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arXiv:physics/0507060v1 [physics.atom-ph] 7 Jul 2005

EPJ manuscript No.

(will be inserted by the editor)

Influence of optical aberrations in an atomic gyroscope

J´ erˆ ome Fils1, Florence Leduc1, Philippe Bouyer2, David Holleville1, No¨ el Dimarcq1, Andr´ e Clairon1and Arnaud

Landragin1

1BNM-SYRTE, UMR 8630, Observatoire de Paris, 61 avenue de l’Observatoire, 75014 Paris, France

2Laboratoire Charles Fabry, UMR 8501, Centre Scientifique d’Orsay, bˆ at. 503, BP 147, 91403 Orsay, France

Received: date / Revised version: date

Abstract. In atom interferometry based on light-induced diffraction, the optical aberrations of the laser

beam splitters are a dominant source of noise and systematic effect. In an atomic gyroscope, this effect is

dramatically reduced by the use of two atomic sources. But it remains critical while coupled to fluctuations

of atomic trajectories, and appears as a main source of noise to the long term stability. Therefore we measure

these contributions in our setup, using cold Cesium atoms and stimulated Raman transitions.

PACS. PACS-03.75.Dg Atom and neutron interferometry – PACS-42.15.Fr Aberrations – PACS-32.80.Pj

Optical cooling of atoms; trapping

1 Introduction

Since the pioneering demonstrations of interferometry with

de Broglie atomic waves using resonant light [1,2] and

nanofabricated structures [3] as atomic beam splitters, a

number of new applications have been explored, including

measurements of atomic and molecular properties, funda-

mental tests of quantum mechanics, and studies of various

inertial effects [4]. Using atom interferometers as inertial

sensors is also of interest for geophysics, tests of general

relativity [5], and inertial guidance systems.

Atom interferometers based on light-induced beam split-

ters have already demonstrated considerable sensitivity to

inertial forces. Sequences of optical pulses generate the

atom optical elements (e.g., mirrors and beam splitters)

for the coherent manipulation of the atomic wave pack-

ets [6]. The sensitivity and accuracy of light-pulse atom

interferometer gyroscopes [7], gravimeters [8] and gravity

gradiometers [9] compare favorably with the performances

of state-of-the-art instruments. Furthermore, this type of

interferometer is likely to lead to a more precise direct

determination of the fundamental constant α from the

measurement of ¯ h/M [10]. In the case of rotation mea-

surements, the sensitivity reaches that of the best labo-

ratory ring laser gyroscope [11]. Indeed the Sagnac phase

shift, proportional to the total energy of the interfering

particle, is much larger for atoms than for photons. This

compensates for the smaller interferometer area and the

lower flux.

In this paper we focus on the effect of the fluctuations

of the atomic trajectory, which might affect the long term

stability of atomic gyroscopes when coupled with local

phase variations induced by optical aberrations. We will

introduce this problem in paragraph 2 and illustrate it

quantitatively in the case of our setup in paragraph 3.

Our experiment consists in an almost complete iner-

tial measurement unit [12], using cold Cesium atoms that

enable for a drastic reduction of the apparatus dimen-

sions while reaching a sensitivity of 30 nrad.s−1.Hz−1/2to

rotation and 4x10−8m.s−2.Hz−1/2to acceleration. Its op-

eration is based on recently developed atom interference

and laser manipulation techniques. Two interferometers

with counter-propagating atomic beams discriminate be-

tween rotation and acceleration [13]. Thanks to the use of

a single pair of counter-propagating Raman laser beams,

our design is intrinsically immune to uncorrelated vibra-

tions between the three beam splitters, usually limiting

such devices. This configuration is made possible by the

use of a reduced launch velocity, inducing a reasonable in-

teraction time between the pulses. However, as any atomic

gyroscope, our sensor’s scheme remains sensitive to local

phase variations, a limitation that has already been en-

countered in optical atomic clocks [14].

