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Journal articles on the topic 'Non-Hermitian Hamiltonian'

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Published: 6 September 2023

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1

Yeşiltaş, Özlem. "Non-Hermitian Dirac Hamiltonian in Three-Dimensional Gravity and Pseudosupersymmetry." Advances in High Energy Physics 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/484151.

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The Dirac Hamiltonian in the(2+1)-dimensional curved space-time has been studied with a metric for an expanding de Sitter space-time which is two spheres. The spectrum and the exact solutions of the time dependent non-Hermitian and angle dependent Hamiltonians are obtained in terms of the Jacobi and Romanovski polynomials. Hermitian equivalent of the Hamiltonian obtained from the Dirac equation is discussed in the frame of pseudo-Hermiticity. Furthermore, pseudosupersymmetric quantum mechanical techniques are expanded to a curved Dirac Hamiltonian and a partner curved Dirac Hamiltonian is generated. Usingη-pseudo-Hermiticity, the intertwining operator connecting the non-Hermitian Hamiltonians to the Hermitian counterparts is found. We have obtained a new metric tensor related to the new Hamiltonian.

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2

Samsonov, Boris F. "Hermitian Hamiltonian equivalent to a given non-Hermitian one: manifestation of spectral singularity." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1989 (April 28, 2013): 20120044. http://dx.doi.org/10.1098/rsta.2012.0044.

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One of the simplest non-Hermitian Hamiltonians, first proposed by Schwartz in 1960, that may possess a spectral singularity is analysed from the point of view of the non-Hermitian generalization of quantum mechanics. It is shown that the η operator, being a second-order differential operator, has supersymmetric structure. Asymptotic behaviour of the eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result, the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible presence of a spectral singularity in the spectrum of the non-Hermitian Hamiltonian may be detected as a resonance in the scattering cross section of its Hermitian counterpart. Nevertheless, just at the singular point, the equivalent Hermitian Hamiltonian becomes undetermined.

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3

Grimaudo, Roberto, Antonino Messina, Alessandro Sergi, Nikolay V. Vitanov, and Sergey N. Filippov. "Two-Qubit Entanglement Generation through Non-Hermitian Hamiltonians Induced by Repeated Measurements on an Ancilla." Entropy 22, no. 10 (October 20, 2020): 1184. http://dx.doi.org/10.3390/e22101184.

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In contrast to classical systems, actual implementation of non-Hermitian Hamiltonian dynamics for quantum systems is a challenge because the processes of energy gain and dissipation are based on the underlying Hermitian system–environment dynamics, which are trace preserving. Recently, a scheme for engineering non-Hermitian Hamiltonians as a result of repetitive measurements on an ancillary qubit has been proposed. The induced conditional dynamics of the main system is described by the effective non-Hermitian Hamiltonian arising from the procedure. In this paper, we demonstrate the effectiveness of such a protocol by applying it to physically relevant multi-spin models, showing that the effective non-Hermitian Hamiltonian drives the system to a maximally entangled stationary state. In addition, we report a new recipe to construct a physical scenario where the quantum dynamics of a physical system represented by a given non-Hermitian Hamiltonian model may be simulated. The physical implications and the broad scope potential applications of such a scheme are highlighted.

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4

Sharma, Preet. "𝒫𝒯-Symmetric Quantum Mechanics Basics & Zeeman Effect." Reports in Advances of Physical Sciences 04, no. 03 (September 2020): 2050006. http://dx.doi.org/10.1142/s2424942420500061.

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The non-Hermitian aspect of Quantum Mechanics has been of great interest recently. There have been numerous studies on non-Hermitian Hamiltonians written for natural processes. Some studies have even expressed the hydrogen atom in a non-Hermitian basis. In this paper, the principles of non-Hermitian quantum mechanics are applied to the time independent perturbation theory and compared with the Zeeman effect. Here, we have also shown the condition under which the Zeeman Effect results will still be true even though the Hamiltonian taken into consideration is non-Hermitian.

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5

Hallford, Randal, and Preet Sharma. "Non-Hermitian Hamiltonian Treatment of Stark Effect in Quantum Mechanics." Emerging Science Journal 4, no. 6 (December 1, 2020): 427–35. http://dx.doi.org/10.28991/esj-2020-01242.

