Atomic Density Matrix Research Articles

Two-particle density matrix calculation for atom by modified perturbation theory

The two-particle density matrix of hydrogen atoms moving in a solid film is calculated. The probability of surviving a bound state between electron and proton in the course of the atomic motion in the film is discussed. The surviving fraction of the initial beam possesses the increased energy of the Coulomb attraction and lowered energy losses. This fraction can be responsible for the observed number of Rydberg states in the exit beam.

Using ion imaging to analyze higher-order moments of the angular distributions of photofragments

We investigate the effect of the hexadecapole (K=4) polarization moment on the spatial distributions of angular momenta for atoms produced during the photodissociation of diatomic or triatomic molecules by polarized radiation. We derive general expressions for the angular distributions of the atomic density matrix for K = 2, 4 and expressions for the corresponding anisotropy parameters that contain all information on the photodissociation dynamics. We show that these anisotropy parameters can be experimentally determined by using ion imaging. We consider oxygen atoms in the 1 D 2 state aligned with respect to the orbital angular momentum as an example and provide ion images of the signals that correspond to the population of the atomic magnetic sublevels ¦m¦ = 0, 1, 2. We show the contributions from the second-and fourth-rank state multipoles to the angular distributions of the atomic density matrix to be comparable in magnitude and significantly different in form.

The Hanle effect observed in solar prominences: interpretation of the 1974-1982 Pic-du-Midi observations, and new perspectives

This paper is devoted to review the development and the results of the program solar prominences that has been aimed to observe the Hanle effect at the Pic-du-Midi during the ascending phase of Cycle XXI (1974-1982). This aim had been defined and the observations have been performed by Jean-Louis Leroy. The Hanle effect is the effect of a weak magnetic field on the scattering linear polarization: its main features are, for some field orientations, a depolarization and eventually a rotation of the polarization direction. The magnetic field diagnostic from polarization measurements requires a modelling of the polarized line formation, that has been achieved in Meudon in the well-adapted formalism of the atomic density matrix. It is shown how the program has been developed to determine the 3 components of the field vector and the electron density, by setting multi-line polarimetric observations. Particular attention has been devoted on the solution of the 180 degrees ambiguity, which has been solved by 3 independent methods. By using this solution, one unique average magnetic field vector has been determined in each of 296 quiescent prominences, leading to results on the field strength, direction, vertical gradient, cyclic variations. The future perspective opened by the low scattered light level of THEMIS and other spectropolarimeters is to increase the spatial resolution of the measurements.

Limit of maximum entropy for the damped Jaynes Cummings model

Employing projection methods from kinetic theory, we study the non-Markovian quantum evolution of a two-level atom that is coupled to a single mode of the electromagnetic field. The interaction between the atom and field mode is described by the damped Jaynes–Cummings model. The general case is considered, in which dissipation is generated by both a photonic and an atomic reservoir of finite temperature. Only one special choice is made. The frequencies of the atom and field mode are in the same ratio as the temperatures of the atomic and the photonic reservoirs. Making use of Laplace transformation, we show that the atomic density matrix evolves to the state of maximum von Neumann entropy if the time, the cube of the initial electromagnetic energy density, the inverse of the photonic damping parameter and the inverse of the atomic damping parameter tend to infinity equally fast. We propose a large class of states from which the full density operator for the atom and field may start. This class includes entangled states. Expansion of the time-dependent exponential of the Laplace backtransform enables us to derive the limit of maximum entropy directly, without explicit evaluation of the atomic density matrix. We interchange limits and sums without proof, so our derivation is not entirely rigorous. Next, we remove the photonic reservoir. Then the damping process gets a sequential character. The field mode is assumed to start from a photon-number state. For the special choice of zero temperature and detuning we verify that the limit of maximum entropy survives the qualitative modification of the model. The only consequence is a slightly different scaling between the parameters that become large. Finally, we argue that our case study could be of value in finding out to what extent quantum dissipative processes obey the general principles of thermodynamics.

