The eigenvalue problem of the spherically symmetric oscillator Hamiltonian is revisited in the context of canonical raising and lowering operators. The Hamiltonian is then factorized in terms of two not mutually adjoint factorizing operators which, in turn, give rise to a non-Hermitian radial Hamiltonian. The set of eigenvalues of this new Hamiltonian is exactly the same as the energy spectrum of the radial oscillator and the new square-integrable eigenfunctions are complex Darboux-deformations of the associated Laguerre polynomials.
A harmonic oscillator Hamiltonian augmented by a non-Hermitian \pt-symmetric part and its su(1,1) generalizations, for which a family of positive-definite metric operators was recently constructed, are re-examined in a supersymmetric context. Quasi-Hermitian supersymmetric extensions of such Hamiltonians are proposed by enlarging su(1,1) to a ${\rm su}(1,1/1) \sim {\rm osp}(2/2, \R)$ superalgebra. This allows the construction of new non-Hermitian Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.
A generalized non-Hermitian oscillator Hamiltonian is proposed that consists of additional linear terms which break PT-symmetry explicitly. The model is put into an equivalent Hermitian form by means of a similarity transformation and the criterion of N-fold supersymmetry with a position-dependent mass is shown to reside in it.
In this parer, q-deformed oscillator for pseudo-Hermitian systems is investigated and pseudo-Hermitian appropriate coherent and squeezed states are studied. Also, some basic properties of these states is surveyed. The over-completeness property of the para-Grassmannian pseudo-Hermitian coherent states (PGPHCSs) examined, and also the stability of coherent and squeezed states discussed. The pseudo-Hermitian supercoherent states as the product of a pseudo-Hermitian bosonic coherent state and a para-Grassmannian pseudo-Hermitian coherent state introduced, and the method also developed to define p…
We study generalized Dirac oscillators with complex interactions in $(1+1)$ dimensions. It is shown that for the choice of interactions considered here, the Dirac Hamiltonians are $η$ pseudo Hermitian with respect to certain metric operators $η$. Exact solutions of the generalized Dirac Oscillator for some choices of the interactions have also been obtained. It is also shown that generalized Dirac oscillators can be identified with Anti Jaynes Cummings type model and by spin flip it can also be identified with Jaynes Cummings type model.
The family of metric operators, constructed by Musumbu {\sl et al} (2007 {\sl J. Phys. A: Math. Theor.} {\bf 40} F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian $\cal PT$-symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily $\cal PT$-symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some exa…
We associate the stationary harmonic oscillator with time-dependent systems exhibiting non-Hermiticity by means of point transformations. The new systems are exactly solvable, with all-real spectrum, and transit to the Hermitian configuration for the appropriate values of the involved parameters. We provide a concrete generalization of the Swanson oscillator that includes the Caldirola-Kanai model as a particular case. Explicit solutions are given in both, the classical and quantum pictures.
A non-Hermitian generalized oscillator model, generally known as the Swanson model, has been studied in the framework of R-deformed Heisenberg algebra. The non-Hermitian Hamiltonian is diagonalized by generalized Bogoliubov transformation. A set of deformed creation annihilation operators is introduced whose algebra shows that the transformed Hamiltonian has conformal symmetry. The spectrum is obtained using algebraic technique. The superconformal structure of the system is also worked out in detail. An anomaly related to the spectrum of the Hermitian counterpart of the non-Hermitian Hamiltoni…
By modifying and generalizing known supersymmetric models we are able to find four different sets of one-dimensional Hamiltonians for the inverted harmonic oscillator. The first set of Hamiltonians is derived by extending the supersymmetric quantum mechanics with reflections to non-Hermitian supercharges. The second set is obtained by generalizing the supersymmetric quantum mechanics valid for non-Hermitian supercharges with the Dunkl derivative instead of $\frac{d}{dx}$. Also, by changing the derivative $\frac{d}{dx}$ by the Dunkl derivative in the creation and annihilation-type operators of …
Partition Functions of Hermitian and PT-Symmetric Oscillators from Integrable Models
Institute for Astrophysics, Zhengzhou University（郑州大学天体物理研究所）；Interdisciplinary Center for Theoretical Study, University of Science and Technology of China（中国科学技术大学理论交叉研究中心）；Peng Huanwu Center for Fundamental Theory（彭桓武基础研究中心）；Department of Physics, Institute of Science Tokyo（东京科学大学物理系）
The applicability of the factorization method is extended to the case of quantum fractional-differential Hamiltonians. In contrast with the conventional factorization, it is shown that the `factorization energy' is now a fractional-differential operator rather than a constant. As a first example, the energies and wave-functions of a fractional version of the quantum oscillator are determined. Interestingly, the energy eigenvalues are expressed as power-laws of the momentum in terms of the non-integer differential order of the eigenvalue equation.
