A phase shift formulation for N-light-pulse atom interferometers: application to inertial sensing

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by  Malo Cadoret

A phase shift formulation for N-light-pulse atom interferometers:
application to inertial sensing
Malo Cadoret1,2*, Nassim Zahzam2, Yannick Bidel2, Clément Diboune2, Alexis Bonnin2,
Fabien Théron2, and Alexandre Bresson2
1Laboratoire Commun de Métrologie, CNAM ,61 rue du Landy,93210 La Plaine Saint-Denis,
France
2ONERA–The French Aerospace Lab, F-91123 Palaiseau cedex, France
7
1 *Corresponding author: [email protected]
0
2

n
a Abstract
J
WereportonanoriginalandsimpleformulationofthephaseshiftinN-light-pulseatominterferometers.
7
Weconsideratomicinterferometers based ontwo-photontransitions (Ramantransitions orBragg pulses).
1
Starting from the exact analytical phase shift formula obtained from the atom optics ABCD formalism,

] we use a power series expansion in time of the position of the atomic wave packet with respect to the
h initial condition. The result of this expansion leads to a formulation of the interferometer phase shift
p where the leading coefficient in the phase terms up to Tk dependences (k ≥ 0) in the time separation T
-
m betweenpulses,canbesimplyexpressedintermsofaproductbetweenaVandermondematrix,andavector
characterizing the two-photon pulse sequence of the interferometer. This simple coefficient dependenceof
o
the phase shift reflects very well the atom interferometer’s sensitivity to a specific inertial field in the
t
a presence of multiple gravito-inertial effects. Consequently, we show that this formulation is well suited
s. when looking for selective atomic sensors of accelerations, rotations or photon recoil only, which can be
c obtained by simply zeroing some specific coefficients. We give a theoritical application of our formulation
i tothe photon recoil measurement.
s
y
h 1 Introduction of two-path interferences in the presence of multiple
p
gravito-inertial fields. The formulation presented in
[
Since the first demonstration of cold atom interfero- this letter could be of practical interest both for the

1 metric inertial sensors [1, 2] in the 1990s, light-pulse understanding of contribution of inertial terms to the
v Atom Interferometers (AIs) have become very stable phase shift, as well as for applications to the devel-
9
and extremely accurate sensors for the measurements opment of selective inertial sensors. This simple and
2
6 of fundamental constants such as gravitational con- exact formulation is obtained starting from an exact
4 stant [3] or the fine structure constant [4, 5], and analytical phase shift formula valid for any pulse se-
0 inertial forces like gravity acceleration [6, 7], Earth’s quence andtaking into accountgravity,gravitygradi-
1. gravitygradient[8,9]orrotations[10–12]findingvari- ent and rotations [22].
0 ety of applications in geodesy, geophysics, metrology, The article is organizedas follows. In section 2 we in-
7 inertial navigation among others. In addition, high troducethe generalframeworkofourcalculationsand
1 sensitivity AIs have become promising candidates for give a simple formulation of the phase shift up to Tk
:
v laboratorytests of General Relativity (GR) [13] espe- dependences. We explicitely derive the phase shift up
i ciallyfor gravitational-waves(GW) detection[14,15], to order k = 4 in the presence of multiple gravito-
X
short-range forces [16] and tests of the weak equiva- inertial fields. In section 3 we give a physical analysis
r
a lence principle (WEP) [17]. For all applications, high of the leading coefficients of the phase shift obtained
resolution measurements must take into account high with our formulation. In section 4 we consider the ef-
order terms in the phase shift to increase the sensors fects of wave vector change and photon recoil on the
accuracy and sensitivity. atomic phase shift in the presence of inertial forces.
In section 5 we give the expression of the total phase
Phase shift calculations in light-pulse AIs in the shift for any N-light pulse AI according to our for-
presenceofmultiplegravito-inertialfieldshavealready mulation. We then apply our formula to the case of
been deeply investigated [18–21]. However, the phase a Mach-Zehnder atomic interferometer and evaluate
shiftformulationsarenotalwaysconvenientwhenone the phase shift terms of our compact cold rubidium
wants to use them for practical applications such as atom gravimeter. We show that our results are con-
the development of atomic sensors that could be se- sistent with previous work of Antoine et al., Dubet-
lectively sensitive to a particular inertial field such as sky et al., and Wolf et al. [19, 22, 24]. In section 6
a rotation, acceleration as well as cross terms. In this we demonstrate the benefit of our formulation when
letter we derive a novel formulation for phase shift searchingforaparticularselectiveinertialatomicsen-
calculations in N light-pulse AIs considering the case
1
sor. We give theoritical examples of N-light-pulse AIs 2.1.2 Interferometric variables
dedicated to the photon recoil measurement.
Any of the AI’s that we consider will consist of time
sequences of two kinds of pulses [25]:
2 Calculations
π/2pulsesthatwillplaytheroleofmatter-wave
2.1 Notations •
beam splitters.
