# Matter-wave interferometers using TAAP rings

P. Navez1, S. Pandey2,3, H. Mas2,4, K. Poulios2, T. Fernholz5, and W. von
Klitzing2 1 Crete Center for Quantum Complexity and Nanotechnology, Depart. of
Physics, University of Crete, Heraklion 70113, Greece
2 Institute of Electronic Structure and Laser, Foundation for Research and
Technology - Hellas, Heraklion 70013, Greece
3 Depart. of Materials, Science and Technology, University of Crete, Heraklion
70113, Greece
4 Depart. of Physics, University of Crete, Heraklion 70113, Greece
5 School of Physics and Astronomy, University of Nottingham, Nottingham NG7
2RD, United Kingdom<EMAIL_ADDRESS>
###### Abstract

We present two novel matter-wave Sagnac interferometers based on ring-shaped
time-averaged adiabatic potentials (TAAP). For both the atoms are put into a
superposition of two different spin states and manipulated independently using
elliptically polarized rf-fields. In the first interferometer the atoms are
accelerated by spin-state-dependent forces and then travel around the ring in
a matter-wave guide. In the second one the atoms are fully trapped during the
entire interferometric sequence and are moved around the ring in two spin-
state-dependent ‘buckets’. Corrections to the ideal Sagnac phase are
investigated for both cases. We experimentally demonstrate the key atom-
optical elements of the interferometer such as the independent manipulation of
two different spin states in the ring-shaped potentials under identical
experimental conditions.

###### pacs:

07.60.Ly, 03.75.-b, 03.75.Dg, 32.80.Pj, 03.75.Lm, 42.50.Gy

##  1 Introduction

###  1.1 Interferometry with Atomic clocks

Since its first demonstration about 20 years ago [1, 2, 3, 4] atom
interferometry is one of the most sensitive and accurate forms of inertial
sensing. It carries the promise of greatly enhanced sensitivity and precision,
which could be exploited for fundamental physics [5], geo-sensing [6] and
inertial navigation [7]. Most atom interferometers use free-space atomic beams
[8, 4] with Raman or Bragg beam splitters. The sensitivity of the Sagnac
interferometer scales with the enclosed area, which is limited by the momentum
difference between the two arms and the available time-of-flight of the free-
falling atoms. Guiding the atoms through an atomtronic circuit can provide
very large areas and thus pave the way towards compact, ultra-sensitive
matter-wave interferometers. A number of such circuits have been proposed and
some demonstrated, e.g. using dipole potentials [9, 10, 11], magnetic traps on
atom-chips [12, 13, 14], and using time-averaged adiabatic potentials (TAAPs)
[15, 16, 17, 18]. Michelson type interferometers have also been demonstrated
over distances of tens of micrometers [13, 19, 20, 21]. However, an
experimental demonstration of coherent atom guiding over macroscopic distances
with interferometric stability has proven to be an elusive goal. The main
reason has been the corrugation of the guiding potentials [22, 23, 24].

Another challenge for atomtronic circuits is the coherent splitting of the
atomic cloud inside the waveguide. In the case of the standard Bragg beam
splitter, the reference with respect to which the interferometer measures the
rotation, is set by the phase difference of the optical beams. Unfortunately,
a drift of the atomic waveguide relative to the Bragg beams can result in a
phase shift at the readout of the interferometer, which would be interpreted
as an acceleration. Coherent splitting of Bose-Einstein condensates has
previously been demonstrated on atomchips by transforming a magnetic trap into
a double-well in the direction of tight confinement [13]. Since the splitting
process has to be very slow compared to the trapping frequency in the
direction of the splitting, this method would be prohibitively slow in the
longitudinal direction. Similarly, the state-dependent manipulation of atoms
can be achieved using direct microwave dressing on magnetic atom-chips [25,
26]. Recently, a clock-type interferometer has been proposed, based on state-
dependent manipulation of atoms on an atom-chip [27, 28]. A superposition of
two atomic hyperfine (clock) states is created using an rf/microwave pulse.
The hyperfine components are then moved separately in opposite directions
around an enclosed area using rf fields. They are finally recombined using a
second rf/microwave pulse, with the interferometric signal being the
population difference between these hyperfine states. In order to realize a
full loop Sagnac interferometer in this way [27] a ring-shaped structure on an
atomic micro-chip can be used, which is dressed using strong rf-fields with a
doughnut shaped polarisation structure [28].

In the present paper, we study a guided clock-type interferometer based on
ultra-cold thermal atoms or Bose-Einstein Condensates (BECs) in time-averaged
adiabatic potentials (TAAPs) [15]. By clock-type interferometer we mean an
interferometer, which utilizes two distinct states having different
eigenenergies, These potentials have the advantage of being extremely smooth,
controllable and flexible. We will examine two types of interferometers, one
where thermal or Bose-condensed atoms propagate freely along a ring-shaped
TAAP waveguide and a second where the atoms are fully confined and moved along
a circular path.

The paper is structured in the following way: Section 2 reviews time averaged
potentials and extends them to arbitrary polarization states for
interferometry. In section 3 the Sagnac phase is briefly introduced and the
different interferometric schemes are presented. TAAP interferometers based on
Bragg beams splitters are discussed in section 3.3 while section 3.4 deals
with TAAP interferometers based on rf/microwave beam splitters with subsequent
state-dependent acceleration (section 3.4.1) or use of moving atom buckets
(section 3.4.2). The paper concludes in section 4.

Figure 1: Sketch of the Sagnac interferometer sequence. The $\pi$ and $\pi/2$
refers to the two photon pulses.

##  2 The trapping and guiding potentials

###  2.1 Time averaged adiabatic potentials (TAAPs)

TAAPs can produce extremely controllable and smooth potentials, making them
ideal candidates for matter-wave interferometry. They have been proposed in
[15] and realised in [11, 16, 17]. In the TAAPs used here, a strong rf-field
$(B_{\text{rf}})$ dresses atoms in a static magnetic quadrupole field
$(B_{\text{Q}})$, leading to adiabatic potentials [29]. An oscillating
homogeneous magnetic field $(B_{\text{m}})$ is then added, which effectively
leads to a displacement of the quadrupole. If the oscillation frequency is low
compared to the Larmor frequency but fast compared to the center of mass
motion of the atoms, then the external potential confining the atoms is the
time-average over the modulated adiabatic potentials. The trapping and guiding
structures of TAAPs are generated using large coils located far way from the
waveguide itself, thus virtually eliminating any corrugation in the potential.
This makes TAAP rings extremely smooth 111Random corrugations in a potential
decay exponentially over the length scale of their characteristic size., and
therefore ideal candidates for the coherent matter-wave guiding over
macroscopic distances and hence for matter-wave interferometry.

Let us now consider the Hamiltonian

$\hat{H}_{\pm}=\pm g\mu_{\text{B}}{\hat{\bf F}}.({\bf B}({\bf r},t)+{\bf
B}_{\text{rf}}(t))\,,$ (1)

where $\hat{\bf F}$ is the total spin of the atoms, $\mu_{\text{B}}$ the Bohr
magneton, and $g$ the Landé factor. We focus on the
$\left|F,m_{\text{F}}\right>=\left|1,1\right>$ and $\left|2,1\right>$ states
of Rubidium 87 and assume that the fields are small enough to neglect the
quadratic Zeeman effect. Note that even for larger fields the differential
energy shift due to quadratic Zeeman effect can be cancelled [30]. We combine
a static quadrupole field ${\bf B}_{\text{Q}}({\bf
r})=\big{(}B_{\text{x}},B_{\text{y}},B_{\text{z}}\big{)}=\alpha\big{(}x,y,-2z\big{)}$
with a homogeneous modulation field which can be tilted by an angle $\delta$
from the vertical (z) axis towards the x axis: ${\bf
B}_{\text{m}}(t)=B_{\text{m}}\sin(\omega_{\text{m}}t)\big{(}\sin(\delta),0,\cos(\delta)\big{)}$.
The total field is then ${\bf B}({\bf r},t)={\bf B}_{\text{Q}}({\bf r})+{\bf
B}_{\text{m}}(t)$. Finally, we add an elliptically polarized rf field ${\bf
B}_{\text{rf}}(t)=\big{(}(B_{+}-B_{-})\cos(\omega_{\text{rf}}t),0,(B_{+}+B_{-})\sin(\omega_{\text{rf}}t)\big{)}$.
Note that $\omega_{\text{rf}}$ itself can be time dependent, i.e.
$\omega_{\text{rf}}(t)$, with its frequency
$\omega_{\text{rf}}\gg\omega_{\text{m}}$ typically being in the MHz range. We
define the ellipticity parameter $s=(B_{+}-B_{-})/(B_{+}+B_{-})$, where $s=0$
denotes a linear rf in the $z$-axis, and $s=1$ means $B_{-}=0$ and thus a pure
circular rf in the $x-z$ plane. In this paper, the polarization will be almost
purely linear with a small admixture of circular rf $(s\,\langle\langle\,1)$.
The rf field can be understood as the sum of a clockwise $(B_{+})$ and an
anticlockwise $(B_{-})$ rotating field. We can calculate the coupling rates
for both the $F=2$ and the $F=1$ hyperfine levels by decomposing the rf field
into a basis, which is co-moving with the modulation field. The resulting
dressed states will henceforth be referred to as $\left|2,1\right>$ and
$\left|1,1\right>$ states respectively. In the rotating wave approximation
(RWA), we neglect the component of the rf parallel to the B-field and
decompose the orthogonal components into two counter-rotating fields, of which
we only use the component that adheres to the conservation of angular momentum
222This can be calculated conveniently using the Jones formalism developed for
optical polarizations.. The resulting Rabi equation is then:

