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arXiv:cond-mat/0106493

Pulsed Quantum Tunneling with Matter Waves

Luca Salasnich

Published Jun 24, 2001• 3 citations
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Abstract

In this report we investigate the macroscopic quantum tunneling of a Bose condensate falling under gravity and scattering on a Gaussian barrier that could model a mirror of far-detuned sheet of light. We analyze the effect of the inter-atomic interaction and that of a transverse confining potential. We show that the quantum tunneling can be quasi-periodic and in this way one could generate coherent Bose condensed atomic pulses. In the second part of the report, we discuss an effective 1D time-dependent non-polynomial nonlinear Schrodinger equation (NPSE), which describes cigar-shaped condensates. NPSE is obtained from the 3D Gross-Pitaevskii equation by using a variational approach. We find that NPSE gives much more accurate results than all other effective 1D equations recently proposed.

DOI
10.48550/arxiv.cond-mat/0106493

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In this report we investigate the macroscopic quantum tunneling of a Bose condensate falling under gravity and scattering on a Gaussian barrier that could model a mirror of far-detuned sheet of light. We analyze the effect of the inter-atomic interaction and that of a transverse confining potential. We show that the quantum tunneling can be quasi-periodic and in this way one could generate coherent Bose condensed atomic pulses. In the second part of the report, we discuss an effective 1D time-dependent non-polynomial nonlinear Schrodinger equation (NPSE), which describes cigar-shaped condensates. NPSE is obtained from the 3D Gross-Pitaevskii equation by using a variational approach. We find that NPSE gives much more accurate results than all other effective 1D equations recently proposed.
We study the tunnelling properties of a cigar-shaped Bose-Einstein condensate by using an effective 1D nonpolynomial nonlinear Schrödinger equation (NPSE). First we investigate a mechanism to generate periodic pulses of coherent matter by means of a Bose condensate confined in a potential well with an oscillating height of the energy barrier. We show that it is possible to control the periodic emission of matter waves and the tunnelling fraction of the Bose condensate. We find that the number of emitted particles 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 centre of mass of the condensate inside the potential well. Then we use NPSE to analyse the periodic tunnelling of a Bose-Einstein condensate in a double-well potential which has an oscillating energy barrier. We show that the dynamics of the Bose condensate critically depends on the frequency of the oscillating energy barrier. The macroscopic quantum self-trapping (MQST) of the condensate can be suppressed under the condition of parametric resonance between the frequency of the energy barrier and the frequency of oscillation through the barrier of the very small fraction of particles which remain untrapped during MQST.
We investigate macroscopic quantum tunneling of a Bose-Einstein condensate, and how it is affected by the interatomic interaction. We study the dynamics of a condensate falling under gravity and scattering on a Gaussian potential barrier that models a mirror formed by a far-detuned sheet of light. We observe bouncing, interference, and quantum tunneling of the condensate. We find that the tunneling rate is very sensitive to the interatomic interaction and to the shape of the condensate. Under many conditions the tunneling rate is strongly enhanced by the interaction as achieved, for instance, by increasing the number of condensed particles. In a quasi one-dimensional situation the tunneling pulse displays two peaks. The quantum tunneling can be quasiperiodic, and in this way one could generate coherent Bose-Einstein condensed atomic pulses.
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.
We investigate the dynamics of Bose-Einstein condensate bright solitons made of alkali-metal atoms with negative scattering length and under harmonic confinement in the transverse direction. Contrary to the one-dimensional (1D) case, the 3D bright soliton exists only below a critical attractive interaction that depends on the extent of confinement. Such a behavior is also found in multisoliton condensates with box boundary conditions. We obtain numerical and analytical estimates of the critical strength beyond which the solitons do not exist. By using an effective 1D nonpolynomial nonlinear Schr\"odinger equation, which accurately takes into account the transverse dynamics of cigarlike condensates, we numerically simulate the dynamics of the ``soliton train'' reported in a recent experiment [Nature (London) 417, 150 (2002)]. Then, analyzing the macroscopic quantum tunneling of the bright soliton on a Gaussian barrier, we find that its interference in the tunneling region is strongly suppressed with respect to nonsolitonic case; moreover, the tunneling through a barrier breaks the shape invariance of the matter wave. Finally, we show that the collapse of the soliton is induced by the scattering on the barrier or by the collision with another matter wave when the density reaches a critical value, for which we derive an accurate analytical formula.
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.
Starting from the three-dimensional (3D) Gross-Pitaevskii equation and using a variational approach, we derive an effective 1D wave equation that describes the axial dynamics of a Bose condensate confined in an external potential with cylindrical symmetry. The trapping potential is harmonic in the transverse direction and generic in the axial one. Our equation, that is a time-dependent nonpolynomial nonlinear Schr\"odinger equation (1D NPSE), can be used to model cigar-shaped condensates, whose dynamics is essentially 1D. We show that 1D NPSE gives much more accurate results than all other effective equations recently proposed. By using 1D NPSE we find analytical solutions for bright and dark solitons, which generalize the ones known in the literature. We deduce also an effective 2D nonpolynomial Schr\"odinger equation (2D NPSE) that models disk-shaped Bose condensates confined in an external trap that is harmonic along the axial direction and generic in the transverse direction. In the limiting cases of weak and strong interaction, our approach gives rise to Schr\"odinger-like equations with different polynomial nonlinearities.