Raman transitions between hyperfine clock states in a magnetic trap

We present our experimental investigation of an optical Raman transition between the magnetic clock states of ${}^{87}\mathrm{Rb}$ in an atom chip magnetic trap. The transfer of atomic population is induced by a pair of diode lasers which couple the two clock states off-resonantly to an intermediate state manifold. This transition is subject to destructive interference of two excitation paths, which leads to a reduction of the effective two-photon Rabi frequency. Furthermore, we find that the transition frequency is highly sensitive to the intensity ratio of the diode lasers. Our results are well described in terms of light shifts in the multilevel structure of ${}^{87}\mathrm{Rb}$. The differential light shifts vanish at an optimal intensity ratio, which we observe as a narrowing of the transition linewidth. We also observe the temporal dynamics of the population transfer and find good agreement with a model based on the system's master equation and a Gaussian laser beam profile. Finally, we identify several sources of decoherence in our system, and discuss possible improvements.

Figure 1

(a) Saturated absorption spectrum in a Rb vapor cell as used for polarization spectroscopy. The hyperfine splitting of the ${}^{87}\mathrm{Rb}$ ground state is visible as frequency offset between the two outermost Doppler profiles. The master laser is locked to the $F=2\to F^{\prime }=3$ transition of ${}^{85}\mathrm{Rb}$, effectively introducing a detuning of $\approx 2.7\mathrm{GHz}$ for both master and slave laser to the hyperfine manifold of the $5p_{3/2}$ state of ${}^{87}\mathrm{Rb}$. (b) Zeeman manifold of the atomic states involved in the $D2$ line of ${}^{87}\mathrm{Rb}$. The ground states $|F=1,m_F=-1\rangle$ and $|F=2,m_F=1\rangle$ (marked in red) constitute the magnetic clock states at a magnetic field of $B=3.23\mathrm{G}$. The Raman laser pair has a difference frequency matching the ground state hyperfine splitting of $\sim 6.834\mathrm{GHz}$. With reference to the quantization axis, the laser beams have both ${\sigma }_{+}$ and ${\sigma }_{-}$ polarization components, introducing two possible transfer paths of atomic populations through $|F^{\prime }=1,m_{F^{\prime }}=0\rangle$ and $|F^{\prime }=2,m_{F^{\prime }}=0\rangle$ (red, dashed). In addition, coupling to several other states (marked in light gray) induces ac Stark shifts. (c) Sketch of the optical setup. The master and slave laser beams are overlapped, guided through an AOM for pulse generation, and coupled into an optical fiber. After the fiber, the laser beam is expanded by a lens and guided to the atom chip experiment. A lens focuses the laser beam to a $1/e^2$ radius of $100\mu \mathrm{m}$ at the atomic cloud, which is confined at $30\mu \mathrm{K}$ in a magnetic $z$-wire trap with the Ioffe-field pointing parallel to the chip surface. A change in the beam path leads to the two configurations labeled 1 and 2, where the first corresponds to a traveling wave and the latter to a standing wave induced by the reflection of the atom chip.

Figure 2

Optimal intensity ratio $I_M/I_S$ of master and slave laser as a function of detuning ${\mathrm{\Delta }}_3$ from $5p_{3/2},F=3$ for positive (a) and negative (b) detuning. The optimal ratio is defined by the intensity ratio at which the differential light shift $\mathrm{\Delta }E=\mathrm{\Delta }{E}_1-\mathrm{\Delta }{E}_0$ given by Eq. (3) vanishes.

Figure 3

(a) Raman spectrum obtained for the beam path 1 (copropagating) as depicted in Fig. 1. The curve shows the fit of a Lorentzian function with a FWHM of $8.8\mathrm{kHz}$. (b) Spectrum obtained for beam path 2 (counterpropagating), where the incoming and reflected laser beams partially overlap. The spectrum is clearly broadened as compared to (a), indicating Doppler broadening. The curve is based on a Gaussian function with FWHM of $321\mathrm{kHz}$. Note also the change in vertical scale for the two figures.

Figure 4

(a) Raman spectra taken for different intensity ratios $R=I_M/I_S$, which are shown next to the corresponding spectrum. A clear shift in frequency is visible, as well as a broadening effect for ratios different than $R=0.40$. The spectra are fitted to extract the center frequency. (b) The center frequencies are plotted against the intensity ratio. The shift in frequencies is clearly a linear function of the intensity ratio as expected from Eq. (3), with fit parameters ${\mathrm{\Delta }}_3=-2\pi \times 2781\mathrm{MHz}$ and $P_M=0.77\mu \mathrm{W}$.

Figure 5

(a): Relative atomic population transferred to $|1\rangle$ after a Raman excitation pulse of $5\mathrm{ms}$ taken for different intensity ratios $R=I_M/I_S$ (marked at the corresponding curve) at overall higher intensities compared to Fig. 4. For ratios different from $R=0.40$ we see an asymmetric broadening to either negative or positive frequencies for lower or higher ratios, respectively. (b) Simulated spectra for the same intensity ratios (marked at the corresponding curve), $P_M=25\mu \mathrm{W}$, and a $5\mathrm{ms}$ excitation pulse. The simulation is based on the dynamics of the effective two-level system $\left\{\right|0\rangle ,|1\rangle \}$ for a Gaussian intensity profile of the excitation beam. Here, $\delta =0$ refers to the frequency offset at the magic field and for vanishing differential light shifts.

Figure 6

(a) Total atomic population detected in the magnetic trap after a Raman pulse with varying pulse length. The blue (dashed, dotted) data set is measured without the Raman lasers and shows the typical fluctuations of detected atoms in the magnetic trap. The red (solid, dotted) data set is measured using $100\mu \mathrm{W}$ for the master and slave laser power, and shows a decay in the amount of detected atoms with increasing pulse time. This loss results from atoms being optically pumped into nontrappable states. The red (dashed) line shows a theoretical fit of $\mathrm{Tr}(\rho$) for ${\gamma }_{\text{loss}}=2\pi \times 164\mathrm{Hz}$. (b) The blue (dotted) line shows the atomic population in the $|1\rangle$ state for varying pulse length of the Raman lasers at the optimal power ratio. The red (dashed) line is a simulation assuming a master laser power of $4\mu \mathrm{W}$.

Figure 7

Simulated temporal evolution of the atomic population in $|1\rangle$ assuming a uniform intensity distribution of the Raman laser beams at $P_M=35\mu \mathrm{W}$. The simulation is performed for different intensity ratios, expressed as $\{1.00,1.01,1.02,1.05\}$ times the optimal ratio $R$. It reveals that even a small deviation from the optimal ratio leads to a significant reduction of the Rabi flopping between $|0\rangle$ and $|1\rangle$. As a comparison, the total remaining atomic population is shown (upper line).