zotero/storage/8ARH3R9C/.zotero-ft-cache

APS/123-QED
Inﬂuence of the Coriolis force in atom interferometry
Shau-Yu Lan,1, ∗ Pei-Chen Kuan,1 Brian Estey,1 Philipp Haslinger,2 and Holger Mu¨ller1, 3 1Department of Physics, University of California, Berkeley, California 94720, USA
2VCQ, Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria 3Lawrence 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 Coriolis force from Earth’s rotation and to characterize conﬁguration space wave packets. For interferometers with large momentum transfer and large pulse separation time, we improve the contrast by up to 350% and suppress systematic eﬀects. We also reach what is to our knowledge the largest spacetime area enclosed in any atom interferometer to date. We discuss implications for future high performance instruments.

arXiv:1110.6910v2 [quant-ph] 19 Jan 2012

Light-pulse atom interferometers use atom-photon interactions to coherently split, guide, and recombine freely falling matter-waves [1]. They are important in measurements of local gravity [2], the gravity gradient [3], the Sagnac eﬀect [4], Newton’s gravitational constant [5], the ﬁne structure constant [6], and tests of fundamental laws of physics [7–9]. Recent progress in increased momentum transfer led to larger areas enclosed between the interferometer arms [10–12] and, combined with commonmode noise rejection between simultaneous interferometers [13, 14], to strongly increased sensitivity. With these advances, what used to be a minuscule systematic eﬀect now impacts interferometer performance: The Coriolis force caused by Earth’s rotation has long been known to cause systematic eﬀects [2]. In this Letter, we not only demonstrate that it causes severe loss of contrast in large space-time atom interferometers, but also use a tip-tilt mirror [15] to compensate for it, improving contrast (by up to 350%), pulse separation time, and sensitivity, and characterize the conﬁguration space wave packets. In addition, we remove the systematic shift arising from the Sagnac eﬀect. This leads to the largest space-time area enclosed in any atom interferometer yet demonstrated, given by a momentum transfer of 10 k, where k is the momentum of one photon, and a pulse separation time of 250 ms.
Fig. 1 shows the atom’s trajectories in our apparatus. We ﬁrst consider the upper two paths: At a time t0, an atom of mass m in free fall is illuminated by a laser pulse of wavenumber k. Atom-photon interactions coherently transfer the momentum of a number 2n of photons to the atom with about 50% probability, placing the atom into a coherent superposition of two quantum states that separate with a relative velocity of 2nvr, where vr = k/m is the recoil velocity. An interval T later, a second pulse stops that relative motion and another interval T ′ later, a third pulse directs the wave packets towards each other. The packets meet again at t4 = t1 + 2T + T ′ when a ﬁnal pulse overlaps the atoms. The probability of detecting an atom in a particular out-
∗ sylan@berkeley.edu

FIG. 1. Simultaneous conjugate Ramsey-Bord´e interferometers. Left: atomic trajectories. Beam splitters (π/2 pulses) split and recombine the wave packets. Right: plotting the populations A through D at the outputs of the interferometers versus each other yields an ellipse whose shape is determined by ∆φ+ − ∆φ− = 16n2( k2/2m)T .
put of the interferometer is given by cos2(∆φ/2), where ∆φ is the phase diﬀerence accumulated by the matter wave between the two paths. It can be calculated to be ∆φ± = 8n2( k2/2m)T ± nkgT (T + T ′), the sum of a recoil-induced term 8n2( k2/2m)T and a gravity induced one, nkgT (T + T ′), where g is the acceleration of free fall and ± correspond to upper and lower interferometer, respectively (Fig. 1) [14].
Because of Earth’s rotation, however, the interferometer does not close precisely. We adopt cartesian coordinates in an inertial frame, one that does not rotate with Earth. 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 t1, see Fig. 2. Later, at t2, t3, and t4, the laser is rotated relative to the inertial frame, changing the direction of the momentum transfer. As a result, the wave packet’s relative velocities during the intervals [t1, t2], [t2, t3], [t3, t4] and [t4, ∞] are, to ﬁrst order in Ω⊕,
v12 = 2nvr(0, 0, 1), v23 = 2nvr (Ω⊕T cos ϑ, 0, 0) , v34 = 2nvr (Ω⊕(2T + T ′) cos ϑ, 0, −1) , v4∞ = 0, (1)
respectively, where Ω⊕ is the angular frequency of Earth’s rotation and ϑ = 37.87◦, the latitude of the laboratory in Berkeley, California. Thus, at t4, the wave

2

FIG. 2. Left: Location of the experiment relative to Earth’s rotation. Right: Setup. Red, yellow and green arrows represent the cooling, Bragg, and detection beams, respectively. The wave packets in the interferometer are separated by up to 8.8 mm with 2n = 10, T = 250 ms.

packets miss each other by

δ = 4nvrΩ⊕T (T + T ′) cos ϑ(1, 0, 0).

