atom interferometry ii - uni-hannover.de · v u b goal: measure planck constant h with a 10-8...

F. Pereira Dos Santos

Atom Interferometry II

Organization of the lecture

1 : Applications of inertial sensors/gravimetry

2 : A case study : the SYRTE atom gravimeter

3 : Towards more compact sensors

4 : Other instruments : gravimeters, gradiometers

5 : Atomic gyroscope

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

Geophysics ground or space

Earth’s rotation rate, tidal effects, gravity field mapping and monitoring

g

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/s2 = 980 Gal

✓ Looking for small (temporal and local) changes 1 - 100 µGal

✓ Variation of g with height : 3 µGal/cm

Simple examples in geophysics

✓ Measurement of (Bouguer) gravity anomaly to map influence of mass distribution, detect changes …

✓ Volcanology: 4D mapping of mass distribution evolutions

✓ Deformation and constraints:

➡ accumulation during seismic cycle...

Measurements at all scales

World gravity map

Fundamental metrology: watt balance

Static phase Dynamic phase

B.i.L = m.g B.i.L = m.g

U = B.L.v U = B.L.v

m g v = U i m g v = U i

Planck

constant

Accurate measurement of g needed

B Fg = m.g

FLaplace = i L ´ B

j.dL

v U

B

Goal: Measure Planck constant h with a 10-8 relative accuracy Interest: Replace the Kg etalon by a definition linked to h

Quantum Hall and

Josephson effects

Organization of the lecture

1 : Applications of inertial sensors/gravimetry

2 : A case study : SYRTE atom gravimeter

3 : Towards more compact sensors

4 : Other instruments : gravimeters, gradiometers

5 : Atomic gyroscope

Interferometer configuration

3-Raman pulses interferometer = Mach Zehnder/ Chu-Bordé interferometer

T T

Acceleration phase shift

2. Takeff

DF = F1(t1) – 2F2 (t2) + F3 (t3) =

2

33 )2(.2

1)( Takt eff

=F

0)( 11 =F t 2

22 .2

1)( Takt eff

=F

a

1F

2F3F

T T)(.)( trkt eff

=F

2

2

1ta

Gravimeter : Implementation of Raman lasers

Miroir

0=z

2

2

1)( g TTz =

22)2( g TTz =

Laser 2

Laser 1

Pulse 1

Pulse 2

Pulse 3

Interferometer measurement = relative displacement atoms/mirror

• Vertical Raman lasers

• Retroreflect two (copropagating) Raman lasers

Reduces influence of path fluctuations (common mode) 4 laser beams 2 pairs of counterpropragating Raman lasers with opposite keff wavevectors

• Position of planes of equal phase difference attached to position of retroreflecting mirror

DF = keff.g.T2

• Titanium vacuum chamber (non magnetic)

• 14 + 2 + 4 viewports

• Indium seals

• Pumps : 2 × getter pumps 50 l/s 1 × ion pump 2 l/s 4 × getter pills

• Two layers magnetic shield

• Retroreflecting mirror under vacuum

SYRTE Gravimeter vacuum chamber

Drop chamber

16

Commercial fiber splitters

Fibered angled MOT collimators

Symmetric detection

Laser requirements

Detection Pusher

Cooling Repumper

5 different frequencies : Cooling+Repumper + Raman 1&2 + Detection

Optical bench « Compact » laser system : 60 x 90 cm

3 ECDL, 2 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

Sequence

• 2D + 3DMOT and far detuned molasses : ~ 100 ms Loading rate ~ 109 atoms/s 108 atoms at T~2µK

• State preparation : ~ 15 ms Use of a combination of MW/pusher/Raman pulses ~107 atoms in F=1, mF=0 δv~1vr

• Interferometer : 2T~140 ms Raman pulses of typically ~10-20-10 µs

• Detection : ~ 80 ms

• Total cycle time : ~ 360 ms

λ/4

Repumper

Detection

Diaphragm Atoms blocker

Mirror

Collection efficiency : 1% Saturation parameter : 2 x 0.5

0 10 20 30 40 50 60

-100

0

100

200

300

400

500

600

700

800

Flu

ore

secn

ce S

ign

al (

a.u

.)

