Narrow Results

Here we discuss interaction of a single two-level atom with a single mode of interacting electromagnetic field in the Jaynes–Cummings model with the rotating wave approximation.

In this paper we study quantum mechanical phase distribution of some nonlinear optical phenomena in a general setting of interacting Fock space. We have investigated the optical phenomena of propagation through a nonlinear medium as in optical fiber and the process of photon absorption from a thermal beam. The input and output phase distribution have been investigated analytically in these two cases.

In this paper, we study Susskind–Glogower phase distribution of entangled state in the framework of interactions involving two modes of interacting Fock space which in its turn provides an interesting insigth into the structure of entangled state.

Starting with the interaction of interacting field and a two level atom we observe that the atom-interacting field system acquires a space parameter dependent Berry phase after the phase parameter slowly changes and ultimately returns to its initial form.

An experimentally realizable scheme is considered for manipulating quantum states using a general superposition (SUP) of products of interacting annihilation and creation operators. Application of such an operation on states with classical features introduces strong nonclassicality. This provides the possibility of engineering quantum states with nonclassical features.

We define a natural transformation of certain symmetric probability measures, and show that it brings the Gaussian-law as well as several deformed analogue of it to the Poisson-law or to the corresponding deformed Poisson-law, respectively. An important example is the q-Gaussian-law with its transform being the q-Poisson-law of Saitoh and Yoshida. By analogy we define an even more general class of deformed Poisson distributions, and show that, as in the classical and the q-case, these deformed Poisson-laws can be achieved as the distributions of certain operators.

The adjacency matrix of a comb graph is decomposed into a sum of monotone independent random variables with respect to the vacuum state. The vacuum spectral distribution is shown to be asymptotically the arcsine law as a consequence of the monotone central limit theorem. As an example the comb lattice is studied with explicit calculation.

In Ref. 2 the authors introduced field operators in one-mode type Interacting Fock Spaces whose spectral measures have common symmetric Jacobi recurrence coefficients but differ in the nonsymmetric ones. We show that the convolution of measures arising from addition of such field operators is the universal convolution of Accardi and Bożejko. We also present the associated central limit theorem in a more general form than in Ref. 2 and give it a proof based on the properties of the convolution.

Consider the Lévy–Meixner one-mode interacting Fock space {Γ_LM, 〈 ⋅, ⋅ 〉_LM}. Inspired by a derivative formula appearing in 〈 ⋅, ⋅ 〉_LM, we define scalar products 〈 ⋅, ⋅ 〉_{LM, n} in symmetric n-particle spaces. Then, we introduce a class of one-mode type interacting Fock spaces naturally associated to the one-dimensional infinitely divisible distributions with Lévy–Meixner type {μ_r; r > 0}. The Fourier transform in generalized joint eigenvectors of a family of Lévy–Meixner Jacobi fields provides a way to explicit a unitary isomorphism 𝔘_LM between and the so-called Lévy–Meixner white noise space . We derive a chaotic decomposition property of the quadratic integrable functionals of the Lévy–Meixner white noise processes in terms of an appropriate Wick tensor product. For their stochastic regularity, we give explicit form and sharp estimate of the associated Donsker's delta function.

We prove that the quadratic second quantization of an operator p on L²(ℝ^d) ∩ L^∞(ℝ^d) is an orthogonal projection on the quadratic Fock space if and only if p is a multiplication operator by a characteristic function χ_I, I ⊂ ℝ^d.

The time evolution of one mode interacting field which interacts with a vee-configuration three-level atom has been discussed. Generated nonclassical state has been utilized to study population inversion of atom and the total noise of the system. Also nonclassicality of the system is shown with the help of Mandel's parameter.

We consider Glauber dynamics on finite cycles. By introducing a vacuum state we consider an algebraic probability space for the generator of the dynamics. We obtain a quantum decomposition of the generator and construct an interacting Fock space. As a result we obtain a distribution of the generator in the vacuum state. We also discuss the monotonicity of the moments of spectral measure as the couplings increase. In particular, when the couplings are assumed to be uniform, as the cycle grows to an infinite chain, we show that the distribution (under suitable dilation and translation) converges to a Kesten distribution.

We prove that, each probability meassure on $ℝ$, with all moments, is canonically associated with (i) a ∗-Lie algebra; (ii) a complexity index labeled by pairs of natural integers. The measures with complexity index $(0,K)$ consist of two disjoint classes: that of all measures with finite support and the semi-circle-arcsine class (the discussion in Sec. 4.1 motivates this name). The class $C(\mu )=(0,0)$ coincides with the $\delta$-measures in the finite support case and includes the semi-circle laws in the infinite support case. In the infinite support case, the class $C(\mu )=(0,1)$ includes the arcsine laws, and the class $C(\mu )=(0,2)$ appeared in central limit theorems of quantum random walks in the sense of Konno. The classes $C(\mu )=(0,K)$, with $K\ge 3$, do not seem to be present in the literature. The class $(1,0)$ includes the Gaussian and Poisson measures and the associated ∗-Lie algebra is the Heisenberg algebra. The class $(2,0)$ includes the non-standard (i.e. neither Gaussian nor Poisson) Meixner distributions and the associated ∗-Lie algebra is a central extension of $\mathrm{\text{sl}}(2,ℝ)$. Starting from $n=3$, the ∗-Lie algebra associated to the class $(n,0)$ is infinite dimensional and the corresponding classes include the higher powers of the standard Gaussian.

