Damping of local Rabi oscillations in the presence of thermal motion
ABSTRACT We investigate both theoretically and experimentally the effect of thermal
motion of laser cooled atoms on the coherence of Rabi oscillations induced by
an inhomogeneous driving field. The experimental results are in excellent
agreement with the derived analytical expressions. For freely falling atoms
with negligible collisions, as those used in our experiment, we find that the
amplitude of the Rabi oscillations decays with time $t$ as $\exp[(t/\tau)^4]$,
where the coherence time $\tau$ drops with increasing temperature and field
gradient. We discuss the consequences of these results regarding the fidelity
of Rabi rotations of atomic qubits. We also show that the process is equivalent
to the loss of coherence of atoms undergoing a Ramsey sequence in the presence
of static magnetic field gradients  a common situation in many applications.
In addition, our results are relevant for determining the resolution when
utilizing atoms as field probes. Using numerical calculations, our model can be
easily extended to situations in which the atoms are confined by a potential or
to situations where collisions are important.
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Article: Quantum computers.
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ABSTRACT: Over the past several decades, quantum information science has emerged to seek answers to the question: can we gain some advantage by storing, transmitting and processing information encoded in systems that exhibit unique quantum properties? Today it is understood that the answer is yes, and many research groups around the world are working towards the highly ambitious technological goal of building a quantum computer, which would dramatically improve computational power for particular tasks. A number of physical systems, spanning much of modern physics, are being developed for quantum computation. However, it remains unclear which technology, if any, will ultimately prove successful. Here we describe the latest developments for each of the leading approaches and explain the major challenges for the future.Nature 03/2010; 464(7285):4553. · 38.60 Impact Factor  SourceAvailable from: Daniel Varela Magalhães[Show abstract] [Hide abstract]
ABSTRACT: We have demonstrated the possibility for a compact frequency standard based on a sample of cold cesium atoms. In a cylindrical microwave cavity, the atoms are cooled and interrogated during a free expansion and then detected. The operation of this experiment is different from conventional atomic fountains, since all the steps are sequentially performed in the same position of space. In this paper we report the analysis of a Ramsey pattern observed to present a 47 Hz linewidth and a stability of 5x10^13 for an integration time longer than 100s. Some of the main limitations of the standard are analyzed. This present report demonstrates considerable improvement of our previous work (S. T. Muller et al, J. Opt. Soc. Am. B, v.25, p.909, 2008) where the atoms were in a free space and not inside a microwave cavity.Journal of the Optical Society of America B 08/2011; 28(11). · 2.21 Impact Factor
Page 1
PHYSICAL REVIEW A 87, 063402 (2013)
Damping of local Rabi oscillations in the presence of thermal motion
Anat Daniel, Ruti Agou, Omer Amit, David Groswasser, Yonathan Japha, and Ron Folman*
Department of Physics, BenGurion University of the Negev, Be’er Sheva 84105, Israel
(Received 10 April 2013; published 4 June 2013)
We investigate both theoretically and experimentally the effect of thermal motion of lasercooled atoms on
the coherence of Rabi oscillations induced by an inhomogeneous driving field. The experimental results are in
excellent agreementwiththederivedanalytical expressions.Forfreelyfallingatomswithnegligiblecollisions,as
thoseusedinourexperiment,wefindthattheamplitudeoftheRabioscillationsdecayswithtimet asexp[−(t/τ)4],
where the coherence time τ drops with increasing temperature and field gradient. We discuss the consequences
of these results regarding the fidelity of Rabi rotations of atomic qubits. We also show that the process is
equivalent to the loss of coherence of atoms undergoing a Ramsey sequence in the presence of static magnetic
fieldgradients—acommonsituationinmanyapplications.Inaddition,ourresultsarerelevantfordeterminingthe
resolution when utilizing atoms as field probes. Using numerical calculations, our model can be easily extended
to situations in which the atoms are confined by a potential or to situations where collisions are important.
DOI: 10.1103/PhysRevA.87.063402PACS number(s): 37.10.Gh, 32.70.Cs, 03.67.−a, 37.10.De
I. INTRODUCTION
A twolevel system is a key element in understanding the
structure of matter and its interaction with electromagnetic
fields. Twolevel systems manipulated by electromagnetic
waves are the fundamental building blocks in many appli
cations, such as nuclear magnetic resonance (NMR) [1] and
electron paramagnetic resonance (EPR) microscopy, atomic
clocks [2,3] and interferometers [4,5], magnetometry with
atoms [6] or nitrogenvacancy (NV) centers in diamonds
[7], and quantum information processing with atoms, ions,
quantum dots, or superconducting qubits [8–14].
