Chaos, fractals, and atomic flights in cavities

A semiclassical study is carried out of the nonlinear interaction dynamics between two-level atoms and a standing-wave field in a high-finesse cavity. As a result of atomic movement or wave amplitude modulation, a dynamic local instability occurs in a strongly coupled atom-field system. The appearance of dynamical Hamiltonian chaos, fractals, and Lévy flights is demonstrated for the models of two experimental devices: a (micro)maser with thermal Rydberg atoms and a microlaser with cold atoms. Numerical simulation showed that the manifestations of classical chaos, atomic fractals, and flights can be observed in the appropriate real experiments. Attention is drawn to the prospects provided by work on the atom-field systems in the coupling-modulated high-finesse cavities for further investigation of the quantum-classical correspondence, quantum chaos, and decoherence.     

Semiclassical interaction of moving two-level atoms with a cavity field: from integrability to Hamiltonian chaos

The dynamics of an ensemble of two-level atoms moving through a single-mode lossless cavity is investigated in the semiclassical and rotating-wave approximations. The dynamical system for the expectation values of the atomic and field observables is considered as a perturbation to one of the following integrable versions: (i) a model with atoms moving through a spatially inhomogeneous resonant field, and (ii) a model with atoms interacting with a nonresonant eigenmode which is assumed to be homogeneous on the cavity size. We find the general exact solutions for both the models and show that they contain special solutions describing a coherent effect of population and radiation trapping. Using the Melnikov method, we prove analytically transverse intersections of stable and unstable manifolds of a hyperbolic fixed point under a small modulation of the vacuum Rabi frequency. These intersections are believed to provide the Smale horseshoe mechanism of Hamiltonian chaos. The analytical results are accompanied with direct computation of topographical maps of maximal Lyapunov exponents that give a representative image of regularity and chaos in the atom-field system in different ranges of its control parameters--the frequency detuning, the number, and the velocity of atoms.     

A mechanism of emergence of Hamiltonian chaos in a basic quantum-optical model

A mechanism of emergence of Hamiltonian chaos is considered for the model describing the interaction between two-level atoms and their own radiation field in an ideal single-mode cavity. The analysis of the semiclassical Maxwell-Bloch equations shows that the Hamiltonian terms that are neglected in the rotating-wave approximation (RWA) give rise to the formation of a stochastic layer near the RWA-system separatrix. The Mel’nikov method is used to prove that the splitting of the separatrix takes place for arbitrarily small vacuum Rabi frequencies Ω_N. The computation of Poincare sections shows that the stochastic layer, which is exponentially narrow for small Ω _N, expands with increasing Ω_N, and at Ω_N ? 1, the system exhibits global chaos that manifests itself in irregular oscillation of the atomic population inversion and the broadening of the power spectrum. Promising candidates for observing manifestations of dynamic chaos in this basic quantum-optical system are Rydberg atoms placed in a high-Q superconducting microwave cavity.     

A re-examination of fermi's hyperfine contact interaction

Summary The hyperfine coupling between spin of electrons ins states and nuclear spin is generally represented by a contact Hamiltonian in which a δ(r) factor appears. Utilizing relativistic equations and considering pointlike nuclei, we show that the δ(r) factor must be replaced by a steeply decreasing radial function of half-maximum width δr=5.8·10⁻¹⁴Z cm. For hydrogen, the correction with respect to the contact Hamiltonian turns out to be small, but for high-Z nuclei this correction acquires substantial importance. For iron 1s states, it rises up to 9.6%.     

Spreading of atomic wave packets and semiclassical chaos

The correspondence between the statistical properties of the evolution of a quantum system and Lyapunov instability and the chaos of its semiclassical analog has been demonstrated. The results of the analyses of atomic motion in a laser field in the semiclassical approximation (dynamics is described by several nonlinear equations) and without this approximation (dynamics is described by an infinite system of linear equations) are compared. In the ranges of the parameters for which the semiclassical dynamics of point-like atoms is unstable, the fast “spreading” of quantized wave packets in the momentum space is observed. Thus, deterministic chaos “imitates” the statistics of the quantum nondeterministic effects, although the semiclassical and quantum solutions are fundamentally different.     

