Showing papers on "Ladder operator published in 2014"

Radial quantum number of Laguerre-Gauss modes

Abstract: We introduce an operator linked with the radial index in the Laguerre-Gauss modes of a two-dimensional harmonic oscillator in cylindrical coordinates. We discuss ladder operators for this variable, and confirm that they obey the commutation relations of the su(1,1) algebra. Using this fact, we examine how basic quantum optical concepts can be recast in terms of radial modes.

What is the relativistic spin operator

Abstract: Although the spin is regarded as a fundamental property of the electron, there is no universally accepted spin operator within the framework of relativistic quantum mechanics. We investigate the properties of different proposals for a relativistic spin operator. It is shown that most candidates are lacking essential features of proper angular momentum operators, leading to spurious zitterbewegung (quivering motion) or violation of the angular momentum algebra. Only the Foldy–Wouthuysen operator and the Pryce operator qualify as proper relativistic spin operators. We demonstrate that ground states of highly charged hydrogen-like ions can be utilized to identify a legitimate relativistic spin operator experimentally.

Combined state-adding and state-deleting approaches to type III multi-step rationally extended potentials: Applications to ladder operators and superintegrability

Abstract: Type III multi-step rationally extended harmonic oscillator and radial harmonic oscillator potentials, characterized by a set of k integers m1, m2, ⋯, mk, such that m1 < m2 < ⋯ < mk with mi even (resp. odd) for i odd (resp. even), are considered. The state-adding and state-deleting approaches to these potentials in a supersymmetric quantum mechanical framework are combined to construct new ladder operators. The eigenstates of the Hamiltonians are shown to separate into mk + 1 infinite-dimensional unitary irreducible representations of the corresponding polynomial Heisenberg algebras. These ladder operators are then used to build a higher-order integral of motion for seven new infinite families of superintegrable two-dimensional systems separable in cartesian coordinates. The finite-dimensional unitary irreducible representations of the polynomial algebras of such systems are directly determined from the ladder operator action on the constituent one-dimensional Hamiltonian eigenstates and provide an algebr...

Operator Systems from Discrete Groups

Abstract: We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlations. To do this we formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group \({\mathbb{F}_n}\) on n generators, as well as the operator systems of the free products of finitely many copies of the two-element group \({\mathbb{Z}_2}\). We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems.

Curvature operator for loop quantum gravity

Abstract: We introduce a new operator in Loop Quantum Gravity - the 3D curvature operator - related to the 3-dimensional scalar curvature. The construction is based on Regge Calculus. We define it starting from the classical expression of the Regge curvature, then we derive its properties and discuss some explicit checks of the semi-classical limit.

Hagedorn Wavepackets in Time-Frequency and Phase Space

Abstract: The Hermite functions are an orthonormalbasis of the space of square integrable functions with favourable approximation properties. Allowing for a flexible localization in position and momentum, the Hagedorn wavepackets generalize the Hermite functions also to several dimensions. Using Hagedorn’s raising and lowering operators, we derive explicit formulas and recurrence relations for the Wigner and FBI transform of the wavepackets and show their relation to the Laguerre polyomials.

Generating nonclassical states from classical radiation by subtraction measurements

Abstract: We describe the creation of nonclassical states of microwave radiation via ideal dichotomic single photon detection, i.e., a detector that only indicates presence or absence of photons. Ideally, such a detector has a back action in the form of the subtraction operator (bare lowering operator). Using the non-linearity of this back action, it is possible to create a large family of nonclassical states of microwave radiation, including squeezed and multi-component cat states, starting from a coherent state. We discuss the applicability of this protocol to current experimental designs of Josephson photomultipliers.

Operator Deformations in Quantum Measurement Theory

Abstract: In this paper, we develop a rigorous observable- and symmetry generator-related framework for quantum measurement theory by applying operator deformation techniques previously used in noncommutative quantum field theory. This enables the conventional observables (represented by unbounded operators) to play a role also in the more general setting. In addition, it gives a way of explicitly calculating the so-called instrument of the measurement process.

Charge Quantization from a Number Operator

Abstract: We explain how an unexpected algebraic structure, the division algebras, can be seen to underlie a generation of quarks and leptons. From this new vantage point, electrons and quarks are simply excitations from the neutrino, which formally plays the role of a vacuum state. Using the ladder operators which exist within the system, we build a number operator in the usual way. It turns out that this number operator, divided by 3, mirrors the behaviour of electric charge. As a result, we see that electric charge is quantized because number operators can only take on integer values. Finally, we show that a simple hermitian form, built from these ladder operators, results uniquely in the nine generators of SU c ( 3 ) and U em ( 1 ) . This gives a direct route to the two unbroken gauge symmetries of the standard model.