WeconsiderN-light-pulseAIsinwhichatomsundergo π pulses that will act as matter-wave reflectors
•
two-photon transitions (Raman transition or Bragg (or mirrors).
pulse) and where the two laser beams are counter-
One path of an N-pulse AI is described by a vector
propagating or co-propagating with pulse sequences
~ǫ = (ǫ ,ǫ , ,ǫ , ,ǫ ) where ǫ accounts for the
1 2 i N i
only separated in time. We consider AIs in the limit ··· ···
angularsplittinginducedbytheatom-lightinteraction
of short pulses. A two-photon transition is treated
at time t , with ǫ = +1 and ǫ = 1 for an upward
i i i
as a single photon transition with effective frequency and downward momentum transfer−∆p = ǫ ~k re-
ω = ω ω and effective wave vector k = ~k ~k i i i
1 − 2 | 1 − 2| spectively,andwherekiisalongthezdirection. When
correspondingto the difference in frequencyand wave
theatomremainsinthesamemomentumstate,ǫ =0.
i
vector respectively. In our formalism, 2M-photon
Anexampleofpulsebeam-splitter(i.e. π/2pulse)and
Braggtransitions(withM beingtheBraggdiffraction
mirror pulse (i.e. π pulse) are given in Fig. 1.
order, M 1), could be considered by simply giving
≥
to the effective wave vector a magnitude Mk. In the
following calculations, we will consider M =1, unless
specified. Thus, a Bragg transition will be equivalent
to a Raman transition if one does not matter about
the quantum internal state. We will precise wether
the internal quantum state changes (Raman pulse) or
not (Bragg pulse) if necessary.
2.1.1 Time and laser field notations
We first consider AIs for which light pulses are all
equally separated by time T,(extension to the case Figure 1: Recoil diagram of a two-photon transition:
of an arbitrary pulse sequence is possible (see section the two-photon transition (Raman or Bragg) couples
3 3.2)). The AIs are cut into as many slices as there two momentum states. During the process the mo-
are interactions. mentum transfer ∆p = ǫ ~k at time t ; (a) Beam-
i i i i
Hence, the time of the i-th light-pulse is defined as : splitter pulse : the atom is put in a coherent superpo-
sition of two momentum states; (b) Mirror pulse: the
t =(i 1)T (i 1) (1)
i − ≥ atom’s momentum state is changed with momentum
∆p. Solid and dashed lines correspond to two differ-
Theeffectivefrequenciesandwavevectorswillbecon-
ent momentum states correponding to the same inter-
sidereddifferentforeachpulseinordertocompensate
nal quantum state (Bragg pulse) or different internal
for Doppler shift. Hence, the effective laser frequency
quantumstates(Ramantransition). Thetwopathsare
is chirped linearly so as to maintain the resonance
described with vector (~ǫ) and (ǫ~).
condition for each light-pulse as it is in the case of ′
gravity, and resonance condition, will be satisfied.
The effective wave vector k and frequency ω of the We will consider two-path AIs. Each path is de-
i i
two-photon transition of the i-th light pulse are de- scribed by a vector Gk (upper path) and G′k (lower
fined as: path) which takes into account the two-photon pulse
sequence:
N
G = (i 1)kǫ
k i
−
i=1
k =k +∆k(i 1) (2) X (5)
i 1 − N
ωi =ω1+∆ω(i 1) (3) G′k = (i−1)kǫ′i
− i=1
X
Where∆kaccountsforthedifferenceineffectivewave- with k an integer (k 0).
vector and ∆ω is the frequency difference due to the We introduce: ≥
laser chirp, independent of the laser pulse number
N
i. For instance, ∆k is equal to zero when chirping ∆G =G G = (i 1)k∆ǫ
symmetrically two counter propagative laser beams, k k− ′k − i (6)
i=1
whereas ∆ω is not. X
=V∆ǫ
i
We introduce an effective laser phase:
where V is a Vandermonde matrix with coefficient
T v =(n 1)m 1andwhere∆ǫ =ǫ ǫ isthediffer-
φi =φ1+ω1(i 1)T +∆ω(i 1)2 (4) mn − − i i− ′i
− − 2 enceofinteractionbetweentwopathsattimet . Asan
i
2
example, we give in Fig. 2 the case of the well-known [22]:
two-path Mach-Zehnder atomic interferometer, and
N N
focusingononeexitportoftheinterferometeronegets z +z
for the upper path ~ǫ = (1,−1,0) and ǫ~′ = (0,1,−1). ∆ΦTotal = i=1ki∆ǫi i 2 i′ +i=1φi∆ǫi (7)
Consequently one finds ∆~ǫ = (1, 2,1). One could X X
−
have chosen the other exit port of the interferometer ∆Φinertial ∆Φt
which would not affect ∆~ǫ. The right hand sid|e of eq{uzation (}7) h|as{tzwo s}eparate
contributions. The first summation term accounts for
the inertial phase shift, that we will denote ∆Φ
inertial
with z (and z ) being the positionof the wave packet
i i′
at time t of the upper path (and lower path) respec-
i
tively, and where one assumes no offset in the initial
positions of the wavepacket, ∆z =z z =0. This
1 1− 1′
inertialphase shift depends onthe classicalmid-point
position of the atoms at time t . The second summa-
i
tiontermisatime-dependentlaserphaseshiftthatwe
willdenote∆Φ . Accordingto(4)thetimedependent
t
laserphaseshiftforanyAIconsistingofN-light-pulse
can be expressed as follows:
N
T
∆Φ = ∆ǫ φ +ω (i 1)T +∆ω(i 1)2
t i 1 1
− − 2
i=1 (cid:18) (cid:19) (8)
X
T
Figure 2: Space-time recoil diagram in the absence of =∆G φ +∆G ω T +∆G ∆ω
0 1 1 1 2
2
gravity of a Mach-Zehnder interferometer consisting
of pulse-sequence π/2 π π/2. where ∆G ,∆G ,∆G are defined in (6).
