pseudo-Hermitian Hamiltonians Research Articles

Generalized unitary evolution for symplectic scalar fermions

The theory of symplectic scalar fermion of LeClair and Neubert is studied. The theory evades the conventional spin-statistics theorem because its Hamiltonian is pseudo Hermitian. The definition of pseudo Hermiticity is examined in the interacting and the Heisenberg picture. For states that evolve under pseudo Hermitian Hamiltonians, we define the appropriate inner-product and matrix element of operators that preserve time translation symmetry. The resulting S-matrix is shown to satisfy the generalized unitarity relation. We clarify the derivation of the symplectic currents and charges. By demanding the currents and charges to be pseudo Hermitian, the global symmetry of the free Lagrangian density reduces from Sp(2, ℂ) to SU(2). By explicit calculations, we show that the LeClair-Neubert model of N quartic self-interacting scalar fermions admits generalized unitary evolution.

Strengthening the atom-field coupling through the deep-strong regime via pseudo-Hermitian Hamiltonians

We present a strategy for strengthening the atom-field interaction through a pseudo-Hermitian Jaynes-Cummings Hamiltonian. Apart from the engineering of an effective non-Hermitian Hamiltonian, our strategy also relies on the accomplishment of short-time measurements on canonically conjugate variables. The resulting fast Rabi oscillations may be used for many quantum optics purposes and specially to shorten the processing time of quantum information.

THE METRIC OPERATORS FOR PSEUDO-HERMITIAN HAMILTONIAN

AbstractThe Hamiltonian of a conventional quantum system is Hermitian, which ensures real spectra of the Hamiltonian and unitary evolution of the system. However, real spectra are just the necessary conditions for a Hamiltonian to be Hermitian. In this paper, we discuss the metric operators for pseudo-Hermitian Hamiltonian which is similar to its adjoint. We first present some properties of the metric operators for pseudo-Hermitian Hamiltonians and obtain a sufficient and necessary condition for an invertible operator to be a metric operator for a given pseudo-Hermitian Hamiltonian. When the pseudo-Hermitian Hamiltonian has real spectra, we provide a new method such that any given metric operator can be transformed into the same positive-definite one and the new inner product with respect to the positive-definite metric operator is well defined. Finally, we illustrate the results obtained with an example.

Generalized gauge transformation and the corresponding Hermitian counterparts of SU(1, 1), SU(2) pseudo-Hermitian Hamiltonians

We study in this paper both the stationary and time-dependent pseudo-Hermitian Hamiltonians consisting respectively of SU(1, 1), SU(2) generators. The pseudo-Hermitian Hamiltonians can be generated from kernel Hermitian-Hamiltonians by a generalized gauge transformation with a non-unitary but Hermitian operator. The metric operator of the biorthogonal sets of eigenstates is simply the square of the transformation operator, which is formulated explicitly. The exact solutions of pseudo-Hermitian Hamiltonians are obtained in terms of the eigenststates of the Hermitian counterparts. We observe a critical point G c of coupling constant, where all eigenstates of the stationary Hamiltonians are degenerate with a vanishing eigenvalue. This critical point is modified as G c (ω) in the time-dependent case including the frequency of external field. Returning to the original gauge we obtain analytically the wave functions and associated non-adiabatic Berry phase, which diverges at the critical point for the SU(2)Hamiltonian. Beyond the critical point Berry phase becomes a complex domain.

Pseudo-Hermitian Levin–Wen models from non-semisimple TQFTs

We construct large classes of exactly solvable pseudo-Hermitian 2D spin Hamiltonians. The ground states of these systems depend only on the spatial topology of the system. We identify the ground state system on a surface with the value assigned to the surface by a non-semisimple TQFT generalizing the Turaev–Viro model. A non-trivial example arises from a non-semisimple subcategory of representations of quantum sl(2) where the quantum parameter is specialized to a root of unity.

Topological study of a Bogoliubov–de Gennes system of pseudo-spin- 1/2 bosons with conserved magnetization in a honeycomb lattice

