Table Of ContentNoise-Immune Conjugate Large-Area Atom Interferometers
Sven Herrmann,1 Sheng-wey Chiow,1 Steven Chu,1,2,3 and Holger Mu¨ller1,2,3,∗
1Physics Department, Stanford University, 382 Via Pueblo Mall, Stanford, California 94305, USA
2Department of Physics, 366 Le Conte Hall, University of California, Berkeley, CA 94720-7300
3Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley, CA 94720.
(Dated: January 13, 2009)
WepresentapairofsimultaneousconjugateRamsey-Bord´eatominterferometers(SCI)usinglarge
(20h¯k)-momentumtransfer(LMT)beamsplitters,where¯hkisthephotonmomentum. Simultaneous
9 operation allows for common-mode rejection of vibrational noise. This allows us to surpass the
0 enclosed space-time area of previous interferometers with a splitting of 20¯hk by a factor of 2,500.
0 Amongapplications, wedemonstratea3.4ppbresolution inthefinestructureconstantanddiscuss
2 tests of fundamental laws of physics.
n
a
13 J sebiniglcLlnieioagnlfhriti(n-npptgpoueblsas).ermaeFltooaodrtmuiveleaxilntayitmoelnrpafrleiegnr,eomliopncehataRelsreegsf.rscahav[in1iftt]y,coolafneavt3hderesptatinaorttessamrfp1eae%rlr-l Height p2 1 p2 p2,p2 2 3 p2,p2 ab (a-b)/(a+b) interferometer 1
4
]
shift of the interference fringe. They thus make excel-
h 6 5
p lent microscopes for small signals that have been ap-
- plied in many cutting-edge precision measurements [1–
m c interferometer 2
10]. Large-momentum transfer (LMT) beam splitters, d
o whichhavebecomepracticalrecently[11],promisetoin- (c-d)/(c+d)
at crease the sensitivity further, by factors of 10s to 100s, t1 T t2 T’t3 T t4
Time
. by increasing the space-time area enclosed between the
s
c interferometer arms. But just as vibrations blur micro- FIG.1: Correlating thefringesoftwointerferometerscreates
si scopic images,they blur the interference fringes in inter- an ellipse whose eccentricity allows to determine the relative
y ferometers. This becomes more pronounced as the sen- phase.
h
sitivity is increased until eventually interferences can no
p
longer be discerned. This has so far limited the use of
[
LMT beamsplitters toveryshortpulse separationtimes atomswithamovingopticalmolasseslaunchandRaman
1 T (Fig. 1) of 1ms, thwarting the potential gain in sensi- sidebandcoolinginanopticallattice,asdescribedin[17].
v
tivity. As beam splitters, we use multiphoton Bragg diffraction
9
1 Cancellation of vibrations has been demonstrated by of matter waves at an optical lattice [11, 18–20]. The
8 using the same laser light to simultaneously address opticallattice is formedby twocounterpropagatinglaser
1 twosimilarinterferometersatseparatelocations [12–14]. beams that we may call the top and bottom beam (Fig.
1. This method, however, is restricted to situations where 2). Bragg diffraction can be described in the initial rest
0 the differential signal is small (in this case, the gravity frameofthe atom. For example,the atommay absorbn
9 gradient), so that the interferometers are similar enough photons at ω1 from the bottom beam and be stimulated
0 to be addressable by the same laser. to emit n at ω2 into the top beam. The atom emerges
:
v In this work, we explore the full potential of LMT by at the same internal quantum state with a momentum
Xi cancelling vibrations between dissimilar interferometers, of 2n¯hk, where k is the wavenumber, and a kinetic en-
the conjugate Ramsey-Bord´e interferometers shown in ergy of (n¯hk)2/(2M), where M is the mass of the atom.
r
a Fig. 1. Theideaistouselaserpulsesthatcontainapair This energy has to match the energy n¯h(ω1 −ω2) lost
of frequencies whose phase noise is extremely well corre- by the laser field, which allows us to choose the Bragg
lated. We use this to demonstrate a 2,500-fold increase diffraction order n by the difference frequency ω1−ω2.
