Macroscopic superposition of ultracold atoms with orbital degrees of freedom

We introduce higher dimensions into the problem of Bose-Einstein condensates in a double-well potential, taking into account orbital angular momentum. We completely characterize the eigenstates of this system, delineating new regimes via both analytical high-order perturbation theory and numerical exact diagonalization. Among these regimes are mixed Josephson- and Fock-like behavior, crossings in both excited and ground states, and shadows of macroscopic superposition states.

Figure 1

Schematic of the potential. (a) Equipotential surfaces of the double-well potential and (b) zoom of single-well and single-particle eigenfunctions, where $l$ is the orbital angular momentum and $m$ its $z$ projection. The first excited level is triply degenerate in 3D.

Figure 2

Characterization of the crossings and validity of the model in the $V_0$-$g$ plane (both dimensionless), for the Duffing potential. Different criteria that characterize the first crossing and the first crossing involving the ground state, as well as the limits of validity of the model for $N=8$ atoms. Three main areas are distinguished: (i) the area in which no crossing occurs; (ii) the area in which eigenstates with occupation of the excited level emerge among the first $N+1$ eigenvectors; and (iii) the area in which even the ground state shows occupation of the excited level. The one-level approximation is valid in area (i).

Figure 3

Characterization of the different regimes in the $V_0$-$g$ plane (both dimensionless). Different criteria that characterize the Josephson and Fock regimes for both levels when $N=8$. Numbered points represent the examples given in subsequent figures. For completeness, the three regions distinguished in Fig. 2 are also represented.

Figure 4

Eigenstates for three examples of the first crossing in the Josephson, intermediate, and Fock regimes. Probability amplitudes $\left|c_{i\ell m}^{(k)}\right|{}^2$ of the first $N+1+6N$ eigenstates for (a) $V_0=20$, $g\approx {10}^{-4}$, (b) $V_0=20.6$, $g\approx 2\times {10}^{-3}$, and (c) $V_0=49$, $g\approx {10}^{-2}$, when $N=8$ atoms. Above the dashed horizontal line one atom occupies the excited level. Dotted horizontal lines separate the Fock vectors for which this atom shows $m=-1$, $m=0$, and $m=1$, respectively.

Figure 5

Eigenstates for three typical examples of the eigenvectors from the Josephson, intermediate, and Fock regimes. Probability amplitudes $\left|c_{i\ell m}^{(k)}\right|{}^2$ for the first $N+1+6N$ eigenstates for $N=8$ atoms, $V_0=32.4$, and (a) $g\approx {10}^{-4}$, (b) $g\approx 8\times {10}^{-4}$, and (c) $g\approx 7\times {10}^{-3}$. Above the dashed horizontal line one atom occupies the excited level. Dotted horizontal lines separate the Fock vectors for which this atom shows $m=-1$, $m=0$, and $m=1$, respectively.

Figure 6

Eigenstates for the mixed regime. Probability amplitudes $\left|c_{i\ell m}^{(k)}\right|{}^2$ (a) for the first $N+1$ eigenstates, in which only the bottom level is occupied, and (b) for all eigenstates corresponding to occupation only of the excited level with $m=0$ when $N=8$ atoms, $V_0=37$, and $g\approx {10}^{-3}$. Note that, while the former eigenstates are MS states, the latter are HO-like states.

Figure 7

Shadows of excited MS states and of the ground state. (a) Probability amplitudes $\left|c_{i\ell m}^{(k)}\right|{}^2$ for the first $(N+1)+6N+21(N-1)$ Fock vectors and eigenstates when $V_0=38.4$, $g\approx 3\times {10}^{-2}$, and $N=8$ atoms. One atom occupies the excited level above the dashed line at $i=N+1$. Two atoms occupy the excited level above the second dashed line, at $i=N+1+6N$. (b) Coefficients for the 120th excited state, the most excited state, with all atoms in the bottom level. This MS state shows non-negligible coupling to Fock vectors with two atoms in the excited level, which are its shadows. (c) Probability amplitudes $\left|c_{i\ell m}^{(k)}\right|{}^2$ for the first $(N+1)+6N+21(N-1)$ Fock vectors and eigenstates when $V_0=49.8$, $g\approx 7\times {10}^{-2}$, and $N=8$. (d) Coefficients for the ground state for this case. Here, even the ground state shows non-negligible coupling to Fock vectors with two atoms in the excited level, these being its shadows.

Figure 8

Change of the criteria in the high- and low-barrier limit with $N$. (a) Coupling constant $g$ (dimensionless) given by criteria (36), (37), (40), and (43) for a large value of $V_0$, as a function of $N$. (b) Barrier height $V_0$ (dimensionless) given by criteria (34), (35), and (42) for a small value of $g$, as a function of $N$. In both cases, the boundaries between different regimes are displaced accordingly for increasing $N$.

Figure 9

Eigenfunctions in the high- and low-barrier limits. (a–c) Analytical functions for the harmonic oscillator approximation, for $V_0=20$; (f–h) numerically calculated ones for the Duffing potential, for $V_0=20$. (d, e) One-dimensional analytical eigenfunctions; (i, j) numerical ones.