$\displaystyle V_{\pm}({\bf r},t)$ $\displaystyle=$
$\displaystyle\hbar\sqrt{\left[\Omega_{\text{L}}({\bf
r},t)-\omega_{\text{rf}}(t)\right]^{2}+\Omega_{\pm}^{2}({\bf r},t)}$ (2)
$\displaystyle\simeq$ $\displaystyle\hbar\Omega_{\pm}({\bf
r},t)+\hbar\frac{\left[\Omega_{\text{L}}({\bf
r},t)-\omega_{\text{rf}}(t)\right]^{2}}{2\Omega_{\pm}({\bf r},t)}\,\,,$

where $\Omega_{\text{L}}({\bf r},t)=g_{\text{F}}\mu_{\text{B}}|{\bf B}|/\hbar$
is the Larmor frequency. The Rabi coupling (see Appendix 6) is:

$\hbar\Omega_{\pm}({\bf r},t)=\hbar\Omega_{0}\sqrt{\left(1\pm
s\frac{B_{\text{y}}}{|\bf{B}|}\right)^{2}-(1-s^{2})\frac{B_{\text{z}}^{2}}{|\bf{B}|^{2}}}\,\,,$
(3)

where $\hbar\Omega_{0}=\frac{1}{2}g_{\text{F}}\mu_{\text{B}}(B_{+}+B_{-})$,
the ‘$+$’ refers to the dressed $\left|2,1\right>$ state and the ‘$-$’ to the
$\left|1,1\right>$ state. $\bf B$, $B_{\text{y}}$ and $B_{\text{z}}$ depend
both on $\bf r$ and $t$.

###  2.2 Experimental Setup

The potentials for the ring-TAAPs are formed from a quadrupole field,
homogeneous time averaging fields, and rf-dressing fields. The experiments are
performed within a glass vacuum chamber with an estimated pressure in the low
$10^{-11}$ Torr range with a vacuum limited life time of more than two
minutes. The quadrupole field of up to 400 G/cm is generated using a pair of
water cooled coils in anti-Helmholtz configuration. The modulation field
oscillates at 5 kHz and can have up to 48 G in amplitude. It is generated by
two pairs of rectangular Helmholtz coils in the horizontal and a pair of round
coils in the vertical direction. The coils are driven from an eight channel 24
bit sound card, which is amplified using a standard two kilo-watt audio
amplifier. In order to control the impedance seen by the amplifier we place
capacitors and water cooled $2\,\Omega$ resistors in series with the coils.
The three interlocking rf-coils each consist of two single loops in Helmholtz
configuration. Impedance matching is achieved by placing the pair of coils and
the capacitors in parallel. The coils are driven by a 25 W amplifier
(Amplifier Research 25A250A), which is controlled by two arbitrary waveform
generators (Tektronix AFG3022). Both the rf-coils and modulation coils can
produce arbitrarily polarized fields in three dimensions. However, for reasons
of experimental simplicity, we keep the frequency of the rf-fields constant.
The modulation fields can be changed in amplitude and polarization on a time-
scale of 3 ms.

The experimental sequence starts by loading a 3D-MOT with $2\times 10^{9}$
atoms $\left({}^{87}Rb\right)$ from a 2D-MOT. We transfer the atoms to a
matched quadrupole time-orbiting potential (TOP). We then compress the trap by
first increasing the quadrupole gradient and then lowering the amplitude of
the modulation field. At the same time we cool the atoms using radio-frequency
(rf) evaporation to a temperature of about $3T_{c}$ (with $T_{c}$ being the
critical temperature). When the modulation field has reached 5 G, we switch on
the rf-dressing fields at a frequency of 2.62 MHz and load the atoms into the
dressed potential, which we then deform into the ring-shaped trap. Finally, we
take resonant absorption images of the atomic clouds using a $4f$ custom
objective and a CCD camera after a short time of flight.

###  2.3 The ring waveguide ($s=0$, $\delta=0$)

Figure 2: Experimental realisation of a ring-shaped TAAP waveguide
$(s=0,\delta=0)$ with Rubidium atoms in the $\left|2,2\right>$ state. The
quadrupole gradient is $\alpha=$50 G/cm with $\omega_{\text{rf}}/2\pi=2.62$
MHz. The measured Rabi frequency is $\Omega_{\text{rf}}/2\pi=215$ kHz. The
radius of the ring is $R=570$ µm.

In order to trap the atoms in the vertical direction as well, we add the
modulation field ${\bf B}_{\text{m}}(t)$. If the frequency of the modulation
field $(\omega_{\text{m}})$ is fast compared to the trapping frequency but
slow compared to the Larmor frequency, then we can time-average the dressed,
modulated quadrupole potential:

$V_{\pm}^{\text{eff}}({\bf
r})=\frac{\omega_{\text{m}}}{2\pi}\int_{0}^{2\pi/\omega_{\text{m}}}\\!\\!\\!dt\,V_{\pm}({\bf
r},t)$ (4)

For a purely linearly polarized rf-field ($s=0$) and a purely vertical
modulation field ($\delta=0$) the field is cylindrically symmetric with
respect to the z-axis and identical for the $\left|2,1\right>$ and
$\left|1,1\right>$ states. The potential then forms in the x-y plane a ring-
shaped trap of radius $R=\hbar\omega_{\text{rf0}}/g\mu_{B}\alpha$. For
mathematical simplicity we modulate the frequency $(\omega_{\text{rf}})$ and
amplitude $(B_{+}(t)=B_{-}(t))$ of the rf-field such that at ${\bf r}=(R,0,0)$
it stays resonant $(\Omega_{\text{L}}({\bf r},t)-\omega_{\text{rf}}(t)=0)$ at
a constant Rabi-frequency $(\Omega_{\text{rf}}({\bf
r},t)=\Omega_{0c}={\text{c}onst.})$. This can be achieved using the following
time dependencies
$\omega_{\text{rf}}(t)=\omega_{\text{rf0}}\sqrt{1+\beta^{2}\sin^{2}(\omega_{m}t)}$
and $\Omega_{0}=\Omega_{0c}\sqrt{1+\beta^{2}\sin^{2}(\omega_{\text{m}}t)}$. In
the current experiments constant rf-frequency and amplitude are used. The ring
can then be described close to the trap bottom as [15]:

$V_{\text{ring}}^{\text{eff}}(r,z)=\hbar\Omega_{0c}+\frac{1}{2}m\omega_{\text{r}}^{2}(r-R)^{2}+\frac{1}{2}m\omega_{\text{z}}^{2}z^{2}$
(5)

with trapping frequencies in the radial and the z axis directions:

$\displaystyle\omega_{\text{r}}$
$\displaystyle=\omega_{0}(1+\beta^{2})^{-1/4}$ (6)
$\displaystyle\omega_{\text{z}}$
$\displaystyle=2\omega_{0}\left[1-(1+\beta^{2})^{-1/2}\right]^{1/2},$

where $\beta=g_{\text{F}}\mu_{\text{B}}B_{\text{m}}/\hbar\omega_{\text{rf0}}$
is the normalized modulation index and
$\omega_{0}=m_{\text{F}}g_{\text{F}}\mu_{\text{B}}\,\alpha\,\left(m\,\hbar\Omega_{\text{0c}}\right)^{-1/2}$.

Figure 2 shows an absorption image of $4.8\times 10^{5}$ atoms in a TAAP ring
of 1.15 mm diameter. Note that the noise in the image is purely due to the
imaging system.

One of the most useful features of the TAAP rings — next to their exceptional
smoothness — is the ability to tune the sensitivity of the interferometer
simply by changing either the dressing frequency or the gradient of the static
quadrupole field $(\alpha)$, which determine the ring’s radius and therefore
the enclosed area. We have demonstrated rings from 400 µm to 2.6 mm in
diameter filled with up to $6\times 10^{5}$ atoms at temperatures down to 2
µK. The life time of atoms in such a trap is of the order of ten seconds.

Figure 3: Radius $R$ of the TAAP rings as a function of the inverse magnetic
gradient of the quadrupole $(1/\alpha)$ for a constant rf-frequency of
$\omega_{\text{rf}}/2\pi=2.5$ MHz. The polarization of the dressing RF is
circular in the x-y plane, the vertical modulation magnitude is 1.9 G and the
Rabi frequency $\Omega_{\text{rf}}/2\pi=300$ kHz. Note that we do not modulate
the frequency of the rf, and that therefore the radius of the ring is smaller
than one might expect $(R<\hbar\omega_{\text{rf0}}/g\mu_{B}\alpha)$. The solid
red line serves as a guide to the eye. The error bars correspond to the
standard deviation of repeated experiments and is typically $\sim 0.3\%$ of
the radius.

