A displacement operator d_ζis introduced, verifying commutation relations [d_ζ, a_f^\dagger]=[d_ζ, a_f]=ζ(f)d_ζwith field creation and annihilation operators that verify [a_f,a_g]=0, [a_f,a_g^\dagger]=(g,f), as usual. f and g are test functions, ζis a Poincare invariant real-valued function on the test function space, and (g,f) is a Poincare invariant Hermitian inner product. The *-algebra generated by all these operators, and a state defined on it, nontrivially extends the *-algebra of creation and annihilation operators and its Fock space representation. If the usual requirement for linearity is weakened, as suggested in quant-ph/0512190, we obtain a deformation of the free quantum field.
A $q$-deformed Weyl-Heisenberg algebra is used to define a deformed displacement operator giving rise to a naturally normalized nonlinear coherent states type. Robust maximally entangled deformed coherent states are studied and the effect of such a deformation on the amount of the entanglement is discussed. The analogy between environment decoherence and algebra deformation is made through the deformation parameter.
We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator, associated with a position-dependent effective mass, by means of the supersymmetric formalism. From the supersymmetric partner Hamiltonians and the shape invariance technique we obtain the eigenstates and the eigenvalues along with the ladders operators, thus showing a preservation of the supersymmetric structure in terms of the deformed counterpartners. The deformed space in supersymmetry allows to characterize position-dependent effective mass, uniform field interactions and to ob…
Using the Wigner distribution function, we analyze the behavior on phase space of generalized coherent states associated with the Morse potential (Morse-like coherent states). Within the f-deformed oscillator formalism, such states are constructed by means of the two following definitions: {\it i)} as deformed displacement operator coherent states (DOCSs) and {\it ii)} as deformed photon-added coherent states (DPACSs).
We construct Deformed Photon Added Nonlinear Coherent States (DPANCSs) by application of the deformed creation operator upon the Nonlinear Coherent States obtained as eigenstates of the deformed annihilation operator and by application of a deformed displacement operator upon the vacuum state. We evaluate some statistical properties like the Mandel parameter, Husimi and Wigner functions for these states and analyze their differences, we give closed analytical expressions for them. We found a profound difference in the statistical properties of the DPANCSs obtained from the two above mentioned …
On the basis of the f-deformed oscillator formalism, we propose to construct nonlinear coherent states for Hamiltonian systems having linear and quadratic terms in the the number operator by means of the two following definitions: i) as deformed annihilation operator coherent states (AOCS) and ii) as deformed displacement operator coherent states (DOCS). For the particular cases of the Morse and Modified Pöschl-Teller potentials, modeled as f-deformed oscillators (both supporting a finite number of bound states), the properties of their corresponding nonlinear coherent states, viewed as DOCS, …
We obtained analytical expressions for Entangled of Formation (EoF) and Quantum Discord (QD) of Werner states and Generalized Werner-Like states. The optimization problem involved under the exact analytical form is obtained for both quantities. In order to illustrate the importance of our results we studied the EoF and the QD of these states. Using discrete formulation of continue states with the $f-$deformed coherent states obtained as deformed annihilation operator coherent states and as deformed displacement operator coherent states. The EoF and QD of bipartite Werner-Like states $f-$deform…
We consider canonically conjugated generalized space and linear momentum operators $\hat{x}_q$ and $ \hat{p}_q$ in quantum mechanics, associated to a generalized translation operator which produces infinitesimal deformed displacements controlled by a deformation parameter $q$. A canonical transformation $(\hat{x}, \hat{p}) \rightarrow (\hat{x}_q, \hat{p}_q)$ leads the Hamiltonian of a position-dependent mass particle in usual space to another Hamiltonian of a particle with constant mass in a conservative force field of the deformed space. The equation of motion for the classical phase space $(…
The aim of these two papers (I and II) is to try to give fundamental concepts of quantum kinematics to q-deformed quantum spaces. Paper I introduces the relevant mathematical concepts. A short review of the basic ideas of q-deformed analysis is given. These considerations are continued by introducing q-deformed analogs of Fourier transformations and delta functions. Their properties are discussed in detail. Furthermore, q-deformed versions of sesquilinear forms are defined, their basic properties are derived, and q-analogs of the Fourier-Plancherel identity are proved. In paper II these reason…
The interaction of a quantum deformed oscillator with the environment is studied deriving a master equation whose form strongly depends on the type of deformation.