Supersymmetric partners of the harmonic oscillator with an infinite potential barrier

Abstract: Supersymmetry transformations of first- and second-order are used to generate Hamiltonians with known spectra departing from the harmonic oscillator with an infinite potential barrier. Also studied is the way in which the eigenfunctions of the initial Hamiltonian are transformed. The first- and certain second-order supersymmetric partners of the initial Hamiltonian possess third-order differential ladder operators. Since systems with this kind of operators are linked with the Painleve IV equation, several solutions of this nonlinear second-order differential equation will be simply found.

Generalized Uncertainty Principle and Self-Adjoint Operators

Abstract: In this work we explore the self-adjointness of the GUP-modified momentum and Hamiltonian operators over different domains. In particular, we utilize the theorem by von-Newmann for symmetric operators in order to determine whether the momentum and Hamiltonian operators are self-adjoint or not, or they have self-adjoint extensions over the given domain. In addition, a simple example of the Hamiltonian operator describing a particle in a box is given. The solutions of the boundary conditions that describe the self-adjoint extensions of the specific Hamiltonian operator are obtained.

Quantum harmonic oscillator with position-dependent mass in the displacement operator formalism

Abstract: The position-dependent effective mass quantum harmonic oscillator problem is considered within the displacement operator framework. Using the analytic and the algebraic approaches, exact expressions for quantum mechanical quantities of the system have been obtained. In the limit of no deformation, results of the constant mass oscillator are recovered.

Harmonic weak Maass forms of integral weight: a geometric approach

Abstract: The purpose of the present work is to provide a geometric framework for the study of the Fourier coefficients of harmonic weak Maass forms, a space of smooth modular forms first introduced by Bruinier and Funke in the context of singular theta lifts. In this geometric framework harmonic weak Maass forms arise from the construction of differentials whose classes are exact in certain de Rham cohomology groups attached to modular forms. We show how this new interpretation naturally leads to strengthenings of the theorems of Bruinier, Ono and Rhoades, by answering in the affirmative conjectures about the field of definitions of Fourier coefficients of harmonic weak Maass forms. Moreover, as part of our geometric framework, we describe a geometric interpretation for the Shimura-Maass lowering operator analogous to the description of the Shimura-Maass raising operator given by Katz. We also produce Eichler-Shimura-style isomorphisms for the de Rham cohomology attached to modular forms, generalizing results of Bringmann, Guerzhoy, Kent and Ono to any level and field of definition.

The analyticity of a generalized Ruelle's operator

Abstract: In this work we propose a generalization of the concept of Ruelle’s operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle’s operator. Our operator generalizes both the Ruelle operator proposed in [2] and the Perron Frobenius operator defined in [7]. We suppose the alphabet is given by a compact metric space, and consider a general a-priori measure to define the operator. We also consider the case where the set of symbols that can follow a given symbol of the alphabet depends on such symbol, which is an extension of the original concept of transition matrices from the theory of subshifts of finite type. We prove the analyticity of the Ruelle’s operator and present some examples.

Thermal balance and photon-number quantization in layered structures

Abstract: The quantization of the electromagnetic field in lossy and dispersive dielectric media has been widely studied during the last few decades. However, several aspects of energy transfer and its relation to consistently defining position-dependent ladder operators for the electromagnetic field in nonequilibrium conditions have partly escaped the attention. In this work we define the position-dependent ladder operators and an effective local photon-number operator that are consistent with the canonical commutation relations and use these concepts to describe the energy transfer and thermal balance in layered geometries. This approach results in a position-dependent photon-number concept that is simple and consistent with classical energy conservation arguments. The operators are formed by first calculating the vector potential operator using Green's function formalism and Langevin noise source operators related to the medium and its temperature, and then defining the corresponding position-dependent annihilation operator that is required to satisfy the canonical commutation relations in arbitrary geometry. Our results suggest that the effective photon number associated with the electric field is generally position dependent and enables a straightforward method to calculate the energy transfer rate between the field and the local medium. In particular, our results predict that the effective photon number in a vacuum cavity formed between two lossy material layers can oscillate as a function of the position suggesting that also the local field temperature oscillates. These oscillations are expected to be directly observable using relatively straightforward experimental setups in which the field-matter interaction is dominated by the coupling to the electric field. The approach also gives further insight on separating the photon ladder operators into the conventional right and left propagating parts and on the anomalies reported for the commutation relations of the corresponding operators within optical cavities.

Spectral approximation of quadratic operator polynomials arising in photonic band structure calculations

Abstract: Galerkin spectral approximation theory for non-self-adjoint quadratic operator polynomials with periodic coefficients is considered The main applications are complex band structure calculations in metallic photonic crystals, periodic waveguides, and metamaterials We show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzero imaginary part The spectral problem is equivalent to a non-compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator Convergence rates of finite element discretizations are considered and numerical experiments with the $$p$$ p -version and the $$h$$ h -version of the finite element method confirm the theoretical convergence rates

A Laplace transform approach to find the exact solution of the N-dimensional Schr\"odinger equation with Mie-type potentials and construction of Ladder operators

Abstract: The second order N-dimensional Schrodinger equation with Mie-type potentials is reduced to a first order differential equation by using the Laplace transformation. Exact bound state solutions are obtained using convolution or Faltungs theorem. The Ladder operators are also constructed for the Mie-type potentials in N- dimensions. Lie algebra associated with these operators are studied and it is found that they satisfy the commutation relations for the SU(1,1) group.

