Variational approach to the optimal control of coherently driven, open quantum system dynamics

Quantum coherence inherently affects the dynamics and the performances of a quantum machine. Coherent control can, at least in principle, enhance the work extraction and boost the velocity of evolution in an open quantum system. Using advanced tools from the calculus of variations and reformulating the control problem in the instantaneous Hamiltonian eigenframe, we develop a general technique for minimizing a wide class of cost functionals when the external control has access to full rotations of the system Hamiltonian. The method is then applied both to time and heat loss minimization problems and explicitly solved in the case of a two-level system in contact with either bosonic or fermionic thermal environments.

Figure 1

Pictorial representation of two possible strategies to control a quantum system in the time interval $[0,\tau ]$: on the left the eigenvectors of the Hamiltonian are fixed, i.e., $[H\left(t_1\right),H\left(t_2\right)]=0$; on the right the Hamiltonian can rotate and $[H\left(t_1\right),H\left(t_2\right)]\ne 0$. In the following we will provide a set of necessary conditions for an optimal control specifically in this last case.

Figure 2

Representation in the Bloch sphere of the minimum time trajectories for the Gibbs mixing channel (22) and the bosonic master equation (29). We suppose that the initial Bloch vectors are in the $x\text{−}z$ plane for ease of representation. A possible optimal trajectory is always composed by two unitary quenches [blue dashed arrows in (a) and (b)] separated by a semiclassical open evolution. The latter depends on the modulus of the initial and final Bloch vectors, as explained in the main text. (a) The final state (orange star) has $|{\tilde{a}}_z\left(\tau \right)|$ smaller than $|{\tilde{a}}_z\left(0\right)|$ of the initial state (green pentagon), so the open evolution (black dashed arrow) occurs, respectively, with $\epsilon \to -\infty$ when Eq. (22) holds and with $\epsilon =0$ when Eq. (29) holds. (b) The opposite case, in which $|{\tilde{a}}_z\left(\tau \right)|>|{\tilde{a}}_z\left(0\right)|$, the open evolution is such that $\epsilon \to \infty$ (red dashed arrow).