Quantum thermodynamics with local control

We investigate the limitations that emerge in thermodynamic tasks as a result of having local control only over the components of a thermal machine. These limitations are particularly relevant for devices composed of interacting many-body systems. Specifically, we study protocols of work extraction that employ a many-body system as a working medium whose evolution can be driven by tuning the on-site Hamiltonian terms. This provides a restricted set of thermodynamic operations, giving rise to alternative bounds for the performance of engines. Our findings show that those limitations in control render it, in general, impossible to reach Carnot efficiency; in its extreme ramification it can even forbid to reach a finite efficiency or finite work per particle. We focus on the one-dimensional Ising model in the thermodynamic limit as a case study. We show that in the limit of strong interactions the ferromagnetic case becomes useless for work extraction, while the antiferromagnetic case improves its performance with the strength of the couplings, reaching Carnot in the limit of arbitrary strong interactions. Our results provide a promising connection between the study of quantum control and thermodynamics and introduce a more realistic set of physical operations well suited to capture current experimental scenarios.

Figure 1

(a) A Carnot cycle in an entropy-temperature diagram. When the working-medium is put in contact with the heat bath, it is already at the same temperature as the bath. Hence, no heat flows from the heat bath until one starts the isothermal expansion. (b) If the bath's temperature cannot be reached by an adiabatic expansion and compression, then there is an unavoidable dissipation (red arrows) reducing the efficiency.

Figure 2

Efficiency at maximum work density (black) for the Ising model as a function of $J$ in the thermodynamic limit for the parameters ${\beta }_h=0.5,{\beta }_c=1$. The gray dashed line shows a protocol that is independent of $J$. Carnot efficiency is 0.5.

Figure 3

(a) Carnot-like protocol; (b) protocol that is not Carnot-like. The figure shows that a non-Carnot-like protocol can always be understood as a concatenation of small Carnot-like protocols connected by isothermals at the cold bath.

Figure 4

Efficiency at maximum work for a system of six spins as a function of $J$ and different imprecisions on the external fields $\epsilon =0.05,0.1,0.25,0.5$ (black to light gray). For larger system sizes, the local minimum on the ferromagnetic side $J>0$ moves to larger values of $J$ and the value at $J=\infty$ decreases exponentially with the system size for any fixed precision.

Figure 5

The optimal magnetic field as a function of $J$ for inverse temperatures $\beta =1$ (blue), $\beta =2$ (orange), and $\beta =3$ (green). It is clearly visible that at the critical point $1/\left(2\beta \right)$ the function is not analytic, similarly to a second order phase transition.