work we study the macroscopic quantum tunnelling effect using the effective inverted Caldirola - Kanai Hamiltonian. Assuming that our system is initially prepared in a coherent state, we prove that the presence of dissipation increases the sojourn time and permits low-noise measurements of signals by squeezed states. Their corresponding wavepackets satisfy the Heisenberg uncertainty relation and move along a classical trajectory with definite position and velocity.

DOI
10.1088/1355-5111/8/5/005

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In the present work we study the macroscopic quantum tunnelling effect using the effective inverted Caldirola - Kanai Hamiltonian. Assuming that our system is initially prepared in a coherent state, we prove that the presence of dissipation increases the sojourn time and permits low-noise measurements of signals by squeezed states. Their corresponding wavepackets satisfy the Heisenberg uncertainty relation and move along a classical trajectory with definite position and velocity.
In order to study the behaviour of the inverted Caldirola-Kanai Hamiltonian in the quantum tunnelling effect, the authors consider a wavepacket as the initial state and they calculate exactly the probability density. They also obtain the transmission and reflection probability, the expectation value of the particle's energy and the sojourn time which appears to be an increasing function of the dissipation parameter gamma . A similar result is also obtained by considering a normalized plane wave as the initial state.
Using the phenomenological quantum friction models introduced by P. Caldirola [Nuovo Cimento 18, 393 (1941)] and E. Kanai [Prog. Theor. Phys. 3, 440 (1948)], M. D. Kostin [J. Chem. Phys. 57, 3589 (1972)], and K. Albrecht [Phys. Lett. B 56, 127 (1975)], we study quantum tunneling of a one-dimensional potential in the presence of energy dissipation. To this end, we calculate the tunneling probability using a time-dependent wave-packet method. The friction reduces the tunneling probability. We show that the three models provide similar penetrabilities to each other, among which the Caldirola-Kanai model requires the least numerical effort. We also discuss the effect of energy dissipation on quantum tunneling in terms of barrier distributions.
We begin by revisiting the so-called Caldirola - Kanai Hamiltonian and discuss its inherent ambiguity: does it represent a dissipative harmonic oscillator (HO) subject to a friction force, or does it describe an HO with a time-dependent mass (TDM)? Although classically both descriptions do coexist, in the quantum domain the solution of the Schrödinger equation (or Heisenberg equations of motion) with a TDM does not present inconsistencies, however, the dissipative Hamiltonian shows violation of the Heisenberg uncertainty principle. This violation is avoided by introducing a stochastic force in the equations of motion, which will take care of the fluctuations due to the environment. Once the distinction between the dissipative and amplifying Hamiltonian is made clear, we consider the problem of the quantum TDM HO subject to dissipation, showing that both phenomena may be merged and described by a single Hamiltonian, the amplifying - dissipative Hamiltonian . We obtain the solutions of the Heisenberg equations of motion for the canonical momentum and position; next, we specialize on the weak damping limit and analyse the effects of the amplifying - dissipative process on the mean values of the physical variables.