Interference in the resonance fluorescence of two incoherently coupled transitions

The fluorescence light emitted by a four-level system in $J=1∕2$ to $J=1∕2$ configuration driven by a monochromatic laser field and in an external magnetic field is studied. We show that the spectrum of resonance fluorescence emitted on the $\pi$ transitions shows a signature of spontaneously generated interference effects. The degree of interference in the fluorescence spectrum can be controlled by means of the external magnetic field, provided that the Landé $g$ factors of the excited and the ground state doublet are different. For a suitably chosen magnetic field strength, the relative weight of the Rayleigh line can be completely suppressed, even for low intensities of the coherent driving field. The incoherent fluorescence spectrum emitted on the $\pi$ transitions exhibits a very narrow peak whose width and weight depend on the magnetic field strength. We demonstrate that the spectrum of resonance fluorescence emitted on the $\sigma$ transitions shows an indirect signature of interference. A measurement of the relative peak heights in the spectrum from the $\sigma$ transitions allows us to determine the branching ratio of the spontaneous decay of each excited state into the $\sigma$ channel.

Figure 1

Schematic representation of the four-level atom of interest. The two upper and lower levels are Zeeman sublevels with $m_j=\pm \frac{1}{2}$. Each upper state can decay by a dipole allowed transition to both ground states. The Zeeman splitting of the magnetic sublevels is not to scale.

Figure 2

Plot of the correlation function $G_{12}$ in relation to its long-time limit $G_{12}\left(\infty \right)=-\sqrt{{\gamma }_1{\gamma }_2}{⟨{\tilde{S}}_1^{+}⟩}_{\mathrm{st}}{⟨{\tilde{S}}_2^{-}⟩}_{\mathrm{st}}$ for the degenerate system. The parameters are $\Omega =3\times {10}^7{\mathrm{s}}^{-1}$, $\Delta =5\times {10}^6{\mathrm{s}}^{-1}$ and $\gamma ={10}^7{\mathrm{s}}^{-1}$. $G_{12}$ has to vanish at $\tau =0$ since the ground states are orthogonal.

Figure 3

Plot of the relative weight of the interference terms $C\left(\delta \right)$ for different values of the detuning $\Delta$ of the laser field from the $1\leftrightarrow 3$ transition. The parameters are given by $\gamma ={10}^7{\mathrm{s}}^{-1}$, $\Delta =-4\times {10}^7{\mathrm{s}}^{-1}$ (solid line), and $\Delta =-5\times {10}^6{\mathrm{s}}^{-1}$ (dashed line).

Figure 4

Incoherent spectrum of resonance fluorescence according to Eq. (36). Plot (a) shows $S_{\mathrm{inc}}^{\pi }$ for the degenerate system (dashed line) and for $\delta =-4\times {10}^6{\mathrm{s}}^{-1}$ (solid line), the other parameters are $\gamma ={10}^7{\mathrm{s}}^{-1}$, $\Delta =-4\times {10}^7{\mathrm{s}}^{-1}$, and $\Omega =6\times {10}^6{\mathrm{s}}^{-1}$. In (b) and (c) the values of $\delta$ are given by $\delta ={\delta }_0$ and $\delta ={\delta }_{\min }$, respectively; the other parameters are the same than in (a). Plot (d) shows the incoherent spectrum for the set of parameters $\Delta =-5\times {10}^6{\mathrm{s}}^{-1}$, $\Omega =6\times {10}^7{\mathrm{s}}^{-1}$, $\gamma ={10}^7{\mathrm{s}}^{-1}$, and $\delta =-8\times {10}^7{\mathrm{s}}^{-1}$.

Figure 5

Dressed state analog of the bare state system in Fig. 1. The frequency of the laser field is labeled by ${\omega }_L$. For $\delta \ne 0$, the detuning of the laser field will be different on each of the $\pi$ transitions. There are thus two effective Rabi frequencies ${\Omega }_1$ and ${\Omega }_2$ involved. The splitting of the dressed states for fixed $N$ is not to scale.

