High-intensity two-frequency photoassociation spectroscopy of a weakly bound molecular state: Theory and experiment

We investigate two-frequency photoassociation of a weakly bound molecular state, focusing on a regime where the AC Stark shift is comparable to the halo-state energy. In this “high-intensity” regime, we observe features absent in low-intensity two-frequency photoassociation. We experimentally measure the spectra of ${}^{86}\mathrm{Sr}$ atoms coupled to the least bound state of the ${}^{86}\mathrm{Sr}_2$ ground electronic channel through an intermediate electronically excited molecular state. We compare the spectra to a simple three-level model that includes a two-frequency drive on each leg of the transition. With numerical solution of the time-dependent Schrödinger equation, we show that this model accurately captures (1) the existence of experimentally observed satellite peaks that arise from nonlinear processes, (2) the locations of the two-photon peak in the spectrum, including AC Stark shifts, and (3) in some cases, spectral line shapes. To better understand these numerical results, we develop an approximate treatment of this model, based on Floquet and perturbation theory, that gives simple formulas that accurately capture the halo-state energies. We expect these expressions to be valuable tools to analyze and guide future two-frequency photoassociation experiments.

Figure 1

(a) A schematic of the two-frequency PA process. $E_1$ is the intermediate state energy, ${\mathrm{\Delta }}_1$ is the detuning of laser ${\omega }_A$ from the $0\to 1$ transition, and ${\mathrm{\Delta }}_2$ is the two-photon detuning with respect to the $0\to 2$ transition. (b) Possible resonance processes at different PA laser frequencies, ${\omega }_A$ and ${\omega }_B$, that correspond to different peaks observed in experimental PA spectra, as labeled in (c). The second-order peaks only become possible if we assume both lasers drive each leg of the transition. (c)–(f) Atom loss spectrum obtained from PA experiment (circles) and numerical solutions of the three-level model (solid line) as a function of PA laser frequency difference, $\mathrm{\Delta }\omega ={\omega }_B-{\omega }_A$, (c) with detuning ${\mathrm{\Delta }}_1/2\pi =-1.5\mathrm{MHz}$ and intensity $I=0.66\mathrm{W}{\mathrm{cm}}^{-2}$, (d) with detuning ${\mathrm{\Delta }}_1/2\pi =-1.5\mathrm{MHz}$ and intensity $I=0.17\mathrm{W}{\mathrm{cm}}^{-2}$, (e) with detuning ${\mathrm{\Delta }}_1/2\pi =-3\mathrm{MHz}$ and intensity $I=0.66\mathrm{W}{\mathrm{cm}}^{-2}$, and (f) with detuning ${\mathrm{\Delta }}_1/2\pi =-3\mathrm{MHz}$ and intensity $I=0.18\mathrm{W}{\mathrm{cm}}^{-2}$. Note that the value of $k$ [Eq. (4)] used in the three-level model is different for each spectrum. Trap parameters, densities, and other experimental conditions are provided in Ref. [10].

Figure 2

(a) Difference in laser frequency, $\mathrm{\Delta }\omega$, at peak atom loss, which we identify as the energy difference between state $|0\rangle$ and $|2\rangle$. This corresponds to the energy (unperturbed energy $+$ AC Stark shift) of the halo ground-state molecule. The frequency at peak loss as a function of intensity is obtained from numerical simulation of the three-level model, given by Eq. (2), and the PA experiment at various detunings, ${\mathrm{\Delta }}_1$. We obtain a global $k=850\mathrm{kHz}{(\mathrm{W}/{\mathrm{cm}}^2)}^{-1/2}$ by fitting the ${\mathrm{\Delta }}_1/2\pi =-1.5\mathrm{MHz}$ data points. Note that the large detuning data is not used to determine $k$, but it contributes to the analysis in Fig. 3. (b) Same plot of the ${\mathrm{\Delta }}_1/2\pi =-1.5\mathrm{MHz}$ data as for (a), except that higher-intensity data are included, for which the three-level model predictions deviate from experimental results. Quadratic fits are plotted to guide the eye.

Figure 3

Plot of susceptibility as a function of detuning for low-intensity data (for which light shift varies almost linearly). Note that in addition to the data from Fig. 2, data for detunings above $6\mathrm{MHz}$ and below $-9\mathrm{MHz}$ are included. The data is fitted with the susceptibility predicted from the Floquet treatment of the four-level model [first derivative of Eq. (13) with respect to intensity in the low-intensity limit], with the additional state having an energy of approximately $E_1+2\pi \times 44\mathrm{MHz}$. Similarly, the susceptibility obtained from the Floquet treatment of the three-level model, given by Eq. (11), is also presented.

Figure 4

Binding energy as a function of intensity obtained from the full simulation, given by Eq. (2), and analytic solution of Floquet theory, given by Eq. (11), at various detuning for the three-level model.

Figure 5

Energy levels and couplings of the four-level model. State X is added to the three-level model and it couples to both state 0 and state 2, with laser A detuning from the $0\to X$ transition defined as ${\mathrm{\Delta }}_X={\omega }_A-E_X$.

Figure 6

PA spectrum obtained from experiment at ${\mathrm{\Delta }}_1/2\pi =-1.5\mathrm{MHz}$ and the three-level model fits at intensities (a) $I=0.66\mathrm{W}{\mathrm{cm}}^{-2}$ and (b) $I=0.17\mathrm{W}{\mathrm{cm}}^{-2}$ (by varying $x$ and ${\mathrm{\Gamma }}_1$). $k$ is adjusted to match the model and experimental binding energies.

Figure 7

Plot of (a) FWHM and (b) area of main negative peak at various intensities for experimental data (blue circles) and the three-level model with parameters $k=850\mathrm{kHz}{(\mathrm{W}/{\mathrm{cm}}^2)}^{-1/2}, x=37.5, {\mathrm{\Gamma }}_1=1.6\times {10}^6{\mathrm{s}}^{-1}$, and ${\mathrm{\Gamma }}_2=0{\mathrm{s}}^{-1}$ (red triangles). The line in (a) is a linear fit to the experimental data. In (b), the line serves to guide the eye through the experimental data.