Single energy measurement Integral Fluctuation theorem and non-projective measurements

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 $\hat{\rho }(t,f)$ 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.

arXiv comment14 pages, 3 figures

References in corpus (30)

• A nonequilibrium equality for free energy differences
• Stochastic thermodynamics, fluctuation theorems, and molecular machines
• Quantifying Coherence
• The Entropy Production Fluctuation Theorem and the Nonequilibrium Work Relation for Free Energy Differences
• Quantum Coherence as a Resource
• Dynamical Ensembles in Nonequilibrium Statistical Mechanics
• Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems
• Equilibrium free energy differences from nonequilibrium measurements: a master equation approach
• Colloquium. Quantum Fluctuation Relations: Foundations and Applications
• Dynamical ensembles in stationary states
• Fluctuation theorems: Work is not an observable
• Continuous quantum measurement and Itô formalism
• No-go theorem for the characterisation of work fluctuations in coherent quantum systems
• Comparison of far-from-equilibrium work relations
• Work measurement as a generalized quantum measurement
• The role of quantum coherence in non-equilibrium entropy production
• Quantum work and the thermodynamic cost of quantum measurements
• Fluctuation theorems for quantum processes
• A Mini-Introduction To Information Theory
• Quantum work statistics of linear and nonlinear parametric oscillators
• Measuring work and heat in ultracold quantum gases
• Heat and work along individual trajectories of a quantum bit
• Quantum Jarzynski Equality in Open Quantum Systems from the One-Time Measurement Scheme
• Quantum fluctuation theorems and generalized measurements during the force protocol
• The role of quantum coherence in energy fluctuations
• Generalized energy measurements and modified transient quantum fluctuation theorems
• Quantum Fluctuation Relations for Ensembles of Wave Functions
• Jarzynski equality for conditional stochastic work
• A quantum fluctuation theorem for any Lindblad master equation
• Quantum thermodynamics under continuous monitoring: a general framework