Observation of Faraday Waves in a Bose-Einstein Condensate

0.17 MB·English

by  P. Engels

Observation of Faraday Waves in a Bose-Einstein Condensate
∗
P. Engels and C. Atherton
Washington State University, Department of Physics and Astronomy, Pullman, WA, 99164
M. A. Hoefer
National Institute of Standards and Technology, Boulder, Colorado 80305†
(Dated: February 6, 2008)
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Faraday waves in a cigar-shaped Bose-Einstein condensate are created. It is shown that periodi-
0
callymodulatingthetransverse confinement,andthusthenonlinearinteractionsintheBEC,excites
2
small amplitude longitudinal oscillations through a parametric resonance. It is also demonstrated

n that even without the presence of a continuous drive, an initial transverse breathing mode excita-
a tionofthecondensateleadstospontaneouspatternformationinthelongitudinaldirection. Finally,
J theeffectsofstrongly drivingthetransversebreathingmodewithlarge amplitudeareinvestigated.

1 In this case, impact-oscillator behavior and intriguing nonlinear dynamics, including the gradual
emergence of multiplelongitudinal modes, are observed.

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r
e PACSnumbers: 03.75.Kk,67.90.+z,05.45.-a,47.54.-r
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In 1831, Faraday studied the behavior of liquids that typical for parametrically driven systems. Floquet anal-
.
t are contained in a vessel subjected to oscillatory verti- ysis reveals that a series of resonances exist, consisting
a
m cal motion [1]. He found that fluids including alcohol, of a main resonance at half the driving frequency, and
white of egg, ink and milk produce regular striations on higher resonance tongues at integer multiples of half the
-
d their surface. These striations oscillate at half the driv- driving frequency [4].
n ing frequency and are termed Faraday waves. They are
In our experiment we exploit this transverse modula-
o
consideredto be animportantdiscovery. Since then, the
c tion scheme for three different applications. First, we
moregeneraltopicofpatternformationindrivensystems
[ apply a relatively weak continuous modulation, demon-
hasbeenmetwithgreatinterest,andpatternshavebeen
strate the emergence of longitudinal Faradaywaves,and
1
observed in hydrodynamic systems, nonlinear optics, os-
v study their behavior as a function of the excitation fre-
cillatory chemical reactions, and biological media [2].
8 quency. Second,weinvestigatelongitudinalpatternsthat
2 In this paper we study pattern formation by mod- emerge as a consequence of an initial transverse breath-
0
ulating the nonlinearity in a Bose-Einstein condensate ingmodeexcitationwithoutthepresenceofacontinuous
1
(BEC). Nonlinear dynamics arise from the interatomic drive. Thishasimportantconsequencesinthecontextof
0
7 interactions in this ultracold gas. In the past the ob- damped BEC oscillations and has been studied theoret-
0 servation of interesting phenomena has motivated re- ically in [6]. Since the first experiments with BECs, the
/ searchers to propose and implement various techniques study of collective excitations has been a central theme
t
a to manipulate the nonlinearity. Such control has been [7, 8]. The transverse breathing mode, which we exploit
m
accomplished for example by exploiting Feshbach reso- in our experiments, plays a prominent role: Chevy et al.
- nances [3]. In our experiment we investigate an alterna- [9] showed that this mode exhibits unusual properties,
d
n tive technique, namely periodically modulating the non- namelyanextremelyhighqualityfactorandafrequency
o linearity by changing the radial confinement of an elon- nearly independent of temperature. Finally, in a third
c gated, cigar-shaped BEC held in a magnetic trap. The set of measurements we study the situation of a rela-
:
v radial modulation leads to a periodic change of the den- tively strong modulation, resonantly driving the trans-
i sity of the cloud in time, which is equivalent to a change verse breathing mode. We show that the condensate re-
X
ofthenonlinearinteractionsandthespeedofsound. This sponds as an impact oscillator,which leads to intriguing
r
a can,inturn,leadtotheparametricexcitationoflongitu- multimode dynamics.
dinal sound-like waves in the direction of weak confine-
The experiments were carried out in a newly con-
ment. ThisprocessisanalogoustoFaraday’sexperiment
structed BEC machine that produces cigar-shaped con-
wheretheverticalmotionofthevesselproducedpatterns densates of 87Rb atoms in the |F = 1,m = −1i
F
that were laterally spread out. state. The typical atom number in the BEC is 5·105,
IthasbeenshowntheoreticallythatforaBEC,aFara- and the atoms are evaporatively cooled until no thermal
day type modulation scheme in the case of small driv- cloud surrounding the condensate is visible any more.
ing frequencies leads one to the same type of analysis as The atoms are held in a cylindrically symmetric Ioffe-
would the direct modulation of the interatomic interac- Pritchard type magnetic trap with the harmonic trap-
tion, e.g., by a Feshbach resonance [4, 5]. In both cases, ping frequencies of {ω /(2π),ω /(2π)} = {160.5,7}Hz.
xy z
thedynamicsaregovernedbyaMathieuequationthatis The weakly confined z-direction is oriented horizontally.