Figure 6

Fluorescence spectrum for the degenerate system according to Eq. (24). The solid line (dashed line) shows the spectrum with (without) the interference terms proportional to ${\gamma }_{12}$, ${\gamma }_{21}$. The Rayleigh peak (the vertical line at $\omega ={\omega }_L$) is present both with and without interference terms. Note that its weight is larger if the interference terms are taken into account. However, the sums of the integrated coherent and incoherent spectra with and without the interference terms are identical, making the total intensity independent of the interference terms. In (a), the parameters are $\Omega =5\times {10}^7{\mathrm{s}}^{-1}$, $\Delta =0$, and $\gamma ={10}^7{\mathrm{s}}^{-1}$. For (b), we have $\Omega ={10}^7{\mathrm{s}}^{-1}$, $\Delta =2\times {10}^7{\mathrm{s}}^{-1}$, and $\gamma ={10}^7{\mathrm{s}}^{-1}$.

Figure 7

Spectrum of resonance fluorescence emitted on the $\sigma$ transitions (solid line) in comparison with the fluorescence spectrum of a two-level atom (dotted line). The parameters in (a) are $\Omega =5\times {10}^6{\mathrm{s}}^{-1}$, $\Delta =6\times {10}^6{\mathrm{s}}^{-1}$, and $\gamma ={10}^7{\mathrm{s}}^{-1}$. In (b), the parameters are $\Omega =6\times {10}^7{\mathrm{s}}^{-1}$, $\Delta =0$, and $\gamma ={10}^7{\mathrm{s}}^{-1}$. Note that $S^{\sigma }$ deviates slightly from the Mollow triplet in the region of the sideband peaks.

Figure 8

Radiative cascade in the dressed states of the system [see Eqs. (42, 43)]. The splitting of the dressed states for a fixed number of laser photons $N$ is not to scale. Each of the two indicated cascades involves the emission of a $\pi$ photon and a $\sigma$ photon with wave vector ${\mathit{k}}_{\pi }$ and ${\mathit{k}}_{\sigma }$, respectively. Depending on the time order of their emission, the $\pi$ photon is either emitted on transition $\mid 4\left(N\right)⟩\to \mid 4(N-1)⟩$ or $\mid 1(N-1)⟩\to \mid 1(N-2)⟩$, corresponding to the bare state transitions $\mid 2⟩\to \mid 4⟩$ and $\mid 1⟩\to \mid 3⟩$, respectively. Since the final and initial states of the two cascades are identical, the corresponding transition amplitudes may interfere.

Figure 9

The solid lines show the fluorescence spectra recorded with a finite frequency resolution $\lambda$. The dashed curves are the spectra without the interference terms proportional to ${\gamma }_{12}$, ${\gamma }_{21}$ in Eq. (61). The parameters are $\Omega =7\times {10}^6{\mathrm{s}}^{-1}$, $\Delta =2\times {10}^7{\mathrm{s}}^{-1}$, $\gamma ={10}^7{\mathrm{s}}^{-1}$, and $B=\delta =0$. This corresponds to a saturation parameter of $s=0.235$ and a mean number of photons per unit time of approximately $9.4\times {10}^5{\mathrm{s}}^{-1}$. The filter bandwidths are given by (a) $\lambda ={10}^2{\mathrm{s}}^{-1}$, (b) $\lambda ={10}^4{\mathrm{s}}^{-1}$, (c) $\lambda =1.9\times {10}^6{\mathrm{s}}^{-1}$, and (d) $\lambda ={10}^7{\mathrm{s}}^{-1}$.

Figure 10

Schematic representation of elastic scattering events into the $3\to 1\to 4$ (solid arrows) and $4\to 2\to 3$ channels (dotted arrows). These processes account for the sharp peak in the fluorescence spectrum $S^{\sigma }$ emitted on the $\sigma$ transitions.