Time (ms)

F=2

F=1

Time of flight signals

Horizontal detection

Collection efficiency : 1% Saturation Parameter : 2 x 25

Raman Vertical Beam back to resonance during the detection

1) First pulse at low intensity to slow down the F=2 atoms (cooling beam only) 2)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

F=2

F=1

PD1

PD2

0 10 20 30 40

0

200

400

600

800

1000

1200

3rd Pulse

Flu

ore

scen

ce S

ign

al (

u.a

.)Time (ms)

F=2

F=12nd Pulse

Vertical detection

DF = keff.g.T2 - aT2

Fringe pattern

π/2 π π /2

t0 a =

23,5 24,0 24,5 25,0 25,5 26,0 26,5

-0,2

-0,1

0,0

0,1

0,2

a/2 (MHz/s)

25,140 25,142 25,144 25,146 25,148 25,150

-0,2

-0,1

0,0

0,1

0,2

a/2 (MHz/s)

• Free fall → Doppler shift of the resonance condition of the Raman transition

• Ramping of the frequency difference to stay on resonance :

-25.1435 -25.14300.2

0.3

0.4

0.5

0.6

0.7

-125.718 -125.716 -125.714

Pro

bab

ilit

é d

e t

ran

sit

ion

a (MHz.s-1)

DDS1

(Hz)

Principle of measurement, interferometer fringes



π/2 π π /2

ta =0

C=45%

m

ktvk

eff

fe eff

2)(

2

21

=

22 TTk eff a=DF g

23

-25.1435 -25.14300.2

0.3

0.4

0.5

0.6

0.7

-125.718 -125.716 -125.714

Pro

bab

ilit

é d

e t

ran

sit

ion

a (MHz.s-1)

DDS1

(Hz)

Principle of measurement : interferometer fringes



π/2 π π /2

ta =0

C=45%

m

ktvk

eff

efffe2

)(

2

21

=

• Dark fringe : independent of T

g =a0

keff

DF = keff g T2 a T2

24

The measurement of g is a frequency measurement

25,1435 25,1440 25,1445

0,3

0,4

0,5

0,6

0,7

0,8

Pro

bab

ilité

de t

ransitio

n

a/2 (MHz/s)

a

)( = PPKa

0aa

P+

0aa

P-

Digital Lock :

a > aa

Principle of measurement : interferometer lock

P

a

Sensitivity

The sensitivity depends : ➡ on the noise on the measurement of P ➡ contrast of the fringes ➡ scaling factor (keff T2)

2P

C

F =

22

CTk

Pg

eff

=

cos( )2

CP A= DF

Parameters 2T=140 ms t = 20 µs sv ~ vr

Ndet = 5 106 Tc = 360 ms Contrast ~ 40-50 %

Sources of noise - Detection noise - Laser phase noise - Mirror vibrations

σΦ = 1/SNR = 160 mrad/shot

sg/g ~ 2 10-7 /shot

Noise

-25.1435 -25.14300.2

0.3

0.4

0.5

0.6

0.7

-125.718 -125.716 -125.714

Pro

bab

ilit

é d

e t

ran

sit

ion

a (MHz.s-1)

DDS1

(Hz)

104

105

106

107

10-3

10-2

sP Vertical Detection

sP Horizontal Detection

Quantum Projection Noise

Detection Noise

Detection Noise

sP

Number of atoms

C ~ 50%

σΦ=2σP/C

104 atoms : 40 mrad

5 10-8g/shot 3 10-8g at 1s

106 atoms : 2 mrad

2.5 10-9g/shot 1.5 10-9g at 1s

QPN

T = 70 ms & Tc= 360 ms

Detection noise

2

2

1

4P

AC

N Ns =

Quantum Projection Noise Electronic Noise

Sensitivity function g(t)

Ptg

0lim2)(

=

t

Quantifies the sensitivity of the interferometer to phase fluctuations

Hypothesis : << , Plane waves Neglect spatial phase inhomogeneities at the scale of the separation between wavepackets

2/t = RR

0 τ

T

T+τ T+2τ -T-2τ

T τ


=

Ptg

0lim2)(

Method of calculation : Evolution matrix

Expression for the matrix :

Raman pulse Free evolution

Total evolution in the interferometer : product of such matrices :

Result :


t

Split a matrix in two : Example if the phase jump occurs dring the first pulse

In the case of a phase jump :

tt

tt

t

2

0

))(sin(

1

)sin(

)(

=

TtT

Tt

t

Tt

t

tg

R

R

Result (for t>0) :


