Fast-forward scaling in a finite-dimensional Hilbert space

Time evolution of quantum systems is accelerated by the fast-forward scaling. We reformulate the method to study systems in a finite-dimensional Hilbert space. For several simple systems, we explicitly construct the acceleration potential. We also use our formulation to accelerate the adiabatic dynamics. Applying the method to the transitionless quantum driving, we find that the fast-forward potential can be understood as a counterdiabatic term.

Figure 1

Protocols (26) and (27) at $\overline{\alpha }=2.0$ and $t_0=10$. The dashed line represents the time before the scaling.

Figure 2

Acceleration potential $V(\sigma =\pm 1,t)$ in Eq. (25). $V(-1,t)$ at $t=10$ goes to $-\infty$ from the left and $\infty$ from the right.

Figure 3

Probability of the up-spin state ${\left|\langle \sigma =1|\psi \left(t\right)\rangle \right|}^2$. The solid line represents the result without the fast-forward scaling. The bold and dashed lines represent the numerical and analytical results with the scaling, respectively. $\overline{\alpha }=2.0$; $t_0=10$.

Figure 4

Overlap of the state with the initial state ${\left|\langle \psi \left(0\right)|\psi \left(t\right)\rangle \right|}^2$ (upper panel) and with the final state $|\langle \psi \left(t_{\mathrm{f}}\right)|\psi \left(t\right)\rangle {|}^2$ (lower). The solid line represents the result without the fast-forward scaling and the bold with the scaling. $\overline{\alpha }=2.0$; $t_0=10$.