Excitations of Bose-Einstein condensates in a one-dimensional periodic potential

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by  N. Fabbri

Excitations of Bose-Einstein condensates in a one-dimensional periodic potential
N. Fabbri*, D. Cl´ement, L. Fallani, C. Fort, M. Modugno, K. M. R. van der Stam, M. Inguscio
European laboratory for Non-linear Spectroscopy (LENS),
Dipartimento di Fisica - Universit`a di Firenze and INFM-CNR,
via N. Carrara 1, 50019 Sesto Fiorentino (FI), Italy
Wereportontheexperimentalinvestigationoftheresponseofathree-dimensionalBose-Einstein
condensate (BEC) in the presence of a one-dimensional (1D) optical lattice. By means of Bragg
spectroscopyweprobethebandstructureoftheexcitationspectruminthepresenceoftheperiodic
potential. We selectively induce elementary excitations of the BEC choosing the transferred mo-
9
mentumandweobservedifferentresonancesintheenergytransfer,correspondingtothetransitions
0
to different bands. The frequency, the width and the strength of these resonances are investigated
0
as a function of theamplitude of the1D optical lattice.
2

n PACSnumbers: 67.85.Hj,67.85.De
a
J
3 The knowledge of the linear response of a complex 120 a)
1 system gives crucial information about its many-body 100
r] btherheaev-dioirm.enFsoiorneaxla(m3Dp)leB, otshee-Esiunpseteriflnucidonpdreonpseartteie(sBEofCa) σ μ(m) 8600
e are related to the linear part of the phononic dispersion Δ
40
h
relation at low momenta [1]. The presence of optical
t 20
o lattices enriches the excitation spectrum of a BEC in a
0
. remarkableway. Fordeepthree-dimensionallattices,the
t
a gas enters the strongly correlated Mott insulator phase 0 20 40 60 80 100
m ν (kHz)
andthe spectrum exhibits a gapatlow energies[2]. The
- response of a BEC in the superfluid phase is also drasti-
d callymodifiedbythe presenceofaone-dimensional(1D) E )R b)
n optical lattice [3, 4, 5, 6, 7]. Indeed, as in any periodic s ( 20
o e
c system, energy gaps open in the spectrum at the multi- ergi 15
[ ples of the lattice momentum and it is possible to excite n
severalstates corresponding to different energy bands at e e 10
1 c
v a given value of the momentum transfer [8, 9]. In addi- nan 5
5 tion, the linear dispersion relation of the superfluid, and o
s
0 thus its sound velocity, is changed. In the mean-field Re 0
8 regime ofinteractionsthese peculiar features ofthe exci- -1.0 -0.5 0.0 0.5 1.0
1 tations of a superfluid BEC in the presence of an optical q/qL
.
1 lattice are captured by the Bogoliubov theory [1].
FIG. 1: (a) Measured BEC excitation spectrum in the pres-
0 Braggspectroscopyrepresentsanexcellentexperimen- enceofalatticewithheights=(22±2). Theincrease∆σ of
9
taltooltoinvestigatethelinearresponseofgaseousBECs
thewidthoftheatomicdensitydistributionismonitoredasa
0
: [10]. It has allowedto measure the dispersion relationof function of the relative detuning ν between the two counter-
v interactingBECsinthemean-fieldregime[11,12,13],to propagatingBraggbeams. ThedataarefittedwithGaussian
Xi characterize the presence of phase fluctuations in elon- functions (black line). The arrows below the resonances in-
gated BECs [14], to study signatures of vortices[15] and dicate the corresponding bands, represented in (b) with the
r
a more recently to study strongly interacting 3D Bose [16] samecolors. (b)Bandstructureoftheexcitationspectrumof
andFermi[17]gasesclosetoFeshbachresonancesaswell aBECina1Dopticallatticewiths=22: first,second,third
and fourth bands are represented (gray, green, blue and red
as 1D Bose gases across the superfluid to Mott insulator
linesrespectively). Thearrowsindicatetheprocessesstarting
transition [18].
