Optimal local work extraction from bipartite quantum systems in the presence of Hamiltonian couplings

We investigate the problem of finding the local analog of the ergotropy, which is the maximum work that can be extracted from a system if we can only apply local unitary transformation acting on a given subsystem. In particular, we provide a closed formula for the local ergotropy in the special case in which the local system has only two levels, and we give analytic lower bounds and semidefinite programming upper bounds for the general case. As nontrivial examples of application, we compute the local ergotropy for an atom in an electromagnetic cavity with Jaynes-Cummings coupling and the local ergotropy for a spin site in an XXZ Heisenberg chain, showing that the amount of work that can be extracted with a unitary operation on the coupled system can be greater than the work obtainable by quenching off the coupling with the environment before the unitary transformation.

Figure 1

Local ergotropy of the state $cos\alpha |n,+\rangle +sin\alpha |n,-\rangle e^{i\mathrm{\Omega }t}$ as a function of the time $t$, for the choice of parameters ${\omega }_S=1, {\omega }_E=1.2, \mathrm{\Omega }=0.1, n=10$, and $\alpha =0.4\pi$.