Semiclassical catastrophes and cumulative angular squeezing of a kicked quantum rotor

We present a detailed theory of spectacular semiclassical catastrophes happening during the time evolution of a kicked quantum rotor [Phys. Rev. Lett. 87, 163601 (2001)]. Both two- and three-dimensional rotational systems are analyzed. It is shown that the wave function of the rotor develops a cusp at certain delay after a kick, which results in a sharply focused rotational wave packet. The cusp is followed by a fold-type catastrophe manifested in the rainbow-type moving singularities. In the three-dimensional case, the rainbows are accompanied by additional singular features similar to the glory structures known in the wave optics. These catastrophes in the time-dependent angular wave function are well described by the appropriate tools of the quasiclassical wave mechanics, i.e., by Airy and Bessel approximations and Pearcey’s functions. A scenario of “cumulative squeezing” is also presented in which a specially designed train of short kicks produces an unlimited narrowing of the rotor angular distribution. This scenario is relevant to the molecular alignment by short laser pulses, and also to the atom lithography schemes in which cold atoms are focused by an optical standing wave.

Figure 1

2D kicked rotor. Classical map representing the final angle $\theta$ as a function of the initial angle ${\theta }_0$ for (a) $\tau =0.5{\tau }_f,$ (b) $\tau ={\tau }_f,$ (c) $\tau =3{\tau }_f,$ and (d) $\tau =10{\tau }_f$.

Figure 2

Angular distribution of the initially uniform classical ensemble of 2D rotors after a single $\delta$ kick. (a) shows focusing at $\tau ={\tau }_f$ and (b) displays rainbows at $\tau =1.84{\tau }_f$.

Figure 3

Classical maps and critical angles for (a) 2D and (b) 3D rotors. In the 2D case, the final angle is plotted modulo $2\pi$ in order to fold it into the interval $[0,2\pi ]$. In (a) $P\tau =3$, and in (b) $P\tau =1.4$.

Figure 4

Angular distribution of a linear molecule (rigid rotor) with a permanent dipole moment. In (a), (b), and (c) the probability distribution is shown for $P\tau =1$, $P\tau =2$, and $P\tau =4$, respectively. In the top row, the probability of finding a classical rotor inside the solid angle element $\sin \theta d\theta d\phi$ is plotted on a sphere. The middle row shows the classical distribution function as a function of $\theta$, and the bottom row displays the corresponding quantum-mechanical probability distribution for a kick strength of $P=75$.

Figure 5

Angular distribution of a linear molecule (rigid rotor) interacting with the external field due to the induced polarization. In (a), (b), and (c) the probability distribution is shown for $P\tau =0.5$, $P\tau =2$, and $P\tau =2.4$, respectively. In the top row, the probability of finding a classical rotor inside the solid angle element $\sin \theta d\theta d\phi$ is plotted on a sphere. The middle row shows the classical distribution function as a function of $\theta$, and the bottom row displays the corresponding quantum-mechanical probability distribution for a kick strength of $P=75$.

Figure 6

Angular distribution of a 2D quantum rotor excited by a strong $\delta$ kick $(P=85)$. The graphs correspond to (a) $\tau =0.5{\tau }_f,$ (b) $\tau ={\tau }_f,$ (c) $\tau =2{\tau }_f$, (d) $\tau ={\tau }_f+T_{rev}∕2,$ (e) $\tau ={\tau }_f+T_{rev}∕3,$ (f) $\tau ={\tau }_f+T_{rev}∕4,$ (g) $\tau =1.8{\tau }_f+T_{rev}∕2,$ (h) $\tau =1.8{\tau }_f+T_{rev}∕3,$ and (i) $\tau =1.8{\tau }_f+T_{rev}∕4,$ respectively.

Figure 7

Probability distribution in $\theta$ for the 3D kicked rotor (solid line) and probability distribution for the planar model (dashed line). The figure shows the probability distribution for $\tau =1∕P$ (a), $\tau =1.2∕P$ (b), $\tau =2∕P$ (c), and $\tau =4∕P$ (d). The kick strength is $P=50$.

Figure 8

(a) shows the probability distribution (2D case) at focal time $\tau =1∕P$. In (b), (c), and (d), $\tau =1.1∕P$, $\tau =1.2∕P$, and $\tau =1.4∕P$, respectively. The kick strength is $P=50$. The solid line represents the exact quantum-mechanical probability distribution, while the dashed line describes the probability distribution in Pearcey approximation.

Figure 9

Glory and oscillations close to the focusing in the kicked 3D rotor. (a) shows the probability distribution at focal time $\tau =1∕P$. In (b), (c), and (d), $\tau =1.1∕P$, $\tau =1.2∕P$, and $\tau =1.4∕P$, respectively. The kick strength is $P=50$. The solid line represents the exact quantum-mechanical probability distribution, while the dashed line shows the probability distribution obtained in the approximation given by Eq. (39) based on Pearcey function.

Figure 10

Airy approximation for 2D WP evolution for times (a) $\tau P$=4, (b) 4.7, and (c) 6, $P=75$. The solid line represents exact quantum result; the dashed one shows the Airy approximation. The arrow points to the rainbow angle ${\theta }_r$.

Figure 11

Uniform Airy approximation in the 3D case [Eq. (47)] is compared to the quantum result for $P\tau$=4 and $P$=75. The arrow points to the rainbow angle. (a) shows the probability density as a function of $\theta$, and (b) and (c) show the real and the imaginary part of the wave function.

Figure 12

Absolute square of the wave function in the uniform Bessel approximation (solid line) compared to the exact wave function (dashed line) for $P\tau =4$ and $P=75$ (3D case).

Figure 13

Angular distribution of an ensemble of classical rotors (in 3D) with the finite initial temperature for $P^{\prime }{t}^{\prime }=1$ (a), $P^{\prime }{t}^{\prime }=3$ (b), and $P^{\prime }{t}^{\prime }=4.5$ (c). The solid line corresponds to the kick strength $P^{\prime }=10$, the dashed line to $P^{\prime }=5$, and the dotted line to $P^{\prime }=1$.

Figure 14

Cumulative squeezing of a classical ensemble of 3D rotors. (a) shows the minimal orientation factor and (b) shows the minimal alignment factor as a function of the number of kicks. The dots (.) correspond to an ensemble of rotors with zero initial temperature $(P^{\prime }=\infty )$. The crosses $\left(\times \right)$ and pluses $(+)$ correspond to $P^{\prime }=5$ and $P^{\prime }=1$, respectively. The orientation factor and alignment factor were calculated with the help of Monte Carlo simulation using 10 000 particles.