non-Hermitian Operators Research Articles

Generation of photon-added coherent states via photon-subtracted generalised coherent states

Based on previous work [A. Dehghani, B. Mojaveri, J. Phys. A 45, 095304 (2012)], we introduce photon-subtracted generalised coherent states (PSGCSs) |z,m⟩ r : = a m |z⟩ r , where m is a nonnegative integer and |z⟩ r denote the generalised coherent states (GCSs). We have shown that the states |z,m⟩ r are eigenstates of a non-Hermitian operator f(n ,m)â, where f(n ,m) is a nonlinear function of the number operator N . Also, the states | z, − m ⟩ r can be considered as another set of eigenstates for negative values of m. They span the truncated Fock space without the first m lowest-lying basis states: | 0 ⟩ , | 1 ⟩ , | 2 ⟩ ,...,| m − 1 ⟩ which are reminiscent of the so-called photon-added coherent states. The resolution of the identity property, which is the most important property of coherent states, is realised for |z,m⟩ r as well as for |z, − m⟩ r . Some nonclassical features such as sub-Poissonian statistics and quadrature squeezing of the states |z, ± m⟩ r are compared. We show that the annihilation operator diminishes the mean number of photons of the initial state |z⟩ r . Finally we show that |z,m⟩ r can be produced through a simple theoretical scheme.

Linearized self-consistent GW approach satisfying the Ward identity

We propose a linearized self-consistent GW approach satisfying the Ward identity. The vertex function derived from the Ward-Takahashi identity in the limit of $\mathbit{q}=0$ and $\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}^{\ensuremath{'}}=0$ is included in the self-energy and the polarization function as a consequence of the linearization of the quasiparticle equation. Due to the energy dependence of the self-energy, the Hamiltonian is a non-Hermitian operator and quasiparticle states are nonorthonormal and linearly dependent. However, the linearized quasiparticle states recover orthonormality and fulfill the completeness condition. This approach is very efficient, and the resulting quasiparticle energies are greatly improved compared to the nonlinearized self-consistent GW approach, although its computational cost is not much increased. We show the results for atoms and dimers of Li and Na compared with other approaches. We also propose convenient ways to calculate the Luttinger-Ward functional $\ensuremath{\Phi}$ based on a plasmon-pole model and calculate the total energy for the ground state. As a result, we conclude that the linearization improves overall behaviors in the self-consistent GW approach.

Gribov ambiguities at the Landau-maximal Abelian interpolating gauge

In a previous work, we presented a new method to account for the Gribov ambiguities in non-Abelian gauge theories. The method consists on the introduction of an extra constraint which directly eliminates the infinitesimal Gribov copies without the usual geometric approach. Such strategy allows to treat gauges with non-hermitian Faddeev-Popov operator. In this work, we apply this method to a gauge which interpolates among the Landau and maximal Abelian gauges. The result is a local and power counting renormalizable action, free of infinitesimal Gribov copies. Moreover, the interpolating tree-level gluon propagator is derived.

Quasilocal conservation laws in XXZ spin-1/2 chains: Open, periodic and twisted boundary conditions

A continuous family of quasilocal exact conservation laws is constructed in the anisotropic Heisenberg (XXZ) spin-1/2 chain for periodic (or twisted) boundary conditions and for a set of commensurate anisotropies densely covering the entire easy plane interaction regime. All local conserved operators follow from the standard (Hermitian) transfer operator in fundamental representation (with auxiliary spin s=1/2), and are all even with respect to a spin flip operation. However, the quasilocal family is generated by differentiation of a non-Hermitian highest weight transfer operator with respect to a complex auxiliary spin representation parameter s and includes also operators of odd parity. For a finite chain with open boundaries the time derivatives of quasilocal operators are not strictly vanishing but result in operators localized near the boundaries of the chain. We show that a simple modification of the non-Hermitian transfer operator results in exactly conserved, but still quasilocal operators for periodic or generally twisted boundary conditions. As an application, we demonstrate that implementing the new exactly conserved operator family for estimating the high-temperature spin Drude weight results, in the thermodynamic limit, in exactly the same lower bound as for almost conserved family and open boundaries. Under the assumption that the bound is saturating (suggested by agreement with previous thermodynamic Bethe ansatz calculations) we propose a simple explicit construction of infinite time averages of local operators such as the spin current.

