Ultracold Bosons in a Tilted Multilevel Double-Well Potential

The $N$-body problem in a tilted double well requires new features for macroscopic quantum superposition in ultracold atoms. In particular, one needs to go beyond the single-particle ground state in each well. We provide explicit criteria for when two energy levels are needed to describe the state space. For typical experimental parameters, two levels are indeed required for the creation of macroscopic superposition states. Furthermore, we show that a small tilt causes the collapse of such states. However, partial macroscopic superposition states reappear when the tilt can be compensated by atom-atom interactions.

Figure 1

Energy eigenstates of a symmetric double well. Shown are the probability amplitudes $|c_n^{(k)}|$ for a small range of Fock index $n$ and energy eigenstate index $k$. Amplitude is indicated by hue. The total number of atoms is $N=20$; above the dotted line at Fock index $n=20$, one atom occupies the higher energy level; above the dotted line at $n=60$, two atoms occupy the higher level, etc. Inset: schematic of the double-well potential.

Figure 2

Breakdown of the one-level approximation. Energy eigenvalues versus (a) interaction energy with $\Delta V=0$ and (b) tilt with $J^0/U^0=0.1$ for a ten-atom system. The thick dashed lines in panels (a) and (b) correspond to $U^0=U_{\mathrm{crit}}^0$ and $\Delta V=\Delta V_{\mathrm{crit}}$, respectively. Eigenvalue crossings cause states with nonzero occupation of the excited energy level to emerge among the $N+1$ lowest-lying eigenstates.

Figure 3

Potential decoherence and tunneling resonances. Eigenstate probability amplitudes for a small range of Fock index $n$ and eigenstate index $k$ for $N=10$, $J^0/U^0=0.1$, and (a) $\Delta V/U^0=0$, (b) $\Delta V/U^0={10}^{-2}$, and (c) $\Delta V/U^0=6$. (b) A small tilt collapses the stationary macroscopic superposition (MS) states. (d) Avoided crossings in the energy eigenvalues indicate a reappearance of partial MS states, as in (c).