Table Of ContentAtom interferometry in the presence of an external test mass
B. Dubetsky,1 S. B. Libby,2 P. R. Berman3
[email protected]
2Physics Division, Physical and Life Sciences,
Lawrence Livermore National Laboratory,
Livermore, California 94550, USA
3Physics Department, University of Michigan,
Ann Arbor, Michigan 48109-1040
6 (Dated: March 15, 2016)
1
0 The influence of an external test mass on the phase of the signal of an atom interferometer
2 is studied theoretically. Using traditional techniques in atom optics based on the density matrix
equations in the Wigner representation, we are able to extract the various contributions to the
r
a phase of the signal associated with the classical motion of the atoms, the quantum correction to
M this motion resulting from atomic recoil that is produced when the atoms interact with Raman
field pulses, and quantum corrections to the atomic motion that occur in the time between the
1 Raman field pulses. By increasing theeffectivewave vector associated with theRaman field pulses
1 using modified field parameters, we can increase the sensitivity of the signal to the point where
such quantum corrections can be measured. The expressions that are derived can be evaluated
] numerically to isolate the contribution to the signal from an external test mass. The regions of
h
validity of the exact and approximateexpressions are determined.
p
-
m PACSnumbers: 03.75.Dg,37.25.+k,04.80.-y
o
t
a I. INTRODUCTION
.
s
c Since its birth about30 yearsago[1], the field of atom interferometry(AI) has maturedsignificantly. Experiments
i
s based on AI have been used to measure fundamental constants [2–5], the acceleration of gravity near the Earth’s
y surface [6–9], the gradient of the Earth’s gravitational field [4, 10, 11], and the curvature of the gravitational field
h
produced by source masses [12]. Atom interferometer gyroscopes allow one to measure rotation rates; experiments
p
haveutilizedopticalfields[13],nanofabricatedstructures[14],andthreeorfourspatiallyortemporallyseparatedsets
[
of fields that drive Raman transitions to split and recombine the matter waves [15–18]. The frequency shift arising
4 from a quadratic Zeeman effect was also measured precisely [19]. There have been limits set on a non-Newtonian
v
Yukawa-typefifthforce[20]andondarkenergy[21]using AI,aswellastheoreticalproposalsforusing AI tomeasure
5
general relativity effects [22, 23], including gravitational waves [24]. A detailed theoretical analysis of the combined
5
effect of rotation and gravity on the AI signal has been given [17], based on three- and four-pulse Raman schemes.
3
0 Atom interferometry has also been used to probe the gravitational field produced by a heavy test mass [4, 5, 12,
0 20, 21]. Using a double-difference technique [4] one can extract that part of the phase of the AI signal caused by the
. gravitational field of the test mass. This article provides a theoretical calculation of this contribution to the phase,
1
based on an atom interferometer using three Raman field pulses. The results can be used to optimize measurements
0
6 oftheNewtoniangravitationalconstantGandtoprovideacompletederivationofresultsoutlinedinapreviouspaper
1 [25]. Additionally, recently, an analytic, semi-classicalexpressionfor the phase response of an atom interferometer to
: an arbitrarily placed, stationary point mass has been derived in [26].
v
i
X
r A. Estimated Phase Corrections resulting from the Test Mass
a
The phase in an atom interferometer depends on the interactions of the atoms with the applied Raman fields as
well as the motion of the atoms between and following the applied Raman pulses. The Raman pulses couple two
hyperfine sublevels, g and e, in the atomic ground state manifold and it is the phase associated with the Raman
coherenceρ thatis measuredusing the interferometer.Thepresenceof agravitationalpotential modifies the atomic
eg
trajectories,leading to a modificationof the AI phase. It is this modification ofthe phase thatservesas a measureof
the sensitivity of AI to gravitationaleffects. Since the Earth’s gravitationalpotential is only slightly inhomogeneous
over the physicalextent of the atom interferometer, it can be approximatedby a Taylor series in which only the lead
andgradienttermsareretained[27]. ApproximatesolutionsfortheatomictrajectorieswereobtainedinRefs. [17,29],
where effects related to the Earth’s rotation(centripetal and Coriolis forces) were also included. An exact expression
for the atom trajectories with these combination of forces has also been derived for a non-spherical gravitational
source (i.e. for an arbitrary gravity-gradienttensor), rotating with constant angular velocity [31].
2
a b
t=Τ =Τ +T
2 1
MM
Τ
Τ 3
x 2 x t=Τ t=Τ =Τ +2T
® Τ 1 3 1
v 1 ®
0 M v0
y y
®
g ®
g
z z
FIG. 1: Schematic representation of an atom interferometer in the presence of a test mass M. The atom cloud of the
interferometer is launched at t = 0 with velocity v and interacts with Raman pulses at times t = τ , t = τ ≡ τ +T, and
0 1 2 1
t=τ ≡τ +2T indicated by thestars in thediagram. (a) A generic interferometer. (b) The fountain geometry used for the
3 1
numericalcalculations. Inthiscase themass isapointmass orspherical masshavingradium y that iscenteredat position
min
(x,y,z)=(xm0,ym0,zm0) at time t=0. The case of a stationary test mass (x(m1)(t)=0) and a test mass moving at constant
velocity (x(m2)(t)=xm0+vmt) are considered. The modification of thesignal produced by thetest mass would bea maximum
if the atom cloud were to touch thetest mass at thetop of thecloud’s trajectory.
