non-Hermitian Quantum Mechanics Research Articles

Quantization of pseudo-hermitian systems

Abstract This work is a generalization of \cite{baldiotti2021} to Grassmann algebras
of arbitrary dimensions. Here we present a covariant quantization scheme for
pseudoclassical theories focused on non-hermitian quantum mechanics. The
quantization maps canonically related pseudoclassical theories to
equivalent quantum realizations in arbitrary dimensions. We apply the
formalism to the problem of two coupled spins with Heisenberg interaction.

Non-Hermitian Casimir effect of magnons

There has been a growing interest in non-Hermitian quantum mechanics. The key concepts of quantum mechanics are quantum fluctuations. Quantum fluctuations of quantum fields confined in a finite-size system induce the zero-point energy shift. This quantum phenomenon, the Casimir effect, is one of the most striking phenomena of quantum mechanics in the sense that there are no classical analogs and has been attracting much attention beyond the hierarchy of energy scales, ranging from elementary particle physics to condensed matter physics, together with photonics. However, the non-Hermitian extension of the Casimir effect and the application to spintronics have not yet been investigated enough, although exploring energy sources and developing energy-efficient nanodevices are its central issues. Here we fill this gap. By developing a magnonic analog of the Casimir effect into non-Hermitian systems, we show that this non-Hermitian Casimir effect of magnons is enhanced as the Gilbert damping constant (i.e., the energy dissipation rate) increases. When the damping constant exceeds a critical value, the non-Hermitian Casimir effect of magnons exhibits an oscillating behavior, including a beating one, as a function of the film thickness and is characterized by the exceptional point. Our result suggests that energy dissipation serves as a key ingredient of Casimir engineering.

Features, Paradoxes and Amendments of Perturbative Non-Hermitian Quantum Mechanics

Quantum mechanics of unitary systems is considered in quasi-Hermitian representation and in the dynamical regime in which one has to take into account the ubiquitous presence of perturbations, random or specific. In this paper, it is shown that multiple technical obstacles encountered in such a context can be circumvented via just a mild amendment of the so-called Rayleigh–Schrödinger perturbation–expansion approach. In particular, the quasi-Hermitian formalism characterized by an enhancement of flexibility is shown to remain mathematically tractable while, on the phenomenological side, opening several new model-building horizons. It is emphasized that they include, i.a., the study of generic random perturbations and/or of multiple specific non-Hermitian toy models. In parallel, several paradoxes and open questions are shown to survive.

Exploring the impact of fluctuation-induced criticality on non-Hermitian skin effect and quantum sensors

In this paper we present a concrete example comparing the results predicted by non-Hermitian quantum mechanics with those of a more comprehensive description that considers environment-induced fluctuations. Our results highlight inaccuracies in the non-Hermitian model. Specifically, we investigate the non-Hermitian skin effect and sensor in the Hatano-Nelson model, contrasting it with a more precise Lindblad description. Our analysis reveals that these phenomena can undergo breakdown when environmental fluctuations come to the forefront, resulting in a nonequilibrium phase transition from a localized skin phase to a delocalized phase. Beyond this specific case study, we engage in a broader discussion regarding the interpretations and implications of non-Hermitian quantum mechanics. This examination serves to broaden our understanding of these phenomena and their potential consequences. Published by the American Physical Society 2024

Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics

Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbol-like operators, and the Schroedinger equation as a parallel transport in this formalism. We then derive the evolution equations for the states and metrics along the emergent dimensions and find that the curvature of the Hilbert space bundle for any given closed system is locally flat. Finally, we show that the fidelity susceptibilities and the Berry curvatures of states are related to these emergent parallel transports.

On Unitarity of the Scattering Operator in Non-Hermitian Quantum Mechanics

On Unitarity of the Scattering Operator in Non-Hermitian Quantum Mechanics

Wave beams, packets and pulses in inhomogeneous non-Hermitian media with dispersive gain or damping