− − 0 1 2
Thus we will mainly focus on the inertial phase shift
calculation. Forsimplicitywewillfirstassumenowave
vector change (∆k = 0), as well as no photon recoil
effect. These high order terms will be taken into ac-
2.2 Atom Interferometer phase shift
count in section 4. Thus, the absence of recoil effect
calculation
leads to z =z in equation (7) which can be written:
i i′
The phase shift in a light-pulse AI is often calculated N
in three steps. In a first step one identifies the differ- ∆Φinertial ki∆ǫizi (9)
≃
entarmsoftheinterferometerandcalculatesthephase i=1
X
shift due to atom-light interaction and free propaga-
In our method, we calculate the inertial phase shift,
tion of the atoms along each independent arm. In
by making a Taylor expansion in power of T of the z
a second step one makes the difference between the
position of the wave packet with respect to the first
phase shift obtained in each arm. Finally, in a third
light-pulse (t = 0) of the interferometer, assuming
1
step, a phase shift due to spatial separation of the
z =0:
1
atoms is added. To calculate the totalphase shift one
canusetheFeynmanpathintegralformalism[26],con- z =z(t )= a tk = a Tk(i 1)k (10)
sidering atoms as plane waves. One limitation of this i i k i k −
k=0 k=0
X X
approach is that it is difficult to perform when one
wants to account for simultaneous gravito-inertial ef- where a = 1 dkz .
k k! dtk
fects such as gravity, rotation, gravity gradient and t=t1
Aftersomealge(cid:16)bra(cid:17)the inertialphaseshiftisgivenby:
their interplay, as addressed in this paper.
In this context, the starting point of our approach is ∆Φ ∆G k a Tk
inertial k 1 k
basedontheatomopticsABCD formalismdeveloped ≃
k=0
X
abtyomBosradséw[2a1v]e. pTachkisetms,eatshowdelallaloswtsobcaoltchultaotecothnesidexer- ≃ ∆kG!kk1 ddktkz Tk (11)
act interferometric phase shift arising from multiple k=0 (cid:18) (cid:19)t=t1
X
inertial effects.
where ∆G is defined in equation (6).
k
One can rewrite (11) considering the action of any
2.2.1 N-light-pulse AI phase shift derivation
gravito-inertialfields onthe atoms locatedatposition
R~, leading to the final expression:
For any time-dependent hamiltonian at most
quadraticinpositionz andmomentump,whichisthe ∆G
∆Φ kTk
case for essentially all atom interferometric applica- inertial ≃ k!
tions,theexactphaseshiftformulaforaN-light-pulse Xk (12)
AI with wave vector k along the z = ~k.R~ direction is ~k1 ~k R~1,~v1,+inertialforces
k × ·F
(cid:16) (cid:17)
3
where we introduce as a notation the function by:
~ = dkR~ . Thisfunctiondependsontheatomic
Fk dtk t=t1 (b)
initial(cid:16)posit(cid:17)ion R~ and velocity ~v , and any gravito-
1 1 d2R~ dR~
inertial forces experienced by the atoms. Moreover, ~g(R~)= +2Ω~
it is independent of the AI geometry. We will apply dt2 ! z × }|dt ! {
′ ′
this novel formulation of the phase shift obtained in R R (13)
(a)
equation (12) in the next section.
| {z } dΩ~
+Ω~ Ω~ R~ + R~
× × dt ×
(cid:16) (cid:17)
(c)
2.3 Phase shift due to gravito-inertial
with (a) the |relative ac{czeleration, (}b) the Coriolis
fields
acceleration and (c) the centrifugal acceleration and
where we have introduce gravity at position R~ as:
In this section we use our formulation to derive the
phase shift when the atom is submitted to simul- ~g(R~)=~g(R~ )+~~Γ. R~ R~ (14)
1 1
taneous time independent gravito-inertial fields like −
(cid:16) (cid:17)
gravity, gravity gradient and rotation. We consider where ~~Γ is the gravity gradient tensor with its origin
the same two cases (A and B) treated in the previous
taken at position R~ =R~(t=t ).
work of [22]. 1 1
iInngcwasiethAr,otthaetiAonIirsafitexeΩ~dwtoitthhreeEspaerctthtforatmoethRe′i,nreorttaiat-l Itno opredrfeorrtmotciamlceuldaetreivF~aktio=n(cid:16)odfdkt(Rk~13(cid:17))t.=tM1 o(kre≥ov2er),,oonneechaans
reference frame (i-frame), (see Fig. 3). In case B, noticethattimederivationof(14)isstrictlydepending
R
the AI is fixed to a rotating platform of rotation rate oncaseAorB.Inordertoprovideaunifiedtreatment
Ω~′ =Ω~ for convenience. weintroduceparameterη todistinguishbetweenboth
cases. Assuming η =1 for case A, and η = 0 for case
B, one obtains:
d ~g(R~ )+~~Γ.(−→R −R→) =
1 1
dt −
(cid:18) (cid:19)
R
~~Γ. dR~ +ηΩ~ ~g
dt ! ′ × (15)
R
+ηΩ~ ~~Γ.(R~ R~
1
× −
(cid:18) (cid:19)
+(1 η)~~Γ. Ω~ R~
− ×
(cid:16) (cid:17)
One interesting feature of equation (15) is that it
takesintoaccountbothcasesAandB simultaneously.
Choosing η = 0, or η = 1 and calculating successive
higherordertimederivativesusing(13)anditsderiva-
tivesandsubstitutinginequation(12)allowstocalcu-
latetheglobalinertialphaseshiftforanyN light-pulse
AIs.
