Parity-time ($\mathcal{PT}$) symmetry is one of the most important accomplishments in optics over the past decade. Here the concept of $\mathcal{PT}$ mode-locking of a laser is introduced, in which active phase locking of cavity axial modes is realized by asymmetric mode coupling in a complex time crystal. $\mathcal{PT}$ mode-locking shows a transition from single to double pulse emission as the $\mathcal{PT}$ symmetry breaking point is crossed. The transition can show a turbulent behavior, depending on a dimensionless modulation parameter that plays the same role as the Reynolds number in hydrodynamic flows.
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…
Mode locking in lasers is a collective effect, where due to a weak coupling a large number of frequency modes lock their phases to oscillate in unison, forming an ultrashort pulse in time. We demonstrate an analogous collective effect in coupled parametric oscillators, which we term "pairwise mode-locking", where many pairs of modes with twin frequencies (symmetric around the center carrier) oscillate simultaneously with a locked phase-sum, while the phases of individual modes remain undefined. Thus, despite being broadband and multimode, the emission is not pulsed and lacks first-order cohere…
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 $…
The parity-time ($\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 $\mathcal{PT}$ symmetr…
Non-Hermitian systems with anti-parity-time ($\mathcal{APT}$) symmetry have revealed rich physics beyond conventional systems. Here, we study optomechanics in an $\mathcal{APT}$-symmetric spinning resonator and show that, by tuning the rotating speed to approach the exceptional point (EP) or the non-Hermitian spectral degeneracy, nonreciprocal light transmission with a high isolation ratio can be realized. Accompanying this process, nonreciprocal group delay or advance is also identified in the vicinity of EP. Our work sheds new light on manipulating laser propagation with optomechanical EP de…
Phase transitions in Wick-rotated $\mathcal{PT}$-symmetric optics
$\mathcal{PT}$ -对称模型与威克旋转的时间（$ t \rightarrow \pm i t$）显示光谱相变，是类似的耗散系统驱动的平衡。光学可以提供一个可访问的测试床，以探索威克旋转$\mathcal{PT}$对称模型的光谱相变。这是通过考虑激光在具有可变反射率和倾斜反射镜的光学腔中的横向动力学而示出的。作为例子，我们详细讨论了流体动力学的Squire涡量模型和Wick旋转的$\mathcal{PT}$对称非线性二聚体模型的光学模拟。
We consider wave transport phenomena in a $\mathcal{PT}$-symmetric extension of the periodically-kicked quantum rotator model and reveal that dynamical localization assists the unbroken $\mathcal{PT}$ phase. In the delocalized (quantum resonance) regime, $\mathcal{PT}$ symmetry is always in the broken phase and ratchet acceleration arises as a signature of unidirectional non-Hermitian transport. An optical implementation of the periodically-kicked $\mathcal{PT}$-symmetric Hamiltonian, based on transverse beam propagation in a passive optical resonator with combined phase and loss gratings, is …
In this letter we discuss a phase transition-like anomalous behavior of Faraday rotation angles in a simple parity-time ($\mathcal{P}\mathcal{T}$) symmetric model with two complex $δ$-potential placed at both boundaries of a regular dielectric slab. In anomalous phase, the value of one of Faraday rotation angles may turn negative, and both angles suffer spectral singularities and yield strong enhancement near singularities.
We review the generalization of tunneling time and anomalous behaviour of Faraday and Kerr rotation angles in parity and time ($\mathcal{P}\mathcal{T}$)-symmetric systems. Similarities of two phenomena are discussed, both exhibit a phase transition-like anomalous behaviour in certain range of model parameters. Anomalous behaviour of tunneling time and Faraday/Kerr angles in $\mathcal{P}\mathcal{T}$-symmetric systems is caused by the motion of poles of scattering amplitudes in energy/frequency complex plane.
We study a periodically driven qubit coupled to a quantized cavity mode. Despite its apparent simplicity, this system supports a rich variety of exotic phenomena, such as topological frequency conversion as recently discovered in [Martin et al, PRX 7, 041008 (2017)]. Here we report on a qualitatively different phenomenon that occurs in this platform, where the cavity mode's oscillations lock their frequency to a rational fraction $r/q$ of the driving frequency $Ω$. This phenomenon, which we term quantum frequency locking, is characterized by the emergence of $q$-tuplets of stationary (Floquet)…
Lattice models with non-hermitian, parity and time-reversal ($\mathcal{PT}$) symmetric Hamiltonians, realized most readily in coupled optical systems, have been intensely studied in the past few years. A $\mathcal{PT}$-symmetric dimer lattice consists of dimers with intra-dimer coupling $ν$, inter-dimer coupling $ν'$, and balanced gain and loss potentials $\pm iγ$ within each dimer. This model undergoes two independent transitions, namely a $\mathcal{PT}$-breaking transition and a topological transition. We numerically and analytically investigate the signatures of these transitions in the tim…
The recent reports of the replica symmetry breaking (RSB) phenomenon in photonic experiments [1-5] boosted the understanding of the role of disorder in multimode lasers, as well as helped to settle enlightening connections [6-13] with the statistical physics of complex systems. RSB manifests when identically-prepared system replicas reach distinct states, yielding different measures of observable quantities [14]. Here we demonstrate the RSB in the spontaneous mode-locking regime of a conventional multimode Nd:YAG laser in a closed cavity. The underlying mechanism is quite distinct from that of…
The nonlinear dynamics of a balanced parity-time symmetric optical microring arrangement are analytically investigated. By considering gain and loss saturation effects, the pertinent conservation laws are explicitly obtained in the Stokes domain-thus establishing integrability. Our analysis indicates the existence of two regimes of oscillatory dynamics and frequency locking, both of which are analogous to those expected in linear parity-time symmetric systems. Unlike other saturable parity time symmetric systems considered before, the model studied in this work first operates in the symmetric …
We investigate $\mathcal{PT}$-symmetric coherent perfect absorbers (CPAs) in the TE mode solution of a linear homogeneous optical system surrounded by graphene sheets. It is revealed that presence of graphene sheets contributes the enhancement of absorption in a coherent perfect absorber. We derive exact analytic expressions, and work through their possible impacts on lasing threshold and CPA conditions. We point out roles of each parameter governing optical system with graphene and show that optimal conditions of these parameters give rise to enhancement and possible experimental realization …
We study light propagation through cyclic arrays, composed by copies of a given $\mathcal{PT}$-symmetric dimer, using a group theoretical approach and finite element modeling. The theoretical mode-coupling analysis suggest the use of these devices as output port replicators where the dynamics are controlled by the impinging light field. This is confirmed to good agreement with finite element propagation in an experimentally feasible necklace of passive $\mathcal{PT}$-symmetric dimers constructed from lossy and lossless waveguides.