We show that the quantum logic gates, {\it viz.} the single qubit Hadamard and Phase Shift gates, can also be realised using q-deformed angular momentum states constructed via the Jordan-Schwinger mechanism with two q-deformed oscillators. {\it Keywords :} quantum logic gates ; q-deformed oscillators ; quantum computation {\it PACS:} 03.67.Lx ; 02.20.Uw
We study two-level q-deformed angular momentum states and us- ing q-deformed harmonic oscillators, we provide a framework for con- structing qubits and quantum gates. We also present the construction of some basic quantum gates including CNOT, SWAP, Toffoli and Fredkin.
A particle bounded in a potential with finite range is described by using an $f$-deformed quantum oscillator approach. Finite range of this potential can be considered as a controllable deformation parameter. The non-classical quantum statistical properties of this deformed oscillator can be manipulated by nonlinearities associated to the finite range.
Deforming the algebraic structure of geometric algebra on the phase space with a Moyal product leads naturally to supersymmetric quantum mechanics in the star product formalism.
We study the dissipative dynamics of deformed coherent states superposition. We find that such kind of superposition can be more robust against decoherence than the usual Schrodinger cat states.
We show that various kinds of one-photon quantum states studied in the field of quantum optics admit ladder operator formalisms and have the generally deformed oscillator algebraic structure. The two-photon case is also considered. We obtain the ladder operator formalisms of two general states defined in the even/odd Fock space. The two-photon states may also have a generally deformed oscillator algebraic structure. Some interesting examples of one-photon and two-photon quantum states are given.
A master equation for the deformed quantum harmonic oscillator interacting with a dissipative environment, in particular with a thermal bath, is derived in the microscopic model by using perturbation theory. The coefficients of the master equation and of equations of motion for observables depend on the deformation function. The steady state solution of the equation for the density matrix in the number representation is obtained and the equilibrium energy of the deformed harmonic oscillator is calculated in the approximation of small deformation.
This is the third part of a paper about non-relativistic Schroedinger theory on q-deformed quantum spaces like the braided line or the three-dimensional q-deformed Euclidean space. Propagators for the free q-deformed particle are derived and their basic properties are discussed. A time-dependent formulation of scattering is proposed. In this respect, q-analogs of the Lippmann-Schwinger equation are given. Expressions for their iterative solutions are written down. It is shown how to calculate S-matrices and transition probabilities. Furthermore, attention is focused on the question what become…
Quantum correlations in compound systems are of great importance, and they are fundamental resource for the development of quantum computation protocols and quantum information. In this work we construct bipartite pure coherent states using the $f$-deformed oscillator formalism and the Barut-Girardello deformation function. Also we construct the extension of these systems to mixed states using Werner like states. For the states constructed, we study the dependence of the entanglement and quantum discord upon the deformation and mixed parameters.
It is shown that the two qubit CNOT (controlled NOT) gate can also be realised using q-deformed angular momentum states via the Jordan-Schwinger mechanism.Thus all the three gates necessary for universality i.e. Hadamard, Phase Shift and the two qubit CNOT gate are realisable with q-deformed oscillators.
A new kind of deformed calculus (the D-deformed calculus) that takes place in fractional-dimensional spaces is presented. The D-deformed calculus is shown to be an appropriate tool for treating fractional-dimensional systems in a simple way and quite analogous to their corresponding one-dimensional partners. Two simple systems, the free particle and the harmonic oscillator in fractional- dimensional spaces are reconsidered into the framework of the D-deformed quantum mechanics. Confined states in a D-deformed quantum well are studied. D-deformed coherent states are also found.
The aim of these three papers (I, II, and III) is to develop a q-deformed version of non-relativistic Schroedinger theory. Paper I introduces the fundamental mathematical and physical concepts. The braided line and the three-dimensional q-deformed Euclidean space play the role of position space. For both cases the algebraic framework is extended by a time element. A short review of the elements of q-deformed analysis on the spaces under consideration is given. The time evolution operator is introduced in a consistent way and its basic properties are discussed. These reasonings are continued by…
The aim of Part II of this paper is to try to describe wave functions on q-deformed versions of position and momentum space. This task is done within the framework developed in Part I of the paper. In order to make Part II self-contained the most important results of Part I are reviewed. Then it is shown that q-deformed exponentials and q-deformed delta functions play the role of momentum and position eigenfunctions, respectively. Their completeness and orthonormality relations are derived. For both bases of eigenfunctions matrix elements of position and momentum operators are calculated. A q-…
In this paper, first we explain what are the `quantum displacements'. We establish a group of bases, which contains the coupled bases coupling a ququart and a bipartite qubit systems. By these bases, we can realize the quantum displacements. We discuss some possible forms of them. At last, we point out that a so-call ''non-imprecisely-cloning theorem'' also holds.