Chaotic dynamics in a nonautonomous Dicke model without the rotating-wave approximation

The properties of a system consisting of two-level atoms interacting with a mode of the electromagnetic field in a cavity are considered for the case when the cavity detuning or the coefficient of the atom-field interaction is modulated. The consideration is performed with account taken of the Hamiltonian terms that are neglected in the rotating-wave approximation. It is shown that in the semiclassical equations for such a model, the effect of extension (compared to the autonomous system) of the range of variation of the quantity characterizing the number of photons can manifest itself; in this case, the energy oscillations have a chaotic character. The dependence of this phenomenon on parameters characterizing the model is studied. It is numerically demonstrated that with account taken for the relaxation, the system studied can have attractors different from the equilibrium positions, i.e., the number of photons in the mode does not tend to a constant value. The limits of validity of the rotating-wave approximation in the parametrically perturbed Dicke model are discussed.     

Interaction Dynamics of the External and Internal Degrees of Freedom of an Atom in the Resonant Field of a Standing Light Wave

The theory of the dynamic interaction of the external (translational) and internal (electronic) degrees of freedom of a twolevel atom in the field of a standing light wave in a perfect cavity of the Fabry–Perot type was developed. The theory describes the energy exchange between three subsystems, namely, translational, electronic, and field subsystems, as opposed to the theories of the parametric interaction (in the approximations of Raman–Nath and/or large resonance detuning) and of the atomic motion in free space. In the semiclassical approximation, the corresponding Heisenberg equations of motion were shown to form a closed Hamiltonian dynamic system with two degrees of freedom, namely, translational and collective electron–field degrees of freedom. This system is integrated in terms of the elliptic Jacobian functions in the resonance limit. The solutions obtained describe the effects of trapping of an atom in the periodic potential of the standing light wave, and its cooling and heating, as well as the effect of the dynamic Rabi oscillations. The latter is caused by the interaction of the internal and external atomic degrees of freedom through the radiation field.     

Sudden Death and Long-Lived Entanglement Between Two Atoms in a Double JC Model System

Considering a double JC model with different coupling constants, we investigate the entanglement between the two two-level atoms, and discuss dependence of the atom-atom entanglement on the different coupling constants, and the detuning between the atomic transition frequency and the cavity field frequency. The results show when Δ=δ/g is small, with the increase of the relative difference of the two atom-cavity coupling constants γ, the atom-atom entanglement periodically evolves with the amplitude slowly and periodically modulated. What’s more interesting is that long-lived entanglement between the two atoms can be obtained when atom A non-resonantly interacts with the cavity field a, and atom B has no coupling with the cavity field b. In this case, the concurrence C ^AB (t) of the two atoms evolves in form of cosine, and is invariant and equals the initial value when far off resonance. In addition, we find that the so-called entanglement sudden death can occur under appropriate conditions on the detunings and the different coupling constants for different initial atomic states.     

Chaotic Rabi vacuum oscillations in cavity quantum electrodynamics

It is shown in numerical simulations with two-level atoms moving through a single-mode high-Q cavity that spontaneous emission of a new type — chaotic Rabi vacuum oscillations — arises in the strong atom-field coupling regime. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 11, 801–806 (10 June 1997)     

Calculation of the Kirchhoff coefficients for the Helmholtz resonator

A Helmholtz resonator is a shell Ω_shell separating a compact cavity Ω_int from a noncompact outer domain Ω_out. A small opening Ω ^δ in the shell connects the cavity with the outer domain, causing the transformation of real eigenfrequencies of the Neumann Laplacian in the cavity into the complex scattering frequencies of the full spectral problem for the Neumann Laplacian on Ω = ℝ³\Ω_shell. The Kirchhoff model of 1882, see [21], gives a convenient ansatz

Implementation of quantum controlled phase gate and preparation of multiparticle entanglement in cavity QED

Schemes are presented for realizing quantum controlled phase gate and preparing an N-qubit W-like state, which are based on the large-detuned interaction among three-state atoms, dual-mode cavity and a classical pulse. In particular, a class of W states that can be used for perfect teleportation and superdense coding is generated by only one step. Compared with the previous schemes, cavity decay is largely suppressed because the cavity is only virtually excited and always in the vacuum state and the atomic spontaneous emission is strongly restrained due to a large atom-field detuning.     

