Limits to the sensitivity of a low noise compact atomic gravimeter
J. Le Gouët, Tanja Mehlstäubler, Jaewan Kim, Sébastien Merlet, Andre Clairon, Arnaud Landragin, Franck Pereira Dos Santos
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Limits to the sensitivity of a low noise compact atomic gravimeter
J. Le Gou¨et, T. E. Mehlst¨aubler, ∗ J. Kim, † S. Merlet,A. Clairon, A. Landragin, and F. Pereira Dos Santos ‡ LNE-SYRTE, CNRS UMR 8630, UPMC, Observatoire de Paris61 avenue de l’Observatoire, 75014 Paris, France (Dated: May 29, 2018)
Abstract
A detailed analysis of the most relevant sources of phase noise in an atomic interferometer iscarried out, both theoretically and experimentally. Even a short interrogation time of 100 msallows our cold atom gravimeter to reach an excellent short term sensitivity to acceleration of1 . × − g at 1s. This result relies on the combination of a low phase noise laser system, efficientdetection scheme and good shielding from vibrations. In particular, we describe a simple androbust technique of vibration compensation, which is based on correcting the interferometer signalby using the AC acceleration signal measured by a low noise seismometer. ∗ Physikalisch-Technische Bundesanstalt (PTB), Bundesallee 100, D-38116 Braunschweig, Germany † Physics Department, Myongji University, Korea. ‡ Electronic address: [email protected] . INTRODUCTION Over the last fifteen years, atom interferometry techniques [1] have been used to developnovel inertial sensors, which now compete with state of the art “classical” instruments [2].After the first demonstration experiments in the early 90’s [3, 4], the performance of thistechnology has been pushed and highly sensitive instruments have been realized. As theinertial phase shifts scale quadratically with the interrogation time, high sensitivies can bereached using either cold atoms along parabolic trajectories [3, 5], such as in microwavefountain clocks, or very long beam machines. Best short term sensitivities to acceleration of0 . − . × − m s − at 1s , and to rotations of 6 × − rad s − at 1s have been reachedin the experiments developed in the groups of S. Chu [6] and M. Kasevich [7]. Moreover, akey feature of these instruments is to provide an absolute measurement with improved longterm stability compared to other sensors, due to the intrinsic stability of their scale factorand a good control of the environment of the atomic samples.Applications of this kind of inertial interferometers are growing, from the measurementof fundamental constants, such as the gravitational constant G [8, 9], to the developmentof transportable devices for navigation, gravity field mapping, detection of undergroundstructures and finally for space missions [10], where ultimate performances can be met,thanks to the absence of gravity and a low vibration environment.We are currently developing a cold atom gravimeter, within the frame of the watt bal-ance project, conducted by the Laboratoire National de M´etrologie et d’Essais (LNE) [11].A watt balance allows to link the unit of mass, the kilogram, to electrical units and pro-vide a measurement of the Planck constant. Watt Balances developed at NIST and NPLpresently reach relative accuracies of a few parts in 10 [12, 13]. During one of the phasesof this experiment, the weight of a test mass is balanced with an electric force. An absolutemeasurement of gravity experienced by the test mass is thus required, which will be realizedwith our atom interferometer with a targeted relative accuracy of 1 ppb.In this paper, we describe the realization of this sensor, which has been designed to berelatively compact, in order to be easily transportable. Our gravimeter reaches, despiterather small interaction times, a sensitivity of 1 . × − g at 1s, better than state of the art“classical” gravimeters, and comparable to much larger atomic fountain gravimeters [6].In this paper, we first describe our experimental setup, and we investigate in detail in the2ext sections the contributions of the different sources of noise which affect the sensitivity.In particular, we describe in section III D an original, simple and efficient technique ofvibration compensation, which allows to improve the sensitivity of our measurement byrejecting residual vibrational noise with the help of a low-noise seismometer [14]. II. EXPERIMENTAL SETUP
The principle of the gravimeter is based on the coherent splitting of matter-waves bythe use of two-photon Raman transitions [3]. These transitions couple the two hyperfinelevels F = 1 and F = 2 of the S / ground state of the Rb atom. An intense beam ofslow atoms is first produced by a 2D-MOT. Out of this beam 10 atoms are loaded within50 ms into a 3D-MOT and subsequently cooled in a far detuned (-25 Γ) optical molasses.The lasers are then switched off adiabatically to release the atoms into free fall at a finaltemperature of 2 . µ K. Both lasers used for cooling and repumping are then detuned fromthe atomic transitions by about 1 GHz to generate the two off-resonant Raman beams. Forthis purpose, we have developed a compact and agile laser system that allows us to rapidlychange the operating frequencies of these lasers, as described in [15]. Before entering theinterferometer, atoms are selected in a narrow vertical velocity distribution ( σ v ≤ v r = 5 . | F = 1 , m F = 0 i state, using a combination of microwave and optical Ramanpulses.The interferometer is created by using a sequence of three pulses ( π/ − π − π/ P from one hyperfine state to the other isgiven by the well-known relation for a two wave interferometer : P = (1 + C cos ∆Φ), where C is the interferometer contrast, and ∆Φ the difference of the atomic phases accumulatedalong the two paths. The difference in the phases accumulated along the two paths dependson the acceleration ~a experienced by the atoms. It can be written as ∆Φ = φ (0) − φ ( T ) + φ (2 T ) = − ~k eff · ~aT [16], where φ (0 , T, T ) is the difference of the phases of the lasers,at the location of the center of the atomic wavepackets, for each of the three pulses. Here ~k eff = ~k − ~k is the