Balanced Gain Research Articles

Non-Hermitian disorder-driven topological transition in a dimerized Kitaev superconductor chain

Recently, topological phases in non-Hermitian systems have attracted great attention. In most of these works, disorders are added to the chemical potential or hopping term and the topological transition driven by disordered loss and gain has not been found. In this paper, we investigate the one-dimensional dimerized Kitaev superconductor chain with balanced loss and gain. In the clean limit, the phase diagram is given by calculating the Bloch winding number and real-space energy spectrum. Furthermore, by calculating the disorder-averaged real-space winding number and disorder-averaged real-space energy spectrum, we identify a topological transition driven by the disordered balanced loss and gain. The disorder-averaged inverse participate ratio of each eigenstate, mean inverse participate ratio, and density distribution of the edge mode are also calculated to support the existence of the topological Anderson phase. Finally, we compare our model with the Hermitian disordered Kitaev chain and discuss the possible experimental scheme of our model. Our work offers a minimal start point to explore the interplay between the dissipation and Majorana zero modes in the topological superconductor.

Bounds for the extremal eigenvalues of gain Laplacian matrices

A complex unit gain graph (T-gain graph), Φ=(G,φ) is a graph where the function φ assigns a unit complex number to each orientation of an edge of G, and its inverse is assigned to the opposite orientation. A T-gain graph Φ is balanced if the product of the edge gains of each cycle (with a fixed orientation) is 1. Signed graphs are special cases of T-gain graphs.The adjacency matrix of Φ, denoted by A(Φ) is defined canonically. The gain Laplacian for Φ is defined as L(Φ)=D(Φ)−A(Φ), where D(Φ) is the diagonal matrix with diagonal entries are the degrees of the vertices of G. The minimum number of vertices (resp., edges) to be deleted from Φ in order to get a balanced gain graph is the frustration number (resp, frustration index). We show that frustration number and frustration index are bounded below by the smallest eigenvalue of L(Φ). We provide several lower and upper bounds for extremal eigenvalues of L(Φ) in terms of different graph parameters such as the number of edges, vertex degrees, and average 2-degrees. The signed graphs are particular cases of the T-gain graphs, all the bounds established in this paper hold for signed graphs. Most of the bounds established here are new for signed graphs. Finally, we perform comparative analysis for all the obtained bounds in the paper with the state-of-the-art bounds available in the literature for randomly generated Erdős-Reýni graphs.Some of the major highlights of our paper are the gain-dependent bounds, limit convergence of the bounds to the extremal eigenvalues, and optimal extremal bounds obtained by posing optimization problems to achieve the best possible bounds.

Scattering of partially coherent vortex beams by a $\mathcal {PT}$-symmetric dipole.

We investigated the statistical properties of partially coherent optical vortex beams scattered by a $\mathcal {PT}$ dipole, consisting of a pair of point particles having balanced gain and loss. The formalism of second-order classical coherence theory is adopted, together with the first Born approximation, to obtain the cross-spectral density of the scattered field. It is shown that the radiated pattern depends strongly on the coherence properties of the incident beam and on the non-Hermitian properties of the dipole. The spectral density for the scattered radiation is ruled by two terms, one associated to the vortex structure and the other independent of the topological charge, and the competition between these terms dictates the directional properties of the scattered radiation. When they have same order of magnitude, the scattered profile resembles that of an incoherent system, with radiation being emitted in all directions in the three-dimensional space, regardless of the dipole's gain and loss properties. Depending on the gain and loss present in the dipole, the system may scatter light in some preferable directions. All of these effects are accompanied by a change in the spectral degree of coherence of the scattered field.

Constructing solvable models of vector non-linear Schrödinger equation with balanced loss and gain via non-unitary transformation

We consider vector Non-linear Schrödinger Equation (NLSE) with balanced loss-gain (BLG), linear coupling (LC) and a general form of cubic nonlinearity. We use a non-unitary transformation to show that the system can be exactly mapped to the same equation without the BLG and LC, and with a modified time-modulated nonlinear interaction. The nonlinear term remains invariant, while BLG and LC are removed completely, for the special case of a pseudo-unitary transformation. The mapping is generic and may be used to construct exactly solvable autonomous as well as non-autonomous vector NLSE with BLG. We present an exactly solvable two-component vector NLSE with BLG which exhibits power-oscillation. An example of a vector NLSE with BLG and arbitrary even number of components is also presented.