The identification mentioned in the title allows a formulation of the multidimensional Favard lemma different from the ones currently used in the literature and which parallels the original 1-dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of positive definite kernels. The above result opens the way to the program of a purely algebraic classification of probability measures on ${ℝ}^d$ with moments of any order and more generally of states on the polynomial algebra on ${ℝ}^d$.

The quantum decomposition of classical real-valued random variables with all moments is one of the main ingredients in the proof.

We prove that, replacing the left Jordan–Wigner $q$-embedding by the symmetric $q$-embedding described in Sec. 2, the result of the corresponding central limit theorem changes drastically with respect to those obtained in Ref. 5. In fact, in the former case, for any $q\in ℂ$, the limit space is precisely the $1$-mode Interacting Fock Space (IFS) that realizes the canonical quantum decomposition of the limit classical random variable. In the latter case, this happens if and only if $q=\pm 1$. Furthermore, as shown in Sec. 4, the limit classical random variable turns out to coincide with the $1$-mode version of the $q^{\mathrm{\Lambda }}$-deformed quantum Brownian introduced by Parthasarathy${}^{8,9}$, and extended to the general context of bi-algebras by Schürman${}^{10,11}$. The last section of the paper (Appendix) describes this continuous version in white noise language, leading to a simplification of the original proofs, based on quantum stochastic calculus.

The $qq$-bit is the $q$-deformation of the $q$-bit. It arises canonically from the quantum decomposition of Bernoulli random variables and the $q$-parameter has a natural probabilistic and physical interpretation as asymmetry index of the given random variable. The connection between a new type of $q$-deformation (generalizing the Hudson–Parthasarathy bosonization technique and different from the usual one) and the Azema martingale was established by Parthasarathy. Inspired by this result, Schürmann first introduced left and right $q$-JW-embeddings of $M_2$ ($2\times 2$ complex matrices) into the infinite tensor product ${\otimes }_{n\in \mathbb{N}}{M}_2$, proved central limit theorems (CLT) based on these embeddings in the context of ∗-bi-algebras and constructed a general theory of $q$-Levy processes on ∗-bi-algebras. For $q=-1$, left $q$-JW-embeddings define the Jordan–Wigner transformation, used to construct a tensor representation of the Fermi anti-commutation relations (bosonization). For $q=1$, they reduce to the usual tensor embeddings that were at the basis of the first quantum CLT due to von Waldenfels. The present paper is the first of a series of four in which we study these theorems in the tensor product context. We prove convergence of the CLT for all $q\in ℂ$. The moments of the limit random variable coincide with those found by Parthasarathy in the case $q\in [-1,+1)$. We prove that the space where the limit random variable is represented is not the Boson Fock space, as in Parthasarathy, but the monotone Fock space in the case $q=0$ and a non-trivial deformation of it for $q\ne -1,0,+1$. The main analytical tool in the proof is a non-trivial extension of a recently proved multi-dimensional, higher order Cesaro-type theorem. The present paper deals with the standard CLT, i.e. the limit is a single random variable. Paper¹ deals with the functional extension of this CLT, leading to a process. In paper² the left $q$-JW–embeddings are replaced by symmetric $q$-embeddings. The radical differences between the results of the present paper and those of² raise the problem to characterize those CLT for which the limit space provides the canonical decomposition of all the underlying classical random variables (see the Introduction, Lemma 4.5 and Sec. 5 of the present paper for the origin of this problem). This problem is solved in the paper³ for CLT associated to states satisfying a generalized Fock property. The states considered in this series have this property.

In the first part of the present paper, we have proved that, in order to find an explicit expression for the action, on the $\mu$-orthogonal polynomials, of the 1-parameter unitary groups $e^{itP}$ and $e^{it{P}^2/2}$, the solution of the inverse normal order problem on the quantum algebra canonically associated to the classical semi-circle random variable is required. In this paper we solve this problem. The solution is obtained in two steps. First we determine the explicit form of normal order in the more general framework of $1$-mode-type interacting Fock spaces. Then this result is applied to solve the inverse normal order problem in the semi-circle case.

The dynamical behaviour of Quantum Electrodynamics(QED) in Stochastic Medium is formulated by state equation in Interacting Fock Space which is a generalization of Langevin dynamics in Boson Fock Space.

We introduce a one-mode type interacting Fock space naturally associated to the negative binomial distribution μ_r,α. The Fourier transform in generalized joint eigenvectors of a family {J_ϕ ; ϕ ∈ ε} of Pascal Jacobi fields provides a way to explicit a unitary isomorphism between and the so-called Pascal white noise space L²(ε′, Λ_r,α). Then, we derive a chaotic decomposition property of the quadratic integrable functionals of the Pascal white noise process in terms of an appropriate wick tensor product.