Coulomb analogy for non-Hermitian degeneracies near quantum phase transitions

Spectral kissing and its dynamical consequences in the squeeze-driven Kerr oscillator

Transmon qubits are the predominant element in circuit-based quantum information processing, such as existing quantum computers, due to their controllability and ease of engineering implementation. But more than qubits, transmons are multilevel nonlinear oscillators that can be used to investigate fundamental physics questions. Here, they are explored as simulators of excited state quantum phase transitions (ESQPTs), which are generalizations of quantum phase transitions to excited states. We show that the spectral kissing (coalescence of pairs of energy levels) experimentally observed in the effective Hamiltonian of a driven SNAIL-transmon is an ESQPT precursor. We explore the dynamical consequences of the ESQPT, which include the exponential growth of out-of-time-ordered correlators, followed by periodic revivals, and the slow evolution of the survival probability due to localization. These signatures of ESQPT are within reach for current superconducting circuits platforms and are of interest to experiments with cold atoms and ion traps.

Spin transport in non-Hermitian quantum systems

Transport in non-Hermitian quantum systems is studied. The goal is a better understanding of transport in non-Hermitian systems like the Lieb lattice due to its flat bands and the integrability of the Ising chain which allows transport in that model to be computed analytically. This is a very special feature that is not present in a generic non-Hermitian system. We obtain the behaviour of the spin conductivity as a function of the non-Hermitian parameters of each system with aim to verify the influence of variation them on conductivity. For all models analyzed: Ising model as well as noninteracting fermion models, we obtain a little influence of the non-Hermitian parameters on conductivity and thus, a small effect over transport coefficients. Furthermore, we obtain an influence of opening of the gap in the spectrum in these models on longitudinal conductivity as well.

Complex analysis of divergent perturbation theory at finite temperature

We investigate the convergence properties of finite-temperature perturbation theory by considering the mathematical structure of thermodynamic potentials using complex analysis. We discover that zeros of the partition function lead to poles in the internal energy and logarithmic singularities in the Helmholtz free energy that create divergent expansions in the canonical ensemble. Analyzing these zeros reveals that the radius of convergence increases at higher temperatures. In contrast, when the reference state is degenerate, these poles in the internal energy create a zero radius of convergence in the zero-temperature limit. Finally, by showing that the poles in the internal energy reduce to exceptional points in the zero-temperature limit, we unify the two main mathematical representations of quantum phase transitions.

Constant of Motion Identifying Excited-State Quantum Phases

We propose that a broad class of excited-state quantum phase transitions (ESQPTs) gives rise to two different excited-state quantum phases. These phases are identified by means of an operator C[over ^], which is a constant of motion in only one of them. Hence, the ESQPT critical energy splits the spectrum into one phase where the equilibrium expectation values of physical observables crucially depend on this constant of motion and another phase where the energy is the only relevant thermodynamic magnitude. The trademark feature of this operator is that it has two different eigenvalues ±1, and, therefore, it acts as a discrete symmetry in the first of these two phases. This scenario is observed in systems with and without an additional discrete symmetry; in the first case, C[over ^] explains the change from degenerate doublets to nondegenerate eigenlevels upon crossing the critical line. We present stringent numerical evidence in the Rabi and Dicke models, suggesting that this result is exact in the thermodynamic limit, with finite-size corrections that decrease as a power law.

Perturbation theory in the complex plane: exceptional points and where to find them

We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree-Fock approximation and Rayleigh-Schrödinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within Møller-Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Padé and quadratic approximants) that can improve the overall accuracy of the Møller-Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.

Superradiance in finite quantum systems randomly coupled to continuum

We study the effect of superradiance in open quantum systems, i.e., the separation of short- and long-living eigenstates when a certain subspace of states in the Hilbert space acquires an increasing decay width. We use several Hamiltonian forms of the initial closed system and generate their coupling to continuum by means of the random matrix theory. We average the results over a large number of statistical realizations of an effective non-Hermitian Hamiltonian and relate robust features of the superradiance process to the distribution of its exceptional points. We show that the superradiance effect is enhanced if the initial system is at the point of quantum criticality.