###  2.4 The tilted ring trap $(s=0$, $\delta\neq 0)$

Figure 4: _A tilted ring trap:_ Experimental realisation of gravity tilted
TAAP buckets in $\left|1,1\right>$ state ($s=0$) with $\alpha=55$ G/cm,
$\omega_{\text{rf}}/2\pi=2.62$ MHz. The radius is $R=500$ µm and the atom
number is about $3\times 10^{5}$ atoms. In a) $\delta=0.28$ rad $=16^{\circ}$.
In b,c) we tilt the ring in the y—z plane, in opposite directions, by the same
amount of $16^{\circ}$.

The plane of the ring is defined by the direction that the time-averaging
field $(B_{\text{m}})$ is oscillating in. For $\delta=0$ the modulation field
is vertical and the ring horizontal. For $\delta\neq 0$ the field tilts
towards the x-axis and gravity creates a half-moon shaped trap in the
direction of the tilt (see Figure 4). For $\delta\langle\langle 1$, the
combination with the gravity field shifts the center of the z-confinement
according to $z\rightarrow
z^{\prime}=z-R\delta\cos\phi/2-g/\omega_{\text{z}}^{2}$, where $\phi$ is the
azimuthal angle. By slightly tilting the modulation field ${\bf B}_{\text{m}}$
from the purely vertical ($0<\delta\langle\langle 1$) we can tilt the ring
thus creating a crescent shaped trap with a potential of

$V_{\text{g}}^{\text{eff}}({\bf
r})=V_{\text{ring}}^{\text{eff}}(r,z^{\prime})-\frac{mg^{2}}{{2}\omega_{\text{z}}^{2}}-\delta\frac{1}{2}mgR\cos(\phi).$
(7)

Figure 4(a) shows an experimental realisation of a tilted trap of 1 mm
diameter with $3\times 10^{5}$ Rubidium atoms in the $\left|1,1\right>$ state
with $\delta=0.28$ rad and Figs. 4(b),(c) are ring traps with the same number
of atoms and trap conditions, tilted in two opposite directions along the y—z
plane.

###  2.5 The state dependent trap $(s\neq 0$, $\delta=0)$

Figure 5: Experimental realisation of a state dependent TAAP bucket $(s\neq
0$, $\delta=0)$ for the $\left|2,2\right>$ state at a quadrupole gradient of
$\alpha=55$ G/cm and $\omega_{\text{rf}}/2\pi=2.62$ MHz. $3\times 10^{5}$
atoms were trapped in the bucket and the radius is $R=490$ µm.

The Rabi coupling for circular rf polarisation depends on the selection rules
for the transition and thus on the state of the atoms. The pure ring trap of
Section 2.3 has a strong vertically polarized rf field. Adding to it a small,
circularly polarized rf will increase or decrease the coupling strength
$\Omega_{\text{rf}}$ of Eq. 3 depending on which state the atoms are in (e.g.
see Fig. 5 for the particular demonstration of a state dependent TAAP bucket
in the $\left|2,2\right>$ state).

For a small positive $s$, the circular rf component lies in the x-z plane and
increases the Rabi frequency of the $\left|2,1\right>$ state in the positive
y-direction while at the same time decreases it for the $\left|1,1\right>$
state. This creates a trap located at ${\bf r}=(-R,0,0)$ for the
$\left|2,1\right>$ state and a trap located at ${\bf r}=(+R,0,0)$ for the
$\left|1,1\right>$ state. For a slightly elliptic rf $(0<s\langle\langle 1)$
we obtain to first order:

$V_{\pm}^{\text{eff}}({\bf
r})=V_{\text{ring}}^{\text{eff}}(r,z)\pm\hbar\Omega_{0c}\frac{2s}{\pi}E(i\beta)\sin\phi\,,$
(8)

where $E(x)$ is the complete elliptic function.

###  2.6 Arbitrary traps $(s\neq 0$, $\delta\neq 0)$

Figure 6: Experimental realisation of arbitrary traps with $(s\neq 0$,
$\delta\neq 0)$ for the two states $\left|2,2\right>$ ($5\times 10^{5}$ atoms)
and $\left|1,1\right>$ ($3\times 10^{5}$ atoms) at
$\omega_{\text{rf}}/2\pi=2.62$ MHz. The fitted radius is $440$ µm and $450$ µm
respectively. The quadrupole gradient is $\alpha=55$ G/cm. Note that a) and b)
are taken with identical experimental conditions and differ only in the state
of the atoms. The axis of the circular rf component and the one of the tilted
modulation are not orthogonal.

Combining the state dependent potentials $(s\neq 0)$ with a tilt of the
modulation field $(\delta\neq 0)$, we create two independent traps each at an
arbitrary position along the ring (e.g. see Figure 6 for a demonstration with
the $\left|1,1\right>$ and $\left|2,2\right>$ states)

The resulting potential is then:

$\displaystyle V^{\text{eff}}_{\pm}({\bf r})$
$\displaystyle=V_{\text{ring}}^{\text{eff}}(r,z^{\prime})-\frac{mg^{2}}{{2}\omega_{\text{z}}^{2}}-V_{0}\cos(\phi\mp\phi_{0})\,\,,$
(9) $\displaystyle V_{0}$
$\displaystyle=\sqrt{\left(\frac{2}{\pi}E(i\beta)\hbar\Omega_{0c}s\right)^{2}+\left({\frac{1}{2}}mgR\delta\right)^{2}}\,\,,$
$\displaystyle\tan(\phi_{0})$
$\displaystyle=\frac{4s\hbar\Omega_{0c}E(i\beta)}{\pi mgR\delta}.$

By adjusting $s$ and $\delta$, we can adjust the trap position $\phi_{0}$ to
take any value between $0$ and $2\pi$.

The two traps are anti-symmetric with respect to the x-z plane. Note also that
in principle the tilt of the modulation and the weak circular rf component can
point in any direction.

The corresponding azimuthal trap frequency

$\omega_{\phi}=\frac{1}{R}\sqrt{\frac{V_{0}}{m}}\,.$ (10)

governs the speed at which one can move the two traps around the ring without
exciting the trapped atomic cloud.

Figure 6 shows a TAAP with two traps in arbitrary positions. Both images are
taken at identical experimental conditions, but with different states: in
figure 6a the atoms are in the $\left|2,2\right>$ state and in figure 6b in
the $\left|1,1\right>$ state. The axis of the circular rf component and of the
tilt are shifted by $\pi/4$.

###  2.7 Landau Zener losses

From equation (6) it is clear that one has an interest in reducing
$\Omega_{0}$ and thus $B_{\text{rf}}$ in order to achieve a maximal radial
confinement. This however is limited by Majorana spin flips, which occur when
the rf-coupling between the energy levels is too weak for the atoms to follow
the adiabatic states. According to the Landau-Zener criterion the probability
of such transition is given by $P=\exp(-2\pi\Gamma_{\pm})$. Since the time
variation of the potential experienced by the atoms depends mainly on the
modulation, rather than the atom’s motion, we obtain according to [15]:

$\Gamma_{\pm}({\bf r},t)=\hbar\frac{\Omega_{\pm}^{2}({\bf
r},t)}{\partial_{\text{t}}\left(\Omega_{\text{L}}({\bf
r},t)-\omega_{\text{rf}}(t)\right)}\gg 1$ (11)

Assuming a small tilt $\delta\langle\langle 1$ and a low amplitude circular rf
$s\leq 1/\sqrt{1+\beta^{2}}$ then spin flip losses are suppressed when:

$\omega_{\text{m}}~{}\langle\langle~{}\frac{\Omega_{0c}^{2}R}{\,\omega\,a_{\text{ho}}}\,\,,$
(12)

where $a_{\text{ho}}=(\hbar/m\omega_{\phi})^{1/2}$ is the azimuthal harmonic
oscillator length.

##  3 TAAP Interferometers

Here we discuss the different types of matter-wave interferometers made
possible by the TAAP potentials. We start with interferometers using the
matter-wave guides of section 2.3 and introduce two different types of beam
splitters: one based on Bragg beams (section 3.3) and one based on
rf/microwave two photon transitions and state selective manipulation of the
atoms (section 3.4). We then show that the latter scheme can also be used in
the fully trapped case (section 3.4.2). In order to avoid any contribution
other than the Sagnac phase $(\phi^{0}_{\text{S}})$, we ensure that the state-
dependent trajectories of the atoms are perfectly symmetric for $\Omega=0$. We
achieve this by constructing the interferometric sequence such that every
point in the trajectory is traversed by the two states in exactly the same
internal state. The only difference between the trajectories of the two clouds
is then the direction in which the atoms travel.

###  3.1 The Sagnac phase

We will shortly review the Sagnac phase as it applies to our guided
interferometers. The Sagnac phase occurs in atom trajectories that enclose an
area. It is often assumed that the two wave packets overlap perfectly at the
second beam-splitter. Insufficient overlap is usually taken into account only
as a reduction in contrast. For coherent matter-waves such as those
originating from a condensate one can achieve a large contrast even when the
overlap is limited [31, 32]. We will show, that in this case there is
nevertheless a correction to the Sagnac phases.

Figure 7: Representation of the area covered by the atoms together with the
correction to non adiabaticity.

This can be formulated under the following rather general theorem: We assume
non interacting atoms evolving from point $A$ to $B$ with phase accumulation
given by $\Phi_{\pm}(t)$.

