We report the bright solitons of the generalized Gross-Pitaevskii (GP) equation with some types of physically relevant parity-time- ( PT-) and non- PT-symmetric potentials. We find that the constant momentum coefficient Γ can modulate the linear stability and complicated transverse power-flows (not always from the gain toward loss) of nonlinear modes. However, the varying momentum coefficient Γ(x) can modulate both unbroken linear PT-symmetric phases and stability of nonlinear modes. Particularly, the nonlinearity can excite the unstable linear mode (i.e., broken linear PT-symmetric phase) to stable nonlinear modes. Moreover, we also find stable bright solitons in the presence of non- PT-symmetric harmonic-Gaussian potential. The interactions of two bright solitons are also illustrated in PT-symmetric potentials. Finally, we consider nonlinear modes and transverse power-flows in the three-dimensional (3D) GP equation with the generalized PT-symmetric Scarff-II potential.

On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT- and non- PT-symmetric potentials.

We report our realization of a parity-time (PT) symmetric non-Hermitian many-body system using cold atoms with dissipation. After developing a theoretical framework on PT-symmetric many-body systems using ultracold atoms in an optical lattice with controlled dissipation, we describe our experimental setup utilizing one-body atom loss as dissipation with special emphasis on calibration of important system parameters. We discuss loss dynamics observed experimentally.

PT-symmetric non-Hermitian quantum many-body system using ultracold atoms in an optical lattice with controlled dissipation

We report the bright solitons of the generalized Gross-Pitaevskii (GP) equation with some types of physically relevant parity-time-(PT-) and non-PT-symmetric potentials. We find that the constant momentum coefficient can modulate the linear stability and complicated transverse power-flows (not always from the gain toward loss) of nonlinear modes. However, the varying momentum coefficient Gamma(x) can modulate both unbroken linear PT-symmetric phases and stability of nonlinear modes. Particularly, the nonlinearity can excite the unstable linear mode (i.e., broken linear PT-symmetric phase) to stable nonlinear modes. Moreover, we also find stable bright solitons in the presence of non-PT-symmetric harmonic-Gaussian potential. The interactions of two bright solitons are also illustrated in PT-symmetric potentials. Finally, we consider nonlinear modes and transverse power-flows in the three-dimensional (3D) GP equation with the generalized PT-symmetric Scarf-II potential

On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT-and non-PT-symmetric potentials

We discover that the physically interesting PT-symmetric Dirac delta-function potentials can not only make sure that the non-Hermitian Hamiltonians admit fully-real linear spectra but also support stable peakons (nonlinear modes) in the Kerr nonlinear Schrodinger equation. For a specific form of the delta-function PT-symmetric potentials, the nonlinear model investigated in this paper is exactly solvable. However, for a class of PT-symmetric signum-function double-well potentials, a novel type of exact flat-top bright solitons can exist stably within a broad range of potential parameters. Intriguingly, the flat-top solitons can be characterized by the finite-order differentiable waveforms and admit the novel features differing from the usual solitons. The excitation features and the direction of transverse power flow of flat-top bright solitons are also explored in detail. These results are useful for the related experimental designs and applications in nonlinear optics and other related fields.

Stable flat-top solitons and peakons in the PT-symmetric δ-signum potentials and nonlinear media.

We present an alternative type of parity-time (PT)-symmetric generalized Scarf-II potentials, which makes possible for non-Hermitian Hamiltonians in the classical linear Schrodinger system to possess fully real spectra with unique features such as the multiple PT-symmetric breaking behaviors and to support one-dimensional (1D) stable PT-symmetric solitons of power-law waveform, namely power-law solitons, in focusing Kerr-type nonlinear media. Moreover, PT-symmetric high-order solitons are also derived numerically in 1D and 2D settings. Around the exactly obtained nonlinear propagation constants, families of 1D and 2D localized nonlinear modes are also found numerically. The majority of fundamental nonlinear modes can still keep steady in general, whereas the 1D multipeak solitons and 2D vortex solitons are usually susceptible to suffering from instability. Likewise, similar results occur in the defocusing Kerr-nonlinear media. The obtained results will be useful for understanding the complex dynamics of nonlinear waves that form in PT-symmetric nonlinear media in other physical contexts.

Soliton formation and stability under the interplay between parity-time-symmetric generalized Scarf-II potentials and Kerr nonlinearity.

A Bose-Einstein condensate in a double-well potential features stationary solutions even for attractive contact interaction as long as the particle number and therefore the interaction strength do not exceed a certain limit. Introducing balanced gain and loss into such a system drastically changes the bifurcation scenario at which these states are created. Instead of two tangent bifurcations at which the symmetric and antisymmetric states emerge, one tangent bifurcation between two formerly independent branches arises [D. Haag et al., Phys. Rev. A 89, 023601 (2014)]. We study this transition in detail using a bicomplex formulation of the time-dependent variational principle and find that in fact there are three tangent bifurcations for very small gain-loss contributions which coalesce in a cusp bifurcation.

Cusp bifurcation in the eigenvalue spectrum of PT -symmetric Bose-Einstein condensates

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.

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

A quantum system exhibiting $\mathcal{PT}$ 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 novel 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.

Realization of balanced gain and loss in a time-dependent four-mode Bose-Hubbard model

The experimental realization of balanced gain and loss in a quantum system has been a long standing goal in quantum mechanics since the introduction of the concept of $\mathcal{PT}$ symmetry and has only recently been achieved. In this paper we analyze balanced gain and loss in Gaussian multi-well potentials with either only gain or loss in each well. By means of symmetrization via matrix models we can construct asymmetric extended potentials with partially real or complex conjugate spectra. This will be demonstrated explicitly for double-well and triple-well systems. Such systems can be realized with Bose-Einstein condensates in optical trapping potentials in the presence of localized particle gain and loss. The usage of asymmetric potentials in the process is more versatile and is considered beneficial in real experimental implementations.

Balanced gain and loss in spatially extended non-PT-symmetric multiwell potentials

Bose-Einstein condensates with balanced gain and loss in a double-well potential have been shown to exhibit PT-symmetric states. As proposed by Kreibich et al [Phys. Rev. A 87, 051601(R) (2013)], in the mean-field limit the dynamical behaviour of this system, especially that of the PT-symmetric states, can be simulated by embedding it into a four-well system with time-dependent parameters. In this paper we go beyond the mean-field approximation and investigate many-body effects in this system, which are in lowest order described by the single-particle density matrix. The conditions for PT symmetry in the single-particle density matrix cannot be completely fulfilled by using pure initial states. Here we show that it is mathematically possible to achieve exact PT symmetry in the four-well many-body system in the sense of the dynamical behaviour of the single-particle density matrix. In contrast to previous work, for this purpose, we use impure initial states fulfilling certain constraints and use them to calculate the dynamics.

Daniel Dizdarevic

Papers

Cusp bifurcation in the eigenvalue spectrum of PT -symmetric Bose-Einstein condensates

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

Realization of balanced gain and loss in a time-dependent four-mode Bose-Hubbard model

Balanced gain and loss in spatially extended non-PT-symmetric multiwell potentials

Using mixed many-body particle states to generate exact $\mathcal{PT} $ -symmetry in a time-dependent four-well system