Multicritical quantum sensors driven by symmetry-breaking

TL;DR

The paper demonstrates that symmetry-breaking can serve as an independent resource for quantum-enhanced sensing near multicritical points, outside of gap-closing mechanisms. In the 1D Kitaev chain, single-parameter estimation of the pairing amplitude $ Delta$ achieves Heisenberg scaling with $ F_{ Delta Delta} \sim L^2$ along a gapless line at the multicritical point, while multiparameter sensing of $(\nmu,\nDelta)$ yields $\nG \sim L^6$, indicating super-Heisenberg scaling in a narrow parameter window where both symmetry-breaking and gap closing influence the QFIM. The work also analyzes practical constraints, showing that even with finite-state preparation times, advantageous regions exist where quantum sensitivity remains superior to classical limits. Overall, multicritical systems emerge as a promising platform for high-precision, multiparameter quantum metrology, highlighting symmetry-breaking as a distinct metrological resource.

Abstract

Quantum criticality has been demonstrated as a useful quantum resource for parameter estimation. This includes second-order, topological and localization transitions. In all these works reported so far, gap-to-gapless transition at criticality has been identified as a crucial resource for achieving the quantum-enhanced sensing, although there are several important concepts associated with criticality, such as long-range correlation, symmetry breaking. In this work, we show that symmetry-breaking alone can drive a quantum-enhanced sensing, even without any gap-to-gapless transition. We analytically demonstrate that the estimation of the superconducting pairing amplitude in the one-dimensional Kitaev model achieves Heisenberg scaling when the system is prepared near a multicritical point and is varied along a gapless critical line, implying symmetry breaking as a standalone metrological resource. Extending our analysis in the realm of simultaneous multiparameter estimation of both the pairing term and the chemical potential, we show that it is possible to obtain $L^6$ scaling in a narrow parameter range, but with definite observable consequence, where the quantum advantage is assisted by gap-to-gapless transition as well. Our work thus identifies a new resource for criticality-enhanced quantum sensing, and also suggests multicritical systems as useful platform for multiparameter sensing.

Figures (12)