A potential scattering theory from deterministic and random $\mathcal{PT}$ collections of particles with gain and loss is introduced and the forms of their structure and pair-structure factors are elucidated. An example relating to light scattering from a random distribution of a pair of particles with gain and loss is considered.
In this work, we study the transmission properties of one dimensional finite periodic systems with $\mathcal{PT}$-symmetry. A simple closed form expression is obtained for the total transmittance from a lattice of N cells, that allows us to describe the transmission minima (maxima) when the system is in the $\mathcal{PT}$-unbroken (broken) phase. Utilizing this expression, we provide the necessary conditions, \textit{independent} of the number of cells, for the occurrence of a CPA-laser for any finite $\mathcal{PT}$-symmetric periodic potential. Under this conditions, we provide a recipe for b…
Unidirectional reflectionless propagation (or transmission) is an interesting wave phenomenon observed in many $\mathcal{PT}$-symmetric optical structures. Theoretical studies on unidirectional reflectionless transmission often use simple coupled-mode models. The coupled-mode theory can reveal the most important physical mechanism for this wave phenomenon, but it is only an approximate theory, and it does not provide accurate quantitative predictions with respect to geometric and material parameters of the structure. In this paper, we rigorously study unidirectional reflectionless transmission…
We theoretically demonstrate soliton steering in $\mathcal{PT}$-symmetric coupled nonlinear dimers. We show that if the length of the $\mathcal{PT}$-symmetric system is set to $2π$ contrary to the conventional one which operates satisfactorily well only at the half-beat coupling length, the $\mathcal{PT}$ dimer remarkably yields an ideal soliton switch exhibiting almost 99.99% energy efficiency with an ultra-low critical power.
Families of coupled solitons of $\mathcal{PT}$-symmetric physical models with gain and loss in fractional dimension and in settings with and without cross-interactions modulation (CIM), are reported. Profiles, powers, stability areas, and propagation dynamics of the obtained $\mathcal{PT}$-symmetric coupled solitons are investigated. By comparing the results of the models with and without CIM, we find that the stability area of the model with CIM is much broader than the one without CIM. Remarkably, oscillating $\mathcal{PT}$-symmetric coupled solitons can also exist in the model of CIM with t…
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 nonl…
We theoretically investigate the solitary waves and their switching dynamics in a $\mathcal{PT}$-symmetric directional fiber coupler, exhibiting Kerr nonlinearity, by developing a variational analysis. We analyze the fundamental switching characteristics of the $\mathcal{PT}$-symmetric Kerr coupler in the picosecond timescale by considering the coupled-mode equation for the unperturbed nonlinear Schrödinger equation, which we compare to its conventional counterpart. The impacts of higher-order perturbations (intrapulse Raman scattering, self-steepening, and third-order dispersion) are investig…
We study collisions of moving nonlinear-Schrödinger solitons with a $\mathcal{PT}$-symmetric dipole embedded into the one-dimensional self-focusing or defocusing medium. Accurate analytical results are produced for bright solitons, and, in a more qualitative form, for dark ones. In the former case, an essential aspect of the approximation is that it must take into regard the intrinsic chirp of the soliton, thus going beyond the framework of the simplest quasi-particle description of the soliton's dynamics. Critical velocities separating reflection and transmission of the incident bright solito…
We study spectral and transport properties of one-dimensional tight-binding $\mathcal{PT}$-symmetric chains with alternating couplings. Based on the transfer matrix method, we have analytically developed the expressions for the transmission and reflection coefficients for any values of control parameters. These expressions are obtained in a very compact form which separately imbed the generic energy dependence valid for any periodic structure, as well as specific properties of a unit cell composing the scattering setup. Out main interest is in specific properties of the left/right reflections …
We consider $\mathcal{PT}$-symmetric ring-like arrays of optical waveguides with purely nonlinear gain and loss. Regardless of the value of the gain-loss coefficient, these systems are protected from spontaneous $\mathcal{PT}$-symmetry breaking. If the nonhermitian part of the array matrix has cross-compensating structure, the total power in such a system remains bounded -- or even constant -- at all times. We identify two-, three-, and four-waveguide arrays with cross-compensatory nonlinear gain and loss that constitute completely integrable Hamiltonian systems.