The situation can change dramatically if a massive test object is brought close to the interferometer (see Fig. 1).
The accumulated phase produced by the test mass’ gravitational field, δg(x,t), increases with decreasing distance
y between the test mass and the trajectories of the atoms in the interferometer and also increases with increasing
min
delaytimes T betweenthe Ramanpulses. ForsufficientlylongT andsmally ,itisnolongeragoodapproximation
min
to retainonlythe leadandgradienttermswhenconsideringthe gravitationalpotentialassociatedwith the testmass.
The maximum value of T is limited by experimental considerations; the largest delay time that has been achieved
is T =1.15 s [32]. Even for smaller delay times, the inhomogeneity of the field can be significant. For example, with
T =200 ms, in a symmetric fountain geometry [33], the length of the atomic trajectory is longer than
1
L= gT2 =0.196 m, (1)
2
where g is Earth’s gravitationalfield. With ymin <∼0.1 m [see Eqs. (103) and (104) in Sec. III], the usually accepted
assumptionthatthegravitationalaccelerationisconstantorslightlyinhomogeneousalongtheatomtrajectorybecomes
invalid.
In calculating the atomic trajectories, we can assume [34] that the magnitude δg of the gravitational field of the
test mass at the position of the atoms in the interferometer is much less than that of the Earth’s field [35],
δg ≪g. (2)
Nevertheless, both the average field and field gradient associated with the test mass can modify the phase of the AI
signal. Let us denote the averagefield ofthe testmass overthe interferometricpath asδg. The interferometricphase
δφ associated with this average field strength is of order [6–9]
δφ∼kδgT2, (3)
where k is an effective wave vector of the Raman field and T is the time delay between Raman pulses. This phase
change arises owing to the accelerationof the atoms produced by the field of the test mass.
In addition to this ”classical”contribution to the phase, there are quantum corrections whose effect we would now
like to estimate. Atom interferometers that make use of copropagating optical fields or copropagating Raman pulses
as their beam splitters and combiners have a signal phase that is insensitive to quantum corrections if the gravity
field is homogeneous. Quantum corrections arise as a result of rotation [36] or inhomogeneous field terms [28, 29].
Quantum corrections φ to the phase from an inhomogeneous gravitationalfield are of order
q
~k2
φ ∼ γT3, (4)
q M
a
3
where M is an atomic mass and γ is the magnitude of the relevant terms in the gravity-gradient tensor. One can
a
understand the estimate (4) as the quantum part of the phase addition kγvT3 [28–30] associated with the change of
atomic velocity v =~k/M owing to recoil[38] after interactionwith a Raman pulse. When the length of the atomic
a
trajectory L becomes comparable with the characteristic distance over which the gravitational potential of the test
mass changes, a reasonable estimate for γ is γ ∼δg/L. As a consequence, we find
φ ~kT
q
∼ . (5)
δφ M L
a
For Rb87 and k ≈1.61×107 m−1
φ
q ∼1.2×10−2. (6)
δφ
Calculations [25] indicate that δφ can be as large as 1 rad, implying that φ can be as large as 10−2 rad. Since a
q
lower limit for the phase noise in the interferometer is of order [32]
φ =10−3rad, (7)
err
one sees that quantum corrections φ are small but measurable; we will include them in our considerations.
q
Another type of quantum correction is produced during the free evolution of the atomic coherence between the
Raman pulses. We formulate the problem in terms of the Wigner representation [37] for the atomic density matrix,
ρ(x,p,t). This is a standard approach for studying phenomena related to quantization of the atomic center of mass
motion[38]andlasercooling[39]. However,toourknowledge,ithasbeenusedsparinglyinthecontextofAI[17,40].
The convenience of this approach is that, for the time between Raman pulses, ρ(x,p,t) obeys an equation that is
similar to the classical Liouville equation for the distribution function [37].
The Wigner distribution function can be written as,
1 1 1
ρ(x,p,t) = dsρ x+ s,x− s,t exp(−ip·s/~) (8a)
(2π~)3 x 2 2
Z (cid:18) (cid:19)
1 1 1
= duρ p+ u,p− u,t exp(ix·u/~) (8b)
(2π~)3 p 2 2
Z (cid:18) (cid:19)
where ρ (x,x′,t) is the density matrix in the coordinate representation and ρ (p,p′,t) is the density matrix in the
x p
momentum representation. To estimate the quantum corrections we start from the time evolution equation for the
Wigner function for times between the application of the Raman pulses. In the absence of the Earth’s rotation, this
equation can be written as [17]
p
∂ + ∂ −∂ U∂ +Q ρ(x,p,t)=0, (9a)
t x x p
M
(cid:26) a (cid:27)
1 1
Q=−(i~)−1 U x+ i~∂ −U x− i~∂ +∂ U∂ (9b)
p p x p
2 2
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
where U(x) is the gravitational potential. For nearly homogeneous fields, such as the Earth’s field, the potential
functions in Eq. (9b) can expanded to first order in ~. In that limit, one finds that Q ∼ 0 and that the Wigner
functionobeysthe sameLiouvilleequationasthe classicaldensitymatrixinthe time betweenpulses. Inthe presence
ofatestmass,however,thegravitationalpotentialisstronglyinhomogeneousandhigherordertermsintheexpansion
are needed.
