Photon Distribution Function Research Articles

Photon distribution in the dynamical Casimir effect with an account of dissipation

A theory of quantum damped oscillator with arbitrary time dependence of the frequency and damping coefficient, based on the Heisenberg-Langevin equations with delta-correlated stochastic force operators, is applied to the case of the dynamical Casimir effect in a cavity with a periodically photo-excited semiconductor boundary. Accompanying results for the mean number of created photons, its variance, and the photon distribution function are given. In the asymptotical regime, the field mode goes to the so-called superchaotic quantum state.

Study on the solutions of the Sunyaev-Zeldovich effect for clusters of galaxies

Based upon the rate equations for the photon distribution function obtained in the previous paper, we study the formal solutions in three different representation forms for the Sunyaev-Zeldovich effect. By expanding the formal solution in the operator representation in powers of both the derivative operator and electron velocity, we derive a formal solution that is equivalent to the Fokker-Planck expansion approximation. We extend the present formalism to the kinematical Sunyaev-Zeldovich effect. The properties of the frequency redistribution functions are studied. We find that the kinematical Sunyaev-Zeldovich effect is described by the redistribution function related to the electron pressure. We also solve the rate equations numerically. We obtain the exact numerical solutions, which include the full-order terms in powers of the optical depth.

Dynamical Casimir effect: Some theoretical aspects

This is a brief description of some recent achievements in the theory of dynamical Casimir effect, mainly in connection with the experiment which is under preparation in the University of Padua. The first part of this paper is devoted to the theory of quantum damped oscillator with arbitrary time dependence of the frequency and damping coefficient. New results for the mean number of created photons, its variance and photon distribution function are given. The second part is devoted to calculations of the time-dependent shift of resonance frequency of an electromagnetic cavity due to strong variations of dielectric properties in a thin layer near an ideally conducting wall. A simple analytical formula for this shift is derived. It generalizes the known Schwinger-Bethe-Casimir result. The influence of different parameters on the photon generation rate is discussed. A brief review of recent publications on the subject is also included.

Manifestation of single macromolecule quantum dynamics in photon distribution function of blinking fluorescence

Distribution function w(N)(T) for photons created by three-level nanoparticle in time interval T under cw laser excitation is calculated for various methods of photon counting. It is found that each exponential process exp(-lambda(i)t) in quantum dynamics of three-level nanoparticle manifests itself via Poissonian function P(N)(lambda(i)t)=(lambda(i)t)(N) exp(-lambda(i)t)/N! in the photon distribution function w(N)(T). The distribution function w(N)(T) is expressed via two or three integrals of two or three Poissonian functions P(N)(lambda(i)t). The simple mathematical expression for w(N)(T) enables one to calculate photon distribution in blinking fluorescence with on and off intervals. A scaling between photon distribution function w(N)(T) and photoelectric pulse distribution function w(n)(T) is found. Comparison of the theoretical distribution w(n)(T) and the distribution measured in blinking fluorescence of single polymer molecule dPPV-PPyV and complex organic molecule 1,1(')-didodecyl-3,3,3('),3(')-tetramethylindocarbocyanine perchlorate (DiI) is carried out. The theoretical distributions are able to describe those found in an experiment.

Identification of photon sources, stochastically embedded in an interstellar cloud

Photon transport is considered in an interstellar cloud containingone or several photon sources (stars), defined by$q_i\delta( x- x_{\i})\,i=1,2,\ldots,$ where thelocations $x_i$'s are given in a stochastic way. First, the caseis examined of a single source of intensity $q_1$ and located at$x_1$ with a probability density $p_1 = \p(x_1)$, such that$\p(x_1)\geq 0$ and $\int_V \p(x_1)\dx_1 = 1$, where $V\subset \R^3$ is the region occupied by the cloud. Then, aBoltzmann-like equation for the average photon distribution function$(x,u;x_1)$ is derived and it is shown that$\p(x_1)$ can be evaluated starting from a far-field measurementof . Finally, the case of two or more photon sources isdiscussed: the corresponding results are reasonably simple if$\p(x_1,x_2) = \p_1(x_1)\p_2(x_2)$, i.e. if thelocations of the two photon source are 'independent'.