Supersymmetric (SUSY) transformation operators corresponding to complex factorization constants are analyzed as operators acting in the Hilbert space of functions square integrable on the positive semiaxis. Obtained results are applied to Hamiltonians possessing spectral singularities which are non-Hermitian SUSY partners of selfadjoint operators. A new regularization procedure for the resolution of the identity operator in terms of continuous biorthonormal set of the non-Hermitian Hamiltonian eigenfunctions is proposed. It is also shown that the continuous spectrum eigenfunction has zero bino…
The non-Hermitian quadratic oscillator studied by Swanson is one of the popular $PT$-symmetric model systems. Here a full classical description of its dynamics is derived using recently developed metriplectic flow equations, which combine the classical symplectic flow for Hermitian systems with a dissipative metric flow for the anti-Hermitian part. Closed form expressions for the metric and phase-space trajectories are presented which are found to be periodic in time. Since the Hamiltonian is only quadratic the classical dynamics exactly describes the quantum dynamics of Gaussian wave packets.…
By adding an imaginary interacting term proportional to ip_1p_2 to the Hamiltonian of a free anisotropic planar oscillator, we construct a new model which is described by the PT-pseudo-Hermitian Hamiltonian with the permutation symmetry of two dimensions. We prove that our model is equivalent to the Pais-Uhlenbeck oscillator and thus establish a relationship between our PT-pseudo-Hermitian system and the fourth-order derivative oscillator model. We also point out the spontaneous breaking of permutation symmetry which plays a crucial role in giving a real spectrum free of interchange of positiv…
In the traditional quantum theory, one-dimensional quantum spin models possess a factorization surface where the ground states are fully separable having vanishing bipartite as well as multipartite entanglement. We report that in the non-Hermitian counterpart of these models, these factorization surfaces either can predict the exceptional points where the unbroken-to-broken transition occurs or can guarantee the reality of the spectrum, thereby proposing a procedure to reveal the unbroken phase. We first analytically demonstrate it for the nearest-neighbour rotation-time RT-symmetric XY model …
We consider a spin half particle in the external magnetic field which couples to a harmonic oscillator through some pseudo-hermitian interaction. We find that the energy eigenvalues for this system are real even though the interaction is not PT invariant.