Let us estimate the correction from these higher order terms. If the term ∂ U∂ in the Liouville equation (9a) is
x p
responsible for the phase δφ in Eq. (3), then the Q−term results in a quantum correction
Q
φ ∼ δφ. (10)
Q ∂ U∂
x p
The density matrix depends on atomic momentum p in two characteristic ways. There is both a thermal momentum
p = 2M k T (11)
0 a B C
(k is Boltzmann constant, T is atom cloud temperatpure) and a momentum associated with the Doppler phase,
B C
M
a
p ∼ . (12)
D
kT
4
For Rb at temperature T ≈1µK, k =1.61×107 m−1, and T =200 ms,
C
p 1
D = ∼2.2×10−5, (13)
p kv T
0 0
where v = p /M = 0.014 m/s is the thermal velocity. In qualitative terms, we can think of the dependence of
0 0 a
ρ(x,p,t) on momentum to vary as
e−p2/p20eip/pD,
where the first factor gives the thermal distribution and the second a phase factor resulting from the accumulated
Doppler phase between Ramanpulses. Explicit forms for the Doppler phase acquiredby the Ramancoherence ρ in
eg
a time T are derived in the next section, but they are typically of order kvT =p/p .
D
If ρ(x,p,t) ∼ e−p2/p20eip/pD, it follows that the Doppler phase factor makes the dominant contribution to the
momentum gradient since
∂ ∼p−1 ≫p−1. (14)
p D 0
To estimate the quantum corrections, we expand, Eq. (9b) to second order in ~ to obtain
2~2 ∂ 3
Q∼ ∂3U p . (15)
3! x 2
(cid:18) (cid:19)
Replacing ∂nU by U/Ln and ∂ by p−1, we find
x p D
φ Q 1 ∂2U 1 ~kT 2
Q ∼ ∼ x = =5.8×10−6. (16)
δφ ∂ U∂ 24Up2 24 LM
x p D (cid:18) a(cid:19)
Consistentwith the phase noise givenin Eq. (7), one should ignore the Q−termin Eq. (9a). Howeverif one uses the
AI technique to measure the Newtonian gravitationalconstantG with anaccuracyofseveralppm(the levelachieved
is already 150ppm [5]) then the Q−term should be included. Anticipating innovations capable of reducing the phase
noise to φ ∼ 3×10−7rad [41], one has to include the Q−term. Consequently, we will include the corrections
err
resulting from this term.
To summarize, there are two types of quantum corrections to the AI phase that are to be considered. The first,
φ , arises from inhomogeneous gravitationalfield modifications of the Doppler phase associated the recoil the atoms
q
undergo on interacting with the Raman fields. The ratio φ /δφ is of order ~kT/M L. The second, φ , arises from
q a Q
quantum corrections to the off-diagonal elements of the Wigner distribution during periods of free evolution. The
ratio φ /δφ is of order (~kT/M L)2.
Q a
It is possible to increase both δφ and the quantum corrections φ and φ using larger values of the effective wave
q Q
vector k. Moreover, since δφ ∝ k, φ ∝ k2 and φ ∝ k3 [see Eqs. (3, 4, 16)], the relative weight of the
q Q
quantum corrections also increases with increasing k. There are at least five ways to increase k: production
of higher order atomic density harmonics in a standing wave field in the Raman-Nath regime [see Eq. (4) in [1]],
higher order Bragg scattering [42], sequential Bragg scattering technique [2], multicolor techniques [43], and Raman
standing wave techniques [44]. For example, standing wave pulses in the Raman-Nath regime were used to produce
the 10th harmonic of the atomic density without excessive loss of signal magnitude and without sub-recoil cooling
[45]. A4~kbeamsplitterwasdemonstratedusinganextensionoftheRamanstandingwavetechnique[46]anda51~k
beam splitter has been produced using higher order Bragg scattering [47]. Recently a high order Bragg scattering
atom interferometer was used to determine the fine structure constant with a resolution 0.25ppb [3]. A 45~k beam
splitter has been utilized for atom interference using sequential Bragg scattering [48]. On the theoretical side, it was
shownthat, with a properchoiceof fieldpolarization,Ramanstanding wavesin the Raman-Nathregime canbe used
to createa 4~k beam splitter without increasingthe number ofseparatedRamanpulses [44, 49]. To accountfor such
enhancements, ourcalculations ofthe AI’s phase arecarriedoutfor aneffective k−vectorthatis scaledby aninteger
factor n .
k
This article is arrangedas follows. In the next section we derive exact and approximate expressions for the phases
δφ, φ , and φ . The results of numerical calculations of the phases are given in Sec. III for a stationary test mass
q Q
and a test mass moving at constant velocity. The calculations enable us to establish the regions of validity of the
approximate expressions for the phases.