(2)

An estimate of the size of the atomic wave packets√is provided by the thermal de Broglie wavelength h/ 2πmkBT , where kB is the Boltzmann constant. For cesium atoms at a temperature T of 2 µK (typical of a moving molasses launch), this is about 100 nm. For typical parameters, e.g., T = 100 ms, T ′ = 5 ms, and 2n = 2, we ﬁnd δ = 13 nm. Even though this will not lead to a substantial loss of contrast, it will still lead to systematic errors that we discuss below. For large momentum transfer beam splitters and longer pulse separation times, however, δ = 0.33 µm (at 2n = 10, T = 250 ms), giving rise to a signiﬁcant contrast reduction. (Use of condensed atoms increases wave-packet size [16], but does not reduce the systematic eﬀects arising from rotation.)
Our experiment is based on a 1.5 m tall fountain of cesium atoms in the F = 3, mF = 0 quantum state, launched ballistically using a moving optical molasses. The launched atoms have a 3-dimensional temperature of 1.2 µK, determined by a time-of-ﬂight measurement. A Doppler-sensitive two-photon Raman process selects a group of atoms having a subrecoil velocity distribution along the vertical launching axis.
Because of the extreme sensitivity of interferometers with large momentum transfer and long pulse separation time, suppression of the sensitivity to vibrations is important. For this reason, we operate a pair of simultaneous conjugate 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 their phases ∆φ±. The inﬂuence of gravity and vibrations cancels out, and the signal can be extracted even when vibrations lead to zero visibility of the fringes for each interferometer. For beam splitting, we use multiphoton Bragg diﬀraction [10, 17]: An atom absorbs a number
n of photons from a ﬁrst laser beam with wavevector k1 while being stimulated to emit the same number of pho-

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

0.2 0.4 0.3 0.2 0.1 0.0 0.1

0.2 0.4 0.3 0.2 0.1 0.0 0.1

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 difference as shown in Fig. 1. The contrast of upper interferometer are 20% and 27% for left and right ﬁgure, respectively.

tons having a wavevector k2 by a second, antiparallel, laser beam, without changing its internal quantum state.
The process transfers a momentum n (k1 − k2) to the atom and thus a kinetic energy of n2 2(k1 − k2)2/(2m). Energy-momentum conservation selects one particular Bragg diﬀraction order n, depending on the laser’s frequency diﬀerence ω1 − ω2. We generate the two laser beams from a common 6 W titanium:sapphire laser and use acousto-optical modulators to shift the frequency of the laser [18] and optimize the eﬃciency of the Bragg diﬀraction beam splitter by adjusting Gaussian pulses width to about 100 µs [11, 14]. The beam is collimated at an 1/e2 intensity radius of 3.6 mm and sent vertically upwards to a retroreﬂection mirror inside the vacuum chamber. The Doppler eﬀect due to the free fall of the atoms singles out one pair of counterpropagating frequencies that satisfy the above resonance condition.
The retroreﬂection mirror is ﬂexibly mounted on the top of the vacuum chamber with a bellows and can be rotated by piezoelectric actuators, see Fig. 2. The rotation 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 sensitivity of the actuators has been calibrated against an Applied Geomechanics 755-1129 tilt sensor. We can use
this to give the momentum transfer k1 − k2 a constant direction as seen from the inertial frame, in spite of Earth’s rotation, to compensate for the Coriolis force.
Fig. 3 shows data obtained with and without Coriolis compensation. The increase in contrast is obvious. We ﬁt the data with an ellipse [14] and determine the contrast by the length of the projection of the ﬁtted ellipse onto the axes, separately for the upper and lower interferometer. For the remainder of the paper, data is quoted for the upper interferometer. By grouping the data into bins of 20 points, the contrast and its standard error is determined by statistics over the bins. Fig. 4 shows the contrast as a function of the tip-tilt rotation rate around the y′ axis for various pulse separation times. A Gaussian function of the rotation rate (with the center Ωopt, width σΩ, amplitude and oﬀset as ﬁt parameters) ﬁts the

