First-order Correlation Function Research Articles

Correlations and coherence in the microwave instability

We expand upon the solvable model for the microwave instability in an electron storage ring as proposed in Wang (Phys. Rev. E 58 (1998) 984). The Vlasov–Maxwell equations are reduced to an integral eigenvalue problem similar to that which arises in the Karhunen–Loéve expansion theory for stochastic processes (Goodman, Statistical Optics, Wiley, New York, 1985). We derive first- and second-order correlation functions for the electron beam line density for the microwave instability. Using a set of coherent modes the correlation functions are diagonalized. A relationship between the first- and second-order correlation functions is obtained and it is shown to be similar to that of the Hanbury-Brown Twiss experiment (Born and Wolf, Principles of Optics, 6th Edition, Pergamon press, New York, 1980; Loudon, The Quantum Theory of Light, Clarendon press, Oxford, 1983). We obtain an expression for the Wigner distribution in terms of the first-order correlation function and relate this to the power in the electric field. We introduce an entropy-like quantity to characterize the coherence of the instability.

Nuclear Creation and Annihilation Operators and Electromagnetic Multipole Fields

The frequency spectrum of electromagnetic radiation can be written as the Fourier transform of the first-order correlation function of the vector potential. If nuclei are coupled to the radiation field, the Heisenberg equations of motion of the field operators contain nuclear operators and vice versa. Under plausible assumptions the equations of motion for the nuclear operators can be integrated and hence, the equations of the field operators can be solved. The vector potential of the radiated field can then be expressed as a function of solely nuclear quantities. The first-order correlation function deduced from it contains only two-times and one-time averages of simple nuclear creation and annihilation operators. The theory can be used to explain homogeneous line broadening for long-lived nuclei submitted to small fluctuating interactions.

Probing first-order spatial coherence of a Bose-Einstein condensate

Abstract We probe the spatial coherence properties of a magnetically trapped Bose gas. Two matter wave beams are extracted from two spatially separated regions of the trap and overlap outside the trapping region. The visibility of the resulting interference pattern measures the phase coherence between the regions of extraction. By varying the spatial separation between the two regions the first-order spatial correlation function of the trapped Bose gas can be measured. The location of the minima of the interference pattern is reproducible, which experimentally confirms that the trapped Bose-Einstein condensate is not fragmented into individual condensates.

A nonperturbative density functional analysis for nonuniform Lennard-Jones fluid

A nonperturbative density functional analysis for nonuniform Lennard-Jones (LJ) fluid is proposed in the framework of weighted density approximation. The required inputs (the second-order and first-order direct correlation functions for uniform LJ fluid) are obtained by numerically solving the OZ equation and using a equation of state for uniform Lennard-Jones fluid. The density distribution resulting from the present analysis is compared favorably with the computer simulation results for two cases: (1) a Lennard-Jones fluid confined in a spherical cavity with a hard wall, and (2) a Lennard-Jones fluid confined between two hard walls.

Probing first-order spatial coherence of a Bose-Einstein condensate

We probe the spatial coherence properties of a magnetically trapped Bose gas. Two matter wave beams are extracted from two spatially separated regions of the trap and overlap outside the trapping region. The visibility of the resulting interference pattern measures the phase coherence between the regions of extraction. By varying the spatial separation between the two regions the first-order spatial correlation function of the trapped Bose gas can be measured. The location of the minima of the interference pattern is reproducible, which experimentally confirms that the trapped Bose-Einstein condensate is not fragmented into individual condensates.

Dephasing of Josephson oscillations between two coupled Bose-Einstein condensates

We study the dynamics of the relative phase between two atomic Bose-Einstein condensates coupled via collisions and via a Josephson-like coupling. We derive the equations of the motion for the relative phase and the relative number operators from the second-quantized Hamiltonian of the system using a quantum field theoretical approach. We distinguish the cases in which the two condensates are in the same trap or in two different traps and study the influence of this difference on the first-order correlation function of atomic fields. In identical traps this function does not undergo dephasing. We calculate the dephasing time for the case of different traps.

Line broadening and line shift of long-lived states of nuclei in a lattice

Long-lived states of nuclei embedded in a lattice are continuously perturbed by the surrounding atoms. The perturbations can be considered as stochastic processes described by a stochastic variable. The first-order correlation function, from which the frequency spectrum radiated by the nuclei can be derived, depends, among other things, on the ensemble average of a particular function of the stochastic variable. This function depends on the actual perturbation mechanisms. Making use of the ergodic theorem allows for the calculation of this ensemble average and, hence, of the radiated frequency spectrum. The spectrum is shown to be Lorentzian and homogeneously broadened. Also a frequency shift with respect to the unperturbed frequency has been found. An order of magnitude of the broadening and shift is given.