effective wave vector (with | ~k eff | = k + k for counter-propagating3eams), and T is the time interval between two consecutive pulses.The Raman light sources are two extended cavity diode lasers based on the design of[17], which are amplified by two independent tapered amplifiers. Their frequency differenceis phase locked onto a low phase noise microwave reference source. The two Raman laserbeams are overlapped with a polarization beam splitter cube, resulting in two orthogonalpolarized beams. First, a small part of the overlapped beams is sent onto a fast photodetectorto measure the optical beat. This beat-note is mixed down with the reference microwaveoscillator, and compared to a stable reference RF frequency in a Digital Phase FrequencyDetector. The phase error signal is then used to lock the laser phase difference at the veryposition where the beat is recorded. The phase lock loop reacts onto the supply currentof one of the two lasers (the “slave” laser), as well as on the piezo-electric transducer thatcontrols the length of its extended cavity. Finally, the two overlapped beams are injectedin a polarization maintaining fiber, and guided towards the vacuum chamber. We obtaincounter-propagating beams by placing a mirror and a quarterwave plate at the bottom of theexperiment. Four beams are actually sent onto the atoms, out of which only two will drivethe counter-propagating Raman transitions, due to conservation of angular momentum andthe Doppler shift induced by the free fall of the atoms. III. SHORT TERM SENSITIVITYA. Detection Noise
The transition probability is deduced from the population in each of the two hyperfinestates, which are measured by fluorescence. Ultimately, noise on this measurement is limitedby the quantum projection noise σ P = 1 / √ N [26], where σ P is the standard deviation ofthe transition probability, and N the number of detected atoms. Other sources of noise,such as electronic noise in the photodiodes, laser frequency and intensity noise, will affectthe measurement, and might exceed the quantum limit depending on the number of atoms[27]. 4 IG. 1: Scheme of the experimental set-up. The interferometer is realized in a stainless steelvacuum chamber, shielded form magnetic field fluctuations with two layers of mu metal. Thechamber is sustained with three legs onto a passive isolation platform (Minus-K BM). Atoms arefirst trapped in a MOT, cooled with optical molasses and released. The interferometer is thenrealized during their free fall, with vertical Raman laser beams, which enter the vacuum chamberfrom the top, and are retro-reflected on a mirror located below. This mirror is attached to alow noise seismometer. Finally, the populations in the two hyperfine states are measured byfluorescence, which allows to determine the interferometer phase shift.
1. Basic scheme
The detection system implemented at first in the gravimeter is similar to the one de-veloped for atomic fountain clocks. It consists of two separated horizontal sheets of light.The first detection zone consists in a laser beam circularly polarized, tuned on the cyclingtransition ( F = 2 → F = 3). This beam is retro-reflected on a mirror, in order to gen-erate a standing wave. It allows to measure the number of atoms in the F = 2 state byfluorescence. The atoms which have interacted with the laser light are then removed by apusher beam, obtained by blocking the lower part of the retro-reflected beam. The secondzone is a repumper beam tuned on the ( F = 1 → F = 2), which pumps the atoms in F = 1 into F = 2. The last zone is a standing wave tuned on the cycling transition, which5llows to measure the number of atoms initially in F = 1. The fluorescence emitted in theupper and lower zones is detected by two distinct photodiodes that collect 1 % of the totalfluorescence. Using a π/ P = 0 .
5) has been measured in the gravimeter as a function ofthe number of atoms, see figure 2. The saturation parameter at the center of the detectionbeams is close to 1. For less than 10 atoms, the detection noise is limited by the noise ofthe photodiodes and the electronics. The standard deviation of the fluctuations of the tran-sition probability σ P is then inversely proportional to the number of atoms. The detectionnoise is equivalent to about 900 atoms per detection zone. For atom number larger than5 × , technical noise, arising from intensity and frequency noise of the detection laser,limits σ P to about 3 × − . The noise in the measured transition probability σ P convertsinto phase noise σ φ = 2 /C × σ P , with C being the contrast of the interferometer. A sensi-tivity close to 1 mrad per shot for the interferometer measurement can thus be obtained forthe interferometer when the number of atoms is larger than 5 × .
2. Improved scheme
In our experiment, the same laser system is used for the Raman interferometer and forthe atom trapping. After the interferometer sequence the lasers are brought back close toresonance in order to trap the atoms during the beginning of the next experimental cycle.In principle, they can also be used to detect the atoms by pulsing the vertical beam, whenthe atoms are located in the detection region.The detection sequence we use has been inspired by the detection system [28] of thegradiometer of [29]. (1) When the atoms are located in front of the top photodiode, a lowintensity pulse, slightly red detuned to the cycling transition is induced in order to stop the F = 2 atoms. (2) One then waits for the atoms in F = 1 to reach the position in frontof the bottom photodiode. (3) A second pulse (10 ms long) is then applied at full power.Cooling and repumper beams are both present during this pulse. (4) A third pulse, 10 mslater, finally serves for background substraction of stray light. The areas of the fluorescencesignals collected by the two photodiodes are thus proportional to the number of atoms ineach of the two hyperfine states. This detection scheme has several advantages. First,the intensity in the vertical beam is much higher than in the standard detection beam, the6 -3 -2 -1 Horizontal detection Vertical detection P Number of atoms