We propose a deformed version of the commutation rule introduced in 1967 by Buchdahl to describe a particular model of the truncated harmonic oscillator. The rule we consider is defined on a $N$-dimensional Hilbert space $\Hil_N$, and produces two biorhogonal bases of $\Hil_N$ which are eigenstates of the Hamiltonians $h=\frac{1}{2}(q^2+p^2)$, and of its adjoint $h^\dagger$. Here $q$ and $p$ are non-Hermitian operators obeying $[q,p]=i(\1-Nk)$, where $k$ is a suitable orthogonal projection operator. These eigenstates are connected by ladder operators constructed out of $q$, $p$, $q^\dagger$ an…
The recently introduced by us two- and three-parameter ($p,q$)- and ($p,q,μ$)-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 Hamiltonian build in terms of non-Hermitian position and momentum operators obeying definite $η(N)$-pseudo-Hermiticity properties. A generalized nonlinear (with the coefficients depending on the excitation number operator $N$) one-mode Bogolyubov transformation is developed as main tool for the correspond…
In this article we will apply the first- and second-order supersymmetric quantum mechanics to obtain new exactly-solvable real potentials departing from the inverted oscillator potential. This system has some special properties; in particular, only very specific second-order transformations produce non-singular real potentials. It will be shown that these transformations turn out to be the so-called complex ones. Moreover, we will study the factorization method applied to the inverted oscillator and the algebraic structure of the new Hamiltonians.
We perform a perturbative calculation of the physical observables, in particular pseudo-Hermitian position and momentum operators, the equivalent Hermitian Hamiltonian operator, and the classical Hamiltonian for the PT-symmetric cubic anharmonic oscillator, $ H=p^1/(2m)+μ^2x^2/2+iεx^3 $. Ignoring terms of order $ ε^4 $ and higher, we show that this system describes an ordinary quartic anharmonic oscillator with a position-dependent mass and real and positive coupling constants. This observation elucidates the classical origin of the reality and positivity of the energy spectrum. We also discus…
We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian of a fourth-order differential equation. In particular we consider the classical problem of the Pais-Uhlenbeck oscillator and write down the accompanying Hamiltonian. We then compare such an expression with an alternative derivation of the Hamiltonian that makes use of the Ostrogradski's method and show that a mapping from the one to the other is achievable by variable transformations. Assuming canonical quantization procedure to be valid we go for the operator version of the Hamiltonian that represents a pair of unco…
When a non-hermitian hamiltonian has a certain symmetry, such as the PT pseudo-hermiticity, it is still possible that the hamiltonian has a real spectrum. In this note, by adding an imaginary potential proportional to ip_1p_2 to the hamiltonian of an anisotropic planar oscillator, we construct a PT pseudo-hermitian hamiltonian and obtain its real spectrum. In addition, we find that the reality of the energy spectrum remains when the model is extended to the canonical noncommutative space.
We show that the standard techniques that are utilized to study the classical like properties of the pure states for Hermitian systems can be adjusted to investigate the classicality of pure states for non-Hermitian systems. The method is applied to the states of complex-valued potentials that are generated by Darboux transformations and can model both non-PT-symmetric and PT-symmetric oscillators exhibiting real spectra.
A set of Hamiltonians that are not self-adjoint but have the spectrum of the harmonic oscillator is studied. The eigenvectors of these operators and those of their Hermitian conjugates form a bi-orthogonal system that provides a mathematical procedure to satisfy the superposition principle. In this form the non-Hermitian oscillators can be studied in much the same way as in the Hermitian approaches. Two different nonlinear algebras generated by properly constructed ladder operators are found and the corresponding generalized coherent states are obtained. The non-Hermitian oscillators can be st…
An unusual type of the exact solvability is reported. It is exemplified by the Coulomb plus harmonic oscillator in D dimensions after a complexification of its Hamiltonian which keeps the energies real. Infinitely many bound states are found in closed form which generalizes the even-parity harmonic-oscillator states at zero charge. Apparently, the model is halfway between exact and quasi-exact.
A system of two independent Bosonic Harmonic Oscillators is converted into the respective fourth-order derivative Pais-Uhlenbeck oscillator model. The conversion procedure displays transparently how the quantization of the fourth-order derivative Pais-Uhlenbeck oscillator has to be performed in order not to suffer from the divergence problems of the vacuum state and path integrals as conjectured most recently by P. D. Mannheim in his article ``Determining the normalization of the quantum field theory vacuum, with implications for quantum gravity" [arXiv:2301.13029 [hep-th]]. In order to make t…
In this study, we introduce a two dimensional complex harmonic oscillator potential with space and time reflection symmetries. The corresponding time independent Schrödinger equation yields real eigenvalues with complex eigenfunctions. We also construct the coherent state of the system by using a superposition of 12 eigenfunctions. Using the complex correspondence principle for the probability density we investigate the possible modifications in the probability densities due to the non-Hermitian aspect of the Hamiltonian.
