PHYSICAL REVIEW A 81, 053632 (2010)

Thermodynamics of Bose-Einstein-condensed clouds using phase-contrast imaging

R. Meppelink, R. A. Rozendaal, S. B. Koller, J. M. Vogels, and P. van der Straten*

Atom Optics and Ultrafast Dynamics, Utrecht University, Post Office Box 80,000, NL-3508 TA Utrecht, The Netherlands(Received 4 September 2009; published 27 May 2010)

Phase contrast imaging is used to observe Bose-Einstein condensates at finite temperature in situ. The imagingtechnique is used to accurately derive the absolute phase shift of a probe laser beam due to both the condensateand the thermal cloud. The accuracy of the method is enhanced by using the periodicity of the intensity signal as afunction of the accumulated phase. The measured density profiles can be described using a two-relevant-parameterfit, in which only the chemical potential and the temperature are to be determined. This allows us to directlycompare the measured density profiles to different mean-field models in which the interaction between thecondensed and the thermal atoms is taken into account to various degrees.

DOI: 10.1103/PhysRevA.81.053632 PACS number(s): 03.75.Hh, 67.85.Bc, 87.64.mh

I. INTRODUCTION

In the field of ultracold atomic physics there are threecommonly used techniques to image the atomic cloud, namely,fluorescence, absorption and phase-contrast imaging (PCI).All these techniques rely on the interaction of near-resonantlight with ultracold atoms. In most cases the techniques areonly used qualitatively. In the study of interactions whereboth the condensed and the noncondensed parts of the atomiccloud play an important role, like hydrodynamic excitations,propagation of first and second sound, and thermal conductionbelow the critical temperature, the densities of the twocomponents have to be measured quantitatively. In that casethe techniques can also be used in a quantitative way andthe temperature, chemical potential, and number of atoms areextracted from the spatial extent and intensity of the resultingimages. The fluorescence and absorption technique are notvery accurate for quantitative measurements due to the highoptical density of the sample, which precludes a linear responseof the signal to the atomic density. This can be avoided byexpanding the cloud before imaging, but this suffices fromthe fact that the expansion at finite temperatures cannot bedescribed accurately. In the case of PCI the high optical densityis avoided by detuning the probe light sufficiently. However,this also reduces the induced phase shift, which makes it hardto probe in one image both the dense, condensed part and thedilute, noncondensed part.

In this paper we have extended the phase-contrast techniqueto measure the in situ density distribution of dense Bose-Einstein condensates (BECs) with the surrounding thermalcloud at finite temperature absolutely and accurately forsingle shots. For this extension we have selected a smallerdetuning compared to the nondestructive PCI schemes andthe accumulated phase goes through 2π several times, whichextends the dynamic range of the technique considerable. Thisallows us to extract the density of the thermal cloud and theBEC accurately for all typical temperatures and allows usto directly study the interplay between the condensed and thethermal atoms. Our analysis shows that the interaction betweenthese two components alters the width, length, and totalaccumulated phase of the image substantially, which makes

*p.vanderstraten@uu.nl

these parameters unsuitable for an accurate determination ofthe number of condensed atoms. We also study the influence ofthe interaction on the transition temperature for condensation.

After a short discussion of the limitations of the commonlyused imaging techniques, the paper describes our extended PCItechnique and the way it is applied to a partially condensedBose gas. In this description the emphasis is on how the numberof atoms can be extracted from the images. Next, the accuracyis demonstrated in two experiments. In the first experiment,described in Sec. VI C1, the number of condensed atoms ismeasured with an accuracy below 5% in a single shot. In thesecond experiment, described in Sec. VI D, the growth of thecondensate is studied as a function of the temperature. In thismeasurement the interaction between the condensed and thethermal atoms is studied directly. Furthermore, measurementsaround the temperature where a BEC is formed are used todetermine the transition temperature.

II. LIMITATION OF THE EXISTING METHODS

In absorption imaging the shadow cast in a probe beamby the cloud of atoms is recorded on a CCD camera. Fromthe resulting image, the spatial extent of the cloud and theoptical density, proportional to the column density alongthe line of sight of the probe beam, can be derived. Fordense clouds absorption imaging becomes unreliable, sincefor optical densities of the order of 4, the dynamic range of theabsorption imagining technique is insufficient to accuratelydetermine the intensity in the shadowed region [1]. For BECsthe typical in situ optical density is of the order of 500 and theimage gets completely “blacked out,” making it impossible toextract the column density from the image. The fluorescenceimaging technique critically depends on the intensity and thefrequency of the probe beam. Furthermore, for high densitiesthe cloud becomes opaque and atoms in the center of the cloudare not subjected to the same intensity as atoms on the edge ofthe cloud.

One way to reduce the optical density is to reduce theatom-photon cross section by choosing a large probe beamdetuning. However, for nonzero detuning the real part of theindex of refraction of the atoms becomes nonzero and thecloud behaves like a gradient-index lens. This results in arefraction of the probe beam and leads to distortion of theimages. The resulting images cannot be used for a relative or

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R. MEPPELINK et al. PHYSICAL REVIEW A 81, 053632 (2010)

absolute column density measurement, since the magnitudeof this effect has a spatial dependence. Detuning the probebeam farther from resonance reduces the index of refractionbut leaves the cloud virtually transparent before the refractionis sufficiently reduced.

The optical density can successfully be decreased byreducing the atomic density. The cloud is allowed to expandduring a specific time of flight and an image is recorded assoon as the optical density is of the order of 2. Althoughabsorption imaging after expansion is the commonly usedtechnique in this field, expansion of the cloud introducessome difficulties as well. First, if the cloud is a BEC at finitetemperature, there is no accurate description of the expansionof the cloud available. Second, the quantization axis of thecloud is no longer well defined since the magnetic fields areturned off. As a result, the effective cross section for thepolarized probe beam becomes ambiguous. As a consequence,both absorption imaging and fluorescence imaging cannot beapplied to accurately determine the column density of denseclouds.

Some techniques have been developed to image denseclouds by using strong saturation in both absorption [2] andfluorescence [3] imaging. However, the high-intensity regimehas to be calibrated based on the low-intensity regime and thelatter regime is imaged after expansion. As a consequence, theissues raised previously about imaging after expansion applyto the strong saturation techniques as well, although cloud canbe imaged in situ.

Alternatively, PCI has been successfully applied to imagedense atomic clouds [4–6]. In PCI the real part of the indexof refraction is measured. The detuning can be chosen in sucha way that the refraction is small but the phase shift due tothe index of refraction remains substantial. PCI can be usedto image BECs in a nondestructive way, since the numberof scattered photons becomes negligible for large detuning.Other techniques based on dispersive light scattering, such asdiffraction-contrast imaging of cold atoms [7] and nondestruc-tive spatial heterodyne imaging of cold atoms [8], have beendemonstrated in dense thermal clouds.

