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Register a new userLandau-Zener-St{u}ckelberg Interferometry in $mathcal{PT}$-symmetric Non-Hermitian models
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We systematically investigate the non-Hermitian generalisations of the Landau-Zener (LZ) transition and the Landau-Zener-St{u}ckelberg (LZS) interferometry. The LZ transition probabilities, or band populations, are calculated for a generic non-Hermitian model and their asymptotic behaviour analysed. We then focus on non-Hermitian systems with a real adiabatic parameter and study the LZS interferometry formed out of two identical avoided level crossings. Four distinctive cases of interferometry are identified and the analytic formulae for the transition probabilities are calculated for each case. The differences and similarities between the non-Hermitian case and its Hermitian counterpart are emphasised. In particular, the geometrical phase originated from the sign change of the mass term at the two level crossings is still present in the non-Hermitian system, indicating its robustness against the non-Hermiticity. We further apply our non-Hermitian LZS theory to describing the Bloch oscillation in one-dimensional parity-time $(mathcal{PT})$ reversal symmetric non-Hermitian Su-Schrieffer-Heeger model and propose an experimental scheme to simulate such dynamics using photonic waveguide arrays. The Landau-Zener transition, as well as the LZS interferometry, can be visualised through the beam intensity profile and the transition probabilitiess measured by the centre of mass of the profile.
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Employing the time-dependent variational principle combined with the multiple Davydov $mathrm{D}_2$ Ansatz, we investigate Landau-Zener (LZ) transitions in a qubit coupled to a photon mode with various initial photon states at zero temperature. Thanks to the multiple Davydov trial states, exact photonic dynamics taking place in the course of the LZ transition is also studied efficiently. With the qubit driven by a linear external field and the photon mode initialized with Schrodinger-cat states, asymptotic behavior of the transition probability beyond the rotating-wave approximation is uncovered for a variety of initial states. Using a sinusoidal external driving field, we also explore the photon-assisted dynamics of Landau-Zener-St{u}ckelberg-Majorana interferometry. Transition pathways involving multiple energy levels are unveiled by analyzing the photon dynamics.
Eigenspectra of a spinless quantum particle trapped inside a rigid, rectangular, two-dimensional (2D) box subject to diverse inner potential distributions are investigated under hermitian, as well as non-hermitian antiunitary $mathcal{PT}$ (composite parity and time-reversal) symmetric regimes. Four sectors or stripes inscribed in the rigid box comprising contiguously conjoined parallel rectangular segments with one side equaling the entire width of the box are studied. The stripes encompass piecewise constant potentials whose exact, complete energy eigenspectrum is obtained employing matrix mechanics. Various striped potential compositions, viz. real valued ones in the hermitian regime as well as complex, non-hermitian but $mathcal{PT}$ symmetric ones are considered separately and in conjunction, unraveling among typical lowest lying eigenvalues, retention and breakdown scenarios engendered by the $mathcal{PT}$ symmetry, bearing upon the strength of non-hermitian sectors. Some states exhibit a remarkable crossover of symmetry `making and `breaking: while a broken $mathcal{PT}$ gets reinstated for an energy level, higher levels may couple to continue with symmetry breaking. Further, for a charged quantum particle a $mathcal{PT}$ symmetric electric field, furnished with a striped potential backdrop, also reveals peculiar retention and breakdown $mathcal{PT}$ scenarios. Depictions of prominent probability redistributions relating to various potential distributions both under norm-conserving unitary regime for hermitian Hamiltonians and non-conserving ones post $mathcal{PT}$ breakdown are presented.
$mathcal{PT}$-symmetric quantum mechanics has been considered an important theoretical framework for understanding physical phenomena in $mathcal{PT}$-symmetric systems, with a number of $mathcal{PT}$-symmetry related applications. This line of research was made possible by the introduction of a time-independent metric operator to redefine the inner product of a Hilbert space. To treat the dynamics of generic non-Hermitian systems under equal footing, we advocate in this work the use of a time-dependent metric operator for the inner-product between time-evolving states. This treatment makes it possible to always interpret the dynamics of arbitrary (finite-dimensional) non-Hermitian systems in the framework of time-dependent $mathcal{PT}$-symmetric quantum mechanics, with unitary time evolution, real eigenvalues of an energy observable, and quantum measurement postulate all restored. Our work sheds new lights on generic non-Hermitian systems and spontaneous $mathcal{PT}$-symmetry breaking in particular. We also illustrate possible applications of our formulation with well-known examples in quantum thermodynamics.
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.
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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