2.3.1 Application
Figure 3: Non rotating inertial geocentric reference
Considering the rotating Earth frame, one can define
frame (i-frame) defined by the fixed stars having
R dR~(t=t1) =~v′ as the initial velocity of the atoms
its origin at the center of the Earth and Earth ref- dt ′ 1
erence frame with local coordinate system (x,y,z) (cid:16)and accele(cid:17)rRation
′
R
where the z-axis is chosen to point away from the
Earth center, with rotation rate components Ω~ = ~a=~g(R~1)−Ω~ × Ω~ ×R~1 (16)
(0,Ωcosλ,Ωsinλ). (cid:16) (cid:17)
as the sum of gravity and centrifugal acceler-
ations. One can now calculate the function
~ R~ ,~v ,~a,Ω~,~~Γ up to order k.
Fk 1 1′
Togivethephaseshiftforeachcase(AandB),we (cid:18) (cid:19)
Calculating the inertial phase shift up to terms pro-
calculate the F~k = ddktRk~ t=t1 functions of equation portional to T2 (k =2) leads to :
(12). For k = 0,1 th(cid:16)ese f(cid:17)unctions simply correspond
to initial atomic position R~1, and velocity ~v1 respec- ∆Φinertial = ∆0G!0~k1·R~1+ ∆1G!1~k1·~v1′T
(17)
tively. To calculate higher order derivatives (k 2)
we use the classical equations of atomic motion g≥iven + ∆2G!2~k1· ~a−2Ω~ ×~v1′ T2+O(T3)
(cid:16) (cid:17)
4
Equation (17) exhibits the total inertial phase shift related to the closure in space-time of the atom in-
to the second order in the time seperation beetween terferometer. In the Ramsey-Raman interferometer
pulses. In this case, one can see that ~ is only a (∆G = 1), thus the total phase shift is dependent
2 1
F
function of constant acceleration~a, initial velocity ~v on the initial atomic velocity and on the time depen-
1′
and constant rotation Ω~. dent phase shift term. In order to build interferome-
In Table 1 we extend the phase shift calculation up tersbothclosedinmomentumandpositionspace,one
to terms proportional to T4 for whatever the case is needs to look for N 3 pulse interferometers. Con-
≥
(A or B). In this case, cross-terms between gravity sidering a three-pulse interferometer one has to solve
gradient ~~Γ, acceleration ~a and rotation Ω~ appear in for:
the phase shift expression.
One can notice from Table 1 that the coupling be-
tween the gravity gradient and rotation is case de-
pendent. When the AI is fixed to a mobile platform,
there is a coupling between gravity gradient and ro-
tation whereas this coupling vanishes in case A, when
the sensor is fixed to the Earth frame.
Moreover, Table 1 highlights the benefit of our for-
mulationwherethephaseshifttermsuptoTk depen-
dencesappearasasimpleproductbetweentheinertial
fields and a specific ∆Gk coefficient. ∆G0 =∆ǫ1+∆ǫ2+∆ǫ3 =0
(19)
Thephysicalmeaningof∆G coefficientswillbegiven
k (∆G1 =∆ǫ2+2∆ǫ3 =0
in section 3 for well-known AIs. Moreover,one would
liketoemphasizethatthisformulationmayalsoadress
the case of an optical corner cube gravimeter such as
the FG-5 [23], by simply nulling the laser chirp∆ω in
Table 1 and considering a 2 pulse interferometer with
∆~ǫ=(0,1).
3 Analysis and link with well
known AIs
In this section we givea physicalinterpretationof the
For ∆ǫ = 1 one finds the well-known single-loop,
1
leading terms ∆G of the phase shift which are di-
k Mach-Zehnder configuration [27, 28] consisting in the
rectly related to the two-photon light pulse sequence
pulse sequence (π/2,t =0) (π,t =T) (π/2,t =
1 2 3
of the interferometer through equation (6). − −
2T). The interferometer is depicted in (Fig. 3.b)).
Considering two-photon Raman transitions, the first
3.1 Impact of ∆G coefficients and AI π/2pulseputstheatominacoherentsuperpositionof
k
symmetry ground and excited states and acts as a beam-splitter
bytransferring~kmomentumtothewavepacketmak-
Assuming no inertial effects and focusing on the first ing the transition to the excited state. Then, the two
∆G (k = 0) term of the phase shift, one can show wave packets propagate freely during time T drifting
0
thatthistermgivesaninformationonthe interferom- apart with relative momentum ~k. A mirror pulse is
eter’s closure in momentum space. This information applied at time T interchanging the ground and ex-
is related to the space-time geometry of the atom in- cited states and reversing their relative momenta. Fi-
terferometer. nally, the two wave packets drift back towards each
Searching for the first N-light-pulse interferometer other during time T and a final beam splitter pulse is
closedinmomentumspace(i.e. ∆G =0)with∆ǫ = applied at time 2T recombining the two wave packets
0 1
1, one finds out that N 2. For N = 2, the sim- and interfering them. The interferometer’s closure in
≥
plest atom interferometer closed in momentum space position space can be related to the time-symmetry
is depicted in (Fig. 3.(a). It is the atomic clock con- through:
figuration, also called Ramsey-Raman interferometer
consisting of two pulses (π/2,t = 0) (π/2,t = T)
1 2
−
separated by a free precession time T = t t . One
2 1
−
canseethatthisinterferometeristime-antisymmetric,
meaning that ∆ǫ = ∆ǫ . This closure in momen-
1 2
−
tum space can be expressed as:
∆ǫ = ∆ǫ (18)
i N+1 i
− −
leading to ∆G =0.
