Vortex macroscopic superpositions in ultracold bosons in a double-well potential

We study macroscopic superpositions in the orbital rather than the spatial degrees of freedom, in a three-dimensional, double-well system. We show that the ensuing dynamics of $N$ interacting excited ultracold bosons, which, in general, requires at least eight single-particle modes and $\left(\genfrac{}{}{0pt}{}{N+7}{N}\right)$ Fock vectors, is described by a surprisingly small set of many-body states. An initial state with half the atoms in each well, and purposely excited in one of them, gives rise to the tunneling of axisymmetric and transverse vortex structures. We show that transverse vortices tunnel orders of magnitude faster than axisymmetric ones and are therefore more experimentally accessible. The tunneling process generates macroscopic superpositions only distinguishable by their orbital properties and within experimentally realistic times.

Figure 1

Schematic of the 3D DW. The eight modes are represented as distorted spherical harmonics (magenta) and the processes among them by arrows. In the right well, the arrows represent both tunneling and interacting processes. The blue surface is the DW equipotential surface.

Figure 2

Axisymmetric and transverse vortex tunneling. (a) Average occupation of well $j=1$ for unexcited atoms (red solid curve) and excited atoms in an axisymmetric vortex with $m=1$ (blue dash-dotted curve). (b) Probability densities $P_i\left(t\right)$ showing the dominant Fock vectors $i$ contributing to dynamics. The excited and unexcited atoms slosh between wells with the same period. (c) Average occupation of well $j=1$ for excited atoms with $m=0$ (blue thin curve) and with $m=1$ (red thick curve) for the transverse vortex. (d) Probability densities showing that two many-body Fock vectors dominate the dynamics. The small coupling to an extra pair of vectors is due to the vortex-antivortex intralevel transitions.

Figure 3

Entropies and angular momentum for the axisymmetric and transverse vortices. (a) Spatial (blue thick curve) and angular momentum for $m=1$ (red thin curve) entanglement entropy normalized to their maximal possible value, for the axisymmetric vortex. (b) Same for the transverse vortex, with angular momentum entropy for $m=0$. (c) For the axisymmetric vortex, the total ${\mathbb{L}}^2$ (blue dash-dotted curve) and ${\mathbb{L}}_z$ (red solid curve) are conserved, while on-site ${\mathbb{L}}_z$ (black dashed curve) is not. (d) For the transverse vortex, both total ${\mathbb{L}}_z$ and on-site ${\mathbb{L}}_z$ (superposed) are conserved, while total ${\mathbb{L}}^2$ is not.

Figure 4

Energy splittings for vortex macroscopic superpositions (VMSs). Axisymmetric (analytical, solid blue curve; numerical, blue crosses) and transverse [analytical, dashed (small $1/\zeta$ limit), dotted (large $1/\zeta$ limit with $\chi =1/80$), and dash-dotted (same with $\chi =1/10$) red curves; numerical, asterisks ($\chi =1/80$) and circles ($\chi =1/10$)] energy splittings. In the transverse case there are two limits: small and large $1/\zeta$. The analytical small $1/\zeta$ limit does not depend on $\chi$. The analytical large $1/\zeta$ limit depends on $\chi$.

Figure 5

Schematic of the Fock vectors. Red circles represent atoms. (a),(b) The Fock vectors that correspond to the initial conditions for the axisymmetric and transverse cases, respectively. (c) The Fock vector to which the transverse vortex is coupled along evolution due to the creation or annihilation of vortex-antivortex pairs.

Figure 6

Logarithm of the time cost (in seconds) of the exact diagonalization algorithm as a function of the logarithm dimension of the Hilbert space $\Omega$. Crosses correspond to the numerical calculation for $N=4$ to $N=12$ atoms. The solid line corresponds to the fitting to a straight line of slope 2.9, thus showing the $\mathcal{O}\left({\Omega }^3\right)$ behavior.