Collective emission of photons from dense, dipole-dipole interacting atomic ensembles

We study the collective radiation properties of cold, trapped ensembles of atoms. We consider the high-density regime with the mean interatomic distance being comparable to, or smaller than, the wavelength of the resonant optical radiation emitted by the atoms. We find that the emission rate of a photon from an excited atomic ensemble is strongly enhanced for an elongated cloud. We analyze collective single-excitation eigenstates of the atomic ensemble and find that the absorption-emission spectrum is broadened and shifted to lower frequencies as compared to the noninteracting (low-density) or single-atom spectrum. We also analyze the spatial and temporal profile of the emitted radiation. Finally, we explore how to efficiently excite the collective superradiant states of the atomic ensemble from a long-lived storage state in order to implement matter-light interfaces for quantum computation and communication applications.

Figure 1

Internal states and spatial configuration of an ensemble of cold atoms with random positions in an elongated harmonic trap. With all the atoms initially in the ground state $|g\rangle$, a weak microwave or Raman transition (light blue arrow) creates a single collective excitation in the storage state $|s\rangle$. The atoms can be transferred from the storage state to the excited state $|e\rangle$ by a laser pulse with Rabi frequency $\mathrm{\Omega }$ and wave vector ${\mathit{k}}_c$. The excited-state atoms decay to the ground state by emitting a photon into the free-space radiation field $E$.

Figure 2

Dynamics of the population $p_{\mathrm{TD}}\left(t\right)={\left|\langle E_{\mathrm{TD}}|\mathrm{\Psi }\left(t\right)\rangle \right|}^2$ of the initially prepared timed-Dicke state $|E_{\mathrm{TD}}\rangle$ of the atoms in a harmonic trap with different aspect ratios ${\sigma }_{x,y}/{\sigma }_z$. The progressively lower curves correspond to decreasing ${\sigma }_{x,y}$ and increasing ${\sigma }_z$, with the product ${\sigma }_x{\sigma }_y{\sigma }_z$ kept constant. We place $N=1000$ atoms at random positions in an effective volume $V_{\mathrm{eff}}={\left(2\pi \right)}^{3/2}{\sigma }_x{\sigma }_y{\sigma }_z={\left(2\pi \right)}^{3/2}8\mu {\mathrm{m}}^3$, i.e., the mean interatomic separation $\langle r_{ij}\rangle =\sqrt[3]{V_{\mathrm{eff}}/N}\simeq 0.5 \mu \mathrm{m}$. Each curve corresponds to a single realization of the ensemble of atoms at random positions, but different realizations for the same atom number and trap geometry give very similar results (for large enough $N$ as here). The wavelength of the resonant transition $|e\rangle \to |g\rangle$ is ${\lambda }_e=780$ nm, decay rate $\mathrm{\Gamma }=2\times {10}^7 {\mathrm{s}}^{-1}$, and the transition dipole moment is along $\widehat{\mathbf{\wp }}=\frac{\widehat{\mathit{x}}+i\widehat{\mathit{y}}}{\sqrt{2}}$ ($\mathrm{\Delta }M=1$ transition). In the lower panel, we fit the population decay curves with the sum of three exponential terms, $p_{\mathrm{TD}}\left(t\right)\simeq p_1{e}^{-{\mathrm{\Gamma }}_{S}t}+p_2{e}^{-\mathrm{\Gamma }t}+p_3{e}^{-{\mathrm{\Gamma }}_{s}t}$, having superradiant ${\mathrm{\Gamma }}_S>\mathrm{\Gamma }$, single-atom $\mathrm{\Gamma }$, and subradiant ${\mathrm{\Gamma }}_s<\mathrm{\Gamma }$ decay rates.

Figure 3

Same as in Fig. 2 (thin solid lines are for reference, with the upper-left, upper-right, lower-left, and lower-right panels corresponding to decreasing ${\sigma }_{x,y}$ and increasing ${\sigma }_z$), but for $N=2000$ atoms, in traps with the same volume $V_{\mathrm{eff}}={\left(2\pi \right)}^{3/2}8\mu {\mathrm{m}}^3$ but twice the atom density (dashed lines), or in traps with rescaled dimensions ${\sigma }_{x,y,z}\to \sqrt[3]{2}{\sigma }_{x,y,z}$ and twice the volume $V_{\mathrm{eff}}={\left(2\pi \right)}^{3/2}16\mu {\mathrm{m}}^3$ but the same density as in Fig. 2 (dotted lines). The insets in the lower panels show the decay dynamics for $N=2000$ atoms in the elongated traps with the same lengths ${\sigma }_z=4,8\mu \mathrm{m}$ but larger widths ${\sigma }_{x,y}\to \sqrt{2}{\sigma }_{x,y}$ and twice the volume $V_{\mathrm{eff}}={\left(2\pi \right)}^{3/2}16\mu {\mathrm{m}}^3$, i.e., the same density as in Fig. 2.

