Broken PT Research Articles

Proposal for Observing Yang-Lee Criticality in Rydberg Atomic Arrays.

Yang-Lee edge singularities (YLES) are the edges of the partition function zeros of an interacting spin model in the space of complex control parameters. They play an important role in understanding non-Hermitian phase transitions in many-body physics, as well as characterizing the corresponding nonunitary criticality. Even though such partition function zeroes have been measured in dynamical experiments where time acts as the imaginary control field, experimentally demonstrating such YLES criticality with a physical imaginary field has remained elusive due to the difficulty of physically realizing non-Hermitian many-body models. We provide a protocol for observing the YLES by detecting kinked dynamical magnetization responses due to broken PT symmetry, thus enabling the physical probing of nonunitary phase transitions in nonequilibrium settings. In particular, scaling analyses based on our nonunitary time evolution circuit with matrix product states accurately recover the exponents uniquely associated with the corresponding nonunitary CFT. We provide an explicit proposal for observing YLES criticality in Floquet quenched Rydberg atomic arrays with laser-induced loss, which paves the way towards a universal platform for simulating non-Hermitian many-body dynamical phenomena.

Fractionality and PT symmetry in a square lattice

We study the spectral stability of a 2D discrete Schr\"{o}dinger equation on a square lattice, in the simultaneous presence of a fractional Laplacian and $\cal{PT}$ symmetry. For that purpose, we compute the plane-wave spectrum in closed form, as a function of the gain/loss parameter and the fractional exponent. Examination of the spectrum reveals that an increase of the gain/loss parameter favors the early appearance of complex eigenvalues, thus is, the onset of a broken ${\cal PT}$ symmetry. On the other hand, as the fractional exponent decreases from unity, at a critical value a gap opens up separating the upper and lower bands, and the spectrum becomes real. Further decrease of the exponent increases the width of the gap and the system remains in the $\cal{PT}$-symmetric phase down to a vanishing value of the fractional exponent. Examination of the density of states and the participation ratio reinforce these observations and lead one to conclude that, unlike the standard, non-fractional case where the binary lattice is always in the broken $\cal{PT}$ phase, for the fractional case it is possible to have a symmetric $\cal{P}{\cal T}$ phase in the presence of a finite gain/loss parameter and a small enough fractional exponent.

PT-symmetry in mutually coupled semiconductor lasers with a non-zero time delay

Existence of PT symmetric regime in a system of two semiconductor lasers coupled face to face is being reported. The coupling involves the field of one laser coupled to the field of the other with a time delay. We investigate the occurrence of PT symmetry and PT transition for non-zero, but small delay times. Eigen values are obtained theoretically for the field transformation matrix formulated using normalized Lang–Kobayashi rate equations. We show that for certain delay times, the system exhibits PT symmetry and a PT transition to broken PT symmetry as the detuning frequency and coupling strength are varied. A control on the coupling coefficient and frequency detuning between the lasers is shown to be an efficient mechanism towards fine control of the PT symmetry in coupled lasers.

Nonlinear topological edge states in a non-Hermitian array of optical waveguides embedded in an atomic gas

We propose a scheme comprising an array of anisotropic optical waveguides, embedded in a gas of cold atoms, which can be tuned from a Hermitian to an odd-PT -- symmetric configuration through the manipulation of control and assistant laser fields. We show that the system can be controlled by tuning intra -- and inter-cell coupling coefficients, enabling the creation of topologically distinct phases and linear topological edge states. The waveguide array, characterized by a quadrimer primitive cell, allows for implementing transitions between Hermitian and odd-PT -symmetric configurations, broken and unbroken PT -symmetric phases, topologically trivial and nontrivial phases, as well as transitions between linear and nonlinear regimes. The introduced scheme generalizes the Rice-Mele Hamiltonian for a nonlinear non-Hermitian quadrimer array featuring odd-PT symmetry and makes accessible unique phenomena and functionalities that emerge from the interplay of non-Hermiticity, topology, and nonlinearity. We also show that in the presence of nonlinearity the system sustains nonlinear topological edge states bifurcating from the linear topological edge states and the modes without linear limit. Each nonlinear mode represents a doublet of odd-PT -conjugate states. In the broken PT phase, the nonlinear edge states may be effectively stabilized when an additional absorption is introduced into the system.

