Atom Interferometry Measurements on Noisy Platforms
Luigi Cacciapuoti SRE-SA
DRAFT 04 August 2010
Introduction
This Technical Note describes the performance levels achievable in a differential measurement by atom interferometry when the instrument is operated on noisy platforms.
We will consider two specific applications:
Test of the Weak Equivalence Principle (WEP);
Gravity gradient measurement.
We will discuss the achievable performance depending on the acceleration noise levels of the measurement platform.
Atom interferometry instruments reach their ultimate performance in freely falling laboratories, where long interaction times between atoms and excitation field can be achieved.
Nowadays, compact systems are being developed for applications on reduced gravity platforms, such as drop towers, zero-g parabolic flights, or orbiting platforms (ISS or other).
A longer interaction time improves the measurement sensitivity, but it also increases the sensitivity of the instrument to the acceleration noise of the hosting platform. Vibration noise can indeed range from ~510-2 (m/s2)/Hz for zero-g parabolic flights, to ~210-4 (m/s2)/Hz for the ISS, or sub (m/s2)/Hz in free flyers or dedicated drag-free platforms.
However, the acceleration noise of the platform can be rejected up to large scale factors by performing a differential measurement, where two atomic samples are simultaneously interrogated by the same excitation field. High rejection factors for the common-mode vibration noise of the platform have already been demonstrated in a gravity gradiometer based on atom interferometry. In an experiment testing the Weak Equivalence Principle, the two interferometers would not share the same sensitivity to inertial effects therefore the evaluation of the common-mode noise rejection factor requires some additional considerations.
Table of content
Introduction ... 2
Table of content... 3
A. The measurement principle ... 4
A-1. Acceleration measurements by atom interferometry ... 4
A-2. Differential measurements by atom interferometry ... 4
Gravity gradient measurements... 5
WEP test... 6
B. Phase noise due to random vibrations of the instrument platform... 8
B-1. Single interferometer... 8
B-2. Differential interferometer ... 8
B-3. Some numbers ... 9
Gravity gradient measurement ... 10
WEP test... 10
C. Conclusions ... 11
References ... 12
List of acronyms... 13
A. The measurement principle
A-1. Acceleration measurements by atom interferometry
In this section, we will briefly discuss the basic principles of an acceleration measurement by atom interferometry. For clarity, we limit the discussion to a Raman-pulse interferometer based on samples of 87Rb atoms. In this set-up, Raman pulses are used to stimulate freely falling samples of 87Rb atoms on the two-photon Raman transition between the hyperfine levels F1 and F2 of the ground state [1].
The light field is generated by two counter-propagating laser beams with wave vectors k1
and k2~k1 aligned along the vertical direction. The laser frequencies 1 and 2 match the resonance condition 1−2=0, where h0 is the energy associated to the F1→ F2 transition.
During a cycle of absorption and stimulated emission, the resonant light field exchanges with atoms a total momentum of ħkħ(k1k2)~2ħk1, coupling the two atomic states 1|1,p
and 2|2,p+ħk , where p is the initial momentum of the atom. The interferometer sequence is composed of a combination of three Raman pulses that drive the atoms on two independent paths along which the quantum mechanical phases of the atomic wavepackets independently evolve: At tt1, a first /2 pulse acts as a beam splitter, preparing the atoms, initially in the state 1, in an equal superposition of 1 and 2; at tt2T, a pulse, playing the role of a mirror, redirects the wavepackets inverting state 1 with 2 and vice versa; finally, at tt32T, the final /2 pulse recombines the wavepackets producing the interference. At the end of the sequence, the probability of detecting the atoms in the state 2 is
1 cos , 21
2
P
1where is the phase difference accumulated by the wavepackets along the two interferometer arms I and II.
In the presence of an acceleration field a, the atoms experience a phase shift depending on the local acceleration a:
(I,t1) (I,t2)
(II,t2) (II,t3)
kaT2.
In other terms, the measurement process can be represented as marking the position of freely falling atoms by using the wavefronts of the Raman lasers as a ruler. Therefore, a measurement of the phase corresponds to a direct measurement of the acceleration at the measurement platform.
