Ultra-cold alkali atoms trapped in two distinct hyperfine states in an external magnetic field can mimic magnetic systems of spin-1/2 particles. We describe the spin-dependent effective interaction as a spin–spin interaction. As a consequence of the zero range, the interaction of spin-1/2 bosons can be described as an Ising or, alternatively, as an XY -coupling. We calculated the spin–spin interaction parameters as a function of the external magnetic field in the degenerate internal state (DIS) approximation. We illustrate the advantage of the spin–spin interaction form by mapping the system of N spin-1/2 bosons confined by a tight trapping potential on that of N spin-1/2 spins coupled via an infinite range interaction.
Ultra-cold alkali atoms trapped in two distinct hyperfine states in an external magnetic field can mimic magnetic systems of spin-1/2 particles. We describe the spin-dependent effective interaction as a spin–spin interaction. As a consequence of the zero range, the interaction of spin-1/2 bosons can be described as an Ising or, alternatively, as an XY -coupling. We calculated the spin–spin interaction parameters as a function of the external magnetic field in the degenerate internal state (DIS) approximation. We illustrate the advantage of the spin–spin interaction form by mapping the system of N spin-1/2 bosons confined by a tight trapping potential on that of N spin-1/2 spins coupled via an infinite range interaction.
An important feature of cold atomic vapours is that particle separations, which are typically of order 10 2 nm, are usually an order of magnitude larger than the length scales associated with the atom–atom interaction. Consequently, the two-body interaction between atoms dominates, and three-and higher-body interactions are unimportant. Moreover, since the atoms have low velocities, many properties of these systems may be calculated in terms of a single parameter, the scattering length. An alkali atom in its electronic ground state has several different hyperfine states, as we have seen in Secs. 3.1 and 3.2. Interatomic interactions give rise to transitions between these states and, as we described in Sec. 4.6, such processes are a major mechanism for loss of trapped atoms. In a scattering process, the internal states of the particles in the initial or final states are described by a set of quantum numbers, such as those for the spin, the atomic species, and their state of excitation. We shall refer to a possible choice of these quantum numbers as a channel . At the temperatures of interest for Bose–Einstein condensation, atoms are in their electronic ground states, and the only relevant internal states are therefore the hyperfine states. Because of the existence of several hyperfine states for a single atom, the scattering of cold alkali atoms is a multi-channel problem.
We introduce a theoretical approach to determine the spin structure of harmonically trapped atoms with two-body zero-range interactions subject to an equal mixture of Rashba and Dresselhaus spin-orbit coupling created through Raman coupling of atomic hyperfine states. The spin structure of bosonic and fermionic two-particle systems with finite and infinite two-body interaction strength g is calculated. Taking advantage of the fact that the N-boson and N-fermion systems with infinitely large coupling strength g are analytically solvable for vanishing spin-orbit coupling strength kso and vanishing Raman coupling strength $\mathrm{\ensuremath{\Omega}}$, we develop an effective spin model that is accurate to second order in $\mathrm{\ensuremath{\Omega}}$ for any kso and infinite g. The three- and four-particle systems are considered explicitly. It is shown that the effective spin Hamiltonian, which contains a Heisenberg exchange term and an anisotropic Dzyaloshinskii-Moriya exchange term, describes the transitions that these systems undergo with the change of kso as a competition between independent spin dynamics and nearest-neighbor spin interactions.
Synthetic spin-orbit coupling in cold atoms couples the pseudospin and spatial degrees of freedom, and therefore, the inherent spin symmetry of the system plays an important role. In systems of two pseudospin degrees, two particles contain symmetric states and antisymmetric states, but the spin symmetry can be mixed for more particles. This mixed spin symmetry has been overlooked and has not been investigated thoroughly. We study the role of mixed spin symmetry in the presence of spin-orbit coupling and consider a system of three bosons with two hyperfine states trapped in a harmonic potential. We investigate the ground state and the energy spectrum by implementing exact diagonalization. Similar to that in two-particle systems, the interplay between spin-orbit coupling and repulsive interactions between antialigned pseudospins increases the population of the unaligned spin components in the ground state. Also, the emergence of mixed-spin symmetric states compensates for the rise of the interaction energy. In contrast to two-particle systems, the pair correlation of the ground state is analogous to the Tonks-Girardeau gas even for relatively small contact interactions, and this feature is enhanced by the spin-orbit coupling.
Cold-atom developments suggest the prospect of measuring scaling properties and long-range fluctuations of continuous phase transitions at zero temperature. We discuss the conditions for characterizing the phase separation of Bose-Einstein condensates of boson atoms in two distinct hyperfine spin states. The mean-field description breaks down as the system approaches the transition from the miscible side. An effective spin description clarifies the ferromagnetic nature of the transition. We show that a difference in the scattering lengths for the bosons in the same spin state leads to an effective internal magnetic field. The point at which the internal magnetic field vanishes ( i.e. , equal values of the like-boson scattering lengths) is a special point. We show that the long-range density fluctuations are suppressed near that point, while the effective spin exhibits the long-range fluctuations that characterize critical points. The zero-temperature system exhibits critical opalescence with respect to long-wavelength waves of impurity atoms that interact with the bosons in a spin-dependent manner.
Using cold bosonic atoms with two (hyperfine) ground states, we introduce a spin-boson mixture that allows one to implement the quantum Ising model in a tunable dissipative environment. The first specie lies in a deep optical lattice with tightly confining wells and forms a spin array; spin-up (spin-down) corresponds to occupation by one (no) atom at each site. The second specie forms a superfluid reservoir. Different species are coupled coherently via laser transitions and collisions. Whereas the laser coupling mimics a transverse field for the spins, the coupling to the reservoir sound modes induces a ferromagnetic (Ising) coupling as well as dissipation. This gives rise to an order-disorder quantum phase transition where the effect of dissipation can be studied in a controllable manner.
From a theoretical point of view, one of the appealing features of clouds of alkali atom vapours is that particle separations, which are typically of order 10 2 nm, are large compared with the scattering length a which characterizes the strength of interactions. Scattering lengths for alkali atoms are of the order of 100 a 0 , where a 0 , is the Bohr radius, and therefore alkali atom vapours are dilute, in the sense that the dominant effects of interaction are due to two-body encounters. It is therefore possible to calculate properties of the gas reliably from a knowledge of two-body scattering at low energies, which implies that information about atomic scattering is a key ingredient in work on Bose–Einstein condensates. An alkali atom in its electronic ground state has several different hyperfine states, as we have seen in Secs. 3.1 and 3.2. Interatomic interactions give rise to transitions between these states and, as we described in Sec. 4.6, such processes are a major mechanism for loss of trapped atoms. In a scattering process, the internal states of the particles in the initial or final states are described by a set of quantum numbers, such as those for the spin, the atomic species, and their state of excitation. We shall refer to a possible choice of these quantum numbers as a channel.1 At the temperatures of interest for Bose-Einstein condensation, atoms are in their electronic ground states, and the only relevant internal states are therefore the hyperfine states. Because of the existence of several hyperfine states for a single atom, the scattering of cold alkali atoms is a multi-channel problem.