2418
J. Opt. Soc. Am. B / Vol. 28, No. 10 / October 2011
Stoner et al.
Analytical framework for dynamic light pulse atom
interferometry at short interrogation times
Richard Stoner,1,* David Butts,1,2 Joseph Kinast,1 and Brian Timmons1
1
C. S. Draper Laboratory, Inc., 555 Technology Square, Cambridge, Massachusetts 02139, USA
2
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology,
77 Massachusetts Avenue, 33-207, Cambridge, Massachusetts 02139, USA
*Corresponding author: [email protected]
Received May 10, 2011; revised August 5, 2011; accepted August 5, 2011;
posted August 9, 2011 (Doc. ID 147352); published September 13, 2011
High-precision inertial sensing demonstrations with light pulse atom interferometry have typically used
Raman pulses having durations orders of magnitude shorter than the dwell time between interferometer pulses.
Environmentally robust sensors operating at high-bandwidth will be required to operate at short (millisecond
scale) dwell times between Raman pulses. In such an operational mode, the Raman pulse duration becomes
an appreciable fraction of the dwell time between pulses. In addition, high-precision inertial sensing applications
have typically been demonstrated in mildly dynamic or nondynamic environments having low rate of change of
inertial input, ensuring that applied Raman pulses satisfy the Raman resonance condition. Application of nonresonant pulses will be inevitable in sensors registering time-varying inertial input. We present a diagrammatic
technique for calculation of atomic output state populations for multipulse atom optics manipulations that
explicitly account for the effects of finite pulse duration and finite Raman detuning effects on the laser-induced
atomic phase. We analyze several atom interferometer sequences. We report accelerometer and gyroscope phase
evolution for fixed Raman laser frequency difference incorporating corrections in powers of the ratio of pulse
duration to time interval between interferometer pulses. Our accelerometer result agrees with other published
results. © 2011 Optical Society of America
OCIS codes: 020.0020, 020.1335, 120.3180.
1. INTRODUCTION
Raman pulse atom interferometry has been successfully
applied to high-precision inertial sensing applications (see,
e.g., [1–7]). High-precision inertial sensing demonstrations
have typically used Raman pulses with duration orders of
magnitude shorter than the interrogation time between
interferometer pulses. Toward the goal of developing environmentally robust, high-bandwidth sensors, several recent
experiments have comprised Raman pulse atom interferometers operating at short (millisecond scale) interrogation
times [8–10]. In such an operational mode, the Raman pulse
duration becomes an appreciable fraction of the dwell time
between pulses, in the sense that it must be accounted
for in calculating the atomic phase accumulated during an
interferometer pulse sequence. Also, high-precision inertial
sensing applications have typically been demonstrated in
mildly dynamic or nondynamic environments having low rate
of change of inertial input, ensuring that Raman pulses are
applied at the Raman resonance condition. Application of
nonresonant pulses will be inevitable in sensors registering
variable inertial input. Nonresonant Raman pulses in an inertially sensitive interferometer sequence result in loss of interferometer contrast and alterations to interferometer phase.
Theoretical treatments of Raman pulse interferometry have
provided general derivations for the Raman pulse operator,
but descriptions of multiple-pulse interferometer sequences
have been restricted to the short pulse limit, near the Raman
resonance condition [11–13]. In [14], Peters presents a density
matrix treatment of a sequence of two Raman pulses that accounts for finite pulse duration and nonzero Raman detuning
0740-3224/11/102418-12$15.00/0
(evolving linearly in time), to first order in a perturbation
expansion (the solutions are valid for Raman detunings much
smaller than the effective two-state Rabi rate). Peters then
concatenates two such sequences to model a three-pulse
sequence in which the second pulse duration is twice that
of the third and first pulses. The phase evolution under acceleration is calculated for a Doppler sensitive Raman pulse
interferometer operating at fixed Raman laser frequency difference; the result accounts for the effect of finite pulse duration, valid to first order in the ratio of pulse duration to time
interval between pulses.
In this paper, we present a comprehensive extension of
existing analytical frameworks, which enables modeling of
Raman pulse inertial sensors in dynamic environments. Our
analysis employs a simple method for computing interferometer output probability amplitudes using concatenated
scattering diagrams. This technique enables easy formulation
of models comprising many Raman pulses. We will illustrate
the method with two examples: a two-pulse atomic clock and
an inertially sensitive three-pulse atom interferometer using
counterpropagating Raman beams in accelerometer and
gyroscope implementations. We report accelerometer and
gyroscope scale factors (i.e., proportionality of interferometer
phase to inertial input) for interferometers operating with
fixed Raman laser frequency difference that include corrections to second order in the ratio of pulse duration to time
interval between interferometer pulses. Our accelerometer
result agrees with the analysis of Peters [14] to first order in
the ratio of the pulse duration to time interval between pulses.
© 2011 Optical Society of America
Stoner et al.
We preface our treatment with a brief sketch of Raman
pulse physics and a discussion of finite atomic coherence.
The stimulated Raman pulse interaction is a two-photon process that is depicted in Fig. 1 for a simplified three-level atom
(e.g., an atom starting in state jgi absorbs a photon with frequency ωA from Raman laser A and then emits a photon with
frequency ωB via stimulated emission to Raman laser B). It is
assumed that the frequencies ωB and ωA are far detuned from
the quantum transitions jei⇔jii and jgi⇔jii, respectively, so
that the population of excited state jii is small and spontaneous emission from jii can be neglected. In the process,
the internal quantum state of the atom changes from jgi to
a coherent superposition of jgi and jei. The process is coherent in nature, and the jei probability amplitude acquires a
change in momentum proportional to the difference of the
wave vectors associated with the two electric fields: e.g.,
the momentum transfer associated with a Raman light pulse
is ℏkeff ¼ ℏðkA − kB Þ, where kA and kB are the wave vectors
associated with the ωA and ωB electric field frequency components, respectively, and ℏ is the reduced Planck constant.
In the following, we consider the case of lasers tuned to the
cesium D2 transition. For the case of counterpropagating Raman beams (the Doppler sensitive case), the magnitude of the
momentum change is approximately twice the spontaneous
recoil momentum; for copropagating beams (the Dopplerinsensitive case), the momentum change is about five orders
of magnitude smaller. The velocities imparted by co- and
counterpropagating beam Raman light pulses are 92 nm=s
and 7 mm=s, respectively, for Cs atoms. The momentum
change in the counterpropagating Raman beam case provides
the basis for Raman light pulse atom optics. We frequently refer to π and π=2 pulses: for an atom that begins in the pure
state jgi, a π=2 Raman light pulse leaves the atom in an equal
superposition of the jgi and jei states. π=2 pulses act as atom
optics beam splitters, the momentum difference between jgi
and jei probability amplitudes effecting phase separation of
Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. B
2419
initially colocated wavefronts. Likewise, applying a π pulse
to a pure state jgi will yield an atom in the pure state jei, with
a corresponding momentum change. π pulses act as mirrors.
We account for a continuum of atomic coherence scenarios. Our model is based on plane wave interferometer theory
[15]. We assume that interferometer input comprises a thermal velocity distribution of plane wave momentum states,
and it is presumed that probability densities will be averaged
over thermal distributions. Although atoms are not localized
in this picture, this approach still provides proper accounting
for interferometer output wavefront coherence for any value
of time delay between interferometer pulses and for both
Doppler-sensitive and non-Doppler-sensitive Raman beam
configurations. This is most readily seen by noting that the
assumption of a thermal plane wave distribution is essentially
equivalent to the assumption of thermally localized atomic
wave packets. We elaborate on issues of atomic coherence
via the examples in Section 4.
Our model interferometer sequences will, of course, account for Raman laser-induced atomic phase. We have chosen
the reference frame comoving with the undeflected ground
state of the atom in which to compute the action of the Raman
lasers. In this frame, only propagation phase associated with
internal state evolution accrues; inertial effects are impressed
upon the laser phase [15]. Use of the comoving frame also
forces the assumption that inertial effects are constant over
wave packet separation distances realized in the interferometer sequence (i.e., no gravity gradients); however, this is an
appropriate assumption for high-bandwidth operation. Assuming zero inertial input spatial variation ensures that interferometer separation phase vanishes [15]. Raman fields are
presumed to be uniform over characteristic wave packet
separation distances, likewise a condition realized at short
interrogation times.
