The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
D. Lacroix - One of the best experts on this subject based on the ideXlab platform.
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stochastic Schroedinger Equation from optimal observable evolution
Annals of Physics, 2007Co-Authors: D. LacroixAbstract:Abstract In this article, we consider a set of trial wave-functions denoted by |Q〉 and an associated set of operators Aα which generate transformations connecting those trial states. Using variational principles, we show that we can always obtain a quantum Monte-Carlo method where the quantum evolution of a system is replaced by jumps between density matrices of the form D = |Qa〉〈Qb|, and where the average evolutions of the moments of Aα up to a given order k, i.e., 〈 A α 1 〉 , 〈 A α 1 A α 2 〉 , … , 〈 A α 1 ⋯ A α k 〉 , are constrained to follow the exact Ehrenfest evolution at each time step along each stochastic trajectory. Then, a set of more and more elaborated stochastic approximations of a quantum problem is obtained which approach the exact solution when more and more constraints are imposed, i.e., when k increases. The Monte-Carlo process might even become exact if the Hamiltonian H applied on the trial state can be written as a polynomial of Aα. The formalism makes a natural connection between quantum jumps in Hilbert space and phase-space dynamics. We show that the derivation of stochastic Schroedinger Equations can be greatly simplified by taking advantage of the existence of this hierarchy of approximations and its connection to the Ehrenfest theorem. Several examples are illustrated: the free wave-packet expansion, the Kerr oscillator, a generalized version of the Kerr oscillator, as well as interacting bosons or fermions.
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Stochastic Schroedinger Equation from optimal observable evolution
Annals of Physics, 2007Co-Authors: D. LacroixAbstract:In this article, we consider a set of trial wave-functions denoted by |Q> and an associated set of operators A$_{alpha}$ which generate transformations connecting those trial states. Using variational principles, we show that we can always obtain a quantum Monte-Carlo method where the quantum evolution of a system is replaced by jumps between density matrices of the form D = |Qa>, are constrained to follow the exact Ehrenfest evolution at each time step along each stochastic trajectory. Then, a set of more and more elaborated stochastic approximations of a quantum problem is obtained which approach the exact solution when more and more constraints are imposed, i.e. when k increases. The Monte-Carlo process might even become exact if the Hamiltonian H applied on the trial state can be written as a polynomial of A$_{alpha}$ The formalism makes a natural connection between quantum jumps in Hilbert space and phase-space dynamics. We show that the derivation of stochastic Schroedinger Equations can be greatly simplified by taking advantage of the existence of this hierarchy of approximations and its connection to the Ehrenfest theorem. Several examples are illustrated: the free wave-packet expansion, the Kerr oscillator, a generalized version of the Kerr oscillator, as well as interacting bosons or fermions
Spyridon Kamvissis - One of the best experts on this subject based on the ideXlab platform.
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long time behavior for the focusing nonlinear Schroedinger Equation with real spectral singularities
Communications in Mathematical Physics, 1996Co-Authors: Spyridon KamvissisAbstract:We consider the effect of real spectral singularities on the long time behavior of the solutions of the focusing Nonlinear Schroedinger Equation. We find that for each spectral singularity λ′ ∈ℝ, such an effect is limited to the region of the (x,t)-plane in which λ′ is close to the point of stationary phase $$\lambda _0 = \tfrac{{ - x}}{{4t}}$$ (the phase here being defined in a standard way by, say, the evolution of the Jost functions). In that region, the solution performs decaying oscillations of the same form as in the other regions, but with different parameters. The order of decay is $$O\left( {\left( {\tfrac{{\log t}}{t}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right)$$ . We prove our result by using the Riemann-Hilbert factorization formulation of the inverse scattering problem. We recover our asymptotics by transforming our problem to one which is equivalent for large time, and which can be interpreted as the one corresponding to the genus 0 algebro-geometric solution of the Equation.
Claudio Conti - One of the best experts on this subject based on the ideXlab platform.
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solitonization of the anderson localization
Physical Review A, 2012Co-Authors: Claudio ContiAbstract:We study the affinities between the shape of the bright soliton of the one-dimensional nonlinear Schroedinger Equation and that of the disorder induced localization in the presence of a Gaussian random potential. With emphasis on the focusing nonlinearity, we consider the bound states of the nonlinear Schroedinger Equation with a random potential; for the state exhibiting the highest degree of localization, we derive explicit expressions for the nonlinear eigenvalue and for the localization length by using perturbation theory and a variational approach following the methods of statistical mechanics of disordered systems. We numerically investigate the linear stability and "superlocalizations". The profile of the disorder averaged Anderson localization is found to obey a nonlocal nonlinear Schroedinger Equation
M Koshiba - One of the best experts on this subject based on the ideXlab platform.
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algebraic solitary wave solutions of a nonlinear schrodinger Equation
Physical Review E, 1995Co-Authors: K Hayata, M KoshibaAbstract:We present algebraic solitary-wave solutions of a nonlinear Schroedinger Equation that includes two terms with power-law nonlinearity. Both bright and dark types are found. Numerical stability analyses show that the solitary wave exhibits a solitonic feature.
David J Tannor - One of the best experts on this subject based on the ideXlab platform.
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loading a bose einstein condensate onto an optical lattice an application of optimal control theory to the nonlinear schrodinger Equation
Physical Review A, 2002Co-Authors: Shlomo E Sklarz, David J TannorAbstract:Using a set of general methods developed by Krotov [A. I. Konnov and V. A. Krotov, Automation and Remote Control 60, 1427 (1999)], we extend the capabilities of optimal control theory to the nonlinear Schroedinger Equation (NLSE). The paper begins with a general review of the Krotov approach to optimization. Although the linearized version of the method is sufficient for the linear Schroedinger Equation, the full flexibility of the general method is required for treatment of the nonlinear Schroedinger Equation. Formal Equations for the optimization of the NLSE, as well as a concrete algorithm are presented. As an illustration, we consider a Bose-Einstein condensate (BEC) initially at rest in a harmonic trap. A phase develops across the BEC when an optical lattice potential is turned on. The goal is to counter this effect and keep the phase flat by adjusting the trap strength. The problem is formulated in the language of optimal control theory (OCT) and solved using the above methodology. To our knowledge, this is the first rigorous application of OCT to the nonlinear Schroedinger Equation, a capability that is bound to have numerous other applications.