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.
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.
We discuss the application of techniques of quantum estimation theory and quantum metrology to thermometry. The ultimate limit to the precision at which the temperature of a system at thermal equilibrium can be determined is related to the heat capacity when global measurements are performed on the system. We prove that if technical or practical limitations restrict our capabilities to local probing, the highest achievable accuracy to temperature estimation reduces to a sort of mesoscopic version of the heat capacity. Adopting a more practical perspective, we also discuss the relevance of qubit systems as optimal quantum thermometers, in order to retrieve the temperature, or to discriminate between two temperatures, characterizing a thermal reservoir. We show that quantum coherence and entanglement in a probe system can facilitate faster, or more accurate measurements of temperature. While not surprising given this has been demonstrated in phase estimation, temperature is not a conventional quantum observable, therefore these results extend the theory of parameter estimation to measurement of non-Hamiltonian quantities. Finally we point out the advantages brought by a less standard estimation technique based on sequential measurements, when applied to quantum thermometry.
Temperature estimation, known as thermometry, is a critical sensing task for physical systems operating in the quantum regime. Indeed, thermal fluctuations can significantly degrade quantum coherence. Therefore, accurately determining the system's operating temperature is a crucial first step toward distinguishing thermal noise from other sources of decoherence. In this work, we estimate the unknown temperature of a collection of identical and independent two-level atoms dispersively probed by a single-mode quantized electromagnetic field. In contrast to previous works, we present an analytical sensing analysis demonstrating that the joint atom-field evolution -- without any assumptions or approximations -- can achieve, at best, the standard quantum limit of precision concerning the number of field excitations. To investigate our analysis further, we propose and implement a quantum thermometer based on a nonlinear Mach-Zehnder interferometer, which we realize through quantum digital simulation. Our simulation is highly flexible regarding atomic state preparation, allowing the initialization of atomic ensembles with positive and effective negative temperatures. This makes our platform a promising and versatile testbed for benchmarking thermometric capabilities in current quantum simulators.
Abstract Temperature estimation, known as thermometry, is a critical sensing task for physical systems operating in the quantum regime. Indeed, thermal fluctuations can significantly degrade quantum coherence. Therefore, accurately determining the system’s operating temperature is a crucial first step toward distinguishing thermal noise from other sources of decoherence. In this work, we estimate the unknown temperature of a collection of identical and independent two-level atoms dispersively probed by a single-mode quantized electromagnetic field. In contrast to previous works, we present an analytical sensing analysis demonstrating that the joint atom-field evolution—without any assumptions or approximations—can achieve, at best, the standard quantum limit of precision concerning the number of field excitations. To investigate our analysis further, we propose and implement our thermometry scheme on a nonlinear Mach–Zehnder interferometer, which we realize through quantum digital simulation. Our proposal is highly flexible regarding atomic state preparation, allowing the initialization of atomic ensembles with positive and effective negative temperatures. This makes our proposal a promising and versatile testbed for benchmarking thermometric capabilities in current quantum simulators.
Quantum thermometry aims to measure temperature in nanoscale quantum systems, paralleling classical thermometry. However, temperature is not a quantum observable, and most theoretical studies have therefore concentrated on analyzing fundamental precision limits set by the quantum Fisher information through the quantum Cramer-Rao bound. In contrast, whether a direct temperature readout can be achieved in quantum thermometry remains largely unexplored, particularly under the nonequilibrium conditions prevalent in real-world applications. To address this, we develop a direct temperature readout scheme based on a thermodynamic inference strategy. The scheme integrates two conceptual developments: (i) By applying the maximum entropy principle with the thermometer's mean energy as a constraint, we assign a reference temperature to the nonequilibrium thermometer. We demonstrate that this reference temperature outperforms a commonly used effective temperature defined through equilibrium analogy. (ii) We obtain positive semi-definite error functions that lower-bound the deviation of the reference temperature from the true temperature-in analogy to the quantum Cramer-Rao bound for the mean squared error-and vanish upon thermalization with the sample. Combining the reference temperature with these error functions, we introduce a notion of corrected dynamical temperature which furnishes a postprocessed temperature readout under nonequilibrium conditions. This corrected dynamical temperature can be evaluated adaptively without prior knowledge of the actual temperature. We validate the corrected dynamical temperature in a qubit-based thermometer under a range of nonequilibrium initial states, confirming its capability to estimate the true temperature. Importantly, we find that increasing quantum coherence can enhance the precision of this readout.
The precise measurement of low temperatures is significant for both the fundamental understanding of physical processes and technological applications. In this work, we present a method for low-temperature measurement that improves thermal range and sensitivity by generating quantum coherence in a thermometer probe. Typically, in temperature measurements, the probes thermalize with the sample being measured. However, we use a two-level quantum system, or qubit, as our probe and prevent direct probe access to the sample by introducing a set of ancilla qubits as an interface. We describe the open system dynamics of the probe using a global master equation and demonstrate that while the ancilla-probe system thermalizes with the sample, the probe per se evolves into a nonthermal steady state due to nonlocal dissipation channels. The populations and coherences of this steady state depend on the sample temperature, allowing for precise and wide-range low-temperature estimation. We characterize the thermometric performance of the method using quantum Fisher information and show that the quantum Fisher information can exhibit multiple and higher peaks at different low temperatures with increasing quantum coherence and the number of ancilla qubits. Our analysis reveals that the proposed approach, using a nonthermal qubit thermometer probe with temperature-dependent quantum coherence generated by a multiple qubit interface between a thermal sample and the probe qubit, can enhance the sensitivity of temperature estimation and broaden the measurable low-temperature range.
We investigate the temperature uncertainty relation in the nonequilibrium probe-based temperature estimation process. We demonstrate that it is the fluctuation of heat that fundamentally determines temperature precision through the temperature-heat uncertainty relation. Specifically, we find that heat is divided into trajectory heat and correlation heat, which are associated with the heat exchange along the thermometer's evolution and the correlation between the thermometer and the sample, respectively. Based on two types of thermometers, we show that both of these heat terms are resources for enhancing temperature precision. By clearly distinguishing the resources for enhancing estimation precision, our findings not only explain why various quantum features are crucial for accurate temperature sensing but also provide valuable insights for designing ultrahigh-sensitive quantum thermometers. Additionally, we demonstrate that the temperature-heat uncertainty relation is consistent with the well-known temperature-energy uncertainty relation in thermodynamics. It establishes a connection between the information theory and the thermodynamics.