Session A55: Non-Hermitian Quantum Systems

Loss-induced phenomena in cavity QED systems

Cavity quantum electrodynamical (QED) systems, in which a confined light in a cavity interacts with a quantum emitter, hold great promise for a wide range of applications, from quantum information [Rev. Mod. Phys. 87, 1379 (2015)] to polaritonic chemistry [Chemical Reviews 123.16, 9786-9879 (2023)]. These systems are typically lossy due to the imperfection of the cavity that enables leakage of the light outside the cavity. While these losses often limit the applicability of cavity QED systems, here we focus on exploring novel phenomena arising from the lossy nature of cavity QED systems. For that, we use the non-Hermitian formalism, which has attracted much attention recently and enables revealing effects that are hidden in the standard formalism [Nature (London) 537, 76 (2016)]. We show that the losses can be utilized to generate single-photon emission through a novel mechanism [Phys. Rev. Lett. 130, 243601 (2023)], converting a non-Markovian bath to Markovian, etc.   

Activity-induced ferromagnetism in one-dimensional quantum many-body systems

In this talk, we present our study on a non-Hermitian quantum many-body model in one dimension analogous to the Vicsek model or active spin models [1]. The model consists of two-component hard-core bosons with ferromagnetic interactions and activity, i.e., spin-dependent asymmetric hopping, which is realizable with dissipative optical lattices in principle. Numerical results show the emergence of a ferromagnetic order induced by the activity, a quantum counterpart of flocking, that even survives in the absence of ferromagnetic interaction. We confirm this phenomenon by proving that activity generally increases the ground state energies of the paramagnetic states, whereas the ground state energy of the ferromagnetic state does not change. By solving the two-particle case, we find that the effective alignment is caused by avoiding the bound state formation due to the non-Hermitian skin effect in the paramagnetic state. We employ a two-site mean-field theory based on the two-particle result and qualitatively reproduce the phase diagram. We further numerically study a variant of our model with the hard-core condition relaxed, and confirm the robustness of ferromagnetic order emerging due to activity.

[1] KT*, Kyosuke Adachi*, Kyogo Kawaguchi, arXiv:2308.04382 (*co-first)   

Entanglement dynamics in a many-body Hatano-Nelson model

Non-Hermitian quantum systems exhibit unusual features, which have no counterpart in Hermitian quantum systems. In this work, as a concrete example, we consider a one-dimensional tight-binding model with non-reciprocal hopping amplitudes (many-body Hatano-Nelson (HN) model) and focus on how non-Hermiticity (non-reciprocity of hopping term) affects entanglement dynamics [1]. Generally, in the case of the Hermitian system, the quasi-particle picture provides an intuitive picture of entanglement dynamics. However, in the case of the many-body HN model, quasi-particles tend to be suppressed/amplified due to the imaginary part of the eigenenergy, leading to the non-monotonic time dependence of entanglement entropy in a specific parameter regime. We numerically and analytically reveal the role of the imaginary part of eigenenergy on entanglement dynamics and why non-monotonic time dependence of entanglement dynamics appears [2].   

Measuring the complex geometric phase in the dynamics of a non-Hermitian system

When the Hamiltonian of a linear system is slowly tuned around a closed circuit in parameter space, the adiabatic theorem guarantees that an eigenstate of the system returns to itself up to an overall phase. This phase has two dominant components: a dynamical component that depends on the time taken to traverse the circuit, and a geometric component that depends only on the shape of the circuit. For non-Hermitian systems it has been predicted that, for circuits in which the adiabatic theorem is applicable, the geometric phase may be complex, i.e. it may contain both a real phase and a gain/loss component. We measure the fully complex geometric phase accumulated by a set of coupled oscillators, whose non-Hermitian Hamiltonian is parametrically tuned in real time via optomechanical dynamical back-action. We show that this phase is well predicted by only the circuit geometry for a variety of circuits.   

Optical PT-symmetry in the absence of gain or loss

Our conventional understanding of physics is based on the fundamental concept of energy conservation that manifests itself in the real-valued energy spectra of Hermitian Hamiltonians. However, so-called open systems can exchange energy with their environment, and, as subsystems of a larger whole, may exhibit non-Hermitian dynamics. Along these lines, non-Hermiticity established by local attenuation and coherent amplification has been firmly established on a variety of platforms. More recently, the concept of parity-time symmetry has been associated with the wave-mechanical interplay of gain and loss in complex-valued potentials. Here, by contrast, we present a projective approach to synthesize genuinely non-Hermitian dynamics without any injection or removal of intensity from the system. To demonstrate the capabilities of our technique, we observe nonorthogonal modes in femtosecond laser-written waveguide arrays and leverage the fluorescent properties of the waveguide cores to selectively probe the on-site intensities. Projective parity-time symmetry makes features of non-Hermiticity accessible in scenarios where actual amplification and/or attenuation may be unavailable or would disrupt the desired physics, e.g. in quantum optics or nonlinear systems.   

