Raman transitions driven by phase-modulated light in a cavity atom interferometer

Atom interferometers in optical cavities benefit from strong laser intensities and high-quality wave fronts. The laser frequency pairs that are needed for driving Raman transitions (often generated by phase modulating a monochromatic beam) form multiple standing waves in the cavity, resulting in a periodic spatial variation of the properties of the atom-light interaction along the cavity axis. Here, we model this spatial dependence and calculate two-photon Rabi frequencies and AC Stark shifts. We compare the model to measurements performed with varying cavity and pulse parameters such as cavity offset from the carrier frequency and the longitudinal position of the atom cloud. We show how setting cavity parameters to optimal values can increase the Raman transition efficiency at all positions in the cavity and nearly double the contrast in a Mach-Zehnder cavity atom interferometer in comparison to the unoptimized case.

Figure 1

(a) Cesium energy levels driven by the frequency components of a phase-modulated laser beam. Raman transitions are driven between the hyperfine ground states $|F=3\rangle$ and $|F=4\rangle$ via the excited state $|e\rangle$. Multiple pairs of frequency components, separated by ${\omega }_{\mathrm{mod}}$, simultaneously satisfy the Raman resonance. The carrier frequency, ${\omega }_L={\omega }_{m=0}$, is red-detuned by $\mathrm{\Delta }\approx 2\pi \times 49$ GHz from the $|F=3\rangle$ to $|e\rangle$ transition. (b) Spectrum of incident phase-modulated light with center frequency ${\omega }_{\text{L}}$. The output of the fiber EOM is coupled into a vertically oriented optical cavity with length $L=37.5$ cm and FSR $\mathrm{\Delta }{\nu }_{\text{FSR}}=399.845$ MHz. The gravitational acceleration is denoted by $\overline{\mathrm{g}}$. (c) Multichromatic light, depicted with Lorentzians colored according to their detuning from the carrier frequency (red or blue detuned), is resonant with multiple longitudinal modes of the optical cavity, depicted with gray shaded Lorentzians. Since the linewidth of the cavity, $\gamma =2\pi \times 3.03$ MHz, is much smaller than ${\omega }_{\text{mod}}\approx {\mathrm{\Delta }}_{\text{hfs}}^{\text{Cs}}=2\pi \times 9.2$ GHz necessary to drive Raman transitions in cesium, the first pair of sidebands is resonant with the cavity 23 FSR away from the carrier. The detuning between the first-order sidebands when ${\omega }_{\text{mod}}={\mathrm{\Delta }}_{\text{hfs}}^{\text{Cs}}$ and the cavity resonance 23 FSR away is defined as ${\delta }_{\text{offset}}^{\text{hf}}$.

Figure 2

(a) Simulation of the Rabi frequency ${\mathrm{\Omega }}_R$ vs ${\delta }_{\text{cav}}$ and time of flight. The colored lines correspond to the data presented in (b), (c), and (d). (b) On the left axis, Rabi frequency vs time of flight (solid purple line). The spatial beat note of the Raman beam pairs can be seen, as well as a general downward trend as ${\omega }_{\text{mod}}$ is linearly ramped to account for the Doppler shift, moving the sidebands out of the cavity line shape, resulting in lower light intensity in the cavity. On the right axis, the atom cloud position is plotted as a function of time of flight (dashed light blue line). The cloud passes the same Rabi dead zone twice. (c, d) Measured Rabi frequency as a function of cavity offset compared to simulation, near a (c) maximum ($t=2.0$ ms) and (d) minimum ($t=16.5$ ms) of the ${\delta }_{\text{cav}}$ = 0 MHz spatial beat note, respectively.

Figure 3

(a) Pulse sequence of the in situ AC Stark shift measurement using a microwave Ramsey interferometer. (b) Ramsey fringes measured for different pulse amplitudes. The frequency of the microwave $\pi /2$ pulses is scanned by an amount ${\delta }_{\text{Ramsey}}$ to obtain interference fringes which accrue phase at a rate of $2\pi /T$, as observed in the population oscillations of atoms between states $|F=3\rangle$ and $|F=4\rangle$. (c) Phase shift, in radians, as a function of pulse amplitude $A_{\text{p}}$ in arbitrary units. The AC Stark phase shift, measured as the phase offset of fringes in (b), increases linearly with the pulse amplitude at a rate determined by the duration of the AC Stark pulse $\tau$ and the AC Stark energy-level shift $\mathrm{\Delta }{\varphi }_{\text{ac}}={\delta }_{\text{ac}}\tau$. (d) AC Stark shift measured as a function of cavity offset ${\delta }_{\text{cav}}$ for two different modulation depths $\beta$. Different modulation frequencies were used as well, as seen from the sideband peak locations (${\omega }_{\text{mod}}={\mathrm{\Delta }}_{\text{hfs}}^{\text{Cs}}-2\pi \times 1.500\left[1.000\right]$ MHz for $\beta =1.08\left[1.55\right]$).

Figure 4

(a) Schematic of the Mach-Zehnder atom interferometer with $T=10$ ms. Atoms fall along the cavity axis ($g$ denotes the direction of gravity), and $T$ was chosen such that the beamsplitter pulses occur at different extrema along the spatial Rabi beat note [see Fig. 2]. (b) Theoretical AC Stark phase shift ${\phi }^{\text{ac}}=\frac{\pi }{2}\frac{{\delta }_{\text{ac}}}{{\mathrm{\Omega }}_R}$ of a $\pi /2$ pulse as a function of cavity offset and time of flight. The first $\pi /2$ pulse at 20 ms is near the Rabi dead zone for ${\delta }_{\text{cav}}=0$, causing it to induce a large phase shift as the pulse is long. However, by changing ${\delta }_{\text{cav}}$ for all pulses it is possible to reduce ${\phi }_1^{\text{ac}}$ to match ${\phi }_3^{\text{ac}}$ more efficiently. (c) Interferometer contrast vs cavity offset ${\delta }_{\text{cav}}$. The contrast measurements are plotted as red circles, while the solid cyan line represents the calculated residual AC Stark shift phase from the first and third beamsplitter pulses $|{\phi }_1^{\text{ac}}-{\phi }_3^{\text{ac}}|$. The normalized cavity line shape $s{\left({\delta }_{\text{cav}}\right)}^2$ is plotted for ${\delta }_{\text{cav}}=0$ (dashed purple line). We observe that the contrast decreases significantly at ${\delta }_{\text{cav}}\approx 0$, where $|{\phi }_1^{\text{ac}}-{\phi }_3^{\text{ac}}|$ is large.

Figure 5

Fitting the Rabi floppings for different time of flights (TOFs). (a), (b), and (d) are at spatial positions with medium to high Rabi frequencies, while (c) is in the dead zone. For such a low Rabi frequency, decoherence across the atomic cloud leads to a very low peak excitation fraction (note the different $y$ scaling on the plots).