Superadiabatic thermalization of a quantum oscillator by engineered dephasing

Fast nonadiabatic control protocols known as shortcuts to adiabaticity have found a plethora of applications, but their use has been severely limited to speeding up the dynamics of isolated quantum systems. We introduce shortcuts for open quantum processes that make possible the fast control of Gaussian states in nonunitary processes. Specifically, we provide the time modulation of the trap frequency and dephasing strength that allow preparing an arbitrary thermal state in a finite time. Experimental implementation can be done via stochastic parametric driving or continuous measurements, readily accessible in a variety of platforms.

Figure 1

(a) Control frequency ${\omega }_t^2/{\omega }_0^2$ for protocols with duration $t_f=2$ (blue) and 6 (red), and reference frequency ${\tilde{\omega }}_t^2/{\omega }_0^2$ (dashed black). Fast protocols can require an inversion of the trap. (b) Dissipation rate ${\gamma }_t/{\gamma }_{\max }$ (solid lines) used to control the dynamics and relative entropy $S({\rho }_t\parallel {\sigma }_t)/S_{\max }$ [Eq. (20), dotted lines], showing the distance of the state to a reference thermal state. The shape of the dephasing strength and the relative entropy are independent of the duration of the protocol, which only influences their maxima. Protocols correspond to cooling, ${\beta }_f/{\beta }_0=2$, with compression (top, ${\omega }_f/{\omega }_0=3$) or expansion (bottom, ${\omega }_f/{\omega }_0=1/4$), with ${\omega }_0={\beta }_0=1$.

Figure 2

(a) Schematic representation of the phase-space map of final thermal states (${\beta }_f,{\omega }_f$) reachable from an initial thermal state (${\beta }_0$, ${\omega }_0$), highlighted with the black dot, through STA control processes. States for which ${\beta }_f{\omega }_f={\beta }_0{\omega }_0$, chosen along the gray line, can be obtained from unitary, phase-space-preserving processes, with no entropy change. STA in open processes allow preparing arbitrary thermal states, while stochastic driving with ${\gamma }_t>0$ gives access to states for which ${\beta }_f{\omega }_f<{\beta }_0{\omega }_0$ (region highlighted in orange). (b) Associated change of the von Neumann entropy for ${\omega }_0={\beta }_0=1$.

Figure 3

Maximum dephasing strength ${\gamma }_{\max }$: (a) Contour plot for an initial state ${\beta }_0=1={\omega }_0$. We verify that ${\gamma }_{\max }$ is negative for ${\beta }_f{\omega }_f>{\beta }_0{\omega }_0$, and nonnegative otherwise; (b) as function of ${\omega }_f$ for fixed ${\beta }_f=1$; (c) as function of ${\beta }_f$, keeping the trap frequency unchanged (${\omega }_f=1$), showing a plateau for large values of ${\beta }_f$; (d) on a line of constant ${\beta }_f{\omega }_f=0.5$ showing that ${\gamma }_{\max }$ depends on the specific choice of ${\beta }_f$ and ${\omega }_f$.