Quantum Jump Approach for Work and Dissipation in a Two-Level System

Abstract

We apply the quantum jump approach to address the statistics of work in a driven two-level system coupled to a heat bath. We demonstrate how this question can be analyzed by counting photons absorbed and emitted by the environment in repeated experiments. We find that the common non-equilibrium fluctuation relations are satisfied identically. The usual fluctuation-dissipation theorem for linear response applies for weak dissipation and/or weak drive. We point out qualitative differences between the classical and quantum regimes.

Heat due to system-reservoir correlations in thermal equilibrium

“…with a linear rise instead of a quadratic one in (17) due to an unbounded energy spectrum. The factorizing assumption seems thus to be better justified for two-level systems than for systems with an infinite number of accessible states.…”

“…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.…”

Maxwell's demon based on a single qubit

“…The heat dissipated into the bath is given by (−)∆E when the system relaxes (gets excited) 22 , see also 23 . We analyze the MD cycle and errors quantitatively assuming that the measurement and the conditional π-pulse are fast enough (duration Γ −1 Σ ) so that no heat is exchanged in this time interval.…”