Influence of the Coriolis force in atom interferometry
Shau-Yu Lan, Pei-Chen Kuan, Brian Estey, Philipp Haslinger, Holger Mueller
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Influence of the Coriolis force in atom interferometry
Shau-Yu Lan, ∗ Pei-Chen Kuan, Brian Estey, Philipp Haslinger, and Holger M¨uller
1, 3 Department of Physics, University of California, Berkeley, California 94720, USA VCQ, Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley, California 94720, USA (Dated: January 23, 2012)In a light-pulse atom interferometer, we use a tip-tilt mirror to remove the influence of the Coriolisforce from Earth’s rotation and to characterize configuration space wave packets. For interferometerswith large momentum transfer and large pulse separation time, we improve the contrast by up to350% and suppress systematic effects. We also reach what is to our knowledge the largest space-time area enclosed in any atom interferometer to date. We discuss implications for future highperformance instruments.
Light-pulse atom interferometers use atom-photon in-teractions to coherently split, guide, and recombine freelyfalling matter-waves [1]. They are important in measure-ments of local gravity [2], the gravity gradient [3], theSagnac effect [4], Newton’s gravitational constant [5], thefine structure constant [6], and tests of fundamental lawsof physics [7–9]. Recent progress in increased momen-tum transfer led to larger areas enclosed between the in-terferometer arms [10–12] and, combined with common-mode noise rejection between simultaneous interferome-ters [13, 14], to strongly increased sensitivity. With theseadvances, what used to be a minuscule systematic effectnow impacts interferometer performance: The Coriolisforce caused by Earth’s rotation has long been known tocause systematic effects [2]. In this Letter, we not onlydemonstrate that it causes severe loss of contrast in largespace-time atom interferometers, but also use a tip-tiltmirror [15] to compensate for it, improving contrast (byup to 350%), pulse separation time, and sensitivity, andcharacterize the configuration space wave packets. In ad-dition, we remove the systematic shift arising from theSagnac effect. This leads to the largest space-time areaenclosed in any atom interferometer yet demonstrated,given by a momentum transfer of 10 ~ k , where ~ k is themomentum of one photon, and a pulse separation timeof 250 ms.Fig. 1 shows the atom’s trajectories in our appara-tus. We first consider the upper two paths: At a time t , an atom of mass m in free fall is illuminated by alaser pulse of wavenumber k . Atom-photon interactionscoherently transfer the momentum of a number 2 n ofphotons to the atom with about 50% probability, plac-ing the atom into a coherent superposition of two quan-tum states that separate with a relative velocity of 2 nv r ,where v r = ~ k/m is the recoil velocity. An interval T later, a second pulse stops that relative motion and an-other interval T ′ later, a third pulse directs the wavepackets towards each other. The packets meet again at t = t + 2 T + T ′ when a final pulse overlaps the atoms.The probability of detecting an atom in a particular out- ∗ [email protected] FIG. 1. Simultaneous conjugate Ramsey-Bord´e interferome-ters. Left: atomic trajectories. Beam splitters ( π/ A through D at the outputs of the interferometersversus each other yields an ellipse whose shape is determinedby ∆ φ + − ∆ φ − = 16 n ( ~ k / m ) T . put of the interferometer is given by cos (∆ φ/ φ is the phase difference accumulated by the matterwave between the two paths. It can be calculated tobe ∆ φ ± = 8 n ( ~ k / m ) T ± nkgT ( T + T ′ ), the sum of arecoil-induced term 8 n ( ~ k / m ) T and a gravity inducedone, nkgT ( T + T ′ ), where g is the acceleration of free falland ± correspond to upper and lower interferometer, re-spectively (Fig. 1) [14].Because of Earth’s rotation, however, the interferome-ter does not close precisely. We adopt cartesian coordi-nates in an inertial frame, one that does not rotate withEarth. We take the x axis horizontal pointing west, the y axis pointing south, and the z axis such that the laser,pointing vertically upwards, coincides with it at t , seeFig. 