Showing papers on "Coherent states published in 2000"

Coherent States, Wavelets, and Their Generalizations

Abstract: This second edition is fully updated, covering in particular new types of coherent states (the so-called Gazeau-Klauder coherent states, nonlinear coherent states, squeezed states, as used now routinely in quantum optics) and various generalizations of wavelets (wavelets on manifolds, curvelets, shearlets, etc.). In addition, it contains a new chapter on coherent state quantization and the related probabilistic aspects. As a survey of the theory of coherent states, wavelets, and some of their generalizations, it emphasizes mathematical principles, subsuming the theories of both wavelets and coherent states into a single analytic structure. The approach allows the user to take a classical-like view of quantum states in physics.Starting from the standard theory of coherent states over Lie groups, the authors generalize the formalism by associating coherent states to group representations that are square integrable over a homogeneous space; a further step allows one to dispense with the group context altogether. In this context, wavelets can be generated from coherent states of the affine group of the real line, and higher-dimensional wavelets arise from coherent states of other groups. The unified background makes transparent an entire range of properties of wavelets and coherent states. Many concrete examples, such as coherent states from semisimple Lie groups, Gazeau-Klauder coherent states,coherent states forthe relativity groups, and several kinds of wavelets, are discussed in detail. The book concludes with a palette of potentialapplications, from the quantum physically oriented,likethe quantum-classical transition or the construction of adequate states in quantum information, to the most innovative techniques to be used in data processing.Intended as an introduction to current research for graduate students and others entering the field, the mathematical discussion is self-contained. With its extensive references to the research literature, the first edition of the book is already a proven compendium for physicists and mathematicians active in the field, and with full coverage of the latest theory and results the revised second edition is even more valuable.

