Controlled formation and reflection of a bright solitary matter-wave

0.49 MB·English

by  A. L. Marchant

Controlled formation and reflection of a bright solitary matter-wave
A. L. Marchant,1,∗ T. P. Billam,2 T. P. Wiles,1 M. M. H. Yu,1 S. A. Gardiner,1 and S. L. Cornish1
1Joint Quantum Centre (JQC), Durham - Newcastle, Department of Physics,
Durham University, Durham DH1 3LE, United Kingdom
2Jack Dodd Centre for Quantum Technology, Department of Physics,
University of Otago, Dunedin 9016, New Zealand
(Dated: January 25, 2013)
Solitons are non-dispersive wave solutions that etry is usually accompanied by the presence of weak ax-
arise in a diverse range of nonlinear systems, sta- ial harmonic trapping which removes the integrability of
3
blised by a focussing or defocussing nonlinear- the system and prevents the appearance of true solitons.
1
0 ity. First observed in shallow water [1], solitons Nevertheless, solitary wave solutions remain which re-
2 have subsequently been studied in many other tain many similarities to the classical soliton solutions
fields including nonlinear optics, biophysics, as- [9], such as propagation without dispersion.
n
a trophysics, plasma and particle physics [2]. They Previously, bright solitary matter-waves have been re-
J are characterised by well localised wavepackets alisedinthreeseparateexperiments[10–12]. Ineachcase

4 that maintain their initial shape and amplitude aFeshbachresonancewasusedtoswitchtheinteractions
2 for all time, even following collisions with other from repulsive (a >0) to attractive (a <0) in order to
s s
solitons. Here we report the controlled forma- formsolitarywavesoutofthecollapseinstability[13]. In
h] tion of bright solitary matter-waves, the 3D ana- twooftheseexperiments,multiplewavepacketswerecre-
p log to solitons, from Bose-Einstein condensates of ated,allowingthestudyofthedynamicsduringcollisions
- 85Rb and observe their propagation in an optical inthetrap. Theobservationofsolitarywavesraisesmany
m
waveguide. These results pave the way for new interesting questions regarding the relationship between
o experimental studies of bright solitary matter- the mathematical ideal and the experimental reality. It
at wave dynamics to elucidate the wealth of existing is unclear how soliton-like the solitary waves created in
. theoreticalworkandtoexploreanarrayofpoten- experimentswithfiniteradialtrappingandharmonicax-
s
c tial applications including novel interferometric ial confinement are. An answer to this question needs to
i devices [3], the study of short-range atom-surface beestablishedbeforepotentialapplicationsutilisingsoli-
s
y potentials [4] and the realisation of Schr¨odinger- tary waves can be realised. At a more fundamental level
h cat states [5, 6]. it remains to be tested whether or not the GPE treat-
p ment fully describes the solitary waves created in exper-
[ Bose-Einstein condensates formed from dilute atomic
1 gasessupportbrightsolitonsolutionsforattractiveinter- icmonetnatisn. (cid:46)So1l0it0a0ryatwoamvse,sprleaacliinsegdtehxepmerwimelelnotaultlsyidteypoifcatlhlye
atomic interactions (focussing nonlinearity), manifesting
v thermodynamic limit and potentially outside the reach
themselves as localized humps in the field amplitude. In
9 of the mean-field description. Several theoretical studies
contrast, dark solitons appear as localized reductions in
5
of bright solitary waves beyond the mean-field descrip-
7 an otherwise uniform field amplitude, preserved by a de-
tionhavenowbeenperformed, eitherincludingeffectsof
5 focussing nonlinearity (repulsive interactions). The con-
quantum noise using the truncated Wigner method [14]
. trol with which these systems can be manipulated, com-
1 orusingapproximateanalyticandnumericalmethodsto
0 binedwiththeuniquepropertiesofmatter-wavesolitons,
simulate the full many-body problem [5, 15]. These gen-
3 leadstoarichtestinggroundfortheoreticaldescriptions
erate results potentially in conflict with the behaviour
1 of quantum many-body systems. Condensates are com-
predicted by the GPE treatment.
