PT-symmetric Systems Research Articles

Entanglement of Pseudo-Hermitian Random States.

In a recent paper (A. Fring and T. Frith, Phys. Rev A 100, 101102 (2019)), a Dyson scheme to deal with density matrix of non-Hermitian Hamiltonians has been used to investigate the entanglement of states of a PT-symmetric bosonic system. They found that von Neumann entropy can show a different behavior in the broken and unbroken regime. We show that their results can be recast in terms of an abstract model of pseudo-Hermitian random matrices. It is found however that although the formalism is practically the same, the entanglement is not of Fock states but of Bell states.

PT-symmetric non-Hermitian quantum many-body system using ultracold atoms in an optical lattice with controlled dissipation

Abstract We report our realization of a parity–time (PT)-symmetric non-Hermitian many-body system using cold atoms with dissipation. After developing a theoretical framework on PT-symmetric many-body systems using ultracold atoms in an optical lattice with controlled dissipation, we describe our experimental setup utilizing one-body atom loss as dissipation with special emphasis on calibration of important system parameters. We discuss loss dynamics observed experimentally.

Spectra of PT-symmetric fractional Schrödinger equations with multiple quantum wells

We evaluate the eigenvalues of a type of one-dimensional PT-symmetric fractional Schrodinger equation with multiple quantum wells potential profile. By using a finite-difference scheme, we solve the fractional Schrodinger equation and present the algorithm. We study the effects of different parameters on the pairwise coalescence of eigenvalues. We show that by using the mentioned parameters, we can tune the position of the pairwise coalescence of the eigenvalues and the surface area between the two eigenvalues that intersect. An interesting phenomenon is that a small value of the fractionality as much as 0.15 can destroy the pairwise coalescence of eigenvalues and produce a single energy level. We also, consider the Hofstadter butterfly of a PT-symmetric one-dimensional system and show that by increasing the intensity of the potential imaginary part, we can kill the butterfly.

Enhanced sensitivity of optical gyroscope in a mechanical parity-time-symmetric system based on exceptional point

High-sensitivity gyroscope is widely used for rotation detection in several practical applications. Recently, exceptional points (EPs) have garnered considerable attention for enhancing the sensitivity of sensors based on optical cavities. Here, we propose an EP-enhanced optical gyroscope based on mechanical parity-time (PT) symmetry in a microcavity system. We demonstrate that by pumping the two optical modes with different colors, i.e., blue and red detuning, an effective mechanical PT-symmetric system can be obtained, and the system can be prepared at EP with appropriate parameters. The sensitivity of gyroscope at EP was enhanced by more than one order of magnitude in the weak perturbation regime as compared to that at diabolic point. This indicates that the sensitivity of gyroscope can be effectively enhanced by monitoring the mechanical modes rather than the optical modes. Overall, our work provides a promising approach to design high-sensitivity gyroscopes in optical microcavities and is potentially useful in a variety of research fields including fundamental physics and precision measurement.

Coexistence of a new type of bound state in the continuum and a lasing threshold mode induced by PT symmetry.

Some photonic systems support bound states in the continuum (BICs) that have infinite lifetimes, although their frequencies and momenta are matched to vacuum modes. Using a prototypical system that can be treated analytically, we show that each of these BICs always splits into a pair of new type BIC and lasing threshold mode when a parity-time (PT)-symmetric perturbation is introduced. The radiation loss at the lasing threshold is exactly balanced by the net gain of the particles. These PT symmetry-induced BICs are different from ordinary BICs, as they can be excited by an external source but do not radiate, and they carry a different quality factor divergence rate from that of the ordinary BICs. While most of the attention of PT-symmetric systems is captured by the coalescence of modes at exceptional points, the splitting of ordinary BICs is a new phenomenon that illustrates the rich physics embedded in PT-symmetric systems.

Efficient Quantum Simulation of an Anti-P-Pseudo-Hermitian Two-Level System.

Besides Hermitian systems, quantum simulation has become a strong tool to investigate non-Hermitian systems, such as PT-symmetric, anti-PT-symmetric, and pseudo-Hermitian systems. In this work, we theoretically investigate quantum simulation of an anti-P-pseudo-Hermitian two-level system in different dimensional Hilbert spaces. In an arbitrary phase, we find that six dimensions are the minimum to construct the anti-P-pseudo-Hermitian two-level subsystem, and it has a higher success probability than using eight dimensions. We find that the dimensions can be reduced further to four or two when the system is in the anti-PT-symmetric or Hermitian phase, respectively. Both qubit-qudit hybrid and pure-qubit systems are able to realize the simulation, enabling experimental implementations in the near future.

