Magnetization, squeezing, and entanglement in dipolar spin-1 condensates

We study the static and dynamic responses of an $F=1$ dipolar spinor condensate with magnetic dipole moment under external magnetic fields, focusing on the effects induced by the magnetic dipolar interactions between atoms. A rich set of phenomena are investigated such as the ground state quantum phases, the magnetization steps, spin squeezing, and macroscopic entanglement generation.

Figure 1

Magnetic phases of dipolar spin-1 condensate. Region A and B corresponds to the ferromagnetic phases with, respectively, the in-plane and the uniaxial anisotropy. Region C represents roughly an anti-ferromagnetic phase.

Figure 2

(Color online) The squeezing parameter $\xi$ as a function of transverse field $H_x$ and $c$ for $c_2>0$ and $N=40$. The dashed and solid lines on the contour plot represent the saturation field, which corresponds to the local minimum of $\xi$, in regions B and C, respectively.

Figure 3

(Color online) The dynamic response of the magnetization to a magnetic field sweep. The parameters we used here are $N=5$, $c=-0.33$, $H_x=3$, and $v={10}^{-3}$, which represents a condensate in region B. The arrows denote the positions of the magnetization jumps. After the third jump, the amplitude of level $\mid N,-N⟩$ in the condensate wave function is essentially zero, therefore, no more jumps are observed at higher field. The inset shows the energy spectrum as a function of applied longitudinal field $H_z$, the arrows in the inset mark the first three anti-crossings which match with the arrows in the main figure.

Figure 4

The scheme for creating entangled atomic state. (a) Field strength $H_x$ as a function of time. (b) The field dependence of the overlap of the condensate wave function with the maximally entangled state ∣GHZ⟩ for $N=20$ and a linear field sweeping rate $d{H}_x∕dt=-0.06$.