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Register a new userExactly solvable $mathcal{PT}$-symmetric models in two dimensions
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Non-hermitian, $mathcal{PT}$-symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-dimensional, $mathcal{PT}$ potentials for a non-relativistic particle confined in a circular geometry. We show that the $mathcal{PT}$ symmetry threshold can be tuned by introducing a second gain-loss potential or its hermitian counterpart. Our results explicitly demonstrate that $mathcal{PT}$ breaking in two dimensions has a rich phase diagram, with multiple re-entrant $mathcal{PT}$ symmetric phases.
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We investigate vortex excitations in dilute Bose-Einstein condensates in the presence of complex $mathcal{PT}$-symmetric potentials. These complex potentials are used to describe a balanced gain and loss of particles and allow for an easier calculation of stationary states in open systems than in a full dynamical calculation including the whole environment. We examine the conditions under which stationary vortex states can exist and consider transitions from vortex to non-vortex states. In addition, we study the influences of $mathcal{PT}$ symmetry on the dynamics of non-stationary vortex states placed at off-center positions.
We study the case of $mathcal{PT}$-symmetric perturbations of Hermitian Hamiltonians with degenerate eigenvalues using the example of a triple-well system. The degeneracy complicates the question, whether or not a stationary current through such a system can be established, i.e. whether or not the $mathcal{PT}$-symmetric states are stable. It is shown that this is only the case for perturbations that do not couple to any of the degenerate states. The physical explanation for the inhibition of stable currents is discussed. However, introducing an on-site interaction restores the capability to support stable currents.
Over the past decade, non-Hermitian, $mathcal{PT}$-symmetric Hamiltonians have been investigated as candidates for both, a fundamental, unitary, quantum theory, and open systems with a non-unitary time evolution. In this paper, we investigate the implications of the former approach in the context of the latter. Motivated by the invariance of the $mathcal{PT}$ (inner) product under time evolution, we discuss the dynamics of wave-function phases in a wide range of $mathcal{PT}$-symmetric lattice models. In particular, we numerically show that, starting with a random initial state, a universal, gain-site location dependent locking between wave function phases at adjacent sites occurs in the $mathcal{PT}$-symmetry broken region. Our results pave the way towards understanding the physically observable implications of time-invariants in the non-unitary dynamics produced by $mathcal{PT}$-symmetric Hamiltonians.
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 $mathcal{PT}$-symmetric double-delta potential. We study its linear variant with a supersymmetric extension. It is shown that both in the $mathcal{PT}$-symmetric as well as in the $mathcal{PT}$-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 $mathcal{PT}$-broken states introducing an instability to the stationary $mathcal{PT}$-symmetric states.
Two coupled two-level systems placed under external time-dependent magnetic fields are modeled by a general Hamiltonian endowed with a symmetry that enables us to reduce the total dynamics into two independent two-dimensional sub-dynamics. Each of the sub-dynamics is shown to be brought into an exactly solvable form by appropriately engineering the magnetic fields and thus we obtain an exact time evolution of the compound system. Several physically relevant and interesting quantities are evaluated exactly to disclose intriguing phenomena in such a system.
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