• Figure 1: Schematic for quantum-enhanced sensing assisted by symmetry-breaking: The phase diagram shows different topological phases in the $\mu$-$\Delta$ plane. The red and the green lines are gapless. In the left panel, the protocol of sensing $\Delta$ is presented. $\mathcal{F}_{\Delta\Delta}$ has an quantum-enhanced scaling when the system is prepared at the critical line of $\mu=\mu_c$ (near the multicritical point, at A), and driven to, say B. We note that this is enhancement of scaling is not because of gap-to-gapless transition, but rather due to breaking of $U(1)$ symmetry. In the multi-parameter regime, one can envision a case where system is prepared at C, and both the parameters are maneuvered along the directed arrows. Since, we are in a two-dimensional plane, the point C can be approached from multiple directions. This leads to a quantum-enhanced scaling for the multi-parameter precision estimator. This involves, both, a gap-to-gapless transition and the symmetry-breaking.
• Figure 2: Variation of $\mathcal{F}_{\Delta\Delta}$ near the multicritical point: (a) $\mathcal{F}_{\Delta\Delta}$ for system sizes $L$ = 400 (solid blue line), 600 (dot-dashed orange line), 800 (dotted green line), 1000 (dashed red line) are presented with respect to $\Delta$ keeping a fixed $\mu = 2$. This is a representative case, we observe similar nature for various other system-sizes as well. The finite-size scaling of $\mathcal{F}_{\Delta\Delta}$ is presented in (b). Setting $\mu = 2$, $\mathcal{F}_{\Delta\Delta}$ at $\Delta = 10^{-7}$ (circular markers) scale as $\mathcal{F}_{\Delta\Delta}^a \sim L^2$ while at $\Delta = 0.7$ (square markers) they scale as $\mathcal{F}_{\Delta\Delta}^b \sim L$. The straight lines are the best fit. (c) We present the scaling exponents $\beta$ of $\mathcal{F}_{\Delta\Delta}$ against the parameter $\Delta$. We identify three regions, (i) low $\Delta$ regime: $\Delta < 5\times10^{-4}$ is represented by light blue color where $\mathcal{F}_{\Delta\Delta} \sim L^2$; (ii) intermediate $\Delta$ regime : $5\times10^{-4}\leq\Delta\leq0.02$ represented by light green color where $\beta$ transitions; (iii) large $\Delta$ regime: $\Delta > 0.02$ is represented by light pink color where $\mathcal{F}_{\Delta\Delta} \sim L$.
• Figure 3: The variation of $\mathcal{F}_{\mu\mu}$ around the multicriticality : (a) $\mathcal{F}_{\mu\mu}$ for system sizes $L$ = 400 (solid blue line), 600 (dot-dashed orange line), 800 (dotted green line), 1000 (dashed red line) are presented against $\mu$, where in the x-axis $\mu_c = 2$ has been subtracted from $\mu$. $\Delta = 0.001$ is set for all cases. This is a representative case, we observe a similar nature for various other system-sizes. (b) The finite-size scaling of $\mathcal{F}_{\mu\mu}$ is presented at $\mu = 2$ and two different $\Delta$. Considering $\Delta = 10^{-7}$, we observe $\mathcal{F}_{\mu\mu}^a \sim L^{6}$ (circular markers) and $\Delta = 0.7$, we observe $\mathcal{F}_{\mu\mu}^b \sim L^{2}$ (square markers). The straight lines are the best fit. (c) We present the scaling exponents $\beta$ of $\mathcal{F}_{\mu\mu}$ against the parameter $\Delta$. We identify three regions, (i) low $\Delta$ regime: $\Delta < 10^{-3}$ represented by light blue color where $\mathcal{F}_{\mu\mu} \sim L^6$; (ii) intermediate $\Delta$ regime: $10^{-3}\leq\Delta\leq10^{-2}$ is represented by light green color, where $\beta$ transitions from $6$ to $2$; (iii) large $\Delta$ regime: $\Delta > 10^{-2}$ is represented by light pink color, where $\mathcal{F}_{\mu\mu} \sim L^2$.
• Figure 4: Variation of $\mathcal{G}$ near multicriticality: (a) The quantity $\mathcal{G}$ is presented with respect to the tuning parameters $\mu$ and $\Delta$ in the vicinity of the multicriticality. The system size is $L = 1000$. (b) The finite-size scaling of $\mathcal{G}$ is presented. Setting $\mu = 2$, $\mathcal{G}$ at $\Delta = 10^{-7}$ (circular markers) scale as $\mathcal{G}^a \sim L^6$, while at $\Delta = 0.7$ (square markers) we have $\mathcal{G}^b \sim L$. The straight lines are the best fit. (c) We present the scaling exponents $\beta$ of $\mathcal{G}$ against the parameter $\Delta$. We identify three regions, (i) low $\Delta$ regime: $\Delta < 10^{-5}$ is represented by light blue color where $\mathcal{G} \sim L^6$; (ii) intermediate $\Delta$ regime : $10^{-5}\leq\Delta\leq0.03$ represented by light green color where $\beta$ transitions for both $\mathcal{F}_{\Delta\Delta}$ and $\mathcal{G}$; (iii) large $\Delta$ regime: $\Delta > 0.03$ is represented by light pink color where $\mathcal{F}_{\Delta\Delta}, \mathcal{G} \sim L$.
• Figure 5: Region of good sensor: The QFIM elements (a)$\mathcal{F}_{\mu \mu}^{-1}$, (b)$\mathcal{F}^{-1}_{\Delta \Delta}$ and (c)$\mathcal{G}^{-1}$ are presented. The value of $\mu$ is fixed at $\mu = 2$. According to quantum Cramér-Rao bound, $\delta {\theta}^2 \geq (M F_{\theta})^{-1}$, where $M$ is the number of measurements. Thus for a effective sensor, the values of $(M F_{\theta})^{-1}$ should be at-least lesser by an order than the parameter $\theta$ that is being sensed, viz. $1/\sqrt{MF_{\theta}} \ll |\theta|$. Typically, the experiments are repeated for few thousand times, i.e., $M=10^3-10^4$. Hence, practical parametric regime for effective sensing can be identified by validity of the constraint, ${F_{\theta}}^{-1/2} < |\theta|$. We shade the regions gray, where this is not the case. In (a) and (b), we consider single-parameter estimation case, while in (c) we consider the multi-parameter estimation.