Photon distribution function in blinking fluorescence of individual molecules

A photon distribution function wN(T) for blinking fluorescence with bright on- and dark off-intervals is derived. The function wN(T) is expressed via few Poissonian functions each of which relates to corresponding exponential process in quantum dynamics of a given individual molecule. The distribution of photons is calculated for short, middle and long time intervals as compared to off-intervals. The distributions are much broader than Poissonian distribution and have rather complicated shape. If time resolution of an experiment does not permit us to see off-interval and, therefore, fluorescence looks like CW emission, the distribution of photons gives a signal about existence of hidden off- intervals in such CW fluorescence.

Analytical Solutions for Expanding Fireballs

Many models of gamma-ray bursts (GRBs) as well as of soft gamma repeaters (SGRs) involve a fireball — an optically thick concentration of radiation energy with a high ratio of energy density to rest mass. We study the asymptotic behavior of an ultrarelativistic fireball consisting of electron-positron pairs and photons. We show that in the ultrarelativistic limit, after photons decouple from the pairs, the photon distribution function remains a blackbody spectrum in some appropriate Lorentz frame, allowing us to define an effective Lorentz factor and temperature for the photon gas. We also study the freezing out of electron-positron pairs and their asymptotic Lorentz factor γ∞. The dependence of these quantities on initial conditions can be described by simple scaling laws. We apply our results to SGR 1806-20 and find that the energy carried by electron-positron pairs is higher than calculated by former estimates, but is still an order of magnitude short of the minimum energy required to produce the observed afterglow. A viable solution of the energy budget is that the fireball is loaded by baryons or electromagnetic flux.

Relativistic Photon Mediated Shocks

A system of equations governing the structure of a steady, relativistic radiation-dominated shock is derived, starting from the general form of the transfer equation obeyed by the photon distribution function. Closure is obtained by truncating the system of moment equations at some order. The anisotropy of the photon distribution function inside the shock is shown to increase with increasing shock velocity, approaching nearly perfect beaming at upstream Lorentz factors Gamma(-) >>1. Solutions of the shock equations are presented for some range of upstream conditions. These solutions are shown to converge as the truncation order is increased.

Uncertainty relation and photon-number distribution for one-and two-mode squeezed light

We study the one-and two-mode photon distribution functions for Gaussian quantum states. The dependence on the photon-quadrature parameters is discussed within the framework of violation of the uncertainty relation. We derive the one-and two-mode photon distribution functions and present plots demonstrating that the expressions for the distribution functions take negative or complex values in the domain of quadrature parameters violating the uncertainty relation.

Beam wandering in the atmosphere: The effect of partial coherence

The effect of a random phase screen on laser beam wander in a turbulent atmosphere is studied theoretically. The photon distribution function method is used to describe the photon kinetics of both weak and strong turbulence. By bringing together analytical and numerical calculations, we have obtained the variance of beam centroid deflections caused by scattering on turbulent eddies. It is shown that an artificial distortion of the initial coherence of the radiation can be used to decrease the wandering effect. The physical mechanism responsible for this reduction and the applicability of our approach are discussed.

Einstein and Jordan frames reconciled: A frame-invariant approach to scalar-tensor cosmology

Scalar-tensor theories of gravity can be formulated in different frames, most notably, the Einstein and the Jordan one. While some debate still persists in the literature on the physical status of the different frames, a frame transformation in scalar-tensor theories amounts to a local redefinition of the metric, and then should not affect physical results. We analyze the issue in a cosmological context. In particular, we define all the relevant observables (redshift, distances, cross sections, \dots{}) in terms of frame-independent quantities. Then, we give a frame-independent formulation of the Boltzmann equation, and outline its use in relevant examples such as particle freeze-out and the evolution of the cosmic microwave background photon distribution function. Finally, we derive the gravitational equations for the frame-independent quantities at first order in perturbation theory. From a practical point of view, the present approach allows the simultaneous implementation of the good aspects of the two frames in a clear and straightforward way.

