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arXiv:2110.04976 · Physical Review A

Numerical investigation of the logarithmic Schrödinger model of quantum decoherence

Rory van Geleuken, Andrew V. Martin

Published Mar 25, 2022• 1 citation
arXivHTMLPDF

Abstract

A logarithmic Schr\"odinger equation with time-dependent coupling to the nonlinearity is presented as a model of collisional decoherence of the wave function of a quantum particle in position space. The particular mathematical form of the logarithmic Schr\"odinger equation has been shown to follow from conditional wave theory, but the validity of the logarithmic Schr\"odinger equation has not yet been investigated numerically for general initial conditions. Using an operator-splitting approach, we solve the nonlinear equation of motion for the wave function numerically and compare it to the solution of the standard Joos-Zeh master equation for the density matrix. We find good agreement for the time-dependent behavior of the ensemble widths between the two approaches, but note curious zero-pinning behavior of the logarithmic Schr\"odinger equation, whereby the zeros of the wave function are not erased by continued propagation. By examining the derivation of the logarithmic Schr\"odinger equation from conditional wave theory, we indicate possible avenues of resolution to this zero-pinning problem.

DOI
10.1103/physreva.105.032210

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A logarithmic Schr"odinger equation with time-dependent coupling to the nonlinearity is presented as a model of collisional decoherence of the wave function of a quantum particle in position space. The particular mathematical form of the logarithmic Schr"odinger equation has been shown to follow from conditional wave theory, but the validity of the logarithmic Schr"odinger equation has not yet been investigated numerically for general initial conditions. Using an operator-splitting approach, we solve the nonlinear equation of motion for the wave function numerically and compare it to the solution of the standard Joos-Zeh master equation for the density matrix. We find good agreement for the time-dependent behavior of the ensemble widths between the two approaches, but note curious zero-pinning behavior of the logarithmic Schr"odinger equation, whereby the zeros of the wave function are not erased by continued propagation. By examining the derivation of the logarithmic Schr"odinger equation from conditional wave theory, we indicate possible avenues of resolution to this zero-pinning problem.
The quantum nonlinear Schrödinger equation, which models the propagation of light through dispersive nonlinear waveguides, provides an integrable quantum field theory that has been solved by a number of methods. Most recently, Gutkin [ J. Func. Anal.77, 327 ( 1988)] has developed an interwining operator technique for obtaining the time evolution of the field operators. Using Gutkin’s formalism, we show how to obtain the exact time evolution of an initial Schrödinger state vector. As an example, the two-particle case is carried out explicitly. Numerical results for the time evolution of the position probability distribution are presented for the case when the two particles initially occupy the same Gaussian wave packet.
In wave mechanics an appropriate description of a system under the influence of a linearly velocity dependent frictional force can be given by a nonlinear Schrödinger equation (NLSE) with logarithmic nonlinearity. However, the particular logarithmic form of the dissipative nonlinear frictional term in the NLSE is connected with the definition of the momentum- or velocity-operator in position-space. Therefore, in momentum-space, this form of the NLSE is no longer correct to describe the same physical sitution. This can be seen, e.g., from the fact that, in contrast to the linear case, the Fourier transform of the solution of the NLSE in position-space does not fulfill anymore the logarithmic NLSE in momentum-space. It will be shown, using results obtained from the theory in position-space, that it is possible to find a form of the nonlinear dissipative frictional term which is valid in position- as well as in momentum-space. Using this form, the NLSE looks like a diffusion equation with complex diffusion coefficient, i.e., a combination of a diffusion and a Schrödinger equation. The solution of this NLSE in momentum-space will be discussed. © 1994 John Wiley & Sons, Inc.
This work is aiming to show the advantage of using the Lie algebraic decomposition technique to solvefor Schrödinger’s wave equation for a quantum model, compared with the direct method of solution. The advantageis a two-fold: one is to derive general form of solution, and, two is relatively manageable to deal with the case oftime-dependent system Hamiltonian. Specifically, we consider the model of 2-level optical atom and solve for thecorresponding Schrödinger’s wave equation using the Lie algebraic decomposition technique. The obtained formof solution for the wave function is used to examine computationally the atomic localization in the coordinate space.For comparison, the direct method of solution of the wave function is analysed in order to show its complicationwhen dealing with time-dependent Hamiltonian.The possibility of using the Lie algebraic method for a qubit model(a driven quantum dot model) is briery discussed, if Schrödinger’s wave function is to be examined for the qubitlocalization.
A nonlinear Schrodinger equation describing how a quantum particle interacts with its surrounding reservoir is derived from the Wigner–Fokker–Planck equation (WFPE) via stochastic (Nelsonian) mechanical techniques. This model can be reduced just to a logarithmic Schrodinger equation through a suitable gauge transformation that allows one to explore its steady state dynamics and makes its mathematical and numerical analysis more tractable. The transient behavior of the standard deviation from the mean position associated with its solutions is also studied numerically and compared with that stemming from the WFPE.