We consider a Bogolibov-de Geenes (BdG) Hamiltonian, which is a non-Hermitian Hamiltonian with pseudo-Hermiticity, for a system of (pseudo) spin-$1/2$ bosons in a honeycomb lattice under the condition that the population difference between the two spin components, i.e., magnetization, is a constant. Such a system is capable of acting as a topological amplifier, under time-reversal symmetry, with stable bulk bands but unstable edge modes which can be populated at an exponentially fast rate. We quantitatively study the topological properties of this model within the framework of the 38-fold way for non-Hermitian systems. We find, through the symmetry analysis of the Bloch Hamiltonian, that this model is classified either as two copies of symmetry class AIII+$\eta_-$ or two copies of symmetry class A+$\eta$ depending on whether the (total) system is time-reversal-symmetric, where $\eta$ is the matrix representing pseudo-Hermiticity and $\eta_-$ indicates that pseudo-Hermiticity and chiral symmetry operators anticommute. We prove, within the context of non-Hermitian physics where eigenstates obey the bi-orthonormality relation, that a stable bulk is characterized by a single topological invariant, the Chern number for the Haldane model, independent of pairing interactions. We construct a convenient analytical description for the edge modes of the Haldane model in semi-infinite planes, which is expected to be useful for models built upon copies of the Haldane model across a broad array of disciplines. We adapt the theorem in our recent work [Phys. Rev. A 104, 013305 (2021)] to pseudo-Hermitian Hamiltonians that are less restrictive than BdG Hamiltonians and apply it to highlight that the vanishing of an unconventional commutator between number-conserving and number-nonconserving parts of the Hamiltonian indicates whether a system can be made to act as a topological amplifier.

Generalized Gauge Transformation with Non-Unitary Operator and Realization of Hermitian Counterparts of Time-Dependent Pseudo-Hermitian Hamiltonians

Generalized Gauge Transformation with Non-Unitary Operator and Realization of Hermitian Counterparts of Time-Dependent Pseudo-Hermitian Hamiltonians

Preface PT Seminars 2020

Non-Hermitian theories that are explicitly PT-symmetric and/or quasi/pseudo-Hermitian have become the subject of an active research field for the last two decades. They constitute well-defined classical and quantum theories possessing real eigenvalue spectra and unitary time-evolution. Since the underlying concepts are generic, the general principles have been applied in essentially all areas of physics and have unravelled new theoretical insights and experimental realisations especially in optical applications.Beginning in 2003 a regular conference series entitled “Pseudo-Hermitian Hamiltonians in Quantum Physics” has taken place annually to bring together researchers in order to take stock of the current state of the field and to assist participants to collaborate in developing new ideas. Unfortunately, the 20th meeting in this series, which was scheduled originally for 2020 to be held in Los Alamos had to be postponed to 2022 due to the outbreak of the Covid 19 pandemic. To bridge the gap and to provide a platform to allow for a continued exchange of ideas and novelties a virtual seminar series via Zoom with a dedicated website (https://vphhqp.com) was organised by Andreas Fring and Francisco Correa. With currently more than 50 seminars scheduled and well over 200 registered participants, the series is still ongoing. The main motivation behind this special issue has been to allow speakers and participants in this series to present their results in a coherent written in the form of mini-reviews of certain subtopics in the field and new research results.We hope that this special issue will become a valuable reference and inspiration for a broad scientific community, experts and newcomers alike, working in mathematical and theoretical physics. Carl Bender, Department of Physics, Washington University, St. Louis, MO 63130, USA;cmb@wuphys. wustl. edu Francisco Correa, Institute de e Ciencias Físicas y Matemáticas , Universidad Austral de ChileCasilla 567, Valdivia, Chile; francisco.correa@uach.cl Andreas Fring, Department of Mathematics, City, University of London, Northampton SquareLondon EC1V 0HB, UK; a.fring@city.ac.uk

A time-dependent pseudo-Hermitian Hamiltonian for a cavity mode with pure imaginary frequency

In this work we first present a general treatment for a time-dependent (TD) pseudo-Hermitian quadratic Hamiltonian, considering an equally TD and non-Hermitian Dyson map. Then we particularize our study to a TD pseudo-Hermitian Hamiltonian describing a cavity mode with pure imaginary frequency. Our goal is to produce squeezed states of the radiation field with an infinite degree of squeezing at a finite time interval. This mechanism is prevented by Hermitian Hamiltonians which allow an infinite degree of squeezing only asymptotically in time. Beyond the specifics of the technical treatment of the problem, we also present a discussion of its experimental implementation in the domain of the radiation–matter interaction. Such a discussion, is crucial to put into perspective the relevance of the quantum mechanics of pseudo-Hermitian Hamiltonians, whose effects can go beyond those coming from Hermitian Hamiltonians. Basically, we conclude that pseudo-Hermitian Hamiltonians enable much higher pumping rates than those of Hermitian processes.