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The Non-Hermitian aspect of Quantum Mechanics has been of great interest recently. There have been numerous studies on non-Hermitian Hamiltonians written for natural processes. Some studies have even expressed the hydrogen atom in a non-Hermitian basis. In this paper the principles of non-Hermitian quantum mechanics is applied to both the time independent perturbation theory and to the time dependant theory to calculate the Stark effect. The principles of spherical harmonics has also been used to describe the development in the non-Hermitian case. Finally, the non-Hermitian aspect has been introduced to the well known Stark effect in quantum mechanics to find a condition in which the Stark effect will still be true even if a non-Hermitian Hamiltonian is used. This study completes the understanding at a fundamental level to understand the well known Stark effect. Doi: 10.28991/esj-2020-01242 Full Text: PDF

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6

BERMAN, GENNADY P., and ALEXANDER I. NESTEROV. "NON-HERMITIAN ADIABATIC QUANTUM OPTIMIZATION." International Journal of Quantum Information 07, no. 08 (December 2009): 1469–78. http://dx.doi.org/10.1142/s0219749909005961.

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We propose a novel non-Hermitian adiabatic quantum optimization algorithm. One of the new ideas is to use a non-Hermitian auxiliary "initial" Hamiltonian that provides an effective level repulsion for the main Hamiltonian. This effect enables us to develop an adiabatic theory which determines ground state much more efficiently than Hermitian methods.

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7

Militello, Benedetto, and Anna Napoli. "Evanescent Wave Approximation for Non-Hermitian Hamiltonians." Entropy 22, no. 6 (June 4, 2020): 624. http://dx.doi.org/10.3390/e22060624.

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The counterpart of the rotating wave approximation for non-Hermitian Hamiltonians is considered, which allows for the derivation of a suitable effective Hamiltonian for systems with some states undergoing decay. In the limit of very high decay rates, on the basis of this effective description we can predict the occurrence of a quantum Zeno dynamics, which is interpreted as the removal of some coupling terms and the vanishing of an operatorial pseudo-Lamb shift.

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8

SINHA, A., and P. ROY. "DARBOUX TRANSFORMATION FOR THE ONE-DIMENSIONAL STATIONARY DIRAC EQUATION WITH NON-HERMITIAN INTERACTION." International Journal of Modern Physics A 21, no. 28n29 (November 20, 2006): 5807–22. http://dx.doi.org/10.1142/s0217751x0603312x.

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The Darboux algorithm is applied to an exactly solvable one-dimensional stationary Dirac equation, with non-Hermitian, pseudoscalar interaction V0(x). This generates a hierarchy of exactly solvable Dirac Hamiltonians, [Formula: see text], defined by new non-Hermitian interactions V1(x), which are also pseudoscalar. It is shown that [Formula: see text] are isospectral to the initial Hamiltonian h0, except for certain missing states.

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9

Mannheim, Philip D. "PT symmetry as a necessary and sufficient condition for unitary time evolution." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1989 (April 28, 2013): 20120060. http://dx.doi.org/10.1098/rsta.2012.0060.

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While Hermiticity of a time-independent Hamiltonian leads to unitary time evolution, in and of itself, the requirement of Hermiticity is only sufficient for unitary time evolution. In this paper, we provide conditions that are both necessary and sufficient. We show that symmetry of a time-independent Hamiltonian, or equivalently, reality of the secular equation that determines its eigenvalues, is both necessary and sufficient for unitary time evolution. For any -symmetric Hamiltonian H , there always exists an operator V that relates H to its Hermitian adjoint according to V HV −1 = H † . When the energy spectrum of H is complete, Hilbert space norms 〈 ψ 1 | V | ψ 2 〉 constructed with this V are always preserved in time. With the energy eigenvalues of a real secular equation being either real or appearing in complex conjugate pairs, we thus establish the unitarity of time evolution in both cases. We also establish the unitarity of time evolution for Hamiltonians whose energy spectra are not complete. We show that when the energy eigenvalues of a Hamiltonian are real and complete, the operator V is a positive Hermitian operator, which has an associated square root operator that can be used to bring the Hamiltonian to a Hermitian form. We show that systems with -symmetric Hamiltonians obey causality. We note that Hermitian theories are ordinarily associated with a path integral quantization prescription in which the path integral measure is real, while in contrast, non-Hermitian but -symmetric theories are ordinarily associated with path integrals in which the measure needs to be complex, but in which the Euclidean time continuation of the path integral is nonetheless real. Just as the second-order Klein–Gordon theory is stabilized against transitions to negative frequencies because its Hamiltonian is positive-definite, through symmetry, the fourth-order derivative Pais–Uhlenbeck theory can equally be stabilized.

