Generating high-order quantum exceptional points

Physical Review A, 2019

Exceptional points (EPs) correspond to degeneracies of open systems. These are attracting much interest in optics, optoelectronics, plasmonics, and condensed matter physics. In the classical and semiclassical approaches, Hamiltonian EPs (HEPs) are usually defined as degeneracies of non-Hermitian Hamiltonians such that at least two eigenfrequencies are identical and the corresponding eigenstates coalesce. HEPs result from continuous, mostly slow, nonunitary evolution without quantum jumps. Clearly, quantum jumps should be included in a fully quantum approach to make it equivalent to, e.g., the Lindblad master-equation approach. Thus, we suggest to define EPs via degeneracies of a Liouvillian superoperator (including the full Lindbladian term, LEPs), and we clarify the relations between HEPs and LEPs. We prove two main theorems: Theorem 1 proves that, in the quantum limit, LEPs and HEPs must have essentially different properties. Theorem 2 dictates a condition under which, in the "semiclassical" limit, LEPs and HEPs recover the same properties. In particular, we show the validity of Theorem 1 studying systems which have (1) an LEP but no HEPs, and (2) both LEPs and HEPs but for shifted parameters. As for Theorem 2, (3) we show that these two types of EPs become essentially equivalent in the semiclassical limit. We introduce a series of mathematical techniques to unveil analogies and differences between the HEPs and LEPs. We analytically compare LEPs and HEPs for some quantum and semiclassical prototype models with loss and gain.

Physical review, 2019

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 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.

Epl, 2014

We consider a generalization of recently proposed non-Hermitian model for resonant cavities coupled by a chiral mirror by taking into account number non-conservation and nonlinear interactions. We analyze non-Hermitian quantum dynamics of populations and entanglement of the cavity modes. We find that an interplay of initial coherence and non-Hermitian coupling leads to a counterintuitive population transfer. While an initially coherent cavity mode is depleted, the other empty cavity can be populated more or less than the initially filled one. Moreover, presence of nonlinearity yields population collapse and revival as well as bipartite entanglement of the cavity modes. In addition to coupled cavities, we point out that similar models can be found in PT symmetric Bose-Hubbard dimers of Bose-Einstein condensates or in coupled soliton-plasmon waveguides. We specifically illustrate quantum dynamics of populations and entanglement in a heuristic model that we propose for a soliton-plasmon system with soliton amplitude dependent asymmetric interaction. Degree of asymmetry, nonlinearity and coherence are examined to control plasmon excitations and soliton-plasmon entanglement. Relations to PT symmetric lasers and Jahn-Teller systems are pointed out.

Progress of Theoretical and Experimental Physics, 2016

We study the physical influence of exceptional points in the spectrum of the Liouville-von Neumann operator (Liouvillian). The system that we consider in this paper is a weakly coupled 1D quantum perfect Lorentz gas, which has exceptional points on the real axis of the wavenumber space in the Liouvillian spectrum at which two eigenvalues coalesce. In order to clarify how the exceptional points affect the system dynamics, we analyze the spatial time evolution of the Wigner distribution function relying on a complex spectral decomposition of the Liouville-von Neumann equation. We show that the exceptional points represent a threshold between two qualitatively different dynamical regimes; one is a diffusive motion due to pure imaginary eigenvalues of the Liouvillian for wavenumbers on one side of the transition, while the other is a ballistic motion due to the existence of the real part of the eigenvalues for wavenumbers on the other side.

Physical Review A, 2019

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 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.