2
For the experiments described below, the following col-
112200 HHzz
lective mode frequencies are of particular importance:
first, there exists a high-frequency transverse breathing
mode. For ourtrapgeometry,this mode hasa frequency
115500 HHzz
of ω⊥/(2π) = 321 Hz [10, 11], very close to the limit of
vanishing axialconfinement 2·ω /(2π). The secondset
xy
ofmodesinwhichweareinterestedhereconsistsofaxial 222200 HHzz
modes which, for largequantum numbers, correspondto
sound waves in the z-direction. The frequencies of this
discretesetofmodes canbe approximatelycalculatedas 332211 HHzz
given in [12, 13].
In order to investigate the parametric driving pro-
112255 mmmm
cess mentioned above, we first performed a set of ”spec-
troscopy” experiments in which we continuously mod-
ulated the transverse trapping confinement at a fixed FIG. 1: In-trap absorption images of Faraday waves in a
modulationfrequencyandobservedthesubsequentemer- BEC. Frequency labels for each image represent the driving
genceoflongitudinalFaradaywaves. Foreachexcitation frequencyat which thetransversetrap confinement is modu-
lated.
frequency the modulation amplitude was adjusted such
thatthe longitudinalpatterns emergedtypicallyatsome
point after 10 to 30 oscillations. On the breathing mode
resonance,a trap modulation of 3.6% was used, while at 30
]
s
many other frequencies trapmodulations of up to 42.5% n
o
were chosen to obtain clearly visible patterns [14]. This cr 25
range of modulation depths is similar to the range used mi
[
in numerical simulations in [5]. Representative exam- y 20
t
ples for the resulting patterns are shown in Fig. 1. All ci
experimental images in this manuscript were taken by odi 15
destructive in-trap imaging. ri
e
p
The averagespacingofadjacentmaxima inthe result-
n 10
ing pattern is plotted against the driving frequency in er
t
Fig. 2. The data lie on a clear curve, with the exception at
p 5
of the points near a driving frequency of 160.5 Hz, cor-
responding to the transversedipole mode resonance (i.e.
100 200 300 400
transverse slosh motion)[10]. However, inspection of our
experimental images reveals that, at this frequency, we drivingfrequency[Hz]
alsoexcitethetransversebreathingmodeat321Hz. Ex-
citation of the breathing mode is a very effective way of
creating longitudinal patterns. Therefore the patterns
FIG.2: Averagespacingofadjacentmaximaofthelongitudi-
obtained at160.5 Hz are actually the same as those pro-
nal patterns plotted versus the transverse driving frequency.
ducedat321Hz. Inordertorationalizethedata,wefirst
Points are experimental data, while the line shows the theo-
note that parametric excitation with a certain driving
reticalvaluescalculated for thelongitudinalmodesclosest to
frequency excites oscillations predominantly at half the half the drivingfrequency.
driving frequency, the main resonance also observed in
Faraday’s experiments. The dispersion relation of longi-
tudinal collective modes that become sound-likefor high duringthepresenceofourcontinuousdriveduetoexper-
quantumnumbersisgivenin[12,13]andisusedtocalcu- imentalimperfections[14]. Forthis,weexcitedthetrans-
late the expected spacing between density maxima. The versebreathingmodeat321Hzbydrivingthetransverse
resulting spacings are plotted as the step-like curve in trap confinement for a few cycles, and then let the con-
Fig. 2 and are in excellent agreement with our experi- densate evolve without the presence of the drive. For
mental data, corroborating the assumption and theory a gentle excitation, we do not see longitudinal patterns
of a parametric driving process. immediately after the end of the drive, but can observe
In a second set of experiments we show that the lon- them emerge at later times. A weaker excitation delays
gitudinal modes are driven by the transverse breathing the emergence of longitudinal modes out to later times,
motion even without the presence of a continuous exter- while in the case of strong excitations the patterns can
nal drive. In particular, this disproves the influence of emerge within the first three cycles. In order to follow
any tiny residual axial trap modulation that has existed the evolution, we quantify the presence of longitudinal
3
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a.a.11550000
0.018 n[n[
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0.016 sese11000000
ossoss
visibility 00..001124 ntegratedcrntegratedcr 550000
pattern 0.010 ii 0000 5500 11llee00nn00ggtthh [[mmiiccrr11oo55nn00ss]] 220000 225500
0.008
FIG. 4: Pattern domains observed in a very elongated BEC.
