Bosonic Oscillator Research Articles

Intermediate-statistics quantum bracket, coherent state, oscillator, and representation of angular momentum [su(2)] algebra

In this paper, we first discuss the general properties of an intermediate-statistics quantum bracket, ${[u,v]}_{n}=uv\ensuremath{-}{e}^{i2\ensuremath{\pi}∕(n+1)}vu$, which corresponds to intermediate statistics in which the maximum occupation number of one quantum state is an arbitrary integer, $n$. A further study of the operator realization of intermediate statistics is given. We construct the intermediate-statistics coherent state. An intermediate-statistics oscillator is constructed, which returns to bosonic and fermionic oscillators respectively when $n\ensuremath{\rightarrow}\ensuremath{\infty}$ and $n=1$. The energy spectrum of such an intermediate-statistics oscillator is calculated. Finally, we discuss the intermediate-statistics representation of angular momentum [su(2)] algebra. Moreover, a further study of the operator realization of intermediate statistics is given in the Appendix.

Foundation for Quantum Computing II

In this continuation of an earlier paper we develop further the theme of quantum logical specification and derive from it some apparently physically viable instantiations of potential quantum computing devices. Specifically, in the case of a one-parameter set of terms (or labels)—read as instants of time—we find, emerging quite naturally from the algebraic setup, the paradigm for a single qubit epitomized by the case of a two-state fermion interacting with an external single mode boson. This covers the cases: cavity QED, trapped ions, and, when the qubits are multiplexed appropriately, NMR based systems. (This case degenerates to one in which only bosons are relevant as in the case of pure bosonic harmonic oscillator models in the “dual rail” representation. Such models fly in the face of the logic itself, thus clearly revealing even at this level their well-known shortcomings as practical quantum computing devices. Here as elsewhere logical constraints apparently dominate physical ones.) In a final section we indicate briefly how this process exactly generalizes, in the case of a manifold of terms more general than the one-parameter case, to yield the notion of holonomic quantum computation. In the course of this investigation we find an interpretation of path integrals as limits of sequences of logical CUTS, thus establishing a link—though still tenuous—between ensembles of acts of quantum computation and Lagrangians.

From gravitons to giants

We discuss exact quantization of gravitational fluctuations in the half-BPS sector around AdS$_5 \times $S$^5$ background, using the dual super Yang-Mills theory. For this purpose we employ the recently developed techniques for exact bosonization of a finite number $N$ of fermions in terms of $N$ bosonic oscillators. An exact computation of the three-point correlation function of gravitons for finite $N$ shows that they become strongly coupled at sufficiently high energies, with an interaction that grows exponentially in $N$. We show that even at such high energies a description of the bulk physics in terms of weakly interacting particles can be constructed. The single particle states providing such a description are created by our bosonic oscillators or equivalently these are the multi-graviton states corresponding to the so-called Schur polynomials. Both represent single giant graviton states in the bulk. Multi-particle states corresponding to multi-giant gravitons are, however, different, since interactions among our bosons vanish identically, while the Schur polynomials are weakly interacting at high enough energies.

Entangling oscillators through environment noise

We consider two independent bosonic oscillators immersed in a common bath, evolving in time with a completely positive, markovian, quasi-free (Gaussian) reduced dynamics. We show that an initially separated Gaussian state can become entangled as a result of a purely noisy mechanism. In certain cases, the dissipative dynamics allows the persistence of these bath induced quantum correlations even in the asymptotic equilibrium state.

Generalized Coherent States: A Novel Approach

We define generalized coherent states for oscillator-like systems connected with orthogonal polynomials (classical, q-deformed, etc.). In considered cases such polynomials play the same role as the Hermite polynomials in the case of the usual boson oscillator. Bibliography: 26 titles.

Barut–Girardello Coherent States for the Gegenbauer Oscillator

We define the Gegenbauer oscillator and introduce a family of Barut–Girardello coherent states (eigenstates of a relevant annihilation operator) for this oscillator. Gegenbauer (ultraspherical) polynomials play the same role for this oscillator as Hermite polynomials do in the case of the usual boson oscillator. We establish the validity of unity decomposition for the introduced states and evaluate their overlap. These results reproduce similar results obtained earlier by the authors for Legendre and Chebyshev polynomials. Bibliography: 38 titles.

Harmonic Oscillator Realization of the Deformed Bogoliubov (p,q)-Transformations without First Finite Fock States

Considering a simple generalization of the (p, g)-deformed boson oscillator algebra, which leads to a two-parameter deformed bosonic algebra in an infinite dimensional subspace of the harmonic oscillator Hilbert space without first, finite Fock states, we establish a new harmonic oscillator realization of the deformed boson operators based on the Bogoliubov (p, q)-transformations. We obtain exact expressions for the transformation coefficients and show that they depend on arbitrary functions of p and q which can be interpreted as the parameters of the (p, q)-deformed GL(2,C) group. We also examine the existence and structure of the corresponding deformed Fock-space representation for our problem.

Cliffordization, spin, and fermionic star products

Deformation quantization is a powerful tool for quantizing theories with bosonic and fermionic degrees of freedom. The star products involved generate the mathematical structures which have recently been used in attempts to analyze the algebraic properties of quantum field theory. In the context of quantum mechanics they provide a quantization procedure for systems with either bosonic or fermionic degrees of freedom. We illustrate this procedure for a number of physical examples, including bosonic, fermionic, and supersymmetric oscillators. We show how non-relativistic and relativistic particles with spin can be naturally described in this framework.