0
The ∆G (k = 1) coefficient of the phase shift is ∆ǫ =∆ǫ (20)
1 i N+1 i
−
5
Theseparationbetweentheatomicwavepacketsleav-
ing the first beam splitter is increased by a factor of
two, leading to a difference of momenta between the
two arms of ∆p = 2~k, hence increasing the space-
time area of the apparatus by a factor of two. One
cansee that for these two configurationsclosedin po-
sition space, the phase shift does not depend on the
initialatomic velocityto firstorder(i.e. whenone ne-
glects inertial forces).
When looking at four-pulse AIs, assuming ∆G =
1
∆G = 0 one finds the Ramsey-Bordé interferome-
0
ter [29–32] and the double-diffraction Ramsey-Bordé
interferometer (M =1) depicted in (Fig. 5.a,b).
Figure 4: Space-time recoil diagrams in the absence
of gravity: (a) Ramsey interferometer consisting of
two π/2 pulses separated by a free precession time
T =t t . Theinterferometerisclosedinmomentum
2 1
−
space ∆G = 0; (b) The Mach-Zehnder interferome-
0
terconsistingof pulsesequence(π/2,t =0) (π,t =
1 2
−
T) (π/2,t = 2T). This single-loop interferometer
3
−
is closed in both momentum and position space. Its
sensitivity to acceleration is proportional to the space
time area enclosed by the loop. (c) Double-diffraction
atom interferometer: two pairs of Raman beams with
effective wave vectors k are shined simultaneously
i
±
on the atoms at three moments in times seperated by
timeT. DDP1andDDP3aredoublediffractionpulses
which act as splitters whereas DDP2 acts as a mirror
pulse transferring momentum ∆p = 2~k to ~k
i i
∓ ±
wave packets. The interferometer is totally symmetric
and the scale factor is improved by a factor of 2. In
cases (b),(c) the phase shift does not depend on the
initial velocity of the atoms.
Solving the set of equations (19) and assuming
(max ∆ǫ = 2M,M 1), any three-pulse AI closed
1 Figure 5: Space time recoil diagrams in the absence
| | ≥
in momentum and position space obeys:
of gravity: (a) Symmetric four-pulse Ramsey-Bordé
interferometer. (b) Ramsey-Bordé interferometer in
2
a double-diffraction scheme (M = 1).(c) Five-pulse
−∆→ǫ=M 4 (21)
 −  double-loop interferometer closed in both momentum
2
and position space. In this configuration, ∆G = 0
2
 
making the interferometer insensitive to homogeneous
For M = 1, one finds the double-diffraction scheme
acceleration.
using for example two-photon Raman transitions [27]
and depicted in Fig. 4.c). In this configuration two
pairs of Raman beams simultaneously couple the ini-
tial ground state with zero momentum g,0~k to the Finally, cancellation of ∆G0,1,2 coefficients can be
excited state with two symmetric mom|entumistates, obtainedwithafive-pulseAIgeometrywherethemid-
e, ~k . The interferometer is time and space sym- dle pulse is not shined. This interferometer which
|me±tricias ǫ = ǫ . This spatial symmetry can be is depicted in (Fig. 5.c), is the well known double-
expressed foir any−N′i-pulse AI as: loop interferometer consisting of the pulse sequence
(π/2,t = 0) (π,t = T) (Nopulse,t = 2T)
1 2 3
− − −
∆ǫ =2ǫ (22) (π,t =3T) (π/2,t =4T). This double-loop inter-
i i 4 5
−
6
ferometer is well suited when one wants for example to ∆G = 1 √5 t2 as was found in the previous
2 − 2 4
to build a sensor insensitive to homogeneous acceler-
work of [19](cid:16)where p(cid:17)hase shift calculations were done
ation for rotation or direct gravity-gradient measure-
using density matrix formalism.
ments [8]. We give in Table 2 the absolute values of
the phase coefficient ∆Gk for the interferometers de-
k!
scribed above.
Considering the simple expression of ∆G given in
equation (23), one could think of finding any interfer-
ometer configurations (i.e. any sets of vector −∆−→ǫ ) for
i
which the interferometer couldbe selectively sensitive
togravity,acceleration,gravitygradientorrotationby
fixing to zero the value of a particular ∆G coefficient
and solving:
∆ǫ=V−1∆G (23)
where V 1 is the inverse Vandermonde Matrix. We
−
will give practical examples of some N-light-pulse in-
terferometerscorrespondingto the resolutionofequa-
Figure 6: Space-time diagram of a double loop inter-
tion (23), in section 6.
ferometer with non identical loops. Assuming t = 0,
1
and total interrogation time t = 1, in order to elim-
4
3.2 Arbitrary temporal pulse-sequence inate all t3 terms (i.e. ∆G = 0) to the phase shift,
3
onefindst = √5 1. Solidanddashlinescorrespond
Our approach remains consistent when considering 2,3 4∓
to momentum states separated by ~k.
non-equally spaced in time laser pulses. In this gen-
eral case, equation (6) has to be re-written including
the instant t of the i-th laser pulse leading to:
i
N
∆G(t) = ∆ǫ tk
k i i (24) 4 Effect of wave vector change
i=1
X
=V(t1,t2,t3,···,tN)∆ǫ and photon recoil on the phase
shift
Consequently, ∆G coefficients are now time-
k
dependentbutstillrelatedtothetwo-photonpulsese-
quenceoftheAIthroughaso-calledVandermondema-
4.1 Wave vector change
trixV(t1,t2,...,ti,...,tN) withtime-dependentcoefficients.
