Showing papers on "Coherent states published in 1991"

Nonclassical properties of states generated by the excitations on a coherent state

Abstract: In the past few years there has been considerable interests in attempts to produce non-classical states of light such as squeezed states and photon number states. The squeezed states have reduced fluctuations in one field quadrature when compared with the coherent states.1 In this paper we consider the state obtained by repeated application of the photon creation operator on the coherent state. Such a state has a nonzero field amplitude and is shown to exhibit non-classical properties like the squeezing in one of the quadratures of the field, and sub- Poissonian photon statistics. We calculate different quasiprobability functions for fields in such states and also the distribution function for one of the field quadratures. In the last section we discuss how such states can be generated in nonlinear processes in cavities.

Quantum Optics of Dielectric Media

Abstract: Because quantum fluctuations impose fundamental limits on the accuracy of measurements, much attention is now being devoted to the problem of moderating or supressing their effect. If a quantity to be measured can be regarded as one of a pair of conjugate variables, for example, then its variance can usually be made arbitrarily small, but only at the expense of increasing the variance of the unmeasured variable. This technique requires putting the system being observed in a special sort of quantum state, referred to as “squeezed” 1.

Quantum electrodynamics with unstable vacuum

Abstract: 1 Introduction.- 1.1 General Introduction and Definition of Notation.- 1.1.1 Chronological Review.- 1.1.2 Basic Notation.- 1.2 Quantum Electrodynamics with an External Field.- 2 Scattering Processes in Which Interactions with the External Field are Taken into Account Exactly.- 2.1 The Quantized Charged Field in an External Electromagnetic Field.- 2.1.1 Spinor Field.- 2.1.2 Scalar Field.- 2.2 Perturbation Expansion in the Radiative Interaction in QED with an External Field.- 2.3 Green's Functions. The Generating Functional.- 2.4 Appendix. Extension of the Normal-Ordering Process to Theories with an Unstable Vacuum.- 3 Expectation Values.- 3.1 Green's Functions for the Calculation of Expectation Values.- 3.2 Perturbations in Powers of Radiative Interaction for Expectation Values.- 3.3 Equation for the Average Electromagnetic Field. Effective Action.- 3.4 Density Matrix of Particles Created by an External Field.- 4 Total Probabilities of Radiative Processes in an External Field.- 4.1 Total Irradiation Probability.- 4.1.1 Total Probability of Irradiation from the Vacuum Accompanied by Pair Creation.- 4.1.2 Differential Probability for Photon Irradiation from the Vacuum Accompanied by Pair Creation.- 4.1.3 Probability of Photon Irradiation from the Vacuum Accompanied by the Creation of a Single Pair.- 4.1.4 Total Probability of Irradiation from a One-Electron State Accompanied by Pair Creation.- 4.1.5 Differential Probability of a One-Photon Emission from a One-Electron State Accompanied by Pair Creation.- 4.1.6 Probability of One-Photon Emission from a One-Electron State Without Pair Creation.- 4.1.7 Radiative Processes with a Photon in the Initial State.- 4.2 Unitarity Relation and the Optical Theorem.- 4.3 Generating Functional for the Total Probabilities of Radiation Processes.- 4.4 Decay Probability of States.- 4.4.1 Vacuum Decay Probability.- 4.4.2 Decay Probability of a One-Electron State.- 4.4.3 Decay Probability of a One-Photon State.- 5 Calculations of Zero-Order Processes in External Electromagnetic Fields.- 5.1 Processes in an Electric Field.- 5.1.1 Constant Electric Field.- 5.1.2 Alternating Electric Field.- 5.2 A Constant Field Combined with that of a Plane Wave.- 5.2.1 Solutions of the Klein-Gordon Equation.- 5.2.2 Solutions of the Dirac Equation.- 5.2.3 Calculation of Zero-Order Processes. Spinor QED.- 5.2.4 Calculation of Zero-Order Processes. Scalar QED.- 5.3 Creation of Particles from the Vacuum in Coherent States.- 5.4 Calculation of the Density Matrix for Particles Created in an External Field.- 6 Propagators of Particles in External Electromagnetic Fields.- 6.1 Introduction.- 6.2 Determination of Propagators by Summing Solutions of the Relativistic Wave Equations.- 6.2.1 Constant Electric Field.- 6.2.2 A Constant Field Combined with that of a Plane Wave.- 6.2.3 Eigenfunction Method.- 6.3 Schwinger's Proper-Time Method.- 6.4 Calculation of the Green's Functions by the Functional Integration Method.- 6.4.1 Path Integral Representation of the Green's Functions.- 6.4.2 Combination of a Constant Field with a Plane Wave.- 6.4.3 Stationary Phase Method.- 7 Calculations of Radiative Processes in External Electromagnetic Fields.- 7.1 Effective Action in the One-Loop Approximation.- 7.2 Vacuum Processes.- 7.2.1 Mean Current of Created Particles.- 7.2.2 Probability of Photon Emission from the Vacuum Accompanied by Pair Creation.- 7.2.3 Total Probability of Photon Emission from the Vacuum Accompanied by Pair Creation.- 7.2.4 Probability of a Photon Emission from the Vacuum Accompanied by the Creation of a Single Pair. Vacuum Decay Probability.- 7.3 Processes with an Electron in the Initial State. Mass Operator.- 7.3.1 Probability of Transition from a One-Electron State with the Emission of a Photon and the Creation of Pairs.- 7.3.2 Total Transition Probability from a One-Electron State with Photon Emission and Pair Creation.- 7.3.3 Probability of Photon Emission from a One-Electron State without Pair Creation. Decay Probability of a One-Electron State.- 7.4 Radiative Processes with a Photon in the Initial State. Polarization Operator.- 8 Green's Function in Non-Abelian Theories.- 8.1 Introduction.- 8.2 Calculation of Green's Functions by the Functional Integration Method.- 8.3 Calculation of Green's Functions in an Abelian-like External Field.- References.

