Influence of the External Classical Field on the Entanglement of a Two-Level Atom

The atomic and the field entropies of a two-level atom, which is additionally driven by an external classical field are investigated. Under a certain canonical transformation for the excited and ground states the system is transformed into the usual JCM. Using the equations of motion in the Heisenberg picture exact solutions for the time-dependent dynamical operators are obtained. The entanglement between atom-field system is studied by using the atomic and the field entropies. Also we use the concurrence to detect the sudden death phenomenon and the relationship between entropies and the concurrence of the entanglement are discussed. It is shown that the amount of entanglement, the atomic and the field entropies of the subsystem can be improved by controlling the external classical field.     

Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators

Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on ℝ ^N . The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In the case of the symmetric group S _N , our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments. Received: 10 March 1997 / Accepted: 7 July 1997     

The squeezing properties in the Jaynes-Cummings model with arbitrary intensity-dependent coupling

Summary We have studied the squeezing properties of the atom and the radiation field in arbitrary intensity-dependent-coupling Jaynes-Cummings model when it is restricted to the following initial condition: the atom in its coherent state and the field in the vacuum state. The influence of virtual-photon processes on the atomic squeezing predicted by the Jaynes-Cummings model (JCM) has been examined. The relationship between the field and atomic squeezing in the resonant multi-photon JCM has been discussed. The symmetry between the field and atomic squeezing (SFAS) has been exposed in the resonant vacuum one-photon JCM, and the influence of non-resonant interaction and virtual-photon processes on the SFAS has also been discussed.     

Spin squeezing in finite temperature Bose-Einstein condensates: Scaling with the system size

We perform a multimode treatment of spin squeezing induced by interactions in atomic condensates, and we show that, at finite temperature, the maximum spin squeezing has a finite limit when the atom number N →∞ at fixed density and interaction strength. To calculate the limit of the squeezing parameter for a spatially homogeneous system we perform a double expansion with two small parameters: 1/N in the thermodynamic limit and the non-condensed fraction ⟨N _nc⟩/N in the Bogoliubov limit. To test our analytical results beyond the Bogoliubov approximation, and to perform numerical experiments, we use improved classical field simulations with a carefully chosen cut-off, such that the classical field model gives for the ideal Bose gas the correct non-condensed fraction in the Bose-condensed regime.     

Correspondence principle in the self-energy problem

It has been shown by [2b.] that in the classical limit, , there is a correspondence between the quantum and classical electron self-energies in spinor electrodynamics. In the present work this result is extended to the cases of scalar electrodynamics and scalar meson theory. The apparent violation of the correspondence principle in the self-energy problem, noted in the literature, can be ascribed to the inadequacy of the usual expansion procedure in terms of the parameters and as (here Λ is the usual cutoff parameter, r₀ is the cutoff radius).     

Revival‐collapse phenomenon in the quadrature squeezing of the multiphoton Jaynes‐Cummings model with the binomial states

In this paper we study the interaction between a two‐level atom and a quantized single‐mode field, namely, the Jaynes‐Cummings model (JCM). The field and the atom are initially prepared in the binomial state and the excited atomic state, respectively. For this system we prove that the revival‐collapse phenomenon exhibited in the atomic inversion of the standard JCM can be numerically (naturally) manifested in the evolution of the squeezing factor of the three‐photon (standard) JCM provided that the initial photon‐number distribution of the radiation field has a smooth envelope.     

Excitation of coherent correlated states in the Jaynes-Cummings model by external deterministic forces: II

In terms of excitation creation and annihilation operators of the Jaynes-Cummings model, acting in the representation of dressed states, the Hamiltonian is written which describes the character of the spectrum of excitations of two modes, representing a quantum analog of the classical behavior of two interacting one-dimensional anharmonic oscillators, namely, the field and atomic oscillators. The anharmonicity is caused by the nonlinearity of the oscillator interaction and manifests itself in the dependence of the frequencies of both modes on the number of excitations, i.e., on the energy. It is shown that an external deterministic force, acting on the system during a certain time t ₀, transfers it from a vacuum state to a coherent state or from one of the coherent states to another coherent state. The probability of the transition from the vacuum state to the coherent state with a given number of excitations represents the Poissonian distribution for the number of excitations formed in the (atom + field) system by the end of action of the external force. It was found to be proportional to the excitation time t ₀.     

Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum-state limit

A frequency tunable source of squeezed light has been developed which is suitable for a variety of spectroscopic applications. In initial experiments continuous tunability over a range of 2 GHz has been achieved with a directly observed nonclassical noise reduction of 6 dB relative to the vacuum-state limit in a balanced homodyne detector. A process of light-induced absorption in the nonlinear crystal has been identified as the principal loss mechanism which prevents the observation of yet larger degrees of squeezing. Although our source is potentially broadly tunable over the range of wavelengths from 840 to 970 nm, the current research centers on the performance at 852 nm for spectroscopy of the D ₂ line of atomic cesium. For frequency-modulated (FM) saturation spectroscopy in a vapor cell, an improvement of 3.1 dB in sensitivity relative to the usual quantum limit is demonstrated for the detection of Doppler-free resonances. When corrected for the thermal noise of the detector, the enhancement in signal-to-noise ratio brought by the squeezed field is 3.8 dB relative to the shot-noise limit set by the vacuum fluctuations of the probe field.     

A cellular ligand-field model for ‘l-l’ spectral intensities

A model for the calculation of the intensity distribution in ligand-field ‘l-l’ spectra is presented. The approach builds upon the eigenvectors of prior ligand-field analysis and incorporates the superposition of contributions associated with individual metal-ligand interactions in a complete complex. Local transition moments are parameterized by quantities {t _λ} which bear some resemblance to the {e _λ} of the cellular model of the ligand field itself. Equivalence between matrix elements of the electric-dipole operators acting within mixed-parity, local orbitals and of even-parity, effective transition-moment operators acting within pure l orbitals is exploited to construct a scheme that allows the computation of ‘l-l’ spectral intensities sequentially upon ligand-field diagonalization and within a similar basis of atomic eigenfunctions.     

Generalized Lamé Operators

We introduce a class of multidimensional Schrödinger operators with elliptic potential which generalize the classical Lamé operator to higher dimensions. One natural example is the Calogero–Moser operator, others are related to the root systems and their deformations. We conjecture that these operators are algebraically integrable, which is a proper generalization of the finite-gap property of the Lamé operator. Using earlier results of Braverman, Etingof and Gaitsgory, we prove this under additional assumption of the usual, Liouville integrability. In particular, this proves the Chalykh–Veselov conjecture for the elliptic Calogero–Moser problem for all root systems. We also establish algebraic integrability in all known two-dimensional cases. A general procedure for calculating the Bloch eigenfunctions is explained. It is worked out in detail for two specific examples: one is related to the B₂ case, another one is a certain deformation of the A₂ case. In these two cases we also obtain similar results for the discrete versions of these problems, related to the difference operators of Macdonald–Ruijsenaars type.On leave of absence from: Advanced Education and Science Centre, Moscow State University, Moscow 119899, Russia     

Wavevector and frequency dependent dielectric function dynamic structure factor and the instability of plasma waves in two component hot rare quantum and classical plasmas

The full wavevector and frequency dependent complex dielectric function for two component classical and quantum rare hot plasmas have been derived. The real part of dielectric function is obtained in the form of a series. Difference between quantum and classical real and imaginary parts of dielectric function have been brought out by making explicit calculations. The quantum nature of the plasma brings about significant changes in both parts depending upon the magnitude of quantum parameter,R (= 8.93(λ_th)/λ). Expressions for the dynamic structure factors for both two component classical and quantum plasma have been evaluated for different values of the mass of the positive componentm ₊, temperature T₊ and wavevector k. It is found that the plasma exhibits well defined collective modes for certain values of |k| accompanied by varying disorder which depends upon the values of m₊ as well as on |k| and T₊. For the quantum case the collective modes are less well defined as compared to the corresponding classical case, thus proving that quantum nature introduces inherent disorder in the system. But for both the cases, increase in temperature destroys collective modes. Another feature is the appearance of a hump near Ω = 0 which becomes smaller and vanishes as the quantum parameter is decreased. Instability of plasma modes in the presence of constant electric field has also been worked out for the quantum case.     