We begin in Section 2 by summarizing the derivation of the
Raman pulse operator; in the derivation, we carefully describe
composition of interferometer pulse sequences. In Section 3,
we use a diagrammatic technique to derive the output probability amplitudes for two- and three-pulse interferometers.
Section 4 presents an analysis of a two-pulse clock, a threepulse accelerometer, and a three-pulse gyroscope, in which
we account for finite Raman pulse duration effects in powers
of the ratio of pulse duration to time interval between pulses.
We show agreement with previous results in the cases of the
clock and accelerometer, and present a new result for the
gyroscope case.
2. DERIVATION OF OPERATORS
Fig. 1. Energy level structure for a simplified three-level atom undergoing a stimulated Raman transition. The magnitude of the detunings
Δ and δ are exaggerated for clarity.
In this section, we derive operators describing the evolution of
a three-level atom in both the presence and absence of a twophoton Raman manipulation. Following the operator derivations, we carefully describe the composition of interferometer
pulse sequences. The derivation of the Raman pulse operator
is provided here for completeness; the simplifying assumptions incorporated into the derivation are consistent with
previous treatments [11–14]. The principal simplifying assumption is that adiabatic elimination can be applied to the
excited state to yield an effective two-state problem. We have
assumed that only a single intermediate excited state is accessible to the ground state atoms. Also, the atom is treated quantum mechanically, but the electric field will be represented
2420
J. Opt. Soc. Am. B / Vol. 28, No. 10 / October 2011
classically. Only electric dipole transitions are permitted, and
we neglect the impact of spontaneous emission, which is assumed to be negligible for conditions of interest. We assume
that there is no cross-coupling between the quantum states
and the two electric field frequency components: i.e., each
electric field is assumed to have a negligible impact on one
of the two allowed electric dipole transitions. Although not
considered here, the inclusion of cross-coupling terms introduces corrections to the effective Rabi rate as well as additional terms accounting for AC Stark shifts arising from
off-resonant electric field components [16]. Finally, we apply
Stoner et al.
the rotating wave approximation in which very high frequency
time dependence is neglected.
We consider a three-state system coupled by two coherent
laser fields, as depicted in Fig. 1. Unless otherwise noted, the
expressions presented below apply to a reference frame corotating with the Raman pulse drive field in which the undeflected atomic ground state is at rest. Parameter definitions
are provided in Table 1. (While a number of parameters are
also defined in the text, the reader is advised that some parameters are defined only in Table 1.) For our three-level
system, the Schrödinger equation takes the form
Table 1. Definitions of Parameters and Variables
Parameter
cc
m
^eA , ^eB
εA , εB
ωA ðtÞ, ωB ðtÞ
~ A ðtÞ, ϕ
~ B ðtÞ
ϕ
ϕA , ϕB
ϕ
ϕj
kA , k B
keff
ΩA , ΩB
q
ωi
δðtÞ
Definition/Description
Complex conjugate of the preceding addend
Atomic mass
Complex polarization vectors for the Raman laser fields
Complex amplitudes of Raman laser fields
Time-dependent optical frequencies for
R lasers A and B 0
R
~ A ðtÞ ¼ t ½ωA ðt0 Þ − kA · zðt
~ B ðtÞ ¼ t ½ωB ðt0 Þ − kB · ℏkeff
_ Þ^ez dt0 þ ϕA , ϕ
Time-dependent laser phases ϕ
0
0
m
0
0
_ Þ^ez dt þ ϕB
− kB · zðt
Laser optical phases (not time-dependent): ϕA ¼ ϕA;o − kA · ^ez zðt ¼ 0Þ, ϕB ¼ ϕB;o − kB · ^ez zðt ¼ 0Þ
Effective Rabi phase: ϕ ≡ ϕA − ϕB ¼ ðϕA;o − ϕB;o Þ − keff · ^ez zðt ¼ 0Þ
Effective Rabi phase for the jth Raman pulse: ϕj ≡ ðϕA − ϕB Þj ¼ ðϕA;o − ϕB;o Þj − keff · ^ez zðt ¼ 0Þ
Propagation vectors of lasers A and B
keff ≡ kA − kB , vector difference of the laser propagation vectors
Complex Rabi frequencies associated with laser fields coupling the lower ground state level to the
excited state and coupling the upper ground state level to the excited state:
ΩA ¼ −qhgj^eA εA · rjii ¼ −jΩA j expðiϕA Þ, ΩB ¼ −qhej^eB εB · rjii ¼ −jΩB j expðiϕB Þ
Elementary charge; the absolute value of the electric charge of a single electron
Energy of the excited state in frequency units
h
i
ℏk2
ℏ
_
Combined Raman laser detuning: δðtÞ ¼ ½ωA ðtÞ − ωB ðtÞ − ωo þ 2meff − 2m
ðk2A − k2B Þ þ keff · ^ez zðtÞ
δj ðtÞ
Combined Raman laser detuning for the jth Raman pulse: δj ðtÞ ¼ ½ωA ðtÞ − ωB ðtÞj
i
h
ℏk2eff;j
ℏ
_
k2A;j − k2B;j þ keff;j · ^ez zðtÞ
− ωo þ 2m
− 2m
Δ
Detuning of the mean laser frequency from the difference between the perturbed ground state energy and
h 2
io
n
ℏk
B ðtÞ
_
the excited state energy: Δ ¼ ωA ðtÞþω
− ωi − 12 2meff − ðkA þ kB Þ · ^ez zðtÞ
2
AC
ΩAC
g , Ωe
δAC
Ωeff
Ωj ðtÞ
θj ðtÞ
2
AC
2
AC Stark shifts of the hyperfine ground states: ΩAC
g ¼ jΩA j =ð4ΔÞ, Ωe ¼ jΩB j =ð4ΔÞ
AC
Differential AC Stark shift: δAC ¼ ΩAC
e − Ωg
Effective Rabi frequency of the Raman transition: Ωeff ¼ ΩA ΩB =ð2ΔÞ ¼ jΩA jjΩB j expðiϕÞ=ð2ΔÞ
Generalized effective Rabi frequency of the jth Raman pulse: Ωj ðtÞ ¼ fjΩeff;j j2 þ ½δAC;j − δj ðtÞ2 g1=2
State space mixing angle parameter of the jth Raman pulse: cos½θj ðtÞ ¼ ½δAC;j − δj ðtÞ=Ωj ðtÞ,
sin½θj ðtÞ ¼ jΩeff;j j=Ωj ðtÞ
^ j ðtÞ ¼ cos½θj ðtÞ^ez þ sin½θj ðtÞ×
Raman pulse operator unit vector for the jth Raman pulse: Ω
ðcos ϕj ^ex þ sin ϕj ^ey Þ
^ j ðtÞ
Raman pulse operator for the jth Raman pulse: Ωj ðtÞ ¼ Ωj ðtÞΩ
C j ≡ cos½τj Ωj ðM j Þ=2 þ i cos½θj ðM j Þ sin½τj Ωj ðM j Þ=2
S j ≡ expðiϕj Þ sin½θj ðM j Þ sin½τj Ωj ðM j Þ=2
0 1
0 −i
1 0
Pauli spin matrix operator: σ ¼ σ x ^ex þ σ y ^ey þ σ z ^ez , σ x ¼
, σy ¼
, σz ¼
1 0
i 0
0 −1
1 0
Identity operator, 1 ¼
0 1
ϕint ≡ Φ2→3 − Φ1→2
ϕloss ≡ Φ2→3 þ Φ1→2
T p ≡ T − τ2 =2
Temporal midpoint of jth pulse: M j ≡ tj þ τj =2
R t þτ
Rt
0
0
Total integrated phase between pulses j and j þ 1: Φj→jþ1 ≡ 0j j dt0 ½δjþ1 ðt0 Þ − δj ðt0 Þ þ tjjþ1
þτj dt δjþ1 ðt Þ
iτ
ð0Þ
Raman pulse evolution operator for the jth pulse: Rj ≡ exp − 2j Ωj · σ
^ j ðtÞ
Ω
Ωj ðtÞ
Cj
Sj
σ
1
ϕint
ϕloss
Tp
Mj
Φj→jþ1
Rj
ð0Þ
Ωj
ð0Þ
Midpulse value of the generalized effective Rabi rate for the jth pulse: Ωj ≡ ½δAC − δðM j Þ^ez þ
Ωeff ðcos ϕ^ex þ sin ϕ^ey Þ
Stoner et al.
Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. B
2
3
3
Ψ
Ψ
e
e
6
7
7
∂6
6
7
7
H6
4 Ψi 5 ¼ iℏ ∂t 4 Ψi 5;
Ψg
Ψg
2
ð1Þ
where the Hamiltonian H ¼ H 0 þ H 1 consists of a term (H 0 )
describing rest frame energies arising from hyperfine splitting
and static magnetic fields and an interaction term (H 1 ). In matrix form, the components of the Hamiltonian are given by
ωo =2 0
H0 ¼ ℏ 0
ωi
0
0
2
0
6 H1 ¼ 6
4 H 1B
0
H 1B
0
H 1A
0 ;
0
−ωo =2
0
ð2Þ
7
H 1A 7
5;
ð3Þ
0
where
H 1A ¼
ℏ
~ A ðtÞ;
jΩ j exp½iϕ
2 A
ð4Þ
H 1B ¼
ℏ
~ B ðtÞ:
jΩ j exp½iϕ
2 B
ð5Þ
In the preceding expressions, we have established an energy
origin at the average energy of the ground states (excluding
any recoil energy). Our chosen reference frame falls freely
with the undeflected atomic ground state. In the absence of
gravity gradients, this represents an inertial frame of reference. At the time origin t ¼ 0, the reference frame is coincident with the instrument frame; the laser sources are assumed
to be at rest in the instrument frame. (Note that t ¼ 0 is the
time of atom release into free propagation and not the time at
which the Raman pulse begins.) The classical particle trajectory zðtÞ describes the motion of the origin of the reference
frame relative to the instrument frame. In Eqs. (4) and (5),
we present the phases
~ A ðtÞ ¼
ϕ
Z
t
_ 0 Þ^ez dt0 þ ϕA ;
½ωA ðt0 Þ − kA · zðt
ð6Þ
0
~ B ðtÞ ¼
ϕ
Z t
0
ωB ðt0 Þ − kB ·
ℏkeff
_ 0 Þ^ez dt0 þ ϕB :
− kB · zðt
m
~ B ðtÞ undergoes similar simplification, but also includes a
ϕ
phase contribution proportional to keff , which accounts for
momentum recoil effects. We express the phases more generally as integrals of time-dependent parameters to accommodate scenarios in which the laser frequencies have
time-varying profiles. This generalization is essential, as atomic kinematics are registered as time-dependent Doppler shifts
in laser frequencies for our chosen reference frame. Finally,
we also note that ΩA and ΩB are complex quantities in which
the phases depend on the atom’s initial position and velocity.
We proceed by applying the following transformation:
ð7Þ
The parameters ϕA ≡ ϕA;o − kA · ^ez zðt ¼ 0Þ and ϕB ≡ ϕB;o −
kB · ^ez zðt ¼ 0Þ capture the laser phase at position zðt ¼ 0Þ
at the time t ¼ 0. We emphasize that the trajectory zðtÞ does
not include position variation arising from photon recoil deflections: recoil momentum effects are accounted for in the
Doppler shift of laser B in Eq. (5). The expressions for the
~ A ðtÞ and ϕ
~ B ðtÞ are most readily understood by conphases ϕ
sidering the limit where the laser frequencies are timeinvariant and the wave vectors are parallel to ^ez . In that case,
~ A ðtÞ → ωA t − kA zðtÞ þ ϕA;o reduces to the familiar phase exϕ
pression for a traveling wave. Under the same conditions,
2
n
o3
~ A ðtÞ − ϕ
~ B ðtÞ − ðϕA − ϕB Þ
be ðtÞ exp − 2i ½ϕ
Ψ ðtÞ
6
7
6 e 7 6
n
o7
6
7 6
i
~ A ðtÞ þ ϕ
~ B ðtÞ − ðϕA þ ϕB Þ 7
6 Ψi ðtÞ 7 ¼ 6 bi ðtÞ exp − 2 ½ϕ
7: ð8Þ
4
7
5 6
n
o 5
4
Ψg ðtÞ
~ A ðtÞ − ϕ
~ B ðtÞ − ðϕA − ϕB Þ
bg ðtÞ exp 2i ½ϕ
2
3
2421
3
Following this transformation, the three-level system can
be reduced to an effective two-level system via adiabatic elimination (see, e.g., [17]). Provided that the laser detuning Δ is
much larger than the effective Rabi rate describing the transfer of population between the ground states, we can argue that
b_ i ðtÞ ≈ 0. One can then solve for bi ðtÞ in terms of bg ðtÞ and
be ðtÞ, and eliminate the intermediate excited state from the
equations of motion. We introduce an additional coordinate
transformation
"
be ðtÞ
bg ðtÞ
#
#
"
AC
ce ðtÞ
ΩAC
e þ Ωg
¼ exp −i
;
ðt − tj Þ
2
cg ðtÞ
ð9Þ
2
AC
2
where ΩAC
e ¼ jΩB j =ð4ΔÞ and Ωg ¼ jΩA j =ð4ΔÞ represent the
AC Stark shifts of the upper and lower atomic ground states,
respectively. As noted earlier, these expressions for the AC
Stark shifts neglect energy level shifts due to cross coupling
from the far detuned electric field components. In Eq. (9), we
have included a subscript j in preparation for the treatment of
sequences involving multiple Raman pulses. Throughout this
paper, the j subscript denotes parameters associated with the
jth pulse of multiple-pulse sequences. Following these transformations, the equations of motion can be expressed in
concise form using matrix operators
"
"
#
#
ce ðtÞ
d ce ðtÞ
i
¼ − ΩðtÞ · σ
:
dt cg ðtÞ
2
cg ðtÞ
ð10Þ
In the most general case, the generalized effective Rabi rate
ΩðtÞ, which depends on the Raman detuning,
δðtÞ ¼ ½ωA ðtÞ − ωB ðtÞ
ℏk2
ℏ 2
_
ðkA − k2B Þ þ keff · ^ez zðtÞ
− ωo þ eff −
;
2m 2m
ð11Þ
will be time dependent, and Eq. (10) can be solved numerically or via perturbation theory. We will exhibit a perturbation
expansion and display a solution to first order in that expansion. The zeroth-order solution is obtained by solving Eq. (10)
in the case where ΩðtÞ and δðtÞ are constant, and is simply the
exponentiation of the matrix operator that appears on the
2422
J. Opt. Soc. Am. B / Vol. 28, No. 10 / October 2011
Stoner et al.
right-hand side of the equation [18]. In that case, the system
state at the conclusion of a pulse with duration τj beginning at
time tj is
"
ce ðtj þ τj Þ
#ð0Þ
cg ðtj þ τj Þ
iτj
¼ exp − Ω · σ
2
" c ðt Þ #
e j
cg ðtj Þ
:
Ωj ðtÞ ¼
c ðt ¼ t Þ ð0Þ
e
j
cg ðt ¼ tj Þ
c ðt ¼ t Þ ð1;2;3;…Þ
e
j
ð12Þ
cg ðt ¼ tj Þ
This solution is exact when the Raman detuning is time independent, which occurs in the absence of inertial input, or in
the operational mode in which the laser difference frequency
is chirped to nullify any motional effects.
