1,357 research outputs found
Solving the nonlinear Schrödinger equation using energy conserving Hamiltonian Boundary Value Methods
In this paper we study the use of energy-conserving methods, in the class of Hamiltonian Boundary Value Methods, for the numerical solution of the nonlinear Schrödinger equation
Energy-conserving methods for the nonlinear Schrödinger equation
In this paper, we further develop recent results in the numerical solution of Hamiltonian partial differential equations (PDEs) (Brugnano et al., 2015), by means of energy-conserving methods in the class of Line Integral Methods, in particular, the Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). We shall use HBVMs for solving the nonlinear Schrödinger equation (NLSE), of interest in many applications. We show that the use of energy-conserving methods, able to conserve a discrete counterpart of the Hamiltonian functional, confers more robustness on the numerical solution of such a problem
Scattering into one-dimensional waveguides from a coherently-driven quantum-optical system
We develop a new computational tool and framework for characterizing the
scattering of photons by energy-nonconserving Hamiltonians into unidirectional
(chiral) waveguides, for example, with coherent pulsed excitation. The temporal
waveguide modes are a natural basis for characterizing scattering in quantum
optics, and afford a powerful technique based on a coarse discretization of
time. This overcomes limitations imposed by singularities in the
waveguide-system coupling. Moreover, the integrated discretized equations can
be faithfully converted to a continuous-time result by taking the appropriate
limit. This approach provides a complete solution to the scattered photon field
in the waveguide, and can also be used to track system-waveguide entanglement
during evolution. We further develop a direct connection between quantum
measurement theory and evolution of the scattered field, demonstrating the
correspondence between quantum trajectories and the scattered photon state. Our
method is most applicable when the number of photons scattered is known to be
small, i.e. for a single-photon or photon-pair source. We illustrate two
examples: analytical solutions for short laser pulses scattering off a
two-level system and numerically exact solutions for short laser pulses
scattering off a spontaneous parametric downconversion (SPDC) or spontaneous
four-wave mixing (SFWM) source. Finally, we note that our technique can easily
be extended to systems with multiple ground states and generalized scattering
problems with both finite photon number input and coherent state drive,
potentially enhancing the understanding of, e.g., light-matter entanglement and
photon phase gates.Comment: Numerical package in collaboration with Ben Bartlett (Stanford
University), implemented in QuTiP: The Quantum Toolbox in Python, Quantum
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Energy conservation issues in the numerical solution of the semilinear wave equation
In this paper we discuss energy conservation issues related to the numerical
solution of the nonlinear wave equation. As is well known, this problem can be
cast as a Hamiltonian system that may be autonomous or not, depending on the
specific boundary conditions at hand. We relate the conservation properties of
the original problem to those of its semi-discrete version obtained by the
method of lines. Subsequently, we show that the very same properties can be
transferred to the solutions of the fully discretized problem, obtained by
using energy-conserving methods in the HBVMs (Hamiltonian Boundary Value
Methods) class. Similar arguments hold true for different types of Hamiltonian
Partial Differential Equations, e.g., the nonlinear Schr\"odinger equation.Comment: 41 pages, 11 figur
Mean-field dynamics of a non-Hermitian Bose-Hubbard dimer
We investigate an -particle Bose-Hubbard dimer with an additional
effective decay term in one of the sites. A mean-field approximation for this
non-Hermitian many-particle system is derived, based on a coherent state
approximation. The resulting nonlinear, non-Hermitian two-level dynamics, in
particular the fixed point structures showing characteristic modifications of
the self-trapping transition, are analyzed. The mean-field dynamics is found to
be in reasonable agreement with the full many-particle evolution.Comment: 4 pages, 3 figures, published versio
Bright solitary waves and non-equilibrium dynamics in atomic Bose-Einstein condensates
In this thesis we investigate the static properties and non-equilibrium dynamics of bright solitary waves in atomic Bose-Einstein condensates in the zero-temperature limit, and we investigate the non-equilibrium dynamics of a driven atomic Bose-Einstein condensate at finite temperature.
Bright solitary waves in atomic Bose-Einstein condensates are non-dispersive and soliton-like matter-waves which could be used in future atom-interferometry experiments. Using the mean-field, Gross-Pitaevskii description, we propose an experimental scheme to generate pairs of bright solitary waves with controlled velocity and relative phase; this scheme could form an important part of a future atom interferometer, and we demonstrate that it can also be used to test the validity of the mean-field model of bright solitary waves. We also develop a method to quantitatively assess how soliton-like static, three-dimensional bright solitary waves are; this assessment is particularly relevant for the design of future experiments.
In reality, the non-zero temperatures and highly non-equilibrium dynamics occurring in a bright solitary wave interferometer are likely to necessitate a theoretical description which explicitly accounts for the non-condensate fraction. We show that a second-order, number-conserving description offers a minimal self-consistent treatment of the relevant condensate -- non-condensate interactions at low temperatures and for moderate non-condensate fractions. We develop a method to obtain a fully-dynamical numerical solution to the integro-differential equations of motion of this description, and solve these equations for a driven, quasi-one-dimensional test system. We show that rapid non-condensate growth predicted by lower-order descriptions, and associated with linear dynamical instabilities, can be damped by the self-consistent treatment of interactions included in the second-order description
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