Atomic Density Matrix Research Articles

Exact solution of the Jaynes-Cummings model with cavity damping

Operating in Laplace language and making use of a representation based on photon-number states, we find the exact solution for the density operator that belongs to the Jaynes-Cummings model with cavity damping. The detuning parameter is set equal to zero and the optical resonator does not contain any thermal photons. It is shown that the master equation for the density operator can be replaced by two algebraic recursion relations for vectors of dimension 2 and 4. These vectors are built up from suitably chosen matrix elements of the density operator. By performing an iterative procedure, the exact solution for each matrix element is found in the form of an infinite series. We demonstrate that all series are convergent and discuss how they can be truncated when carrying out numerical work. With the help of techniques from function theory, it is proved that our solutions respect the following conditions on the density operator: conservation of trace, Hermiticity, convergence to the initial state for small times, and convergence to the ground state for large times. We compute some matrix elements of the density operator for the case of weak damping and find that their analytic structure becomes much simpler. Finally, it is shown that the exact atomic density matrix converges to the state of maximum von Neumann entropy if the time, the square of the initial electromagnetic energy density, and the inverse of the cavity-damping parameter tend to infinity equally fast. The initial condition for the atom can be chosen freely, whereas the field may start from either a coherent or a photon-number state.

Polarization-sensitive correlation spectroscopy of an atomic medium polarized in angular momentum: II. Application to the spin ground state

We discuss the possibilities for using the correlation spectroscopy technique to study an atomic medium polarized in angular momentum. The general results of our previous paper are applied here to the analysis of the density matrix fluctuations of an atomic medium with a spin ground state. Such systems are characterized by a long ground state relaxation time in comparison with the excited state natural lifetime, so that any fluctuation of the atomic density matrix polarization is slowly dependent on the time process. We show how the quantum behaviour of low-frequency fluctuations can be observed via quantum features of scattered light statistics. In particular, we discuss the squeezing of transmitted light caused by the spin angular momentum orientation in the direction of an external magnetic field. This squeezing effect is explained by quantum non-commutation of angular momentum projections which are transverse to the magnetic field. Other applications of the polarization-sensitive correlation spectroscopy technique, such as the possible observation of the average atomic density matrix components and the measurement of different spectroscopic characteristics of the atomic ensemble, are also discussed.

Polarization-sensitive correlation spectroscopy of an atomic medium polarized in angular momentum: I. General formalism

We discuss general polarization-sensitive correlation measurements of light fluctuations based on the Hanbury Brown and Twiss interference detection scheme. Such measurements are considered as a basis for correlation spectroscopy, which allows us to observe the correlations of the fluctuations of all polarization components (Stokes parameters) of the probe light. The interferometer scheme is discussed in application to the study of the polarization-dependent interaction of light with an atomic ensemble with polarized angular momentum and put into an external magnetic field. We describe the general procedure for evaluating the photocurrent correlation function in the case of an optically thin atomic medium. Our calculations are based on the Keldysh diagram technique which is usually used for the study of non-equilibrium systems by means of the second quantized formalism. It allows us to express the Raman field-correlation functions and parametric scattering processes describing the output light fluctuations in terms of a different type of time-ordered electric field and atomic Green functions. We show that with definite conditions the photocurrent correlation function can be written as the sum of two terms proportional to the correlation functions of atomic density matrix fluctuations in symmetric and antisymmetric forms. This expression of the photocurrent correlation function can be used in various different applications.

Optical pumping in a L-system by parametric luminescence light

We examine the optical pumping effect in an ensemble of three-level atoms with a Λ configuration of the energy sublevels excited by parametric luminescence light in the squeezed state. We derive quantum kinetic equations that describe the evolution of the density matrix of atoms irradiated by low-intensity squeezed light with a finite-width spectrum. In particular, we show that because of the quantum statistical properties of the squeezed light there can be a redistribution of atoms among the lower energy sublevels, despite the equality of the intensities of the spectral components of the light that resonantly excites optical transitions in the Λ-system. The relation of the optical pumping effect to the correlation and spatial-temporal spectral properties of squeezed light is discussed in detail. Finally, we show that the effects are closely linked to the finiteness of the width of the squeezed-light spectrum.

Derivation of cooperative-dynamic equations for an ensemble of three-level atoms in a coherent electromagnetic field

Equations for the density matrix of three-level atoms interacting collectively with a two-mode radiation field are derived for level schemes of three types within the framework of a formalism with secondary quantization of the electromagnetic field.

