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Many-Body State and Dynamic Behaviour of the Pair-Correlation Function of a Small Bose–Einstein Condensate Confined in a Ring Potential

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Abstract

We investigate the many-body state and the static and the dynamic behaviour of the pair-correlation function of a Bose–Einstein condensate with a finite atom number, which is confined in a quasi-one-dimensional toroidal/annular potential, both for repulsive, and for attractive interactions. We link the dynamic pair-correlation function that we evaluate with the problem of quantum time crystals. For weak repulsive interatomic interactions and a finite number of atoms the pair-correlation function shows a periodic temporal behaviour, which disappears in the limit of a large atom number, in agreement with general arguments. Finally we provide some insight into older results of attractive interactions, where the time-crystalline behaviour exists only in the limit of a large atom number.

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Data Availability Statement

The presented data are available on request from the authors.

Notes

  1. Very generally, from Eq. (20), when \(\Phi (0, 0)\) acts on \(|\Psi _1 \rangle\) an atom is taken out of the system. This operation creates a linear superposition of states (having various values of the angular momentum, but all of them having \(N-1\) atoms). When the exponential \(e^{i H t}\) acts on this superposition of states, each of these states will have an exponential \(e^{i {{\mathcal {E}}}_n t}\) (here we do not write explicitly the dependence of \({{\mathcal {E}}}_n\) on the atom number and on the angular momentum). The same will happen when the second exponential, \(e^{-i H t}\), acts on \(\Phi ^{\dagger }(\theta , 0) \Phi (\theta , 0) e^{i H t} \Phi (0, 0) | \Psi _1 \rangle\), which will give another exponential factor \(e^{-i {{\mathcal {E}}}_m t}\). If the energy spectrum is equidistant (for all values of the angular momentum, in general), these exponentials will involve an integer multiple of only one energy and as a result \(n^{(2)}(\theta , t; \theta '=0, t'=0)\) will be a periodic function (in time).

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Acknowledgements

GMK wishes to thank Chris Pethick, Krzysztof Sacha, and Wolf von Klitzing for useful discussions.

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Authors and Affiliations

  1. Hellenic Mediterranean University, P.O. Box 1939, 71004, Heraklion, Greece

    A. Roussou, J. Smyrnakis, M. Magiropoulos & G. M. Kavoulakis

  2. Department of Physics, Chalmers University of Technology, 412 96, Göteborg, Sweden

    A. Roussou

  3. HMU Research Center, Institute of Emerging Technologies, P.O. Box 1939, 71004, Heraklion, Greece

    G. M. Kavoulakis

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  1. A. Roussou
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Correspondence to G. M. Kavoulakis.

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Roussou, A., Smyrnakis, J., Magiropoulos, M. et al. Many-Body State and Dynamic Behaviour of the Pair-Correlation Function of a Small Bose–Einstein Condensate Confined in a Ring Potential. J Low Temp Phys 210, 51–67 (2023). https://doi.org/10.1007/s10909-022-02831-8

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