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Physical Review A

Resonance solutions of the nonlinear Schrödinger equation: Tunneling lifetime and fragmentation of trapped condensates

Nimrod Moiseyev, Lorenz S. Cederbaum

Published Sep 9, 2005• 47 citations
Publisher

Abstract

It is shown here how the lifetimes and energies of resonance states can be calculated by applying the complex scaling transformation to the nonlinear Schr\"odinger equation. It is essential to first apply the complex scaling transformation to the full Hamiltonian and to subsequently derive from the result the correct complex scaled nonlinear Schr\"odinger equation. The latter equation is physically relevant and amenable to numerical calculations. To analyze the results obtained by solving this equation, it is necessary to realize the close association of resonance phenomena with fragmentation of the system. As an illustrative example, we apply this theory to the Gross-Pitaevskii nonlinear equation to calculate the tunneling lifetime of a condensate inside an external (either optical or magnetic) trap. We show that by varying the scattering length, the external potential acts like a ``selective membrane'' which controls the direction of the flux of the cold atoms through the barriers and, thereby, controls the size of the stable condensate inside the trap.

DOI
10.1103/physreva.72.033605

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It is shown here how the lifetimes and energies of resonance states can be calculated by applying the complex scaling transformation to the nonlinear Schr"odinger equation. It is essential to first apply the complex scaling transformation to the full Hamiltonian and to subsequently derive from the result the correct complex scaled nonlinear Schr"odinger equation. The latter equation is physically relevant and amenable to numerical calculations. To analyze the results obtained by solving this equation, it is necessary to realize the close association of resonance phenomena with fragmentation of the system. As an illustrative example, we apply this theory to the Gross-Pitaevskii nonlinear equation to calculate the tunneling lifetime of a condensate inside an external (either optical or magnetic) trap. We show that by varying the scattering length, the external potential acts like a ``selective membrane'' which controls the direction of the flux of the cold atoms through the barriers and, thereby, controls the size of the stable condensate inside the trap.
The fundamental question of how Bose-Einstein condensates tunnel into a barrier is addressed. The cubic nonlinear Schr\"odinger equation with a finite square well potential, which models a Bose-Einstein condensate in a trap of finite depth, is solved for the complete set of localized and partially localized stationary states, which the former evolve into when the nonlinearity is increased. An immediate application of these different solution types is tunable tunneling. Magnetically tunable Feshbach resonances can change the scattering length of certain Bose-condensed atoms, such as 85Rb,{}^{85}\mathrm{Rb},85Rb, by several orders of magnitude, including the sign, and thereby also change the mean field nonlinearity term of the equation and the tunneling of the wave function. We find both linear-type localized solutions and uniquely nonlinear partially localized solutions where the tails of the wave function become nonzero at infinity when the nonlinearity increases. The tunneling of the wave function into the nonclassical regime (and thus its localization) therefore becomes an external experimentally controllable parameter.
We present numerical results from solving the time-dependent nonlinear Schr\"odinger equation (NLSE) that describes an inhomogeneous, weakly interacting Bose-Einstein condensate in a small harmonic trap potential at zero temperature. With this method we are able to find solutions for the NLSE for ground state condensate wave functions in one dimension or in three dimensions with spherical symmetry. These solutions corroborate previous ground state results obtained from the solution of the time-independent NLSE. Furthrmore, we can examine the time evolution of the macroscopic wave function even when the trap potential is changed on a time scale comparable to that of the condensate dynamics, a situation that can be easily achieved in magneto-optical traps. We show that there are stable solutions for atomic species with both positive and negative s-wave scattering lengths in one-dimensional (1D) and 3D systems for a fixed number of atoms. In both the 1D and 3D cases, these negative scattering length solutions have solitonlike properties. In 3D, however, these solutions are only stable for a modest range of nonlinearities. We analyze the prospects for diagnosing Bose-Einstein condensation in a trap using several experiments that exploit the time-dependent behavior of the condensate.
The mean-field dynamics of a Bose-Einstein condensate is studied in the presence of a microscopic trapping potential from which the condensate can escape via tunneling through finite barriers. We show that the method of complex scaling can be used to obtain a quantitative description of this decay process. A real-time propagation approach that is applied to the complex-scaled Gross-Pitaevskii equation allows us to calculate the chemical potentials and lifetimes of the metastably trapped Bose-Einstein condensate. The method is applied to a one-dimensional harmonic confinement potential combined with a Gaussian envelope, for which we compute the lowest symmetric and antisymmetric quasibound states of the condensate. A comparison with alternative approaches using absorbing boundary conditions as well as complex absorbing potentials shows good agreement.
Numerical solution of nonlinear Schrodinger equation (NLSE) can be used to describe an inhomogeneous,weakly interact Bose-Einstein condensate in a small harmonic trap potential at zero temperature. In one dimension situation, we find solutions for the NLSE for ground-state condensate wave function, and obtain the ground-state condensate energy per particle. Furthermore, with the time-dependent NLSE, we can examine the time evolution of the macroscopic wave function even when the trap potential is changed on a time scale comparable to that of the condensate dynamics, a situation that can be easily achieved in magneto-optical traps. We show that there are stable solutions for atomic species with both positive and negative s-wave scattering length in one-dimensional (1D) system for a fixed number of atoms, and the negative scattering length solutions have solitonlike properties.
We investigate the quantum tunneling of a Bose-Einstein condensate confined in a optical trap. We show that periodic pulses of coherent matter are emitted from the trap by using an oscillating energy barrier. Moreover, the emitted fraction of condensed atoms strongly increases if the period of oscillation of the height of the energy barrier is in parametric resonance with the period of oscillation of the center of mass of the condensate inside the potential well. Our model is analyzed by numerically solving the nonpolynomial Schrodinger equation (NPSE), an effective one-dimensional equation which describes the macroscopic wavefunction of Bose condensates under transverse harmonic confinement. The range of validity of NPSE is discussed and compared with that of Gross-Pitaevskii equation.