In this paper we use the PCI technique to quantitativelyextract parameters of the images by choosing the probe laserfrequency closer to resonance, such that the induced phaseshift goes through 2π several times. In this way we can extendthe dynamic range of the method significantly, allowing forsimultaneous detection of both the BEC and the thermal cloud,which differ in density by roughly 1 order of magnitude.

III. THEORETICAL DESCRIPTION OFPHASE-CONTRAST IMAGING

A. Imaging the phase

It is possible to image a cloud of atoms by illuminating thecloud with a probe beam and recording the shadow cast due tothe absorption of the atoms on a CCD camera. Since the opticaldensity is measured, this technique probes the imaginary partof the complex index of refraction N of the atomic cloud. Incontrast, the imaging technique described in this paper probesthe real part of N . This technique is used in the phase-contrastmicroscope developed by Zernike in 1933. In 1953 Zernike

was awarded the Nobel prize “for his demonstration of thephase contrast method, especially for his invention of the phasecontrast microscope” [9].

In general, the complex phase φ accumulated by the light ina medium such as a cloud of atoms can be expressed as φatoms =φ′ + iφ′′/2, with φ′′ the optical density. Writing the electricfield of the probe beam as Eprobe = E0 exp(iφ), with E0 theamplitude of the light field, the electric field after passing acloud of atoms can be written as

E = Eprobe + Eatoms = Eprobe + Eprobe(eiφatoms − 1), (1)

where the electric field is split into two parts: the part thatis diffracted by the atoms, Eatoms, and the part that is not,Eprobe. In absorption imaging the phase information is loston a CCD camera, since the intensity I is measured as I =cε0EE∗/2 = I0e

−φ′′, where I0 = cε0|E0|2/2, c is the speed

of light, and ε0 is the vacuum permittivity. So the real phaseterm φ′ cancels and only the absorption is measured. In thesimplest form of PCI, the nondiffracted part of the probe light[first term on the right-hand side of Eq. (1)] is blocked. Thelight that arrives at the camera is the light which is diffractedby the atoms, E = Eprobe[exp(iφatoms) − 1]. Now the phase in-formation no longer cancels and the intensity on the camera isgiven by

I = 2I0[1 − cos (φatoms)], (2)

where the arbitrary phase of the probe beam cancels. This tech-nique is referred to as dark-field imaging and was successfullyused to image 23Na atoms in the early days of experimentalBEC physics [6]. An important experimental drawback of thismethod is the absence of probe light on the camera, preventingthe normalization of the intensity profile, which makes thistechnique unsuitable for measuring the absolute phase shift.However, the relative signal can be used to study the spatialdistribution.

A more elaborate way of converting the accumulated phaseinto an intensity profile is by phase-shifting the nondiffractedlight instead of blocking it, in the same way that the phase-contrast microscope works. The nondiffracted light is theplane-wave part of the beam, which can be phase-shifted byplacing a small transparent object with a refractive index inthe probe beam in such a way that only the plane-wave partof the beam propagates through it (see Fig. 1). This object iscalled a phase spot. The plane-wave part propagating throughthe phase spot accumulates a phase θ , changing Eq. (1) to

E = Eprobeeiθ + Eprobe(eiφatoms − 1). (3)

Now the intensity is given by

I = I0[3 − 2 cos(θ ) + 2 cos(θ − φatoms) − 2 cos(φatoms)],

(4)

where the arbitrary phase of the probe beam cancels. Theprobe beam is not blocked and it is possible to derive theabsolute phase shift, since the intensity profile I0 can bemeasured and used for normalization. This technique hasalso been successfully applied to image 23Na atoms [5]. Theimplementation of PCI described in Ref. [5] uses a detuning of

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THERMODYNAMICS OF BOSE-EINSTEIN-CONDENSED . . . PHYSICAL REVIEW A 81, 053632 (2010)

LAtoms 1 PS CCD

f1 d

L2

f2

FIG. 1. (Color online) Schematic representation of the PCI setup.The atoms located at the center of the trap are imaged with lens L1

(with focal distance f1), which is placed at a distance f1 from thecenter. A sharp image is created by placing lens L2 (with focal lengthf2) at a distance d from L1 and at a distance f2 from the CCD camera.The phase spot (PS) is placed in the focal plain of the nondiffractedprobe beam.

the probe beam that is large compared to the atomic resonance,and the absorption of the light by the atoms is therefore reducedto a degree that it is no longer destructive. This allows for takingmultiple images of the same BEC.

In our experiment we do not focus on the ability to make anondestructive image but we use the periodicity of the intensityas a function of φatoms [see Eq. (4)] to accurately determine theaccumulated phase of both the BEC and the thermal cloud.In recent years, different schemes have been proposed toderive the phase information of a cloud of cold atoms [8,10],but the accumulated phase has never been used for accuratequantitative measurements of the density distribution of BECsat finite temperature.

B. Index of refraction of a cold gas

We continue with a detailed description of the phase shiftof an electromagnetic wave propagating through a cloud ofspin-polarized atoms to relate φatoms to the atomic density (seealso Ref. [11]). The phase shift is caused by the modificationof the probe field by the atomic dipoles, so we start with anexpression of the polarizability of one atom. The polarizabilitytensor

←→α in the low-intensity limit and δ � ω is given by [12]

←→α =

∑g,g′,e

i

h

1

γ /2 − iδ〈g| �µge|e〉〈e| �µeg|g′〉〈g′|σgg|g〉, (5)

where the natural linewidth γ is given by

γ = ω3µ2

3πε0hc3. (6)

Here, �µeg,ge are the electric-dipole operators, σgg is the densitymatrix, δ is the detuning, and ω is the probe frequency. Inthe experiment various linear polarizations of the light fieldhave to be dealt with, so the dependence of the polarizability←→α on the polarization has to be determined. We define the

angle β as the angle between the z axis of the atomic cloudand the (linear) polarization direction of the light field whichis in the (x,z) plane. In this system, the line of sight is alongthe y axis. To evaluate Eq. (5) for different polarizations, afixed polarization axis is chosen parallel to the polarization

of the light field and Eq. (5) is evaluated in this frame, inwhich, conveniently, only �M = 0 transitions are induced.Furthermore, only one component of the polarizability tensorwill contribute in this frame and the polarizability becomes ascalar. Since all atoms in the trap are in the |Fg,Mg〉 = |1,−1〉state with respect to the z axis of the magnetic field confiningthe atoms, the density matrix σgg only has one nonzero element,which makes the calculation of the rotation from the magneticz axis to the polarization axis of the light field straightforward.