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10

Chen Zeng-Jun and Ning Xi-Jing. "Physical meaning of non-Hermitian Hamiltonian." Acta Physica Sinica 52, no. 11 (2003): 2683. http://dx.doi.org/10.7498/aps.52.2683.

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11

Fring, Andreas, and Rebecca Tenney. "Infinite series of time-dependent Dyson maps." Journal of Physics A: Mathematical and Theoretical 54, no. 48 (November 4, 2021): 485201. http://dx.doi.org/10.1088/1751-8121/ac31a0.

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Abstract We propose and explore a scheme that leads to an infinite series of time-dependent Dyson maps which associate different Hermitian Hamiltonians to a uniquely specified time-dependent non-Hermitian Hamiltonian. We identify the underlying symmetries responsible for this feature respected by various Lewis–Riesenfeld invariants. The latter are used to facilitate the explicit construction of the Dyson maps and metric operators. As a concrete example for which the scheme is worked out in detail we present a two-dimensional system of oscillators that are coupled to each other in a non-Hermitian PT -symmetrical fashion.

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12

Znojil, Miloslav. "Hermitian–Non-Hermitian Interfaces in Quantum Theory." Advances in High Energy Physics 2018 (2018): 1–12. http://dx.doi.org/10.1155/2018/7906536.

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In the global framework of quantum theory, the individual quantum systems seem clearly separated into two families with the respective manifestly Hermitian and hiddenly Hermitian operators of their Hamiltonian. In the light of certain preliminary studies, these two families seem to have an empty overlap. In this paper, we will show that whenever the interaction potentials are chosen to be weakly nonlocal, the separation of the two families may disappear. The overlapsaliasinterfaces between the Hermitian and non-Hermitian descriptions of a unitarily evolving quantum system in question may become nonempty. This assertion will be illustrated via a few analytically solvable elementary models.

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13

Gavrilik, Alexandre, and Ivan Kachurik. "Pseudo-Hermitian position and momentum operators, Hermitian Hamiltonian, and deformed oscillators." Modern Physics Letters A 34, no. 01 (January 10, 2019): 1950007. http://dx.doi.org/10.1142/s021773231950007x.

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The recently introduced by us, two- and three-parameter (p, q)- and (p, q, [Formula: see text])-deformed extensions of the Heisenberg algebra were explored under the condition of their direct link with the respective (nonstandard) deformed quantum oscillator algebras. In this paper, we explore certain Hermitian Hamiltonians build in terms of non-Hermitian position and momentum operators obeying definite [Formula: see text](N)-pseudo-hermiticity properties. A generalized nonlinear (with the coefficients depending on the particle number operator N) one-mode Bogoliubov transformation is developed as main tool for the corresponding study. Its application enables to obtain the spectrum of “almost free” (but essentially nonlinear) Hamiltonian.

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14

De Carlo, Martino, Francesco De Leonardis, Richard A. Soref, Luigi Colatorti, and Vittorio M. N. Passaro. "Non-Hermitian Sensing in Photonics and Electronics: A Review." Sensors 22, no. 11 (May 24, 2022): 3977. http://dx.doi.org/10.3390/s22113977.

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Recently, non-Hermitian Hamiltonians have gained a lot of interest, especially in optics and electronics. In particular, the existence of real eigenvalues of non-Hermitian systems has opened a wide set of possibilities, especially, but not only, for sensing applications, exploiting the physics of exceptional points. In particular, the square root dependence of the eigenvalue splitting on different design parameters, exhibited by 2 × 2 non-Hermitian Hamiltonian matrices at the exceptional point, paved the way to the integration of high-performance sensors. The square root dependence of the eigenfrequencies on the design parameters is the reason for a theoretically infinite sensitivity in the proximity of the exceptional point. Recently, higher-order exceptional points have demonstrated the possibility of achieving the nth root dependence of the eigenfrequency splitting on perturbations. However, the exceptional sensitivity to external parameters is, at the same time, the major drawback of non-Hermitian configurations, leading to the high influence of noise. In this review, the basic principles of PT-symmetric and anti-PT-symmetric Hamiltonians will be shown, both in photonics and in electronics. The influence of noise on non-Hermitian configurations will be investigated and the newest solutions to overcome these problems will be illustrated. Finally, an overview of the newest outstanding results in sensing applications of non-Hermitian photonics and electronics will be provided.

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15

SERGI, ALESSANDRO, and KONSTANTIN G. ZLOSHCHASTIEV. "NON-HERMITIAN QUANTUM DYNAMICS OF A TWO-LEVEL SYSTEM AND MODELS OF DISSIPATIVE ENVIRONMENTS." International Journal of Modern Physics B 27, no. 27 (October 15, 2013): 1350163. http://dx.doi.org/10.1142/s0217979213501634.

