Phase Matching in Lower Dimensions

Phase matching refers to a process in which atom-field interactions lead to the creation of an output field that propagates coherently through the interaction volume. By studying light scattering from arrays of cold atoms, we show that conditions for phase matching change as the dimensionality of the system decreases. In particular, for a single atomic chain, there is phase-matched reflective scattering in a cone about the symmetry axis of the array that scales as the square of the number of atoms in the chain. For two chains of atoms, the phase-matched reflective scattering can be enhanced or diminished as a result of Bragg scattering. Such scattering can be used for mapping collective states within an array of neutral atoms onto propagating light fields and for establishing quantum links between separated arrays.

Figure 1

Bragg scattering from a one-dimensional atomic chain. (a) A linear chain with $N=10$ atoms separated by $d=7.49\mu \mathrm{m}$ is aligned along the $z$ axis. An excitation laser with wave vector ${\mathbf{k}}_{\mathrm{exc}}$ is directed onto the chain at an angle ${\theta }_{\mathrm{exc}}=4°$ with respect to the $z$ axis. The scattered light with wave vector $\mathbf{k}$ is detected as a function of spherical angles $(\theta ,\varphi )$. (b) Structure factor $S(\theta ,\varphi )$ for equal separation of atoms, and (c) in the presence of disorder of atomic positions with standard deviations $({\sigma }_x,{\sigma }_y,{\sigma }_z)=(0,0,0.3\lambda )$, and (d) $({\sigma }_x,{\sigma }_y,{\sigma }_z)=(0.3\lambda ,2.4\lambda ,0.3\lambda )$. Gray panels represent the $x\text{−}z$ plane in which the excitation laser propagates. To measure the scattered signal, we use a detector whose axis has polar angles $({\theta }_{\det },0)$.

Figure 2

Phase-matched emission from a single chain. (a) Averaged atomic fluorescence images of a linear chain containing $N_t=20$ traps. The left image shows a configuration in which the atoms are aligned along with the detection mode axis, i.e., ${\theta }_{\det }=0°$. For the middle (right) image, the chain is prepared such that ${\theta }_{\det }\approx 4°(8°)$. (b) The measured photocount rate as a function of ${\theta }_{\det }$ for $N=4$, 8, 12. Each point is an average of randomly filled chains with a given $N$. Error bars represent one standard deviation of the observed photoelectric counting events. The green lines are the numerical results based on a Monte Carlo simulation with 1000 runs. The shading on the line represents the standard deviation of the simulation divided by the square root of the averaged number of trials in the experiments. (c) The peak count rate as a function of $N$. Each point and its error bar represent the observed value at ${\theta }_{\det }={\theta }_0=4°$. For the single-shot measurement of $N=15$, we associate a $\sqrt{M}$ Poissonian error for $M$ photoelectric count events. The solid (dashed) line represents the numerical simulation with (without) displacement of atomic positions due to imperfect positioning of the traps and finite temperature effects. The black dotted lines in (b),(c) show the detection background measured without loading atoms.

Figure 3

Observation of constructive and destructive Bragg scattering from two chains. (a) Averaged fluorescence images of two linear chains under three different values of the chain separation $L=L_{\parallel }\sin {\theta }_0+L_{\perp }\cos {\theta }_0$. (b) Measured photon count rate as a function of $L_{\parallel }$ for $N=8$, 12, 16. Error bars represent one standard deviation for $M$ photoelectric counting events. The green lines are the numerical simulation based on a Monte Carlo simulation with 1000 runs. The shading on the line represents the standard deviation of the simulation divided by the square root of the averaged number of trials in the experiments. The black dotted lines show the detection background which is measured without loading atoms. (c) The observed scaling of the interference visibilities $\mathcal{V}$ as a function of the total number of atoms $N$. Each point is obtained by fitting the observed fringe by a sinusoidal function. Error bars indicate the fitting errors with 68% confidence intervals. The solid line is the result of the numerical simulation together with the standard error of the mean (shaded area).