January 1, 2015 / Vol. 40, No. 1 / OPTICS LETTERS
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Investigating the frequency-dependent amplification of a tapered amplifier in atom interferometers Su Zhan,1 Xiao-Chun Duan,1,2 Min-Kang Zhou,1 Hui-Bin Yao,1 Wen-Jie Xu,1 and Zhong-Kun Hu1,* 1
Key Laboratory of Fundamental Physical Quantities Measurement of Ministry of Education, School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China 2 e-mail: [email protected] *Corresponding author: [email protected]
Received October 1, 2014; revised November 9, 2014; accepted November 21, 2014; posted November 24, 2014 (Doc. ID 222423); published December 17, 2014 We present the investigation on the frequency-dependent amplification (FDA) of a tapered amplifier (TA) and the corresponding influence on Raman-type atom interferometers. In our interferometer, the output of two phase-locked diode lasers is injected into a TA to generate Raman beams. The frequency of one laser is chirped during the interfering process, which induces a variance of the Raman lasers power as a result of the FDA of the TA. The corresponding power ratio variation of the Raman lasers is measured by beat note method, which shows a linear dependence with a slope of −0.0874∕GHz when the laser frequency changes over 2 GHz at 780 nm. The corresponding error related to AC Stark effect due to this frequency-dependent variation is estimated for our atom interferometer. The investigation presented here may provide hints for other experiments involving TAs. © 2014 Optical Society of America OCIS codes: (140.0140) Lasers and laser optics; (020.3320) Laser cooling; (020.6580) Stark effect. http://dx.doi.org/10.1364/OL.40.000029
Atom interferometers [1,2] have important applications in fundamental physics, such as gravitational wave detection [3], Newton’s gravitational constant determination [4,5] and equivalence principle test [6]. They are also widely used for inertial sensing, for example, measurements of gravity [7–12], gravity gradient [13,14], and the rotation of the Earth [15]. In these applications, it is popular to utilize stimulated Raman transitions to coherently manipulate the atom wave-packets [2]. Therefore, possible systematic errors related to the Raman lasers should be carefully evaluated for improving the corresponding measurement accuracy. This work focuses on AC Stark effect due to the frequency chirp of Raman lasers in atom-interferometrybased inertial sensors. AC Stark effect is a fundamental effect in Raman-type interferometers, which involves the interaction between Raman lasers and atoms [2,7]. The induced differential AC Stark shift between the two ground hyperfine levels involved in stimulated Raman transitions is linearly dependent on the power of both the two beams in Raman lasers. In conventional consideration, the differential AC Stark shift can be eliminated by setting an appropriate power ratio between the two beams of Raman lasers [2,7,16]. However, in actual experiments, it is necessary for the effective frequency of Raman lasers to be chirped to compensate the Doppler shift due to free falling of the atoms in gravity field. For example, the chirp rate for Raman lasers at 780 nm is about 25 MHz∕s for our 87 Rb atom gravimeter, and the effective frequency varies by about 15 MHz within an interfering process of 600 ms. If there is a power variation of the Raman lasers due to this frequency chirp, then the differential AC Stark shift cannot always be eliminated by setting a constant power ratio. The Raman lasers can be generated by injecting the output of two phase-locked external cavity diode lasers (ECDLs) into a tapered amplifier (TA), which is an effective way to obtain abundant laser power, especially in cases of multiple beams with different frequencies in 0146-9592/15/010029-04$15.00/0
demand [17–20]. The amplification of a TA is usually nonlinear and frequency-related, which has been explored in the generation of terahertz radiation or multi-frequency laser source [21–25]. However, in our case, it may cause a power variation of the Raman lasers due to the frequency chirp. In this report, we have measured the power ratio variation between TA’s two output beams by beat note method [25–27] with one laser frequency changing over a range of 2 GHz at 780 nm, and the corresponding error related to AC Stark effect due to this frequency chirp for our atom interferometer is estimated. This experiment is performed in the Raman laser system for our atom gravimeter [11]. Simply speaking, two ECDLs (Toptica, DL 100), denoted as master laser and slave laser, respectively, in Fig. 1(a), are used as the seed lasers. The master laser is grating stabilized to the 87 Rb D2 line of j52 S 1∕2 ; F 2i to j52 P 3∕2 ; F 2i by modulation transfer (MT) method [28], and the slave laser is phase-locked to the master laser by an optical phaselocked loop (OPLL) [7,29]. The frequency of the slave laser is about 6.834 GHz higher than that of the master laser, which corresponds to the energy splitting (denoted as ωsplit here) between the two ground levels of 52 S 1∕2 .