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We consider a non-Hermitian Hamiltonian in order to effectively describe a two-level system (TLS) coupled to a generic dissipative environment. The total Hamiltonian of the model is obtained by adding a general anti-Hermitian part, depending on four parameters, to the Hermitian Hamiltonian of a tunneling TLS. The time evolution is formulated and derived in terms of the normalized density operator of the model, different types of decays are revealed and analyzed. In particular, the population difference and coherence are defined and calculated analytically. We have been able to mimic various physical situations with different properties, such as dephasing, vanishing population difference and purification.

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16

Ramos, B. F., I. A. Pedrosa, and K. Bakke. "Effects of a non-Hermitian potential on the Landau quantization." International Journal of Modern Physics A 34, no. 12 (April 30, 2019): 1950072. http://dx.doi.org/10.1142/s0217751x19500726.

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In this work, we solve the time-independent Schrödinger equation for a Landau system modulated by a non-Hermitian Hamiltonian. The system consists of a spinless particle in a uniform magnetic field submitted to action of a non-[Formula: see text] symmetric complex potential. Although the Hamiltonian is neither Hermitian nor [Formula: see text]-symmetric, we find that the Landau problem under study exhibits an entirely real energy spectrum.

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17

Nininahazwe, Ancilla. "Non Hermitian Matrix Quasi-Exactly Solvable Hamiltonian." Open Journal of Microphysics 08, no. 03 (2018): 15–25. http://dx.doi.org/10.4236/ojm.2018.83003.

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18

Chamberlain, S. R., J. G. Tucker, J. M. Conroy, and H. G. Miller. "Waxman’s algorithm for non-Hermitian Hamiltonian operators." Journal of Physics Communications 2, no. 2 (February 21, 2018): 025026. http://dx.doi.org/10.1088/2399-6528/aaaea3.

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19

Mazharimousavi, S. Habib. "Non-Hermitian Hamiltonian versusE= 0 localized states." Journal of Physics A: Mathematical and Theoretical 41, no. 24 (June 3, 2008): 244016. http://dx.doi.org/10.1088/1751-8113/41/24/244016.

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20

Bebiano, N., J. da Providência, S. Nishiyama, and J. P. da Providência. "Fermionic Model with a Non-Hermitian Hamiltonian." Brazilian Journal of Physics 50, no. 2 (January 6, 2020): 143–52. http://dx.doi.org/10.1007/s13538-019-00729-7.

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21

Faria, C. Figueira de Morisson, and A. Fring. "Time evolution of non-Hermitian Hamiltonian systems." Journal of Physics A: Mathematical and General 39, no. 29 (July 5, 2006): 9269–89. http://dx.doi.org/10.1088/0305-4470/39/29/018.

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22

Grigorenko, A. N. "Quantum mechanics with a non-Hermitian Hamiltonian." Physics Letters A 172, no. 5 (January 1993): 350–54. http://dx.doi.org/10.1016/0375-9601(93)90116-h.

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23

Lee, Dean. "THE ROLE OF DIAGONALIZATION WITHIN A DIAGONALIZATION/MONTE CARLO SCHEME." International Journal of Modern Physics A 16, supp01c (September 2001): 1245–47. http://dx.doi.org/10.1142/s0217751x01009430.

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We discuss a method called quasi-sparse eigenvector diagonalization which finds the most important basis vectors of the low energy eigenstates of a quantum Hamiltonian. It can operate using any basis, either orthogonal or non-orthogonal, and any sparse Hamiltonian, either Hermitian, non-Hermitian, finite-dimensional, or infinite-dimensional. The method is part of a new computational approach which combines both diagonalization and Monte Carlo techniques.

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24

Bender, Carl M., Alexander Felski, Sandra P. Klevansky, and Sarben Sarkar. "PT symmetry and renormalisation in quantum field theory." Journal of Physics: Conference Series 2038, no. 1 (October 1, 2021): 012004. http://dx.doi.org/10.1088/1742-6596/2038/1/012004.

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Abstract Quantum systems governed by non-Hermitian Hamiltonians with PT symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that PT symmetry may also be important and present at the level of Hermitian quantum field theories because of the process of renormalisation. In some quantum field theories renormalisation leads to PT -symmetric effective Lagrangians. We show how PT symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework. From the study of examples PT -symmetric interpretation is naturally built into a path integral formulation of quantum field theory; there is no requirement to calculate explicitly the PT norm that occurs in Hamiltonian quantum theory. We discuss examples where PT -symmetric field theories emerge from Hermitian field theories due to effects of renormalisation. We also consider the effects of renormalisation on field theories that are non-Hermitian but PT -symmetric from the start.

