基于双模相干-纠缠态表象的算符恒等式构造法

利用最近引进的相干-纠缠态|x,α〉表象和有序算符内的积分技术,给出了关于双模光场正交分量（X₁+X₂）/2^1/2和参数化的正交分量（μX₁+νX₂）/λ的函数的若干正规乘积展开式,这对于研究场的高阶压缩行为和构建广义压缩态是有用的. 关键词： 双模相干-纠缠态 算符恒等式 有序算符内的积分技术     

Spin relaxation. The renormalized operator cumulant approach

Two-level systems under the influence of external noise are generic models of quantum relaxation processes. The spin 1/2 particle in a stochastic magnetic field is one of the best known examples. Interesting effects are observed if the noise is colored as is the case in most applications. The time-ordered operator cumulant expansion forms a convenient approach to this problem. Recently, a renormalization procedure has been proposed which corresponds to a partial summation of the naive expansion and hence constitutes an essential improvement over the second cumulant considered usually. For colored noise of intermediate strength and correlation time this approach is used to derive renormalized expressions for the transverse (T ₂) and longitudinal (T ₁) relaxation time and the frequency shift. In this parameter region, the ratioT ₁/T ₂= is found to deviate substantially from its value =2 valid for white noise exactly and for colored noise in the second cumulant approximation. This corroborates and extends results of Budimir and Skinner obtained by including the second and fourth cumulant only. However, their truncated expansion is shown to lead to unphysical results in the intermediate parameter region treated still correctly by the renormalized expansion.     

Comments on the Stress-Energy Tensor Operator in Curved Spacetime

 The technique based on a *-algebra of Wick products of field operators in curved spacetime, in the local covariant version proposed by Hollands and Wald, is strightforwardly generalized in order to define the stress-energy tensor operator in curved globally hyperbolic spacetimes. In particular, the locality and covariance requirement is generalized to Wick products of differentiated quantum fields. Within the proposed formalism, there is room to accomplish all of the physical requirements provided that known problems concerning the conservation of the stress-energy tensor are assumed to be related to the interface between the quantum and classical formalism. The proposed stress-energy tensor operator turns out to be conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion. The definition is based on the existence of convenient counterterms given by certain local Wick products of differentiated fields. These terms are independent from the arbitrary length scale (and any quantum state) and they classically vanish on solutions of the Klein-Gordon equation. Considering the averaged stress-energy tensor with respect to Hadamard quantum states, the presented definition turns out to be equivalent to an improved point-splitting renormalization procedure which makes use of the nonambiguous part of the Hadamard parametrix only that is determined by the local geometry and the parameters which appear in the Klein-Gordon operator. In particular, no extra added-by-hand term g _αβQ and no arbitrary smooth part of the Hadamard parametrix (generated by some arbitrary smooth term ``ω ₀ ') are involved. The averaged stress-energy tensor obtained by the point-splitting procedure also coincides with that found by employing the local ζ-function approach whenever that technique can be implemented. Received: 24 September 2001/Accepted: 14 May 2002 Published online: 22 November 2002     

Breakdown of weak-turbulence and nonlinear wave condensation

The formation of a large-scale coherent structure (a condensate) as a result of the long time evolution of the initial value problem of a classical partial differential nonlinear wave equation is considered. We consider the nonintegrable and unforced defocusing NonLinear Schrödinger (NLS) equation as a representative model. In spite of the formal reversibility of the NLS equation, the nonlinear wave exhibits an irreversible evolution towards a thermodynamic equilibrium state. The equilibrium state is characterized by a homogeneous solution (condensate), with small-scale fluctuations superposed (uncondensed particles), which store the information necessary for “time reversal”. We analyze the evolution of the cumulants of the random wave as originally formulated by D.J. Benney and P.G. Saffman [D.J. Benney, P.G. Saffman, Proc. Roy. Soc. London A 289 (1966) 301] and A.C. Newell [A.C. Newell, Rev. Geophys. 6 (1968) 1]. This allows us to provide a self-consistent weak-turbulence theory of the condensation process, in which the nonequilibrium formation of the condensate is a natural consequence of the spontaneous regeneration of a non-vanishing first-order cumulant in the hierarchy of the cumulants’ equations. More precisely, we show that in the presence of a small condensate amplitude, all relevant statistical information is contained in the off-diagonal second order cumulant, as described by the usual weak-turbulence theory. Conversely, in the presence of a high-amplitude condensate, the diagonal second-order cumulants no longer vanish in the long time limit, which signals a breakdown of the weak-turbulence theory. However, we show that an asymptotic closure of the hierarchy of the cumulants’ equations is still possible provided one considers the Bogoliubov’s basis rather than the standard Fourier’s (free particle) basis. The nonequilibrium dynamics turns out to be governed by the Bogoliubov’s off-diagonal second order cumulant, while the corresponding diagonal cumulants, as well as the higher order cumulants, are shown to vanish asymptotically. The numerical discretization of the NLS equation implicitly introduces an ultraviolet frequency cut-off. The simulations are in quantitative agreement with the weak turbulence theory without adjustable parameters, despite the fact that the theory is expected to breakdown nearby the transition to condensation. The fraction of condensed particles vs energy is characterized by two distinct regimes: For small energies (H?H_c) the Bogoliubov’s regime is established, whereas for H?H_c the small-amplitude condensate regime is described by the weak-turbulence theory. In both regimes we derive coupled kinetic equations that describe the coupled evolution of the condensate amplitude and the incoherent field component. The influence of finite size effects and of the dimensionality of the system are also considered. It is shown that, beyond the thermodynamic limit, wave condensation is reestablished in two spatial dimensions, in complete analogy with uniform and ideal 2D Bose gases.     

