Shortcut to Equilibration of an Open Quantum System

We present a procedure to accelerate the relaxation of an open quantum system towards its equilibrium state. The control protocol, termed the shortcut to equilibration, is obtained by reverse-engineering the nonadiabatic master equation. This is a nonunitary control task aimed at rapidly changing the entropy of the system. Such a protocol serves as a shortcut to an abrupt change in the Hamiltonian, i.e., a quench. As an example, we study the thermalization of a particle in a harmonic well. We observe that for short protocols the accuracy improves by 3 orders of magnitude.

Figure 1

Scheme of the shortcut to equilibration protocol (curved red line) and the quench protocol (blue step line), transforming an initial thermal state at temperature $T$ and frequency ${\omega }_i$ to a final thermal state with an equivalent temperature and frequency ${\omega }_f$.

Figure 2

Control protocols as a function of the scaled time $t/t_f$: (a),(b) the oscillator frequency $\omega$ and (c),(d) energy for the STE (red line), quench (dashed blue line), and adiabatic (dot-dashed green line) protocols. (a),(c) Expansion, and (b),(d) compression protocols. The dynamics of the STE and quenched systems are shown for $t_f=8\mathrm{a}.\mathrm{u}.$, and the adiabatic dynamics are obtained in the limit $t_f\to \infty$. (c),(d) Inset: details of the final approach to the target state. Model parameters (atomic units): $\omega (0)/{\omega }_f=5/10$ for the compression, and reverse for the expansion and bath temperature $T=2$.

Figure 3

The fidelity of the final state relative to the target thermal state for the shortcut to equilibration (red) and quench (blue) protocols. (a) Expansion protocol and (b) compression protocol. The error of the STE is dominated by the accuracy of the inertial solution (Supplemental Material VI [51]). The fidelity was therefore estimated from the deviation between the inertial solution and the exact free propagation. The inset shows the accuracy $\mathcal{A}=-{\log }_{10}(1-\mathcal{F})$, highlighting the three-digit accuracy of the STE protocol. Model parameters are the same as in Fig. 2.

Figure 4

Work required to perform the driving protocol as a function of the normalized time. Model parameters are identical to Fig. 3. Upper part, compression; lower part, expansion.