Long-range atom-metal-surface interaction and interference of atomic states

The properties of two-dimensional magnetic traps for laser-cooled atoms are analysed using complex functions. The two components of the magnetic field from a series of parallel, infinitely long, current-carrying wires are represented by a single complex number. The regions of the field where paramagnetic atoms can be trapped occur where the magnetic field is zero. The locations of the zeroes of the field are obtained as the solution to a polynomial and the multiplicity m of the solution determines both the 2(m + 1)-pole nature of the trap and the field gradient through the centre. The zeroes of the field can be merged or split by varying the locations of the currents, their strengths or by applying a uniform magnetic field. The theory is applied to magnetic traps created from long thin wires or permanent magnets on a substrate. The properties of a number of magnetic trap configurations used for atom guides are discussed. Received 28 February 2001 and Received in final form 6 July 2001     

Density of Complex Critical Points of a Real Random <Emphasis Type="Italic">SO</Emphasis>(<Emphasis Type="Italic">m</Emphasis>+1) Polynomial

We study the density of complex critical points of a real random SO(m+1) polynomial in m variables. In a previous paper (Macdonald in J. Stat. Phys. 136(5):807, 2009), the author used the Poincaré-Lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials, the SU(m+1) polynomials. In this paper, we use the Kac-Rice formula to prove an analogous result: the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial. In one variable, we give an exact formula and a scaling limit formula for the density of critical points of the real random SO(2) polynomial as well as for the density of critical points of the corresponding complex random SU(2) polynomial.     

Einstein metrics onS 3,R 3 andR 4 bundles

Starting from a 4n-dimensional quaternionic Kähler base space, we construct metrics of cohomogeneity one in (4n+3) dimensions whose level surfaces are theS ² bundle space of almost complex structures on the base manifold. We derive the conditions on the metric functions that follow from imposing the Einstein equation, and obtain solutions both for compact and non-compact (4n+3)-dimensional spaces. Included in the non-compact solutions are two Ricci-flat 7-dimensional metrics withG ₂ holonomy. We also discuss two other Ricci-flat solutions, one on theR ⁴ bundle overS ³ and the other on anR ⁴ bundle overS ⁴. These haveG ₂ and Spin(7) holonomy respectively.     

On the Connection between Particles and Poles of the S-Matrix

In axiomatic S-matrix theory it is usually assumed that stable particles give rise to simple poles of the S-matrix for real negative energies while unstable particles give rise to poles close to the real axis on an unphysical sheet of the energy Riemann surface. The stable particle — pole association has been known for a long time not to be always true. For example in potential scattering what is relevant in this case in fact is not the S-matrix but the Jost function. The zeroes of this function for real negative energies are in fact in one-to-one correspondence with the bound states, while the correspondence may break down for the poles of the S-matrix. On the other hand it has recently been pointed out that there also is in general no connection between unstable particles and poles of the S-matrix.     

Correlation Functions for Random Complex Zeroes: Strong Clustering and Local Universality

We prove strong clustering of k-point correlation functions of zeroes of Gaussian Entire Functions. In the course of the proof, we also obtain universal local bounds for k-point functions of zeroes of arbitrary nondegenerate Gaussian analytic functions. In the second part of the paper, we show that strong clustering yields the asymptotic normality of fluctuations of some linear statistics of zeroes of Gaussian Entire Functions, in particular, of the number of zeroes in measurable domains of large area. This complements our recent results from the paper “Fluctuations in random complex zeroes”.     

Integrability Conditions for Complex Homogeneous Kukles Systems

In this paper we study the existence of local analytic first integrals for complex polynomial differential systems of the form ? = x + P_n(x, y), ? = ?y, where P_n(x, y) is a homogeneous polynomial of degree n, called the complex homogeneous Kukles systems of degree n. We characterize all the homogeneous Kukles systems of degree n that belong to the Sibirsky ideal. Finally, we provide necessary and sufficient conditions when n = 2,?. . .?, 7 in order that the complex homogeneous Kukles system has a local analytic first integral computing the saddle constants and using Gröbner bases to find the decomposition of the algebraic variety into its irreducible components.     

Ba原子6pnd(J=1, 3)自电离光谱的实验研究

采用三台可调谐激光实施孤立实激发,分三步将处于基态的Ba原子激发到6p_1/2nd(J=1,3)和6p_3/2nd(J=1,3)自电离态上,获得了分别从6snd¹D₂(n=7—15)和6snd³D₂(n=7—12) 激发而得到的6p_1/2nd(J=1,3)和6p_3/2nd (J=1,3)自电离光谱,重点对主量子数n较低的自电离态进行了实验研究. 通过光谱的线形拟合得到了上述能级的位置和宽度等数据,进而获得了量子亏损和约化宽度等信息. 通过对不同系列的自电离光谱的分析和比较,详细讨论了这些自电离态的光谱特征及其复杂光谱结构的成因. 关键词： 孤立实激发 组态相互作用 自电离态     

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

The eigenvalue problem for arbitrary linear combinations kα + μα^? of a boson annihilation operator α and a boson creation operator α^? is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ ? μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ? 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α^? to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α^? are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ ? μ/k as a parameter.     

