A $γ$-deformed version of $\mathfrak{su}(2)$ algebra has been obtained from a bi-orthogonal system of vectors in $\bf{C^2}$. Fusion of Jordan-Schwinger realization of complexified $\mathfrak{su}(2)$ with Dyson-Maleev representation gives a 3-boson realization of Higgs algebra of cubic polynomial type. The non-hermitian Hamiltonian thus obtained is found to have real eigenvalues and eigen states with symmetry induced orthogonality. The notion of partial ${\mathcal {PT}}$-symmetry (henceforth $\partial_{\mathcal { PT}}$) has been introduced as a characteristic feature of these multi-boson realizations. The Hamiltonian along with its eigenstates have been studied in the light of $\partial_{\mathcal { PT}}$-symmetry. The possibility of $\partial_{\mathcal { PT}}$-symmetry breaking is also discussed. The deformation parameter $γ$ plays a crucial role in the entire formulation and non-trivially modifies the eigenfunctions under consideration.
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…
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…
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…
Starting on the basis of the non-commutative q-differential calculus, we introduce a generalized q-deformed Schrödinger equation. It can be viewed as the quantum stochastic counterpart of a generalized classical kinetic equation, which reproduces at the equilibrium the well-known q-deformed exponential stationary distribution. In this framework, q-deformed adjoint of an operator and q-hermitian operator properties occur in a natural way in order to satisfy the basic quantum mechanics assumptions.
PT-/non-PT-symmetric and non-Hermitian deformed Morse and Poschl-Teller potentials are studied first time by quantum Hamilton-Jacobi approach. Energy eigenvalues and eigenfunctions are obtained by solving quantum Hamilton-Jacobi equation.
The solutions of trigonometric Scarf potential, PT/non-PT-symmetric and non-Hermitian q-deformed hyperbolic Scarf and Manning-Rosen potentials are obtained by solving the Schrodinger equation. The Nikiforov-Uvarov method is used to obtain the real energy spectra and corresponding eigenfunctions.
The supersymmetric solutions of PT-/non-PT-symmetric and non-Hermitian deformed Morse and Pöschl-Teller potentials are obtained by solving the Schrödinger equation. The Hamiltonian hierarchy method is used to get the real energy eigenvalues and corresponding eigenfunctions. \newline {PACS:05.20.-y; 05.30.-d; 05.70. Ce; 03.65.-w}\newline {\it Keywords}{\small : Supersymmetric quantum mechanics, Hamiltonian Hierarchy Method, Morse potential, Pöschl-Teller potential}
For the first time, we have observed the annihilation of multiple eigenstates of the parent potentials and redistribution of the energy in the deformed potentials in the system with spontaneously breaking of parity-time symmetry while preserving the SUSY of the system. We have observed that the deformation of the potentials is sensitive to initial conditions. This first experimental observation enables the localization of eigenspectra in lateral space (so-called time) and the localization of eigenstates in longitudinal space in the deformed potentials by experimentally applying the SUSY higher…
We propose to uncover the topology of a pseudo-Hermitian Chern insulator by quantum quench dynamics. The Bloch Hamiltonian of the pseudo-Hermitian Chern insulator is defined in the basis of the q-deformed Pauli matrices, which are related to the representation of the deformed algebras. We show the bulk-surface duality of the pseudo-Hermitian phases, then further build a concrete relation between the static band topology and quench dynamics, in terms of the time-averaged spin textures. The results are also generalized into a fully nonequilibrium case where the postquench evolution is governed b…
Among non-Hermitian systems, pseudo-Hermitian phases represent a special class of physical models characterized by real energy spectra and by the absence of non-Hermitian skin effects. Here, we show that several pseudo-Hermitian phases in two and three dimensions can be built by employing $q$-deformed matrices, which are related to the representation of deformed algebras. Through this algebraic approach we present and study the pseudo-Hermitian version of well known Hermitian topological phases, raging from two-dimensional Chern insulators and time-reversal-invariant topological insulators to …
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…
In this paper, we investigate the relation between the curvature of the physical space and the deformation function of the deformed oscillator algebra using non-linear coherent states approach. For this purpose, we study two-dimensional harmonic oscillators on the flat surface and on a sphere by applying the Higgs modell. With the use of their algebras, we show that the two-dimensional oscillator algebra on a surface can be considered as a deformed one-dimensional oscillator algebra where the effect of the curvature of the surface is appeared as a deformation function. We also show that the cu…
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…
Deformed Heisenberg algebra and its Hilbert space representations
African Institute for Mathematical Sciences (AIMS)（非洲数学科学研究所）；Universite Thomas Sankara（托马斯·桑卡拉大学）；Université Felix Houphouët Boigny-Abidjan（费利克斯·乌弗埃-博瓦尼大学）
