1. Introduction

Stimulated Raman transitions that couple atomic ground states with counterpropagating laser beams are resonant only within a narrow velocity band. This atomic velocity selection [1] has proven to be a useful tool for a variety of experiments, including subrecoil Raman cooling [2], atom interferometry [3], atom velocimetry [4], and magnetometry [5, 6].

The population distribution among atomic magnetic sublevels is generally measured using stimulated Raman spectroscopy. In this technique, the frequency difference between two copropagating, phase-locked lasers is scanned in a magnetic field [7, 8], and the number of atoms excited as a function of frequency difference is monitored by hyperfine-state-selective imaging. Knowledge of the sublevel distribution is important for determining, e.g., the efficiency of optical pumping to a dark state or the degree of spin polarization of the sample. Because Raman spectroscopy relies on coherent transitions, the experimental overhead can be high, given the requirement of phase-locked lasers with controllable frequency difference. Furthermore, each point in the sublevel spectrum is obtained by scanning the Raman frequency difference, which makes the data collection intrinsically serial.

In this paper, we describe a single-shot imaging technique for rapid assessment of the magnetic sublevel distributions. The technique relies on velocity-selective Raman transitions, but no scanning of the relative laser frequencies is required. Instead, a range of relative frequency differences is provided simultaneously by using Doppler shifts in a thermally broad sample, and each magnetic sublevel is resonant in a different velocity class. As will be shown, the resonances are easily identified by imaging the atom cloud after ballistic expansion disperses these velocity classes.

2. Background

When an atom cloud is imaged after expansion, the spatial profile is a map of the average velocity distribution of the cloud. Thus, a velocity-dependent process that occurs during the expansion can be observed at a corresponding location in the expansion image. In previous work [5, 6], velocity-dependent transitions within a single hyperfine manifold were used to measure local magnetic fields.

In this work, the transitions occur between magnetic sublevels of two hyperfine manifolds, allowing discrimination of transitions between specific magnetic sublevels. In a magnetic field, the Zeeman shifts of the two hyperfine manifolds are opposite in sign, so that the two-photon transition frequencies vary depending on the magnetic sublevels involved. We record images using state-selective illumination so that fluorescence only occurs at locations where transitions occurred; each of these locations corresponds to a different magnetic sublevel, providing an image of the sublevel distribution. The process is briefly outlined below.

The lab-frame energy level diagram for velocity-selective resonances (VSR) is shown in Fig. 1. An atom in the lower hyperfine level, F, with velocity v along the x-axis is exposed to a light field composed of two counterpropagating laser beams with frequencies ω ₁ and ω ₂ such that ω ₁=ω ₂+ω_HF, where ω_HF is the hyperfine splitting. These beams have corresponding wavevectors k ₁=2π/ω ₁ x̂ and k₂=−2π/ω₂x̂, and the polarization configuration is lin ⊥ lin. The one-photon detuning, Δ≫ω_HF. For an arbitrary B-field, this polarization configuration couples m_f −m_i=Δm=0,±1,±2 magnetic sublevels, where we choose our quantization axis along B. The initial energy of an atom in this field is:

where E^LS_i is the light shift, g_F is the gyromagnetic ratio, µ_B is the Bohr magneton, B is the scalar magnetic field, p_i is the initial momentum, and M is the mass. Raman transitions to F+1, i.e., photon absorption from the higher frequency beam and emission into the lower frequency beam, results in a linear momentum change of h̄(k₁−k₂)≈2h̄k=Mυ_rx̂, where υ_r is the recoil velocity h̄k/M. The final energy E_f is then

The Raman transitions occur between two momentum states with average momentum pavg=p_i+h̄k. With this substitution, and setting E_i=E_f, we obtain

The light shifts are negligible compared to the Zeeman shifts so we can neglect them for this technique. Then, with p=Mv, a resonant transition occurs for atoms with v=v_avg±υ_rx̂ such that

From the atom’s center-of-mass frame, this is equivalent to the requirement that the Doppler shift offset the relative Zeeman energy for that transition. Also note that since gF≈−gF+1, the resonant velocities are uniquely determined by the sum m_i+m_f ; with that approximation, the relationship becomes simply

with ω_L=gFu_BB/h̄. Thus, in the limit that all atoms derive from a point source, these transitions will appear at locations defined by

with T_i the imaging time after release of the cloud. In Fig.1b, we show the possible pathways for Δm=0,±1 transitions. For Δm=0, there are 2F+1 unique energy splittings. Both Δm=+1 and Δm=−1 also have 2F+1 transition pathways, but only 2F+2 of these are unique, for a total of 4F+3 different energy splittings. Note that because the positions of resonances are proportional to m_i+m_f, Δm=±2 transitions would appear at the same locations as Δm=0 transitions (e.g. an m_i=1 to m_f=3 transition would appear at the same location as an m_i=2 to m_f=2 transition). As we will discuss later, however, the Δm=2 transitions are generally weak for large Δ and are not shown.