0.006 The graph shows an integrated cross section of theBEC dis-
played in the inset.
0 5 10 15 20
timeafterendofexcitation[Tbreathing]
emerge in spatially localized domains, rather than uni-
formly across the whole cloud. To show this effect, we
FIG. 3: Onset of Faraday wave formation after driving the produced condensates in a very elongated cigar-shaped
transverse breathing mode for only two cycles. Pattern visi- trap with trapping frequencies of {ωxy/(2π),ωz/(2π)}=
bility is quantifiedas described in [15]. {286.1,2.8}Hz. We temporarily jumped to a different
trap of {ω /(2π),ω /(2π)} = {88.4,5.1}Hz for the du-
xy z
ration of 1.3 ms, then jumped back to the first trap and
patterns as described in [15]. Fig. 3 shows the evolution let the cloud evolve. After about 10 ms, we observed
after the end ofa moderate excitation. For this data, we BECs in which a perfectly periodic density modulation
excited the condensate for only two cycles, varying the wasstretchingoveralmostthe entirecloud; butit isalso
transverse trap frequency by 9% with a modulation fre- not uncommon to observe patterns in several separate
quency of 321 Hz during the modulation. Immediately domains, as shown in Fig. 4. Considering that the speed
aftertheexcitation,theobtainedimagesshowednolongi- of sound in the elongated cloud is only about 2 mm/s
tudinalwaves,andourpatternvisibilitymeasure,plotted [13, 16], it is plausible that over the time scale of this
in the figure, is initially picking up high frequency noise experiment different regions of the BEC can get inde-
along the longitudinal axis. A main contribution to this pendently excited and evolve into longitudinal patterns
noiseis the imagingnoise ofourdetectionsystem. Weak independently from each other.
pattern formation is observedstarting at five periods af- Intheexperimentsdescribedsofar,ourparametricex-
ter the end of the modulation, and strong longitudinal citation has led to longitudinal modes oscillating at half
patternsthenappearafteraboutnine periods. Asimilar the frequency of the parametric driving frequency. In
behavior is known for example from parametric ampli- the case ofstrongparametricamplification,it is theoret-
fiers in optics: if no input signal amplitude is present, ically expected that modes at other frequencies (higher
a signal emerging from noise (or zero point energy) can resonance tongues), in particular modes at the driv-
form if the amplification is large enough. This behavior ing frequency, can be excited, too [4]. This motivated
is also found by our numerical simulations based on the our third set of experiments in which we started again
Gross-Pitaevskii equation. In simulations with similar with a condensate in a trap with trap frequencies of
parametersasintheexperiment,nopatternformationis {ω /(2π),ω /(2π)} = {160.5,7}Hz. We then continu-
xy z
observed for several breathing mode periods. From the ously modulated our transversetrap frequency by about
onsetofpatternformation,ittakesjustthreeperiodsfor 19%with a modulationfrequencyof 321Hz. The result-
patterns to grow to their full strength. We find that the ing breathing motion is seen in Fig. 5 where we plot the
onsettimeofpatternformationinthesimulationsisear- Thomas-Fermiradiusofthecloudinthetransversedirec-
lier when more noise is added to the initial relaxedwave tionversustime. Thegraphclearlyshowsthatthecloud,
function. upon strong excitation, gradually starts behaving as an
Thisexperimentiscloselyrelatedtothetheoreticalsit- impactoscillator,i.e., asanoscillatorbouncingoffa stiff
uation described in [6], where a single and sudden jump wallduringeachperiod. Aclassicalimpactoscillator,re-
in the transverse trap frequency was used to excite the alizedforexamplebyaballbouncingoffastiffsurface,is
breathing mode, instead of a sinusoidal trap frequency aparadigmfornonlineardynamics,nonlinearresonances
modulation. We have used a trap jumping excitation andchaoticbehavior. In the presentcase,the roleofthe
in the case of very elongated BECs to demonstrate a stiff wall was played by the strong mean-field repulsion
second observation about the onset of longitudinal pat- duringtheslimphaseoftheoscillationswhentheBECis
terns, namely the fact that these patterns can start to stronglycompressedinthetransversedirection. Theoret-
4
aa)) resonances lead to Faraday waves along the long BEC
axis. These results advance the understanding of col-
lective mode behavior in a condensate, which is one of
4400
112255mmmm the key tools to study BEC dynamics. In addition, we
3355 bb)) have shown that strongly driving the transverse breath-
s]s] ing mode leads to an instability whereupon the mode
nn
oo 3300 amplitude increases exponentially, accompanied by the
crcr
mimi strong excitation of multiple sound-like modes.