Scale-free localization and PT symmetry breaking from local non-Hermiticity

We show that a local non-Hermitian perturbation in a Hermitian lattice system generically induces scale-free localization for the continuous-spectrum eigenstates. When the perturbation lies at a finite distance to the boundary, the scale-free eigenstates are promoted to exponentially localized modes, whose number is proportional to the distance. Furthermore, when the local non-Hermitian perturbation respects parity-time (PT) symmetry, the PT symmetry breaking is always accompanied by the emergence of scale-free or exponential localization. Intriguingly, we find a concise band-structure condition which tells not only when the continuous-spectrum PT breaking of scale-free modes can occur but also the precise PT-breaking energy window. Our results uncover a series of unexpected generic phenomena induced by a local non-Hermitian perturbation, which has interesting interplay with PT symmetry.   

Real-time dissipation-engineering-assisted quantum state transfer near exceptional points

Dissipation engineering has provided an effective approach to realize non-Hermitian quantum systems. The complex energies of these systems are described by Riemann structures in the vicinity of non-Hermitian degeneracies—also known as exceptional points—and enable chiral quantum state transfer through dynamically tuning system parameters along a closed trajectory. While dissipation plays a critical role in this application, it also leads to unwanted decoherence for quantum states. Therefore, it is desirable to have full control of dissipation to minimize decoherence effect while maintaining state transfer. In this work, we implement cavity assisted bath engineering to a transmon superconducting circuit and realize real-time control of its dissipation rate. Through dynamically tuning the dissipation and other relevant parameters along a closed trajectory, we investigate quantum state transfer in a two-qubit device. This protocol is expected to transfer superposition states of single qubit and entangled states of two qubits with high fidelity. Our study opens new avenues to harness dissipation engineering and non-Hermitian physics in quantum technologies.   

Is there a generic quantum theory for non-Hermitian Hamiltonians?

Unlike in the Hermitian case, the quantum mechanical formalism for the non-Hermitian operators is not unique, and additionally, suffers from singularities and instabilities. Here we propose a framework which is generic, unique and resolves these singularities. We find a Hilbert space of a dynamically generated Hermitian operator, namely the computational space, in which exceptional points are pushed to the vacua, leading to an analytic span of the energy eigenstates. In addition, we also discover a unique dynamical `space-time’ transformation as the dual space map, incorporating physical properties such as decoherence and spectral flow. The real energy condition manifests in the limit when the dynamical transformation becomes a static symmetry. Our framework also provides insights into other features such as gauge obstructions, symmetry-based classifications, dynamical metric manifolds, spectral flattening, among others.   

Universal Spectral Moment Theorem And Its Applications In Non-Hermitian Systems

The high sensitivity of the spectrum to boundary conditions poses significant obstacles for understanding bulk physical observables in non-Hermitian systems. In this paper, we propose the spectral moments in the tight-binding Hamiltonian with finite hopping range as intrinsic bulk quantities, independent of boundary conditions. Utilizing the invariance of the spectral moments, we demonstrate that in the continuum limit under open-boundary conditions, the upper and lower bounds of the imaginary parts of the spectrum remain constant regardless of the boundary geometry. This leads to the conclusion that the outer boundary of the continuum spectrum is insensitive to the shape of boundary geometry. We further establish a criterion for the reality of the spectrum and employ it to determine the $mathcal{PT}$-symmetry-breaking threshold in $mathcal{PT}$-symmetric systems in arbitrary dimensions. Finally, we show that in the long-time limit, the universal dynamical behavior under open boundary conditions converges to that of an infinite system, remarkably different from the behavior under periodic boundary conditions.   