The basic operation in twolevel system manipulation is
Rabi rotation (also called Rabi flopping or Rabi oscillation),
which appears whenever a twolevel system is subjected to a
constant nearly resonant driving field. Measurement of Rabi
oscillations and their damping provides information about
the coherence of the system. Decoherence may follow from
spontaneousemission[15],externalorintrinsicnoise[16–19],
and spatial inhomogeneities across the sample, which may
be due to inhomogeneities of external fields or due to the
dynamics of the twolevel systems themselves during the
oscillations (e.g., dipoledipole interactions) [19,20].
A process which is analogous to the damping of Rabi
oscillations is the decoherence (dephasing) of free phase
oscillations of twolevel systems, which are prepared in a
superposition of the two energy eigenstates. In NMR, this
process is called freeinduction decay (FID) and is used for
characterizing the environment, and in atomic clocks and
interferometers, it involves the loss of visibility of Ramsey
fringes. This decoherence is usually caused by fluctuations
or inhomogeneities in the energy splitting between the two
levels. If these inhomogeneities are time independent or vary
slowlyintime,thenthisincoherence maybereversedbyusing
a spinecho technique which reverses the evolution of the
relative phase (equivalent to the direction of spin precession).
In this way, it is possible to distinguish between the effect of
static inhomogeneities and other sources of decoherence.
*folman@bgu.ac.il
In dilute gases of alkalimetal atoms, which are used for
atomic clocks, interferometers, and magnetic sensors, dipole
dipoleinteractionsarenegligiblesuchthatspindecoherenceis
usuallycausedbyfluctuationsandinhomogeneitiesofexternal
magnetic or electromagnetic fields, and by atomic collisions
[21]. These hindering effects of inhomogeneous fields are also
relevant for single trapped atoms and ions if the fields vary
significantly over the length scale of the particle localization.
In this context, it is important to understand the effect of
temperature.
On the other hand, such inhomogeneous fields may be
useful in the case of lowvelocity cold atoms, which can be
locally manipulated by these fields. Furthermore, cold atoms
maybeusedformicronscalesensingoflocalforcesandfields.
For example, in Ref. [22], local forces were probed, while
in Refs. [23] and [24], it was shown that probing local Rabi
oscillationsofultracoldatomsdrivenbyinhomogeneousfields
can serve as a tool for mapping the intensity and direction
of electromagnetic waves in the microscopic scale. In this
context, it is important to understand the resolution limits of
such methods of local manipulation or sensing when thermal
motion mixes measurements at neighboring locations and
reduces the visibility of spatial and temporal modulations of
the atomic population.
Here we consider the damping of Rabi oscillations in
a sample of lasercooled thermal atoms subjected to an
inhomogeneous driving field. Beyond spatially dependent
Rabi frequencies, which imply the observation of internal
state population modulation (“fringes”) across the applied
electromagneticfield,weobservedampingofRabioscillations
atanygivenlocationwithaconstantfieldintensity(seeFig.1).
This effect is shown to be sensitive to the atomic temperature
and we attribute it to the thermal motion of the atoms.
We analyze both theoretically and experimentally a model
systemoffreelypropagatingtwolevelatomsinthepresenceof
gradients of driving fields or stateselective potentials, which
are weak enough not to affect the dynamics of the motional
degrees of freedom of the atoms. The atomic motion is then
mainly governed by the initial thermal velocity distribution.
In this case, we obtain simple analytical expressions for the
damping of Rabi oscillations or Ramsey phase oscillations.