Photon distribution functions of the fluorescence photons from a single nanoparticle in various photon counting methods

Simple expressions have been derived for three photon distribution functions wNM(T), wNZ(T), and wNO(T) corresponding to three different methods for counting fluorescence photons from a single nanoparticle excited by continuous laser radiation. In contrast to the previously derived expressions represented in the form of N multiple integrals, the new expressions contain only single or double integrals of Poisson functions, which makes it possible to easily perform the numerical calculation of the photon distribution. The simplest photon counting method corresponds to the lengthiest function wNM(T); on the contrary, the simplest function wNO(T) corresponds to the most complex photon counting method. The functions wNM(T), wNZ(T), and wNO(T) are noticeably different in short time intervals T; however, the distributions calculated using these functions are almost indistinguishable from each other in long T intervals. This circumstance makes it possible to use the simplest function wNO(T) to consider the photon statistics measured by the simplest method. This possibility is particularly important for investigating the fluorescence photon statistics, where the intensity fluctuates.

Entropy and efficiency in laser cooling of solids

The thermodynamics of laser cooling of solids is analyzed. Using the general theory of radiation entropy, the important roles of the optical frequency and the photon distribution function in determining the radiation entropy are identified. The usefulness of a narrowband approximation is established for a wide range of radiant sources. This approximation is then applied to compare the entropies of different light sources, including blackbody radiation, lasers, fluorescence, and the emerging class of random lasers. Based on these results, the Carnot efficiency for laser cooling of solids is determined, for emission fields with various entropy characteristics. It is shown that fluorescent emission is the most efficient form of the radiated field for laser cooling of solids, and cooling schemes based on any stimulated emission process (including random laser action) are inherently less efficient. The influence of luminescence quantum yield on cooling is also considered.Copyright © 2007 by ASME

Analytical cumulant solution of the vector radiative transfer equation investigates backscattering of circularly polarized light from turbid media

The backscattering of circularly polarized light pulses from an infinite uniform scattering medium is studied as a function of helicity of the incident light and size of scatterers in the medium. The approach considers a polarized short pulse of light incident on the scattering medium, and uses an analytical cumulant solution of the vector radiative transfer equation with the phase matrix obtained from the Mie theory to calculate the temporal profile of scattered polarized photons for any position and any angle of detection. The general expression for the scattered photon distribution function is an expansion in spatial cumulants up to an arbitrary high order. Truncating the expansion at the second-order cumulant, a Gaussian analytical approximate expression for the temporal profile of scattered polarized photons is obtained, whose average center position and half width are always exact. The components of scattered light copolarized and cross polarized with that of the incident light can be calculated and used for determining the degree of polarization of the scattered light. The results show that circularly polarized light of the same helicity dominates the backscattered signal when scatterer size is larger than the wavelength of light. For the scatterers smaller than the wavelength, the light of opposite helicity makes the dominant contribution to the backscattered signal. The theoretical estimates are in good agreement with our experimental results.

Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere

The photon density operator function is used to calculate light beam propagation through turbulent atmosphere. A kinetic equation for the photon distribution function is derived and solved using the method of characteristics. Optical wave correlations are described in terms of photon trajectories that depend on fluctuations of the refractive index. It is shown that both linear and quadratic disturbances produce sizable effects for long-distance propagation. The quadratic terms are shown to suppress the correlation of waves with different wave vectors. We examine the intensity fluctuations of partially coherent beams (beams whose initial spatial coherence is partially destroyed). Our calculations show that it is possible to significantly reduce the intensity fluctuations by using a partially coherent beam. The physical mechanism responsible for this pronounced reduction is similar to that of the Hanbury-Braun--Twiss effect.

The stochastic radiative transfer equation: Quantum damping, Kirchoffs law and NLTE

A method based on the theory of quantum damping is presented, for deriving a self consistent but approximate form of the quantum transport for photons interacting with a fully ionized electron plasma. Specifically, we propose in this paper a technique of approximately including the effects of background plasma on a photon distribution function without directly solving any kinetic equations for the plasma itself. The result is a quantum Langevin equation for the photon number operator; the quantum radiative transfer equation. A dissipation term appears which is the imaginary part of the dielectric function for an electron gas with photon mediated electron–electron interactions due to absorption and re-emission. It depends only on the initial state of the plasma. A quantum noise operator also appears as a result of spontaneous emission of photons from the electron plasma. The thermal expectation value of this noise operator yields the emissivity which is exactly of the form of the Kirchoff–Planck relation. This non-zero thermal expectation value is a direct consequence of a fluctuation–dissipation relation (FDR).