Quantized-Energy Equation for N-Level Atom in the Probability Representation of Quantum Mechanics

For Hermitian and non-Hermitian Hamiltonian matrices H, we present the Schr¨odinger equation for qudit (spin-j system, N-level atom) with the state vector |ψ〉 in a new form of the linear eigenvalue equation for the matrix ℋ = (H ⊗ 1N) and the probability eigenvector |p〉 identified with quantum states in the probability representation of quantum mechanics. We discuss the possibility to experimentally detect the difference between the system states described by the solutions, corresponding to the Schrodinger equation with Hermitian and non-Hermitian Hamiltonians, by measuring the probabilities of artificial spin-1/2 projections m = ± 1/2, sets of which are identified with qudit states. We show that different symmetries of systems, including 𝒫𝒯 -symmetry and broken 𝒫𝒯 -symmetry, are determined by a set of N complex eigenvalues of the Hamiltonian matrix H.

Floquet higher-order exceptional points and dynamics in PT-symmetric quadrimer waveguides

We investigate the effect of periodic modulation on PT-symmetry recovery, higher-order exceptional points (EPs) and dynamics for a periodically modulated PT-symmetric quadrimer waveguide system. It is analytically and numerically revealed that the periodic modulation not only can restore the broken PT symmetry, but also can manipulate the PT symmetry by tuning modulation amplitude. Furthermore, we found that multiple EPs can emerge, and as the modulation amplitude vary, these EPs can collide and merge, leading to higher-order Floquet singularities. By defining average relative light intensity distribution, we explore the dynamical characteristics nearby these EPs. It is found that we can control the light tunneling direction of optical quadrimer waveguide via periodic modulation and proper choice of gain/loss parameters.

PT-symmetric periodic structures with the modulation of the Kerr nonlinearity.

We study the PT-symmetric periodic layered structure with the modulated Kerr nonlinearity. We demonstrate that such systems can be transformed from the full to the broken PT symmetry or in the opposite direction. These transitions in the periodic structure with a finite length may be observed indirectly with the help of the dependence of the transmitted wave intensity on the incident wave intensity. Furthermore, the bistability properties depend on a gain and loss value and the choice of an input surface. The reduction of a feedback between counterpropagating waves is the main impact of the complex PT-symmetric refractive index variation on a bistability curve.

Superoscillatory PT -symmetric potentials

We introduce the one-dimensional PT-symmetric Schrodinger equation, with complex potentials in the form of the canonical superoscillatory and suboscillatory functions known in quantum mechanics and optics. While the suboscillatory-like potential always generates an entirely real eigenvalue spectrum, its counterpart based on the superoscillatory wave function gives rise to an intricate pattern of PT-symmetry-breaking transitions, controlled by the parameters of the superoscillatory function. One scenario of the transitions proceeds smoothly via a set of threshold values, while another one exhibits a sudden jump to the broken PT symmetry. Another noteworthy finding is the possibility of restoration of the PT symmetry, following its original loss, in the course of the variation of the parameters.

Vacuum induced transparency and photon number resolved Autler-Townes splitting in a three-level system

We study the absorption spectrum of a probe field by a Λ-type three-level system, which is coupled to a quantized control field through the two upper energy levels. The probe field is applied to the ground and the second excited states. When the quantized control field is in vacuum, we derive a threshold condition to discern vacuum induced transparency (VIT) and vacuum induced Autler-Townes splitting (ATS). We also find that the parameter changing from VIT to vacuum induced ATS is very similar to that from broken PT symmetry to PT symmetry. Moreover, we find the photon number resolved spectrum in the parameter regime of vacuum induced ATS when the mean photon number of the quantized control field is changed from zero (vacuum) to a finite number. However, there is no photon number resolved spectrum in the parameter regime of VIT even that the quantized control field contains the finite number of photons. Finally, we further discuss possible experimental realization.

Quantum work relations and response theory in parity-time-symmetric quantum systems.