Interference of radiating states and ion dynamics in spectral line broadening

The influence of the plasma coupling between the populations and the coherences on the lineshape is investigated. Ion dynamics is taken into account. The present research is performed within the atomic density matrix formalism. Ion microfield dynamics is simulated by the kangaroo - Poisson stochastic process (model microfield method). Numerical calculations of both lifetimes of radiating states and lineshapes are performed for the spectral doublet (1s - 2s) - (1s - 4p), (1s - 2) - (1s - 4d) of helium-like multicharged ions in hot dense plasmas. It is found that the ion microfield essentially influences the difference of populations of radiating states. Calculation of the lineshape of the doublet (1s - 2p) - (1s - 4d), (1s - 2p) - (1s - 4f) of neutral helium at astrophysical plasma conditions is also performed. The contribution of nonlinear interference effects (NIEF) in both allowed and forbidden components is calculated at various plasma conditions and a comparison with binary adiabatic theory is made. The results demonstrate that it is essential to take account of NIEF in the calculation of lineshapes of multicharged ions, but not essential in the case of neutral helium.

Semiclassical approach to resonance-optics polarization effects in coherent and squeezed fields

Equations are derived for the atomic density matrix and relaxation operator for a broadband squeezed field in an arbitrary polarization state and resonance atomic energy levels with an arbitrary degree of degeneracy. It is shown that suppression of the relaxation of the quadrature component of the atomic polarization depends strongly on the type of resonance transition and the polarization state of the squeezed and coherent perturbing fields. When the resonance levels are strongly degenerate, the relaxation of the quadrature component of the atomic polarization under conditions of maximum suppression is nonexponential in character. The mathematical apparatus developed here makes it possible to calculate polarization-related aspects of the multifrequency optical behavior of atomic and molecular systems resonantly excited both by coherent light and by broadband squeezed fields.

Lineshapes in laser spectroscopy of three-level systems

The lineshapes originated in a magnetic resonance process for atomic alignment are compared to the lineshapes associated to the resonance of coherent population trapping in a three-level system. An approach based on the tensorial components of the atomic density matrix is applied to analyze both resonances and to evidence the strict relations between the lineshapes.

Atom in an elliptically polarized resonance field: Exact stationary solution

An analytical method is developed for finding the exact stationary solution for the density matrix of atoms in a monochromatic resonance field of arbitrary ellipticity and intensity. A solution for the transitions Fg=F → Fe=F+1 (Fg and Fe are the angular momenta of the ground and excited states) is obtained in an analytical invariant form. The properties of this solution are investigated.

Loss of spatial coherence in atomic deflection from an off-resonant quantized standing-wave field.

We show that the transverse spatial coherence of the atomic density matrix is destroyed to some extent by the deflection of the atom from an off-resonant quantized standing-wave field. The collapse and revival of the atomic spatial coherence are demonstrated. \textcopyright{} 1996 The American Physical Society.

Selective reflection from a dense atomic vapor.

A theory of selective reflection (SR) of a light beam from the interface of a dielectric medium and a dense atomic vapor is presented. Following another earlier work on this subject [J. Guo et al., Opt. Commun. 110, 732 (1994)], we calculate the atomic density matrix and the transmitted field in the vapor self-consistently. We extend our previous theory to include an attractive atom-dielectric wall interaction and multiple atomic transitions. We examine the limits at which the conventional SR theories, which are based on the assumption of a plane-wave field in the vapor, start to fail. It is shown that at or above vapor densities of order ${10}^{17}$/${\mathrm{cm}}^{3}$, the frequency shifts of the SR line shape due to the local-field correction (Lorentz-Lorenz shift), the atom-wall interaction, and the nonexponential attenuation of the field in the vapor can have comparable orders of magnitude, and the proper determination of all these effects is essential for calculating the correct SR signals. A comparison of the theoretical SR spectra with an experimental result for a Rb vapor with a density of order ${10}^{17}$/${\mathrm{cm}}^{3}$ is also presented. \textcopyright{} 1996 The American Physical Society.

Generation of transient Rabi sidebands in pulse propagation: A possible source for cone emission.

We study the one-dimensional propagation of a laser pulse through an atomic vapor of finite length L. Both the temporal and the time-dependent spectral profiles of the exiting laser pulse are calculated. It is shown that, when the pulse duration is comparable with or shorter than the effective relaxation time of the atomic density matrix, the spectrum of the final pulse exhibits frequency sidebands located in the vicinity of the time-varying generalized Rabi frequency, which can contain up to a few percent of the total laser power. The possible connection between this result and cone emission in the pulsed regime is addressed.

