Showing papers on "Coherent states in mathematical physics published in 2012"
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05 Feb 2012
TL;DR: The standard coherent states of quantum mechanics were defined and analyzed in this article, where the Weyl symbols of the metaplectic operators were represented as Weyl-Heisenberg group.
Abstract: The standard coherent states of quantum mechanics.- The Weyl-Heisenberg group and the coherent states of arbitrary profile.- The coherent states of the Harmonic Oscillator.- From Schrodinger to Fock-Bargmann representation.- Weyl quantization and coherent states: Classical and Quantum observables.- Wigner function.- Coherent states and operator norm estimates.- Product rule and applications.- Husimi functions, frequency sets and propagation.- The Wick and anti-Wick quantization.- The generalized coherent states in the sense of Perelomov.- The SU(1,1) coherent states: Definition and properties.- The squeezed states.- The SU(2) coherent states.- The quantum quadratic Hamiltonians: The propagator of quadratic quantum Hamiltonians.- The metaplectic transformations.- The propagation of coherent states.- Representation of the Weyl symbols of the metaplectic operators.- The semiclassical evolution of coherent states.- The van Vleck and Hermann-Kluk approximations.- The semiclassical Gutzwiller trace formula using coherent states decomposition.- The hydrogen atom coherent states: Definition and properties.- The localization around Kepler orbits.- The quantum singular oscillator: The two-body case.- The N-body case.
262 citations
TL;DR: In this paper, a review of entanglement in quantum systems is presented, focusing on the mathematical and physical aspects of entangling coherent states, which are in a sense the most classical states of a dynamical system.
Abstract: We review entangled coherent state research since its first implicit use in 1967 to the present. Entangled coherent states are important to quantum superselection principles, quantum information processing, quantum optics and mathematical physics. Despite their inherent fragility, entangled coherent states have been produced in a conditional propagating-wave quantum optics realization. Fundamentally the states are intriguing because they entangle the coherent states, which are in a sense the most classical of all states of a dynamical system.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
210 citations
TL;DR: The stochastic master equations, that is to say, quantum filters, and master equations for an arbitrary quantum system probed by a continuous-mode bosonic input field in two types of non-classical states are derived.
Abstract: We derive the stochastic master equations, that is to say, quantum filters, and master equations for an arbitrary quantum system probed by a continuous-mode bosonic input field in two types of nonclassical states. Specifically, we consider the cases where the state of the input field is a superposition or combination of (1) a continuous-mode, single-photon wave packet and vacuum, and (2) any continuous-mode coherent states.
121 citations
TL;DR: In this article, a two-level atom interacts with a single-mode quantized cavity field (by using an intensity-dependent Jaynes-Cummings model) and at the same time a strong external classical field.
Abstract: In this paper, we first suggest a scheme for the generation of a particular class of Gilmore–Perelomov-type SU(1,1) coherent states, which may be established as nonlinear coherent states. The proposal employs a two-level atom that interacts with a single-mode quantized cavity field (by using an intensity-dependent Jaynes–Cummings model) and at the same time a strong external classical field. The time evolution of the system first leads to the generation of a superposition of SU(1,1) coherent states. Depending on the initial states of the atom and the field which may be appropriately prepared, and also under the conditions in which the atom is detected (in the excited or ground state) after the occurrence of the interaction, the field will be collapsed to arbitrary combinations or a single class of Gilmore–Perelomov-type SU(1,1) coherent states. Then, it is shown that, following a similar procedure, our proposed scheme can successfully generate various superpositions and, in particular, a single class of SU(2) coherent states, too.
38 citations
TL;DR: In this article, a modification to the standard construction, based on the recently introduced (non-commutative) flux representation, was introduced, and the resulting quantum states have some welcome features, in particular, concerning peakedness properties, when compared to other coherent states in the literature.
Abstract: As part of a wider study of coherent states in (loop) quantum gravity, we introduce a modification to the standard construction, based on the recently introduced (non-commutative) flux representation. The resulting quantum states have some welcome features, in particular, concerning peakedness properties, when compared to other coherent states in the literature.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?Coherent states: mathematical and physical aspects?.
33 citations
TL;DR: In this paper, the authors investigate the construction of coherent states for quantum theories of connections based on graphs embedded in a spatial manifold, as in loop quantum gravity, and discuss the many subtleties of the construction, mainly relatedtothediffomorphisminvariance of the theory.
