A Bose‐Einstein condensate in a 𝒫𝒯 symmetric double well

Dast, Dennis; Haag, Daniel; Cartarius, Holger; Wunner, Günter; Eichler, Rüdiger; Main, Jörg

doi:10.1002/prop.201200080

Cited by 61 publications

(125 citation statements)

References 35 publications

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“…Previous works showed that the nonanalytic nonlinearity of the GPE, which is proportional to the square modulus of the wave function |ψ| 2 , is responsible for the change in the number of solutions at the bifurcation points [28,35,36].…”

Section: Analytic Continuationmentioning

confidence: 99%

Eigenvalue structure of a Bose–Einstein condensate in a $\mathcal {PT}$-symmetric double well

Dast¹,

Haag²,

Cartarius³

et al. 2013

J. Phys. A: Math. Theor.

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Abstract. We study a Bose-Einstein condensate in a PT -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 PT 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 analyze 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.

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Section: Analytic Continuationmentioning

confidence: 99%

Eigenvalue structure of a Bose–Einstein condensate in a $\mathcal {PT}$-symmetric double well

Dast¹,

Haag²,

Cartarius³

et al. 2013

J. Phys. A: Math. Theor.

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show abstract

“…Figure 4a shows Fig. 4 (a) Energy spectrum E (2) 0 of the only existing state φ (2) 0 in the system described by the potential V 2 for removed ground state. For comparison the energy E (1) 1 of the original system is also shown.…”

Section: Removal Of Pt -Symmetric Statesmentioning

confidence: 99%

“…The only existing state is the former excited state, which is the ground state of the new system, and hence is labelled φ (2) 0 . Its energy must be positive since it must be above that …”

Section: Removal Of Pt -Symmetric Statesmentioning

confidence: 99%

“…a system of which the Hamiltonian commutes with the combined action of the parity and time-reversal operators, [PT , H ] = 0 [1][2][3]. These systems are of special interest since they allow for the existence of real eigenvalues despite the presence of gain and loss of the probability amplitude, which is described by the non-Hermitian contributions to the Hamiltonian [4].…”

Section: Introductionmentioning

confidence: 99%

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Supersymmetric Model of a Bose-Einstein Condensate in a 𝓟𝓣-Symmetric Double-delta Trap

Abt

Cartarius

2015

Int J Theor Phys

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The most important properties of a Bose-Einstein condensate subject to balanced gain and loss can be modelled by a Gross-Pitaevskii equation with an external PT -symmetric double-delta potential. We study its linear variant with a supersymmetric extension. It is shown that both in the PT -symmetric as well as in the PT -broken phase arbitrary stationary states can be removed in a supersymmetric partner potential without changing the energy eigenvalues of the other state. The characteristic structure of the singular delta potential in the supersymmetry formalism is discussed, and the applicability of the formalism to the nonlinear Gross-Pitaevskii equation is analysed. In the latter case the formalism could be used to remove PT -broken states introducing an instability to the stationary PT -symmetric states.

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“…This potential is extended with asymmetric imaginary parts, representing a particle drain in one well and a particle source in the other. Theoretical investigations revealed that this PT -symmetric system has observable eigenstates [19][20][21][26][27][28][29][30][31]. However, an experimental confirmation of these results cannot be made as long as there is no way to realize a non-Hermitian potential.…”

Section: Introductionmentioning

confidence: 99%

Coupling approach for the realization of aPT-symmetric potential for a Bose-Einstein condensate in a double well

et al. 2014

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We show how non-Hermitian potentials used to describe probability gain and loss in effective theories of open quantum systems can be achieved by a coupling of the system to an environment. We do this by coupling a Bose-Einstein condensate (BEC) trapped in an attractive double-δ potential to a condensate fraction outside the double well. We investigate which requirements have to be imposed on possible environments with a linear coupling to the system. This information is used to determine an environment required for stationary states of the BEC. To investigate the stability of the system we use fully numerical simulations of the dynamics. It turns out that the approach is viable and possible setups for the realization of a PT -symmetric potential for a BEC are accessible. Vulnerabilities of the whole system to small perturbations can be attributed to the singular character of the simplified δ-shaped potential used in our model.

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A Bose‐Einstein condensate in a 𝒫𝒯 symmetric double well

Cited by 61 publications

References 35 publications

Eigenvalue structure of a Bose–Einstein condensate in a $\mathcal {PT}$-symmetric double well

Eigenvalue structure of a Bose–Einstein condensate in a $\mathcal {PT}$-symmetric double well

Supersymmetric Model of a Bose-Einstein Condensate in a 𝓟𝓣-Symmetric Double-delta Trap

Coupling approach for the realization of aPT-symmetric potential for a Bose-Einstein condensate in a double well

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