Optimal thermodynamic control in open quantum systems

We apply advanced methods of control theory to open quantum systems and we determine finite-time processes which are optimal with respect to thermodynamic performances. General properties and necessary conditions characterizing optimal drivings are derived, obtaining bang-bang-type solutions corresponding to control strategies switching between adiabatic and isothermal transformations. A direct application of these results is the maximization of the work produced by a generic quantum heat engine, where we show that the maximum power is directly linked to a particular conserved quantity naturally emerging from the control problem. Finally we apply our general approach to the specific case of a two-level system, which can be put in contact with two different baths at fixed temperatures, identifying the processes that minimize heat dissipation. Moreover, we explicitly solve the optimization problem for a cyclic two-level heat engine driven beyond the linear-response regime, determining the corresponding optimal cycle, the maximum power, and the efficiency at maximum power.

Figure 1

Pictorial representation of the two-bath model: a system $S$ whose Hamiltonian ${\widehat{H}}_{\mathbf{u}\left(t\right)}$ is driven via a collection of external control fields $\mathbf{u}\left(t\right)$, evolves in time while being coupled with a cold bath of inverse temperature ${\beta }_c$ and with a hot bath of inverse temperature ${\beta }_h$ through the couplings parameters ${\gamma }_c\left(t\right)$ and ${\gamma }_h\left(t\right)$ which can also be externally controlled.

Figure 2

(a) Form of the cold (blue) isotherm when ${\beta }_h=0.3{\beta }_c$ and $\mathcal{K}=-0.05\mathrm{\Gamma }/{\beta }_c$. (b) Form of the hot (red) isotherm for the same set of parameters. The area below the function $p\left({\beta }_{c}u\right)$ is equal to the heat released [(a) light blue] or the heat absorbed [(b) orange] for the cold and hot isotherm respectively, measured in units of ${\beta }_c^{-1}$. The arrows show the direction of the dynamics.

Figure 3

(a) Profile of the zero contour of $f(p,\mathcal{K})$ when ${\beta }_h=0.3{\beta }_c$ determining the condition for adiabatic jumps. For ease of representation we plot the contour of $f$ as a function of $-\mathcal{K}$ and we omit the dependence of $p_{ad,1}\left(\mathcal{K}\right)$ and $p_{ad,2}\left(\mathcal{K}\right)$ by $\mathcal{K}$. (b) General form of the optimal trajectories for ${\beta }_h=0.3{\beta }_c$ and $\mathcal{K}=-0.05\mathrm{\Gamma }/{\beta }_c$. Curved blue and red lines (respectively, passing through E,A,B and C,D,F) are cold and hot isotherms. The two horizontal lines correspond to the values $p_{ad,1}$ and $p_{ad,2}$ when the system undergoes intermediate adiabatic jumps (represented with horizontal black arrows). The area enclosed by the cycle is colored in orange and is equal to the heat absorbed during the cycle.

Figure 4

(a) Dimensionless function $g$ proportional to the minimum heat dissipation rate according to ${\mathcal{K}}^{*}=-(\mathrm{\Gamma }/{\beta }_c)g({\beta }_h/{\beta }_c)$. For cyclic processes, $g({\beta }_h/{\beta }_c)$ corresponds to the maximum power (in units of $\mathrm{\Gamma }/{\beta }_c$) achievable by the two-level system heat engine. (b) Efficiency at maximum power ${\eta }^{*}$ (dark blue) derived in Eq. (24) compared with the Carnot efficiency ${\eta }_C$ (gray dashed line) and the Curzon-Ahlborn efficiency ${\eta }_{CA}$ (light green).

Figure 5

(a) Optimal excitation probability $p^{*}=p_{ad,1}\left({\mathcal{K}}^{*}\right)=p_{ad,2}\left({\mathcal{K}}^{*}\right)$ as a function of the temperature ratio. (b) The two extrema of the quenches characterizing the maximum power heat engine $u_c^{*}$ (dark blue) and $u_h^{*}$ (light red), measured in units of ${\beta }_c^{-1}$ as a function of ${\beta }_h/{\beta }_c$. The case of ${\beta }_h/{\beta }_c=0.3$ is singled out. (c) The optimal driving protocol for the parameter $u^{*}\left(t\right)$, measured in units of ${\beta }_c^{-1}$. The driving is composed by a succession of infinitesimal Otto cycles, where the extrema $u_c^{*}$ and $u_h^{*}$ can be found directly from the left panel of the figure. The value $\mathrm{\Delta }\tau \ll \tau$ is an arbitrarily small unit of time expressing the duration of each degenerate cycle.