We now consider an approximate solution to Eq. (10) in the
case where the Raman detuning and Rabi rate are time dependent. We begin by Taylor expanding the generalized effective
Rabi rate about the temporal midpoint of the jth pulse
1 €
ðt − M j Þ^ez − δ
ðt − M j Þ2^ez þ …;
t¼M j
2 t¼M j
− δ_ ð0Þ
Ωj
in which we specify the initial conditions
"
ce ðtÞ
#
cg ðtÞ
"
¼
ce ðtÞ
#ð0Þ
cg ðtÞ
"
þ
ce ðtÞ
#ð1Þ
"
þ
cg ðtÞ
ce ðtÞ
#ð2Þ
cg ðtÞ
þ…
ð14Þ
and substitute Eqs. (13) and (14) into Eq. (10):
("
d
dt
ce ðtÞ
#ð0Þ
"
þ
ce ðtÞ
#ð1Þ
"
þ
ce ðtÞ
#ð2Þ
)
þ…
cg ðtÞ
cg ðtÞ
cg ðtÞ
i
1 €
ð0Þ
_
¼ − Ωj − δ ðt − M j Þ^ez − δ
ðt − M j Þ2^ez þ … · σ
t¼M j
2
2 t¼M j
#ð0Þ "
#ð1Þ "
#
)
("
ce ðtÞ
ce ðtÞ ð2Þ
ce ðtÞ
þ
þ
þ… :
ð15Þ
×
cg ðtÞ
cg ðtÞ
cg ðtÞ
Equation (15) can be expressed as a sum of solutions to the
following equations:
"
#ð0Þ
"
#ð0Þ
ce ðtÞ
d ce ðtÞ
i ð0Þ
¼− Ωj ·σ
;
dt cg ðtÞ
2
cg ðtÞ
"
#ð1Þ
#
"
ce ðtÞ ð1Þ
d ce ðtÞ
i ð0Þ
¼− Ωj ·σ
dt cg ðtÞ
2
cg ðtÞ
"
#
ce ðtÞ ð0Þ
i _ þ δ
ðt−M j Þ^ez ·σ
;
2 t¼M j
cg ðtÞ
"
#ð2Þ
"
#
ce ðtÞ ð2Þ
d ce ðtÞ
i ð0Þ
¼− Ωj ·σ
dt cg ðtÞ
2
cg ðtÞ
"
#
ce ðtÞ ð1Þ
i _ þ δ
ðt−M j Þ^ez ·σ
2 t¼M j
cg ðtÞ
"
#
ce ðtÞ ð0Þ
i €
2^
þ δ
ðt−M j Þ ez ·σ
;etc:;
4 t¼M j
cg ðtÞ
c ðt ¼ t Þ e
j
cg ðt ¼ tj Þ
0
¼
:
0
;
ð17Þ
[We note that the sum of Eqs. (16), with initial conditions (17),
is equivalent to Eq. (15).] Equations (16) are a hierarchical
perturbation expansion in which lower-order solutions
comprise inhomogeneities for equations of higher order. The
solution to the first-order equation is
ce ðtÞ
cg ðtÞ
ð1Þ
Z
t
¼
tj
i
ð0Þ
dt0 exp − ðt − t0 ÞΩj · σ
2
i_
· δðM
ez ðt0 − M j Þ · σ
j Þ^
2
i
ce ðtj Þ
ð0Þ
;
· exp − ðt0 − tj ÞΩj · σ
cg ðtj Þ
2
ð13Þ
where the time derivatives are evaluated at M j ≡ tj þ τj =2 and
we define the midpulse value of the generalized effective Rabi
ð0Þ
rate Ωj ≡ ½δAC − δðM j Þ^ez þ Ωeff ðcos ϕ^ex þ sin ϕ^ey Þ. We now
express the solution of Eq. (10) as a perturbation expansion
¼
ð18Þ
as can be shown by direct substitution of Eq. (18) into the firstorder equation of Eqs. (16). Although we do not undertake it
here, this process can be continued to account for higherorder time derivatives in the expansion of the Raman
detuning.
The output state resulting from application of the jth
Raman pulse to first order in the perturbation expansion is
the sum of results (12) and (18) evaluated at t ¼ tj þ τj :
c ðt þ τ Þ e j
j
cg ðtj þ τj Þ
≃
c ðt þ τ Þ ð0Þ
e j
j
þ
c ðt þ τ Þ ð1Þ
e j
j
cg ðtj þ τj Þ
cg ðtj þ τj Þ
iτj ð0Þ
i
_ t¼M
¼ exp − Ωj · σ
1−
δj
j
ð0Þ
2
2½Ωj 2
^ ð0Þ × ^ez Þ sinðΩð0Þ τj Þ
× σ · ðΩ
j
j
ð0Þ
−
Ωj τ j
2
ð0Þ
½cosðΩj τj Þ − 1
ð0Þ
Ωj τj
ð0Þ
sinðΩj τj Þ
2
c ðt Þ e j
ð0Þ
þ cosðΩj τj Þ − 1
;
ð19Þ
cg ðtj Þ
^ · ^ez Þ − ^ez ^ ðΩ
þ σ · ½Ω
j
j
ð0Þ
ð16Þ
ð0Þ
in which we evaluated the integral in Eq. (18). The bold 1 deð0Þ
^ ð0Þ are the magnitude
notes the identity matrix, and Ωj and Ω
j
ð0Þ
and unit vector operator, respectively, of Ωj . In the limit
ð0Þ
2
2
_
fδðM j Þτj =½2Ωj τj g ≪ 1, operator (19) is unitary, and this limit should be regarded as a condition of validity for Eq. (19).
_ j ÞFor simplicity of presentation, we do not include δðM
proportional terms and consider only the zeroth-order terms
in subsequent analyses; however, inclusion of first-order
terms and derivation of higher-order terms is straightforward.
It remains to transform the solution back to the nonrotating
frame. The solution in the nonrotating frame is
Stoner et al.
Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. B
Z t þτ
j
j
iσ
¼ exp − z
½δðt0 Þ þ ωo dt0
2 0
Ψg ðtj þ τj Þ
AC ΩAC
iτj ð0Þ
e þ Ωg
× exp −i
τj × exp − Ωj · σ
2
2
Z
Ψe ðtj Þ
iσ z tj
0
0
× exp
: ð20Þ
½δðt Þ þ ωo dt
2 0
Ψg ðtj Þ
Ψe ðtj þ τj Þ
Z
Φj→jþ1 →
δðtÞ
d be ðtÞ
b ðtÞ
¼i
:
−
σz e
bg ðtÞ
2
dt bg ðtÞ
Z t þτ
d
d
σ
be ðtd þ τd Þ
be ðtd Þ
¼ exp i z
:
dt0 δðt0 Þ
bg ðtd þ τd Þ
bg ðtd Þ
2 td
ð27Þ
The first pulse is followed by evolution in the dark
The action of the second Raman pulse beginning at time t2 and
the second period of evolution in the dark beginning at time
t2 þ τ2 are respectively described by
ð22Þ
As expected, in the absence of external interactions with the
Raman laser fields, atomic evolution is driven solely by the
internal Hamiltonian.