5
II. BASIC FORMALISM
The working medium of the atom interferometer consists of a cloud of atoms that are launched with some initial
velocity at t = 0. The cloud interacts with three Raman pulses that are separated in time; these pulses couple
two hyperfine sublevels in the atomic ground state manifold. In the time intervals between the pulses, the atoms
move under the influence of a gravitationalpotential U(x,t). The cloud is assumed to be characterizedby a Wigner
distribution f(x,p) at time t = 0 and the cloud is assumed to be sufficiently localized such that, at any time, the
gravitational field is the same for all atoms in the cloud. In other words, the cloud can be considered as a point
insofar as its interactions with both the Earth’s and the test mass’ gravitational fields. The problem can be broken
down into periods of ”free evolution” of density matrix elements before the first Raman pulse is applied and for the
time intervals between subsequent Raman pulses and into time intervals in which the Raman fields are applied. By
”free evolution,” we mean evolution in the absence of applied radiation fields, but including the effects produced by
U(x,t). We consider each region separately and then piece together the total response.
We will see that the quantum corrections leading to φ originate in the recoil the atoms undergo as a result of
q
their interactionwith the Raman pulses. Following the interactions this recoil leads to a contribution to the Doppler
phase of the off-diagonal density matrix elements ρ (g and e are sublevels of the atoms’ ground state manifold) in
eg
the time intervals between the pulses. In addition, the momentum derivatives of the Doppler phase factors give rise
to the Q−termcorrections;as such,the Q−termcorrectionsdepend only onthe free evolutionofoff-diagonaldensity
matrix elements between the pulses.
A. Density matrix evolution between the Raman pulses
Between the Raman pulses, the Wigner function evolves according to Eqs. (9). When the distance L over which
the gravitationalpotential energy varies significantly is much largerthan ~ divided by the characteristic width ∆p of
the momentum distribution, i.e.
~
≪1, (17)
∆pL
we can expand Q [Eq. (9b)] in a power series in ~ to obtain
~2
Q≈− χ′ (x,t)∂ ∂ ∂ , (18)
24 ikl pi pk pl
where
χ′ (x,t)=−∂ ∂ ∂ U(x,t). (19)
ikl xi xk xl
A summation conventionimplicit in Eq. (18) willbe usedin all subsequentequations. Repeated indices and symbols
appearingontheright-hand-side(rhs)ofanequationaretobesummedover,unlesstheyalsoappearontheleft-hand
side (lhs) of that equation.
We have already shown in Eq. (16) that the Q−term can be considered as a small perturbation, allowing us to
write
ρ(x,p,t)=ρ (x,p,t)+ρ (x,p,t), (20)
0 Q
where ρ (x,p,t) is the unperturbed density matrix obeying the equation
0
p
∂ + ∂ −∂ U(x,t)∂ ρ (x,p,t)=0 (21)
t M x x p 0
(cid:26) a (cid:27)
and ρ (x,p,t) is a perturbation whose evolution is governedby the equation
Q
p
∂ + ∂ −∂ U(x,t)∂ ρ (x,p,t)=−Qρ (x,p,t). (22)
t M x x p Q 0
(cid:26) a (cid:27)
The ρ (x,p,t) term contains the φ corrections, while the ρ (x,p,t) term provides the φ corrections.
0 q Q Q
Equation (21) has been studied in Ref. [17] for the Earth’s gravitational field. In this article we obtain a solution
of Eq. (21) in the presence of a test mass and solve Eq. (22) to get the contribution to the AI phase arising from the
6
Q−term. We assume that density matrix is known at some preceding time t′ and arbitrarily set ρ (x,p,t′) = 0 at
Q
this time, such that, at t = t′, ρ (x,p,t′) = ρ(x,p,t′). The solution of the homogeneous Eq. (21) is then given by
0
[17]
ρ (x,p,t)=ρ[X(x,p,t′,t),P(x,p,t′,t),t′], (23)
0
where{X(x,p,t ,t ),P(x,p,t ,t )}areatomicclassicalpositionandmomentumattimet subjecttotheconstraint
1 2 1 2 1
thatthepositionandmomentumarespecifiedby{x,p}attimet .Inotherwords,inEq. (23),welookforthe values
2
{X(x,p,t′,t),P(x,p,t′,t)} for which {X(t′),P(t′)} will lead to values {X(t),P(t)} = {x,p} under the influence
of the applied fields.