On a family of photon-added deformed coherent states associated with two-parameter deformed boson oscillator algebra

By introducing a generalization of the (p, q)-deformed boson oscillator algebra, we establish a two-parameter deformed oscillator algebra in an infinite-dimensional subspace of the Hilbert space of a harmonic oscillator without first finite Fock states. We construct the associated coherent states, which can be interpreted as photon-added deformed states. In addition to the mathematical characteristics, the quantum statistical properties of these states are discussed in detail analytically and numerically in the context of conventional as well as deformed quantum optics. Particularly, we find that for conventional (nondeformed) photons the states may be quadrature squeezed in both cases Q = pq < 1, Q = pq > 1 and their photon number statistics exhibits a transition from sub-Poissonian to super-Poissonian for Q < 1 whereas for Q > 1 they are always sub-Poissonian. On the other hand, for deformed photons, the states are sub-Poissonian for Q > 1 and no quadrature squeezing occurs while for Q < 1 they show super-Poissonian behavior and there is a simultaneous squeezing in both field quadratures.PACS Nos.: 42.50.Ar, 03.65.–w

A NOTE ON QUANTUM OSCILLATORS COUPLED BY NONRESONANT INTERACTION: ENVIRONMENT MODIFYING DYNAMICS

We study the dynamics of a single fermionic system interacting non-resonantly with a single mode of a bosonic oscillator coupled to a reservoir. It is shown that the damping rates for both systems are modified when the action of their environments are frequency-dependent. The interest of such demand is discussed.

An alternative formulation of light-cone string field theory on the plane wave

We construct a manifestly SO(4) x SO(4) invariant, supersymmetric extension of the closed string cubic interaction vertex and dynamical supercharges in light-cone string field theory on the plane wave space-time. We find that the effective vertex for states built out of bosonic creation oscillators coincides with the one previously constructed in the SO(8) formalism and conjecture that in general the two formulations are physically equivalent. Further evidence for this claim is obtained from the discrete Z_2-symmetry of the plane wave and by computing the mass-shift of the simplest stringy state using perturbation theory. We verify that the leading non-planar correction to the anomalous dimension of the dual gauge theory operators is correctly recovered.

Open String--BMN Operator Correspondence in the Weak Coupling Regime

We reproduce the correspondence between open string states ∣n〉 and BMN operators On in supersymmetric gauge theory using a worldsheet computation with the relation 〈B∣n〉 = On, where 〈B∣ is a boundary state of D3-branes. We regard the NS ground states and their excitations with respect to a bosonic oscillator as the states ∣n〉 and obtain a chiral operator and BMN operators, respectively, in correspondence to them. Because we take the coupling constant to be weak and the background spacetime to be flat Minkowski, not AdS, the string states ∣n〉 are circular waves around the D3-branes, instead of KK modes on S5. We also discuss the 1/J correction to the correspondence.

SU (N) coherent states

We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labeled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.

Supercurves

The TST-dual of the general 1/4-supersymmetric D2-brane supertube is identified as a 1/4-supersymmetric IIA ‘supercurve’: a string with arbitrary transverse displacement travelling at the speed of light. A simple proof is given of the classical upper bound on the angular momentum, which is also recovered as the semi-classical limit of a quantum bound. The classical bound is saturated by a ‘superhelix’, while the quantum bound is saturated by a bosonic oscillator state in a unique SO(8) representation.

Schwinger's oscillator method, supersymmetric quantum mechanics and massless particles

We consider the Schwinger's method of angular momentum addition using the SU(2) algebra with both a fermionic and a bosonic oscillator. We show that the total spin states obtained are: one boson singlet state and an arbitrary number of spin-1/2 states, the later ones are energy degenerate. It means that we have in this case supersymmetric quantum mechanics and also the addition of angular momentum for massless particles. We review too the cases of two bosonic and fermionic oscillators.

Schwinger's oscillator method, supersymmetric quantum mechanics and massless particles

We consider Schwinger's method of angular momentum addition using the SU(2) algebra with both a fermionic and a bosonic oscillator. We show that the total spin states obtained are: one boson singlet state and an arbitrary number of spin-1/2 states, the later ones are energy degenerate. It means that we have in this case supersymmetric quantum mechanics and also the addition of angular momentum for massless particles. We review too the cases of two bosonic and two fermionic oscillators.

A Two-Parameter Deformed N = 2 SUSY Algebra

A two-parameter deformed N = 2 SUSY algebra is constructed by using the q-deformed bosonic and fermionic Newton oscillator algebras. The Fock space representation of the (q1,q2)-deformed N = 2 SUSY algebra is analyzed. The comparison between the algebra constructed and earlier versions of deformed N = 2 SUSY algebras is also made.

TRANSFORMATIONS OF q-BOSON AND q-FERMION ALGEBRAS

We investigate the algebras satisfied by q-deformed boson and fermion oscillators, in particular the transformations of the algebra from one form to another. Based on a specific algebra proposed in recent literature, we show that the algebra of deformed fermions can be transformed to that of undeformed standard fermions. Furthermore, we also show that the algebra of q-deformed fermions can be transformed to that of undeformed standard bosons.

Multi-parameter deformations and multi-particle representations of the bosonic oscillator

In this paper, we consider a multi-parameter deformation of the bosonic oscillator algebra and determine the consistency conditions on the parameters. For a d-dimensional oscillator we find 2d parameters. Finally, we present the Fock representation of this oscillator.

Representations of the deformed oscillator algebra for different choices of generators

For the q-deformed boson oscillator algebra, different commutation relations corresponding, to different choices of generators are considered in the general context. By the standard technique, the problems of construction and classification of the 1-deformed boson oscillator algebra representations in different bases are solved. The possibility of introducing the Hopf algebra structure in the q-deformed oscillator algebra is considered. The possibility of constructing the universal R-matrix for the algebra under consideration is briefly discussed. Bibliography: 20 titles.