A-2. Differential measurements by atom interferometry
In a differential measurement by atom interferometry, two distinct atomic clouds (same species, but spatially separated or two different atomic species) are simultaneously interrogated by the same /2−−/2 Raman-pulse sequence. The difference of the phase shifts detected by each interferometer provides a direct measurement of the differential acceleration or, equivalently, of the geodesic deviation between the two samples of freely flying atoms.
Let’s label the two atomic samples with A and B. As said before, they could be two spatially separated samples of the same atomic species or two samples of two different atomic species. We are here interested in a measurement of the differential acceleration aaA-aB
along the propagation direction of the Raman lasers. Let’s also indicate with kA and kB the effective wave vectors for the two-photon Raman transitions of species A and B respectively.
The differential phase as measured by the instrument is given by
ABkAaATA2kBaBTB2, where TA and TB are the interrogation times of the interferometers A and B respectively. Therefore, by adjusting the interrogation times to have the same calibration factor SkATA2kBTB2 on both interferometers, the differential phase shift
Sacan be used to provide a direct measurement of the differential acceleration
aaAaB experienced by the atomic samples A and B.
The two atom interferometers will provide a measurement of the population of A and B atoms in the hyperfine levels of the ground state. Our instrument will then simultaneously measure (see Eq. 1)
) cos(
) cos(
B B B
A A
A
F E n
D C
n
where the capital letters indicate fluctuating quantities. In particular, A and B are related to the phase noise perturbing the measurement of the interferometers A and B.
In this section, we will briefly discuss how a gravity gradient measurement or a measurement testing the WEP can physically be implemented using atom interferometry techniques.
Gravity gradient measurements
In a gravity gradiometer, the two probes of the gravitational field are two samples of the same atomic species (A=B), but spatially separated by the distance l, representing the baseline over which the gravity gradient measurement is performed. The two atomic clouds are simultaneously interrogated by the same /2−−/2 pulse sequence. The instrument measures the difference of the phase shifts detected on each interferometer, providing a direct measurement of the differential acceleration induced by gravity on the two atomic samples.
This method has the major advantage of being highly insensitive to noise sources appearing in common-mode on both interferometers. In particular, any spurious acceleration induced by vibrations or seismic noise on the common reference frame identified by the vertical Raman beams is efficiently rejected by the differential measurement technique.
Figure 1 shows some measurement taken by the gravity gradiometer MAGIA operated at the Physics Department of the Firenze University [2]. Even if the phase noise induced by vibrations completely washes out the atom interference fringes detected by the two interferometers (top), the signals simultaneously detected on the upper and lower accelerometer remain coupled and preserve a fixed phase relation. Therefore, when the trace of the upper accelerometer is plotted as a function of the lower one (bottom), experimental points distribute along an ellipse. The differential phase shift can then be obtained from the eccentricity and the rotation angle of the ellipse fitting the experimental data.
In this configuration, the two atom interferometers share the same effective vector kkAkB
and the same interrogation time TTATB. Therefore, they will have the same sensitivity to acceleration noise, resulting (see Sec. B) in a high rejection factor for the common-mode vibration noise of the instrument platform.
Figure 1: Gravity gradient measurements performed by the MAGIA instrument at the Physics Department of the Fiirenze University [2].
WEP test
An atom interferometry test of the Weak Equivalence Principle is based on a differential measurement performed by two atom interferometers simultaneously probing the acceleration experienced by two clouds of two different atomic species.
The WEP violation can be expressed by using the Eötvos parameter 2(aAaB)/(aAaB), with AB (e.g. 39K-87Rb or 85Rb-87Rb). In this case, we are interested in the differential acceleration aaAaB of the freely falling atomic samples A and B. However, as kA is in
general different from kB, matching the sensitivities SkATA2kBTB2 of the two interferometers will result in this case into different interrogation times, TATB, and into different sensitivity functions for A and B.
In the following, we will explicitly focus on the phase noise introduced by the mechanical vibrations on the instrument platform, neglecting the phase noise introduced by the Raman lasers that, using appropriate techniques, can be significantly reduced. In this case, A,B can also be expressed in terms of the random spatial displacement X(t) of the mirror retroreflecting the Raman lasers:
)
B (
A, kX t
.