Quantum mechanical calculation of Suryan’s line broadening in nuclear magnetic resonance

The first quantum theory of the classical radiation damping in nuclear magnetic resonance is presented. Relaxation times and life times arising from the interaction of nuclear spin with the radio-frequency radiation field are calculated. Second-order line shifts are predicted and the existence ofI _z andI _z ² -type operators due to photons is pointed out. The predicted line shifts as well as relaxation are found to be measurably large. Numerical estimates are given for protons in water.     

The effect of environment induced pure decoherence on the generalized Jaynes-Cummings model

We have studied the effect of environment induced pure decoherence on the generalized Jaynes-Cummings model (JCM). This generalized JCM is introduced to take into account both atom-field interaction and a class of spin-orbit interactions in the same framework. For generalized JCM with atom-field interaction, it is shown that along with the suppression of the oscillatory behaviour of the atomic and field variables, in the steady state, atomic energy is transferred to the field or vice versa through the dressed state coherence depending on the initial condition of the atom-field system and the model under consideration. It is also shown that initial Poissonian field acquires a sub-Poissonian character in the steady state and thus all the nonclassical properties are not erased by the decoherence in JCM. An interesting effect of this decoherence mechanism is that it affects the population and coherence properties of the individual subsystem in a different way. As an example of generalized JCM with spin-orbit interaction, the dynamics of spin of the hydrogen atom in a magnetic field is studied to show the effect of decoherence.     

Fluid-dynamical representations of the Dirac equation

A relative kinetic mass operator is defined bym =c ⁻²·(E −eΦ), and it is shown that bt using it in a symmetric form one can correlate the (charge) velocity operatorα in the Dirac theory exactly with the general quantum mechanical momentum —ih∇. Then the net force, defined as the rate of change of the relative momentum with time, is exactly equal to the Lorentz force. The contribution due to the time variation of mass equals the negative of space variation of the scalar potential, the Newtonian force, whereas the time variation of the charge current absorbs the entire vector potential dependence. The analogous Euler equations can be written either in terms of the charge current or in terms of the mass current. For a many particle system one needs the usual net single particle parameters and the consideration of both the direct and exchange contributions of the two particle interaction. These Euler equations yield two different conditions of the stationary state. It is shown that the charge-current condition is necessary but not sufficient, whereas the mass-current condition retains the appropriate scalar potential dependence. These two conditions are compared for the spherically symmetric case. The charge density, charge current and relative mass current are tabulated for atomic spinors. Differences between the quantum and classical forces for the H ₂ ⁺ molecular ion exhibit the inadequacy of ordinary atomic spinor basis in forming molecular spinors.     

Modulation of atomic exit and injection rates on the phase-dependent gain without inversion in a Doppler broadened open four-level system

It is shown that in a Doppler broadened open N-type four-level atomic system with spontaneously generated coherence (SGC), the gain without inversion (GWI) is very sensitive to the variation of the relative phase between the probe field and the driving field; the atomic exit rate (R0) and the ratio (S) of the atomic injection rates have a considerable modulation effect on the phase-dependent GWI. GWI first increases and then decreases with R0 increasing; in a certain value range of S, GWI increases monotonically with S increasing; by adjusting the values of R0 and S, in an open system a much larger GWI can be obtained than in the corresponding closed system [2011 Phys. Rev. A 83 043805]. The modulation effects of R0 and S on the phase-dependent GWI in the case with the counter-propagating probe and driving fields are stronger than those in the co-propagating case, GWI in the co-propagating case is much larger than that in the counter-propagating case.     