fromaBECatq=0andinducingthecreationofexcitations
In this work we use Bragg spectroscopy to probe the
in the different bandsat a quasi-momentum transfer 0.12qL.
excitation spectrum of a 3D BEC loadedin a 1D optical
lattice. Previous experimental studies have so far in-
vestigated the excitations of superfluid BECs within the
lowest energy band of a 3D optical lattice by means of widths ofthe transitionsto differentbands ofthe 1Dop-
lattice modulation [19] and Bragg spectroscopy [18, 20]. tical lattice. The measurements are quantitatively com-
This paper presentsa detailedexperimentalstudy ofthe paredwithBogoliubovmean-fieldcalculationsforourex-
differentbandsintheexcitationspectrumofaninteract- perimental system [7].
ing 3D BEC in the presence of a 1D optical lattice. We We produce a 3D cigar-shaped BEC of N≃ 3 × 105
measuretheresonancefrequencies,thestrengthsandthe 87Rb atoms in a Ioffe-Pritchard magnetic trap whose
2
axial and radial frequencies are ω = 2π ×8.9 Hz and tice of relatively small amplitude (s = 5), the density
y f
ω = ω = 2π×90 Hz respectively, corresponding to a distributionexhibits aninterferencepattern[21]. We ex-
x z
chemicalpotentialµ≃h×1kHz,withhbeingthePlanck tract the rms width σ of the centralpeak of this density
constant. The condensate is loaded in an optical lattice distribution by fitting it with a Gaussian function. The
along the longitudinal direction (yˆ axis). Two counter- increase∆σ of this quantity is usedasa measurementof
propagating laser beams with wavelength λ = 830 nm the energy transfer [19]. For a given value of the trans-
L
create the lattice potential V(y) = sE sin2(q y) where ferred momentum q and amplitude s of the lattice, this
R L
q = 2π/λ = 7.57 µm−1 is the wave-number of the procedure is repeated varying the energy hν of the exci-
L L
beams, s measures the height of the lattice in units of tation in order to obtain the spectrum. In our regime of
the recoilenergyE =h2/2mλ2 ≃h×3.3kHz, misthe weak inter-atomic interactions, the excitation spectrum
R L
massofa87Rbatom. TheloadingoftheBECinthe lat- ofthe BEC inthe presence ofa1Dopticallattice canbe
tice is performed by slowly increasing the laser intensity described by the mean-field Bogoliubov approach [3, 5],
up to a height s with a 140 ms long exponential ramp by which we calculate the resonance frequencies ν and
j
with time constant τ =30 ms. the transition strengths Z to create an excitation in the
j
After a holding time of typically 20 ms in the lattice Bogoliubov band j.