0634021
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DANIEL, AGOU, AMIT, GROSWASSER, JAPHA, AND FOLMANPHYSICAL REVIEW A 87, 063402 (2013)
FIG. 1. (Color online) (a) Absorption image of87Rb atoms at
temperature T = 43 μK in the hyperfine state F = 2 after applying
a spatially inhomogeneous microwave (MW) field for 3.5 ms. The
atomsareinfreefallandtheobservedspatialaxesareperpendicularto
gravity.TheMWhornantenna,whichislocatedabout10cmtotheleft
of the cloud center, radiates electromagnetic waves whose amplitude
decreases with distance, thus generating multiple Rabi oscillation
frequencies which can be viewed simultaneously in the form of an
internal state population modulation (a fringelike pattern) along the
cloud. (b) Longitudinal density n2(x) of atoms in the F = 2 state
as a function of position along the axis of the MW beam, obtained
by vertical averaging over a 100pixelwide strip along the center
of the cloud in (a). The data (black curve) is fitted to a Gaussian
functionmultipliedby1 + v cos(∂x?tx + φ),wherev isthevisibility
ofthefringelikepatternandφ isapositionindependentphase.Thefit
(red curve, χ2= 0.98) provides an estimation of the Rabi frequency
gradient ∂x? = −1.77 ± 0.04 (ms mm)−1. (c) Rabi oscillations as
a function of the MW pulse duration, measured at the center of the
cloud. The data points, representing an average over a 100pixel
long vertical strip at x = 0, are fitted by Eq. (18) and scaled to the
background density nb. The coherence decay is due to the gradient
of the MW field and the thermal velocity of the atoms. (d) The
dependence of the observed Rabi frequency on the position across
the cloud. Each point is a result of a fit to an oscillation measurement
as in (c). The slope of the linear fit (solid curve) is ∂x? = −1.74 ±
0.13(msmm)−1,ingoodagreementwiththegradientmeasuredin(b).
The model can be easily extended to cases where the
atoms move in a potential (as long as their motion may be
treated classically) or to the case where atomic collisions are
important. In such a case, the theoretical solution may need to
involve numerical integration. This part is not included here
because the simple version of the model is sufficient for a
quantitative understanding of the experimental results.
The structure of the paper is as follows: in Sec. II, we
present the theoretical model and its simple solutions for the
collisionless potentialfree case. In Sec. III, we present the
specific experimental realization with cold atoms at different
temperatures and analyze the results with the help of the
theoretical model of Sec. II. In Sec. IV, we discuss some
fundamental and practical implications.
II. THEORETICAL MODEL
Consider an ensemble of twolevel atoms in the presence
of inhomogeneous fields. We assume that the two levels 1?
and 2? are not coupled by an electric dipole transition, such
that spontaneous emission is negligible over the time of the
experiment. The singleatom Hamiltonian is then
H = Hextˆ1 −1
Here, the first term is the stateindependent part, Hext= p2/
2m + V(x), which governs the external (motional) degrees of
freedom,withˆ1beingthe2 × 2unitymatrix.Thesecond term
describes a timeindependent energy splitting, ¯ hω12(x), which
may depend on position due to static inhomogeneous fields.
The last term describes the coupling of the atom to a driving
field with frequency ω, with ˆ σ ≡ (ˆ σx,ˆ σy,ˆ σz) being the vector
of Pauli matrices and ?(x) being a vector representing the
amplitudes of atomfield coupling corresponding to angular
frequencies of rotation about the axes of the Bloch sphere.
In the rotatingwave approximation, only terms which
oscillate with frequencies that are nearly resonant with the
atomiclevelsplittingω12areretained,whilerapidlyoscillating
terms are dropped. The effective Hamiltonian becomes
?−ω12(x)
where ?(x) ≡ ?x(x) + i?y(x) is typically complex.
In general, the spatially dependent Hamiltonian of Eq. (2)
determines the dynamics of the internal state as well as the
motional degrees of freedom. However, here we consider
driving frequencies in the microwave (MW) regime and field
gradients that are too small to affect the atomic motion in
the time scale of the experiment, namely, ∇ω12,∇? ?
mvT/¯ ht, where vT is the average thermal velocity and t is
the time scale of the experiment. In this case, the atomic
sample may be approximated by an ensemble of atoms with
classicaltrajectories ¯ x(t),whichareindependentoftheinternal
dynamics. The internal wave function of a single atom in
the frame of reference moving with the atom along a given
trajectory is then
?
where the coefficients a¯ x and b¯ x satisfy the Schr¨ odinger
equations,
2¯ hω12(x)ˆ σz+ ¯ hˆ σ · ?(x)cosωt.
(1)
HRWA= Hextˆ1 +¯ h
2
eiωt?(x)
ω12(x)
e−iωt?∗(x)
?
,
(2)
ψ¯ x(t)? = a¯ x(t)1? + b¯ x(t)exp
−i
?t
0
ω12[¯ x(t?)]dt?
?
2?, (3)
˙ a¯ x= −i
2?(¯ x(t))exp
?
i
?t
0
?(t?)dt?
?
b¯ x,
(4)
˙b¯ x= −i
2?∗(¯ x(t))exp
?
− i
?t
0
?(t?)dt?
?
a¯ x,
(5)
where ?[¯ x(t)] = ω − ω12[¯ x(t)] is the local detuning of the
driving field frequency from the energy splitting. In principle,
Doppler shifts can also be included in the detuning frequency.