We now compose a sequence of Raman pulses separated by
periods of evolution in the dark. The example will include
three Raman pulses and two dark evolution intervals, but it
will be obvious how to extend the composition method to
an arbitrary number of pulses. We consider a general case that
can accommodate possible multiple Raman laser beam pairs
having different frequencies at any given time. As noted earlier, to accommodate variations in these parameters for different pulses, we enumerate parameters such as the Raman
detuning and generalized effective Rabi rate with a subscript
j that indicates the pulse number. Further, we introduce the
simplifying notation
iτj ð0Þ
Rj ≡ exp − Ωj · σ ;
ð24Þ
2
Φj→jþ1 ≡
0
tj þτj
0
0
0
dt ½δjþ1 ðt Þ − δj ðt Þ þ
Z
tjþ1
tj þτj
dt0 δjþ1 ðt0 Þ;
σ
Ψe ðt2 Þ
Ψe ðt1 þ τ1 Þ
¼ exp −i z ωo ½t2 − ðt1 þ τ1 Þ
:
Ψg ðt2 Þ
Ψg ðt1 þ τ1 Þ
2
ð28Þ
In the nonrotating frame, the operator for evolution in the
dark becomes
σ
Ψe ðtd þ τd Þ
Ψe ðtd Þ
:
ð23Þ
¼ exp −i z ωo τd
Ψg ðtd þ τd Þ DARK
Ψg ðtd Þ
2
Z
ð26Þ
Z
σ z t1 þτ1 0
Ψe ðt1 þ τ1 Þ
0
¼ exp −i
dt ½δ1 ðt Þ þ ωo R1
Ψg ðt1 þ τ1 Þ
2 0
Z t
1
σ
Ψe ðt1 Þ
:
× exp i z
dt0 ½δ1 ðt0 Þ þ ωo Ψg ðt1 Þ
2 0
ð21Þ
Since the Raman field unit vector is independent of time,
we can write the solution in terms of an integral of a timedependent detuning over a dwell time τd , which begins at
time td :
dt0 δðt0 Þ:
In the following, we consider the general case in which each
Raman beam pair is independently generated and, therefore,
has its own detuning profile δj ðtÞ. The first pulse begins at time
t1 and acts as
Atom interferometer sequences utilizing Raman pulses are
composed of two or more pulses separated by dwell times
during which the atom evolves free of any interactions with
the Raman laser beams. The free evolution operator can be
represented as a Raman pulse operator in which the effective
Rabi rate vanishes, i.e., Ωeff ¼ 0. Setting the complex Rabi
frequencies ΩA ¼ ΩB ¼ 0, one can show that in the bðtÞ basis,
the equations of motion become
tjþ1
tj þτj
2423
ð25Þ
where Rj is the Raman pulse operator for the jth pulse and
Φj→jþ1 is the total integrated phase accrued during the dwell
time between pulses j and j þ 1. In the case where all Raman
pulses are produced by the same laser beam pair, the total
integrated phase simplifies to
Z t þτ
2
2
σ
Ψe ðt2 þ τ2 Þ
¼ exp −i z
dt0 ½δ2 ðt0 Þ þ ωo R2
Ψg ðt2 þ τ2 Þ
2 0
Z t
2
σ
Ψe ðt2 Þ
;
× exp i z
dt0 ½δ2 ðt0 Þ þ ωo Ψg ðt2 Þ
2 0
ð29Þ
σ
Ψe ðt3 Þ
Ψe ðt2 þ τ2 Þ
¼ exp −i z ωo ½t3 − ðt2 þ τ2 Þ
:
Ψg ðt3 Þ
Ψg ðt2 þ τ2 Þ
2
ð30Þ
Finally, the third Raman pulse acts according to
Z t þτ
3
3
σ
Ψe ðt3 þ τ3 Þ
dt0 ½δ3 ðt0 Þ þ ωo R3
¼ exp −i z
Ψg ðt3 þ τ3 Þ
2 0
Z t
3
σ
Ψe ðt3 Þ
:
× exp i z
dt0 ½δ3 ðt0 Þ þ ωo Ψg ðt3 Þ
2 0
ð31Þ
Combining Eqs. (27)–(31) yields
Z t þτ
3
3
σ
¼ exp −i z
dt0 ½δ3 ðt0 Þ þ ωo R3
2 0
Ψg ðt3 þ τ3 Þ
iσ z
iσ z
× exp
Φ2→3 R2 × exp
Φ1→2 R1
2
2
Z
σ z t1 0
× exp i
dt ½δ1 ðt0 Þ þ ωo 2 0
Ψe ðt1 Þ
×
:
ð32Þ
Ψg ðt1 Þ
Ψe ðt3 þ τ3 Þ
In Section 4, we also consider a two-pulse interferometer
sequence for use in clock applications. Using the same methodology as for the three-pulse sequence, we can immediately
model a two-pulse Raman sequence using Eqs. (27)–(29):
2424
J. Opt. Soc. Am. B / Vol. 28, No. 10 / October 2011
Stoner et al.
Z t þτ
2
2
σ
¼ exp −i z
dt0 ½δ2 ðt0 Þ þ ωo R2
2 0
Ψg ðt2 þ τ2 Þ
Z t
1
iσ z
σ
× exp
dt0 ½δ1 ðt0 Þ
Φ1→2 R1 × exp i z
2
2 0
Ψe ðt1 Þ
þ ωo :
ð33Þ
Ψg ðt1 Þ
Ψe ðt2 þ τ2 Þ
In nearly all circumstances the initial input amplitudes will
correspond to a pure state in which either Ψe ðt1 Þ or Ψg ðt1 Þ is
unity and the remaining state amplitude is zero. In this case,
Eqs. (32) and (33) can be simplified by noting that the initial
and final rotation operators introduce only trivial phase factors that can be neglected, i.e., for a three-pulse sequence
using pure input states,
iσ z
iσ z
¼ R3 exp
Φ2→3 R2 exp
Φ1→2
2
2
Ψg ðt3 þ τ3 Þ
Ψe ðt1 Þ
;
× R1
Ψg ðt1 Þ PURE
Ψe ðt3 þ τ3 Þ
where
Ψe ðt1 Þ
Ψg ðt1 Þ
¼
PURE
1
0
;
:
0
1
ð34Þ
ð35Þ
Likewise, a two-pulse sequence involving pure input states is
represented by
iσ z
Ψe ðt2 þ τ2 Þ
Ψe ðt1 Þ
¼ R2 exp
: ð36Þ
Φ1→2 R1
Ψg ðt2 þ τ2 Þ
Ψg ðt1 Þ PURE
2
Two- or three-pulse sequence output amplitudes can be calculated by direct multiplication of operators in Eqs. (34) and
(36). Direct multiplication becomes cumbersome for longer
pulse sequences, however. In Section 3, we present a diagrammatic technique for computing probability amplitudes that
enables ready analysis of long pulse sequences.
3. CALCULATION OF PROBABILITY
AMPLITUDES
In this section, we compute output probability amplitudes for
two- and three-pulse interferometer sequences using a novel
diagrammatic approach. The diagrammatic technique is based
on a parsing of Raman pulse action on the two pure basis spinors presented in Eq. (35). It can be shown that the action of a
single Raman pulse operator on the pure basis states yields
Rj
Rj
1
1
0
¼ C j
− iS j
;
0
0
1
ð37Þ
0
0
1
¼ Cj
− iS j
;
1
1
0
ð38Þ
where the parameters S j and C j are defined in Table 1. We
represent these relations graphically in Fig. 2. Quantum states
are represented by solid arrows, horizontal arrows correspond to the lower ground state [the second pure state listed
in Eq. (35)], while arrows with an upward slant correspond to
the upper ground state [the first pure state listed in Eq. (35)].
Fig. 2. Diagrammatic representation of a Raman pulse operator
applied at time t ¼ t1 . An initial state purely in the lower ground state
is shown on the left; a pure state beginning in the upper ground state is
shown on the right.
Vertical dashed lines are associated with Raman pulses. Pure
input states acted upon by Raman pulses emerge with output
amplitudes specified by Eqs. (37) and (38). Figure 2 depicts
this action graphically. In addition to tracking the impact of
Raman pulses, evolution in the dark must also be captured.
By inspection of Eqs. (34) and (36), we find that, during evolution in the dark, a phase exp½iðΦj→jþ1 Þ=2 accrues to the
upper ground state and exp½−iðΦj→jþ1 Þ=2 accrues to the lower state, for a dwell between the end of pulse j and the onset of
pulse j þ 1.
We can then represent a Raman interferometer pulse sequence with a series of diagrams, each constructed around
a vertical line representing the respective Raman pulse.
The first pulse comprises a single node denoting the scattering
of the input state as shown in Fig. 2. If we assume a pure initial
state in which the lower ground level is occupied, the action of
a second pulse on the time-evolved output amplitudes of the
first pulse is shown in Fig. 3. Figure 4 represents the action of
a third Raman pulse on the time-evolved outputs of the second
pulse. This procedure can be repeated ad infinitum to compose 2N output probability amplitudes for an N pulse Raman
sequence with arbitrary detunings separated by evolution in
the dark of arbitrary duration. Figures 3 and 4 comprise a
complete and general derivation of the output amplitudes
for the two- and three-pulse Raman interferometers, respectively. The net probability amplitude for each ground state
is composed as the sum of output probability amplitudes
for that state. The output interferometer probability densities
are composed for the corresponding amplitudes; these densities are then averaged over the initial velocity distribution to
obtain the interferometer response for a thermal atom cloud
or beam.