Bidirectional acoustic negative refraction based on a pair of metasurfaces with both local and global PT-symmetries

Negative refraction plays an important role in acoustic wave manipulation and imaging. However, conventional systems based on acoustic metamaterials suffer from the limits induced by loss-related and resolution issues. In this work, a parity-time (PT)-symmetric system is introduced to realize loss-free bidirectional acoustic negative refraction. The system is composed of a pair of locally PT-symmetric multi-layer metasurfaces sandwiching a region of free space, which also forms a global PT symmetry. The property of bidirectional negative refraction, which is rare for general PT-symmetric structures, is related to the coexistence of amplification and absorption in the locally PT-symmetric metasurfaces at their PT-broken phases. Such metasurfaces can freely switch their states between coherent perfect absorber (CPA) and amplifier depending on the direction of incidence. Our results provide a physical mechanism for realizing bidirectional functions in acoustic PT-symmetric systems.

Parity-time symmetry in wavelength space within a single spatial resonator

We show a parity-time (PT) symmetric microwave photonic system in the optical wavelength space within a single spatial resonator, in which the gain and loss modes can perfectly overlay spatially but are distinguishable in the designated parameter space. To prove the concept, a PT-symmetric optoelectronic oscillator (OEO) in the optical wavelength space is implemented. The OEO has a single-loop architecture, with the microwave gain and loss modes carried by two optical wavelengths to form two mutually coupled wavelength-space resonators. The operation of PT symmetry in the OEO is verified by the generation of a 10-GHz microwave signal with a low phase noise of −129.3 dBc/Hz at 10-kHz offset frequency and small sidemodes of less than −66.22 dBc/Hz. Compared with a conventional spatial PT-symmetric system, a PT-symmetric system in the wavelength space features a much simpler configuration, better stability and greater resilience to environmental interferences.

PT -symmetric chiral metamaterials: Asymmetric effects and PT -phase control

We investigate the influence of chirality on the PT-symmetric and PT-broken phase of PT-symmetric chiral systems. Starting from the point that transverse magnetic (TM) and transverse electric (TE) waves have different exceptional points, we show that with circularly polarized waves (which are linear combinations of TM and TE waves) mixed PT-symmetric phases can be realized and the extent of these phases can be highly controlled by either or both the chirality and the angle of incidence. Additionally, while the transmission of both TM and TE waves in non-chiral PT-symmetric systems is the same for forward and backward propagation, we show that with chirality this symmetry can be broken. As a result, it is possible to realize asymmetric, i.e. side-dependent, rotation and ellipticity in the polarization state of the transmitted wave. Our results constitute a simple example of a chiral PT-symmetric optical system in which the various phases (full PT, mixed, broken) and the asymmetric effects can be easily tuned by adjusting the chirality parameter and/or the angle of incidence.

Circuit Theory Analysis of Parity-Time-symmetric Wireless Power Transfer System

Wireless power transfer (WPT) technology reduces the risks brought by connection between electrical equipment and power source, it has been widely used in various fields in recent years. To overcome issues of low transfer efficiency and poor robustness when coupling coefficient varies, a WPT system based on parity-time (PT) symmetric circuit is proposed, which consists of two RLC oscillators. The system state equation is obtained by circuit theory, then be analyzed to derive the system resonance frequency, transmission efficiency, and phase difference between Tx and Rx. A simulation based on PSIM is established to verify the theoretical derivation of transmission characteristics. The simulation results illustrate that the resonance frequency of the WPT system is adjusted automatically in the strong coupling region when the coupling coefficient changes, the output voltage across the load resistance always equals source voltage on the transmitter. Compared with the non-parity-time symmetric system, PT-symmetric WPT system could achieve higher transfer efficiency over a longer distance, this scheme can transfer power with constant efficiency of over 80% in a certain region. A set of simulations with variation load resistance are considered to verify the system robustness. All results are consistent with theoretical derivation and analysis.Wireless power transfer (WPT) technology reduces the risks brought by connection between electrical equipment and power source, it has been widely used in various fields in recent years. To overcome issues of low transfer efficiency and poor robustness when coupling coefficient varies, a WPT system based on parity-time (PT) symmetric circuit is proposed, which consists of two RLC oscillators. The system state equation is obtained by circuit theory, then be analyzed to derive the system resonance frequency, transmission efficiency, and phase difference between Tx and Rx. A simulation based on PSIM is established to verify the theoretical derivation of transmission characteristics. The simulation results illustrate that the resonance frequency of the WPT system is adjusted automatically in the strong coupling region when the coupling coefficient changes, the output voltage across the load resistance always equals source voltage on the transmitter. Compared with the non-parity-time symmetric system, PT-symmetric WPT system could achieve higher transfer efficiency over a longer distance, this scheme can transfer power with constant efficiency of over 80% in a certain region. A set of simulations with variation load resistance are considered to verify the system robustness. All results are consistent with theoretical derivation and analysis.

