Trapped Ions in Laser Fields: a Benchmark for Deformed-Quantum Oscillators

Some properties of the non--linear coherent states (NCS), recognized by Vogel and de Matos Filho as dark states of a trapped ion, are extended to NCS on a circle, for which the Wigner functions are presented. These states are obtained by applying a suitable displacement operator $D_{h}(\alpha)$ to the vacuum state. The unity resolutions in terms of the projectors $| \alpha, h> < \alpha, h|$. $D_{h}(\alpha)$ is also used for introducing the probability distribution funtion $\rho_{A,h}(z)$ while the existence of a measure is exploited for extending the P-representation to these states. The weight of the n-th Fock state of the NCS relative to a trapped ion with Lamb-Dicke parameter $\eta ,$ oscillates so wildly as $n$ grows up to infinity that the normalized NCS fill the open circle $\eta ^{-1}$ in the complex $\alpha$-plane. In addition this prevents the existence of a measure including normalizable states only. This difficulty is overcome by introducing a family of deformations which are rational functions of n, each of them admitting a measure. By increasing the degree of these rational approximations the deformation of a trapped ion can be approximated with any degree of accuracy and the formalism of the P-representation can be applied

A Quantum Group Approach to some Exotic States in Quantum Optics

This subject of this thesis is the physical application of deformations of Lie algebras and their use in generalising some exotic quantum optical states. We begin by examining the theory of quantum groups and the q-boson algebras used in their representation theory. Following a review of the properties of conventional coherent states, we describe the extension of the theory to various deformed Heisenberg-Weyl algebras, as well as the q-deformations of su(2) and su(1,1). Using the Deformed Oscillator Algebra of Bonatsos and Daskaloyannis, we construct generalised deformed coherent states and investigate some of their quantum optical properties. We then demonstrate a resolution of unity for such states and suggest a way of investigating the geometric effects of the deformation. The formalism devised by Rembielinski et al is used to consider coherent states of the q-boson algebra over the quantum complex plane. We propose a new unitary operator which is a q-analogue of the displacement operator of conventional coherent state theory: This is used to construct q-displaced vacuum states which are eigenstates of the annihilation operator. Some quantum mechanical properties of these states are investigated and it is shown that they formally satisfy a Heisenberg-type minimum uncertainty relation. After briefly reviewing the theory of conventional squeezed states, we examine the various q-generalisations. We propose a q-analogue of the squeezed vacuum state, and use this in conjunction with the unitary q-displacement operator to construct a general q-squeezed state, parameterised by noncommuting variables.. It is shown that, like their conventional counterparts, such states satisfy the Robertson-Schrodinger Uncertainty Relation. We conclude with a brief discussion about the appearance of noncommuting variables in the states that have been considered

Multiphoton Quantum Optics and Quantum State Engineering

We present a review of theoretical and experimental aspects of multiphoton quantum optics. Multiphoton processes occur and are important for many aspects of matter-radiation interactions that include the efficient ionization of atoms and molecules, and, more generally, atomic transition mechanisms; system-environment couplings and dissipative quantum dynamics; laser physics, optical parametric processes, and interferometry. A single review cannot account for all aspects of such an enormously vast subject. Here we choose to concentrate our attention on parametric processes in nonlinear media, with special emphasis on the engineering of nonclassical states of photons and atoms. We present a detailed analysis of the methods and techniques for the production of genuinely quantum multiphoton processes in nonlinear media, and the corresponding models of multiphoton effective interactions. We review existing proposals for the classification, engineering, and manipulation of nonclassical states, including Fock states, macroscopic superposition states, and multiphoton generalized coherent states. We introduce and discuss the structure of canonical multiphoton quantum optics and the associated one- and two-mode canonical multiphoton squeezed states. This framework provides a consistent multiphoton generalization of two-photon quantum optics and a consistent Hamiltonian description of multiphoton processes associated to higher-order nonlinearities. Finally, we discuss very recent advances that by combining linear and nonlinear optical devices allow to realize multiphoton entangled states of the electromnagnetic field, that are relevant for applications to efficient quantum computation, quantum teleportation, and related problems in quantum communication and information.Comment: 198 pages, 36 eps figure

A new class of coherent states and it's properties.