Electromagnetic Noise and Quantum Optical Measurements

Abstract: 1. Maxwell's Equations, Power, and Energy.- 1.1 Maxwell's Field Equations.- 1.2 Poynting's Theorem.- 1.3 Energy and Power Relations and Symmetry of the Tensor.- 1.4 Uniqueness Theorem.- 1.5 The Complex Maxwell's Equations.- 1.6 Operations with Complex Vectors.- 1.7 The Complex Poynting Theorem.- 1.8 The Reciprocity Theorem.- 1.9 Summary.- Problems.- Solutions.- 2. Waveguides and Resonators.- 2.1 The Fundamental Equations of Homogeneous Isotropic Waveguides.- 2.2 Transverse Electromagnetic Waves.- 2.3 Transverse Magnetic Waves.- 2.4 Transverse Electric Waves.- 2.4.1 Mode Expansions.- 2.5 Energy, Power, and Energy Velocity.- 2.5.1 The Energy Theorem.- 2.5.2 Energy Velocity and Group Velocity.- 2.5.3 Energy Relations for Waveguide Modes.- 2.5.4 A Perturbation Example.- 2.6 The Modes of a Closed Cavity.- 2.7 Real Character of Eigenvalues and Orthogonality of Modes.- 2.8 Electromagnetic Field Inside a Closed Cavity with Sources.- 2.9 Analysis of Open Cavity.- 2.10 Open Cavity with Single Input.- 2.10.1 The Resonator and the Energy Theorem.- 2.10.2 Perturbation Theory and the Generic Form of the Impedance Expression.- 2.11 Reciprocal Multiports.- 2.12 Simple Model of Resonator.- 2.13 Coupling Between Two Resonators.- 2.14 Summary.- Problems.- Solutions.- 3. Diffraction, Dielectric Waveguides, Optical Fibers, and the Kerr Effect.- 3.1 Free-Space Propagation and Diffraction.- 3.2 Modes in a Cylindrical Piecewise Uniform Dielectric.- 3.3 Approximate Approach.- 3.4 Perturbation Theory.- 3.5 Propagation Along a Dispersive Fiber.- 3.6 Solution of the Dispersion Equation for a Gaussian Pulse.- 3.7 Propagation of a Polarized Wave in an Isotropic Kerr Medium.- 3.7.1 Circular Polarization.- 3.8 Summary.- Problems.- Solutions.- 4. Shot Noise and Thermal Noise.- 4.1 The Spectrum of Shot Noise.- 4.2 The Probability Distribution of Shot Noise Events.- 4.3 Thermal Noise in Waveguides and Transmission Lines.- 4.4 The Noise of a Lossless Resonator.- 4.5 The Noise of a Lossy Resonator.- 4.6 Langevin Sources in a Waveguide with Loss.- 4.7 Lossy Linear Multiports at Thermal Equilibrium.- 4.8 The Probability Distribution of Photons at Thermal Equilibrium.- 4.9 Gaussian Amplitude Distribution of Thermal Excitations.- 4.10 Summary.- Problems.- Solutions.- 5. Linear Noisy Multiports.- 5.1 Available and Exchangeable Power from a Source.- 5.2 The Stationary Values of the Power Delivered by a Noisy Multiport and the Characteristic Noise Matrix.- 5.3 The Characteristic Noise Matrix in the Admittance Representation Applied to a Field Effect Transistor.- 5.4 Transformations of the Characteristic Noise Matrix.- 5.5 Simplified Generic Forms of the Characteristic Noise Matrix.- 5.6 Noise Measure of an Amplifier.- 5.6.1 Exchangeable Power.- 5.6.2 Noise Figure.- 5.6.3 Exchangeable Power Gain.- 5.6.4 The Noise Measure and Its Optimum Value.- 5.7 The Noise Measure in Terms of Incident and Reflected Waves.- 5.7.1 The Exchangeable Power Gain.- 5.7.2 Excess Noise Figure.- 5.8 Realization of Optimum Noise Performance.- 5.9 Cascading of Amplifiers.- 5.10 Summary.- Problems.- Solutions.- 6. Quantum Theory of Waveguides and Resonators.- 6.1 Quantum Theory of the Harmonic Oscillator.- 6.2 Annihilation and Creation Operators.- 6.3 Coherent States of the Electric Field.- 6.4 Commutator Brackets, Heisenberg's Uncertainty Principle and Noise.- 6.5 Quantum Theory of an Open Resonator.- 6.6 Quantization of Excitations on a Single-Mode Waveguide.- 6.7 Quantum Theory of Waveguides with Loss.- 6.8 The Quantum Noise of an Amplifier with a Perfectly Inverted Medium.- 6.9 The Quantum Noise of an Imperfectly Inverted Amplifier Medium.- 6.10 Noise in a Fiber with Loss Compensated by Gain.- 6.11 The Lossy Resonator and the Laser Below Threshold.- 6.12 Summary.- Problems.- Solutions.- 7. Classical and Quantum Analysis of Phase-Insensitive Systems.- 7.1 Renormalization of the Creation and Annihilation Operators.- 7.2 Linear Lossless Multiports in the Classical and Quantum Domains.- 7.3 Comparison of the Schrodinger and Heisenberg Formulations of Lossless Linear Multiports.- 7.4 The Schrodinger Formulation and Entangled States.- 7.5 Transformation of Coherent States.- 7.6 Characteristic Functions and Probability Distributions.- 7.6.1 Coherent State.- 7.6.2 Bose-Einstein Distribution.- 7.7 Two-Dimensional Characteristic Functions and the Wigner Distribution.- 7.8 The Schrodinger Cat State and Its Wigner Distribution.- 7.9 Passive and Active Multiports.- 7.10 Optimum Noise Measure of a Quantum Network.- 7.11 Summary.- Problems.- Solutions.- 8. Detection.- 8.1 Classical Description of Shot Noise and Heterodyne Detection.- 8.2 Balanced Detection.- 8.3 Quantum Description of Direct Detection.- 8.4 Quantum Theory of Balanced Heterodyne Detection.- 8.5 Linearized Analysis of Heterodyne Detection.