Rotating the density matrix of the ground state σgg isachieved using

σgg(β) = Ry(β)†σggRy(β), (7)

where the rotation matrix Ry(β) for rotating a system aroundthe y axis over an angle β for a system with Jg = 1 can befound using Wigner’s formula as [13]

Ry(β) =

⎛⎜⎜⎝

cos2(

β

2

) sin(β)√2

sin2(

β

2

)− sin(β)√

2cos(β) sin(β)√

2

sin2(

β

2

) − sin(β)√2

cos2(

β

2

)⎞⎟⎟⎠ . (8)

Since the two dipole moment operators �µeg do not changethe M state of the atoms, only the diagonal elements of Eq. (5)play a role, and we end up with

α = 2i

hγ

∑g,e

〈Fg,Mg| �µge|Fe,Me〉2〈Fg,Mg|σgg(β)|Fg,Mg〉1 − 2iδe/γ

.

(9)

Furthermore, we can define the square of the Clebsch-Gordancoefficients CFe,Mg

(transition strength) as

Cg,e = 〈Fg,Mg| �µge|Fe,Me〉2/µ2, (10)

and obtain, using Eq. (6),

α = iε0cσλ

ω

∑g,e

Cg,e〈Fg,Mg|σgg(β)|Fg,Mg〉1 − 2iδe/γ

, (11)

where δe is the detuning with respect to transition Fg → Fe

and σλ is the cross section for absorption of light:

σλ ≡ 3λ2

2π. (12)

For sodium atoms in the |Fg,Mg〉 = |1, − 1〉 state, thisresults in

α = iε0cσλ

ω

∑e

DFe(β)

1 − 2iδe/γ, (13)

where

D0(β) = 424 sin2(β), (14a)

D1(β) = 524 [1 + cos2(β)] + 6

24 , (14b)

D2(β) = 624 + 1

24 sin2(β). (14c)

For alkali metal atoms the polarizability is independent ofthe angle β if the detuning δ is large compared to the hyperfinesplitting of the excited state. In that limit the numerator of the

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sum in Eq. (13) is 23 with no angular dependence. Finally, the

complex index of refraction N is given by

N 2 = 1 + ρα

ε0, (15)

with ρ the density. Note that for atoms confined in a potential,the density is not homogeneous and ρ depends on the position,ρ = ρ(x,y,z).

The index of refraction given by Eq. (15) is only valid forlow densities. For higher densities, the dipole moment of anatom is influenced by the internal field of the surroundingatoms. This results in a modification of the refractive index,which is accounted for by the Lorentz-Lorenz equation [14],

N 2 = 1 + ρα/ε0

1 + C, (16)

where C is given by

C = − 13ρα/ε0. (17)

Since the atoms being imaged are bosons instead ofclassical particles, the index of refraction will be modifiedfurther due to bunching of the atoms. Two terms are addedto Eq. (17), as discussed in Ref. [15]. The first term accountsfor modifications due to the enhanced photon scattering crosssection caused by the quantum statistics of the atoms. Thesecond term describes the modified refractive index inducedby resonant Van der Waals interaction. For the parameters usedin Ref. [15] it results in an increase of almost 10% in both thereal part and the imaginary part of the refractive index.

We have derived the modifications along the lines ofRef. [15] and have confirmed the result for the detuning anddensities reported in that paper. We found that modificationsto the complex index of refraction strongly depend on thechosen detuning and, therefore, on α. In Ref. [15], δ ∼ γ ,while we are in the regime δ � γ . For our parameters, the realpart of the index of refraction does not change significantly.However, we do find an enhancement of the absorption by afactor of 3. Since the overall absorption is very small, up to 4%at the detuning used, enhanced absorption does not influencethe intensity profile significantly. As a result, we expect themeasured phase to change less than 1% due to this effect.

C. Imaging the atomic density

The image of a cloud of atoms recorded on a CCD camerayields the column density of the cloud, that is, the densityprofile integrated along the probe line of sight. The columndensity can be linked to the complex index of refraction inthe following way. In general, light passing through a mediumwith an index of refraction N over a distance l accumulates acomplex phase φ given by

φ = k(N − 1) l, (18)

where k = 2π/λ. Here it is assumed that the light is notdiffracted inside the condensate through the inhomogeneousdensity distribution of the cloud. This issue is addressed in

Sec. VI A. Since the index of refraction depends on theposition, the accumulated phase is written as

φ(x,z) = k

∫ (√1 + ρ(x,y,z)α/ε0

1 + C− 1

)dy. (19)

Since |ρα/ε0| � 1, the index of refraction can be approxi-mated as

N =√

1 + ρα/ε0

1 + C≈ 1 + ρα

2ε0. (20)

Under typical experimental conditions |ρα/ε0| is 10−2, intro-ducing only a small error by using this approximation. Now,Eq. (19) can be written as

φ(x,z) = kα

2ε0

∫ρ(x,y,z)dy = k

α

2ε0ρc(x,z), (21)

where the integration is along the line of sight y and ρc(x,z)is the column density.

The real part of φ(x,z), proportional to Re (α), yields thephase shift, whereas the imaginary part, proportional to Im (α),yields the absorption. Since α is constant for fixed detuning δ,the phase shift is directly proportional to the column density.Since Eq. (13) satisfies the Kramers-Kronig relation,

Re (N ) − 1 =(

2δ

γ

)Im (N ) , (22)

the ratio between the phase shift and the absorption is constantfor a given detuning.

IV. PHASE-CONTRAST IMAGING SETUP

To create a phase-contrast image of the atoms, the plane-wave part of the probe field has to be phase-shifted asdescribed in Sec. III A. This is commonly achieved by Fouriertransformation using a lens. The concept of the imaging setupused in our experiment is shown schematically in Fig. 1.A two-lens setup is used to generate a sharp image of theatoms on a camera. The first achromatic lens L1, with a focaldistance f1 = 250 mm is placed at a distance f1 from thecenter of the magnetic trap. The second achromatic lens L2,with focal distance f2 = 750 mm is placed at a distance f2

from the CCD camera. The distance between the two lenses isd = 150 mm. The combination of both lenses creates a sharpimage of the cloud on the CCD camera at a magnificationM = f2/f1 = 3.0 ± 0.05, where the uncertainty is the resultof the uncertainty in the focal length of the lenses.

To shift the phase of the plane-wave part of the light, aphase spot is placed in the back focal plane (Fourier plane)of the two lenses, which lies, for these parameters, beyondthe second lens, L2. This position can be found easily in theexperiment, since the parallel probe beam has its waist at theback focal plane and therefore the phase spot is placed at thisposition.