Fig. 1. (a) Optic scheme of the generation of Raman lasers in the experiment. (b) Scheme of measuring the power ratio by beat note method. There is only one reference beam to take in beating at one time with the help of the mechanical shutter. © 2015 Optical Society of America
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This frequency difference is adjustably controlled by the reference frequency chain in the OPLL. In actual experiment, the chirp of the effective frequency of the Raman lasers is usually realized by chirping the frequency of the slave laser with the help of the OPLL, while the frequency of the master laser keeps fixed. Partial output from the two lasers is overlapped and coupled into a fiber, and then injected into a TA. The main output of the TA is used as Raman lasers, which is far red detuned from resonance by afterward passing through a 1.5-GHz acoustic optical modulator (AOM) to decrease the spontaneous emission. And a fraction of the TA output is sampled for the power ratio measurement of the two Raman beams in this experiment. However, due to the nonlinear amplification of TA, the practical power ratio of the two Raman beams must be measured with both the injected beams present. Beat note method is adopted here to measure the power ratio. As shown in Fig. 1(b), two reference beams, which are also branches from the two ECDLs, are used to beat with the TA output. The subscripts M and S, respectively, refer to the master and slave lasers. Before coupled into a fiber, the reference beams pass twice through an AOM in a typical double-pass (DP) configuration. At one time, there is only one beam from the reference beams to take in beating, while the other is blocked by a mechanical shutter. When the reference beam with the frequency of f M − 2f AOM and a power of P rM (which is directly measured by a power meter) beats with the TA output, the corresponding power of the 2f AOM frequency component in the beat note, denoted as BM , is measured by a frequency spectrum analyzer (FSA). Theoretically, BM is proportional to the product of P rM and the power of the Raman beam with the frequency of f M (denoted as P M here), namely BM ∝ P M P rM . Similarly, BS ∝ P S P rS when the f S − 2f AOM reference beam beats with the TA output. The power ratio α of the two beams from the TA output can thus be measured as α ≡ P M ∕P S BM ∕BS ∕P rM ∕P rS ;
(1)
where possible difference of the photo sensitivities of the photodetector (PD) for the master and slave laser is neglected for current experiment precision (see the test experiment below). Several notable issues are considered to achieve high precision of measuring the power ratio in our setup. First, the concerned beat component is always generated by two beams that arise from the same laser, reducing the influence of possible laser frequency jitter in repeated measurements. Second, the concerned beat component is always at the frequency of 2f AOM , minimizing the influence of possible frequency-dependent transmission of the beat note signal from the PD to the spectrum analyzer. Third, the fiber used to transmit the reference beams guarantees the two beams irradiate at the same spot of the PD, thus the power ratio P rM ∕P rS measured with an ordinary laser power meter at the fiber output can honestly reflect the practical power ratio of the two beams that take in the beat. We also note that the knowledge of the absolute characteristics of related optical and RF components, for example, the photo
sensitivity of the PD, is not required in this measurement of the power ratio. Typically, in this work, the output of the TA is about 300 mW at a total injection power of 25 mW with an operating current of 2 A. The AOM is working at a frequency of 119 MHz, and the signal-tonoise ratio (SNR) of the 238-MHz component of beat note signal is typically better than 65 dB with a resolution bandwidth of 1 kHz. The power ratio measurement system is first tested by measuring the calibrated power ratio of two test beams. These test beams are from a small part of the two ECDLs output and then overlapped using a polarization beam splitter (PBS). The power of each test beam can be directly measured by a power meter, from which