Abstract
While the ability to measure low temperatures accurately in quantum systems is important in a wide range of experiments, the possibilities and the fundamental limits of quantum thermometry are not yet fully understood theoretically. Here we develop a general approach to low-temperature quantum thermometry, taking into account restrictions arising not only from the sample but also from the measurement process. {We derive a fundamental bound on the minimal uncertainty for any temperature measurement that has a finite resolution. A similar bound can be obtained from the third law of thermodynamics. Moreover, we identify a mechanism enabling sub-exponential scaling, even in the regime of finite resolution. We illustrate this effect in the case of thermometry on a fermionic tight-binding chain with access to only two lattice sites, where we find a quadratic divergence of the uncertainty}. We also give illustrative examples of ideal quantum gases and a square-lattice Ising model, highlighting the role of phase transitions.
Highlights
While the ability to measure low temperatures accurately in quantum systems is important in a wide range of experiments, the possibilities and the fundamental limits of quantum thermometry are not yet fully understood theoretically
If the precision is quantified by the relative error, it is not surprising that thermometry becomes more challenging as temperature is reduced
Our approach clearly identifies regimes featuring an exponentially diverging error, and those where sub-exponential scaling is possible even with finite measurement resolution. We show that the latter is possible in a physical system, namely a fermionic tight-binding chain with access to only two lattice sites
Summary
Before we consider bounds that are imposed by limited experimental access, it is illustrative to consider the limitations imposed by the system itself. For any system with a finite energy gap ∆ between the ground and the first excited state, Eq (1) implies that the absolute error in any temperature measurement diverges exponentially as absolute zero is approached (i.e., kBT ∆) [24]. The main goal of this work is to provide a detailed investigation of low-temperature thermometry under limited access, resulting in a number of novel bounds on the associated measurement error. To this end, we consider measurements with finitely many outcomes.
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