Fractional resonances in the atom-optical $\delta$-kicked accelerator

We consider resonant dynamics in a dilute atomic gas falling under gravity through a periodically pulsed standing-wave laser field. Our numerical calculations are based on a Monte Carlo method for an incoherent mixture of noninteracting plane waves, and we show that quantum resonances are highly sensitive to the relative acceleration between the atomic gas and the pulsed optical standing wave. For particular values of the atomic acceleration, we observe fractional resonances. We investigate the effect of the initial atomic momentum width on the fractional resonances and quantify the sensitivity of fractional resonances to thermal effects.

Figure 1

(Color online) (Left-hand panel) Momentum distributions with a resolution of $\hslash K$ and (right-hand panel) momentum moments of order $q=2$ (○) and $q=4$ $(\times )$ for a $\delta$-kicked accelerator. The initial condition is a zero-momentum eigenstate and parameters are $\mathcal{N}=1$, $T=T_T$ $\left(\ell =2\right)$, ${\varphi }_d=0.8\pi$, (a) and (b) $\Omega =0$, (c) and (d) $\Omega =1/8$, (e) and (f) $\Omega =1/4$, (g) and (h) $\Omega =1/2$, (i) and (j) $\Omega =1$, and (k) and (l) $\Omega =\left(1+\sqrt{5}\right)/2$. In the right-hand panel the solid and dashed lines correspond to Eqs. (31, 32), respectively. The vertical lines in (d), (f), and (h), indicate where $n$ is an integer multiple of $s$ (as taken from $\Omega =1/s$).

Figure 2

(Color online) ${⟨{\widehat{p}}^2⟩}_n^{1/2}$ in units of $\hslash K$ for a $\delta$-kicked accelerator initially in a zero-momentum eigenstate. Parameters are $\mathcal{N}=1$, ${\varphi }_d=0.8\pi$, (a) $T=T_T/2$ $\left(\ell =1\right)$, and (b) $T=T_T$ $\left(\ell =2\right)$. The markers highlight particular features for $T=T_T$ $\left(\ell =2\right)$, which are shown in more detail in (c) $\Omega =0$, (d) $\Omega =1/8$, (e) $\Omega =1/4$, (f) $\Omega =1/2$, (g) $\Omega =1$, and (h) $\Omega =\left(1+\sqrt{5}\right)/2$.

Figure 3

(Color online) (Left-hand panel) Momentum distributions with a resolution of $\hslash K$ and (right-hand panel) momentum moments of order $q=2$ (○) and $q=4$ $(\times )$, for a $\delta$-kicked accelerator. The initial atomic momenta are distributed according to Eq. (28) and parameters are $\mathcal{N}=\mathrm{10\hspace{0.17em}000}$, $w=2.5$, $T=T_T$ $\left(\ell =2\right)$, ${\varphi }_d=0.8\pi$, (a) and (b) $\Omega =0$, (c) and (d) $\Omega =1/8$, (e) and (f) $\Omega =1/4$, (g) and (h) $\Omega =1/2$, (i) and (j) $\Omega =1$, and (k) and (l) $\Omega =\left(1+\sqrt{5}\right)/2$. In the right-hand panel, the solid lines correspond to Eq. (33). The dashed lines in (b) and (j) correspond to Eq. (34), and the dashed line in (h) corresponds to Eq. (35). The vertical lines in (d), (f), and (h), indicate where $n$ is an integer multiple of $s$ (as taken from $\Omega =1/s$).

Figure 4

(Color online) ${⟨⟨{\widehat{p}}^2⟩⟩}_n^{1/2}$ in units of $\hslash K$ for a $\delta$-kicked accelerator where the system is initially prepared in the momentum eigenstate $|\beta ⟩$. Parameters are $\mathcal{N}=1$, ${\varphi }_d=0.8\pi$, $T=T_T$ $\left(\ell =2\right)$, (a) $\Omega =0$, (b) $\Omega =1/8$, (c) $\Omega =1/4$, (d) $\Omega =1/2$, (e) $\Omega =1$, and (f) $\Omega =\left(1+\sqrt{5}\right)/2$.

