Aspects of the decoherence in high spin environments: Breakdown of the mean-field approximation

The study of the decoherence of qubits in spin systems is almost restricted to environments whose constituents are spin-$\frac{1}{2}$ particles. In this paper we consider environments that are composed of particles of higher spin, and we investigate the consequences on the dynamics of a qubit coupled to such baths via Heisenberg $XY$ and Ising interactions. It is shown that while the short time decay in both cases gets faster as the magnitude of the spin increases, the asymptotic behavior exhibits an improvement of the suppression of the decoherence when the coupling is through Heisenberg $XY$ interactions. In the case of a transverse Ising model, we find that the mean field approximation breaks down for high values of the spin.Comment: Preprint; 27 pages, 8 figure

Diffusion coefficients preserving long-time correlations: Consequences on the Einstein relation and on entanglement in a bosonic Bogoliubov system

We analytically derive the diffusion coefficients that drive a system of $N$ coupled harmonic oscillators to an equilibrium state exhibiting persistent correlations. It is shown that the main effect of the latter consists in a renormalization of the natural frequencies and the friction coefficients of the oscillators. We find that the Einstein relation may be satisfied at low temperatures with frequency-dependent effective friction coefficients, provided that the physical constraints are fulfilled. We also investigate the entanglement evolution in a bipartite bosonic Bogoliubov system initially prepared in a thermal squeezed state. It is found that, in contrast to what one may expect, strong coupling slows down the entanglement sudden death, and for initially separable states, entanglement generation may occur

On the analytical evaluation of the magnetization of ferromagnetic lattices

We investigate analytically the magnetization of Heisenberg ferromagnetic lattices in one and two dimensions, and we derive approximate expressions that are valid at high and low temperatures. In the case of the spin-$\frac{1}{2}$ Heisenberg XX chain in a transverse field, we show that, when the applied magnetic field $h$ exceeds its critical value $h_c=J$, where $J$ is the exchange coupling constant, the magnetization per site deviates at low temperatures from its saturation value, $\frac{1}{2}$, following a power series with terms involving the power laws $T^{1/2}$, $T^{3/2}$, $T^{5/2}$ ..etc. When $h<h_c$, the zero temperature magnetization per spin turns out to be equal to $\frac{1}{\pi}\arcsin(\frac{h}{h_c})$. In this case, the temperature dependence of the magnetization is given by a series expansion with power laws of the form $T$, $T^3$, $T^{5}$,...etc. In both cases, the coefficients of the expansions are temperature-dependent and are explicitly derived. Using he properties of the Eulerian polynomials, we show that, because of the fast convergence of the derived series for the Fermi-Dirac and the Bose-Einstein distributions, it is possible (in particular in strong magnetic fields) to express the magnetization of the Heisenberg model in a simple analytical form. Furthermore, the analytical results are compared with the exact numerical ones.Comment: 28 pages; 10 figure

Quantum mean-field treatment of the dynamics of a two-level atom in a simple cubic lattice

The mean field approximation is used to investigate the general features of the dynamics of a two-level atom in a ferromagnetic lattice close to the Curie temperature. Various analytical and numerical results are obtained. We first linearize the lattice Hamiltonian, and we derive the self-consistency equation for the order parameter of the phase transition for arbitrary direction of the magnetic field. The reduced dynamics is deduced by tracing out the degrees of freedom of the lattice, which results in the reduction of the dynamics to that of an atom in an effective spin bath whose size is equal to the size of a unit cell of the lattice. It is found that the dephasing and the excited state occupation probability may be enhanced by applying the magnetic field along some specific directions. The dependence on the change of the temperature and the magnitude of spin is also investigated. It turns out that the increase of thermal fluctuations may reduce the occupation probability of the excited state. The entanglement of two such atoms that occupy non-adjacent cells is studied and its variation in time is found to be not much sensitive to the direction of the magnetic field. Entanglement sudden death and revival is shown to occur close to the critical temperature.Comment: New analytical results adde

On the partial trace over collective spin-degrees of freedom

We derive analytical properties for the degeneracy $\nu(N,j)$ occurring in the decomposition $\bigoplus\limits_{j}^\frac{N}{2}\nu(N,j)\mathbb C^{2j+1}$ of the state space $\mathbb C^{2\otimes N}$. We also investigate the dynamics of two qubits coupled via Ising interactions to separate spin baths, and we study the thermodynamic limit.Comment: 14 pages, 6 figures, minor typographical errors fixe