Achieving sub-shot-noise sensing at finite temperatures

We investigate sensing of magnetic fields using quantum spin chains at finite temperature and exploit quantum phase crossovers to improve metrological bounds on the estimation of the chain parameters. In particular, we start by analyzing the $XX$ spin chain. The magnetic sensitivity of this system is dictated by its magnetic susceptibility, which scales extensively (linearly) in the number of spins $N$. We introduce an iterative feed-forward protocol that actively exploits features of quantum phase crossovers to enable superextensive scaling of the magnetic sensitivity. Furthermore, we provide experimentally realistic observables to saturate the quantum metrological bounds. Finally, we extend our analysis on magnetic sensing to the Heisenberg $XY$ spin chain.

Figure 1

Specific QFI (i.e., $\mathcal{F}\left(h\right)/N$) for the estimation of the magnetic field $h$ in the $XX$ model as a function of $h/J$, at three different temperatures. The shaded area corresponds to the ferromagnetic region. In the plot $N={10}^5$ and $J=1$.

Figure 2

Solid black line: Specific QFI for magnetic field sensing in the $XX$ model versus $h/J$. All the plotted area lies within the ferromagnetic phase. Dashed red line: Low-temperature approximation of the magnetic sensitivity ${\mathcal{F}}_{\text{app}}\left(h\right)$ from Eq. (15). The region $k_{B}T>J-h$, where the approximation breaks down appears in shaded gray. Inset: Closeup of the neighborhood of the critical point. The temperature was set to $\beta =100, N={10}^4$, and $J=1$.

Figure 3

Solid black line: Specific QFI for the estimation of $J$ in the $XX$ model versus $J/h$. As in Fig. 1 the ferromagnetic phase has been shaded. Dashed red line: Specific $J$ sensitivity $F(J;{\widehat{J}}_z)/N$ of the total magnetization ${\widehat{J}}_z$. The temperature was set to $\beta =100, h=1$, and $N={10}^3$. Inset: Same as in the main plot for the much larger temperature $\beta =2$. Note that, unlike in the main plot, the vertical axis of the inset is not scaled by the ${10}^3$ factor.

Figure 4

(a) Specific QFI (in logarithmic scale) in the $XY$ model as a function of $h/J$ and $\gamma$. The critical line appears highlighted in white. Note that $\gamma =0$ corresponds to the $XX$ model and $\gamma =1$ to the Ising model. The sensitivity increases with the asymmetry parameter $\gamma$ in the paramagnetic region, whereas it decreases with $\gamma$ in the ferromagnetic phase. For this illustration $N={10}^3, \beta ={10}^3$, and $J=1$. (b) Solid black line: Specific QFI in the Ising model. Dashed red line: Specific $h$ sensitivity $F(h;{\widehat{J}}_z)/N$ of the total magnetization in the $z$ direction. Dotted green line: Specific $h$ sensitivity $F(h;{\widehat{J}}_x^2)/N$. Inset: Zoom-in view of the high-sensitivity region, not shown in the main plot. The ferromagnetic phase is highlighted in shaded gray, and $N=40, \beta =100$, and $J=1$.