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arXiv:1812.04672 · Physical Review Letters

Quantum Time Crystals and Interacting Gauge Theories in Atomic Bose-Einstein Condensates

Patrik Öhberg, Ewan M. Wright

Published Dec 20, 2019• 41 citations
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Abstract

We study the dynamics of a Bose-Einstein condensate trapped circumferentially on a ring, and which is governed by an interacting gauge theory. We show that the associated density-dependent gauge potential and concomitant current nonlinearity permits a ground state in the form of a rotating chiral bright soliton. This chiral soliton is constrained to move in one direction by virtue of the current nonlinearity, and represents a time crystal in the same vein as Wilczek's original proposal.

DOI
10.1103/physrevlett.123.250402

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In a recent Letter [Phys. Rev. Lett. 123, 250402PRLTAO0031-900710.1103/PhysRevLett.123.250402], Öhberg and Wright describe a Bose-Einstein condensate trapped on a ring in the presence of the density-dependent gauge potential. It is claimed that the ground state of the system corresponds to a rotating chiral bright soliton and consequently it forms a genuine time crystal which minimizes its energy by performing periodic motion. We show that the energy of the chiral soliton in the laboratory frame is not correctly calculated in the Letter. The correct energy becomes minimal if the soliton does not move.

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We study the dynamics of a Bose-Einstein condensate trapped circumferentially on a ring, and which is governed by an interacting gauge theory. We show that the associated density-dependent gauge potential and concomitant current nonlinearity permits a ground state in the form of a rotating chiral bright soliton. This chiral soliton is constrained to move in one direction by virtue of the current nonlinearity, and represents a time crystal in the same vein as Wilczek's original proposal.
In a recent Letter [Phys. Rev. Lett. 123, 250402PRLTAO0031-900710.1103/PhysRevLett.123.250402], Öhberg and Wright describe a Bose-Einstein condensate trapped on a ring in the presence of the density-dependent gauge potential. It is claimed that the ground state of the system corresponds to a rotating chiral bright soliton and consequently it forms a genuine time crystal which minimizes its energy by performing periodic motion. We show that the energy of the chiral soliton in the laboratory frame is not correctly calculated in the Letter. The correct energy becomes minimal if the soliton does not move.
In a recent comment [1] on our paper [2] Syrwid, Kosior, and Sacha (SKS) asserted that we did not correctly calculate the chiral soliton energy ELAB{\cal E}_{LAB}ELAB​ in the lab frame, and further that the system we proposed is not capable of supporting a genuine quantum time crystal. We concede that we did not correctly calculate ELAB{\cal E}_{LAB}ELAB​ but we illustrate below that our system can nonetheless display time crystal behavior with a ground state that is spatially localized and rotating.
We show how density dependent gauge potentials can be induced in dilute gases of ultracold atoms using light-matter interactions. We study the effect of the resulting interacting gauge theory and show how it gives rise to novel topological states in the ultracold gas. We find in particular that the onset of persistent currents in a ring geometry is governed by a critical number of particles. The density-dependent gauge potential is also found to support chiral solitons in a quasi-one-dimensional ultracold Bose gas.
Abstract We investigate chiral quantum dynamics of a Bose-Einstein condensate confined in a finite-depth potential well under a density-dependent gauge field. The interplay of the finite-depth potential well and the gauge field results in rich chiral dynamics. When the external potential is strong, coherent chiral collective dynamics, i.e. , chiral breathing dynamics accompanied by the dipole dynamics emerges. The wave packet in the well undergoes periodic directional-dependent compression/expansion. Notably, this chiral dynamics leads to the emergence of a Rabi-like oscillation of particles between the inside and outside well, providing an effective route to investigate the Rabi oscillation in the system. In a weak external potential well, the gauge field dominates and chiral tunneling and diffusion dynamics take place. Particularly, when diffusion and collapse coexist, a chiral moving soliton is generated, which plays a key role for coherent chiral quantum transport.
We study the collective modes of a Bose-Einstein condensate subject to an optically induced density-dependent gauge potential. The corresponding interacting gauge theory lacks Galilean invariance, yielding an exotic superfluid state. The nonlinear dynamics in the presence of a current nonlinearity and an external harmonic trap are found to give rise to dynamics which violate Kohn's theorem; where the frequency of the dipole mode strongly depends on the strength of the mass current in the gas. The linearised spectrum reveals how the centre of mass and shape oscillations are coupled, whereas in the strongly nonlinear regime the dynamics is irregular.
We show how the chiral properties of Bose-Einstein condensates subject to current-density interactions and loaded in optical lattices can be observed in the realization of nonlinear Bloch states, whose spectrum lacks the usual periodic structure. Chirality is also manifested by spatially localized states, or gap solitons, which are found for positive rotation rates of the lattice at the energy gaps between the linear energy bands, whereas for negative rotations they appear in the semi-infinite gap of the linear spectrum. The stability of extended and localized states is checked through the spectrum of linear excitations and nonlinear time evolution of perturbed states, and the phenomenon of Bloch oscillations is explored. Our results are obtained in quasi-one-dimensional ring geometries with feasible experimental parameters.