Pseudo-Hermitian trimer: Asymmetric resonances and exceptional points

A pseudo-Hermitian (PH) system made up of three coupled waveguides (trimer) with an overall balanced gain and loss is investigated in a parity time (PT)-symmetric-like regime. Remarkably, the pseudo hermiticity parameter controls the asymmetric spatial field modulation between left and right propagating waves whereby, a complex coupling arises that induces an additional source of non-Hermiticity, which in the system acts as a relative gain/loss Hellmann-Feynman correction. In this situation, we show that, transmissionless backscatterings are observed for the PH-trimer in the non PT broken phase. Interestingly, the scattering properties of PH-trimer result in multiple lasing states with different mechanisms. In particular, we show that in the PT-symmetric case, the lasing modes associated to the coherent perfect-absorption (CPA), occur from the spectral singularity induced by the Hermitian chain coupling and give rise to unidirectional invisibility in the direction of incident wave. On the contrary, additional emerging lasing modes in a PH regime result from the scattering maxima resonance. We demonstrate that this specific cases suppress the CPA-laser phenomenon and rather exhibit exceptional points characterized by reflectionless but also the points of transmissionless. Numerical beam propagations realized to emphasize the results are in good agreement with the scattering investigation until gain/loss parameter is around it’s value giving resonance.

Evolution of exceptional points and anisotropic transmission resonance phenomenon for one-dimensional local resonant phononic crystal

In this study, a spring–mass physical model is proposed to study the complex band structure of a one-dimensional parity-time (PT)-symmetric local resonant phononic crystal. By solving the kinetic equations, the analytical solutions of the dispersion relation and effective mass are obtained. As is known, the infinite effective mass would appear at the resonant frequency in a Hermitian system without any gain or loss. Once the balanced gain and loss are added to form a PT-symmetric system, the infinite effective mass would become finite, and the exceptional points can be observed in the subwavelength realm. With the increase in gain and loss, exceptional points would coalesce and form a higher order one. The numerical simulations in a practical structure agree well with the analytical analysis. In addition, the simulated transmission/reflection spectrum and field distribution clearly demonstrate the anisotropic transmission resonances. Our investigation enriches the physical connotation of local resonant phononic crystals in non-Hermitian systems.

Parity-time symmetric systems with memory

Classical open systems with balanced gain and loss, i.e. parity-time ($\mathcal{PT}$) symmetric systems, have attracted tremendous attention over the past decade. Their exotic properties arise from exceptional point (EP) degeneracies of non-Hermitian Hamiltonians that govern their dynamics. In recent years, increasingly sophisticated models of $\mathcal{PT}$-symmetric systems with time-periodic (Floquet) driving, time-periodic gain and loss, and time-delayed coupling have been investigated, and such systems have been realized across numerous platforms comprising optics, acoustics, mechanical oscillators, optomechanics, and electrical circuits. Here, we introduce a $\mathcal{PT}$-symmetric (balanced gain and loss) system with memory, and investigate its dynamics analytically and numerically. Our model consists of two coupled $LC$ oscillators with positive and negative resistance, respectively. We introduce memory by replacing either the resistor with a memristor, or the coupling inductor with a meminductor, and investigate the circuit energy dynamics as characterized by $\mathcal{PT}$-symmetric or $\mathcal{PT}$-symmetry broken phases. Due to the resulting nonlinearity, we find that energy dynamics depend on the sign and strength of initial voltages and currents, as well as the distribution of initial circuit energy across its different components. Surprisingly, at strong inputs, the system exhibits self-organized Floquet dynamics, including $\mathcal{PT}$-symmetry broken phase at vanishingly small dissipation strength. Our results indicate that $\mathcal{PT}$-symmetric systems with memory show a rich landscape.

Low coherence-induced resonance in double-layer structures having parity-time symmetry.

We derive simple formulas for the transmittance T and reflectance R of Gaussian-Schell beams incident upon any stratified dielectric structure by using second-order classical coherence theory in the space-frequency picture. The formalism is applied to a particular structure consisting of a double layer, with balanced gain and loss, satisfying parity-time symmetry conditions. It is shown that sources with a low degree of spatial coherence, on the order of the wavelength, can induce large resonant peaks in the transmitted and reflected amplitudes. The resonance peaks vanish as the spatial coherence increases.

Pseudo‐Hermitian Systems Constructed by Transformation Optics with Robustly Balanced Loss and Gain

Pseudo-Hermitian Systems In article number 2000081, Yun Lai, Jie Luo, and co-workers propose an optical analog of quantum field theory of vacuum, where virtual particles are created and annihilated. Based on transformation optics, optical loss and gain are separated in physical space, forming a new class of non-Hermitian systems with unprecedented robustness in energy balance, thus denoted as pseudo-Hermitian systems.

