We present a technique to excite Raman transitions with minimum phase noise. A phase modulator generates the Raman beams and a long calcite crystal rotates the polarization of the sidebands by 90° with respect to the carrier. That polarization converts the destructive interference of the Raman pairs into constructive interference, opening the possibility to drive both co-propagating and counter-propagating transitions at high detuning with the same setup. The technique has low phase noise and a low sensitivity to vibrations or temperature fluctuations. We apply it to drive velocity insensitive Raman transitions. The crystal can be also configured to filter out one of the sidebands.
© 2017 Optical Society of America
Atomic interferometers are quite appropriate for a wide range of applications. They are used at the laboratory level to determine the Newtonian constant of gravitation G , the fine structure constant α [2, 3], variations of gravity at short distances  or to verify the universality of free fall [5, 6], among others. One method to do atomic interferometry makes use of Raman transitions with two phase locked beams with a frequency difference close to the ground state hyperfine transition in the case of alkali atoms . The traditional way to generate these beams is by an optical phase lock loop of two independent lasers [8–13]. This technique is more susceptible to vibrations and air fluctuations since both beams follow independent paths, and the phase noise is determined by the quality of the feedback system. Alternatively, the two frequencies can be generated from a single laser using modulators [14–20]. This technique is gaining popularity since it provides a compact low phase noise Raman system [21–24].
Phase electro-optical modulators introduce more than two co-propagating frequency components with the same polarization, and therefore the interference between all the Raman pairs must be carefully taken into account. In the case of co-propagating Raman transitions for example, there is a cancellation between the Raman pairs produced at high detuning [18, 25]. A high detuning is desirable to minimize decoherence effects from spontaneous emission and AC Stark shifts . The cancellation can be avoided in the case of counter-propagating configurations by changing to circular polarization . Another solution is to convert the phase modulation into amplitude modulation by the use of a Mach-Zehnder interferometer . The stabilization of the interferometer adds extra noise that is improved using a Sagnac interferometer instead . In this work we demonstrate a technique to change the relative polarization of the frequency components so that the carrier becomes perpendicular to the sidebands. This has the effect of transforming the destructive interference in co-propagating Raman transitions into a constructive one. This configuration enables the possibility of driving both co-propagating and counter-propagating Raman transitions with a single setup.
There are ways to remove the extra frequency components altogether in order to eliminate the perturbations they introduce . An elegant solution uses serrodyne modulation to generate only a single sideband when there is enough bandwidth available , and narrow optical filters have been also used to eliminate a particular frequency . The system we present can be reconfigured also as a narrow frequency filter. Since the filter is not interferometric, it is less sensitive to vibrations and temperature fluctuations. With an electro-optical modulator it is possible to generate multiple isotope traps  and generate the Raman beams required for gravimetry, all with a single laser . The modulator we use allows for high optical power eliminating the need for additional optical amplification stages.
2. Generation of Raman beams with a phase modulator and a birefringent crystal
In this section we explain how to combine a phase modulator with a birefringent crystal to obtain the phase locked beams with the required polarization for Raman transitions. The Rabi frequency of the Raman transition is given by Appendix A) Eq. (2) the Raman transition will be suppressed at high detuning. In the case of phase modulation, ϕ = β sin (ωmt), we obtain 18, 25].
Figure 1 shows the system for the generation of Raman beams with minimum phase noise. The light from a Ti-sapphire laser goes through an acousto-optical modulator to have pulses from the first diffraction order and then to a phase modulator to generate the sidebands. Then it goes through a long (13.13 cm) birefringent calcite crystal to have a frequency dependent polarization change. The long length of the crystal makes it highly dispersive to produce a different polarization change for the carrier and sidebands. We tune the crystal to keep the same input linear polarization for the carrier and at the same time rotate the polarization of the two first order sidebands (±1) by 90°. This produces the appropriate perpendicular polarization for each Raman pair [Eq. (2)] and, as we show below, it converts the destructive interference between the two co-propagating Raman pairs into constructive interference.
To understand the action of the calcite crystal consider a single frequency beam like the one in Eq. (4) (with ϕ=0) going through the crystal with its optical axis perpendicular to the beam propagation and rotated 45° with respect to the input polarization. The electric field at the output is given by33]). Calcite crystals this long are hard to obtain; instead we do quadruple pass through a crystal of length L = 3.28 ± 0.01 cm [Fig. 1] corresponding to a retardation of n = 28, 281 complete wavelengths.Eq. (2)] of pair 1 is of the form and the one for pair 2 has the two vectors reversed . This minus sign compensates the π phase difference between the two pairs and produces constructive interference of the two contributions to the Rabi frequency. There is a π/2 phase difference in Eq. (8) between the +1 sideband and the carrier in pair 1 and also between the carrier and -1 sideband in pair 2, that shows up at the end as the same π/2 phase in the Rabi frequency for both pairs. There is a similar polarization rotation for all the sidebands [Eq. (6)], for example, the +2 sideband gets a polarization along x. The Raman pair formed by the +1 and +2 sidebands (or any other Raman pair between sidebands) also interferes constructively to the total Rabi frequency.