- 8.6 Heterodyne Detection of a Multimodal Signal.- 8.7 Heterodyne Detection with Finite Response Time of Detector.- 8.8 The Noise Penalty of a Simultaneous Measurement of Two Noncommuting Observables.- 8.9 Summary.- Problems.- Solutions.- 9. Photon Probability Distributions and Bit-Error Rate of a Channel with Optical Preamplification.- 9.1 Moment Generating Functions.- 9.1.1 Poisson Distribution.- 9.1.2 Bose-Einstein Distribution.- 9.1.3 Composite Processes.- 9.2 Statistics of Attenuation.- 9.3 Statistics of Optical Preamplification with Perfect Inversion.- 9.4 Statistics of Optical Preamplification with Incomplete Inversion.- 9.5 Bit-Error Rate with Optical Preamplification.- 9.5.1 Narrow-Band Filter, Polarized Signal, and Noise.- 9.5.2 Broadband Filter, Unpolarized Signal.- 9.6 Negentropy and Information.- 9.7 The Noise Figure of Optical Amplifiers.- 9.8 Summary.- Problems.- Solutions.- 10. Solitons and Long-Distance Fiber Communications.- 10.1 The Nonlinear Schrodinger Equation.- 10.2 The First-Order Soliton.- 10.3 Properties of Solitons.- 10.4 Perturbation Theory of Solitons.- 10.5 Amplifier Noise and the Gordon-Haus Effect.- 10.6 Control Filters.- 10.7 Erbium-Doped Fiber Amplifiers and the Effect of Lumped Gain.- 10.8 Polarization.- 10.9 Continuum Generation by Soliton Perturbation.- 10.10 Summary.- Problems.- Solutions.- 11. Phase-Sensitive Amplification and Squeezing.- 11.1 Classical Analysis of Parametric Amplification.- 11.2 Quantum Analysis of Parametric Amplification.- 11.3 The Nondegenerate Parametric Amplifier as a Model of a Linear Phase-Insensitive Amplifier.- 11.4 Classical Analysis of Degenerate Parametric Amplifier.- 11.5 Quantum Analysis of Degenerate Parametric Amplifier.- 11.6 Squeezed Vacuum and Its Homodyne Detection.- 11.7 Phase Measurement with Squeezed Vacuum.- 11.8 The Laser Resonator Above Threshold.- 11.9 The Fluctuations of the Photon Number.- 11.10 The Schawlow-Townes Linewidth.- 11.11 Squeezed Radiation from an Ideal Laser.- 11.12 Summary.- Problems.- Solutions.- 12. Squeezing in Fibers.- 12.1 Quantization of Nonlinear Waveguide.- 12.2 The x Representation of Operators.- 12.3 The Quantized Equation of Motion of the Kerr Effect in the x Representation.- 12.4 Squeezing.- 12.5 Generation of Squeezed Vacuum with a Nonlinear Interferometer.- 12.6 Squeezing Experiment.- 12.7 Guided-Acoustic-Wave Brillouin Scattering.- 12.8 Phase Measurement Below the Shot Noise Level.- 12.9 Generation of Schrodinger Cat State via Kerr Effect.- 12.10 Summary.- Problems.- Solutions.- 13. Quantum Theory of Solitons and Squeezing.- 13.1 The Hamiltonian and Equations of Motion of a Dispersive Waveguide.- 13.2 The Quantized Nonlinear Schrodinger Equation and Its Linearization.- 13.3 Soliton Perturbations Projected by the Adjoint.- 13.4 Renormalization of the Soliton Operators.- 13.5 Measurement of Operators.- 13.6 Phase Measurement with Soliton-like Pulses.- 13.7 Soliton Squeezing in a Fiber.- 13.8 Summary.- Problems.- Solutions.- 14. Quantum Nondemolition Measurements and the "Collapse" of the Wave Function.- 14.1 General Properties of a QND Measurement.- 14.2 A QND Measurement of Photon Number.- 14.3 "Which Path" Experiment.- 14.4 The "Collapse" of the Density Matrix.- 14.5 Two Quantum Nondemolition Measurements in Cascade.- 14.6 The Schrodinger Cat Thought Experiment.- 14.7 Summary.- Problems.- Solutions.- Epilogue.- Appendices.- A.1 Phase Velocity and Group Velocity of a Gaussian Beam.- A.2 The Hermite Gaussians and Their Defining Equation.- A.2.1 The Defining Equation of Hermite Gaussians.- A.2.2 Orthogonality Property of Hermite Gaussian Modes.- A.2.3 The Generating Function and Convolutions of Hermite Gaussians.- A.3 Recursion Relations of Bessel Functions.- A.4 Brief Review of Statistical Function Theory.- A.5 The Different Normalizations of Field Amplitudes and of Annihilation Operators.- A.5.1 Normalization of Classical Field Amplitudes.- A.5.2 Normalization of Quantum Operators.- A.6 Two Alternative Expressions for the Nyquist Source.- A.7 Wave Functions and Operators in the n Representation.- A.8 Heisenberg's Uncertainty Principle.- A.9 The Quantized Open-Resonator Equations.- A.10 Density Matrix and Characteristic Functions.- A.10.1 Example 1. Density Matrix of Bose-Einstein State.- A.10.2 Example 2. Density Matrix of Coherent State.- A.11 Photon States and Beam Splitters.- A.12 The Baker-Hausdorff Theorem.- A.12.1 Theorem 1.- A.12.2 Theorem 2.- A.12.3 Matrix Form of Theorem 1.- A.12.4 Matrix Form of Theorem 2.- A.13 The Wigner Function of Position and Momentum.- A.14 The Spectrum of Non-Return-to-Zero Messages.- A.15 Various Transforms of Hyperbolic Secants.- A.16 The Noise Sources Derived from a Lossless Multiport with Suppressed Terminals.- A.17 The Noise Sources of an Active System Derived from Suppression of Ports.- A.19 The Heisenberg Equation in the Presence of Dispersion.- References.

