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arXiv:cond-mat/0102291 · Physical Review A

Tunable tunneling: An application of stationary states of Bose-Einstein condensates in traps of finite depth

Lincoln D. Carr, Khan W. Mahmud, William P. Reinhardt

Published Aug 6, 2001• 43 citations
arXivHTMLPDF

Abstract

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 ${}^{85}\mathrm{Rb},$ 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.

DOI
10.1103/physreva.64.033603

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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 study tunneling of a Bose-Einstein condensates confined in a effective double-well potential (a single well with a spatially magnetic modulated scattering length, actually), called pseudo double-well trap, in which the interaction of atoms characterized by the s-wave scattering length a s can be widely tuned with a magnetic-field Feshbach resonance. As a result, corresponding to different nonlinear parameters, the energy levels of the nonlinear Schrödinger equation can have complex structures in their dependence on the bias between the wells. We discuss the emergence of looped levels, which lead to a breakdown of adiabaticity that the Landau-Zener transition probability does not vanish even in the adiabatic limit. Moreover, we also find that the Landau-Zener tunneling in the pseudo trap show many striking properties distinguished from that of the real trap model (where the barrier is created by the external potential). Possible experimental observation in an opticallyinduced photonic lattices in a photorefractive material is suggested.
The variational JWKB method is used to determine experimentally accessible macroscopic quantum tunneling regimes of quasibound Bose-Einstein condensates in two quasi-one-dimensional trap configurations. The potentials can be created by magnetic and optical traps: a symmetric trap from two offset Gaussian barriers and a tilt trap from a linear gradient and Gaussian barrier. Scaling laws in barrier parameters, ranging from inverse polynomial to square root times exponential, are calculated and used to elucidate different dynamical regimes, such as when classical oscillations dominate tunneling rates in the symmetric trap. The symmetric trap is found to be versatile, with tunneling times at and below 1 s, able to hold 103−−104{10}^{3}--{10}^{4}103−−104 atoms, and realizable for atoms ranging from rubidium to lithium, with unadjusted scattering lengths. The tilt trap produces subsecond tunneling times, is able to hold a few hundred atoms of lighter elements such as lithium, and requires the use of Feshbach resonance to reduce scattering lengths. To explore a large parameter space, an extended Gaussian variational ansatz is used, which can approximate large traps with Thomas-Fermi profiles. Nonlinear interactions in the Gross-Pitaevskii equation are shown to produce additional effective mean-field barriers, affecting scaling laws.
The realization of Bose-Einstein condensation in dilute atomic gases opens an exciting way to quantum mechanics and begins a new area of quantum simulation. As a macroscopic quantum object and a many-body bosonic system, the Bose-Einstein condensates can show numerous exotic quantum effects and have naturally attracted great attention. One of the simplest quantum many-body systems to be realized experimentally and studied theoretically is ultra-cold atoms in a double-well potential. This system can exhibit a great variety of quantum interference phenomena such as tunneling oscillation, self-trapping and the entanglement of macroscopic superpositions. Specifically, the double-well potentials built by optical or magnetic fields are easy to change and the many-body interaction between ultra-cold atoms can be changed by the method of Feshbach resonance, enabling the precise quantum control of the double-well dynamics of the condensates. In the present work, we study the dynamics of a condensate in a trapping potential consisting of an unalterable double-well trap and an additional moving optical lattice. If the lattice space is much smaller than the size of the double-well trap, the system can be simplified into a double-well trapped condensate with a tunable effective mass. Using the mean-field factorization assumption, together with a two-mode approximation, we obtain the analytic expressions for the dependence of the tunneling rate and the self-collision strength on the effective mass. The tunneling rate decays and the collision strength grows up with the increase of the effective mass. As a consequence of their different changes, we conclude that the adjustment of the effective mass of the ultra-cold atoms, rather than the changing of the trap barrier or adjusting of the atomic scattering length, is an alternative approach to controlling the double-well dynamics of the condensate. Via numerical simulations of the mean-field dynamical equations with some realistic parameters, we show that a transition between the quantum coherent tunneling and the self-trapping behaviors is experimentally realizable with the mass-control approach. Specifically, we show that the approach is still valid for the case of negative mass. Moreover, we find that the negative-mass case can be used even to stimulate the double-well dynamics of the condensate with a negative atomic scattering length.
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.
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 present a new family of stationary solutions to the cubic nonlinear Schrödinger equation with an elliptic function potential. In the limit of a sinusoidal potential our solutions model a quasi-one-dimensional dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided that the ratio of the height of the variations of the condensate to its dc offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Recent developments allow for experimental investigation of these predictions.