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Register a new userTime-dependent $mathcal{PT}$-symmetric quantum mechanics in generic non-Hermitian systems
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Added by
Jiangbin Gong Prof.
Publication date
2019
fields
Physics
and research's language is
English
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$mathcal{PT}$-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.
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Time-dependent $mathcal{PT}$-symmetric quantum mechanics is featured by a varying inner-product metric and has stimulated a number of interesting studies beyond conventional quantum mechanics. In this paper, we explore geometric aspects of time-dependent $mathcal{PT}$-symmetric quantum mechanics. We not only find a geometric phase factor emerging naturally from cyclic evolutions of $mathcal{PT}$-symmetric systems, but also formulate a series of differential geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. Our findings constitute a useful, perhaps indispensible, tool to tackle physical problems involving $mathcal{PT}$-symmetric systems with time-varying systems parameters. To exemplify the application of our findings, we show that the unconventional geometrical phase [Phys. Rev. Lett. 91, 187902 (2003)], consisting of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase identified in this work.
A series of geometric concepts are formulated for $mathcal{PT}$-symmetric quantum mechanics and they are further unified into one entity, i.e., an extended quantum geometric tensor (QGT). The imaginary part of the extended QGT gives a Berry curvature whereas the real part induces a metric tensor on systems parameter manifold. This results in a unified conceptual framework to understand and explore physical properties of $mathcal{PT}$-symmetric systems from a geometric perspective. To illustrate the usefulness of the extended QGT, we show how its real part, i.e., the metric tensor, can be exploited as a tool to detect quantum phase transitions as well as spontaneous $mathcal{PT}$-symmetry breaking in $mathcal{PT}$-symmetric systems.
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.
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.
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.
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