Optimizing Thermalization

We present a rigorous approach, based on the concept of continuous thermomajorization, to algorithmically characterize the full set of energy occupations of a quantum system accessible from a given initial state through weak interactions with a heat bath. The algorithm can be deployed to solve complex optimization problems in out-of-equilibrium setups and it returns explicit elementary control sequences realizing optimal transformations. We illustrate this by finding optimal protocols in the context of cooling, work extraction, and catalysis. The same tools also allow one to quantitatively assess the role played by memory effects in the performance of thermodynamic protocols. We obtained exhaustive solutions on a laptop machine for systems with dimension $d\le 7$, but with heuristic methods one could access much higher $d$.

Figure 1

Markovian thermalization in $d=3$. Simplex representing the state space of all three-dimensional probability distributions. The gray area denotes the set of achievable states from $\mathit{p}$ under arbitrary Markovian thermal processes when the thermal state is $\mathit{\gamma }=[1/2,1/3,1/6]$. The elementary thermalizations $T^{i,j}$, which fully thermalize the pair of levels $i$ and $j$ and whose action is indicated by red and blue arrows, play a central role in the sequence of required controls.

Figure 2

$\epsilon$-deterministic work extraction from a two-level system. Minimal error $\epsilon$ as a function of the work $W$ extracted from a two-level system with energy splitting $\mathrm{\Delta }$ prepared in a thermal state at temperature $1/{\beta }_S$ smaller than the environmental temperature $1/{\beta }_E$. System-environment interactions are modeled by TP or MTP. Parameters: ${\beta }_S\mathrm{\Delta }=2$ and ${\beta }_E\mathrm{\Delta }=1$.

Figure 3

Optimal cooling of a four-level system. Optimal change of the ground state population $\delta p_0^{\mathrm{opt}}$ for a four-level quantum system initially at equilibrium with inverse temperature $\beta$ in one round of algorithmic cooling consisting of a unitary inversion of the populations, followed by the optimal TP or MTP. The system has equidistant energy spectrum with smallest energy difference $\mathrm{\Delta }$.