at the height s, we excite the gas by shining two off-
resonant laser beams (Bragg beams) for a time ∆tB =3 25 6.5
ms. The Bragg beams induce a two-photon transition E ) R 65..05
transferring momentum and energy to the atomic sam- s (20 5.0
atporleea.tywTpahivceeai-rlnlyuwmadvbeetelurennqegBdth=byis2π3λ5/B0λB=G=H7z880.w05intmhµmrce−osrp1r,eecastpnodtnotdhtinheyge ce energie15 44..500 1 2 3 4 5 6
n
D2 transition of 87Rb. To change the transferred mo- na10
o
mentum we use two different geometries of the Bragg es a)
R
beams. In the first configuration the two beams are 5
counter-propagatingalongthe yˆdirectionandthe trans- 0 5 10 15 20 25 30
Amplitude s of the 1D lattice (E )
ferred momentum is q =2q =2.12q , that corresponds R
B L
to a quasi-momentum 0.12qL. In the second configura- 1.0
tion the angle between the Bragg beams is smaller and
0.8
the measured value of the transferred momentum (and s
h
quasi-momentum, in this case) along the yˆ direction is ngt0.6
e
qtu=ned0.f9r6oqmL.eaIcnhbooththertbhye acafsreesq,utehnecytwdoiffbereeanmces νaruesdineg- ative str0.4
twophase-lockedacousto-opticmodulators. Wequantify Rel0.2
b)
the response to the excitation by measuring the energy
0.0
transferredtothegaseousBEC.Themeasurementofthe
0 5 10 15 20 25 30
energy transfer E(ν,q) is connected with the dynamical
Amplitude s of the 1D lattice (E )
R
structure factor S(ν,q) (giving information on the exci-
tation spectrum) by the relation [1]
FIG. 2: (a) Band spectroscopy of a BEC in the presence of
E(ν,q)∝νS(ν,q), (1) a 1D optical lattice: theenergy of theresonances is reported
as a function of the height s of the lattice. The experimen-
tal points (green squares, blue circles and red diamonds) are
where ν and q are the frequency and the momentum of
compared with the numerical calculation of the Bogoliubov
the excitation. In particular, this result applies for long
spectrum in the presenceof a 1D lattice (solid lines) and the
enough Bragg pulses, namely ν∆t ≫ 1, which is the
B single-particle Bloch spectrum (dashed lines). The lines cor-
case in our experiment since ν is of the order of several
respond to the energy of an excitation in the second (green
kHz. line),third(blueline)andfourth(redline)Bogoliubovbands.
In order to get an estimate of the transferred energy In the inset of (a): zoom of the graph (a) for low values of
E(ν,q), we adopt the following procedure. We linearly s. (b) Relative strength of the excitations in the j=2, j=3
rampin15msthelongitudinalopticallatticefroms(the and j=4 bands. Symbols and colors are the same as in (a).
lattice height at which we have applied the Braggpulse)
to the fixed value s = 5. Then we let the excitation Wefirstdiscusstheresultsobtainedwiththeconfigura-
f
being redistributed over the entire system by means of tion of counter-propagatingbeams, i.e. for a transferred
the inter-atomic collisions for 5 ms. After this interval momentum q = 2.12q . The induced two-photon tran-
L
time we abruptly switch off both the optical lattice and sition is characterizedby a measured Rabi frequency for
the magnetic trap, letting the cloud expand for a time- theBECintheabsenceoftheopticallatticeΩ ≃2π×1
R
of-flight t = 20 ms and we then take an absorption kHzforthetypicalpoweranddetuningofthebeamsused
TOF
image of the density distribution integrated along the xˆ intheexperiment. AtypicalBraggspectrumispresented
axis. Since the atoms are released from an optical lat- inFig.1(a)correspondingtoalatticeheights=(22±2).
3
Thespectrumexhibitsmultipleresonancescorresponding Fig. 2(b)).
to the creationof excitations in the different Bogoliubov
bands as shown in Fig. 1(b). From Gaussian fit of each
resonance we extract the central frequency, the width 4000
and the relative strength of the transition towards the
correspondingband. InFig.2(a),weplottheenergyval- Hz) 3000
uescorrespondingtothemeasuredcentralfrequenciesas h (
dt
a function of s. The vertical error bars come from the s wi 2000
result of the fitting procedure while the horizontal error m
r
bars correspond to possible systematic errors in the lat-
1000
ticecalibration(estimatedwithin10%). Forlargeenough
amplitude s of the periodic potential we observe up to
0
threedifferentbands. Wefindagoodagreementbetween
0 5 10 15 20 25 30
the experimental data and the numerical results of the Amplitude s of the 1D lattice (E )
R
Bogoliubov calculation (solid lines in Fig. 2(a)). In par-
ticular, for low amplitudes of the 1D lattice (s < 6) the FIG. 3: Rms width of the resonances j = 2 (green squared)
agreementoftheresonanceenergieswiththeBogoliubov and j = 3 (blue circles) as a function of s. The gray region
bands (full lines) is better than with the single-particle corresponds to the experimental rms width (with its uncer-
tainty) for the BEC in the absence of the lattice (s = 0).