QPN
FIG. 2: Allan standard deviation of the excitation probability σ P for both methods described. Opencircles correspond to horizontal detection, full circles to vertical detection. QPN is the Quantumprojection Noise limit. The phase noise introduced onto the atom interferometer is σ φ = 2 /C × σ P . saturation parameter being close to 50. Atoms will thus remain resonant despite the heatinginduced by photon recoils. Second, atoms spend a longer time in front of the photodiodes.Finally, repumper is present on both clouds during the whole duration of the detection pulse.With this scheme, the detection noise is now equivalent to only 150 atoms per zone, thanksto the increase of the fluorescence signal, see figure 2. The detection is at the quantumprojection noise limit with about 10 atoms. The same limit of σ P ≈ × − is foundfor large number of atoms. This detection scheme is thus more efficient for low number ofatoms, for instance when using Bose Einstein condensates or narrower velocity selection.Limits to the signal to noise ratio for large numbers of atoms still have to be identified.They could arise from laser intensity and frequency fluctuations, as well as fluctuations ofthe normalization. In principle, the second detection scheme should be insensitive to laserfluctuations, which are common mode for the two populations, as the measurements areperformed simultaneously. B. Phase noise
In our interferometer, the noise in the phase difference of the two Raman lasers inducesfluctuations of the interferometer phase. In a previous publication [18], we have introduced7 useful tool to calculate the influence of the different sources of noise onto the stabilityof the interferometer phase measurement, the sensitivity function, g ( t ) [19]. This functionquantifies the influence of a relative laser phase shift δφ occurring at time t onto the phaseof the interferometer δ Φ( δφ, t ). When operating the interferometer at mid-fringe, it can bewritten as: g ( t ) = lim δφ → δ Φ( δφ, t ) δφ . (1)The interferometer phase shift Φ induced by fluctuations of φ is then given by:Φ = Z + ∞−∞ g ( t ) dφ ( t ) dt dt. (2)For completeness we briefly recall the expression of the sensitivity function. With a sequenceof three pulses π/ − π − π/ τ R − τ R − τ R and a time origin chosen at thecenter of the π pulse, g is an odd function whose expression was first derived in [18]: g ( t ) = sin Ω R t for 0 < t < τ R τ R < t < T + τ R − sin Ω R ( T − t ) for T + τ R < t < T + 2 τ R (3)where Ω R is the Rabi frequency.The noise of the interferometer is characterized by the Allan variance of the interfero-metric phase fluctuations, σ ( τ ), defined as: σ ( τ ) = 12 h ( ¯ δ Φ k +1 − ¯ δ Φ k ) i (4)= 12 lim n →∞ ( n n X k =1 ( ¯ δ Φ k +1 − ¯ δ Φ k ) ) . (5)Here ¯ δ Φ k is the average value of δ Φ over the interval [ t k , t k +1 ] of duration τ . The Allanvariance is equal, within a factor of two, to the variance of the differences in the successiveaverage values ¯ δ Φ k of the interferometric phase. Our interferometer operates sequentially ata rate f c = 1 /T c , where τ is a multiple of T c : τ = mT c . Without loosing generality, we canchoose t k = − T c / kmT c .For large averaging times τ , the Allan variance of the interferometric phase is given by[18]: σ ( τ ) = 1 τ ∞ X n =1 | H (2 πnf c ) | S φ (2 πnf c ) . (6)8 IG. 3: Scheme of the microwave synthesis and phase locked lasers. The 100 MHz quartz signalenters a frequency chain which generates the reference microwave frequency. Its microwave outputis mixed with the lasers beatnote, and the IF is compared with a DDS signal. The error signal isused to phase lock the Raman lasers. ECL Extended Cavity Laser, DDS Direct Digital Synthesizer,PhC Photoconductor, SRD Step Recovery Diode, PBS Polarizing beam splitter, DRO DielectricResonator Oscillator.
The transfer function is thus given by H ( ω ) = ωG ( ω ), where G ( ω ) is the Fourier transformof the sensitivity function: G ( ω ) = Z + ∞−∞ e − iωt g ( t ) dt. (7)Equation 6 shows that the sensitivity of the interferometer is limited by an aliasingphenomenon similar to the Dick effect in atomic clocks [19]: only the phase noise at multiplesof the cycling frequency appears in the Allan variance, weighted by the Fourier componentsof the transfer function.
1. Reference oscillator
The previous formalism is used to analyze the specifications required for the referencemicrowave frequency [18]. We choose to generate the reference microwave signal by mul-tiplication of an ultra-stable quartz. Assuming perfect multiplication of state of the artultra-stable quartz oscillators, we find that their phase noise at low frequency will limit the9ensitivity of the interferometer phase measurement at the mrad per shot level, for our totalinterferometer time of 2 T = 100 ms. A noise of 1 mrad/shot corresponds to a sensitivityto acceleration of 1 . × − g at 1 s, for a repetition rate of 4 Hz and interaction time2 T = 100ms.Moreover, the relatively short duration of the Raman pulses makes the interferometerparticularly sensitive to high frequency noise. If we consider a Raman pulse duration of 10 µ s, a white noise floor of the reference microwave source, with a PSD of -120 dBrad /Hz,contributes to the interferometer phase noise at the level of 1 mrad/shot.The required phase noise specifications for such a reference oscillator cannot be met bothat low and high frequency by a single quartz, but can be achieved by phase locking twoquartz oscillators. We use a combination of two quartz : a Ultra Low Noise 10 MHz quartz(Blue Top from Wenzel) is multiplied up to 100 MHz, to which a 100 MHz SC Premium(Wenzel) is phase locked with a bandwidth of about 400 Hz. This system, whose performanceis indicated as trace (a)t in figure 4, was realized by Spectradynamics Inc. Measurementsperformed in our laboratory on two independent such systems confirmed these specifications.If this oscillator is multiplied to 6.8 GHz without any degradation - see trace (b) in figure 4- we calculate that its phase noise degrades the sensitivity of the interferometer at the levelof 1.2 mrad per shot.