TurningourattentiontoEq. (22),weseethatthecurlybracketsinthatequationisatotaltimederivative,enabling
us to write
dρ (x,p,t′′)
Q =−Qρ (x,p,t′′) (24)
dt′′ 0
Integrating this equation from t′′ equals t′ to t, using the fact that ρ (x,p,t′) = 0, and making use of Eqs. (23),
Q
(18), and (19), we find
~2 t
ρ (x,p,t)= dt′′[χ′ (ξ,t′′)∂ ∂ ∂ ρ (ξ,π,t′′)] . (25)
Q 24Zt′ ikl πi πk πl 0 ξ=X(x,p,t′′,t),π=P(x,p,t′′,t)
Using Eq. (23) one more time, we arrive at
~2 t
ρ (x,p,t)= dt′′[χ′ (ξ,t′′)∂ ∂ ∂ ρ (X(ξ,π,t′,t′′),P(ξ,π,t′,t′′),t′)] . (26)
Q 24Zt′ ikl πi πk πl 0 ( πξ )=( XP((xx,,pp,,tt′′′′,,tt)))
B. Changes in Density Matrix Elements Produced by the Raman Pulses
Considernowacloudofatomshavingtwohyperfine sublevelsg ande inthe groundstate manifold. Theatomsare
π π
prepared in level g at time t = 0 and they proceed to interact with a −π− sequence of Raman pulses applied
2 2
at times
τ ={τ ,τ =τ +T,τ =τ +2T}, (27)
1 2 1 3 1
where τ is time delay between cloud launch and the first Raman pulse and T is the time delay between pulses. The
1
initial atomic density matrix (8) is given by
ρ (x,p,0)=f(x,p), (28a)
gg
ρ (x,p,0)=ρ (x,p,0)=0, (28b)
eg ee
where f(x,p) is the Wigner distribution at t=0.
If a π/2−pulse applied at time τ , the density matrix elements undergo changes given by [17]
j
1 ~k
ρ (x,p,τ )= ρ (x,p,τ )+ρ (x,p−~k,τ ) +Re iexp −i k·x−δ(j)τ −φ ρ x,p− ,τ ,
ee j+ 2 ee j− gg j− 12 j j eg 2 j−
(cid:2) (cid:3) (cid:26) h (cid:16) (cid:17)i (cid:18) (29a(cid:19))(cid:27)
1 ~k
ρ (x,p,τ )= ρ (x,p+~k,τ )+ρ (x,p,τ ) −Re iexp −i k·x−δ(j)τ −φ ρ x,p+ ,τ ,
gg j+ 2 ee j− gg j− 12 j j eg 2 j−
(cid:2) (cid:3) (cid:26) h (cid:16) (cid:17)i (cid:18) (29b(cid:19))(cid:27)
i ~k ~k
ρ (x,p,τ )= exp i k·x−δ(j)τ −φ ρ x,p+ ,τ −ρ x,p− ,τ
eg j+ 2 12 j j ee 2 j− gg 2 j−
h (cid:16) (cid:17)i(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
1
+ ρ (x,p,τ )+exp 2i k·x−δ(j)τ −φ ρ (x,p,τ ) ,. (29c)
2 eg j− 12 j j ge j−
n h (cid:16) (cid:17)i o
7
Similarly, for π−pulse applied at time τ ,
j
ρ (x,p,τ )=ρ (x,p−~k,τ ), (30a)
ee j+ gg j−
ρ (x,p,τ )=ρ (x,p+~k,τ ), (30b)
gg j+ ee j−
ρ (x,p,τ )=exp 2i k·x−δ(j)τ −φ ρ (x,p,τ ), (30c)
eg j+ 12 j j ge j−
h (cid:16) (cid:17)i
In these equations, k is an effective wave vector (assumed to be the same for all the pulses), δ(j) is the detuning
12
between the hyperfine transition frequency and the effective frequency of the Raman fields (that is the frequency
difference of the two fields used to create the Raman pulse), φ is the phase difference between traveling components
j
of the Raman field, and τ are times just after and before the pulse. We allow pulses to have different detunings
j±
(j)
and phases δ ,φ (j =1,2,3).
12 j
ItisassumedthatthetemporalwidthoftheRamanpulsesaresufficientlyshorttoguaranteethatallphasesrelated
tothedetuning,Dopplershifts,andthegravitationalfieldsareeffectivelyfrozenduringtheapplicationofthepulses. In
additionweassumethattheRamanfieldamplitudeandphaseareconstantoverthesizeoftheatomiccloud,allowing
us to neglect corrections arising from the ac-Stark effect and wave front curvature of the Raman fields. In principle
most of these assumptions are not necessary. One can derive and explore the analogue of Eqs. (29, 30) considering
extended atom clouds at finite temperature, including corrections arising from Doppler broadening, ac-Stark effects
and gravitational acceleration produced during the Raman pulses. In this case, however, the corrections depend on
the initial atomic distribution f(x,p). Since this distribution is usually not known accurately, it is preferable for
high precision atomic interferometry to use Raman pulses of sufficiently short duration, sufficiently large diameter
and sufficiently flat wave fronts to avoid such corrections.
If a π/2 pulse acts on a ground state atom, it produces a superposition of ground and excited states. If there
was a momentum p associated with the ground state amplitude a (p) before the pulse is applied, the excited state
g
amplitudea (p)dependsona (p−~k). Asaconsequence,theoff-diagonaldensitymatrixelementfollowingthepulse
e g
involves the product of state amplitudes evolving with different momenta. It is this difference in momentum that
leads to the Q−term correction in periods of free evolution.