The study of coherent states (CS) for a quantum mechanical system has received a lot of attention. The definition, applications, generalizations of such states have been the subject of work by researchers. A common starting point of all these approaches is the observation of properties of the original CS for the harmonic oscillator. It is well-known that they are described equivalently as (a) eigenstates of the usual annihilation operator, (b) from a displacement operator acting on a fundamental state and (c) as minimum uncertainty states. What we observe in the different generalizations proposed is that the preceding definitions are no longer equivalent and only some of the properties of the harmonic oscillator CS are preserved. In this thesis we propose to study a new class of coherent states and its properties. We note that in one example our CS coincide with the ones proposed by Glauber where a set of three requirements for such states has been imposed. The set of our generalized coherent states remains invariant under the corresponding time evolution and this property is called temporal stability. Secondly, there is no state which is orthogonal to all coherent states (the coherent states form a total set). The third property is that we get all coherent states by acting on one of these states [¿fiducial vector¿] with operators. They are highly non-classical states, in the sense that in general, their Bargmann functions have zeros which are related to negative regions of their Wigner functions. Examples of these coherent states with Bargmann function that involve the Gamma and also the Riemann ¿ functions are represented. The zeros of these Bargmann functions and the paths of the zeros during time evolution are also studied.Libyan Cultural Affair

Particle Correlations in Bose-Einstein Condensates

The impact of interparticle correlations on the behavior of Bose-Einstein Condensates (BECs) is discussed using two approaches. In the first approach, the wavefunction of a BEC is encoded in the N-particle sector of an extended catalytic state\u27\u27. Going to a time-dependent interaction picture, we can organize the effective Hamiltonian by powers of N -1/2. Requiring the terms of order N1/2 to vanish, we get the Gross-Pitaevskii Equation. Going to the next order, N0, we obtain the number-conserving Bogoliubov approximation. Our approach allows one to stay in the Schrödinger picture and to apply many techniques from quantum optics. Moreover, it is easier to track different orders in the Hamiltonian and to generalize to the multi-component case. In the second approach, I consider a state of N=l×n bosons that is derived by symmetrizing the n-fold tensor product of an arbitrary l-boson state. Particularly, we are interested in the pure state case for l=2, which we call the Pair-Correlated State (PCS). I show that PCS reproduces the number-conserving Bogoliubov approximation; moreover, it also works in the strong interaction regime where the Bogoliubov approximation fails. For the two-site Bose-Hubbard model, I find numerically that the error (measured by trace distance of the two-particle RDMs) of PCS is less than two percent over the entire parameter space, thus making PCS a bridge between the superfluid and Mott insulating phases. Amazingly, the error of PCS does not increase, in the time-dependent case, as the system evolves for longer times. I derive both time-dependent and -independent equations for the ground state and the time evolution of the PCS ansatz. The time complexity of simulating PCS does not depend on N and is linear in the number of orbitals in use. Compared to other methods, e.g, the Jastrow wavefunction, the Gutzwiller wavefunction, and the multi-configurational time-dependent Hartree method, our approach does not require quantum Monte Carlo nor demanding computational power

Extremal quantum states

The striking differences between quantum and classical systems predicate disruptive quantum technologies. We peruse quantumness from a variety of viewpoints, concentrating on phase-space formulations because they can be applied beyond particular symmetry groups. The symmetry-transcending properties of the Husimi $Q$ function make it our basic tool. In terms of the latter, we examine quantities such as the Wehrl entropy, inverse participation ratio, cumulative multipolar distribution, and metrological power, which are linked to intrinsic properties of any quantum state. We use these quantities to formulate extremal principles and determine in this way which states are the most and least "quantum;" the corresponding properties and potential usefulness of each extremal principle are explored in detail. While the extrema largely coincide for continuous-variable systems, our analysis of spin systems shows that care must be taken when applying an extremal principle to new contexts.Comment: 30 pages, 2 figures; comments welcome