Figure 4

Eigenvalues of the effective Hamiltonian (19) for $N=1000$ atoms in four different traps as in Fig. 2 (same color code, with the upper-left, upper-right, lower-left, and lower-right panels corresponding to decreasing ${\sigma }_{x,y}$ and increasing ${\sigma }_z$). The main panels show the eigenvalues, as obtained for a single realization of the ensemble of atoms at random positions in the trapping volume; each eigenvalue ${\lambda }_n=({\omega }_e+{\delta }_n)-i{\gamma }_n$ is shown as a circle centered at the corresponding $[{\delta }_n,{\gamma }_n]$ with the radius equal to the norm $|\langle E_{\mathrm{TD}}|{\mathrm{\Psi }}_n\rangle {|}^2$ of the Franck-Condon overlap of the eigenstate $|{\mathrm{\Psi }}_n\rangle$ with the single-excitation state $|E_{\mathrm{TD}}\rangle$ of Eq. (16). The upper inset in each panel shows the spectrum of eigenvalues averaged over ${10}^3$ random realizations of the ensemble, while the lower inset shows the same spectrum with each eigenvalue weighted by the corresponding FC factor (the shading is in arbitrary units, for best visibility).

Figure 5

Density of eigenvalues weighted by the corresponding FC factors $|\langle E_{\mathrm{TD}}|{\mathrm{\Psi }}_n\rangle {|}^2$ averaged over ${10}^3$ random realizations of the ensemble of $N=1000$ atoms in the elongated harmonic traps with lengths ${\sigma }_z=4,8,...,24 \mu \mathrm{m}$ and widths ${\sigma }_{x,y}=\sqrt{8/{\sigma }_z}$, i.e., the same effective volume $V_{\mathrm{eff}}={\left(2\pi \right)}^{3/2}{\sigma }_x{\sigma }_y{\sigma }_z={\left(2\pi \right)}^{3/2}8 \mu {\mathrm{m}}^3$, and mean interatomic separation $\langle |{\mathit{r}}_{ij}|\rangle =\sqrt[3]{V_{\mathrm{eff}}/N}\simeq 0.5 \mu \mathrm{m}$ as in Figs. 2 and 4.

Figure 6

Excitation spectrum $S\left(\mathrm{\Delta }\right)$ of Eq. (20) for $N=1000$ atoms in harmonic traps with different length ${\sigma }_z$ and width ${\sigma }_{x,y}=\sqrt{8/{\sigma }_z}$, i.e., the same effective volume $V_{\mathrm{eff}}={\left(2\pi \right)}^{3/2}{\sigma }_x{\sigma }_y{\sigma }_z={\left(2\pi \right)}^{3/2}8 \mu {\mathrm{m}}^3$. The spectrum is progressively broadened and shifted towards the negative detuning $\mathrm{\Delta }$ with increasing ${\sigma }_z$. For each geometry, the shown spectrum is averaged over ${10}^3$ random realizations of the atomic ensemble. Insets show the peak position and full width at half maximum (FWHM) of each spectrum.

Figure 7

Angular probability distribution $U(\theta ,\varphi )$ of the photon emitted in the $z$ direction, as a function of ${\theta }_x=\theta cos\left(\varphi \right)$ and ${\theta }_y=\theta sin\left(\varphi \right)$, with $\theta$ the polar and $\varphi$ the azimuthal angles, for $N=1000$ atoms in elongated harmonic traps with the same effective volume $V_{\mathrm{eff}}={\left(2\pi \right)}^{3/2}{\sigma }_x{\sigma }_y{\sigma }_z$ and different aspect ratios ${\sigma }_{x,y}/{\sigma }_z$. The black solid line in the upper inset of each density plot shows $U({\theta }_x,{\theta }_y=0)$ while the red dashed line is the Gaussian of Eq. (22) with the corresponding width $\mathrm{\Delta }\theta =\frac{\sqrt{2}}{k_e{\sigma }_{x,y}}$. Only the interval of $\left|\theta \right|\le 0.2\pi$ is shown as in the remaining solid angle only a weak noisy signal is present.