Unconventional scaling at non-Hermitian critical points

Critical phase transitions contain a variety of deep and universal physics, and are intimately tied to thermodynamic quantities through scaling relations. Yet, these notions are challenged in the context of non-Hermiticity, where spatial or temporal divergences render the thermodynamic limit ill-defined. In this work, we show that a thermodynamic grand potential can still be defined in pseudo-Hermitian Hamiltonians, and can be used to characterize aspects of criticality unique to non-Hermitian systems. Using the non-Hermitian Su-Schrieffer-Heeger (SSH) model as a paradigmatic example, we demonstrate the fractional order of topological phase transitions in the complex energy plane. These fractional orders add up to the integer order expected of a Hermitian phase transition when the model is doubled and Hermitianized. More spectacularly, gap preserving highly degenerate critical points known as non-Bloch band collapses possess fractional order that are not constrained by conventional scaling relations, testimony to the emergent extra length scale from the skin mode accumulation. Our work showcases that a thermodynamic approach can prove fruitful in revealing unconventional properties of non-Hermitian critical points.

Adiabatic theorem and generalized geometrical phase in the case of pseudo-Hermitian systems

A generalization of the adiabatic theorem for quantum systems governed by pseudo-Hermitian Hamiltonians and details of its demonstration are given. Introducing a modified time-dependent metric giving a precise description of the quantum unitary evolution where the obtained effective Hamiltonian is observable because it is mean value is real. We show that an eigenstate of a pseudo-Hermitian Hamiltonian slowly transported will acquire a real generalized geometrical phase factor which contains two contributions: the first one corresponds to the conventional Berry’s phase as expected and a new geometrical term that we call the metric geometrical phase. We will apply our results in the cases of the famous time dependent brachistochrone problem and a non-Hermitian displaced harmonic oscillator.

Optimal time evolution for pseudo-Hermitian Hamiltonians

If an initial state $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ and a final state $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ are given, then there exist many Hamiltonians under whose action $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ evolves into $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ . In this case, the problem of the transition of $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ to $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ in the least time is very interesting. It was previously shown that for a Hermitian Hamiltonian, there is an optimum evolution time if $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ and $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ are orthogonal. But for a $$PT$$ -symmetric Hamiltonian, this time can be arbitrarily small, which seems amazing. We discuss the optimum time evolution for pseudo-Hermitian Hamiltonians and obtain a lower bound for the evolution time under the condition that the Hamiltonian is bounded. The optimum evolution time can be attained in the case where two quantum states are orthogonal with respect to some inner product. The results in the Hermitian and pseudo-Hermitian cases coincide if the evolution is unitary with some well-defined inner product. We also analyze two previously studied examples and find that they are consistent with our theory. In addition, we give some explanations of our results with two examples.

Towards Generalized Riesz Systems Theory

Pseudo-Hermitian Hamiltonians have recently become a field of wide investigation. Originally, the generalized Riesz systems (GRS) have been introduced as an auxiliary tool in this theory. In contrast, the current paper, GRSs are analysed in terms of basis theory. The relationship between semi-regular sequences and GRSs is provided. Various characterizations of GRSs are discussed.

Pseudo-Hermitian Systems with P T $\mathcal {P}\mathcal {T}$ -Symmetry: Degeneracy and Krein Space

We show in the present paper that pseudo-Hermitian Hamiltonian systems with even PT-symmetry admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd PT-symmetric systems which is appropriate to the fermions as shown by Jones-Smith and Mathur [1] who extended PT-symmetric quantum mechanics to the case of odd time-reversal symmetry. We establish that the pseudo-Hermitian Hamiltonians with even PT-symmetry admit a degeneracy structure if the operator PT anticommutes with the metric operator {\eta} which is necessarily indefinite. We also show that the Krein space formulation of the Hilbert space is the convenient framework for the implementation of unbroken PT-symmetry. These general results are illustrated with great details for four-level pseudo-Hermitian Hamiltonian with even PT-symmetry.

PT symmetry and a dynamical realization of the SU(1, 1) algebra

We show that the elementary modes of the planar harmonic oscillator can be quantized in the framework of quantum mechanics based on pseudo-hermitian Hamiltonians. These quantized modes are demonstrated to act as dynamical structures behind a new Jordan–Schwinger realization of the SU(1, 1) algebra. This analysis complements the conventional Jordan–Schwinger construction of the SU(2) algebra based on hermitian Hamiltonians of a doublet of oscillators.

Elementary modes of coupled oscillators as whispering-gallery microresonators

We obtain the elementary modes of a system of parity-time reversal (PT)-symmetric coupled oscillators with balanced loss and gain. These modes are used to give a physical picture of the phase transition recently reported [C. M. Bender, M. Gianfreda, B. Peng, S. K. Özdemir and L. Yang, Phys. Rev. A 88, 062111 (2013); L. Yang, S. K. Özdemir and B. Peng, 12th Int. Workshop and Conf. Pseudo-Hermitian Hamiltonians in Quantum Physics, Istanbul, Turkey, July 2013; B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender and L. Yang, Nat. Phys. 10, 394 (2014)] in experiments with whispering-gallery microresonators.