General-dyne unravelling of a thermal master equation

We analyse the unravelling of the quantum optical master equation at finite temperature due to direct, continuous, general-dyne detection of the environment. We first express the general-dyne Positive Operator Valued Measure (POVM) in terms of the eigenstates of a non-hermitian operator associated to the general-dyne measurement. Then, we derive the stochastic master equation obtained by considering the interaction between the system and a reservoir at thermal equilibrium, which is measured according to the POVM previously determined. Finally, we present a feasible measurement scheme which reproduces general-dyne detection for any value of the parameter characterising the stochastic master equation.

Skew projectors and generalized observables in polarization optics: a biorthogonal analyses [Invited

This paper constitutes an application of the polarization optics in the problem of quantum measurement. The non-Hermitian operators of the nonorthogonal multilayer optical polarizers represent observables in the sense of the generalized quantum theory of measurement. The intimate spectral structure of these polarizers can be disclosed in the frame of skew-angular vector bases and biorthonormal vector systems. We show that these polarizers correspond to skew projectors; their operators are "generated" by skew projectors in the sense of the spectral theorem of linear operators theory. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their "nuclei" are (orthogonal or skew) projectors--the generating projectors.

Time-dependent -symmetric quantum mechanics

The parity-time-reversal ()-symmetric quantum mechanics (QM) (PTQM) has developed into a noteworthy area of research. However, to date, most known studies of PTQM focused on the spectral properties of non-Hermitian Hamiltonian operators. In this work, we propose an axiom in PTQM in order to study general time-dependent problems in PTQM, e.g., those with a time-dependent -symmetric Hamiltonian and with a time-dependent metric. We illuminate our proposal by examining a proper mapping from a time-dependent Schrödinger-like equation of motion for PTQM to the familiar time-dependent Schrödinger equation in conventional QM. The rich structure of the proper mapping hints that time-dependent PTQM can be a fruitful extension of conventional QM. Under our proposed framework, we further study in detail the Berry-phase generation in a class of -symmetric two-level systems. It is found that a closed path in the parameter space of PTQM is often associated with an open path in a properly mapped problem in conventional QM. In one interesting case, we further interpret the Berry phase as the flux of a continuously tunable fictitious magnetic monopole, thus highlighting the difference between PTQM and conventional QM despite the existence of a proper mapping between them.

Bifurcations and exceptional points in dipolar Bose–Einstein condensates

Bose–Einstein condensates are described in a mean-field approach by the nonlinear Gross–Pitaevskii equation and exhibit phenomena of nonlinear dynamics. The stationary states can undergo bifurcations in such a way that two or more eigenvalues and the corresponding wavefunctions coalesce at critical values of external parameters. For example, in condensates without long-range interactions a stable and an unstable state are created in a tangent bifurcation when the scattering length of the contact interaction is varied. At the critical point, the coalescing states show the properties of an exceptional point. In dipolar condensates fingerprints of a pitchfork bifurcation have been discovered by Rau et al (2010 Phys. Rev.A 81 031605). We present a method to uncover all states participating in a pitchfork bifurcation, and investigate in detail the signatures of exceptional points related to bifurcations in dipolar condensates. For the perturbation by two parameters, namely the scattering length and a parameter breaking the cylindrical symmetry of the harmonic trap, two cases leading to different characteristic eigenvalue and eigenvector patterns under cyclic variations of the parameters need to be distinguished. The observed structures resemble those of three coalescing eigenfunctions obtained by Demange and Graefe (2012 J. Phys. A: Math. Theor. 45 025303) using perturbation theory for non-Hermitian operators in a linear model. Furthermore, the splitting of the exceptional point under symmetry breaking in either two or three branching singularities is examined. Characteristic features are observed when one, two or three exceptional points are encircled simultaneously.

Exterior integrability: Yang–Baxter form of non-equilibrium steady-state density operator

A new type of quantum transfer matrix, arising as a Cholesky factor for the steady-state density matrix of a dissipative Markovian process associated with the boundary-driven Lindblad equation for the isotropic spin-1/2 Heisenberg (XXX) chain, is presented. The transfer matrix forms a commuting family of non-Hermitian operators depending on the spectral parameter, which is essentially the strength of dissipative coupling at the boundaries. The intertwining of the corresponding Lax and monodromy matrices is performed by an infinitely dimensional Yang–Baxter R-matrix, which we construct explicitly and is essentially different from the standard 4 × 4 XXX R-matrix. We also discuss a possibility to construct Bethe ansatz for the spectrum and eigenstates of the non-equilibrium steady-state density operator. Furthermore, we indicate the existence of a deformed R-matrix in the infinite dimensional auxiliary space for the anisotropic XXZ spin-1/2 chain, which in general provides a sequence of new, possibly quasi-local, conserved quantities of the bulk XXZ dynamics.

