TGS15 tune out gyro

Atom Interferometer Gyroscope with Spin-Dependent Phase Shifts
Induced by Light near a Tune-Out Wavelength
Raisa Trubko,1
James Greenberg,2
Michael T. St. Germaine,2
Maxwell D. Gregoire,2
William F. Holmgren,2
Ivan Hromada,2
and Alexander D. Cronin1,2
1
College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA
2
Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
(Received 31 October 2014; published 9 April 2015)
Tune-out wavelengths measured with an atom interferometer are sensitive to laboratory rotation rates
because of the Sagnac effect, vector polarizability, and dispersion compensation. We observed shifts in
measured tune-out wavelengths as large as 213 pm with a potassium atom beam interferometer, and we
explore how these shifts can be used for an atom interferometer gyroscope.
DOI: 10.1103/PhysRevLett.114.140404 PACS numbers: 03.75.Dg, 32.10.Dk, 06.30.Gv
Atom interferometers have an impressive variety of
applications ranging from inertial sensing to measurements
of fundamental constants, measurements of atomic proper-
ties, and studies of topological phases [1]. In particular,
making a better gyroscope has been a long-standing goal in
the atom optics community because atom interferometers
have the potential to outperform optical Sagnac gyroscopes.
Advances in the precision and range of applications for atom
interferometry have been realized by using interferometers
with multiple atomic species [2–5], multiple atomic veloc-
ities [6–10], multiple atomic spin states [11–13], and
multiple atomic path configurations [14–18]. Here, we
use atoms with multiple spin states to demonstrate a new
method for rotation sensing. Our atom interferometer
gyroscope shown in Fig. 1 reports the absolute rotation rate
Ω in terms of an optical wavelength, using a spin-dependent
phase echo induced by light near a tune-out wavelength.
A tune-out wavelength, λzero, occurs where the dynamic
polarizability of an atom changes sign between two reso-
nances [19–29]. Since atomic vector polarizability depends
on spin [30–32], theoretical tune-out wavelengths usually
describe atoms with spin mF ¼ 0. The same λzero should be
found, on average, for atoms in a uniform distribution of spin
states.However, in this Letter, we show that the Sagnac effect
breaks the symmetry expected from the vector polarizability
in a way that makes tune-out wavelengths remarkably
sensitive to the laboratory rotation rate. We measured
tune-out wavelengths λzero;lab using a potassium atom inter-
ferometer and circularly polarized light, and found that our
measurements were shifted by 0.213 nm from the theoretical
tune-out wavelength of λzero ¼ 768.971 nm [19]. This shift
is more than 100 times larger than the uncertainty with which
λzero can bemeasured[29],andthissuggeststhepossibilityof
creating a sensitive gyroscope using tune-out wavelengths.
The purpose of this Letter is therefore to explain how an atom
interferometergyroscopecanmeasurethe laboratory rotation
rate Ω with the aid of atomic spin-dependent phase shifts
induced by light near a tune-out wavelength. This is a new
application of tune-out wavelengths and a new method for
atom interferometry that could improve sensors needed for
navigation, geophysics, and tests of general relativity.
Atom interferometer gyroscopes [1,6–9,33–39] can
sense changes in rotation rate (ΔΩ) because of the
Sagnac effect. Some atom interferometers [7–9,34,35]
can also report the absolute rotation rate (Ω) with respect
to an inertial frame of reference since the Sagnac phase
depends on atomic velocity. Because the Sagnac phase is
dispersive, Ω can affect the interference fringe contrast.
References [6–9,34,35] applied auxiliary rotations to an
atom interferometer to compensate for the earth’s rotation
Ωe and thus maximize contrast. References [34,35] even
used contrast as a function of applied rotation rate in order to
measure Ωe.
In comparison, here we demonstrate optical and static
electric field gradients that compensate for dispersion in the
Sagnac phase. This is a general example of dispersion
compensation [40,41] in which one type of phase compen-
sates for dispersion in another. Furthermore, we show that
circularly polarized light at λzero makes an observable
Ω-dependent phase shift for our unpolarized atom beam
atom beam
nanogratings Ωlaser
system
x
z
polarizationcontrol
FIG. 1 (color online). Apparatus diagram. The branches of a
3-nanograting Mach-Zehnder atom interferometer [1] are illumi-
nated asymmetrically by laser light propagating perpendicular to
the page. An optical cavity (not shown) recycles the light to
increase the phase shift. A single mode optical fiber (yellow)
guides the laser light into the atom beam vacuum chamber, and
the loops in the fiber are used to control the optical polarization.
