Consequences of optical pumping and interference for excitation spectra in a coherently driven molecular ladder system

This paper presents observations and an interpretation of laser-induced excitation and fluorescence in a ladder $g\text{−}e\text{−}f$ of three molecular levels [$X{}^1\Sigma _g^{+}(v^{″}=0,J^{″}=7)$, $A{}^1\Sigma _u^{+}(v^{\prime }=10,J^{\prime }=8)$, and $5{}^1\Sigma _g^{+}(v=10,J=9)$, respectively] observed in a supersonic molecular beam of ${\mathrm{Na}}_2$. The $g\text{−}e$ coupling, by cw laser $P$, is strong. A weak cw laser $S$ couples levels $e$ and $f$. The basic observations are of level-$f$ fluorescence as a function of the detuning of the $S$ laser from resonance. The signal profile does not appear as the typical Autler-Townes doublet, but as a spectral structure, whose position, width, and shape depend upon several laser parameters. We interpret these results using a simple model of three nondegenerate quantum states coherently excited while undergoing population loss to states outside the three-level system. We invoke the mechanism of optical pumping and evolution along adiabatic states, together with Landau-Zener transition probabilities. We also present results from numerical studies, which include all quantum states, all radiative couplings, coherent and incoherent, as well as convolutions with the relevant distribution functions (velocities and Zeeman sublevels). Although no adjustable parameters are involved, excellent agreement with the experiment is found. Since successive avoided crossings of adiabatic eigenvalues occur, interference effects may be relevant. Such effects are not expected to be visible in the present experiment, for reasons that are discussed. However, we discuss conditions which would allow resolving the interference structure experimentally. We also suggest possible interesting applications of the interference to rapidly switch off Rydberg state population or to control its spatial distribution.

Figure 1

Three-level ladder linkage in ${\mathrm{Na}}_2$. Levels $g$ and $e$ are coupled by a strong $P$ laser, nominally of wavelength $633\mathrm{nm}$ and detuned from resonance by ${\Delta }_P$. A weak $S$ laser, nominally $587\mathrm{nm}$ and detuned from the $e\text{−}f$ resonance by ${\Delta }_S$, brings population into level $f$. Spontaneous emission, observed as spectrally unresolved fluorescence, occurs from levels $e$ and $f$, as indicated by wiggling arrow-headed lines. Our observations of excitation spectra consist of measuring the fluorescence from level $f$ as a function of the probe detuning ${\Delta }_S$.

Figure 2

Geometry of the experiment. The collinear laser beams define the $x$ direction. The molecular beam axis and linear polarization directions of laser fields are parallel to the $z$ axis, which serves as the quantization axis for the molecular orientation.

Figure 3

Fluorescence signal $S_f$ from the upper level $f$ as a function of the $S$-laser detuning ${\Delta }_S∕2\pi$ for various choices of the $P$-laser detuning ${\Delta }_P∕2\pi$ (a) $0\mathrm{MHz}$, (b) $-80\mathrm{MHZ}$ (or ${\Delta }_P∕{\Omega }_P=-0.21$), (c) $-170\mathrm{MHz}$ (or ${\Delta }_P∕{\Omega }_P=-0.45$), (d) $-260\mathrm{MHz}$ (or ${\Delta }_P∕{\Omega }_P=-0.69$). Circles represent the experimental results, and solid lines show the results of the numerical simulations described in Sec. 5B. The spectra for positive detunings of ${\Delta }_P$ (not shown) are mirror reflections of the spectra for ${\Delta }_P<0$ with respect to ${\Delta }_S=0$.