The starting and end points of the classical trajectory in the absence of
rotation are $\bf r_{A}$ and $\bf r_{B}^{(0)}$. The presence of the rotation
changes the classical Lagrangian with a linear correction term $\delta L(t)$
that induces the change in the classical action governing the phase
accumulation. If $\delta\bf r_{B}$ is the linear correction to the external
rotation ${\bf\Omega}$ (see Fig. 7), then the linear contribution due to this
external rotation is given by:

$\displaystyle\Phi_{BA}$ $\displaystyle=\int_{t_{A}}^{t_{B}}\\!dt\,\delta
L(t)/\hbar$ (13)
$\displaystyle=\frac{m}{\hbar}\left[2{\bf\Omega}.\int_{A}^{B}{\bf r}\times
d{\bf r}+{\bf\dot{r}}_{B}^{(0)}.\delta{\bf r}_{B}\right]\,.$

A proof of this relying only on the classical dynamics of the atom motion can
be found in Appendix 7. The first term of equation (13) gives the area
enclosed by the atoms while the second gives the correction due to non-
adiabaticity. Note that the second term can always be made to vanish by an
appropriate choice of origin of the reference frame such that the external
rotation motion cancels the final velocity (by adding a term
${\bf\Omega}\times{\bf a}$, where ${\bf a}$ is the vector of the new origin).
The two clouds accumulate two different corrective phases
$\delta\Phi_{B^{\pm}A}$. If we assume that without the external rotation they
end up in the same position, the net result is (see also Fig. 8):

$\displaystyle\delta\Phi$ $\displaystyle=\Phi_{B^{+}A}-\Phi_{B^{-}A}$ (14)
$\displaystyle=\frac{m}{\hbar}\left(2{\bf\Omega}.\int_{\cal A}{\bf
dS}+{\bf\dot{r}}_{B^{+}}^{(0)}.\delta{\bf
r}_{B^{+}}-{\bf\dot{r}}_{B^{-}}^{(0)}.\delta{\bf r}_{B^{-}}\right).$

where $\cal A$ is the surface enclosed by the path $B^{-}AB^{+}OB^{-}$ and
${\bf dS}$ is the differential surface vector. Please note that the choice of
origin $O$ does not change the value of the integral, since the change in area
is exactly compensated by the change in ${\bf\dot{r}}_{B^{+}}^{(0)}.\delta{\bf
r}_{B^{+}}-{\bf\dot{r}}_{B^{-}}^{(0)}.\delta{\bf r}_{B^{-}}$. We retrieve the
standard Sagnac formula $\Phi_{\text{S}}^{0}=\frac{4\pi\,\Omega{\cal
A}}{h/m}$, if at the end of the interferometer sequence the atoms are at rest
and the area is closed, i.e. the atomic clouds overlap fully.

Figure 8: The ‘area‘ encircled by the atoms of the two states if the atom
trajectories $\bf B^{-}AB^{+}$ are not entirely closed. $O$ represents the
origin of the coordinate system and is entirely arbitrary.

In most Sagnac interferometers the atoms travel along the identical path in
opposite directions. In this case the area of the interferometer is the sum of
the areas surrounded by each arm and thus twice the area of the single path.
Taking the particular case of a ring-shaped interferometer of radius $R$ and
radial trapping frequency $\omega_{\rho}$ and assuming that the atoms are
strongly confined to the ring $(\omega_{\rho}\gg\Omega)$ the area is ${\cal
A}=2\pi R^{2}$. Since ${\bf r}_{B}^{2}=R^{2}$ we find ${\bf
r}_{B\pm}^{(0)}.\delta{\bf r}_{B\pm}=0$ and due to the tight confinement
$\delta{\bf r}_{B\pm}=0$.

Therefore, we obtain a much simpler expression

$\displaystyle\Phi_{\text{S}}(T)=\frac{\delta\phi(T)}{h/m}\Omega{\cal
A}=\frac{\delta\phi(T)}{4\pi}\,\Phi_{\text{S}}^{0}\,\,,$ (15)

where $\delta\phi(T)=\phi_{+}(T)-\phi_{-}(T)$ is the difference in angle
between the cloud centers accumulated during their trajectory along the ring
interferometer, with $\delta\phi(T)=4\pi$ signifying therefore the closed
area. The Sagnac phase is therefore only affected by the final position of the
mass center of the atoms.

For a finite radial confinement we have to take into account that the atoms
will experience a centrifugal force during their trajectory, which will be
compensated by the radial trapping potential. The area of the interferometer
(and thus its sensitivity) will therefore increase slightly. An upper bound
for this is

$\frac{\delta
A}{A}\leq\text{Max}\left[2\left(\frac{\dot{\phi_{0}}}{\omega_{\rho}}\right)^{2}\right].$
(16)

Taking a transit time of 1 s and a radial trapping frequency of 400 Hz, we
find a negligible correction of $10^{-5}$. Note that this correction affects
only the sensitivity and not the zero-fringe.

Finally, care needs to be taken to minimise the spread of the wavepackets
during their trajectory as well as the influence of the interatomic
interactions. For a thermal cloud this can always be done simply by reducing
the number of atoms. For a Bose condensed cloud, the phase diffusion time
$T_{\phi}$ can be estimated assuming a Poissonian atom number fluctuations and
the chemical potential $\mu$ taken for a Thomas-Fermi profile:

${T_{\phi}}=\frac{1}{\hbar}\sqrt{N}{\left.{\frac{{\partial\mu}}{{\partial
N}}}\right|_{N/2}}={{\rm{N}}^{-10}}{\left({\frac{{72a^{2}\omega_{\phi}\omega_{\rho}^{2}}}{{125a_{{\rm{Ho}}}^{2}}}}\right)^{1/5}},$
(17)

where $a_{\text{Ho}}$ is the harmonic oscillator length, $a$ the s-wave
scattering length. Surprisingly enough this implies that—as oopposed to the
thermal case—the phase coherence time of a BEC interferometer increases for
increasing atom number even in the case of absence of squeezing, much in
contrast to the thermal case.

###  3.2 Guided Sagnac Interferometry

The basic idea of a Sagnac interferometer is to let atoms travel in opposite
directions around an enclosed area, completing a full loop. Due to the
resulting common path, the interferometer becomes independent of static
acceleration (gravity) but will be sensitive to rotation.

The main challenge for guided Sagnac interferometers is the smoothness of the
waveguides. In the case of magnetic guiding potentials the shape of the
waveguide is defined by the shape of the field generating wires. Therefore,
imperfections in the wires translate directly into imperfections of the
waveguides. This is the main reason why interferometry in waveguides has only
been demonstrated over microscopic distances so far [19, 13, 33]. The TAAPs of
this paper are generated from coils that are large and far away. The resulting
waveguides are therefore inherently smooth (see section 2.1 and [15]).

A second challenge is the beam splitter. As will be discussed in section 3.3,
the standard beam splitters of atom interferometry (Raman or Bragg), can also
be used in TAAP interferometers. A completely different beam splitter will be
discussed in section 3.4, where we propose to use state-dependent manipulation
of a superposition of atoms in the rf-dressed
$\left|F,m_{\text{F}}\right>=\left|1,1\right>$ and $\left|2,1\right>$ states.

With an appropriate choice of parameters the linear Zeeman energy can be
nearly identical for both states 333By carefully choosing the trapping
parameters, the quadratic field dependence of the energy difference between
the two states can also be eliminated. This is the case for a static magnetic
field at 3.23 Gauss [34]. Similar results have been predicted for rf-dressed
potentials [30]. The ideal dressing and modulation fields of the TAAPs will be
investigated separately.. However, they interact differently with different
polarisations of the rf dressing fields. A $\sigma_{+}$ polarized rf field
will couple the $\left|1,-1\right>$ to the $\left|1,+1\right>$ state, but not
the $\left|2,+1\right>$ to the $\left|2,-1\right>$ state. The opposite is true
for a $\sigma_{-}$ polarized rf. We can therefore use the _degree of circular
polarisation_ of the rf dressing field in order to manipulate the atoms in a
state-dependent fashion.

Combining an elliptically polarized rf-field with a tilt of the ring-shaped
waveguide, we can move the two states independently to anywhere on the ring
(e.g. Fig. 6) including moving the two clouds around the ring in opposite
directions. For such a case the standard Sagnac phase is defined as

$\Phi_{\text{S}}^{0}=\frac{4\pi\,\Omega{\cal A}}{h/m},$ (18)

where $\Omega$ is the angular velocity of the rotation to be determined, $m$
the mass of the atom and $\cal A$ is the area enclosed by the arms of the
interferometer [4, 35]. For the case of the ring-shaped wave guide discussed
in this paper, where the two wavepackets travel once around the ring in
opposite directions, the area is $A=2\pi\rho$, where $\rho$ is the radius of
the ring.

The Sagnac formula still holds both for wave-guided interferometers and
3D-trapped atomic clouds. We extend the concept of the area to cases where the
paths are not fully closed and consider corrections to the area taking into
account the centrifugal forces. As shown previously, the phase accumulation
corresponds to the classical action (in units of $\hbar$) taken over the
trajectory [35].