2. Microwave frequency synthesis
The microwave chain generates the 6.834 GHz reference signal, used to lock the laserto the Raman transition, out of a stable reference quartz (see figure 3). Details on thissynthesis and its performance have been published in a previous publication [20]. We brieflyrecall its architecture in the following paragraph.In a first stage the 100 MHz output of the quartz system is multiplied by 2. Then, the200 MHz output is amplified and sent to a Step Recovery Diode, which generates a combof multiples of 200 MHz. The 35th harmonic is selected with a passive filter, and comparedin a mixer with a Dielectric Resonator Oscillator (DRO) operating at 7.024 GHz. The 24MHz intermediate frequency is mixed again with a Direct Digital Synthesizer (DDS) usinga digital phase/frequency detector. Using the phase error signal, the DRO is finally phaselocked onto the comb with an offset frequency controlled by the DDS (DDS2 on figure 3).10
10 100 1000 10000 100000-160-140-120-100-80-60 c) and d)b) S ( d B r a d / H z ) Frequency (Hz)a)
FIG. 4: Power spectral density of phase fluctuations of the reference 100 MHz oscillator at 100MHz (trace a) and at 6.8 GHz (trace b) assuming no degradation. c) and d) display the powerspectral density of the phase noise generated by the synthesis, with a digital PLL (trace c, in grey)and analog PLL (trace d, in black)
Figure 4 diplays the phase noise power spectral density of the microwave chain, which hasbeen measured by comparing the outputs of two identical chains, that shared a common 100MHz input. The degradation generated by the system on the phase noise has been measured.It contributes to 0.6 mrad/shot to the sensitivity of the interferometer measurement. Figure4 also displays the phase noise obtained when replacing the Digital phase detector by ananalog mixer, which allows to reach a lower white phase noise floor at high frequency. Still,we currently use the digital phase detector, as the lock loop is then more robust, the DROstays locked even if the DDS frequency is changed rapidly by several MHz.
3. Laser phase lock
The phase difference between the two Raman lasers is locked onto the phase of thereference microwave signal with an electronic phase lock loop (PLL) [21]. We have experi-mentally measured the residual phase noise power spectral density of our phase lock system.The measurement was performed by mixing the intermediate signal at 190 MHz and thelocal oscillator DDS1 onto an independent RF mixer, whose output phase fluctuations were11 -130-120-110-100-90-80 S ( d B r a d / H z ) Frequency (Hz)
FIG. 5: Power spectral density of laser phase noise. The measurement is performed by comparingthe laser phase with respect to the reference oscillator, with a FFT analyzer below 100 kHz, andwith a spectrum analyzer above. Note that the intrinsic phase noise of our spectrum analyzer isabout -110 dBrad /Hz at 100 kHz, which is above the lasers phase noise. The bandwidth of thePLL is 3.5 MHz. analyzed with a FFT analyzer (for frequencies less than 100 kHz) and a RF spectrum ana-lyzer (for frequencies above 100 kHz). The result of the measurement is displayed in figure 5.The phase noise decreases at low frequencies down to a minimum value of -121 dBrad / Hzat about 30 kHz. At this frequency, we found that the residual noise was not limited bythe finite gain of the PLL, but by the intrinsic noise of the PLL circuit. Above 60 kHz, thenoise increases up to -90 dBrad / Hz at 3.5 MHz, which is the natural frequency of our servoloop. The contribution of the residual noise is dominated by this high frequency part. Wecalculate a contribution to the interferometer phase noise of 1.5 mrad/shot.
4. Propagation in the fiber
In our experiment, the Raman beams are generated by two independent laser sources.The beams are finally overlapped by mixing them on a polarizing beam splitter cube, sothe beams have orthogonal polarizations. A small fraction of the total power is sent to oneof the two exit ports of the cube, where a fast photodetector detects the beat frequency.12 -1 -13 -11 -9 -7 -5 -3 With foam Without foam D SP F i b e r N o i s e (r a d † / H z ) Frequency (Hz)
FIG. 6: Power spectral density of phase fluctuations induced by the propagation in the fiber, withfoam (black trace) and without (grey foam). The two Raman lasers travel with two orthogonalpolarizations in the same polarization maintaining fiber. The noise, dominated by low frequencynoise due to acoustic noise and thermal fluctuations, gets significantly reduced by shielding thefiber with foam.
The beat note is compared with the reference signal produced by the microwave chain, inorder to phase lock the lasers. The laser beams, sent to the atoms, are diffracted through anacousto-optical modulator, used as an optical shutter to produce the Raman pulses. Duringthe interferometer, the total power is diffracted in the first order to produce the verticalRaman beams. Both beams are guided towards the atoms with a polarization maintainingfiber. Since the Raman beams have orthogonal polarization, any fiber length fluctuation willinduce phase fluctuations, due to its birefringence. We measured the phase noise induced bythe propagation in the fiber by comparing the beat signal measured after the fiber with theone we use for the phase lock. Figure 6 displays the power spectral density of the phase noiseinduced by the propagation, which is dominated by low frequency noise due to acoustic noiseand thermal fluctuations. This source of noise was reduced by shielding the fiber from theair flow of the air conditioning, surrounding it with some packaging foam. The calculatedcontribution to the interferometer phase noise is 1.0 mrad/shot. This source of noise can besuppressed by using two identical linear polarizations for the Raman beams.13 . Retro-reflection delay
The phase lock loop guaranties the stability of the phase difference between the two Ra-man lasers at the particular position, where their beatnote is measured on the fast photode-tector. Between this position and the atoms, this phase difference is affected by fluctuationsof the respective paths of the two beams over the propagation distance. In our experiment,the influence of path length variations is minimized by overlapping the two beams, and mak-ing them propagate as long as possible over the same path. However, for the interferometerto be sensitive to inertial forces, the two beams need to be counter-propagating. The twooverlapped beams are thus directed to the atoms and finally retro-reflected on a mirror.As a consequence, the reflected beam is delayed with respect to the other one. The phasedifference at the atoms position is then affected by the phase noise of the lasers accumulatedduring this reflection delay. This effect has been described in detail in [22], where the influ-ence of the frequency noise of the Raman lasers onto the interferometer phase was studiedquantitatively. From the measurement of the power spectral density of the laser frequencyfluctuations, we derived a contribution of this effect of 2.0 mrad/shot. This effect can besignificantly reduced by reducing the linewidth of the reference laser, and/or reducing thedelay by bringing the mirror closer to the atoms.