Figure 8

Probability $P$ of Eq. (24) for collecting the emitted photon into the Gaussian mode of waist $w_0=\sqrt{2}{\sigma }_{x,y}$ as a function of length ${\sigma }_z$ (lower horizontal axis) or width ${\sigma }_{x,y}$ (upper horizontal axis) of a cloud of $N=1000$ interacting atoms with fixed effective volume $V_{\mathrm{eff}}={\left(2\pi \right)}^{3/2}{\sigma }_x{\sigma }_y{\sigma }_z={\left(2\pi \right)}^{3/2}8$ (red solid line with open circles). Also shown is the total probability of photon emission into the $z$ direction within the solid angle ${\mathrm{\Omega }}_f=\pi {\left(2\mathrm{\Delta }\theta \right)}^2$ (blue dashed line with filled triangles). For comparison, we also show the approximate analytic results of Eq. (15) for an ensemble of noninteracting atoms (black dotted line).

Figure 9

Dynamics of population transfer of the atoms from the storage state to the excited state that decays to the ground state with the emission of a photon. The upper panel shows the time dependence of the coupling field Rabi frequency $\mathrm{\Omega }\left(t\right)={\mathrm{\Omega }}_0\frac{1}{2}[1+\mathrm{erf}\left(\frac{t-t_0}{\sqrt{2}{\sigma }_t}\right)]$ with $t_0=1 \mu \mathrm{s}$ and ${\sigma }_t=0.4 \mu \mathrm{s}$. The lower panel shows the populations of the storage $p_S$, excited $p_E$, and ground $p_G$ states for three different detunings ${\mathrm{\Delta }}_c$ of the coupling field (with the detunings ${\mathrm{\Delta }}_c/2\pi =\pm 3\mathrm{MHz}$ leading to slower or faster transfer compared to the ${\mathrm{\Delta }}_c=0$ case), as obtained from averaging over ${10}^2$ random realizations of the ensemble for $N=1000$ atoms in a harmonic trap with dimensions ${\sigma }_z=8\mu \mathrm{m}$ and ${\sigma }_{x,y}=1\mu \mathrm{m}$. Also shown are the populations for an ensemble of noninteracting atoms under the same driving and decay conditions (light gray curves). The inset shows the transfer probability to the ground state $p_G$ at time $t=2\mu \mathrm{s}$ as a function of detunings ${\mathrm{\Delta }}_c$, as obtained from a single realization of the random atomic ensemble (thick solid brown line), and as obtained from the approximate analytic solution of Eqs. (28) with parameters ${\mathrm{\Omega }}_{\mathrm{eff}}=0.92\mathrm{\Omega }, {\delta }_E=-1.0\mathrm{\Gamma }$, and ${\mathrm{\Gamma }}_S=6.0\mathrm{\Gamma }$ (thin dotted black line); note that this value for ${\mathrm{\Gamma }}_S$ was obtained in Fig. 2 by fitting the decay curve for an initially excited atomic ensemble.

Figure 10

Angular probability distribution $U(\theta ,\varphi )$ of the photon emitted in the $z$ direction for Raman excitation of the atomic cloud with all the parameters the same as in Fig. 9. The left panel shows $U(\theta ,\varphi )$ vs ${\theta }_x=\theta cos\left(\varphi \right)$ and ${\theta }_y=\theta sin\left(\varphi \right)$ as obtained for a single realization of the random atomic ensemble. For large integration times, the photon emission pattern is the same for different detunings ${\mathrm{\Delta }}_c$ of the coupling field, and the probability of cooperative emission into a Gaussian mode of waist $w_0=\sqrt{2}{\sigma }_{x,y}$ is $P\simeq 0.583$, with the remaining radiation incoherently scattered into all $4\pi$ directions. The right panel shows $U(\theta ,\varphi )$ for a noninteracting atomic ensemble under the otherwise identical conditions, leading to $P\simeq 0.727$ [for comparison, the analytic result of Eq. (15) and Fig. 8 is $P=0.794$]. The black solid line in the upper inset of each density plot shows $U({\theta }_x,{\theta }_y=0)$ while the red dashed line is the Gaussian of Eq. (22) with the corresponding width $\mathrm{\Delta }\theta =\frac{\sqrt{2}}{k_e{\sigma }_{x,y}}$.