Quasi-Classical and Classical Cross Sections for the Scattering of Electrons on the Coulomb Cut-Off Potential

The elastic scattering of electrons on the cut-off Coulomb potential U_c(r) = ?1/r + 1/r_c, for r ≦ r_c and U_c(r) = 0, for r > r_c, has been considered. It has been shown that for ε > 0.5/r_c (ε is the energy of free electrons in atomic units) the analytical quasi-classical expressions describe quite well the behaviour of transport, differential and total cross sections for elastic scattering. It has been shown moreover, that in the energy range considered, transport cross section could be determined with practically the same accuracy already by means of classical, analytical expressions. Born approximations show larger deviations from exact quantum calculations.     

Ursell operators in statistical physics of dense systems: the role of high order operators and of exchange cycles

The purpose of this article is to discuss cluster expansions in dense quantum systems, as well as their interconnection with exchange cycles. We show in general how the Ursell operators of order l≥ 3 contribute to an exponential which corresponds to a mean-field energy involving the second operator U₂, instead of the potential itself as usual - in other words, the mean-field correction is expressed in terms of a modification of a local Boltzmann equilibrium. In a first part, we consider classical statistical mechanics and recall the relation between the reducible part of the classical cluster integrals and the mean-field; we introduce an alternative method to obtain the linear density contribution to the mean-field, which is based on the notion of tree-diagrams and provides a preview of the subsequent quantum calculations. We then proceed to study quantum particles with Boltzmann statistics (distinguishable particles) and show that each Ursell operator U_n with n≥ 3 contains a “tree-reducible part”, which groups naturally with U₂ through a linear chain of binary interactions; this part contributes to the associated mean-field experienced by particles in the fluid. The irreducible part, on the other hand, corresponds to the effects associated with three (or more) particles interacting all together at the same time. We then show that the same algebra holds in the case of Fermi or Bose particles, and discuss physically the role of the exchange cycles, combined with interactions. Bose condensed systems are not considered at this stage. The similarities and differences between Boltzmann and quantum statistics are illustrated by this approach, in contrast with field theoretical or Green's functions methods, which do not allow a separate study of the role of quantum statistics and dynamics. Received 18 October 2001     

开放的V型三能级系统中原子注入和退出速率对位相相关的无反转激光增益和强度空间演化的调制作用

利用数值计算结果, 研究了具有自发辐射诱导相干的开放V 型三能级原子系统中的传播效应. 研究表明: 改变探测场和驱动场之间的相对位相对无反转激光(LWI)增益和 强度随传播距离的变化即空间演化具有重要的影响; 而原子的注入速率比(S)和退出速率(r₀)的改变将对位相相关的LWI增益和强度的空间演化产生明显的调制作用. 在S(r₀)的一定取值范围内, S(r₀)的值越大, LWI增益和强度越大且强度达到极大值需要的传播距离越长; 相应封闭系统中LWI增益和强度的最大值及能产生增益的传播距离都小于开放系统. Doppler效应对LWI增益和强度的空间演化也具有明显的影响, 存在Doppler效应时得到的LWI增益和强度明显小于无Doppler效应时的值.     

原子运动和场模结构对原子动力学特性的影响

研究了双光子Jaynes-Cummings(J-C)模型中原子的动力学特性,讨论了在不同初态下,原子运动和场模结构对体系中原子反转的演化特性及原子偶极矩k阶压缩的影响.研究结果表明,原子的动力学特性不仅决定于原子和场的初态参量,而且通过时间因子g^Θ(t)=2^-1gt-(4_p)^-1sin(2_pgt)决定于原子的运动速度与场模结构参量.     

AN-Type Dunkl Operators and New Spin Calogero–Sutherland Models

A new family of A _N -type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero–Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin Hamiltonians. Received: 17 February 2001 / Accepted: 8 March 2001