Abstract: We investigate the construction of coherent states for quantum theories of connections based on graphs embedded in a spatial manifold, as in loop quantum gravity. We discuss the many subtleties of the construction, mainly relatedtothediffeomorphisminvarianceofthetheory.Aimingatapproximating a continuum geometry in terms of discrete, graph-based data, we focus on coherent states for collective observables characterizing both the intrinsic and extrinsic geometry of the hypersurface, and we argue that one needs to revise accordingly the more local definitions of coherent states considered in the literature so far. In order to clarify the concepts introduced, we work through a concrete example that we hope will be useful in applying coherent state techniques to cosmology.
28 citations
TL;DR: In this article, the authors describe connections between the localization technique introduced by I. B. Simonenko and operator covariant transform produced by nilpotent Lie groups, and show how the two techniques can be combined.
Abstract: We describe connections between the localization technique introduced by I. B. Simonenko and operator covariant transform produced by nilpotent Lie groups.
25 citations
TL;DR: In this article, a class of hypergeometric type of generalized displacement operators, 1Fr([0], [0, 1,?, r? 1], za], act on the vacuum state of the harmonic oscillator and generate normalized quantum states of the Fock space which admit a resolution of the identity through a positive definite measure on the complex plane.
Abstract: The main goal of this paper is to present an alternative method to construct new kinds of nonlinear coherent states. To do this, we first establish a class of hypergeometric type of generalized displacement operators, 1Fr([0], [0, 1, ?, r ? 1], za?), act on the vacuum state of the harmonic oscillator and generate normalized quantum states of the Fock space which admit a resolution of the identity through a positive definite measure on the complex plane. Furthermore, realization of the compact form of these states, as functions of the position coordinate x for r = 2, leads to a generating function of the Hermite polynomials in terms of the modified Bessel function. Finally, studying some statistical characters reveals that they have indeed non-classical features such as squeezing, an anti-bunching effect and sub-Poissonian statistics, too.
21 citations
TL;DR: Several advances have extended the power and versatility of coherent state theory to the extent that it has become a vital tool in the representation theory of Lie groups and their Lie algebras as discussed by the authors.
Abstract: Several advances have extended the power and versatility of coherent state theory to the extent that it has become a vital tool in the representation theory of Lie groups and their Lie algebras. Representative applications are reviewed and some new developments are introduced. The examples given are chosen to illustrate special features of the scalar and vector coherent state constructions and how they work in practical situations. Comparisons are made with Mackey's theory of induced representations. For simplicity, we focus on square integrable (discrete series) unitary representations although many of the techniques apply more generally, with minor adjustment.
19 citations
TL;DR: In this article, the reproducing kernel property of coherent states is used to ensure the resolution of the identity of a coherent state, which is a fundamental feature of coherent state construction.
Abstract: In the process of generalizing coherent states, the situation when the measure—which is customarily incorporated in their definition—is indeterminate becomes unavoidable. A more dramatic situation may occur if there is no measure which makes the reproducing kernel Hilbert space, involved in the construction of coherent states, isometrically included in an space. Therefore, it appears that there is a need to redefine coherent states making the definition measure-free. Starting out with the reproducing kernel property, we ensure the basic feature of coherent states—resolution of the identity—to be maintained, cf (15) and remark 2. The only investment in the whole undertaking is a sequence (Φn)dn = 0 satisfying (4); the rest, including the aforesaid resolution of the identity, is a consequence of our choice. The approach is supported by examples which make the circumstances clear under which the sequence (Φn)dn = 0 appears; moment problems or rather orthogonal polynomials are one of them.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
17 citations
TL;DR: In this paper, the authors explore some orthogonal polynomials which are naturally associated with certain families of coherent states, often referred to as nonlinear coherent states in the quantum optics literature.
Abstract: We explore in this paper some orthogonal polynomials which are naturally associated with certain families of coherent states, often referred to as nonlinear coherent states in the quantum optics literature. Some examples turn out to be known orthogonal polynomials but in many cases we encounter a general class of new orthogonal polynomials for which we establish several qualitative results.
TL;DR: In this article, the coherent states of the harmonic oscillator in the framework of the generalized (gravitational) uncertainty principle (GUP) were presented, which is consistent with various theories of quantum gravity such as string theory, loop quantum gravity, and black-hole physics.