6. Overall laser phase noise contribution
Adding all the laser phase noise contributions described above, leads to a sensitivity of 3mrad/shot, which corresponds to an acceleration sensitivity of 3 . × − g/Hz / , for ourinterrogation time of 2 T = 100 ms. C. Short term fluctuations of frequency dependent shifts
Due to its intrinsic symmetry, the phase of the interferometer is not sensitive to a shift ofthe resonance frequency δν as long as this shift is constant. On the contrary, a time depen-dent frequency shift will in general lead to a phase shift of the interferometer. For instance,sensitivity to acceleration can be seen as arising from a time dependent Doppler shift. Suchfrequency shifts are also caused by Stark shifts, Zeeman shifts, cold atom interactions. Theformalism of the sensitivity function can be used to determine the influence of these effects1424], as equation (2) can be written asΦ = Z + ∞−∞ g ( t )2 πδν ( t ) dt (8)In this section we detail the effects of the dominant contributions : AC Stark shifts andquadratic Zeeman shift.
1. Light shifts
Each of the Raman lasers, as they are detuned with respect to the electronic transition5 S / − P / , induces a light shift on the two-photon Raman transition. This one-photonlight shift (OPLS) can be expressed as a linear combination of the laser intensities, δν = αI + βI . For a detuning ∆ ν smaller than the hyperfine transition frequency, the OPLS canbe cancelled by adjusting the ratio between the two laser beams : αI + βI = 0 [23]. Still,intensity fluctuations occurring on times scales shorter than the interferometer duration 2 T can lead to noise in the interferometer phase, given byΦ = Z + ∞−∞ g ( t ) h ( t )2 π ( αI ( t ) + βI ( t )) dt, (9)where h(t) = 1, during the Raman pulses, and 0 during free evolution times.Following the same formalism as used in [18], the degradation of the sensitivity can beexpressed as σ ( τ ) = 1 τ (2 παI ) ∞ X n =1 | G ′ (2 πnf c ) | ( S I /I (2 πnf c ) + S I /I (2 πnf c )) , (10)where G ′ is the Fourier transform of g ( t ) · h ( t ) and S Ii/Ii is the power spectral density ofrelative intensity fluctuations of the i-th laser [24].We measured the relative intensity noise (RIN) of both laser beams at the output of theoptical fiber. The power spectral densities of the lasers RINs are displayed in figure 7. Fromthe measurement of the resonance condition as a function of laser intensities, we determined αI = 70 kHz, from which we finally calculate a contribution of 0.8 mrad per shot.In the geometry of our experiment, where a pair of co-propagating beams is retro-reflectedin order to generate the counter-propagating laser beams, another frequency shift occurs dueto non-resonant two photons transitions. The main contribution arises from the two-photonstransition with inverted k eff , additional contributions are also present from co-propagating15 .1 1 10 100 1000 10000 10000010 -15 -13 -11 -9 -7 PS D o f R e l a ti v e I n t e n s it y N o i s e Frequency (Hz)
RAMAN LASER 1 RAMAN LASER 2
FIG. 7: Power spectral density of relative intensity noise (RIN) of the Raman lasers measured atthe output of the fiber. and magnetic field sensitive transitions, ∆ m = 2. This two photons shift, as detailed in [24]and [25], can be expressed as Φ T P LS = Z + ∞−∞ g ( t ) h ( t ) X i ¯ h Ω i δ i , (11)where Ω i and δ i are the Rabi frequency and detuning with respect to the two photonstransition. Here again, fluctuations in the intensities will induce noise on the interferometerphase. As δ i increases with time due to the increasing Doppler shift, the influence of thesecond and third Raman pulse can be neglected, and eq. 11 approximated byΦ T P LS = 1Ω X i ¯ h Ω i δ i . (12)As Ω i , Ω ∝ √ I I , this leads to δ Φ T P LS = Φ
T P LS ( δI I + δI I ) . (13)As this effect scales linearly with the Rabi frequency, it is measured with a differen-tial measurement, alternating measurements with two different Rabi frequencies. We findΦ T P LS = 40 mrad. Shot to shot fluctuations of the relative intensity are measured to be3 × − . The contribution is thus about 0.1 mrad/shot.16 . Magnetic fields In order to reduce the sensitivity to magnetic field fluctuations, the atoms are selected inthe m F = 0 state. Still, the Raman resonance condition exhibits a quadratic Zeeman shiftof δν = KB where B is the amplitude of the magnetic field and K = 575Hz/G . Magneticfield gradients will thus induce a shift of the interferometer phase given byΦ = Z + ∞−∞ g ( t )2 πKB ( t ) dt . (14)In our experiment we observe a large magnetic field gradient induced by residual magnetiza-tion of the vacuum chamber, which is made out of stainless steel. It causes a constant phaseshift of 320 mrad in our interferometer, which we reject at a level of 1 per 300 by consecutivedifferential measurements with reversed k eff -vectors. For it’s influence on the short termstability, we determine the stability of the magnetic fields by recording fluctuations of theresonance condition of a field-sensitive transition. The relative stability of the field is 10 − per shot, which induces a negligible phase shift fluctuation of 3 × − mrad per shot. D. Vibrations
As the interferometer phase shift is a measurement of the relative acceleration betweenfree falling atoms and the “optical ruler” attached to the phase planes of the Raman lasers,vibrations of the experimental setup will add noise to the measurement. With the retro-reflected geometry, the phase difference between the laser beams depends only on the po-sition of a single element, the retro-reflecting mirror. For optimal performances, it is thusmandatory to shield this element from external vibrational noise.The degradation of the sensitivity due to parasitic vibrations can easily be derived fromequation 6, by replacing S φ ( ω ) by k eff S z ( ω ) = k eff S a ( ω ) ω , where S z and S a are power spectraldensities of position and acceleration fluctuations. The transfer function of the interferom-eter thus acts as a second order low pass filter, which reduces drastically the influence ofhigh frequency noise. The sensitivity of the interferometer is finally given by: σ ( τ ) = k eff τ ∞ X n =1 | H (2 πnf c ) | (2 πnf c ) S a (2 πnf c ) . (15)17 . Isolation of the vibration noise We tested two different vibration isolation tables and compared their performances. Thetables were loaded up to their nominal load with lead bricks. Acceleration noise on theplatforms were measured with a low noise seismometer Guralp T40. Figure 8 displaysthe PSD of the residual vibration noise, compared to the noise measured directly on theground. We finally selected the passive platform (Minus K BM), which displays a betternoise between 0.5 and 50 Hz, where our interferometer is most sensitive. The experimentalsetup was then assembled on it. With respect to the spectrum of figure 8, the vibrationnoise of the experiment increased at frequencies higher than 50 Hz, due to several structuralresonances excited by acoustic noise. We therefore enclosed the experimental setup witha wooden box, whose walls were covered with dense isolation acoustic foam. The gain onthe vibration power spectrum was about 20 dB above 50 Hz. The contribution of residualvibrations to the interferometer can be calculated by weighting the vibration spectrum withthe transfer function of the interferometer [18]. We then foresee a sensitivity of 6 . × − gat 1 s. Vibrations are typically less during the night (from 1 to 5 AM) when there isno underground traffic, and the sensitivity is then 5 × − g at 1 s. Measurements of theinterferometer phase noise are in excellent agreement with the inferred vibration noise, whichsurpasses all other sources of phase noise.