Figure 2 shows the intensity I/I0 as a function of theaccumulated phase φatoms for different values of θ based onEq. (4). In all experiments described here, θ = π/3 is used.This value is chosen since it yields a maximum visibility(Imax − Imin)/(Imax + Imin) = 1 and a dynamic range Imax −Imin = 4I0. A visibility of 1 is reached only for θ = π/3, as

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THERMODYNAMICS OF BOSE-EINSTEIN-CONDENSED . . . PHYSICAL REVIEW A 81, 053632 (2010)

00

2

4

6

8

10

φatoms (rad)

I/I 0

π 2π 3π

ππ/2π/3π/4

FIG. 2. (Color online) The phase-contrast signal I/I0 as afunction of the accumulated phase φatoms for different phases θ =π,π/2,π/3,π/4 of the phase spot.

shown in Fig. 2. Furthermore, the response is approximatelylinear for small accumulated phases.

For technical reasons, the π/3 spot consists of a glassplate with a dimple instead of a spot on top. This dimpleis made by dry-etching a 50-µm-diameter round dimple witha depth of 5λ/[6(N − 1)] = 1071 ± 15 nm in a 50-mm roundfused silica glass plate of 4-mm thickness. Light propagatingthrough the dimple accumulates 5π/3 less phase than lightpropagating through the full thickness of the plate, and thusθ = +π/3.

The atoms are imaged using an Apogee AP1E camera witha Kodak KAF-0401E chip camera with a pixel size of 9 ×9 µm2. At the magnification M = 3.0, the effective cameraresolution of 3.0 µm per pixel is comparable to the diffractionlimit xres ≈ 1.22λf1/(2r) ≈ 3.6 µm of the imaging lens L1

with radius r = 25 mm for probe light with a wavelengthλ = 589 nm.

V. DENSITY DISTRIBUTION

A. Three models for Bose-Einstein condensatesat finite temperature

A phase-contrast image of a BEC at nonzero temperaturecontains the column density distribution of the two-componentcloud, that is, a BEC and a thermal cloud which can bedistinguished from each other due to their distinct densitydistributions. To derive the relevant parameters from such animage, like the thermal (excited) density nex, the condensate(ground state) density nc, and the temperature T , the measuredprofiles are fitted to a theoretical model describing the cloud.Various models exist in literature in which the interactionbetween the condensed and the thermal atoms are taken intoaccount to various degrees. In this paper we compare ourmeasurements to three of these models. The first model is thecommonly used bimodal distribution, in which all interactionsbetween the two components are ignored. This model is usedto describe the vast majority of the experiments on cold Bosegases [1]. In the second model, only the effect of the condensedatoms on the thermal atoms is taken into account. The third

model incorporates the effect of the density distribution ofthe thermal atoms on the condensed atoms as well. Here, thedensity distributions of the thermal cloud and the BEC in eachof these models are described consecutively.

In the first model, referred to as the ideal model, thechemical potential is given by µ = Vext(�r) + nc(�r)U0, whereU0 = 4πh2a/m is the interaction parameter for bosons withs-wave scattering length a and mass m [16]. Since there areno interactions between condensed atoms with density nc andthermal atoms with density nex,µ does not depend on nex. Inthe ideal model nex is given by

nex(�r) =∫

d �p(2πh)3

1

e[ε(�r)−µ]/kBT − 1, (23)

where ε is given by

ε(�r) = p2

2m+ Vext(�r). (24)

In this approximation both nex and nc can be derivedanalytically, yielding the bimodal density distribution, the sumof a Maxwell-Bose distribution modeling the thermal cloud,and a Thomas-Fermi (TF) distribution modeling the BEC. Inthis approximation nex is given by

nex(�r) = Li3/2(ze−Vext(�r)/kBT )/λ3dB, (25)

where the de Broglie wavelength is given by λdB =√(2πh2)/(mkBT ) and z = exp[µ/(kBT )] is the fugacity.

Li3/2(z) is the Bose function, given by Lin(z) ≡ ∑∞k=1 zk/kn,

and

Vext = 12m

(ω2

radr2x + ω2

radr2y + ω2

axr2z

)(26)

is the external potential. In the TF approximation nc is givenby

nc(�r) = µ

U0

[1 −

(rx

Rrad

)2

−(

ry

Rrad

)2

−(

rz

Rax

)2]

, (27)

where Rrad,ax are the TF radii of the condensate, given by

Rrad =√

2µ

mω2rad

, (28a)

Rax =√

2µ

mω2ax

. (28b)

Since ωrad/ωax ≈ 50 in the experiments described here, andthus Rrad � Rax, the clouds are cigar shaped. The number ofcondensed atoms Nc is found by integrating nc over �r andyields

Nc =∫

nc(�r)d�r = 8π

15

(2µ

mω2

)3/2µ

U0. (29)

Integration of the density over the line of sight yields thecolumn density, which is the property being measured in theexperiments.

The second model, denoted the semi-ideal model, doesinclude interactions, but only the contribution of the mean-fieldpotential of the condensate to the thermal density distributionis taken into account [17]. Therefore, µ remains unchanged,

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-2 -1 0 1 20.0

0.1

0.2

0.3

0.4

0.5

-2 -1 0 1 20

1

2

3

4

Den

sity

(1020

m−3

)

Position (mm)

FIG. 3. The density distribution of a thermal cloud nex calculatedusing the noninteracting model (dashed line) and the semi-idealmodel (solid line), for T = 0.9 µK, N = 1 × 109 atoms, for thetypical experimental trap parameters. The inset shows the total densitydistribution nex + nc for both models.

µ = Vext(�r) + nc(�r)U0, but the energy of the thermal atomsis changed due to the mean-field potential of the condensedatoms and is written as

ε2(�r) = [p2/(2m) + 2nc(�r)U0 +Vext(�r) − µ]2 − [nc(�r)U0]2.

(30)

We calculate the density distribution in this semi-idealmodel by numerical integration of Eq. (23), where ε(�r) isgiven by Eq. (30) [16]. The shape of the density distributionof the condensate remains unchanged, but the repulsion ofthe mean-field potential of the condensate causes the thermalatoms to be repelled from the center, which results in theshape shown in Fig. 3. To have the same total number of atomsand temperature in Fig. 3, the chemical potential is differentin both models. The decrease in density in the center is theresult of the repulsion due to the mean-field potential of thecondensed atoms and, therefore, absent in the noninteractingdistribution. Due to the lower central density the resultingcondensate fraction is lower in the semi-ideal model comparedto the noninteracting model for the same total number of atoms,N = Nex + Nc, and temperature T .

