Diagrammatic treatment of coherent backscattering of intense light by cold atoms with degenerate energy levels

We present a generalization of the diagrammatic pump-probe approach to coherent backscattering (CBS) of intense laser light for atoms with degenerate energy levels. We employ this approach for a characterization of the double-scattering signal from optically pumped atoms with the transition $J_g\to J_e=J_g+1$ in the helicity-preserving polarization channel. We show that, in the saturation regime, the internal degeneracy becomes manifest for atoms with $J_g\ge 1$, leading to a faster decrease of the CBS enhancement factor with increasing saturation parameter than in the nondegenerate case.

Figure 1

Double-scattering processes surviving the disorder average: (a) ladder, or background, contribution, describing copropagating amplitudes; (b) crossed, or interference, contribution, resulting from the interference between counterpropagating amplitudes. The incident laser field at frequency ${\omega }_L$ is assumed to be strong enough to induce nonlinear inelastic scattering processes, whereupon the frequencies ${\omega }_p,{\omega }_p^{\prime }$, and ${\omega }_D$ may differ from the incident laser field's frequency.

Figure 2

Graphical definitions of the elementary single-atom spectral responses, together with our notation for the corresponding correlation functions. (a)–(d) Complex scattering amplitudes associated with the perturbative solutions of Eq. (3); (e)–(h) blocks associated with the perturbative solutions for the fluctuating part of the atomic dipole correlation function $\langle \Delta {\sigma }^{+}\left(t\right)\Delta {\sigma }^{-}\left(t^{\prime }\right)\rangle$ [see Eq. (5)]. Blank and hatched shapes denote elastic and inelastic spectral responses, respectively (see text).

Figure 3

An example of a double-scattering diagram contributing to the elastic crossed spectrum. The spectral response of the left atom is elastic and constructed as a product (denoted by the “X”) of the block which is the complex conjugate of the one shown in Fig. 2 and of the block Fig. 2 (one should bear in mind that, for circles, the direction of the outgoing arrow is immaterial for the definition of the spectral response [24]). The spectral response of the right atom is represented by the block in Fig. 2. There is no hatching of the square since, as discussed in the text, the outgoing arrow that represents the detected field is elastic with respect to the laser frequency. The overall mathematical expression for this double-scattering process is given in Eq. (7).

Figure 4

Double-scattering diagrams which survive the disorder average for the case of vector electric fields and atoms with degenerate dipole transitions: (a) ladder spectrum; (b) crossed spectrum. The meaning of the labels ${\omega }_L,{\omega }_p,{\omega }_p^{\prime }$, and ${\omega }_D$ is the same as in Fig. 1. Indices $q_L,q,q^{\prime },r,r^{\prime }$, and $q_D$ refer to the polarization indices of the corresponding arrows in the spherical basis (see text for further details).

Figure 5

A linearly polarized laser wave [thick red (light gray) arrow] excites the $\pi$ transitions [double red (light gray) thin arrows] of two atoms with ground- and excited-state angular momenta $J_g$ and $J_e$ equal to $1/2$. Atom 1 emits a ${\sigma }_{-}$ polarized amplitude $\left(q=-1\right)$ towards atom 2, and, after projection of the polarization vector onto the plane perpendicular to the line connecting both atoms, excites a ${\sigma }_{+}$ transition thereof $\left(q^{\prime }=+1\right)$. Finally, a $\pi$-polarized amplitude $\left(q^{\prime \prime }=0\right)$ is emitted by atom 2 towards the detector.

Figure 6

Diagrammatic expansion of the double-scattering process depicted in Fig. 4 into its elastic (open circles) and inelastic (hatched boxes) components. Left (A), (B) Single-atom building blocks contributing to the ladder spectrum. Right (a1)–(b5) Expansion of the single-atom building blocks into elementary building blocks.

Figure 7

Diagrammatic expansion of the double-scattering process depicted in Fig. 4 into the elastic (open circles) and inelastic (hatched boxes) components. Left (C), (D) Single-atom building blocks contributing to the crossed spectrum. Right (c1)–(d3) Expansion of the single-atom building blocks into elementary building blocks.

Figure 8

Elementary single-atom building blocks, together with the corresponding spectral response functions in the vector case. (a)–(d) Complex scattering amplitudes associated with the perturbative solutions of Eq. (17); (e)–(h) blocks associated with the perturbative solutions for the fluctuating part of the atomic dipole correlation function; see Eq. (19). The notation ${\omega }^{\left[q\right]}$ for a positive- (negative-) frequency wave implies a wave with frequency $\omega$ and polarization described by the unit vector ${\widehat{\mathbf{e}}}_q \left({\widehat{\mathbf{e}}}_q^{*}\right)$. Blank and hatched shapes denote elastic and inelastic spectral responses, respectively, in full analogy to the scalar case; see Fig. 2.