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25

MOSTAFAZADEH, ALI. "PSEUDO-HERMITIAN REPRESENTATION OF QUANTUM MECHANICS." International Journal of Geometric Methods in Modern Physics 07, no. 07 (November 2010): 1191–306. http://dx.doi.org/10.1142/s0219887810004816.

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A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as [Formula: see text], the true meaning and significance of the so-called charge operators [Formula: see text] and the [Formula: see text]-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.

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26

Ruschhaupt, Andreas, Miguel A. Simon, Anthony Kiely, and J. Gonzalo Muga. "The Role of Symmetry in Non-Hermitian Scattering¹." Journal of Physics: Conference Series 2038, no. 1 (October 1, 2021): 012020. http://dx.doi.org/10.1088/1742-6596/2038/1/012020.

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Abstract We review recent work on asymmetric scattering by Non-Hermitian (NH) Hamiltonians. Quantum devices with an asymmetric scattering response to particles incident from right or left in effective ID waveguides will be important to develop quantum technologies. They act as microscopic equivalents of familiar macroscopic devices such as diodes, rectifiers, or valves. The symmetry of the underlying NH Hamiltonian leads to selection rules which restrict or allow asymmetric response. NH-symmetry operations may be organized into group structures that determine equivalences among operations once a symmetry is satisfied. The NH Hamiltonian posseses a particular symmetry if it is invariant with respect to the corresponding symmetry operation, which can be conveniently expressed by a unitary or antiunitary superoperator. A simple group is formed by eight symmetry operations, which include the ones for Parity-Time symmetry and Hermiticity as specific cases. The symmetries also determine the structure of poles and zeros of the S matrix. The ground-state potentials for two-level atoms crossing properly designed laser beams realize different NH symmetries to achieve transmission or reflection asymmetries.

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27

NESTEROV, ALEXANDER I., GENNADY P. BERMAN, and ALAN R. BISHOP. "NON-HERMITIAN DESCRIPTION OF A SUPERCONDUCTING PHASE QUBIT MEASUREMENT." International Journal of Quantum Information 08, no. 06 (September 2010): 895–904. http://dx.doi.org/10.1142/s0219749910006630.

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We present an approach based on a non-Hermitian Hamiltonian to describe the process of measurement by tunneling of superconducting phase qubit states. We derive simple analytical expressions which describe the dynamics of measurement, and compare our results with those experimentally available. In particular, we show that even for a single qubit, the analytical expressions simplify the analysis of the dynamics in comparison with the density matrix approach. We also demonstrate that the effect of the interference of tunneling channels can be easily described by using the approach based on the non-Hermitian Hamiltonian.

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28

Maamache, Mustapha. "NON-UNITARY TRANSFORMATION OF QUANTUM TIME-DEPENDENT NON-HERMITIAN SYSTEMS." Acta Polytechnica 57, no. 6 (December 30, 2017): 424. http://dx.doi.org/10.14311/ap.2017.57.0424.

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We provide a new perspective on non-Hermitian evolution in quantum mechanics by emphasizing the same method as in the Hermitian quantum evolution. We first give a precise description of the non unitary transformation and the associated evolution, and collecting the basic results around it and postulating the norm preserving. This cautionary postulate imposing that the time evolution of a non Hermitian quantum system preserves the inner products between the associated states must not be read naively. We also give an example showing that the solutions of time-dependent non Hermitian Hamiltonian systems given by a linear combination of SU(1,1) and SU(2) are obtained thanks to time-dependent non-unitary transformation.

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29

Kecita, F., A. Bounames, and M. Maamache. "A Real Expectation Value of the Time-dependent Non-Hermitian Hamiltonians*." Physica Scripta 96, no. 12 (December 1, 2021): 125265. http://dx.doi.org/10.1088/1402-4896/ac3dbd.

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Abstract With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent   -symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the  ( t ) PT −inner product, where  ( t ) is a obtained from the charge conjugation operator  through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the  ( t ) PT normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.

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30

Geyer, H. B., W. D. Heiss, and F. G. Scholtz. "The physical interpretation of non-Hermitian Hamiltonians and other observables." Canadian Journal of Physics 86, no. 10 (October 1, 2008): 1195–201. http://dx.doi.org/10.1139/p08-060.