Non-Hermitian Hydrogen atom

We have constructed a set of non-Hermitian operators that satisfy the commutation relations of the \(SO(3)\)-Lie algebra. Using these set of operators we have constructed a non-Hermitian Hamiltonian corresponding to the Hydrogen atom that includes a complex term but with the same spectra as in the Hermitian case. It is also found a non-Hermitian Runge–Lenz vector that represents a conserved quantity. In this way, we obtain a set of non-Hermitian operators that satisfy the commutation relations of the \(SO(4)\)-Lie algebra.

Preface: Quantum Physics with non‐Hermitian Operators: Theory and Experiment

Fortschritte der PhysikVolume 61, Issue 2-3 p. 57-58 PrefaceFree Access Preface: Quantum Physics with non-Hermitian Operators: Theory and Experiment J. Bird, J. BirdSearch for more papers by this authorR. Kaiser, R. KaiserSearch for more papers by this authorI. Rotter, I. RotterSearch for more papers by this authorG. Wunner, G. WunnerSearch for more papers by this author J. Bird, J. BirdSearch for more papers by this authorR. Kaiser, R. KaiserSearch for more papers by this authorI. Rotter, I. RotterSearch for more papers by this authorG. Wunner, G. WunnerSearch for more papers by this author First published: 01 February 2013 https://doi.org/10.1002/prop.201306111Citations: 1AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article.Citing Literature Volume61, Issue2-3Special Issue: Quantum Physics with non‐Hermitian Operators: Theory and ExperimentFebruary 2013Pages 57-58 RelatedInformation

New application of non-Hermitian Hamiltonian operator in solving master equation for laser process

We find that laser process can be equivalently described by a symplectic evolution in the context of thermo field dynamics, and the corresponding coherent state evolution for the corresponding master equation is recognized. More interestingly, this embodies a new application of non-Hermitian Hamiltonian operator which can well expose the entanglement between the system and its environment.

Random-matrix theory of amplifying and absorbing resonators with or symmetry

We formulate Gaussian and circular random-matrix models representing a coupled system consisting of an absorbing and an amplifying resonator, which are mutually related by a generalized time-reversal symmetry. Motivated by optical realizations of such systems we consider a or a time-reversal symmetry, which impose different constraints on magneto-optical effects, and then focus on five common settings. For each of these, we determine the eigenvalue distribution in the complex plane in the short-wavelength limit, which reveals that the fraction of real eigenvalues among all eigenvalues in the spectrum vanishes if all classical scales are kept fixed. Numerically, we find that the transition from real to complex eigenvalues in the various ensembles display a different dependence on the coupling strength between the two resonators. These differences can be linked to the level spacing statistics in the Hermitian limit of the considered models.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

Asymmetric wave propagation through nonlinear PT-symmetric oligomers

In the present paper, we consider nonlinear PT-symmetric dimers and trimers (more generally, oligomers) embedded within a linear Schrödinger lattice. We examine the stationary states of such chains in the form of plane waves, and analytically compute their reflection and transmission coefficients through the nonlinear PT symmetric oligomer, as well as the corresponding rectification factors which clearly illustrate the asymmetry between left and right propagation in such systems. We examine not only the existence but also the dynamical stability of the plane wave states and interestingly find them to be unstable except in the vicinity of the linear limit. Lastly, we generalize our numerical considerations to the more physically relevant case of Gaussian initial wavepackets and confirm that the asymmetry in the transmission properties persists in the case of such wavepackets, as well. The role of potential asymmetries in the nonlinearity or in the gain/loss pattern is also considered.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

