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arXiv:1905.07945 · Physical Review A

Realization of PT -symmetric and PT -symmetry-broken states in static optical-lattice potentials

Felix Kogel, Sebastian Kotzur, Daniel Dizdarevic, Jörg Main, Günter Wunner

Published Jun 17, 2019• 11 citations
arXivHTMLPDF

Abstract

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 $\mathcal{PT}$-symmetric quasistationary and $\mathcal{PT}$-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.

DOI
10.1103/physreva.99.063610

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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.
More than a decade ago, it was shown that non-Hermitian Hamiltonians with combined parity (P) and time-reversal (T ) symmetry exhibit real eigenvalues over a range of parameters. Since then, the field of PT symmetry has seen rapid progress on both the theoretical and experimental fronts. These effective Hamiltonians are excellent candidates for describing open quantum systems with balanced gain and loss. Nature seems to be replete with examples of PT -symmetric systems; in fact, recent experimental investigations have observed the effects of PT symmetry breaking in systems as diverse as coupled mechanical pendula, coupled optical waveguides, and coupled electrical circuits. Recently, PT -symmetric Hamiltonians for tight-binding lattice models have been extensively investigated. Lattice models, in general, have been widely used in physics due to their analytical and numerical tractability. Perhaps one of the best systems for experimentally observing the effects of PT symmetry breaking in a one-dimensional lattice with tunable hopping is an array of evanescently-coupled optical waveguides. The tunneling between adjacent waveguides is tuned by adjusting the width of the barrier between them, and the imaginary part of the local refractive index provides the loss or gain in the respective waveguide. Calculating the time evolution of a wave packet on a lattice is relatively straightforward in the tight-binding model, allowing us to make predictions about the behavior of light propagating down an array of PT -symmetric waveguides. In this thesis, I investigate the the strength of the PT -symmetric phase (the region over which the eigenvalues are purely real) in lattices with a variety of PT - symmetric potentials. In Chapter 1, I begin with a brief review of the postulates of quantum mechanics, followed by an outline of the fundamental principles of PT - symmetric systems. Chapter 2 focuses on one-dimensional uniform lattices with a pair of PT -symmetric impurities in the case of open boundary conditions. I find that the PT phase is algebraically fragile except in the case of closest impurities, where the PT phase remains nonzero. In Chapter 3, I examine the case of periodic boundary conditions in uniform lattices, finding that the PT phase is not only nonzero, but also independent of the impurity spacing on the lattice. In addition, I explore the time evolution of a single-particle wave packet initially localized at a site. I find that in the case of periodic boundary conditions, the wave packet undergoes a preferential clockwise or counterclockwise motion around the ring. This behavior is quantified by a discrete momentum operator which assumes a maximum value at the PT -symmetry- breaking threshold. In Chapter 4, I investigate nonuniform lattices where the parity-symmetric hop- ping between neighboring sites can be tuned. I find that the PT phase remains strong in the case of closest impurities and fragile elsewhere. Chapter 5 explores the effects of the competition between localized and extended PT potentials on a lattice. I show that when the short-range impurities are maximally separated on the lattice, the PT phase is strengthened by adding short-range loss in the broad-loss region. Consequently, I predict that a broken PT symmetry can be restored by increasing the strength of the short-range impurities. Lastly, Chapter 6 summarizes my salient results and discusses areas which can be further developed in future research.
A PT-symmetric Bose-Einstein condensate can be theoretically described using a complex optical potential, however, the experimental realization of such an optical potential describing the coherent in- and outcoupling of particles is a nontrivial task. We propose an experiment for a quantum mechanical realization of a PT-symmetric system, where the PT-symmetric currents of a two-well system are implemented by coupling possibly time-dependently varied additional wells to the system, which act as particle reservoirs and thus form a Hermitian system. We map the time-dependence of the amplitudes of a frozen Gaussian variational ansatz to a Schrödinger equation with a Hamiltonian matrix. This relates the parameters of a realistic external potential to the matrix elements of a matrix model. On one side, we can use the matrix model as a computationally cheap model to obtain results, on the other hand we can then map the time-dependence of the matrix elements back to the parameters of the potential, which would serve as an input for an experimental setup. In terms of these simple matrix model we derive conditions under which two wells of the Hermitian multi-well system behave exactly as the two wells of the PT-symmetric system. It turns out that the matrix elements of the Hermitian system must be time-dependent to fulfill the conditions for a sufficient time. These results are applied to calculate the time-dependencies of the matrix elements, first, by means of a four-well system, then consequently building up until we arrive at a system with a large number of wells, which can be analyzed and interpreted in terms of optical lattices. As a second method we use a full time-dependent Gaussian variational ansatz, where every variational parameter of the Gaussian functions is chosen to be time-dependent. This should give more accurate results in that this ansatz can describe significant more degrees of freedom. We derive conditions analogous to that of the matrix models that must be fulfilled such that PT-symmetry can be realized. This method is applied and results are obtained for the important case of an adiabatic current ramp which is proposed as a realistic experimental method to create a PT-symmetric ground state. Finally, the results to both methods are compared and in particular the approximations that lead to the simple matrix models are justified.
A quantum system exhibiting PT\mathcal{PT}PT symmetry is a Bose-Einstein condensate in a double-well potential with balanced particle gain and loss, which is described in the mean-field limit by a Gross-Pitaevskii equation with a complex potential. A possible experimental realization of such a system by embedding it into a Hermitian time-dependent four-mode optical lattice was proposed by Kreibich et al. [Phys. Rev. A 87, 051601(R) (2013)], where additional potential wells act as reservoirs and particle exchange happens via tunneling. Since particle influx and outflux have to be controlled explicitly, a set of conditions on the potential parameters had to be derived. In contrast to previous work, our focus lies on a full many-body description beyond the mean-field approximation using a Bose-Hubbard model with time-dependent potentials. This gives rise to additional quantum effects, such that the differences between mean-field and many-body dynamics are of special interest. We further present stationary analytical solutions for the embedded wells in the mean-field limit, different approaches for the embedding into a many-body system, and a very efficient method for the evaluation of hopping terms to calculate exact Bose-Hubbard dynamics.
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.
The observation of PT\mathcal{PT}PT symmetry in a coupled optical waveguide system that involves a complex refractive index has been demonstrated impressively in the experiment by R\"uter et al. [Nat. Phys. 6, 192 (2010)]. This is, however, only an optical analog of a quantum system, and it would be highly desirable to observe the manifestation of PT\mathcal{PT}PT symmetry and the resulting properties also in a real, experimentally accessible, quantum system. Following a suggestion by Klaiman et al. [Phys. Rev. Lett. 101, 080402 (2008)], we investigate a PT\mathcal{PT}PT-symmetric arrangement of a Bose-Einstein condensate in a double-well potential, where in one well cold atoms are injected while in the other particles are extracted from the condensate. We investigate, in particular, the effects of the nonlinearity in the Gross-Pitaevskii equation on the PT\mathcal{PT}PT properties of the condensate. To study these effects we analyze a simple one-dimensional model system in which the condensate is placed into two PT\mathcal{PT}PT-symmetric $\ensuremath{\delta}$-function traps. The analysis will serve as a useful guide for studies of the behavior of Bose-Einstein condensates in realistic PT\mathcal{PT}PT-symmetric double wells.