Atom Interferometry II. F. Pereira Dos Santos

Transcription

1 Atom Interferometry II F. Pereira Dos Santos

2 Organization of the lecture 1 : Applications of inertial sensors/gravimetry : A case study : the SYRTE atom gravimeter 3 : Towards more compact sensors 4 : Other instruments : gravimeters, gradiometers 5 : Atomic gyroscope

3 Organization of the lecture 1 : Applications of inertial sensors/gravimetry : A case study : the SYRTE atom gravimeter 3 : Towards more compact sensors 4 : Other instruments : gravimeters, gradiometers 5 : Atomic gyroscope

4 Inertial sensors Inertial navigation satellite, submarine Fundamental physics measurement of a, G... watt balance (gravimeter) test of general relativity Lense-Thirring effect (gyroscope) WEP, anomalous gravity (accelerometer) gravitational waves g Geophysics ground or space Earth s rotation rate, tidal effects, gravity field mapping and monitoring

5 Gravimetry Deduce from local variations of gravity density variations in the ground Measure small variations with transportable sensors microgravimetry : 1 µgal ~ 10-9 g with g ~9.8 m/s = 980 Gal Looking for small (temporal and local) changes µgal Variation of g with height : 3 µgal/cm

6 Simple examples in geophysics Measurement of (Bouguer) gravity anomaly to map influence of mass distribution, detect changes Deformation and constraints: accumulation during seismic cycle... Volcanology: 4D mapping of mass distribution evolutions

7 Measurements at all scales

8 World gravity map

9

10 Fundamental metrology: watt balance Goal: Measure Planck constant h with a 10-8 relative accuracy Interest: Replace the Kg etalon by a definition linked to h Static phase Dynamic phase B F Laplace = i L B j.dl B.i.L = m.g F g = m.g U m g v = U i B v U = B.L.v Quantum Hall and Josephson effects Planck constant Accurate measurement of g needed

11 Organization of the lecture 1 : Applications of inertial sensors/gravimetry : A case study : SYRTE atom gravimeter 3 : Towards more compact sensors 4 : Other instruments : gravimeters, gradiometers 5 : Atomic gyroscope

12 Interferometer configuration 3-Raman pulses interferometer = Mach Zehnder/ Chu-Bordé interferometer T T

13 Acceleration phase shift F( t) = k. r( t) eff 1 at T F T F 3 a F 1 F 1 ( t 1 ) = 0 1 F ( t) = keff. at. at DF = F 1 (t 1 ) F (t ) + F 3 (t 3 ) = k eff F 1 3 ( t3) = keff. a (T )

14 Gravimeter : Implementation of Raman lasers Laser 1 Vertical Raman lasers DF = k eff.g.t Pulse 1 Pulse z = 0 1 z ( T ) = gt Retroreflect two (copropagating) Raman lasers Reduces influence of path fluctuations (common mode) 4 laser beams pairs of counterpropragating Raman lasers with opposite k eff wavevectors Pulse 3 z ( T ) = gt Position of planes of equal phase difference attached to position of retroreflecting mirror Miroir Laser Interferometer measurement = relative displacement atoms/mirror

15 SYRTE Gravimeter vacuum chamber Titanium vacuum chamber (non magnetic) viewports Indium seals Pumps : getter pumps 50 l/s 1 ion pump l/s 4 getter pills Two layers magnetic shield Retroreflecting mirror under vacuum

16 Drop chamber Commercial fiber splitters Fibered angled MOT collimators Symmetric detection 16

17 Laser requirements 5 different frequencies : Cooling+Repumper + Raman 1& + Detection Cooling Repumper Detection Pusher

18 Optical bench «Compact» laser system : 60 x 90 cm 3 ECDL, TA Key feature : Use the same lasers for Cooling/Repumping and Raman beams Frequency control thanks to offset locks (using FVC) and/or PLL REFERENCE/ DETECTION REPUMPER/ RAMAN1 COOLING/ RAMAN1

19 Sequence D + 3DMOT and far detuned molasses : ~ 100 ms Loading rate ~ 10 9 atoms/s 10 8 atoms at T~µK State preparation : ~ 15 ms Use of a combination of MW/pusher/Raman pulses ~10 7 atoms in F=1, mf=0 δv~1v r Interferometer : T~140 ms Raman pulses of typically ~ µs Detection : ~ 80 ms Total cycle time : ~ 360 ms

20 Fluoresecnce Signal (a.u.) Horizontal detection Diaphragm Atoms blocker Mirror Repumper Detection λ/4 Collection efficiency : 1% Saturation parameter : x Time of flight signals Time (ms) F= F=1

