Interacting Fock spaces and subproduct systems

We present a new more flexible definition of interacting Fock space that allows to resolve in full generality the problem of embeddability. We show that the same is not possible for regularity. We apply embeddability to classify interacting Fock spaces by squeezings. We give necessary and sufficient criteria for when an interacting Fock space has only bounded creators, giving thus rise to new classes of non-selfadjoint and selfadjoint operator algebras.

The qq-bit (III): Symmetric q-Jordan–Wigner embeddings

We prove that, replacing the left Jordan–Wigner [Formula: see text]-embedding by the symmetric [Formula: see text]-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 [Formula: see text], the limit space is precisely the [Formula: see text]-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 [Formula: see text]. Furthermore, as shown in Sec. 4, the limit classical random variable turns out to coincide with the [Formula: see text]-mode version of the [Formula: see text]-deformed quantum Brownian introduced by Parthasarathy[Formula: see text], and extended to the general context of bi-algebras by Schürman[Formula: see text]. 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 (I): Central limits with left q-Jordan–Wigner embeddings, monotone interacting Fock space, Azema random variable, probabilistic meaning of q

The [Formula: see text]-bit is the [Formula: see text]-deformation of the [Formula: see text]-bit. It arises canonically from the quantum decomposition of Bernoulli random variables and the [Formula: see text]-parameter has a natural probabilistic and physical interpretation as asymmetry index of the given random variable. The connection between a new type of [Formula: see text]-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 [Formula: see text]-JW-embeddings of [Formula: see text] ([Formula: see text] complex matrices) into the infinite tensor product [Formula: see text], proved central limit theorems (CLT) based on these embeddings in the context of ∗-bi-algebras and constructed a general theory of [Formula: see text]-Levy processes on ∗-bi-algebras. For [Formula: see text], left [Formula: see text]-JW-embeddings define the Jordan–Wigner transformation, used to construct a tensor representation of the Fermi anti-commutation relations (bosonization). For [Formula: see text], 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 [Formula: see text]. The moments of the limit random variable coincide with those found by Parthasarathy in the case [Formula: see text]. 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 [Formula: see text] and a non-trivial deformation of it for [Formula: see text]. 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 [Formula: see text]-JW–embeddings are replaced by symmetric [Formula: see text]-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.

The q-Gamma White Noise

For 0 < q < 1 and 0 < α < 1, we construct the infinite dimensional q-Gamma white noise measure γα,q by using the Bochner-Minlos theorem. Then we give the chaos decomposition of an L2 space with respect to the measure γα,q via an isomorphism with the 1-mode type interacting Fock space associated to the q-Gamma measure.

An experimental scheme to generate nonclassical states in interacting Fock space

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.

Glauber dynamics on the cycles: Spectral distribution of the generator

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.

NONCLASSICAL STATES IN INTERACTING FOCK SPACE

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.