In this work, we show that a universal quantum work relation for a quantum system driven arbitrarily far from equilibrium extends to a parity-time- (PT-) symmetric quantum system with unbroken PT symmetry, which is a consequence of microscopic reversibility. The quantum Jarzynski equality, linear response theory, and Onsager reciprocal relations for the PT-symmetric quantum system are recovered as special cases of the universal quantum work relation in a PT-symmetric quantum system. In the regime of broken PT symmetry, the universal quantum work relation does not hold because the norm is not preserved during the dynamics.

PT phase control in circular multi-core fibers.

We consider light dynamics in a circular multi-core fiber with balanced gain and loss core distribution, and show that transition from unbroken to broken PT phases can be conveniently controlled by geometric twist of the fiber. The twist introduces Peierls phases in the coupling constants and, thus, acts as an artificial gauge field. As an application, we discuss twist-induced tuning of optical transmission in a six-core fiber with one lossy core.

Origin of folded bands in metamaterial crystals

Recently it has been found numerically that the spectra of metamaterial crystals may contain pairs of bands which disappear inside the Brillouin zone. We observe that the wave equations for such systems are essentially non-Hermitian, but PT-symmetric. We show that the real-frequency spectra correspond to PT-symmetric solutions of the wave equation. At those momenta in the Brillouin zone where apparently no solutions exist, there appear pairs of complex-frequency solutions with spontaneously broken PT symmetry.

Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarff-II potentials.

We present a unified theoretical study of the bright solitons governed by self-focusing and defocusing nonlinear Schrödinger (NLS) equations with generalized parity-time- (PT) symmetric Scarff-II potentials. Particularly, a PT-symmetric k-wave-number Scarff-II potential and a multiwell Scarff-II potential are considered, respectively. For the k-wave-number Scarff-II potential, the parameter space can be divided into different regions, corresponding to unbroken and broken PT symmetry and the bright solitons for self-focusing and defocusing Kerr nonlinearities. For the multiwell Scarff-II potential the bright solitons can be obtained by using a periodically space-modulated Kerr nonlinearity. The linear stability of bright solitons with PT-symmetric k-wave-number and multiwell Scarff-II potentials is analyzed in detail using numerical simulations. Stable and unstable bright solitons are found in both regions of unbroken and broken PT symmetry due to the existence of the nonlinearity. Furthermore, the bright solitons in three-dimensional self-focusing and defocusing NLS equations with a generalized PT-symmetric Scarff-II potential are explored. This may have potential applications in the field of optical information transmission and processing based on optical solitons in nonlinear dissipative but PT-symmetric systems.

APICAL SEALING EFFICIENCY OF DIFFERENT OBTURATION TECHNIQUES OVER APICALLY BROKEN ROTARY NICKEL-TITANIUM FILES (AN IN VITRO STUDY)

Introduction: A broken instrument within the root canal is a panic problem which one should never wish to occur during routine endodontic therapy. However, the decision regarding how to manage this problem is complex. Therefore, the clinician must evaluate carefully the options of treatment; retrieving the instrument; bypassing the instrument; or leaving the fractured instrument in the canal as part of the final root canal filling. Objective: The objective of this study is to compare the sealing ability of three obturation techniques over apically broken nickel-titanium (Ni-Ti) ProTaper file. Materials and methods: Forty eight extracted human single-rooted mandibular premolars were prepared using ProTaper files (PT) and subjected to breakage of 3mm from F5 tip at the apical one third of the canal. The canals were randomly divided into three groups of 12 teeth each according to type of obturation technique; E&Q plus obturation, using a Thermafil obturator, or lateral compaction technique. The remaining 12 teeth served as positive controls which contained canals with apically broken files only without any type of obturation. Apical leakage was assessed by the dye penetration method in which all teeth were immersed in 2% methylene blue dye for 48 hours. Then the specimens were longitudinally split into two halves. Linear dye penetration was measured in millimeters under a stereomicroscope for each specimen. Statistical analysis of the results was performed using Kruskal-Wallis and post hoc tests. Results: The results showed no statistically significant difference in apical leakage between the three obturation techniques used over apically broken F5 files. Conclusions: All obturation techniques used over apically broken PT files played a significant role in the sealing ability

Jarzynski Equality in PT-Symmetric Quantum Mechanics.

