Anti-Critical Quantum Metrology

TL;DR

This work rethinks quantum metrology by showing that enhanced parameter estimation need not rely on a vanishing energy gap. By explicitly accounting for the interrogation time, the authors distinguish genuine resource gains from critical slowing down and introduce anti-critical metrology, which opens the gap to accelerate dynamics while preserving or enhancing sensitivity. Using the quantum Rabi model as a concrete platform, they derive effective Hamiltonians for both critical and anti-critical sectors and analyze the quantum Fisher information, showing that near criticality $\mathcal{I}_\omega$ diverges but the required time also diverges, whereas in the anti-critical regime the gap opens and the timing shortens, yielding comparable or superior precision per unit time. They extend the analysis to many-body settings (LMG and Ising variants) and discuss practical considerations, highlighting that gap engineering can broaden the toolkit for metrology in realistic interacting systems.

Abstract

Critical quantum metrology exploits the dramatic growth of the quantum Fisher information near quantum phase transitions to enhance the precision of parameter estimation. Traditionally, this enhancement is associated with a closing energy gap, which causes the characteristic timescales for adiabatic preparation or relaxation to diverge with increasing system size. Consequently, the apparent growth of the quantum Fisher information largely reflects the increasing evolution time induced by critical slowing down, rather than a genuine gain in metrological performance, thereby severely limiting the practical usefulness of such protocols. Here we show that the relationship between energy-gap variations, quantum Fisher information, and achievable precision is far more subtle in interacting quantum systems: enhanced sensitivity does not require a vanishing gap, and, perhaps more surprisingly, a decreasing quantum Fisher information does not necessarily imply reduced precision once the time is properly taken into account. Building on this insight, we introduce an anti-critical metrology scheme that achieves enhanced precision while the energy gap increases. We illustrate this mechanism using the quantum Rabi model, thereby identifying a route to metrological advantage that avoids the critical slowing down associated with conventional criticality.

Figures (5)

• Figure 1: Schematic illustrating the concepts of critical and anti-critical metrology. Top panels represent the phase space ground state of the associated potentials depicted in the bottom panel. Starting from the non-interacting spectrum and a corresponding vacuum state (b), interactions can close the energy gap—realizing critical metrology (a)—or open it—realizing anti-critical metrology (c). In both cases the final state is squeezed, but the relevant timescales differ dramatically. In critical metrology, a vanishing gap lengthens the timescales, artificially enhancing the quantum Fisher information, whereas in anti-critical metrology, a growing gap shortens them, artificially suppressing the quantum Fisher information. This behavior reflects the fact that, in many interacting quantum systems, the energy gap—and thus the characteristic timescales—scale with the system size.
• Figure 2: Comparison between the quantum Fisher information (solid gray line) and the quantum Fisher information multiplied by the energy gap—equivalently, divided by the adiabatic time required to transform neighboring ground states (dashed black line). While the quantum Fisher information diverges near the critical point ($g^{2}/g_{c}^{2}\approx1$), accounting for the transformation time reveals that the sensitivity per unit time can be equally large away from criticality, where the required evolution time is much shorter due to the absence of critical slowing down. Exploiting the opening of the energy gap, and thus enabling an anti-critical metrology approach, may however require significantly stronger interactions. Simulations are performed using the effective description of the quantum Rabi model from Eq. \ref{['eq:effectiveH']} for $g^2/g_c^2 >0$, and from Eq. \ref{['eq:antiH']} for $g^2/g_c^2<0$.
• Figure 3: Lipkin-Meshkov-Glick model. (a) Variances of collective spin components as a function of $g/g_c$ ($g_c\equiv \omega$). The variances of $\hat{S}_x$ (blue line) and $\hat{S}_y$ (orange line) are flipped with respect to each other which shows that quantum correlations can be generated close and away from the critical point. The variance of $\hat{S}_z$ (green line) grows both close and away from the critical point similarly as its expectation value (red line shifted by $N/2$ for the matter of presentation). (b) The associated energy gap (blue line) and its inverse (orange line) are very closely related to variances of $\hat{S}_x$ and $\hat{S}_y$ which points at a close connection between these quantities. Finally, in (c), we show the quantum Fisher information (blue line) as a function of $g/g_c$ which is related to the expectation value of $\hat{S}_z$ weighted by the squared inverse of the energy gap (dashed orange line). Naively it seems that the quantum Fisher information is superior close to the critical point, however, this is caused by the fact that the characteristic time scale artificially elevates the quantum Fisher information. Once the characteristic time is factored out (green line), it turns out that quantum Fisher information can be equally large far away from the critical point and achieved much faster due to decreasing energy gap.
• Figure 4: Transverse field Ising model. (a) Variances of $\hat{S}_x$ (blue line), $\hat{S}_y$ (orange line), and $\hat{S}_z$ (green line) as well as expectation value of $\hat{S}_z$ (red line) as a function of $g/g_c$). As can be seen, correlations can be equally large close and away from the critical point $g/g_c=1$. However the energy gap (b) is symmetric with respect to $g/g_c=0$ point so the gap never increases with increasing coupling. In (c), we see the effect on the quantum Fisher information (blue line) which is symmetric because the correlations depend on $|g/g_c|$, and once divided by the energy gap (orange line) is also symmetric because the gap also depends on $|g/g_c|$ and not on its sign. In this case, anti-critical metrology approach cannot work. In the simulations we have set $N=10$.
• Figure 5: Transverse field Ising model with transverse interactions. (a) Variances of $\hat{S}_x$ (blue line), $\hat{S}_y$ (orange line), and $\hat{S}_z$ (green line) as well as expectation value of $\hat{S}_z$ (red line) as a function of $g/g_c$). As can be seen, again correlations can be equally large close and away from the critical point $g/g_c=1$. However, the energy gap (b) is asymmetric with respect to $g/g_c=0$ point so the gap can increase with the increasing coupling. In (c), we see the effect on the quantum Fisher information (blue line) which is now asymmetric because the correlations depend on $g/g_c$ in a different way than the gap. In this case, anti-critical metrology approach can work as evidenced by the quantum Fisher information divided by the energy gap (orange line). In the simulations we have set $N=10$.