Physical symmetries in a theory of several scalar real fields

Let us consider a theory ofn scalar, real, local, Poincaré covariant quantum fields forming an irreducible set and giving rise to one particle states belonging to the same mass different from zero. The vacuum is unique. It is shown under fairly weak assumptions that every Poincaré and TCP invariant symmetry of the theory, implemented unitarily, which mapps localized elements of the field algebra into operators almost local with respect to the former (such a symmetry we call a physical one) can be defined uniquely in terms of the incoming or outgoing fields and ann-dimensional (real) orthogonal matrix. The symmetry commutes with the scattering matrix. Incidentally we show also that the symmetry groups are compact. A special case of these symmetries are the internal symmetries and symmetries induced by locally conserved currents local with respect to the basic fields and transforming under the same representation of the Poincaré group. We may make linear combinations out the original fields resulting in complex fields and its complex conjugate in a suitable way. The inspection of the representations of the groupsSO(n) and their subgroups sheds some light on the s.c. generalized Carruthers Theorem concerning the self- and pair-conjugate multiplets.     

Density of Complex Zeros of a System of Real Random Polynomials

We study the density of complex zeros of a system of real random SO(m+1) polynomials in m variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case. We also show that the behavior the scaled density of complex zeros near ℝ ^m of the system of real random polynomials is different in the m≥2 case than in the m=1 case: the density approaches infinity instead of tending linearly to zero.     

On a q-analog of the Wallach–Okounkov Formula

We obtain a q-analog of the well-known result on a joint spectrum of invariant differential operators with polynomial coefficients on a prehomogeneous vector space of complex n × n-matrices. We are motivated by applications to the problems of harmonic analysis in the quantum matrix ball: our main theorem can be used to prove the Plancherel formula (to be published).     

Counterexamples to the Maximal p-Norm Multiplicativity Conjecture for all p > 1

The quantum marginal problem asks what local spectra are consistent with a given spectrum of a joint state of a composite quantum system. This setting, also referred to as the question of the compatibility of local spectra, has several applications in quantum information theory. Here, we introduce the analogue of this statement for Gaussian states for any number of modes, and solve it in generality, for pure and mixed states, both concerning necessary and sufficient conditions. Formally, our result can be viewed as an analogue of the Sing-Thompson Theorem (respectively Horn’s Lemma), characterizing the relationship between main diagonal elements and singular values of a complex matrix: We find necessary and sufficient conditions for vectors (d ₁,..., d _n ) and (c ₁,..., c _n ) to be the symplectic eigenvalues and symplectic main diagonal elements of a strictly positive real matrix, respectively. More physically speaking, this result determines what local temperatures or entropies are consistent with a pure or mixed Gaussian state of several modes. We find that this result implies a solution to the problem of sharing of entanglement in pure Gaussian states and allows for estimating the global entropy of non-Gaussian states based on local measurements. Implications to the actual preparation of multi-mode continuous-variable entangled states are discussed. We compare the findings with the marginal problem for qubits, the solution of which for pure states has a strikingly similar and in fact simple form.     

Proof of the Cases <Emphasis Type="Italic">p</Emphasis>≤ 7 of the Lieb-Seiringer Formulation of the Bessis-Moussa-Villani Conjecture

It is shown that the polynomial λ (t)=Tr[(A+ tB)^p] has nonnegative coefficients when p≤ 7 and A and B are any two complex positive semidefinite n× n matrices with arbitrary n. This proves a general nontrivial case of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture which is a long standing problem in theoretical physics.     

Dynamic hyperpolarizabilities of excited states of hydrogen

On the basis of the generalized Sturm expansion of the radial part of the Coulomb Green function, a computational method is proposed and numerical results are presented for the dynamic hyperpolarizability γ and the corrections E ⁽⁴⁾ (quadratic in the light intensity) to the quasi-energy of the ground and excited states of hydrogen with principal quantum numbers n≤5 in a monochromatic light field. In this approach, the problem is reduced to the summation of well-convergent double series of the hypergeometric kind, which ensures reliable numerical results both for states with a large n, and in a wide range of field frequencies ω, including the above-threshold frequency range of ?ω?|E _n | (|E _n | is the ionization potential of the state |nlm〉 under investigation). We consider the frequency dependence of γ and E ⁽⁴⁾, their differences for the cases of linear and circular polarizations of the field, and the relation between their real and imaginary parts, which determine the laser field-induced corrections to the position and width of energy levels. For n=5, the significant role of mixing the |nlm〉 states with different values of l by a laser field in the region of resonances on intermediate bound states is demonstrated. The linear (in intensity) corrections to the photoionization cross section for excited states are analyzed and the threshold intensity corresponding to the onset of atomic level stabilization is estimated for a number of states with n=3 and n=5.