in the enclosed space-time area of interferometers with Fig. 1 (left) shows a space-time diagram of our
20-photon momentum transfer, without a reduction in Ramsey-Bord´e interferometers. Let us specialize to the
contrast. Thispavesthepathtowardsstronglyenhanced lower one, whose outputs are labelled c and d. An atom
sensitivity in measurements of fundamental constants enters on its way upwards. At a time t , a “π/2” laser
1
[8,9,15], tests ofgeneralrelativity[1]orthe equivalence pulse transfers a momentum of 2n¯hk with a probability
principle [10], and detection of gravitationalwaves [16]. of 50%. Depending on whether momentum was trans-
ferred or not, the atom follows trajectory 1 or 6. At
Atom interferometers basically consist of a source of t +T, a second π/2 pulse stops the relative motion of
1
atomsandbeamsplittersforthematterwaves. Theatom them. After two more pulses, the paths are recombined
source for our interferometers is a fountain of cesium into the outputs c andd where they interfere. A second,
2
upper,interferometerisformedbyrecombiningtheother 180MHz ~ ~
outputs of the beam splitter at t2. fm~ DBM2 DBM3 R82a-m94pMHz PBS
l/4
faitsohtroeTtymhheex’eisnaptmphferrarpeofselebeera,eedbv.iiisofflilTetugyrhtieviinotsehcnnecaotbboneyfttthatwciheneoeessan2itanφottm,echroewfneahtrbrroereirimbaveumeetstφieosarnp=taliφrotmφFtueFtsrposwu+ftahtnφehcndIe, TmC8i9:oos9hda-eip2frip1eehdnt AOM1 Fiber AOM2 l/4EOM1 AOMAO4M3 EOM2 Magnetic shield ~1mT(wo2p, bwe3a)ms
PD1 Cesium atoms
olunteioonfpthhaeseinφtera=ctiSon/w¯hitihs gtihveenligbhyttφhIe.clTashseicafrleaecetvioon- DBM1 AWG Ampl. stab. PPBDS21 l/4 B(wo1tt)om beam
F Cl 80MHz ~ Servo
SCl = R(Ekin − Epot)dt, where Ekin and Epot are the l/4
Retroreflector
kinetic and potential energy. The interaction phase φ
I
is because whenever a photon is absorbed, its phase is FIG. 2: Setup.
addedtothematterwavephaseandsubtractedforemis-
sion of a photon [21]. This phase is different for the two
paths because of the respective spatial separation of the 1 form an ellipse. The common phase moves the data
interactions at t and t . Summing up, [8, 11] points around the ellipse, but the differential phase can
2 3
be extractedby ellipse-specific fitting. One waythat has
φ± =±8n2ωrT +2nkg(T +T′)T +nφ±L, (1) been realized, without LMT [12–14], is to use the same
radiationtoaddressbothinterferometers,whichtrivially
where ω = h¯k2/(2M) is the recoil frequency and g
r leads to δφ = 0. However, the use of this method is
the local gravitational acceleration. The plus and mi- L
restricted to measurements of very small differential sig-
nus signs are for the upper and lower interferometer, re-
spectively, and φ± =φ −φ −φ±+φ± is given by the nals,sothat the interferometersaresimilar;whereasour
L 2 1 4 3 SCIs are dissimilar and can be sensitive to the relatively
phases φ1−4 of the laser pulses at t1−4. This equation large 16n2ω T differential signal.
shows that LMT beam splitters can increase the sensi- r
One essential requirement for the experimental setup
tivity of the phase towards gravity by a factor of n and
the one towards the recoil by n2. (Fig. 2)is to satisfyδφL =0aswellastechnicallypossi-
ble. Moreover, our laser system is optimized for driving
Because of the motion of the atoms, which gives rise
LMT beam splitters based on high-order Bragg diffrac-
to a Doppler frequency shift, addressing the upper and
tion [11, 20], which requires laser pulses having smooth
lower interferometer requires two separate laser frequen-
envelope functions with an optimized duration and high
cies ω ,ω in the top beam. The phases of these respec-
2 3
± ± power.
tive frequencies are denoted φ and φ .