Exceptional points and enhanced sensitivity in PT-symmetric continuous elastic media

We investigate non-Hermitian degeneracies, also known as exceptional points, in continuous elastic media, and their potential application to the detection of mass and stiffness perturbations. Degenerate states are induced by enforcing parity-time symmetry through tailored balanced gain and loss, introduced in the form of complex stiffnesses and may be implemented through piezoelectric transducers. The introduction of external perturbations leads to a splitting of the eigenvalues, which is explored as a sensitive approach to the detection of such perturbations. Numerical simulations on one-dimensional waveguides illustrate the presence of several exceptional points in their vibrational spectrum, and conceptually demonstrate their sensitivity to point mass inclusions. Second order exceptional points are shown to exhibit a frequency shift in the spectrum with a square root dependence on the perturbed mass, which is confirmed by a perturbation approach and by frequency response analyses. Elastic domains supporting guided waves are then investigated, where exceptional points are formed by the hybridization of Lamb wave modes. After illustrating a similar sensitivity to point mass inclusions, we also show how these concepts can be applied to surface wave modes for sensing crack-type defects. The presented results describe fundamental vibrational properties of PT-symmetric elastic media supporting exceptional points, whose sensitivity to perturbations goes beyond the linear dependency commonly encountered in Hermitian systems. The findings are thus promising for applications involving sensing of perturbations such as added masses, stiffness discontinuities and surface cracks.

Pseudo‐Hermitian Systems Constructed by Transformation Optics with Robustly Balanced Loss and Gain

Non‐Hermitian systems with parity‐time symmetry are found to exhibit real spectra of eigenvalues, indicating a balance between the loss and gain. However, such a balance is not only dependent on the magnitude of loss and gain but also easily broken due to external disturbance. Herein, a transformation‐optics approach is proposed to construct a unique class of non‐Hermitian systems with robustly balanced loss and gain, irrespective of the magnitude of loss/gain and the environmental disturbance. Through transformation‐optics operators such as space folding and stretching, loss and gain can be generated and separated in the real space. However, in the virtual space, the loss and gain are still combined to each other, rendering a balance of energy that is far more robust than other non‐Hermitian systems. This amazing feature is verified by finite‐element simulations. This work reveals a class of non‐Hermitian systems in which loss and gain are balanced robustly, thereby denoted as pseudo‐Hermitian systems.

Widely Wavelength-Tunable Parity-Time Symmetric Single-Longitudinal-Mode Fiber Ring Laser With a Single Physical Loop

A wavelength-tunable single-longitudinal-mode fiber ring laser based on parity-time (PT) symmetry for mode selection is proposed and experimentally demonstrated. The PT symmetry is implemented using a single physical fiber-optic loop, which functions equivalently to two mutually coupled optical loops with an identical geometry but balanced gain and loss. In the proposed system, a linearly polarized light wave is split into two, which are traveling in a Sagnac loop incorporating a polarization beam splitter (PBS) and two polarization controllers (PCs). By controlling the polarization states of the light waves injected into the PBS via tuning the PCs, the gain and loss of the two equivalent loops corresponding to the light waves traveling in the clockwise (CW) and counter-clockwise (CCW) directions are controlled. When the gain and loss are balanced and exceed the coupling coefficient, PT-symmetry breaking condition is met, and single-longitudinal-mode lasing is achieved. The proposed PT-symmetric fiber ring laser is experimentally demonstrated. Single-longitudinal-mode lasing with a tunable wavelength from 1530 to 1565 nm and a narrow linewidth as small as 390 Hz is achieved. The sidemode suppression ratio is 41.9 dB confirming the effectiveness of using PT symmetry for mode selection.

Forward kinematic modeling of fault-bend folding

We present a forward numerical modeling approach for fault-bend folding based on a velocity description of deformation. The approach incorporates algorithms capable of modeling multiple fault bends of different geometries (e. g. fault bends not stepping up from a detachment), imbricates, and variable velocity-boundary orientations, with corresponding varying slip ratios. When modeling contraction, the approach is capable of reproducing rounded-hinges and parallel folds with localized bed thinning or thickening commonly observed in natural structures. Extensional fault-bend folds can be modeled using the same set of equations, with the minor modification that velocity boundary orientations are defined independently of the fault shape. The modeled structures conserve area, and commonly observed features of extensional fault-bend folds, such as rollover structures with growth, are produced. Thus, we present a unified inclined-shear and flexural-slip general transformation associated with displacement over bends in faults, describing the theoretical framework which we have implemented in the associated program, fbfFor. We show the utility of this kinematic approach by matching seismic reflection examples, analog models, and mechanical models of fault-bend folds to create progressive, balanced kinematic interpretations and gain further insight into the formation of these structures.

