Quantum Critical Metrology

Quantum metrology fundamentally relies upon the efficient management of quantum uncertainties. We show that under equilibrium conditions the management of quantum noise becomes extremely flexible around the quantum critical point of a quantum many-body system: this is due to the critical divergence of quantum fluctuations of the order parameter, which, via Heisenberg’s inequalities, may lead to the critical suppression of the fluctuations in conjugate observables. Taking the quantum Ising model as the paradigmatic incarnation of quantum phase transitions, we show that it exhibits quantum critical squeezing of one spin component, providing a scaling for the precision of interferometric parameter estimation which, in dimensions $d>2$, lies in between the standard quantum limit and the Heisenberg limit. Quantum critical squeezing saturates the maximum metrological gain allowed by the quantum Fisher information in $d=\infty$ (or with infinite-range interactions) at all temperatures, and it approaches closely the bound in a broad range of temperatures in $d=2$ and 3. This demonstrates the immediate metrological potential of equilibrium many-body states close to quantum criticality, which are accessible, e.g., to atomic quantum simulators via elementary adiabatic protocols.

Figure 1

Squeezing at the QCP. (a) Scaling of the variance of the collective spin component $J^y=\sum _i{S}_i^y$ in $d=1$, 2, 3, and $d=\infty$. Data are taken at $g=0.62$ and $T=0$ from the exact solution on a system with open boundaries in $d=1$ and at $g=g_c$ for all other dimensions. The data for $d=\infty$ are the $T=0$ exact solution, while the data for $d=2$ and $d=3$ come from QMC simulations on an $N=L^d$ lattice with periodic boundaries, at temperatures sufficiently low to eliminate thermal effects [as low as $k_{B}T=J/(6L)$ in $d=2$ and $k_{B}T=J/(5L)$ in $d=3$]. Dashed lines are power-law fits to the form $a\times N^{{\zeta }^{\prime }}$. The solid line indicates the shot-noise limit $\text{Var}(J^y)/N=1/4$. (b) Scaling of the spin-squeezing parameter ${\xi }_R^2$. Same significance of symbols as in panel (a).

Figure 2

Quantum correlations along the quantum critical trajectory. Squeezing parameter ${\xi }_R^2$, ${\chi }^{-2}$ parameter, and its bounds provided by the quantum variance for the TFI model as a function of the temperature at $g=g_c$: (a) $d=1$, $N=50$; (b) $d=2$, $N=64^2$; (c) $d=3$, $N=28^3$; (d) $d=\infty$, $N=1000$.

Figure 3

Squeezing around the QCP. Squeezing parameter ${\xi }_R^{-2}$ (in dB) across the phase diagram of the TFI model close to the QCP: (a) $d=2$, $N=64^2$; (b) $d=\infty$, $N=500$. The gray circle marks the QCP, and the dashed blue lines indicate the critical temperatures $T_c$ on the ordered side ($T_c$ values for $d=2$ from Ref. [46], and for $d=\infty$ from Ref. [47]). In the white region, ${\xi }_R^{-2}\le 1$ (absence of squeezing).