3 4
The laser light originates from a 6W injection-locked
The influence of vibrations is because the atoms de-
titanium:sapphire laser at 852nm wavelength [11, 23].
fine a freely falling inertial frame. Any vibrations of the
Its frequency is referenced to the F = 3 → F′ = 2 D2
laboratory translate into phase shifts of the laser beams
line of a modulation transfer spectroscopy in a Cs va-
in this frame, and if the distribution of φ has a width
L
por cell (not shown); an offset of up to ±20GHz can
that is comparable to π/n, they render the interferences
be set by means of an offset lock. For intensity con-
invisible. With non-LMT beam splitters, vibrations can
trol and forming the Gaussian envelope functions of the
be suppressed to acceptable levels by state of the art vi-
beam splitting-pulses, we use an acousto-optical modu-
bration isolation [22]. This becomes difficult, however,
with n≫1 LMT beam splitters. lator AOM1 within a feedback loop, see Fig. 2.
As a result of the different velocities, the resonance
Theideaunderlyingthecancellationofvibrationsisto
conditions for the third and fourth beam splitter are
run the upper and lower interferometers simultaneously
shifted by 16nω between the two interferometers. To
(“simultaneous conjugate interferometers,” SCIs) and to r
satisfy both, AOM2 is driven by two rf signals of equal
use
amplitude at frequencies of 180MHz±f . It thus gener-
m
Φ≡φ+−φ− =16n2ω T +nφ , (2) ates two optical frequencies in its deflected output that
r L
differ by 2f . They follow the same optical path; thus,
m
where
lengthfluctuations suchas causedby vibrations,air cur-
φ =(φ+−φ−)+(φ−−φ+) (3) rents, etc., are common-mode and do not degrade the
L 3 3 4 4
phase noise in the difference frequency. We have previ-
depends only upon the difference between laser phases ously shownthat a phase variance ofσ2 ≈(160µrad)2 is
at the last two beam splitters. Thus, the requirement possible [24]. The powerin eachcomponentis setto 1/8
of absolute phase stability in one individual interferome- ofthepowerattheinputofAOM2,whichmaximizesthe
ter has been reduced to one of relative stability between Rabi frequency of driving the atoms.
two: if δφ = 0, the overall phase Φ =const., indepen- To generate the counterpropagating beam, we use the
L
dent of vibrations. Then, their fringes as plotted in Fig. undeflected power from AOM2. Use of this radiation,
3
which would otherwise be lost, allows us to make good
use of our available laser power. This, however, varies
between 1/2 and 1 of AOM2’s input power at the beat
frequency 2f of the two rf signals. AOM3 is used to
m
take out this undesired modulation. It is driven by a
“conjugate” rf signal, which is strong when the rf drive
of AOM2 is weak and vice versa. The amplitude modu-
lation is thus suppressed - residual sidebands are below FIG. 3: Left: 12¯hk, 1ms, 25% contrast. Middle: 14¯hk, 1ms,
0.0018 (or -27dB) of the carrier power. 25% contrast; Right: 20¯hk, 1ms, 27%.
Due to the free fall of the atoms, the resonance condi-
tion in the laboratory frame changes at a rate of about
23MHz/s. Weaccountforthisbyrampingthefrequency
of the bottom beam using the double-passed AOM4.
manoTd1h/ee,e2pboienlaatmerniszsaiattryieonrba-rmdoiauuigsnhtotafitnaobinotghuetfib3eexmrpsmerainbmdyenacotcllovimmiamastieenrdgcliteao-l Contrast /%10 Contrast /%10 n = 1 03.3 GHz
12 GHz
fiber port (Thorlabs, Inc., Newton, NJ) or 12.5mm by a 1260 GGHHzz
1 1
tripletlens,acombinationofanaplanaticmeniscus(CVI 1 10 100 1 10 100
Pulse separation time /ms Pulse separation time T/ms
Melles Griot 01 LAM 225/076) and an achromatic dou-
blet(ThorlabsAC508-200B).Thepolarizationsaremade
FIG.4: Left: Contrastversuspulseseparationtimefora10¯hk
circularto(σ+−σ+)byzeroth-orderquarterwaveretar- interferometer. Theimprovementat T =100mswasreached
dation plates. The bottom beam can have a maximum by improved optics and shielding from air currents (see Fig.
powerof1.15Watthefiberoutput;thetopbeamapeak 5 for the ellipse underlying this datum). Right: same for a
power of 1.6W, i.e., 0.4W per frequency. Alternatively, 20h¯k interferometer with different detunings.
we overlapboth beams ata polarizingbeamsplitter and
sendthemthroughthesamefiberwithorthogonalpolar-
izations. The upper fiber collimation optics is then re- in Fig. 4. A certain decrease for long T is expected be-
placedbyahollowcorner-cuberetroreflectorandaquar- cause then a fraction of the atoms leave the area of the
ter wave plate. This method simplifies beam alignment. laser beams due to their thermal velocity. Nevertheless,
Also, it was found important to shield the beams from a contrast of 21% can be obtained for T = 100ms and
air currents to prevent a loss of contrast. 10¯hk. For a 20h¯k device, contrast is 10% at T = 50ms.