The beams in Fig. 1 are automatically phase locked since they come from a single laser, and given that they follow the same trajectory (except for the final retro-reflecting mirror in the counter-propagating configuration) they are less sensitive to vibrations [34, 35]. The calcite crystal works better than other interferometric solutions [16, 19] in terms of sensitivity to vibrations or changes in temperature as will be shown below. The phase modulator we use (Advr WPM-P78P78-ALO) takes up to 250 mW, providing enough power to drive the transition at high detuning without any subsequent amplification. This avoids the spontaneous emission pedestal introduced by tapered amplifiers [36, 37] or the complications involved in injection locking [38–40]. To make the system as simple as possible, we modulate directly at 780 nm rather than modulating at lower frequencies and then doubling the frequency .
3. Calcite crystal characterization
In this section we characterize the performance of the calcite crystal. The crystal works as a multiple order retarder as described by Eq. (7). To measure the polarization change as a function of frequency we use the single frequency polarized light from the laser and we send it through the crystal in quadruple pass. We analyze the polarization by sending the output light through a polarizer that transmits the horizontal component into a detector. Figure 2 shows the fringes we measure as we scan the laser frequency with the optical axis of the crystal rotated by 45° with respect to the input polarization. The signal at the detector is
The visibility of the fringes at a particular crystal angle (θ) is given byFigure 3 shows the measured fringe visibility obtained with the calcite crystal in single (black) and quadruple (red) pass as we rotate the crystal keeping always the optical axis perpendicular to the light propagation. The solid line corresponds to Eq. (10) with an offset and scale factor as fitting parameters, that give a result very close to the ideal case of Eq. (10). From the fit we extract a maximum visibility of 1.00 ± 0.04 (0.88 ± 0.04) for single (quadruple) pass at an angle of θ =45°, indicating that we get a polarization very close to linear with that configuration. Setting the carrier frequency to the maximum of the fringe and the sidebands to the minimum with the crystal at 45° gives us the desired polarization rotation of the light while keeping the polarization linear.
We arrange the mirrors used for multiple passes in a way to minimize any tilting on the beams that deteriorates the visibility of the fringes. It is also possible to use a prism to displace and retro-reflect the beam, but it is important to keep the plane of incidence with respect to the prism aligned with the optical axis of the crystal. The crystal has an anti reflection coating to minimize both the power loss after multiple passes and the relative reflection difference at the interface for both polarizations. The transmission after single pass is 98.5 ± 0.1 and 99.1 ± 0.1 for the ordinary and extraordinary modes.
In order to verify the retardation introduced by the crystal (ϕ) we use fringes similar to those in Fig. 2. We calibrate the frequency of the scan by sending part of the light through a Fabry Perot cavity of known Free Spectral Range. The measurement was done in quadruple pass through the crystal and gives a fringe spacing of 12.4 ± 0.1 GHz, that is a little bit smaller than the expected 2 × 6.8 GHz = 13.6 GHz. Most likely this difference comes from an slightly incorrect value for the birefringence of the crystal. The difference is still small and we get beams that are almost perpendicular to each other, for instance, if we set the crystal to have a carrier with linear polarization , then the left and right sidebands have a polarization given by Eq. (7) with φ = ±1.102π. This polarization is a little bit elliptical (ϵ = 0.16) with the long axis perpendicular to the carrier.
The calcite crystal can also be used as a frequency filter to eliminate one of the sidebands when used in double pass. This would give an alternative way to eliminate the extra sideband that introduces the destructive interference. To implement the filter we set the -1 sideband frequency to have vertical linear polarization and in this case the +1 sideband has horizontal linear polarization and the carrier has circular polarization since for double pass it would be at the middle of the fringe of Fig. 2. The -1 sideband is eliminated by sending the beam through a horizontal polarizer, leaving only the carrier and the +1 sideband (pair 1). The highly dispersive nature of the crystal gives a spectrally narrow filter (12.4 GHz wide) but we lose half of the power in the process. The filter reduces the -1 sideband by −19.6 ± 1.5 dB. We send the remaining Raman pair 1 through the crystal again but in quadruple pass to make the linear polarization of two frequencies orthogonal to each other [Fig. 4(a)]. Figure 4(b) shows the spectrum measured in a scanning Fabry Perot cavity (Thorlabs SA200-5B). The lower trace was obtained with a horizontal polarizer in front of the Fabry Perot cavity and we see only the carrier (repeated again after the 1.5 GHz Free Spectral Range of the cavity). The polarizer for the upper trace is vertical and shows only the +1 sideband, whereas the middle trace has the polarizer at 45° and shows both frequencies simultaneously. The +1 sideband is separated from the carrier by 6.8 GHz, but appears closer in a Fabry Perot scan due to the repetition of the signal every Free Spectral Range of the cavity. The -1 sideband is absent from the three traces since it has been filtered out. Figure 4 shows in a clear graphical way both the use of the calcite crystal as a narrow frequency filter and as a device to rotate by 90° the relative polarization of two frequencies separated by 6.8 GHz. We do not put much power on the second order sidebands or higher as can be seen from the absence of those components in Fig. 4(b).