Non-Hermitian edge burst: dynamics, steady states, and interactions

Many intriguing dynamical phenomena arise in non-Hermitian systems that have no analogs in Hermitian cases. In this talk, I will first introduce a novel non-Hermitian quantum dynamical phenomenon called non-Hermitian edge burst, which means that a particle in a lossy lattice exhibits an unexpectedly large probability for the particle loss occurring at the boundary. The edge burst originates from a universal bulk-edge scaling relation of loss probability, which is a consequence of the interplay between two unique non-Hermitian properties: non-Hermitian skin effect and dissipative gap closing. This interesting dynamical phenomenon has been observed recently in photonic platforms. In the second part of this talk, I will introduce a novel dynamical-steady correspondence where the edge burst in non-Hermitian dynamics is mapped to many-body steady states in open quantum systems. Therefore, a universal bulk-edge scaling relation can also exist in the steady-state edge burst. Moreover, adding dissipative interactions further enriches the content of many-body edge burst in steady states, providing a new mechanism of the many-body non-Hermitian skin effect.   

Large deviation theory for Green's functions in non-Hermitian disordered systems

The competition between non-Hermitian skin effect and Anderson localization leads to various intriguing phenomena concerning spectrums and wavefunctions. Here, we study the linear response of disordered non-Hermitian systems, which is precisely described by the Green's function. We find that the average maximum value of matrix elements of Green's functions, which quantifies the maximum response against an external perturbation, exhibits different phases characterized by different scaling behaviors with respect to the system size. Whereas the exponential-growth phase is also seen in the translation-invariant systems, the algebraic-growth phase is unique to disordered non-Hermitian systems. We explains the found results by using the large deviation theory, which provides analytical insights into the algebraic scaling factors of non-Hermitian disordered Green's functions. Furthermore, we show that these scaling behaviors can be observed in the steady states of disordered open quantum systems, offering a quantum-mechanical avenue for their experimental detection. Our work highlights an unexpected interplay between non-Hermitian skin effect and Anderson localization.   

Non-Bloch Self-energy

The non-Hermitian skin effect describes the exponential localization of single-particle eigenstates near the boundaries of the system. We tackle its generalization to the many-body regime by investigating interacting fermions in open quantum systems, and present an exact integral formula to calculate the self-energy based on the non-Bloch band theory. We employ a bi-base mapping to represent the fermionic Liouvillian by two types of quasi-particle excitations from the vacuum (steady state). We then utilize the perturbation theory to calculate the self-energy of these quasi-particles and develop a Feynman-diagrammatic representation. By imposing complex momentum conservation on the vertices, we simplify the formula to a double integral over the Brillouin zone. We demonstrate the high accuracy of our formula by comparing it to numerical results. Our integral formula provides a quantitative tool for investigating interacting fermions in open quantum systems with the non-Hermitian skin effect.   

Operator-space fragmentation and integrability in Lindblad open quantum systems

We investigate the extension of Hilbert-space fragmentation to Lindblad open quantum systems, in which it manifests in the dynamics of observables in the Hilbert-Schmidt space. Using bond and commutant algebras of superoperators, we formalise this general framework and discuss the different cases revealed, similar to the classification of classical and quantum fragmentation in the closed system case.

We highlight a particularly interesting example where the operator space decomposes into exponentially many fragments which range in sizes up to the square-root of the full Hilbert-Schmidt dimension. At the boundary, there are a number of edge modes and in the bulk the operator dynamics is controlled by the fragments. Each of these fragments is equivalent to a non-Hermitian Lee-Yang theory with perturbed boundary fields, which may be exactly solved. This model has a phase transition governed by a non-unitary conformal field theory with a negative central charge. In the full Lindbladian, these transitions correspond to a series of exceptional points in the spectrum. We study the consequence of this for the underlying open quantum system, dynamics of operators and correlation functions, and for the stability of the edge modes.   

A Criterion for localization phenomena inherent in non-Hermitian systems

When a Hamiltonian is effectively non-Hermitian, its eigenstates can be localized at the boundary of the system, which is known as the non-Hermitian skin effect(NHSE). The NHSE was initially characterized as a phenomenon in which all the eigenstates of an open chain are localized at the end. However, this counting is not obvious in higher dimensions. A question of interest is how many localized eigenstates are needed to characterize the NHSE.

In this talk, we propose a criterion for localization phenomena inherent in non-Hermitian systems just using some eigenstates. It is applicable to one- or higher-dimensional systems and even the generic geometry systems. For this purpose, we define ``localization'' and ``localization length'' of state vectors in our manner. We rigorously show that if the number of eigenstates localized at the same location exceeds a certain threshold, the corresponding Hamiltonian must be non-Hermitian.