2. Later, at t , t , and t , the laser is rotated rel-ative to the inertial frame, changing the direction of themomentum transfer. As a result, the wave packet’s rel-ative velocities during the intervals [ t , t ] , [ t , t ] , [ t , t ]and [ t , ∞ ] are, to first order in Ω ⊕ , v = 2 nv r (0 , , , v = 2 nv r (Ω ⊕ T cos ϑ, , ,v = 2 nv r (Ω ⊕ (2 T + T ′ ) cos ϑ, , − , v ∞ = 0 , (1)respectively, where Ω ⊕ is the angular frequency ofEarth’s rotation and ϑ = 37 . ◦ , the latitude of the lab-oratory in Berkeley, California. Thus, at t , the wave FIG. 2. Left: Location of the experiment relative to Earth’srotation. Right: Setup. Red, yellow and green arrows rep-resent the cooling, Bragg, and detection beams, respectively.The wave packets in the interferometer are separated by upto 8.8 mm with 2 n = 10 , T = 250 ms. packets miss each other by ~δ = 4 nv r Ω ⊕ T ( T + T ′ ) cos ϑ (1 , , . (2)An estimate of the size of the atomic wave pack-ets is provided by the thermal de Broglie wavelength h/ √ πmk B T , where k B is the Boltzmann constant. Forcesium atoms at a temperature T of 2 µ K (typical of amoving molasses launch), this is about 100 nm. For typi-cal parameters, e.g., T = 100 ms, T ′ = 5 ms, and 2 n = 2,we find δ = 13 nm. Even though this will not lead to asubstantial loss of contrast, it will still lead to system-atic errors that we discuss below. For large momentumtransfer beam splitters and longer pulse separation times,however, δ = 0 . µ m (at 2 n = 10 , T = 250 ms), givingrise to a significant contrast reduction. (Use of condensedatoms increases wave-packet size [16], but does not re-duce the systematic effects arising from rotation.)Our experiment is based on a 1.5 m tall fountain ofcesium atoms in the F = 3 , m F = 0 quantum state,launched ballistically using a moving optical molasses.The launched atoms have a 3-dimensional temperatureof 1 . µ K, determined by a time-of-flight measurement.A Doppler-sensitive two-photon Raman process selects agroup of atoms having a subrecoil velocity distributionalong the vertical launching axis.Because of the extreme sensitivity of interferometerswith large momentum transfer and long pulse separationtime, suppression of the sensitivity to vibrations is impor-tant. For this reason, we operate a pair of simultaneousconjugate Ramsey-Bord´e interferometers [14], see Fig. 1.The direction of the recoil is reversed between them,reversing the sign of the gravity-induced term in theirphases ∆ φ ± . The influence of gravity and vibrationscancels out, and the signal can be extracted even whenvibrations lead to zero visibility of the fringes for eachinterferometer. For beam splitting, we use multiphotonBragg diffraction [10, 17]: An atom absorbs a number n of photons from a first laser beam with wavevector ~k while being stimulated to emit the same number of pho- - - - - - - - - - - FIG. 3. Raw data obtained without Coriolis compensation(left) and with (right) at T = 180 ms, T ′ = 2 ms, and 10 ~ k momentum transfer. The axes are normalized population dif-ference as shown in Fig. 1. The contrast of upper interferom-eter are 20% and 27% for left and right figure, respectively. tons having a wavevector ~k by a second, antiparallel,laser beam, without changing its internal quantum state.The process transfers a momentum n ~ ( ~k − ~k ) to theatom and thus a kinetic energy of n ~ ( ~k − ~k ) / (2 m ).Energy-momentum conservation selects one particularBragg diffraction order n , depending on the laser’s fre-quency difference ω − ω . We generate the two laserbeams from a common 6 W titanium:sapphire laser anduse acousto-optical modulators to shift the frequency ofthe laser [18] and optimize the efficiency of the Braggdiffraction beam splitter by adjusting Gaussian pulseswidth to about 100 µ s [11, 14]. The beam is collimatedat an 1 /e intensity radius of 3.6 mm and sent verticallyupwards to a retroreflection mirror inside the vacuumchamber. The Doppler effect due to the free fall of theatoms singles out one pair of counterpropagating frequen-cies that satisfy the above resonance condition.The retroreflection mirror is flexibly mounted on thetop of the vacuum chamber with a bellows and can berotated by piezoelectric actuators, see Fig. 2. The rota-tion axes are roughly pointing west ( x ′ ) and south ( y ′ ),enclosing an angle of 82 ◦ . In order to rotate the mirror,a linear electrical ramp is applied to the piezos. The sen-sitivity of the actuators has been calibrated against anApplied Geomechanics 755-1129 tilt sensor. We can usethis to give the momentum transfer ~k − ~k a constant di-rection as seen from the inertial frame, in spite of Earth’srotation, to compensate for the Coriolis force.Fig. 3 shows data obtained with and without Corioliscompensation. The increase in contrast is obvious. Wefit the data with an ellipse [14] and determine