4. Comparison with other interferometric solutions
A Michelson interferometer can be used as a frequency filter to eliminate one of the sidebands. A small displacement (ΔL) of one of the mirrors translates mainly into amplitude noise in the Raman transition. Any displacement of order λ ≈ 780 nm is enough to change the Raman amplitude considerably. A better choice would be to use a birefringent Sagnac interferometer [16, 41]. Here the beams going in opposite directions have perpendicular polarization and travel at a different speed in the birefringent medium. This introduces a frequency dependence that has been exploited to implement thermometers  or vibration sensors . The sensitivity to vibrations in the Sagnac interferometer is suppressed by the birefringence (B) of the material. Using the calcite crystal we avoid the interferometer altogether. Since it is a transmissive element (rather than reflective as in the mirrors of an interferometer) the vibrations would be important only when they have a wavelength of the order of the size of the crystal or smaller. This corresponds to frequencies above 150 kHz that are in the upper acoustic range. The change in temperature introduces an expansion of the material ΔL and a change in birefringence ΔB. This last one dominates in the case of calcite with dB/dT ≈ 10−5/°C . The effect scales with L and we expect a worse sensitivity with low birefringence materials since they need to be longer to keep φ = BLω/c constant.
To compare the filters, we implemented the Michelson and Sagnac interferometers in addition to the calcite crystal we present. The Michelson interferometer had short arms of L1 = 5 cm and L2 = 6.1 cm respectively. The difference in length of 1.1 cm was required to have a Free Spectral Range similar to that of the quadruple pass calcite crystal. The Sagnac interferometer used a polarization maintaining fiber (Thorlabs p1-780pm-fc) with a specified B = 3.5 × 10−4. To obtain the correct Free Spectral Range the required fiber length was 60 m, but we did the measurement with a 15 m long fiber which should have smaller temperature sensitivity. We measured the temperature variation required to shift the optical fringes of the interferometer or the crystal by one complete fringe. We obtained 0.03 ± 0.01, 0.03 ± 0.004 and 0.5 ± 0.08 °C for the Michelson and Sagnac interferometers and calcite crystal respectively. The frequency filter with the calcite crystal gives the best result, it is simpler to use and it allows to also obtain the desired polarization of the beams for a Raman transition.
5. Phase noise measurements
We characterized the phase noise properties of the beams produced by the modulator. To measure the noise we send the light out of the modulator to the calcite crystal in double pass and then through a polarizer to eliminate the -1 sideband (double pass part in Fig. 4(a)). The signal goes into a fast detector (Vescent IDS-160) to measure the beat note between the two frequency components. If the -1 sideband is not eliminated, the beat note of pair 1 and pair 2 cancel each other in a way similar to the Raman transition. We measure the noise spectrum in two ways. First we send the beat note signal directly to an spectrum analyzer (Agilent EXA N9010A) (gray line in Fig. 5). To overcome the limit in sensitivity of the spectrum analyzer we used homodyne detection by mixing the detector signal with the original modulating signal and sending it to an FFT analyzer (SRS SR760) (red line in Fig. 5). The Power Spectral Density (PSD) measured is very similar to the one we obtain by measuring the noise of the microwave source (Phase Matrix FSW-0010) directly. For this last measurement we split the signal of the synthesizer and we mix it back with unequal cable lengths (length difference of 60 cm) before sending it to the FFT analyzer. A better measurement would be obtained if two identical independent synthesizers were available. These measurements use a fixed frequency source, but the frequency tuning required for the counter-propagating Raman transitions can be added by mixing a low frequency signal onto the microwaves .
The noise spectrum decreases steadily by about 10 dBrad2/Hz every frequency decade to reach a value of −116 dBrad2/Hz at 100 kHz. The phase variance integrated from 1 Hz to 100 kHz gives 2 × 10−6 rad2. The noise at lower frequencies is the more relevant for interferometry since the atomic response works as a band pass filter [10, 46]. The specified phase noise of the synthesizer  stays constant from 100 kHz to 1 MHz and then continues to decrease above 1 MHz, giving an improved integrated variance compared with OPLL that have excess noise at high frequencies [8–13]. Our system is very simple to implement and should allow us to reach a precision in gravimetry of Δg/g ≈ 5 × 10−9 when using 100 μs long π/2 interferometric pulses and 30 ms between them.