: monly described by a mean-field treatment [7, 8] leading
v
In this work we report the controlled formation of
i to the well-known Gross-Pitaevskii equation (GPE) in
X bright solitary matter-waves from a 85Rb Bose-Einstein
which the atomic interactions are described by a nonlin-
r ear term proportional to the s-wave scattering length a condensate. The experimental geometry is such that the
a s velocityofthewavepacketscanbepreciselycontrolled, a
andthecondensatedensity. Intheone-dimensional(1D),
keyfactorinfacilitatingthefutureexplorationofsolitary
homogeneous limit the GPE takes the form of a nonlin-
wave interactions and collisions. In addition, we observe
ear Schr¨odinger equation which supports a spectrum of
andmodelthecontrolledreflectionofsolitarywavesfrom
mathematically ideal soliton solutions. Experiments ap-
a broad Gaussian potential barrier, demonstrating their
proach this theoretically ideal scenario by confining the
particle-like nature.
condensate in an elongated, prolate trap typically with
tight radial confinement. However, this quasi-1D geom- 85Rb is a prime candidate for solitary wave experi-
ments owing to the existence of a broad Feshbach res-
onance at ∼155 G in collisions between atoms in the
F = 2,m = −2 state. We use this resonance to form
F
∗Electronicaddress: [email protected] a stable, repulsively interacting condensate in a crossed
2
a b
Bias coils Offset coil BEC formed in
crossed dipole
Quadrupole coils trap, a s ~ 300 a 0 Magnetic potential
along waveguide
BEC released into waveguide
Change value of as
Crossed dipole
trap
Waveguide
1.0
140 c d
ontal width (µm)ontal width (m)µ 1102680000 sion rate (mm/s)sion rate (mm/s) 000...468
HorizHoriz 2400 xpanExpan 0.2
E 0.0
0
0 20 40 60 80 100 120 140 -100 0 100 200 300 400 500 600
ProEpxapgaantsiioonn ttimime e(m (sm)s) NNaa/ a/ 1(x010003) a
ss 0 0
FIG. 1: Expansion in the waveguide: (a) Experimental setup showing the glass science cell, the crossed dipole trap used to
createtheBEC,theopticalwaveguideandthequadrupole,biasandoffsetcoils. Alsoshowninthecellisasuper-polishedDove
prism (blue), mounted on a macor support, to be used for future experiments. (b) Schematic of the release of the condensate
from the crossed dipole trap into the waveguide. (c) Condensate expansion in the waveguide for a = 165 a (black), 23 a
s 0 0
(red), 4 a (blue), -7 a (green) and -11 a (purple). Solid lines are linear fits to the experimental data where the widths
0 0 0
are rms values. (d) Condensate expansion rate in the waveguide as a function of atom number and scattering length. The
solid line is the theoretical expansion rate calculated from a zero-temperature simulation of the experimental expansion using
a cylindrically-symmetric, 3D Gross-Pitaevskii equation. As in the data, the expansion rate is defined using the change in the
widthoftheBECbetween10msand100msafterreleaseintothewaveguidepotential,whichisapproximatelylinearoverthis
time interval in all cases
opticaldipoletrap,showninFig.1(a). Thecondensateis the repulsive wavepacket propagates the axial expansion
then loaded into a quasi-1D waveguide, better suited ge- causesasignificantdropinopticaldepthnotseenforthe
ometrically to the observation of solitary waves. At the solitary wave. We observe a solitary wave propagating
point of release into the waveguide, the magnetic bias over a distance of 1.1 mm in a time of ∼150 ms with
field controlling the atomic scattering length is jumped very little distortion.
