We investigate the effects of a time-periodic, non-hermitian, PT-symmetric perturbation on a system with two (or few) levels, and obtain its phase diagram as a function of the perturbation strength and frequency. We demonstrate that when the perturbation frequency is close to one of the system resonances, even a vanishingly small perturbation leads to PT symmetry breaking. We also find a restored PT-symmetric phase at high frequencies, and at moderate perturbation strengths, we find multiple frequency windows where PT-symmetry is broken and restored. Our results imply that the PT-symmetric Rabi problem shows surprisingly rich phenomena absent in its hermitian or static counterparts.

http://arxiv.org/abs/1407.4535

Optics (physics.optics); Other Condensed Matter (cond-mat.other); Quantum Physics (quant-ph)

We investigate the parity- and time-reversal (PT)-symmetry breaking in lattice models in the presence of long-ranged, non-hermitian, PT-symmetric potentials that remain finite or become divergent in the continuum limit. By scaling analysis of the fragile PT threshold for an open finite lattice, we show that continuum loss-gain potentials \(V_a(x)\sim i|x|^a{\rm sign}(x)\) have a positive PT-breaking threshold for \(\alpha>−2\), and a zero threshold for α≤−2. When α<0 localized states with complex (conjugate) energies in the continuum energy-band occur at higher loss-gain strengths. We investigate the signatures of PT-symmetry breaking in coupled waveguides, and show that the emergence of localized states dramatically shortens the relevant time-scale in the PT-symmetry broken region.

http://arxiv.org/abs/1403.4204
Quantum Physics (quant-ph); Optics (physics.optics)

In systems with “balanced loss and gain”, the PT-symmetry is broken by increasing the non-hermiticity or the loss-gain strength. We show that finite lattices with oscillatory, PT-symmetric potentials exhibit a new class of PT-symmetry breaking and restoration. We obtain the PT phase diagram as a function of potential periodicity, which also controls the location complex eigenvalues in the lattice spectrum. We show that the sum of PT-potentials with nearby periodicities leads to PT-symmetry restoration, where the system goes from a PT-broken state to a PT-symmetric state as the average loss-gain strength is increased. We discuss the implications of this novel transition for the propagation of a light in an array of coupled waveguides.

http://arxiv.org/abs/1402.2544
Quantum Physics (quant-ph); Optics (physics.optics)

Over the last two decades, advances in fabrication have led to significant progress in creating patterned heterostructures that support either carriers, such as electrons or holes, with specific band structure or electromagnetic waves with a given mode structure and dispersion. In this article, we review the properties of light in coupled optical waveguides that support specific energy spectra, with or without the effects of disorder, that are well-described by a Hermitian tight-binding model. We show that with a judicious choice of the initial wave packet, this system displays the characteristics of a quantum particle, including transverse photonic transport and localization, and that of a classical particle. We extend the analysis to non-Hermitian, parity and time-reversal (\(\mathcal{PT}\)) symmetric Hamiltonians which physically represent waveguide arrays with spatially separated, balanced absorption or amplification. We show that coupled waveguides are an ideal candidate to simulate \(\mathcal{PT}\)-symmetric Hamiltonians and the transition from a purely real energy spectrum to a spectrum with complex conjugate eigenvalues that occurs in them.

http://arxiv.org/abs/1305.3565
Optics (physics.optics); Disordered Systems and Neural Networks (cond-mat.dis-nn); Quantum Physics (quant-ph)

In recent years, open systems with equal loss and gain have been investigated via their symmetry properties under combined parity and time-reversal (\(\mathcal{PT}\)) operations. We numerically investigate \(\mathcal{PT}\)-symmetry properties of an incompressible, viscous fluid with “balanced” inflow-outflow configurations. We define configuration-dependent asymmetries in velocity, kinetic energy density, and vorticity fields, and find that all asymmetries scale quadratically with the Reynolds number. Our proposed configurations have asymmetries that are orders of magnitude smaller than the asymmetries that occur in traditional configurations at low Reynolds numbers. Our results show that \(\mathcal{PT}\)-symmetric fluid flow configurations, which are defined here for the first time, offer a hitherto unexplored avenue to tune fluid flow properties.