A theorem concerning the positive metric

It is proved that if then-point correlation functions of a system vanish for alln>N then they vanish for alln>2. The theorem is valid for a wide variety of formalisms and an explicit proof is given for a Bose system with the canonical commutation relations; a proof is sketched out for a relativistic field theory of the Wightman type. The essential property used in the proof is the positive definite metric.     

Larc: A State Reduction Theory of Quantum Measurement

This proposes a new theory of Quantum measurement; a state reduction theory in which reduction is to the elements of the number operator basis of a system, triggered by the occurrence of annihilation or creation (or lowering or raising) operators in the time evolution of a system. It is from these operator types that the acronym ‘LARC’ is derived. Reduction does not occur immediately after the trigger event; it occurs at some later time with probability P _t per unit time, where P _t is very small. Localisation of macroscopic objects occurs in the natural way: photons from an illumination field are reflected off a body and later absorbed by another body. Each possible absorption of a photon by a molecule in the second body generates annihilation and raising operators, which in turn trigger a probability per unit time P _t of a state reduction into the number operator basis for the photon field and the number operator basis of the electron orbitals of the molecule. Since all photons in the illumination field have come from the location of the first body, wherever that is, a single reduction leads to a reduction of the position state of the first body relative to the second, with a total probability of mP _t , where m is the number of photon absorption events. Unusually for a reduction theory, the larc theory is naturally relativistic.     

Kinetic equations and brownian motion for a solid in a stationary temperature gradient

A kinetic equation for the distribution function for a subsystem s interacting with a bath b maintained in a stationary non-isothermal state by reservoirs is derived by using a projection operator formalism and a perturbation expansion parameter λ appropriate to some brownian motion problems. Thus, when s is a heavy particle of mass m ₀ and b a lattice of light particles of mass m then λ = (m m ₀)^1/2. By means of an assumption about the decay of correlations of b variables in the field of s, the terms are classified as destruction terms which vanish to arbitrary order in λ for long enough times and collision terms which give well-defined integrals for long times. For the heavy particle in a lattice the leading collision terms, after linearization in the temperature gradient, lead to an equation equivalent to the generalized nonisothermal Fokker-Planck equation of Nicolis.     

Thermodynamics of an interacting Fermi system in the static fluctuation approximation

We suggest a new method of calculation of the equilibrium correlation functions of an arbitrary order for the interacting Fermi-gas model in the framework of the static fluctuation approximation method. This method based only on a single and controllable approximation allows obtaining the so-called far-distance equations. These equations connecting the quantum states of a Fermi particle with variables of the local field operator contain all necessary information related to the calculation of the desired correlation functions and basic thermodynamic parameters of the many-body system. The basic expressions for the mean energy and heat capacity for the electron gas at low temperatures in the high-density limit were obtained. All expressions are given in the units of r _s , where r _s determines the ratio of a mean distance between electrons to the Bohr radius a ₀. In these expressions, we calculate terms of the respective order r _s and r _s ². It is also shown that the static fluctuation approximation allows finding the terms related to higher orders of the decomposition with respect to the parameter r _s .     