21 Fluorescence Signal (u.a.) Vertical detection Raman Vertical Beam back to resonance during the detection 1) First pulse at low intensity to slow down the F= atoms (cooling beam only) ) Wait for the F=1 atoms to reach the bottom position 3) Second pulse at full power (cooling + repumper beams) 4) Third pulse for background substraction 100 nd Pulse F= F= F= PD rd Pulse F=1 400 PD Time (ms) Collection efficiency : 1% Saturation Parameter : x 5

22 Fringe pattern Free fall Doppler shift of the resonance condition of the Raman transition Ramping of the frequency difference to stay on resonance : = a t 0 DF = k eff.g.t - at π/ π π / 0, 0,1 0,0-0,1-0, 5,140 3,5 4,0 5,14 4,55,144 5,0 5,1465,5 5,148 6,0 5,150 6,5 a/ (MHz/s)

23 Probabilité de transition Principle of measurement, interferometer fringes Free fall Doppler shift of the resonance condition of the Raman transition 1 = e k v( t) m Ramping of the frequency difference to stay on resonance : f DDS1 (Hz) a t π/ π π / eff k eff = DF = k eff gt at C=45% a (MHz.s -1 ) 3

24 Probabilité de transition Principle of measurement : interferometer fringes Free fall Doppler shift of the resonance condition of the Raman transition 1 = e k v( t) Ramping of the frequency difference to stay on resonance : f DDS1 (Hz) a t π/ π π / eff k eff m = DF = keff g T a T C=45% 0.4 Dark fringe : independent of T g = a 0 k eff The measurement of g is a frequency measurement a (MHz.s -1 ) 4

25 Probabilité de transition Principle of measurement : interferometer lock 0,8 a a 0 a a 0 0,7 0,6 a Digital Lock : a > aa 0,5 0,4 P - P + a 0,3 5,1435 5,1440 5,1445 a/ (MHz/s) P a = K( P P )

26 Sensitivity P C = A cos( DF) F = P C g = k eff P CT The sensitivity depends : on the noise on the measurement of P contrast of the fringes scaling factor (keff T )

27 Probabilité de transition Noise DDS1 (Hz) Parameters T=140 ms t = 0 µs s v ~ v r N det = T c = 360 ms Contrast ~ % a (MHz.s -1 ) σ Φ = 1/SNR = 160 mrad/shot s g /g ~ 10-7 /shot Sources of noise - Detection noise - Laser phase noise - Mirror vibrations

28 Detection noise A 1 s = P N 4N C Electronic Noise Quantum Projection Noise s P 10 - s P Vertical Detection s P Horizontal Detection Quantum Projection Noise Detection Noise Detection Noise σ Φ =σ P /C C ~ 50% T = 70 ms & Tc= 360 ms 10 4 atoms : 40 mrad g/shot g at 1s 10-3 QPN Number of atoms 10 6 atoms : mrad g/shot g at 1s

29 Sensitivity function g(t) Quantifies the sensitivity of the interferometer to phase fluctuations t g( t) = lim 0 P Hypothesis : << R, Rt = / Plane waves Neglect spatial phase inhomogeneities at the scale of the separation between wavepackets T T τ -T-τ 0 τ T+τ T+τ

30 g( t) = lim 0 Sensitivity function g(t) P Method of calculation : Evolution matrix Expression for the matrix : Raman pulse Free evolution Total evolution in the interferometer : product of such matrices : Result :

31 Sensitivity function g(t) t Split a matrix in two : Example if the phase jump occurs dring the first pulse In the case of a phase jump : t t t t t 0 )) ( sin( 1 ) sin( ) ( = T t T T t t T t t t g R R Result (for t>0) :

32 Sensitivity function g(t) The sensitivity function is an odd function

33 Experimental demonstration P. Cheinet et al., "Measurement of the sensitivity function in time-domain atomic interferometer", IEEE Transactions on Instrumentation and Measurement 57, (008)

34 ( 0 ) v gt k v k dt dz k dt d eff eff eff = = = Gives the scale factor : = dt dt d t g P t ) ( 1 ) ( = = F s ) ( ) ( ) ( ) ( d S H d S G In the spectral domain : )) sin( ) ) ( )(cos( ) ( sin( 4 ) ( T T T i H R R R t t = Response to a time dependent perturbation Example : gravity!! : ) 4 ( ) ( t t = D T T g k eff Time dependent phase fluctuations

35 <H(f) > Transfer function H(ω) 4i R H( ) = R ( T sin( t ) ( T )(cos( t ) ) R T sin( )) H(f) 10 t Frequency (Hz) Insensitive to frequencies multiples of 1/(T+τ) /(f) 1E Frequency (Hz) 3 = Finite duration of the pulses Low pass filtering 0