 

Fig. 1. a) Relevant energy levels for the Raman transitions. Magnetic sublevels are split by the Larmor precession frequency, ω_L. The polarization configuration is lin⊥lin. The energy scale is exaggerated for clarity. The detuning, Δ, is measured from the D2 line. b) The possible transitions for Δm=0 (black), 1 (red), and -1 (blue). As discussed in the text, the ±2 orders are weak. Numerical values are shown for ⁸⁵ Rb.

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Fig. 2. Experimental setup for observing velocity-selective resonances in a magnetic field.

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Fig. 3. Two techniques for phase-locking. (a) uses a triple-passed AOM. The deflection caused by the AOM occurs out of the plane of the figure, while the multiple-passing occurs in the plane as drawn. (b) uses an EOM to impose two sidebands. Only one of the sidebands is resonant by making the sidebands asymmetric.

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The Raman pulse initiates coherent oscillations between the lower and upper hyperfine levels. The two-photon detuning is velocity dependent, so that atoms satisfying Eq. 5, with no two-photon detuning, exhibit stronger and slower oscillations than those at slightly higher or lower speeds along x̂. After the oscillations dephase, the steady-state signal can be used as a measure of atom number. As we will show, the steady-state population in the upper hyperfine level can be imaged using cycling transition light, and transitions among each of the 4F+3 different pathways will be recorded at different locations according to Eq. 6 to provide an image of the magnetic sublevel distribution.

3. Experimental Setup

The experimental setup and coordinate system used is shown in Fig. 2. The atom sample is a vapor cell MOT containing 10^{7 85}Rb or ⁸⁷Rb atoms (we demonstrate the technique with both atomic species). Raman beams travel along x̂, the camera viewing axis is along ŷ, and gravity is along −ẑ. Helmholtz coils for magnetic field bias adjustment are also along cartesian axes.

 

Fig. 4. Images of the expanded atom cloud as a function of angle between the Raman beam direction and the magnetic field for both ⁸⁷Rb and ⁸⁵Rb. We show cross-sections for ⁸⁷Rb to indicate the labeling scheme. Odd numbers are derived from Δm=±1 transitions; Even numbers are from Δm=0 transitions. Arrows in the ⁸⁷Rb image mark the locations of the 1/e points of the Gaussian cloud. Images and plots have been normalized to this distribution.

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The Raman beams are derived from the same single-frequency laser source (New Focus StableWave). The hyperfine frequency shift was done in two different ways, shown in Fig. 3. In the first method, a single-frequency beam is amplified to 500 mW by a tapered ampliflier, 250 mW of which is coupled into polarization maintaining fiber. We use 50–100 mW for one of the Raman beams, while the remaining 150–200 mW is triple-passed in a high-frequency AOM with 25% diffraction efficiency. The frequency-shifted light is amplified by injection seeding a laser diode, from which ≈ 25–50 mW is used as the other Raman beam.

The preceding technique is complicated due to the AOM alignment and injection seeding, but has the benefit of producing two spatially-separated, independently-controlled beams. A second, simpler technique is also used, as shown in Fig. 3b. In this case, part of the input beam is shifted by an amount ν_AOM, while the rest passes through an electro-optic modulator (EOM) running at ν_HF+νAOM. The EOM creates multiple sidebands, in addition to the carrier frequency, but only one has the necessary frequency difference of ν_HF to induce transition between hyperfine levels [9]. Both beams are further amplified by independent tapered amplifiers and coupled into PM fibers. The benefit of this technique is that it is easily reconfigured for any atomic species. Because of possible unwanted interactions by the non-resonant sidebands, its use was compared to the AOM technique and no difference was observed. Therefore, all results reported here use the EOM.