PRL 114, 140404 (2015) P H Y S I C A L R E V I E W L E T T E R S
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interferometer. This works because spin-dependent
dispersion compensation causes higher contrast for one spin
state. Thus, using spin as a degree of freedom and light near a
tune-out wavelength, we made a gyroscope that reports the
absoluterotationrateΩintermsofalight-inducedphaseshift.
Our gyroscope, shown in Fig. 1, uses material nano-
gratings that permit interferometry with distributions of
atomic spin and velocity, both of which are needed in order
to cause the shifts in λzero;lab that are sensitive to Ω. An atom
interferometer like ours was previously shown to monitor
changes in rotation rate ΔΩ [33]. We now show that an
atom interferometer gyroscope with material nanogratings
can measure absolute rotation rates smaller than Ωe. This is
significant because nanogratings offer some advantages
such as simplicity, reliability, and spin-independent and
nearly velocity-independent diffraction amplitudes that
may enable more robust and economical Ω sensors.
We studied the light-induced phase shift ϕ for an
ensemble of atoms, which we model as
ϕ ¼ ϕon − ϕoff ð1Þ
where ϕon is the measured phase when the light is on and
ϕoff is the measured phase when the light is off. For an atom
beam with a velocity distribution PvðvÞ and a uniform
distribution of spin states PsðF; mFÞ ¼ 1=8, the contrast
Con and phase ϕon for the ensemble are described by
Coneiϕon ¼ Co
X
F;mF
PsðF; mFÞ
Z ∞
0
PvðvÞeiΦtotal dv ð2Þ
where Φtotal ¼ΦL þΦS þΦa þΦo. Here, ΦL is the velocity-
dependent and spin-dependent phase caused by light, ΦS
is the velocity-dependent Sagnac phase, Φa is the velocity-
dependent phase induced by an acceleration or gravity,
Φo is the initial phase, and Co is the initial contrast of
the interferometer. A similar equation can be written for
Coffeiϕoff with the light off so that ΦL ¼ 0. Our atom beam
has a velocity distribution PvðvÞ adequately described by
PvðvÞ ¼ Av3exp½−ðv − v0Þ2=ð2σ2
vÞŠ, where A is a normali-
zation constant [42].
The Sagnac phase [33,34]
ΦS ¼
4πL2
Ω
vdg
ð3Þ
is a function of atomic velocity v and the rotation rate Ω
along the normal of the interferometer’s enclosed area. L is
the distance between gratings, and dg is the period of the
gratings. In our interferometer, dg ¼ 100 nm and
L ¼ 0.94 m, so ΦS ¼ 2.7 rad for a 1600 m=s atom beam
in our laboratory at 32° N latitude due to Ωe.
The gravity phase Φa [33] is
Φa ¼
2πL2
g sinðθÞ
v2dg
ð4Þ
where g sinðθÞ is the gravitational acceleration along the
grating wave vector direction. As we discuss later, θ and Φa
are small, but nonzero.
The light phase is
ΦL ¼
αðωÞ
2ϵocℏv
Z
s

d
dx
Iðr; ωÞ

dz ð5Þ
where the dynamic polarizability αðωÞ depends on the
atomic state jF; mFi and the laser polarization [30–32].
Near the second nanograting, we shine 50 mWof laser light
perpendicular to the plane of the interferometer. The laser’s
irradiance gradient in a beam with a 100 μm diameter waist
asymmetrically illuminates the atom beam paths as
sketched in Fig. 1. The irradiance gradient ðd=dxÞI is
integrated along the atom beam paths in the z direction. The
path separation s is proportional to v−1
. Hence, for laser
beams much wider than s, the light phase ΦL approxi-
mately depends on v−2
. The fact that this does not exactly
match the v−1
dispersion of the Sagnac phase means the
dispersion compensation is imperfect, which is why we see
caustics in Fig. 2. Figure 2 presents modeled phase shifts
for ground-state potassium atoms with several different
velocities and five different spin states. Figure 2 illustrates
how spin-dependent dispersion compensation works and
how it can make λzero;lab ≠ λzero.
The way ΦS affects the light-induced phase ϕðλÞ leads to
several testable predictions that we experimentally verified.