(dashed lines) Bloch bands (see inset in Fig. 2(a)). For
The green and blue lines are respectively the bandwidth of
larger amplitude of the 1D lattice, we can not explic-
the second and the third band, calculated in the mean-field
itly distinguish between the Bogoliubov and Bloch re-
Bogoliubov approach.
sults. This comes from the experimental uncertainty on
the calibration of the lattice amplitude.
FromtheGaussianfitofthe experimentalspectra(see
Intheentirerangeofsvaluesusedinthiswork,weob-
Fig. 1(c)), we also extract the rms width of the reso-
servea resonancecorrespondingto an excitationcreated
nances j=2 and j=3 with the results plotted in Fig. 3.
inthethirdbandj=3(bluecirclesonFig.2). Forlarger
Differentsourcescontributetobroadenthe observedres-
lattice amplitudes two other resonances appear, respec-
onances. TheinhomogeneousdensityofthetrappedBEC
tively for s > 4 and s > 20, corresponding to an excita-
isafirstsource[11]. Fromthe measuredspectrumofthe
tioninthej=2band(greensquaredinFig.2(a))andin
BEC in the absence of the optical lattice (s = 0) we
the j=4band(reddiamondsinFig.2(a)). This demon-
extract this contribution being (0.36±0.11) kHz, con-
stratesthepossibilitytoexcite,inaperiodicsystem,sev-
sistent with the expected value ≃ 0.26 kHz [11]. The
eral states for a given momentum transfer [8]. For weak
othercontributionstothewidtharerelatedtotheBragg
optical lattices creating an excitation in the j = 3 band
spectroscopic scheme. For our experimental parameters
is the most efficient process since the excitation energy
the largest contribution comes from the power broaden-
of this band is continuously connected as s → 0 to that
ing (∆ν ≃ 1 kHz), whereas the atom-light interaction
of the BEC in the absence of the 1D optical lattice at P
time broadening (∆ν ≃ 167 Hz) is much smaller. The
the transferredmomentum q =2.12q . On the contrary, t
L
total resonance width can be obtained by quadratically
the possibility to excite states in the second and fourth
adding up all these rms contributions. In the presence
bands of the optical lattice requires a large enough am-
of the optical lattice we observe that the widths of the
plitude s. These observations can be quantified in terms
resonancescorrespondingto the excitations inthe bands
of the strength Z of the different excitations, which can
j
j = 2 and j = 3 lie within the experimental range of
beextractedfromtheenergyspectrum. ThestrengthsZ
j the resolution as expected for a coherent system, except
areproportionaltotheintegral dν S (q,ν)withS (q,ν)
j j
in the case of j = 3 for large amplitudes of the lattice
being the structure factor correRsponding to the creation
(s > 20) where the width is much larger. We attribute
ofanexcitationintheBogoliubovbandj[5]. FromEq.1
these larger widths at high amplitude of the 1D lattice
and assuming that ν is much larger than the width of
j to the long tunneling times (0.11 s for s = 20) implying
the resonances of S (q,ν),we obtain
j
that the system is not fully coherent along the yˆ direc-
tiononthetimescaleoftheexperiment. Indeed,theloss
1 of coherence spreads the population of quasi-momenta
Z(q)∝ dν S(q,ν)∝ dν E(q,ν)≡g. (2)
j Z j ν Z j j across a larger fraction of the Brillouin zone. This re-
j
sults in a wider range of resonant energies in the system
In the experiment, we extract the quantity g from a and, for large amplitude s, one expects the width of the
j
Gaussianfit ofthe different resonances. Normalizingthe resonances in the energy spectrum to be equal to the
sumofthesequantitiestooneforthefirstthreeobserved bandwidths. In Fig. 3 we have plotted the bandwidths
resonances (g +g +g = 1) allows direct comparison of the j = 2 and j = 3 bands (green and blue lines).
