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arXiv:1302.2008 · Physical Review A

Hermitian four-well potential as a realization of a PT -symmetric system

Manuel Kreibich, Jörg Main, Holger Cartarius, Günter Wunner

Published May 7, 2013• 76 citations
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Abstract

A $\mathcal{PT}$-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 $\mathcal{PT}$-symmetric system, where the $\mathcal{PT}$-symmetric currents of a two-well system are implemented by coupling two additional wells to the system, which act as particle reservoirs. In terms of a simple four-mode model we derive conditions under which the two middle wells of the Hermitian four-well system behave exactly as the two wells of the $\mathcal{PT}$-symmetric system. We apply these conditions to calculate stationary solutions and oscillatory dynamics. By means of frozen Gaussian wave packets we relate the Gross-Pitaevskii equation to the four-mode model and give parameters required for the external potential, which provides approximate conditions for a realistic experimental setup.

DOI
10.1103/physreva.87.051601

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A PT\mathcal{PT}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\mathcal{PT}PT-symmetric system, where the PT\mathcal{PT}PT-symmetric currents of a two-well system are implemented by coupling two additional wells to the system, which act as particle reservoirs. In terms of a simple four-mode model we derive conditions under which the two middle wells of the Hermitian four-well system behave exactly as the two wells of the PT\mathcal{PT}PT-symmetric system. We apply these conditions to calculate stationary solutions and oscillatory dynamics. By means of frozen Gaussian wave packets we relate the Gross-Pitaevskii equation to the four-mode model and give parameters required for the external potential, which provides approximate conditions for a realistic experimental setup.
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.
Realizing PT{\mathcal{P}}{\mathcal{T}}PT-symmetric BEC subsystems in closed Hermitian systems
In open double-well Bose–Einstein condensate systems which balance in- and outfluxes of atoms and which are effectively described by a non-Hermitian -symmetric Hamiltonian -symmetric states have been shown to exist. -symmetric states obey the combined parity and time reversal symmetry. We tackle the question of how the in- and outfluxes can be realized and introduce a Hermitian system in which two -symmetric subsystems are embedded. This system no longer requires an in- and outcoupling to and from the environment. We show that the subsystems still have -symmetric states. In addition we examine what degree of detail is necessary to correctly model the -symmetric properties and the bifurcation structure of such a system. We examine a four-mode matrix model and a system described by the full Gross–Pitaevskii equation in one dimension. We see that a simple matrix model correctly describes the qualitative properties of the system. For sufficiently isolated wells there is also quantitative agreement with the more advanced system descriptions. We also investigate which properties the wave functions of a system must fulfil to allow for -symmetric states. In particular the requirements for the phase difference between different parts of the system are examined.
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.
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.
In open double-well Bose-Einstein condensate systems which balance in- and outuxes of atoms and which are eectively described by a non-hermitian PT - symmetric HamiltonianPT -symmetric states have been shown to exist. We tackle the question of how these in- and outuxes can be realized and introduce model systems in whichPT -symmetric subsystems are embedded into a closed hermitian system. We show that such subsystems still havePT -symmetric states. In addition we examine what degree of detail is necessary to describe the PT -symmetric properties and the bifurcation structure of such a system correctly by investigating a four-mode matrix model and the full Gross-Pitaevskii equation in one dimension. We also investigate which properties the wave functions of a system must full to allow for PT -symmetric states. In particular the requirements for the phase dierence
Eigenvalue structure of a Bose–Einstein condensate in a PT\mathcal {PT}PT-symmetric double well
We study a Bose–Einstein condensate in a -symmetric double-well potential where particles are coherently injected in one well and removed from the other well. In mean-field approximation the condensate is described by the Gross–Pitaevskii equation thus falling into the category of nonlinear non-Hermitian quantum systems. After extending the concept of symmetry to such systems, we apply an analytic continuation to the Gross–Pitaevskii equation from complex to bicomplex numbers and show a thorough numerical investigation of the four-dimensional bicomplex eigenvalue spectrum. The continuation introduces additional symmetries to the system which are confirmed by the numerical calculations and furthermore allows us to analyse the bifurcation scenarios and exceptional points of the system. We present a linear matrix model and show the excellent agreement with our numerical results. The matrix model includes both exceptional points found in the double-well potential, namely an EP2 at the tangent bifurcation and an EP3 at the pitchfork bifurcation. When the two bifurcation points coincide the matrix model possesses four degenerate eigenvectors. Close to that point we observe the characteristic features of four interacting modes in both the matrix model and the numerical calculations, which provides clear evidence for the existence of an EP4.