Quantum properties of a binary bosonic mixture in a double well

This work contains a detailed analysis of the properties of the ground state of a two-component two-site Bose-Hubbard model, which captures the physics of a binary mixture of Bose-Einstein condensates trapped in a double-well potential. The atom-atom interactions within each species and among the two species are taken as variable parameters, while the hopping terms are kept fixed. To characterize the ground state, we use observables such as the imbalance of population and its quantum uncertainty. The quantum many-body correlations present in the system are further quantified by studying the degree of condensation of each species, the entanglement between the two sites, and the entanglement between the two species. The latter is measured by means of the Schmidt gap, the von Neumann entropy, or the purity obtained after tracing out a part of the system. A number of relevant states are identified, e.g., Schrödinger catlike many-body states, in which the outcome of the population imbalance of both components is completely correlated, and other states with even larger von Neumann entropy which have a large spread in Fock space.

Figure 1

Spectral decomposition of the ground state, $|C_{k_A,k_B}{|}^2$, for repulsive intraspecies interaction ($\mathrm{\Lambda }>0$), plotted for different values of ${\mathrm{\Lambda }}_{AB}$. We observe the transition from (a) a regime dominated by the attraction between the different species to (g) a regime dominated by the repulsion between species $A$ and $B$, going through intermediate regimes in (b)–(f). For all panels, $\mathrm{\Lambda }=4$, $N_A=N_B=20$, $J=20$, and $\epsilon /J={10}^{-10}$.

Figure 2

Spectral decomposition of the ground state, $|C_{k_A,k_B}{|}^2$, for attractive intraspecies interaction ($\mathrm{\Lambda }<0$), plotted for different values of ${\mathrm{\Lambda }}_{AB}$. We observe the transition from (a) a regime dominated by the attraction between the different species to (g) a regime dominated by the repulsion between species $A$ and $B$, going through intermediate regimes in (b)–(f). For all panels, $\mathrm{\Lambda }=-3$, $N_A=N_B=20$, $J=20$, and $\epsilon /J={10}^{-10}$.

Figure 3

Properties of the ground state, varying $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}_{AB}$. (a) Population imbalance, $z_A$. The particles of this type are all on the left $z_A=1$ (yellow), all on the right $z_A=-1$ (black), or $z_A=0$ (red). (b) Population imbalance, $z_B$. The particles of this type are all on the left $z_B=1$ (yellow) and $z_B=0$ (red). (c) Dispersion of population imbalance, $\sigma z_B$. (d) Condensed fraction, $n_1^B$. For ${\mathrm{\Lambda }}_{AB}=0$, the results obtained in [13] are reproduced. Green spots mark where the states of Figs. 1 and 2 are found, except for the cases ${\mathrm{\Lambda }}_{AB}=-0.01$ and $0.01$ which are not marked. For all panels, $N_A=N_B=20$, $J=20$, and $\epsilon /J={10}^{-10}$.

Figure 4

Properties of the ground state, varying $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}_{AB}$. (a) Energy gap, $\mathrm{\Delta }{E}_{1,0}$. The black region is where there is degeneracy. (b) The von Neumann entropy for subsystem $A\left(B\right)$, $S_{A\left(B\right)}$, normalized to its maximum value. (c) Trace of the density matrix squared for subsystem $A\left(B\right)$, $P_{A\left(B\right)}$. (d) Schmidt gap for subsystem $A\left(B\right)$, $\mathrm{\Delta }{\lambda }^{A\left(B\right)}$. The $L-R$ von Neumann entropy for (e) the whole system, $S_{LR}$, and (f) for subsystem $A\left(B\right)$, $S_{LR}^{A\left(B\right)}$, both normalized to their maximum values. Green spots mark where the states of Figs. 1 and 2 are found, except for the cases ${\mathrm{\Lambda }}_{AB}=-0.01$ and $0.01$ which are not marked. For all panels, $N_A=N_B=20$, $J=20$, and $\epsilon /J={10}^{-10}$.

Figure 5

Entanglement properties for $\mathrm{\Lambda }=4$ depending on the interspecies interaction ${\mathrm{\Lambda }}_{AB}$. (a) Schmidt gap for subsystem $A\left(B\right)$. (b) Trace of the density matrix squared for subsystem $A\left(B\right)$, $P_{A\left(B\right)}$. (c) The von Neumann entropy for subsystem $A\left(B\right)$, $S_{A\left(B\right)}$, normalized to its maximum value. Dashed green lines mark where the states of Fig. 1 are found. (a)–(c) Vertical cuts in (b)–(d) of Fig. 4, respectively. For all panels, $N_A=N_B=20$, $J=20$, and $\epsilon /J={10}^{-10}$.

Figure 6

Spectral decomposition of the ground state, $|C_{k_A,k_B}{|}^2$, plotted for different values of ${\mathrm{\Lambda }}_{AB}$. Here there is repulsion between $A$-type bosons (${\mathrm{\Lambda }}_A>0$) and attraction between $B$-type bosons (${\mathrm{\Lambda }}_B<0$). We observe the transition from (a) a regime dominated by the attraction between the different species to (e) a regime dominated by the repulsion between species $A$ and $B$, going through intermediate regimes in (b)–(d). For all panels, ${\mathrm{\Lambda }}_A=4$, ${\mathrm{\Lambda }}_B=-5$, $N_A=N_B=20$, $J=20$, and $\epsilon /J={10}^{-10}$.