Measure of the nonclassicality of nonclassical states

Abstract: A continuous parameter introduced into the convolution transformation between P and Q functions leads to a measure of how nonclassical quantum states are with values ranging from 0 to 1: For photon-number states, the value is 1, the maximum possible. For squeezed vacuum states, it is a monotonically increasing function of the squeeze parameter with values varying from 0 to 1/2. This measure is identical to the minimum number of thermal photon necessary to destroy whatever nonclassical effects existing in the quantum states.

Nonclassical state from two pseudoclassical states

Abstract: The quantum-mechanical superposition of two coherent states of identical mean photon number but different phases yields a state that can exhibit sub-Poissonian and oscillatory photon statistics, as well as squeezing.

Atom- and field-state evolution in the Jaynes-Cummings model for large initial fields.

Abstract: An asymptotic result is derived for the Jaynes-Cummings model of a two-level atom interacting with a quantized single-mode field, which is valid when the field is initially in a coherent state with a large average photon number. It is shown that for certain initial atomic states the joint atom-field wave function factors into an atomic and a field part throughout the interaction, so that each system remains separately in a pure state. The atomic part of the wave function displays a crossing of trajectories in the atom Hilbert space that leads to a unique state for the atom, independent of its initial state, at a specific time to (equal to half the revival time). The field part of the wave function resembles a crescent squeezed state. The well-known collapses and revivals are investigated from this perspective. The collapse appears to be associated with a "measurement" of the initial state of the atom with the field as the measuring apparatus. The measurement is not complete for finite average photon number: the system is instead left in a coherent superposition of macroscopically distinct states. At the half-revival time to this superposition of states is entirely in the field part of the state vector, so that the (pure) state of the field at that time is of the form sometimes referred to as a "Schrodinger cat. " The revivals of the population inversion are seen to be entirely due to the fact that the linear superposition of the two macroscopically distinct field states is coherent (i.e., a pure state), as opposed to an incoherent mixture.

Quasiprobability distributions for the Jaynes-Cummings model with cavity damping.