the power ratio of the two test beams is obtained. On the other hand, the power ratio of the test beams is also measured by the beat note method, and the result is shown in Fig. 2. According to the result of the linear fitting, the slope 1.008 0.014 is very close to the expected value of 1, where 1.4% is the statistics uncertainty, most probably contributed from the instability of the laser power from the ECDLs. This test experiment shows that the power ratio can be measured with a relative precision 1.4% by this beat note measurement system. In our experiment, the total power of the two injected laser beams is nearly invariable, but the power ratio needs to be adjusted in order to eliminate the differential AC Stark shift. However, the output power ratio of the TA may be not equal to the power ratio of the injected beams. The ratio between the output power ratio and the input power ratio of the TA is denoted as κ here. And κ is first measured with the frequency of the slave laser keeping fixed with a 6.834-GHz frequency difference relative to the master laser, while the input power ratio is adjusted from 0.1 to 8.8. The result is shown in Fig. 3, which shows a linear dependence, thus κ is independent of the input power ratio in this input power range. For this frequency difference at ωsplit 6.834 GHz, κ 0 1.344, which is obtained through the linear fitting. The result that κ deviates from 1, which is the expected value for a linear optical component, is supposedly due to the frequency-dependent amplification (FDA) of the TA. Finally, the variation of the Raman beams power ratio due to frequency chirp is investigated. However,
Fig. 2. Test of the power ratio measuring system. The test beams are generated by simply overlapping the branches of the two ECDLs by a PBS, which is a linear optical component, and thus the power ratio can be directly measured by a power meter.
January 1, 2015 / Vol. 40, No. 1 / OPTICS LETTERS
ΔφAC
AC δAC 3 − δ1 ; Ωeff
31
(2)
where Ωeff denotes the effective two-photon Rabi freand δAC are the differential AC Stark quency, and δAC 1 3 shifts of the first and third interfering pulses, respectively. In the approximation that the 52 P 3∕2 hyperfine splitting is neglected, the differential AC Stark shift is expressed as
Fig. 3. Power ratio at the output of the TA versus that at the input with the frequency difference of the two ECDLs keeping fixed at 6.834 GHz. The input power ratio is directly measured by a power meter, and the absolute power of single beam varies in a range from about 2 to 23 mW.
the frequency of the slave laser due to this chirp is only several MHz for practical experiment, and the induced variation of the power ratio of the Raman beams is supposed to be tiny, presenting a stringent requirement on the precision of measuring the power ratio. Alternatively, the amplification over a frequency change of 2 GHz is measured here, and then the actual amplification change over several MHz frequency chirp can be obtained by interpolation. The frequency of the slave laser is adjusted from 6 to 8 GHz relative to the master laser frequency, and the corresponding output power ratio is measured by beat note method with the input power ratio keeping at 1:1. The frequency change step is 200 MHz, and for each frequency difference, the measurement is repeated for 500 times. The result is shown in Fig. 4, which is reported by the ratio κ. It indicates that the frequency change does cause an amplification variation of the TA, and the output power ratio change is approximately proportional to the frequency change over this range. The slope is −0.0874∕GHz according to the linear fitting, namely ∂κ∕∂f S −0.0874∕GHz over this range. The corresponding change of κ is −1.316 × 10−3 with a frequency increment of 15 MHz. Though the amplification relative change is only at the level of 10−3 in actual frequency scan range of several MHz, its influence should still be carefully examined in a high-precision atom interferometer. For the atom gravimeter based on three-pulse Raman type atom interferometry, the phase shift due to AC Stark effect is [7]
Fig. 4. Measurement of the FDA of the TA. The input power ratio keeps fixed at 1∶1, namely 12.5 mW for each beam.
