Characterization and generation of a SQL-beating catlike state through repetitive measurements

TL;DR

This work addresses metrological sensitivity limits by introducing SQL-beating catlike states with macroscopic coherence quantified by $q>1.5$, attainable through repetitive measurements on a spin ensemble coupled to a superconducting flux qubit. The proposed protocol leverages measurement backaction to steer a thermal spin state toward states with $ ext{Var}(\,\hat{S}_z\,) = \Theta(N^2)$, enabling Heisenberg-like scaling in ideal conditions and Zeno-like performance under dephasing. The authors provide analytical bounds showing $\delta\omega \le 1/\Theta(N^{q-1})$ for $1.5<q<2$, and substantiate the emergence of SQL-beating states via detailed numerical simulations demonstrating $q$ approaching 2 (up to $\approx 1.94$) as measurements increase. The results establish a practical route to entanglement-enhanced quantum metrology in solid-state hybrids, with implications for high-precision magnetic sensing and scalable quantum sensing architectures.

Abstract

Sensitivity in metrology without entanglement is limited by the standard quantum limit (SQL). Recent studies have found that the Heisenberg-limited scaling, the ultimate sensitivity in quantum metrology, can be achieved by generalized cat states, which are characterized by an index that indicates coherence among macroscopically distinct states and are associated with additive observables. Although generalized cat states include diverse states, encompassing classical mixtures of exponentially large numbers of states, the preparation of large generalized cat states has not been demonstrated yet. Here we characterize SQL-beating catlike states using the index $q$ indicating macroscopic coherence and prove that any state with $q>1.5$ has a potential to surpass the SQL when used as a sensor. We propose a protocol to generate them through repetitive measurements on a quantum spin system of $N$ spins, which we call a spin ensemble. Starting from a thermal equilibrium state of the spin ensemble, we demonstrate that we can increase the coherence among the spin ensemble via repetitive weak measurements of its total magnetization, which is indirectly measured through an ancillary qubit collectively coupled to the ensemble. Notably, our method for creating the SQL-beating catlike states requires no dynamical control over the spin ensemble. As a potential experimental realization, we discuss a hybrid system composed of a superconducting flux qubit and donor spins in silicon. Our results pave the way for the realization of entanglement-enhanced quantum metrology in state-of-the-art technology.

Figures (5)

• Figure 1: Schematic of the spin ensemble and the FQ. The electron spins (white dots) are situated on the substrate (gray oval), which is placed within the loop of the FQ (navy parallelogram). An external magnetic field (red arrow) is applied along the $z$ axis to induce the Zeeman energy $-\omega_{\rm P}\hat{S}_z$. The interaction between the spin ensemble and the FQ is represented by a green arrow.
• Figure 2: Schematic of the time dependence of $g(t)$. To induce an interaction between the FQ and the spin ensemble, we modulate the coupling strength to oscillate.
• Figure 3: Plot of the probability $p(k)$ of obtaining the trajectories in which $\hat{W}_+$ is applied $k$ times, as described in Eq. (\ref{['withoutappr']}), against $k$. In (a) the parameters are $gt=0.2$, $m=400$, and $N=3$ (blue) and $N=4$ (yellow), satisfying $gtN< \pi/2$. In this case, $p(k)$ is approximately given by the sum of the sharp normal-distribution-like peaks, whose heights become larger when they are closer to $k=m/2$. In (b) the parameters are $gt=1.0$, $m=100$, and $N=3$ (blue) and $N=4$ (yellow), satisfying $gtN\geq \pi/2$.
• Figure 4: Visualization of the creation of generalized cat states through repetitive measurements with $N=15$ qubits. The horizontal axis represents the number of measurements with the FQ, while the vertical axis denotes the value of $C_{\mathrm cat}(\hat{S}_z, \hat{\rho}_\mathrm{P}(m))$. Due to the probabilistic nature of each measurement, the values of $C_{\mathrm cat}(\hat{S}_z, \hat{\rho}_\mathrm{P}(m))$ exhibit temporal fluctuations within a single trajectory. The inset illustrates the results averaged over $3000$ runs. In the inset, the black vertical line shows $m=82\simeq m_\mathrm{relax}$. Purple, green, red, blue and orange lines in the inset indicate $m=10, 50, 100, 600, 1000$, respectively. This average reveals a clear, gradual increase in$C_{\mathrm cat}(\hat{S}_z, \hat{\rho}_\mathrm{P}(m))$. The initial state is $\exp(-\beta h\hat{S}_z)/\mathrm{Tr}\exp(-\beta h\hat{S}_z)$ with $1/\beta=0.1$, $h=0.5$, and a coupling strength multiplied by the interaction time given by $gt=0.222$.
• Figure 5: A log-log plot of the value of $C_{\mathrm cat}(\hat{S}_z, \hat{\rho}_\mathrm{P}(m))$ against $N$ for different $m$values, along with two reference plots. From the bottom to the top, the $m$ values are$m=10$ (purple), $50$ (orange), $100$ (cyan), $600$ (red), and $1000$ (blue). The initial state is given by$\exp(-\beta h\hat{S}_z)/\mathrm{Tr}\exp(-\beta h\hat{S}_z)$ with $1/\beta=0.1$, $h=0.5$, and a coupling strength of$gt=0.222$. The thick gray line represents the reference obtained from Eq. (\ref{['ideal']}), which is $\mathcal{C}_{\mathrm cat}$ of a generalized cat state. The green dotted line represents the reference value of $\tilde{\mathcal{C}}_{\mathrm cat}$ obtained from Eq. (\ref{['maminekoexpect']}). Error bars were omitted as they were smaller than the widths of the dots.