C. AI Signal
Our goal is to calculate ρ (x,p,τ ), the excited state atomic density matrix element following the 3rd Raman
ee 3+
pulse, since ρ (x,p,τ ) can be related to experimentally measurable quantities. To carry out the calculation, we
ee 3+
use Eqs. (23, 26) for the ”free evolution” of density matrix elements before the first Raman pulse is applied and
for the time intervals between subsequent Raman pulses and use Eqs. (29, 30) for changes in the density matrix
elements resulting fromthe applicationof the Ramanpulses. In these free evolutionregions,density matrix elements
are affected by the presence of a gravitationalpotential that ultimately contributes to the phase of the AI signal.
From the time the cloud is launched at t = 0 to the time τ that the first Raman pulse is applied, the only non-
1
vanishing density matrix element is ρ (x,p,t). In the time interval between t = 0 and t = τ , this density matrix
gg 1
element evolves to
ρ (x,p,τ )=f(X(x,p,0,τ ),P(x,p,0,τ )). (31)
gg 1− 1 1
For reasons to be discussed below, corrections from the Q term can be neglected in this time interval. After the first
π/2−pulse, the density matrix elements change to
1
ρ (x,p,τ ) = f(X(x,p−~k,0,τ ),P(x,p−~k,0,τ )), (32a)
ee 1+ 2 1 1
1
ρ (x,p,τ ) = f(X(x,p,0,τ ),P(x,p,0,τ )), (32b)
gg 1+ 2 1 1
i ~k ~k
ρ (x,p,τ ) = − exp i k·x−δ(1)τ −φ f X x,p− ,0,τ ,P x,p− ,0,τ . (32c)
eg 1+ 2 12 1 1 2 1 2 1
h (cid:16) (cid:17)i (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
One uses these density matrix elements as initial values for the free evolution between the 1st and 2nd pulses of the
unperturbed density matrix; that is,
ρ (x,p,τ )=ρ(x,p,τ ). (33)
0 1+ 1+
8
We now consider the modifications produced by the Q−term in the time interval between the 1st and 2nd pulses.
The modifications produced by the Q−term (26) in the atomic coherence before the second pulse acts, ρ can be
Qeg
calculated from Eqs. (26, 32c, 33) as
~2 τ2
ρ (x,p,τ )=−i dt
Qeg 2− 48
Zτ1
exp i k·X(ξ,π,τ ,t)−δ(1)τ −φ
1 12 1 1
×χ′ikl(ξ,t)∂πi∂πk∂πl×f(cid:18)XP(cid:0)XXh(((cid:16)ξξ,,ππ,,ττ11,,tt)),,PP((ξξ,,ππ,,ττ11,,tt))−−~~22kk,,0(cid:17)0,,iττ11(cid:1), (cid:19) ξ = X (x,p,t,τ2). (34)
( π ) ( P )
(cid:0) (cid:1)
In Eq. (34), the π derivatives lead to two types of terms. The first of these originates from the thermal distribution
and is of order
∂ ∼p−1, (35)
πiThermal 0
where p is thermal momentum defined in Eq. (11). The second arises from the phase factor
0
exp i k·X(ξ,π,τ ,t)−δ(1)τ −φ in Eq. (34), evaluated at t − τ ∼ T. To estimate this contribution, we
1 12 1 1 1
”turhn(cid:16)off” the gravitationalfield. In t(cid:17)hiis approximation
X(ξ,π,τ ,t)=ξ−π(t−τ )/M (36)
1 1 a
and the Doppler phase becomes equal to k· π(t−τ )/M . This phase factor is a rapidly oscillating function of
1 a
momentum π having period of order p defined by Eq. (12), from which we find
D
∂ ∼p−1. (37)
πiDoppler D
In the limit that
kv T ≫1, (38)
0
the thermal derivative is smaller than Doppler derivative by the ratio given in Eq. (13) and can be neglected.
When inequality (38) holds, the time separation between pulses T is sufficiently large to insure that the dominant
contributionstotheQ−termcomesfromthemomentumderivativesoftheDopplerphasefactor. Aswewillshow,the
atomic levels’populations (ρ andρ ) haveno phase factor for 0<t<τ ; therefore the Q−termcorrectionsarise
ee gg 3−
only from the atomic coherence ρ . As a consequence, we can neglect any contribution to the Q− term corrections
eg
from atomic state populations. It was for this reason we did not include any Q−term corrections to the Wigner
distribution for the time interval 0<t<τ . In the Doppler limit defined by Eq. (38), the AI phase is pretty much
1−
independent of the atomic momentum and spatial distributions.
