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arXiv:1107.2523 · Journal of Physics A Mathematical and Theoretical

Dynamics of localized states in extended supersymmetric quantum mechanics with multi-well potentials

Victor P. Berezovoj, M. I. Konchatnij

Published May 17, 2012• 8 citations
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Abstract

In this paper we propose a self-consistent approach to the description of temporal dynamics of localized states. This approach is based on exactly solvable quantum mechanical models with multi-well potentials and their propagators. States of Hamiltonians with multi-well potentials form a suitable basis for the expansion of wave packets with different shapes and localization degrees. We also consider properties of the tunneling wave packets, taking into account all states of Hamiltonians with symmetric and asymmetric potentials. The study of the dynamics of initially localized states shows that application of the two-state approximation for the description of tunneling is considerably limited, especially for systems which have several states in the under-barrier region, as for example in modern superconducting quantum interference devices and traps for cold atoms.

DOI
10.1088/1751-8113/45/22/225302

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In this paper we propose a self-consistent approach to the description of temporal dynamics of localized states. This approach is based on exactly solvable quantum mechanical models with multi-well potentials and their propagators. States of Hamiltonians with multi-well potentials form a suitable basis for the expansion of wave packets with different shapes and localization degrees. We also consider properties of the tunneling wave packets, taking into account all states of Hamiltonians with symmetric and asymmetric potentials. The study of the dynamics of initially localized states shows that application of the two-state approximation for the description of tunneling is considerably limited, especially for systems which have several states in the under-barrier region, as for example in modern superconducting quantum interference devices and traps for cold atoms.
Inspired by new trends in atomtronics, cold atom devices and Bose–Einstein condensate dynamics, we apply a general technique of N = 4 extended supersymmetric quantum mechanics to isospectral Hamiltonians with triple-well potentials, i.e. symmetric and asymmetric. Expressions of quantum-mechanical propagators, which take into account all states of the spectrum, are obtained, within the N = 4 SQM approach, in the closed form. For the initial Hamiltonian of a harmonic oscillator, we obtain the explicit expressions of potentials, wavefunctions and propagators. The obtained results are applied to tunneling dynamics of localized states in triple-well potentials and for studying its features. In particular, we observe a Josephson-type tunneling transition of a wave packet, the effect of its partial trapping and a non-monotonic dependence of tunneling dynamics on the shape of a three-well potential. We investigate, among others, the possibility of controlling tunneling transport by changing parameters of the central well, and we briefly discuss potential applications of this aspect to atomtronic devices.
Quantum wave dynamics in systems with potential wells, bathers and outer impenetrable walls has been studied using solutions of time-dependent Schrodinger equation. Numerical integration of Schrodinger equation, calculation of dynamical variables and its average values, Fourier-spectra were carried out for different initial conditions and different relations between wave packet energy and bather height. In non-symmetric wells the oscillations of wave packets are accompanied by multistep discrete tunneling of probability fluid from well through barrier. In double well potential (symmetric wells) the temporal scales of oscillations and tunneling were investigated in details. Influence of additional time-dependent potential on quantum wave packet dynamics in above listed systems is also 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.
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.
Dynamical tunneling of a quantum wave packet in the system with distributed potential and impenetrable walls is discussed. This potential is specified as algebraic difference of quadratic and cubed functions. It is limited on finite length and incorporates local minimum and maximum. Potential distribution in the vicinity of local minimum forms asymmetric well, and barrier originates into domain with maximum. Initial Gaussian wave packet is located into the potential minimum and tunneling comes through potential maximum. Numerical integration of Schrodinger equation was carried out at zero conditions on system walls. For high barrier, the wave packet oscillates in the well for a long time and its small portions passes through the barrier. If the barrier height is small, tunneling takes place in a short time interval. Expectation coordinate and field velocity, probability density at separate points as function of time, control of uncertainty relation and normalization condition were calculated in detail.