###  3.3 A waveguided interferometer with a Bragg beam splitter

Section 2.3 demonstrates the ring-shaped TAAP trapping potentials $(s=0$,
$\delta=0)$, that will serve as waveguides for the interferometer. In order to
place an atom cloud into this waveguide we load a cold thermal cloud or a BEC
into the harmonic trap created by tilting the modulation fields $(s=0$,
$\delta\neq 0)$ as discussed in section 2.4. We then switch suddenly to the
flat ring $(s=0$, $\delta=0)$ and apply a near resonant Bragg pulse, which
splits the atomic cloud into two momentum states $\pm 2\hbar k$ in the
laboratory frame, with $k=2\pi/\lambda$ being the wave vector of the light
[36]. The atoms then move along the ring with a constant (angular) velocity
${\dot{\phi}}_{0\pm}(t)=\pm 2\pi/T$, where $T=2\pi R/(2v_{\text{rec}})$ is the
half round trip time and $v_{\text{rec}}$ is the recoil velocity of the Bragg
beams. After a full trip around the ring we apply a second Bragg pulse, which
then produces three atomic clouds: one at rest $p=0$ and two with a momentum
of $p=\pm 2\hbar k$. The interferometric signal is then calculated by
subtracting the sum of the populations of the $p=\pm 2\hbar k$ wave packets
from the population of the $p=0$ wave packet. We assume tight radial and axial
confinement in the ring, so that the atoms move according to ${\bf
r}_{0\pm}(t)=\big{(}R\cos[(\pm 2\pi/T+\Omega)t],R\sin[(\pm
2\pi/T+\Omega)t],0\big{)}$. Since the atoms remain at the trap minimum only
the kinetic energy contributes to the phase difference. We obtain a phase
shift for a round trip of the two clouds:

$\Phi_{\pm}(T)=\frac{mR^{2}}{2}\int_{0}^{T}\frac{dt}{\hbar}\left(\Omega\pm\frac{2\pi}{T}\right)^{2}.$
(19)

By calculating the phase difference between the two trajectories we recover
the standard Sagnac phase of equation (18):
$\Phi_{\text{S}}(T)=\Phi_{+}(T)-\Phi_{-}(T)=\Phi_{\text{S}}^{0}$. In the more
general case, however, the components have different angular velocities
$\phi_{\pm}(t)={\dot{\phi}}_{\pm}t$, and thus ${\bf
r}_{0\pm}(t)=\big{(}R\cos[({\dot{\phi}}_{\pm}+\Omega)t],R\sin[({\dot{\phi}}_{\pm}+\Omega)t],0\big{)}$.
An example is the use of Raman beamsplitters, where the two beams may have
different frequencies. This results in the modified Sagnac signal:

$\Phi_{\text{S}}=\frac{2\pi
R^{2}T}{h/m}\left[({\dot{\phi}}_{+}-{\dot{\phi}}_{-})\Omega+({\dot{\phi}}_{+}^{2}-{\dot{\phi}}_{-}^{2})/2\right]$
(20)

Raman pulses in free space offer very precise control over the difference in
the velocity of the split atomic clouds. For the case of a TAAP waveguide,
however, the alignment of the potential with respect to the optical Raman
beams has to be taken into account. In practice this means that the square
terms in the Sagnac equation (20) severely limit the precision in the
determination of $\Omega$. It is therefore desirable to choose the Bragg
scheme, where the two velocities are identical and the non-linear terms
vanishes.

###  3.4 Interferometry based on state dependent manipulation of atoms in a
ring

Even though the simplicity of the Bragg scheme is conceptually very appealing,
it has the major drawback that the reference for the Sagnac phase is the
standing wave of the Bragg beams and thus the retro-reflecting mirror which
generates it. Whereas in free-space interferometers this is a very suitable
reference, in wave-guided interferometers this poses the problem that this can
cause a spurious phase shift when the position of the wave guide moves
relative to the one of the retro-reflecting mirror. We therefore have to look
for interferometers, where the phase reference is defined by the trapping
fields themselves. This is the case for clock type interferometers [27] using
as a beam splitter a two photon rf/microwave pulse. The position of the beam
splitter is then simply the position of the atoms in the trap.

We start with atoms in the $\left|2,1\right>$ state in the tilted ring
described in section 2.4 at the initial angular position
$\phi_{\pm}=\phi_{0}(t=0)=0$. The atoms are at rest in the rotating frame,
therefore the initial velocity in the inertial frame is $\Omega R$. We then
apply a two-photon $\pi/2$ pulse, which puts the atoms into an equal
superposition of the two dressed hyperfine states $\left|2,1\right>$ and
$\left|1,1\right>$. We then move the atoms half way around the ring 444Note
that the experimental conditions are such that these two components experience
the same trapping potential.. After half of a round trip we apply a $\pi$
pulse, which turns the $\left|2,1\right>$ state into the $\left|1,1\right>$
state. We then complete the round trip for each of the two states and apply a
second $\pi/2$ pulse. Due to the $\pi$ pulse the experimental sequence of the
second half can be exactly the time reversal of the first half, thus ensuring
perfect symmetry of the whole interferometric sequence. There are two
principle options to move the atoms around the ring interferometer: a) the
atoms are subjected to an initial state-dependent acceleration and then
propagate freely in a TAAP matter-wave guide the static or dynamic
_accelerator rings_ described below or b) the atoms are fully confined in
three dimensions using the _moving buckets_ of section 3.4.2.

####  3.4.1 The Accelerator Ring:

After preparing the atoms in a superposition of $\left|2,1\right>$ and
$\left|1,1\right>$, we switch to a state-dependent trap $(s\neq 0$ and
$\delta=0$, as discussed in section 2.5) with the two trap centers at
$\phi_{0\pm}=\pm\pi/2$. The $\left|2,1\right>$ and $\left|1,1\right>$
components of the atom clouds then start to accelerate symmetrically in
opposite direction until they reach $\phi_{0\pm}=\pm\pi/2$, at which point
they start to slow down. Due to the symmetry of the potential the clouds come
to a standstill after they have concluded a half turn $(\phi_{0\pm}=\pi)$. We
then apply a $\pi$-pulse, which turns $\left|2,1\right>$ into
$\left|1,1\right>$ and vice versa. The atom clouds, now with their identity
exchanged, accelerate again and continue to travel in the same direction as
before, until they reach $\phi_{0\pm}=\mp\pi/2$ and finally come to a
standstill at $\phi_{0\pm}=0$. A further $\pi/2$ pulse then closes the
interferometer, with the readout being the difference in populations of the
$\left|2,1\right>$ and $\left|1,1\right>$ states. Conveniently, these two
components separate again and are easily imaged when they reach their maximum
separation at $\phi_{0\pm}=\pm\pi/2$. The angle $\phi_{\pm}(t)=\pm\phi(t)$
follows the nonlinear classical equation (see Appendix B):

$mR^{2}{\ddot{\phi}}(t)=V_{0}(t)\sin(\phi_{0}(t)-\phi(t))$ (21)

and formula (15) still holds.

Since the TAAP waveguides are extremely smooth [15], no transverse modes are
excited during the transit and equation (21) yields

$T=2\sqrt{\frac{mR^{2}}{2V_{0}}}\int_{0}^{\pi}\frac{d\phi}{\sqrt{\sin\phi}}\simeq
7.4/\omega_{\phi}.$ (22)

With the atoms overlapping and being at rest during the second $\pi/2$ pulse,
the interferometer is well described by the Sagnac phase $\Phi_{\text{S}}$.

The fact that the atoms come to a standstill at $\phi_{0\pm}=\pi$ causes an
unnecessarily slow transfer of the atoms. The transit time can be reduced by
dynamically adjusting the arbitrary ring potentials of section 2.6 such that
the acceleration in the first half of the cycle and the deceleration in the
second half both remain maximum. Care should be taken for the relative
velocity of the two clouds to remain small or densities sufficiently low so as
to avoid s-wave scattering [37].

####  3.4.2 The moving buckets:

In most inertial atom-interferometric measurements the atoms are in free fall
for most of the interferometer sequence. Even in the case of the waveguided
interferometer described above the atoms are free to travel around the ring.
However, as is evident e.g. from equation (14) and described in [27, 28], this
is not a strict requirement: atoms in a matter-wave guide or even in a 3-D
trap still reproduce the Sagnac phase. Even macroscopic atom clocks flown once
around the globe display the Sagnac phase [38]. In the following sections, we
describe Sagnac interferometers where the atoms are for most of their
trajectories fully 3D-trapped. One important advantage of this is that this
prevents the atomic cloud from expanding excessively during its trajectory.

Figure 9: A sketch of the isopotential surfaces for the ‘bucket’ traps showing
the different stages of transport during the interferometer sequence. $\bf
s^{\prime}$ and $\bf\delta^{\prime}$ stand for the degree of tilt and circular
polarisation respectively.