A Possible Method for Non-Hermitian and Non-PT-Symmetric Hamiltonian Systems

Abstract: A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator η+ and defining the annihilation and creation operators to be η+ -pseudo-Hermitian adjoint to each other. The operator η+ represents the η+ -pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-PT-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator η+ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution, are obtained for the non-Hermitian and non-PT-symmetric Hamiltonian. Moreover, this Hamiltonian turns out to be coupled when it is extended to the canonical noncommutative space with noncommutative spatial coordinate operators and noncommutative momentum operators as well. Our method is applicable to the coupled Hamiltonian. Then the first and second order noncommutative corrections of energy levels are calculated, and in particular the reality of energy spectra, the positive-definiteness of inner products, and the related properties (the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution) are found not to be altered by the noncommutativity.

Semiclassical asymptotic spectrum of a Hartree-type operator near the upper boundary of spectral clusters

Abstract: We consider the problem for eigenvalues of a perturbed two-dimensional oscillator in the case of a resonance frequency. The exciting potential is given by a Hartree-type integral operator with a smooth self-action potential. We find asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundary of spectral clusters, which form around energy levels of the nonperturbed operator. To calculate them, we use asymptotic formulas for quantum means.

Polynomial stability of semigroups generated by operator matrices

Abstract: In this paper, we study the stability properties of strongly continuous semigroups generated by block operator matrices. We consider triangular and full operator matrices whose diagonal operator blocks generate polynomially stable semigroups. As our main results, we present conditions under which also the semigroup generated by the operator matrix is polynomially stable. The theoretical results are used to derive conditions for the polynomial stability of a system consisting of a two-dimensional and a one-dimensional damped wave equation.

The Generic Superintegrable System on the 3-Sphere and the 9j Symbols of su(1,1)

Abstract: The 9j symbols of su(1; 1) are studied within the framework of the generic superintegrable system on the 3-sphere. The canonical bases corresponding to the binary coupling schemes of four su(1; 1) representations are constructed explicitly in terms of Jacobi polynomials and are seen to correspond to the separation of variables in different cylindrical coordinate systems. A triple integral expression for the 9j coefficients exhibiting their symmetries is derived. A double integral formula is obtained by extending the model to the complex three-sphere and taking the complex radius to zero. The explicit expression for the vacuum coefficients is given. Raising and lowering operators are constructed and are used to recover the relations between contiguous coefficients. It is seen that the 9j symbols can be expressed as the product of the vacuum coefficients and a rational function. The recurrence relations and the difference equations satisfied by the 9j coefficients are derived.

Ladder operators for solvable potentials connected with exceptional orthogonal polynomials

Abstract: Exceptional orthogonal polynomials constitute the main part of the bound-state wavefunctions of some solvable quantum potentials, which are rational extensions of well-known shape-invariant ones. The former potentials are most easily built from the latter by using higher-order supersymmetric quantum mechanics (SUSYQM) or Darboux method. They may in general belong to three different types (or a mixture of them): types I and II, which are strictly isospectral, and type III, for which k extra bound states are created below the starting potential spectrum. A well-known SUSYQM method enables one to construct ladder operators for the extended potentials by combining the supercharges with the ladder operators of the starting potential. The resulting ladder operators close a polynomial Heisenberg algebra (PHA) with the corresponding Hamiltonian. In the special case of type III extended potentials, for this PHA the k extra bound states form k singlets isolated from the higher excited states. Some alternative constructions of ladder operators are reviewed. Among them, there is one that combines the state-adding and state-deleting approaches to type III extended potentials (or so-called Darboux-Crum and Krein-Adler transformations) and mixes the k extra bound states with the higher excited states. This novel approach can be used for building integrals of motion for two-dimensional superintegrable systems constructed from rationally-extended potentials.

Chern-Simons interactions in AdS$_3$ and the current conformal block

Abstract: We compute the four point function of scalar fields in AdS$_3$ charged under $U(1)$ Chern-Simons fields using the bulk version of the operator state mapping. Then we show how this four point function is reproduced from a CFT$_2$ with a global $U(1)$ symmetry, through the contribution of the corresponding current operator in the operator product expansion, i.e. through the conformal block of the current operator. We work in a "probe approximation" where the gravitational interactions are ignored, which corresponds to leaving out the energy momentum tensor from the operator product expansion.