Multi-Parameter Multi-Critical Metrology of the Dicke Model

Abstract

Critical quantum metrology exploits the hypersensitivity of quantum systems near phase transitions to achieve enhanced precision in parameter estimation. While single-parameter estimation near critical points is well established, the simultaneous estimation of multiple parameters, which is essential for practical sensing applications, remains challenging. This difficulty arises from sloppiness, a phenomenon that typically renders the QFIM singular or nearly singular. In this work, we demonstrate that multiparameter critical metrology is not only feasible but can also retain divergent precision scaling, provided one accepts a trade-off in the scaling exponent. Using the GS of the single-cavity DM, we show that two Hamiltonian parameters can be simultaneously estimated with a scalar variance bound scaling as the square root of the critical parameter. This overcomes the inherent sloppiness by leveraging higher-order contributions to the QFIM. To recover the optimal quadratic scaling, we introduce the DD with photon hopping. In this model, a triple point in the phase diagram enables the simultaneous closure of two excitation gaps, which effectively increases the rank of the QFIM and restores the ideal single-parameter scaling for specific parameter pairs. Furthermore, we extend our to a lossy scenario. We prove that the SSs of both the DM and the DD still support multiparameter estimation with diverging precision, exhibiting linear scaling in the DM and quadratic scaling near the triple point in the DD. Finally, we establish a connection between the derived critical scalings and the fundamental state preparation time, providing a unified framework to operationally compare different sensing strategies. Our results demonstrate that critical quantum metrology can be made robust against dissipation, paving the way for practical quantum sensors operating near phase transitions.

Figures (7)

• Figure 1: Pictorial representation of the cavity model considered. Namely: (a) single cavity Dicke model, (b) Dicke dimer, (c) single cavity Dicke model subject to photon loss, (d) Dicke dimer subject to photon loss.
• Figure 2: Plot of the scalar variance bound $C_S^{\{i,j\}} = \operatorname{Tr}[(Q^{\{i,j\}})^{-1}]$ for all pairwise combinations of parameters, as a function of the normalized coupling $g/g_c$. The simulation is performed employing the GS of the DM at fixed $\omega_c = 1$ and $\omega_a = 0.7$. The inset shows the prefactor $\mathcal{T}^{\{ 1,2\}}$, given by Eq. \ref{['eq:prefactor']}, as a function of $\omega_c$ for fixed $\omega_a = 0.7$.
• Figure 3: (a) Plot of the scalar variance bound $C_S^{\{1,2,3\}}$ for the triplet $\{\omega_c, g, \omega_a\}$ as a function of $g/g_c$ for fixed $\xi \in \{0.1, 0.2, 0.4\}$, employing the GS of the DD as a probe. (b) Plot of the scalar variance bound $C_S^{\{i,4\}} = \operatorname{Tr}[(Q^{\{i,4\}})^{-1}]$ for the simultaneous estimation of $\xi$ and one other parameter ($\omega_c, g,$ or $\omega_a$), as a function of $g/g_t$. The simulation has been performed employing the GS of DD as a probe and dynamically tuning both $g$ and $\xi$ toward the TP over a trajectory defined by Eq. \ref{['eq:traj']} with fixed $k = 1$. The inset represents the phase diagram of the DD in the $g-\xi$ plane, with the trajectory highlighted. Both panels use fixed $\omega_c = 1, \omega_a = 0.7$.
• Figure 4: Plot of the scalar variance bound $C_S^{\{i,j,k\}}$ for the SS of the DM, as a function of the normalized coupling $g/g_c$. We fixed $\omega_a = 0.7$, $\omega_c = 1$ and $\kappa = 0.1$. The main panel shows the estimation of the decay rate $\kappa$ simultaneously with two other parameters. The inset panel shows the simultaneous estimation of the Hamiltonian parameters $\omega_c, g, \omega_a$.
• Figure 5: Plot of the scalar variance bound $C_S$ for various parameter subsets as a function of the normalized coupling $g/g_c$, employing the SS of the DD as a probe. Other parameters are fixed at $\omega_c = 1$, $\omega_a = 0.7$, $\kappa = 0.1$ and $\xi = 0.4$.