Cavity-aided quantum parameter estimation in a bosonic double-well Josephson junction

We describe an apparatus designed to make nondemolition measurements on a Bose-Einstein condensate trapped in a double-well optical cavity. This apparatus contains, as well as the bosonic gas and the trap, an optical cavity. We show how the interaction between the light and the atoms, under appropriate conditions, can allow for a weakly disturbing yet highly precise measurement of the population imbalance between the two wells and its variance. We show that the setting is well suited for the implementation of quantum-limited estimation strategies for the inference of the key parameters defining the evolution of the atomic system and based on measurements performed on the cavity field. This would enable de facto Hamiltonian diagnosis via a highly controllable quantum probe.

Figure 1

Schematic representation of the system. A double-well potential loaded with ultracold atoms is coupled to an optical cavity that accommodated only one of the wells. The cavity is pumped by a classical field and the signal leaking out of the resonator is measured to infer the key features of the double-well system.

Figure 2

Evolution of the Wigner function of a high-$Q$ cavity, interacting with a BEC in a double well, out of the self-trapping regime. The parameter values are $\eta =\beta =\kappa =\gamma =R=4\times {10}^{-34}\text{J}$ and $N=30$. The initial state is coherent with amplitude $\alpha =3$.

Figure 3

Time-dependence of $\langle {\widehat{n}}_1\rangle$ [panels (a) and (c)] and $\langle {\widehat{n}}_1^2\rangle$ [panels (b) and (d)]. The solid red line represents their numerical evolution, while the blue dashed lines show the results of numerical estimations obtained using Eqs. (11) and (12), and the parameters $N=30,\eta =\kappa \simeq 4\times {10}^{-34}\text{J},\gamma /\kappa =500,\beta /\kappa =1/16$, which are valid for a system of ${}^{87}\mathrm{Rb}$ atoms. In panels (a) and (b) we have used a value of the tunneling rate comparable to the self-interaction energy $\left(R/\kappa =1\right)$, while panels (c) and (d) are for a tunneling dominated situation $\left(R/\kappa =30\right)$.

Figure 4

Distribution of values taken by the parameters ${\xi }_m$ [panel (a)] and ${\xi }_v$ [panel (b)] obtained for 100 randomly generated initial states of the BEC. Here we have taken the parameters used in panels (a) and (b) of Fig. 3. We have taken $t_0=0.07/\kappa$ and $t_1=0.8/\kappa$.

Figure 5

Distribution of values taken by the parameters ${\xi }_m$ [panel (a)] and ${\xi }_v$ [panel (b)] obtained for 100 randomly generated initial states of the BEC. Here we have taken the parameters used in panels (c) and (d) of Fig. 3. We have taken $t_0=0.07/\kappa$ and $t_1=0.8/\kappa$.

Figure 6

(a) Single-parameter estimation of the self-interaction rate $\kappa$: We plot the single-parameter quantity $\Lambda \left(\kappa \right)=ln[{\mathcal{H}}^{-1}\left(\kappa \right)/{\beta }^2]$ at a set value of the tunneling rate $R/\beta =0.5$ against the evolution time and the actual self-interaction energy $\kappa$. In panel (b) we plot $\Lambda \left(R\right)=ln[{\mathcal{H}}^{-1}\left(R\right)/{\beta }^2]$ against $R$ and $\beta t$ at $\kappa /\beta =0.5$. In both panels we have taken ${\omega }_a/\beta =0.1,{\omega }_c/\beta =0.1,{\omega }_p/\beta =0.1,\eta /\beta =0.1$, and $\gamma /\beta =1$.

Figure 7

(a) Multiparameter estimation approach. (b) Sequential estimation approach. In both panels we have used the parameters $\gamma /\beta =1,\eta /\beta =0.1,{\omega }_p/\beta =0.1,{\omega }_c/\beta =0.1,{\omega }_a/\beta =0.1$. The evolution time has been set at $t=10/\beta$ and we have considered a two-atom initial state with mode 1 initially fully populated and an empty cavity.

Figure 8

Multiparameter vs sequential estimation approach. We compare ${\Lambda }_{mp}(R,\kappa )$ and ${\Lambda }_{se}(R,\kappa )$ in various parameter regimes. In panel (a) we plot the two figures of merit at $R/\beta =0.15$ and investigate the estimation performance through the two strategies as the value of $\kappa$ is changed. Panel (b) displays a similar comparison performed against the tunneling rate $R$ at $\kappa /\beta =0.15$. The straight lines $({\overline{\Lambda }}_{se},{\overline{\Lambda }}_{mp})$ show the average of the two figures of merit over the range of values taken by $\kappa$ [panel (a)] and $R$ [panel (b)]. In both panels we have taken $\gamma /\beta =1,\eta /\beta =0.1,{\omega }_p/\beta =0.1,{\omega }_c/\beta =0.1$, and ${\omega }_a/\beta =0.1$. The evolution time has been set at $t=10/\beta$ and we have considered a two-atom initial state with mode 1 initially fully populated and an empty cavity.