Measuring the temperature of a quantum system is an essential task in almost all aspects of quantum technologies. Theoretically, an optimal strategy for thermometry requires measuring energy which demands full accessibility over the entire system as well as complex entangled measurement basis. In this paper, we take a different approach and show that single qubit sequential measurements in the computational basis not only allows precise thermometry of a many-body system but may also achieve precision beyond the theoretical bound, avoiding demanding energy measurements at equilibrium. To obtain such precision, the time between the two subsequent measurements should be smaller than the thermalization time so that the probe never thermalizes. Therefore, the non-equilibrium dynamics of the system continuously imprint information about temperature in the state of the probe. This allows the sequential measurement scheme to reach precision beyond the accuracy achievable by complex energy measurements on equilibrium probes.
Measuring the temperature of a quantum system is an essential task in almost all aspects of quantum technologies. Theoretically, an optimal strategy for thermometry requires measuring energy which demands full accessibility over the entire system as well as complex entangled measurement basis. In this paper, we take a different approach and show that single qubit sequential measurements in the computational basis not only allows precise thermometry of a many-body system but may also achieve precision beyond the theoretical bound, avoiding demanding energy measurements at equilibrium. To obtain such precision, the time between the two subsequent measurements should be smaller than the thermalization time so that the probe never thermalizes. Therefore, the non-equilibrium dynamics of the system continuously imprint information about temperature in the state of the probe. This allows the sequential measurement scheme to reach precision beyond the accuracy achievable by complex energy measurements on equilibrium probes.
Measuring the temperature of a quantum system is an essential task in almost all aspects of quantum technologies. Theoretically, an optimal strategy for thermometry often requires measuring energy, which demands full accessibility over the entire system as well as a complex entangled measurement basis. In this paper, we take a different approach and show that single-qubit sequential measurements in the computational basis not only allow for precise thermometry of a many-body system, but may also achieve precision beyond the thermometry capacity of the probe at equilibrium, given by the Cram\'er-Rao bound. Thus, using consecutive single-qubit measurements of the probe out of equilibrium is, in most cases, very beneficial, as it achieves lower-temperature uncertainties and avoids demanding energy measurements when compared with probes at thermal equilibrium. To obtain such precision, the time between the two subsequent measurements should be smaller than the thermalization time so that the probe never thermalizes. Therefore, the nonequilibrium dynamics of the system continuously imprint information about temperature in the state of the probe. To demonstrate the generality of our findings, we consider thermometry in both spin chains and the Jaynes-Cummings model.
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.
Abstract Controlling and measuring the temperature in different devices and platforms that operate in the quantum regime is, without any doubt, essential for any potential application. In this review, we report the most recent theoretical developments dealing with accurate estimation of very low temperatures in quantum systems. Together with the emerging experimental techniques and developments of measurement protocols, the theory of quantum thermometry will decisively impinge and shape the forthcoming quantum technologies. While current quantum thermometric methods differ greatly depending on the experimental platform, the achievable precision, and the temperature range of interest, the theory of quantum thermometry is built under a unifying framework at the crossroads of quantum metrology, open quantum systems, and quantum many-body physics. At a fundamental level, theoretical quantum thermometry is concerned with finding the ultimate bounds and scaling laws that limit the precision of temperature estimation for systems in and out of thermal equilibrium. At a more practical level, it provides tools to formulate precise, yet feasible, thermometric protocols for relevant experimental architectures. Last but not least, the theory of quantum thermometry examines genuine quantum features, like entanglement and coherence, for their exploitation in enhanced-resolution thermometry.
Equilibrium quantum thermometry seeks to enhance temperature estimation for reservoirs at thermal equilibrium using quantum resources, yet achieving the Heisenberg scaling in a physical protocol remains a significant challenge. Moreover, quantum measurements face an intrinsic limitation arising from an exponential suppression of thermometric sensitivity as the temperature approaches zero. This limitation severely hinders access to high precision in the low-temperature regime. Here, we demonstrate a simple but powerful quantum engineering protocol to overcome these limitations and reach high precision for temperature estimation. By engineering a small subset of central probes and their linear coupling to a larger array of adjacent probes, we show that the composite partition function can be tailored to yield Heisenberg-limited precision for temperature estimation. Critically, we show that this protocol can simultaneously saturate the Landau bound at low temperatures, an inverse quadratic scaling with the temperature, thereby fully exploiting the potential of equilibrium thermometry and circumventing the exponential suppression of thermometric sensitivity in the low-temperature regime. This scheme relies solely on linear interactions, avoiding complicated many-body couplings and ensuring scalability with the number of probes. Moreover, we find that the comparable precision can be achieved using only local measurements on the central probes, substantially reducing the implementation complexity compared to global measurement schemes and offering a promising approach toward high-precision equilibrium thermometry in physical systems.
Abstract High-precision low-temperature thermometry is a challenge for experimental quantum physics and quantum sensing. Here we consider a thermometer modeled by a dynamically-controlled multilevel quantum probe in contact with a bath. Dynamical control in the form of periodic modulation of the energy-level spacings of the quantum probe can dramatically increase the maximum accuracy bound of low-temperatures estimation, by maximizing the relevant quantum Fisher information. As opposed to the diverging relative error bound at low temperatures in conventional quantum thermometry, periodic modulation of the probe allows for low-temperature thermometry with temperature-independent relative error bound. The proposed approach may find diverse applications related to precise probing of the temperature of many-body quantum systems in condensed matter and ultracold gases, as well as in different branches of quantum metrology beyond thermometry, for example in precise probing of different Hamiltonian parameters in many-body quantum critical systems.
We address a particular instance where open quantum systems may be used as quantum probes for an emergent property of a complex system, as the temperature of a thermal bath. The inherent fragility of the quantum probes against decoherence is the key feature making the overall scheme very sensitive. The specific setting examined here is that of quantum thermometry, which aims to exploit decoherence as a resource to estimate the temperature of a sample. We focus on temperature estimation for a bosonic bath at equilibrium in the Ohmic regime (ranging from sub-Ohmic to super-Ohmic), by using pairs of qubits in different initial states and interacting with different environments, consisting either of a single thermal bath or of two independent ones at the same temperature. Our scheme involves pure dephasing of the probes, thus avoiding energy exchange with the sample and the consequent perturbation of temperature itself. We discuss the role of correlations among the probes and the presence of a local versus a global bath. We show that entanglement improves thermometry at short times if the two qubits are embedded in a common bath, whereas if the interaction time is not constrained, then coherence rather than entanglement is the key resource in quantum thermometry.