Temporally stable coherent states for infinite well and P\"oschl-Teller potentials

Abstract: This paper is a direct illustration of a construction of coherent states which has been recently proposed by two of us (JPG and JK). We have chosen the example of a particle trapped in an infinite square-well and also in Poschl-Teller potentials of the trigonometric type. In the construction of the corresponding coherent states, we take advantage of the simplicity of the solutions, which ultimately stems from the fact they share a common SU(1,1) symmetry a la Barut--Girardello. Many properties of these states are then studied, both from mathematical and from physical points of view.

Decoherence and Decay of Motional Quantum States of a Trapped Atom Coupled to Engineered Reservoirs

Abstract: We present results from an experimental study of the decoherence and decay of quantum states of a trapped atomic ion’s harmonic motion interacting with several types of engineered reservoirs. We experimentally simulate three types of reservoirs: a high-temperature amplitude reservoir, a zero-temperature amplitude reservoir, and a high-temperature phase reservoir. Interaction with these environments causes the ion’s motional state to decay or heat, and in the case of superposition states, to lose coherence. We report measurements of the decoherence of superpositions of coherent states and two-Fock-state superpositions into these reservoirs, as well as the decay and heating of Fock states. We confirm the theoretically well-known scaling laws that predict that the decoherence rate of superposition states scales with the square of the ‘‘size’’ of the state.

Unambiguous state discrimination in quantum cryptography with weak coherent states

Abstract: The use of linearly independent signal states in realistic implementations of quantum key distribution (QKD) enables an eavesdropper to perform unambiguous state discrimination. We explore quantitatively the limits for secure QKD imposed by this fact taking into account that the receiver can monitor, to some extent the photon-number statistics of the signals even with todays standard detection schemes. We compare our attack to the beam-splitting attack and show that security against the beam-splitting attack does not necessarily imply security against the attack considered here.

Cloning of continuous quantum variables

Abstract: The cloning of quantum variables with continuous spectra is analyzed. A Gaussian quantum cloning machine is exhibited that copies equally well the states of two conjugate variables such as position and momentum. It also duplicates all coherent states with a fidelity of 2/3. More generally, the copies are shown to obey a no-cloning Heisenberg-like uncertainty relation.

Gauge Field Theory Coherent States (GCS) : III.