In the third model the mean-field effect of the thermalcloud on the condensate is taken into account as well. Thismodel corresponds to the Popov approximation in the TFlimit, so we refer to it as the Popov model [16]. The chemicalpotential is now given by µ = Vext(�r) + [nc(�r) + 2nex(�r)]U0.The expression for the density of the excited atoms in thePopov approximation is given by

nex(�r) =∫

d �p(2πh)3

p2/2m + 2n(�r)U0 + Vext(�r) − µ

ε(�r)

× 1

eε(�r)/kBT − 1, (31)

-2 -1 0 1 20

1

2

3

4

-1.0 -0.9 -0.8 -0.7 -0.60.0

0.5

1.0

1.5

2.0

-0.2 -0.1 0.0 0.1 0.22.80

2.85

2.90

2.95

3.00

Den

sity

(1020

m− 3

)

Position (mm)

FIG. 4. The combined density distribution of the thermal cloudand condensate nex + nc calculated using the semi-ideal model(dashed line) and the Popov model (solid line) for T = 0.9 µK,N = 1 × 109 atoms, and the typical experimental trap parameters.The left and right insets show an enlargement of the edges and thecenter of the condensate, respectively.

with n = nex + nc and where, in the semiclassical approxima-tion, ε(�r) is given by

ε2(�r) = [p2/(2m) + 2n(�r)U0 + Vext(�r) − µ]2 − [nc(�r)U0]2.

(32)

Now the resulting density distribution of both nex and nc

is found by numerical integration of Eq. (31), where aniterative procedure is used to find a self-consistent solution.The difference between the semi-ideal model and the Popovmodel is shown in Fig. 4, focusing on the density distributionof the condensed atoms. The density distribution of the thermalatoms as shown in Fig. 3 results in a higher effective potentialfor the condensed atoms. This effect results in a compressionof the density distribution of the condensate compared to thedistribution calculated using the semi-ideal model.

B. Determining the chemical potential in the experiment

In the theoretical models the density profile is determinedby the parameters µ and T in a harmonic confinement givenby Vext. Therefore, it seems evident to analyze the measureddensity profiles by fitting them to a theoretical model inwhich only µ and T are to be determined. However, thisapproach can be applied only if the size and the height ofthe measured profiles can be determined absolutely. Since theaforementioned PCI method is expected to do so, we can applysuch a fit procedure here.

In absorption imaging the size and height of the densityprofiles are not measured directly due to the high opticaldensity, but after a finite time of flight. The initial conditionshave to be determined by fitting either the size or the height ofthe signal of both the BEC and the thermal cloud separately.In that case, µ and T are derived from either the height orthe size of the measured density profiles and the experimentalconditions favor the use of one above the other.

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Being able to describe the complete cloud by µ and T , as isdone in PCI, is expected to yield a higher accuracy, since thedifferent measures of the cloud, like the size and height, arefitted simultaneously and both measures depend differently onexperimental parameters such as the magnification, detuning,and trap frequencies. Furthermore, it allows for taking theinteraction between the condensed and the thermal atomsdirectly into account. This interaction is ignored in the vastmajority of experiments in this field, in contrast to thetheoretical description of a condensate at finite temperaturefound in many textbooks. Our PCI setup is expected to yieldan accurate and absolute determination of both the densityand the size of the cloud, and measuring in situ prevents theneed to account for the expansion of the cloud. Therefore,since all measures are absolute, µ and T are expected tofully describe the measured density profiles, given the fixedexperimental parameters such as the magnification and the trapfrequencies. This allows us to directly compare our measureddensity profiles to the three mean-field models described inthe previous section.

The PCI method not only yields absolute density profiles,but is expected to yield a more accurate determinationof the thermal fraction for low temperatures compared toabsorption imaging as well. Since the accumulated phase forthe dense BEC can be multiple times π , a significant phaseis accumulated in the thermal cloud, without saturating thesignal in the BEC. In absorption imaging, on the contrary,the optical density of the thermal cloud is strongly reduced inorder to reduce the optical density of the BEC to the order of 4.Especially for low temperatures, when only the low-density tailof the thermal distribution extends the BEC spatially, it turnsout to be difficult to distinguish between the two components inthe experiment. Various schemes have been used to enhancethe contrast between the thermal cloud and the condensate,for example, by making a two-pass fit routine, first only onthe thermal cloud, followed by the condensate and thermalcloud, while fixing the temperature found in the first pass [18].Another scheme used is spatial separation of both componentsusing Bragg spectroscopy and fitting both parts separately [18].In the PCI method the contrast between the two components issufficient for all measured temperatures and prevents the needto introduce elaborate schemes to enhance the contrast.

VI. EXPERIMENTAL RESULTS

A. Experimental parameters

Imaging the atoms is done in such a way that the periodicityof the intensity of the PCI technique is used. Since the intensitysignal varies periodically as a function of φatoms [Eq. (4)],the intensity signal shows rings in the intensity profile forsufficiently large values of φatoms (see Fig. 2). The number ofrings depends on φatoms, which scales with the density and thedetuning δ as can be seen from Eqs. (21) and (13). The detuningis a parameter which can be tuned accurately in a wide range(up to 400 MHz) using acoustic-optical modulators, allowingus to make images with an adjustable number of rings. Dueto the ring pattern the imaging lens can be put in positionvery sensitively, allowing us to resolve features down to 4 µm.Furthermore, δ can be chosen in such a way that the intensity

profile of the BEC shows rings and the less dense thermalcloud yields a significant intensity signal as well, even for lowtemperatures. The resulting effective dynamic range of thismethod is therefore increased by the periodic dependence ofthe intensity on the accumulated phase.

A drawback of this method lies in the fact that the BECacts as a gradient-,index lens, the lensing being stronger forsmaller δ. The effect of the lensing on the imaging resolutionis estimated by a computer simulation in which the Fresnel-Kirchhoff diffraction integral is numerically integrated. In thissimulation the finite size of the imaging lenses is explicitlytaken into account. The mutual distances among the BEC,lenses, phase spot and image plane are chosen identical to thevalues used in the experiment. The intensity distribution at theimage plane is used to estimate the effects of the refractionof the BEC on the imaging resolution. If the diffraction is toolarge, the higher diffraction orders can miss the aperture of theimaging lenses and will degrade the resolution of the image.This simulation yields an upper limit for the allowed phaseshift of the BEC and, therefore, a lower limit of the applieddetuning given the condensate density and dimensions. In theexperiment we are in the regime in which the aperture of thelenses does not limit the resolution, although the onset of aslight deformation of the density profile is observed for thehighest condensate densities. For the typical parameters, themaximum focal length of the BEC is found to be of the orderof 100 µm. Even though the diffraction is not too large for theapertures in the imaging path, the imaging lenses have to beplaced in focus within a few tens of micrometers to preventdistortions of the imaged intensity profile, since the imagedobject itself acts like a lens.