We start with atoms in the $\left|2,1\right>$ state at the initial angular
position $\phi_{\pm}=\phi_{0}(t=0)=0$ and initial radius $R$. The atoms are at
rest in the rotating frame, therefore the initial velocity in the inertial
frame is $\Omega R$. We apply a two-photon $\pi/2$ pulse, which puts the atoms
into an equal superposition of the two dressed hyperfine states
$\left|2,1\right>$ and $\left|1,1\right>$. The two states are then moved in
opposite directions for a half of a turn around the ring
($\phi_{\pm}(t=T)=\pm\pi$). When the atoms are exactly at the opposite side of
the ring, i.e. after a transit time $t=T/2$ we apply a $\pi$-pulse, which
swaps the two hyperfine states. When the full turn is complete ($t=T$) the
atomic clouds are recombined by a second $\pi/2$ pulse at $\phi_{\pm}(t=T)=\pm
2\pi=0$. Note that each of the two wave packets travel through the exactly
same regions of space in the same hyperfine state. In a static inertial frame,
the trajectory of the atoms is therefore symmetric with respect to time
reversal and with respect to the plane spanned by $\phi=0$ and the z-axis. The
interferometer is therefore not sensitive to any static (inhomogeneous)
external field or acceleration (gravity). We realize three dimensional
trapping using a combination of the state-dependent and gravitational
potentials as described in equation (9). The position of the trap minimum
angle $\phi_{0}(t)$ expressed in equation (9) is tuned slowly by a variation
of the ellipticity ($s$ parameter) and the rotation ($\delta$ parameter) of
the modulation field so that the condensates follow the trap center.

Let us first consider an adiabatic transfer of the atoms around the ring. In
order for the atoms to remain at the ground state of the trap, the trap
frequencies have to be much higher than the typical time variation associated
with $\phi_{0}(t)$, i.e. $\ddot{\phi_{0}}(t)<<\omega_{\phi}^{2}$. The position
of the atoms is then ${\bf r}_{\pm}(t)={\bf r}_{0\pm}(t)$ and the equations
(50-52) yield ${\bf\dot{r}}_{\pm}(t)={\bf\dot{r}}_{0\pm}(t)=\hbar{\bf
k}_{\pm}(t)/m$. Assuming a slow rotation $\Omega$, tight radial confinement,
and that the final velocity of the two atom clouds is zero
$({\dot{\phi}_{0}}(T)=0)$ we get the usual Sagnac phase shift
$\Phi_{\text{S}}(T)=\Phi_{\text{S}}$.

In a real experiment the adiabatic transfer would require an extremely slow
acceleration of the trap ($R\ddot{\phi_{0}}(t)<<\omega_{\phi}^{2}$) and a very
long transit time $(T\langle\langle 2\pi/\omega_{\phi})$. A faster motion
would cause the center of mass of the atoms to oscillate with respect to the
center of the trap. However, for a harmonic trap this oscillation can be
optimally controlled [39]

Let us now look at the case when the trap moves too fast for the adibaticity
to be preserved, but slow enough for the atoms to remain within the harmonic
limit of the trap. For a trap moving at a constant speed $\phi_{0\pm}(t)=\pm
2\pi\,t/T$ and assuming the atoms start in the laboratory frame at position
$\phi(0)=0$ and are initially at rest ($\dot{\phi}(0)=0$) we can write down
the classical motion of the atoms as

$\displaystyle\phi_{\pm}(t)$ $\displaystyle=\pm
2\pi\left[\frac{t}{T}-\frac{\sin(\omega_{\phi}t)}{\omega_{\phi}T}\right]$ (23)
$\displaystyle\dot{\phi}_{\pm}(t)$
$\displaystyle=\pm\frac{2\pi}{T}\left[1-\cos(\omega_{\phi}t)\right]$

If we assume tight radial confinement then equation (15) yields the Sagnac
phase

$\Phi_{\text{S}}(T)=\left[1-\frac{\sin(\omega_{\phi}T)}{\omega_{\phi}T}\right]\Phi^{0}_{\text{S}}$
(24)

Therefore, we recover the standard Sagnac phase $(\Phi^{0}_{\text{S}})$ if the
transit time $T$ is a multiple of the trap oscillation time
$(2\pi/\omega_{\phi})$.

We can also avoid the oscillation by starting with static traps at the
position $\phi_{0\pm}(0)=\pm\phi_{\text{a}}$ rather than at $\phi_{0}(0)=0$. A
quarter of oscillation period later $t=\pi/2\omega_{\phi}$ the atoms will then
be exactly at the trap bottom and move at an angular velocity
$\dot{\phi}_{\pm}=\pm\omega_{\phi}\phi_{\text{a}}$. If at that moment we start
to move the trap at the same angular velocity, then the atoms are transported
in the ground state of the trap. Stopping the trap at
$\phi_{0\pm}=\mp\phi_{\text{a}}$ and waiting a further quarter of an
oscillation period perfectly overlaps the two atom clouds at zero relative
velocity. We therefore measure the standard Sagnac phase. The transit time for
this trajectory is then
$T=\frac{2\pi}{\omega_{\phi}}(1/2-1/\pi+1/\phi_{\text{a}})$. This trajectory
guarantees that for most of the transit the atoms are at rest in the moving
frame and is much faster than the adiabatic requirement $(T\langle\langle
2\pi/\omega_{\phi})$.

###  3.5 Sensitivity

We have demonstrated rings with radius’ ranging from $R=200$ µm to 1.5 mm.
Using equation (18) with ${\cal A}=2\pi R^{2}$ we calculate the transfer
function or scale factor of a ring-interferometer with a radius $R=1.5$ mm to
be $\Phi_{\text{S}}/\Omega=4\times 10^{4}$ rad/(rad/s). The earth’s rotation
of $\sim 72$ µrad/s therefore results in a phase shift of
$\Phi_{\text{S}}\simeq 3$ rad.

Numerical simulations for a condensate in our experimental conditions indicate
that the Sagnac phase can be preserved for atom numbers as large as a few
thousand [40], resulting in a shot noise limited phase resolution of tens of
milli radians. Given our repetition rate of a few tens of seconds, we expect a
sensitivity of $10^{-5}$ rad/s/Hz1/2. According to [27] also thermal atoms can
be used. In that case, using $10^{6}$ thermal atoms at a repetition rate of 1
Hz and pushing the ring diameter to one centimeter a sensitivity of $10^{-9}$
rad/s/Hz1/2 will be reached.

##  4 Conclusions

We demonstrated and analysed ring-shaped potentials for ultra-cold atoms and
propose different schemes for its use in Sagnac interferometry. We
demonstrated that using an elliptical polarization of the rf-field in
combination with gravity we can arbitrarily and dynamically manipulate two
different spin states at the same time.

We proposed a number of different Sagnac interference experiments based both
on matter-wave guiding and fully trapped manipulation of the atoms. We
analysed the contributions to the Sagnac phase due to dynamic effects, and
found that the standard formula also holds for non adiabatic situations
provided the atomic clouds fully overlap at the final beam splitter. An
extension of the standard Sagnac phase has been formulated and generalized for
atoms evolving in an extended class of external potentials. This can be used
as a tool for designing improved potential configurations for Sagnac
interferometry measurements.

Acknowledgments: PN gratefully acknowledges that this work was supported by
the European Union’s Seventh Framework Programme (FP7-REGPOT-2012-2013-1)
under grant agreement number 316165. This research has also been supported by
the EU-FET Grant FP7-ICT-601180 (MatterWave) and by FP7-PEOPLE-2012-ITN-317485
(QTea). We gratefully acknowledge useful discussions with the members of the
MatterWave consortium.

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##  6 Appendix A: Determination of the potential

###  6.1 Coupling field calculation

To determine the effective potential felt by the atoms, we use the rotating
wave approximation (RWA) i.e. we assume that only the polarized component of
this field rotating perpendicularly to the quadrupole field effectively
interacts with the atoms. This perpendicular component is given by:

${\bf B}_{\text{rf}}^{\perp}({\bf r},t)={\bf B}_{\text{rf}}(t)-\frac{{\bf
B}_{\text{rf}}(t).{\bf B}({\bf r},t)}{|{\bf B}({\bf r},t)|^{2}}{\bf B}({\bf
r},t)~{}.$ (25)

This component rotates at the frequency $\omega_{\text{rf}}$ with an
elliptical polarization:

$\displaystyle{\bf B}_{\text{rf}}^{\perp}({\bf r},t)$ $\displaystyle={\bf
B}_{1}\cos(\omega_{\text{rf}}t)+{\bf B}_{2}\sin(\omega_{\text{rf}}t)$ (26)

where

$\displaystyle{\bf
B}_{1}=(B_{+}+B_{-})\left(-\frac{B_{\text{z}}B_{\text{x}}}{|{\bf
B}|^{2}},-\frac{B_{\text{z}}B_{\text{y}}}{|{\bf
B}|^{2}},1-\frac{B_{\text{z}}^{2}}{|{\bf B}|^{2}}\right)~{},$ (27)
$\displaystyle{\bf B}_{2}=(B_{+}-B_{-})\left(1-\frac{B_{\text{x}}^{2}}{|{\bf
B}|^{2}},-\frac{B_{\text{y}}B_{\text{x}}}{|{\bf
B}|^{2}},-\frac{B_{\text{z}}B_{\text{x}}}{|{\bf B}|^{2}}\right)~{}.$ (28)

It can be rewritten in the equivalent form:

$\displaystyle{\bf B}_{\text{rf}}^{\perp}({\bf r},t)$ $\displaystyle={\bf
B^{\prime}}_{1}\cos(\omega_{\text{rf}}t+\chi)+{\bf
B^{\prime}}_{2}\sin(\omega_{\text{rf}}t+\chi)$ (29)

with a phase shift $\chi$ chosen such as to fulfill the orthogonality of the
amplitudes ${\bf B^{\prime}}_{1}.{\bf B^{\prime}}_{2}=0$. This last condition
together with the comparison between the two last expressions Eq.(26) and
Eq.(29) allow to find the following identities:

$\displaystyle\left(\begin{array}[]{c}{\bf B^{\prime}}_{1}\\\ {\bf
B^{\prime}}_{2}\end{array}\right)=\left(\begin{array}[]{cc}\cos\chi&-\sin\chi\\\
\sin\chi&\cos\chi\end{array}\right).\left(\begin{array}[]{c}{\bf B}_{1}\\\
{\bf B}_{2}\end{array}\right)$ (36)

and

$\displaystyle\tan(2\chi)=\frac{2{\bf B}_{1}.{\bf B}_{2}}{{\bf B}_{2}^{2}-{\bf
B}_{1}^{2}}~{}.$ (37)

Rewriting the amplitude of these field as $|{\bf
B^{\prime}}_{1}|=B^{\prime}_{+}+B^{\prime}_{-}$ and $|{\bf
B^{\prime}_{2}|}=B^{\prime}_{+}-B^{\prime}_{-}$, we can identify the new
parameters as the effective Rabi coupling $\hbar\Omega_{\pm}({\bf
r},t)=g\mu_{\text{B}}{B^{\prime}_{\pm}}$ and recover the equation (3).