These would lead to the broadening of the transition between
the two states. In the MW range of frequencies used in our
experiment (more specifically, 6.8GHz) and the range of
temperatures used (T < 100 μK), Doppler shifts are of the
orderofafewHz,whiletheRabifrequencyalongthesampleis
of the order of kHz (see Fig. 1). For this reason, we neglect the
effects of Doppler shifts in what follows. Doppler broadening
effects would be important at room temperature, where they
reach the order of a few kHz.
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PHYSICAL REVIEW A 87, 063402 (2013)
The density matrix of the internal state at a given position
x is obtained by averaging over the pure density matrices of
all the atoms with different trajectories:
?
?a¯ x(t)2
where P(¯ x) is the probability for an atom to be in a given
trajectory.
In principle, the density matrix can be found by solving
Eqs. (4) and (5) numerically for all the possible trajectories
for a given external potential and initial conditions. Such
a simulation of the trajectories may also include collisional
effects. However, here we consider two simple cases in which
Eqs. (4) and (5) have a simple analytical solution, which is
relevant to common experimental conditions, including our
experiment which is described in Sec. III.
ρ(x,t) =
¯ x
P(¯ x)δ[¯ x(t) − x]
×
a¯ x(t)b∗
b¯ x(t)2
¯ x(t)
a∗
¯ x(t)b¯ x(t)
?
,
(6)
A. Resonant Rabi oscillations
In the absence of static gradients, i.e., if ω12is constant and
ω = ω12everywhere in space, the solution of the Schr¨ odinger
equation may be represented by a simple trajectory on the
Bloch sphere. If we further assume for simplicity that the axis
of rotation is constant everywhere in space, we may, without
loss of generality, take ? to be real and obtain the analytic
solution
a¯ x(t) = cos[θ¯ x(t)/2]a¯ x(0) − i sin[θ¯ x(t)/2]b¯ x(0),
b¯ x(t) = −i sin[θ¯ x(t)/2]a¯ x(0) + cos[θ¯ x(t)/2]b¯ x(0),
where
?t
is the Bloch sphere angle relative to the z axis.
Ifcollisionsarerareduringthetimescaleoftheexperiment,
then each atomic trajectory is characterized by a constant
velocity v. An atom in a position x and velocity v at time
t has gone through the trajectory ¯ x(t?) = x − v(t − t?). If the
Rabi frequency along the trajectory changes linearly such
that?[¯ x(t?)] ≈ ?[x] − v · (∇?)(t − t?),thentheBlochsphere
angle at time t for this trajectory is given by
θ¯ x(t) = ?(x)t −1
We consider an initial atomic cloud having a Gaussian
position distribution of width ?x(0) along the gradient of the
driving field intensity and a thermal velocity distribution of
width ?v=√kBT/m. At time t = 0, the cloud is released
and freely expands with negligible collisions. The distribution
at time t > 0 is
1
2π?x?v
This corresponds to a timedependent spatial width ?x(t) =
α(t)?x(0) and velocity width ?v(t) = ?v(0)/α(t), where
α(t) =
initially in state 1? and use the solution in Eq. (8) to determine
the probability for an atom in a given trajectory ¯ x to be in
(7)
(8)
θ¯ x(t) =
0
?[¯ x(t?)]dt?
(9)
2v · (∇?)t2.
(10)
P(x,v,t) =
exp
?
−x − vt2
2?2
x
−
v2
2?2
v
?
.
(11)
?1 + ?v(0)2t2/?x(0)2. We assume that the atoms are
the state 2? at time t, namely, sin2(θ¯ x/2) =1
By inserting this into Eq. (6) and integrating over all the
trajectories that pass through the point x, we obtain the
following expression for the probability distribution of atoms
in the state 2?:
ρ22(x,t) =e−x2/2?x(t)2
?
where˜? = ? −1
to the Rabi frequency at x due to averaging over the Rabi
frequenciesduringtheexpansion.Itfollowsthattheamplitude
of Rabi oscillations decays as exp[−t4/τ4
temperaturedependent coherence time is
2(1 − cosθ¯ x).
√2π?x(t)
1
2
?
1 − cos[˜?(x,t)t]
× exp
−1
8∇?2?v(t)2t4
??
,
(12)
2x · ∇??v(0)2t2/?x(t)2is shifted relative
vα2(t)], where the
τv=
81/4
[?v(0)∇?]1/2= 2
?
m
2kBT∂x?2
?1/4
.