Physical Review E, 1998

The second-quantization of a scalar eld in an open cavity is formulated, from rst principles, in terms of the quasinormal modes (QNMs), which are the eigensolutions of the evolution equation that decay exponentially in time as energy leaks to the outside. This formulation provides a description involving the cavity degrees of freedom only, with the outside acting as a (thermal or driven) source. Thermal correlation functions and cavity F eynman propagators are thus expressed in terms of the QNMs, labeled by a discrete index rather than a continuous momentum. Single-resonance domination of the density of states and the spontaneous decay rate is then given a proper foundation. This is a rst essential step towards the application of QNMs to cavity QED phenomena, to be reported elsewhere. 05.30.-d, 03.70.+k, 42.50.-p, 02.90.+p Typeset using REVT E X

International Journal of Theoretical Physics, 2012

We propose an analysis technique for the exceptional points (EPs) occurring in the discrete spectrum of open quantum systems (OQS), using a semi-infinite chain coupled to an endpoint impurity as a prototype. We outline our method to locate the EPs in OQS, further obtaining an eigenvalue expansion in the vicinity of the EPs that gives rise to characteristic exponents. Finally, we offer a heuristic QPT analogy for the emergence of the resonance (giving rise to irreversibility via exponential decay) in which the decay width plays the role of the order parameter; the associated critical exponent is then determined by the above eigenvalue expansion.

Physical Review B

We present a novel approach and a theoretical framework for generating high order exceptional points of degeneracy (EPD) in photonic structures based on periodic coupled resonators optical waveguides (CROWs). Such EPDs involve the coalescence of Floquet-Bloch eigenwaves in CROWs, without the presence of gain and loss, which is in contrast to the requirement of Parity-Time (PT) symmetry to develop exceptional points based on gain and loss balance. The EPDs arise here by introducing symmetry breaking in a conventional chain of coupled resonators through periodic coupling to an adjacent uniform optical waveguide, which leads to unique modal characteristics that cannot be realized in conventional CROWs. Such remarkable characteristics include high quality factors (Q-factor) and strong field enhancement, even without any mirrors at the two ends of a cavity. We show for the first time the capability of CROWs to exhibit EPDs of various order; including the degenerate band edge (DBE) and the stationary inflection point (SIP). The proposed CROW of finite length shows enhanced quality factor when operating near the DBE, and the Q-factor exhibits an unconventional scaling with the CROW's length. We develop the theory of EPDs in such unconventional CROW using coupled-wave equations, and we derive an analytical expression for the dispersion relation. The proposed unconventional CROW concepts have various potential applications including Q-switching, nonlinear devices, lasers, and extremely sensitive sensors. I.

Journal of Physics: Conference Series, 2021

We review some recent work on the occurrence of coalescing eigenstates at exceptional points in non-Hermitian systems and their influence on physical quantities. We particularly focus on quantum dynamics near exceptional points in open quantum systems, which are described by an outwardly Hermitian Hamiltonian that gives rise to a non-Hermitian effective description after one projects out the environmental component of the system. We classify the exceptional points into two categories: those at which two or more resonance states coalesce and those at which at least one resonance and the partnering anti-resonance coalesce (possibly including virtual states as well), and we introduce several simple models to explore the dynamics for both of these types. In the latter case of coalescing resonance and anti-resonance states, we show that the presence of the continuum threshold plays a strong role in shaping the dynamics, in addition to the exceptional point itself. We also briefly discuss...

Physical Review A, 2012

We construct families of symmetric, antisymmetric, and asymmetric solitary modes in onedimensional bichromatic lattices with the second-harmonic-generating (χ (2)) nonlinearity concentrated at a pair of sites placed at distance l. The lattice can be built as an array of optical waveguides. Solutions are obtained in an implicit analytical form, which is made explicit in the case of adjacent nonlinear sites, l = 1. The stability is analyzed through the computation of eigenvalues for small perturbations, and verified by direct simulations. In the cascading limit, which corresponds to large mismatch q, the system becomes tantamount to the recently studied single-component lattice with two embedded sites carrying the cubic nonlinearity. The modes undergo qualitative changes with the variation of q. In particular, at l ≥ 2, the symmetry-breaking bifurcation (SBB), which creates asymmetric states from symmetric ones, is supercritical and subcritical for small and large values of q, respectively, while the bifurcation is always supercritical at l = 1. In the experiment, the corresponding change of the phase transition between the second and first kinds may be implemented by varying the mismatch, via the wavelength of the input beam. The existence threshold (minimum total power) for the symmetric modes vanishes exactly at q = 0, which suggests a possibility to create the solitary mode using low-power beams. The stability of solution families also changes with q.