Darboux-deformations of short range one-dimensional potentials are constructed by means of Gamow-Siegert functions (resonance states). Results include both Hermitian and non-Hermitian short range potentials which are exactly solvable. As illustration, the method is applied to square wells and barriers for which the transmission coefficient is a superposition of Fock-Breit-Wigner distributions. Resonance levels are calculated in the long lifetime limit by means of analytical and numerical approaches. The new complex potentials behave as an optical device which both refracts and absorbs light wa…
We demonstrate that the recent paper by Jana and Roy entitled ''Non-Hermitian quantum mechanics with minimal length uncertainty'' [SIGMA 5 (2009), 083, 7 pages, arXiv:0908.1755] contains various misconceptions. We compare with an analysis on the same topic carried out previously in our manuscript [arXiv:0907.5354]. In particular, we show that the metric operators computed for the deformed non-Hermitian Swanson models differs in both cases and is inconsistent in the former.
Recently a $f$-deformed Fock space which is spanned by $|n>_λ$ has been introduced. These bases are indeed the eigen-states of a deformed non-Hermitian Hamiltonian. In this contribution, we will use a rather new non-orthogonal basis vectors for the construction of coherent and squeezed states, which in special case lead to the earlier known states. For this purpose, we first generalize the previously introduced Fock space spanned by $|n>_λ$ bases, to a new one, spanned by an extended two-parameters bases $|n>_{λ_{1},λ_{2}}$. These bases are now the eigen-states of a non-Hermitian Hamiltonian $…
Deformations of the canonical commutation relations lead to non-Hermitian momentum and position operators and therefore almost inevitably to non-Hermitian Hamiltonians. We demonstrate that such type of deformed quantum mechanical systems may be treated in a similar framework as quasi/pseudo and/or PT-symmetric systems, which have recently attracted much attention. For a newly proposed deformation of exponential type we compute the minimal uncertainty and minimal length, which are essential in almost all approaches to quantum gravity.
The theory of non-Hermitian systems and the theory of quantum deformations have attracted a great deal of attention in the past decades. In general, non-Hermitian Hamiltonians are constructed by an ad hoc manner. Here, we study the (2+1) Dirac oscillator and show that in the context of the $κ$-deformed Poincaré-Hopf algebra its Hamiltonian is non-Hermitian but has real eigenvalues. The non-Hermiticity stems from the $κ$-deformed algebra. From the mapping in Bermudez et al., Phys. Rev. A 76, 041801(R) (2007), we propose the $κ$-Jaynes-Cummings and $κ$-anti-Jaynes-Cummings models, which describe…
Genuinely non-Hermitian topological phases can be realized in open systems with sufficiently strong gain and loss; in such phases, the Hamiltonian cannot be deformed into a gapped Hermitian Hamiltonian without energy bands touching each other. Comparing Green functions for periodic and open boundary conditions we find that, in general, there is no correspondence between topological invariants computed for periodic boundary conditions, and boundary eigenstates observed for open boundary conditions. Instead, we find that the non-Hermitian winding number in one dimension signals a topological pha…
We consider wave dynamics for a Schrödinger equation with a non-Hermitian Hamiltonian $\mathcal{H}$ satisfying the generalized (anyonic) parity-time symmetry $\mathcal{PT H}= \exp(2 i φ) \mathcal{HPT}$, where $\mathcal{P}$ and $ \mathcal{T}$ are the parity and time-reversal operators. For a stationary potential, the anyonic phase $φ$ just rotates the energy spectrum of $\mathcal{H}$ in complex plane, however for a drifting potential the energy spectrum is deformed and the scattering and localization properties of the potential show intriguing behaviors arising from the breakdown of the Galilea…
We note that, though nonanticommutative (NAC) deformations of Minkowski supersymmetric theories do not respect the reality condition and seem to lead to non-Hermitian Hamiltonians H, the latter belong to the class of ``cryptoreal'' Hamiltonians considered recently by Bender and collaborators. They can be made manifestly Hermitian via the similarity transformation H -> exp{R} H exp{-R} with a properly chosen R. The deformed model enjoys the same supersymmetry algebra as the undeformed one, though being realized differently on the involved canonical variables. Besides quantum-mechanical models, …
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…
We provide a unified approach for finding the coherent states of various deformed algebras, including quadratic, Higgs and q-deformed algebras, which are relevant for many physical problems. For the non-compact cases, coherent states, which are the eigenstates of the respective annihilation operators, are constructed by finding the canonical conjugates of these operators. We give a general procedure to map these deformed algebras to appropriate Lie algebras. Generalized coherent states, in the Perelomov sense, follow from this construction.