Abstract: In the preceding paper of this series of articles we established peakedness properties of a family of coherent states that were introduced by Hall for any compact gauge group and were later generalized to gauge eld theory by Ashtekar, Lewandowski, Marolf, Mour~ ao and Thiemann. In this paper we establish the \Ehrenfest Property" of these states which are labelled by a point (A;E), a connection and an electric eld, in the classical phase space. By this we mean that i) The expectation value of all elementary quantum operators ^ O with respect to the coherent state with label (A;E) is given to zeroth order in h by the value of the corresponding classical function O evaluated at the phase space point (A;E) and ii) The expectation value of the commutator between two elementary quantum operators [ ^ O1; ^ O2]=(i h) divided by i h with respect to the coherent state with label (A;E) is given to zeroth order in h by the value of the Poisson bracket between the corresponding classical functionsfO1;O2g evaluated at the phase space point (A;E). These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. It follows that the innitesimal quantum dynamics of quantum general relativity is to zeroth order in h indeed given by classical general relativity.

Hilbert-Schmidt distance and non-classicality of states in quantum optics

Abstract: The Hilbert—Schmidt distance between two arbitrary normalizable states is discussed as a measure of the similarity of the states. Unitary transformations of both states with the same unitary operator (e.g. the displacement of both states in the phase plane by the same displacement vector and squeezing of both states) do not change this distance. The nearest distance of a given state to the whole set of coherent states is proposed as a quantitative measure of non-classicality of the state which is identical when considering the coherent states as the most classical ones among pure states and the deviations from them as non-classicality. The connection to other definitions of the non-classicality of states is discussed. The notion of distance can also be used for the definition of a neighbourhood of considered states. Inequalities for the distance of states to Fock states are derived. For given neighbourhoods, they restrict common characteristics of the state as the dispersion of the number operato...

Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime

Abstract: A model of a four-wave mixer operated in a nonlinear regime, studied by Yurke and Stoler [Phys. Rev. A 35, 4846 (1987)], is reexamined. Yurke and Stoler have shown that this device, under a certain condition, acts as an even-odd filter, switching even-photon-number states from the pump mode to the signal mode. An initial coherent state in the pump is converted into an entanglement of even and odd coherent states with vacuum states in the output signal and pump modes. We point out that under a different condition and with an even-photon-number state initially in the pump and a vacuum in the signal, the device creates a maximally entangled state between the number state and the vacuum. Using the device to replace the first beamsplitter of a Mach-Zehnder interferometer, phase uncertainties at the Heisenberg limit $(\ensuremath{\Delta}\ensuremath{\varphi}=1/n)$ can be obtained. Since number states are difficult to generate, we point out that an even coherent state obtained from the output of one device can be used as input to a second to achieve the phase uncertainties $\ensuremath{\Delta}\ensuremath{\varphi}{=1/n}_{e},$ where ${n}_{e}$ is the average photon number of the even coherent state.

Entangled coherent states for systems with SU(2) and SU(1,1) symmetries

Abstract: Entangled SU(2) and SU(1,1) coherent states are developed as superpositions of multiparticle SU(2) and SU(1,1) coherent states. As a special example of entangled SU(2) coherent states, entangled binomial states are introduced and these entangled binomial states enable the contraction from entangled SU(2) coherent states to entangled harmonic oscillator coherent states. Entangled SU(2) coherent states are discussed in the context of pairs of qubits. We also introduce the entangled negative binomial states and entangled squeezed states as examples of entangled SU(1,1) coherent states. A method for generating the entangled SU(2) and SU(1,1) coherent states is discussed and degrees of entanglement calculated. Two types of SU(1,1) coherent states are discussed in each case: Perelomov coherent states and Barut-Girardello coherent states.

Decoherence, pointer engineering and quantum state protection

Abstract: We present a proposal for protecting states against decoherence, based on the engineering of pointer states. We apply this procedure to the vibrational motion of a trapped ion, and show how to protect qubits, squeezed states, approximate phase eigenstates and superpositions of coherent states.

