Bridging thermodynamics and metrology in nonequilibrium quantum thermometry

Abstract

Single-qubit thermometry presents the simplest tool to measure the temperature of thermal baths with reduced invasivity. At thermal equilibrium, the temperature uncertainty is linked to the heat capacity of the qubit, however the best precision is achieved outside equilibrium condition. Here, we discuss a way to generalize this relation in a non-equilibrium regime, taking into account purely quantum effects such as coherence. We support our findings with an experimental photonic simulation.Introduction:-Identifying strategies for improving the measurement precision by means of quantum resources is the purpose of Quantum Metrology [1][2][3]. In particular, through the Quantum Cramér-Rao Bound (QCRB), it sets ultimate limits on the best accuracy attainable in the estimation of unknown parameters even when the latter are not associated with observable quantities. These considerations have attracted an increasing attention in the field of quantum thermodynamics, where an accurate control of the temperature is highly demanding [4][5][6][7][8]. Besides the emergence of primary and secondary thermometers based on precisely machined microwave resonators [9,10], recent efforts have been made aiming at measuring temperature at even smaller scales, where nanosize thermal baths are higly sensitives to disturbances induced by the probe [11][12][13][14][15][16][17]. Some paradigmatic examples of nanoscale thermometry involve nanomechanical resonators [19], quantum harmonic oscillators [20] or atomic condensates [21][22][23] (also in conjunction with estimation of chemical potential [24]). In this context the analysis of quantum properties needs to be taken into account in order to establish, and eventually enhance, metrological precision [18,[25][26][27][28][29].In a conventional approach to thermometry, an external bath B at thermal equilibrium is typically indirectly probed via an ancillary system, the thermometer S, that is placed into weak-interaction with the former. Assuming hence that the thermometer reaches the thermal equilibrium configuration without perturbing B too much, the Einstein Theory of Fluctuations (ETF) can be used to characterize the sensitivity of the procedure in terms of the heat capacity of S which represents its thermal susceptibility to the perturbation imposed by the bath [30][31][32]. Since this last is an equilibrium property, one should not expect it to hold in non-equilibrium regimes. However thermometry schemes that do not need a full thermalization of the probe have been recognized to offer higher sensitivities in temperature estimation [33].Thus, if on the one hand the QCRB can still be used as the proper tool to gauge the measurement uncertainty on the bath temperature, on the other hand establishing a direct link between this approach and the thermodynamic properties of the probe is still an open question. Furthermore, the advantages pointed out in [33] are conditional on precisely addressing the probe during its evolution, a task which might be demanding in real experiments [28]. Here S i...

Quantum Probes for Ohmic Environments at Thermal Equilibrium

“…On the other hand, thermal fluctuations often spoil the effectiveness of quantum metrological protocols, the most dramatic case being represented by quantum interferometry, where an infinitesimal amount of noise is enough to kill Heisenberg scaling and reinstates the shot noise limit [ 19 ]. In turn, the effect of temperature has been analyzed in different metrological contexts, for example the out-of-equilibrium regimes [ 20 ] and phase estimation in Gaussian states [ 21 ]. For these reasons, we extend here the analysis to the more realistic case of complex systems at thermal equilibrium and discuss in detail the interplay among thermal fluctuations and time evolution in making the qubit an effective probe for the cutoff frequency of its environment.…”

Quantum metrology out of equilibrium

“…On the other hand, it has been recently shown that optimal estimation by quantum probes may be achieved also at finite time, i.e. when the probe has not reached stationarity, and it is still in an outof-equlibrium state [18][19][20][21][22][23][24]. Following this results, we address here open quantum systems out-of-equilibrium as possible quantum probes for the characterisation of their environment.…”

Numerical Approach to Nonequilibrium Quantum Thermodynamics: Nonperturbative Treatment of the Driven Resonant Level Model Based on the Driven Liouville von-Neumann Formalism

“…The superscripts in Eqs. (11) and (12) denote the order in ̇ of the calculated quantity. Specifically, the WBL expressions for -dependent contributions to the particle number, energy, and entropy are given by (See Ref.…”

“…Next, consider shifting the dot's energy at a finite rate. Some preliminary notes are in place: (i) As mentioned above, recent analytical studies [4][5][6][7][10][11][12][13][14][15] have evaluated the lowest order (in the driving rate) corrections to thermodynamic functions and fluxes. To approach this limit in the numerical simulations we need to use relatively slow shift rates, for which deviations from the quasistatic limit are small, implying relatively large numerical errors.…”

“…[2] and references therein). Studies that consider the implications of strong system-bath coupling have recently emerged [4][5][6][10][11][12][13][14][15]. In the latter, the use of weak system-reservoir(s) coupling is replaced by the assumption that the dynamics imposed on the system is slow.…”