Impurity in a Bose-Einstein condensate in a double well

We compare and contrast the mean-field and many-body properties of a Bose-Einstein condensate trapped in a double-well potential with a single impurity atom. The mean-field solutions display a rich structure of bifurcations, as parameters such as the boson-impurity interaction strength and the tilt between the two wells are varied. In particular, we study a pitchfork bifurcation in the lowest mean-field stationary solution, which occurs when the boson-impurity interaction exceeds a critical magnitude. This bifurcation, which is present for both repulsive and attractive boson-impurity interactions, corresponds to the spontaneous formation of an imbalance in the number of particles between the two wells. If the boson-impurity interaction is large, the bifurcation is associated with the onset of a Schrödinger-cat state in the many-body ground state. We calculate the coherence and number fluctuations between the two wells, and also the entanglement entropy between the bosons and the impurity. We find that the coherence can be greatly enhanced at the bifurcation.

Figure 1

Eigenvalues of the many-body Hamiltonian as a function of the tilt $\Delta \epsilon$ $(=\Delta {\epsilon }^a)$ for $N=6$ bosons and a single impurity. Each panel has a different value of the boson-impurity coupling: (a) $W/U=-4$, (b) $W/U=-2$, (c) $W/U=0$, (d) $W/U=2$, (e) $W/U=4$. The values of the other parameters are $J/U=J^a/U=1.5$.

Figure 2

Energies of the static solutions to the mean-field equations (19, 20, 21, 22) as a function of the tilt $\Delta \epsilon$ ($=\Delta {\epsilon }^a$). The various solutions are characterized by their phase differences: $\alpha =\beta =0$ (black circles); $\alpha =\pi$, $\beta =0$ (orange squares); $\alpha =0$, $\beta =\pi$ (blue diamonds); and $\alpha =\beta =\pi$ (red triangles). Each panel has a different value of the boson-impurity coupling: (a) $W/U=-4$, (b) $W/U=-2$, (c) $W/U=0$, (d) $W/U=2$, (e) $W/U=4$. All panels have $J/U=J^a/U=1.5$. We have also included the many-body eigenvalues (thin lines) for $N=6$ bosons and a single impurity.

Figure 3

A zoom-in of Fig. 2 showing the birth of the loop in the lowest band as the boson-impurity interaction strength $W$ is varied: (a) $W/U=2.0$, (b) $W/U=2.122$, (c) $W/U=2.2$. The solid lines are given by solving $F\left(\theta \right)=0$, as given in Eq. (40), and the dots are given by solving the third-order Taylor expansion $F_3\left(\theta \right)$, as given in Eq. (41). The other parameters are set at $N=6$, $J=J^a=1.5U$.

Figure 4

The atom number difference between the right- and left-hand wells for the lowest band plotted as a function of the tilt $\Delta \epsilon$ ($=\Delta {\epsilon }^a$). The left-hand column gives relative atom number difference $Z/N=(N_R-N_L)/2N$ for the BEC, and the right-hand column gives the probability difference $Y=(M_R-M_L)/2$ for the impurity. Each row has a different value of the boson-impurity interaction $W$, and corresponds to the equivalent panel of Fig. 3: (a) $W/U=2.0$, (b) $W/U=2.122$, (c) $W/U=2.2$. The other parameters are set at $N=6$, $J=J^a=1.5U$.

Figure 5

The atom number difference between the right- and left-hand wells for the lowest band plotted as a function of the boson-impurity interaction strength $W$. The left-hand column gives relative atom number difference $Z/N=(N_R-N_L)/2N$ for the BEC, and the right-hand column gives the probability difference $Y=(M_R-M_L)/2$ for the impurity. The top row has zero tilt $\Delta \epsilon =\Delta {\epsilon }^a=0$, and the bottom row has a finite tilt $\Delta \epsilon =\Delta {\epsilon }^a=U$. The other parameters are set at $N=6$, $J=J^a=1.5U$.

Figure 6

Dependence of the many-body and mean-field energies upon the boson-impurity interaction $W$ for a finite value of the tilt. The solid curves are many-body eigenvalues, and the dotted curves are the static solutions to the mean-field equations: $\alpha =\beta =0$ (black circles); $\alpha =\pi$, $\beta =0$ (orange squares); $\alpha =0$, $\beta =\pi$ (blue diamonds). This figure only shows the low energies and so the $\alpha =\beta =\pi$ solution does not appear. The other parameters are $N=30$, $J=J^a=1.5U$, $\Delta \epsilon =\Delta {\epsilon }^a=7U$.