Equation (2) led us to predict a new wavelength λzero;lab for
which ϕ is zero. A simulation of this prediction is shown in
-6
-5
-4
-3
-2
-1
0
1
phase(rad)
769.6769.2768.8768.4
wavelength (nm)
mF = 0 mF = 1 mF = -1
mF = 2 mF = -2 ensemble
zero,lab
zero
zero,lab
mF= +2mF= -2
FIG. 2 (color online). Light-induced phase spectra demonstrate
dispersion compensation. The phase ΦLðλ; vÞ þ ΦSðvÞ − ϕoff is
plotted for 95% circularly polarized light interacting with five
atomic spin states (colors) and a range of atomic velocities
spanning 80% to 120% of v0 ¼ 2000 m=s. Black curves show
spectra for velocity v0 for each spin state. Curves for each spin
state coalesce in caustics at a different λ where spin-dependent
ΦLðλ; vÞ compensates for dispersion in ΦSðvÞ. The ensemble
phase shift (green line) shows the root in ϕ at λzero;lab, which is
shifted by −120 pm from λzero. The phasor diagram (inset)
illustrates how ΦL compounds with ΦS to increase dispersion
for one spin state and decrease dispersion for another spin state.
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140404-2

Fig. 2, and data demonstrating þ203 to −213 pm shifts in
λzero;lab are shown in Fig. 3.
Higher irradiance on the left interferometer path when
looking from the source towards the detector would cause a
longer λzero;lab (if the grating tilt were zero so that Φa ¼ 0).
This is because attraction towards light on the left compen-
sates for ΦS in the northern hemisphere, and only spin states
with roots in αðωÞ at longer wavelengthsare attracted to light
at λzero. These states therefore contributewith moreweight to
ϕðλzeroÞ because of dispersion compensation. On the other
side, if the irradiance is stronger on the right-hand interfer-
ometer path, then repulsion from the light compensates for
ΦS, and spin states with roots in αðωÞ at shorter wavelengths
contribute more to ϕðλzeroÞ. Grating tilt θ and the gravity
phase Φa complicate this picture. In our experiment, the
dispersion dΦa=dv is opposite and slightly larger in mag-
nitude than the dispersion dΦS=dv, so higher irradiance on
the left path of the atom interferometer causes a shorter
λzero;lab. Figure 3 shows data verifying this prediction.
We predict that the wavelength difference Δλ ¼
λzero;lab − λzero will not change if the optical k vector
reverses direction, nor if the optical circular polarization
reverses handedness, nor if the magnetic field parallel to the
optical k vector reverses direction. None of these reversals
changes the fact that a potential gradient that is attractive
towards the left side (or repulsive from the right side) is
needed to compensate for the Sagnac phase dispersion in
the northern hemisphere. Therefore, the magnitude jΔλj
can increase if the laser is simply reflected over the atom
beam path. We tested this prediction by constructing an
optical cavity with plane mirrors to recycle light so that
the same interferometer path is exposed to upward and
downward propagating laser beams for several passes. This
increased the magnitude of ϕðλzeroÞ as predicted.
External magnetic fields also affect λzero;lab. A uniform
magnetic field parallel or antiparallel to the optical k
vector maximizes the sensitivity to optical polarization.
Alternatively, a magnetic field perpendicular to the optical
k vector reduces Δλ because the atomic spin states precess
about the field so the resulting spin-dependent differences
in light shift time-average to zero. Data in Fig. 4 show that
λzero;lab is closer to λzero when we apply a perpendicular
magnetic field. Residual differences between λzero;lab and
λzero are due to imperfect alignment of the magnetic field
perpendicular to the k vector and the limited (15 G) strength
of the magnetic field.
On the basis of the work presented thus far, deducing Ω
from measurements of Δλ is challenging because it requires
knowing the magnetic field, the laser power, laser polari-
zation, laser beam waist, and the atom beam velocity spread.
To solve this problem, we used a static electric field gradient
to induce additional phase shifts that mimic the effect of
auxiliary rotation on the atom interferometer (to first order
in v). A measurement of light-induced phase shift as a
function of electric-field-induced phase shift can serve to
calibrate the relationship between Δλ and Ω. Furthermore,
we can determine the absolute rotation rate of the laboratory
by measuring the additional phase shift needed to make
λzero;lab ¼ λzero. The phase due to a static electric field
gradient is
Φ∇E ¼
αð0Þ
2ℏv
Z
s
d
dx
E2
dz ð6Þ
where αð0Þ is the static electric dipole polarizability [5].
