Fast forward to the classical adiabatic invariant

We show how the classical action, an adiabatic invariant, can be preserved under nonadiabatic conditions. Specifically, for a time-dependent Hamiltonian $H=p^2/2m+U(q,t)$ in one degree of freedom, and for an arbitrary choice of action $I_0$, we construct a so-called fast-forward potential energy function $V_{\mathrm{FF}}(q,t)$ that, when added to $H$, guides all trajectories with initial action $I_0$ to end with the same value of action. We use this result to construct a local dynamical invariant $J(q,p,t)$ whose value remains constant along these trajectories. We illustrate our results with numerical simulations. Finally, we sketch how our classical results may be used to design approximate quantum shortcuts to adiabaticity.

Figure 1

Illustration of the classical adiabatic invariant. Fifty trajectories evolving under a slowly varying Hamiltonian are shown at an initial time and a later time. The closed curves are instantaneous energy shells—level curves of $H$—with identical values of the action $I=\oint pdq$. Trajectories were generated using $H(q,p,t)$ given by Eq. (22), setting $\tau =10.0$ to achieve slow driving.

Figure 2

The region of phase space enclosed by the adiabatic energy shell $\mathcal{E}\left(t\right)$ is divided by line segments at $\left\{q_n\left(t\right)\right\}$ into vertical strips of equal phase space volume. The motion of these lines is described by velocity and acceleration fields $v(q,t)$ and $a(q,t)$.

Figure 3

Initial (a) and final [(b), (c)] conditions for trajectories launched from a single energy shell $\mathcal{E}\left(0\right)$. The trajectories in panel (b) evolved under $H(z,t)$ [Eq. (22)], while those in panel (c) evolved under $H_{\mathrm{FF}}=H+V_{\mathrm{FF}}$, with $\tau =1.0$. The solid black curves show the adiabatic energy shell $\mathcal{E}\left(t\right)$ at initial and final times.

Figure 4

A snapshot, at $t=\tau /2$, of 50 trajectories evolving under $H_{\mathrm{FF}}(z,t)$ using a rapid protocol, with $\tau =0.2$ (see text). The closed black loop is the adiabatic energy shell $\mathcal{E}\left(t\right)$, and the red loop above it is constructed by displacing each point on the lower loop by an amount $mv(q,t)$ along the $p$ axis. As predicted by Eq. (21), the trajectories coincide with the red loop.