Kinetics and Optical Properties of the Strongly Driven Gas Medium of Interacting Atoms

This paper investigates stimulated emission and absorption near resonance for a driven system of interacting two-level atoms. Microscopic kinetic equations for the density matrix elements of N-atom states including atomic motion are built, taking into account atom-field and atom-atom interactions. Analytical solutions are given for the resulting macroscopic equations in different limits, for a system composed of a strong coherent “pump” field and a weak counter-propagating “probe” field. It was shown that the existence of a dipole-dipole (long-range) interaction between atoms separated by distance less than the pump wave-length can cause the formation of periodic polarization and population structures (gratings in time and space) in the pumped medium without a probe field. The magnitude of pump induced population grating can have a strong dependence on the relation between the pump field detuning and the polarization decay rate. The “interaction” between pump and probe induced polarization/population gratings through a dipole-dipole interaction mechanism causes the absorption line shape asymmetry. Under certain conditions, this asymmetry is revealed in increasing probe gain for the “red”-shifted (relative to pump) probe and suppressing the gain for the “blue”-shifted probe field when pump is “red”-shifted relative to the ensemble averaged resonant frequency. The theoretical results are consistent with experimental data for the probe gain or absorption as the function of frequency and the dependance of the gain on atomic density for sodium vapor when the pump laser is tuned near the D ₂ line. Here the dependance of gain on particle density was explained in the terms of the long-range interaction between the atoms.     

Electromagnetically induced transparency in Λ-system involving D 2 line of Rb atoms confined in sub-micron columns

The effect of electromagnetically induced transparency (EIT) in a Λ-system formed by rubidium atoms contained in thin (10–60 μm) and extremely thin (0.3–5 μm) cells was studied experimentally. It was found that parameters of the EIT resonance degrade slowly in the case where the frequency of the coupling laser is in resonance with the D ₂ transition of rubidium, which enabled the registration of the EIT resonance in a record thin cell with a thickness of L = 390 nm. The specific features of EIT in extremely thin cells reveal themselves when the coupling laser has a frequency detuning Δ from the atomic transition. In this case, the width of the EIT resonance rapidly increases upon an increase in Δ at fixed L (an opposite effect takes place in centimeter-scale cells). It is shown that the width of the EIT resonance is inversely proportional to L in the case of fixed large detuning Δ. The nearly tenfold broadening of the EIT resonance for large values of detuning Δ is caused by the influence of atomic collisions with cell windows on dephasing rate of coherence. The expressions that allow the estimation of the EIT-resonance width for various values of detuning Δ and small values of thickness L are found.     

Born Reciprocity and the 1/r Potential

Many structures in nature are invariant under the transformation pair, (p,r)→(b  r,−p/b), where b is some scale factor. Born’s reciprocity hypothesis affirms that this invariance extends to the entire Hamiltonian and equations of motion. We investigate this idea for atomic physics and galactic motion, where one is basically dealing with a 1/r potential and the observations are very accurate, so as to determine the scale b≡mΩ. We find that an Ω∼1.5×10⁻¹⁵ s⁻¹ has essentially no effect on atomic physics but might possibly offer an explanation for galactic rotation, without invoking dark matter.     

Interaction of nonlinear resonances in cavity QED

Strong coupling of the internal and external degrees of freedom of a cold atom to each other and to the spatially periodic field of the standing light wave in a high-finesse cavity is responsible for the dynamic instability of the atomic center-of-mass motion. Due to a weak interaction of the internal nonlinear resonances in the standard model of cavity QED, a stochastic layer appears, whose width in the semiclassical approximation is estimated in terms of the main parameters of the system: atomic recoil frequency, mean number of excitations, and detuning from the resonance. As a result, the atomic motion in the absolutely regular potential has the fractal character, with long Lévy flights alternating with small chaotic oscillations in potential wells.     

Thermal Entanglement of the Two-Qubit Heisenberg Spin Chain Coupled to a Single-Mode Cavity Field

Thermal entanglement of a two-qubit Heisenberg spin chain coupled to a single-mode cavity field is investigated. It is found that (1) thermal entanglement without the rotating-wave approximation (RWA) is explicitly smaller than that obtained with the RWA, which means that the counter-rotating terms have a large impact on thermal entanglement, therefore they cannot be neglected; (2) the case (ω≪Ω) is more beneficial for enhancing thermal entanglement than the resonance case (ω=Ω), the near-resonant case (ω≈Ω) and the case (ω≫Ω); (3) for thermal entanglement, there is a competition process between the exchange coupling J (the direct-coupling between the two two-level atoms) and the coupling constant g (which deduces the indirect effect between the two two-level atoms); the critical value of g increases with the spin coupling strength J.     