3

Contrast Contrast Contrast

0.40

0.35

0.30

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0.05

-10 -5

0

130ms 160ms 180ms 220ms 250ms

5

10 15 20 25 30 35 40

-5
[10 rad/s]

FIG. 4. Contrast versus tip-tilt mirror rotation rate for various pulse separation times (T = 130, 160, 180, 220, 250 ms; T ′ = 2 ms). The y′-axis rotation rate is varied, the x′-axis rotation rate is ﬁxed at -26.2 µrad/s. The loss of contrast for larger T is mainly due to the thermal expansion of the atomic cloud and wavefront distortion in the interferometer beam.

TABLE I. Ωopt and σΩ are the ﬁtting center and width from Fig. 4. σ is calculated from σΩ and Eq. (4).

T [ms] Ωopt [µrad/s] σΩ [µrad/s] σ [nm]

130

49 ± 4

124 ± 8 106 ± 7

160

51 ± 2

81 ± 4 105 ± 5

180

50 ± 2

66 ± 3 108 ± 5

220

52 ± 2

38 ± 5 92 ± 12

250

54 ± 2

34 ± 4 107 ± 13

data within the standard error. The ﬁt results are tab-
ulated in Tab. I. A weighted average for the optimum
tip-tilt rotation rate is Ωopt=(51.3 ± 0.8) µrad/s. We also performed a similar measurement for the x′ axis, Fig. 5
(left). From both measurements, we compute the magni-
tude of the rotation rate, (58.5 ± 1.0) µrad/s (taking into account the actual angle of 82◦ between x′ and y′). This
agrees with Ω⊕ cos θ = 57.4 µrad/s within a ∼ 1σ error.
To model this loss of contrast, we calculate the overlap integral ψ′(x)|ψ(x) of the interfering wave pack-
ets at t = t4. Since the free time evolution of a wave packet is given by a unitary operator U (t, t0), the overlap integral of the wave packet ψ′(t)|ψ(t) = ψ′(t0)U †(t, t0)|U (t, t0)ψ(t0) = ψ′(t0)|ψ(t0) is independent of the free time evolution and depends on the
relative position only. For example, the atom may ini-
tially be represented by a Gaussian wave packet

ψ=

det A π3

1/4

e−

1 2

xAx,

(3)

where the matrix A can be taken as symmetric. In its

principal and σ3−2.

frame, A is The overlap

diagonal with elements σ1−2, σ2−2, integral is independent of time,

d3rψ∗(r

+

δ)ψ(r)

=

e−

1 4

δAδ ,

(4)

0.30

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0.05

-15 -10

-5

0

5

10

-5
[10 rad/s]

0.25 0.20 0.15 0.10 0.05
-60 -40 -20 0 20 40 60 Delay [ s]

FIG. 5. Left: Contrast versus tip-tilt mirror rotation rate in x′ direction (T = 180 ms, T ′ = 2 ms). The y′-axis rotation is ﬁxed at 69.8 µrad/s. The Gaussian ﬁt has its maximum at (−22 ± 1) µrad/s with a width of 53 ± 4 µrad/s, which leads to an estimate of σy = (86 ± 7) nm. Right: Contrast versus delay of t4 in Fig. 1. The width of the ﬁt is (23.1 ± 0.6) µs.

where δ is given by Eq. (2). The experiment validates this model: Figs. 4, 5 show that the data is well described by Gaussian functions. According to Tab. I, the measured widths of the overlap integral agree with one another for all measured T . From the data, we can determine the parameters of the overlap integral. The symmetry of the atomic fountain suggests that the principal axes of the matrix A coincide with the x, y, z laboratory frame. In what follows, we neglect the small diﬀerence of the x, x′ and y, y′ directions. The weighted average of the numbers in the last column of Tab. I is σx = (105±3) nm. The ﬁt shown in Fig. 5, left, yields σy = (86 ± 7) nm. To determine σz, we vary the time interval between t3 and t4 (Fig. 1), see Fig. 5, right. The ﬁtted width corresponds to σz = (813 ± 21) nm.
Because each atom interferes only with itself, these measured quantities are properties of individual atoms, averaged over the atomic ensemble. They need not be related to the temperature of the ensemble. This is illustrated by the data: The expectation value of the squared momentum along the ith coordinate p2i of the wave packet Eq. (3) allows one to compute an expectation value ψ|p2i /2m|ψ = 2/(2mσi2). If we set this equal to kBTi/2, where Ti has the dimension of temperature, we obtain Tx = (0.33 ± 0.02) µK and Ty = (0.49 ± 0.07) µK. Since our atomic ensemble is not a Bose-Einstein condensate, these values are unrelated to (speciﬁcally, lower than) the 1.2 µK ensemble temperature. For Tz, we obtain (5.5 ± 0.3) nK. This low value results from the velocity sele√ction in our atomic fountain: The Fourier width of the 1/ e intensity duration σvs of the Gaussian velocity selection pulse corresponds to a Doppler velocity width of c/(ωσvs) and, per ∆x∆p = /2 (the minimum uncertainty product applies to Gaussian √wave packets), a position envelope function having a 1/ e width of vrσvs/2, where vr = k/m. For σvs = 500 µs, this evaluates to 880 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 = 2Ω⊕ · (v0 × k)T 2, where v0 is the atom’s initial velocity. If the interferometer is used for gravity measurements, the corresponding grav-