Studies on nonlinear coherent states

Abstract: Nonclassical states of light are of fundamental importance in quantum optics. The properties of these states are due to the quantal nature of the electromagnetic field. In recent years there have been many experimental demonstrations of the realizability of nonclassical states in various physical schemes such as resonance fluorescence, four-wave mixing, centre-of-mass motion of a trapped ion and manipulation of field in a cavity. It is of interest to introduce new classes of nonclassical states and investigate their properties. In this context, generalization of the concept of coherent states (CSs) has played a major role. New concepts such as the nonlinear coherent states (NCSs) and interference in phase space have emerged from such studies. In this tutorial review we are concerned with the construction of new classes of nonclassical states by generalizing the notion of CSs. Our investigations on the photon-added coherent states (PACSs) indicate that these states can be interpreted as NCSs. Also, we introduce a new class of nonclassical states related to the PACSs. Having introduced a realizable example of NCSs, we extend the notion of even and odd CSs to the case of NCSs by introducing even and odd NCSs. With this new definition we interrelate some of the well known states of light. We suggest a scheme to generate a class of even and odd NCSs in the centre-of-mass motion of a trapped, laser-cooled, two-level ion.

New nonlinear coherent states and some of their nonclassical properties

Abstract: We construct a displacement-operator-type nonlinear coherent state and examine some of its properties. In particular, it is shown that this nonlinear coherent state exhibits nonclassical properties such as squeezing and sub-Poissonian behaviour.

Entangled Coherent State Qubits in an Ion Trap

Abstract: We show how entangled qubits can be encoded as entangled coherent states of two-dimensional center-of-mass vibrational motion for two ions in an ion trap. The entangled qubit state is equivalent to the canonical Bell state, and we introduce a proposal for entanglement transfer from the two vibrational modes to the electronic states of the two ions in order for the Bell state to be detected by resonance fluorescence shelving methods.

Time dependent quantum propagation in phase space

Abstract: Numerical solutions of the quantum time-dependent integro-differential Schrodinger equation in a coherent state Husimi representation are investigated. Discretization leads to propagation on a grid of nonorthogonal coherent states without the need to invert an overlap matrix, with the further advantage of a sparse Hamiltonian matrix. Applications are made to the evolution of a Gaussian wave packet in a Morse potential. Propagation on a static rectangular grid is fast and accurate. Results are also presented for a moving rectangular grid, guided at its center by a mean classical path, and for a classically guided moving grid of individual coherent states taken from a Monte Carlo ensemble.

Entangled SU(2) and SU(1,1) coherent states

Abstract: Entangled SU(2) and SU(1,1) coherent states are developed as superpositions of multiparticle SU(2) and SU(1,1) coherent states. In certain cases, these are coherent states with respect to generalized su(2) and su(1,1) generators, and multiparticle parity states arise as a special case. As a special example of entangled SU(2) coherent states, entangled binomial states are introduced and these entangled binomial states enable the contraction from entangled SU(2) coherent states to entangled harmonic oscillator coherent states. Entangled SU(2) coherent states are discussed in the context of pairs of qubits. We also introduce the entangled negative binomial states and entangled squeezed states as examples of entangled SU(1,1) coherent states. A method for generating the entangled SU(2) and SU(1,1) coherent states is discussed and degrees of entanglement calculated. Two types of SU(1,1) coherent states are discussed in each case: Perelomov coherent states and Barut-Girardello coherent states.

The Semiclassical propagator for spin coherent states

Abstract: We use a continuous-time path integral to obtain the semiclassical propagator for minimal-spread spin coherent states. We pay particular attention to the “extra phase” discovered by Solari and Kochetov, and show that this correction is related to an anomaly in the fluctuation determinant. We show that, once this extra factor is included, the semiclassical propagator has the correct short time behavior to O(T2), and demonstrate its consistency under dissection of the path.

New nonlinear coherent states and some of their nonclassical properties

Abstract: We construct a displacement-operator-type nonlinear coherent state and examine some of its properties. In particular, it is shown that this nonlinear coherent state exhibits nonclassical properties such as squeezing and sub-Poissonian behaviour.

Quantum limits in the measurement of very small displacements in optical images.

