[Paper Review] Pulsed Quantum Tunneling with Matter Waves

Luca Salasnich|arXiv (Cornell University)|Jun 24, 2001

Cold Atom Physics and Bose-Einstein Condensates1 references3 citations

TL;DR

This paper investigates pulsed macroscopic quantum tunneling (MQT) in Bose-Einstein condensates falling under gravity and scattering off a Gaussian potential barrier, modeling a far-detuned optical mirror. Using a variational approach to derive a 1D non-polynomial nonlinear Schrödinger equation (NPSE) from the 3D Gross-Pitaevskii equation, the authors show that MQT is quasi-periodic and can generate coherent atomic pulses, with NPSE providing superior accuracy over existing 1D models for cigar-shaped condensates.

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.

Motivation & Objective

• To investigate macroscopic quantum tunneling (MQT) of a Bose condensate falling under gravity and scattering on a Gaussian potential barrier that models an optical mirror.
• To analyze the effects of inter-atomic interactions and transverse confinement on MQT dynamics.
• To develop an accurate effective 1D equation for describing the axial dynamics of cigar-shaped Bose condensates.
• To validate the new non-polynomial nonlinear Schrödinger equation (NPSE) against 3D Gross-Pitaevskii solutions for both ground state and dynamical properties.

Proposed method

• Numerical solution of the 3D Gross-Pitaevskii equation (3D GPE) using a predictor-corrector splitting method with imaginary time to find the initial ground state.
• Use of a Gaussian variational ansatz for the transverse wavefunction to reduce the 3D GPE to an effective 1D equation via action minimization.
• Derivation of a time-dependent non-polynomial nonlinear Schrödinger equation (NPSE) that includes nonlinearity from inter-atomic interactions and transverse confinement effects.
• Validation of NPSE against 3D GPE, standard 1D GPE, and a corrected 1D GPE (CGPE) in harmonic and non-harmonic traps.
• Analysis of tunneling probability and quasi-periodic behavior in the presence of gravity and a Gaussian barrier.
• Comparison of results across different interaction strengths and geometries to assess the role of dimensionality and confinement.

Experimental results

Research questions

• RQ1 Can macroscopic quantum tunneling in a falling Bose condensate be quasi-periodic due to repeated bouncing and interference?
• RQ2 How do inter-atomic interactions and transverse confinement affect the tunneling probability in 3D and 1D geometries?
• RQ3 Does the derived NPSE accurately describe both ground-state and dynamical properties of cigar-shaped condensates compared to the 3D GPE?
• RQ4 What is the role of the aspect ratio and transverse trapping frequency in determining tunneling efficiency?
• RQ5 Can the NPSE be used as a reliable alternative to existing 1D models for simulating strong nonlinearity and large density variations?

Key findings

• MQT in the system is a quasi-periodic phenomenon due to the interference of repeatedly reflected and expanding condensate fragments, enabling the generation of coherent Bose-Einstein condensed atomic pulses.
• For initially spherical condensates under strong transverse confinement, inter-atomic repulsion reduces the tunneling fraction, while for cigar-shaped condensates, it enhances tunneling, consistent with theoretical predictions.
• The derived NPSE provides significantly more accurate results than standard 1D GPE and corrected 1D GPE (CGPE), especially in the strongly interacting regime.
• In the weakly interacting limit, the NPSE reduces to the standard 1D GPE with a renormalized nonlinear coefficient, confirming consistency with known limits.
• In the strongly interacting limit, the NPSE yields a 1D Thomas-Fermi density profile that is quadratic in (μ′ − V(z)), matching the 3D Thomas-Fermi approximation with a 2/9 factor, close to the 1/4 factor from direct 3D integration.
• Numerical comparisons show that NPSE closely matches 3D GPE results for both ground-state density profiles and dynamical evolution, validating its use in simulating complex, non-equilibrium dynamics in cigar-shaped condensates.