(13)
At a short enough time, where the spatial width of the atomic
cloud has not yet grown considerably, we find that the decay
of the visibility of the oscillations has a quartic exponential
dependence and becomes Gaussian when the cloud expands to
a few times its original size.
The t4exponential dependence of the decay of Rabi
oscillations follows from the Gaussian velocity distribution in
the thermal cloud. Atoms with higher velocity travel a larger
distance along the field gradient, thereby acquiring a larger
phase difference relative to atoms at rest in the detection point.
This additional phase is an integral over time along the way
which was traversed by an atom with a given velocity, namely,
vt,suchthatthetotalphasedependsquadraticallyontime.The
combination of the quadratic dependence of the phase on time
and the quadratic exponential dependence of the distribution
on velocity provides the t4exponential dependence of the
coherence on time. We may consider τvas a critical time such
that at smaller times, the Rabi rotation is unaffected by the
velocity distribution.
B. Ramsey fringes
Anequivalentsituationthatcanbesolvedanalyticallyisthe
decayofRamseyfringes,whosevisibilityisdeterminedbythe
coherence of free phase oscillations of the energy eigenstates.
Consider a π/2 Rabi pulse at time t = 0, which prepares the
system in a superposition (1? + 2?)/√2. In the presence of
inhomogeneous fields that induce an inhomogeneous energy
shift of the levels 1? and 2?, Eqs. (4) and (5) yield the trivial
solution
1
√2[1? + exp[−iφ¯ x(t)]2?],
where the Bloch sphere angle φ¯ x(t) for the given trajectory ¯ x
is given by
?t
in analogy with Eq. (10), where we have made the same
assumptions regarding the inhomogeneity of ω12(x) as we did
above for ?(x).
ψ¯ x(t)? =
(14)
φ¯ x(t) =
0
ω12[¯ x(t?)]dt?= ω12(¯ x)t −1
2(v · ∂xω12)t2,
(15)
0634023
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DANIEL, AGOU, AMIT, GROSWASSER, JAPHA, AND FOLMANPHYSICAL REVIEW A 87, 063402 (2013)
The Ramsey sequence is terminated by a second π/2
pulse at time t, after which the populations of the two
energy eigenstates are determined by the phase difference
φ(t) accumulated during the free phase oscillation time. The
coherence of the state after the Ramsey sequence is given
by the offdiagonal component of the density matrix ρ12just
beforethesecondπ/2pulse.ByinsertingEq.(15)intoEq.(6),
we obtain the following result for the coherence:
ρ12(x,t) =e−x2/2?x(t)2
√2π?x(t)e−i ¯ ω12(x)texp
?
−1
8∇ω122?v(t)2t4
?
,
(16)
such that the populations after the Ramsey sequence are
determined by the phase ω12(x)t and the visibility of the
Ramsey fringes decays equivalently to the decay of Rabi
oscillations, with ∇ω12replacing ∇? in the expression for τv.
It is interesting to examine the effect of a spinecho
technique on the coherence of the atomic sample in the
presence of thermal motion. In this process, a π pulse which
flips the atomic population is applied at half the time interval
between the π/2 pulses of the Ramsey sequence. In this case,
the total phase that is accumulated for an atom with velocity v
and position x is given by
?t/2
= −(v · ∇ω12)
=1
In contrast to Eq. (15), here the term proportional to ω12(x),
which implies a spatially dependent phase during the Ramsey
sequence and spatial modulation of the population after the
sequence,hasdropped.Weareleftwithapositionindependent
termproportionaltothegradientoftheinternalenergysplitting
and the velocity. After summation over trajectories to obtain
theoffdiagonaldensitymatrixρ12(x),onefindsthatfollowing
the second π/2 pulse at time t, the internal state population
is homogeneous over the atomic cloud, similarly to what
happens for a spin echo in a zero atom velocity sample in an
inhomogeneous environment. However, in analogy with the
derivation following Eq. (10), we find that the coherence of
the atomic population will decrease by a factor exp[−(t/τv)4]
as before. Due to the factor of 1/4 appearing in the last line
of Eq. (17), the value of the coherence time τvis larger by a
factorof√2relativetoitsvalueforasimpleRamseysequence
without spin echo. It follows that the spin echo removes the
population inhomogeneity due to the spatial inhomogeneity of
the field, but does not cancel the decoherence caused by the
thermal velocity distribution.
φ¯ x(t) =
0
dt?ω12[¯ x(t?)] −
??t/2
4(v · ∇ω12)t2.