Considering a four-pulse double-loop interferome-
ter ∆~ǫ = (1, 2, 2, 1), one can look for solu- Considering the wave vector change then one has to
− − −
tions in time t of the laser-pulse sequence assuming add a correction to equation (11) leading to:
i
∆G =0. Applying equation (23) with respect to
k=0,1
V(t1,t2,t3,t4) leads to:
∆Φ
inertial
≃
∆ǫ=(V(t1,t2,t3,t4))−1∆G (25) ∆Gk+ ∆k1k∆Gk+1 Tk ~k ~ R~ ,~v ,~a,Ω~,~~Γ (26)
1 k 1 1
k! · ·F
ThedetailedcalculationsaregiveninAppendixBcon- k=0 (cid:18) (cid:19)
X
sidering two cases and assuming t = 0 and t = 1 in
1 4
dimensionless unit.
where one can see that the wave vector change con-
In the first case we assumed ∆G = 0. If one
2 tribution to the phase shift at the order k in the time
refers to Table 1, in this case, the phase dependent
separationbetweenpulses,issimplyobtainedfromthe
terms which scale quadratically with time t are elim-
calculation of ∆G coefficient.
k+1
inated. This is interesting when one wants to mea-
sure gravity-gradients or rotations. It then comes
out that the pulse-sequence is t = 1t ;t = 3t
{ 2 4 4 3 4 4}
leading to ∆G = 3 t3. This pulse-sequence corre-
3 16 4 4.2 Recoil phase shift
sponds exactly to the case of the identical double-
loop interferometer found in section 3.A. when one
was not varying the inter-pulse time (Fig 5.c). In The two-photontransitionscontributeto transfertwo
the second case we assumed ∆G3 = 0, hence elim- photon momenta to the atoms. Thus, the inertial
inating all phase terms scaling as t3. This case is phase shift is modified and one needs to account for
interesting when one wants to increase the accuracy z =z . Assuming no rotation and no inhomogeneous
to acceleration measurement by eliminating T3 cor- aicc6 eleri′ation (i.e. gravity gradient), one can express
rections to the phase. We represent in Fig. 6 the simplythepositionoftheatomsunderfree-fallattime
pulse sequence t = √5 1;t = √5+1 corresponding t considering the recoil velocity term ~k/m for an
{ 2 4− 3 4 } i
to a non-identical double-loop interferometer leading atom of atomic mass m:
7
where the first term is the inertial phase shift given
by equation (26) in the absence of photon recoil, and
~ i 1
− the second term is the recoil phase shift contribution
z =z +v (i 1)T + T (i j)k ǫ
i 1 1 − m − j j that can be expressed in the time separation between
j=1
X pulses as:
(a)
~ ~ ~
+ 1g(i 1|)2T2+O{z(T3), } ∆Φrec = k2T∆A1+ k∆kT∆B1+ ∆k2T∆C1+O(T2)
2 − 2m 2m 2m
(30)
~ i 1
− where we introduce :
z =z +v (i 1)T + T (i j)k ǫ
i′ 1 1 − m − j ′j
Xj=1 N
∆A = (ǫ ǫ )
(b) k i− ′i ×
+ 1g(i| 1)2T2{+z O(T3)}, (27) Xi=1 (31)
2 − i−1
(i j)k(ǫ +ǫ )
where (a) and (b) are the recoil dependent terms  − j ′j 
j=1
associated to the two-photon transition for the upper X
 
and lower path respectively. Hence, calculating the
classical mid-point position of the atoms one finds:
N
∆B = (ǫ ǫ )
zrec = zi+zi′ =z +v (i 1)T + 1g(i 1)2T2 k i− ′i ×
i 2 1 1 − 2 − Xi=1 (32)
+ 2~mT i−1(i−j)kj(ǫj +ǫ′j) (28) i−1(i−1+j−1)(i−j)k(ǫj +ǫ′j)
Xj=1 Xj=1
 
(c)
where (c) represe|nts the reco{izl effect cont}ribution to N
∆C = (ǫ ǫ )
the inertial phase shift. In Fig 7 we represent the k i− ′i ×
space-time recoil interferometer paths and the classi- Xi=1 (33)
calmid-pointpositionoftheatomsinaMach-Zehnder i 1
−
geometry with pulse-sequence (π/2,t1 =0)−(π,t2 =  (j−1)(i−j)k(i−1)(ǫj +ǫ′j)
T) (π/2,t =2T). j=1
3 X
−
 
for k 0.
≥
Thefirsttermofequation(30)isthewellknownrecoil
shift associated to the kinetic energy of an atom ab-
sorbing a photon of momentum ~k, and the two last
terms depend on the wave vector change ∆k to the
first and second order respectively. These terms may
account for corrections to gravity measurement when
one considers the variation of the laser frequencies to
compensate the Doppler shift of freely falling atoms
[24].
One can generalize the recoil phase shift to higher or-
Figure 7: Space-time diagram showing the Mach-
ders in the time separation between pulses by simply
Zehnderinterferometerpathsinthepresenceofgravity
writing:
acceleration. Solid lines: upper and lower interferom-
eter paths (zi′ and zi). Dash line: classical mid-point ∆Φrec =
positions of the atoms taking into account the recoil
∆A + ∆k∆B +(∆k)2∆C
effect; Solid line (in blue): parabola in the absence of k k1 k k1 k Tk
photon recoil. The trajectories are plotted assuming k! · (34)
k
X
k =k =k .
