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Quantum Electronics and Laser Science Conference

Monte Carlo wave functions for nonlinear master equations

Elena V. Goldstein, K. Plattner, P. Maystre

Published Jun 7, 1996• 0 citations
OpenAlex

Abstract

Summary form only given. Nonlinear Schrodinger equations are ubiquitous in physics, and appear in problems ranging from Bose liquids and light propagation in nonlinear media to quantum chains, nonlinear atom optics, and atomic Bose condensation. A generic situation leading to a nonlinear Schrodinger equation description involves the effective single-particle dynamics of a many-particle system coupled by a two-body interaction, and with the manybody wave function truncated via a Hartree-Fock Ansatz. However, if this system is in addition coupled to one, or possibly several reservoirs, such a description is no longer appropriate, and must be superseded by a master equation.

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Summary form only given. Nonlinear Schrodinger equations are ubiquitous in physics, and appear in problems ranging from Bose liquids and light propagation in nonlinear media to quantum chains, nonlinear atom optics, and atomic Bose condensation. A generic situation leading to a nonlinear Schrodinger equation description involves the effective single-particle dynamics of a many-particle system coupled by a two-body interaction, and with the manybody wave function truncated via a Hartree-Fock Ansatz. However, if this system is in addition coupled to one, or possibly several reservoirs, such a description is no longer appropriate, and must be superseded by a master equation.
We present a many-body theory of nonlinear atom optics, and discuss some of its physical implications in the coherent regime. Considering a system of N identical two-level atoms interacting with classical and quantum-mechanical electromagnetic fields, we derive a Fock-space many-particle master equation. Introducting a Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy and a Hartree-Fock factorization to truncate this hierarchy, we obtain an effective nonlinear single-particle master equation that forms the basis of nonlinear atom optics. In the second part of the paper, we concentrate on the coherent part of that master equation, and derive an effective single-atom nonlinear Schr\"odinger equation. This equation leads to the prediction of a number of effects, and, in particular, several kinds of atomic solitons. We discuss and numerically study two such kinds of solitons, Thirring solitons and gap solitons. Finally, the axial containment of an atomic gap soliton is illustrated.
Nonlinear effects lie amongst the most interesting phenomena in physical, chemical and biological systems. In quantum physics nonlinearity is provided by inter-particle interactions. A theoretical many-body description can be made in the framework of the mean field approximation, which yields the Hartree-Fock equations for a system of interacting fermions and the Gross-Pitaevskii equation for interacting bosons. The latter is widely applied to Bose-Einstein condensates (BECs) of cold atoms and satisfactorily describes a variety of nonlinear phenomena, such as the superfluid transition, formation of single vortices and vortex lattices. From the mathematical point of view the Gross-Pitaevskii equation is equivalent to the nonlinear Schrödinger equation, used in nonlinear optics for the description of self-focusing and soliton propagation.
Although quantum mechanics is linear, there are nevertheless quantum systems with multiple interacting particles in which the effective evolution of a single particle is governed by a nonlinear equation. This includes Bose-Einstein condensates, which are governed by the Gross-Pitaevskii equation, which is a cubic nonlinear Schrodinger equation with a term proportional to /[psi]/ ²[psi] scaling of Grover's algorithm. Since the Gross- Pitaevskii equation effectively approximates the multi- particle Schrodinger equation, for which Grover's algorithm is optimal, our result leads to a quantum information-theoretic bound on the number of particles needed for this approximation to hold, asymptotically. The Gross-Pitaevskii equation is not the only nonlinearity of the form f(/[psi]/²)[psi] that arises in effective equations for the evolution of real quantum physical systems, however : The cubic-quintic nonlinear Schrodinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schrodinger equation describes Bose liquids under certain conditions. Analysis of computation with such systems yields some surprising results; for example, when time-measurement precision is included in the resource accounting, searching a database'' when there is a single correct answer may be easier than searching when there are multiple correct answers. These results lead to quantum information-theoretic bounds on the physical resources required for these effective nonlinear theories to hold, asymptotically. Furthermore, strongly regular graphs, which have no global symmetry, are sufficiently complete for quantum search on them to asymptotically behave like unstructured search. Certain sufficiently complete graphs retain the improved runtime and resource scalings for some nonlinearities, so our scheme for nonlinear, analog quantum computation retains its benefits even when some structure is introduced
A numerical method is given for affecting nonlinear Schr\"odinger evolution on an initial wave function, applicable to a wide range of problems, such as time-dependent Hartree, Hartree-Fock, density-functional, and Gross-Pitaevskii theories. The method samples the evolving wave function at Chebyshev quadrature points of a given time interval. This achieves an optimal degree of representation. At these sampling points, an implicit equation, representing an integral Schr\"odinger equation, is given for the sampled wave function. Principles and application details are described, and several examples and demonstrations of the method and its numerical evaluation on the Gross-Pitaevskii equation for a Bose-Einstein condensate are shown.
We discuss Monte Carlo simulation of the quantum master equation in terms of system wave functions undergoing a sequence of quantum jumps based on the delay function for the occurrence of the “next jump”.1-3 The formalism can be applied to calculate the density operator and correlation functions (spectra) in quantum systems with a large number of degrees of freedom. We present the formalism and applications to (i) a multilevel atom undergoing spontaneous emission including mechanical light effects (in particular, laser cooling in the quantum limit), (ii) coupling of an atomic system to a squeezed vacuum, and (iii) cavity QED problems with coupling to thermal and squeezed baths.