The standard deviation of the random spatial displacement variable X(t) will then be
X/kA,B.
B. Phase noise due to random vibrations of the instrument platform In this section, we will evaluate the noise contribution to the atom interferometry phase measurement induced by the acceleration noise of the instrument platform. Our analysis will follow Ref. 3.
B-1. Single interferometer
In the case of a single interferometer, the phase noise induced by vibrations can be expressed in terms of the random displacement at the retroreflecting mirror of the atom interferometer:
dt dt t kX t d
h
( ( ))
)
( 2
where h(t) is the sensitivity function of the atom interferometer [4]. h(t) is an odd and piecewise function, equal to zero out of the window [T2, T2] that for t>0 is given by:
2 0
)]
( sin[ 1
) sin(
)
( T tt Tt T
T t
t t
h
R R
where R is the Rabi frequency of the two-photon Raman transition and is the duration of the /2 pulse: R/2.
If we neglect the duration of the Raman pulses (<<T), h(t) is an odd and piecewise function, equal to zero out of the window [T, T], 1 in [T, 0[, and 1 in ]0, T]. Therefore, integrating by parts Eq. 2, we obtain
dt t t
kf
( )( ) , f(t) being the primitive of the sensitivity function h(t).
The variance of the interferometer phase noise is then given by the following expression
2 2 kX 2 dt1dt2kf(t1)kf(t2)(t1)(t2)
, 3
which for white acceleration noise, (t1)(t2) S(t1t2), simplifies into
2 k2T3S
3
2
. 4
B-2. Differential interferometer
As discussed in Sec. A-2, in a differential interferometer based on two distinct atomic species, we have kAkB and TATB and therefore two different sensitivity functions. In this case, the relative interferometer phase noise AB can be calculated from Eq. 3 by replacing kf(t) with kAfA(t)kB fB(t), which for white acceleration noise leads to
2 3
2 2
2 3
2
k
S k T
k
. 5
Using Eq. 4 and Eq. 5, it is possible to evaluate the rejection factor of common-mode vibration noise with respect to the single atom interferometer:
k k 2
.
Therefore, the closer the effective k-vectors of the Raman transitions for the two atomic species are, the lower is the sensitivity of the differential accelerometer to the vibration noise introduced by the instrument platform. In Table 1, we consider the following atomic species and transitions: the potassium D2 line at 767 nm, the potassium D1 line at 770 nm, and the D2 lines of 85Rb and 87Rb at 780 nm, which are separated by only 0.03 nm.
(nm) / 39K(D2) 87Rb(D2) 13 910-3
39K(D1) 87Rb(D2) 10 610-3
85Rb(D2) 87Rb(D2) 0.03 1.010-7
Table 1
In a differential interferometer based on two spatially separated clouds of the same atomic species (k0), extremely high rejection factors for common-mode vibration noise can be achieved. In this case, the two atom interferometers share exactly the same sensitivity function (kAkB and TATB) resulting, to first order, in /0.
However, as the two atomic samples are now spatially separated, the vibration noise on the two atom interferometers is not perfectly correlated due to the finite propagation time of the Raman lasers from the first to the second atomic sample. This effect can be estimated by replacing kf(t) with kf(t)kf(td) in Eq. 3. Here, d=cd represents the time needed by the Raman laser beams to propagate along the distance d separating the first from the second atomic cloud.
In case of white acceleration noise, the variance of the relative interferometer phase can be estimated as
2 2 2 2k TS d
,
corresponding to a rejection factor for common-mode vibration noise of T
d
3
.
For a measurement baseline d1m and for an interferometer time T=1s, /610-9. Therefore, the gravity gradiometer measurement offers an extremely high rejection factor for common mode noise sources. The sensitivity of a gravity gradiometer to linear accelerations has been tested in [5] by shaking the instrument platform on which the retro-reflecting mirror was mounted. For frequencies in the range 1–100 Hz no significant degradation of the SNR was observed for drive amplitudes as high as to 2.510-2g. In this experiment, the common- mode rejection factor was found to be better than 140 dB, the measurement being limited by the sensitivity of the gravity gradiometer.