to a new value (see Fig. 1(b)). As the BEC propagates To probe the stability of the solitary wave we investi-
along the waveguide the value of a determines the rate gate reflection of the wavepacket from a repulsive Gaus-
s
ofexpansionofthecondensateintheaxialdirection. We sian barrier with a 1/e2 radius of 130 µm, shown in
probethisexpansionbymeasuringthecondensatesizeas Fig. 3(a). Figures 3(b) and (c) show the position of the
a function of time for different values of a as shown in solitary wave as a function of time in the presence of a
s
Fig. 1(c). Fitting the experimental data we can extract 760 nK barrier potential. In this case the barrier height
an expansion rate for the BEC, dependent on a and N. isgreaterthanthekineticenergyofthesolitarywaveand
s
ThisisshowninFig.1(d), alongwitha3DGPEsimula- the wavepacket is cleanly reflected.
tionoftheexpansion(thesolidline). Atas =−11a0and Usingabarriermuchwiderthanthesolitarywavesize
N =2,000weseetheexpansionrateoftheBECbecomes the atomic center-of-mass coordinate behaves classically,
consistent with zero. This lack of dispersion with time with the solitary wave acting as a single particle rolling
indicates the formation of a bright solitary matter-wave. upapotentialhill.Byvaryingtheheightofthepotential
barrier it is possible to select whether the solitary wave
Figure 2 shows the propagation of this solitary wave, is reflected or allowed to travel over the barrier. The po-
contrasted to that of a repulsively interacting BEC. As sition of the solitary wave after 150 ms is shown in Fig.
3
a Optical depthOptical depth 0000....02468 b Optical depthOptical depth 0000....02468
140 ms
120 ms
100 ms
80 ms
60 ms
0 mm 0 mm 40 ms
0.5 mm 0.5 mm 20 ms
1.0 mm 1.0 mm
1.5 mm 1.5 mm
FIG. 2: Propagation in the waveguide: (a) As a repulsive BEC propagates along the waveguide the atomic interactions cause
the condensate to spread, leading to a drop in optical depth. (b) In contrast, the attractive interactions present in a bright
solitary matter-wave hold the atomic wavepacket together as it propagates, maintaining its shape with time. Crosscuts shown
are the horizontal optical depth profiles of the condensates after 140 ms propagation time along the waveguide.
3(d) as a function of barrier height. The solid line is a multiplewavepacketscanthenbeusedtoinvestigatethe
theoretical trajectory, calculated using classical mechan- phasedependenceofbinarycollisions[17], thebehaviour
icswithnofreeparameters,showingexcellentagreement of collisions of two solitary waves on a barrier [16, 18]
with the data. and would provide a solid first step towards the reali-
In Fig. 3(e) we compare the effect of reflection from sation of a bright solitary wave interferometer. In the
the barrier for a solitary wave and a repulsive BEC and limitoflowkineticenergyamean-fieldGPEtreatmentof
contrast the change in width to the case of a repulsive the problem begins to break down [22] and quantum be-
BEC propagating along the waveguide in the absence of haviour, described by the Lieb-Liniger Hamiltonian [23],
thebarrier. Thesolidlinesarethetheoreticalcondensate becomesmoresignificant. Here,splittingofthesolitonis
widths calculated by solving the 3D (cylindrically sym- energetically forbidden and it becomes possible to create
metric) GPE. As expected, the solitary wave is robust Schr¨odinger-cat states [5, 6].