The sensitivity function is an odd function

Experimental demonstration

P. Cheinet et al., "Measurement of the sensitivity function in time-domain atomic interferometer", IEEE Transactions on Instrumentation and Measurement 57, 1141 - 1148 (2008)

)( 0vgtkvkdt

dzk

dt

deffeffeff ===

Gives the scale factor :

= dtdt

dtgPt

)(

2

1)(

==F

s

2)()(

2)()( 2222 d

SHd

SGIn the spectral domain :

))2

sin()2

)2()(cos(

2

)2(sin(

4)(

22

TTTiH R

R

R

tt

=

Response to a time dependent perturbation

Example : gravity !! :

)4

()2(

tt =D TTgkeff

Time dependent phase fluctuations

Transfer function H(ω)

))2

sin()2

)2()(cos(

2

)2(sin(

4)(

22

TTTiH R

R

R

tt

=

103

104

105

106

1E-3

0.01

0.1

1

10

<H

(f)2

>Frequency (Hz)

2/(f) 2

10-1

100

101

102

0.01

0.1

1

10

H(f

)2

Frequency (Hz)

Finite duration of the pulses Low pass filtering

Insensitive to frequencies multiples of 1/(T+2τ)

2

2

t

02 321 =

Stability characterized by the Allan standard deviation Sampled measurement : fc=1/Tc => Aliasing For long enough averaging time, only components at frequencies at multiples of the cycling frequency contribute

)2()2(1

)(

2

1

2

c

n

c

m

m nfSnfH t

ts

=

F =

m

cm

TS

tt

ts

02

2

2)(

=F

White phase noise ( ) => 0

S

Degradation of the stability

Derivation of the stability

where δΦk is the k-th average of the interferometer phase over duration τm

Using the sensitivity function, it can be expressed as

where

Allan standard deviation

Derivation of the stability

The difference between consecutive averages can be expressed as :

Assuming uncorrelated averaged samples

with

As

We finally get


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

- …

Laser phase stability

DDS

190 MHz

PLL

6,834 GHz

2L ~ 1 m

ECL1

ECL2

PhC

100 MHz

HF synthesis

7,024 GHz

DDS

190 MHz

PLL

DDS

190 MHz

PLL

DDS

190 MHz

PLL

6,834 GHz

2L ~ 1 m

ECL1

ECL2

PhC

100 MHz

HF synthesis

7,024 GHz

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

Laser phase stability : influence of ULN Quartz

0.1 1 10 100 1000 10000 100000-180

-160

-140

-120

-100

-80

-60

SF(d

Bc)

Frequency (Hz)1 10 100 1000

10-10

10-9

10-8

10-7

10-6

Co

ntr

ibutio

n to s

F

2

Frequency (Hz)

T=70 ms

Total : 6.4 10-7

=> 0.8 10-3 rad @1s

=> ~1 µGal @1s

100 MHz reference signal realized by phase locking an ULN 100 MHz oscillator onto an ULN 5 MHz oscillator

Contribution to the noise of the interferometer dominated by low frequencies

)2()2(1

)(

2

1

2

c

n

c

m

m nfSnfH t

ts

=

F =

Influence of laser phase noise

DDS

190 MHz

PLL

6,834 GHz

2L ~ 1 m

ECL1

ECL2

PhC

100 MHz

HF synthesis

7,024 GHz

DDS

190 MHz

PLL

DDS

190 MHz

PLL

DDS

190 MHz

PLL

6,834 GHz

2L ~ 1 m

ECL1

ECL2

PhC

100 MHz

HF synthesis

7,024 GHz

Source σΦ

(mrad/coup)

Lasers

Référence 100 MHz 1,0

Synthèse HF 0,7

Résidu PLL 1,6

Fibre optique 1,0

Rétro-réflexion 2,0

Total 3,1

Détection 4

Total {lasers+détection} 5,0

σg (g/Hz1/2)

1,3·10-9

0,9·10-9

2,0·10-9

1,3·10-9

2,6·10-9

3,9·10-9

5 · 10-9

6,5·10-9

(Evaluation of noise sources for T=50 ms)

