Quantum interferometric measurements of temperature

We provide a detailed description of the quantum interferometric thermometer, which is a device that estimates the temperature of a sample from the measurements of the optical phase. We rigorously analyze the operation of such a device by studying the interaction of the optical probe system prepared in a single-mode Gaussian state with a heated sample modeled as a dissipative thermal reservoir. We find that this approach to thermometry is capable of measuring the temperature of a sample in the nanokelvin regime. Furthermore, we compare the fundamental precision of quantum interferometric thermometers with the theoretical precision offered by the classical idealized pyrometers, which infer the temperature from a measurement of the total thermal radiation emitted by the sample. We find that the interferometric thermometer provides a superior performance in temperature sensing even when compared with this idealized pyrometer. We predict that interferometric thermometers will prove useful for ultraprecise temperature sensing and stabilization of quantum optical experiments based on the nonlinear crystals and atomic vapors.

Figure 1

A general parameter estimation scheme in which a probe system prepared in an initial quantum state ${\rho }_0$ evolves under the quantum channel ${\mathrm{\Lambda }}_{\theta }$ to the output quantum state ${\rho }_{\theta }$. Note that ${\mathrm{\Lambda }}_{\theta }$ takes into account all decoherence processes that may be present in the setup. Following the evolution stage, the probe system is subjected to a general quantum measurement described by a POVM $\left\{{\mathrm{\Pi }}_x\right\}$, which produces the measurement results $x$. These measurement results are then used to make an estimate of the parameter $\theta$ via the estimator function $\tilde{\theta }\left(x\right)$.

Figure 2

Basic scheme for quantum interferometric thermometry. A single-mode Gaussian state of light ${\rho }_0$ is sent through a sample with temperature $T$ and transmissivity $\eta$, resulting in a mixed output state ${\rho }_{\phi }$, which is then measured using a general POVM $\left\{{\mathrm{\Pi }}_x\right\}$ measurement. Assuming that the phase shift $\phi$ is temperature dependent, the measurement outcomes are then used to find an estimated value of $T$ via the estimator function $\tilde{T}\left(x\right)$. The bright yellow beam depicts a classical reference, which allows us to define the phase shift in a meaningful way [37]. The propagation of light through a heated sample with transmissivity $\eta$ can be decomposed into two distinct processes: first the single-mode Gaussian beam undergoes the phase shift $\phi$ relative to the reference beam and then it undergoes a photon loss process, which is modeled with the help of a beam splitter with transmissivity $\eta$, where the second port is filled with light in the thermal state ${\rho }_N$.

Figure 3

The precision of the quantum interferometric thermometer near $T=298$ K for a nonlinear PPKTP crystal with a length of 1 cm plotted as a function of the mean number of photons $\overline{N}$ with $\lambda =1064$ nm ($\omega =1.77\times {10}^{15}$ Hz). In our model for the PPKTP crystal, we set $M=3$ g, $C_s=688\text{J(kg}{\text{K)}}^{-1},n=1.74,{\alpha }_T=1.1\times {10}^{-5}{\text{K}}^{-1},n^{\prime }=0.6\times {10}^{-5}{\text{K}}^{-1}$, and $\eta =exp[-L{\alpha }_{\mathrm{abs}}]=0.9998$, with ${\alpha }_{\mathrm{abs}}=0.0002 {\mathrm{cm}}^{-1}$ [47]. The red solid curve represents the exact lower bound for the optimal single-mode Gaussian state obtained via the QCRB with $k=1$ and $Q_T$ given in Eq. (25), with the tail correcting for a possible heating up of the crystal. The gray areas depict the precision regions lying below the asymptotic bound for the single-mode squeezed-vacuum state given in Eq. (28) and lying above the asymptotic bound for the single-mode coherent state given in Eq. (29). The black dashed line represents the fundamental precision for the idealized pyrometer given in Eq. (8) with $S=1{\text{cm}}^2$ and $\delta t=10$ ms.