Hamiltonian chaos in the interaction of moving two-level atoms with cavity vacuum

We study nonlinear dynamics of the fundamental cavity quantum-electrodynamical system consisting of a point-like collection of identical two-level atoms moving through a lossless single-mode cavity. Taking into account the interatomic and the atom-field quantum correlations of the first order, we go beyond the semiclassical model and derive a dynamical system that is able to describe the vacuum Rabi oscillations with atoms moving in a spatially inhomogeneous cavity field. A simple expression for the equilibrium points of this system provides a class of initial conditions for atoms and a cavity mode under which the atomic population and radiation may be trapped. In the strong-coupling limit and the rotating-wave approximation, the model is shown to be integrable with atoms moving through a resonant cavity with an arbitrary spatial profile of the mode along the propagation axis. The general exact solution is derived in an explicit form in terms of Jacobian elliptic functions. Numerical simulation confirms that perturbations, that are produced by a modulation of the coupling between moving atoms and a cavity mode, provide, out of resonance, a mechanism responsible for Hamiltonian chaos in the interaction of two-level atoms with cavity vacuum. These chaotic vacuum Rabi oscillations may be considered as a new kind of reversible spontaneous emission.     

Quantum chaos and fractals with atoms in cavities

We study the coupled translational, electronic, and field dynamics of the combined system “a two-level atom + a single-mode quantized field + a standing-wave ideal cavity”. In the semiclassical approximation with a point-like atom, interacting with the classical field, the dynamics is described by the Heisenberg equations for the atomic and field expectation values which are known to produce semiclassical chaos under appropriate conditions. We derive Hamilton–Schrödinger equations for probability amplitudes and averaged position and momentum of a point-like atom interacting with the quantized field in a standing-wave cavity. They constitute, in general, an infinite-dimensional set of equations with an infinite number of integrals of motion which may be reduced to a dynamical system with four degrees of freedom if the quantized field is supposed to be initially prepared in a Fock state. This system is found to produce semiquantum chaos with positive values of the maximal Lyapunov exponent. At exact resonance, the semiquantum dynamics is regular. At large values of detuning |δ|1, the Rabi atomic oscillations are usually shallow, and the dynamics is found to be almost regular. The Doppler–Rabi resonance, deep Rabi oscillations that may occur at any large value of |δ| to be equal to |αp₀|, is found numerically and described analytically (with α to be the normalized recoil frequency and p₀ the initial atomic momentum). Two gedanken experiments are proposed to detect manifestations of semiquantum chaos in real experiments. It is shown that in the chaotic regime values of the population inversion z_out, measured with atoms after transversing a cavity, are so sensitive to small changes in the initial inversion z_in that the probability of detecting any value of z_out in the admissible interval [−1,1] becomes almost unity in a short time. Chaotic wandering of a two-level atom in a quantized Fock field is shown to be fractal. Fractal-like structures, typical with chaotic scattering, are numerically found in the dependence of the time of exit of atoms from the cavity on their initial momenta.     

Measures on diffeomorphism groups for non-archimedean manifolds: Group representations and their applications

Nondegenerate σ-additive measures with ranges in ℝ and ℚ_q (q≠p are prime numbers) that are quasi-invariant and pseudodifferentiable with respect to dense subgroups G′ are constructed on diffeomorphism and homeomorphism groups G for separable non-Archimedean Banach manifolds M over a local fieldK,K ⊃ ℚ_q, where ℚ_q is the field of p-adic numbers. These measures and the associated irreducible representations are used in the non-Archimedean gravitation theory. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 381–396, June, 1999.     

Quantum integrability and quantum chaos in the micromaser

The time-dependent quantum Hamiltonians

Microscopic theory of superconductivity in MgB<Subscript>2</Subscript>-type systems in a magnetic field: A neighborhood of <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis>2</Subscript>

We calculate the upper critical magnetic field H _c2 in the framework of a microscopic superconductivity theory with two energy bands of different dimensions on the Fermi surface with the cavity topology typical of the compound MgB₂ taken into account (an anisotropic system). We assume an external magnetic field parallel to the crystallographic z axis. We obtain analytic formulas in the low-temperature range (T/T_c ≪ 1) and also near the critical temperature ((T-T_c)/T_c ≪ 1). We compare the temperature dependence of H_c2 for a two-band anisotropic system with that of H _c2 ⁰ corresponding to a two-band isotropic system (with Fermi-surface cavities of the same topology). We determine the role of the band-structure anisotropy, the positive curvature of the upper critical field near the critical temperature, and the important role of the ratio v₁/v₂ of the velocities on the Fermi surface in determining H_c2. We also obtain the values of the parameters Δ₁ and Δ₂ along the line of the critical magnetic field. This paper is dedicated to the 90th birthday of Professor D. N. Zubarev __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 113–128, January, 2008.     

