On the Squeezing of Coherent States by the Multiphoton Jaynes-Cummings Hamiltonian with an Intensity-dependent Coupling

Abstract

The squeezing properties of the multiphoton Hamiltonian with intensity-dependent coupling are evaluated for the [xcirc] and [pcirc] _x quadratures, for the initial state of a coherent electromagnetic field and an atom in the ground state. Two measures of squeezing: the percentage of total squeezing and the squeezing time-period percentage, are introduced. Interesting squeezing properties with respect to [xcirc] are observed for real coherent states when the time evolution of the above measures and of the time-averaged squeezing are analysed. The multiphoton intensity-dependent coupling Hamiltonian is found to be almost independent of the specific powers of the annihilation and creation operators, as long as the sum of the powers is kept constant.     

Squeezing with Rydberg Atoms

Abstract

It is shown that some 17 Rydberg Na atoms initially placed into a coherent atomic state and super-radiating into a low-Q microwave cavity at temperatures T ? 0·4 K will show modest squeezing in the fluorescence field, the squeezing arising from terms oscillating at twice the cavity frequency. It is also shown that similar numbers of Rydberg atoms undergoing resonance fluorescence in a coherent single-mode microwave driving field and without any cavity will show more substantial squeezing in the fluorescence field.     

Squeezed states in spontaneous Raman and in stimulated Raman processes

The Hamiltonian and the equations of motion for various modes involving spontaneous and stimulated Raman processes are derived. The coupled differential equations involving these modes are solved analytically by using an intuitive approach (IA). The solution of the present paper does not require any short time approximation. As a matter of fact, the present solution for field operators and hence the squeezing for various modes are valid for all interaction time t. In this way, the present paper is more general compared to the earlier investigations where short-time approximations were found a must. By exploiting the solutions for the field operators, we obtain the squeezing effects of input coherent light for pure and for mixed field modes. The IA employs the perturbation theory and hence we came across the secular terms in the solution. Of course, for small coupling constants these secular terms could be summed for all orders. To establish our claim, we remove the secular terms at least for one occasion.     

Cavity field spectrum for multiphoton Jaynes-Cummings model

Abstract

The field spectra in an ideal cavity for the multiphoton Jaynes-Cummings model are studied. The analytical expression for the spectrum is obtained from the finite double-Fourier transform of the two-time field correlation function. The spectral differences between the initial coherent and initial thermal states are discussed, and the comparisons between the field spectra and atomic emission spectra for k = 2, 3 and 4 are presented. It is shown that the kth power of the photon annihilation operator and atomic lowering operator are subjected to identical forms of a second-order differential equation, in which the coefficients consist of the constants of motion. The field spectrum in the cavity and the emission spectrum of the atom for the initial vacuum state are compared.     

Two derivations of the Feynman formulae for the fields of a moving charge

Abstract

The Feynman expressions for the electric and magnetic fields of a charge moving on an arbitrary trajectory are found by two simple methods. The first involves a direct integration of Maxwell's time-dependent equations. The second is based on the formalism of quantum optics which uses the first order equations of motion for creation and annihilation operators. As an illustration of the simplicity of these methods, the fields of an oscillating electric dipole are obtained.     

Quantum travelling-wave analysis of a quasi-phase-matched parametric amplifier

The paper describes a phase-mismatched Fokker–Planck equation formulation for parametric amplification in periodically inverted nonlinear media. An analytical solution of the phase-mismatched Fokker–Planck equation and the amplitude quantum fluctuation after passing through the quasi-phase-matched (QPM) device are obtained. The calculated results for the QPM device conform to that of the Langevin equation in the case of no loss k=0, and give a general solution for k≠0. From this one can derive knowledge of the dependence of squeezing on the loss coefficient k and make a comparison between the QPM device and a phase matched device at the threshold k=ε₀.     

Macroscopic canonical quantization in quantum optics: Properties of quasi mode annihilation and creation operators

Abstract

Macroscopic canonical quantization of the EM field and radiative atom systems occurring in quantum optics experiments involving linear classical optics devices can be carried out via expansion of the vector potential either in terms of true mode functions for the optical device or in terms of approximate or quasi mode functions. The relationship between the true mode and quasi mode annihilation, creation operators is determined and shown to involve a Bogolubov transformation. Analytic properties are also examined and it is found that the annihilation, creation operators times the square root of the angular frequency are analytic functions of the variables specifying the modes.     

