Dynamical phase transitions as a resource for quantum enhanced metrology

We consider the general problem of estimating an unknown control parameter of an open quantum system. We establish a direct relation between the evolution of both system and environment and the precision with which the parameter can be estimated. We show that when the open quantum system undergoes a first-order dynamical phase transition the quantum Fisher information (QFI), which gives the upper bound on the achievable precision of any measurement of the system and environment, becomes quadratic in observation time (cf. “Heisenberg scaling”). In fact, the QFI is identical to the variance of the dynamical observable that characterizes the phases that coexist at the transition, and enhanced scaling is a consequence of the divergence of the variance of this observable at the transition point. This identification makes it possible to establish the finite time scaling of the QFI. Near the transition the QFI is quadratic in time for times shorter than the correlation time of the dynamics. In the regime of enhanced scaling the optimal measurement whose precision is given by the QFI involves measuring both system and output. As a particular realization of these ideas, we describe a theoretical scheme for quantum enhanced estimation of an optical phase shift using the photons being emitted from a quantum system near the coexistence of dynamical phases with distinct photon emission rates.

Figure 1

(a) Open quantum system with a dynamics that features two dynamical phases of different activity and depends on the unknown parameter $g$. Near a first-order DPT the output (e.g., photons) shows strong intermittency where the temporal length of active and inactive periods is on average proportional to the correlation time $\tau$. (b) In the vicinity of a DPT the QFI of the combined system-output state scales quadratically for observation times $t\ll \tau$. In the example of photon counting this regime features a bimodal count distribution; i.e., the two phases can be resolved. For $t\gg \tau$ this is no longer the case and the distribution becomes unimodal. Consequently, the QFI acquires a linear scaling with $t$. (c) Wigner distribution $W(Q,P)$ of the state (12) after being projected on an appropriate system state, e.g., $|\mathrm{I}\rangle +|\mathrm{A}\rangle$. The two peaks are located at radii that correspond to the count rates ${\mu }_{\mathrm{I},\mathrm{A}}$ of the inactive and active phases. The count distribution is not sensitive to the parameter $\varphi$, hence, counting is not an appropriate measurement for its estimation. Still, this state features an enhanced QFI with respect to changes in the parameter $\varphi$ due to the highly oscillatory fringe pattern [with period $\propto t({\mu }_{\mathrm{A}}-{\mu }_{\mathrm{I}})]$ between the two peaks, which is characteristic for a Schrödinger cat state.