A small distance expansion of the resulting effective potential Eq.(4)
quadratic around the ring center ($r\sim R$ and $z\sim 0$) allows to identify
the trap frequencies (6) along the radial and z-direction respectively and the
hyperfine dependent azimuthal components in Eq.(8).

##  7 Appendix B: Non adiabatic interferometry

In this appendix we discuss how to calculate the phase evolution of the atoms
during their trajectory inside the ring. We show that, if the atoms are at
rest in the beginning and end of the interferometric sequence, then the
resulting phase depends only on the area enclosed and is essentially
independent of the dynamic evolution of the atoms within the interferometer.
We generalize the Sagnac formula (13) for the case that the trajectories do
not fully enclose an area.

###  7.1 The Sagnac phase in a travelling harmonic trap

We assume that the interactions between the atoms are negligible so that the
atomic clouds in the two states $\left|2,1\right>$ and $\left|1,1\right>$
evolve according to the Schrödinger equation for the atoms in the inertial
frame:

$\displaystyle i\hbar\partial_{\text{t}}\psi_{\pm}({\bf
r},t)=-\frac{\hbar^{2}}{2m}\nabla_{\bf r}^{2}\psi_{\pm}({\bf
r},t)+V_{\pm}\left({\bf r}-{\bf r}_{0\pm}(t),t\right)\psi_{\pm}({\bf r},t)\,,$
(38)

where the atoms are trapped in the moving harmonic potential $V_{\pm}({\bf
r}-{\bf r}_{0\pm}(t),t)$ centered at the position ${\bf r}_{0\pm}(t)$ whose
dynamics can be chosen arbitrarily. For the particular ring trap
configuration, we use the dynamics ${\bf
r}_{0\pm}(t)=\left\\{R\cos\left[\pm\Omega t+\phi_{0}(t)\right],\pm
R\sin\left[\pm\Omega t+\phi_{0}(t)\right],0\right\\}$ so that the potential
equation (9) is now rewritten in the inertial frame by taking into account of
the external rotation angle $\Omega t$ that is added or subtracted to the
angle of the trap minimum $\phi_{0}(t)$ in the laboratory frame. In the trap
center frame, the harmonic potential has the local form:

$V_{\pm}({\bf r},t)=\frac{m}{2}{\bf
r}.\underline{\underline{{\mathbf{\omega}}}}^{2}_{\pm}(t).{\bf r}^{T}$ (39)

and depends on the trap position $\phi_{0}(t)$ through the tensor of the trap
frequencies:

$\displaystyle\underline{\underline{{\mathbf{\omega}}}}^{2}_{\pm}(t)=\underline{\underline{{\mathbf{O}}}}(\Omega
t\pm\phi_{0}(t)).\left(\begin{array}[]{ccc}\omega^{2}_{r}&0&0\\\
0&\omega^{2}_{\phi}&0\\\
0&0&\omega_{\text{z}}^{2}\end{array}\right).\underline{\underline{{\mathbf{O}}}}^{T}(\Omega
t\pm\phi_{0}(t))$ (43)

where we used the rotation matrix:

$\displaystyle\underline{\underline{{\mathbf{O}}}}(\phi)=\left(\begin{array}[]{ccc}\cos(\phi)&-\sin(\phi)&0\\\
\sin(\phi)&\cos(\phi)&0\\\ 0&0&1\end{array}\right)$ (47)

This description can also be used for a ring shaped waveguide, with
$\omega_{\phi}=0$ and the trajectory of the trap center $\phi_{0}(t)$ being
chosen such that it follows the free evolution of atom cloud. We use the
following transformations:

$\displaystyle\psi_{\pm}({\bf r},t)=e^{i[{\bf k}_{\pm}(t).({\bf r}-{\bf
r}_{\pm}(t))+\Phi_{\pm}(t)]}\psi_{0\pm}({\bf r}-{\bf r}_{\pm}(t),t)$ (48)

to rewrite the Schrödinger equation in the moving frame of the position of the
atom wave packet center ${\bf r}_{\pm}(t)$ as

$\displaystyle i\hbar\partial_{\text{t}}\psi_{0\pm}({\bf
r},t)=[-(\hbar^{2}/2m)\nabla_{\bf r}^{2}+V_{\pm}({\bf r},t))]\psi_{0\pm}({\bf
r},t)$ (49)

This is achieved if this position ${\bf r}_{\pm}(t)$ obeys the dynamic
equations:

$\displaystyle{\bf\dot{r}}_{\pm}(t)$ $\displaystyle=$ $\displaystyle\hbar{\bf
k}_{\pm}(t)/m$ (50) $\displaystyle\hbar{\bf\dot{\bf k}}_{\pm}(t)$
$\displaystyle=$ $\displaystyle-\frac{\partial V({\bf r}_{\pm}(t)-{\bf
r}_{0\pm}(t),t)}{\partial{\bf r}_{\pm}(t)}$ (51) $\displaystyle\Phi_{\pm}(t)$
$\displaystyle=$ $\displaystyle\int_{0}^{t}\frac{dt^{\prime}}{\hbar}L({\bf
r}_{\pm}(t^{\prime}),{\bf\dot{r}}_{\pm}(t^{\prime}),t)$ $\displaystyle
L_{\pm}({\bf r}_{\pm}(t),{\bf\dot{r}}_{\pm}(t),t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}m{\bf\dot{r}}^{2}_{\pm}(t)-V({\bf r}_{\pm}(t)-{\bf
r}_{0\pm}(t),t).$ (52)

This result is strictly only valid for a harmonic potential and is thus a good
approximation for atoms located close to the trap center $|{\bf
r}_{\pm}(t)-{\bf r}_{0\pm}(t)|\langle\langle R$ (see Appendix 7.3). As a
consequence of the Kohn transformation that extends the translational
invariance transformation concept to the case of the harmonic trap [41], the
center of mass position of the condensate center evolves according to the
classical Lagrangian $L_{\pm}({\bf r}_{\pm}(t),{\bf\dot{r}}_{\pm}(t),t)$ [35].
The transformation applied here to cartesian coordinates can be generalized to
any time-dependent harmonic potential for any coordinates choice like the
cylindrical coordinates developed in Appendix 7.3 provided the resulting
potential can be assumed quadratic in these coordinates.

###  7.2 Proof of the non adiabatic area theorem

Let us suppose the rotation of angular velocity ${\bf\Omega}$ of the
laboratory frame. Then the laboratory coordinates ${\bf r}_{L}(t)$ are related
to the inertial coordinates through the relation ${\bf
r}^{T}(t)=\underline{\underline{{\mathbf{O}}}}({\bf\Omega}t){\bf
r}^{T}_{L}(t)$ where we define the rotation matrix
$\underline{\underline{{\mathbf{O}}}}({\bf\Omega}t)$ about the angular
velocity at angle $|{\bf\Omega}t|$. In this frame, the classical motion
equation of an atom under an external potential $V({\bf r}(t),t)$ becomes:

$\displaystyle m({\bf\ddot{\bf r}}_{L}(t)+2{\bf\Omega}\times{\dot{\bf
r}}_{L}(t)+{\bf\Omega}\times({\bf\Omega}\times{\bf r}_{L}(t))$
$\displaystyle=$ $\displaystyle-\frac{\partial V({\bf
r}_{L}(t),t)}{\partial{\bf r}_{L}(t)}$ (53)

Making the perturbative development ${\bf r}_{L}(t)={\bf r}^{(0)}(t)+{\bf
r}^{(1)}(t)+\dots$ up to the first order in the angular velocity and applying
it to Eq.(53), we determine the correction to the action perturbatively.
Suppose a classical motion between time $t_{A}$ and $t_{B}$, we calculate
successively:

$\displaystyle\int_{t_{A}}^{t_{B}}dt$ $\displaystyle S({\bf r}_{L}(t),\dot{\bf
r}_{L}(t),t)$ (54)
$\displaystyle=\int_{t_{A}}^{t_{B}}dt\left[\frac{m}{2}({\dot{\bf
r}}_{L}(t)+{\bf\Omega}\times{\bf r}_{L}(t))^{2}-V({\bf r}_{L}(t),t)\right]$
$\displaystyle=\int_{t_{A}}^{t_{B}}dtS^{(0)}({\bf r}^{(0)}(t),\dot{\bf
r}^{(0)}(t),t)+2m{\bf\Omega}.\int_{{\bf r}^{(0)}(t_{A})}^{{\bf
r}^{(0)}(t_{B})}{\bf r}^{(0)}\times d{\bf r}^{(0)}+m{\bf\dot{r}}^{(0)}(t).{\bf
r}^{(1)}(t)|_{t_{A}}^{t_{B}}$

Setting ${\bf r}^{(1)}(t_{A})=0$, we find the result (13).

