For spin-1 condensates, the spatial degrees of freedom can be considered as being frozen at temperature zero, while the spin-degrees of freedom remain free. Under this condition, the entanglement entropy has been derived exactly with an analytical form. The entanglement entropy is found to decrease monotonically with the increase of the magnetic polarization as expected. However, for the ground state in polar phase, an extremely steep fall of the entropy is found when the polarization emerges from zero. Then the fall becomes a gentle descent after the polarization exceeds a turning point.
Abstract Condensate of spin-1 atoms frozen in a unique spatial mode may possess large internal degrees of freedom. The scattering amplitudes of polarized cold atoms scattered by the condensate are obtained with the method of fractional parentage coefficients that treats the spin degrees of freedom rigorously. Channels with scattering cross sections enhanced by the square of the atom number of the condensate are found. Entanglement between the condensate and the propagating atom can be established by scattering. Entanglement entropy is analytically obtained for arbitrary initial states. Our results also give a hint for the establishment of quantum thermal ensembles in the hyperfine space of spin states.
We study the spin entanglement in a spin-1 Bose-Einstein condensate with antiferromagnetic atomic interactions using the Hartree-Fock approach. Based on the isomorphism between symmetric
N-qubit states and spin-
j=N/2 states, we analyze the negativity of the spin-1 ground state of the condensate viewed as a two spins
1/2, and explore its dependence with the temperature and external Zeeman fields. The scope of this type of entanglement is highlighted and contrasted with other types of entanglement, as the mode and particle entanglement. It is shown that, at finite temperatures, there is a strong dependence of the negativity with respect to the strengths of quadratic Zeeman fields and the spin-spin interactions of the condensate. Interestingly, in the antiferromagnetic ground-state phase, the negativity and the linear Zeeman field are connected quadratically through the equation of a simple circle, in which its radius depends on the temperature. On the other hand, for the polar phase, being the phase that exhibits the highest degree of entanglement, we were able to identify a clear dependency on the spin-spin interactions and the Zeeman fields that can be expressed analytically in a closed form as a function of the temperature. This results might be relevant for applications in quantum information and metrology.
Thermodynamic features of spin-1 condensates with a fixed magnetization M at extremely low temperature T are studied. Three turning points T1 to T3 are found. After the freeze of the spatial degrees of freedom (DF) and when T T1, spin-fluctuation becomes significant and the internal energy U relative to the ground state increases. However, U, which is contributed by all the spin-DF, is found always < (3/2)kBT, the energy assigned to the spatial motion of only a single particle. T2 marks the maximization of the entropy, while T3 marks that the interference from the spatial DF is no more negligible. In particular, the strongly bound condensates with N small and M close to N may serve as low-dimensional and nearly pure spin-systems with relative higher T1 and T2(< T3). The energy involved in them is extremely low, thus they might be promising for application.
The variation of the hyperfine populations $N{\ensuremath{\rho}}_{\ensuremath{\mu}}$ of the ground state (g.s.) of spin-2 condensates against a magnetic field
B is studied. A
N-body theory which is rigorous in dealing with the spin degrees of freedom has been adopted. The conservation of the total magnetization
M is assumed. A combined hyperfine population $N{\ensuremath{\rho}}_{\text{comb}}=N({\ensuremath{\rho}}_{0}\ensuremath{-}3{\ensuremath{\rho}}_{2}\ensuremath{-}3{\ensuremath{\rho}}_{\ensuremath{-}2})$, which is an observable, is defined. The lower bound for this population has been derived analytically. It turns out that the lower bound is identical to the actual population when
B=0 and $\ensuremath{\infty}$, and the population is monotonically uprising in between. Thus, via the analytical form of the lower bound, the variation of the population against
B can be roughly known in advance. Numerical examples are given to demonstrate the applicability of the bound. When
M=0, the g.s. in the polar phase was found to be highly unstable against
B because the neighboring level density is extremely high. When
M=N, the g.s. in the cyclic phase was also found to be highly unstable. This is because the appearance of
B, even if it is very weak, will break the high degeneracy of the g.s. and result in having numerous levels emerging in the neighborhood of the g.s.
We consider the ground state of the XY model on an infinite chain at zero temperature. Following Bennett, Bernstein, Popescu, and Schumacher we use entropy of a sub-system as a measure of entanglement. Vidal, Latorre, Rico and Kitaev conjectured that von Neumann entropy of a large block of neighboring spins approaches a constant as the size of the block increases. We evaluated this limiting entropy as a function of anisotropy and transverse magnetic field. We used the methods based on integrable Fredholm operators and Riemann-Hilbert problem. The entropy is singular at phase transitions.
An isolated spin system that is in internal thermodynamic equilibrium and that has an upper limit to its allowed energy states can possess a negative temperature. We calculate the thermodynamic characteristics and the concurrence in this system over the entire range of positive and negative temperatures. Our calculation was performed for different real structures, which can be used in experiments. It is found that the temperature dependence of the concurrence is substantially asymmetrical similarly to other thermodynamic characteristics. At a negative temperature the maximum concurrence and the absolute temperature of the entanglement appearance are significantly larger than those at a positive temperature. The concurrence can be characterized by two dimensionless parameters: the ratio between the Zeeman and dipolar energies and the ratio of the thermal and dipolar energies. It was shown that for all considered structures the dimensionless temperatures of the transition between entanglement and separability of the first and second spins are independent of spin structure and the number of spins.