2. Seismometer correction
To further improve the sensitivity of the sensor, we use the signal of the seismometerto correct the interferometer phase from the fluctuations induced by the vibration noise.An efficient rejection requires that the seismometer measures the vibrations of the retro-reflecting mirror as accurately as possible. We placed the mirror directly on top of theseismometer, which is underneath the vacuum chamber. Figure 9 displays the measuredtransition probability as a function of the phase shift calculated from the seismometer outputsignal. To obtain large variations of the interferometer phase, we deliberately increased forthis measurement the vibrational noise by having the passive platform non-floating. Thisfigure illustrates the good correlation between the seismometer signal and the interferometerphase. In the conditions of minimal noise, with a floating isolation platform, the correlation18 ,1 1 10 10010 -9 -8 -7 -6 -5 -4 v i b r a ti on no i s e a m p lit ud e s p ec t r a l d e n s it y ( g / H z - / ) Frequency (Hz)
Passive platform Active platform On the ground
FIG. 8: Comparison of the performance of two isolation platforms. We found the passive platformbehaves better in the low frequency range, where our interferometer is most sensitive to vibrations. -60 -40 -20 0 20 40 600.30.40.50.60.7 T r a n s iti on p r ob a b ilit y Phase shift calculated from the seismometer (a.u.)
FIG. 9: Interferometer signal as a function of the phase shift calculated from the seismometersignal with the passive isolation platform down. Experimental determination of the transitionprobabilities are displayed as dots. The line displays a sinusoidal fit to the data, with a constrainedamplitude. coefficient between sismometer acceleration noise and interferometer phase noise is found tobe as high as 0.94.We then studied two types of vibration compensation. The first one is a post-correction:19
IG. 10: Scheme of the real-time compensation set-up. The phase shift induced by vibrations issubtracted from the phase error signal at the output of the phase comparator. The laser phasedifference counterbalances vibrational noise, so that in the inertial frame, the laser phase planesare steady. PhC : photoconductor. the velocity signal from the seismometer is recorded during the interferometer, and wesubstract from the measured interferometer phase the calculated phase shifts due to therecorded vibrations. As the correction is applied to the transition probability, the interfer-ometer should be measuring at mid-fringe, in order to have a simple (linear) relationshipbetween the change in the transition probability and the fluctuation of the interferometerphase. This technique requires that peak to peak phase noise fluctuations remain less thana few tens of degrees and that the contrast remains constant.The second method is a real time compensation, where the seismometer velocity signal isamplified and integrated by an analog circuit. The integrated signal is subtracted from thephase error signal at the output of the comparator in the phase lock loop of the Raman lasersystem. In this feed-forward loop, the laser phase difference counterbalances vibrationalnoise, so that in the inertial frame, the laser phase planes are steady. A scheme of the setupis displayed on figure 10.In principle, the digital phase detector ensures the linearity of its voltage output withrespect to the input phase difference. Here, a maximum voltage output values ± . V corre-sponds to ± π input phase difference. For the phase lock loop to remain active, the voltageat the output of the integrator has to correspond to a phase that remains within this range.This was ensured by resetting the integrator at each cycle and by compensating the intrinsic20ffset of the seismometer output. As this offset fluctuates, due to either electronic noise orlow frequency vibrations, the compensation is realized by i) acquiring the seismometer signalwith a digital card at the end of the previous measurement cycle, ii) calculating its averagevalue, iii) ouputting it with an analog voltage board at the beginning of the next cycle, andiv) subtracting this last value from the seismometer signal before being integrated. Despiteall these precautions, the range of the phase compensation had to be increased by a factor2 (dividing the intermediate frequency signal by 2 before the phase comparator), in orderfor the phase lock loop to remain active.In principle, this last technique is more powerful, as it remains efficient even if the contrastchanges, and can compensate large phase fluctuations, if the effective dynamic of the mixeris large enough. In practice we found that the digital phase detector had two drawbacks.i) The residual non-linearity of the output signal is enough to induce a large bias to theinterferometer phase (the linearity should be at the mrad level for typical phase excursionsof about 1 rad over 100 ms.) ii) Operation far from null output increases the phase noise.To circumvent these problems, the phase compensation could be performed in the phaselock loop of a second quartz onto the 100 MHz signal, with an analog mixer. This increasesthe dynamics by more than one order of magnitude and guarantees the linearity. But, as onour platform, the vibration noise is low, we finally chose the post-correction method, whichis simpler and more robust.In both cases, the rejection efficiency was limited to a factor of 3, corresponding to atypical sensitivity of 2 × − g at 1s.
