# 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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**ABSTRACT:**We have studied the coherent evolution of ultracold atomic rubidium clouds subjected to a microwave field driving Rabi oscillations between the stretched states of the F=1 and F=2 hyperfine levels. A phase winding of the two-level system pseudo-spin vector is encountered for elongated samples of atoms exposed to an axial magnetic field gradient and can be observed directly in state-selective absorption imaging. When dispersively recording the sample-integrated spin population during the Rabi drive, we observe a damped oscillation directly related to the magnetic field gradient, which we quantify using a simple dephasing model. By analyzing such dispersively acquired data from millimeter sized atomic samples, we demonstrate that field gradients can be determined with an accuracy of $\sim25$ nT/mm. The dispersive probing of inhomogeneously broadened Rabi oscillations in prolate samples opens up a path to gradiometry with bandwidths in the kilohertz domain.Physical Review A 08/2013; · 3.04 Impact Factor

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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, Ben-Gurion 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 laser-cooled 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 two-level system is a key element in understanding the

structure of matter and its interaction with electromagnetic

fields. Two-level 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 nitrogen-vacancy (NV) centers in diamonds

[7], and quantum information processing with atoms, ions,

quantum dots, or superconducting qubits [8–14].

The basic operation in two-level system manipulation is

Rabi rotation (also called Rabi flopping or Rabi oscillation),

which appears whenever a two-level 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 two-level systems themselves during the

oscillations (e.g., dipole-dipole interactions) [19,20].

A process which is analogous to the damping of Rabi

oscillations is the decoherence (dephasing) of free phase

oscillations of two-level systems, which are prepared in a

superposition of the two energy eigenstates. In NMR, this

process is called free-induction 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 spin-echo 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 alkali-metal 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 low-velocity cold atoms, which can be

locally manipulated by these fields. Furthermore, cold atoms

maybeusedformicron-scalesensingoflocalforcesandfields.

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 laser-cooled 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

systemoffreelypropagatingtwo-levelatomsinthepresenceof

gradients of driving fields or state-selective 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.

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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 100-pixel-wide 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 isaposition-independentphase.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 100-pixel-

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 potential-free 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 two-level 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 single-atom Hamiltonian is then

H = Hextˆ1 −1

Here, the first term is the state-independent part, Hext= p2/

2m + V(x), which governs the external (motional) degrees of

freedom,withˆ1beingthe2 × 2unitymatrix.Thesecond term

describes a time-independent 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 atom-field coupling corresponding to angular

frequencies of rotation about the axes of the Bloch sphere.

In the rotating-wave 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 time-dependent 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 − vt|2

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

temperature-dependent 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)

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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 off-diagonal 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|∇ω12|2?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 spin-echo

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.Weareleftwithaposition-independent

termproportionaltothegradientoftheinternalenergysplitting

and the velocity. After summation over trajectories to obtain

theoff-diagonaldensitymatrixρ12(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 magneto-optical 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,

Accubeat-Rubidium frequency standard). The signal is then

passed through a MW shutter (SWNND-2184-1DT AMC

Inc.), which provides accurate microwave pulses. The pulse

is amplified by a 3 W MW amplifier (ZVE-3W-83, 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 iso-Rabi-frequency 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 on-resonance 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.

063402-4

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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 30-cm-focal-length 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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DANIEL, AGOU, AMIT, GROSWASSER, JAPHA, AND FOLMANPHYSICAL REVIEW A 87, 063402 (2013)

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 time-of 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 single-shot 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 ?ψv|ψ?target= cos(v∂x?t2

the overlap over the different velocities, we obtain F2=

?dvP(v)|?ψv|ψtarget?|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 state-selective field gradients. For a sample

offreelypropagatingthermalatoms,wehaveshownthatinthe

presence of gradients of driving fields, local Rabi oscillations

of two-level 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 laser-cooled 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.

jwj|ψj??ψj|. The fidelity is then

jwj|?ψj|ψ?target|2]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“matter-wave

interferometry” (601180).

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