?t
t?dt?−
t/2
dt?ω12[¯ x(t?)]
0
?t
t/2
t?dt?
?
(17)
III. EXPERIMENT
To demonstrate the theoretical model, we experimentally
investigate a cloud of freely propagating atoms in free fall. We
start our experiment with a cloud of 106cold87Rb atoms. The
atoms are cooled by a standard magnetooptical trap (MOT)
followed by laser molasses to the required temperature, of the
order of a few μK. The atoms are prepared in the F = 1
hyperfine state and are then released into free fall for the
duration of the experiment, typically 10–30 ms. At the time
of release, the atomic cloud has a nearly circular Gaussian
distribution of half width√2?x= 1.55 mm (at 1/e of max
imum density). During the free fall, the atoms are subjected
for a time t to a MW field generated by a horn antenna, which
is tuned to the 6.8GHz, F,mF? = 1,0? ≡ 1? → 2,0? ≡ 2?
clock transition. The MW field induces Rabi oscillations.
TheRabifrequencyis
?2(μB/¯ h)(gSˆS + gIˆI) · BMW(x)1?, where gS and gI are the
Land´ e factors of the electronic and nuclear spins, respectively,
ˆSandˆIarethecorrespondingspinoperators,μBisBohr’smag
neton,andBMW(x)isthemagneticfieldoftheMWradiation.In
the case of a single clock transition, the Rabi frequency matrix
element reduces to ? =
is the component of the MW magnetic field which is parallel
to the quantization axis of the Zeeman sublevels, determined
by a static magnetic field.
Figure 2 illustrates the experimental setup. A 6.8 GHz
radiation is generated by a signal generator (SMR20, Rohde
and Schwarz) synchronized with an atomic clock (AR40A,
AccubeatRubidium frequency standard). The signal is then
passed through a MW shutter (SWNND21841DT AMC
Inc.), which provides accurate microwave pulses. The pulse
is amplified by a 3 W MW amplifier (ZVE3W83, Mini
Circuits) before being transmitted to a horn antenna. After
time t, the MW field is switched off and the population of
the atoms in the F = 2 hyperfine state is determined by on
resonance absorption imaging directed along the gravitational
axis. As the horn antenna produces a spatially inhomogeneous
MWfield,agradientofRabifrequenciesisproducedalongthe
cloud (frequency decreasing with growing distance from the
antenna); this is exhibited in Fig. 1(a) as a fringelike pattern
[23]. In other words, the fringes are isoRabifrequency bands,
which vary smoothly to yield multiple Rabi oscillations that
can be viewed simultaneously, n2(x) ∝ sin2[?(x)t/2]. This
observationmaybeviewedasameasurementofthelocalRabi
frequency and hence the local amplitude of the driving field
component BMW
?
the matrixelement
? =
1
2¯ hμB(gS− gI)BMW
?
(x), where BMW
?
(x). This is illustrated in Fig. 1(b), where we
FIG. 2. (Color online) Illustration of the experimental setup. The
atoms arecooled and trapped withina vacuum chamber by astandard
MOT. The atoms are prepared in the F = 1 state by turning off the
repumper beam before the cooling beams. Once the MOT beams are
turned off, the atoms fall due to gravity and are subjected to a MW
field generated by a horn antenna. The MW shutter is controlled by a
trigger pulse sent from the experimental control. After the MW field
is switched off, an onresonance imaging beam directed along the
gravitational axis is applied. The beam passes through the cloud and
is collected by a CCD camera. The population of the atoms in F = 2
is extracted from the absorption image.
0634024
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DAMPING OF LOCAL RABI OSCILLATIONS IN THE ...
PHYSICAL REVIEW A 87, 063402 (2013)
deduce the Rabi frequency gradient from a fit of the horizontal
dependence of the atomic density to a Gaussian multiplied
by a sinusoidal function. In Fig. 1(c), we present a typical
Rabi oscillation over time, where each data point represents
the population of the atoms in the F = 2 state averaged over a
vertical strip of camera pixels, one pixel wide and 100 pixels
long (perpendicular to the direction of the driving field
gradient). The graph may be fitted to find the Rabi frequency
and the damping constants, as we show below. Figure 1(d)
shows the local Rabi frequencies deduced from time evolution
curves as in Fig. 1(c). The gradient of the Rabi frequencies is
deducedbyfittingthespatialdependenceofthemeasuredRabi
frequenciestoalinearslope.Farfromtheantenna,theradiated
magnetic field is expected to fall like 1/r with an approximate
linear dependence in the relevant range 10 < r < 10.6 cm.
