Transient energy excitation in shortcuts to adiabaticity for the time-dependent harmonic oscillator

We study for the time-dependent harmonic oscillator the transient energy excitation in speed-up processes (“shortcuts to adiabaticity”) designed to reproduce the initial populations at some predetermined final frequency and time. We provide lower bounds and examples. Implications for the limits imposed to the process times and for the principle of unattainability of the absolute zero, in a single expansion or in quantum refrigerator cycles, are drawn.

Figure 1

Contour plot of the lower bound for time-averaged energy [in units of $E_0(0)=\hslash {\omega }_0/2$] as a function of $t_f$ and ${\omega }_f$. ${\omega }_0=2\pi \times 250$ Hz.

Figure 2

Dependence of the time-averaged energy for the ground state $\overline{E_0}/E_0(0)$ on the (a) short time $t_f$ (${\omega }_f=2\pi \times 0.25$ Hz) and (b) final frequency ${\omega }_f$ ($t_f=2$ ms). In both cases ${\omega }_0=2\pi \times 250$ Hz. Bound given by Eq. (14) (solid red line); asymptotic expression Eq. (15) (dot-dashed magenta line), and time-averaged energy for polynomial $b$ (dotted blue line). The $\delta$’s are the asymptotic exponents of $t_f$ and ${\omega }_f$, respectively, as they go to zero.

Figure 3

Example of hybrid $b$ combining the quasi-optimal trajectory in the central segment $[\tau ,1-\tau ]$ and “cap polynomials” with the right boundary conditions, where ${\omega }_0=2\pi \times 250$ Hz, ${\omega }_f=2\pi \times 0.25$ Hz, $t_f=2$ ms, and $\tau =0.4$ (solid red line); quasi-optimal square-root $b$ of Eq. (13) (dotted black line); and polynomial trajectory (blue dashed line).

Figure 4

Time-averaged energy versus $\tau$ for polynomial $b$ (solid blue line), bound in Eq. (14) (red dashed line), and hybrid $b$ (dot-dashed magenta line). The contribution from the “cap polynomials” (orange line with “$*$”) and the central segment (black line with “$•$”) are also depicted. Parameters are the same as in Fig. 3.

Figure 5

Dependence of the time-averaged uncertainty of energy $\overline{\Delta H_0}$ on the (a) short time $t_f$ and (b) final frequency ${\omega }_f$, where the parameters are the same as Fig. 2, polynomial trajectory (dotted blue line), AA relation (solid red line).