Collective dipole-dipole interactions in an atomic array

The coherent photon scattering of atoms in an atomic array is studied. It is found that due to cooperative photon exchanges, the excitation probability of an atom in an array parallel to the direction of laser propagation ($\widehat{\mathit{k}}$) will either grow or decay along $\widehat{\mathit{k}}$, depending on the detuning of the laser. The symmetry of the system for atomic separations of $a=j\lambda /2$ causes the excitation distribution and scattered radiation to abruptly become symmetric about the center of the array, where $j$ is an integer and $\lambda$ is the transition wavelength. For atomic separations of $a<\lambda /2$, a collection of nonradiating states ($\mathrm{\Gamma }\sim 0$) disrupts the described trend. In order to interpret this surprising result, a band-structure calculation in the $N\to \infty$ limit is conducted, where the decay rates and the collective Lamb shifts of the eigenmodes versus quasimomentum are obtained. This calculation shows that the collective exchange of photons in an array strongly affects its scattered radiation, allowing one to easily manipulate the collective Lamb shift and directly excite either superradiant or subradiant eigenmodes by correctly choosing the angle of the driving laser.

Figure 1

When a red detuned laser drives an array of atoms, the probability of excitation grows logarithmically down the line of the array. This effect and the others discussed in this paper, are due to the spatially dependent phase correlations between the driving laser and the dipole radiation.

Figure 2

(a) Probability of excitation of atom $n \left[P\right(n\left)\right]$ divided by the single-atom excitation probability ($P_0$) for that detuning ($\mathrm{\Delta }$). For the top two plots, $P\left(n\right)/P_0$ is shown for a detuning of $-0.6\mathrm{\Gamma }$ using 50 and 100 atoms. Note that except for small oscillations, both plots lie on top of each other for up to $n=50$. This is due to the highly forward character of the coherent interactions. For the bottom two plots, $P\left(N\right)/P_0$ is shown for a detuning of $+0.6\mathrm{\Gamma }$ using 50 and 100 atoms. The noninteracting probability for the same Rabi frequency and $\left|\mathrm{\Delta }\right|$ is shown for reference. (b) The symmetric probability distribution present for integer and half-integer values of $a$ is shown for red and blue detunings.

Figure 3

Photon scattering rate ($\gamma$) in the $-\widehat{\mathit{k}}$ direction versus array spacing $a$ for an array of 100 atoms for values of detuning $\mathrm{\Delta }=0.5\mathrm{\Gamma }$ (green solid line) and $\mathrm{\Delta }=-0.5\mathrm{\Gamma }$ (red dotted line). The inset shows the same graph zoomed in around $a=2\lambda$. Note that the difference in the heights of the peaks is mainly due to the fact that the collective Lamb shift of the driven eigenmode changes with $a$.

Figure 4

The probability of excitation of atom $n \left[P\right(n\left)\right]$ divided by the probability of excitation of the first atom $\left[P\right(1\left)\right]$ for an array containing 500 atoms. (a) Comparison between the full numerical calculation (green), Eq. (13) (red), and our analytic model (blue) for $a=2.3\lambda$. Since the approximations made in the analytic derivation hold only for $ka\gg 1$, the analytic model is only qualitatively accurate in this regime. Note that when $P\left(n\right)$ is generated from Eq. (13), it lies on top of the full calculation (excluding small oscillations at large $n$) for all $a>\lambda$. (b) Comparison between the full numerical calculation (green) and the analytic model (blue) for $ka\gg 1$. This illustrates the fact that the analytic model approaches the quantitative numerical result for this condition, as well as the fact that the logarithmic growth holds for very large array spacings. Note that these results are only true for $a\ne \frac{j\lambda }{2}$.

Figure 5

The probability of excitation of atom $n \left[P\right(n\left)\right]$ divided by the probability of excitation of atom 1 $\left[P\right(1\left)\right]$. (a) Excitation distribution for a laser resonant with the collection of subradiant states (${\mathrm{\Gamma }}_{\mu }\sim 0$) for an array of 100 atoms separated by a distance of $0.4\lambda$. Over the range of laser detunings $\mathrm{\Delta }=-0.4\mathrm{\Gamma }-0.9\mathrm{\Gamma }$, the behavior is drastically altered due to the almost complete lack of decay by photon emission. (b) The same array of atoms for laser detunings that do not lie within ($\mathrm{\Delta }\sim -0.4\mathrm{\Gamma }-0.9\mathrm{\Gamma }$) show qualitatively similar results to those described previously.

Figure 6

The decay rate (${\mathrm{\Gamma }}_{\mu }$) and the collective Lamb shift (${\delta }_{\mu }$) of the eigenmodes of an array divided by the single-atom decay rate ($\mathrm{\Gamma }$), where $a=0.4\lambda , \mathrm{\Delta }=0$, and $N=100$. Note the collection of eigenvalues near the ${\mathrm{\Gamma }}_{\mu }=0$ axis, despite the fact that the length of the array is $40\lambda$.

Figure 7

The values of ${\mathrm{\Gamma }}_q$ and of ${\delta }_q$ (the collective Lamb shift) are plotted versus values of $0\le qa/2\pi <0.5$. (a) Shows this for array spacings of $a=0.3\lambda$, (b) $a=0.7\lambda$, (c) $a=1.3\lambda$, and (d) $a=1.7\lambda$. Note that for all four graphs the collective Lamb shift diverges at $qa/2\pi =0.3$, while ${\mathrm{\Gamma }}_q$ gives a discontinuous jump caused by the appearance or disappearance of a Bragg diffraction peak.

Figure 8

The dependance of the scattered photon emission on the laser angle $\theta$, which is the angle between an array of 1000 atoms and the driving laser. (a) The scattered radiation versus $acos\theta /\lambda$ for a laser with $\mathrm{\Delta }=0$ and arrays with spacings of $1.7\lambda$ and $2.3\lambda$. Note that the large shifts in scattered radiation occur when the eigenmode that the laser drives crosses a band-structure discontinuity. (b) The photon scattering rate versus detuning of an array where $a=1.3\lambda$ for three different laser angles ($\theta$): {0, $arccos(0.3/1.3)$}, {$\pi /2, 1.0/1.3$}, and $arccos(0.5/1.3)$. The scattering rates for all three angles fit to a Lorentzian line shape well, with decay rates and line shifts that correspond to those given by their $N\to \infty$ limit band-structure calculations seen in Fig. 7.