Scattering Properties of PT-Symmetric Chiral Metamaterials

The combination of gain and loss in optical systems that respect parity–time (PT)-symmetry has pointed recently to a variety of novel optical phenomena and possibilities. Many of them can be realized by combining the PT-symmetry concepts with metamaterials. Here we investigate the case of chiral metamaterials, showing that combination of chiral metamaterials with PT-symmetric gain–loss enables a very rich variety of phenomena and functionalities. Examining a simple one-dimensional chiral PT-symmetric system, we show that, with normally incident waves, the PT-symmetric and the chirality-related characteristics can be tuned independently and superimposed almost at will. On the other hand, under oblique incidence, chirality affects all the PT-related characteristics, leading also to novel and uncommon wave propagation features, such as asymmetric transmission and asymmetric optical activity and ellipticity. All these features are highly controllable both by chirality and by the angle of incidence, making PT-symmetric chiral metamaterials valuable in a large range of polarization-control-targeting applications.

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.

QES solutions of a two-dimensional system with quadratic nonlinearities

We consider a one-parameter family of a PT symmetric two-dimensional system with quadratic nonlinearities. Such systems are shown to perform periodic oscillations due to existing centers. We describe this system by constructing a non-Hermitian Hamiltonian of a particle with position-dependent mass. We further construct a canonical transformation which maps this position-dependent mass system to a QES system. First few QES levels are calculated explicitly by using Bender–Dunne polynomial method.

Robust extended-range wireless power transfer using a higher-order PT-symmetric platform

This study introduces a real and constant eigenfrequency in the higher-order PT-symmetric electronic systems that can be used to realize a robust wireless power transfer platform without hopping the frequency of operation.

Synthetic Anti-PT Symmetry in a Single Microcavity.

Non-Hermitian systems based on parity-time (PT) and anti-PT symmetry reveal rich physics beyond the Hermitian regime. So far, realizations of such symmetric systems have been limited to the spatial domain. Here we theoretically and experimentally demonstrate synthetic anti-PT symmetry in a spectral dimension induced by nonlinear Brillouin scattering in a single optical microcavity, where Brillouin scattering induced transparency or absorption in two spectral resonances provides the optical gain and loss to observe a phase transition between two symmetry regimes. This scheme provides a new paradigm towards the investigation of non-Hermitian physics in a synthetic photonic dimension for all-optical signal processing and quantum information science.

Hartman effect from layered PT-symmetric system

The time taken by a wave packet to cross through a finite-layered PT-symmetric system is calculated by stationary phase method. We consider the PT-symmetric system of fix spatial length L consisting of N units of the potential system ‘$${+}\,iV$$’ and ‘$${-}\,iV$$’ of equal width ‘b’ such that $$L=2Nb$$. In the limit of large ‘b’, the tunneling time is found to be independent of L and therefore, the layered PT-symmetric system displays the Hartman effect. The interesting limit of $$N \rightarrow \infty $$ such that L remains finite is investigated analytically. In this limit, the tunneling time matches with the time taken to cross an empty space of length L. The result of this limiting case $$N \rightarrow \infty $$ also shows the consistency of phase space method of calculating the tunneling time, despite the existence of controversial Hartman effect. The reason of Hartman effect is unknown to present day; however, the other definitions of tunneling time that indicate a delay which depends upon the length of traversing region have been effectively ruled out by recent attosecond measurements.