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FerFer 1155 ∗ Electronic address: [email protected]
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TT 55 [1] The study is contained in the appendix of the paper M.
Faraday,Philos.Trans.R.Soc.London121,299(1831).
00 [2] For a comprehensive review, see M. C. Cross and P. C.
00 55 1100 1155 2200 2255 3300 Hohenberg, Rev.Mod. Phys.65, 8511112 (1993).
[3] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D.
ttiimmee [[mmss]]
M. Stamper-Kurn, and W. Ketterle, Nature (London)
392, 151 (1998).
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the tight direction vs. time when continuously driving the Rev. Lett.89, 210406 (2002).
transverse breathing mode. Points show experimental data, [5] K. Staliunas, S. Longhi, and G. J. de Valc´arcel, Phys.
while the curve corresponds to numerical simulation. Inset: Rev. A 70, 011601(R) (2004).
(a) Image of BEC after 5.2 driving periods, i.e. 16.2 ms. (b) [6] Yu.KaganandL.A.Maksimov,Phys.Rev.A64,053610
Fourier transform of theimage in (a). (2001).
[7] D.S.Jin,J.R.Ensher,M.R.Matthews,C.E.Wieman,
and E. A.Cornell, Phys.Rev.Lett. 77, 420 (1996).
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Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle,
drivingandthe impactoscillatorbehaviorhavebeenan-
Phys. Rev.Lett. 77, 988 (1996).
alyzed in [17]. The behavior of the cloud radius in the
[9] F.Chevy,V.Bretin,P.Rosenbusch,K.W.Madison,and
transverse direction was also reproduced in our numer- J. Dalibard, Phys. Rev.Lett. 88, 250402 (2002).
ical simulation of the azimuthally symmetric 3D Gross- [10] S. Stringari, Phys.Rev.Lett. 77, 2360 (1996).
Pitaevskii equation, displayed as the solid line in Fig. 5. [11] V. M. P´erez-Garc´ıa, H. Michinel, J. I. Cirac, M. Lewen-
However,the numerics, when starting with a thoroughly stein, and P. Zoller, Phys.Rev. Lett. 77, 5320 (1996).
relaxed wave function in the initial trap, show a sign of [12] M. Fliesser, A. Csord´as, P. Sz´epfalusy, and R. Graham,
Phys. Rev.A 56, R2533 (1997).
longitudinalpatternformationonlyafter18ms,while in
[13] S. Stringari, Phys.Rev.A 58, 2385 (1998).
the experiment, longitudinal patterns clearly formed al-
[14] Sinusoidally modulating our bias field strength led to a
ready during the third period (9 ms). This, again, hints
modulation of the transverse confinement ωxy/(2π) be-
at the importance of initial noise in the condensate that tween 155 Hz and 166.7 Hz in the case of a 3.6% modu-
seeds the parametric amplification. In the experiment, lation depth, and between 122 Hz and 303 Hz for our
thepatternsstartoutsimilartothosedisplayedinFig.1 strongest modulation depth of 42.5%. Even with our
for the case of a weak drive at 321 Hz. But, upon the strongest transverse modulation, the longitudinal con-
finement was modulated byless than 0.6%.
action of the strong drive, they quickly evolve into more
[15] We integrate the observed density along the tight direc-
complicated patterns, involving the excitation of several
tion,anddeterminethestandarddeviationoftheresult-
other modes. The inset of Fig. 5 shows an image taken
ingcrosssection from aThomas-Fermishapeinthecen-
after 5.2 driving periods (a), together with the Fourier tral region. We normalize the result by dividing by the
transform(b). TheFourierspectrumrevealsthatseveral numberofatomsinthesameregion.TheThomas-Fermi
modescorrespondingtothefirstresonancetongueoflon- shapeisobtainedbyeliminatingallbutthelowfrequency
gitudinal modes with nearly half the driving frequency Fouriercomponentsthatcorrespondtotheoverallshape
oftheBEC.Thisprocedureprovidesagood measurefor
are excited. In addition, modes at twice the distance
the excitation of the longitudinal modes, which we have
from the central Fourier peak are visible. Those modes
checked also by visual inspection of thedata.
belong to the second resonance tongue of the main reso- [16] E. Zaremba, Phys. Rev.A 57, 518 (1998).
nance. [17] J.J.G.RipollandV.M.P´erez-Garc´ıa,Phys.Rev.A59,
Inconclusion,wehaveexperimentallyobservedthe ef- 2220(1999);J.J.Garc´ıa-Ripoll,V.M.P´erez-Garc´ıa,and
fects of parametric resonances in a BEC. The observed P. Torres, Phys. Rev.Lett. 83, 1715 (1999).