Fundamental limits on low-temperature quantum ewline thermometry

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 tightbinding 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 squarelattice Ising model, highlighting the role of phase transitions. A sizable part of contemporary physics is focused on phenomena that occur at low temperatures [1]. While for some of these phenomena, finite temperatures constitute an undesirable nuisance (usually in the form of noise), in other scenarios temperature gradients are at the heart of the physics under investigation [2, 3]. In both cases, accurately determining temperature is desirable for characterizing the examined processes and ultimately leads to an increase in control and a better understanding of the underlying physics [4, 5]. If the precision is quantified by the relative error, it is not surprising that thermometry becomes more challenging as temperature is reduced. However, the state of the matter is even worse as it turns out that also the absolute error usually increases as temperature is reduced. Sensitively measuring low temperatures is therefore challenging. But how hard is it exactly? A number of previous studies considered specific physical models and found that it is exponentially hard, in the sense that the absolute error diverges exponentially at low temperatures [6–17]. In fact, this Patrick P. Hofer: patrick.hofer@teorfys.lu. se exponential scaling can be derived from very general arguments [18–21], based on the energy spectra of the sample and the probe, and could therefore appear to represent a fundamental limit on thermometry. However, it was recently shown that (using a measurement with infinite resolution) a sub-exponential scaling is possible in a system of strongly coupled harmonic oscillators [22]. This raises the question of what the fundamental limits to low-temperature quantum thermometry really are. In particular, when and how exponential scaling can be overcome. In addition to fundamental insight, addressing this question is also highly desirable given the importance of thermometry for quantum experiments [23]. With this goal in mind, we present a general approach to low-temperature quantum thermometry. We consider not only limitations imposed by the system itself, but also limitations imposed by the experimentally accessible measurements. We derive a fundamental bound on quantum thermometry assuming only that measurements have finite resolution, as they do in any realistic experiment. Alternatively the same bound can in fact be recovered from the third law of thermodynamics, without making any assumptions on the measurements. More generally, 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. In this case, the uncertainty diverges only quadratically. Hence the same scaling as in the model of Ref. [22] is achieved, but here requiring only finite resolution. Furthermore, we apply our formalism to illustrative examples of ideal quantum gases and a square-lattice Ising model, highlighting the role of phase transitions.