This is the second part of a paper about a q-deformed analog of non-relativistic Schroedinger theory. It applies the general ideas of part I and tries to give a description of one-particle states on q-deformed quantum spaces like the braided line or the q-deformed Euclidean space in three dimensions. Hamiltonian operators for the free q-deformed particle in one as well as three dimensions are introduced. Plane waves as solutions to the corresponding Schroedinger equations are considered. Their completeness and orthonormality relations are written down. Expectation values of position and moment…
We demonstrate that superpositions of coherent and displaced Fock states, also referred to as generalized Schrödinger cats cats, can be created by application of a nonlinear displacement operator which is a deformed version of the Glauber displacement operator. Consequently, such generalized cat states can be formally considered as nonlinear coherent states. We then show that Glauber-Fock photonic lattices endowed with alternating positive and negative coupling coefficients give rise to classical analogs of such cat states. In addition, it is pointed out that the analytic propagator of these d…
With a q-deformed quantum mechanical framework, features of the uncertainty relation and a novel formulation of the Schrödinger equation are considered.
In this work we make use of deformed operators to construct the coherent states of some nonlinear systems by generalization of two definitions: i) As eigenstates of a deformed annihilation operator and ii) by application of a deformed displacement operator to the vacuum state. We also construct the coherent states for the same systems using the ladder operators obtained by traditional methods with the knowledge of the eigenfunctions and eigenvalues of the corresponding Schrödinger equation. We show that both methods yield coherent states with identical algebraic structure.
In this paper, the quantization and generalized uncertainty relation for some quantum deformed algebras are investigated. For several deformed algebras, the commutation relation between the position and the momentum operator is shown to be the same as the one in the Planck scale deformed quantum mechanics.
We construct the exact position representation of a deformed quantum mechanics which exhibits an intrinsic maximum momentum and use it to study problems such as a particle in a box and scattering from a step potential, among others. In particular, we show that unlike usual quantum mechanics, the present deformed case delays the formation of bound states in a finite potential well. In the process we also highlight some limitations and pit-falls of low-momentum or perturbative treatments and thus resolve two puzzles occurring in the literature.
I extend the three-dimensional q-deformed Euclidean space by a time element and discuss the algebraic structure of this quantum space together with its differential calculi. Using the star-product formalism, I will give basic operations of q-deformed analysis for the q-deformed Euclidean space with a time element. I show that the time-evolution operator of a quantum system living in the q-deformed Euclidean space is of the same form as in the undeformed case. The reasonings also show that the well-known methods of quantum dynamics apply to quantum systems living in the q-deformed Euclidean spa…
Within the formulation of a q-deformed Quantum Mechanics a qualitative undercut of the q-deformed uncertainty relation from the Heisenberg uncertainty relation is revealed. When $q$ is some fixed value not equal to one, recovering of ordinary quantum mechanics and the corresponding recovering condition are discussed.
A new version of the q-deformed supersymmetric quantum mechanics (q-SQM), which is inspired by the Tamm--Dankoff-type (TD-type) deformation of quantum harmonic oscillator, is constructed. The obtained algebra of q-SQM is similar to that in Spiridonov's approach. However, within our version of q-SQM, the ground state found explicitly in the special case of superpotential yiealding q-superoscillator turns out to be non-Gaussian and takes the form of special (TD-type) q-deformed Gaussian.
We develop a new quantum neural network layer designed to run efficiently on a quantum computer but that can be simulated on a classical computer when restricted in the way it entangles input states. We first ask how a classical neural network architecture, both fully connected or convolutional, can be executed on a quantum computer using quantum phase estimation. We then deform the classical layer into a quantum design which entangles activations and weights into quantum superpositions. While the full model would need the exponential speedups delivered by a quantum computer, a restricted clas…
Starting on the basis of $q$-symmetric oscillator algebra and on the associate $q$-calculus properties, we study a deformed quantum mechanics defined in the framework of the basic square-integrable wave functions space. In this context, we introduce a deformed Schroedinger equation, which satisfies the main quantum mechanics assumptions and admits, in the free case, plane wave functions that can be expressed in terms of the q-deformed exponential, originally introduced in the framework of the basic-hypergeometric functions.