ity oﬀset is ∆g = 2Ω⊕ · (v0 × kˆ), where kˆ is a unit vector pointing along the laser beams. This is zero when the launch has no horizontal velocity component, but in practice a small horizontal component is inevitable due to alignment error. If, e.g., we assume a horizontal velocity of 1 cm/s typical of a laser-cooled atomic fountain, ∆g = 6 × 10−8g due to Earth’s rotation, a dominant contribution to the accuracy of atom intererometers [2]. Coriolis compensation as employed here can remove it without a need to know v0. The accuracy from our rotation measurement, ∆Ω/Ω ∼ 0.017, would reduce ∆g to 1 × 10−9g and thus below the precision of state-ofthe-art instruments. A tip-tilt mirror using actuators with active feedback could easily increase this accuracy further, and maximizing the contrast provides an independent veriﬁcation of successful compensation. Possible remaining imperfections of the overlap of the wave packets are due to the vibration of the retroreﬂection mirror and the gravity gradient. We note that Coriolis compensation removes the leading order eﬀect of Earth’s rotation but higher order eﬀects remain [15]. However, they are negligible here.
We have used a tip-tilt mirror to compensate the inﬂuence of Earth’s rotation in atom interferometry, and also to characterize the overlap integral of the interfering atomic wave packets. The observations are well described by Gaussian wave packets, whose properties were determined from the data. Coriolis compensation allows us to reach better contrast, larger space-time enclosed area and reduce systematic eﬀects in atom interferometry. For example, from the measured width of the overlap integral (Tab. I) together with the displacement Eq. (2), an uncompensated Coriolis force would reduce the contrast by a factor of exp[−(Ω⊕ cos ϑ)2/(2σ2)] = 0.28, for 2n = 10

4
and T = 250 ms. At T = 130 ms, we reach a contrast of 40%. Coriolis compensation is thus crucial for the most sensitive large-area, large momentum transfer atom interferometers. We also note that Coriolis compensation has allowed us to experimentally demonstrate the atom interferometer with the largest enclosed space-time area thus far: The gravitationally-induced phase 2nkgT (T + T ′) in our interferometer is 6.3 × 107 rad (2n = 10 and T = 250 ms), compared to 3.2 × 107 rad in Ref. [9]. (Other work [19] has reached higher momentum transfer but substantially smaller T and thus lower overall phase shift.) The recoil-induced phase 16n2( k2/2m)T between our simultaneous conjugate interferometers is 1.2 × 106 rad, compared to 5 × 105 rad in Ref. [14]. Such a measurement can be used to determine the fundamental constants /m and α, the ﬁne structure constant. Our data allows a r√esolution in /m of 12 ppb within 42 minutes (10 ppb hr), twice as good as in Ref. [14]. We expect that Coriolis-compensation will enhance future high-performance interferometers, e.g., in gravity wave detection [20], measurements of /m, α [6], Avogadro constant NA, new tests of general relativity [21], and inertial sensing, with applications in navigation and geophysics. The technique will be especially important for achieving high performance in mobile and space-borne atom interferometers [22, 23], which must cope with rotation rates that are orders of magnitude larger than Earth’s rotation.
We thank Justin Brown, Paul Hamilton, Michael Hohensee, Gee-Na Kim, and Achim Peters for discussions and the Alfred P. Sloan Foundation, the David and Lucile Packard foundation, the National Aeronautics and Space Administration, the National Institute of Standards and Technology, and the National Science Foundation for support.

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