Exact dynamics in the inhomogeneous central-spin model

We study the dynamics of a single spin-1/2 coupled to a bath of spins-1/2 by inhomogeneous Heisenberg couplings including a central magnetic field. This central-spin model describes decoherence in quantum bit systems. An exact formula for the dynamics of the central spin is presented, based on the Bethe ansatz. This formula is evaluated explicitly for initial conditions such that the bath spins are completely polarized at the beginning. For this case we find, after an initial decay, a persistent oscillatory behaviour of the central spin. For a large number of bath spins $N_b$, the oscillation frequency is proportional to $N_b$, whereas the amplitude behaves like $1/N_b$, to leading order. No asymptotic decay due to the non-uniform couplings is observed, in contrast to some recent studies.Comment: 7 pages, 3 figure

Dynamics and decoherence in the central spin model using exact methods

The dynamics and decoherence of an electronic spin-1/2 qubit coupled to a bath of nuclear spins via hyperfine interactions in a quantum dot is studied. We show how exact results from the integrable solution can be used to understand the dynamic behavior of the qubit. It is possible to predict the main frequency contributions and their broadening for relatively general initial states analytically, leading to an estimate of the corresponding decay times. Furthermore, for a small bath polarization, a new low-frequency time scale is observed.Comment: 4 pages, 2 figures. Published version. See also http://www.physik.uni-kl.de/eggert/papers/index.htm

Discretized vs. continuous models of p-wave interacting fermions in 1D

We present a general mapping between continuous and lattice models of Bose- and Fermi-gases in one dimension, interacting via local two-body interactions. For s-wave interacting bosons we arrive at the Bose-Hubbard model in the weakly interacting, low density regime. The dual problem of p-wave interacting fermions is mapped to the spin-1/2 XXZ model close to the critical point in the highly polarized regime. The mappings are shown to be optimal in the sense that they produce the least error possible for a given discretization length. As an application we examine the ground state of a interacting Fermi gas in a harmonic trap, calculating numerically real-space and momentum-space distributions as well as two-particle correlations. In the analytically known limits the convergence of the results of the lattice model to the continuous one is shown.Comment: 7 pages, 5 figure

Spin- and entanglement-dynamics in the central spin model with homogeneous couplings

We calculate exactly the time-dependent reduced density matrix for the central spin in the central-spin model with homogeneous Heisenberg couplings. Therefrom, the dynamics and the entanglement entropy of the central spin are obtained. A rich variety of behaviors is found, depending on the initial state of the bath spins. For an initially unpolarized unentangled bath, the polarization of the central spin decays to zero in the thermodynamic limit, while its entanglement entropy becomes maximal. On the other hand, if the unpolarized environment is initially in an eigenstate of the total bath spin, the central spin and the entanglement entropy exhibit persistent monochromatic large-amplitude oscillations. This raises the question to what extent entanglement of the bath spins prevents decoherence of the central spin.Comment: 8 pages, 2 figures, typos corrected, published versio

Characterisation of multiple hindered settling regimes in aggregated mineral suspensions

Aqueous suspensions of magnesium hydroxide are shown to exhibit low ζ-potential behavior and highly complex settling dynamics. Two distinct regimes of hindered settling behavior are observed on either side of a threshold concentration, ϕ*, of 2.38% v/v, which is considerably below the gel point, ϕg, observed at 5.4 ± 1.6% v/v. The low-concentration regime was characterized by a very large Richardson and Zaki exponent of 146, a factor of 10 larger than that of the high-concentration regime. Michaels and Bolger analysis of the low-concentration regime implies settling governed by large, low-density macroaggregates of 138–147 μm diameter and low intraaggregate packing fractions on the order of 0.05, which is in good agreement with in situ particle characterization undertaken using particle vision and measurement (PVM) and focused-beam reflectance measurements (FBRM). The large macroaggregates must undergo some shear densification within the higher-concentration hindered settling regime in order for the suspension to gel at a concentration of 5.4% v/v. Consequently, fluid flow past small, shear-resistant primary agglomerates, observed within the aggregates using scanning electron microscopy and flow particle image analysis, during aggregate densification may represent the limiting step for dewatering within the high-concentration regime

The open XXZ-chain: Bosonisation, Bethe ansatz and logarithmic corrections

We calculate the bulk and boundary parts of the free energy for an open spin-1/2 XXZ-chain in the critical regime by bosonisation. We identify the cutoff independent contributions and determine their amplitudes by comparing with Bethe ansatz calculations at zero temperature T. For the bulk part of the free energy we find agreement with Lukyanov's result [Nucl.Phys.B 522, 533 (1998)]. In the boundary part we obtain a cutoff independent term which is linear in T and determines the temperature dependence of the boundary susceptibility in the attractive regime for $T\ll 1$. We further show that at particular anisotropies where contributions from irrelevant operators with different scaling dimensions cross, logarithmic corrections appear. We give explicit formulas for these terms at those anisotropies where they are most important. We verify our results by comparing with extensive numerical calculations based on a numerical solution of the T=0 Bethe ansatz equations, the finite temperature Bethe ansatz equations in the quantum-transfer matrix formalism, and the density-matrix renormalisation group applied to transfer matrices.Comment: 35 pages, 8 figure

Two-spinon dynamic structure factor of the one-dimensional S=1/2 Heisenberg antiferromagnet

The exact expression derived by Bougourzi, Couture, and Kacir for the 2-spinon contribution to the dynamic spin structure factor $S_{zz}(q,\omega)$ of he one-dimensional $S$=1/2 Heisenberg antiferromagnet at $T=0$ is evaluated for direct comparison with finite-chain transition rates ($N\leq 28$) and an approximate analytical result previously inferred from finite-$N$ data, sum rules, and Bethe-ansatz calculations. The 2-spinon excitations account for 72.89% of the total intensity in $S_{zz}(q,\omega)$. The singularity structure of the exact result is determined analytically and its spectral-weight distribution evaluated numerically over the entire range of the 2-spinon continuum. The leading singularities of the frequency-dependent spin autocorrelation function, static spin structure factor, and $q$-dependent susceptibility are determined via sum rules.Comment: 6 pages (RevTex) and 5 figures (Postscript