The phase spot has to be aligned accurately as well, since allof the plane-wave light has to propagate through the phase spot.In the long axis of the cloud the diffraction is weak, and fortoo large a phase spot the diffracted light propagates throughthe phase spot as well. To make a trade-off between the twocriteria that the phase spot should be large enough that thenondiffracted light is phase-shifted, but small enough that thediffracted light is not, the phase spot has a size only a few timesthe waist of the probe beam. As a consequence, the phase spotis easily misaligned. Misalignment of the phase spot leads toreflection of the probe beam on the edges of the phase spot andpossible shifting of the phase of the light, which is diffractedby the atoms. As a result of the misalignment, we notice ahigher, tilted intensity profile, which is easily detected, sincewe have several maxima in the intensity profile. The commonway to apply PCI is by using a large detuning and the intensitysignal never reaches the first maximum value. Without a well-defined maximum in the image, (slightly) more intensity dueto misalignments may be unnoticed and the method can nolonger be used to determine the absolute phase.

As shown in Fig. 2, the intensity as a function of the phasevaries rapidly in certain ranges. Choosing a detuning in such away that the maximum phase lies in this range, a slight increasein the density, leading to a slight increase in the maximumphase, leads to a large change in the intensity in the center ofthe cloud, making the method very sensitive to small densitychanges.

In the experiment, imaging the atoms consists of taking asequence of three images. The first image obtains the intensity

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FIG. 5. (Color online) A set of four phase-contrast images during the final stage of the evaporation process: (a) µ/h = 2.6 kHz, T =1.01 µK, N = 7 × 108 atoms, Nc/N < 1%; (b) µ/h = 4.4 kHz, T = 0.95 µK, N = 6 × 108 atoms, Nc/N ≈ 15%; (c) µ/h = 5.2 kHz,T = 0.58 µK, N = 4 × 108 atoms, Nc/N ≈ 50%; (d) µ/h = 4.8 kHz, T = 0.25 µK, N = 2 × 108 atoms, Nc/N > 95%.

profile Iatoms(x,z) of the probe field and the atoms. The secondimage contains the intensity profile I0(x,z) of the probe field inthe absence of the atoms. This image is generally shot after themagnetic trap has been turned off for 2 s and no atoms are left.The third image is the intensity profile Ibg(x,z), with neitherthe probe field nor the atoms, and is used as a backgroundimage. The normalized intensity profile I (x,z) of the atoms isgiven by

I (x,z) = Iatoms(x,z) − Ibg(x,z)

I0(x,z) − Ibg(x,z), (33)

where the subtraction and division are performed on a pixel-by-pixel basis. The magnification scheme used causes the probebeam to get blown up to such an extent that the beam profileis no longer smooth on the camera due to interference effectscaused by dust and small scratches on the imaging optics. Inprinciple, these imperfections are canceled in the final images,but since the imaging path is not interferometrically stablebetween the acquisition of the three images, some distortionsremain.

B. Phase-contrast images

To demonstrate the PCI method, atoms are cooled belowthe critical temperature Tc in a 95.57 × 95.57 × 2.234-Hz trapand imaged with a probe beam with an intensity I0 = 6 ×

10−2 Isat for 50 µs. The probe beam is detuned 28.1γ belowthe |Fg = 1〉 → |Fe = 1〉 transition and results in a typicalintensity profile I (x,z) as shown in Fig. 5. The high-frequencynoise found in these images is caused by the distorted profileof the probe beam. Small fringes close to the BEC are theresult of the lenses being slightly out of focus.

The images are fitted using the models in Sec. V A. Afit yields two parameters, namely, the chemical potential µ

and the temperature T . With these two parameters, Eq. (31)is used to calculate nex(�r), and the expression for µ justabove Eq. (31) is used to calculate nc(�r). An iterativeprocedure is used to find a self-consistent solution of thesetwo equations. The total number of atoms is found bynumerically integrating the densities over the whole cloud.Note that raw summing the intensity of the image to get anestimate for the number of atoms is not possible, sincethe intensity of each pixel is not a unique measure of theaccumulated phase, since the phase goes through 2π multipletimes.

C. Accurate determination of the numberof Bose-condensed atoms

1. Accuracy using absorption imaging

The usual way to determine the number of condensed atomsis by taking a series of absorption images in time of flight.

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Absorption imaging turns out to become unreliable when theoptical density exceeds 4, due to the limited dynamic range ofthe CCD camera [1]. If the cloud of atoms is a BEC, the typicaloptical density is of the order of 500 when the probe is on-resonance. Imaging the atoms off-resonance to take advantageof the reduced photon scattering cross section turns out to becomplicated since the cloud behaves as a gradient-index lens inthis regime. The optical density can be lowered by turning offthe confinement, causing the atoms to expand during a certaintime of flight, until the density is low enough that the opticaldensity is of the order of 4. However, expansion complicatesthe interpretation of the measured density profiles, since itconsists of the convoluted momentum and spatial distributionof the atoms and the expansion of the cloud cannot be describedexactly.

Furthermore, the switching of the magnetic confinementcomplicates the retrieval of the column density due to the finitetime needed to switch the magnetic fields. In the absence of themagnetic field, the quantization axis of the atoms is no longerwell defined, causing the atoms to align along small residualmagnetic fields during time of flight. This changes the effectivecross section of the atoms for the applied polarized probe light.The magnitude and direction of the residual magnetic fieldsare expected to be spatially dependent. Since the atoms fallduring time of flight due to gravity, the effective cross sectionis also expected to depend on the time-of-flight duration.Furthermore, the imaging lenses have to be repositioned foreach time-of-flight duration. A final complication is that theaccuracy of detuning of the probe light is limited to 1 MHz,corresponding to a reduction of the absorption up to 4%. Allthese effects cause uncertainties in the measured number ofatoms up to 20%.

A technique is proposed based on the nonlinear responseof atoms on the applied probe intensity to lower the opticaldensity of the cloud without the use of expansion. Thisallows for quantitative in situ absorption imaging, althoughthe precise calibration required has to be done after time offlight, reducing the accuracy of this method [2]. Moreover,the required reduction of the optical density of the condensateresults in a reduction in the optical density of the thermalcloud as well, complicating the determination of both densityprofiles simultaneously. The high intensity needed to reach thestrong saturation regime, I ∝ do × Isat ≈ 8 W/cm2, with do

the optical density and Isat the saturation intensity, complicatesthe implementation of the method as well.

2. Accuracy using phase-contrast imaging

Using the PCI technique circumvents the issues raised here.The detuning is large compared to the uncertainty, δ = 281 ± 1MHz, and since the measurement is conducted in situ, theconfinement is not switched off and the quantization axis ofthe atoms remains well defined. The Zeeman shift caused bythe confinement is less than 2 MHz and therefore negligiblecompared to the detuning.