2 3 4
with the relative strengths Z /(Z +Z +Z ) for j = When the system becomes incoherent the width of the
j 2 3 4
2,3,4. The comparison between the experimental data resonance j=3 is equal to the bandwidth. This effect is
and the calculation reveals a reasonable agreement (see not observable for the band j = 2 where the bandwidth
4
(green line in Fig. 3) is smaller than the experimental of height s =11 is depicted. Note that a first resonance
resolution. atlowfrequencyisvisiblecorrespondingtoanexcitation
with non-zeromomentum within the lowestenergyband
50 a) (j=1). Such a resonance is not observed using counter-
propagating Bragg beams because the strength of this
40
m) transition is negligible for q =2.12q . Due to the varia-
L
μ 30
Δσ ( 20 ptiroenvioofutshceaster,atnhsefefrrreeqdumenocmyeonfttuhmerqeswonitahnrceesspaercetsthoiftthede
10 according to the dispersion relation of the different en-
0 ergy bands of the system. In Fig. 4(b) we report the
frequencyofthe resonancesj=1,j=2andj=3 forthe
0 10 20 30 40
ν (kHz) two values of quasi-momentum used in the experiment
(0.12q and 0.96q ). We use the region comprised be-
L L
E )R12 b) ltiwneeesn) ttohetackaelcuinlatotedacbcoaunndts tfohres10=%10eraronrdisn=th1e2la(stotilcide
es ( 10 calibration. The experimental points are in good agree-
ergi 8 ment with the numerical calculation of the Bogoliubov
en 6 bands for s = 11 (dotted lines in the Figure) within the
e experimental uncertainty.
nc 4
a
on 2 In conclusion, Bragg spectroscopy has been used to
s
e probe the response of a Bose-Einstein condensate in the
R 0
presence of a 1D optical lattice. Changing the angle of
0.0 0.2 0.4 0.6 0.8 1.0 theBraggbeamsallowedustoinvestigateexcitationsfor
q/q
L a transferred quasi-momentum close to the center and
to the edge of the reduced Brillouin zone. We have ob-
FIG.4: (a)ExcitationspectrumofaBECinthepresenceofa
serveddifferentresonancesintheresponsefunctionofthe
1Dlatticewithheights=11atatransferredmomentumq=
system corresponding to the different bands of the peri-
0.96qLalongtheyˆdirection. Thearrowsbelowtheresonances
odic potential. Being the system in a weakly interact-
indicatethecorrespondingbands,representedin(b)withthe
ing regime, the experimental results are in quantitative
same colors. (b) Energy of the resonances corresponding to
an excitation in the band j = 1 (gray circle), j = 2 (green agreement with the Bogoliubov bands approach. This
squared)andj=3(bluecircle)asafunctionofthetransferred work opens the way to investigate the modification of
quasi-momentum q for a fixed value of the lattice height s= the excitation spectrum in the presence of an additional
(11±1). The experimental points are compared with the lattice with different wavelength (bichromatic potential)
numerical calculation of the energy bands in the Bogoliubov [22] and eventually to study the localization of the exci-
approach for s = 11 (gray, green and blue dotted lines); the tations in a true disordered potential [23].
solid lines correspond to the bands for s =10 and s =12 to
takeinto account the10% uncertaintyof s.
This work has been supported by UE contract No.
RII3-CT-2003-506350,MIURPRIN2007andEnteCassa
We also perform the experiment with a different con- di Risparmio di Firenze, DQS EuroQUAM Project,
figuration of the Bragg beams corresponding to a trans- NAMEQUAM Project and Integrated Project SCALA.
ferred momentum along yˆ q = 0.96q . In Fig. 4(a) an We acknowledge all the colleagues of the Quantum De-
L
excitation spectrum in the presence of an optical lattice generate Group at LENS for fruitful comments.
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