Wave beams, packets or pulses are known to be subject to a drift if the properties of the medium change across their extension. This effect is often analyzed considering the dispersive properties of the oscillation, related to the real part of the dispersion relation. The evolution of Gaussian beams/packets/pulses in nonuniform media in the presence of gain or damping is investigated in detail, with particular emphasis on the role of dispersion on both the real and the imaginary part of the dispersion relation. In the paraxial limit, the influence of a non-Hermitian medium on the evolution of the wave can be treated employing the equations derived by Graefe and Schubert in the frame of non-Hermitian quantum mechanics (Phys. Rev. A 83 060101(R)). Analytic solutions of the corresponding paraxial equations are obtained here for a one-dimensional complex dispersion relation characterized by a linear or quadratic dependence on the transverse coordinate (a space coordinate for beams and packets, the time in the co-moving frame for a pulse). In the presence of a constant gradient in both the real and the imaginary part of the dispersion relation, the contribution of the latter can lead to a faster or slower propagation with respect to the Hermitian case, depending on the parameters. In focusing media, a constant gain can counteract dispersive or inhomogeneous damping producing packets of asymptotically constant shape. The analytic formulas derived in this paper offer a way to predict or control the properties of beams/packets/pulses depending on their initial conditions and on the characteristics of the medium.

On the η pseudo symmetry theory for non-Hermitian Hamiltonians: time-dependent systems

In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo symmetry, i.e. the non-Hermitian Hamiltonian H is related to its adjoint H † via the relation, . We propose a derivation of pseudo symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new symmetry operator that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1, 1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new symmetry operator and discuss a concrete physical applications of our results.Dedicated to the memory of my mother Djabou Zoulikha and to Djabi Smail

Computation of the complex energy shift of decaying states in intense laser fields

We present a specific calculation concerning laser-assisted decaying systems with non-Hermitian spectral calculations merged with the (t,t’)-formalism to estimate the possible intense laser-field-induced corrections to the lifetime of quasi-stationary states. The perturbed complex energy shift caused by the external vector potential which represents the linearly and circularly polarized laser field is calculated in details. This wavefunction-centered method implies the unsuitability of the radiation gauge and length gauge when quasi-stationary states are considered and described within the framework of non-hermitian quantum mechanics since in this case the A2 term cannot be transformed out.

Interatomic and intermolecular Coulombic decay rates from equation-of-motion coupled-cluster theory with complex basis functions.

When a vacancy is created in an inner-valence orbital of a dimer of atoms or molecules, the resulting species can undergo interatomic/intermolecular Coulombic decay (ICD): the hole is filled through a relaxation process that leads to a doubly ionized cluster with two positively charged atoms or molecules. Since they are subject to electronic decay, inner-valence ionized states are not bound states but electronic resonances whose transient nature can only be described with special quantum-chemical methods. In this work, we explore the capacity of equation-of-motion coupled-cluster theory with two techniques from non-Hermitian quantum mechanics, complex basis functions and Feshbach-Fano projection with a plane wave description of the outgoing electron, to describe ICD. To this end, we compute the decay rates of several dimers: Ne2, NeAr, NeMg, and (HF)2, among which the energy of the outgoing electron varies between 0.3 and 16eV. We observe that both methods deliver better results when the outgoing electron is fast, but the characteristic R-6 distance dependence of the ICD width is captured much better with complex basis functions.

Nonequilibrium diffusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator.

The nonequilibrium Fokker-Planck dynamics in an arbitrary force field f[over ⃗](x[over ⃗]) in dimension N is revisited via the correspondence with the non-Hermitian quantum mechanics in a real scalar potential V(x[over ⃗]) and in a purely imaginary vector potential [-iA[over ⃗](x[over ⃗])] of real amplitude A[over ⃗](x[over ⃗]). The relevant parameters of irreversibility are then the N(N-1)/2 magnetic matrix elements B_{nm}(x[over ⃗])=-B_{mn}(x[over ⃗])=∂_{n}A_{m}(x[over ⃗])-∂_{m}A_{n}(x[over ⃗]), while it is enlightening to explore the corresponding gauge transformations of the vector potential A[over ⃗](x[over ⃗]). This quantum interpretation is even more fruitful to study the statistics of all the time-additive observables of the stochastic trajectories, since their generating functions correspond to the same quantum problem with additional scalar and/or vector potentials. Our main conclusion is that the analysis of their large deviations properties and the construction of the corresponding Doob conditioned processes can be drastically simplified via the choice of an appropriate gauge for each purpose. This general framework is then applied to the special time-additive observables of Ornstein-Uhlenbeck trajectories in dimension N, whose generating functions correspond to quantum propagators involving quadratic scalar potentials and linear vector potentials, i.e., to quantum harmonic oscillators in constant magnetic matrices. As simple illustrative example, we finally focus on the Brownian gyrator in dimension N=2 to compute the large deviations properties of the entropy production of its stochastic trajectories and to construct the corresponding conditioned processes having a given value of the entropy production per unit time.