J. Le Gouët et al., Appl. Phys. B 92, 133–144 (2008)

Vibration noise

2 2

4

( )( ) ( )eff z eff

SaS k S k

= =

A fluctuation δz of the lasers equiphase induces a change δφ=keff δz

the contribution to the stability can be expressed as

As

2T=100 ms

Transfer function for acceleration noise Sensitivity to low frequencies

Vibration isolation

Acoustic + air flow isolation

Passive isolation platform : Natural frequency 0.5 Hz )2(

)sin(1)(

1

4

2

ca

k c

cg kfS

Tkf

Tkf

tts

=

=

Corresponding sensitivities @ 1s : 2,9 · 10-6 g when OFF (day) 7,6 · 10-8 g when ON (day)

=> Gain : factor 40

• Measure vibrations with a seismometer at the same time

• Convert the seismometer signal into an interferometer phase shift

Residual vibration corrections

Interferometer

Seismometer

Passive platform

Mirror

Good correlations with the calculated phase shift

⇒ the seismometer is used to correct the atomic measurement

vibs = keff Ks gs(t)Us(t) dtT

T

Residual vibration corrections

Typical sensitivity

σg/g = 2. 10-8 at 1s Resolution ~ 10-9g after 500-1000 s

Post correction Seismometer PC

v(t) → φvibS

keffgT² + φvibS Interferometer

keffgT²

Gain : from 2 to ~10

Seismometer transfer function

Phase shift between seismometer signal and mirror vibrations → Rejection at highfrequency will be limited

10 100-270

-180

-90

0

Ph

ase

(deg

rés)

Fréquence (Hz)

Mesure fabricant

Mesure interféromètre

fc =50 Hz

High pass filter : 30 s (0.03 Hz) Low pass filter (mechanical resonances) : fc = 50 Hz

( ) 1( )

( ) 1 ( )c

GH

F= =

F

Correction :

Gain :

( ) ( ) H( ) ( ) (1 ( )) ( )c H F = F F = F

Non linear filtering

2

1

0

)/(1

1

/1

/1)(

cfffjf

fjffF

=

Implement a digital filter Goal : compensate the phase shift of the seismometer

1 10 100

-40

-30

-20

-10

0

10

Réj

ecti

on

(d

B)

Fréquence (Hz)

Sans filtrage

Avec filtrage

0 10 20 30

0.5

0.6

Pro

bab

ilit

é d

e t

ran

siti

on

Temps (s)

Aucune correction

Post-correction sans filtrage

Post-correction avec filtrage

Expected gain of a factor of ~ 3 => σg < 10-8 g Hz-1/2

Filtre NL, results

1 10

10-8

10-7

1D correction

3D correction

sg/g

Averaging Time (s)

Alternative technique (simpler) : compensate for the delay Same results when implementing a delay of 4.6 ms

Best short term sensitivity : 1.4 10-8 g @ 1s

Cross-couplings : Acquisition of 3 axes is necessary

Modest gain (<expected 3 ) Self noise of the seismometer : 4·10-9 g.Hz-1/2 (according specs)

Post corrections Seismometer PC

v(t) → φvibS

keffgT² + φvibS Interferometer

keffgT²

Acquisition without isolation platform

S. Merlet, et al., Metrologia 46, 87 (2009)

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

Fringe fitting

vibs = keff Ks gs(t)Us(t) dtT

T

Post correction of mirror vibrations with

Fringe fitting )cos(,,

, =

= zyxj

S

jvibjbaPand mod ±/2

Day Night

-5 0 5

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

Tra

nsitio

n p

roba

bili

ty

s

vib(rad)

-5 0 5

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

Tra

nsitio

n p

roba

bili

ty

S

vib(rad)

1.8 × 10-7 g @ 1s 5.5 × 10-8 g @ 1s Gain : 20 Gain : 30

In the presence of high anthropic noise, we have reached gains as high as 100 !!