6. Raman transitions with co-propagating beams
In this section we use the laser system to induce co-propagating Raman transitions in a sample of laser cooled 87Rb atoms. We perform Rabi oscillations to compare the Rabi frequency obtained using only the carrier and the +1 sideband (pair 1) to that obtained by the constructive interference of the two pairs once they go through the calcite crystal [Eq. (8)]. In order to obtain only pair 1 with perpendicular linear polarizations we use the calcite crystal first in double pass to filter out the -1 sideband and a second time in quadruple pass [Fig. 4(a)]. The crystal is temperature stabilized to better than 0.01 °C to the value needed to eliminate the -1 sideband at a particular detuning. We apply a Raman transition on the F = 1, m = 0 to F = 2, m = 0 clock transition in 87Rb at a magnetic field of 400 mG. We determine the fractional population on the F = 2 level after the Raman pulse using a florescence detection sequence on the free falling atoms. We determine the Rabi frequency as a function of the detuning [Fig. 6] and we find the expected 1/δ scaling for the Raman transition [Eqs. (2) and (3)]. The beam has a total power of 1.2 mW (with 16% of that power in the +1 sideband) and it is focused to a 0.5 cm waist.
Using the calcite crystal as proposed in this work [Fig. 1] to create constructive interference between the two pairs, we measured an increase on the Rabi frequency by a factor of 1.15 ± 0.04 with respect to having only one pair at a detuning of 11 GHz, with the same carrier and sideband powers. The increase in ΩR should not be by a factor of 2 but instead by a factor of 1.62 because the detuning of the two Raman pairs is not the same. Also the slightly elliptical polarization inferred from the imperfect visibility [Fig. 3] brings the expected value down to 1.43. The observed ratio of 1.15 is close to the expected value of 1.43, and the deviation might be due to differences in light shifts, even when care was taken to avoid having a resonant higher order sideband. The contributions of higher order sidebands to the Rabi frequency are at the 3% level. Our method introduces several improvements: it gives a bigger Rabi frequency, it avoids the loss of power introduced by the frequency filter, it is less sensitive to vibrations or temperature fluctuations and it gives the appropriate perpendicular polarization required at high detuning. To extend the method to velocity selective Raman transitions one can add a polarizing cube to split the carrier and sidebands to send them in counter-propagating configuration. The system is more robust to vibrations since all the beams travel along the same path for most of their trajectory.
We present a technique to generate the phase locked beams required for a Raman transition with minimum phase noise. The system is intrinsically less noisy because both beams come from a common laser. There is no need to have amplification after the modulator since this last one is capable of handling up to 250 mW of power. We use a long calcite crystal to obtain beams with perpendicular polarization to drive Raman transitions and to convert the destructive interference that appears at high detuning into a constructive one. This configuration increases the Rabi frequency of the Raman transition and it is less sensitive than other solutions to vibrations and temperature fluctuations. Given the highly dispersive nature of the crystal, it can be used also as a narrow frequency filter. The system can be arranged in both co-propagating and counter-propagating Raman configurations with minimum sensitive to vibrations since the beams share the same path for most of their trajectory. The phase noise we observe is close to the one measured by the microwave synthesizer alone. With this system it is possible to produce complex excitation patterns that would be impossible to do in the traditional configuration of two independent lasers.
In this appendix we derive a formula for the Rabi frequency for a Raman transition at a detuning larger than the hyperfine separation but small compared to the fine splitting. We consider a common detuning (δn ≈ δ) for all hyperfine levels. We define the matrix element vector as followsEq. (1) as
We apply a separation of the atomic part in the irreducible scalar, vector and second rank tensor components using 
We consider a transition between hyperfine levels of an alkali atom |j〉 = |F1, m1〉 and |k〉 = |F2, m2〉. In this case the scalar and the second rank tensor components vanish and only the vector part survives. This last one can be written as a cross product
The Rabi frequency of the Raman transition is obtained by contracting these two last vectors using again Eq. (13)Eq. (2). The scalar and second rank tensor components decay as 1/δ2 and they become negligible at high detuning. In contrast to light shifts , the scalar part does not contribute here to first order because for the Raman transition there is a change of hyperfine level (ΔF = 1).
CONACyT (Frontiers of Science, Infrastructure, Catedras and the Research for Education Fund); UASLP; and Marcos Moshinsky Foundation.
We thank Alexander Franco and Yasser Jeronimo for useful discussions.
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