N-level Atom Interacting with Single-mode Radiation Field: An Exactly Solvable Model with Multiphoton Transitions and Intensity-dependent Coupling

Abstract

We study the dynamics of an N-level atom coupled in a lossless cavity to a single-mode near-resonant quantized field. The atomic levels are coupled by the multiphoton transitions and the coupling constants between the field and the atomic levels are supposed to be intensity dependent. We find the exact solution for the state vector describing the dynamics of the atom-plus-field system. As an illustration we use the model for studying (i) the time evolution of the atomic occupation probability with the initially coherent field and (ii) the light squeezing, when the cavity field is initially in the vacuum state and the atom is prepared in the atomic ‘coherent state’ (a superposition of atomic states).     

Negative binomial states of the quantized radiation field

Abstract

Negative binomial states of a single field mode are introduced and their properties are compared with those of the (positive) binomial states. We find that the two types of state have similar properties if the roles of the creation and annihilation operators are interchanged.     

Orthonormalized Eigenstates of Operator a k and Their Squeezing Properties

Abstract

The orthonormalized eigenstates of the higher powers, a ^k (k  3), of the annihilation operator a were obtained. We discuss the mathematical properties and the squeezing properties of the orthonormalized eigenstates of a ^k. We prove that these orthonormalized eigenstates of a ^k construct a complete representation, that none of them have the squeezed effect in the usual sense, and that higher-order squeezing can be found.     

Intracavity Optical Bistability Driven by Squeezed Light from a Parametric Amplifier

Abstract

We describe the theory of optical bistability when atoms are collectively excited within the cavity of a parametric oscillator. Both optical bistability and parametric amplification can squeeze significantly the cavity-field quantum noise. When they are coupled together we find significant changes both on the mean value bistability and on the spectrum of squeezing as the parametric coupling increases. These are calculated directly from the appropriate master equation for the density matrix using a quantum distribution function (positive P) to develop Fokker—Planck and Ito equations.     

Field quantization,photons and non-Hermitean modes

Field quantization in unstable optical systems is treated by expanding the vector potential in terms of non-Hermitean (Fox-Li) modes. We define non-Hermitean modes and their adjoints in both the cavity and external regions and make use of the important bi-orthogonality relationships that exist within each mode set. We employ a standard canonical quantization procedure involving the introduction of generalized coordinates and momenta for the electromagnetic (EM) field. Three-dimensional systems are treated, making use of the paraxial and monochromaticity approximations for the cavity non-Hermitean modes. We show that the quantum EM field is equivalent to a set of quantum harmonic oscillators (QHOs), associated with either the cavity or the external region non-Hermitean modes, and thus confirming the validity of the photon model in unstable optical systems. Unlike in the conventional (Hermitean mode) case, the annihilation and creation operators we define for each QHO are not Hermitean adjoints. It is shown that the quantum Hamiltonian for the EM field is the sum of non-commuting cavity and external region contributions, each of which can be expressed as a sum of independent QHO Hamiltonians for each non-Hermitean mode, except that the external field Hamiltonian also includes a coupling term responsible for external non-Hermitean mode photon exchange processes. The non-commutativity of certain cavity and external region annihilation and creation operators is associated with cavity energy gain and loss processes, and may be described in terms of surface integrals involving cavity and external region non-Hermitean mode functions on the cavity-external region boundary. Using the essential states approach and the rotating wave approximation, our results are applied to the spontaneous decay of a two-level atom inside an unstable cavity. We find that atomic transitions leading to cavity non-Hermitean mode photon absorption are associated with a different coupling constant to that for transitions leading to photon emission, a feature consequent on the use of non-Hermitean mode functions. We show that under certain conditions the spontaneous decay rate is enhanced by the Petermann factor.     

Quantization of coupled modes propagation in integrated optical waveguides

Abstract

A quantum mechanical analysis of the propagation of coupled modes in integrated optical waveguides is given. The modal orthonormalization property on a cross-section of an optical waveguide, the vector structure of the guided optical modes and the reversal-time symmetry are taken into account to derive the quantum momentum operator and Heisenberg's equations giving a quantum-consistent formulation of the coupled mode propagation as a function of forward and backward creation and annihilation operators.     

