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arXiv:1901.01174 · Physical Review A

Interaction effects on the PT -symmetry-breaking transition in atomic gases

Ziheng Zhou, Zhenhua Yu

Published Apr 10, 2019• 22 citations
arXivHTMLPDF

Abstract

Non-Hermitian systems having parity-time ($\mathcal{PT}$) symmetry can undergo a transition, spontaneously breaking the symmetry. Ultracold atomic gases provide an ideal platform to study interaction effects on the transition. We consider a model system of $N$ bosons of two components confined in a tight trap. Radio-frequency and laser fields are coupled to the bosons such that the single-particle Non-Hermitian Hamiltonian ${h}_{\mathcal{P}T}=\ensuremath{-}i\mathrm{\ensuremath{\Gamma}}{\ensuremath{\sigma}}_{z}+J{\ensuremath{\sigma}}_{x}$, which has $\mathcal{PT}$ symmetry, can be simulated in a passive way, namely, in the physical system there is only incurred atom loss but no gain. We show that when interatomic interactions are tuned to maintain the symmetry, the $\mathcal{PT}$-symmetry-breaking transition is affected only by the SU(2) variant part of the interatomic interactions. We find that the transition point ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{tr}}$ decreases as the strength of this interaction part or $N$ increases; in the strong strength limit for this interaction part, ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{tr}}$ scales as the strength to the power of $\ensuremath{-}(N\ensuremath{-}1)$. We also give signatures of the $\mathcal{PT}$-symmetric and the symmetry-breaking phases for the interacting bosons in experiment.