-symmetric electronics

We show both theoretically and experimentally that a pair of inductively coupled active LRC circuits (dimer), one with amplification and another with an equivalent amount of attenuation, display all the features which characterize a wide class of non-Hermitian systems which commute with the joint parity-time operator: typical normal modes, temporal evolution, and scattering processes. Utilizing a Liouvilian formulation, we can define an underlying -symmetric Hamiltonian, which provides important insight for understanding the behavior of the system. When the -dimer is coupled to transmission lines, the resulting scattering signal reveals novel features which reflect the -symmetry of the scattering target. Specifically we show that the device can show two different behaviors simultaneously, an amplifier or an absorber, depending on the direction and phase relation of the interrogating waves. Having an exact theory, and due to its relative experimental simplicity, -symmetric electronics offers new insights into the properties of -symmetric systems which are at the forefront of the research in mathematical physics and related fields.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

Weak measurements in non-Hermitian systems

‘Weak measurements’—involving a weak unitary interaction between a quantum system and a meter followed by a projective measurement—are investigated when the system has a non-Hermitian Hamiltonian. We show in particular how the standard definition of the ‘weak value’ of an observable must be modified. These studies are undertaken in the context of bound-state scattering theory, a non-Hermitian formalism for which the involved Hilbert spaces are unambiguously defined and the metric operators can be explicitly computed. Numerical examples are given for a model system.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

Nonlinear n-pseudo fermions

Nonlinear pseudo-fermions of degree n (n-pseudo-fermions) are introduced as (pseudo) particles with creation and annihilation operators b and a, b ≠ a†, obeying the simple nonlinear anticommutation relation ab + bnan = 1. The (n + 1)-order nilpotency of these operators follows from the existence of unique (up to a bi-normalization factor) a-vacuum. Supposing appropriate (n + 1)-order nilpotent para-Grassmann variables and integration rules the sets of n-pseudo-fermion number states, ‘right’ and ‘left’ ladder operator bi-overcomplete sets of coherent states are constructed. Explicit examples of n-pseudo-fermion ladder operators are provided, and the relation of pseudo-fermions to finite level pseudo-Hermitian systems is briefly considered.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

Stationary states of a PT symmetric two-mode Bose–Einstein condensate

The understanding of nonlinear PT symmetric quantum systems, arising for example in the theory of Bose–Einstein condensates in PT symmetric potentials, is widely based on numerical investigations, and little is known about generic features induced by the interplay of PT symmetry and nonlinearity. To gain deeper insights it is important to have analytically solvable toy models at hand. In the present paper the stationary states of a simple toy model of a PT symmetric system previously introduced in [1, 2] are investigated. The model can be interpreted as a simple description of a Bose–Einstein condensate in a PT symmetric double well trap in a two-mode approximation. The eigenvalues and eigenstates of the system can be explicitly calculated in a straightforward manner; the resulting structures resemble those that have recently been found numerically for a more realistic PT symmetric double delta potential. In addition, a continuation of the system is introduced that allows an interpretation in terms of a simple linear matrix model.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

Quantum catastrophes: a case study

The bound-state spectrum of a Hamiltonian H is assumed real in a non-empty domain of physical values of parameters. This means that for these parameters, H may be called crypto-Hermitian, i.e. made Hermitian via an ad hoc choice of the inner product in the physical Hilbert space of quantum bound states (i.e. via an ad hoc construction of the operator Θ called the metric). The name quantum catastrophe is then assigned to the N-tuple-exceptional-point crossing, i.e. to the scenario in which we leave the domain along such a path that at the boundary of , an N-plet of bound-state energies degenerates and, subsequently, complexifies. At any fixed N ⩾ 2, this process is simulated via an N × N benchmark effective matrix Hamiltonian H. It is being assigned such a closed-form metric which is made unique via an N-extrapolation-friendliness requirement.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

An optimal expression of a Kraus operator as a linear combination of unitary matrices

Kraus operators are widely used in describing the evolution of an open quantum system. In this paper, we study the properties of a Kraus operator as a linear combination of unitary matrices and demonstrate that every single Kraus operator can be realized in an interference quantum circuit. We determine the minima of both l1 and l0 norm of the combination coefficients, where l1 norm means the sum of the absolute values of the coefficients and l0 norm means the number of non-zero coefficients. We find that both of them have clear physical meanings. The l1 minimum signifies the most constructive interference, and the l0 minimum provides the simplest way to construct a Kraus operator in a quantum circuit. These results may be useful in understanding interference and Kraus operators as well as in designing a quantum algorithm with a quantum computer in an open environment.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.