We show that the quantum Jarzynski equality generalizes to PT-symmetric quantum mechanics with unbroken PT symmetry. In the regime of broken PT symmetry, the Jarzynski equality does not hold as also the CPT norm is not preserved during the dynamics. These findings are illustrated for an experimentally relevant system-two coupled optical waveguides. It turns out that for these systems the phase transition between the regimes of unbroken and broken PT symmetry is thermodynamically inhibited as the irreversible work diverges at the critical point.

Coherent Perfect Absorption in 𝓟𝓣 Symmetric Optical Lattice

In this work we study scattering in a 𝓟𝓣 symmetric periodic optical lattice, when two coherent beams are incident on it from two opposite directions. We focus our attention on both 𝓟𝓣 broken as well as 𝓟𝓣 unbroken phase, with special emphasis on spectral singularity. This is another example of an analytically solvable model, apart from the 𝓟𝓣 symmetric Scarf II potential discussed in ref. Ahmed (Phys. Rev. A 81, 022102 (2014)), which exhibits the phenomenon of Coherent Perfect Absorption (CPA) with lasing, in the domain of broken 𝓟𝓣 symmetry. At the same time, CPA is neither possible for unbroken 𝓟𝓣 symmetry, nor at the transition point.

Resonant mode conversion in the waveguides with unbroken and broken PT symmetry

We study resonant mode conversion in parity time (PT)-symmetric multimode waveguides, where symmetry breaking manifests itself in the sequential destabilization (i.e., the appearance of complex eigenvalues) of the pairs of adjacent guided modes. We show that efficient mode conversion is possible even in the presence of the resonant longitudinal modulation of the complex refractive index. The distinguishing feature of the resonant mode conversion in the PT-symmetric structure is a drastic growth of the width of the resonance curve when the gain/losses coefficient approaches a critical value, at which symmetry breaking occurs. We found that in the system with broken symmetry, the resonant coupling between the exponentially growing mode and the stable higher-order one effectively stabilizes dynamically coupled pairs of modes and remarkably diminishes the average rate of the total power growth.

Bound states in the continuum in PT-symmetric optical lattices

This Letter shows that bound states in the continuum (BIC), which are normalizable modes with energy embedded in the continuous spectrum of scattered states, exist in certain optical waveguide lattices with PT-symmetric defects. There are two distinct types of BIC modes: BIC states that exist in the broken PT phase, and correspond to exponentially localized modes with either exponentially damped or amplified optical power; and BIC modes with sub-exponential spatial localization that also can exist in the unbroken PT phase. These two types of BIC modes at the PT symmetry breaking point behave differently: in the former case, spatial localization is lost and the defect coherently radiates outgoing waves with an optical power that linearly increases with the propagation distance; in the latter case, localization is maintained and the optical power increase is quadratic.

Wavepacket Dynamics in a Non-Hermitian PT Symmetric Tight-Binding Chain

We examine the time development of a wavepacket in a Non-Hermitian PT symmetric tight-binding chain based on the Bethe ansatz solutions. The complete orthogonal sets in both unbroken and broken PT symmetry regions are established with respect to an inner product in terms of CPT conjugation. The dynamical development of a stationary wavepacket is exhibited, showing the impacts from additional imaginary potentials within unbroken and broken symmetric regimes. Our result indicates that there are distinct dynamic characteristics for both two regions. The unbroken symmetric imaginary end potentials can be regarded as a special kind of boundary condition in the sense of Hermitian quantum mechanics.

New quasi-exactly solvable sextic polynomial potentials

A Hamiltonian is said to be quasi-exactly solvable (QES) if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in closed form. An entirely new class of QES Hamiltonians having sextic polynomial potentials is constructed. These new Hamiltonians are different from the sextic QES Hamiltonians in the literature because their eigenfunctions obey PT-symmetric rather than Hermitian boundary conditions. These new Hamiltonians present a novel problem that is not encountered when the Hamiltonian is Hermitian: It is necessary to distinguish between the parametric region of unbroken PT symmetry, in which all of the eigenvalues are real, and the region of broken PT symmetry, in which some of the eigenvalues are complex. The precise location of the boundary between these two regions is determined numerically using extrapolation techniques and analytically using WKB analysis.