http://arxiv.org/abs/1304.5348

Fluid Dynamics (physics.flu-dyn); Quantum Physics (quant-ph)

Recently, open systems with balanced, spatially separated loss and gain have been realized and studied using non-Hermitian Hamiltonians that are invariant under the combined parity and time-reversal (\(\mathcal{PT}\)) operations. Here, we model and investigate the effects of a local, two-state, quantum degree of freedom, called a pseudospin, on a one-dimensional tight-binding lattice with position-dependent tunneling amplitudes and a single pair of non-Hermitian, \(\mathcal{PT}\)-symmetric impurities. We show that if the resulting Hamiltonian is invariant under exchange of two pseudospin labels, the system can be decomposed into two uncoupled systems with tunable threshold for \(\mathcal{PT}\) symmetry breaking. We discuss implications of our results to systems with specific tunneling profiles, and open or periodic boundary conditions.

http://arxiv.org/abs/1302.4314
Quantum Physics (quant-ph); Optics (physics.optics)

We study the properties of an N-site tight-binding ring with parity and time-reversal (PT) symmetric, Hermitian, site-dependent tunneling and a pair of non-Hermitian, PT-symmetric, loss and gain impurities \(\pm i\gamma\). The properties of such lattices with open boundary conditions have been intensely explored over the past two years. We numerically investigate the PT-symmetric phase in a ring with a position-dependent tunneling function \(t_\alpha(k)=[k(N-k)]^{\alpha/2}\) that, in an open lattice, leads to a strengthened PT-symmetric phase, and study the evolution of the PT-symmetric phase from the open chain to a ring. We show that, generally, periodic boundary conditions weaken the PT-symmetric phase, although for experimentally relevant lattice sizes \(N \sim 50\), it remains easily accessible. We show that the chirality, quantified by the (magnitude of the) average transverse momentum of a wave packet, shows a maximum at the PT-symmetric threshold. Our results show that although the wavepacket intensity increases monotonically across the PT-breaking threshold, the average momentum decays monotonically on both sides of the threshold.

http://arxiv.org/abs/1207.1945
Quantum Physics (quant-ph); Optics (physics.optics)

We investigate the effects of competition between two complex, \(\mathcal{PT}\)-symmetric potentials on the \(\mathcal{PT}\)-symmetric phase of a “particle in a box”. These potentials, given by \(V_Z(x)=iZ\mathrm{sign}(x)\) and \(V_\xi(x)=i\xi[\delta(x-a)-\delta(x+a)]\), represent long-range and localized gain/loss regions respectively. We obtain the \(\mathcal{PT}\)-symmetric phase in the \((Z,\xi)\) plane, and find that for locations \(\pm a\) near the edge of the box, the \(\mathcal{PT}\)-symmetric phase is strengthened by additional losses to the loss region. We also predict that a broken \(\mathcal{PT}\)-symmetry will be restored by increasing the strength \(\xi\) of the localized potential. By comparing the results for this problem and its lattice counterpart, we show that a robust \(\mathcal{PT}\)-symmetric phase in the continuum is consistent with the fragile phase on the lattice. Our results demonstrate that systems with multiple, \(\mathcal{PT}\)-symmetric potentials show unique, unexpected properties.

http://arxiv.org/abs/1206.3310
Quantum Physics (quant-ph)

We investigate the properties of an \(N\)-site tight-binding lattice with periodic boundary condition (PBC) in the presence of a pair of gain and loss impurities \(\pm i\gamma\), and two tunneling amplitudes \(t_0,t_b\) that are constant along the two paths that connect them. We show that the parity and time-reversal PT-symmetric phase of the lattice with PBC is robust, insensitive to the distance between the impurities, and that the critical impurity strength for PT-symmetry breaking is given by \(\gamma_{PT}=|t_0-t_b|\). We study the time-evolution of a typical wave packet, initially localized on a single site, across the PT-symmetric phase boundary. We find that it acquires chirality with increasing \(\gamma\), and the chirality reaches a universal maximum value at the threshold, \(\gamma=\gamma_{PT}\), irrespective of the initial location of the wave packet or the lattice parameters. Our results imply that PT-symmetry breaking on a lattice with PBC has consequences that have no counterpart in open chains.

