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Physical Review A

Tunneling dynamics of squeezed states in a potential well

Alan Lenef, Stephen Colby Rand

Published Jan 1, 1994• 8 citations
Publisher

Abstract

WKB calculations have revealed that tunneling dynamics of localized wave packets in simple potential wells depend sensitively on the degree of squeezing of the wave function. Squeezed states are shown to tunnel more slowly in general than eigenstates of the well and states which are initially localized near the bottom of a well have much lower tunneling rates than states localized at the sides of the well. The dependence of tunneling time on energy also exhibits steps related to quasi-bound-state energies in a complex way. Additionally, modulations in the tunneling probability correlate with the motion of the squeezed wave packet in the well. These results have implications both for the dynamics of squeezed radiation fields and squeezed matter states, and examples of applications to controlled vibrational-state chemistry, ultralow-noise measurements of weak signals with Josephson junctions, and optical crystals are discussed.

DOI
10.1103/physreva.49.32

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WKB calculations have revealed that tunneling dynamics of localized wave packets in simple potential wells depend sensitively on the degree of squeezing of the wave function. Squeezed states are shown to tunnel more slowly in general than eigenstates of the well and states which are initially localized near the bottom of a well have much lower tunneling rates than states localized at the sides of the well. The dependence of tunneling time on energy also exhibits steps related to quasi-bound-state energies in a complex way. Additionally, modulations in the tunneling probability correlate with the motion of the squeezed wave packet in the well. These results have implications both for the dynamics of squeezed radiation fields and squeezed matter states, and examples of applications to controlled vibrational-state chemistry, ultralow-noise measurements of weak signals with Josephson junctions, and optical crystals are discussed.
The motion of the center of a wave packet localized in one of the wells of an asymmetric double-well potential is studied. The asymmetric potential is obtained from a solvable symmetric potential using the Gel'fand–Levitan formalism, by keeping the energy eigenvalues unchanged, but changing the normalization of the ground-state wave function. While in a symmetric double well the wave packet tunnels back and forth between the two wells, for an asymmetric well, in general, tunneling does not take place. This is independent of the initial location of the wave packet, i.e., whether it is centered at the minimum of the shallower or of the deeper well. However, when such a system is coupled to a dissipative force, then the tunneling becomes possible. For instance it is shown that if, as a model of dissipative coupling, one chooses Gisin's nonlinear evolution equation, then the center of the wave packet ends up in the deeper well (decay of false vacuum). This result depends on the particular model of dissipation, for instance, an optical potential model yields a different result.
The roles of spatial symmetry and strength of external time-dependent perturbation on the dynamics of a quantum particle, initially localized in one of the wells of an asymmetric double-well potential are studied using the recently developed techniques incorporating quantum theory of motion and time-dependent Fourier grid Hamiltonian methods. The model used here includes a mimic of the related experimental situations which is considered as a perturbation to the static double-well potential. Analysis of localized and delocalized phase space structures and corresponding time-profile of tunneling probability reveal the recipe toward controlling the tunneling oscillations by modulating the parameters of applied perturbation. A study on a stochastic pulsating potential also reveals the root to the quantum localization, even in moderate field strength.
We introduce a general formalism, based on the stochastic formulation of quantum mechanics, to obtain localized quasi-classical wave packets as dynamically controlled systems, for arbitrary anharmonic potentials. The control is in general linear, and it amounts to introduce additional quadratic and linear time-dependent terms to the given potential. In this way one can construct for general systems either coherent packets moving with constant dispersion, or dynamically squeezed packets whose spreading remains bounded for all times. In the standard operatorial framework our scheme corresponds to a suitable generalization of the displacement and scaling operators that generate the coherent and squeezed states of the harmonic oscillator.
For an asymmetric double-well potential system, it is shown that, if the potential is quadratic until it reaches several times of the zero-point energies from the bottoms in each well, the energy eigenvalues of the low lying excited states of the double-well system must be close to the eigenvalues of the quadratic potentials. These eigenvalue structures suggest a method for the coherent control of the tunneling as well as realizing almost complete localization of the wave packet in one of the wells, by handling the double-well asymmetry. Numerical examples are included to indicate that the method could be useful also in a more general potential, and to propose experimental confirmations.
We examine the dynamics of a wave packet that initially corresponds to a coherent state in the model of quantum kicked rotator. This main model of quantum chaos, which allows for a transition from regular to to chaotic behavior in the classical limit, may be realized in experiments with cold atoms. We study the generation of squeezed states in the quasiclassical limit and in a time interval when quantum-classical correspondence is yet well-defined. We find that the degree of squeezing depends on the degree of local instability in the system and increases with the Chirikov parameter of stochasticity. We discuss the dependence of the degree of squeezing on the initial width of the packet, the problems of stability and observability of squeezed states at the transition to quantum chaos, as well as the dynamics of wave packet destruction.