Moments of work in the two-point measurement protocol for a driven open quantum system

Abstract

We study the distribution of work induced by the two-point measurement protocol for a driven open quantum system. We first derive a general form for the generating function of work for the total system, bearing in mind that the Hamiltonian does not necessarily commute with its time derivative. Using this result, we then study the first few moments of work by using the master equation of the reduced system, invoking approximations similar to the ones made in the microscopic derivation of the reduced density matrix. Our results show that already in the third moment of work, correction terms appear that involve commutators between the Hamiltonian and its time derivative. To demonstrate the importance of these terms, we consider a sinusoidally, weakly driven and weakly coupled open two-level quantum system, and indeed find that already in the third moment of work, the correction terms are significant. We also compare our results to those obtained with the quantum jump method and find a good agreement.

Heat due to system-reservoir correlations in thermal equilibrium

“…When a weak external driving is exerted to the system, according to the first law of thermodynamics, part of its energy is deposited into the system (internal energy) and part of it is transferred to the bath in the form of heat. Weak coupling approaches such as master equations obtain work and heat based on separable thermal equilibria [17][18][19]. For example, for a monochromatic pulse with frequency and amplitude λ 0 in resonance with a two-level system, they are applicable as long as work and heat are on the order of (λ 0 / ) 2 1 which basically matches the impact of system-bath correlations.…”

“…We derive general expressions and discuss specific results for systems of possible experimental interest [3][4][5]16,20], namely, two-level systems and harmonic modes. It turns out that even in the weak coupling regime, this heat flow is substantial at low temperatures and may become comparable to typical predictions for the work based on conventional weak coupling approaches [3,[17][18][19]. It further depends sensitively on non-Markovian features of the reservoir such that the commonly made simplification of a strictly Ohmic environment [8] is always unphysical.…”

“…In many theoretical studies the simplified situation of (at least initially) factorizing thermal equilibria is taken for granted [15,16]. This way, predictions for work, work distributions, and heat due to the presence of weak classical driving sources have been obtained based on conventional master equations or related approaches [3,[17][18][19]. However, in actual measurements, specifically in solid state structures, quantum correlations between system and reservoir may be of relevance not only far from but also close to and in thermal equilibrium.…”

Comparison between quantum jumps and master equation in the presence of a finite environment

“…For direct master equation calculations, the moments of work can be alternatively calculated with the power operator approach, which can be derived by starting from the two measurement protocol of an isolated system whose evolution is unitary during the protocol [24]. With the power operator approach, the first two work moments for the total isolated system are given by [23,24] …”

“…We also show that both methods produce the same work moments when the work is defined using the two-measurement protocol (TMP) [37][38][39] without the interaction energy. For master equation calculations, we additionally show that the power operator definition of work [23,24] does not produce exactly the same work moments due to a different order of approximations. Finally, we numerically study a qubit coupled to an calorimeter consisting of ten two-level systems with an energy gap equivalent to that of the qubit.…”