Quantum phase transitions of atom-molecule Bose mixtures in a double-well potential

The ground state and spectral properties of Bose gases in double-well potentials are studied in two different scenarios: (i) an interacting atomic Bose gas, and (ii) a mixture of an atomic gas interacting with diatomic molecules. A ground state second-order quantum phase transition is observed in both scenarios. For large attractive values of the atom-atom interaction, the ground state is degenerate. For repulsive and small attractive interaction, the ground state is not degenerate and is well approximated by a boson coherent state. Both systems depict an excited state quantum phase transition. In both cases, a critical energy separates a region in which all the energy levels are degenerate in pairs, from another region in which there are no degeneracies. For the atomic system, the critical point displays a singularity in the density of states, whereas this behavior is largely smoothed for the mixed atom-molecule system.

Figure 1

Panels (a) and (c) show the variational parameters ${\gamma }_R$ and ${\gamma }_L$, as a function of the control parameter $U$. Panels (b) and (d) show the normalized expectation value of the invalance operator $\widehat{I}$ in solid line, and the square root of the Fisher information, obtained by numerical diagonalization, in open symbols. Panels (a) and (b) are for the mixed atom-molecule system [two numerical calculations for different system sizes are shown in (b), (black) circles correspond to $N=100$, and (orange) squares, to $N=200\text{]}$ whereas panels (c) and (d) present the system without molecule.

Figure 2

Gap between ground state and first-excited state, for different system sizes: system with molecule, $N=20$ (triangles, red online), 40 (circles, green online), 80 (diamonds, blue online), 160 (squares, magenta online) and 320 (hexagons, cyan online), in panel (a); and system without molecule, $N=250$ (triangles, red online), 1000 (circles, green online), 4000 (diamonds, blue online) and 16000 (squares, magenta online), panel (b). In all the cases, system sizes increase from left to right.

Figure 3

Scaling of the finite-size precursor of the critical point.

Figure 4

In panel (a) the ground state energy per particle of the model with the diatomic is plotted as a function of the control parameter $U$ for $N=200$. Panel (b) shows the difference between the exact ground state energy per particle obtained numerically and the values obtained by using the two analytic approximations considered in this work in a small region around the critical point for a system with $N=200$. The rest of the control parameters in the Bose-Hubbard Hamiltonian are $J=1,\omega =5$, and $g=5$.

Figure 5

First few excited-state energy gaps as a function of the control parameter $U$. Symbols are the exact results for $N=200$, while lines provide the beyond mean-field approximation. The rest of the parameters are fixed to $J=1,\omega =5$, and $g=5$ in panel (a), and $J=1$ in panel (b). The vertical dotted line marks the separation between the two phases.

Figure 6

Complete phase diagram. Panel (a) depicts the case with molecule, obtained with $N=320$ particles, and $J=1,\omega =5$, and $g=5$. Panel (b) depicts the case without molecule, with $N=1000$ particles and $J=1$. In both cases, the dark filled region (blue online) indicates the symmetry-breaking phase, in which the eigenstates are doubly degenerate, and the light filled region (yellow online) is the normal phase, in which there are no degeneracies and the parity is a good quantum number for all the eigenstates. Vertical dotted lines enclose the region in which there is no ESQPT.

Figure 7

Density of states $\rho \left(E\right)$: Panel (a) for $N=320,U=9$ and a molecule at resonance. Panel (b) for $N=2000$ without molecule.

Figure 8

Ratio of degenerate energy levels as a function of the energy: panel (a) for $N=320,U=9$ and a molecule at resonance. Panel (b) for $N=2000$ without molecule.

Figure 9

Fisher information as a function of the energy: Panel (a) for $N=320,U=9$ and a molecule at resonance. Panel (b) for $N=2000$ without molecule.

Figure 10

Spectral statistics for $U=9$ and $N=360$, even-parity states, as a function of the scaled energy $E/N$.