The Raman beams are collimated to 1/e ² beam waists of ≈4.5mm, and we typically use Δ=25-50 GHz to eliminate spontaneous scattering during the (≈0.5 ms) Raman pulse. The pulse is applied to a freely expanding atom cloud ≈15 ms after shutting off the MOT trapping fields. No polarization gradient cooling was applied, and the sample has a temperature of ≈100µK. Atoms are initialized in the lower hyperfine state either by extinguishing the repump light prior to the cooling light, or explicitly through optically pumping to a desired sublevel in the lower F level. The expanded atom cloud is imaged with MOT cooling light ≈30 ms after releasing the MOT so that only those atoms excited to F+1 are observed in the fluorescence images.

4. Results

In general, in the lin⊥lin configuration, the Raman beams couple strongly only to Δm=0,±1, while the Δm=±2 transitions are weak. We previously used this fact to compensate and image magnetic fields [5, 6], but in the present technique, the 4F+3 possible transitions result in images with many more features. To best understand the type of images and transition pathways that are observed, we begin with results obtained for an atom cloud in a uniform magnetic field as a function of B·k. B is varied in 10 degree steps from B⊥ k to B‖ k as B=B ₀[sin(θ)ẑ+cos(θ)x̂], where B ₀ is ≈300 mG. The results are shown in Fig. 4 for unpolarized samples of both ⁸⁷Rb and ⁸⁵Rb, with T_i=20ms. Cross sections of the data are also shown for 87Rb. Because the velocity distribution is approximately Gaussian, the resonances in different velocity classes appear with Gaussian-weighted strengths. To account for this, all results in this paper are normalized by the velocity distribution, measured by a separate fluorescence image.

 

Fig. 5. a) Cross-section amplitude as a function of Raman pulse duration. Fits are to a phenomenological model to extract the resonant two-photon Rabi frequency. The steady-state values are different because these measurements were not made on an unpolarized sample. b) Calculated results for our pulse parameters. The relative oscillation frequencies match the experiment very well. Oscillations in the experiment dephase more quickly due to time-varying magnetic fields and nonuniform laser intensity profiles.

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The features for each B-field orientation indicate the allowed transition pathways, and are separated by an amount proportional to the scalar magnetic field, in accordance with Eqs. 5–6. We label each feature according to the sum m_i +m_f, so that odd labels identify Δm=±1 transitions and even labels identify Δm=0 transitions. When B⊥k, only Δm=±1 are allowed, as one Raman beam is polarized along B, coupling only Δm=0, and the other is perpendicular to B, coupling Δm=±1. With this selection rule, 4 (6) features are expected for ⁸⁷Rb (⁸⁵Rb) (see Fig. 1b).

When B ‖ k, only transitions between Δm=0 are observed, so that the label identifies 2m_i. In the B-axis basis, each Raman beam now contains both σ ⁺ and σ ⁻ so that Δm=0,±2 are allowed. Our calculations show that Δm=±2 are weak enough that these transitions are not observed with our pulse areas. Given the Δm=0 selectivity, each of the 3 (5) possible features correspond to specific magnetic sublevels of ⁸⁷Rb (⁸⁵Rb).

For an arbitrary B-field orientation, both sets of transitions are allowed. As observed in Fig. 4 the relative strengths of the Δm=0 transitions to the Δm=±1 transitions varies according to the angle between B and k, but further analysis in an arbitrary magnetic field is outside the scope of this technique, and will be done elsewhere.

5. Measurement of sublevel occupation

As shown above, when B ‖ k the only transitions are Δm=0. In this section, we examine this more closely. The amplitude of these features is related both to the number of atoms present and the strength of the transition. In the limit in which the Raman pulse duration is less than a π-pulse, the feature strengths are due to both the atom number and the transition strength. Conversely, if the Raman pulse duration is larger than the dephasing time of the Rabi oscillations, the feature strengths are due primarily to atom numbers.