Radiation transport equations in non-Riemannian space-times

The transport equations for polarized radiation transfer in non-Riemannian, Weyl-Cartan type space-times are derived, with the effects of both torsion and non-metricity included. To obtain the basic propagation equations we use the tangent bundle approach. The equations describing the time evolution of the Stokes parameters, of the photon distribution function and of the total polarization degree can be formulated as a system of coupled first order partial differential equations. As an application of our results we consider the propagation of the cosmological gamma ray bursts in spatially homogeneous and isotropic spaces with torsion and non-metricity. For this case the exact general solution of the equation for the polarization degree is obtained, with the effects of the torsion and non-metricity included. The presence of a non-Riemannian geometrical background in which the electromagnetic fields couple to torsion and/or non-metricity affect the polarization of photon beams. Consequently, we suggest that the observed polarization of prompt cosmological gamma ray bursts and of their optical afterglows may have a propagation effect component, due to a torsion/non-metricity induced birefringence of the vacuum. A cosmological redshift and frequency dependence of the polarization degree of gamma ray bursts also follows from the model, thus providing a clear observational signature of the torsional/non-metric effects. On the other hand, observations of the polarization of the gamma ray bursts can impose strong constraints on the torsion and non-metricity and discriminate between different theoretical models.

Statistical and Physical Analysis of the External Factors Perturbation on Solar Radiation Exergy

The purpose of this paper is to analyze the external factor perturbations on solar radiation. Firstly, the influence on the photon distribution function in the space of frequencies is analyzed and, later, the modification generated in equations of internal energy, entropy, and radiation is viewed in order to deduce an expression to calculate the spectral exergy of perturbed radiation.

Photon distribution function, tomograms and entanglement in stimulated Raman scattering

Several aspects of the stimulated Raman scattering process are studied using the model of a two-dimensional oscillator. The photons of the Stokes mode are taken as one mode of the oscillator and the phonons of the medium are taken as the other mode of the oscillator. The interaction Hamiltonian of the photons and phonons is taken to be quadratic in the annihilation and creation operators of photons and phonons. The laser is considered as a classical light source and its depletion is neglected. The photon?phonon joint probability distribution function is found and expressed in terms of Hermite polynomials of four variables. The general formula for the photon number probability distribution as a function of the laser-field intensity and the medium parameters is obtained in terms of Hermite polynomials of two variables. It is shown that the phenomenon of entanglement appears in photon?phonon modes due to the interaction of the light with the medium. The tomograms of the photon?phonon states and (averaged over the phonon mode) photon states are investigated within the framework of a symplectic tomography scheme.

The quantum radiative transfer equation: quantum damping, Kirchoff's Law, and the approach to equilibrium of photons in a quantum plasma

A method is presented based on the theory of quantum damping, for deriving a self-consistent but approximate form of the quantum transport for photons interacting with a fully ionized electron plasma. Specifically, we propose in this paper a technique for approximately including the effects of a background plasma on a photon distribution function by replacing the influence of the plasma degrees of freedom with quantum fluctuation and damping terms in the radiation transport equation. We consider the Markov limit where the electron relaxation time scale is short compared to the photon relaxation time scale. The result is a quantum Langevin equation for the photon number operator; the quantum radiative transfer equation. A dissipation term appears which is the imaginary part of the dielectric function for an electron gas undergoing electron scattering due to emission and absorption of photons. It depends only on the initial state of the plasma. A quantum noise operator also appears as a result of spontaneous emission of photons from the electron plasma. The thermal expectation value of this noise operator yields the emissivity which is exactly of the form of the Kirchoff–Planck relation. This non-zero thermal expectation value is a direct consequence of a fluctuation–dissipation relation. The fluctuations of the quantum noise operator yield the deviations from the Kirchoff–Planck relation. Using the quantum radiative transfer equation, transient fluctuations in the photon number are computed.