Figure 7

Properties of the ground state for ${\mathrm{\Lambda }}_A>0$ and varying ${\mathrm{\Lambda }}_B$ and ${\mathrm{\Lambda }}_{AB}$. Population imbalance (a) $z_A$ and (b) $z_B$. Dispersion of population imbalance for each species, (c) $\sigma z_A$ and (d) $\sigma z_B$. Condensed fractions (e) $n_1^A$ and (f) $n_1^B$. Green spots mark where the states of Fig. 6 are found. For all panels, ${\mathrm{\Lambda }}_A=4$, $N_A=N_B=20$, $J=20$, and $\epsilon /J={10}^{-10}$.

Figure 8

Properties of the ground state for ${\mathrm{\Lambda }}_A>0$ and varying ${\mathrm{\Lambda }}_B$ and ${\mathrm{\Lambda }}_{AB}$. (a) Energy gap, $\mathrm{\Delta }{E}_{1,0}$. (b) The von Neumann entropy for subsystem $A\left(B\right)$, $S_{A\left(B\right)}$, normalized to its maximum value. (c) $L-R$ von Neumann entropy for subsystem $A$, $S_{LR}^A$, and (d) for subsystem $B$, $S_{LR}^B$, both normalized to their maximum values. Green spots mark where the states of Fig. 6 are found. For all panels, ${\mathrm{\Lambda }}_A=4$, $N_A=N_B=20$, $J=20$, and $\epsilon /J={10}^{-10}$.

Figure 9

Spectral decomposition of the ground state, $|C_{k_A,k_B}{|}^2$, plotted for different values of ${\mathrm{\Lambda }}_{AB}$. Here there is attraction between $A$-type bosons (${\mathrm{\Lambda }}_A<0$) and repulsion between $B$-type bosons (${\mathrm{\Lambda }}_B>0$). We observe the transition from (a) a regime dominated by the attraction between the different species to (e) a regime dominated by the repulsion between species $A$ and $B$, going through intermediate regimes in (b)–(d). For all panels, ${\mathrm{\Lambda }}_A=-4$, ${\mathrm{\Lambda }}_B=5$, $N_A=N_B=20$, $J=20$, and $\epsilon /J={10}^{-10}$.

Figure 10

Properties of the ground state for ${\mathrm{\Lambda }}_A<0$ and varying ${\mathrm{\Lambda }}_B$ and ${\mathrm{\Lambda }}_{AB}$. Population imbalance (a) $z_A$ and (b) $z_B$. Dispersion of population imbalance for each species, (c) $\sigma z_A$ and (d) $\sigma z_B$. Condensed fraction (e) $n_1^A$ and (f) $n_1^B$. Green spots mark where the states of Fig. 9 are found. For all panels, ${\mathrm{\Lambda }}_A=-4$, $N_A=N_B=20$, $J=20$, and $\epsilon /J={10}^{-10}$.

Figure 11

Properties of the ground state for ${\mathrm{\Lambda }}_A<0$ and varying ${\mathrm{\Lambda }}_B$ and ${\mathrm{\Lambda }}_{AB}$. (a) Energy gap, $\mathrm{\Delta }{E}_{1,0}$. (b) The von Neumann entropy for subsystem $A\left(B\right)$, $S_{A\left(B\right)}$, normalized to its maximum value. (c) $L-R$ von Neumann entropy for subsystem $A$, $S_{LR}^A$, and (d) for subsystem $B$, $S_{LR}^B$, both normalized to their maximum values. Green spots mark where the states of Fig. 9 are found. For all panels, ${\mathrm{\Lambda }}_A=-4$, $N_A=N_B=20$, $J=20$, and $\epsilon /J={10}^{-10}$.

Figure 12

Properties of the ground state, varying $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}_{AB}$. Population imbalance (a) $z_A$ and (b) $z_B$. Dispersion of population imbalance for each species, (c) $\sigma z_A$ and (d) $\sigma z_B$. Condensed fraction (e) $n_1^A$ and (f) $n_1^B$. For all panels, $N_A=8$, $N_B=20$, $J_A=8$, $J_B=20$, and $\epsilon /J_B={10}^{-10}$.

Figure 13

Properties of the ground state, varying $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}_{AB}$. (a) Energy gap, $\mathrm{\Delta }{E}_{1,0}$. (b) The von Neumann entropy for subsystem $A\left(B\right)$, $S_{A\left(B\right)}$, normalized to its maximum value. (c) $L-R$ von Neumann entropy for subsystem $A$, $S_{LR}^A$, and (d) for subsystem $B$, $S_{LR}^B$, both normalized to their maximum values. For all panels, $N_A=8$, $N_B=20$, $J_A=8$, $J_B=20$, and $\epsilon /J_B={10}^{-10}$.