Because of the expected dependence of the signals on the interaction time, we have studied the relative feature strengths as a function of Raman pulse duration. Fig. 5 shows typical results for levels in ⁸⁵Rb. We plot the signal amplitudes for m_i=0,1, 2. (We note that these were taken under different degrees of spin polarization to maximize the signal for the sublevel of interest, so they reach different steady state values). For our Raman beam intensities, the signals undergo Rabi oscillations that are damped within 2–3 cycles. After 200 µs, the signals are constant within the error of the measurement. The error bars of ≈10% are due to both statistical and systematic errors over 10 averages, but systematic errors dominate and are primarily due to technical issues in the CCD camera. We determine approximate Rabi frequencies by fitting these data to a phenomenological, sinusoidally-modulated exponential model. For the detunings and powers used in that plot, the Rabi frequencies for mi=0, 1, and 2 are 2π×16.7(6), 15.8(4), and 12.6(3) kHz, which corresponds to a ratio of 1:0.95:0.75. The calculated resonant Rabi frequencies for these three Δm=0 pathways m_i=0, 1, and 2 are 2π×12.6, 12.0, and 9.4 kHz, corresponding to relative frequencies of 1:0.945:0.747, in excellent agreement with the experiment. The calculated Δm=±2 transitions are 2–3 orders of magnitude smaller and also never appear in any experimental results. The absolute Rabi frequencies differ between theory and experiment due to errors in estimates of the Raman beam intensities. The observed damping likely comes from time-varying magnetic fields, Raman laser beam spatial variations, and the distribution of two-photon detunings that contribute in the velocity selective process. We note that a feature of this technique is that inhomogeneities arising from spatial variations are recorded by the imaging process, so that in principle one can record Rabi frequencies using small regions of the image over which the spatial variations are constant.

The steady-state signals can be used for estimates of the atom number. The proportionality between the steady state signals and the actual atom numbers can always be calibrated against either a known sublevel distribution (e.g. a spin-unpolarized sample) or against traditional Raman spectroscopy. Time-varying fields may skew the proportionalities by changing the resonant velocity classes during the pulse, but these variations are often tied to experimental procedures (e.g. eddy currents from extinguishing magnetic fields) and thus would produce repeatable results that can be calibrated. In our apparatus, the field variation is ≈1mG/ms at the time of the Raman pulse.

We have also modeled the coherent evolution of the atoms in the presence of the counterpropagating laser field to compare with our experimental results. The theoretical results are shown in Fig. 5b for an unpolarized sample of ⁸⁵Rb in a field with our laser parameters. Excitations to the upper hyperfine sublevels were integrated over a 6.6 mm/s velocity width centered on the two-photon resonance. The subsequent range of two-photon detunings introduces the damping observed in the theory, which does not include experimental effects of magnetic field variations and beam inhomogeneities. As mentioned above, the relative oscillation frequency is in excellent agreement with the experimental results. The calculations predict contributions from Δm=±2 transitions that are 2–3 orders of magnitude below those of the Δm=0,±1 transitions, due to canceling Clebsch-Gordon coefficients. This cancellation causes the Δm=±2 transition strengths to scale with 1/Δ2 instead of the 1/Δ scaling calculated for the Δm=0,±1 transitions. The calculated results are for an unpolarized sample, so the fraction of the total atom number that is excited in any velocity class is relatively small. In a completely unpolarized sample, only 20% of the atoms will be resonant, and in steady state only half will be in the upper hyperfine level. Thus we are able to make these Raman spectrographs in a single shot at the expense of the total effective number of atoms participating.

For further verification of the technique, we have compared it with traditional Raman spectroscopy, in which the Raman beams copropagate [7, 8]. This makes the transitions velocity independent, but to record a sublevel spectrum, the relative detuning of the Raman beams must be scanned and the fluorescence at each detuning recorded separately. To go from the counterpropagating to the copropagating measurement is simple in our setup because the beams are fiber-delivered. The copropagating beams are combined on a polarizing beamsplitter so that the polarizations of the two beams are linear and orthogonal. All other aspects (timings, laser powers, etc) of the experiment remain unchanged.

 

Fig. 6. Sublevel spectra recorded using traditional Raman spectroscopy (red circles) with the VSR technique (blue line). Fits to Gaussian profiles are shown as black solid lines. The widths of the Raman transitions for m≠0 are larger than those for m=0 due to time-varying magnetic fields. The integrated areas under the curves agree very well (see text).

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To record the sublevel spectrum, we scan the relative detuning in 4 kHz steps over each of the m_i=m_f transitions in a field of 260 mG for ⁸⁵Rb. The results are shown in Fig. 6 for an arbitrarily-chosen spin-polarization. The widths of the peaks in the copropagating Raman technique depend on m_i because of time-varying magnetic fields in our chamber. To properly compare the two techniques, we integrate the areas under the curves. For the copropagating Raman case, the m=0 : 1 : 2 peaks are in the ratio 0.31:0.56:1, while for the VSR technique, they are in the ratio 0.27:0.57:1, so the agreement is good. In Fig. 6, the x-axis of the VSR technique is plotted both in units of the pixel value on the camera, and in frequency units according to Eq. 6.