We summarize some basics about mathematical tools of analysis for the q-deformed Euclidean space. We use the new tools to examine q-deformed eigenfunctions of the momentum or position operator within the framework of the star product formalism. We show that these two systems of functions are complete and orthonormal. With the q-deformed momentum or position eigenfunctions, we calculate matrix elements of the momentum or position operator. Considerations about expectation values and probability densities conclude the studies.
We show how the Weyl quantum walk derived from principles in Ref. [1], enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras--the usual Poincaré and the k-Poincaré algebras.
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.
In this paper, using the notion of nonlinear coherent states, we define a deformed bosonic dephasing channel modelling the impact of a Kerr medium on a quantum state, as it occurs, for instance, in quantum communication based on optical fibers. We show that, in certain regimes, the Kerr nonlinearity is able to compensate the dephasing. In addition, our studies reveal that the quantum capacity of the deformed bosonic dephasing channel can be greater than that of the undeformed, standard bosonic dephasing channel for certain nonlinearity parameters.
We investigate the quantum state transfer in a chain of particles satisfying q-deformed oscillators algebra. This general algebraic setting includes the spin chain and the bosonic chain as limiting cases. We study conditions for perfect state transfer depending on the number of sites and excitations on the chain. They are formulated by means of irreducible representations of a quantum algebra realized through Jordan-Schwinger maps. Playing with deformation parameters, we can study the effects of nonlinear perturbations or interpolate between the spin and bosonic chain.
We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translationally) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, by an infinite set of partial differential equations can be generalized to deformed shape invariant potentials in deformed supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to $\hbar$-independent superpotentials and to a superpotential explicitly depending on $\…
Generalized $f$-coherent state approach in deformation quantization framework is investigated by using a $\ast $-eigenvalue equation. For this purpose we introduce a new Moyal star product called $f$-star product, so that by using this ${\ast}_{f}$-eigenvalue equation one can obtain exactly the spectrum of a general Hamiltonian of a deformed system.
We present the quantum and classical mechanics formalisms for a particle with position-dependent mass in the context of a deformed algebraic structure (named $κ$-algebra), motivated by the Kappa-statistics. From this structure we obtain deformed versions of the position and momentum operators, which allow to define a point canonical transformation that maps a particle with constant mass in a deformed space into a particle with position-dependent mass in the standard space. We illustrate the formalism with a particle confined in an infinite potential well and the Mathews-Lakshmanan oscillator, …
On the basis of the non-commutative q-calculus, we investigate a q-deformation of the classical Poisson bracket in order to formulate a generalized q-deformed dynamics in the classical regime. The obtained q-deformed Poisson bracket appears invariant under the action of the q-symplectic group of transformations. In this framework we introduce the q-deformed Hamilton's equations and we derive the evolution equation for some simple q-deformed mechanical systems governed by a scalar potential dependent only on the coordinate variable. It appears that the q-deformed Hamiltonian, which is the gener…
In this work, we study the exact solution of Dirac equation in the hyper-spherical coordinate under influence of separable q-Deformed quantum potentials. The q-deformed hyperbolic Rosen-Morse potential is perturbed by q-deformed non-central trigonometric Scarf potentials, where whole of them can be solved by using Asymptotic Iteration Method (AIM). This work is limited to spin symmetry case. The relativistic energy equation and orbital quantum number equation lD-1 have been obtained using Asymptotic Iteration Method. The upper radial wave function equations and angular wave function equations …
Charged and symmetry-resolved Rényi entropies are entanglement measures quantifying the degree of entanglement within different charge sectors of a theory with a conserved global charge. We use holography to determine the dependence of charged Rényi entropies on small shape deformations away from a spherical or planar entangling surface in general dimensions. This dependence is completely characterized by a single coefficient appearing in the two point function of the displacement operator associated with the Rényi defect. We extract this coefficient using its relation to the one point functio…
Fractional supersymmetric quantum mechanics of order $λ$ is realized in terms of the generators of a generalized deformed oscillator algebra and a Z$_λ$-grading structure is imposed on the Fock space of the latter. This realization is shown to be fully reducible with the irreducible components providing $λ$ sets of minimally bosonized operators corresponding to both unbroken and broken cases. It also furnishes some examples of Z$_λ$-graded uniform topological symmetry of type (1, 1, ..., 1) with topological invariants generalizing the Witten index.