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A recent surge of publications about non-Hermitian Hamiltonians has led to considerable controversy and — in our opinion — to some misunderstandings of basic quantum mechanics. The present paper scrutinizes the metric associated with a quasi-Hermitian Hamiltonian and its physical implications. The consequences of the non-uniqueness such as the question of the probability interpretation and the possible and forbidden choices of additional observables are investigated and exemplified by specific illustrative examples. In particular, it is argued that the improper identification of observables lies at the origin of the claimed violation of the brachistchrone transition time between orthogonal states. The need for further physical input to remove ambiguities is pointed out.PACS Nos.: 03.65.–w, 03.65.Ca, 03.65.Ta, 03.65.Xp

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Zloshchastiev, Konstantin G. "Model Hamiltonians of open quantum optical systems: Evolvement from hermiticity to adjoint commutativity." Journal of Physics: Conference Series 2407, no. 1 (December 1, 2022): 012011. http://dx.doi.org/10.1088/1742-6596/2407/1/012011.

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Abstract In the conventional quantum mechanics of conserved systems, Hamiltonian is assumed to be a Hermitian operator. However, when it comes to quantum systems in presence of dissipation and/or noise, including open quantum optical systems, the strict hermiticity requirement is nor longer necessary. In fact, it can be substantially relaxed: the non-Hermitian part of a Hamiltonian is allowed, in order to account for effects of dissipative environment, whereas its Hermitian part would be describing subsystem’s energy. Within the framework of the standard approach to dissipative phenomena based on a master equation for the reduced density operator, we propose a replacement of the hermiticity condition by a more general condition of commutativity between Hermitian and anti-Hermitian parts of a Hamiltonian. As an example, we consider a dissipative two-mode quantum system coupled to a single-mode electromagnetic wave, where we demonstrate that the adjoint-commutativity condition does simplify the parametric space of the model.

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32

VENTRIGLIA, FRANCO. "ALTERNATIVE HAMILTONIAN DESCRIPTIONS FOR QUANTUM SYSTEMS AND NON-HERMITIAN OPERATORS WITH REAL SPECTRUM." Modern Physics Letters A 17, no. 24 (August 10, 2002): 1589–99. http://dx.doi.org/10.1142/s0217732302007946.

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Many problems in theoretical physics are very frequently dealt with non-Hermitian operators. Recently there has been a lot of interest in non-Hermitian operators with real spectra. In this paper, by using the inverse problem for quantum systems, we show that, on finite-dimensional Hilbert spaces, all diagonalizable operators with a real spectrum can be made Hermitian with respect to a properly chosen inner product. In particular this allows the use of standard methods of quantum mechanics to analyze non-Hermitian operators with real spectra.

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33

Jones-Smith, Katherine. "A ‘Dysonization’ scheme for identifying quasi-particles using non-Hermitian quantum mechanics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1989 (April 28, 2013): 20120056. http://dx.doi.org/10.1098/rsta.2012.0056.

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Dyson analysed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of the non-Hermitian quantum mechanics formalism at our disposal when considering Dyson’s work, both technically and contextually. Here, we recast Dyson’s work on ferromagnets explicitly in terms of two inner products, with respect to which the Hamiltonian is always self-adjoint, if not manifestly ‘Hermitian’. Then we extend his scheme to doped anti-ferromagnets described by the t – J model, with hopes of shedding light on the physics of high-temperature superconductivity.

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34

Swanson, Mark S. "Transition elements for a non-Hermitian quadratic Hamiltonian." Journal of Mathematical Physics 45, no. 2 (February 2004): 585–601. http://dx.doi.org/10.1063/1.1640796.

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35

Jin, L., and Z. Song. "A physical interpretation for the non-Hermitian Hamiltonian." Journal of Physics A: Mathematical and Theoretical 44, no. 37 (August 23, 2011): 375304. http://dx.doi.org/10.1088/1751-8113/44/37/375304.

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36

Miniatura, Ch, C. Sire, J. Baudon, and J. Bellissard. "Geometrical Phase Factor for a Non-Hermitian Hamiltonian." Europhysics Letters (EPL) 13, no. 3 (October 1, 1990): 199–203. http://dx.doi.org/10.1209/0295-5075/13/3/002.

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Miniatura, Ch, C. Sire, J. Baudon, and J. Bellissard. "Geometrical Phase Factor for a Non-Hermitian Hamiltonian." Europhysics Letters (EPL) 14, no. 1 (January 1, 1991): 91. http://dx.doi.org/10.1209/0295-5075/14/1/017.

