Optimal strategies for transient and equilibrium quantum thermometry using Gaussian and non-Gaussian probes

Abstract

We study temperature estimation using quantum probes, including single-mode initial states and two-mode states generated via stimulated parametric down-conversion in a nonlinear crystal at finite temperature. We explore both transient and equilibrium regimes and compare the performance of Gaussian and non-Gaussian probe states for temperature estimation. In the non-equilibrium regime, we show that single-mode non-Gaussian probe states - such as Fock, odd cat, and Gottesman-Kitaev-Preskill states - can significantly enhance the speed of estimation, particularly at short interaction times. In the two-mode setting, entangled states such as the two-mode squeezed vacuum, NOON state, and entangled cat state can enable access to temperature information at earlier times. In the equilibrium regime, we analyze temperature estimation using two-mode squeezed thermal states, which outperform single-mode strategies. We evaluate practical measurement strategies and find that energy-based observables yield optimal precision, population difference observables provide near-optimal precision, while quadrature-based measurements are suboptimal. The precision gain arises from squeezing, which suppresses fluctuations in the population difference.

Figures (15)

• Figure 1: Schematic of parametric down conversion process: a coherent pump $|\alpha_p\rangle$ enters a non-linear $\chi^{(2)}$ medium, which is at temperature $T$, converting pump photons into signal ($a_s$) and idler ($a_i$) modes. A detector is used to perform measurements on the two modes for estimation of the temperature $T$.
• Figure 2: (a) Non-equilibrium QFI $F_Q(t)$ as a function of the interaction time $t$ for initial squeezed vacuum state (red dashed curve) and Fock state (green dotted curve). (b) shows the time-dependent ratio $R(t) = F_Q^{\mathrm{Fock}}(t) / F_Q^{\mathrm{SVS}}(t)$, comparing the QFI of the Fock state to that of the squeezed vacuum state at different values of temperature $T$. The parameters are set to $\omega=1$, $T=0.4$, $\gamma=0.2$, and $n_0=4$. The two states have the same initial energy for which $r=\sinh^{-1}(\sqrt{n_0})$.
• Figure 3: Non-equilibrium QFI $F_Q(t)$ for different initial preparations of the single probe state as a function of the interaction time $t$ for estimation of the bath temperature $T$. The probe states are prepared with parameters (coherent amplitude $\alpha$, squeezing $r$, photon number $n_0$, etc.) chosen such that all states have the same average energy $E_0 =4.5$. The other parameters are set to $\omega=1$, $T=0.4$, and $\gamma=0.2$.
• Figure 4: Non-equilibrium QFI for different initial Gaussian and non-Gaussian states as a function of interaction time $t$ for estimation of bath temperature $T$. We use both modes as a probe to measure the temperature of the nonlinear crystal. The parameters are set to $\omega=1$, $\alpha_p=4$, $g=0.08$, $T=0.4$, and $\gamma=0.2$. The rest of the parameters are set to achieve the fixed target energy $E_t=6$.
• Figure 5: Non-equilibrium QFI as a function of time for a single-mode squeezed initial state (red dashed) and two-mode squeezed state (solid blue) for the parameters set to $r=1.0$ and $T=0.8$. The rest of the parameters are the same as in Figs. \ref{['fig:Smode']} and \ref{['fig:2Modes']}.