Controlling quantum chaos: Time-dependent kicked rotor

One major objective of controlling classical chaotic dynamical systems is exploiting the system's extreme sensitivity to initial conditions in order to arrive at a predetermined target state. In a recent Letter [Phys. Rev. Lett. 130, 020201 (2023)], a generalization of this targeting method to quantum systems was demonstrated using successive unitary transformations that counter the natural spreading of a quantum state. In this paper further details are given and an important quite general extension is established. In particular, an alternate approach to constructing the coherent control dynamics is given, which introduces a time-dependent, locally stable control Hamiltonian that continues to use the chaotic heteroclinic orbits previously introduced, but without the need of countering quantum state spreading. Implementing that extension for the quantum kicked rotor generates a much simpler approximate control technique than discussed in the Letter, which is a little less accurate, but far more easily realizable in experiments. The simpler method's error can still be made to vanish as $\hslash \to 0$.

Figure 1

Schematic of optimal coherent chaotic quantum targeting. The circular zone represents the Wigner transform density of a minimal uncertainty state centered at $(q_{\alpha },p_{\alpha })$, which can be slightly shifted (via a unitary operator, ${\widehat{U}}_s$) to be centered on an optimal heteroclinic trajectory starting at $[{\mathbf{p}}_{\gamma }\left(0\right),{\mathbf{q}}_{\gamma }\left(0\right)]$ [designated $(q_0,p_0)]$. As it propagates the density follows this trajectory, but is locally spreading, which must be counteracted by contractions, ${\widehat{U}}_{M^{-1}}$; see Eq. (1). At the end, it can be shifted from $[{\mathbf{p}}_{\gamma }\left(\tau \right),{\mathbf{q}}_{\gamma }\left(\tau \right)]$ [designated $(q_{\tau },p_{\tau })]$ to the centroid of the target state $(q_{\beta },p_{\beta })$ (reprinted with permission from Ref. [10]).

Figure 2

Illustration of initial and target Gaussian wave packet states. The dimensionality of the space is given by $N=200$ and thus $\hslash =1/200\pi$. The width is chosen so that the constant density contours of the Wigner transforms appear circular in Fig. 3 ahead. The initial state has momentum and position centroids of $(p_{\alpha },q_{\alpha })=(0.0,0.5)$ and the final state $(p_{\beta },q_{\beta })=(0.0,0.0)$.

Figure 3

Evolution of the Wigner transform densities of the propagated initial state. The numbers label the time for each propagated density. The outer circle is the $2\sigma$ contour, which encloses an area $h$, i.e., that area occupied by a single quantum state. The $\sigma /2$ contour is also drawn (inner dashed circles).

Figure 4

Quantum propagation along the chosen heteroclinic trajectory of Table 1. The six panels from upper left to lower right mark the evolution at the trajectory's six full time steps. The solid lines are the absolute value, the dashed (green) lines are the real part, and the dotted (blue) lines are the imaginary part of the controlled propagated wave function, $\langle q\left|\alpha \right(t)\rangle$. As time increases, deviation from the ideal Gaussian wave packet becomes more visible. Nevertheless, the state follows the heteroclinic trajectory in Table 1 quite closely.

Figure 5

Accuracy of quantum targeting for the kicked rotor. The deviation of the squared overlap of the target state with the controlled propagated state from unity is plotted versus Planck's constant $h$. As $\hslash \to 0$, the error tends to vanish exponentially. Dimensionalities ranging from $N=50$ to 1400 are included. The asterisks are derived from the dynamics of the control Hamiltonian and various details of Sec. 4. The line is drawn using the results of [10] and the ${\widehat{U}}_{M^{-1}}$ operators.

Figure 6

Accuracy of the quantum targeting propagation for the kicked rotor for $h=0.01$ ($N=200$). The deviation from unity of the squared overlap of a wave packet centered at $(q_t,p_t)$ ($t=n$, see Table 1) with the controlled propagating state is plotted versus number of iterations. Nearly all of the inaccuracy arises from the third and fourth iterations. The final value at $t=6$ matches the point (asterisk) at $h=0.01$ in Fig. 5, as it must.

Figure 7

Evolution of the Wigner transform densities of the propagated initial state for the Hamiltonian of Eq. (B1) using the modification of Eq. (B8) in Eq. (B7). The numbers label the time for each propagated density. The outer circle is the $2\sigma$ contour, which encloses an area $h$, i.e., that area occupied by a single quantum state. The $\sigma /2$ contour is also drawn.

Figure 8

Comparison of the final propagated Wigner transform densities for the Hamiltonians of Eqs. (13) and (B1) using the modification of Eq. (B8) in Eq. (B7). The outer circles are the $2\sigma$ contours, which show a much greater improvement than the $\sigma /2$ contours.

Figure 9

Evolution of the Wigner transform densities of the identical propagated initial state, Fig. 2, under the dynamics of Eq. (B1) with the potentials in Eqs. (B9) and (B10). The outer circle is the $2\sigma$ contour, which encloses an area $h$, i.e., that area occupied by a single quantum state. The $\sigma /2$ contour is also drawn. As previously, $K=8$, and $N=200$.