Abstract: We present the coherent states of the harmonic oscillator in the framework of the generalized (gravitational) uncertainty principle (GUP). This form of GUP is consistent with various theories of quantum gravity such as string theory, loop quantum gravity, and black-hole physics and implies a minimal measurable length. Using a recently proposed formally self-adjoint representation, we find the GUP-corrected Hamiltonian as a generator of the generalized Heisenberg algebra. Then following Klauder's approach, we construct exact coherent states and obtain the corresponding normalization coefficients, weight functions, and probability distributions. We find the entropy of the system and show that it decreases in the presence of the minimal length. These results could shed light on possible detectable Planck-scale effects within recent experimental tests.
TL;DR: In this paper, the authors investigated the pseudo-Hermitian coherent states and their Hermitian counterpart coherent states under the generalized quantum condition in the framework of a position-dependent mass.
Abstract: In the context of the factorization method, we investigate the pseudo-Hermitian coherent states and their Hermitian counterpart coherent states under the generalized quantum condition in the framework of a position-dependent mass. By considering a specific modification in the superpotential, suitable annihilation and creation operators are constructed in order to reproduce the Hermitian counterpart Hamiltonian in the factorized form. We show that by means of these ladder operators, we can construct a wide range of exactly solvable potentials as well as their accompanying coherent states. Alternatively, we explore the relationship between the pseudo-Hermitian Hamiltonian and its Hermitian counterparts, obtained from a similarity transformation, to construct the associated pseudo-Hermitian coherent states. These latter preserve the structure of Perelomov?s states and minimize the generalized position?momentum uncertainty principle.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?Quantum physics with non-Hermitian operators?.
TL;DR: In this article, a method of constructing discrete coherent states for n qubits is proposed, where the coherent states appear as displaced versions of a fiducial vector that is fixed by imposing a number of natural symmetry requirements on its Q-function.
Abstract: We put forward a method of constructing discrete coherent states for n qubits. After establishing appropriate displacement operators, the coherent states appear as displaced versions of a fiducial vector that is fixed by imposing a number of natural symmetry requirements on its Q-function. Using these coherent states, we establish a partial order in the discrete phase space, which allows us to picture some n-qubit states as apparent distributions. We also analyze correlations in terms of sums of squared Q-functions.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
TL;DR: In this article, the coherent and Fock states of a charge moving in a varying homogeneous magnetic field are studied in the tomographic probability representation of quantum mechanics and the states are expressed in terms of quantum tomograms.
Abstract: The coherent and Fock states of a charge moving in varying homogeneous magnetic field are studied in the tomographic probability representation of quantum mechanics. The states are expressed in terms of quantum tomograms. The coherent states tomograms are shown to be described by normal distributions with varying dispersions and means. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments.
TL;DR: In this paper, a general formalism for the construction of various classes of nonlinear coherent states in the context of multi-mode quantum states has been proposed, taking into account the most popular nonlinearity function associated with f-deformed coherent states.
Abstract: Using the nonlinear coherent states approach, a general formalism for the construction of various classes of nonlinear trio coherent states in the context of multi-mode quantum states has been proposed. In particular, taking into account the most popular nonlinearity function associated with f-deformed coherent states, i.e. the nonlinearity function of the centre-of-mass motion of a trapped ion, it is illustrated that the corresponding trio coherent state possesses some interesting nonclassical properties. To establish this observation, sub-Poissonian statistics, three-mode squeezing, the behaviour of Vogel's characteristic function and the existence of entanglement are investigated. Due to the intrinsic non-classicality nature of the considered three-mode states, their physical production may be of high interest in the quantum optics field. Thus, we have finally demonstrated how nonlinear motional trio coherent states can be generated in three-dimensional anisotropic traps, appropriately.
TL;DR: In this paper, a nonlinear deformed algebraic theory for quantum confined systems is constructed using a supersymmetric and shape-invariant approach, and a mathematical Barut-Girardello procedure is presented to lead to nonlinear, deformed coherent states by the successive application of a deformed raising operator on the ground state of these systems.
Abstract: A nonlinear deformed algebraic theory for quantum confined systems is constructed using a supersymmetric and shape-invariant approach. In the framework of this model, we present a mathematical Barut–Girardello procedure which leads to nonlinear deformed coherent states by the successive application of a deformed raising operator on the ground state of these systems. As an application, we use a simple exponential form for the nonlinearity function to study in detail the mathematical and physical properties of such states and their evolution in time.
TL;DR: A simple linear program which solves the task of finding optimal transforms, and a method of characterizing the introduced leak and redundancy in information-theoretic terms are given.