Dynamical quantum phase transitions: a review

Quantum theory provides an extensive framework for the description of the equilibrium properties of quantum matter. Yet experiments in quantum simulators have now opened up a route towards the generation of quantum states beyond this equilibrium paradigm. While these states promise to show properties not constrained by equilibrium principles, such as the equal a priori probability of the microcanonical ensemble, identifying the general properties of nonequilibrium quantum dynamics remains a major challenge, especially in view of the lack of conventional concepts such as free energies. The theory of dynamical quantum phase transitions attempts to identify such general principles by lifting the concept of phase transitions to coherent quantum real-time evolution. This review provides a pedagogical introduction to this field. Starting from the general setting of nonequilibrium dynamics in closed quantum many-body systems, we give the definition of dynamical quantum phase transitions as phase transitions in time with physical quantities becoming nonanalytic at critical times. We summarize the achieved theoretical advances as well as the first experimental observations, and furthermore provide an outlook to major open questions as well as future directions of research.

Exceptional points near first- and second-order quantum phase transitions

We study the impact of quantum phase transitions (QPTs) on the distribution of exceptional points (EPs) of the Hamiltonian in the complex-extended parameter domain. Analyzing first- and second-order QPTs in the Lipkin-Meshkov-Glick model we find an exponentially and polynomially close approach of EPs to the respective critical point with increasing size of the system. If the critical Hamiltonian is subject to random perturbations of various kinds, the averaged distribution of EPs close to the critical point still carries decisive information on the QPT type. We therefore claim that properties of the EP distribution represent a parametrization-independent signature of criticality in quantum systems.

Identifying the order of a quantum phase transition by means of Wehrl entropy in phase space

We propose a method to identify the order of a quantum phase transition by using area measures of the ground state in phase space. We illustrate our proposal by analyzing the well known example of the quantum cusp and four different paradigmatic boson models: Dicke, Lipkin-Meshkov-Glick, interacting boson model, and vibron model.

Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance

Scattering experiments with microwave cavities were performed and the effects of broken time-reversal invariance (TRI), induced by means of a magnetized ferrite placed inside the cavity, on an isolated doublet of nearly degenerate resonances were investigated. All elements of the effective Hamiltonian of this two-level system were extracted. As a function of two experimental parameters, the doublet and the associated eigenvectors could be tuned to coalesce at a so-called exceptional point (EP). The behavior of the eigenvalues and eigenvectors when encircling the EP in parameter space was studied, including the geometric amplitude that builds up in the case of broken TRI. A one-dimensional subspace of parameters was found where the differences of the eigenvalues are either real or purely imaginary. There, the Hamiltonians were found to be PT invariant under the combined operation of parity (P) and time reversal (T) in a generalized sense. The EP is the point of transition between both regions. There a spontaneous breaking of PT occurs.

Exceptional points and non-Hermitian degeneracy of resonances in a two-channel model

We study the mixing and degeneracy of two unbound energy eigenstates (resonances) in a two coupled channel model of scattering and reactions. We derive the necessary and sufficient conditions for existence of an exceptional point in the extended spectrum of bound and resonance energy eigenvalues in this model and show that these are not the same as in the single channel case. When these conditions are satisfied, in the complex energy plane, the two simple resonance poles of the scattering matrix merge into one double pole at the exceptional point. In parameter space, the surface of the eigenenergies has a branch point of square root type and branch cuts in its real and imaginary parts that start at the exceptional point and extend in opposite directions. The rich phenomenology of crossings and anticrossings of energies and widths of the doublet of unbound states, as well as the changes of identity of the poles of the scattering matrix observed when one control parameter is varied while the other is kept constant, is fully explained in terms of sections of the eigenenergy surfaces.

Exceptional points in a microwave billiard with time-reversal invariance violation

We report on the experimental study of an exceptional point (EP) in a dissipative microwave billiard with induced time-reversal invariance (T) violation. The associated two-state Hamiltonian is non-Hermitian and nonsymmetric. It is determined experimentally on a narrow grid in a parameter plane around the EP. At the EP the size of T violation is given by the relative phase of the eigenvector components. The eigenvectors are adiabatically transported around the EP, whereupon they gather geometric phases and in addition geometric amplitudes different from unity.

