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Open systems with gain, loss, or both, described by non-Hermitian Hamiltonians, have been a research frontier for the past decade. In particular, such Hamiltonians which possess parity-time ($mathcal{PT}$) symmetry feature dynamically stable regimes of unbroken symmetry with completely real eigenspectra that are rendered into complex conjugate pairs as the strength of the non-Hermiticity increases. By subjecting a $mathcal{PT}$-symmetric system to a periodic (Floquet) driving, the regime of dynamical stability can be dramatically affected, leading to a frequency-dependent threshold for the $mathcal{PT}$-symmetry breaking transition. We present a simple model of a time-dependent $mathcal{PT}$-symmetric Hamiltonian which smoothly connects the static case, a $mathcal{PT}$-symmetric Floquet case, and a neutral-$mathcal{PT}$-symmetric case. We analytically and numerically analyze the $mathcal{PT}$ phase diagrams in each case, and show that slivers of $mathcal{PT}$-broken ($mathcal{PT}$-symmetric) phase extend deep into the nominally low (high) non-Hermiticity region.
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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.
We theoretically study the dynamics of typical optomechanical systems, consisting of a passive optical mode and an active mechanical mode, in the $mathcal{PT}$- and broken-$mathcal{PT}$-symmetric regimes. By fully analytical treatments for the dynamics of the average displacement and particle numbers, we reveal the phase diagram under different conditions and the various regimes of both $mathcal{PT}$-symmetry and stability of the system. We find that by appropriately tuning either mechanical gain or optomechanical coupling, both phase transitions of the $mathcal{PT}$-symmetry and stability of the system can be flexibly controlled. As a result, the dynamical behaviors of the average displacement, photons, and phonons are radically changed in different regimes. Our study shows that $mathcal{PT}$-symmetric optomechanical devices can serve as a powerful tool for the manipulation of mechanical motion, photons, and phonons.
The parity-time ($mathcal{PT}$) symmetric structures have exhibited potential applications in developing various robust quantum devices. In an optical trimmer with balanced loss and gain, we analytically study the $mathcal{PT}$ symmetric phase transition by investigating the spontaneous symmetric breaking. We also illustrate the single-photon transmission behaviors in both of the $mathcal{PT}$ symmetric and $mathcal{PT}$ symmetry broken phases. We find (i) the non-periodical dynamics of single-photon transmission in the $mathcal{PT}$ symmetry broken phase instead of $mathcal{PT}$ symmetric phase can be regarded as a signature of phase transition; and (ii) it shows unidirectional single-photon transmission behavior in both of the phases but comes from different underlying physical mechanisms. The obtained results may be useful to implement the photonic devices based on coupled-cavity system.
We explore the consequences of incorporating parity and time reversal ($mathcal{PT}$) symmetries on the dynamics of nonreciprocal light propagation exhibited by a class of nonuniform periodic structures known as chirped $mathcal{PT}$-symmetric fiber Bragg gratings (FBGs). The interplay among various grating parameters such as chirping, detuning, nonlinearities, and gain/loss gives rise to unique bi- and multi-stable states in the unbroken as well as broken $mathcal{PT}$-symmetric regimes. The role of chirping on the steering dynamics of the hysteresis curve is influenced by the type of nonlinearities and the nature of detuning parameter. Also, incident directions of the input light robustly impact the steering dynamics of bistable and multistable states both in the unbroken and broken $mathcal{PT}$-symmetric regimes. When the light launching direction is reversed, critical stable states are found to occur at very low intensities which opens up a new avenue for an additional way of controlling light with light. We also analyze the phenomenon of unidirectional wave transport and the reflective bi- and multi-stable characteristics at the so-called $mathcal{PT}$-symmetry breaking point.
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