Bose-Einstein condensates on a ring with periodic scattering length: Spontaneous symmetry breaking and entanglement

We study the properties of Bose-Einstein condensates in a one-dimensional ring with periodically modulated interaction strength. We demonstrate that within the semiclassical mean-field theory, a condensate can undergo a quantum phase transition between a solitonlike state and a spatially periodic condensate that matches the scattering length modulation. However, we show via exact diagonalization of the full many-body Hamiltonian that the solitonic ground states do not exist in the true quantum treatment. Instead, the system is driven into an entangled macroscopic-quantum-superposition-like state and the semiclassical soliton state may be regarded as a decohered macroscopic quantum superposition state. We investigate how the properties of the quantum ground state scales with the total number of atoms. Our work elucidates how semiclassical properties emerge from quantum behavior.

Figure 1

(Color online) Mean-field ground-state wave function with $d=2$ and various values of ${\gamma }_0$, obtained by solving Eq. (4) using imaginary time propagation. ${\gamma }_0<0.52$ results in sinusoidal solutions (the dot-dashed line). $0.52<{\gamma }_0<0.57$ results in asymmetrical two-peaked solutions (the solid line). ${\gamma }_0>0.57$ results in single-peaked solitonlike solutions (the dashed line).

Figure 2

(Color online) Total dimensionless energy $\left(E\right)$, kinetic energy $\left(E_{\mathrm{kin}}\right)$, interaction energy $\left(E_{\mathrm{int}}\right)$ per particle, and chemical potential $\left(\mu \right)$ as functions of ${\gamma }_0$. The inset shows the first derivative of the ground-state energy with respect to ${\gamma }_0$, indicating quantum phase transition at ${\gamma }_0={\gamma }_{\mathrm{crit}}=0.52$. The critical value ${\gamma }_{\mathrm{crit}}$ is indicated by the arrows in both the main figure and the inset.

Figure 3

(Color online) Many-body ground-state energy per particle for $N=2$, 4, and 6. The bottom dashed curve shows the corresponding mean-field result.

Figure 4

(Color online) Density profile of quantum mechanical ground states obtained by exact diagonalization of the many-body Hamiltonian with $N=6$.

Figure 5

(Color online) Left-right spatial correlation function for $N=2$, 4, and 6.

Figure 6

Entropy $S_0$ as a function of ${\gamma }_0$ for $N=2$.

Figure 7

Ground-state overlap as a function of ${\gamma }_0$ for $\delta \gamma =0.1$ and $N=2$.

Figure 8

(Color online) Energy gap between the quantum mechanical ground state and the first excited state as a function of particle number $N$.