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Interaction effect in a $\mathcal{PT}$-symmetric Bose-Einstein Condensate

Xiaoling Cui

Published Aug 10, 2021• 0 citations
OpenAlex

Abstract

We investigate the ground state and excitation properties of a weakly interacting Bose-Einstein Condensate(BEC) under a parity-time($\mathcal{PT}$) symmetric potential with balanced gain and loss. It is shown that the $\mathcal{PT}$-symmetry, as preserved by the macroscopic condensate, can be spontaneously broken by its Bogoliubov quasi-particles. The associated $\mathcal{PT}$-breaking transitions in the Bogoliubov spectrum can be conveniently tuned by the interaction anisotropy of BEC and the strength of $\mathcal{PT}$ potential. In the $\mathcal{PT}$-unbroken regime, the real Bogoliubov modes are generally gapped, in contrast to the Hermitian case with gapless phonon mode. Moreover, it is found that the presence of $\mathcal{PT}$ potential can enhance the mean-field collapse and thereby intrigue the droplet formation after incorporating the repulsive force from quantum fluctuations. These remarkable interplay effects of $\mathcal{PT}$-symmetry and interaction can be directly probed in cold atoms experiments, which shed light on intriguing collective phenomena in other non-Hermitian systems with $\mathcal{PT}$-symmetry.

