Spectroscopy of the hydrogen $1S-3S$ transition with chirped laser pulses

We identify a systematic present in two-photon direct frequency comb spectroscopy (DFCS) which is a result of chirped laser pulses and is a manifestation of the first-order Doppler effect. We carefully analyze this systematic and propose methods for its mitigation within the context of our measurement of the hydrogen $1S-3S$ transition. We also report on our determination of the absolute frequency of this transition, which is comparable to a previous measurement using continuous-wave spectroscopy [O. Arnoult et al., Eur. Phys. J. D 60, 243 (2010)], but was obtained with a different experimental method.

Figure 1

Diagram showing the pulse collision volume (PCV) resulting from colliding pulse trains (frequency combs). Also shown in the diagram are the lengths $w$ and $c\tau$ along with a representative atomic trajectory $\vec{r}\left(t\right)$.

Figure 2

Experimental setup: A mode-locked 820-nm Ti:sapphire laser is frequency quadrupled to 205 nm using two resonant doubling stages. This radiation is then sent to an enhancement cavity where spectroscopy of atomic hydrogen is performed. A delay stage external to the spectroscopy cavity is used to create counterpropagating pulses inside the cavity. pd: photodiode, fc: fiber coupler, $\lambda /4$: quarter-wave plate, $\lambda /2$: half-wave plate.

Figure 3

Hydrogen nozzle and detector. Atomic hydrogen enters the nozzle and exits along the laser beam path. Four aspheric lenses (only two are shown in the figure) with focal lengths of 2.76 mm focus the fluorescence onto multimode optical fibers (1-mm-diameter core, 0.48 NA). Copper grids coated in colloidal graphite surround the pulse collision volume on four sides. By controlling the voltage on two of these grids ($V_x$ and $V_y$) along with the voltage applied to a wire ring ($V_z$), we can adjust the electric field at the pulse collision volume. Four multimode fibers also collect fluorescence on the back side of the nozzle. In that location—away from the pulse collision volume—only the Doppler-broadened signal is present, which is used for normalization. This normalization is required due to fluctuating laser power and atom density. Because there is no need to prevent dc stark shifts on the normalization signal, we forego the electrodes.

Figure 4

Diagram showing the relevant levels and two-photon transitions in hydrogen. The hyperfine splitting of the $3D$ states is unresolved due to the 10.3-MHz natural linewidth of the $3D$ states. The $3P_{3/2}$ and $3D_{3/2}$ states are easily mixed with electric fields due to their small energy difference.

Figure 5

Experimental fluorescence signal as the frequency offset between one frequency comb mode of the Ti:sapphire laser and the stabilized ECDL is scanned. (The solid line represents Lorentzian fits for the individual lines.) The $F$ quantum number labels correspond to the $1S$ state. For the transitions to the $3S$ states, only $\mathrm{\Delta }F=0$ transitions are allowed. The repetition rate of the laser was chosen so that the $1S-3S F=1$ transition is well isolated.

Figure 6

Measured frequency $f$ of the observed $1S-3S F=1$ line center as a function of detector position. The observed resonance was determined by fitting the experimental data with a Lorentzian line shape. Data points of a certain color were all taken on the same day and the change in slope from day to day is due to a variation of the pulse chirp $b$. From our simulations, the observed slopes correspond to a variation of $b$ from $\sim 0.15$ to $\sim 0.7$. The wide stripes represent pointwise 68% confidence bands for a simple linear regression analysis of 1 day of data. The crossing of these confidence bands gives a confidence region for the experimental coordinates of the crossing point on the $(z,f)$ plane. The 68%, 95%, and 99.7% confidence regions for this point are shown on the upper-right plot, together with projections of the resulting likelihood function on the $z$ and $f$ axes. Data points at large detector misalignment ($>1$ mm) lie in a region where we do not expect a linear variation with detector position and so these data points are excluded in the data analysis.

Figure 7

Simulated frequency shifts using the Monte Carlo method for different values of the chirp parameter $b$ as a function of detector offset from the center of the pulse collision volume (PCV). In addition to the CIFODS, the simulations include the second-order Doppler shift. The gray shaded region shows the relative position and size of the pulse collision volume. The position where the curves intersect is insensitive to the chirp parameter $b$, and the remaining frequency shift is due to the second-order Doppler effect.