The first interferometer pulse is typically applied at Inpreviousworkwithout SCIs,8%contrastat20h¯k was
t = 70ms after launch. For 20h¯k beam splitters, we only possible at T ≤ 1ms [11]. Thus, SCIs allow us to
1
use a detuning of about 3-4GHz and peak intensities of improve the pulse separation time to 50ms from 1ms,
0.4W/cm2 inthe bottombeamand0.13W/cm2 per fre- without a reduction in contrast. This corresponds to a
quency in the top beam with a waist of 12.5mm; with 2,500-foldincrease in the enclosed space-time area.
thin (w0 = 3mm) beams a detuning of 16GHz is used. Our dissimilar interferometersexhibit a largedifferen-
Afterelapseofthefullinterferometersequence,theatoms tial signal (the 16n2ω T term due to the photon recoil),
r
in the four interferometer outputs a-d (Fig. 2, left) are allowing for high-resolution measurements. To demon-
separately detected by their fluorescence fa−d as they strate this, Fig. 5 shows 2 data sets, each containing
passa photomultiplier tube infree fall. To takeout fluc- 1,300 data points taken with a 10h¯k interferometer pair
tuations in the atom number, we define the normalized having a pulse separation time of T = 100ms. To ana-
fluorescence Fu =(fa−fb)/(fa+fb) of the upper inter- lyze the data, we use a Bayesian estimation [13], which
ferometerandFl inanalogyfor the lowerinterferometer. shows a better immunity from systematic errors than
Fig. 3 shows examples for ellipses measured by our simpler methods [25]. At a given signal to noise ratio,
SCIs at a short pulse separation time of T = 1ms. A the phase Φ can be best determined if it is near ±π/2,
contrast of around 25-31% is achieved at momentum whentheellipseisclosetoacircle. Therefore,anoffsetof
transfersbetween(8−20)h¯k (the theoreticaloptimum is +π/2 and −π/2,respectively, was used for the two mea-
50%, because each detected interferometer output over- surements shown, and the laser frequency offset f that
m
laps spatially with population lost in the third beam yields Φ = 0 can be determined from the average of the
splitter which does not interfere). It is evident that the phase estimates. From a total of 12,000 such points, we
strong dependence of the contrast upon the momentum obtainaresolutionof6.8ppbwithin7hofmeasurement.
transfer,that wasobservedin previousLMT interferom- Via Eq. 2, ω and thus h¯/M can be determined; corre-
r
eters [11], is absent. This is a first benefit of SCIs. spondingly, our SCIs are sensitive to the fine structure
The dependence of the contrast on the pulse separa- constant α via α = [(2R∞/c)(M/me)(h/M)]1/2, where
tion time T is shown for 10h¯k and 20h¯k interferometers R∞istheRydbergconstantandmetheelectronmass,to
4
interference contrast. By removing the most important
0.0 0.0
limitation on the space-time area, and hence sensitivity,
-0.1 -0.1 of such large momentum transfer interferometers, this
workopensthedoortowardsmanyexcitingexperiments.
-0.2 -0.2
-0.3 -0.3 S.H. and H.M. thank the Alexander von Humboldt
-0.4 -0.4 foundation. This materialis basedupon worksupported
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 by the National Science Foundation under Grant No.
0400866andbytheAirForceOfficeofScientificResearch
FIG. 5: Each graph shows 1300 data points (n = 5,T = under Award Number FA9550-04-1-0040.
100ms,C =20%)takenatΦ=+π/2and−π/2,respectively.
a resolution of 3.4ppb. Without SCIs and LMT, achiev-
ingasimilarresolutionwouldtakeseveralweeks’worthof
∗ Electronic address: [email protected];http://physics.
data[8]. ItissimilartointerferometersusingBlochoscil-
berkeley.edu/research/mueller/
lation for ∼ 2000¯hk common-mode momentum transfer,
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1 1 2 2 1,2
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using 20h¯k momentum transfer, without a reduction in