An image of a BEC taken with the PCI technique yieldsthe accumulated phase and the size of the condensate. Ana-lyzing all known systematic and statistical errors shows thatthe largest contribution to the error in size originates fromthe magnification of the imaging system. We determined the

magnification to be M = 3.05 ± 0.05. Since the condensateradius R is proportional to M , the number of condensedatoms scales as Nc ∝ M5 [see Eqs. (28a) and (29)]. Thetrap frequencies are derived from center-of-mass oscillationmeasurements, yielding a trap frequency with a statisticaluncertainty below 10−3.

The largest contribution to the error in the accumulatedphase originates from the lensing effect of the condensateand is estimated to be less than 5% in the measured phasebased on the computer simulation we conducted. This effectdoes not influence the axial size of the cloud since thedensity varies slowly in this direction, making the lensingnegligible. Since the number of atoms scales with the accu-mulated phase as N ∝ φ5/3, lensing is expected to yield anerror of up to 8% in the number of atoms for the highestdensities.

We analyze measured clouds for parameters similar to thoseshown in Fig. 5, where the condensate fraction is at least90%, by making a least-squares fit to the ideal model. Weconclude that both the determined size and the phase of thecondensate yield the same chemical potential µ within 3% forthe typical number of condensed atoms Nc ∼ 2.5 × 108. Sincethe thermal fraction is small, interactions are expected to be ofminor importance in this regime.

We expect that the small discrepancy between the measuresfor µ is caused by the lensing effect of the condensate, whichslightly alters the density profile. If the imaging lens is alignedincorrectly, more lensing is observed and the discrepancybetween the chemical potential based on phase versus sizeincreases.

D. Observation of interactions between the thermal cloudand the Bose-Einstein condensate

The effect of interactions between the thermal cloud andthe BEC becomes important if a significant thermal fractionis present. As pointed out in Ref. [19], the thermodynamicbehavior of the cloud is fixed by two parameters: the reducedtemperature t = T/T 0

c and the ratio η, given by

η = µTF0

kBT 0c

≈ 1.57(N1/6 a

a

)2/5. (34)

Here T 0c = hω[N/ζ (3)]1/3 denotes the transition temperature

in the absence of interactions and a = √h/(mω) is the

harmonic oscillator length.Under our typical conditions, η = 0.3, which is less than its

value in other experiments that have studied the interactions[18,20]. There it is shown that interactions between the atomsshift the transition temperature Tc downward. Reference [18]measured the effects of the interactions by spatially separatingthe thermal cloud from the BEC using Bragg spectroscopy toavoid the need to incorporate interactions in the description ofthe measured density profiles. The separation of both cloudsdepends sensitively on the applied Bragg pulse, and introducesan extra uncertainty. Furthermore, the accuracy is limited bythe absence of an exact theory describing the expansion of theBEC.

Using PCI these interactions can be measured directlyand more accurately than in these previous studies, althoughη is smaller. The measurements are conducted as follows.

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Using evaporative cooling we obtain a cold cloud of atomsat a temperature T below Tc under the same experimentalconditions as the measurement shown in Fig. 5. A phase-contrast image of this elongated cloud is taken with eachcloud cooled to different temperatures. Each image is fittedto all three two-dimensional models described in Sec. V A andyields a temperature and chemical potential. We find the best fitcan be made, determined by the smallest sum of squares, usingthe Popov model and the semi-ideal model. The noninteractingmodel gives inferior results and cannot be used to accuratelydescribe the measured profiles. This has been noted in previouswork [18,20].

In this experiment we can also see the effect of themean-field potential of the thermal cloud on the condensedatoms, although it is less prominent than the modificationof the thermal distribution. This effect was hinted at inRef. [18]. The density distribution of the thermal cloud causesan effectively larger potential for the condensed atoms. Theeffect of the resulting distribution is analogous to increasingthe trap frequency: the cloud gets smaller and denser. Sincewe can measure both density and size separately and with ahigh accuracy, we measure the effect of the interactions bycomparing the chemical potential based on the axial size andthe accumulated phase. The axial size is favored above theradial size for two reasons: lensing effects in this direction arenegligible, and the size is large compared to the resolution ofthe imaging setup.

From the fit results we determine the axial size and phaseof the condensed part of the cloud and calculate its chemicalpotential based only on the accumulated phase, yielding µφ ,or the axial size, yielding µax. The ratio µax/µφ is plottedagainst the temperature T in Fig. 6. We find the chemical

0.4 0.6 0.8 1.0

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µ ax/

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FIG. 6. The ratio µax/µφ of the chemical potential based onthe axial size Rax and phase φ as a function of the temperature.(a) η = 0.3; (b) η = 0.25. In the second series η is reduced byreducing the number of atoms.

potential based on the axial size for temperatures close toTc to be approximately 15% too small compared to thechemical potential based on the accumulated phase. As thetemperature decreases, this effect gets smaller until at lowtemperatures (low thermal densities), we find correspondencebetween both measures within a few percent, consistent withthe results found in the previous section. Since our standardfitting procedure uses µ and T to fully describe the densityprofile, we find the sum of squares to be at least 10% smallerat temperatures where a significant thermal fraction is present(condensate fractions up to 80%) for the fit using the Popovmodel compared to that using the semi-ideal model. Since thisshows that the Popov model yields the most consistent results,all measurements in the remainder of this paper are analyzedusing the Popov model.

E. Accuracy and reproducibility

Repeated measurements are done in the regime where three-body losses limit the density of condensed atoms. The totalnumber of condensed atoms Nc is therefore approximatelyconstant and the results are used to determine the sensitivityas well as the reproducibility of the PCI method. Themeasurement series takes over 2 h, and during this time thetemperature of the cloud changes from T = 0.573 ± 0.004 µKto T = 0.387 ± 0.004 µK due to the apparatus heatingup. However, in the density-limited regime the number ofcondensed atoms is roughly constant in this temperaturerange. The measured number of condensed atoms is foundto change less than 3% during the two measurement times ona cloud containing Nc = (250 ± 10) × 108 condensed atoms.For measurements shot back to back, the number of condensedatoms changes approximately 1%. Since we assume thatthe largest contribution is caused by the difference in theenvironmental conditions, this result sets the upper limit of 1%for the shot-to-shot reproducibility of the PCI method usingthe two-parameter fit.

Our fit procedure, in which µ and T are determined byboth phase and size simultaneously using the Popov model,yields statistical errors less than 1%, smaller than the estimatedsystematic error. The uncertainty in the number of condensedatoms, which scales as Nc ∼ µ5/2, is therefore estimated fromthe discrepancy between µax and µφ , which is roughly 3%and yields an uncertainty of 5% in the number of condensedatoms.

