Many-body quantum metrology with scalar bosons in a single potential well

We theoretically investigate the possibility of performing high precision estimation of an externally imposed acceleration using scalar bosons in a single-well trap. We work at the level of a two-mode truncation, valid for weak to intermediate two-body interaction couplings. The splitting process into two modes is in our model entirely caused by the interaction between the constituent bosons and is hence neither due to an externally imposed double-well potential nor due to populating a spinor degree of freedom. The precision enhancement gained by using various initial quantum states using a two-mode bosonic system is well established. Here we therefore instead focus on the effect of the intrinsic dynamics on the precision, where, in a single well, the Hamiltonian assumes a form different from that of the typical double-well case. We demonstrate how interactions can significantly increase the quantum Fisher information maximized over initial states as well as the quantum Fisher information for a fragmented or a coherent state, the two many-body states that can commonly represent the ground state of our system.

Figure 1

Maximal QFI (cQFI) [Eq. (22)] for a noninteracting system (the left plot has $t=1$ and the right plot has $\lambda =1$). The dynamics is $\lambda {\widehat{J}}_x-{\delta }_{\epsilon }{\widehat{J}}_z$ and the parameter to be estimated is $\lambda$. The plots illustrate the detrimental effect of the presence of the ${\delta }_{\epsilon }{\widehat{J}}_z$ term during the phase accumulation (for large ${\delta }_{\epsilon }$ the cQFI becomes very small).

Figure 2

Maximal QFI (cQFI) in the presence of interaction. The magenta dotted line displays the upper bound $N^2{t}^2$ (obtained by using an ideal interferometer). We see in the upper left plot ($\lambda =1$ and $t=1$) how a large enough interaction strength helps to achieve a high cQFI, almost reaching its upper bound. On the upper right plot ($\lambda =1,t=1$, and ${\delta }_{\epsilon }=10$) we illustrate how for larger values of ${\delta }_{\tilde{A}}$ only near $q=0$ the cQFI saturates its upper bound. On the bottom plots we investigate how the behavior as a function of ${\delta }_{\epsilon }$ (left plot, $t=1$ and $g=20$) and $t$ (right plot, $\lambda =1$ and ${\delta }_{\epsilon }=10$) is modified by the presence of interactions (compare with Fig. 1).

Figure 3

QFI for a fragmented (left) and coherent (right) initial state as a function of the interactions strength ($\lambda =1,t=1$). The magenta dotted line corresponds to the QFI for a multiplicative Hamiltonian $\lambda {\widehat{J}}_x$ (ideal protocol regarding the maximal QFI), and helps to visualize how the natural dynamics of the system improves the estimation of the acceleration.