Figure 4

Adiabatic eigenvalues ${\epsilon }_{-}$, ${\epsilon }_{+}$, and ${\epsilon }_f=-{\Delta }_S$, in units of ${\Omega }_P^{\mathit{max}}$ versus $t∕\tau =z∕R_P$. (a) ${\Delta }_P=0$, ${\Delta }_S=-0.5{\epsilon }_{-}^{\mathit{min}}$; (b) ${\Delta }_P=-0.3{\Omega }_P^{\mathit{max}}$, ${\Delta }_S=-0.75{\epsilon }_{-}^{\mathit{min}}$; (c) ${\Delta }_P=-0.6{\Omega }_P^{\mathit{max}}$, ${\Delta }_S=-1.0{\epsilon }_{-}^{\mathit{min}}$. At each crossing of the curves ${\epsilon }_{-}$ and $f$, population transfer occurs. There are two spatially located regions for such transitions: one for $t<0$ and the other for $t>0$.

Figure 5

Anatomy of excitation spectrum of level $f$ as a function of $S$-laser detuning for a $P$-laser detuning ${\Delta }_P∕2\pi =-80\mathrm{MHz}$ and Rabi frequency ${\Omega }_P∕2\pi =339\mathrm{MHz}$. The dotted lines are signals calculated for molecules traveling along the particle beam axis with the flow velocity $v_m$. The transit time across the $P$-laser beam is ${\tau }_{\mathit{tr}}\simeq 75\mathrm{ns}$. The electronic lifetimes are ${\tau }_e=12.45\mathrm{ns}$ and ${\tau }_f=35\mathrm{ns}$. The lines labeled $m_J=0$ and $m_J=7$ are the partial excitation spectra for ladders with the given $m_J$ quantum numbers, while ${\Sigma }_m$ is the total signal related to the sum of all eight separate ladders with $0⩽m_J⩽7$. Finally, the solid line is the total signal, which takes into account the finite width of the velocity distribution of molecules in the beam both parallel and perpendicular to the particle beam axis. The open circles (with a central dot) show experimental results [see Fig. 3b].

Figure 6

Calculated spatial distributions of the population of level $f$ as a function of the spatial coordinate $z$ along the molecular beam axis for the ladder of Zeeman sublevels with $m_J=0$. The detuning of the $P$ laser field is fixed at ${\Delta }_P∕2\pi =80\mathrm{MHz}$. The plots show the spatial distributions calculated using a transit time ${\tau }_{\mathit{tr}}\simeq 37.5\mathrm{ns}$, and electronic lifetimes ${\tau }_e=12.45\mathrm{ns}$ and ${\tau }_f=300\mathrm{ns}$. The lines $\mathbf{a}$ and $\mathbf{b}$ correspond to the $S$-laser frequency tuned to an interference minimum or maximum, respectively.

Figure 7

Calculated excitation spectrum of level $f$ as a function of $S$-laser detuning for various values of $S$-laser Rabi frequency ${\Omega }_S$: trace 1—${\Omega }_S∕2\pi =35\mathrm{MHz}$; trace 2—${\Omega }_S∕2\pi =87\mathrm{MHz}$; trace 3—${\Omega }_S∕2\pi =173\mathrm{MHz}$. The $P$-laser detuning ${\Delta }_P∕2\pi =-80\mathrm{MHz}$. The lifetime ${\tau }_f=100\mathrm{ns}$ and transit time ${\tau }_{\mathit{tr}}=18.75\mathrm{ns}$. The other parameters $(m_J,{\tau }_e)$ are the same as for Fig. 6.

Figure 8

Calculated excitation spectrum of level $f$ as a function of $P$-laser Rabi frequency ${\Omega }_P$. The laser detunings ${\Delta }_P∕2\pi =-80\mathrm{MHz}$ and ${\Delta }_S∕2\pi =186\mathrm{MHz}$ correspond to the first minimum of Fig. 6 for ${\Omega }_P∕2\pi =339\mathrm{MHz}$. The lifetime ${\tau }_f=100\mathrm{ns}$ and the other parameters ($m_J$, ${\tau }_{\mathit{tr}}$, and ${\tau }_e$) are the same as for Fig. 6.