Abstract: The Jaynes-Cummings model with cavity damping is investigated in the rotated-wave approximation. First we introduce six appropriate combinations of the matrix elements of the density operator, which are still operators with respect to the light field. With the help of the s-parametrized quasiprobability distributions of Cahill and Glauber [Phys. Rev. 177, 1882 (1969)] the equations of motion for the density operator transform to six coupled partial-differential equations. By expanding the quasiprobability distributions into two suitable sets, we obtain six tridiagonally coupled differential equations for the expansion coefficients, which are solved by a Runge-Kutta method. Starting with an initial coherent state of the cavity field and the atom in its upper state, we find that the initially one-peak quasiprobability function splits into two peaked functions counterrotating in the complex plane and, depending on the damping constant, spiraling into the origin. Revivals of the inversion oscillation are found for those times, when the two peaks collide. The time dependence of the inversion and the intensity as well as some special distributions of interest are also discussed.

On coherent states for the simplest quantum groups

Abstract: The coherent states for the simplest quantum groups (q-Heisenberg-Weyl, SU q (2) and the discrete series of representations of SU q (1, 1)) are introduced and their properties investigated. The corresponding analytic representations, path integrals, and q-deformation of Berezin's quantization on ℂ, a sphere, and the Lobatchevsky plane are discussed.

Displaced Fock states and their connection to quasiprobabilities

Abstract: The properties of the displaced Fock states mod a,n) identical to d(a,a*) mod n), (a complex numbers, D(a,a*) displacement operators, n=0,1,2,. . .) are systematically investigated with emphasis on the connections to the Heisenberg-Weyl group and to its irreducible representations. The displaced Fock states comprise the coherent states mod a) identical to mod a,0) as well as the Fock states mod n) identical to mod 0,n) as particular cases. An orthocompleteness relation for the displaced Fock states in the form of the area integral of the operators mod a,m)(a,n mod over the complex a-plane is derived. It generalizes for m=n the well known completeness relation for the coherent states and leads for m not=n to identities expressing the overcompleteness of the displaced Fock states. A basic formula is obtained for the convolution of the operators mod a,m)(a,n mod with the class of Gaussian functions that have exponents proportional to aa*. The connection of the displaced Fock states to the transition operators from the density operator for a single boson mode to quasiprobabilities is studied in general form and specified to the class of transition operators with the displaced Fock states as their eigenstates and connected by convolutions with Gaussian functions to the coherent-state quasiprobability, the Wigner quasiprobability, and the Glauber-Sudarshan quasiprobability.

Quantum dynamics of the parametric oscillator

Abstract: We present dynamical calculations for the quantum parametric oscillator using both number-state and coherent-state bases. The coherent-state methods use the positive-P representation, which has a nonclassical phase space an essential requirement in obtaining an exact stochastic representation of this nonlinear problem. This also provides a way to directly simulate quantum tunneling between the two above-threshold stable states of the oscillator. The coherent-state methods provide both analytic results at large photon numbers, and numerical results for any photon number, while our number-state calculations are restricted to numerical results in the low-photon-number regime. The number-state and coherent-state methods give precise agreement within the accuracy of the numerical calculations. We also compare our results with methods based on a truncated Wigner representation equivalent to stochastic electrodynamics, and find that these are unable to correctly predict the tunneling rate given by the other methods. An interesting feature of the results is the much faster tunneling predicted by the exact quantum-theory methods compared with earlier semiclassical calculations using an approximate potential barrier. This is similar to the faster tunneling found when comparing quantum penetration of a barrier to classical thermal activation. The quantum parametric oscillator, which has an exact steady-state solution, therefore provides a useful and accessible system in which nonlinear quantum effects can be studied far from thermal equilibrium.

Squeezing and quantum groups.

Abstract: Generalized quasicoherent states for the Weyl-Heisenberg quantum group have been defined by Biedenharn and MacFarlane. In this Letter other quantum Weyl-Heisenberg coherent states are defined for complex q in the usual Fock space. Such states are shown to exhibit interesting squeezing properties, in particular when \ensuremath{\Vert}q\ensuremath{\Vert}\ensuremath{\approxeq}1, for the q analog to the harmonic oscillator.