Structural features of a Lennard-Jones system at melting and crystallization

The structural features of the crystallization and melting of a system of particles whose pair interaction is described by the Lennard-Jones potential have been considered. The bond order parameter method is used to quantitatively describe the orientational short-range order. The rotational invariants of the second (q _l ) and third (w _l ) orders are calculated for each particle of the system. These calculations require only information on the snapshot of atom positions, which is quite easily obtained in experiments, and provide the distribution functions of particles in q _l and w _l (where l is the rank of an invariant; the results for l = 4, 6 are presented), which are important characteristics of the phase state of the system. It has been shown that the cumulant of the distribution of particles in w ₆ is very sensitive to the destruction/formation of the short-range orientational order in the Lennard-Jones system and, correspondingly, can be used as a criterion of the melting and crystallization of this system.     

Nonclassical effects in a highly nonlinear generalized homogeneous Dicke model

An intensity dependent nonlinear coupling model of N two-level atoms (generalized Dicke model) interacting dispersively with a bimodal cavity field via two-photon transitions is investigated in a scenario where the rotating wave approximation is assumed. The model becomes homogeneous in the sense that the spin transition frequency is the same for all atoms and the coupling constants emerging from the collective interactions of the atomic system with the cavity field depend only on the particular radiation field mode. This allows us to represent the Dicke Hamiltonian entirely in terms of the total angular momentum J. It is assumed that, initially, the atomic system and the field are in a disentangled state where the field modes are in Glauber coherent states and the atomic system is a superposition of states |JM〉 (Dicke states). The model is numerically tested against simulations of normal squeezing variance of the field, squeezing factors based on the Heisenberg uncertainty principle, along with the statistical properties of the light leading to the possible production of nonclassical effects, such as degree of second-order coherence in the modes, degree of intermode correlation, as well as violation of the Cauchy–Schwartz inequality. Analytical expression of the total density operator matrix elements at t>0 shows the present nonlinear model to be strongly entangled, which is reflected in the time evolution of the linear entropy, where the superposition states are reduced to statistical mixtures. Thus, the present generalized Dicke model does not preserve the modulus of the Bloch vector. The computations, performed in the weak coupling and strong field limits, were conducted via second-order Dyson perturbative expansion of the time evolution operator matrix elements for the totality of the angular momentum states of the atomic system.     

Pauli's electron as a dynamic system

A dynamic systemS _P described by the Pauli equation for nonrelativistic electron is investigated merely as a distributed dynamic system. No quantum principles are used. This system is shown to be a statistical ensemble of nonrelativistic stochastic pointlike particles. The electron spin is shown to have a classical analog which is a collective (statistical) property of the ensemble (not a property of a single electron). The magnetic moment of the electron is a quantum property which has no classical analog. The magnetic moment is parallel to the spin only in the stationary state. In the arbitrary state the magnetic moment is not connected with the spin direction.     

Torus n-Poi nt Fu nctio ns for $${\mathbb{R}}$$ -graded Vertex Operator Superalgebras a nd Co nti nuous Fermio n Orbifolds

We consider genus one n-point functions for a vertex operator superalgebra with a real grading. We compute all n-point functions for rank one and rank two fermion vertex operator superalgebras. In the rank two fermion case, we obtain all orbifold n-point functions for a twisted module associated with a continuous automorphism generated by a Heisenberg bosonic state. The modular properties of these orbifold n-point functions are given and we describe a generalization of Fay’s trisecant identity for elliptic functions. Partial support provided by NSF, NSA and the Committee on Research, University of California, Santa Cruz. Supported by a Science Foundation Ireland Frontiers of Research Grant, and by Max-Planck Institut für Mathematik, Bonn.     

Formal structure of kinetic theory

We study the Liouville equation in the domain of small deviations from absolute equilibrium. The solution is expressed in terms of amplitudes ofn-body additive functions which are orthogonal with respect to the Gibbs weight factor. In the memory operator approach the memory operators are formally exact continued fractions. We show that with the isolation in the Liouville operator of a one-body additive operatorL _o, any memory operator can be written alternatively as an exact infinite series, each term of which can be calculated exactly. This yields improvements of the dressed particle approximation. We discuss the choice ofL _o, which is in general time dependent. The theory is developed both for smooth potentials and for hard spheres, where we use pseudo-Liouville operators. The theory can be formulated in an equivalent manner by introducing modified cumulant distributions, which are closely related to the amplitudes. The modified cumulants have the same spatial asymptotic properties as ordinary cumulants, but have superior short-time and small-distance behavior.Work supported by the National Science Foundation.     