Figure 7

Dependence of the many-body and mean-field energies upon the boson-impurity interaction $W$ for a finite value of the tilt $\Delta \epsilon =\Delta {\epsilon }^a=7U$. This picture differs from Fig. 6 in that we have far fewer atoms ($N=6$), and we have decreased the ratio of $J$ to $J=J^a=0.5U$. We have also plotted the energies of the relevant Fock states (dashed lines). The solid curves, which mostly lie on top of the Fock state energies, are the eigenvalues of the many-body Hamiltonian. The black circles give the $\alpha =\beta =0$ static solutions to the mean-field equations (for simplicity, we have not shown the other branches).

Figure 8

The first four many-body eigenstates for $W=0$. The ground state is shown in panel (a), the first excited state in panel (b), and so on. The red circles (blue triangles) are the amplitudes for the impurity to be in the left (right) well. The total number of atoms is $N=100$, and $N_R$ is the number of atoms in the right well. The hopping energy is set to $J=J^a=10U$ and the tilt is zero. According to Hamiltonian (8), the energies of these four states are $-0.0995,-0.0975,-0.0947$, and $-0.0927$ in units of $U{N}^2$.

Figure 9

The ground state for $W/U=-10$ (a) and $W/U=10$ (b). The red circles (blue triangles) are the amplitudes for the impurity to be in the left (right) well. The total number of atoms is $N=100$, and $N_R$ is the number of atoms in the right well. The hopping energy is set to $J=J^a=10U$ and the tilt is zero. According to Hamiltonian (8), both the positive and negative values of $W$ lead to the same ground-state energy of $-0.101$ in units of $U{N}^2$.

Figure 10

The correlation between critical value $W_c$ [see Eq. (42)] of the boson-impurity interaction at which a bifurcation occurs in the lowest band, and the equivalent value $W_{\mathrm{dip}}$ at which the many-body wave function first develops a dip in its center (i.e., at $N_R=N/2$). Each data point corresponds to $N=100$, but to a different value of $J=J^a$, lying in the range $1<J/U<180000$. The solid line is a fit to the data and is given by $W_c=-3.8+5.1W_{\mathrm{dip}}.$

Figure 11

The coherence $\alpha$, and the square of the number fluctuations ${\sigma }^2$, in the ground state as a function of the boson hopping energy $J$, with $N=100$. The boson-impurity interaction $W$ is set to zero, as is the tilt. The dashed lines indicate the boundaries between the Fock, Josephson, and Rabi regimes.

Figure 12

The coherence $\alpha$, and the square of the number fluctuations ${\sigma }^2$, in the ground state as a function of the impurity hopping energy $J^a$. In this figure, $N=100$, $J/U=0.001$, $W/U=2$, and the tilt is set to zero. The three labels indicate the values of $J^a$ used in the different panels of Fig. 13: (a) $J^a/U=0.001$, (b) $J^a/U=1.514$, and (c) $J^a/U=1000$.

Figure 13

The many-body ground-state wave function as a function of $N_R$, the number of atoms in the right well, for $N=100$, $J/U=0.001$, and $W/U=2$. Each panel has a different value of the impurity hopping energy $J^a$ corresponding to the labels shown in Fig. 12: (a) $J^a/U=0.001$, (b) $J^a/U=1.514$, and (c) $J^a/U=1000$. The red circles (blue triangles) are the amplitudes for the impurity to be in the left (right).

Figure 14

The coherence $\alpha$, and the square of the number fluctuations ${\sigma }^2$, in the ground state as a function of the impurity hopping energy $J^a$. In this figure, $N=100$, $J/U=0.001$, $W/U=10$, and the tilt is set to zero. The three labels indicate the values of $J^a$ used in the different panels of Fig. 15: (a) $J^a/U=0.001$, (b) $J^a/U=48$, and (c) $J^a/U=1000$. A zoom-in of this figure is given in Fig. 16.

Figure 15

The many-body ground-state wave function as a function of $N_R$, the number of atoms in the right well, for $N=100$, $J/U=0.001$, and $W/U=10$. Each panel has a different value of the impurity hopping energy $J^a$ corresponding to the labels shown in Fig. 14: (a) $J^a/U=0.001$, (b) $J^a/U=48$, and (c) $J^a/U=1000$. The red circles (blue triangles) are the amplitudes for the impurity to be in the left (right) well.

Figure 16

The coherence $\alpha$, and the square of the number fluctuations ${\sigma }^2$, in the ground state as a function of the impurity hopping energy $J^a$. In this figure, $N=100$, $J/U=0.001$, $W/U=10$, and the tilt is set to zero. This plot is a zoom-in of Fig. 14.

Figure 17

Entanglement entropy $S$ as a function of $W$ for $N=100$ and $J=J^a=1.5U$. The solid orange curve is for zero tilt $\Delta \epsilon =\Delta {\epsilon }^a=0$, and the dashed blue curve is for finite tilt $\Delta \epsilon =\Delta {\epsilon }^a=U$.