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arXiv:cond-mat/0406189

Tunneling lifetime of trapped condensates

Nimrod Moiseyev, Lorenz S. Cederbaum

Published Jun 8, 2004• 0 citations
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Abstract

Using the complex scaling transformation we calculate the tunneling lifetime of a condensate inside a magnetic/optical 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
10.48550/arxiv.cond-mat/0406189

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Using the complex scaling transformation we calculate the tunneling lifetime of a condensate inside a magnetic/optical 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.
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 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.
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.
Bose–Einstein condensates are studied in a potential of finite depth which supports both bound and quasi-bound states. This potential, which is harmonic for small radii and decays as a Gaussian for large radii, models experimentally relevant optical traps. The nonlinearity, which is proportional to both the number of atoms and the interaction strength, can transform bound states into quasi-bound ones. The latter have a finite lifetime due to tunnelling through the barriers at the borders of the well. We predict the lifetime and stability properties for repulsive and attractive condensates in one, two and three dimensions, for both the ground state and excited soliton and vortex states. We show, via a combination of the variational and WKB approximations, that macroscopic quantum tunnelling in such systems can be observed on time scales of 10 ms to 10 s.
The simplicity and versatility of ultra-cold atoms make it an ideal system for studying Quantum Mechanical phenomenon like tunneling. This thesis describes progress towards experimentally measuring the time spent by a tunneling particle in the classically forbidden region. A detailed analysis of the experimental requirements to measure the tunneling time is presented along with the experimental tools developed to meet these requirements. These include a thin \SI{1.3}{\micro m} optical tunnel barrier, a smooth atomic waveguide, and a Larmor clock to measure the tunneling time. Delta-kick cooling is used to achieve the extremely low temperatures required for the tunneling time experiments. A 2.4x reduction in the atomic velocity spread is demonstrated, reducing the rms velocity spread to \SI{0.46(5)}{mm/s}. This corresponds to cooling to an effective temperature of \SI{2.0(4)}{nK}. The thin optical barrier opens up the possibility to study the decay of quasi-bound condensates via tunneling. We develop a novel trapping configuration for this purpose, in which the barrier acts as one of the walls of the trap. Inter-atomic interactions strongly dictate the escape dynamics of the condensate out of this trap, giving rise to three distinct regimes--- classical over the barrier spilling, quantum tunneling driven by interactions, and decay dominated by background losses. We show that in the tunneling regime, the decay rate depends exponentially with the chemical potential of the condensate. Experimental results show good agreement with numerical solutions of the 3D Gross-Pitaevskii equation.
We study the decay dynamics of an interacting Bose-Einstein condensate in the presence of a metastable trap- ping potential from which the condensate can escape via tunnel- ing through finite barriers. The time-dependent decay process is reproduced by means of the instantaneous decay rates of the condensate at a given population of the quasi-bound state, which are calculated with the method of complex scaling. Both for the case of a double-barrier potential as well as for the case of a tilted periodic potential, we find pronounced deviations from a monoexponential decay behavior, which would generally be expected in the absence of the atom-atom interaction.