     

Holomorphic curves in loop groups

It was observed by Atiyah that there is a correspondence between based gauge equivalence classes ofSU _n -instantons overS ⁴ of charged on the one hand, and based holomorphic curves of genus zero inSU _n of degreed on the other hand. In this paper we study the parameter space of such holomorphic curves which have the additional property that they lie entirely in the subgroup _alg SU _n of algebraic loops. We describe a cell decomposition of this parameter space, and compute its complex dimension to be (2n–1)d.     

Excited Two-Mode Generalized Squeezed Vacuum State as a Squeezed Two-Variable Hermite Polynomial Excitation State

A new class of excited two-mode generalized squeezed vacuum states denoted by |r,s,m,n〉 are presented, which are obtained by repeatedly applying creation operators a ^† and b ^† on the two-mode generalized squeezed vacuum state. We find that it is just regarded as a generalized squeezed two-variable Hermite polynomial excitation on the vacuum state and its normalization constant is just a Jacobi polynomial. Their statistical properties are investigated such as squeezing properties, photon number distribution and the violations of Cauchy-Schwartz inequality. Especially, the Wigner function for |r,s,m,n〉 depending on the excitation photon numbers is discussed graphically.     

Moduli space of model real submanifolds

In [6], to a completely nondegenerate germ of a real submanifold of a chosen C R-type (n, K) in a complex space, we assigned a tangent polynomial model of the submanifold. In the present paper, we construct the moduli space M(n, K) of the family of polynomial models, i.e., the space parametrizing the holomorphically nonequivalent polynomial models. The space thus obtained is used to construct C R-characteristic classes. Financially supported by the RFBR grant no. 05-01-0981 and by the grant NSh-2040.2003.1.     

Correlation between zeroes and poles ofS matrix for complex potentials

Using several illustrative examples, the nature of resonance poles and the corresponding zeroes of the s-waveS matrix is examined for several potentials having an absorptive pocket followed by a barrier. It is shown that even though the presence of absorption practically suppresses the manifestation of resonance in the elastic scattering cross section, the effect of the resonances generated by the absorptive pocket is more clearly manifested in the absorption cross section provided the barrier width is not too large. We further find that the signature of barrier top resonances are also more clearly manifested in the absorption cross section rather than in the elastic scattering cross section. These results have been interpreted in terms of complex resonance poles and corresponding zeroes of theS matrix. This implies that in complex potential scattering like heavy ion collisions, the reaction channel cross section peak is a more reliable signature of resonance phenomenon than the variation of the elastic channel cross section with energy.     

Jacobi,Ellipsoidal Coordinates and Superintegrable Systems

Abstract

We describe Jacobi’s method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems to which we then turn. A Schrödinger operator with potential on a Riemannian space is second-order superintegrable if there are 2n ? 1 (classically) functionally independent second-order symmetry operators. (The 2n ? 1 is the maximum possible number of such symmetries.) These systems are of considerable interest in the theory of special functions because they are multiseparable, i.e., variables separate in several coordinate sets and are explicitly solvable in terms of special functions. The interrelationships between separable solutions provides much additional information about the systems. We give an example of a superintegrable system and then present very recent results exhibiting the general structure of superintegrable systems in all real or complex two-dimensional spaces and three-dimensional conformally flat spaces and a complete list of such spaces and potentials in two dimensions.     

Zeroes of chromatic polynomials: A new approach to Beraha conjecture using quantum groups

The number of colourings of a graphG withQ or fewer colors is a polynomial inQ known as the chromatic polynomialP _G (Q). It coïncides with the partition functionF _G of theQ state Potts model onG at zero temperature and in the antiferromagnetic regimee ^K =0. In the planar case, the Beraha conjecture particularizes the numbers \(B_n = 4\cos ^2 \frac{\pi }{n}\) as possible accumulation points of real zeroes ofP _G in the infinite graph limit. We suggest in this work an approach based on recent developments of quantum groups to handle this conjecture. For the square, triangular and honeycomb lattices and systems wrapped on a cylinderl×t, we first exhibit in the (Q, e ^K ) Potts parameter space a critical line, whereF _G(Q,e ^K) has real zeroes converging to and only to theB _n 's asl, t→∞. The analysis is based on the vertex representation of theQ state Potts model, quantum algebraU _qSl (2) properties forq a root of unity, and conformal invariance.U _qSl (2) symmetry is present for anye ^K , including the chromatic polynomial casee ^K =0. Using an additional hypothesis on the eigenvalues structure and knowledge of the Potts parameter space, we then argue that forP _G (Q), real zeros occur and converge toB _n 's, 2≦n≦n ₀ only, wheren ₀ depends on the lattice. Extensions to other kinds of graphs and size dependence of the zeros are discussed. Finally physical applications are also mentioned.