Fluorescence intensity and squeezing in a driven three-level atom: Ladder case.

A theoretical study is made of the steady-state intensity and squeezing properties of the fluorescent light from a three-level-atom system in a ladder configuration, which is subject to spontaneous-emission decay to the electromagnetic-field vacuum. Two cases are examined: in the equispaced (ES) level case, the two atomic transition frequencies are nearly equal and two photon absorption processes occur from a single coherent laser field that couples both transitions and has a small detuning from each; in the non- equispaced (NES) level case, the atomic transition frequencies are rather different and two photon absorption processes occur from two coherent laser fields, each coupled to a single transition and in near resonance with it. In both cases, a situation of small two-photon detuning occurs. Optical Bloch equations for the atomic density matrix in rotating frames are given and matrix expressions for determining the steady-state populations and coherences are obtained. Analytic expressions are also given for special cases. The fluorescent intensity is obtained from the populations of the intermediate and upper states, with the normally ordered variance (NOV) for quadrature components of the fluorescent field involving, in addition, the atomic coherences. The fluorescent intensities are shown graphically as a function of two-photon detuning for a variety of one-photon detunings and Rabi frequencies for the ES and NES cases. In both cases, the fluorescent intensities show the well known resonances from the upper state at zero two-photon detuning and from the intermediate state at zero one-photon detuning for the lower transition. However, further resonances are also found. The intermediate-state intensity displays a resonance at zero two-photon detuning. Also, the upper-state intensity shows a resonance at zero one-photon detuning for the upper transition, but only for the NES case; evidently, the transfer rate from upper to lower transition coherences destroys this resonance for the ES case. The time-averaged NOV is also shown graphically as a function of two-photon detuning for a variety of one-photon detunings and Rabi frequencies for the ES and NES cases. For the ES case, the quadrature frequency is chosen as the single laser frequency and for the NES case, it is chosen as either the average of the two laser frequencies or the lower transition laser frequency. Squeezing occurs near zero two-photon detuning for both ES and NES cases, though only for the quadrature frequency equal to the average laser frequency in the latter case. It also occurs near zero detuning for the lower transition for both cases, though only for the quadrature frequency equal to the lower transition laser frequency in the NES case. The squeezing minima show a splitting effect for large Rabi frequencies corresponding to two-photon or one-photon Rabi splitting of the dressed atom levels for the situation near two-photon or one-photon resonance, respectively. The large squeezing near two-photon resonance for moderate Rabi frequencies and large one-photon detuning corresponds to essentially pure three-level squeezing, while that near one-photon resonance for weak Rabi frequencies involves two-level squeezing.

Effect of a Squeezed Vacuum on Coherent Population Trapping in a Three-level Lambda System

Abstract Master equation methods are used to investigate the effects of a broad-band squeezed vacuum on a three-level atom of the lambda configuration. The two-mode squeezed vacuum is treated as a Markovian reservoir in a non-stationary phase-dependent state. In addition to the squeezed vacuum the atom is driven by two coherent laser fields each of which, depending on the polarization, can couple to one or both of the atomic transitions. We show that in general the optical Bloch equations for the atomic density matrix elements have oscillatory coefficients, thereby necessitating the use of Floquet methods. For the case of equal laser frequencies, which are also equal to the carrier frequency of the squeezed vacuum, the coefficients of the Bloch equations become time independent and stationary solutions for the populations and coherences are obtained by standard matrix methods. For the ordinary vacuum the usual coherent population trapping effect at two-photon resonance is obtained, with the upper state po...

A theoretical consideration of the effects upon a resonance fluorescence spectrum caused by interference between coupled hyperfine excited states

In the context of laser spectroscopy we consider the effects of interference between the spontaneous emission to a common lower level from neighbouring, closely spaced, excited-state levels. In particular the effects on the fluorescence spectrum and on the measured positions of the line centres are described. A formalism, based upon the atomic density matrix, is developed quantitatively to describe these effects within complex hyperfine systems such as the sodium D-lines. Where possible, we have tried to simplify the resultant equations, defining the magnitude of the various fluorescent components and establish a general result for the phase of each component. Finally, the formalism is applied to the particular case of the sodium D1-line, where very different results are obtained depending on whether optical pumping into unobserved states is included, or omitted, from the calculation.