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Interaction effect in a PT\mathcal{PT}PT-symmetric Bose-Einstein Condensate
We investigate the ground state and excitation properties of a weakly interacting Bose-Einstein Condensate(BEC) under a parity-time(PT\mathcal{PT}PT) symmetric potential with balanced gain and loss. It is shown that the PT\mathcal{PT}PT-symmetry, as preserved by the macroscopic condensate, can be spontaneously broken by its Bogoliubov quasi-particles. The associated PT\mathcal{PT}PT-breaking transitions in the Bogoliubov spectrum can be conveniently tuned by the interaction anisotropy of BEC and the strength of PT\mathcal{PT}PT potential. In the PT\mathcal{PT}PT-unbroken regime, the real Bogoliubov modes are generally gapped, in contrast to the Hermitian case with gapless phonon mode. Moreover, it is found that the presence of PT\mathcal{PT}PT potential can enhance the mean-field collapse and thereby intrigue the droplet formation after incorporating the repulsive force from quantum fluctuations. These remarkable interplay effects of PT\mathcal{PT}PT-symmetry and interaction can be directly probed in cold atoms experiments, which shed light on intriguing collective phenomena in other non-Hermitian systems with PT\mathcal{PT}PT-symmetry.
We investigate the effects of quantum fluctuations in a parity-time (PT)(\mathcal{PT})(PT) symmetric two-species Bose-Einstein condensate (BEC). It is found that the PT\mathcal{PT}PT symmetry, though preserved by the macroscopic condensate, can be spontaneously broken by its Bogoliubov quasiparticles under quantum fluctuations. The associated PT\mathcal{PT}PT-breaking transitions in the Bogoliubov spectrum can be conveniently tuned by the interaction anisotropy in spin channels and the strength of PT\mathcal{PT}PT potential. In the PT\mathcal{PT}PT-unbroken regime, the real Bogoliubov modes are generally gapped, in contrast to the gapless phonon mode in Hermitian case. Moreover, the presence of PT\mathcal{PT}PT potential is found to enhance the mean-field collapse and thereby intrigue the droplet formation after incorporating the repulsive force from quantum fluctuations. These remarkable interplay effects of PT\mathcal{PT}PT symmetry and interaction can be directly probed in cold atoms experiments, which shed light on related quantum phenomena in general PT\mathcal{PT}PT-symmetric systems.
Labouvie et al. [Phys. Rev. Lett. 116, 235302 (2016)] have described an experiment with a weakly interacting Bose-Einstein condensate trapped in a one-dimensional optical lattice with localized loss created by a focused electron beam. We show that by setting suitable initial currents between neighboring sites, it is possible to create PT\mathcal{PT}PT-symmetric quasistationary and PT\mathcal{PT}PT-symmetry-broken decaying states in an embedded two-mode subsystem. This subsystem exhibits gain provided by the coupling to the reservoir sites and localized loss due to the electron beam and shows the same dynamics as a non-Hermitian two-mode system with symmetric real and antisymmetric imaginary time-independent potentials, except for a proportionality factor in the chemical potential. We also show that there are three other equivalent scenarios and that the presence of a localized loss term significantly reduces the size of the condensate required for the realization.
Abstract Parity-time ( P T ) symmetry has drawn great research interest in non-Hermitian physics. Recently, there is an emerging model of P T -symmetric spin–orbit coupling (SOC) which could be realized with spin-1/2 atomic Bose–Einstein condensates (BECs) when the two spin components are respectively subjected to momentum-dependent gain and loss (Qin et al 2022 New J. Phys. 24 063025). In this model, inter-atom interaction has no influence on the P T -symmetric plane waves. Thus, collective excitation playing the role of fingerprint of inter-atom interaction is investigated in this paper. For the phonon excitation, it is shown that the repulsive interaction between atoms in different spin states, which tends to drive the atoms to populate in only one spin component, can break the P T -symmetry, and leads to a phonon instability. Whereas, for the case of the phonon-maxon-roton collective excitation spectrum, the repulsive interaction between atoms in different spin states can lead to a maxon instability, which does not occur in Hermitian BEC systems (dipolar BECs or BECs with Raman induced SOC). Simulation of the time evolution of the plane wave solution against small noise shows that the maxon instability can result in the formation of a supersolid-like stripe pattern at the initial stage, but the non-Hermitian nature of the system finally destroys the pattern. The phase diagram of stability shows that the Rabi coupling and repulsive interaction between the atoms in the same spin state can stabilize the system.
Based on Bogoliubov's truncated Hamiltonian H B for a weakly interacting Bose system, and adding a U(1) symmetry breaking term V 1/2 (λ a 0 +λ * a 0 + ) to H B , we show by using the coherent state theory and the mean-field approximation rather than the c-number approximations, that the Bose–Einstein condensation(BEC) occurs if and only if the U(1) symmetry of the system is spontaneously broken. The real ground state energy and the justification of the Bogoliubov c-number substitution are given by solving the Schrödinger eigenvalue equation and using the self-consistent condition.
We study topologically nontrivial excitations of a weakly interacting, spin-orbit coupled Bose-Einstein condensate in a two-dimensional square optical lattice, a system recently realized in experiment [Phys. Rev. Lett. 121, 150401 (2018)]. We focus on situations where the system is not subjected to a Zeeman field and thus does not exhibit nontrivial single-particle band topology. Of special interest then is the role of particle interaction as well as its interplay with the symmetry properties of the system in producing topologically nontrivial excitations. We find that the noninteracting system possesses a rich set of symmetries, including the PT\mathcal{PT}PT symmetry, the modified dihedral point-group symmetry ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}}_{4}$ and the nonsymmorphic symmetry. These combined symmetries ensure the existence of pairs of degenerate Dirac points at the edge of Brillouin zone for the single-particle energy bands. In the presence of particle interaction and with sufficient spin-orbit coupling, the atoms condense in a ground state with net magnetization which spontaneously breaks the PT\mathcal{PT}PT and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}}_{4}$ symmetry. We demonstrate that this symmetry breaking leads to a gap opening at the Dirac point for the Bogoliubov spectrum and consequentially topologically nontrivial excitations. We confirm the nontrivial topology by calculating the Chern numbers of the lowest excitation bands and show that gapless edge states form at the interface of systems characterized by different values of the Chern number.
We employ mean-field, Bogoliubov and many-body theories to study critical fluctuations in the position and momentum of a Bose-Einstein condensate whose translation symmetry is spontaneously broken due to attractive interactions. In a homogeneous system, the many-body ground state of the symmetry-preserving Hamiltonian is very fragile against superposition of low-lying states, while the mean-field theory predicts a stable bright soliton which spontaneously breaks translation symmetry. We show that weak symmetry-breaking perturbations cause the translation-symmetric many-body ground state to cross over to a many-body bright soliton. We argue that the center-of-mass fluctuations in the soliton state arise primarily from the depletion of the condensate to translation modes. We develop an extended mean-field theory to analytically reproduce these results obtained by the exact diagonalization method.