Systematics

DFint= keff g T2 DFLS2 DFCoriolis DFAb DFRF DFLS1 DFZeeman

of keff orientation

Dependent Independent

1- Some effects are keff orientation dependent

|f,p

keff

g

52

→ Reject independent effects by switching between keff / : need two measurements

|f,p

keff

keff

g

dependentT DF=DFDF

2

2geffk

of keff orientation




53

Systematics

|f,p

keff

keff

g

effLSDF

22- Reject DFLS2 bias alternating measurements eff and eff/2

DF, / 2 DF, / 2

DF, DF

,

2= keffgT

2 DFCoriolis DFAb

of keff orientation



dependentT DF=DFDF

2

2geffk


54

The algorithm degrades the sensitivity by √10

Systematics

12 days of continuous measurement

Gal ? 1 µGal = 10-8 m.s-2 ≈ 10-9 g

Gray: 400 drops (170 s) Black: 10 000 s

55

Long term sensitivity

Sensitivity flickers at 7-8 × 10-10 g Limited by tidal model? Water table? Soil moisture? Instrument?

g (4 conf) (-)/2 (+)/2

56

Coriolis acceleration

For T = 2 µK, σv ~ 1 cm.s-1 = 100 v0 !!! Average shift is null if 1) velocity distribution symmetric 2) detection symmetric 3) mean velocity is zero

Solutions Control v0 (CCD imaging, Raman velocimetry/selection) Colder atoms Alternate ±v0 (rotate the experiment by 180°)

Non zero transverse velocity → sensitivity to the Sagnac effect

Ω keff

v┴

g

Ω keff

v┴

g

Δg = 10-9 g for v0 = 100 µm.s-1

Opposite trends for the two orientations signature of Coriolis shift

Trend is not so linear changes in the aberration shift ? why linear anyway? At the crossing point : Null (mean) Coriolis shift

g (µ

Gal)

• Handle on the mean velocity : power imbalance EO molasses beams • Alternate two opposite orientations within 2 hours


-0.4 -0.2 0.0 0.2 0.4 0.6980890720

980890740

980890760

980890780

Imbalance in the molasses beams i=(P2-P1)/(P2+P1)

ORIENTATION N

ORIENTATION S

Changes in Coriolis Vs changes in the wavefront distortions shift? Half difference : Coriolis Half sum : changes in the aberration shift


-0.4 -0.2 0.0 0.2 0.4 0.6-30

-20

-10

0

10

20

30-0.4 -0.2 0.0 0.2 0.4 0.6

-30

-20

-10

0

10

20

30

Half sum

Dg (

µG

al)

Imbalance in the molasses beams i=(P2-P1)/(P2+P1)

Half difference

Null Coriolis bias

Uncertainty in the correction of the Coriolis shift : 0.4 µGal

Wavefront aberrations

Δg < 10-9 g with T = 2 µK

R > 10 km !

→ flatness better than λ/300 !!!

Case of a curvature → δφ = K.r2 (with K = k1/2R)

Measure aberrations with wavefront sensor + excellent optics + colder atoms

t = 0

t = T

t = 2T

(v = 0)

(v = 0)

D = 0D ≠ 0

vr≠0

1 =(vr = 0)

2> 1

3 >

R

t = 0

t = T

t = 2T

(v = 0)

(v = 0)

D = 0D ≠ 0

vr≠0

1 =(vr = 0)

2> 1

3 >

R

Wavefronts are not flat : gaussian beams, flatness of the optics …

• 40mm diameter

• PV= l/10

• RMS =l/100

Mirror

Simulation :

• T = 2.5mK

• s = 1.5mm

g/g = 1.4 10-9

g/g = 8 10-9

l/4 PV

Characterization of the optics

Mirror + l/4 plate under vacuum

Accuracy

0 1 2 3 4 5 6 7

-4

-2

0

2

4

0 1 2 3 4 5 6 7

-4

-2

0

2

4

n=108

6

4

n=2

Dg

(µ

Ga

l)

Tat (µK)

0 1 2 3 4 5 6 7

-4

-2

0

2

4

0 1 2 3 4 5 6 7

-4

-2

0

2

4

Dg

(µ

Ga

l)

Tat (µK)

Exploration of the wavefronts : Differential measurements for different cloud temperatures and extrapolation to zero Analysis with a set of Zernike polynomials => No reliable extrapolation

Need to limit the expansion of the cloud: colder atoms and/or use horizontal Raman selection

Self gravity effect

Dg = (1.3 ± 0.1) × 10-8 m.s-2

64

G. D’Agostino et al, Metrologia 48 (2011)

Accuracy budget

65

atom interferometry ii - uni-hannover.de · v u b goal: measure planck constant h with a 10-8...

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