On regular and chaotic dynamics of a non--symmetric Hamiltonian system of a coupled Duffing oscillator with balanced loss and gain

A non--symmetric Hamiltonian system of a Duffing oscillator coupled to an anti-damped oscillator with a variable angular frequency is shown to admit periodic solutions. The result implies that -symmetry of a Hamiltonian system with balanced loss and gain is not necessary in order to admit periodic solutions. The Hamiltonian describes a multistable dynamical system—three out of five equilibrium points are stable. The dynamics of the model is investigated in detail by using perturbative as well as numerical methods and shown to admit periodic solutions in some regions in the space of parameters. The phase transition from periodic to unbounded solution is to be understood without any reference to -symmetry. The numerical analysis reveals chaotic behaviour in the system beyond a critical value of the parameter that couples the Duffing oscillator to the anti-damped harmonic oscillator, thereby providing the first example of Hamiltonian chaos in a system with balanced loss and gain. The method of multiple time-scales is used for investigating the system perturbatively. The dynamics of the amplitude in the leading order of the perturbation is governed by an effective dimer model with balanced loss and gain that is non--symmetric Hamiltonian system. The dimer model is solved exactly by using the Stokes variables and shown to admit periodic solutions in some regions of the parameter space.

Emergence of PT-symmetry breaking in open quantum systems

The effect of \mathcal{PT}𝒫𝒯-symmetry breaking in coupled systems with balanced gain and loss has recently attracted considerable attention and has been demonstrated in various photonic, electrical and mechanical systems in the classical regime. However, it is still an unsolved problem how to generalize the concept of \mathcal{PT}𝒫𝒯 symmetry to the quantum domain, where the conventional definition in terms of non-Hermitian Hamiltonians is not applicable. Here we introduce a symmetry relation for Liouville operators that describe the dissipative evolution of arbitrary open quantum systems. Specifically, we show that the invariance of the Liouvillian under this symmetry transformation implies the existence of stationary states with preserved and broken parity symmetry. As the dimension of the Hilbert space grows, the transition between these two limiting phases becomes increasingly sharp and the classically expected \mathcal{PT}𝒫𝒯-symmetry breaking transition is recovered. This quantum-to-classical correspondence allows us to establish a common theoretical framework to identify and accurately describe \mathcal{PT}𝒫𝒯-symmetry breaking effects in a large variety of physical systems, operated both in the classical and quantum regimes.

A non-Hermitian PT-symmetric kicked top

A non-Hermitian PT-symmetric version of the kicked top is introduced to study the interplay of quantum chaos with balanced loss and gain. The classical dynamics arising from the quantum dynamics of the angular momentum expectation values are derived. It is demonstrated that the presence of PT-symmetry can lead to ‘stable’ mixed regular chaotic behaviour without sinks or sources for subcritical values of the gain–loss parameter. This is an example of what is known in classical dynamical systems as reversible dynamical systems. For large values of the kicking strength a strange attractor is observed that also persists if PT-symmetry is broken. The intensity dynamics of the classical map is investigated, and found to provide the main structure for the Husimi distributions of the subspaces of the quantum system belonging to certain ranges of the imaginary parts of the quasienergies. Classical structures are also identified in the quantum dynamics. Finally, the statistics of the eigenvalues of the quantum system are analysed and it is shown that if most of the eigenvalues are complex (which is the case already for fairly small non-Hermiticity parameters) the nearest-neighbour distances of the (unfolded) quasienergies follow a two-dimensional Posisson distribution when the classical dynamics is regular. In the chaotic regime, on the other hand they are in line with recently identified universal complex level spacing distributions for non-Hermitian systems, with transpose symmetry ÂT = Â. It is demonstrated how breaking this symmetry (by introducing an extra term in the Hamiltonian) recovers the more familiar universality class for non-Hermitian systems given by the complex Ginibre ensemble. Both universality classes display cubic level repulsion. The PT-symmetry of the system does not seem to influence the complex level spacings. Similar behaviour is also observed for the spectrum of a PT-symmetric extension of the triadic Baker map.