36 Degradation of the stability Stability characterized by the Allan standard deviation Sampled measurement : f c =1/T c => Aliasing For long enough averaging time, only components at frequencies at multiples of the cycling frequency contribute s ( t ) = F m t 1 m n= 1 H ( nf c ) S ( nf c ) White phase noise ( S 0 ) => s F ( t ) = m S t 0 T t c m

37 Derivation of the stability Allan standard deviation where δφ k is the k-th average of the interferometer phase over duration τ m Using the sensitivity function, it can be expressed as where

38 Derivation of the stability The difference between consecutive averages can be expressed as : Assuming uncorrelated averaged samples with As We finally get

39 Sensitivity function g(t) The sensitivity function is a simple and powerful tool Allows to calculate the influence of all kinds of perturbations : - Laser/microwave reference phase/frequency noise - Magnetic field fluctuations - Light shift fluctuations - Vibrations -

40 Laser phase stability The two Raman lasers are phase locked onto an ultra-low noise µwave oscillator The µwave oscillator is phase locked onto an ULN Quartz oscillator The Raman phase is also perturbed by non common perturbations during propagation ECL1 ECL PLL DDS 190 MHz PhC 6,834 GHz 7,04 GHz HF synthesis L ~ 1 m 100 MHz

41 S F (dbc) Laser phase stability : influence of ULN Quartz 100 MHz reference signal realized by phase locking an ULN 100 MHz oscillator onto an ULN 5 MHz oscillator s ( t ) = F m t 1 m n= 1 H ( nf c ) S ( nf c ) Contribution to s F T=70 ms Total : => => ~ Frequency (Hz) Frequency (Hz) Contribution to the noise of the interferometer dominated by low frequencies

42 Influence of laser phase noise ECL1 Source σ Φ (mrad/coup) σ g (g/hz 1/ ) ECL Référence 100 MHz 1,0 1, DDS 190 MHz PLL PhC 6,834 GHz L a s e r s Synthèse HF 0,7 Résidu PLL 1,6 Fibre optique 1,0 Rétro-réflexion,0 0,9 10-9, ,3 10-9, ,04 GHz L ~ 1 m Total 3,1 3, HF synthesis Détection MHz Total {lasers+détection} 5,0 6, (Evaluation of noise sources for T=50 ms) J. Le Gouët et al., Appl. Phys. B 9, (008)

43 Vibration noise A fluctuation δz of the lasers equiphase induces a change δφ=k eff δz As S ( ) = k S ( ) = k eff z eff Sa( ) 4 the contribution to the stability can be expressed as T=100 ms Transfer function for acceleration noise Sensitivity to low frequencies

44 Vibration isolation 1 s g ( t ) = t k = 1 sin( kf kf c c T T) 4 S a (kf c ) Passive isolation platform : Natural frequency 0.5 Hz Acoustic + air flow isolation Corresponding 1s :, g when OFF (day) 7, g when ON (day) => Gain : factor 40

45 Residual vibration corrections Mirror Seismometer Interferometer Measure vibrations with a seismometer at the same time Convert the seismometer signal into an interferometer phase shift s vib = k eff K T s g s (t) U s (t) dt T Passive platform Good correlations with the calculated phase shift the seismometer is used to correct the atomic measurement

46 Residual vibration corrections Seismometer PC k eff gt² Post correction v(t) φ vib S Interferometer k eff gt² + φ vib S Gain : from to ~10 Typical sensitivity σ g /g = at 1s Resolution ~ 10-9 g after s

47 Phase (degrés) Seismometer transfer function High pass filter : 30 s (0.03 Hz) Low pass filter (mechanical resonances) : f c = 50 Hz Phase shift between seismometer signal and mirror vibrations Rejection at highfrequency will be limited Mesure fabricant Mesure interféromètre Correction : F ( ) = F( ) H( ) F ( ) = (1 H( )) F( ) c f 100 c =50 Hz Fréquence (Hz) Gain : G( ) F( ) 1 = ( ) = F 1 ( ) c H

48 Probabilité de transition Réjection (db) Non linear filtering Implement a digital filter Goal : compensate the phase shift of the seismometer F( f ) = 1 1 jf jf / / f f ( f 1 / f c ) 0.6 Aucune correction Post-correction sans filtrage Post-correction avec filtrage 10 0 Sans filtrage Avec filtrage Temps (s) Fréquence (Hz) Expected gain of a factor of ~ 3 => σ g < 10-8 g Hz -1/

49 Filtre NL, results Best short term sensitivity : s Modest gain (<expected 3 ) D correction 3D correction Self noise of the seismometer : g.hz -1/ (according specs) Cross-couplings : Acquisition of 3 axes is necessary s g /g Averaging Time (s) Alternative technique (simpler) : compensate for the delay Same results when implementing a delay of 4.6 ms