2 Principle

We first briefly review the basic light-pulse method in the

case of a symmetric Ramsey-Bord´ e interferometer scheme

[15], where three travelling-wave pulses of light resonantly

couple two long-lived electronic states.The two-photon stim-

ulated Raman transitions between ground state hyperfine

levels are driven by two lasers with opposite propaga-

tion vectors ke and kg (ke ≃ −kg). First, at t = t1 a

beam splitting pulse puts the atom into a coherent super-

position of its two internal states. Because of conserva-

tion of momentum during the atom-light interaction, this

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Fig. 1. Time-pulsed Ramsey-Bord´ e atom interferometer us-

ing stimulated Raman transitions induced by two counter-

propagating laser beams of wave vectors ke and kg. Cesium

atoms are launched on the same trajectory but in opposite di-

rections with velocities vL,R= {0,±vy,vz}, from right to left

(R) and left to right (L). The interactions with light pulses

occur at times ti=1,2,3 at three different locations. The detec-

tion consists in measuring the probability of presence in each

output port after the last pulse.

pulse introduces a relative momentum ¯ hkeff= ¯ hkg− ¯ hke

between the atomic wave packets corresponding to each

state. These wave packets drift apart for a time T, after

which a mirror pulse is applied at t2= t1+ T to redirect

the two wave packets. After another interval of duration

T, the wave packets physically overlap, and a final beam

splitting pulse recombines them at t3= t1+2T. The mea-

surement of the probabilities of presence in both internal

states at the interferometer output leads to the determi-

nation of the difference of accumulated phases along the

two paths. In general, atoms are launched with a velocity

v so that each stimulated Raman transition occurs at a

particular position {xi,yi,zi}i=1,2,3that can be evaluated

from the classical trajectories associated with the atomic

wave packets [16], as shown fig. 1. In our setup, Raman

laser beams propagate in the (Ox) direction and atoms

are launched in the (y,z) plane. We define ui={yi,zi} the

atomic cloud positions in this plane at time ti.

In the absence of any external forces, atoms initially

prepared in a particular state (6S1/2,F = 3,mF= 0 in the

present setup) will return to this state with unit proba-

bility. A uniform external acceleration or rotation induces

a relative phase shift between the interfering paths. This

phase shift modifies the transition probability between the

two Cesium internal states6S1/2,F = 3,mF = 0 and

6S1/2,F = 4,mF= 0 (noted |3? and |4? in the following).

Hence the transition probability measurement leads to the

determination of the phase shift and finally the evaluation

of the perturbing forces.

It can be shown that the only contribution to the

phase shift results from the interaction with the laser light

fields [16]. In the limit of short, intense pulses, the atomic

phase shift associated with a transition |3? → |4? (resp.

|4? → |3?) is +φi (resp. -φi), where φi is the phase dif-

ference between the two Raman laser beams. We then

find that the transition probability from |3? to |4? at the

exit of the interferometer is simply1

∆φ = φ1− 2φ2+ φ3. The three quantities correspond to

2[1 − cos(∆φ)] where

the phase imparted to the atoms by the initial beam split-

ting pulse, the mirror pulse, and the recombining pulse

where φi = φg(ui,ti) − φe(ui,ti) = keff · xi+ Φ(ui).

The sensitivity to rotation and acceleration arises from

the first term keff· xiand simplifies to ∆φacc= axkeffT2

and ∆φrot = −2keffvyΩzT2for the present setup. The

phase Φ(ui) for the pulse at time ticorresponds to the lo-

cal phase in the (y,z) plane due to wavefront distortions of

both laser beams1. It induces a residual phase error at the

exit of the interferometer δΦ = Φ(u1) − 2Φ(u2) + Φ(u3).

Acceleration cannot be discriminated from rotation in

a single atomic beam sensor, as stated above. This limita-

tion can be circumvented by installing a second, counter-

propagating, cold atomic beam (fig. 1) [13]. When both

atomic beams perfectly overlap, the area vectors for the re-

sulting interferometer loops have opposite directions. The

corresponding rotational phase shifts ∆φrothave opposite

signs while the acceleration phase shifts ∆φaccare identi-

cal. Consequently, acceleration is calculated by summing

the two interferometer’s phase shifts: ∆φ+ ∼ 2∆φacc;

while taking the difference rejects the contribution of uni-

form accelerations so that ∆φ−∼ 2∆φrot. In addition, the

residual phase error δΦ vanishes in ∆φ−, but remains in

∆φ+as an absolute phase bias 2 × δΦ.