Density Matrix Elements and Moments for Generalized Gaussian State Fields

Abstract

A single-mode radiation field with Gaussian Wigner functions (Gaussian state field) is studied. The single-mode Gaussian-field generating function for both the density matrix elements and the moments of the creation and annihilation operators is calculated. It is shown that the matrix elements as well as the moments of different orders can be expressed in terms of Hermitian polynomials of two variables. Furthermore, the cumulants of the Gaussian state field are calculated. Finally, we generalize the Gaussian state field and compute the corresponding moments and density matrix elements.     

Fluctuation conductivity of Bi<Subscript>2</Subscript>Sr<Subscript>2</Subscript>CaCu<Subscript>2</Subscript>O<Subscript>8 + δ</Subscript> in a two-band superconductivity model

A modified analytical Aslamazov-Larkin formula for the temperature dependence of the fluctuation conductivity near the critical temperature T _c is obtained for two-band superconducting compounds on the basis of linearized equations of the two-band Ginzburg-Landau theory. The results are compared to experimental data for the high-T _c superconductor compound Bi₂Sr₂CaCu₂O_{8 + δ}.     

Four-photon Coherent States

Abstract

We present a detailed discussion of a type of four-photon coherent state defined as right eigenstates of the operator â ⁴ where â is the usual annihilation operator. There are actually four sets of states that need to be considered, namely those containing as the lowest number states ¦0>, ¦1>, ¦2>, or ¦3>. These correspond to the possible unique superpositions of the ordinary coherent states ¦±α> and ¦±iα>. We discuss the nonclassical properties of these states such as photon antibunching and squeezing. The usual second order squeezing does not exist for these states but higher order squeezing and square field amplitude squeezing do exist. Also discussed are the quasiprobability distributions, namely the P-function, the Q-function and the Wigner function. Finally, a method of generating these states based on the competition between a four-photon parametric process and incoherent losses from four-photon absorption is presented.     

On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment

A reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L² inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well‐posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well‐posed and stable far‐field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty‐like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd.     

Equation-of-motion approach to the critical dynamics of liquid helium at the lambda point

A complete treatment of dynamic scaling in liquid helium at the lambda point is developed straightforwardly from explicit application of the nonlinear mode-coupling equations of motion, without Langevin noise terms. Special attention is given to the ratio w of the order-parameter and entropy relaxation rates. Various estimates indicate w = O(0.1), which produces a characteristic double-humped fluctuation spectrum, amenable to experimental verification. Further details are a thorough study of the transient solutions as well as the presentation of a new expansion scheme similar to the expansion but of superior accuracy.Research supported in part by the U.S. National Science Foundation under grants DMR 76-82345 and DMR 76-24472.     

Equations for magnetization processes based on the laws of thermodynamics

A series of logarithmic equations founded on thermodynamic principles are used to rationalize the process of magnetization. These equations yield precise power-law fits for the virgin magnetization curves of single-crystal and polycrystalline specimens. Using these equations, values ofM _s are obtained without using laborious extrapolation procedures. This characteristic, together with their extreme simplicity and their ability to follow and account for the magnetization of different products over all field values, makes them very useful. These equations, their derivations and supporting data are presented in this report.     

Diffusion of Impurity Excitations in Superfluid and Solid 3He–4He Mixtures

Starting from the system of linearized kinetic equations, the exact expressions for the spin and mass diffusion coefficients and thermal diffusion ratio in quantum mixtures of helium isotopes with any degree of degeneration is obtained. From the general expression the various limiting expressions have been obtained, which are determined by various relations between collision rates of quasiparticles. The results of the calculations are compared with available experimental data. It is shown that in the low-temperature region, where an impuriton gas is one-component, the relaxation of concentration of ³ He in superfluid ³ He– ⁴ He mixtures is determined by the acoustic and dissipative collective modes with an effective diffusion coefficient. From the general relation, the expression for a spin diffusion coefficient in solid ³ He– ⁴ He quantum mixtures is derived. The comparison of the results obtained with the experimental data makes it possible to find a numerical value for the width of the impuriton energy zone.