Shortcuts to Adiabaticity in Driven Open Quantum Systems: Balanced Gain and Loss and Non-Markovian Evolution

A universal scheme is introduced to speed up the dynamics of a driven open quantum system along a prescribed trajectory of interest. This framework generalizes counterdiabatic driving to open quantum processes. Shortcuts to adiabaticity designed in this fashion can be implemented in two alternative physical scenarios: one characterized by the presence of balanced gain and loss, the other involves non-Markovian dynamics with time-dependent Lindblad operators. As an illustration, we engineer superadiabatic cooling, heating, and isothermal strokes for a two-level system, and provide a protocol for the fast thermalization of a quantum oscillator.

Parity-time phase transition in photonic crystals with C_{6v} symmetry

We investigate the parity-time (PT) phase transition in photonic crystals with C_{6v} symmetry, with balanced gain and loss on dielectric rods in the triangular lattice. A two-level non-Hermitian model that incorporates the gain and loss in the tight-binding approximation was employed to describe the dispersion of the PT symmetric system. In the unbroken PT phase, the double Dirac cone feature associated with the C_{6v} symmetry is preserved, with a frequency shift of second order due to the presence of gain and loss. The helical edge states with real eigenfrequencies can exist in the common band gap for two topologically distinct lattices. In the broken PT phase, the non-Hermitian perturbation deforms the dispersion by merging the frequency bands into complex conjugate pairs and forming the exceptional contours that feature the PT phase transition. In this situation, the band gap closes and the edge states are mixed with the bulk states.

Non-Hermitian Floquet second order topological insulators in periodically quenched lattices

Higher-order topological phases are characterized by protected states localized at the corners or hinges of the system. By applying time-periodic quenches to a two-dimensional lattice with balanced gain and loss, we obtain a rich variety of non-Hermitian Floquet second order topological insulating phases. Each of the phases is characterized by a pair of integer topological invariants, which predict the numbers of Floquet corner modes at zero and $\pi$ quasienergies. We establish the topological phase diagram of the model, and find a series of non-Hermiticity induced transitions between different Floquet second order topological phases. We further generalize the mean chiral displacement to two-dimensional non-Hermitian systems, and use it to extract the topological invariants of our model dynamically. This work thus extend the study of higher-order topological matter to more generic nonequilibrium settings, in which the interplay between Floquet engineering and non-Hermiticity yields fascinating new phases.

Localized modes inPT-symmetric sine-Gordon couplers with phase shift

• We study both analytically and numerically the nonlinear localized modes for waveguide arrays with PT -symmetry characterized by the balanced gain and loss parameters. • To investigate the nonlinear modes in PT -symmetry, the system of coupled sine-Gordon equations, together with a variety of nonlinear terms, are considered. • We applied asymptotic analysis together multiple-scale expansion, and obtained analytical amplitude equations for the optical modes, including the fundamental and excited modes. • The nonlinear localized modes are obtained by considering the discrete and spatially continuous spectrum, that is phonon mode. • It is noted that, the obtained amplitude equations of oscillations are affected by the strength of cross-coupling, that is, when the system is coupled weakly, the amplitude in the lossy side is decreased continuously, while on the gain side is increased for a long time. • It is, further, noted that, in the presence of strong coupling parameters, when the damping is present, the rate of blow and decay re-balance waveguides, and the system oscillates for a long period. • We have also discussed the waveguides for sinusoidal and quadratic coupling. • It is also observed that, when the waveguide is coupled strongly, the interaction in kink and anti-kink oscillation form a breather. • We conclude that such breather formation can also be feasible if it is on the amplified field of the PT -symmetric system. • The analytical results are compared with obtained numerical simulations in the numerical section, where good agreements are obtained. In this article, we address the nonlinear localized modes for waveguide arrays with parity-time ( PT )-symmetry characterized by the balanced gain and loss parameters. To investigate the PT -symmetric nonlinear modes, the system of coupled sine-Gordon equations together with a variety of nonlinear terms are considered. We applied asymptotic analysis together with multiple-scale expansion and obtained analytical amplitude equations for the optical modes, including the fundamental and excited modes. The nonlinear localized modes are obtained by considering the discrete and spatially continuous spectrum. It is noted that the obtained amplitude equations of oscillations are affected by the strength of cross-coupling, that is, when the system is coupled weakly, the amplitude in the lossy side decreases continuously while it increases on the gain side of the waveguide for a long time. It is further noted that in the presence of strong and sinusoidally coupling when the damping is present, the rate of blow and decay re-balance the waveguides, and the system oscillates for a long period. It is observed that, in the case of quadratic coupling, the amplitude in the lossy side decreases while the attenuated field increases. It is also observed that when the waveguide is coupled strongly, the interaction in kink and anti-kink oscillations form a breather. The analytical results are compared with the obtained numerical simulations in the numerical section, where good agreements are obtained.