δAC Ωeff
ωsplit α − α0 ωsplit p Δκ p ∼ Ωeff α0 ; (3) ωsplit − Δ ωsplit − Δ κ0 α
where α0 ≡ ωsplit − Δ∕ωsplit Δ is the power ratio between the two Raman beams at which the differential AC Stark shift is canceled. Δ denotes the −1.5 GHz far red detuning. In the simplest case, the power ratio of the Raman beams is set as α0 for the first Raman pulse, which is about 1.56 for our experiment. For a frequency increment of 15 MHz, Δκ is −1.316 × 10−3 and the corresponding phase shift is about −1.008 mrad, where the minus sign corresponds to the case that the effective wave vector of the Raman lasers is in the same direction of the local gravity. Consider a pulse separation of T 300 ms and an effective wave vector of keff ∼ 1.6 × 107 m−1 , the resulting systematic error is about −0.695 × 10−10 g for our gravimeter [11], according to the relation δg δφ∕keff T 2 . The modified error is −0.64 × 10−10 g if the 52 P 3∕2 hyperfine splitting is considered. This error is about two orders of magnitude lower than the current accuracy (3 ∼ 4 μGal) of the best atom gravimeters [7,30]. If the effective vector of the Raman lasers is reversed, the frequency of the slave laser decreases by 15 MHz in the chirp and the resulting systematic error will be 0.64 × 10−10 g. This indicates that the AC Stark effect related with the frequency chirp considered here cannot be eliminated by simply reversing the directions of the Raman lasers, which has been conventionally performed to suppress the AC Stark effect (see [7], for example). In conclusion, the FDA of a TA is measured and its influence on our atom interferometer is investigated in this report. In our case, a linear variation of the power ratio of the Raman lasers due to the frequency chirp is found, of which the slope is −0.0874∕GHz over a range of 2 GHz change of the slave laser. The influence of this variation in the actual frequency change range of 15 MHz is at the level of 10−10 g for our gravimeter. Though the induced systematic error is negligible for current μGal absolute atom gravimeters, the influence should be carefully examined in future higher precision measurements. We also note that this FDA may relate to the TA parameters such as saturation, temperature, and operating current, which may cause different FDA for different TAs. And consequently the corresponding influences should be carefully evaluated in other precision measurements where a TA is used. We thank Prof. Jun Luo and Prof. Ze-Huang Lu for enlightening discussions. This work is supported by the National Natural Science Foundation of China (Grants Nos. 41127002, 11204094, and 11205046) and the National Basic Research Program of China (Grant No. 2010CB832806).
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Investigating the frequency-dependent amplification of a tapered amplifier in atom interferometers Su Zhan,1 Xiao-Chun Duan,1,2 Min-Kang Zhou,1 Hui-Bin Yao,1 Wen-Jie Xu,1 and Zhong-Kun Hu1,* 1
Key Laboratory of Fundamental Physical Quantities Measurement of Ministry of Education, School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China 2 e-mail: [email protected] *Corresponding author: [email protected]
Received October 1, 2014; revised November 9, 2014; accepted November 21, 2014; posted November 24, 2014 (Doc. ID 222423); published December 17, 2014 We present the investigation on the frequency-dependent amplification (FDA) of a tapered amplifier (TA) and the corresponding influence on Raman-type atom interferometers. In our interferometer, the output of two phase-locked diode lasers is injected into a TA to generate Raman beams. The frequency of one laser is chirped during the interfering process, which induces a variance of the Raman lasers power as a result of the FDA of the TA. The corresponding power ratio variation of the Raman lasers is measured by beat note method, which shows a linear dependence with a slope of −0.0874∕GHz when the laser frequency changes over 2 GHz at 780 nm. The corresponding error related to AC Stark effect due to this frequency-dependent variation is estimated for our atom interferometer. The investigation presented here may provide hints for other experiments involving TAs. © 2014 Optical Society of America OCIS codes: (140.0140) Lasers and laser optics; (020.3320) Laser cooling; (020.6580) Stark effect. http://dx.doi.org/10.1364/OL.40.000029