the con-trast by the length of the projection of the fitted ellipseonto the axes, separately for the upper and lower inter-ferometer. For the remainder of the paper, data is quotedfor the upper interferometer. By grouping the data intobins of 20 points, the contrast and its standard error isdetermined by statistics over the bins. Fig. 4 shows thecontrast as a function of the tip-tilt rotation rate aroundthe y ′ axis for various pulse separation times. A Gaus-sian function of the rotation rate (with the center Ω opt ,width σ Ω , amplitude and offset as fit parameters) fits the -10 -5 0 5 10 15 20 25 30 35 400.050.100.150.200.250.300.350.40 C on t r a s t [10 -5 rad/s] FIG. 4. Contrast versus tip-tilt mirror rotation rate for var-ious pulse separation times ( T = 130 , , , ,
250 ms; T ′ = 2 ms). The y ′ -axis rotation rate is varied, the x ′ -axisrotation rate is fixed at -26.2 µ rad/s. The loss of contrast forlarger T is mainly due to the thermal expansion of the atomiccloud and wavefront distortion in the interferometer beam.TABLE I. Ω opt and σ Ω are the fitting center and width fromFig. 4. σ is calculated from σ Ω and Eq. (4). T [ms] Ω opt [ µ rad/s] σ Ω [ µ rad/s] σ [nm]130 49 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± data within the standard error. The fit results are tab-ulated in Tab. I. A weighted average for the optimumtip-tilt rotation rate is Ω opt =(51 . ± . µ rad/s. We alsoperformed a similar measurement for the x ′ axis, Fig. 5(left). From both measurements, we compute the magni-tude of the rotation rate, (58 . ± . µ rad/s (taking intoaccount the actual angle of 82 ◦ between x ′ and y ′ ). Thisagrees with Ω ⊕ cos θ = 57 . µ rad/s within a ∼ σ error.To model this loss of contrast, we calculate the over-lap integral h ψ ′ ( ~x ) | ψ ( ~x ) i of the interfering wave pack-ets at t = t . Since the free time evolution of awave packet is given by a unitary operator U ( t, t ),the overlap integral of the wave packet h ψ ′ ( t ) | ψ ( t ) i = h ψ ′ ( t ) U † ( t, t ) | U ( t, t ) ψ ( t ) i = h ψ ′ ( t ) | ψ ( t ) i is inde-pendent of the free time evolution and depends on therelative position only. For example, the atom may ini-tially be represented by a Gaussian wave packet ψ = (cid:18) det Aπ (cid:19) / e − ~xA~x , (3)where the matrix A can be taken as symmetric. In itsprincipal frame, A is diagonal with elements σ − , σ − , and σ − . The overlap integral is independent of time, Z d rψ ∗ ( ~r + ~δ ) ψ ( ~r ) = e − ~δA~δ , (4) -15 -10 -5 0 5 100.050.100.150.200.250.30 [10 -5 rad/s] C on t r a s t -60 -40 -20 0 20 40 600.050.100.150.200.25 C on t r a s t Delay [ s] FIG. 5. Left: Contrast versus tip-tilt mirror rotation ratein x ′ direction ( T = 180 ms, T ′ = 2 ms). The y ′ -axis rotationis fixed at 69.8 µ rad/s. The Gaussian fit has its maximum at( − ± µ rad/s with a width of 53 ± µ rad/s, which leadsto an estimate of σ y = (86 ±
7) nm. Right: Contrast versusdelay of t in Fig. 1. The width of the fit is (23 . ± . µ s. where δ is given by Eq. (2). The experiment validatesthis model: Figs. 4, 5 show that the data is well de-scribed by Gaussian functions. According to Tab. I, themeasured widths of the overlap integral agree with oneanother for all measured T . From the data, we can de-termine the parameters of the overlap integral. The sym-metry of the atomic fountain suggests that the principalaxes of the matrix A coincide with the x, y, z laboratoryframe. In what follows, we neglect the small difference ofthe x, x ′ and y, y ′ directions. The weighted average of thenumbers in the last column of Tab. I is σ x = (105 ±
3) nm.The fit shown in Fig. 5, left, yields σ y = (86 ±
7) nm. Todetermine σ z , we vary the time interval between t and t (Fig. 1), see Fig. 5, right. The fitted width correspondsto σ z = (813 ±
21) nm.Because each atom interferes only with itself, thesemeasured quantities are properties of individual atoms,averaged over the atomic ensemble. They need not berelated to the temperature of the ensemble. This is illus-trated by the data: The expectation value of the squaredmomentum along the i th coordinate h p i i of the wavepacket Eq. (3) allows one to compute an expectationvalue h ψ | p i / m | ψ i = ~ / (2 mσ i ). If we set this equal to k B T i /
2, where T i has the dimension of temperature, weobtain T x = (0 . ± . µ K and T y = (0 . ± . µ K.Since our atomic ensemble is not a Bose-Einstein con-densate, these values are unrelated to (specifically, lowerthan) the 1 . µ K ensemble temperature. For T z , we ob-tain (5 . ± .