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Bebiano, N., J. da Providência, S. Nishiyama, and J. P. da Providência. "A quantum system with a non-Hermitian Hamiltonian." Journal of Mathematical Physics 61, no. 8 (August 1, 2020): 082106. http://dx.doi.org/10.1063/5.0011098.

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Singh, Ram Mehar. "Real Eigenvalue of a Non-Hermitian Hamiltonian System." Applied Mathematics 03, no. 10 (2012): 1117–23. http://dx.doi.org/10.4236/am.2012.310164.

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Alexandre, Jean. "Foldy-Wouthuysen transformation for a non-Hermitian Hamiltonian." Journal of Physics: Conference Series 631 (July 30, 2015): 012071. http://dx.doi.org/10.1088/1742-6596/631/1/012071.

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Alexandre, Jean. "Non-Hermitian Lagrangian for Quasirelativistic Fermions." Advances in Mathematical Physics 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/527967.

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We present a Lorentz-symmetry violating Lagrangian for free fermions, which is local but not Hermitian, whereas the corresponding Hamiltonian is Hermitian but not local. A specific feature of the model is that the dispersion relation is relativistic in both the IR and the UV but not in an intermediate regime, set by a given mass scale. The consistency of the model is shown by the study of properties expected in analogy with the Dirac Lagrangian.

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BAAQUIE, BELAL E. "ACTION WITH ACCELERATION I: EUCLIDEAN HAMILTONIAN AND PATH INTEGRAL." International Journal of Modern Physics A 28, no. 27 (October 30, 2013): 1350137. http://dx.doi.org/10.1142/s0217751x13501376.

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An action having an acceleration term in addition to the usual velocity term is analyzed. The quantum mechanical system is directly defined for Euclidean time using the path integral. The Euclidean Hamiltonian is shown to yield the acceleration Lagrangian and the path integral with the correct boundary conditions. Due to the acceleration term, the state space depends on both position and velocity — and hence the Euclidean Hamiltonian depends on two degrees of freedom. The Hamiltonian for the acceleration system is non-Hermitian and can be mapped to a Hermitian Hamiltonian using a similarity transformation; the matrix elements of the similarity transformation are explicitly evaluated.

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43

Yang, Frank, Ciril S. Prasad, Weijian Li, Rosemary Lach, Henry O. Everitt, and Gururaj V. Naik. "Non-Hermitian metasurface with non-trivial topology." Nanophotonics 11, no. 6 (February 2, 2022): 1159–65. http://dx.doi.org/10.1515/nanoph-2021-0731.

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Abstract The synergy between topology and non-Hermiticity in photonics holds immense potential for next-generation optical devices that are robust against defects. However, most demonstrations of non-Hermitian and topological photonics have been limited to super-wavelength scales due to increased radiative losses at the deep-subwavelength scale. By carefully designing radiative losses at the nanoscale, we demonstrate a non-Hermitian plasmonic–dielectric metasurface in the visible with non-trivial topology. The metasurface is based on a fourth order passive parity-time symmetric system. The designed device exhibits an exceptional concentric ring in its momentum space and is described by a Hamiltonian with a non-Hermitian Z 3 ${\mathbb{Z}}_{3}$ topological invariant of V = −1. Fabricated devices are characterized using Fourier-space imaging for single-shot k-space measurements. Our results demonstrate a way to combine topology and non-Hermitian nanophotonics for designing robust devices with novel functionalities.

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Bogoliubov, Nikolai M., and Andrei V. Rybin. "The Generalized Tavis—Cummings Model with Cavity Damping." Symmetry 13, no. 11 (November 8, 2021): 2124. http://dx.doi.org/10.3390/sym13112124.

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In this Communication, we consider a generalised Tavis–Cummings model when the damping process is taken into account. We show that the quantum dynamics governed by a non-Hermitian Hamiltonian is exactly solvable using the Quantum Inverse Scattering Method, and the Algebraic Bethe Ansatz. The leakage of photons is described by a Lindblad-type master equation. The non-Hermitian Hamiltonian is diagonalised by state vectors, which are elementary symmetric functions parametrised by the solutions of the Bethe equations. The time evolution of the photon annihilation operator is defined via a corresponding determinant representation.

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Bagchi, Bijan, Rahul Ghosh, and Sauvik Sen. "Analogue Hawking Radiation as a Tunneling in a Two-Level PT-Symmetric System." Entropy 25, no. 8 (August 12, 2023): 1202. http://dx.doi.org/10.3390/e25081202.