Superradiant Rayleigh scattering of light from a Bose-Einstein condensate of dilute atomic gases, taking into account backward recoil momentum

An extension of the semiclassical model is proposed for superradiant Rayleigh scattering of light from a Bose-Einstein condensate of dilute atomic gases. In contrast to the familiar models, this model takes into account the fact that the atom may be excited by the scattered light and may acquire backward recoil momentum. The presented equations are solved analytically for the initial (linear) stage of the process. The intensity of the scattered light and the efficiency of populating atomic states with different momentum values are studied as a function of detuning from resonance. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 73, No. 1, pp. 65–72, January–February, 2006.     

核类似铜酸盐超导模型的sus(1,1)⊕sud(1,1)代数结构

由Iachello提出的核类似铜酸盐超导模型具有su(3)代数结构,其平均场近似下的Hamilton量可以写成su(3)生成元的线性组合.通过代数生成元的实现,该模型约化的Hamilton量具有su_s(1,1)⊕su_d(1,1)代数结构,利用相干算子U(θ,φ)的幺正变换,可得到系统约化Hamilton量的能隙方程及本征值,发现应用不同代数结构求解会得到不同侧重点的分析结果.     

Renormalized Kinetic Theory of Classical Fluids in and out of Equilibrium

We present a theory for the construction of renormalized kinetic equations to describe the dynamics of classical systems of particles in or out of equilibrium. A closed, self-consistent set of evolution equations is derived for the single-particle phase-space distribution function f, the correlation function C=〈δfδf〉, the retarded and advanced density response functions χ ^R,A =δf/δφ to an external potential φ, and the associated memory functions Σ ^R,A,C . The basis of the theory is an effective action functional Ω of external potentials φ that contains all information about the dynamical properties of the system. In particular, its functional derivatives generate successively the single-particle phase-space density f and all the correlation and density response functions, which are coupled through an infinite hierarchy of evolution equations. Traditional renormalization techniques (involving Legendre transform and vertex functions) are then used to perform the closure of the hierarchy through memory functions. The latter satisfy functional equations that can be used to devise systematic approximations that automatically imply the conservation laws of mass, momentum and energy. The present formulation can be equally regarded as (i) a generalization to dynamical problems of the density functional theory of fluids in equilibrium and (ii) as the classical mechanical counterpart of the theory of non-equilibrium Green’s functions in quantum field theory. It unifies and encompasses previous results for classical Hamiltonian systems with any initial conditions. For equilibrium states, the theory reduces to the equilibrium memory function approach used in the kinetic theory of fluids in thermal equilibrium. For non-equilibrium fluids, popular closures of the BBGKY hierarchy (e.g. Landau, Boltzmann, Lenard-Balescu-Guernsey) are simply recovered and we discuss the correspondence with the seminal approaches of Martin-Siggia-Rose and of Rose and we discuss the correspondence with the seminal approaches of Martin-Siggia-Rose and of Rose.     

Dynamic and spectral mixing in nanosystems

In the framework of a simple spin-boson Hamiltonian we study an interplay between dynamic and spectral roots to stochastic-like behavior. The Hamiltonian describes an initial vibrational state coupled to discrete dense spectrum reservoir. The reservoir states are formed by three sequences with rationally independent periodicities 1; 1 ± δ typical for vibrational states in many nanosize systems (e.g., large molecules containing CH₂ fragment chains, or carbon nanotubes). We show that quantum evolution of the system is determined by a dimensionless parameter δΓ, where Γ is characteristic number of the reservoir states relevant for the initial vibrational level dynamics. When δΓ > 1 spectral chaos destroys recurrence cycles and the system state evolution is stochastic-like. In the opposite limit δΓ < 1 dynamics is regular up to the critical recurrence cycle k _c and for larger k > k _c dynamic mixing leads to quasi-stochastic time evolution. Our semi-quantitative analytic results are confirmed by numerical solution of the equation of motion. We anticipate that both kinds of stochastic-like behavior (namely, due to spectral mixing and recurrence cycle dynamic mixing) can be observed by femtosecond spectroscopy methods in nanosystems in the spectral window 10¹¹–10¹³ s⁻¹