Generalized q-bosons and their squeezed states

Abstract: Generalized q-boson operators associated with the simultaneous creation of several q-bosons are introduced. These operators give rise to a realization of the quantum Weyl-Heisenberg algebra and are applied to the construction of corresponding Holstein-Primakoff realizations of the quantum algebras SUq(2) and SUq(1,1). Coherent states of these algebras are defined in the various ways suggested by the equivalent definitions of the harmonic oscillator coherent states, and some of their properties are studied. Particular attention is devoted to the squeezing properties of the quadrature of the electromagnetic field in these states.

Correlated two-mode SU(1, 1) coherent states: nonclassical properties

Abstract: The properties of correlated two-mode SU(1, 1) coherent states are studied. These states are a generalization of the two-mode squeezed vacuum states and are generated by applying the two-mode squeeze operator to a state in which there are initially q photons in one mode, |q, 0〉, the squeezed vacuum (q = 0) obviously being a special case. For these states the photon-number distribution and the nonclassical properties of photon antibunching, enhanced phase fluctuations, violations of the Cauchy–Schwarz inequality, two-mode squeezing, and sum squeezing are studied. A Q-function quasi-probability distribution for these states is also presented.

Photon statistics of two-mode squeezed states and interference in four-dimensional phase-space

Abstract: We derive the joint photon-number distribution for two-mode squeezed states with coherent amplitudes. For certain ranges of the parameters, the joint distribution exhibits nonclassical oscillations, which are the two-mode analog of the oscillations found in the single-mode case. These nonclassical oscillations are absent from the marginal distributions for each mode. The nonclassical features of the joint distribution may be explained using the interference–in–phase-space concept in four-dimensional phase space.

Paraxial quantum propagation

Abstract: A quantum version of the classical paraxial approximation is established. After expanding the classical free-field wave equation in powers of \ensuremath{\theta} (the characteristic opening angle of the paraxial rays), the familiar time-dependent paraxial equation appears in zeroth order. The arguments employed in deriving this approximation scheme are used in turn to identify the subspace of the photon Fock space consisting of paraxial states of the field. The quantum version of the paraxial approximation is then obtained by restricting the Maxwell field operators to this domain. Exact equal-time commutation relations are recovered as expansions in \ensuremath{\theta}. In zeroth order, the theory yields a quantized analog of the classical paraxial wave equation, and formally resembles a nonrelativistic many-body theory. This formalism is applied to show that Mandel's local-photon-number operator and Glauber's photon-counting operator reduce, in zeroth order, to the same true-number operator. In addition, it is shown that the O(${\mathrm{\ensuremath{\theta}}}^{2}$) difference between them vanishes for experiments described by stationary coherent states.

Quasiprobability distributions for the cavity-damped Jaynes-Cummings model with an additional Kerr medium.

Abstract: A cavity-damped Jaynes-Cummings model with a Kerr-like medium filling the cavity is investigated in the rotating-wave approximation. We introduce six operators with respect to the light field whose equations of motion are transformed to six coupled partial differential equations using the s-parametrized quasiprobability distributions of Cahill and Glauber [Phys. Rev. 177, 1882 (1969)]. Equations of motion for expansion coefficients of the distribution functions are solved by a Runge-Kutta procedure for vector tridiagonal relations. Starting with an initial coherent state for the cavity field and the atom in its upper state, we find that revivals of the atomic inversion are more pronounced for a given damping constant compared to the case of no Kerr medium. Also, quadrature squeezing is less affected by weak cavity damping and thermal noise compared to the standard Jaynes-Cummings model. The effect of damping on interesting non-Gaussian structures is also discussed.