1 2 3 ~k ~ ~0,~v = ~~k1,~0,Ω~,~~Γ
1 k r
Considering the two-photonmomenta transfer the × ·F 2m !
inertial phase shift cannot be approximated anymore
by equation (9) and one has to account for the recoil where the ~k function takes into account the recoil
F
effect contribution of (28). Hence, the inertial phase velocity increment~vr in the presence of laser interac-
shift is: tion. Thus the recoil velocity is coupled to inertial
forces. The photon recoil phase shift contribution is
N
giveninTable 3assuming∆k =0forsimplicity. Nev-
∆Φ = k ∆ǫ zrec
inertial i i i ertheless, ∆k = 0 phase shift terms can be obtained
i=1 6
X (29) by calculating ∆B and ∆C coefficients.
N k k
We give in Table 4 the ∆A /k! terms for usual
= k ∆ǫ z +∆Φ k
i i i rec
interferometers.
i=1
X
8
One can note that contrary to the Ramsey-Bordé- Where the ~ functions are given in Table 1 and Ta-
k
F
ChuAI[35]consistingoftwopairsofπ/2pulseswhere ble 3. One can now calculate the total phase shift of
thesecondpairofπ/2pulsespropagateintheopposite any N-light-pulse AI for any cases (A,B) of section2.
direction to the first pair, the Ramsey-Bordé interfer-
ometer(Fig4(a))usedtodeterminethe finestructure
constantthroughameasurementoftheatomrecoilve-
locity in [4, 5] is not sensitive to the recoilshift at the 5 Application to the Mach-
first order in time T between pulses (ie ∆A = 0) al-
1
Zehnder AI
thoughrecoilvelocitydependent terms appearas cor-
rections to the phase in power of k 3 in time sep-
≥
aration between pulses. This main difference comes Asanapplication,weuseourformulationtocalculate
fromtheaditionalkineticenergytermacquiredbythe the total phase shift ∆Φ of the well-known Mach-
MZ
atomsinoneofthe armofthe Ramsey-Bordé-Chuin- Zehnder AI. According to our formulation, the tem-
terferometer, inducing a photon recoil dependence of poralsymmetriesoftheinterferometerimplies∆A =
k
the phase. However, in order to measure the atomic ∆G ;∆B =∆G ;∆C =0. Considering equation
k k k+1 k
recoilin the Ramsey-Bordéinterferometer,one has to (36)thisleadstoasimpleexpressionofthephaseshift
addphotonrecoiltotheatomsinbetweenthetwosets in terms of ~ function:
k
F
ofbeamsplitters. Thiscanbedonebyimplementinga
sequenceofBlochoscillationsasin[4,5], thus leading ∆G + ∆k∆G
to a significant sensitivity increase in the recoil mea- ∆ΦMZ =∆ωT + k k1 k+1 Tk
k! ·
surement. k
X (37)
Moreover, recoil phase shift terms presented in Ta- ~ R~ ,~v + ~~k1,~a,Ω~,~~Γ
ble 3 are responsible for not perfectly closed AIs due k 1 1
×F 2m !
to coupling between the atomic recoil with rotations
and gravity gradient forces. Focusing on the Mach-
where ∆ωT is the time dependent laser phase shift
ZehnderAI usedas agravimeterwithverticallyprop-
whichdepends onthe laserfrequency chirpappliedto
agating optical pulses (along the z-axis), this means
the lasers.
that the classical trajectories of the upper path and
As an application, we evaluate phase terms with pa-
the lower path will not exactly intersect at the fi-
rameterscorrespondingtoourtransportablecold87Rb
nal beam splitter. The separation distance between
atomgravimeterbasedona(π/2 π π/2)two-photon
the wave packets at the last pulse is given by ∆z = − −
∂∆Φ /∂k ∆A3~kzΓzzT3 when one only looks at Raman pulse sequence, described in [33]. The time
rec 1 ≈ 3! 2m betweenpulsesisT =48mswithinitialvelocitycom-
the contribution of gravity gradient force.
ponents ~v = (v = v ,v ), where v 1,2 cm/s
Considering a 87Rb atom interferometer (λ=780 nm, 1′ x,y ⊥ z ⊥ ≃
correspondtothetransversevelocityoftheatomsand
T = 1 s, k = 4π), one finds a separation between
z λ v 15 cm/s corresponding to the vertical velocity
the wave packets of ∆z 18nm. This separation z ≃
≃ at the first light-pulse. All phase terms are calculated
distanceisnotaproblemwhenusingµK-temperature
considering the sensor fixed in the local Earth frame
thermalcloudsourcesobtainedfromusualopticalmo-
defined in section 2. Results are given in Table 5.
lasses. Nevertheless, this separation phase shift could
Comparison of our Mach-Zehnder AI phase shift
becomeanissueforlongbaselineAIs (typicallyT >5
determinationcanbemadewithpreviousworkofAn-
s) where BEC-based AIs are required [34].