Atom interferometers [1,2] have important applications in fundamental physics, such as gravitational wave detection [3], Newton’s gravitational constant determination [4,5] and equivalence principle test [6]. They are also widely used for inertial sensing, for example, measurements of gravity [7–12], gravity gradient [13,14], and the rotation of the Earth [15]. In these applications, it is popular to utilize stimulated Raman transitions to coherently manipulate the atom wave-packets [2]. Therefore, possible systematic errors related to the Raman lasers should be carefully evaluated for improving the corresponding measurement accuracy. This work focuses on AC Stark effect due to the frequency chirp of Raman lasers in atom-interferometrybased inertial sensors. AC Stark effect is a fundamental effect in Raman-type interferometers, which involves the interaction between Raman lasers and atoms [2,7]. The induced differential AC Stark shift between the two ground hyperfine levels involved in stimulated Raman transitions is linearly dependent on the power of both the two beams in Raman lasers. In conventional consideration, the differential AC Stark shift can be eliminated by setting an appropriate power ratio between the two beams of Raman lasers [2,7,16]. However, in actual experiments, it is necessary for the effective frequency of Raman lasers to be chirped to compensate the Doppler shift due to free falling of the atoms in gravity field. For example, the chirp rate for Raman lasers at 780 nm is about 25 MHz∕s for our 87 Rb atom gravimeter, and the effective frequency varies by about 15 MHz within an interfering process of 600 ms. If there is a power variation of the Raman lasers due to this frequency chirp, then the differential AC Stark shift cannot always be eliminated by setting a constant power ratio. The Raman lasers can be generated by injecting the output of two phase-locked external cavity diode lasers (ECDLs) into a tapered amplifier (TA), which is an effective way to obtain abundant laser power, especially in cases of multiple beams with different frequencies in 0146-9592/15/010029-04$15.00/0
demand [17–20]. The amplification of a TA is usually nonlinear and frequency-related, which has been explored in the generation of terahertz radiation or multi-frequency laser source [21–25]. However, in our case, it may cause a power variation of the Raman lasers due to the frequency chirp. In this report, we have measured the power ratio variation between TA’s two output beams by beat note method [25–27] with one laser frequency changing over a range of 2 GHz at 780 nm, and the corresponding error related to AC Stark effect due to this frequency chirp for our atom interferometer is estimated. This experiment is performed in the Raman laser system for our atom gravimeter [11]. Simply speaking, two ECDLs (Toptica, DL 100), denoted as master laser and slave laser, respectively, in Fig. 1(a), are used as the seed lasers. The master laser is grating stabilized to the 87 Rb D2 line of j52 S 1∕2 ; F 2i to j52 P 3∕2 ; F 2i by modulation transfer (MT) method [28], and the slave laser is phase-locked to the master laser by an optical phaselocked loop (OPLL) [7,29]. The frequency of the slave laser is about 6.834 GHz higher than that of the master laser, which corresponds to the energy splitting (denoted as ωsplit here) between the two ground levels of 52 S 1∕2 .
Fig. 1. (a) Optic scheme of the generation of Raman lasers in the experiment. (b) Scheme of measuring the power ratio by beat note method. There is only one reference beam to take in beating at one time with the help of the mechanical shutter. © 2015 Optical Society of America
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This frequency difference is adjustably controlled by the reference frequency chain in the OPLL. In actual experiment, the chirp of the effective frequency of the Raman lasers is usually realized by chirping the frequency of the slave laser with the help of the OPLL, while the frequency of the master laser keeps fixed. Partial output from the two lasers is overlapped and coupled into a fiber, and then injected into a TA. The main output of the TA is used as Raman lasers, which is far red detuned from resonance by afterward passing through a 1.5-GHz acoustic optical modulator (AOM) to decrease the spontaneous emission. And a fraction of the TA output is sampled for the power ratio measurement of the two Raman beams in this experiment. However, due to the nonlinear amplification of TA, the practical power ratio of the two Raman beams must be measured with both the injected beams present. Beat note method is adopted here to measure the power ratio. As shown in Fig. 1(b), two reference beams, which are also branches from the two ECDLs, are used to beat with the TA output. The subscripts M and S, respectively, refer to the master and slave lasers. Before