Quasieigenstate coalescence in an atom-cavity quantum composite

We report the first direct observation of an exceptional point (EP) in an open quantum composite of a single atom and a high-Q cavity mode. The atom-cavity coupling constant was made a continuous variable by utilizing the multisublevel nature of a single rubidium atom when it is optimally coupled to the cavity mode. The spectroscopic properties of quasieigenstates of the atom-cavity composite were experimentally investigated near the EP. Branch-point singularity of quasieigenenergies was observed and its 4pi symmetry was demonstrated. Consequently, the cavity transmission at the quasieigenstate was observed to exhibit a critical behavior at the EP.

Partition function zeros of the square-lattice Ising model with nearest- and next-nearest-neighbor interactions

The distributions of the partition function zeros in the complex a=e2betaJ1 plane of the square-lattice Ising model with nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions are investigated as a function of R=J2/J1. Starting from the well-known two-circle distribution of the zeros a=+/-1+sqrt[2]eitheta for R=0 , finally the partition function zeros lie on the unit circle a=eitheta for R=infinity . Between these two ends, the changes in the zero distributions are described. Using the partition function zeros, the critical point ac(R) and the thermal scaling exponent yt(R) are estimated for the Ising ferromagnet (equivalently, antiferromagnet) and superantiferromagnet. For the special case of R=1/2 , the possible implications of the zero distributions are also discussed.

Classical analysis of quantum phase transitions in a bilayer model

In this Brief Report we extend the classical analysis performed on the schematic model proposed in [T. Moreira, G. Q. Pellegrino, J. G. Peixoto de Faria, M. C. Nemes, F. Camargo, and A. F. R. Toledo Piza, Phys. Rev. E 77, 051102 (2008)] concerning quantum phase transitions in a bilayer system. We show that appropriate integrations along the classical periodic orbits reproduce with excellent agreement both the quantum spectrum and the expected mean value for the number of excitons in the system, quantities which are directly related to the observed boson-fermion quantum phase transition.

Resonance coalescence in molecular photodissociation

We study theoretically the photodissociation dynamics of the H_{2};{+} molecular ion exposed to a linearly polarized laser light. It is shown that it is possible to choose a laser wavelength and intensity so as to produce a coalescence of two photodissociation vibronic resonance states. At such a coalescence point, also called an exceptional point, the photodissociative resonance wave function is self-orthogonal. This unique phenomenon which is presented here for light induced molecular dynamics enables us to transfer completely the nondissociated molecules from one vibronic state to another by varying adiabatically the laser frequency and intensity along a closed contour which encircles the exceptional point.

Observation of PT-symmetry breaking in complex optical potentials

In 1998, Bender and Boettcher found that a wide class of Hamiltonians, even though non-Hermitian, can still exhibit entirely real spectra provided that they obey parity-time requirements or PT symmetry. Here we demonstrate experimentally passive PT-symmetry breaking within the realm of optics. This phase transition leads to a loss induced optical transparency in specially designed pseudo-Hermitian guiding potentials.

Impact of quantum phase transitions on excited-level dynamics

The influence of quantum phase transitions on the evolution of excited levels in the critical parameter region is discussed. The analysis is performed for one- and two-dimensional systems with first- and second-order ground-state transitions. Examples include the cusp and nuclear collective Hamiltonians. Applications in systems of higher dimensions are possible.

Visualization of branch points in PT-symmetric waveguides

The visualization of an exceptional point in a PT-symmetric directional coupler (DC) is demonstrated. In such a system the exceptional point can be probed by varying only a single parameter. Using the Rayleigh-Schrödinger perturbation theory we prove that the spectrum of a PT-symmetric Hamiltonian is real as long as the radius of convergence has not been reached. We also show how one can use a PT-symmetric directional coupler to measure the radius of convergence for non-PT-symmetric structures. For such systems the physical meaning of the rather mathematical term radius of convergence is exemplified.