Sequential Measurements for Quantum-Enhanced Magnetometry in Spin Chain Probes

Quantum sensors outperform their classical counterparts in their estimation precision, given the same amount of resources. So far, quantum-enhanced sensitivity has been achieved by exploiting the superposition principle. This enhancement has been obtained for particular forms of entangled states, adaptive measurement basis change, critical many-body systems, and steady state of periodically driven systems. Here, we introduce a different approach to obtain quantum-enhanced sensitivity in a many-body probe through utilizing the nature of quantum measurement and its subsequent wave function collapse without demanding prior entanglement. Our protocol consists of a sequence of local measurements, without reinitialization, performed regularly during the evolution of a many-body probe. As the number of sequences increases, the sensing precision is enhanced beyond the standard limit, reaching the Heisenberg bound asymptotically. The benefits of the protocol are multifold as it uses a product initial state and avoids complex initialization (e.g., prior entangled states or critical ground states) and allows for remote quantum sensing.

Figure 1

Schematic of the protocol. (a) A spin chain probe is initialized in a product state for measuring a local magnetic field $\mathit{B}$ at site 1. (b) The readout is performed sequentially on the last site, separated by intervals of free evolution.

Figure 2

Inverse of the Fisher information ${\mathcal{F}}^{-1}$ as a function of the number of sequential measurements $n_{\mathrm{seq}}$ performed at site $N$ for $B_x=0.1J$. We consider two cases for the time interval between measurements: (a) $J{\tau }_i=5$ and (b) $J{\tau }_i=N$. A fitting function $g(n_{\mathrm{seq}})$ with exponent $\beta >1$ is shown.

Figure 3

(a) Posterior distribution as a function of $B_x/J$ for several $n_{\mathrm{seq}}$ for sensing $B_x=0.1J$. (b) Average of $\delta B_x^2$ as a function of $B_x/J$ for two values of $n_{\mathrm{seq}}$, where each point is averaged over 100 samples. In both panels, the posterior is obtained by repeating the procedure for $M=1000$ times in a probe of $N=6$ with fixed $J{\tau }_i=N$.

Figure 4

Estimating $B_x=0.1J$ with a probe of $N=6$ and $J{\tau }_i=N$. (a) Averaged squared relative error $\overline{\delta B_x^2}$ versus $n_{\mathrm{seq}}$ for two total execution times $T$. (b) Fitting coefficient $\alpha (T)$ as function of time $T$. (c) Fitting exponent $\beta (T)$ as function of time. (d) Universal behavior of $\overline{\delta B_x^2}$ versus $(JT{)}^{-\nu }{n}_{\mathrm{seq}}^{-\beta }$ for several values of $n_{\mathrm{seq}}$ and total time $T$.

Figure 5

(a) and (b) Posterior distributions for $n_{\mathrm{seq}}=1$ and $n_{\mathrm{seq}}=7$ for the estimation of $\mathit{B}=(0.15,0,0.1)J$, respectively. (c) Averaged squared relative error $\overline{\delta {\mathit{B}}^2}$ as a function of $n_{\mathrm{seq}}$.