The value of the Rabi frequency gradient that we find in the
linear fit in Fig. 1(d) is in good agreement with the value
obtained from a spatial fit of a single image in Fig. 1(b).
Next, we analyze quantitatively the damping of the local
Rabioscillationsasafunctionofsampletemperature.InFig.3,
we present, as an example, four data sets of Rabi oscillations
at different temperatures. We fit each data set to the function
?
n2(t) = Aexp
−
t4/τ4
vt2??2
v
1 + ?2
x
−t2
τ2
x
?
cos(?t + φ) + nb,
(18)
where A is the amplitude of oscillations at the moment t = 0,
? is the local Rabi frequency, φ is an arbitrary constant phase
used to account for possible systematic shifts in the timing of
the MW pulse, and nbis the background population, whose
time dependence due to cloud expansion is neglected.
In Eq. (18), the argument of the first damping exponent is
derived from Eq. (12) and is due to thermal motion. In this
term, ?v=√kBT/m is calculated for each temperature, the
initial width of the cloud, ?x, is extracted from a Gaussian fit
to the image of the initial cloud (?x= 1.1 mm), and τvis left
as a free parameter.
FIG. 3. (Coloronline)Rabioscillationsfordifferenttemperatures
(8to37μK).ThegraphspresentthepopulationinF = 2scaledtothe
background population nb, as a function of the MW pulse duration. It
can be seen that the coherence time of the oscillations increases when
the temperature decreases. Each of the graphs was fitted to the model
of Eq. (18) (see text for details). The resulting τvand χ2are shown.
The second damping parameter τx is obtained when
we consider a finite spatial resolution of the imaging
system, such that the image of the atoms represents a
convolution of Eq. (12) with a Gaussian resolution disk,
(√πσI)−1exp[−(x − x?)2/2σ2
the observed Rabi oscillations is then due to the fact that
the periodicity of the spatial modulation of the internal state
population becomes shorter with time, such that the spatial
visibility of these fringes drops due to the limited optical
resolution. This gives rise to a temporal damping of the
observed local oscillations with a time constant,
I], of radius σI. The decay of
τx=
21/2
σI(∂x?).
(19)
In order to make the fit, we first estimate the value of τx.
When leaving both τxand τvas free parameters, a fit to the
T = 43 μK data set appearing in Fig. 1 returns τx= 8.8 ms
withχ2of0.97.ThiscorrespondstoσIof94microns[Eq.(19),
with ∂x? = 1.7 (mm ms)−1from Fig. 1(d)], which in turn
corresponds to a misalignment of our 30cmfocallength lens
by 1 mm or so along the imaging axis. As we estimate that our
optics alignment error is at least that (as the cloud size itself
is about 1 mm in all directions), we adopt τx= 8.8 ms value
for the rest of the paper, and leave A, B, ?, φ, and τvas free
parameters.
Let us note that the fitting procedure is very robust, and
changing σI by a factor of 2 in each direction returns χ2
values with a mere change of 1%. Finally, we also fit the data
to Eq. (18) while replacing the power of 4 by a free parameter
d andfindthatitconvergestovaluesofd = 3.778–4.1withχ2
values 0.961–0.971. For comparison, we plot in Fig. 4 one set
of Rabi oscillations (T = 43 μK) with a fit to three possible
models: a Gaussian model (t2), an exponential model (t), and
the t4model developed here. It can be clearly seen that the t4
model provides the best fit.
We now use the gradient measured by the fit presented in
Fig. 1(d) to compare the observed coherence times at different
temperatures to the theoretically expected value [Eq. (13)]. As
presented in Fig. 5, we find an excellent agreement between
the theoretical prediction and the experimental data.
FIG. 4. (Color online) A comparison of different model fits to the
data (T = 43 μK). The dotted, dashed, and solid lines represent the
envelope of the exponential (t) model, the Gaussian (t2) model, and
our model presented in Eq. (18), respectively. The last model returns
a χ2value of 0.972, while the Gaussian model returns χ2= 0.947
and the exponential model returns χ2= 0.844.
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FIG. 5. (Color online) Coherence time as a function of the tem
perature (T = 8–102 μK). The data points are the coherence times
extracted from the t4model, as shown in Fig. 3. The temperatures
are measured using timeof flight (TOF). The error values of the
coherence time and temperature are those estimated by the t4model
andTOFfits(confidencelevelof95%).Theregionbetweenthedotted
lines indicates the range of the theoretical prediction calculated from
Eq. (13), while taking into account the errors in the measured Rabi
frequency gradient [Fig. 1(d)]. No free parameters are used. This
confirms, to a high level of confidence, our theoretical model.