CalculatingthederivativesandretainingthosecontributionstothederivativesarisingfromtheDopplerphaseonly,
we arrive at
~2 τ2
ρ (x,p,τ )=− k k k dt
Qeg 2− 48 u v w
Zτ1
×[χ′ (ξ,t)∂ X (ξ,π,τ ,t)∂ X (ξ,π,τ ,t)∂ X (ξ,π,τ ,t)]
ikl πi u 1 πk v 1 πl w 1 ξ X
= (x,p,t,τ2)
(π ) ( P )
exp i k·X(ξ,π,τ ,t)−δ(1)τ −φ
1 12 1 1
×f(cid:18)XP(cid:0)XXh(((cid:16)ξξ,,ππ,,ττ11,,tt)),,PP((ξξ,,ππ,,ττ11,,tt))−−~~22kk,,0(cid:17)0,,iττ11(cid:1), (cid:19) ξ = X (x,p,t,τ2), (39)
( π ) ( P )
(cid:0) (cid:1)
where k is the uth component of the effective k−vector. In this approximation,the derivative no longer acts on the
u
term inside the curly brackets. Therefore we can apply the multiplication law,
X X
(X(x,p,t′,t′′),P(x,p,t′,t′′),t,t′)= (x,p,t,t′′) (40)
P P
(cid:26) (cid:27) (cid:26) (cid:27)
9
to get
X X
(ξ,π,τ ,t) = (x,p,τ ,τ ). (41)
P 1 ξ=X(x,p,t,τ2),π=P(x,p,t,τ2) P 1 2
(cid:26) (cid:27) (cid:26) (cid:27)
The expression inside the curly brackets of Eq. (39) becomes t−independent and the Q−term just before the second
pulse is given by
~2
ρ (x,p,τ )=− exp i k·ξ−δ(1)τ −φ f(ξ,π)
Qeg 2− 48 12 1 1 ξ = X(x,p,τ1,τ2)
n h (cid:16) (cid:17)i o(π ) (P(x,p,τ1,τ2)−~k/2 )
τ2
×k k k dt[χ′ (ξ,t)∂ X (ξ,π,τ ,t)∂ X (ξ,π,τ ,t)∂ X (ξ,π,τ ,t)] . (42)
u v w ikl πi u 1 πk v 1 πl w 1 ξ X
Zτ1 = (x,p,t,τ2)
( π ) ( P )
We stillneedanexpressionforthe timeevolutionofρ (x,p,t)betweenthefirstandsecondpulses. FromEqs. (23,
0
32, 40), we find
1
ρ (x,p,τ )= f(X(ξ,π,0,τ ),P(ξ,π,0,τ )) , (43a)
ee 2− 2 1 1 ξ=X(x,p,τ1,τ2),π=P(x,p,τ1,τ2)−~k
1
ρ (x,p,τ )= f(X(x,p,0,τ ),P(x,p,0,τ )), (43b)
gg 2− 2 2 2
i
ρ (x,p,τ )=− exp i k·ξ−δ(1)τ −φ
0eg 2− 2 12 1 1
×f(X(ξ,π,0,τ ),P(ξ,π,0,nτ ))}h (cid:16) (cid:17)i (43c)
1 1 ξ=X(x,p,τ1,τ2),π=P(x,p,τ1,τ2)−~k/2
At time τ the π pulse acts which, according to Eqs. (30), transforms these density matrix elements at time τ
2 2−
into
1
ρ (x,p,τ ) = f(X(x,p−~k,0,τ ),P(x,p−~k,0,τ )), (44a)
ee 2+ 2 2 2
1
ρ (x,p,τ ) = f(X(ξ,π,0,τ ),P(ξ,π,0,τ )) (44b)
gg 2+ 2 1 1 ξ=X(x,p+~k,τ1,τ2),π=P(x,p+~k,τ1,τ2)−~k
i
ρ (x,p,τ ) = exp i k·(2x−ξ)−2δ(2)τ +δ(1)τ −2φ +φ
0eg 2+ 2 12 2 12 1 2 1
×nf(X(nξ,hπ,0,τ ),P(ξ,π,0,τ ))} io , (44c)
1 1 ξ=X(x,p,τ1,τ2),π=P(x,p,τ1,τ2)−~k/2
~2
ρ (x,p,τ ) = − exp i k·(2x−ξ)−2δ(2)τ +δ(1)τ −2φ +φ
Qeg 2+ 48 12 2 12 1 2 1
×f(Xn(ξ,πn,0h,τ ),P(ξ,π,0,τ ))} io
1 1 ξ=X(x,p,τ1,τ2),π=P(x,p,τ1,τ2)−~k/2
τ2
×k k k dt[χ′ (ξ,t)
u v w ikl
Zτ1
×∂ X (ξ,π,τ ,t)∂ X (ξ,π,τ ,t)∂ X (ξ,π,τ ,t)] . (44d)
πi u 1 πk v 1 πl w 1 ξ X
= (x,p,t,τ2)
( π ) ( P )
The next step is to calculate the Q−term corrections in time interval [τ ,τ ]. Each density matrix element in
2 3
Eqs. (44) produces a Q−term correction. However, the diagonal matrix elements given by Eqs. (44a, 44b) contain