We provide a time-dependent Dyson map and metric for the two dimensional harmonic oscillator with a non-Hermitian $i xy$ coupling term. This particular time-independent model exhibits spontaneously broken $\mathcal{PT}$-symmetry and becomes unphysical in the broken regime, with the spectrum becoming partially complex. By introducing an explicit time-dependence into the Dyson map, we provide a time-dependent metric that renders the model consistent across the unbroken and broken regimes.
We study non Hermitian quantum systems in noncommutative space as well as a \cal{PT}-symmetric deformation of this space. Specifically, a \mathcal{PT}-symmetric harmonic oscillator together with iC(x_1+x_2) interaction is discussed in this space and solutions are obtained. It is shown that in the \cal{PT} deformed noncommutative space the Hamiltonian may or may not possess real eigenvalues depending on the choice of the noncommutative parameters. However, it is shown that in standard noncommutative space, the iC(x_1+x_2) interaction generates only real eigenvalues despite the fact that the Ham…
The generally deformed oscillator (GDO) and its multiphoton realization as well as the coherent and squeezed vacuum states are studied. We discuss, in particular, the GDO depending on a complex parameter q (therefore we call it q-GDO) together with the finite dimensional cyclic representations. As a realistic physical system of GDO the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator. It is pointed out that starting from the q-GDO with q root of unity one can define the hermitian phase operators…
We notice that PT symmetric non-Hermitian one dimensional simple Harmonic Oscillator under simultaneous transformation of co-ordinate and momentum with proper choice of positive oscillating frequency can reflect negative spectrum with well behaved wave function in real space. We also present a suitable comuter programme to realise the negative spectrum directly. PACS(2008) : 03.65.Db
The harmonic oscillator Hamiltonian, when augmented by a non-Hermitian $\cal{PT}$-symmetric part, can be transformed into a Hermitian Hamiltonian. This is achieved by introducing a metric which, in general, renders other observables such as the usual momentum or position as non-Hermitian operators. The metric depends on one real parameter, the full range of which is investigated. The explicit functional dependence of the metric and each associated Hamiltonian is given. A specific choice of this parameter determines a specific combination of position and momentum as being an observable; this ca…
We present a new approach to study a class of non-Hermitian (1+1)-dimensional Dirac Hamiltonian in the presence of local Fermi velocity. We apply the well known Nikiforov-Uvarov method to solve such a system. We discuss applications and explore the solvability of both $\mathcal{PT}$-symmetric and non-$\mathcal{PT}$ symmetric classes of potentials. In the former case we obtain the solution of a harmonic oscillator in the presence of a linear vector potential while in the latter case we solve the shifted harmonic oscillator problem.
We outline the nonlinear transformation in the path integral representation for partition function of O(N) symmetric oscillator systems bringing their duality to certain one-dimensional oscillators with unstable potential shapes. This duality transformation realizes the equivalence between spectra of a Hermitian and a non-Hermitian Hamiltonians. It is exploited to calculate exactly the spectra of quantum rotators . The zero-temperature limit is considered and the dual representation of n-point correlators of coordinate operators is obtained.