3. Seismometer response function
In order to study the transfer function between the seismometer and the retro-reflectingmirror, we induce a platform oscillation at given frequencies (by running ac-currents througha loaded loudspeaker placed onto the isolation platform) and record simultaneously theatomic and seismometer signals. The excitation frequencies f exc were chosen close but notexactly equal to multiple of the cycling frequency f exc = kf c + δf , so that the transitionprobability and its correction calculated from the sismometer signal were modulated intime, at an apparent frequency controlled by δf . The sismometer transfer function can bedetermined from the ratio of the amplitudes of the modulation of the two signals and their21
10 100-180-900 -10-8-6-4-202 G a i n ( d B ) P h a s e ( d e g r ee s ) Frequency (Hz)
FIG. 11: Transfer function between the acceleration measured with the interferometer and theseismometer. The transfer function agrees well with the response function of the seismometer(solid lines). phase difference. The measured transfer function is displayed in figure 11. It correspondsvery well to the response of the seismometer. This response, which is provided by themanufacturer, can also be retrieved by the user using an internal measurement protocol, seesolid lines in figure 11. As one can see from these curves, the seismometer has a built-in lowpass filter to cut mechanical resonances, which occur typically above 450 Hz. This filter hasa 3 dB cut-off frequency of 50 Hz. This adds some phase shifts to the calculated correctionwith respect to the real perturbation for higher frequencies. This corrupts the rejectionefficiency and eventually adds noise. To improve the rejection, we would either need toflatten the transfer function, or to cut frequencies at which the rejection process degradesthe sensitivity.
4. Digital filtering
We apply a digital filter to the seismometer signal to compensate for attenuation andphase shifts. In the ideal situation, where the transfer function would be made perfectlyflat, meaning that the efficiency of the rejection would be 100% for each frequency. Thelimitation of the vibration rejection would arise from the intrinsic noise of the seismometer,22hich strongly depends on the frequency. When taking this noise into account, it appearsuseless to literally flatten the seismometer response. We numerically found that a simplefirst order digital filter compensates the response function well enough to reach a sensitivityof 5 × − g at 1 s, despite the high order of the internal low pass filter of the seismometer.As for the noise spectrum given by the manufacturer, we calculated that its limitation tothe interferometer sensitivity would amount to 2 × − g at 1 s.Since the vibration signal is processed with the computer, a digital filtering seems veryfavorable. We use a recursive IIR (Infinite Impulse Response) filter with the following shape:a unity gain below the lower frequency f , an increasing slope of 20 dB/decade from f to f , and a constant gain above f . We also take benefit of the post-correction process toimplement a non-causal low-pass filter (NCLPF). Such a filtering consists in processing thesampled data in a forward and backward sense with respect to time. A positive phase shiftinduced by the direct reading will be canceled by the reversed one, whereas the attenuationis applied twice. In our case, the NCLPF prevents the IIR filter from amplifying the intrinsicnoise of the seismometer at high frequencies, without affecting the phase advance needed toimprove the rejection. More precisely, the corner frequency of the NCLPF corresponds tothe frequency above which the seismometer signal doesn’t carry any useful information.After combining the IIR and low-pass non causal filter and before implementing themduring the interferometer measurement, we checked their effect on the amplitude and thephase shift of the vibration signal, by exciting the platform again and comparing the seis-mometer signal, with and without filter. Excellent agreement with the expected behaviorwas found.As the atomic signal was also recorded during this measurement, the influence of thefilter on the efficiency of the vibration phase correction could be demonstrated directly onthe interferometer signal. To illustrate the gain on the rejection, the modulation of theinterferometer signal is displayed in figure 12 a) for an excitation frequency of 14 Hz inthe different cases : i) we apply no correction, ii) the correction without filter and iii) thecorrection with the digital filtering. The rejection efficiencies are displayed on figure 12 b) asa function of the excitation frequency. They are obtained by calculating the ratio betweenthe amplitudes with and without correction, for the two cases where the digital filter is usedor not.We then implemented the digital filtering in the interferometer phase correction and23
10 100-40-30-20-10010 R e j ec ti on ( d B ) Frequency (Hz)
Without Filter With Filter T r a n s iti on P r ob a b ilit y Number of measurement
No correction Corrected without filter Corrected with filter
FIG. 12: Left: modulation of the transition probability for an excitation frequency of 14 Hz,without correction, with correction, with and without filter. Right: measured rejection efficienciesversus frequency, with and without digital filter. operated the interferometer with the nominal vibration noise. First measurements withthe interferometer showed a resolved influence of the filter, but the rejection was improvedby only 15%. Putting neoprene rubbers below the seismometer legs, the vibrations above30 Hz are well damped, so that the signal above this frequency reaches the seismometerintrinsic noise. This way, we reduce the contribution of the “high” frequencies, for which itis difficult to well compensate the response of the seismometer. Nevertheless, the filter doesnot improve the rejection efficiency by more than 25%, whereas the calculation predicts animprovement by a factor of 3.At night and with air conditioning switched off, the sensitivity reaches its best level.Considering the standard deviation at one shot, we deduce an equivalent noise of 1 . × − gat 1 s (see figure 13). Deviation from the expected τ − / behavior could be due to thecrosstalk with the horizontal directions (see III D 5), or to fluctuations of the systematics,most probably to intensity fluctuations. In this situation, with low environmental noise,we find that the filter has no influence, which seems to indicate that the sensitivity is notlimited anymore by vibrational noise.This sensitivity corresponds to phase fluctuations of 11 mrad/shot, which exceeds thelevel obtained when summing (quadratically) all other contributions (4 mrad). As themeasurements of laser phase noise were performed in steady state condition, one cannot24 .1 1 10 100 -9 -8 g / g Time (s)
FIG. 13: Allan standard deviation of the interferometer phase fluctuations. The sismometer signalis filtered before calculating the phase correction. The air conditioning is switched off. exclude differences in the noise spectra when the interferometer is operated sequentially, aslaser frequencies undergo abrupt changes and sweeps and Raman lasers are pulsed. Intrinsicnoise of the sismometer, if higher than rated by the manufacturer, could also be responsiblefor the observed higher noise level.
