Elastic versus inelastic coherent backscattering of laser light by cold atoms: A master-equation treatment

We give a detailed derivation of the master-equation description of the coherent backscattering of laser light by cold atoms. In particular, our formalism accounts for the nonperturbative nonlinear response of the atoms when the injected intensity saturates the atomic transition. Explicit expressions are given for total and elastic backscattering intensities in the different polarization channels, for the simplest nontrivial multiple-scattering scenario of intense laser light multiply scattering from two randomly placed atoms.

Figure 1

Level scheme of a $J_g=0\to J_e=1$ dipole transition, with atomic transition frequency ${\omega }_0$, and natural linewidth $2\gamma$. The sublevels $\mid 1⟩$ and $\mid 3⟩$ have magnetic quantum number $m=0$. Sublevels $\mid 2⟩$ and $\mid 4⟩$ correspond to $m=-1$ and $m=1$, respectively.

Figure 2

(Color online) Elementary configuration for the polarization-selective detection of the coherent backscattering signal (CBS) (dashed arrows) from laser light (thick arrows) scattering on two atoms 1 and 2 (black circles). The dipole-dipole coupling strength between the atoms is given by the coupling constant $g$. Backscattered photons are detected in four polarization channels: (a) right circular in, left circular out (with respect to a fixed observation direction; $h\Vert h$); (b) linear in, orthogonal linear out $(\mathrm{lin}\perp \mathrm{lin})$; (c) right circular in, right circular out $(h\perp h)$; (d) linear in, parallel linear out $(\mathrm{lin}\Vert \mathrm{lin})$. Note that in cases (c) and (d) CBS appears on the background of single scattering from independent atoms.

Figure 3

(Color online) Saturation dependence of the total double-scattering contribution $I_2^{\mathrm{tot}}$ to the CBS signal in the $h\Vert h$ channel, decomposed in its ladder and interference parts $L_2^{\mathrm{tot}}$ and $C_2^{\mathrm{tot}}$, according to Eqs. (46, 47), at exact resonance $\delta =0$. At finite saturation $s$, the interference term $C_2^{\mathrm{tot}}$ drops below the ladder contribution $L_2^{\mathrm{tot}}$, indicating a loss of coherence. At large $s\ge 1$, the total double-scattering intensity must decrease with $s$, since the scattering cross section of the emitting atom drops as $s^{-1}$.

Figure 4

(Color online) Numerical enhancement factor $\alpha$ in the helicity preserving channel $h\Vert h$, versus saturation $s$ of the atomic transition, for different detunings $\delta$ from resonant driving. Larger detuning leads to a faster loss of the CBS contrast. At small $s$, one recovers the perturbative prediction [11] linear in $s$ (straight dashed line). At large $s$, $\alpha$ saturates at a $\delta$-dependent value ${\alpha }_{{\delta }_{\infty }}>1$, due to the constructive self-interference of inelastically scattered photons. Remarkably, $\alpha$ also passes through a minimum at $s\simeq 0.5$, for very large detuning $\delta =20\gamma$. This indicates destructive interference, and the physical cause of this observation remains to be identified.

Figure 5

(Color online) Numerical enhancement factor $\alpha$ (numerical solution) in the helicity preserving channel, versus the laser detuning $\delta$, for two values of the driving Rabi frequency $\Omega$. The decrease of $\alpha$ below unity for $\Omega =20\gamma$, in the vicinity of $\mid \delta \mid =20\gamma$, correlates with the minimum displayed by $\alpha \left(s\right)$ in the corresponding plot in Fig. 4, at $s\simeq 0.5$.

Figure 6

(Color online) Total crossed [Eqs. (70)] and ladder [Eq. (69)] terms, together with the double-scattering intensity $I_2^{\mathrm{tot}}=L_2^{\mathrm{tot}}+C_2^{\mathrm{tot}}\left(0\right)$, in the $h\perp h$ channel, as functions of the saturation $s$, at exact resonance $\delta =0$.