In an experiment we set up to determine the accuracy ofthe method in a single shot, we produce clouds at varioustemperatures below Tc by changing the final frequency ofthe evaporation ramp. The resulting number of thermal andcondensed atoms as well as the condensate fraction determinedon the fitted µ and T is shown in Fig. 7, where each pointcorresponds to a single measurement. Since the fluctuationsin these results are only a few percent for the number of boththermal and condensed atoms, the measurements show thatfrom a single measurement we can determine the numberof atoms with a high accuracy over the whole temperaturerange. This can only be achieved if two conditions are fulfilled,namely, that the PCI method produces consistent results overthe whole temperature range and that the reproducibilityof the production of Bose condensed clouds is large. Note

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0.2 0.4 0.6 0.8 1.0 1.2Temperature (µK)

0.00.20.40.60.81.00.00.51.01.52.0

2.5x10802468

10x108

0.0

Nex

Nc

Nc/

(Nc+

Nex

)

FIG. 7. The number of thermal atoms Nex, number of condensedatoms Nc, and condensate fraction as a function of the temperatureT . Each point is the result of a fit to the Popov model for a singlemeasurement. The statistical uncertainty is smaller than the size ofthe symbol.

that the condensation fraction is not given by the simpleideal-gas behavior of [1 − (T/Tc)3]. This is not caused bythe interactions between the condensed and the noncondensedatoms, but due to the fact that Tc depends on the total numberof atoms. Since lower temperatures are reached by evaporativecooling, the total number of atoms decreases upon loweringthe temperature, as can be seen from adding the number ofatoms in the first two graphs.

F. Transition temperature for condensation

Generating clouds above as well as below the transitiontemperature allows us to determine the temperature at whichcondensation sets in. The Popov model yields µ and T in bothregions, where µ becomes negative for temperatures aboveT 0

c . Note that our implementation of the Popov model reducesto the Hartree-Fock model for T > Tc [16].

For T − T 0c � T 0

c the chemical potential as a functionof temperature for a noninteracting Bose gas can be writtenas [16]

µ ≈ 3ζ (3)

ζ (2)kB

(T − T 0

c

), (35)

where ζ is the Riemann ζ function. Equation (35) can be usedto estimate T 0

c . Note that the repulsive interactions lower thecentral density and therefore reduce the transition temperature.Therefore, condensation does not set in at T 0

c , where µ = 0,but at µ = 2U0nex.

In the experiment we generate clouds at different temper-atures above as well as below the transition temperaturesand derive T and µ from a fit to the Popov model. Theresults presented in Fig. 8 show positive values of µ forall temperatures, although no BEC is observed in the fivepoints with the highest temperatures. The seven data pointswith the highest temperatures are used to make a fit toEq. (35), with T 0

c being the only free parameter. This yieldsT 0

c = 0.998 ± 0.007 µK. The thermal density is determinedby averaging the two measurements which are the closest to the

0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

Che

mic

alpo

tent

ial /

h (k

Hz)

Temperature (µK)

FIG. 8. The chemical potential of a cold cloud derived from thefit to the Popov model as a function of temperature, both aboveand below Tc. The line is the result of a fit to Eq. (35) on theseven data points where µ/h < 2.5 kHz. The data point enclosedin the box marks the observation of a small BEC. The data pointmarked by the cross is the point with the lowest temperature, forwhich no bimodal distribution is observed. The dashed line indicatesµ = 2U0nex, the expected value at which condensation will set in,calculated using the average thermal density of the two accentuateddata points.

temperature where a condensate is formed. Both nex and T 0c are

used to find the temperature in Eq. (35), where µ = 2U0nex =h(1.65 ± 0.05) kHz. This yields Tc = 0.972 ± 0.008 µK.

The measured shift due to the interactions is (T 0c −

Tc)/Tc = 0.027 ± 0.001, which agrees well with the theoret-ical prediction based on the Hartree-Fock mean-field theory,�Tc/Tc ≈ −1.33a/aN1/6 ≈ 0.025 [16].

To confirm that the systematic errors are well estimated,we compare the value of Tc to the highest temperature wherewe observe a bimodal distribution in the image, both by fittingthe cloud and by examining line profiles through the center ofthe cloud. The highest temperature where a BEC is observedis determined at T = 0.963 ± 0.005 µK, where the lowesttemperature without a BEC yields T = 0.975 ± 0.003 µKand the determined value of Tc indeed lies in between thesetwo temperatures. The consistency of these numbers suggeststhat the systematic errors are well estimated.

VII. CONCLUSION

In conclusion, this paper describes the accurate determi-nation of the density distribution of condensates at finitetemperatures using PCI. A detailed description is given of theamount of phase a probe beam accumulates when a cloud ofatoms is passed. The resulting images show a periodic variationof the intensity as a function of the accumulated phase, andthe resulting high dynamic range is used to measure thedensity distribution of both a thermal cloud and a condensatesimultaneously with a high accuracy. Since all measurementsare done in situ, there is no need to describe the expansion of thecloud. The size of the condensate, also a measure of its density,can therefore be determined accurately as well. The size of thethermal cloud, the measure for the temperature of the cloud,is also determined from the same image. Only two relevantparameters, µ and T , are needed to describe the measuredclouds given the harmonic confinement, as µ and T determineboth the peak density and the size of both components ofthe cloud. This procedure is used to discriminate among

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three models describing the equilibrium properties of trappedclouds at nonzero temperatures, each model incorporatingthe interactions between thermal atoms and condensed atomsdifferently.

We resolve the effect of the interaction on the densitydistribution of thermal cloud as well as the smaller effect theinteraction has on the density distribution of the condensate.The model accounting for these effects gives consistent resultsover the complete temperature range. The imaging scheme isused to determine the number of atoms within 1% from shotto shot at our typical number of condensed atoms, between2 × 108 and 3 × 108. Systematic errors are estimated bycomparing the different measures of the chemical potential,and we find the uncertainty in the number of condensed atoms

to be roughly 5%. PCI can be used both below and abovethe transition temperature, and we used it to determine Tc

and derived the shift of the transition temperature due to theinteractions. The agreement with the theoretical value of thisshift indicates that the systematic errors are smaller than theuncertainty in the determination of T .

ACKNOWLEDGMENTS

We are grateful to Chris Retif and AMOLF for manufactur-ing the phase spot. This work was supported by the Stichtingvoor Fundamenteel Onderzoek der Materie (FOM) and bythe Nederlandse Organisatie voor Wetenschaplijk Onderzoek(NWO).

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