We study the coupled even number of microcavities with the balanced gain and loss between any pair of their neighboring components. The effective non-Hermitian Hamiltonian for such a structure has the cyclic permutation-time symmetry with respect to the cavity modes, and this symmetry determines the patterns of the dynamical evolutions of the cavity modes. The systems also have multiple exceptional points for the degeneracy of the existing supermodes, exhibiting the ``phase transition'' of system dynamics across these exceptional points. We illustrate the quantum dynamical properties of the systems with the evolutions of cavity photon numbers and correlation functions. Moreover, we demonstrate the effects of the quantum noises accompanying the amplification and dissipation of the cavity modes. The reciprocal light transportation predicted with the effective non-Hermitian models for the similar couplers is violated by the unavoidable quantum noises.
We study the coupled even number of microcavities with the balanced gain and loss between any pair of their neighboring components. The effective non-Hermitian Hamiltonian for such a structure has the cyclic permutation-time symmetry with respect to the cavity modes, and this symmetry determines the patterns of the dynamical evolutions of the cavity modes. The systems also have multiple exceptional points for the degeneracy of the existing supermodes, exhibiting the ``phase transition'' of system dynamics across these exceptional points. We illustrate the quantum dynamical properties of the systems with the evolutions of cavity photon numbers and correlation functions. Moreover, we demonstrate the effects of the quantum noises accompanying the amplification and dissipation of the cavity modes. The reciprocal light transportation predicted with the effective non-Hermitian models for the similar couplers is violated by the unavoidable quantum noises.
We use a non-Lindbladian master equation of the Scully-Lamb laser model for the analysis of light propagation in a parity-time symmetric photonic system composed of coupled active and passive whispering-gallery microresonators. Performing the semiclassical approximation, we obtain a set of two nonlinear coupled differential equations describing the time evolution of intracavity fields. These coupled equations are able to explain the experimentally observed light nonreciprocity [Peng et al., Nat. Phys. 10, 394 (2014); Chang et al., Nat. Photon. 8, 524 (2014)]. We show that this effect arises from the interplay between gain saturation in the active microcavity, intercavity coupling, and losses in the cavities. Additionally, using this approach, we study the effect of the gain saturation on exceptional points, i.e., exotic degeneracies in non-Hermitian systems. Namely, we demonstrate that the inclusion of gain saturation leads to a modification of the exceptional points in the presence of intense intracavity fields. The Scully-Lamb master equation for systems of coupled optical structures, as proposed and applied here, constitutes a promising tool for the study of quantum optical effects in coupled systems with losses, gain, and gain saturation.
A two-mode optical parity-time (PT) symmetric system, with gain and damping, described by a quantum quadratic Hamiltonian with additional small Kerr-like nonlinear terms, is analyzed from the point of view of nonclassical-light generation. Two kinds of stationary states with different types of (in)stability are revealed. Properties of one of these are related to the presence of semiclassical exceptional points, i.e., exotic degeneracies of the non-Hermitian Hamiltonian describing the studied system without quantum jumps. The evolution of the logarithmic negativity, principal squeezing variances, and sub-shot-noise photon-number correlations, considered as entanglement and non-classicality quantifiers, is analyzed in the approximation of linear-operator corrections to the classical solution. Suitable conditions for nonclassical-light generation are identified in the oscillatory regime, especially at and around exceptional points that considerably enhance the nonlinear interaction and, thus, the non-classicality of the generated light. The role of quantum fluctuations, inevitably accompanying attenuation and amplification in the evolution of quantum states, is elucidated. The evolution of the system is analyzed for different initial conditions.
In this work, taking the most general non-Hermitian Hamiltonian without parity-time symmetry and with non-Hermitian off-diagonal elements into account, we study the exceptional points of the model system. We find that there exists an exceptional ellipse in theory, compared with the exceptional ring studied in non-Hermitian symmetry Hamiltonian, in which the parameters in the model Hamiltonian take particular values. We also report on the dynamics of the model Hamiltonian and symmetry Hamiltonian. We find that they display different dynamical behaviors under different parameters. Moreover, there exists an unstable behavior of the system. The unstable behavior has a surprising property by which the gain always dominates at long times no matter how small the gain is. This phenomenon enables the compensation for losses in dissipative systems and opens a wide range of applications in quantum optics, plasmonics, and optoelectronics, where the loss is inevitable but can be compensated by the gain.
The investigation of pseudo-Hermitian systems has garnered widespread attention due to their distinctive property of purely real eigenvalues splitting into complex conjugate pairs. This transition profoundly modifies the system's behavior, specifically giving rise to remarkable dynamical effects. Here we explore the novel characteristics of phase transitions and dynamic currents in a pseudo-Hermitian system of coupled sites with tunable gain and loss. With tunable gain and loss, we verify that the system uniquely supports two pseudo-Hermitian configurations, which exhibit distinct symmetries. The phase transitions from the PT-symmetric phase to the spontaneous broken PT-symmetric phase, along with the associated high-order exceptional points (EPs), are unambiguously demonstrated in both types of symmetry configurations. We measure the dynamic current within the four coupled sites, which allows us to probe the EPs and phase transitions between distinct phases. Our results offer insights into the phase transitions and dynamics of pseudo-Hermitian systems, with potential applications in the design and manipulation of novel quantum phenomena across various fields including photonics and other related quantum systems.
PT-like phase transitions from square roots of supersymmetric Hamiltonians
We introduce a general framework for realizing PT-like phase transitions in non-Hermitian systems without imposing explicit parity--time (PT) symmetry. The approach is based on constructing a Hamiltonian as the square root of a supersymmetric partner energy-shifted by a constant. This formulation naturally leads to bipartite dynamics with balanced gain and loss and can incorporate non-reciprocal couplings. The resulting systems exhibit entirely real spectra over a finite parameter range precisely when the corresponding passive Hamiltonian lacks a zero mode. As the non-Hermitian parameter representing gain and loss increases, the spectrum undergoes controlled real-to-complex transitions at second-order exceptional points. We demonstrate the versatility of this framework through several examples, including well-known models such as the Hatano--Nelson (HN) and complex Su--Schrieffer--Heeger (cSSH) lattices. Extending the formalism to q-commuting matrices further enables the systematic realization of higher-order exceptional points in systems with unidirectional couplings. Overall, this work uncovers new links between non-Hermitian physics and supersymmetry, offering a practical route to engineer photonic arrays with tunable spectral properties beyond what is achievable with explicit PT-symmetry.
A two-mode optical parity-time (PT) symmetric system, with gain and damping, described by a quantum quadratic Hamiltonian with additional small Kerr-like nonlinear terms, is analyzed from the point of view of nonclassical-light generation. Two kinds of stationary states with different types of (in)stability are revealed. Properties of one of these are related to the presence of semiclassical exceptional points, i.e., exotic degeneracies of the non-Hermitian Hamiltonian describing the studied system without quantum jumps. The evolution of the logarithmic negativity, principal squeezing variances, and sub-shot-noise photon-number correlations, considered as entanglement and nonclassicality quantifiers, is analyzed in the approximation of linear-operator corrections to the classical solution. Suitable conditions for nonclassical-light generation are identified in the oscillatory regime, especially at and around exceptional points that considerably enhance the nonlinear interaction and, thus, the nonclassicality of the generated light. The role of quantum fluctuations, inevitably accompanying attenuation and amplification in the evolution of quantum states, is elucidated. The evolution of the system is analyzed for different initial conditions.