5. Seismometer intrinsic noise
First, since coils are used as actuators inside the sismometer, excess noise could be due tomagnetic field fluctuations. We therefore measured the sismometer response to magnetic fieldfluctuations, by modulating the current in a coil placed around it. Having then measuredthe PSD of ambient magnetic field fluctuations in the laboratory, we finally calculated theequivalent vibration noise, and found less than 1 × − g / Hz − / , which rules out magneticfield fluctuations.We then tried to measure the intrinsic noise of the sismometer, by stacking up two suchdevices and subtracting their output signals with a low noise differential amplifier. Figure 14diplays the result of this differential measurement, as well as the vibration noise measuredby the sensors and the intrinsic noise given by the manufacturer. The rejection efficiency athigh frequency (above 10 Hz) is poor, as expected, due to the difference between the transferfunctions of the two sensors, which prevents reaching their intrinsic noise. More surprisingly,25 .1 1 10 10010 -10 -9 -8 -7 -6 -5 V i b r a ti on no i s e a m p lit ud e s p ec t r u m d e n s it y ( g / H z / ) Frequency (Hz)
Difference between the seismometers (0(cid:176)) Vertical noise measured by one seismometer Difference between the seismometers (180(cid:176)) Intrinsic noise (from manufacturer)
FIG. 14: Differential measurements between two sismometers. The grey thick curve displays thedifference between seismometers (with identical orientations in the horizontal plane)and the blackcurve the signal from one of the two sismometers. The dotted curve displays the differential signalfor a relative orientation of 180 degrees. The dotted line displays the intrinsic noise of the sensor,as given by the manufacturer. we find around 2 Hz a poor rejection (only 14 dB) and a broad structure in the spectrumof the differential signal. The output signals from both sismometers being in phase at lowfrequency and the difference in their scale factor being less than 1%, the rejection efficiencyshould reach about 40 dB. The same broad peak also appears in the horizontal accelerationnoise spectrum on the platform, about 30 times higher, up to 6 × − g . Hz − / at 2 Hz. Wethus attribute this residual noise to crosstalk (at the level of a few %) between horizontal andvertical directions. This assumption was further confirmed by noticing that the rejection wasconsiderably spoiled when positioning the sensors with different orientations in the horizontalplane: see on figure 14 the differential signal for an angle between the horizontal axes ofthe two devices of 180 degrees. Still, this parasitic contribution to the vertical accelerationnoise at 0 degree is probably partially rejected in the difference, but not completely, dueto a difference in the amplitude of the crosstalks, or to the fact that the two devices notbeing at the same position don’t see exactly the same horizontal acceleration. This crosstalkcould then constitute the limit in the efficiency of any 1D vibration compensation scheme.A quantitative evaluation of the impact of this effect deserves further studies, and wouldrequire the determination of the crosstalk amplitude and transfer function.26 . Other contributions related to transverse displacement Many other effects can affect the interferometer phase. In particular, the most importantsystematic shifts are related to Coriolis acceleration and wavefront aberrations.The Coriolis acceleration leads to a Sagnac effect. If the atoms are released from themolasses with a transverse velocity as low as 100 µ m/s, the shift on the interferometer signalleads to a bias as large as 10 − g. If the velocity distribution is symmetric and centeredaround zero, then this effect is canceled when detecting the atoms, provided the detectionefficiency is homogeneous across the whole cloud. Experimental inhomogeneities will ingeneral lead to a residual shift. We have measured the fluctuations of the mean velocity ofthe atomic sample in the horizontal plane by performing absorption imaging at two differentdelays after releasing the atoms. We found short term fluctuations on the order of 10 µ m/sshot. The equivalent noise is 0.03 mrad/shot, which is negligible with respect to the othereffects studied above.Wavefront aberrations induce an interferometer phase shift that depends on the trajec-tories of the atoms. A quantitative evaluation of this effect was performed in [30]. Shortterm fluctuations of the positions (and velocities) of atoms released from a moving molasseswere found to limit the sensitivity of their gyro-accelerometer at the level of 0.2 mrad/shot.This limit depends on the details of the wavefront distortions and of the atomic trajectories.It is expected to be different in our geometry, as free falling atoms remain at the center ofthe laser beam during the interferometer. We have investigated the influence of the velocityfluctuations on the interferometer phase by unbalancing the power in the molasses beams.We found phase shifts of 0.4 mrad per percent change in the intensity ratio, which corre-sponds to 40 µ m/s velocity change. Velocity fluctuations thus induce phase instability atthe level of 0.1 mrad/shot. IV. CONCLUSION
We have extensively studied the different sources of noise in an atom interferometer, andtheir influence on the short term sensitivity of the gravity measurement.We have demonstrated that a very high sensitivity can be achieved even with a moderateinterrogation time of only 100 ms. This requires an excellent control of laser phase fluctu-27tions and efficient detection schemes. We also show that the sensitivity can be efficientlyimproved by compensating the phase shifts induced by vibrations, using the signal of a lownoise seismometer, down to a level limited by intrinsic noise of the sensor and/or by crosstalkbetween the different measurement axes.Our final sensitivity is more than twice better than ”classical” corner-cube gravime-ters. The measurement at the same location, and in the same vibration environment, withFG5 × − g.More generally, the work presented here allows to quantify the performances of atominterferometers as a function of interaction time, cycling rate, and sources of perturbations.The same formalism can be used for the design of ultimate sensitivity instruments (such asa space interferometer for instance [20]), as well as for the realization of lower level compactinstruments. In particular, the compensation technique that we have demonstrated in thispaper is particularly attractive for the development of a simple and compact instrument,which could reach high sensitivities without vibration isolation. Acknowledgments
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