Quantum-optical properties of polariton waves

Abstract: We develop a quantum-mechanical Hamiltonian formulation to treat the polariton in the framework of quantum optics. We exploit two specific Hamiltonians: the conventional Hopfield model, and a more general Hamiltonian. For both of these, exciton-polariton quantum states are found to be squeezed (intrinsic polariton squeezing) with respect to states of an intrinsic, nonpolaritonic, mixed photon-exciton boson. The amount and duration of intrinsic squeezing during the polariton period are calculated for exciton polaritons in typical I-VII and III-V semiconductors. Among the noteworthy features is the possibility of tuning the amount of intrinsic squeezing by varying the frequency--wave-vector dispersion of the polariton mode. We further analyze the photon statistics of the electromagnetic component of the polariton. Tunable non-Poissonian photon statistics and squeezing (optical polariton squeezing) are found in the radiative component of the exciton polariton. This entails the reduction of the fluctuations of the polariton electromagnetic field component below the limit set by the vacuum fluctuations. The Mandel Q factor for the number distribution of photons in a polariton coherent state has been evaluated. Although small, for I-VII and III-V materials in the range of modes analyzed, the Q factor could be enhanced for phonon polaritons as well as for other materials. Interpretations of the origin of squeezing in polariton states are presented.

Coherent states for n‐dimensional Euclidean groups E(n) and their application

Abstract: The n‐dimensional Euclidean group—the group of rotations and translations of Rn —is the natural canonical group for quantizing a system whose configuration space is the sphere Sn−1, n≥2. Since every unitary irreducible representation of E(n) is nonsquare integrable it follows that coherent states involving the whole group do not exist for such representations. Instead, coherent states are defined based on unitary, reducible representations of the group, and it will be shown that, with such states, quantum dynamics on the sphere admits a path‐integral representation involving novel first‐order classical actions defined on E(n) as well as integrations over the entire group manifold at each time slice. Irreducibility of the quantization is achieved through a final limiting process.

Thermal Coherent States and Thermal Squeezed States

Abstract: Various definitions of thermal coherent states and thermal squeezed states are compared and interrelated. Different ordering of squeezing, displacement and thermalizing operators is discussed in order to establish a relation between all definitions. A one-to-one correspondence is exhibited between general Gaussian density matrices in coordinate space and thermal squeezed states.

Quasi-probability distribution Q(α, α*) versus phase distribution P(θ) in a description of superpositions of coherent states

Abstract: The generation of discrete superpositions of coherent states in the anharmonic oscillator model is discussed from the point of view of their quasi-probability distribution Q(α, α*) and phase probability distribution P(θ). It is shown that for the superposition with well-distinguishable states both distributions show the same rotational symmetry. The maximum number of well-distinguishable states is estimated. The two functions are illustrated graphically to show explicitly their symmetry and the influence of the interference terms. The similarity between the Q function integrated over the amplitude and the phase distribution P(θ) is shown to exist for the anharmonic oscillator states.

Aspects of q-oscillator quantum mechanics

Abstract: We investigate some aspects of q Heisenberg algebra. We show how su(2) and su(1,1) generators can be constructed in terms of the q creation and annihilation operators. We also construct the coherent states for the q oscillator and show that they can be obtained by the action of a displacement operator on the vacuum. For the multimode case with q=0, corresponding to the infinite statistics of Greenberg [Phys. Rev. Lett. 64, 705 (1990)], we generalize our single-mode construction to obtain the corresponding coherent states. These states, which are eigenstates of the annihilation operators, interestingly, turn out to be degenerate owing to the noncommutativity of the displacement operators.

Phase operators via group contraction

Abstract: The problem of quantization of the classical phase of a harmonic oscillator (HO) is solved here in two steps. First, polar decomposition of the step operators of the u(2) algebra is performed. Second, the method of group contraction is used through which, in the limit j→∞, a,a° is passed to for the quantized HO and its Hermitian phase operators. Also, phase states, i.e., states with sharply defined phase, are constructed and the dynamical aspects of the contraction limit between the Jaynes–Cummings model (JCM) and a finite‐dimensional counterpart with increasing j parameter are studied. Finally, the old problem of the phase operators is discussed in the wider frame of rank‐1 algebras and classify the previous works in this frame.