coupled into a fiber, the reference beams pass twice through an AOM in a typical double-pass (DP) configuration. At one time, there is only one beam from the reference beams to take in beating, while the other is blocked by a mechanical shutter. When the reference beam with the frequency of f M − 2f AOM and a power of P rM (which is directly measured by a power meter) beats with the TA output, the corresponding power of the 2f AOM frequency component in the beat note, denoted as BM , is measured by a frequency spectrum analyzer (FSA). Theoretically, BM is proportional to the product of P rM and the power of the Raman beam with the frequency of f M (denoted as P M here), namely BM ∝ P M P rM . Similarly, BS ∝ P S P rS when the f S − 2f AOM reference beam beats with the TA output. The power ratio α of the two beams from the TA output can thus be measured as α ≡ P M ∕P S BM ∕BS ∕P rM ∕P rS ;
(1)
where possible difference of the photo sensitivities of the photodetector (PD) for the master and slave laser is neglected for current experiment precision (see the test experiment below). Several notable issues are considered to achieve high precision of measuring the power ratio in our setup. First, the concerned beat component is always generated by two beams that arise from the same laser, reducing the influence of possible laser frequency jitter in repeated measurements. Second, the concerned beat component is always at the frequency of 2f AOM , minimizing the influence of possible frequency-dependent transmission of the beat note signal from the PD to the spectrum analyzer. Third, the fiber used to transmit the reference beams guarantees the two beams irradiate at the same spot of the PD, thus the power ratio P rM ∕P rS measured with an ordinary laser power meter at the fiber output can honestly reflect the practical power ratio of the two beams that take in the beat. We also note that the knowledge of the absolute characteristics of related optical and RF components, for example, the photo
sensitivity of the PD, is not required in this measurement of the power ratio. Typically, in this work, the output of the TA is about 300 mW at a total injection power of 25 mW with an operating current of 2 A. The AOM is working at a frequency of 119 MHz, and the signal-tonoise ratio (SNR) of the 238-MHz component of beat note signal is typically better than 65 dB with a resolution bandwidth of 1 kHz. The power ratio measurement system is first tested by measuring the calibrated power ratio of two test beams. These test beams are from a small part of the two ECDLs output and then overlapped using a polarization beam splitter (PBS). The power of each test beam can be directly measured by a power meter, from which the power ratio of the two test beams is obtained. On the other hand, the power ratio of the test beams is also measured by the beat note method, and the result is shown in Fig. 2. According to the result of the linear fitting, the slope 1.008 0.014 is very close to the expected value of 1, where 1.4% is the statistics uncertainty, most probably contributed from the instability of the laser power from the ECDLs. This test experiment shows that the power ratio can be measured with a relative precision 1.4% by this beat note measurement system. In our experiment, the total power of the two injected laser beams is nearly invariable, but the power ratio needs to be adjusted in order to eliminate the differential AC Stark shift. However, the output power ratio of the TA may be not equal to the power ratio of the injected beams. The ratio between the output power ratio and the input power ratio of the TA is denoted as κ here. And κ is first measured with the frequency of the slave laser keeping fixed with a 6.834-GHz frequency difference relative to the master laser, while the input power ratio is adjusted from 0.1 to 8.8. The result is shown in Fig. 3, which shows a linear dependence, thus κ is independent of the input power ratio in this input power range. For this frequency difference at ωsplit 6.834 GHz, κ 0 1.344, which is obtained through the linear fitting. The result that κ deviates from 1, which is the expected value for a linear optical component, is supposedly due to the frequency-dependent amplification (FDA) of the TA. Finally, the variation of the Raman beams power ratio due to frequency chirp is investigated. However,
Fig. 2. Test of the power ratio measuring system. The test beams are generated by simply overlapping the branches of the two ECDLs by a PBS, which is a linear optical component, and thus the power ratio can be directly measured by a power meter.