Abstract: We consider the problem of the measurement of very small displacements in the transverse plane of an optical image with a split photodetector. We show that the standard quantum limit for such a measurement, which is equal to the diffraction limit divided by the square root of the number of photons used in the measurement, cannot be overcome by use of ordinary single-mode squeezed light. We give the form of possible multimode nonclassical states of light, enabling us to enhance by orders of magnitude the resolution of such a measurement beyond the standard quantum limit.

Exact diagonalization of two quantum models for the damped harmonic oscillator

Abstract: The damped harmonic oscillator is a workhorse for the study of dissipation in quantum mechanics. However, despite its simplicity, this system has given rise to some approximations whose validity and relation to more refined descriptions deserve a thorough investigation. In this work, we apply a method that allows us to diagonalize exactly the dissipative Hamiltonians that are frequently adopted in the literature. Using this method, we derive the conditions of validity of the rotating-wave approximation (RWA) and show how this approximate description relates to more general ones. We also show that the existence of dissipative coherent states is intimately related to the RWA. Finally, through the evaluation of the dynamics of the damped oscillator, we notice an important property of the dissipative model that has not been properly accounted for in previous works, namely the necessity of new constraints to the application of the factorizable initial conditions.

Generalized coherent states for SU(n) systems

Abstract: Generalized coherent states are developed for SU(n) systems for arbitrary n. This is done by first iteratively determining explicit representations for the SU(n) coherent states and then determining parametric representations useful for applications. For SU(n), the set of coherent states is isomorphic to a coset space SU(n)/SU(n-1), and thus shows the geometrical structure of the coset space. These results provide a convenient (2n-1)-dimensional space for the description of arbitrary SU(n) systems. We further obtain the metric and measure on the coset space and show some properties of the SU(n) coherent states.

Self-organization in nonlinear wave turbulence

Abstract: We present a statistical equilibrium model of self-organization in a class of focusing, nonintegrable nonlinear Schroedinger (NLS) equations. The theory predicts that the asymptotic-time behavior of the NLS system is characterized by the formation and persistence of a large-scale coherent solitary wave, which minimizes the Hamiltonian given the conserved particle number (L{sup 2}-norm squared), coupled with small-scale random fluctuations, or radiation. The fluctuations account for the difference between the conserved value of the Hamiltonian and the Hamiltonian of the coherent state. The predictions of the statistical theory are tested against the results of direct numerical simulations of NLS, and excellent qualitative and quantitative agreement is demonstrated. In addition, a careful inspection of the numerical simulations reveals interesting features of the transitory dynamics leading up to the long-time statistical equilibrium state starting from a given initial condition. As time increases, the system investigates smaller and smaller scales, and it appears that at a given intermediate time after the coalescense of the soliton structures has ended, the system is nearly in statistical equilibrium over the modes that it has investigated up to that time. (c) 2000 The American Physical Society.

On the coherent states, displacement operators and quasidistributions associated with deformed quantum oscillators

Abstract: The Wigner (W), Husimi-Kano (Q) and Glauber-Sudarshan (P) quasidistributions are generalized to f-deformed quasidistributions which extend the parametric family of s-ordered quasidistributions of Cahill and Glauber. The deformation procedure is obtained via a canonical nonisometric transform of the displacement operators which preserves the form of the standard creation-annihilation commutation relation, hence the Heisenberg-Weyl algebra, but changes the scalar product in the Hilbert space of the oscillator states. A whole class of new resolutions of the identity is introduced. The time evolution equation for the new generalized quasidistributions is derived.

Quantum mechanics on a sphere and coherent states

Abstract: The coherent states for a particle on a sphere are introduced. These states are labelled by points of the classical phase space, i.e. the position on the sphere and the angular momentum of a particle. As with the coherent states for a particle on a circle discussed in Kowalski et al (1996 J. Phys. A: Math. Gen. 29 4149), we deal with a deformation of the classical phase space related to quantum fluctuations. The expectation values of the position and the angular momentum in the coherent states are regarded as the best possible approximation of the classical phase space. The correctness of the introduced coherent states is illustrated by an example of the rotator.