RIGOROUS QUANTUM FORMULATION OF PARITY-TIME SYMMETRIC COUPLED RESONATORS

Rigorous quantum formulation of the Parity-Time (PT) symmetry phenomenon in the RF/microwave regime for a coupled coil resonators with lump elements has been presented. The coil resonator is described by the lump-element model that consists of an inductor (L), a resistor (R) and a capacitor (C). Rigorous quantum Hamiltonian for the coupled LRC coil resonators system has been derived through twice basis transforms of the original basis. The first basis transform rotates the original basis such that off-diagonal terms of the governing matrix of the equation system of the coupled coil resonators reduces to constants. Then a second basis transform obtains the quantum Hamiltonian, including the diagonal effective complex frequencies and the off-diagonal coupling terms, together with the transformed basis. With the obtain quantum Hamiltonian, the eigenvalues and eigenvectors of the coupled coil resonators can be obtained as usual as the quantum Hamiltonian. Finally, numerical simulation verifies the correctness of the theory. The quantum formulation of the coupled coil resonators can provide better guideline to design a better PT-symmetric system.

Parity-time symmetry in a single-loop photonic system

Non-Hermitian photonics based on parity-time (PT) symmetry has been considered an effective solution to achieve mode selection for optical or microwave single-mode oscillation. A PT-symmetric system is usually implemented using two mutually coupled elements with identical geometry or a single element based on precise longitudinal refractive index modulation. In this study, we propose a novel scheme to achieve PT symmetry based on polarization mode manipulation in a single optical loop. By controlling the polarization states of two light waves that are propagating along the clockwise (CW) and counter-clockwise (CCW) directions in the single optical loop, two equivalent mutually-coupled loops with identical geometry and a balanced gain and loss are formed. Based on this concept, a wavelength-tunable single-mode fiber laser with a sub-kHz linewidth is realized. The tunability is achieved by thermally tuning the wavelength of an integrated microdisk resonator (MDR) incorporated in the optical loop. Experimental results show that a continuously tunable single-mode laser with a sub-kHz linewidth of 640 Hz and a wavelength tuning range from 1552.953 to 1554.147 nm is realized. The key advantages of using a single optical loop to implement PT symmetry include greatly simplified implementation and highly improved stability. The demonstration opens new avenues to implement high-performance PT-symmetric systems for applications in photonic and microwave photonic systems.

Mechanical Exceptional-Point-Enhanced Second-Order Sideband Generation

Realizing the enhancement and control of the optomechanical induced second-order sideband generation is of great significance in making a low-power optomechaical amplifier and highly sensitive sensor. Here we analyze theoretically light transmission obtained from a hybrid optomechanical system composed of an optical cavity and a pair of directly coupled PT-symmetric mechanical resonators, and in which the cavity is coherently driven by a bichromatic input field consisting of a strong control field and a weak probe field and that works within the perturbative regime. Using experimentally achievable parameters, we show that the transmission of the probe field changes from single to double transparency window via the transition from a broken mechanical PT-symmetric phase to an unbroken mechanical PT-symmetric phase, correspondingly, the second-order sideband generation can be also split from one frequency center into two frequency centers. Especially, when the assisted mechanical PT-symmetric systems reach the vicinity of the exceptional point, the efficiency of second-order generation can be significantly enhanced, compared to the conventional optomechanical system, what can be improved about four orders of magnitude, which is a manifestation of the unique property of mechanical PT-symmetric systems. The proposed mechanism of reaching new levels of second-order sideband generation for optomechanical systems, offers a promising route towards achieving the multimode higher-order sidebands generation and controlling optical frequency combs with an integrable structure.

General stationary solutions of the nonlocal nonlinear Schrödinger equation and their relevance to the PT-symmetric system.

With the stationary solution assumption, we establish the connection between the nonlocal nonlinear Schrödinger (NNLS) equation and an elliptic equation. Then, we obtain the general stationary solutions and discuss the relevance of their smoothness and boundedness to some integral constants. Those solutions, which cover the known results in the literature, include the unbounded Jacobi elliptic-function and hyperbolic-function solutions, the bounded sn-, cn-, and dn-function solutions, as well as the hyperbolic soliton solutions. By the imaginary translation transformation of the NNLS equation, we also derive the complex-amplitude stationary solutions, in which all the bounded cases obey either the PT- or anti-PT-symmetric relation. In particular, the complex tanh-function solution can exhibit no spatial localization in addition to the dark- and antidark-soliton profiles, which is in sharp contrast with the common dark soliton. Considering the physical relevance to the PT-symmetric system, we show that the complex-amplitude stationary solutions can yield a wide class of complex and time-independent PT-symmetric potentials, and the symmetry breaking does not occur in the PT-symmetric linear system with the associated potentials.