We now state probability densities for the two- and threepulse interferometer sequences in which the initial state is the
second pure state spinor listed in Eq. (35). The output amplitudes displayed in Fig. 3 are those of the two-pulse interferometer: adding the contributions to each ground state probability
amplitude yields
i
Ψe ðt2 þ τ2 Þj2-pulse ¼ −iC 1 S 2 exp − Φ1→2
2
i
ð39Þ
− iS 1 C 2 exp Φ1→2 ;
2
i
Ψg ðt2 þ τ2 Þj2-pulse ¼ C 1 C 2 exp − Φ1→2
2
i
− S 1 S 2 exp Φ1→2 ;
2
ð40Þ
Stoner et al.
Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. B
2425
i
i
Φ1→2 exp Φ2→3
2
2
i
i
− iC 1 C 2 S 3 exp − Φ1→2 exp − Φ2→3 ; ð45Þ
2
2
ðΨe Þloss ¼ −iS 1 C 2 C 3 exp
Ψg ðt3 þ τ3 Þj3-pulse ¼ ðΨg Þint þ ðΨg Þloss ;
Fig. 3. Diagrammatic illustration of the action of a second Raman
pulse on the output amplitudes of a first pulse as shown in Fig. 2.
The lower ground initial state of Fig. 2 is assumed. Input to each node
is scaled as per the rules of Fig. 2 with the resultant output amplitudes
shown. The sum of amplitudes associated with the respective ground
states yields the net probability amplitudes.
i
i
ðΨg Þint ¼ −C 1 S 2 S 3 exp − Φ1→2 exp Φ2→3
2
2
i
i
− S 1 S 2 C 3 exp Φ1→2 exp − Φ2→3 ;
2
2
ð46Þ
ð47Þ
i
i
Φ1→2 exp Φ2→3
2
2
i
i
þ C 1 C 2 C 3 exp − Φ1→2 exp − Φ2→3 : ð48Þ
2
2
ðΨg Þloss ¼ −S 1 C 2 S 3 exp
and the probability densities are
jΨe ðt2 þ τ2 Þj2-pulse j2 ¼ jC 1 j2 jS 2 j2 þ jS 1 j2 jC 2 j2
þ 2Re½C 1 S 1 C 2 S 2 expðiΦ1→2 Þ;
ð41Þ
jΨg ðt2 þ τ2 Þj2-pulse j2 ¼ jC 1 j2 jC 2 j2 þ jS 1 j2 jS 2 j2
− 2Re½C 1 S 1 C 2 S 2 expðiΦ1→2 Þ:
ð42Þ
The sum of the probabilities in Eqs. (41) and (42) is easily
shown to be unity. In Section 4, we examine these results
in greater detail.
The output amplitudes displayed in Fig. 4 are those of the
three-pulse interferometer. It is convenient to break up the
output probability amplitudes into the sum of two terms as
follows:
Ψe ðt3 þ τ3 Þj3-pulse ¼ ðΨe Þint þ ðΨe Þloss ;
ðΨe Þint
i
i
¼ −iC 1 S 2 C 3 exp − Φ1→2 exp Φ2→3
2
2
i
i
þ iS 1 S 2 S 3 exp Φ1→2 exp − Φ2→3 ;
2
2
ð43Þ
We note that, in the preceding expressions, “loss” terms feature sums of integrated phases Φj→jþ1 in the exponentials
while “int” terms are characterized by differences of the integrated phases. The output probability densities on a per atom
basis prior to thermal averaging are
jΨe ðt3 þ τ3 Þj3-pulse j2 ¼ jS 1 j2 jS 2 j2 jS 3 j2 þ jC 1 j2 jS 2 j2 jC 3 j2
− 2Re½expðiϕint ÞC 1 S 1 ðS 2 Þ2 C 3 S 3 þ jS 1 j2 jC 2 j2 jC 3 j2 þ jC 1 j2 jC 2 j2 jS 3 j2
þ 2Re½expðiϕloss ÞC 1 S 1 ðC 2 Þ2 C 3 S 3 þ 2Re½expðiϕint =2Þ
× expð−iϕloss =2ÞC 1 S 1 C 2 S 2 ðjC 3 j2 − jS 3 j2 Þ
þ 2Re½expðiϕint =2Þ expðiϕloss =2Þ
× C 2 S 2 C 3 S 3 ðjC 1 j2 − jS 1 j2 Þ;
ð44Þ
ð49Þ
jΨg ðt3 þ τ3 Þj3-pulse j2 ¼ jC 1 j2 jS 2 j2 jS 3 j2 þ jS 1 j2 jS 2 j2 jC 3 j2
þ 2Re½expðiϕint ÞC 1 S 1 ðS 2 Þ2 C 3 S 3 þ jS 1 j2 jC 2 j2 jS 3 j2 þ jC 1 j2 jC 2 j2 jC 3 j2
− 2Re½expðiϕloss ÞC 1 S 1 ðC 2 Þ2 C 3 S 3 − 2Re½expðiϕint =2Þ expð−iϕloss =2Þ
× C 1 S 1 C 2 S 2 ðjC 3 j2 − jS 3 j2 Þ
− 2Re½expðiϕint =2Þ expðiϕloss =2Þ
× C 2 S 2 C 3 S 3 ðjC 1 j2 − jS 1 j2 Þ:
Fig. 4. Diagrammatic illustration of the action of a third Raman pulse
on the output amplitudes of a second pulse as shown in Fig. 3. The
output amplitudes shown are the output probability amplitudes for the
three-pulse interferometer.
ð50Þ
The sum of the upper and lower state probability densities can
be readily shown to be unity. At this point, these results are
completely general and describe both Doppler-insensitive and
Doppler-sensitive cases. In application of thermal averaging,
we note that certain terms will be aggressively suppressed in
the Doppler sensitive case. This is because the phases ϕloss
and (ϕloss ϕint ) contain dependence on initial atom velocity,
as shown in Section 4.
2426
J. Opt. Soc. Am. B / Vol. 28, No. 10 / October 2011
Stoner et al.
4. EXAMPLES
We first consider the application of a two-pulse Dopplerinsensitive interferometer pulse sequence to a timing application. Microwave-based interrogation of the ground state
hyperfine splitting has long provided a basis for atomic timekeeping. However, Doppler-insensitive Raman transitions can
also be used, and they offer particular advantages for clocks
using laser-cooled atoms. Optical interrogation possesses inherent advantages for reduced device size and power consumption. Additionally, optical methods afford much more
localized addressing of atom clouds than would be possible
with RF interrogation.
In general, the interference term in Eqs. (41) and (42) is
zero after thermal averaging for the Doppler-sensitive case
(where jkeff j ¼ jkA j þ jkB j) for dwell times longer than a
few microseconds. (We discuss this claim more thoroughly
in the context of three-pulse interferometer sequences later.)
We therefore restrict the discussion of the two-pulse sequence
to the Doppler-insensitive case, using a single pair of Raman
beams. Both pulses are π=2 pulses.