IV. DISCUSSION
The damping of local Rabi oscillations sets a limit on the
spatialresolutionofdifferentialmanipulationofthermalatoms
by engineered spatially varying fields, and likewise sets a limit
on the probing accuracy of driving field amplitudes by such
atoms. Suppose that we want to obtain a singleshot measure
ment of the driving field gradient ∂x?. We would then fit the
atomic population to a function n(x) = Acos(ax + b) + B,
wherea = ∂x?t andb = ?(x = 0)t.Ifthemeasurementerror
of the coefficients a and b is constant with time, then it
follows that the accuracy of ?(x = 0) and ∂x? improves
linearly with time. On the other hand, the measurement is
limited by a maximum measurement time of t ∼ τv, as the
visibility of population modulations drops drastically at this
time. It follows that at the optimal measurement time, the
error in the measurement of ?(x) is proportional to 1/τv∝
T1/4(∂x?)1/2. We conclude that detection error grows slowly
with temperature.
Another aspect that can be derived from this work concerns
the limitation of thermal atom manipulation by inhomo
geneous fields where the field gradient is viewed as an
imperfection. For example, our model can be used to infer the
fidelityofaπ/2pulseappliedtoanatomicsamplebyadriving
field which is inhomogeneous (e.g., an atomic cloud passing
through a MW cavity in an atomic clock). Fidelity is defined
by the overlap between a target state ψ?targetand an actual
state ψ?. If the actual state is not pure, then it is described
by a density matrix ρ =?
the target state is ψ?target= cos(π/4)1? − i sin(π/4)2?.
For a given velocity, the actual state is ψv? = cos(1
∂x?vt2
The overlap between the actual state and the target state
is ?ψvψ?target= cos(v∂x?t2
the overlap over the different velocities, we obtain F2=
?dvP(v)?ψvψtarget?2=1
the fidelity drops to a minimum value of F = 1/√2, which
representsthefidelityforatotallyrandomqubitstate.Itfollows
again that τvacts as a critical time for atom manipulation in
the presence of gradients and thermal velocities.
To conclude, we have developed a simple model for the
damping of local Rabi oscillations in the presence of driving
fieldgradientsanddampingofRamseyfringecoherenceinthe
presence of static stateselective field gradients. For a sample
offreelypropagatingthermalatoms,wehaveshownthatinthe
presence of gradients of driving fields, local Rabi oscillations
of twolevel atoms lose their coherence with an exponential
quartic time dependence. Equivalently, in the presence of
gradients of static fields, the coherence of local population
oscillations in a Ramsey sequence reduces in the same way.
The coherence time scales inversely with the square root of
the field gradient and with the fourth root of the temperature.
We have demonstrated the theoretical model in an experiment
with lasercooled atoms and obtained an excellent agreement
between the analytical solutions of the theory and the exper
imental results. Our model and experimental demonstration
lays the groundwork for an understanding of more general
situations in which a sample of atoms interacts with local
fields. On the one hand, the atoms can serve as a measurement
tool for probing the amplitudes of local fields and their spatial
dependence,inwhichcaseourmodelmaybeusedtodetermine
the accuracy limits of such a measurement. On the other hand,
our model may contribute to the understanding of limitations
onlocalqubitmanipulationinsystemsofthermalqubitswhose
external motion may be described classically and when they
are not localized well enough relative to the variation length
scale of the manipulating fields. The model may be extended
to cases where the atomic gas is confined by a potential or in
a vapor cell. Further extensions of the model may also include
the effects of atomic collisions or the behavior of atoms at
ultracold temperatures where a degenerate gas is formed.
jwjψj??ψj. The fidelity is then
jwj?ψjψ?target2]1/2. For a π/2 pulse,given by F = [?
4(π +
0)]1? − i sin[1
4(π + ∂x?vt2
0)]2?, where t0= π/4?0.
0/4). Integrating the square of
2{1 + exp[−(π/4?0τv)4]}. When
?0τv> π/4, the fidelity is almost 1, while if ?0τv< π/4,
ACKNOWLEDGMENTS
For their assistance, we are grateful to the members of
the atom chip group and especially Amir Waxman, Shimon
Machluf, Menachem Givon, and Zina Binshtok. We acknowl
edgesupportfromtheFP7Europeanconsortium“matterwave
interferometry” (601180).
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