The large rejection factors for the common mode acceleration noise of the instrument platform that can be achieved in a gravity gradiometer measurement by atom interferometry make these instruments compatible with the ISS. A gravity gradiometer in space operated on an interferometer time T=10s can realistically reach a sensitivity better than 10-11 m/s2 per shot, down to 10-14 m/s2 after 106 measurements. For these parameters, the noise introduced on the phase measurement by the mechanical vibrations of the measurement platform is negligible.
WEP test
For a Weak Equivalence Principle test, the performance levels realistically achievable depend on the atomic species and on the atomic transitions selected for the two atom interferometers.
A differential measurement performed on the two different isotopes of rubidium (85Rb(D2)
87Rb(D2)) would still benefit from a rejection factor for common mode vibration noise as low as 1.010-7 (see Table 1). A differential accelerometer operated on an interferometer time T=10s can reach a sensitivity better than 10-11 m/s2 per shot, down to 10-14 m/s2 after 106 measurements. For these parameters, the noise introduced on the phase measurement by the mechanical vibrations of the measurement platform would be compatible with the instrument sensitivity. In this case, a measurement of the Eötvos parameter down to an accuracy of 110-15 could be achieved after about 100 days of integration time.
In a WEP test involving two different atomic species, Rb and K, the mismatch between the sensitivity functions of the two atom interferometers is responsible for a non optimised rejection factor for the common mode acceleration noise of the instrument platform (see Table 1). In this case, ensuring a sensitivity to differential accelerations of ~10-11 m/s2 per shot at a T=10s of interferometer time would require the stabilization of the random spatial displacement X of the mirror retroreflecting the Raman lasers to better than X~1m in the relevant frequency band. Such stabilization levels could be reached in the weightlessness conditions of the ISS by efficiently decoupling the retroreflecting mirror from the low- frequency vibrations of the platform.
C. Conclusions
According to our preliminary estimates:
The vibration levels of the ISS would not represent a problem for the measurement of differential accelerations (gravity gradient measurement) at the level of 10-14-10-15 g;
A WEP test to 10-15 performed on 85Rb(D2) 87Rb(D2) is still compatible with the spurious acceleration levels of the ISS;
A WEP test to 10-15 performed on 39K(D2) 87Rb(D2) or 39K(D1) 87Rb(D2) would require the stabilization of the random spatial displacement X of the mirror retroreflecting the Raman lasers to better than X~1m in the relevant frequency band (achievable by proper decoupling from low-frequency vibrations of the platform).
Concerning the extraction of the phase information out of atom interferometry measurements, Ref. 3 shows that the Bayesian estimation technique can be efficiently used to perform a differential measurement between two inertial sensors using atoms with different mass and effective wavelegths. Even for large vibration noise and large interrogation times, the Bayesian estimator rapidly converges allowing for a measurement of the differential phase shift to high accuracy.
It is also worth mentioning here that all these techniques are presently being studied and experimentally tested in parabolic flight experiments (Zero-G Airbus), where a precision
~510-11 is expected at a free-fall time of 4 s and after only 30 experimental data points [3]
(ICE, CNES contract). Compact atom interferometry systems are under construction for future tests in the Bremen drop tower and on sounding rockets (SAI, ESA contract;
QUANTUS, DLR contract).
References
1. A. Peters et al., Metrologia 38, 25 (2001) and references therein.
2. G. Lamporesi et al., Phys. Rev. Lett. 100, 050801 (2008).
3. G. Varoquaux et al., New Journal of Physics 11, 113010 (2009).
4. P. Cheinet et al., IEEE Trans. Instrum. Meas. 57, 1141 (2008).
5. J.M. McGuirk et al., Phys. Rev. A 65, 033608 (2002).
List of acronyms
ICE: Interférométrie à Source Cohérente (“Interefometry with a Coherent Source”) ISS: International Space Station
MAGIA: Misura Accurata di G per Interferometria Atomica (“Atom Interferometry Measurement of G”)
QUANTUS: QUANTengase Unter Schwerelosigkeit (“QUANTum gases Under Microgravity”)
SAI: Space Atom Interferometer SNR: Signal to Noise Ratio
WEP: Weak Equivalence Principle