3) nK. This low value results from the veloc-ity selection in our atomic fountain: The Fourier width ofthe 1 / √ e intensity duration σ vs of the Gaussian velocityselection pulse corresponds to a Doppler velocity widthof c/ ( ωσ vs ) and, per ∆ x ∆ p = ~ / / √ e width of v r σ vs / v r = ~ k/m . For σ vs = 500 µ s, this evaluates to880 nm, in reasonable agreement with the observed value.Uncompensated rotation also causes systematic effects[2]. For a Mach-Zehnder interferometer, e.g. , the result-ing phase shift is given by ∆ φ = 2 ~ Ω ⊕ · ( ~v × ~k ) T , where ~v is the atom’s initial velocity. If the interferometer isused for gravity measurements, the corresponding grav-ity offset is ∆ g = 2 ~ Ω ⊕ · ( ~v × ˆ k ), where ˆ k is a unit vec-tor pointing along the laser beams. This is zero whenthe launch has no horizontal velocity component, but inpractice a small horizontal component is inevitable dueto alignment error. If, e.g. , we assume a horizontal ve-locity of 1 cm/s typical of a laser-cooled atomic fountain,∆ g = 6 × − g due to Earth’s rotation, a dominantcontribution to the accuracy of atom intererometers [2].Coriolis compensation as employed here can remove itwithout a need to know ~v . The accuracy from our ro-tation measurement, ∆Ω / Ω ∼ . g to 1 × − g and thus below the precision of state-of-the-art instruments. A tip-tilt mirror using actuatorswith active feedback could easily increase this accuracyfurther, and maximizing the contrast provides an inde-pendent verification of successful compensation. Possibleremaining imperfections of the overlap of the wave pack-ets are due to the vibration of the retroreflection mirrorand the gravity gradient. We note that Coriolis compen-sation removes the leading order effect of Earth’s rotationbut higher order effects remain [15]. However, they arenegligible here.We have used a tip-tilt mirror to compensate the in-fluence of Earth’s rotation in atom interferometry, andalso to characterize the overlap integral of the interferingatomic wave packets. The observations are well describedby Gaussian wave packets, whose properties were deter-mined from the data. Coriolis compensation allows usto reach better contrast, larger space-time enclosed areaand reduce systematic effects in atom interferometry. Forexample, from the measured width of the overlap integral(Tab. I) together with the displacement Eq. (2), an un-compensated Coriolis force would reduce the contrast bya factor of exp[ − (Ω ⊕ cos ϑ ) / (2 σ )] = 0 .
28, for 2 n = 10 and T = 250 ms. At T = 130 ms, we reach a contrast of40%. Coriolis compensation is thus crucial for the mostsensitive large-area, large momentum transfer atom inter-ferometers. We also note that Coriolis compensation hasallowed us to experimentally demonstrate the atom inter-ferometer with the largest enclosed space-time area thusfar: The gravitationally-induced phase 2 nkgT ( T + T ′ )in our interferometer is 6 . × rad (2 n = 10 and T = 250 ms), compared to 3 . × rad in Ref. [9].(Other work [19] has reached higher momentum trans-fer but substantially smaller T and thus lower overallphase shift.) The recoil-induced phase 16 n ( ~ k / m ) T between our simultaneous conjugate interferometers is1 . × rad, compared to 5 × rad in Ref. [14]. Sucha measurement can be used to determine the fundamen-tal constants ~ /m and α , the fine structure constant.Our data allows a resolution in ~ /m of 12 ppb within42 minutes (10 ppb √ hr), twice as good as in Ref. [14].We expect that Coriolis-compensation will enhance fu-ture high-performance interferometers, e.g. , in gravitywave detection [20], measurements of ~ /m, α [6], Avo-gadro constant N A , new tests of general relativity [21],and inertial sensing, with applications in navigation andgeophysics. The technique will be especially importantfor achieving high performance in mobile and space-borneatom interferometers [22, 23], which must cope with ro-tation rates that are orders of magnitude larger thanEarth’s rotation.We thank Justin Brown, Paul Hamilton, Michael Ho-hensee, Gee-Na Kim, and Achim Peters for discussionsand the Alfred P. Sloan Foundation, the David and Lu-cile Packard foundation, the National Aeronautics andSpace Administration, the National Institute of Stan-dards and Technology, and the National Science Foun-dation for support. [1] A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard,Rev. Mod. Phys.
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