Models for homogeneous line broadening in long-livednuclear states

Nuclei embedded in a lattice are submitted to interactions with their surroundings. This produces a randomly fluctuating microfield at the nuclei, perturbing their energy levels. These perturbations can be considered as stochastic processes. Many of the fluctuating phenomena represent the result of a very large number of very small changes. The standard description of such fluctuations is a Gaussian process, defined as a process for which only the first and second moments are different from zero. If a radiating nucleus is constantly perturbed the radiated field can be regarded as undergoing phase fluctuations around the unperturbed nuclear Bohr frequency ω0. The instant frequency can then be written as ω(t)=ω0-x(t), where x(t) is the value of a stochastic variable X with zero mean. The first-order correlation function, whose Fourier transform gives the radiated frequency spectrum, depends on the probability function governing X. For a Gaussian, stationary, delta-correlated process, it is shown that the frequency spectrum is Lorentzian. For an arbitrary perturbation having a broad spectrum, it is shown that the frequency spectrum is again Lorentzian. For a narrow perturbation spectrum, the corresponding frequency spectrum is Gaussian.

Interactions of coherent electromagnetic radiation with nuclei

When coherent electromagnetic radiation resonantly couples two levels of a nucleus, the system can emit photons by stimulated and by spontaneous emission. The spontaneous emission spectrum is defined as the Fourier transform of the first-order correlation function of the magnetic or electric field operator. This function can be expressed in terms of one-time and two-time averages of nuclear operators. The one-time averages can be calculated by solving the Bloch-type equations for the matrix elements of the nuclear density operator. Application of the quantum regression theorem permits to calculate two-time averages with the aid of one-time averages and hence the radiated spectrum.

Completely quantized theory of homogeneous line broadening in long-lived nuclear states.

Long-lived nuclei incorporated in a lattice are submitted to fluctuating interactions exerted on them by the lattice atoms surrounding them. Two models are considered: Gaussian Markovian processes and Gaussian broadband fluctuations. The frequency spectrum of the electromagnetic radiation emitted by nuclei is the Fourier transform of the first-order correlation function, which can be expressed solely in terms of two-time and one-time averages of nuclear operators. The one-time averages, which depend on the fluctuations, can be calculated. Using the quantum regression theorem, the two-time averages are calculated as well and hence the frequency spectrum. For both models this spectrum is Lorentzian with width equal to the sum of the natural linewidth and a lattice-induced width.

Coherent properties of the stimulated emission from a three-level atom.

First-order correlation functions of the electric field and photon-number probabilities in the case of a three-level atom interacting with two-mode radiation field are obtained to investigate the coherent properties of the stimulated emission under different initial conditions. It is found that double stimulation will cause the field to approach its initial coherent state and the oscillation of the photon-number probabilities to collapse and revive.

Multidetector intensity interferometers

The intensity fluctuations of incoherent radiation are used to determine the position of sources with respect to a detector system. For electromagnetic radiation the system uses square-law detectors and for acoustic waves the detector output is squared before being processed. The theory is developed for n detectors and the results are applied to two-, three-, and four-detector arrangements. The two-detector intensity inteferometer measures the square of the first-order correlation function and can determine the angular size of a source. The three-detector intensity interferometer yields two coordinates of the location of a source, and a modified version involving only one detector functions as a Fourier transform spectrometer. It measures the first-order correlation function and thus can be used to obtain the spectrum of the source. The four-detector intensity interferometer can simultaneously determine the three-dimensional coordinates of several sources. The operation of several acoustic multidetector intensity interferometers are described and a number of applications are discussed.

A Procedure to Determine the Spatial Coherence of a Wave Field

It is shown that the first-order correlation function connected with a scalar wave field, can be determined from a set of intensity distributions in the image plane, generated by different settings of the defocusing and the moment of axial astigmatism. Estimates of the error with which the coherence function can be determined are derived.

A Model for Photodetection of Single-mode Cavity Radiation

A model for photodetection is presented which permits evolution of the detected field during the detection process. The integrated univariate and bivariate statistics of the photocount are obtained exactly for detection of single-mode thermal, coherent and free-field cavity radiation. The scheme encompasses the results of three hitherto disparate photocounting models, and demonstrates explicitly the relation between the univariate photon counting statistics and the photon statistics of the field. First-order photocount correlation functions are derived and compared with the corresponding cavity photon population correlations. Photocount antibunching is examined and found to be possible in this model.