However, an imperfect overlapping of the two counter-

propagating wavepackets trajectories might lead to an im-

perfect common mode rejection of the residual phase error

in ∆φ−. Thus, a phase bias δΦ−= δΦL−δΦRwill appear,

where the notationsLandRconcern the left and right

atom interferometers. While the phase bias δΦ+≃ 2×δΦ

depends on the local value of the phase at the average

position ri=

2

, the phase bias δΦ−depends on the

local phase gradient at the average position ri with the

position offset δri= uL

uL

i+uR

i

i− uR

i:

δΦ−= ∇Φ(r1) · δr1− 2∇Φ(r2) · δr2

+∇Φ(r3) · δr3.

Equation 1 shows that uncorrelated fluctuations of the

wavepackets trajectories from shot to shot causes fluctu-

ations of the phase bias, which amplitude depends on the

local wavefront slope of the phase. If we consider a per-

fect control of the launch velocity2, fluctuations of tra-

jectories are only due to fluctuations of the initial posi-

tions of the atomic clouds. Consequently, we can consider

δr1= δr2= δr3. The phase fluctuation is then simply pro-

portional to the product of the fluctuations of the cloud

initial position (y0,z0) with the phase gradients ∆Φi. As

the phase gradients are time-independent, the Allan vari-

ance of the phase σ2

δΦ−is simply:

(1)

σ2

δΦ−= σ2

+σ2

y0.[∂y(Φ(r1) − 2Φ(r2) + Φ(r3))]2

z0.[∂z(Φ(r1) − 2Φ(r2) + Φ(r3))]2

1The interferometer is also sensitive to time fluctuations of

the Raman laser phases [12]. These fluctuations are identical

for the two interferometers and disappear from the rotation

signal. They will be neglected in this paper.

2We can reach a stability of 10−4m.s−1or better from shot

to shot thanks to the moving molasses technique [17].

(2)

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3

Fig. 2. Front view of our gyroscope; the interaction zone is

located near the top of the atomic trajectories. Atoms are

launched symmetrically at initial velocity v0 = 2.4 m.s−1, mak-

ing an angle of 82owith the horizontal axis. The enclosed ori-

ented areas are equivalent to their projections on the (Oxy)

plane.

where σ2

horizontal and vertical positions. Eq. 2 shows that the

fluctuations of the clouds initial positions, as well as the

wavefront quality of the Raman beams, have to be sys-

tematically investigated in atomic gyroscopes in order to

estimate how it affects its performances.

y0and σ2

z0are the Allan variances of the initial

3 Experimental results

In our setup, the atomic sources are clouds of Cesium

atoms, cooled in magneto-optical traps and launched with

a parabolic flight (fig. 2). As the initial angle reaches 82o,

and the launch velocity 2.4 m.s−1, the horizontal veloc-

ity vyis 0.3 m.s−1. The single pair of Raman laser beams

propagates along the x-axis and is switched on three times

at the top of the atomic trajectories. If the three pulses are

symmetric with respect to the trajectory apogees, the in-

terferometer oriented enclosed areas are equivalent to their

flat horizontal projections: the oriented vertical projection

is naught. The time delay between pulses is typically 45

ms. The positions of the atoms during the three Raman

pulses are given in fig. 2.

In order to investigate the fluctuations of the atomic

initial positions from shot to shot, we image one of the two

clouds. The cycling sequence takes about 1.3 s and consists

on a trap phase of 500 ms, a molasses phase of 20 ms, a

launching phase of 2 ms and a waiting time phase of 800

ms needed to process the image: download of the image,

subtraction of a background image and determination of

the cloud barycenter position in y- and z-axes. The image

is taken just after turning off the trap magnetic field, at

the end of the molasses phase. We calculate the Allan

standard deviations [18] of the barycenter horizontal and

vertical positions (fig. 3) from a one hour acquisition. Two

Fig. 3. Allan standard deviations of the horizontal (black

squares) and vertical (grey triangles) MOT positions as a func-

tion of the integration time τ, plotted in log-log scale. On the

right axis the Allan standard deviation of the intensity ratio

of MOT cooling lasers is plotted in dashed line as a function

of the integration time τ.

peaks, appearing after 10 s and 150 s of integration time,

are characteristic of fluctuations of periods equal to 20 s

and 300 s. After about 10 min integration (630 s), the

position standard deviations reach 10 µm and 5 µm in

the horizontal and vertical directions respectively. This

dissymmetry is consistent with the magnetic field gradient

configuration, which is twice higher on the Z-direction.