We investigate four different types of representations of deformed canonical variables leading to generalized versions of Heisenberg's uncertainty relations resulting from noncommutative spacetime structures. We demonstrate explicitly how the representations are related to each other and study three characteristically different solvable models on these spaces, the harmonic oscillator, the manifestly non-Hermitian Swanson model and an intrinsically noncommutative model with Poeschl-Teller type potential. We provide an analytical expression for the metric in terms of quantities specific to the g…
We discuss non-Hermitian field theories where the spectrum of the Hamiltonian involves only real energies. We make three observations. (i) The theories obtained from supersymmetric theories by nonanticommutative deformations belong in many cases to this class. (ii) When the deformation parameter is small, the deformed theory enjoys the same supersymmetry algebra as the undeformed one. Half of the supersymmetries are manifest and the existence of another half can be deduced from the structure of the spectrum. (iii) Generically, the conventionally defined S--matrix is not unitary for such theori…
The recently introduced two- and three-parameter ($p,q$)- and ($p,q,μ$)-deformed extensions of the Heisenberg algebra were explored under the condition of their connectedness with the respective nonstandard (other than known ones) deformed quantum oscillator algebras. In this paper we show that such connection dictates certain new $η(N)$-pseudo-Hermitian conjugation rule between the creation and annihilation operators, with $η(N)$ depending on the particle number operator $N$. In turn, that leads to the related $η(N)$-pseudo-Hermiticity of the position/momentum operators, though the involved H…
Floquet engineering, modulating quantum systems in a time periodic way, lies at the central part for realizing novel topological dynamical states. Thanks to the Floquet engineering, various new realms on experimentally simulating topological materials have emerged. Conventional Floquet engineering, however, only applies to time periodic non-dissipative Hermitian systems, and for the quantum systems in reality, non-Hermitian process with dissipation usually occurs. So far, it remains unclear how to characterize topological phases of periodically driven non-Hermitian systems via the frequency sp…
We discuss how a q-mutation relation can be deformed replacing a pair of conjugate operators with two other and unrelated operators, as it is done in the construction of pseudo-fermions, pseudo-bosons and truncated pseudo-bosons. This deformation involves interesting mathematical problems and suggests possible applications to pseudo-hermitian quantum mechanics. We construct bi-coherent states associated to $\D$-pseudo-quons, and we show that they share many of their properties with ordinary coherent states. In particular, we find conditions for these states to exist, to be eigenstates of suita…
The construction of exactly-solvable models has recently been advanced by considering integrable $T\bar{T}$ deformations and related Hamiltonian deformations in quantum mechanics. We introduce a broader class of non-Hermitian Hamiltonian deformations in a nonrelativistic setting, to account for the description of a large class of open quantum systems, which includes, e.g., arbitrary Markovian evolutions conditioned to the absence of quantum jumps. We relate the time evolution operator and the time-evolving density matrix in the undeformed and deformed theories in terms of integral transforms w…
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…