For the two-parameter $p,q$-deformed Heisenberg algebra introduced recently and in which, instead of usual commutator of $X$ and $P$ in the l.h.s. of basic relation $[X,P] = {\rm i}\hbar$, one uses the $p,q$-commutator, we established interesting properties. Most important is the realizability of the $p,q$-deformed Heisenberg algebra by means of the appropriate deformed oscillator algebra. Another uncovered property is special extension of the usual mutual Hermitian conjugation of the creation and annihilation operators, namely the so-called $η(N)$-pseudo-Hermitian conjugation rule, along with…
The quantum harmonic oscillator with parity-time ($\mathcal{PT}$) symmetry, obtained from the ordinary (Hermitian) quantum harmonic oscillator by an imaginary displacement of the spatial coordinate, provides an important and exactly-solvable model to investigate non-Hermitian extension of the Ehrenfest theorem. Here it is shown that transverse light dynamics in an optical resonator with off-axis longitudinal pumping can emulate a $\mathcal{PT}$-symmetric quantum harmonic oscillator, providing an experimentally accessible system to investigate non-Hermitian coherent state propagation.
We analyse some PT-symmetric oscillators with $T_{d}$ symmetry that depend on a potential parameter $g$. We calculate the eigenvalues and eigenfunctions for each irreducible representation and for a range of values of $g$. Pairs of eigenvalues coalesce at exceptional points $g_c$; their magnitude roughly decreasing with the magnitude of the eigenvalues. It is difficult to estimate whether there is a phase transition at a nonzero value of $g$ as conjectured in earlier papers. Group theory and perturbation theory enable one to predict whether a given space-time symmetry leads to real eigenvalues…
In a recent paper Cariñena et al analyzed a non-polynomial one-dimensional quantum potential representing an oscillator which they argued was intermediate between the harmonic and isotonic oscillators. In particular they proved that it is Schrödinger soluble, and explicitly obtained the wavefunctions and energies of the bound states. In this paper we show that these results can be obtained much more simply by noting that this potential is a supersymmetric partner potential of the harmonic oscillator. We then use this observation to generate an infinite set of potentials which can be exactly so…
Engineered non-Hermitian systems featuring exceptional points can lead to a host of extraordinary phenomena in diverse fields ranging from photonics, acoustics, opto-mechanics, electronics, to atomic physics. Here we introduce and present non-Hermitian dynamics of coupled optical parametric oscillators (OPOs) arising from phase-sensitive amplification and de-amplification, and show their distinct advantages over conventional non-Hermitian systems relying on laser gain and loss. OPO-based non-Hermitian systems can benefit from the instantaneous nature of the parametric gain, noiseless phase-sen…
Schroedinger equation H ψ=E ψwith PT - symmetric differential operator H=H(x) = p^2 + a x^4 + i βx^3 +c x^2+i δx = H^*(-x) on L_2(-\infty,\infty) is re-arranged as a linear algebraic diagonalization at a>0. The proof of this non-variational construction is given. Our Taylor series form of ψcomplements and completes the recent terminating solutions as obtained for certain couplings δat the less common negative a.
In the Schroedinger formulation of non-Hermitian quantum theories a positive-definite metric operator $η\equiv e^{-Q}$ must be introduced in order to ensure their probabilistic interpretation. This operator also gives an equivalent Hermitian theory, by means of a similarity transformation. If, however, quantum mechanics is formulated in terms of functional integrals, we show that the $Q$ operator makes only a subliminal appearance and is not needed for the calculation of expectation values. Instead, the relation to the Hermitian theory is encoded via the external source $j(t)$. These points ar…
New families of time-dependent potentials related with the stationary singular oscillator are introduced. This is achieved after noticing that a non stationary quantum invariant can be constructed for the singular oscillator. Such invariant depends on coefficients that are related to solutions of an Ermakov equation, the latter becomes essential since it guarantees the regularity of the solutions at each time. In this form, after applying the factorization method to the quantum invariant, rather than the Hamiltonian, one manages to introduce the time parameter into the transformation, leading …