Figure 5

(Color online) ${⟨⟨{\widehat{p}}^2⟩⟩}_n^{1/2}$ in units of $\hslash K$ for a $\delta$-kicked accelerator where the system is initially prepared in the momentum eigenstate $|\beta ⟩$. Parameters are $\mathcal{N}=1$, ${\varphi }_d=0.8\pi$, $T=T_T$ $\left(\ell =2\right)$, (a) $\Omega =0$, (b) $\Omega =1/8$, (c) $\Omega =1/4$, and (d) $\Omega =1/2$.

Figure 6

(Color online) (Left-hand panel) ${⟨{\widehat{p}}^2⟩}_n$ and (right-hand panel) ${⟨{\widehat{p}}^4⟩}_n$ for a Gaussian initial atomic momentum distribution with $w=1/128$ $(\times )$, $w=1/32$ (○), and $w=1/8$ $(+)$. Parameters are $\mathcal{N}=\mathrm{10\hspace{0.17em}000}$, $T=T_T$ $\left(\ell =2\right)$, ${\varphi }_d=0.8\pi$, [(a) and (b) $\Omega =0$, (c) and (d) $\Omega =1/2$, (e) and (f) $\Omega =1/4$, (g) and (h) $\Omega =1/8$, and (i) and (j) $\Omega =\left(1+\sqrt{5}\right)/2$. The dashed lines correspond to the $w=0$ analytic predictions [see (left-hand panel) Eq. (31) and (right-hand panel) Eq. (32)]. The solid lines correspond to the large-$w$ limit lower bound [see (left-hand panel) Eq. (33), (b) Eq. (34), and (d) Eq. (35), all with $w=0$]. The vertical dotted lines indicate $n=n_{\text{FR}}$ [see Eq. (39)], and the solid vertical lines in (c)–(h) indicate where $n$ is an integer multiple of $s$ (as taken from $\Omega =1/s$).

Figure 7

(Color online) ${⟨⟨{\widehat{p}}^4⟩⟩}_n^{1/3}$ for a $\delta$-kicked accelerator with Gaussian initial momentum distribution of standard deviation $w=1/4$ (○), $w=1/16$ (◻), $w=1/32$ $(\times )$, $w=1/64$ (◇), $w=1/128$ (△), and $w=1/1024$ $(+)$. Parameters are $\mathcal{N}=\mathrm{10\hspace{0.17em}000}$, $T=T_T$ $\left(\ell =2\right)$, ${\varphi }_d=0.8\pi$, (a) $\Omega =0$, (b) $\Omega =1/2$, (c) $\Omega =1/4$, (d) $\Omega =1/8$, and (e) $\Omega =\left(1+\sqrt{5}\right)/2$. The dashed lines correspond to Eq. (44), and the solid lines correspond to (a) Eq. (46) and (b) Eq. (47). The vertical lines in (b)–(d) indicate where $n$ is an integer multiple of $s$ (as taken from $\Omega =1/s$).

Figure 8

(Color online) The asymptote gradient $b_{\text{opt}}$ of ${⟨⟨{\widehat{p}}^4⟩⟩}^{1/3}$. Parameters are $\mathcal{N}=\mathrm{10\hspace{0.17em}000}$, $T=T_T$ $\left(\ell =2\right)$, ${\varphi }_d=0.8\pi$, $\Omega =0$ $(\times )$, $\Omega =1/2$ $(+)$, $\Omega =1/4$ (○), $\Omega =1/8$ (◻), and $\Omega =1/16$ (△). The dashed lines are shown to guide the eyes. The horizontal solid lines correspond to the thermal asymptote gradients of Eqs. (46, 47). The vertical dotted lines correspond to Eq. (40).