DOI
10.1103/physreva.99.043412

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Non-Hermitian systems having parity-time (PT\mathcal{PT}PT) symmetry can undergo a transition, spontaneously breaking the symmetry. Ultracold atomic gases provide an ideal platform to study interaction effects on the transition. We consider a model system of NNN bosons of two components confined in a tight trap. Radio-frequency and laser fields are coupled to the bosons such that the single-particle Non-Hermitian Hamiltonian hPT=\ensuremath−i\ensuremathΓ\ensuremathσz+J\ensuremathσx{h}_{\mathcal{P}T}=\ensuremath{-}i\mathrm{\ensuremath{\Gamma}}{\ensuremath{\sigma}}_{z}+J{\ensuremath{\sigma}}_{x}hPT​=\ensuremath−i\ensuremathΓ\ensuremathσz​+J\ensuremathσx​, which has PT\mathcal{PT}PT symmetry, can be simulated in a passive way, namely, in the physical system there is only incurred atom loss but no gain. We show that when interatomic interactions are tuned to maintain the symmetry, the PT\mathcal{PT}PT-symmetry-breaking transition is affected only by the SU(2) variant part of the interatomic interactions. We find that the transition point \ensuremathΓtr{\mathrm{\ensuremath{\Gamma}}}_{\mathrm{tr}}\ensuremathΓtr​ decreases as the strength of this interaction part or NNN increases; in the strong strength limit for this interaction part, \ensuremathΓtr{\mathrm{\ensuremath{\Gamma}}}_{\mathrm{tr}}\ensuremathΓtr​ scales as the strength to the power of \ensuremath−(N\ensuremath−1)\ensuremath{-}(N\ensuremath{-}1)\ensuremath−(N\ensuremath−1). We also give signatures of the PT\mathcal{PT}PT-symmetric and the symmetry-breaking phases for the interacting bosons in experiment.
Trapped ions are an ideal platform to implement quantum simulation. Previously, the parity-time (PT\mathcal{PT}PT)-reversal symmetry-breaking transition in the paradigmatic non-Hermitian Hamiltonian ${h}_{\mathcal{PT}}=J{\ensuremath{\sigma}}_{x}\ensuremath{-}i\mathrm{\ensuremath{\Gamma}}{\ensuremath{\sigma}}_{z}$ was observed in a single-ion experiment in a passive way. In this work we propose to study the interaction effects on the PT\mathcal{PT}PT-symmetry-breaking transition in a chain of NNN trapped interacting ions. We consider an effective Ising interaction ${H}_{\text{int}}^{\text{Ising}\phantom{\rule{4.pt}{0ex}}x}={\ensuremath{\sum}}_{j<k}{U}_{jk}{\ensuremath{\sigma}}_{x}^{j}{\ensuremath{\sigma}}_{x}^{k}$ between the ions on top of hPT{h}_{\mathcal{PT}}hPT​. We find that sufficiently strong interaction strength can enhance the PT\mathcal{PT}PT-symmetric phase for even NNN, while the phase is suppressed in all the other cases. In particular, the suppression can be so strong that even infinitesimal dissipation, quantified by $\mathrm{\ensuremath{\Gamma}}$, can turn the system into the PT\mathcal{PT}PT-symmetry-breaking phase. In addition, we assess the convolved effects due to the coupling and the spin phase shifts. Our findings can be readily tested in ion chain experiments.
Non-Hermitian quantum mechanics with parity-time (PT) symmetry offers a powerful framework for exploring the complex interplay of dissipation and coherent interactions in open quantum systems. While PT-symmetry breaking has been studied in various physical systems, its observation on a quantum many-body level remains elusive. Here, we experimentally realize a non-Hermitian XY model in a strongly-interacting Rydberg-atom array. By measuring the Loschmidt Echo of a fully polarized state, we observe distinct dynamical signatures of a PT-symmetry-breaking phase transition. Dipole interactions are found to play a crucial role, not only determining the transition point but also triggering a non-Hermitian many-body blockade effect that protects the Loschmidt Echo from decay with a non-monotonic dependence on the system size. Our results reveal intricate interaction-induced effects on PT-symmetry breaking and open the door for exploring non-Hermitian many-body dynamics beyond single-particle and mean-field paradigms.
Abstract Open physical systems with balanced loss and gain, described by non-Hermitian parity-time (PT)\left( {{\cal P}{\cal T}} \right)(PT) P T reflection symmetric Hamiltonians, exhibit a transition which could engender modes that exponentially decay or grow with time, and thus spontaneously breaks the PT{\cal P}{\cal T}PT P T -symmetry. Such PT{\cal P}{\cal T}PT P T -symmetry-breaking transitions have attracted many interests because of their extraordinary behaviors and functionalities absent in closed systems. Here we report on the observation of PT{\cal P}{\cal T}PT P T -symmetry-breaking transitions by engineering time-periodic dissipation and coupling, which are realized through state-dependent atom loss in an optical dipole trap of ultracold 6Li atoms. Comparing with a single transition appearing for static dissipation, the time-periodic counterpart undergoes PT{\cal P}{\cal T}PT P T -symmetry breaking and restoring transitions at vanishingly small dissipation strength in both single and multiphoton transition domains, revealing rich phase structures associated to a Floquet open system. The results enable ultracold atoms to be a versatile tool for studying PT{\cal P}{\cal T}PT P T -symmetric quantum systems.
Analysis of fundamentally open systems which exhibit both spatial reflection (parity) and time-reversal symmetry, called PT-symmetric [1] systems, has played an important role in understanding the dynamics of a broad class of non-Hermitian systems. Such systems support distinct regimes of stability and instability respectively, and the transition between these plays an important role in the system dynamics. We investigate a variety of non-Hermitian systems which exhibit a PT-symmetry breaking transition in both classical and quantum contexts using an array of theoretical and numerical approaches which are directly amenable to physical experiments. Organizationally, we divide these studies into those done on classical and quantum systems. Specifically for classical systems we analyze several interesting phenomena in the context of optical waveguide arrays. These provide an important testing ground for much of the discrete, PT-symmetric theory. In our case, we analyze PT-symmetric extensions of several interesting lattice models. Firstly, we explore single gain/loss pair, PT-symmetric extensions of the Su-Schreiffer-Heeger (SSH) model [2] and the Aubry-Andre-Harper (AAH) model [3,4]. In both cases, we find a veiled symmetry which is responsible for the PT-symmetry. Secondly we expand upon the Rudner-Levitov (RL) model [5,6], a non-Hermitian extension of the SSH model with multiple gain/loss pairs, and we explore the dependence of topological phenomena on changes in the initial state as well as their robustness in the presence of gain-induced nonlinearity. For quantum systems, we explore the rich PT-symmetric phase diagram resulting from time-periodic, Floquet PT-symmetry [7,8]. This is experimentally implementable in ultracold atomic systems [9], and features a reentrant PT-symmetric phase. Secondly, we examine quantum multi-photon statistics for a simple non-Hermitian model which is transformed and mapped to a system with unitary evolution via a unitary dilation, and is fully realizable on a small-scale quantum simulator [10].
Inspired by the discovery of real energy spectra of non-Hermitian Hamiltonians obeying parity-time (PT) symmetry, many proposals and relevant experiments have been suggested and implemented for the realization of PT-symmetric Hamiltonians in various physical systems. Among them PT-symmetric atomic gases are of great interest as they possess many unique advantages, including: (i) the possession of active PT symmetry with balanced optical gain and loss, (ii) the ability to be extended to high-dimensional ones, and (iii) the feasibility to acquire enhanced Kerr nonlinearity based on the resonance character between light and atoms. Here, we review the new progresses made on the study of PT symmetry in coherent atomic gases since our last review paper [Advances in Physics: X, 2, 737–783 (2017)]. Particularly, we focus on the newly emerged research topics in this direction, including nonlocal nonlinear effect, non-Hermitian quantum effect, exceptional point and associated phenomenon, and non-Hermitian skin effect with PT symmetry. Perspectives and potential applications of the subject are also discussed.
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.