Coherent states of nonlinear algebras: applications to quantum optics

Abstract: We present a general unified approach for finding the coherent states (CSs) of polynomially deformed algebras such as the quadratic and Higgs algebras, which are relevant for various multiphoton processes in quantum optics. We give a general procedure to map these deformed algebras to appropriate Lie algebras. This is used, for the noncompact cases, to obtain the annihilation operator eigenstates, by finding the canonical conjugates of these operators. Generalized CSs, in the Perelomov sense, also follow from this construction. This allows us to explicitly construct CSs associated with various quantum optical systems.

Nonequilibrium quantum dynamics of second order phase transitions

Abstract: We use the so-called Liouville--von Neumann (LvN) approach to study the nonequilibrium quantum dynamics of time-dependent second order phase transitions. The LvN approach is a canonical method that unifies the functional Schr\"odinger equation for the quantum evolution of pure states and the LvN equation for the quantum description of mixed states of either equilibrium or nonequilibrium. As nonequilibrium quantum mechanical systems we study a time-dependent harmonic and an anharmonic oscillator and find the exact Fock space and density operator for the harmonic oscillator and the nonperturbative Gaussian Fock space and density operator for the anharmonic oscillator. The density matrix and the coherent, thermal, and coherent-thermal states are found in terms of their classical solutions, for which the effective Hamiltonians and equations of motion are derived. The LvN approach is further extended to quantum fields undergoing time-dependent second order phase transitions. We study an exactly solvable model with a finite smooth quench and find the two-point correlation functions. Because of the spinodal instability of long wavelength modes, the two-point correlation functions lead to the ${t}^{1/4}$-scaling relation during the quench and the Cahn-Allen scaling relation ${t}^{1/2}$ after completion of the quench. Further, after the finite quench the domain formation shows a time-lag behavior at the cubic power of the quench period. Finally we study the time-dependent phase transition of a self-interacting scalar field.

Near-field coherent excitation spectroscopy of InGaAs/GaAs self-assembled quantum dots

Abstract: Using near-field optical microscopy, we have performed coherent excitation spectroscopy of self-assembled quantum dots (SAQDs). A pair of coherent pulses with a time delay between them allows measurement of the temporal coherence of the carrier wave function in single quantum dots. The observed decoherence time is about 15 ps and is well explained by resonant Raman scattering of phonons. Furthermore, quantum beats originating from the superposition of two closely spaced coherent states have been observed. This opens up possibilities of quantum mechanical control of the carrier wave function in SAQDs.

Monge Metric on the Sphere and Geometry of Quantum States

Abstract: Topological and geometrical properties of the set of mixed quantum states in the N-dimensional Hilbert space are analysed. Assuming that the corresponding classical dynamics takes place on the sphere we use the vector SU(2) coherent states and the generalised Husimi distributions to define the Monge distance between arbitrary two density matrices. The Monge metric has a simple semiclassical interpretation and induces a non-trivial geometry. Among all pure states the distance from the maximally mixed state \rho_*, proportional to the identity matrix, admits the largest value for the coherent states, while the delocalized 'chaotic' states are close to \rho_*. This contrasts the geometries induced by the standard (trace, Hilbert-Schmidt or Bures) metrics, where the distance from \rho_* is the same for all pure states. We discuss possible physical consequences including unitary time evolution and the process of decoherence. We introduce also a simplified Monge metric, defined in the space of pure quantum states, and more suitable for numerical computation.

Proposal for measurement of harmonic oscillator berry phase in ion traps

Abstract: We propose a scheme for measuring the Berry phase in the vibrational degree of freedom of a trapped ion. Starting from the ion in a vibrational coherent state we show how to reverse the sign of the coherent state amplitude by using a purely geometric phase. This can then be detected through the internal degrees of freedom of the ion. Our method can be applied to preparation of entangled states of the ion and the vibrational mode.

Decoherence via the Dynamical Casimir Effect

Abstract: We derive a master equation for a mirror interacting with the vacuum field via radiation pressure. The dynamical Casimir effect leads to decoherence of a superposition state in a time scale that depends on the degree of "macroscopicity" of the state components, and which may be much shorter than the relaxation time scale. Coherent states are selected by the interaction as pointer states.