Canonical transformation for single-atom resonance fluorescence: The strong-driving-field limit

A canonical transformation, which diagonalizes the single-mode radiation Hamiltonian in the rotating wave approximation, is exploited to dress a two-level atom by the strong monochromatic driving field in the resonance fluorescence phenomenon. Langevin equations of motion are obtained for dressed atomic operators, and their solutions yield the amplitude of waves scattered into the fluorescence field. The power spectrum of the scattered field is obtained from the first-order correlation functions between dressed atomic operators. The results of the present treatment are compared with others, and advantages and disadvantages of the present method are discussed.

Second-order correlation functions and bi-spectra in biological-rhythm research

Some properties of the second-order correlation function R(τ 1,τ 2) have been studied with the help of a model that is believed to mimic certain physiological time series containing more than one periodicity. The model consists of the superposition of two series of oscillatory wave packets, occuring randomly and more or less in synchrony, while the degree of coupling is variable. In contrast to the first-order correlation function R(τ), which is hardly able to differentiate between loose and tight coupling of the two components, R(τ 1,τ 2) is able to do so, at least within a limited range of parameter values. A similar conclusion holds if one compares power spectrum and bi-spectrum.

Correlation functions and the AC Stark effect

A Heisenberg picture QED treatment is presented of the resonance fluorescence of a two-level atom. It is shown that the equations of motion for first-order correlation functions have an asymmetry with respect to the two time arguments. This asymmetry arises in a correct and careful evaluation of the unequal-time commutator between free-field and atomic operators. General solutions of these equations are found without going to the stationary limit.

Theory of dispersion in relation to light shifts

A quantum mechanical system consisting of one photon interacting with N atoms not far from resonance is considered and an (approximately) stationary solution found for the eigenstates and energy of the interacting system. It is shown that the interaction energy (light shift) is simply related to the atomic polarizability. A consideration of the expectation value of the operator representing the first-order correlation function of an optical field yields a quantum-mechanical interpretation of refractive index.

Electron Correlation in Closed Shell Systems. I. Perturbation Theory Using Gaussian-Type Geminals

The calculation of electron correlation energy in closed-shell atoms and molecules is approached using Rayleigh-Schroedinger perturbation theory with the symmetric sum of Hartree-Fock operators for H0. The alleged advantages of using a VN−1 potential are questioned. Variational equations for first-order pair correlation functions are computed for He, Be, B+ and Ne by expansion in linear combinations of correlated Gaussian-type geminal basis functions containing r122 in the exponent. Such functions form mathematically complete sets, have convenient symmetry properties, and are integrable in closed form. An extensive search for optimum exponential parameters yielded trial functions for each of the ss orbital pairs giving better than 99% of the limiting second-order pair energy using only five basis functions per pair. Similar but less thorough studies of sp and pp pairs in neon are also reported. Careful attention is paid to computational accuracy. An upper bound of −0.3428 a.u. is established on the second-order contribution to the correlation energy in neon.

Analysis of Gaussian Light by Clipped Photocount Autocorrelation: The Effect of Finite Sampling Times and Incomplete Spatial Coherence

The theoretical treatment of clipped photocount autocorrelation of Gaussian light, introduced by Jakeman and Pike, has been extended to include the effects of finite-duration sampling intervals and incomplete spatial coherence. A system of ``interval partitioning'' is used to derive theoretical expressions for 〈Nq(0)N(τ)〉 and 〈N0(0) N0(τ)〉, the single- and double-clipped photocount autocorrelation functions. Here N(t) is the number of photons detected in a sampling interval centered at t and Nq(t) is the clipped count at level q. Each original interval is systematically partitioned in space and time. Multidimensional generating functions are applied to the sets of subintervals with the object of expressing g(1) (t), the first-order field correlation function, in terms of measurable quantities. It is shown that under certain well-defined conditions, 〈Nq(0) N(τ)〉−〈Nq〉〈N〉=γq|g(1)(τ)|2 and 〈1−N0〉−2−[1−2〈N0〉+〈N0(0)N0(τ)〉]−1=φ|g(1)(τ)|2, where, for a given set of experimental parameters, γq and φ can be taken as simple multiplicative factors independent of the delay time τ. For comparative purposes, a similar analysis is performed for the unclipped correlation function 〈N(0)N(τ)〉. Finally, computer calculated values of various quantities of interest are presented for several representative cases.