Quantum metrology with nonclassical states of atomic ensembles

Quantum technologies exploit entanglement to revolutionize computing, measurements, and communications. This has stimulated the research in different areas of physics to engineer and manipulate fragile many-particle entangled states. Progress has been particularly rapid for atoms. Thanks to the large and tunable nonlinearities and the well developed techniques for trapping, controlling and counting, many groundbreaking experiments have demonstrated the generation of entangled states of trapped ions, cold and ultracold gases of neutral atoms. Moreover, atoms can couple strongly to external forces and light fields, which makes them ideal for ultra-precise sensing and time keeping. All these factors call for generating non-classical atomic states designed for phase estimation in atomic clocks and atom interferometers, exploiting many-body entanglement to increase the sensitivity of precision measurements. The goal of this article is to review and illustrate the theory and the experiments with atomic ensembles that have demonstrated many-particle entanglement and quantum-enhanced metrology.Comment: 76 pages, 40 figures, 1 table, 603 references. Some figures bitmapped at 300 dpi to reduce file siz

Quantum statistical properties of multiphoton hypergeometric coherent states and the discrete circle representation

S.A. thanks M.C. and J.G. for their hospitality during his stay at the University of Granada where this work was done, and the Coimbra Group for the financial support. This study has been partially financed by the Consejería de Conocimiento, Investigación y Universidad, Junta de Andalucía, and European Regional Development Fund (ERDF) under projects with Ref. Nos. FQM381 and SOMM17/6105/UGR, and by the Spanish MICINN under Project No. PGC2018-097831-B-I00. J.G. thanks the Spanish MICINN for financial support (Grant No. FIS2017-84440-C2-2-P).We review the definition of hypergeometric coherent states, discussing some representative examples. Then, we study mathematical and statistical properties of hypergeometric Schrödinger cat states, defined as orthonormalized eigenstates of kth powers of nonlinear f-oscillator annihilation operators, with f of the hypergeometric type. These “k-hypercats” can be written as an equally weighted superposition of hypergeometric coherent states ∣zl⟩, l = 0, 1, …, k − 1, with zl = ze2πil/k a kth root of zk, and they interpolate between number and coherent states. This fact motivates a continuous circle representation for high k. We also extend our study to truncated hypergeometric functions (finite dimensional Hilbert spaces), and a discrete exact circle representation is provided. We also show how to generate k-hypercats by amplitude dispersion in a Kerr medium and analyze their generalized Husimi Q-function in the super- and sub-Poissonian cases at different fractions of the revival time.Consejería de Conocimiento, Investigación y Universidad, Junta de Andalucía, and European Regional Development Fund (ERDF) under projects with Ref. Nos. FQM381 and SOMM17/6105/UGRSpanish MICINN under Project No. PGC2018-097831-B-I00Spanish MICINN for financial support (Grant No. FIS2017-84440-C2-2-P

Second International Workshop on Harmonic Oscillators

The Second International Workshop on Harmonic Oscillators was held at the Hotel Hacienda Cocoyoc from March 23 to 25, 1994. The Workshop gathered 67 participants; there were 10 invited lecturers, 30 plenary oral presentations, 15 posters, and plenty of discussion divided into the five sessions of this volume. The Organizing Committee was asked by the chairman of several Mexican funding agencies what exactly was meant by harmonic oscillators, and for what purpose the new research could be useful. Harmonic oscillators - as we explained - is a code name for a family of mathematical models based on the theory of Lie algebras and groups, with applications in a growing range of physical theories and technologies: molecular, atomic, nuclear and particle physics; quantum optics and communication theory