Abstract: We investigate probabilistic transformations of quantum states from a 'source' set to a 'target' set of states Such transforms have many applications They can be used for tasks which include state-dependent cloning or quantum state discrimination, and as interfaces between systems whose information encodings are not related by a unitary transform, such as continuous-variable systems and finite-dimensional systems In a probabilistic transform, information may be lost or leaked, and we explain the concepts of leak and redundancy Following this, we show how the analysis of probabilistic transforms significantly simplifies for symmetric source and target sets of states In particular, we give a simple linear program which solves the task of finding optimal transforms, and a method of characterizing the introduced leak and redundancy in information-theoretic terms Using the developed techniques, we analyse a class of transforms which convert coherent states with information encoded in their relative phase to symmetric qubit states Each of these sets of states on their own appears in many well studied quantum information protocols Finally, we suggest an asymptotic realization based on quantum scissors
TL;DR: In this paper, the authors constructed nonlinear coherent states for the generalized isotonic oscillator and studied their non-classical properties in detail by transforming the deformed ladder operators suitably, which generate the quadratic algebra, and obtained Heisenberg algebra.
Abstract: In this paper, we construct nonlinear coherent states for the generalized isotonic oscillator and study their non-classical properties in detail. By transforming the deformed ladder operators suitably, which generate the quadratic algebra, we obtain Heisenberg algebra. From the algebra, we define two non-unitary and one unitary displacement type operators. While the action of one of the non-unitary type operators reproduces the original nonlinear coherent states, the other one fails to produce a new set of nonlinear coherent states (dual pair). We show that these dual states are not normalizable. For the nonlinear coherent states, we evaluate the parameter A3 and examine the non-classical nature of the states through quadratic and amplitude-squared squeezing effect. Further, we derive analytical formula for the P-function, Q-function, and the Wigner function for the nonlinear coherent states. All of them confirm the non-classicality of the nonlinear coherent states. In addition to the above, we obtain the...
TL;DR: In this article, a bipartite coherent-entangled state is introduced in the two-mode Fock space, which exhibits the properties of both a coherent state and an entangled state.
Abstract: A new bipartite coherent-entangled state is introduced in the two-mode Fock space, which exhibits the properties of both a coherent state and an entangled state. The set of coherent-entangled states makes up a complete and partly nonorthogonal representation. A simple experimental scheme to produce the coherent-entangled state using an asymmetric beamsplitter is proposed. Some applications of the coherent-entangled state in quantum optics are also presented.
TL;DR: In this article, a discrete representation of the nonlinearly deformed SU(1,1) algebra was constructed for the cubic algebra related to the conditionally solvable radial oscillator problem.
Abstract: In a previous paper [{\it J. Phys. A: Math. Theor.} {\bf 40} (2007) 11105], we constructed a class of coherent states for a polynomially deformed $su(2)$ algebra. In this paper, we first prepare the discrete representations of the nonlinearly deformed $su(1,1)$ algebra. Then we extend the previous procedure to construct a discrete class of coherent states for a polynomial su(1,1) algebra which contains the Barut-Girardello set and the Perelomov set of the SU(1,1) coherent states as special cases. We also construct coherent states for the cubic algebra related to the conditionally solvable radial oscillator problem.
TL;DR: In this paper, the nonlinear coherent states approach was used to find a relation between the geometric structure of the physical space and the geometry of the corresponding projective Hilbert space. And the geometric phase in the curved space was explored.
Abstract: In this paper, by using the nonlinear coherent states approach, we find a relation between the geometric structure of the physical space and the geometry of the corresponding projective Hilbert space To illustrate the approach, we explore the quantum transition probability and the geometric phase in the curved space
TL;DR: In this paper, the ground state of a quantum system is used as the minimizer of the supersymmetric Heisenberg uncertainty product, which can be used as a basis for bound state calculations.
Abstract: System-specific coherent states are constructed based on the formulation of supersymmetric quantum mechanics for arbitrary quantum systems. By regarding the superpotential as a generalized displacement variable, we identity the ground state of a quantum system as the minimizer of the supersymmetric Heisenberg uncertainty product. A special case is the ground state of the standard harmonic oscillator. One constructs standard coherent states by applying a shift operator to a 'fiducial function', taken as the ground state Gaussian. By analogy, we use the ground state for any other system as a new fiducial function, generating from its shifts new dynamically-adapted, overcomplete coherent states. The discretized system-specific coherent states can serve as a dynamically-adapted basis for bound state calculations. Accurate computational results for the Morse potential, the double well potential and the two-dimensional anharmonic oscillator systems demonstrate that the system-specific coherent states can provide rapidly-converging approximations for excited state energies and wave functions.