Theoretical study of the collisional depolarization and of the Hanle effect in the Na I D$\mathsf{_{2}}$ line observed on the solar limb

In recent years, Landi Degl' Innocenti (1998, 1999) has proposed a model of the polarization spectrum of the Nai D lines observed near the solar limb, based on lower level polarization effects. By so doing, he obtains a remarkable agreement between his model and the observations, so that he comes to a conclusion about a paradox, because the existence of lower level polarization is incompatible with a magnetic field strength higher than 10 mGauss (except vertical), and with possible depolarizing collision effects. In the present paper, we investigate the depolarizing collision effects (collisions with neutral hydrogen) by using the collisional rates computed with ab-initio and quantum chemistry methods by Kerkeni (2001; see also Kerkeni et al. 2000, and Kerkeni 2002). We solve the statistical equilibrium equations for the atomic density matrix, taking into account these depolarizing collisions. We investigate the effect of a weak magnetic field (Hanle effect). Our results indicate that the lower levels should be completely depolarized by the collisions at the depth where the NaI D lines are formed. Furthermore, large values of the lower level alignment such as those introduced by Landi Degl'Innocenti in his model to get a good theoretical fit of the observations, seem to us unlikely, as our computations confirm. Thus, as the agreement between the model by Landi Degl'Innocenti and the observations is however very convincing, the paradox is confirmed, reinforced and increased by our results.

Unification of the Jaynes–Cummings model and Planck’s radiation law

By combining iterative methods with Laplace transformation, we construct the solution of a dissipative Jaynes–Cummings model. The dissipative part of the model is based on the standard Markovian master equation for a harmonic oscillator that is coupled to a heat bath of nonzero temperature. Besides photon loss, we also take into account frequency detuning between atom and field. Before commencing the iteration, we subject the matrix elements of the density operator to a transformation that depends on temperature. As a result, the pole structure of all Laplace transformed matrix elements is improved. It becomes manifest which poles do not contribute to the asymptotic behavior of the density operator. In proving that our iterative process yields convergent results, we assume upper bounds on: the matrix elements of the density operator, the matrix elements of the initial density operator, the damping parameter of the heat bath, and the temperature of the heat bath. All of these bounds are physically acceptable. The photon field may start from a coherent state or a number state. For experiments in a microwave cavity, temperatures of the order of 0.1 [K] are allowed. As an application, the evolution of the atomic density matrix is studied. We propose a limit for which this matrix converges to the state of maximum von Neumann entropy. The time, the cubed initial energy density, and the inverse of the damping parameter must tend to infinity equally fast. The photon field is assumed to be in a number state at time zero, whereas the initial state of the atom can be chosen freely.

Spectral detection of resonance fluorescence as a generalized quantum measurement

Spectrally resolved detection of single atom resonance fluorescence in the limit of well-separated spectral lines is considered. By using a special type of correlation measurements over the fluorescent field, in which a filtered photon detection is followed by an unfiltered photon detection, we obtain a conditioned atomic state following the filtered photon detection. The properties of the atomic state following detection of the reflected photon are studied and interpreted on the basis of quantum interference between the dressed states. Measurement operators associated with the detection of the passed and reflected photons are derived and used to construct the master equation for the atomic density matrix subjected to continuous spectral detection, the filter tuning being arbitrary.

Asymptotic behavior of the one-particle density matrix of atoms at distances $Z^{-1}$ from the nucleus

We prove that the suitably rescaled density matrix of ground states of atomic Schrodinger operators with nuclear chargeZ converges on the scale 1/Z to the projection of the negative spectral subspace of the Schrodinger operator of the hydrogen atom (Z=1).

Theory of an optical dipole trap for cold atoms

The theory of an atom dipole trap composed of a focused, far red-detuned, trapping laser beam, and a pair of red-detuned, counterpropagating, cooling beams is developed for the simplest realistic multilevel dipole interaction scheme based on a model of a (3+5)-level atom. The description of atomic motion in the trap is based on the quantum kinetic equations for the atomic density matrix and the reduced quasiclassical kinetic equation for atomic distribution function. It is shown that when the detuning of the trapping field is much larger than the detuning of the cooling field, and with low saturation, the one-photon absorption (emission) processes responsible for the trapping potential can be well separated from the two-photon processes responsible for sub-Doppler cooling atoms in the trap. Two conditions are derived that are necessary and sufficient for stable atomic trapping. The conditions show that stable atomic trapping in the optical dipole trap can be achieved when the trapping field has no effect on the two-photon cooling process and when the cooling field does not change the structure of the trapping potential but changes only the numerical value of the trapping potential well. It is concluded that the separation of the trapping and cooling processes in a pure optical dipole trap allows one to cool trapped atoms down to a minimum temperature close to the recoil temperature, keeping simultaneously a deep potential well.