50 Acquisition without isolation platform Seismometer PC k eff gt² Post corrections v(t) φ vib S Interferometer k eff gt² + φ vib S Influence of the filtering much larger in the presence of large vibration noise Makes measurements feasible in noisy environment: outside the lab in mobile vehicule S. Merlet, et al., Metrologia 46, 87 (009)

51 Fringe fitting Post correction of mirror vibrations with Fringe fitting s vib P = a b cos( and mod ± / = k eff K s T T gs (t) Us (t) dt j vibs, j ) j=x, y,z Night Transition probability Transition probability Day s (rad) vib s Gain : S 5 (rad) vib s Gain : 30 In the presence of high anthropic noise, we have reached gains as high as 100!!

52 Systematics 1- Some effects are k eff orientation dependent DF int = k eff g T DF LS DF Coriolis DF Ab DF RF DF LS1 DF Zeeman k eff Dependent of k eff orientation Independent f,p g 5

53 Systematics 1- Some effects are k eff orientation dependent DF int = k eff g T DF LS DF Coriolis DF Ab DF RF DF LS1 DF Zeeman k eff Dependent of k eff orientation Independent f,p g DF DF = k eff gt DF dependent k eff Reject independent effects by switching between k eff / : need two measurements 53

54 Systematics 1- Some effects are k eff orientation dependent DF int = k eff g T DF LS DF Coriolis DF Ab DF RF DF LS1 DF Zeeman k eff Dependent of k eff orientation Independent f,p g DF DF = k eff gt DF dependent k eff - DF Reject DF LS eff LS bias alternating measurements eff and eff / DF, / DF, / DF, DF, = k eff gt DF Coriolis DF Ab The algorithm degrades the sensitivity by 10 54

55 1 days of continuous measurement Gal? 1 µgal = 10-8 m.s g Gray: 400 drops (170 s) Black: s 55

56 Long term sensitivity g (4 conf) (-)/ (+)/ Sensitivity flickers at g Limited by tidal model? Water table? Soil moisture? Instrument? 56

57 Coriolis acceleration Non zero transverse velocity sensitivity to the Sagnac effect Δg = 10-9 g for v 0 = 100 µm.s -1 Ω k eff v For T = µk, σ v ~ 1 cm.s -1 = 100 v 0!!! g Average shift is null if 1) velocity distribution symmetric ) detection symmetric 3) mean velocity is zero Solutions Control v 0 (CCD imaging, Raman velocimetry/selection) Colder atoms Alternate ±v 0 (rotate the experiment by 180 )

58 g (µgal) Coriolis acceleration Handle on the mean velocity : power imbalance EO molasses beams Alternate two opposite orientations within hours ORIENTATION N ORIENTATION S Opposite trends for the two orientations signature of Coriolis shift Trend is not so linear changes in the aberration shift? why linear anyway? Imbalance in the molasses beams i=(p-p1)/(p+p1) At the crossing point : Null (mean) Coriolis shift

59 Dg (µgal) Coriolis acceleration Changes in Coriolis Vs changes in the wavefront distortions shift? Half difference : Coriolis Half sum : changes in the aberration shift 30 0 Half difference Null Coriolis bias Half sum Imbalance in the molasses beams i=(p-p1)/(p+p1) Uncertainty in the correction of the Coriolis shift : 0.4 µgal

60 Wavefront aberrations Wavefronts are not flat : gaussian beams, flatness of the optics Case of a curvature δφ = K.r (with K = k 1 /R) t = 0 t = T 1 =(v r = 0) v r 0 > 1 R (v = 0) Δg < 10-9 g with T = µk R > 10 km! flatness better than λ/300!!! t = T 3 > D 0 D = 0 (v = 0) Measure aberrations with wavefront sensor + excellent optics + colder atoms

61 Characterization of the optics Mirror 40mm diameter PV= l/10 RMS =l/100 Simulation : T =.5mK s = 1.5mm g/g = PV l/4 g/g =

62 Mirror + l/4 plate under vacuum

63 Dg (µgal) Dg (µgal) Accuracy Exploration of the wavefronts : Differential measurements for different cloud temperatures and extrapolation to zero Analysis with a set of Zernike polynomials => No reliable extrapolation n= n= T at (µk) T at (µk) Need to limit the expansion of the cloud: colder atoms and/or use horizontal Raman selection

64 Self gravity effect Dg = (1.3 ± 0.1) 10-8 m.s - G. D Agostino et al, Metrologia 48 (011) 64

65 Accuracy budget 65