Figure 9

Degenerate dipole transition $J_g\to J_e=J_g+1$ driven by ${\sigma }_{+}$-polarized light [thick red (light gray) double arrows]. In the notation of the ground- and excited-state sublevels, the first number refers to the angular momentum and the second one to the magnetic quantum number. The black dots show that the populations in the steady-state limit are distributed among the states $|J_g{J}_g\rangle$ and $|J_e{J}_e\rangle$, with $J_e=J_g+1$. The CBS signal in the helicity preserving channel appears due to the double-scattering process on the transition between $|J_g{J}_g\rangle$ and $|J_e{J}_e-2\rangle$ depicted by the thin blue (dark gray) arrow.

Figure 10

Inelastic ladder (solid lines) and crossed (dashed lines) double-scattering CBS spectra in the weakly inelastic regime $\left(\Omega =0.3\gamma \right)$, for four different dipole transitions. (Top) $J_g=0$ and $J_g=1/2$. Both spectra coincide after rescaling the $J_g=1/2$ plots by the factor 1/9. (Middle) $J_g=1$; (bottom) $J_g=3$. (Left column) Exact resonance, $\delta =0$; (right column) detuned driving, $\delta =5\gamma$. Insets magnify narrow resonances that emerge for $J_g\ge 1$, centered at the driving frequency for $\delta =0$, and slightly shifted towards a more pronounced sideband for $\delta \ne 0$. In cases (d) and (f), sidebands at $\delta =5\gamma$ exist, but are not resolved on this scale.

Figure 11

Dressed-state structure for atoms with different ground-state angular momenta (a) $J_g=0,$ (b) $J_g=1/2$, and (c) $J_g\ge 1$. Modified Rabi frequency $\tilde{\Omega }=\sqrt{{\Omega }^2+{\delta }^2}$, and ${\Omega }^{\prime }$ is given by Eq. (31). While in cases (a) and (b) the structure of the levels relevant for double scattering is the same, in case (c) the excited state of the CBS transition is dressed by the laser field.

Figure 12

Examples of double-scattering ladder (top) and crossed (bottom) spectra for three different values of the Rabi frequency $\Omega$ (see legend) at resonance ($\delta =0$; left) and at a detuning $\delta =5\gamma$ (right), for $J_g=0$ and $J_g=1/2$. The spectra for both values of $J_g$ coalesce upon rescaling the $J_g=1/2$ spectra by a factor ${(4J_g+1)}^{-2}=1/9$.

Figure 13

Examples of the double-scattering ladder (top) and crossed (bottom) spectra, for three different values of the Rabi frequency $\Omega$ (see legend) for (a),(b),(e),(f), $J_g=1$; and (c),(d),(g),(h), $J_g=3$. Plots (a), (c), (e), and (g) are obtained at exact resonance, $\delta =0$, while (b), (d), (f), and (h) show the result for finite detuning, $\delta =5\gamma$.

Figure 14

Inelastic ladder (solid lines) and crossed (dashed lines) double-scattering CBS spectra at exact resonance $\left(\delta =0\right)$ in the deep saturation regime $\left(\Omega =18\gamma \right)$, for different dipole transitions: (a) $J_g=30$, (b) $J_g=110$, (c) $J_g={10}^3$, and (d) $J_g={10}^4$. Insets magnify narrow resonances.

Figure 15

Coherent backscattering enhancement factor $\alpha$ vs saturation $s$, at resonant driving, for different values of $J_g$. The enhancement factor initially decreases faster with increasing saturation for higher values of $J_g$.

Figure 16

Residual enhancement factor ${\alpha }_{\infty }$ at $s=162\gg 1$, versus $J_g=0$, 1/2, 1, 3/2, 2, 5/2, 3, 4, 5, 6, 7, 8, 9, 10, 15, etc., with ${\alpha }_{\infty }\left(0\right)={\alpha }_{\infty }(1/2)\approx 1.095$. (Inset) Semilog graph of ${\alpha }_{\infty }\left(J_g\right)$, with ${\alpha }_{\infty }\left({10}^4\right)\approx 1.0016$. The continuous line is a guide for the eye.