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In light of a general scenario of a two-level non-Hermitian PT-symmetric Hamiltonian, we apply the tetrad-based method to analyze the possibility of analogue Hawking radiation. We carry this out by making use of the conventional null-geodesic approach, wherein the associated Hawking radiation is described as a quantum tunneling process across a classically forbidden barrier on which the event horizon imposes. An interesting aspect of our result is that our estimate for the tunneling probability is independent of the non-Hermitian parameter that defines the guiding Hamiltonian.

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B, Tuguldur, Zhenhuan Yi, and Jonathan S. Ben-Benjamin. "Effective Raman Hamiltonian revisited." Физик сэтгүүл 32, no. 553 (March 14, 2022): 1–8. http://dx.doi.org/10.22353/physics.v32i553.561.

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In this paper, the effective Raman Hamiltonian is revisited. A common way to obtain the effective Raman Hamiltonian is by using the time-dependent perturbation method (TDPT) along with Fermi’s golden rule to keep the total energy and probability constant. However, for a non-resonant Raman process the obtained effective Hamiltonian is not convenient because it is not Hermitian. Hence, we present the Magnus expansion method for obtaining the effective Raman Hamiltonian, which has the advantages of being Hermitian and featuring effects absent in the TDPT effective Hamiltonian. To our knowledge, this is the first time that the Magnus expansion is utilized as an alternative method. We compare the our obtained effective Hamiltonian to that obtained from canonical transformation method. We determine the extra piece in second-order time-dependent perturbation theory which causes loss or gain of total probability.

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Cao, Yusong, and Junpeng Cao. "Exact Solution of a Non-Hermitian Generalized Rabi Model." Chinese Physics Letters 38, no. 8 (September 1, 2021): 080202. http://dx.doi.org/10.1088/0256-307x/38/8/080202.

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An integrable non-Hermitian generalized Rabi model is constructed. A twist matrix is introduced to the construction of Hamiltonian and generates the non-Hermitian properties. The Yang-Baxter integrability of the system is proven. The exact energy spectrum and eigenstates are obtained using the Bethe ansatz. The method given in this study provides a general way to construct integrable spin-boson models.

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Fernández, Viviano, Romina Ramírez, and Marta Reboiro. "Swanson Hamiltonian: non-PT-symmetry phase." Journal of Physics A: Mathematical and Theoretical 55, no. 1 (December 7, 2021): 015303. http://dx.doi.org/10.1088/1751-8121/ac3a35.

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Abstract In this work, we study the non-Hermitian Swanson Hamiltonian, particularly the non-parity-time symmetry phase. We use the formalism of Gel’fand triplet to construct the generalized eigenfunctions and the corresponding spectrum. Depending on the region of the parameter model space, we show that the Swanson Hamiltonian represents different physical systems, i.e. parabolic barrier, negative mass oscillators. We also discussed the presence of Exceptional Points of infinite order.

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ANDRIANOV, A. A., M. V. IOFFE, F. CANNATA, and J. P. DEDONDER. "SUSY QUANTUM MECHANICS WITH COMPLEX SUPERPOTENTIALS AND REAL ENERGY SPECTRA." International Journal of Modern Physics A 14, no. 17 (July 10, 1999): 2675–88. http://dx.doi.org/10.1142/s0217751x99001342.

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We extend the standard intertwining relations used in supersymmetrical (SUSY) quantum mechanics which involve real superpotentials to complex superpotentials. This allows us to deal with a large class of non-Hermitian Hamiltonians and to study in general the isospectrality between complex potentials. In very specific cases we can construct in a natural way "quasicomplex" potentials which we define as complex potentials having a global property so as to lead to a Hamiltonian with real spectrum. We also obtained a class of complex transparent potentials whose Hamiltonian can be intertwined to a free Hamiltonian. We provide a variety of examples both for the radial problem (half axis) and for the standard one-dimensional problem (the whole axis), including remarks concerning scattering problems.

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Singh, Ram Mehar, S. B. Bhardwaj, Kushal Sharma, Anand Malik, and Fakir Chand. "Classical invariants for non-Hermitian anharmonic potentials." Canadian Journal of Physics 98, no. 11 (November 2020): 1004–8. http://dx.doi.org/10.1139/cjp-2019-0320.

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Keeping in view the importance of complex dynamical systems, we investigate the classical invariants for some non-Hermitian anharmonic potentials in one dimension. For this purpose, the rationalization method is employed under the elegance of the extended complex phase space approach. The invariants obtained are expected to play an important role in studying complex Hamiltonian systems at the classical as well as quantum levels.

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