Quantum top inside a Bose-Einstein-condensate Josephson junction

We consider an atomic quantum dot confined between two weakly coupled Bose-Einstein condensates, where the dot serves as an additional tunneling channel. It is shown that the thus-embedded atomic quantum dot is a pseudospin subject to an external torque, and therefore equivalent to a quantum top. We demonstrate by numerical analysis of the time-dependent coupled evolution equations that this microscopic quantum top is very sensitive to any deviation from linear oscillatory behavior of the condensates. For sufficiently strong dot-condensate coupling, the atomic quantum dot can induce or modify the tunneling between the macroscopic condensates in the two wells.

Figure 1

(Color online) An atomic quantum dot between two weakly coupled condensates, trapped in a double-well potential $V_{\mathrm{BEC}}$. The dot is a simple two-level $\text{system}\equiv \text{single}$ atom present-not present, is created by the tight potential $V_{\mathrm{AQD}}$ located at the position of the top of the barrier. Atoms can be exchanged between wells either by direct tunneling (dashed arrows) or via the dot coupled to the condensates by a transfer matrix $T$.

Figure 2

Results for weak coupling $T_{\mathrm{rel}}=0.01$ ($N_0=1500$ throughout Figs. 2, 3, 4, as used in experiment [13]) and $\Lambda =10$. $n\left(0\right)=0$ [(a)-solid line], $n\left(0\right)=0.5$ [(a)-dashed line]. (b) The dot occupation for $n\left(0\right)=0$; (c) the precessional behavior of $\mathbf{\omega }$ (in units of $2\kappa$); and (d) the corresponding pseudospin nutation for $n\left(0\right)=0$.

Figure 3

Condensate in the self-trapped MST state, $n\left(0\right)=0.8$, $\Lambda =10$, $T_{\mathrm{rel}}=0.01$. Relative population oscillations are shown in (a) and the dot occupancy in (b). In (c), we display the projection of the frequency $\mathbf{\omega }$ [Eq. (10)] on the $x\text{−}y$ plane in units of $2\kappa$, and in (d) the pseudospin.

Figure 4

(Color online) Results for a $\pi$-junction. In (a), self-trapped MST state with $n\left(0\right)=0.6$, $\Lambda =1.25$; decoupled dot, $T_{\mathrm{rel}}=0$ (dashed line); $T_{\mathrm{rel}}=0.1$ (thick line), $T_{\mathrm{rel}}=1$ (thin wavy line); the pseudospin then nutates, (c) $(T_{\mathrm{rel}}=0.1)$. The AQD induces strong modifications, both for small oscillations between wells and in the MST state, (b) and (d); $n\left(0\right)=0.01$ and $T_{\mathrm{rel}}=0$ (dotted curve), $T_{\mathrm{rel}}=1$ (solid curve). Weak interaction coupling, $\Lambda =0.1$ in (b), strongly coupled MST state, $\Lambda =1.1$, in (d).