Neglecting the position dependent contribution, the phase
term of Eqs. (41) and (42) has the form
Z
Φ1→2 ¼
t2
t1 þτ1
dt0 δðt0 Þ ¼
Z
t2
t1 þτ1
f½ωA ðt0 Þ − ωB ðt0 Þ − ωo gdt0 : ð51Þ
Additional phases arising from the Raman pulse physics can
also be estimated. We first note that the two copropagating
Raman beams are derived from the same seed laser, so that
S 1 S 2 ¼ jS 1 jjS 2 j; also, the factor C 1 C 2 contributes to the net
phase. In the absence of AC Stark shifts, for a π=2 pulse,
C j
δðM j Þ
δðM j Þτj
1
1
≅ pffiffiffi 1 þ i
≅ pffiffiffi exp i
ΩðM j Þ
ΩðM j Þτj
2
2
1
2
≅ pffiffiffi exp i δðM j Þτj ;
π
2
ð52Þ
where we have set ΩðM j Þτj ¼ π=2 and introduced the exponential by invoking the approximation δðM j Þ ≪ ΩðM j Þ. We
can then express the factor C 1 C 2 as
1
2
π
C 1 C 2 ¼ exp i ½δðM 1 Þτ1 þ δðM 2 Þτ2 for ΩðM j Þτj ¼ :
2
π
2
ð53Þ
For the example of small, constant detuning δ ≪ Ωeff and zero
AC Stark shift,
1
4
C 1 C 2 ≅ exp i δτ
for τ1 ¼ τ2 ¼ τ;
2
π
ð54Þ
so that the total phase shift is
4τ
δT 1 þ
;
πT
ð55Þ
where T ¼ t2 − ðt1 þ τ1 Þ is the dwell time between the Raman
pulses (see Fig. 5). Equation (55) is the integrated discrepancy
between the Raman frequency difference and the hyperfine
transition frequency, with a scale factor correction term to
first order in τ=T. Stabilization of the interferometer phase
locks the Raman difference frequency to the hyperfine transition. For a sequence of two near-resonant π=2 pulses at fixed
Raman difference frequency, the output probabilities are
jΨe ðt2 þ τ2 Þj2-pulse j2 ¼ jC 1 j2 jS 2 j2 þ jS 1 j2 jC 2 j2
4τ
;
þ 2jC 1 S 1 C 2 S 2 j cos δT 1 þ
πT
ð56Þ
jΨg ðt2 þ τ2 Þj2-pulse j2 ¼ jC 1 j2 jC 2 j2 þ jS 1 j2 jS 2 j2
4τ
:
− 2jC 1 S 1 C 2 S 2 j cos δT 1 þ
πT
ð57Þ
This result, including the scale factor modification [Eq. (55)]
to first order in τ=T, agrees with results previously obtained
for clocks based on direct microwave excitation [19].
We now examine both Doppler-sensitive and Dopplerinsensitive implementations of three-pulse Raman interferometers comprising a single Raman beam pair. Before starting,
we briefly note that the three-pulse π=2 − π − π=2 sequence
is conceptually analogous to the spin echo effect [20]. In nuclear magnetic resonance applications, the inclusion of a π
pulse following a π=2 pulse permits the rephasing of an ensemble of protons that have experienced dephasing due to
magnetic field inhomogeneities. In the context of Raman
pulse-based interferometers, the π pulse facilitates rephasing
of an atomic ensemble that has experienced dephasing due to
the effective spread in detuning that arises from the thermal
velocity distribution of the atom cloud.
We begin by exhibiting the initial velocity dependence of
ϕloss . Recall that ϕloss ≡ Φ2→3 þ Φ1→2 :
_ ¼ 0Þ½t2 − ðt1 þ τ1 Þ þ t3 − ðt2 þ τ2 Þ
ϕloss ¼ −keff · ^ez zðt
Z t 3
ℏk2eff
0
0
ωA ðt Þ − ωB ðt Þ − ωo þ
þ
dt0
2m
t2 þτ2
Z t 2
ℏk2eff
0
0
ωA ðt Þ − ωB ðt Þ − ωo þ
þ
dt0
2m
t1 þτ1
Z t Z t0
2
−
keff · ^ez €zðt00 Þdt00 dt0
Z
−
t1 þτ1
t3
t2 þτ2
0
Z
t0
keff · ^ez €zðt00 Þdt00
dt0 ;
ð58Þ
0
in which the velocity dependence of the detuning is written in
integral form. This initial velocity may include a deliberately
imposed launch or beam velocity, but it must also contain a
thermal velocity contribution. Even for a 0:5 μK thermal sample and a 1 msp
dwell
time T,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi in the Doppler-sensitive case, we
_ ¼ 0Þ2 iT ≈ 1:4 × 102 . The integration of the
obtain keff · ^ez hzðt
sinusoid with the velocity-dependent phase over the velocity
distribution is thus heavily suppressed. On the other hand,
even for a 120 μK sample and a 100 ms dwell
time, in ffi the
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
_ ¼ 0Þ2 iT ≈
Doppler-insensitive case, we obtain keff · ^ez hzðt
1:9 × 10−3 . Here, the sinusoid with the velocity-dependent
phase is nearly constant over the integration. Thus, we
approximate the Doppler-insensitive case with results (49)
and (50) in which we set keff ≅ 0 and all of the phasedependent terms contribute, even when there is a mismatch
Stoner et al.
Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. B
2427
This phase is either constant or zero and does not depend on
initial atom position. There are also phases associated with
the C j in the various interference terms. As in the clock case,
we consider small detunings, zero AC Stark shifts, and nearideal pulses in order to draw a correspondence with previous
results. Using Eq. (52), we can write,
1
2
C 1 C 3 ¼ exp i ½δðM 3 Þτ3 − δðM 1 Þτ1 2
π
Fig. 5. Timing diagrams for (a) two and (b) three Raman pulse
sequences.
between the first and second dwell times. For the Doppler
sensitive case, we drop all phase-dependent terms that carry
initial velocity dependence; in fact, there is no interference at
all except for closely matched dwell times. Then
jΨe ðt3 þ τ3 Þj3-pulse;DS j2 ¼ jS 1 j2 jS 2 j2 jS 3 j2 þ jC 1 j2 jS 2 j2 jC 3 j2
þ jS 1 j2 jC 2 j2 jC 3 j3 þ jC 1 j2 jC 2 j2 jS 3 j2
− ½expðiϕint ÞC 1 S 1 ðS 2 Þ2 C 3 S 3 þ cc;ð59Þ
jΨg ðt3 þ τ3 Þj3-pulse;DS j2 ¼ jC 1 j2 jS 2 j2 jS 3 j2 þ jS 1 j2 jS 2 j2 jC 3 j2
For the example of small constant detuning and zero AC Stark
shift, C 1 C 3 ≅ 1=2 for τ1 ¼ τ3 . This implies that the net total
interferometer phase for small constant detuning is zero [or
at least constant, in light of Eq. (62)].
Another scenario of interest is considered in [14], which examines the case of constant difference frequency in the presence of constant acceleration, and calculates the resultant
interferometer output. Let us compare with [14] the results
obtained from our framework. For constant acceleration,
_ ¼ zðt
_ ¼ 0Þ þ ao t] and constant
a0 along the ^ez axis [i.e., zðtÞ
difference frequency,
ℏk2
_ 3Þ
δðM 3 Þ − δðM 1 Þ ¼ ðωA − ωB Þ − ωo þ eff − keff · ^ez zðM
2m
ℏk2
_ 1Þ
− ðωA − ωB Þ þ ωo þ eff þ keff · ^ez zðM
2m
¼ −keff ao ðτ þ 2TÞ:
þ jS 1 j2 jC 2 j2 jS 3 j3 þ jC 1 j2 jC 2 j2 jC 3 j2
þ ½expðiϕint ÞC 1 S 1 ðS 2 Þ2 C 3 S 3 þ cc;ð60Þ
in which we also insist that the dwell time mismatch j½t3 −
ðt2 þ τ2 Þ − ½t2 − ðt1 þ τ1 Þj ≤ μsec time scale in order for the
interference term in ϕint to be included unaltered. Any
term containing dependence on ϕloss disappears from the
Doppler-sensitive final state probability, hence the name.
The phase terms require further discussion in order to
compare the present results with previous work [18]. We first
consider terms containing ϕint dependence only
expðiϕint ÞC 1 S 1 ðS 2 Þ2 C 3 S 3 þ cc ¼ expðiϕint Þ exp½iðϕ1 − 2ϕ2 þ ϕ3 Þ
× jS 1 jjS 2 j2 jS 3 jC 1 C 3 þ cc:
π
for ΩðM j Þτj ¼ :
2
ð63Þ
ð61Þ
ð64Þ
The detunings are evaluated at midpulse, which is equivalent
to time-averaged values in the present case of linearly varying
δðtÞ. In obtaining Eq. (64), we assumed that τ1 ¼ τ3 . Using
Eqs. (52) and (64), we obtain
C 1 C 3
1
2
¼ exp −i ½keff ao ðτ þ 2TÞτ :
2
π
ð65Þ
Also, for this case,
Z
ϕint ¼ −keff
t3
t2 þτ2
_ 0 Þdt0 −
zðt
τ2
¼ −keff ao T 2 − 2 :
4
Z
t2
t1 þτ1
_ 0 Þdt0
zðt
ð66Þ
The interference term is then
Recall that the Raman phases are ϕj ≡ ðϕA;o − ϕB;o Þj − keff ·
^ez zðt ¼ 0Þ; where the terms ðϕA;o − ϕB;o Þj are the same for
all pulses j when the Raman pulses are all produced by the
same beam pair as would be the case for an accelerometer
and are, in general, different if each pulse is produced by a
different beam pair as would be the case for a gyroscope.