Iterative Methods for the Non‐LTE Transfer of Polarized Radiation: Resonance Line Polarization in One‐dimensional Atmospheres

This paper shows how to generalize to non-LTE polarization transfer some operator splitting methods that were originally developed for solving unpolarized transfer problems. These are the Jacobi-based accelerated Λ-iteration (ALI) method of Olson, Auer, & Buchler and the iterative schemes based on Gauss-Seidel and successive overrelaxation (SOR) iteration of Trujillo Bueno and Fabiani Bendicho. The theoretical framework chosen for the formulation of polarization transfer problems is the quantum electrodynamics (QED) theory of Landi Degl'Innocenti, which specifies the excitation state of the atoms in terms of the irreducible tensor components of the atomic density matrix. This first paper establishes the grounds of our numerical approach to non-LTE polarization transfer by concentrating on the standard case of scattering line polarization in a gas of two-level atoms, including the Hanle effect due to a weak microturbulent and isotropic magnetic field. We begin demonstrating that the well-known Λ-iteration method leads to the self-consistent solution of this type of problem if one initializes using the exact solution corresponding to the unpolarized case. We show then how the above-mentioned splitting methods can be easily derived from this simple Λ-iteration scheme. We show that our SOR method is 10 times faster than the Jacobi-based ALI method, while our implementation of the Gauss-Seidel method is 4 times faster. These iterative schemes lead to the self-consistent solution independently of the chosen initialization. The convergence rate of these iterative methods is very high; they do not require either the construction or the inversion of any matrix, and the computing time per iteration is similar to that of the Λ-iteration method.

Quantum Interference in Radiation Redistribution of Highly Charged Ions in Plasmas

The influence of quantum interference effects induced by mixing of coherences and populations in the fluctuating plasma microfield on the redistribution of resonance radiation by HCI is studied in terms of the atomic density matrix. The thermal ion motion is considered within the model microfield method. Results show that the thermal ion motion (ion dynamics) induces QIEF as well, and both are essential in line centres and nearest wings of the scattered radiation.

Theory of generalized damping and quantum interference in multi-level atomic systems

When a monochromatic laser resonantly excites a multi-level atom the optical Bloch equations for the reduced atomic density matrix, which describe the evolution of such a system, may exhibit generalized damping terms. We justify the origin of these damping terms within the radiative cascade description of resonance fluorescence in the dressed-atom picture and in the spin-orbit coupling picture. We describe the connection between generalized damping terms and quantum interference in the process of spontaneous emission.

Model microfield method for partial redistribution of resonance radiation in dense plasmas

The influence of nonlinear interference effects (NIEF) on the radiation redistribution functions is investigated in terms of the atomic density matrix formalism. The ion motion is considered within the model microfield method. The particular calculations have been performed for the spectral doublet , of helium-like ion in hot dense plasmas (T = 350 eV, ). Taking account of both NIEF and ion dynamics is essential in the line centre and the nearest wings of the scattered radiation.

Optical pumping of velocity-selective coherent population trapping states in three-level atoms

An analytical theory of optical pumping of velocity-selective coherent population trapping states in three-level Λ-atoms is presented. The theory is based on an atomic density matrix in the superpositional state representation. The representation introduced for the atomic density matrix directly describes coherent population trapping (CPT) states in a three-level Λ-atom and yields an approximate analytical description for the shape of the two-peak momentum (velocity) distribution of atoms. Optical pumping of velocity-selective CPT states is described in the superpositional state representation as a process of redistribution of atoms in the region of small velocities due to the photon recoil. Typical times of formation of velocity-selective CPT states are found to be inversely proportional to the square of the recoil energy.

Tensor structure of the stationary point of the radiative relaxation operator of an atom

We formulate the problem of the stationary point of the operator of radiative relaxation of an atom: the initial distribution among the sublevels of the excited state, whose nonzero eigenvalues (populations) coincide with the populations of the final distribution (after spontaneous decay) among the sublevels of the ground state. We show that these distributions can be expressed in terms of spherical functions of the complex direction. The results are then used to develop a compact analytical representation of the stationary density matrix of atoms interacting with an elliptically polarized monochromatic field.