Bounds for fixed point free elements in a transitive group and applications to curves over finite fields

We investigateV _f , the cardinality of the value set of a polynomialf of degreen over a finite field of cardinalityq. It has been shown that iff is not bijective, thenV _f ≤q−(q−1)/n. Polynomials do exist which essentially achieve that bound. We do prove that if the degree off is prime to the characteristic andf is not bijective, then asymptoticallyV _f ≤(5/6)q. We consider related problems for curves and higher dimensional varieties. This problem is related to the number of fixed point free elements in finite groups, and we prove some results in that setting as well. Both authors partially supported by the NSF.     

High harmonic generation in a two-color field composed of a pump field and a weak subsidiary high frequency field

“Coherent control of high-harmonic generation in a two-color field” has been widely concerned. Using split-operator algorithm, we have calculated the high-harmonic generation for helium ion He+ in a two-color field which is composed of a driving field and a weak subsidiary high frequency field (Is=I0/100, （ω,13ω), …(ω, 120ω)) and found that such a field can produce much higher harmonic intensity, typically increasing the harmonics corresponding to the incident frequency of the subsidiary field. The different effects coming from the different subsidiary fields are calculated and analyzed. It is indicated that one of the important underlying mechanisms is high frequency photon induced radiation.     

Extremum problems in minimax signal detection forl q -ellipsoids with anl p -ball removed

An asymptotic minimax problem of signal detection for signals froml _q -ellipsoids with an l_p-ball removed (p>2 or q<p<2) in a Gaussian white noise is considered. Asymptotically sharp distinguishability conditions and some estimates of the minimax probability of errors of signal detection for ellipsoids with axes a_n≈n^−λ λ>0, are obtained. The proofs use results and techniques developed by Yu. I. Ingster. Bibliography:6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 312–332     

Finite groups in whichp′-classes haveq′-length

IfG is a finite group in which every element ofp′-order centralizes aq-Sylow subgroup ofG, wherep andq are distinct primes, it is shown thatO ^q′ (G) is solvable,l _q (G)≤1 andl _p (O ^q′ (G)) ≤2. Further, the structure ofG is determined to some extent. Work partially supported by MURST research program “Teoria dei gruppi ed applicazioni”.     

Generalization of the fricke theorem on entire functions of finite index

We prove that, for every sequence (a _k) of complex numbers satisfying the conditions Σ(1/|a _k |) < ∞ and |a _k+1| − |a _k | ↗ ∞ (k → ∞), there exists a continuous functionl decreasing to 0 on [0, + ∞] and such thatf(z) = Π(1 −z/|a _k |) is an entire function of finitel-index.     

On the influence of parameters determining anisotropic spaces of smooth functions on characteristics of the embedding theorems

An anisotropic Sobolev and Nikol'skii-Besov space on a domain G is determined by its integro-differential (shortly, ID) parameters. On the other hand, the geometry of G is characterized by the set Λ(G) of all vectors λ=(λ₁,..., λ_n) such that G satisfies the λ-horn condition. We study the dependence of the totality of possible embeddings upon the set Λ(G) and theID-parameters of the space. We consider only embeddings with q≥pⁱ, where pⁱ are the integral parameters of the space and q is the integral embedding parameter. For a given space, we introduce its initial matrix A₀ determined by theID-parameters. A₀ turns out to be a Z-matrix. On the basis of a natural classification of Z-matrices, a classification of anisotropic spaces is introduced. This classification allows one to restate the existence of an embedding with q≥pⁱ in terms of certain specific properties of A₀. Let A₀ be a nondegenerate M-matrix. Any vector λ∈Λ(G) gives rise to a certain set of admissible values of the embedding parameters. We call λ optimal if this set is the largest possible. It turns out that the optimal vector λ _G ^* is determined by Λ(G) and A₀, and may be found by a linear optimization procedure. The following cases are possible: a) , b) , c) λ _G ^* does not exist. In case a) the set of admissible values of the embedding parameters is the biggest, while in case c) no embeddings with q≥pⁱ exist. In case b) the so-called saturation phenomenon occurs, i.e., certain variations of some differential parameters of the space do not change the set of admissible values of the embedding parameters. The latter fact has some applications to the problem of extension of all functions belonging to the given space from G to Eⁿ. Bibliography: 20 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 22–94. Translated by A. A. Mekler.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> norm of spectral clusters for compact manifolds with boundary

We use microlocal and paradifferential techniques to obtain L ⁸ norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal L ^q bounds, in the range 2⩽q⩽∞, for L ² - normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp L ^q estimates in higher dimensions for a range of exponents q̅_n⩽q⩽∞. The authors were supported by the National Science Foundation, Grants DMS-0140499, DMS-0099642, and DMS-0354668.     

Lebesgue measure and Hausdorff dimension of special sets of real numbers from (0,1)

We estimate the Hausdorff dimension and the Lebesgue measure of sets of continued fractions of the type a=[a ₁,a ₂,…] where a _n belongs to a set S _n ⊂ℕ for every n∈ℕ. An upper bound for the Hausdorff dimension of the set of numbers with continued fraction expansions which fulfill some properties of asymptotic densities is also included.     