The observed phase shift for the ensemble of atoms due to
an electric field gradient ϕ∇E is calculated using Eq. (2)
with Φ∇E added to Φtotal (and ΦL ¼ 0). This phase shift can
compensate for the dispersion in theSagnacphaseuniformly
for all atomic spin states.
In Fig. 5, we show that ϕðλzeroÞ depends continuously on
ϕ∇E, just as Δλ would on Ω. Specifically, ϕðλzeroÞ is the
phase shift caused by light at λzero. The data in Fig. 5 are
FIG. 3 (color online). Measured light-induced phase spectra
ϕðλÞ using elliptically polarized light and a magnetic field parallel
to the optical k vector. The open square red data show λzero;lab ¼
768.758ð15Þ nm when the laser beam is on the right side of the
atom interferometer, and the solid circle blue data show λzero;lab ¼
769.174ð7Þ nm when the laser beam is on the left side of the atom
interferometer. Each data point is the average of 40 five-s files,
and the error bars show the standard error of the mean. Broad
band radiation from the tapered amplifier caused a systematic
shift of 15(5) mrad that we accounted for in the ϕ data shown.
The red and blue curves show the theory using Eqs. (1)–(5) with
an additional average over the width of the atom beam. For these
data, the grating tilt θ was −20ð5Þ mrad.
FIG. 4 (color online). Measured tune-out wavelengths for
different orientations of magnetic field and irradiance gradients.
Each data point comes from ϕðλÞ spectra such as those shown in
Fig. 3. For these data, the grating tilt θ was −20ð5Þ mrad.
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140404-3

obtained by alternately turning ∇E on and off, blocking and
unblocking the laser, and then repeating the process with a
new ∇E strength. Importantly, the root in phase ϕðλzeroÞ at
ϕroot
∇E occurs when the electric field gradient compensates
for dispersion in ΦS and Φa. We can interpret this condition
mathematically as
d
dv
ðΦS þ Φa þ Φ∇EÞ ¼ 0; ð7Þ
and then it becomes unnecessary to know the laser power or
to perform the integral over velocity shown in Eq. (2) for
reporting Ω. Using the approximation Φ∇E ¼ ϕroot
∇E ðv0=vÞ2
,
we find
Ω ¼ −
dgv0ϕroot
∇E
2πL2
−
g sinðθÞ
v0
: ð8Þ
Equation (8) does not include ΦL because when Eq. (7) is
satisfied there is no net dispersion to break the symmetry;
so including ΦLðλzeroÞ in Eq. (2) produces zero ensemble
phase shift ϕ. The fact that Eq. (8) does not include ΦL is
convenient because now we can use light at λzero to measure
Ω without precise knowledge of the laser spot size,
polarization or irradiance, or the resultant slope dϕ=dλ.
Those factors affect the precision with which we can find
the root (ϕroot
∇E ), but not the value of the root. We also
emphasize that an electric field gradient can be used to
increase the dynamic range of our gyroscope.
To report Ω, we measured ϕroot
∇E ¼ 1.2ð3Þ rad with data in
Fig. 5, we measured v0 ¼ 1585ð10Þ m=s using phase
choppers [43], and we measured θ ¼ −10ð2Þ mrad by
comparing the nanograting bars to a plumb line. We find
Ω ¼ 0.4ð2ÞΩe, which can be compared to the expected
value 0.5 Ωe (the vertical projection of Ωe at our latitude of
32° N). In Fig. 5, we also show how Coff depends on ϕ∇E.
The phase ϕmax C
∇E that maximizes contrast is another way to
find the static electric field gradient that compensates
for dispersion in the Sagnac phase and acceleration phase.
The value of ϕmax C
∇E ¼ 0.6ð2Þ rad leads to Ω ¼ 0.6ð2ÞΩe.
The dominant source of error in our experiment was the
measurement of the nanograting tilt. Discrepancy between
ϕmax C
∇E and ϕroot
∇E indicates a systematic error, possibly caused
by de Broglie wave phase front curvature induced by the
laser beam [44], optical pumping, magnetic field gradients,
or the broad band component of our laser spectrum.