The long-term variations are due to fluctuations of the

MOT cooling lasers intensity ratio, which Allan standard

deviation is plotted in fig. 3. We see again the oscillation

of period 300 s, appearing for 150 s integration time. We

analyze this as the period of the air conditioning, creating

temperature variations on the fibre splitters delivering the

cooling lasers.

This result has to be coupled to the optical aberra-

tions of the Raman lasers. The main contribution to these

aberrations comes from the vacuum windows used for the

Raman laser beams, which clear diameter is 46 mm. They

have been measured with a Zygo wavefrontanalyzer, which

gives the laser phase distortion created by the windows.

This distortion is projected on the Zernike polynomial

base [19]. As our atomic clouds are about 2 mm wide, the

decomposition is pertinent only up to the 36th polynomial.

Indeed, the upper numbers correspond to high spatial fre-

quencies, so that their effect will be smoothed by averag-

ing on the atomic cloud dimensions. To reduce the stress

on the vacuum windows, essentially due to the mount-

ing, they were glued in place. Thanks to this method, the

wavefront quality reaches λ/50 rms over the whole clear

diameter of 42 mm.

The wavefront measurement allows for evaluation of

the atomic phase shift fluctuations due to the coupling

between aberrations and position fluctuations using eq.2

assuming that the two sources are uncorrelated. Their rel-

ative position fluctuations are√2 times greater than these

observed for one source. The contribution of this phase

fluctuations to the Allan standard deviation of the rota-

tion rate measurement is shown in fig. 4. We compare it

with the ultimate stability of our gyroscope, given by the

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Fig. 4. Allan standard deviation of the rotation measurement,

taking into account the optical aberrations when coupled with

position fluctuations. The dashed curve shows the quantum

projection noise limit, indicating that the optical aberrations

may affect the gyroscope performances at long term.

quantum projection noise. It is estimated to 30/√τ nrad.s−1

(τ is the integration time) from the ultimate signal-to-

noise ratio obtainable with 106atoms.

The rotation noise induced by position fluctuations has

a significant contribution for integration times larger than

100 s. At the present stage of the experiment, this lim-

itation is due to the high temperature sensitivity of the

fibre splitters. This could be the main limitation of the

gyroscope performances.

4 Conclusion

In the present paper we studied the stability of a cold atom

gyroscope based on two symmetrical Ramsey-Bord´ e inter-

ferometers, with respect to optical phase inhomogeneity.

Instability due to aberrations is not a specific problem in-

duced by Raman transitions, but concerns every type of

atom interferometer using light beam splitters. We showed

that the coupling between wavefront distortions of these

lasers and fluctuations of the atomic trajectory becomes

predominant at long term, despite a wavefront quality of

λ/50 rms obtained thanks to glued windows. In our setup,

atomic trajectory fluctuations are mainly due to fluctua-

tions of the intensity ratio of the MOT cooling lasers, in-

duced by the fibre splitters used for their generation.

However several improvements may render their contribu-

tion negligible:

- reduce the atomic trajectory fluctuations, by using dis-

crete optical couplers for the MOT instead of the present

fibre splitters,

- minimize the number of optics which contribute to the

interferometer instability. This can be done by including

the Raman laser beam imposition optics in the vacuum

chamber, in order to remove the aberrations due to the

vacuum windows, or by minimizing the number of non-

common optics for the two Raman lasers, since only the

phase difference between the lasers is imprinted on the

atomic phase shift.

Such techniques open large improvement possibilities,

which will be confirmed directly on the long-term stability

measurement of the atomic signal in our interferometer

setup.

5 Acknowledgements

The authors would like to thank DGA, SAGEM and CNES

for supporting this work, Pierre Petit for the early stage

of the experiment and Christian Bord´ e for helpful dis-

cussions. They also thank Thierry Avignon and Lionel

Jacubowiez from SupOptique for their help in the wave-

front measurement.

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