Equilibrium thermometry in the multilevel quantum Rabi model

TL;DR

This work develops a multilevel adiabatic framework to quantify equilibrium thermometry in a multilevel quantum Rabi model, deriving a closed-form thermal QFI that decomposes into bright-intra-doublet, bright-bright inter-doublet, and bright–dark transfer contributions. By analyzing dark- and bright-manifold saturations, it reveals how large dark degeneracies can produce near-ideal peak sensitivity at high temperatures, while broad, disorder-robust sensitivity emerges from large, fully bright manifolds. The approach enables efficient thermometric optimization in large, structured MQRM systems and connects spectral engineering to practical temperature sensing across wide ranges. The findings offer actionable design principles for cavity-QED thermometry and motivate future work beyond the adiabatic, closed-system limit.

Abstract

The temperature sensitivity of a probe in equilibrium can be gauged by its thermal quantum Fisher information (QFI). It is known that probes exhibiting degeneracy in their energy-level structure can achieve larger sensitivities, while probes with a more uniform spectrum may remain sensitive over a broader temperature range. Here, we study the thermometric performance of a multilevel quantum Rabi model in which two well-separated atomic manifolds of near-degenerate levels couple to a single cavity mode. We generalise the standard quantum Rabi treatment in the adiabatic regime to find an approximate closed-form expression for the thermal QFI. We then characterise two complementary limits. On the one hand, a large dark-state manifold (dark-manifold saturation) produces a robust peak in thermal sensitivity due to bright--dark population transfer. Such increase in sensitivity is further maximised at an intermediate light--matter coupling strength. Maximising instead the number of bright states (bright-manifold saturation) generates a broadband thermal response that becomes increasingly stable under random light--matter couplings as the number of levels is increased. The rich spectral structure of our cavity-QED model thus makes it a versatile and sensitive equilibrium thermometer over a broad range of temperatures.

Figures (9)

• Figure 1: Schematic overview of large--manifold MQRM thermometry and the two limiting cases we consider. A multilevel atom with near-degenerate ground and excited manifolds of sizes $D_g$ and $D_e$ couples to a single cavity mode through a general complex light--matter coupling matrix $\Lambda_{ij}$, and is brought into thermal equilibrium with a sample at temperature $T$. The resulting thermal sensitivity depends strongly on how the spectrum is partitioned into bright and dark sectors. Bright-manifold saturation ($D_g\approx D_e\gg1$) distributes sensitivity across many bright transitions, producing a broad response. Dark-manifold saturation ($D_e\gg D_g$) generates a dominant peak associated with bright--dark population transfer, that can approach the theoretical maximum for a matched effective degeneracy.
• Figure 2: Spectrum comparison between numerical diagonalisation and AA energies for $D_g=2,D_e=4$ MQRM with $\omega_a=0.2\,\omega_f,\varepsilon=0.1\,\omega_a$, as a function of the coupling scale factor $g$. The maximum oscillator basis level for the numerical diagonalisation is chosen to be sufficiently high to avoid any truncation error ($N_\text{max} = 25$). Intraband detunings $\delta_k^{g/e}$ are specifically chosen as to be equally spaced within their respective domains, with $\boldsymbol \delta^g=[-1,1]$, and $\boldsymbol \delta^e=[-1,-1/3,1/3,1]$. The light--matter coupling matrix is drawn from a random distribution (see details in the main text). Our multilevel AA is in excellent agreement with the energy levels of the MQRM, while improving as $g/\omega_f$ increases.
• Figure 3: Plots of AA QFI estimates and numerically exact calculations from brute-force diagonalisation of a $D_g=2$, $D_e=4$ MQRM Hamiltonian. The truncation of the full MQRM Hamiltonian levels is chosen to be high enough to ensure convergence within the whole parameter regime. The lines tracing the locations of the peak QFI of TLSs optimally tuned for each temperature (cf. Eq. \ref{['schottkyformula']}) as well as the QFI of a fixed TLS resonant with the lowest bright doublet are plotted for comparison. In this case, they show that, over the temperature range of interest, the model has similar thermometric capability to the maximum of any simple two-level system, although over a much wider temperature range. System parameters are the same as in Fig. \ref{['fig:AAverificationSPEC']}, including the light--matter couplings $\boldsymbol{\Lambda}$. The 'AA Formula' QFI is calculated from our Eq. \ref{['QFI_formula']}, and we retain terms that correspond to energies up to the 5-th dark manifold. (a) Weak coupling regime, $g=0.8\omega_f$. Despite the fairly large overall detunings $\varepsilon=0.1\,\omega_a$, the AA QFI recovers the exact result for the lowest temperatures, however, it underestimates the true value at higher temperatures due to truncation error. This can be mitigated by increasing the energy cutoff. (b) Strong coupling regime, $g=5\omega_f$. In the strong regime, the behaviour is very similar, with inaccuracy due to truncation at higher temperature. We otherwise succeed in recovering the exact QFI in the low temperature range.
• Figure 4: Schematic energy spectrum of the MQRM with the lowest primary bright doublet ($D_g=1$), and parameters chosen as $\omega_a=0.2\,\omega_f$, $\varepsilon=0$. The solid red arrows indicate the thermometrically relevant transitions between bright and dark states at a given temperature, due to differing equilibrium-state populations. For a given temperature that corresponds to a bright--dark peak of the MQRM QFI, these transitions define an effective energy gap $E_{\mathrm{eff}}^{\mathrm{ideal}}$ for a comparative benchmark ideal thermometer. We additionally show that any $n=0$ and $k>1$ bright doublets exist within the bounds of the ground state and the first dark band (shaded blue area).
• Figure 5: Dark manifold saturation regime and the approach towards the ideal thermometer. (a) Weak--coupling regime $g=0.1\omega_f$. (b) Intermediate coupling $g=1.2\omega_f$. Atomic frequency is chosen as $\omega_a=0.2\omega_f$, with $\varepsilon=0$ intraband detuning. The figures show the ratio between the maximal QFI of the dark-bright manifold MQRM and that of the corresponding ideal probe, $\mathcal{F}_T^{*\,\text{BD}}/\mathcal{F}_T^{*\,\text{ideal}}$, as a function of the dark state degeneracy $D$ for different numbers of ground states $D\gg D_g$ (black: $D_g=1$, green: $D_g=10$). The red dashed line marks the ideal limit $\mathcal{F}_T^{*\,\text{BD}}/\mathcal{F}_T^{*\,\text{ideal}}=1$. We see that, for all $D_g$, increasing the size of the dark manifold systematically raises the peak-QFI ratio $\mathcal{F}_T^{*\,\text{BD}}/\mathcal{F}_T^{*\,\text{ideal}}$ and drives the MQRM towards ideal thermometric behaviour in both the weak and intermediate coupling regimes. In the weak coupling regime the bright doublets compete for thermal weight and, as we would expect, strongly reduce the fraction of population that participates in the desired dominant bright--dark process. This hinders the overall sensitivity. Note also how the atomically ideal but light--matter coupled $D_g=1$ configuration becomes comparatively less efficient. This is due to the fact that stronger couplings populate higher primary bright ladder states, weakening the effective bright--dark transition. In turn, for $D_g=10$, the trend is reversed, with the peak QFI ratio improving slightly with increased coupling. This indicates that additional bright ladders can compensate for the stronger coupling in the intermediate coupling regime.