January 1, 2015 / Vol. 40, No. 1 / OPTICS LETTERS
ΔφAC
AC δAC 3 − δ1 ; Ωeff
31
(2)
where Ωeff denotes the effective two-photon Rabi freand δAC are the differential AC Stark quency, and δAC 1 3 shifts of the first and third interfering pulses, respectively. In the approximation that the 52 P 3∕2 hyperfine splitting is neglected, the differential AC Stark shift is expressed as
Fig. 3. Power ratio at the output of the TA versus that at the input with the frequency difference of the two ECDLs keeping fixed at 6.834 GHz. The input power ratio is directly measured by a power meter, and the absolute power of single beam varies in a range from about 2 to 23 mW.
the frequency of the slave laser due to this chirp is only several MHz for practical experiment, and the induced variation of the power ratio of the Raman beams is supposed to be tiny, presenting a stringent requirement on the precision of measuring the power ratio. Alternatively, the amplification over a frequency change of 2 GHz is measured here, and then the actual amplification change over several MHz frequency chirp can be obtained by interpolation. The frequency of the slave laser is adjusted from 6 to 8 GHz relative to the master laser frequency, and the corresponding output power ratio is measured by beat note method with the input power ratio keeping at 1:1. The frequency change step is 200 MHz, and for each frequency difference, the measurement is repeated for 500 times. The result is shown in Fig. 4, which is reported by the ratio κ. It indicates that the frequency change does cause an amplification variation of the TA, and the output power ratio change is approximately proportional to the frequency change over this range. The slope is −0.0874∕GHz according to the linear fitting, namely ∂κ∕∂f S −0.0874∕GHz over this range. The corresponding change of κ is −1.316 × 10−3 with a frequency increment of 15 MHz. Though the amplification relative change is only at the level of 10−3 in actual frequency scan range of several MHz, its influence should still be carefully examined in a high-precision atom interferometer. For the atom gravimeter based on three-pulse Raman type atom interferometry, the phase shift due to AC Stark effect is [7]
Fig. 4. Measurement of the FDA of the TA. The input power ratio keeps fixed at 1∶1, namely 12.5 mW for each beam.
δAC Ωeff
ωsplit α − α0 ωsplit p Δκ p ∼ Ωeff α0 ; (3) ωsplit − Δ ωsplit − Δ κ0 α
where α0 ≡ ωsplit − Δ∕ωsplit Δ is the power ratio between the two Raman beams at which the differential AC Stark shift is canceled. Δ denotes the −1.5 GHz far red detuning. In the simplest case, the power ratio of the Raman beams is set as α0 for the first Raman pulse, which is about 1.56 for our experiment. For a frequency increment of 15 MHz, Δκ is −1.316 × 10−3 and the corresponding phase shift is about −1.008 mrad, where the minus sign corresponds to the case that the effective wave vector of the Raman lasers is in the same direction of the local gravity. Consider a pulse separation of T 300 ms and an effective wave vector of keff ∼ 1.6 × 107 m−1 , the resulting systematic error is about −0.695 × 10−10 g for our gravimeter [11], according to the relation δg δφ∕keff T 2 . The modified error is −0.64 × 10−10 g if the 52 P 3∕2 hyperfine splitting is considered. This error is about two orders of magnitude lower than the current accuracy (3 ∼ 4 μGal) of the best atom gravimeters [7,30]. If the effective vector of the Raman lasers is reversed, the frequency of the slave laser decreases by 15 MHz in the chirp and the resulting systematic error will be 0.64 × 10−10 g. This indicates that the AC Stark effect related with the frequency chirp considered here cannot be eliminated by simply reversing the directions of the Raman lasers, which has been conventionally performed to suppress the AC Stark effect (see [7], for example). In conclusion, the FDA of a TA is measured and its influence on our atom interferometer is investigated in this report. In our case, a linear variation of the power ratio of the Raman lasers due to the frequency chirp is found, of which the slope is −0.0874∕GHz over a range of 2 GHz change of the slave laser. The influence of this variation in the actual frequency change range of 15 MHz is at the level of 10−10 g for our gravimeter. Though the induced systematic error is negligible for current μGal absolute atom gravimeters, the influence should be carefully examined in future higher precision measurements. We also note that this FDA may relate to the TA parameters such as saturation, temperature, and operating current, which may cause different FDA for different TAs. And consequently the corresponding influences should be carefully evaluated in other precision measurements where a TA is used. We thank Prof. Jun Luo and Prof. Ze-Huang Lu for enlightening discussions. This work is supported by the National Natural Science Foundation of China (Grants Nos. 41127002, 11204094, and 11205046) and the National Basic Research Program of China (Grant No. 2010CB832806).
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