We construct a time-dependent double well potential as an exact spectral equivalent to the explicitly time-dependent negative quartic oscillator with a time-dependent mass term. Defining the unstable anharmonic oscillator Hamiltonian on a contour in the lower-half complex plane, the resulting time-dependent non-Hermitian Hamiltonian is first mapped by an exact solution of the time-dependent Dyson equation to a time-dependent Hermitian Hamiltonian defined on the real axis. When unitary transformed, scaled and Fourier transformed we obtain a time-dependent double well potential bounded from belo…
We propose an alternative method to factorize an integer by using three harmonic oscillators. These oscillators are coupled together via specific Kerr nonlinear interactions. This method can be applied even if two harmonic oscillators are prepared in mixed states. As simple examples, we show how to factorize N=15 and 35 using this approach. The effect of dissipation of the harmonic oscillators on the performance of this method is studied. We also study the realization of nonlinear interactions between the coupled oscillators. However, the probability of finding the factors of a number is inver…
In the popular ${\cal PT}-$symmetry-based formulation of quantum mechanics of closed systems one can build unitary models using non-Hermitian Hamiltonians (i.e., $H \neq H^\dagger$) which are Hermitizable (so that one can write, simultaneously, $H = H^\ddagger$). The essence of the trick is that the reference Hilbert space $\cal R$ (in which we use the conventional inner product $\langle ψ_a|ψ_b\rangle$ and write $H \neq H^\dagger$) is declared unphysical. The necessary Hermiticity of the Hamiltonian $H = H^\ddagger$ can be then achieved by the mere metric-mediated amendment $\langle ψ_a|Θ|ψ_b…
The geometrical Berry phase is key to understanding the behaviour of quantum states under cyclic adiabatic evolution. When generalised to non-Hermitian systems with gain and loss, the Berry phase can become complex, and should modify not only the phase but also the amplitude of the state. Here, we perform the first experimental measurements of the adiabatic non-Hermitian Berry phase, exploring a minimal two-site $\mathcal{PT}$-symmetric Hamiltonian that is inspired by the Hatano-Nelson model. We realise this non-Hermitian model experimentally by mapping its dynamics to that of a pair of classi…
In this investigation, the displacement operator is revisited. We established a connection between the Hermitian version of this operator with the well-known Weyl ordering. Besides, we characterized the quantum properties of a simple displaced harmonic oscillator, as well as of a displaced anisotropic two-dimensional non-Hermitian harmonic oscillator. By constructing the partition functions for both harmonic oscillators, we were able to derive several thermodynamic quantities from their energy spectra. The features of these quantities were depicted and analyzed in details.
New families of time-dependent potentials related to the parametric oscillator are introduced. This is achieved by introducing some general time-dependent operators that factorize the appropriate constant of motion (quantum invariant) of the parametric oscillator, leading to new families of quantum invariants that are almost-isospectral to the initial one. Then, the respective time-dependent Hamiltonians are constructed, and the solutions of the Schrödinger equation are determined from the intertwining relationships and by finding the appropriate time-dependent complex-phases of the Lewis-Ries…
We provide time-evolution operators, gauge transformations and a perturbative treatment for non-Hermitian Hamiltonian systems, which are explicitly time-dependent. We determine various new equivalence pairs for Hermitian and non-Hermitian Hamiltonians, which are therefore pseudo-Hermitian and in addition in some cases also invariant under PT-symmetry. In particular, for the harmonic oscillator perturbed by a cubic non-Hermitian term, we evaluate explicitly various transition amplitudes, for the situation when these systems are exposed to a monochromatic linearly polarized electric field.
Non-Hermitian systems with exceptional points lead to many intriguing phenomena due to the coalescence of both eigenvalues and corresponding eigenvectors, in comparison to Hermitian systems where only eigenvalues degenerate. In this paper, we have investigated entropic uncertainty relation (EUR) in a non-Hermitian system and revealed a general connection between the EUR and the exceptional points of non-Hermitian system. Compared to the unitarity dynamics determined by a Hermitian Hamiltonian, the behaviors of EUR can be well defined in two different ways depending on whether the system is loc…