Radiative anti-parity-time plasmonics

Space and guided electromagnetic waves, as widely known, are two crucial cornerstones in extensive wireless and integrated applications respectively. To harness the two cornerstones, radiative and integrated devices are usually developed in parallel based on the same physical principles. An emerging mechanism, i.e., anti-parity-time (APT) symmetry originated from non-Hermitian quantum mechanics, has led to fruitful phenomena in harnessing guided waves. However, it is still absent in harnessing space waves. Here, we propose a radiative plasmonic APT design to harness space waves, and experimentally demonstrate it with subwavelength designer-plasmonic structures. We observe two exotic phenomena unrealized previously. Rotating polarizations of incident space waves, we realize polarization-controlled APT phase transition. Tuning incidence angles, we observe multi-stage APT phase transition in higher-order APT systems, constructed by using the scalability of leaky-wave couplings. Our scheme shows promise in demonstrating novel APT physics, and constructing APT-symmetry-empowered radiative devices.

Spectroscopic signatures of states in the continuum characterized by a joint experimental and theoretical study of pyrrole.

We report a combined experimental and theoretical investigation of electron-molecule interactions using pyrrole as a model system. Experimental two-dimensional electron energy loss spectra (EELS) encode information about the vibrational states of the molecule as well as the position and structure of electronic resonances. The calculations using complex-valued extensions of equation-of-motion coupled-cluster theory (based on non-Hermitian quantum mechanics) facilitate the assignment of all major EELS features. We confirm the two previously described π resonances at about 2.5 and 3.5eV (the calculations place these two states at 2.92 and 3.53eV vertically and 2.63 and 3.27eV adiabatically). The calculations also predict a low-lying resonance at 0.46eV, which has a mixed character-of a dipole-bound state and σ* type. This resonance becomes stabilized at one quanta of the NH excitation, giving rise to the sharp feature at 0.9eV in the corresponding EELS. Calculations of Franck-Condon factors explain the observed variations in the vibrational excitation patterns. The ability of theory to describe EELS provides a concrete illustration of the utility of non-Hermitian quantum chemistry, which extends such important concepts as potential energy surfaces and molecular orbitals to states embedded in the continuum.

Nonadiabatic dissociation of molecular Bose-Einstein condensates: Competition between chemical reactions

We provide a framework to solve generic models describing the dissociation of multiple molecular Bose-Einstein condensates in a nonadiabatic regime. The competition between individual chemical reactions can lead to non-trivial dependence on critical components such as path interference and symmetries, thus, affecting the final distribution of atomic population. We find an analytical solution for an illustrative example model involving four atomic modes. When the system parameters satisfy $CPT$ symmetry, where $C$ is charge conjugation, $P$ is parity, and $T$ is time-reversal symmetry, our solution predicts a population imbalance between atomic modes that is exponentially sensitive to system parameters. However, a weakly broken symmetry alters the population in each atomic mode and can reverse the population imbalance. Our solution also demonstrates a strong quantum correlation between atomic modes that leads to the spontaneous production of atoms in a multi-mode squeezed state. Moreover, in our framework, a time-dependent non-Hermitian quantum mechanics naturally manifests which can alternatively be realized experimentally in photonic systems.

Non-Hermitian critical dynamics and its application to quantum many-body systems

In recent years, two independent research fields, i.e. non-Hermitian andstrongly correlated systems have been merged, forming an important researchfield in physics. The progress of relevant theories and experiments hasreshaped our understanding of matter. In this field, the research object isnot limited to the influence of non-Hermiticity on the energy spectrum andthe eigenstate properties of many-body systems. Researchers have paid more attentionto the manipulation of quantum states. It is universally received that the exceptional point is the most significant featurethat distinguishes non-Hermitian quantum mechanics from Hermitian quantum mechanics. In addition to the recent advances in non-Hermitian topological band theory and quantum sensing around the exceptional points, this paper concentrates on the non-Hermitian critical dynamical phenomenon and its application to the quantum many-body system. When the system has an exceptional point, an arbitrary initial state belonging to the coalescent subspace will be projected on the coalescent state. Based on the directionality of the evolved quantum state, this paper reviews our several representative researches in recent years, including local-field-induced dynamical magnetization, quantum phase transition in transverse field, Ising model at non-zero temperature, quantum mold casting in the center-environment system, as well as superconducting state preparation in the non-Hermitian strongly correlated system. We also focus on the new preparation methods and detection schemes of non-equilibrium quantum states related to exception points.