toine et al, Dubetsky et al and Wolf et al [19, 22, 24]
withasimplechangeofvariable,consideringcaseAor
4.3 Total phase shift
B of section 2.As an example, in the work of Antoine
The total phase shift of an N-light-pulse AI is simply et al.[22], the wave vector change is not considered
given by: (∆k =0) and gravity is defined as:
∆ΦTotal =∆Φt+∆Φinertial (35) ~g =~g(R~ )+~~Γ (R~ R~ ) (38)
1 1 1
· −
where the total inertial phase shift takes into account
the wave vector mismatch and recoil effect. where g is gravity acceleration defined at the first
1
One finds: light-pulse. In previous work of [19] one has to con-
T sider the rotating Earth frame with initial velocity of
∆ΦTotal =∆G0φ1+∆G1ω1T +∆G2∆ω2 the atoms~v1′ =~v1−Ω~R~1 with acceleration:
∆G + ∆k∆G
+ k k1 k+1 Tk ~g =~a(R~ )+Ω~ (Ω~ R~ ) (39)
k! · 1 1 × × 1
k
X
×~k1·F~k R~1,~v1,~a,Ω~,~~Γ (36) nwahlelyr,eiancctehleerwatoirokno~af iWs odleffientedal.i[n24e]qwuahtoiosntu(d1i6e)d. tFhie-
(cid:18) (cid:19)
∆A + ∆k∆B +(∆k)2∆C effect of wave vector change (∆k =0) one has to con-
+ k k1 k k1 k Tk sider our formulation in case B :6η = 0 in an inertial
k! · ·
frameintheabsenceofrotation. Inallcaseswefound
~k ~ ~0,~v = ~~k1,~0,Ω~,~~Γ allphase shifts to be consistentwith our novelformu-
1 k r
× ·F 2m ! lation.
9
6 Original N-light-pulse AI sen-
sor
Our simple formulation allows to seek for original N-
light-pulse AIs where the phase shift dependences to
specific inertial effects could be cancelled, by simply
searching solutions to equation (23) and zeroing spe-
cific ∆G coefficients. We give hereafter two theorit-
k
ical examples of such AIs that would be dedicated to
the measurement of the photon recoil.
Figure 8: Space-time recoil diagram of the 6-pulse
atom interferometer sensitive to recoil phase shift and
6.1 Photon recoil measurement AIs
insensitive to gravity acceleration. Black line: (upper
path) ~ǫ = (1, 1, 1,1,1, 1) and Red line : (lower
Directandsensitiverecoilfrequencymeasurementsus- path) ~ǫ = (0,−0,1,− 1,0,0)−. S: Beam-splitter pulse ;
′
ingafour-pulseRamsey-Bordé-ChuAIcanbeusedto DDP:Double-Diffra−ction Pulse.
determine the fine structure constant [35, 36]. Con-
sideringTable1,Table3andTable4theatomicphase
shiftuptothepower2intimeT intherotatingEarth
(ie: Case A η =1) is :
∆Φ =∆ωT+M2ω T+M~k (~g 2Ω~ ~v )T2+O(T3) WegiveinTable6thephaseshiftcoefficientvalues
Total r 1 − × 1′ (40) of the interferometer.
where ω = ~~k2/(2m) is the recoil frequency of the
r 1
Onecanverifythatthetotalphaseshifttothefirst
two photon transition of effective wave vector k , and
1
order in T is:
M is the Bragg diffraction order when using Bragg
pulses as LMT beam splitters like in [36] to enhance
the sensitivity to the recoil shift by a factor M2.
One cansee fromequation (40)that the phaseshift is ~k2
∆Φ6pulse = 2 1T = 2ω T +O(T2) (41)
sensitive to inertial forces. Total − 2m − r
In order to make measurements only sensitive to the
recoil shift, conjugate interferometer geometries are
used to remove the sensitivity to local gravitational
whereas all terms scaling quadratically with time T
accelerationandmoreover,simultaneousconjugatein-
areremoved. However,gravitygradientandEarthro-
terferometers (SCIs) help rejecting common-mode vi-
tation rate remains present within the terms scaling
brationalnoise [36]that mayaffect the interferometer
cubically and quartically with time T (ie ∆G and
sensitivity. However,for example, gravitygradientef- 3
∆G coefficients).
fect which scales cubically with time T (see Table 1) 4
cannot be totally rejected as the two conjugate inter-
ferometers are separated in space.
Wegivehereafteranillustrationofhowtouseoursim-
pleformulationtofindatheoriticalN-pulseAIscheme
sensitive to the recoil frequency and independent of
gravity acceleration, gravity gradient and Earth rota-
tion. 6.1.2 Example 2: measurement independent
of gravity, gravity gradient and Earth
rotation
6.1.1 Example 1: measurement independent
of gravity
We lookedfor an AI configurationinsensitive to grav-
OnecanlookforanAIthatwouldmeasurethephoton ity gradient and Earth rotation. For this case we
recoilindependentofthelocalgravityacceleration. In solved∆G =0with∆A =0. Consideringfor
k=0,1,2,3 1
6
this case, one would have to solve equation (23) as- simplicity, optical pulses equally spaced with time T,
suming ∆G = 0 and ∆A = 0. If one consid- and assuming max∆ǫ = 2, we found several 8-light
k=0,1,2 1
6 | |
ersequaltime T betweenthe two-photonlaserpulses, pulse AI configurations. We give two of these config-
andassumingmax∆ǫ =2,onefindstheN =6-light- urationsonFig 9 andFig.10. We give inTable 7 and
| |
pulse sequence of Fig 8 where we recall that the two- Table 8 the phase shift coefficient values of the two
photon wave vector is not adressed in terms of direc- degenerateinterferometerconfigurations. Onecansee
tion. Nevertheless, one can see that for the third and thatthe maindifferenceappearsinthe ∆A /3!coeffi-
3
fourth pulse one needs to realize a double-diffraction cient where rotation and gravity gradient are coupled
scheme. The beam splitters of the interferometer are to the atomic recoil leading to a difference of almost
denoted S. anorderofmagnitudeinbetweenthe twocoefficients.
10