Faraday waves in collisionally inhomogeneous Bose-Einstein condensates

Antun Balaž, Remus Paun, Alexandru I. Nicolin, Sudharsan Balasubramanian, and Radha Ramaswamy

Phys. Rev. A 89, 023609 – Published 11 February 2014

Abstract

We study the emergence of Faraday waves in cigar-shaped collisionally inhomogeneous Bose-Einstein condensates subject to periodic modulation of the radial confinement. Considering a Gaussian-shaped radially inhomogeneous scattering length, we show through extensive numerical simulations and detailed variational treatment that the spatial period of the emerging Faraday waves increases as the inhomogeneity of the scattering length gets weaker, and that it saturates once the width of the radial inhomogeneity reaches the radial width of the condensate. In the regime of strongly inhomogeneous scattering lengths, the radial profile of the condensate is akin to that of a hollow cylinder, while in the weakly inhomogeneous case the condensate is cigar shaped and has a Thomas-Fermi radial density profile. Finally, we show that when the frequency of the modulation is close to the radial frequency of the trap, the condensate exhibits resonant waves which are accompanied by a clear excitation of collective modes, while for frequencies close to twice that of the radial frequency of the trap, the observed Faraday waves set in forcefully and quickly destabilize condensates with weakly inhomogeneous two-body interactions.

Received 3 December 2013
Corrected 6 October 2014

DOI:https://doi.org/10.1103/PhysRevA.89.023609

Corrections

6 October 2014

Erratum

Publisher's Note: Faraday waves in collisionally inhomogeneous Bose-Einstein condensates [Phys. Rev. A 89, 023609 (2014)]

Antun Balaž, Remus Paun, Alexandru I. Nicolin, Sudharsan Balasubramanian, and Radha Ramaswamy

Phys. Rev. A 90, 049902 (2014)

Authors & Affiliations

Antun Balaž^1,*, Remus Paun², Alexandru I. Nicolin², Sudharsan Balasubramanian³, and Radha Ramaswamy³

¹Scientific Computing Laboratory, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
²Department of Computational Physics and Information Technologies, Horia Hulubei National Institute of Physics and Nuclear Engineering (IFIN-HH), P. O. Box MG-6, 077125, Romania
³Centre for Nonlinear Science, Post-Graduate and Research Department of Physics, Government College for Women (Autonomous), Kumbakonam 612001, India

^*antun@ipb.ac.rs

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Vol. 89, Iss. 2 — February 2014

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Figure 1
Weakly inhomogeneous collisions, inhomogeneity parameter $b = 4 b_{0}$ . (a) Radial component of the density profile at $z = 0$ for the condensate ground state. The full red line shows the GPE numerical results, while the dashed blue line corresponds to the variational results. (b) Full $ρ − z$ density profile of the ground state of the condensate. (c) Time evolution of the radially integrated longitudinal density profile obtained with the modulation amplitude $ε = 0.1$ and the modulation frequency $ω = 250 \times 2 π$ Hz. The Faraday wave becomes fully visible after 200 ms. (d) Fast Fourier transform (FFT) spectrum of the longitudinal density profile of the condensate at $t = 250$ ms. The peak at $k_{F}, W = 0.60 μ m^{- 1}$ corresponds to the Faraday wave, yielding a spatial period of $p = 2 π / k_{F}, W = 10.5 μ m$ .
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Figure 2
Strongly inhomogeneous collisions, inhomogeneity parameter $b = b_{0} / 4$ . (a) Radial component of the density profile at $z = 0$ for the condensate ground state. The full red line shows the GPE numerical results, while the dashed blue line corresponds to the variational results. (b) Full $ρ − z$ density profile of the ground state of the condensate. (c) Time evolution of the radially integrated longitudinal density profile obtained with the modulation amplitude $ε = 0.1$ and the modulation frequency $ω = 250 \times 2 π$ Hz. The Faraday wave becomes fully visible after 200 ms. (d) Fast Fourier transform (FFT) Fourier spectrum of the longitudinal density profile of the condensate at $t = 250$ ms. The peak at $k_{F}, S = 1.16 μ m^{- 1}$ corresponds to the Faraday wave, yielding a spatial period of $p = 2 π / k_{F}, S = 5.4 μ m$ .
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Figure 3
Average spatial period of the longitudinal Faraday waves as a function of the inhomogeneity scale $b$ for the modulation amplitude $ε = 0.1$ and the modulation frequency $ω = 250 \times 2 π$ Hz. Black squares depict the full numerical results obtained from the Fourier analysis of the solution of the time-dependent GPE; red triangles and blue circles correspond to the variational prediction of the spatial period obtained as $2 π / \bar{k}$ , with $\bar{k}$ given by Eq. (24). The red triangles are obtained using the dispersion relation (22) for strong inhomogeneity, while the blue circles are obtained from the dispersion relation (15) for weak inhomogeneity.
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Figure 4
Time evolution of the radially integrated longitudinal density profile obtained with the modulation amplitude $ε = 0.1$ and the modulation frequency $ω = 160 \times 2 π$ Hz for (a) $b = 4 b_{0}$ ; (b) $b = b_{0}$ ; (c) $b = b_{0} / 4$ . The excited collective modes soften for smaller values of $b$ .
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Figure 5
Time evolution of the radially integrated longitudinal density profile obtained with the modulation amplitude $ε = 0.1$ and the modulation frequency $ω = 320 \times 2 π$ Hz for (a) $b = b_{0}$ ; (b) $b = b_{0} / 4$ . In (a) the condensate destabilizes violently after the Faraday wave sets in, while in (b) the destabilization is slower and one can clearly see the formation and subsequent evolution of the Faraday wave.
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