Building ground states of the Hubbard model by time-ordered bound-pair injection

According to energy band theory, ground states of a normal conductor and insulator can be obtained by filling electrons individually into energy levels, without any restrictions. It fails when the electron-electron correlation is taken into account. In this work, we investigate the dynamic process of building ground states of a Hubbard model. It is based on time-ordered quantum quenches for unidirectional hopping across a central and an auxiliary Hubbard model. We find that there exists a set of optimal parameters (chemical potentials and pair binding energy) for the auxiliary system, which takes the role of electron-pair reservoir. The exceptional point dynamics in non-Hermitian quantum mechanics allows the perfect transfer of electron pair from the reservoir to the central system, obtaining its ground states at different fillings. The dynamics of time-ordered pair filling not only provides a method for correlated quantum state engineering, but also reveals the feature of the ground state in an alternative way.

Intracavity Measurement Sensitivity Enhancement without Runaway Noise.

A method to increase the sensitivity of an intracavity differential phase measurement that is not made irrelevant by a larger increase of noise is explored. By introducing a phase velocity feedback by way of a resonant dispersive element in an active sensor in which two ultrashort pulses circulate, it is shown that the measurement sensitivity is elevated without significantly increasing the Petermann excess noise factor. This enhancement technique has considerable implications for any optical phase based measurement; from gyroscopes and accelerometers to magnetometers and optical index measurements. Here we describe the enhancement method in the context of past dispersion enhancement studies including the recent work surrounding non-Hermitian quantum mechanics, justify the method with a theoretical framework (including numerical simulations), and propose practical applications.

Hidden Dynamical Symmetry and Quantum Thermodynamics from the First Principles: Quantized Small Environment

Evolution of a self-consistent joint system (JS), i.e., a quantum system (QS) + thermal bath (TB), is considered within the framework of the Langevin–Schrödinger (L-Sch) type equation. As a tested QS, we considered two linearly coupled quantum oscillators that interact with TB. The influence of TB on QS is described by the white noise type autocorrelation function. Using the reference differential equation, the original L-Sch equation is reduced to an autonomous form on a random space–time continuum, which reflects the fact of the existence of a hidden symmetry of JS. It is proven that, as a result of JS relaxation, a two-dimensional quantized small environment is formed, which is an integral part of QS. The possibility of constructing quantum thermodynamics from the first principles of non-Hermitian quantum mechanics without using any additional axioms has been proven. A numerical algorithm has been developed for modeling various properties and parameters of the QS and its environment.

Quantum thermodynamics aspects with a thermal reservoir based on -symmetric Hamiltonians

We present results concerning aspects of quantum thermodynamics under the background of non-Hermitian quantum mechanics for the dynamics of a quantum harmonic oscillator. Since a better control over the parameters in quantum thermodynamics processes is desired, we use concepts from collisional model to introduce a simple prototype of thermal reservoir based on -symmetric Hamiltonians and study its effects under the thermalization process of a single harmonic oscillator prepared in a displaced thermal state. We verify that controlling the -symmetric features of the reservoir allows to reverse the heat flow between system and reservoir, as well as to preserve the coherence over a longer period of time and reduce the entropy production. Furthermore, we considered a modified quantum Otto cycle in which the standard hot thermal reservoir is replaced by the thermal reservoir based on -symmetric Hamiltonians. By defining an effective temperature depending on the -symmetric parameter, it is possible to interchange the quantum Otto cycle configuration from engine to refrigerator by varying the -symmetric parameter. Our results indicate that -symmetric effects could be useful to achieve an improvement in quantum thermodynamics protocols such as coherence protection and entropy production reduction.

Entanglement and Purification Transitions in Non-Hermitian Quantum Mechanics.

A quantum system subject to continuous measurement and postselection evolves according to a non-Hermitian Hamiltonian. We show that, as one increases the strength of postselection, this non-Hermitian Hamiltonian can undergo a spectral phase transition. On one side of this phase transition (for weak postselection), an initially mixed density matrix remains mixed at all times, and an initially unentangled state develops volume-law entanglement; on the other side, an arbitrary initial state approaches a unique pure state with low entanglement. We identify this transition with an exceptional point in the spectrum of the non-Hermitian Hamiltonian, at which PT symmetry is spontaneously broken. We characterize the transition as well as the nontrivial steady state that emerges at late times in the mixed phase using exact diagonalization and an approximate, analytically tractable mean-field theory; these methods yield consistent conclusions.