###  7.3 Anharmonic treatment

We generalize the results of section 7.1 to the case of an anharmonic
equation. In the case of strong radial confinement, the Schroedinger equation
in the inertial frame is simplified into:

$\displaystyle
i\hbar\partial_{\text{t}}\psi_{\pm}(\phi,t)=[-\frac{\hbar^{2}\partial_{\phi}^{2}}{2mR^{2}}-V_{0}(t)\cos(\phi\mp\phi_{0}(t)-\Omega
t)]\psi_{\pm}(\phi,t)~{}.$

Similarly to the harmonic treatment, the azimuthal part of the trap potential
(9) written in the inertial frame is centered about $\pm\phi_{0}(t)+\Omega t$.
Thus according to the hyperfine components, it accounts for the external
rotation angle $\Omega t$ that is added or substracted to the trap minimum
angle $\phi_{0}(t)$. For the symmetric case where the atom components evolve
in opposite directions, the following transformation

$\displaystyle\psi_{\pm}(\phi,t)=e^{i[l_{\pm}(t).(\phi-\phi_{\pm}(t)-\Omega
t)+\Phi_{\pm}(t)]}\psi_{0\pm}(\phi-\phi_{\pm}(t)-\Omega t)$

rewrites the Schroedinger equation in the moving frame of the atoms as:

$\displaystyle i\hbar\partial_{\text{t}}\psi_{0\pm}(\phi,t)$ $\displaystyle=$
$\displaystyle\bigg{\\{}-\frac{\hbar^{2}\partial_{\phi}^{2}}{2mR^{2}}+V_{0}(t)[(1-\cos(\phi))\cos(\phi_{0}(t)\mp\phi_{\pm}(t))$
(57)
$\displaystyle-(\sin\phi-\phi)\sin(\phi_{0}(t)\mp\phi_{\pm}(t))]\bigg{\\}}\psi_{0\pm}(\phi,t)$

provided the following conditions are fulfilled:

$\displaystyle{\dot{\phi}}_{\pm}(t)$ $\displaystyle=$
$\displaystyle\hbar{l}_{\pm}(t)/mR^{2}-\Omega~{},$ (58) $\displaystyle
mR^{2}{\ddot{\phi}}_{\pm}(t)$ $\displaystyle=$ $\displaystyle-
V_{0}(t)\sin(\phi_{\pm}(t)\mp\phi_{0}(t))~{},$ (59)
$\displaystyle\Phi_{\pm}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\frac{dt^{\prime}}{\hbar}L_{\pm}({\phi}_{\pm}(t),{\dot{\phi}}_{\pm}(t),t)~{},$
$\displaystyle L_{\pm}({\phi}_{\pm}(t),{\dot{\phi}}_{\pm}(t),t)$
$\displaystyle=$
$\displaystyle\frac{\hbar^{2}({\dot{\phi}}_{\pm}(t)+\Omega)^{2}}{2mR^{2}}+V_{0}(t)\cos(\phi_{0}(t)\mp\phi_{\pm}(t))~{}.$

With the boundary conditions ${\phi}_{\pm}(0)={\dot{\phi}}_{\pm}(0)=0$, we
recover the formulae (15) whatever the dynamics involved. Let us note that
$l_{\pm}(t)$ should strictly speaking be an integer since the wave function
should be periodic in $\phi$. However as long as the radius is much larger
than the spatial extension of the wave function ($R\gg a_{\text{ho}}$), it can
be extended to the real value domain. Under the condition of a localized wave
function, the equation (57) is simplified into an harmonic oscillator:

$\displaystyle
i\hbar\partial_{\text{t}}\psi_{0\pm}(\phi,t)=\left\\{-\frac{\hbar^{2}\partial_{\phi}^{2}}{2mR^{2}}+V_{0}(t)\cos[\phi_{0}(t)\mp\phi_{\pm}(t)]\frac{\phi^{2}}{2}\right\\}\psi_{0\pm}(\phi,t)\,$
(61)

where the effective trap frequency is time dependent:

$\omega^{\text{eff}}_{\phi\pm}(t)=\sqrt{V_{0}(t)\cos(\phi_{0}(t)\mp\phi_{\pm}(t))/mR^{2}}\,.$

In order to avoid further phase accumulation, strictly speaking, one has to
ensure the adiabaticity of the evolution of the wave function. The main
limitation is the transition to the excited state in the trap which is
negligible when the transit time $T$ is much longer than the time variation of
the trap frequency i.e.
$T{\dot{\omega}^{\text{eff}}_{\phi\pm}(t)}/\omega^{\text{eff}}_{\phi\pm}(t)\langle\langle
1$. All these conditions fulfilled, we recover using (7.3) the formula (15)
whatever the dynamics involved. In practice, these conditions are only
fulfilled in the bucket case where

using Eq.(23) we find $\omega_{\phi}T\gg 1$. Nevertheless, this additional
phase accumulation is not important as it cancels out in the phase difference
if we restrict the calculation in the linear contribution in $\Omega$.

##  8 Appendix C: Contrast

The atom profile after recombination at the second $\pi/2$ pulse at time $t$
is altered when the wave packets coming from the two arms are not
synchronized. The contrast resulting from these imperfections can be
determined quantitatively from the atom density obtained for the two states
after interferometry:

$\displaystyle P_{\pm}({\bf r},T)$ $\displaystyle=$
$\displaystyle\frac{1}{4}|\psi_{+}({\bf r},t)\pm\psi_{-}({\bf r},t)|^{2}~{}.$
(62)

Because of the high confinement in along the radial direction, we neglect the
effect of the internal rotation inside the trap during the interferometry
process. Using Eq.(48) together with the ground state wave function:

$\displaystyle\psi_{0\pm}({\bf
r})=\left(\frac{m\omega_{ho}}{\pi\hbar}\right)^{3/4}\exp[-\frac{m}{2\hbar}{\bf
r}.\underline{\underline{{\mathbf{\omega}}}}_{\pm}(t).{\bf r}^{T}]$ (63)

and $\omega_{ho}=(\omega_{\text{r}}\omega_{\text{z}}\omega_{\phi})^{1/3}$, we
obtain after integration the probability of the two hyperfine states after
interferometry:

$\displaystyle P_{\pm}(T)$ $\displaystyle=$ $\displaystyle\int d^{3}{\bf
r}P_{\pm}({\bf r},T)$ (64) $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{(}1\pm{\cal
V}(T)\cos\left[\Phi_{\text{S}}(T)-\frac{{\bf k}_{+}(T)+{\bf
k}_{-}(T)}{2}.\delta{\bf r}(T)\right]\bigg{)}$

where we define the visibility:

$\displaystyle{\cal V}(T)=\exp\left[-\frac{m^{2}\delta{\bf
r}(T).\underline{\underline{{\mathbf{\omega}}}}_{\pm}(t).\delta{\bf
r}^{T}(T)+\delta{\bf
k}(T).\underline{\underline{{\mathbf{\omega}}}}^{-1}_{\pm}(t).\delta{\bf
k}^{T}(T)}{4\hbar m}\right]$ (65)

depending on the mass center position $\delta{\bf r}(t)={\bf r}_{+}(T)-{\bf
r}_{-}(T)$ and the momentum difference $\delta{\bf k}(t)={\bf k}_{+}(T)-{\bf
k}_{-}(T)$. The contrast of the interferometer will be reduced, if the two
wave packets originating from the two arms of the interferometer have a finite
velocity relative to each other or if they do not overlap perfectly at the
final beam splitter. In the case of a tight confinement within the ring, the
probability simplifies into:

$\displaystyle P_{\pm}(T)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\pm\frac{{\cal
V}(T)}{2}\cos\left[\Phi_{\text{S}}(T)-\frac{mR^{2}}{\hbar}\frac{{\dot{\phi}}_{+}(T)+{\dot{\phi}}_{-}(T)}{2}(\delta{\phi}(T)-4\pi)\right]$
(66)

where

$\displaystyle{\cal
V}(T)=\exp\left[-\frac{mR^{2}[\omega_{\phi}^{2}(\delta{\phi}(T)-4\pi)^{2}+\delta{\dot{\phi}}^{2}(T)]}{4\hbar\omega_{\phi}}\right]~{}.$
(67)

Using the phase dynamics Eq.(23) for the bucket case, the visibility has the
explicit expression:

$\displaystyle\ln{\cal
V}(T)=-\frac{mR^{2}\omega_{\phi}}{\hbar}\left(\frac{4\pi\sin(\omega_{\phi}T/2)}{\omega_{\phi}T}\right)^{2}~{}.$
(68)

Thus unless we are in the resonance region $T=2\pi n/\omega_{\phi}$ or in the
large limit $T\omega_{\phi}\rightarrow\infty$ the visibility is less than one.