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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Low-order qubit representation for two coupled Bose-Einstein condensates (BEC)

George Vahala, Jeffrey Yepez, Linda L. Vahala

Published May 5, 2006• 0 citations
Publisher

Abstract

A quantum lattice algorithm is developed for the solution of the coupled Gross-Pitaevskii equations, which are just the nonlinear Schrodinger equations with an external potential. In the language of solitons, a sufficiently strong external potential can localize the solitons and even the radiation in the case of Manikov solitons. For non-integrable potential cases the radiation is no longer localized and one can also lose the soliton structures themselves. One also finds locked mode structures if the systems is close to being integrable.

DOI
10.1117/12.665333

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A quantum lattice algorithm is developed for the solution of the coupled Gross-Pitaevskii equations, which are just the nonlinear Schrodinger equations with an external potential. In the language of solitons, a sufficiently strong external potential can localize the solitons and even the radiation in the case of Manikov solitons. For non-integrable potential cases the radiation is no longer localized and one can also lose the soliton structures themselves. One also finds locked mode structures if the systems is close to being integrable.
The nonlinear Schrodinger (NLS) equation in a self-defocusing Kerr medium supports dark solitons. Moreover the mean field description of a dilute Bose-Einstein condensate (BEC) is described by the Gross-Pitaevskii equation, which for a highly anisotropic (cigar-shaped) magnetic trap reduces to a one-dimensional (1D) cubic NLS in an external potential. A quantum lattice algorithm is developed for the dark solitons. Simulations are presented for both black (stationary) solitons as well as (moving) dark solitons. Collisions of dark solitons are compared with the exact analytic solutions and coupled dark-bright vector solitons are examined. The quantum algorithm requires 2 qubits per scalar field at each spatial node. The unitary collision operator quantum mechanically entangles the on-site qubits, and this transitory entanglement is spread throughout the lattice by the streaming operators. These algorithms are suitable for a Type-II quantum computers, with wave function collapse induced by quantum measurements required to determine the coupling potentials.
Coupled nonlinear Schrodinger equations (CNLS) with an external elliptic function potential model a quasi one--dimensional interacting two-component Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions of the CNLS with a periodic potential are presented and their stability studied by direct numerical simulations. Some of these solutions allow reduction to Manakov system. From a physical point of view these solutions can be interpreted as exact Bloch states at the edge of the Brillouin zone. Some of them are stable while others are found to be unstable against modulations of long wavelength. The solutions which are modulationally unstable are shown to lead to the formation of localized ground states of the coupled BEC system.
We systematically study the properties of lattice solitons in Bose-Einstein condensates with either attractive or repulsive atom interactions. This is done, by exactly solving the mean-field Gross-Pitaevskii equation in the presence of a periodic potential. We find new families of lattice soliton solutions that are characterized by the position of the energy eigenvalue within the associated band structure. These include lattice solitons in condensates with either attractive or repulsive atom interactions that exist in finite or semi-infinite gaps, as well as nonlinear modes that exhibit atomic population cutoffs.
We discuss localized ground states of Bose-Einstein condensates (BEC's) in optical lattices with attractive and repulsive three-body interactions in the framework of a quintic nonlinear Schr\"odinger equation which extends the Gross-Pitaevskii equation to the one-dimensional case. We use both a variational method and a self-consistent approach to show the existence of unstable localized excitations which are similar to Townes solitons of the cubic nonlinear Schr\"odinger equation in two dimensions. These solutions are shown to be located in the forbidden zones of the band structure, very close to the band edges, separating decaying states from stable localized ones (gap solitons) fully characterizing their delocalizing transition. In this context the usual gap solitons appear as a mechanism for arresting the collapse in low-dimensional BEC's in optical lattices with an attractive real three-body interaction. The influence of the imaginary part of the three-body interaction, leading to dissipative effects in gap solitons, and the effect of atoms feeding from the thermal cloud are also discussed. These results may be of interest for both BEC's in atomic chips and Tonks-Girardeau gas in optical lattices.
Bose-Einstein condensates (BECs) have been one of the most active areas of research since their experimental birth in 1995. The complicated nature of the experiments on BECs suggests to observe them in reduced dimensions. The dependence of the collective excitations of the systems on the spatial degrees of freedom allows the study in lower dimensions. In this thesis, we first study two effectively one-dimensional parallel linearly coupled BECs in the presence of external potentials. The system is modelled by linearly coupled Gross-Pitaevskii (GP) equations. In particular, we discuss the dark solitary waves and the grey-soliton-like solutions representing analogues of superconducting Josephson fluxons which we refer to as the fluxon analogue (FA) solutions. We analyze the existence, stability and time dynamics of FA solutions and coupled dark solitons in the presence of a harmonic trap. We observe that the presence of the harmonic trap destabilizes the FA solutions. However, stabilization is possible by controlling the effective linear coupling between the condensates. We also derive theoretical approximations based on variational formulations to study the dynamics of the solutions semi-analytically.\n\nWe then study multiple FA solutions and coupled dark solitons in the same settings. We examine the effects of trapping strength on the existence and stability of the localized solutions. We also consider the interactions of multiple FA solutions as well as coupled dark solitons. In addition, we determine the oscillation frequencies of the prototypical structures of two and three FA solutions using a variational approach.\n\nFinally, we consider two effectively two-dimensional parallel coupled BECs enclosed in a double well potential. The system is modelled by two GP equations coupled by linear and nonlinear cross-phase-modulations. We study a large set of radially symmetric nonlinear solutions of the system in the focusing and defocusing cases. The relevant three principal branches, i.e. the ground state and the first two excited states, are continued as a function of either linear or nonlinear couplings. We investigate the linear stability and time evolution of these solutions in the absence and presence of a topological charge. We notice that only the chargeless or charged ground states can be stabilized by adjusting the linear or nonlinear coupling between the condensates.
This chapter focuses on exploring localized excitations in the variable coefficient nonlinear Schrodinger (NLS) equation and its generalizations. The motivation comes from the fact that the NLS equation and its variants appear in several branches/topics of physics, including nonlinear optics and Bose-Einstein condensates (BECs). The chapter reviews critically the variety of matter wave solitons and their interactions in single- and multi-component Bose-Einstein condensates. It provides a brief overview of the mean Gross- Pitaevskii (GP) equation, derivation of the one-dimensional GP equation, and mapping of the 1D GP equation to the standard NLS equation. The chapter discusses matter wave single and multi bright and dark solitons of Bose– Einstein condensates with varying trap potentials and atomic interactions. It shows various solitons solutions of multi-component BECs described by the two coupled GP equations. In this case, the coupled GP equation has been mapped onto a two coupled NLS equation.