Driven two-level atom.

The two-level atom in an external field of arbitrary time dependence is solved formally using projection techniques. As a particular result of the general solution, an exact expression is presented for the density matrix of a two-level atom in a sinusoidal field.

Relaxation, generation, and detection of Zeeman coherences in an optically pumped heavy-alkali-metal vapor.

The relaxation and generation of multipole orientations in a heavy-alkali-metal vapor in the presence of a weak static magnetic field are theoretically studied both for populations and for Zeeman coherences by taking into account the effects on the evolution of the atomic density matrix of the relaxation processes (by alkali-atom--buffer-gas, alkali-atom--alkali-atom with spin exchange and alkali-atom--cell-wall collisions) and of the optical pumping with any light polarization and spectral distribution. The dependence of the stationary-state multipole orientations on the pumping characteristic parameters is numerically analyzed. The detection process of these orientations is also studied.

Optimum field squeezing from atomic sources: Three-level atoms.

Light emitted from laser-driven atoms often has squeezed quadrature fluctuations reflecting the phase dependence of the atomic source excited into a coherent superposition by the driving field. In this paper, we study the maximum squeezing in the emitted light fields from three-level atoms, paying particular attention to the role of atomic coherences in governing the optimum squeezing which is possible. Some phase dependent nonlinear optical processes have the potential for substantial quantum noise reduction at useful intensities: for example, four-wave mixing involving atomic systems has already been used to generate quadrature-squeezed light. The threoretical description of such a source is generally complicated with damping processes, coherent and incoherent pumping, and so on, all needing to be taken into account. Fully quantum treatments are available, but the basic physics of the process is often difficult to grasp in the necessarily complicated analysis. Although less important from the point of view of efficient practical sources of squeezed light, studying the generation of squeezed light in simpler systems such as two- or three-level atoms interacting with one or two modes (as in Jaynes-Cummings models) or in spontaneous emission or resonance fluorescence gives greater insight into the fundamentals of the squeezing process. In particular, the relationship between the atomic coherences and the degree of squeezing in the generated fields can be explored. We study the maximum (optimized) squeezing that can be obtained in the emitted fields from the irreversible decay from three-level atoms for any choice of initial conditions, excitations and decay process, for all quadrature components and for all possible configurations (V, , and ladder systems). We show that for V and systems, optimum squeezing is actually associated with a two-level state (in a suitable basis) involving a single one-photon coherence, but for ladder systems the state for maximum noise reduction is not equivalent to a two-level system and involves a single two-photon coherence and no intermediate state population. We pay particular attention to whether the total field is in a minimum uncertainty state, and the nature of the atomic state associated with maximum squeezing, especially whether it is a mixed or pure state. In the case where the initial free field has a zero amplitude at the detector and the source atoms are confined to a region small compared to the wavelength (Dicke source) we show that the squeezing in the total field is given in terms of the squeezing in the source field, and hence related to atomic populations and atomic one-photon and two-photon coherences for the case of three level atom. Previously unconsidered source-free field interferences terms are shown to be zero. The choice of quadrature phase and frequency is optimized to minimize the source field normally ordered variance. We then minimize further with respect to the choice of atomic density matrix elements subject to the constraints that the density matrix is Hermitean, positive, and has a trace equal to unity. In all cases we find that the optimum squeezing is produced when the source field is in a minimum uncertainty state.

Generation of pseudopotentials from correlated wave functions

The density matrix, or equivalently the natural orbitals play an essential role in determining the transferability of pseudopotentials to all orders of perturbation theory. In this work the one-particle density matrix and natural orbitals of Li, C, and Ne atoms are obtained using variational and diffusion Monte Carlo. Using these a pseudopotential is computed for the lithium atom.

ELECTROMAGNETICALLY-INDUCED TRANSPARENCY: STRONG SIGNAL AND ANTI-SYMMETRICAL DETUNING CASE

The equations of atomic density matrix are solved without any approximation when the strong:ignal field and external coherence-driving field are detuned to respective dipole-allowed transition of the Λ atom in such a way that Δ1=-Δ2. Under the equal Rabi frequency (G) condition, it is found that approximate, coherent population trapping appears for small detuning, which leads to the transparency of the medium to the strong signal having an order of Ｏ（Ｇ-3）. If the detuning is G/2, the atom nearly equally populates three levels and the transparency is of only order Ｏ（Ｇ-1）.