Power-law behavior in the quantum-resonant evolution of the $\delta$-kicked accelerator

We consider the atom-optical $\delta$-kicked accelerator when the initial momentum distribution is symmetric. We demonstrate the existence of quantum-resonant dynamics, and derive analytic expressions for the system evolution. In particular, we consider the dynamical evolution of the momentum moments and find that all even-ordered momentum moments exhibit a power-law growth. In the ultracold (zero-temperature) limit the exponent is determined by the order of the moment, whereas for a broad, thermal initial momentum distribution the exponent is reduced by 1. To demonstrate the power-law behavior explicitly we consider the evolutions of the second- and fourth-order momentum moments, and cumulants, for an initially Gaussian momentum distribution corresponding to the Maxwell-Boltzmann distribution of an ideal gas at thermal equilibrium.

Figure 1

(Color online) Parameter $\eta$ as a function of $\Omega$ and $\beta$ for $\ell =2$ at (a) 15 kicks (b) 30 kicks, and (c) 60 kicks. An antiresonance of order $2N_s$, with $\Omega =r∕s$, will initially mimic exactly resonant growth for $N_{s}s$ kicks. The diminishing finite size of the resonances in (a)–(c) is due to nearby high-order antiresonances which become apparent as $n$ increases. Panel (d) shows the location of those resonances (see Eq. (16)) for $\Omega =1∕s$, where $s=1,2,\dots ,20$. The resonances become infinite in number as $\Omega \to 0$, but for $\Omega =0$ there are only three resonances. Units are dimensionless.

Figure 2

(Color online) The first seven even momentum moments of a $\delta$-kicked atomic cloud in the ultracold regime as given by Eq. (31) for integer $\Omega$, with the resonance condition (15) satisfied by choosing any appropriate value of $\ell$. We choose ${\varphi }_d=0.8\pi$ as corresponding to an experimentally typical driving strength [46]. The growth tends asymptotically to a power-law behavior with the power given by the order of the moment. Units are dimensionless.

Figure 3

(Color online) Second-order (a) and fourth-order (b) momentum moments yielded by the quantum-resonant evolution of an atomic cloud, as propagated by Eq. (7). We set $\Omega =0$ (i.e., no net acceleration), $T=T_T$ $(\ell =2)$, and choose ${\varphi }_d=0.8\pi$ [46]. The initial momentum distributions are Gaussian [see Eq. (56)] with standard deviations $w$ given by $(*)w=1∕8$, $\left(△\right)w=1∕32$, $(◻)w=1∕128$, and $(\circ )w=1∕512$. The solid lines correspond to analytic results given by Eqs. (57, 58), and the symbols correspond to data from a Monte Carlo simulation, the details of which are discussed elsewhere [57]. The dash-dotted lines indicate the ultracold limit [see Eq. (55) with $s=1$], and the dashed lines are a lower bound limit for a broad distribution [see Eq. (59) with $w=0$]. The vertical dotted lines in (a) indicate $n_G=1∕\ell \pi w$ for $w=1∕8,1∕32,1∕128$, i.e., the discrete time characterizing when growth in $⟨{\widehat{P}}^2⟩$ deviates from quadratic to linear. Units are dimensionless.

Figure 4

(Color online) Cube roots of the fourth-order momentum cumulants of a resonantly kicked atomic cloud evolved according to Eq. (7). Parameters are $\Omega =0$, $T=T_T$ $(\ell =2)$, and we choose ${\varphi }_d=0.8\pi$ [46]. We consider Gaussian initial momentum distributions with standard deviation $(*)w=1∕8$, $\left(△\right)w=1∕32$, $(◻)w=1∕128$, and $(\circ )w=1∕512$. After a certain number of kicks, the growth becomes linear, i.e., the fourth-order cumulant grows cubically. The inset shows the ideal (assuming perfect $\delta$ kicks) long-time behavior. Units are dimensionless.