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Springer series in solid-state sciences

Quantum Coherence and the Nonlinear Schrödinger Equation

David W. Brown, Katja Lindenberg, Bruce J. West

Published Jan 1, 1987• 0 citations
Publisher
DOI
10.1007/978-3-642-83033-4_24

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By employing supersymmetric quantum mechanics, we present a general algorithm to construct supersymmetric partner potentials and hence derive exact stationary solutions of the inhomogeneous nonlinear Schrödinger equation (INLSE). This is possible due to the connection between the INLSE and the nonlinear Schrödinger equation (NLSE), which can be established from a treatment based on Lie point symmetries and is related with Schrödinger equation, under certain conditions. As an illustrative example, we construct exact solutions for the INLSE through a Pösch-Teller potential with a single bound state.
Weak measurement of a subset of noncommuting observables of a quantum system can be modeled by the open-system evolution, governed by the master equation in the Lindblad form.The open-system density operator can be represented as a statistical mixture over non-unitarily evolving pure states, driven by the stochastic nonlinear Schrödinger equation (sNLSE).The globally stable solution of the sNLSE is obtained in the case where the measured subset of observables comprises the spectrum-generating algebra of the system.This solution is a generalized coherent state (GCS), associated with the algebra.The result is based on proving that the GCS minimizes the trace-norm of the covariance matrix, associated with the spectrum-generating algebra.
In this report we investigate the macroscopic quantum tunneling of a Bose condensate falling under gravity and scattering on a Gaussian barrier that could model a mirror of far-detuned sheet of light. We analyze the effect of the inter-atomic interaction and that of a transverse confining potential. We show that the quantum tunneling can be quasi-periodic and in this way one could generate coherent Bose condensed atomic pulses. In the second part of the report, we discuss an effective 1D time-dependent non-polynomial nonlinear Schrodinger equation (NPSE), which describes cigar-shaped condensates. NPSE is obtained from the 3D Gross-Pitaevskii equation by using a variational approach. We find that NPSE gives much more accurate results than all other effective 1D equations recently proposed.