Single energy measurement Integral Fluctuation theorem and non-projective measurements

TL;DR

This work extends quantum fluctuation theorems by deriving a Jarzynski-type equality for work defined from a single unsharp energy measurement. The authors decompose the resulting relation into contributions from coherences, measurement-induced noise, and information gained, and show how meter resolution and spectral spacing shape the correction terms. Using a two-level system, they illustrate non-overlapping and overlapping measurement regimes, providing analytic expressions and numerical simulations that highlight when the standard projective-measurement result is recovered up to a meter-dependent factor versus when information-measurement coupling arises. The results illuminate the interplay between measurement backaction and quantum coherence in thermodynamic work, with implications for experiments employing realistic, imperfect energy detectors.

Abstract

We study a Jarzysnki type equality for work in systems that are monitored using non-projective unsharp measurements. The information acquired by the observer from the outcome $f$ of an energy measurement, and the subsequent conditioned normalized state $\hat ρ(t,f)$ evolved up to a final time $t$ are used to define work, as the difference between the final expectation value of the energy and the result $f$ of the measurement. The Jarzynski equality obtained depends on the coherences that the state develops during the process, the characteristics of the meter used to measure the energy, and the noise it induces into the system. We analyze those contributions in some detail to unveil their role. We show that in very particular cases, but not in general, the effect of such noise gives a factor multiplying the result that would be obtained if projective measurements were used instead of non-projective ones. The unsharp character of the measurements used to monitor the energy of the system, which defines the resolution of the meter, leads to different scenarios of interest. In particular, if the distance between neighboring elements in the energy spectrum is much larger than the resolution of the meter, then a similar result to the projective measurement case is obtained, up to a multiplicative factor that depends on the meter. A more subtle situation arises in the opposite case in which measurements may be non-informative, i.e. they may not contribute to update the information about the system. In this case, a correction to the relation obtained in the non-overlapping case appears. We analyze the conditions in which such a correction becomes negligible. We also study the coherences, in terms of the relative entropy of coherence developed by the evolved post-measurement state. We illustrate the results by analyzing a two-level system monitored by a simple meter.

Figures (3)

• Figure 1: A scheme of a possible distribution of $G\left(f|\mu_1(t_0)\right)$ and $G\left(f|\mu_2(t_0)\right)$, as functions of the outcome of the measurement $f$.
• Figure 2: The evolution of $\xi$vs$\beta\sigma$ in a measurement process with characteristic function (\ref{['heaviside']}), in a two level system described by the Hamiltonian (\ref{['HTL']}). The dots are the result of the numerical simulation. The blue dashed line corresponds to the theoretical expressions, (\ref{['jarzynskynonoverlap']}) and (\ref{['Jnonov']}), for the non-overlapping case, and the orange dashed line to the expressions, (\ref{['eq:ov2']}) and (\ref{['FT4']}), for the overlapping case. The system parameters, $\omega_q=2 \pi \times 6.541 \times 10^9 \text{Hz}$ and $\Omega_R= 2 \pi \times 10^6 \text{Hz}$, have been taken from naghiloo2020heat. The temperature is $T=0.14 \text{K}$, the final time $t_f=2 \pi/3 \omega_q$ and $\psi=\pi/4$.
• Figure 3: Numerical results for the mean relative entropy of coherence $\langle\Delta C\rangle=\langle C_{H(t)}(\hat{\rho}(t,f))-C_{H(t_0)}(\hat{\rho}(t_0))\rangle$ (blue line), the mean increment of Kullback-Leibler divergence $\langle\Delta D_{KL}\rangle=\langle D_{KL}(\hat{\rho}_D(t,f)||\hat{\rho}_{th}(t))-D_{KL}(\hat{\rho}_D(t_0)||\hat{\rho}_{th}(t_0))\rangle$ (orange line), and mean increment of the relative entropy $\langle\Delta S_R\rangle=\langle S(\hat{\rho}(t,f)||\hat{\rho}_{th}(t_0))-S(\hat{\rho}(t_0)||\hat{\rho}_{th}(t_0))\rangle$ (green line) vs. the frequency ratio $\Omega_R/\omega_q$. Notice how the increment of the relative entropy of coherence saturates at a value $\ln 2$ (see text). The resonant frequency has been set at $\omega_q=2 \pi \times 6.541 \times 10^9 \text{Hz}$. The temperature is $T=0.14 \text{K}$, the final time $t_f=2 \pi/3 \omega_q$, and $\psi=0$ and $\sigma=2 \hbar \omega_q$. The averages $\langle*\rangle$ are taken over $10^5$ realizations.