Quantum Coherence and Ergotropy

Constraints on work extraction are fundamental to our operational understanding of the thermodynamics of both classical and quantum systems. In the quantum setting, finite-time control operations typically generate coherence in the instantaneous energy eigenbasis of the dynamical system. Thermodynamic cycles can, in principle, be designed to extract work from this nonequilibrium resource. Here, we isolate and study the quantum coherent component to the work yield in such protocols. Specifically, we identify a coherent contribution to the ergotropy (the maximum amount of unitarily extractable work via cyclical variation of Hamiltonian parameters). We show this by dividing the optimal transformation into an incoherent operation and a coherence extraction cycle. We obtain bounds for both the coherent and incoherent parts of the extractable work and discuss their saturation in specific settings. Our results are illustrated with several examples, including finite-dimensional systems and bosonic Gaussian states that describe recent experiments on quantum heat engines with a quantized load.

Figure 1

Position of the various states (see main text) in a coherence-versus-average-energy diagram. Gray dots represent quantum states—e.g., arising from the initial state $\widehat{\rho }$ after the transformations ${\widehat{E}}_{\rho }$, $\mathrm{\Delta }$, ${\widehat{V}}_{\tilde{\pi }}$ are performed. The arrows representing transformations are intended merely to point from the initial to the final state, without implying a precise path in the plane. For example, the transformation ${\widehat{V}}_{\tilde{\pi }}$ is represented by a horizontal line because it connects states with the same amount of coherence; however, coherence may change during the transformation. The horizontal distance $\mathrm{\Delta }{\mathcal{E}}_c$ between the thermal state ${\widehat{\rho }}_{{\beta }^{*}}$ and ${\widehat{P}}_{\rho }$ is the bound ergotropy (see Sec. 4). It may be zero, depending on the system at hand (i.e., iff the eigenvalues of $\rho$ and ${\rho }_{{\beta }^{*}}$ are related by a permutation).

Figure 2

In a three-level system, we identify the class of states that allow us to saturate the right inequality in Eq. (7) by looking at a pair of eigenvalues, $r_1$ and $r_2$, for which $\mathrm{\Delta }{\mathcal{E}}_c=0$. The various lines refer to the cases in which the ratio of the second and third energy eigenvalues is given by $R=0$, 0.1, 0.3, 0.5, 0.7, 1 (lighter dashed to darker solid lines).