Exact solutions for non-Hermitian Dirac-Pauli equation in an intensive magnetic field

We consider modified Dirac–Pauli equations that are entered using -mass factorization, , of an ordinary Klein–Gordon operator. We also consider the interaction of fermions with an intensive uniform magnetic field, focusing on their gyromagnetic factor. Due to effective research procedures, we derive the exact solutions of the enregy spectra of pseudo-Hermitian Hamiltonians, taking into account the spin of the fermions. The basic research methods are the elucidation of the new border areas of the unbroken symmetry of non-Hermitian Hamiltonians. In particular, it is shown that the reality energy spectrum of fermions at rest can be expressed by limiting the intensity of the magnetic field, where is an anomalous magnetic moment of particles.

Special Issue “Pseudo-Hermitian Hamiltonians in Quantum Physics in 2014”

The concept as well as the collection of contributions to the present Special Issue found its main inspiration in the success of the ninth continuation of the series of international conferences devoted to the same subject. This inspiring meeting with informative homepage address http://www.math.zju.edu.cn/wjd took place in Zhejiang University in Hangzhou during June 21–24, 2010. As usual, the attention of cca 60 participants of the meeting (cf. the photo at frontispiece) transcended the strict scope of the PHHQP series, this time mainly towards the rich and quickly developing area of the so called effect algebras and other deeper layers of mathematical foundations of quantum physics. Within this overall framework our present special issue of IJTP offers a compact and well balanced sample of the results obtained in the subject areas covered by the meeting. The readership of our Special Issue will find here not only several up-to-date reports on the current state of development of the field but also, and predominantly, multiple carefully refereed samples of the original research written, naturally, not only by the direct participants of the conference. The latter paradox of the participation of non-participants is in fact not too surprising. Indeed, both the main subject areas covered by this Special Issue belong among the fresh and most quickly developing new subfields of quantum physics and related mathematics. For this reason the major part of the presented scientific material should be perceived as an integral part of the mainstream publication activity of the authors. In this sense both the Guest Editors would like to formulate and express their personal gratitude not only to all of the organizers of the ninth PHHQP meeting but, first of all, to all of the authors who accepted the challenge and who did contribute, to our Special Issue, with compact but still high-quality presentations of their most recent and interesting results.

Orthogonal polynomial projection quantization: a new Hill determinant method

Accurate energy eigenvalues are obtained by simply projecting the unknown bound state wave function on, essentially, arbitrary sets of orthogonal polynomials, and setting a subset of these to zero. The projection integrals are represented in terms of the power moments of the wave function, obtained recursively by transforming Schrödinger’s equation into a moment equation. Because unbounded wave functions do not have power moments, all solutions are guaranteed to be L2, resulting in a more robust, rapidly converging and stable method when compared with configuration space Hill determinant methods. More importantly, our approach permits the use of arbitrary, nonanalytic, positive reference functions, including those that manifest the true asymptotic behavior of the discrete states. These advantages are not usually possible with the standard Hill approach. The formulation presented here can be applied to any problem for which Schrödinger’s equation can be transformed into a moment equation. In this regard it is related to the L2 quantization prescription developed by Tymczak et al (1998 Phys. Rev. Lett. 80 3674; 1998 Phys. Rev. A 58 2708) corresponding to the Hill determinant method in momentum space, and to the eigenvalue moment method (Handy and Bessis 1985 Phys. Rev. Lett. 55 931; Handy et al 1988 Phys. Rev. Lett. 60 253), the first use of semidefinite programming related analysis in quantum physics. The latter method can be used to generate the ground state, whose orthogonal polynomials can serve to generate even more rapidly converging estimates for the excited discrete state energies. We demonstrate the power of this new approach on the sextic and quartic anharmonic oscillators, as well as on recently studied one- and two-dimensional pseudo-Hermitian Hamiltonians.

Snyder noncommutativity and pseudo-Hermitian Hamiltonians from a Jordanian twist

Nonrelativistic quantum mechanics and conformal quantum mechanics are deformed through a Jordanian twist. The deformed space coordinates satisfy the Snyder noncommutativity. The resulting deformed Hamiltonians are pseudo-Hermitian Hamiltonians of the type discussed by Mostafazadeh. The quantization scheme makes use of the so-called “unfolded formalism” discussed in previous works. A Hopf algebra structure, compatible with the physical interpretation of the coproduct, is introduced for the universal enveloping algebra of a suitably chosen dynamical Lie algebra (the Hamiltonian is contained among its generators). The multi-particle sector, uniquely determined by the deformed two-particle Hamiltonian, is composed of bosonic particles.