Then,
ϕ1 − 2ϕ2 þ ϕ3 ¼ ðϕA;o − ϕB;o Þ1 − keff · ^ez zðt ¼ 0Þ
¼ 2Refexpðiϕint Þ exp½iðϕ1 − 2ϕ2 þ ϕ3 ÞjS 1 jjS 2 j2 jS 3 jC 1 C 3 g
1
4τ
τ2 2 τ22
2
þ
−
¼ cos keff ao T 1 þ
2
πT T 2 π 4τ2
1
4τ
τ2 2
þ 2
−1
; for τ2 ¼ 2τ;
⇒ cos keff ao T 2 1 þ
2
πT T π
ð67Þ
− 2½ðϕA;o − ϕB;o Þ2 − keff · ^ez zðt ¼ 0Þ
þ ðϕA;o − ϕB;o Þ3 − keff · ^ez zðt ¼ 0Þ
¼ ðϕA;o − ϕB;o Þ1 − 2ðϕA;o − ϕB;o Þ2
þ ðϕA;o − ϕB;o Þ3 :
2Re½expðiϕint ÞC 1 S 1 ðS 2 Þ2 C 3 S 3 þ cc
ð62Þ
where we have retained phase corrections to second order in
τ=T. In the general case, computing the second-order phase
correction requires using the first-order perturbation result
for the Raman pulse operator: the first-order perturbation
2428
J. Opt. Soc. Am. B / Vol. 28, No. 10 / October 2011
Stoner et al.
theory result of Eq. (19) contributes phase correction terms
_ 2 . However, it can be shown that, in the
proportional to δτ
special case of linear variation in δ, those terms sum to zero.
Thus, Eq. (67) is valid to second order in τ=T in the constant
acceleration case. The interference phase of Peters [14] is
cos ϕPeters ¼ cos½keff ao ðT p þ 2τÞT p −
ϕPeters
2keff ao ðT p þ 2τÞ
Ωeff
× sin½keff ao ðT p þ 2τÞT p 4τ
⇒
≅ cos keff ao ðT p þ 2τÞ T p þ
π
4τ
τ2 4
þ 2
−1 ;
¼ keff ao T 2 1 þ
πT T π
ð68Þ
where T p ≡ T − τ2 =2 and we have retained phase correction
terms to second order in τ=T. Additionally, we used the fact
that Ωeff τ ¼ π=2. To first order in τ=T, the present treatment
yields the same scale factor correction as was calculated
in Peters; the second-order scale factor corrections are
discrepant.
We now determine the scale factor (proportionality between rotation rate and interferometer phase) of a Raman
pulse gyroscope comprising three Raman beam pairs operated with the same difference frequency. These beams are applied to discrete clouds of atoms propagating through a
rotating interaction region, such that atoms are exposed only
to a single beam pair at a time. Thus, laser pulse timing is the
same for all atoms, unlike a gyro implementation using a continuous thermal beam of atoms.
As opposed to the single beam pair analysis, we must now
specify a detuning function for each Raman laser beam pair.
For simplicity, we consider only rotation input as per Fig. 6.
The inertial sensitivity of a beam-based atom interferometer
arises from relative instrument motion with respect to the
atom beam when the instrument is rotated or accelerated
along an axis normal to the initial particle direction (see
Fig. 6). Rotation shifts the Raman difference frequencies of
the beams as seen by the propagating atoms; these frequency
shifts register the rate of rotation.
We now determine the rotation sensitivity of the Raman
laser gyroscope. Refer to Fig. 6 for definition of system parameters. The launch point of the atoms is taken to be fixed (rate
of rotation is low enough that centrifugal forces are negligible:
thus, we are free to place the center of rotation in a convenient
location). The time-dependent detuning associated with each
beam is due to the velocity at which the beam is rotating about
the center of rotation (a constant value); we are in the reference frame in which the undeflected atomic wave is stationary. The atoms are launched at a mean velocity vl with a
thermal (random) velocity component vt , where vl is normal
in direction to keff , vl · keff ¼ 0, subjected to a constant rate of
rotation Ωo normal to both keff and vl (see Fig. 6). We also
assume that jvl j ≫ jvt j. The laser beams are moving with respect to a particular atom’s trajectory at a rate
vj ¼ −Ωo ðvo · ^ex ÞM j ^ez ;
ð69Þ
where we represent the total velocity of the atom as
vo ¼ vl þ vt . The time-dependent detuning for beam pair j
for a single atom moving at velocity vo is then
Fig. 6. Rotating interferometer (not to scale): atoms (black dot) are
launched on a linear trajectory at t ¼ 0 and propagate with mean velocity vl . The instrument is taken to be rotating about the point of atom
launch. The rotation induces relative motion of the Raman beams with
respect to the atom trajectory. L is the distance between Raman beam
pairs as shown, d is the diameter of the Raman beam pairs, Ωo is the
rate of rotation. The vector direction of rotation is along þy (points
out of the page). The initial velocity of launch is in the þx direction.
keff is directed from top to bottom in the figure (along z).
δj ðtÞ ¼ ½ωA ðtÞ − ωB ðtÞj − ωo −
ℏk2eff
ℏ 2
ðk − k2B Þ
þ
2m A
2m
− keff Ωo ðvo · ^ex ÞM j :
ð70Þ
We now consider the ensemble average of the interferometer
phase, in which contributions from the random thermal velocity components vt for individual atoms are suppressed.
It is straightforward to show that, for constant laser difference
frequency and the symmetric timing sequence displayed in
Fig. 6, the mean interferometer phase is
τ
ϕint ¼ Φ2→3 − Φ1→2 ¼ −2keff Ωo ðvl · ^ex Þ T þ T:
2
ð71Þ
It is important to note that, in order to eliminate phase dependence on the time of the first pulse t1 , it is necessary to set
τ1 ¼ τ3 ¼ τ to obtain Eq. (71).
We now consider open loop scale factor, i.e., the observed
interferometer phase dependence on rotation rate when the
frequency difference is maintained at a constant value. As
with the calculation of the open loop scale factor for the
accelerometer, we consider midpulse values of the detuning.
For convenience, let us choose a fixed laser frequency difference such that δðM 1 Þ ¼ 0 (the difference in detunings does not
depend on the choice of value for δðM 1 Þ ¼ 0, however): in that
case,
ðωA − ωB Þ1 ¼ ωo þ
δðM 1 Þ ¼ 0:
ℏk2eff
ℏ 2
ðk − k2B Þ þ keff Ωo vl M 1^ez ;
−
2m 2m A
for
ð72Þ
A straightforward argument then shows that δðM 3 Þ ¼
−keff Ωo vl ð2T þ τÞ. Using Eq. (52), we write
Stoner et al.
Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. B
1
2
C 1 C 3 ¼ exp i τ½δðM 3 Þ − δðM 1 Þ
2
π
1
2
¼ exp −i keff Ωo vl ð2T þ τÞτ
2
π
REFERENCES
1.
ð73Þ
2.
so that the total open loop phase is
3.
2
ϕint þ τ½δðM 3 Þ − δðM 1 Þ
π
τ
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:
þ
þ
≅ −2keff Ωo vl T 2 1 þ
2T πT π T
2429
4.
ð74Þ
It can also be shown in the gyro case that the phase contributions arising from the first-order perturbation theory portion
of the Raman pulse operator vanish; thus, Eq. (74) is valid to
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5. CONCLUSION
In this paper, we have presented an analytical framework for
Raman pulse interferometry in dynamic environments. The
framework is applicable to atom optics manipulations that
feature off-resonant light pulses and brief interrogation times
and accounts for finite Raman pulse duration and partial
coherence effects that can become important at very short interrogation times. We have presented a diagrammatic method
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