no rapidly oscillating phase factors in momentum space allowing us to neglect their Q−term corrections. Moreover,
Eq. (44d) is already linear in Q and can produce only higher order corrections that we neglect in this work. As a
consequence,we needconsider only the Q−termcorrectionproduced by the coherence in Eq. (44c), which we denote
10
as ρ′ . From Eq. (26) we find
Qeg
~2 τ3
ρ′ (x,p,τ ) = i dt{χ′ (ξ,t)∂ ∂ ∂
Qeg 3− 48 ikl πi πk πl
Zτ2
×exp{i[k·(2X(ξ,π,τ ,t)−X(ξ,π,τ ,t))
2 1
− 2δ(2)τ +δ(1)τ −2φ +φ
12 2 12 1 2 1
io ~k
×f X X(ξ,π,τ ,t),P(ξ,π,τ ,t)− ,0,τ
1 1 1
2
(cid:18) (cid:18) (cid:19)
~k
P X(ξ,π,τ ,t),P(ξ,π,τ ,t)− ,0,τ (45)
1 1 1
2 ξ X
(cid:18) (cid:19)(cid:19)(cid:27) = (x,p,t,τ3)
( π ) ( P )
where we used the multiplication law (40),
X X
(X(ξ,π,τ ,t),P(ξ,π,τ ,t),τ ,τ )= (ξ,π,τ ,t). (46)
P 2 2 1 2 P 1
(cid:26) (cid:27) (cid:26) (cid:27)
In Eq. (45) the differentiation over momentum π is carried out only for the Doppler phase factors. After differen-
tiation, we apply the multiplication law two more times to the phase factor and distribution f, namely
X X
(ξ,π,τ ,t) = (x,p,τ ,τ ) (47)
P i ξ X P i 3
(cid:26) (cid:27) = (x,p,t,τ3) (cid:26) (cid:27)
( π ) ( P )
for i=1,2, and find that these terms become t−independent. As a result one gets for Q−termρ′ before the third
Qeg
pulse,
~2
ρ′ (x,p,τ ) = − exp i k·(2X(x,p,τ ,τ )−ξ)−2δ(2)τ +δ(1)τ −2φ +φ
Qeg 3− 48 2 3 12 2 12 1 2 1
×f(Xn(ξ,πn,0h,τ ),P(ξ,π,0,τ ))} io
1 1 ξ=X(x,p,τ1,τ3),π=P(x,p,τ1,τ3)−~k/2
τ3 ∂X (ξ,π,τ ,t) ∂X (ξ,π,τ ,t)
×k k k dt χ′ (ξ,t) u 1 −2 u 2
u v w ikl ∂π ∂π
Zτ2 (cid:26) (cid:20) i i (cid:21)
∂X (ξ,π,τ ,t) ∂X (ξ,π,τ ,t)
v 1 v 2
× −2
∂π ∂π
(cid:20) k k (cid:21)
∂X (ξ,π,τ ,t) ∂X (ξ,π,τ ,t)
w 1 w 2
× −2 . (48)
(cid:20) ∂πl ∂πl (cid:21)(cid:27) ξ = X (x,p,t,τ3)
( π ) ( P )
The value of ρ (x,p,τ ) will depend both on ρ′ (x,p,τ ) and ρ (x,p,τ ). In other words, we must also
ee 3+ Qeg 3− 0 3−
calculate the time evolution of ρ (x,p,t) between the second and third pulses. Applying Eq. (23), we find
0
1
ρ (x,p,τ ) = f(X(ξ,π,0,τ ),P(ξ,π,0,τ )) ; (49a)
ee 3− 2 2 2 ξ=X(x,p,τ2,τ3),π=P(x,p,τ2,τ3)−~k
1
ρ (x,p,τ ) = f(X(ξ,π,0,τ ),P(ξ,π,0,τ )) (49b)
gg 3− 2 1 1 {ξ=X(X(x,p,τ2,τ3),P(x,p,τ2,τ3)+~k,τ1,τ2),
π=P(X(x,p,τ2,τ3),P(x,p,τ2,τ3)+~k,τ1,τ2)−~k}
i
ρ (x,p,τ ) = exp i k·(2X(x,p,τ ,τ )−ξ)−2δ(2)τ +δ(1)τ −2φ +φ
0eg 3− 2 2 3 12 2 12 1 2 1
×fn(X(nξ,hπ,0,τ ),P(ξ,π,0,τ ))} io, (49c)
1 1 ξ=X(x,p,τ1,τ3),π=P(x,p,τ1,τ3)−~k/2
~2
ρ (x,p,τ ) = − exp i k·(2X(x,p,τ ,τ )−ξ)−2δ(2)τ +δ(1)τ −2φ +φ
Qeg 3− 48 2 3 12 2 12 1 2 1
×f(Xn(ξ,πn,0h,τ ),P(ξ,π,0,τ ))} io
1 1 ξ=X(x,p,τ1,τ3),π=P(x,p,τ1,τ3)−~k/2
τ2
×k k k dt[χ′ (ξ,t)∂ X (ξ,π,τ ,t)∂ X (ξ,π,τ ,t)
u v w ikl πi u 1 πk v 1
Zτ1
×∂ X (ξ,π,τ ,t)] . (49d)
πl w 1 ξ X
= (x,p,t,τ3)
( π ) ( P )