The shot noise limited sensitivity of our atom interfer-
ometer gyroscope can be estimated from the fact that
ϕðλzeroÞ changes by 0.22 rad due to 0.53Ωe, and the
statistical phase noise is δϕ ¼ ð2=NÞ1=2
½ð1=CoffÞ2
þ
ð1=ConÞ2
Š1=2
which is 0.06 rad=
ffiffiffiffiffiffi
Hz
p
for Coff ¼ 0.2,
Con ¼ 0.08, and N ¼ ð100 000 counts= secÞ × t. This indi-
cates a sensitivity of 0.2Ωe=
ffiffiffiffiffiffi
Hz
p
for measurements of
rotation with respect to an inertial reference frame, which is
competitive with methods presented in Refs. [7–9,34,35].
To make a more sensitive gyroscope, the scale factor
ϕðλzeroÞ=Ω can be somewhat increased by using more laser
power and a broader velocity distribution. However, a limit
to the sensitivity arises from balancing the benefit of an
increased scale factor against the detriment of increased
statistical phase noise. This compromise occurs because
maximizing the scale factor ϕðλzeroÞ=Ω requires significant
contrast loss from the two mechanisms described by
Eq. (2): first, averaging over the spread in ΦS (which is
affected by σv) and second, averaging over the distribution
in ΦL (which is affected by the laser power and polariza-
tion). Optimizing σv and laser power can increase the
sensitivity (for the same flux and contrast) to 0.05Ωe=
ffiffiffiffiffiffi
Hz
p
for Ω measurements.
This work also indicates how to make measurements of
λzero more independent of Ω. Experiments are less sensitive
to Ω if they use linearly polarized light, a narrow velocity
distribution, a perpendicular magnetic field, and an addi-
tional dispersive phase such as Φ∇E to compensate for ΦS.
For example, the λzero measurements in Ref. [29] were not
significantly affected by Ω because there was minimal
contrast loss at λzero. Specifically, the sharp velocity
distribution ðv0=σv ¼ 18Þ caused dispersion in ΦS þ Φa
that reduced Coff by less than 1% of C0, and ΦL reduced
Con by 4% of C0; so shifts in λzero;lab were less than 1pm in
Ref. [29]. To increase sensitivity to Ω for measurements
reported here, in Figs. 3–5 we used a broad velocity
distribution (v0=σv ¼ 7) so ΦS þ Φa reduced Coff by 8%
of C0, and we also used a large irradiance gradient with
circular polarization that reduced Con by 40% of C0.
In summary, an atom beam interferometer with multiple
atomic spin states enabled us to demonstrate systematic
shifts in tune-out wavelength measurements (λzero;lab) that
FIG. 5 (color online). (top) Contrast data as a function of phase
shift induced by an electric field gradient ϕ∇E. A Gaussian fit
(dashed black line) to the red data points shows that a maximum
in contrast occurs at ϕ∇E ¼ 0.6ð2Þ rad due to dispersion
compensation. The solid red curve shows the theory using
Eqs. (1)–(6) with Ω ¼ 0.6Ωe. (bottom) Light-induced phase shift
ϕ as a function of ϕ∇E, using light at λzero ¼ 768.971 nm. An
error function fit (dashed black line) to the blue data points shows
the root ϕroot
∇E ¼ 1.2ð3Þ rad. The solid blue curve shows the theory
using Eqs. (1)–(6) with Ω ¼ 0.4Ωe. For these data, the grating tilt
θ was −10ð2Þ mrad. The solid green curves show contrast and
phase theory for Ω ¼ 0, but the same θ ¼ −10 mrad.
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140404-4

are larger than 200 pm due to rotation and acceleration.
Then, we used the phase induced by light at a theoretical
tune-out wavelength ϕðλzeroÞ as a function of an additional
dispersive phase ϕ∇E applied to report the rotation rate of
the laboratory with an uncertainty of 0.2Ωe. This work is a
new application for tune-out wavelengths, paves the way
for improving precision measurements of tune-out wave-
lengths, and demonstrates a new technique for atom
interferometer gyroscopes. The spin-multiplexing tech-
niques demonstrated here may find uses in other atom
[12,13] and neutron [45,46] interferometry experiments,
NMR gyroscopes, and NMR spectroscopy.
This work is supported by NSF Grant No. 1306308 and a
NIST PMG. Authors R. T. and M. D. G. also thank NSF
GRFP Grant No. DGE-1143953 for support. We thank
Professor Brian P. Anderson for helpful discussions.
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