Entropy studies for the damped and the undamped Jaynes-Cummings model

On the basis of a representation in terms of photon-number states we derive an analytically solvable set of ordinary differential equations for the matrix elements of the density operator belonging to the Jaynes-Cummings model. We allow for atomic detuning, spontaneous emission, and cavity damping, but we do not take into account the presence of thermal photons. The exact results are employed to perform a careful investigation of the evolution in time of atomic inversion and von Neumann entropy. A factorization of the initial density operator is assumed, with the privileged field mode being in a coherent state. We invoke the mathematical notion of maximum variation of a function to construct a measure for entropy fluctuations. In the undamped case the measure is found to increase during the first few revivals of Rabi oscillations. Hence, the influence of the surroundings on the atom does not decrease monotonically from time zero onwards. A further non-Markovian feature of the dynamics is given by the strong dependence of our measure on the initial atomic state, even for times at which damping brings about irreversible decay. For weak damping and high initial energy density the atomic evolution exhibits a crossover between quasireversible revival dynamics and irreversible Markovian decay. During this stage the state of maximum entropy acts as an attractor for the trajectories in atomic phase space. Subsequently, all trajectories follow a unique route to the atomic ground state, for which the off-diagonals of the atomic density matrix equal zero. From our entropy studies one learns what kind of difficulties must be overcome in establishing formulae for entropy production, the use of which is not limited to semigroup-induced dynamics.

Determination of atom-field observables via resonant interaction

We find the observables that-can be determined by two practical schemes based on atom-field resonant interaction. In the first example, the interaction is followed by photon-number and atomic-population measurements. In the second scheme the directly measured quantity is the atomic deflection. In particular, we show that they provide the measurement of the atom-field relative phase. When the initial field state is known, the atomic density matrix can be reconstructed.

Spontaneous emission spectrum of a two-atom system in an optical cavity with injected squeezed light

We study the spontaneous emission spectrum of two identical two-level atoms in an optical cavity. The cavity mode is coupled to a broadband squeezed vacuum. Using the bad-cavity limit, the cavity field is adiabatically eliminated to obtain the equations of motion for atomic density matrix elements. These equations are then used to calculate the emission spectrum. In a cavity environment the two-photon correlation of the injected squeezed vacuum is modified. Here we investigate the effect of this modification on the spontaneous emission spectrum of the two-atom system. We compare our results with those obtained for the atoms interacting with a broadband squeezed vacuum in free space. We show that in the cavity arrangement alteration of the emission spectrum does occur due to selective non-thermal atomic population distribution of the collective atomic states. Nonetheless this spectrum differs appreciably from the spectrum obtained for the free space case. For example, in free space, the emission spectrum shows only one peak, located at frequency − V 12, when atoms are interacting with a squeezed vacuum in a minimum uncertainty state (MUS). Here V 12 is the dipole–dipole interaction between the atoms. In contrast to this, when atoms are enclosed in an optical cavity, the spectrum exhibits two peaks, located at frequencies ± V 12. We also show that in cavity arrangement, for large squeezing photon number, the emission spectrum consists of a single narrow peak at the line centre.

Population, alignment, and coherence of the foil-excitedn=2hydrogen atoms: Polarization observation of the field-dependent quantum beats in the Lyman-α line

We have developed an experimental method for determining the initial density matrix of the foil-excited atoms. The excited hydrogen $(n=2)$ atoms pass through a region of a variable electric field ($\ensuremath{-}200\mathrm{k}\mathrm{V}/\mathrm{m}$ to $+200\mathrm{k}\mathrm{V}/\mathrm{m}$), and the intensity of the Lyman-\ensuremath{\alpha} line (121.6 nm) emitted in a free space downstream is observed. By using a vacuum ultraviolet polarizer, the dependence of the intensities of the polarized components on the electric field, including the field-dependent quantum beats, was measured and it was numerically fitted to a calculation. Thus all the initial density matrix elements were determined in a single measurement. It was found that, in the energy range 100--180 keV, just after the excitation, the electron cloud has higher density in the front side of the ion and the electron probability current is backwards.

Fluorescence into Flat and Structured Radiation Continua: An Atomic Density Matrix without a Master Equation

We investigate an atomic $\Lambda$-system with one transition coupled to a laser field and a flat continuum of vacuum modes and the other transition coupled to field modes near the edge of a photonic band gap. The system requires simultaneous treatment of Markovian and non-Markovian dissipation processes, but the photonic band gap continuum can not be eliminated within a density matrix treatment. Instead we propose a formalism based on Monte-Carlo wavefunctions, and we present results relevant to an experimental characterization of a structured continuum.