TL;DR: In this article, a parameterized family of Toeplitz operators in the context of affine coherent states based on the Calderon reproducing formula and specific admissible wavelets related to Laguerre functions is studied.
Abstract: We study a parameterized family of Toeplitz operators in the context of affine coherent states based on the Calderon reproducing formula (= resolution of unity on ) and the specific admissible wavelets (= affine coherent states in ) related to Laguerre functions. Symbols of such Calderon–Toeplitz operators as individual coordinates of the affine group (= upper half-plane with the hyperbolic geometry) are considered. In this case, a certain class of pseudo-differential operators, their properties and their operator algebras are investigated. As a result of this study, the Fredholm symbol algebras of the Calderon–Toeplitz operator algebras for these particular cases of symbols are described.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
TL;DR: In this paper, a theoretical analysis of the k-boson nonlinear coherent states of a two-level trapped ion interacting with two laser fields is presented, which are both the zero-energy state of the interaction Hamiltonian and the eigenstates of a deformed annihilation operator.
TL;DR: In this paper, the authors extract from measured data normally or anti-normally ordered moments of photon creation and annihilation operators, the set of which contains complete information on the quantum state of an electromagnetic field.
Abstract: The detection of microwave states is complicated by strong thermal noise, which is inevitably introduced by linear amplifiers. We show how to extract from measured data normally or anti-normally ordered moments of photon creation and annihilation operators, the set of which contains complete information on the quantum state of an electromagnetic field. Equations for the evolution of the quantum state are derived in terms of moments. Using this approach, we consider in detail issues of decoherence and thermalization of microwave quantum states. Results are illustrated using the examples of Fock, coherent, squeezed, thermal, and even and odd coherent states (Schrodinger cat states).
TL;DR: An approach with displaced states that can be used for rotations of coherent states that enables to construct local rotations for coherent states, in particular, Hadamard gate being mainframe element for quantum computation with coherent states.
Abstract: We propose an approach with displaced states that can be used for rotations of coherent states. Our approach is based on representation of arbitrary one-mode pure state in free-travelling fields, in particular superposition of coherent states (SCSs), in terms of displaced number states with arbitrary amplitude of displacement. Optical scheme is developed for construction of displacing Hadamard gate for the coherent states. It is based on alternation of single photon additions and displacement operators (in general case, N-singe photon additions and N ? 1-displacements are required) with seed coherent state to generate both even and odd displaced squeezed SCSs regardless of number of used photon additions. The optical scheme is sensitive to the seed coherent state provided that other parameters of the scheme are invariable. Output states approximate with high fidelity either even squeezed SCS or odd SCS shifted relative each other by some value. It enables to construct local rotations for coherent states, in particular, Hadamard gate being mainframe element for quantum computation with coherent states. The effects deteriorating quality of output states are considered.
TL;DR: In this article, the authors present a possible construction of coherent states on the unit circle as configuration space based on Borel quantizations on S1 including the Aharonov-Bohm-type quantum description.
Abstract: We present a possible construction of coherent states on the unit circle as configuration space. Our approach is based on Borel quantizations on S1 including the Aharonov–Bohm-type quantum description. Coherent states are constructed by Perelomov’s method as group-related coherent states generated by Weyl operators on the quantum phase space . Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction, we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was . The coherent states constructed in this work are shown to satisfy the resolution of unity. To compare them with canonical coherent states, some of their further properties are also studied demonstrating similarities as well as substantial differences.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
TL;DR: A Mathematica code is provided for decomposing strongly correlated quantum states described by a first-quantized, analytical wave function into many-body Fock states, which refer to the subset of Fock–Darwin functions with no nodes.
TL;DR: In this paper, the coherent states for a quantum particle on a Mπ{o}bius strip are constructed and their relation with the natural phase space for fermionic fields is shown.
Abstract: The coherent states for a quantum particle on a M\"{o}bius strip are constructed and their relation with the natural phase space for fermionic fields is shown. The explicit comparison of the obtained states with previous works where the cylinder quantization was used and the spin 1/2 was introduced by hand is given, and the relation between the geometrical phase space, constraints and projection operators is analyzed and discussed.