A new representation theorem for completely [O-] simple semigroups

A semigroup is completely [O-] simple iff it is isomorphic to a semigroup ϕ of binary Boolean (I×I)-matrices (where I is a set) such that for every p,q∈I there exists precisely one matrix (a_ij)∈ϕ such that a_pg=1. Every such isomorphic representation of a completely [O-] simple semigroup is an arbitrary inflation of a reduced representation and all reduced representations are equivalent. Dedicated to Lazar Matveevič Gluskin on the occasion of his fiftieth birthday.     

An approximation property of lacunary sequences

Let f ₁ and f ₂ be two positive numbers of the field , and let f _n+2 = f _n+1 + f _n for each n ≥ 1. Let us denote by {x} the fractional part of a real number x. We prove that, for each ξ ∉ K, the inequality {ξf _n } > 2/3 holds for infinitely many positive integers n. On the other hand, we prove a result which implies that there is a transcendental number ξ such that {ξf _n } < 39/40 for each n ≥ 1. Moreover, it is shown that, for every a > 1, there is an interval of positive numbers that contains uncountably many numbers ξ such that {a ⁿ } 6 min 2/(a − 1), (34a ² − 32a + 7)/(9(2a − 1)²) for each n > 1. Here, the minimum is strictly smaller than 1 for each a > 1. In contrast, by an old result of Weyl, for any a > 1, the sequence {ξa ⁿ }, n = 1, 2, ..., is uniformly distributed in [0, 1] (and so everywhere dense in [0, 1]) for almost all real numbers ξ.     

Mode-coupling approximation in a fractional-power generalization: Particle dynamics in supercooled liquids and glasses

We describe the particle dynamics in supercooled liquids in the mode-coupling approximation in a fractional-power generalization, where the kinetic integro-differential equations are exactly derived from the equations of motion of the dynamical variables by projection operator methods. We show that in the case of separated time scales in the particle dynamics and with the nonlinear interaction between the stochastic and translation motion modes taken into account, the solution of the equations gives a well-defined picture of singularities of the one-particle dynamics of supercooled liquids and glasses. Comparison with the data for the metal alloy Fe ₅₀ Cr ₅₀ atomic dynamics simulation demonstrates a good agreement in the entire temperature range corresponding to the supercooled liquid and glass phases.     

Studying the properties of carbon nanotubes with the functional integration method

We obtain an exact, closed, self-consistent system of equations for describing nanotubes that takes electron and oscillation subsystems in the collective variables into account. Collective excitations in nanotubes are described by the quantum numbers n, m, and k, where n − 1 is the number of radial modes, m is the number of azimuthal modes, and k − 1 is the number of longitudinal modes of the wave function. The results obtained approximate the experimental data better than those obtained by the method of linear combinations of atomic orbitals. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 127–144, October, 2006.     

Two-band superconductivity theory beyond the Migdal theorem

We propose a superconductivity theory of two-band nonadiabatic systems with strong electron correlations in the linear approximation in nonadiabaticity. Assuming a weak electron-phonon interaction, we obtain analytic expressions for the vertex and “intersecting” functions for each of the two bands. With the diagrams involving intersections of two electron-phonon interaction lines taken into account (which means going beyond the Migdal theorem), we determine mass operators of the Green’s functions and use them to derive the basic equations of the superconductivity theory for two-band systems. We find an analytic expression for the superconducting transition temperature T_c that differs from the expression in the case of the standard two-band systems by an essential renormalization of the relevant quantities that results from the nonadiabaticity effects and strong electron correlations. We study the dependence of T_c and of the isotopic coefficient α on the Migdal parameter m = ω₀/ε_F and show that accounting for the overlap of energy bands on the Fermi surface and for the nonadiabaticity effects at small values of the transferred momentum (q ≪ 2p_F) allows obtaining high values of T_c even for the weak electron-phonon interaction. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 111–126, October, 2006. An erratum to this article is available at .     

The relativistic operator of interaction of two quasimolecular electrons as a third-order effect of quantum electrodynamics

We solve the problem of interaction two quasimolecular electrons located at an arbitrary separation near different atoms (nuclei). We consider third-order effects in quantum electrodynamics, which include the virtual photon exchange between electrons with emission (absorption) of a real photon. We obtain the general expression for matrix elements of the operator of the effective interaction energy of two quasimolecular electrons with the external radiation field, which allows calculating probabilities of inelastic processes with rearrangement at slow collisions of multicharge ions with relativistic atoms. We demonstrate that consistently taking the natural condition of the interaction symmetry with respect to the two electrons into account results in the appearance of additional terms in the operators of spin-orbit, spin-spin, and retarded interactions compared with the previously obtained expressions for these operators. We construct the operator of the dipole-dipole interaction of two neutral atoms located at an arbitrary separation.     

Continued radicals

If a ₁,a ₂,…,a _n are nonnegative real numbers and , then f ₁○f ₂○⋅⋅⋅○f _n (0) is a nested radical with terms a ₁,…,a _n . If it exists, the limit as n→∞ of such an expression is a continued radical. We consider the set of real numbers S(M) representable as a continued radical whose terms a ₁,a ₂,… are all from a finite set M. We give conditions on the set M for S(M) to be (a) an interval, and (b) homeomorphic to the Cantor set.