Spectrum of coherently backscattered light from two atoms

We present a detailed analytical and numerical analysis of the inelastic coherent backscattering spectrum of laser light incident on cold atoms. We identify frequency domains where the interference contribution can be positive as well as negative—or exhibits dispersive character. These distinctive features are explained by reciprocity arguments and dressed-state two-photon scattering amplitudes.

Figure 1

(Color online) Model of CBS with two atoms. (a) Atoms (black dots) are driven by laser light with right circular polarization, while CBS is observed in the helicity-preserving channel, that is, with flipped polarization. Photons in this channel appear as a result of double scattering. $g$ is the strength of the far-field dipole-dipole coupling responsible for the exchange of photons. (b) Internal atomic structure corresponding to a $J_g=0\to J_e=1$ dipole transition. ${\omega }_0$ is the transition frequency, $2\gamma$ is the radiative linewidth, $\Omega$ is the Rabi frequency. Sublevels $|1⟩$ and $|3⟩$ have magnetic quantum number $m=0$. Sublevels $|2⟩$ and $|4⟩$ correspond to $m=-1$ and $1$, respectively. The thick solid arrow represents the laser field driving the $|1⟩\leftrightarrow |4⟩$ transition, while the dashed arrow indicates the CBS field originating from the $|1⟩\leftrightarrow |2⟩$ transition.

Figure 2

(Color online) Elastic ladder $L_2^{\text{el}}$ (solid), inelastic ladder $L_2^{\text{inel}}$ (dashed), and inelastic crossed $C_2^{\text{inel}}$, (dashed-dotted) CBS contributions vs Rabi frequency $\Omega$, for different values of the detuning $\delta$: $\delta =\left(\text{a}\right)0$; (b) $5\gamma$; (c) $20\gamma$.

Figure 3

(Color online) Normalized inelastic spectra of the ladder ${\tilde{L}}_2^{\text{inel}}\left(\nu \right)$ (solid) and crossed ${\tilde{C}}_2^{\text{inel}}\left(\nu \right)$ (dashed) terms at $\Omega =0.1\gamma$. $\delta =\left(\text{a}\right)0$; (b) $\gamma$; (c) $5\gamma$.

Figure 4

Direct (a) and (time-)reversed (b) scattering paths and amplitudes, giving rise to the CBS signal in the weakly inelastic scattering regime. Gray-shaded and black spots represent inelastically and elastically scattering atoms, respectively. The amplitudes spell out the transformation of ingoing frequencies ${\omega }_L$ into outgoing frequencies $\omega$. These interfering amplitudes are nonreciprocal, since the frequencies of the intermediate photons for direct and reversed paths are different.

Figure 5

(Color online) Two-photon processes describing the laser frequency transformation on the left atom in Fig. 4. (a) corresponds to Fig. 4a: Two laser photons are converted into two scattered photons such that ${\omega }_1\simeq {\omega }_0+2\delta$ and ${\omega }_2\simeq {\omega }_0$. Either one of these photons then travels to the other atom. (b) corresponds to Fig. 4b in the case $\omega \simeq {\omega }_1$: One laser photon and one elastically scattered photon give rise to the CBS photon at $\omega \simeq {\omega }_1$ and the undetected fluorescence photon at ${\omega }_2\simeq {\omega }_0$. The roles of the photons at ${\omega }_1$ and ${\omega }_2$ can be interchanged by flipping their polarizations.

Figure 6

(Color online) Normalized inelastic spectra of the ladder ${\tilde{L}}_2^{\text{inel}}$ (solid) and crossed ${\tilde{C}}_2^{\text{inel}}$ (dashed) terms. Left panel (a)–(c): $\Omega =10\gamma$; right panel (d)–(f): $\Omega =20\gamma$. The detuning increases from top to bottom: $\delta =0$ (top); $5\gamma$ (middle); $20\gamma$ (bottom).

Figure 7

Single- (left) and double- (right) scattering processes. Only the laser-driven and the CBS transition are depicted, respectively. Radiative processes (a) and (b) on the left give rise to the Rayleigh peak of the Mollow triplet with linewidth $\gamma$ centered at $\nu =0$. (c) and (d) give rise to the sidebands at $\nu =\Omega$ and $-\Omega$, respectively, with linewidths $3\gamma ∕2$. Photons emitted by atom 1 (left) propagate to atom 2 (right), and are scattered from either one of the dressed states of the energy manifold with $N-1$ laser photons. The diagrams on the right describe the scattering of (a)–(d) a sideband photon centered at $\nu =-\Omega$; (e)–(h) a photon from the central peak; (i)–(l) a sideband photon at $\nu =\Omega$; (m), (n) resonant photons at $\nu =\pm \Omega ∕2$.

Figure 8

(Color online) Normalized inelastic spectra of the ladder ${\tilde{L}}_2^{\text{inel}}\left(\nu \right)$ (solid) and crossed ${\tilde{C}}_2^{\text{inel}}$ (dashed) terms, in the limit of well-separated spectral lines, at $\Omega =100\gamma$. $\delta =\left(\text{a}\right)0$; (b) $20\gamma$. The numbers near the resonances indicate their areas, such that the overall areas of the ladder and crossed terms give unity and $C_2^{\text{inel}}∕L_2^{\text{inel}}$, respectively.

Figure 9

Qualitative interpretation of the character of the interference. Gray-shaded and black spots represent atoms scattering inelastically and quasielastically, respectively. Solid and dashed arrows correspond to direct and reversed scattering amplitudes, respectively. In examples (a) and (b), the direct and the reversed amplitudes are reciprocal, whereas they are not in cases (c) and (d).