6. Optical Pumping

Because the technique can directly image the magnetic sublevel occupation, it can be used, for example, to determine how well a sample has been optically pumped. We demonstrate this use in both ⁸⁵Rb and ⁸⁷Rb in Fig. 7. The images on the left-hand-side are montages of several different optical pumping pulse durations. In Fig. 7a, the ⁸⁷Rb sample was exposed to a beam connecting F=2→F′=2. A repump beam, connecting F=1→F′=1 was also present, with linear polarization along B. This scheme makes F=1,m_F=0 the dark state. Each of the ≈20 bands represents a 10µs increase in repump pulse duration. A similar pumping scheme was used for ⁸⁵Rb in Fig. 7b. Finally, we demonstrate optical pumping to the stretched state |F=2,m_F=−2〉 using σ- polarized repump light. These optically-pumped samples also indicate that Δm=±2 transitions are very small as predicted above in our calculations. For instance, in Fig. 7a, peaks m=±1 would be present even after optical pumping, through transitions from m_i=0 to m_f=±2, but the small signal at those locations is most likely due to incomplete optical pumping. All results have been normalized to the velocity distribution of the atom cloud, which is approximately Gaussian along each cartesian direction. Without this normalization, the m_i=0 transitions would appear stronger than the m_i=m_F transitions, which can require velocities in the tails of the velocity distribution. We note that this technique assumes that there is no initial velocity dependence to the spin polarization, which is the case for single-photon optical pumping, but if it is combined with other velocity-selective processes, care must be taken in its implementation.

 

Fig. 7. Evolution of VSR images as a function of optical pumping pulse duration (left) and cross-sections (right) for three different cases. The images are montages of increasing pulse duration from 0 to 200 µs pulses. These are fluorescence images, shown in inverse palette for visibility. a) ⁸⁷Rb pumped to m=0; b) ⁸⁵Rb pumped to m=0; c) ⁸⁵Rb pumped to m=−2.

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7. Further Discussion

We note a few other practical aspects to consider when using this technique. First, the width of each peak in these spectrographs is a convolution of the initial MOT size with the resonance width. Typically, this means that the widths will be dominated by the physical size of the initial MOT cloud, so that a small, dense sample is favored over a larger one. The sample should expand so that the resonant velocity classes can be clearly resolved, and ideally a bias magnetic field should be chosen such that the resonant velocities remain within the velocity distribution of the atom cloud. For species with high F, such as cesium, which would have 4F+3=15 transitions in an arbitrary magnetic field, it may be difficult to cleanly separate all transitions simultaneously, but the spectrograph can still be measured by recording two images in which the relative frequencies of the Raman beams have been shifted. This moves extremal resonant velocities back within the velocity distribution of the atom cloud. A similar technique was described and demonstrated in Ref. [5] to calibrate an imaging magnetometer. This extension may also be required for very cold samples, the Doppler width of which may not be large enough to accommodate all possible transitions.

Second, the primary quantity directly measured by this technique is the number of atoms promoted to the upper hyperfine state as a function of atomic velocity. Because the features are well resolved, it is simple to associate each peak with a particular magnetic sublevel or transition. Therefore, the exact locations are relatively unimportant to the measurement. In principle, it is possible to measure other quantities with this technique, such as the two-photon Rabi frequency (as was shown in Fig. 5) or AC Stark shifts, but these values do not affect the measurement of sublevel occupation. This greatly simplifies the interpretation of the results and allows real-time, visual spectrographs for rapid optimization or assessment of optical pumping.

Third, this technique may offer advantages for measuring magnetic fields over our previously published technique [5, 6]. In that technique, the spatial features were also separated by an amount proportional to the magnetic field. Here, the feature locations are also proportional to the sublevel number (Eq. 6), so that larger distances can be recorded for a given magnetic field. The differential light shifts were negligible for our previous work, which coupled states within a single hyperfine level. While these differential light shifts may be of significance, this VSR technique still provides a good estimate of the the Larmor frequency through Eq. 5–6.

8. Conclusion

We have used velocity-selective resonances between magnetic sublevels of the hyperfine levels of both ⁸⁵Rb and ⁸⁷Rb to image magnetic sublevel distributions in a single loading cycle of a magneto-optical trap. This technique has advantages over traditional Raman spectroscopy for rapid assessment of optical pumping and spin-polarization. The results agree with theoretical calculations and with copropagating-beam Raman spectroscopy.

This work was funded by the Office of Naval Research and the Defense Advanced Research Projects Agency.