On the uniqueness of the equilibrium state for Ising spin systems

We show that for an Ising spin system of arbitrary spin with a ferromagnetic pair interaction and a periodic external magnetic field there is a unique equilibrium state if and only if the magnetization is continuous with respect to a uniform change in the external field. Hence, if the critical temperatureT _c is defined as the temperature where the spontaneous magnetization (which is a non-increasing function of the temperature) becomes positive, then the equilibrium state is unique forT>T _c and is non-unique forT<T _c (when the external field is zero). This implies that the correlation functions have a cluster property forT>T _c .We also show that for an anti-ferromagnet consisting of two sublattices there is a unique equilibrium state if and only if the staggered magnetization is continuous with respect to a change in the staggered field.Supported in part by U.S.A.F.O.S.R. under contract F 44620-71-C-0013, P001.     

The rovibrational levels of ammonia

In this paper we report variational rovibrational studies on ammonia and its isotopomers. We use six internal coordinates (one of which describes the umbrella motion). The expansion functions are products of one-dimensional functions of these internal coordinates, multiplied by appropriate functions of the Euler angles to describe the rotational motion. We use a previously published high accuracy six-dimensional potential energy surface [LEONARD, C., HANDY, N. C., CARTER, S., and BOWMAN, J. M., 2002, Spectrachim. Acta, 58, 825]. We derive the full kinetic energy operator for the rovibrational motion in these coordinates. This operator has been completely checked to give a hermitian secular matrix. All matrix elements are evaluated numerically by quadrature. The symmetry of the expansion functions is fully described in D_3h, C_2v and C_s. It is not possible to perform the calculations in D_3h, but complete degeneracy in the appropriate levels is obtained with the C_2v program. The algebraic complexity of this program has been far greater than for any other variational study we have undertaken for a tetra-atomic molecule.

We present J = 0, 1, 2 energy levels for the experimentally observed band origins of NH₃, and J = 0, 1 energy levels for the ground state and fundamentals of NH₂D, ND₂H and ND₃. For the asymmetric isotopomers, identical results are obtained for both C_2v and C_s, thus confirming the validity of the method. The levels we obtain are completely converged. Agreement with observation is of the order of 0.5% (of course being dependent upon the accuracy of the potential energy surface); therefore the ordering of the rotational levels and their splitting is completely predicted and understood.     

Resonant axion generation in a strong magnetic field as a mechanism of forming a cold latent mass

It is demonstrated that considering vacuum polarization in superstrong magnetic field B ≫B₀ = m² / e = 4.41∙10¹³ G leads to giant (by 13 orders of magnitude) intensification of axion generation processes. In this case, for example, the probability of resonant axion radiation is about 10^-12 (B / B₀ ) of the probability of photon radiation, which is by many orders of magnitude greater than usually. In this regard, it is assumed that this mechanism plays the main role in the formation of an axion component of cold latent mass in the Universe.     

Exponential Localization of Hydrogen-like Atoms in Relativistic Quantum Electrodynamics

We consider two different models of a hydrogenic atom in a quantized electromagnetic field that treat the electron relativistically. The first one is a no-pair model in the free picture, the second one is given by the semi-relativistic Pauli-Fierz Hamiltonian. We prove that the no-pair operator is semi-bounded below and that its spectral subspaces corresponding to energies below the ionization threshold are exponentially localized. Both results hold true, for arbitrary values of the fine-structure constant, e ², and the ultra-violet cut-off, Λ, and for all nuclear charges less than the critical charge without radiation field, Z _c  = e ⁻²2/(2/π + π/2). We obtain similar results for the semi-relativistic Pauli-Fierz operator, again for all values of e ² and Λ and for nuclear charges less than e ⁻²2/π.     

Magnetotransport study of in situ magnetic field textured Dy1Ba2Cu3O7−δ superconductors

The electrical resistivity, thermoelectric power and the Nernst effect have been studied as a function of temperature and magnetic field for a typical superconductor DyBa₂Cu₃O_7– magnetically textured in situ at 1035°C. The three transport coefficients show hybrid microscopic features. In particular, we show the anisotropy with respect to the field direction (H//a,H//c) in all transport coefficients. We verify that Tinkham's law is obeyed for the broadening of the resistive transition in a magnetic field. Similarly we obtainanisotropic broadening exponents for each integrated excess property. From Arrhenius plots we obtain orders of magnitude for the activation energies characterizing each property. They are markedly different from each other.