Classical and Quantum Shortcuts to Adiabaticity for Scale-Invariant Driving

A shortcut to adiabaticity is a driving protocol that reproduces in a short time the same final state that would result from an adiabatic, infinitely slow process. A powerful technique to engineer such shortcuts relies on the use of auxiliary counterdiabatic fields. Determining the explicit form of the required fields has generally proven to be complicated. We present explicit counterdiabatic driving protocols for scale-invariant dynamical processes, which describe, for instance, expansion and transport. To this end, we use the formalism of generating functions and unify previous approaches independently developed in classical and quantum studies. The resulting framework is applied to the design of shortcuts to adiabaticity for a large class of classical and quantum, single-particle, nonlinear, and many-body systems.

Popular Summary

The rapid compression of a gas inside a sealed container is a turbulent and dissipative event, as energy is lost irreversibly to the thermal motions of the gas molecules. When the same gas is compressed slowly, it evolves in a far more controlled manner, with negligible dissipation. This distinction is familiar to any student of thermodynamics, but it applies similarly to microscopic systems obeying the laws of quantum mechanics. When the process is slow, or adiabatic, the quantum system evolves predictably, with little or no dissipation. In recent years, there has been much interest—and progress—in developing methods for achieving controlled, dissipationless evolution of quantum systems in processes that occur rapidly rather than slowly. These methods are known collectively as “shortcuts to adiabaticity.”

This question is, in fact, of practical importance in many quantum technologies that depend on accurate and fast quantum-state control. In meeting this need, a set of mathematical tools has been developed to design such shortcuts for driven quantum systems. A particularly powerful technique among them is called “transitionless quantum driving.” The essence of this technique is to apply an additional field that effectively suppresses all excitations due to the driving. However, the mathematical complexity of computing this “counterdiabatic field” has so far severely restricted the practical applicability of the transitionless quantum driving. In this theoretical paper, we greatly broaden the scope and applicability of this technique.

Our work focuses on “scale-invariant” quantum-control processes, in which the scale of the potential energy function varies with time while its overall shape remains fixed. In particular, these processes encompass expansion, compression, and transport of matter waves. Moreover, scale invariance is particularly interesting, as it has a close analogy to classical processes. Taking advantage of this analogy and the dynamical symmetry associated with scale invariance, we have derived the general form of the counterdiabatic field required to mimic adiabatic dynamics in an arbitrary scale-invariant process and illustrated its applicability to a wide range of experimentally relevant systems.

Figure 1

Shortcut to adiabaticity based on dissipationless classical driving. Energy shells for a particle in a one-dimensional box [Eq. (42)], in a time-dependent piston of width $L(t)$ that changes at a constant rate $u$ for $t_0\le t\le t_1$ (inset). Energy shells corresponding to $H_0(\overline{z};L)$ are shown as a pair of parallel, dotted line segments of length $L$, at momenta $\pm \overline{p}$. The solid lines represent a level surface of the adiabatic invariant $I$ corresponding to the full, counteradiabatic Hamiltonian $H(\overline{z};L)$ (43). At $t=t_0$, the force $f(q,t)$ [Eq. (44)] induces a “jump” of trajectories $\overline{z}$ from dashed to solid lines—for $t_0\le t\le t_1$, the adiabatic energy shell is deformed invariantly—and finally, at $t=t_1$, force $f(q,t)$ [Eq. (44)] induces jumps back to $\overline{p}$.

Figure 2

Shortcut to adiabatic transport by local counterdiabatic driving. Whenever the trapping potential $U(\mathbf{q},t)=U_0[\mathbf{q}-f(t)\widehat{\mathbf{n}}]$ [with $\gamma (t)=1$], the auxiliary counterdiabatic term with $\mathbf{F}=-{\ddot{\mathbf{f}}}_F≕{\mathbf{f}}_F\chi (t)$ induces the self-similar evolution $\mathrm{\Phi }(t)=\mathcal{U}\mathrm{\Psi }(t)$, which at $t=\{0,{\tau }_F\}$ reduces to $\mathrm{\Psi }(t)=\mathrm{\Phi }(t)$, and the auxiliary term vanishes. CD guarantees that the density profile of the system is centered at all times at $\mathbf{q}=f(t)\widehat{\mathbf{n}}$. This protocol holds exactly for an arbitrary trapping potential $U_0(\mathbf{q})$ and is valid for an arbitrary single-particle, nonlinear, or many-body system, requiring no modulation of the coupling constant $\epsilon (t)$ in the case of interacting systems, i.e., $\epsilon (t)=1$.

Figure 3

Shortcut to an adiabatic expansion by local counterdiabatic driving. Whenever the trapping potential $U(\mathbf{q},t)=U_0[\mathbf{q}/\gamma (t)]/\gamma (t{)}^2$ [with $\mathbf{f}(t)=0$], the auxiliary counterdiabatic term modulated by $-m\ddot{\gamma }/\gamma (t)$ induces a nonadiabatic self-similar dynamics that reduces to the target state $\mathrm{\Psi }(t)$ at $t=\{0,{\tau }_F\}$, when the auxiliary term vanishes. This protocol holds exactly for an arbitrary trapping potential $U_0(\mathbf{q})$ and can be applied to an arbitrary single-particle, nonlinear, or many-body system. In the case of interacting systems, it requires a modulation of the coupling constant $\epsilon (t)$, as in Eq. (50).

Figure 4

Scale-invariant counterdiabatic driving of the Morse oscillator. Morse potential $U(q,\beta )$ [Eq. (b7)] for $\gamma =1$ (dashed blue line) and $\gamma =1/2$ (solid red line).

Figure 5

Phase-space volume $\mathrm{\Omega }(E,\gamma )$ corresponding to $U(q,\gamma )$ [Eq. (b7)] for $E=-1/2$ and $\gamma =1$ (dashed blue line) and $\gamma =1/2$ (solid red line).