http://arxiv.org/abs/1203.1345
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas)

We investigate the effects of disorder on single particle time-evolution and two-particle correlations in an array of evanescently coupled waveguides with position-dependent tunneling rates. In the clean limit, the energy spectrum of such an array is widely tunable. In the presence of a Hermitian on-site or tunneling disorder, we find that the localization of a wave packet is highly sensitive to this energy spectrum. In particular, for an input confined to a single waveguide, we show that the fraction of light localized to the original waveguide depends on the tunneling profile. We compare the two-particle intensity correlations in the presence of Hermitian, tunneling disorder and non-Hermitian, parity-and-time-reversal \{\mathcal{PT}\} symmetric, on-site potential disorder. We show the two-particle correlation function in both cases is qualitatively similar, since both disorders preserve the particle-hole symmetric nature of the energy spectrum.

http://arxiv.org/abs/1112.4720
Quantum Physics (quant-ph); Optics (physics.optics)

By investigating a parity and time-reversal (PT) symmetric, $N$-site lattice with impurities \(\pm i\gamma\) and hopping amplitudes \(t_0 (t_b)\) for regions outside (between) the impurity locations, we probe the origin of maximal PT-symmetry breaking that occurs when the impurities are nearest neighbors. Through a simple and exact derivation, we prove that the critical impurity strength is equal to the hopping amplitude between the impurities, \(\gamma_c=t_b\), and the simultaneous emergence of \(N\) complex eigenvalues is a robust feature of any PT-symmetric hopping profile. Our results show that the threshold strength \(\gamma_c\) can be widely tuned by a small change in the global profile of the lattice, and thus have experimental implications.

http://arxiv.org/abs/1108.6083
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech)

Recently, waveguide lattices with non-uniform tunneling have been explored due to their myriad tunable properties, many of which arise from the extended nature of their eigenstates. Here, we investigate the dynamics, localization, and parity- and time-reversal-(PT) symmetry breaking in lattices with only localized eigenstates. We propose three families of tunneling profiles that lead to qualitatively different single-particle time evolution, and show that the effects of weak disorder contain signatures of the localized or extended nature of clean-lattice eigenstates. Our results suggest that waveguide lattices with only localized eigenstates will exhibit a wide array of phenomena that are absent in traditional systems.

http://arxiv.org/abs/1108.1402
Optics (physics.optics); Quantum Gases (cond-mat.quant-gas); Quantum Physics (quant-ph)

We investigate the robustness of parity- and time-reversal PT-symmetric phase in an N-site lattice with position-dependent, parity-symmetric hopping function and a pair of imaginary, PT-symmetric impurities. We find that the “fragile” PT-symmetric phase in these lattices is stronger than its counterpart in a lattice with constant hopping. With an open system in mind, we explore the degrees of broken PT symmetry and their signatures in single-particle wavepacket evolution. We predict that when the PT-symmetric impurities are closest to each other, the time evolution of a wavepacket in an even-N lattice is remarkably different from that in an odd-$N$ lattice. Our results suggest that PT-symmetry breaking in such lattices is accompanied by rich, hitherto unanticipated, phenomena.

http://arxiv.org/abs/1104.1666
Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas)

We investigate the eigenvalue distribution $\sigma(x)$ and level-spacing distribution $p(s)$ of random matrices $M=AF\neq M^{\dagger}$ where $F$ is a diagonal inner-product and $A$ is a random, real symmetric or complex Hermitian matrix with independent entries drawn from a probability distribution $q(x)$ with zero mean and finite higher moments. Although not Hermitian, the matrix $M$ is self-adjoint with respect to $F$ and thus has a purely real spectrum. We find that the eigenvalue probability distribution $\sigma_F(x)$ is independent of the underlying distribution $q(x)$, is solely characterized by $F$, and therefore generalizes Wigner’s semicircle distribution $\sigma_W(x)$. We find that the level-spacing distributions $p(s)$ are independent of $q(x)$, are dependent upon the inner-product $F$ and whether $A$ is real or complex, and therefore generalize Wigner’s surmise for level spacing.

http://arxiv.org/abs/1012.1202
Disordered Systems and Neural Networks (cond-mat.dis-nn); Statistical Mechanics (cond-mat.stat-mech)