against collisions with a repulsive Gaussian barrier and The use of a narrow potential to controllably split a
following the reflection maintains its shape, continuing solitary wave presents an opportunity to investigate one
to propagate without dispersion. In the absence of the of the key open questions arising from previous work;
barrier the repulsive BEC expands steadily in time. (We whatgovernsthedynamicsandstabilityofmultiplesoli-
attribute the disagreement between experiment and the- tary waves existing in the same trap? The long lived
oryatlongertimestoasmallthermalcomponentmaking nature of the solitary waves and their apparent stability
the measurement of the condensate width less accurate.) during binary collisions has been the subject of a wealth
In the barrier reflection case, an oscillation in the con- of theoretical work [14, 24–27]. Within the framework of
densate width is induced as a result of the larger spatial the GPE, the observed stability of soliton collisions can
extentoftherepulsiveBECcausingittobestronglycom- only be explained by imposing a relative phase φ = π
pressed as it is reflected from the barrier. Such contrast between neighbouring solitary waves [26] such that the
in the behaviour of the repulsive BEC and the solitary collisions are effectively repulsive in character. Several
wave reflection lends weight to previous theoretical pre- other studies address the apparent stability of solitary
diction regarding the superior characteristics of solitary waves in binary collisions, offering different interpreta-
wavesforobservingquantumreflectionfromsurfaces[4]. tions which do not require the imposition of a relative
phase φ = π between neighbouring solitary waves. The
There is currently much theoretical interest [16–20] in inclusion of quantum noise [14] or accounting for many
the scattering of solitary waves from narrow potential body effects [15] both result in effectively repulsive in-
barriers where, if the barrier width is on the order of teractions between solitary waves, irrespective of initial
the solitary wave width, quantum effects are observable. phase. Interestingly, incoherent, fragmented objects are
At high kinetic energy soliton splitting is energetically alsopredictedtoforminthemanybodyformalism. Fur-
allowed at narrow repulsive barriers. The effect of quan- ther experimental studies are undoubtedly required to
tumtunnellingmeansthebarriercanactasabeamsplit- addresstheroleoftherelativephaseinsolitarywavecol-
ter [16], dividing the soliton into two parts [21]. These lisionsandtotestthedifferenttheoreticaldescriptionsof
4
2.0
a b
K) 1.5
µ
otential/ k T (ential/ kT (K)µBB 01..50 BarrierWaveguide 12104 1m06 1sm08 s2m00 sm0 sms
Pot 100 ms
P 80 ms
0.0 60 ms
40 ms
-0.25 mm 0 mm 20 ms
0.0 0.5 1.0 1.5
0.25 mm
PoPsoistiiotionn ( (mmmm)) 0.5 mm
100
m) c m) 1.2 d e
m 1.5 m m)
stance from XDT (Horizontal position (mm) 01..50 stance from XDT (Distance from XDT (mm) 00001.....24680 Horizontal width (µHorizontal width (m)µ 24680000
Di 0.0 Di 0.0 0
0 50 100 150 200 250 300 0.0 0.2 0.4 0.6 0.8 0 50 100 150
PropEaxpgaantsiioonn t imtime (em s()ms) BarrieBra rhrieeri ghehigt h/t (kµ KT) (μK) ProPproapgaagattiioonn t imtime (em s()ms)
B
FIG. 3: Reflection from a repulsive Gaussian barrier: (a) Potential in the axial direction along the waveguide in the presence
of the repulsive barrier. (Inset, upper: Combined waveguide and Gaussian barrier potential. Lower: Experimental setup.) (b)
False colour images of a solitary wave reflecting from the barrier. The white line shows the location of the barrier centre. (c)
Horizontal position of a solitary wave propagating in the waveguide in the absence (red) and presence (black) of the repulsive
barrier. (d) The position of a solitary wave after 150 ms propagation time as a function of the barrier height. Red (black)
pointscorrespondtothesolitarywavetravellingover(beingreflectedfrom)thebarrier. Solidlinesin(c)and(d): Theoretical
trajectory calculated using a classical particle model with no free parameters. (e) Condensate width following reflection from
the barrier. In the absence of a barrier, a repulsive BEC will expand as it propagates (red). With the barrier in place, an
oscillationinthecondensatewidthissetupfollowingthestrongcompressionofthecondensateatthebarrierduetotheshape
ofthepotential(black). Asolitarywaveundergoingthesamecollisionemergesunaltered(blue). Solidlinesarethetheoretical
condensate widths calculated by solving the 3D (cylindrically symmetric) GPE.
quantum many-body systems. I. METHODS
A. Production of a tunable Bose-Einstein
condensate
We create a Bose-Einstein condensate with tunable
Althoughreflectionandsplittingexperimentsshowthe atomic interactions using the method described in [29].
potentialtosettlethetheoreticaldebateoverthesolitary A magnetic Feshbach resonance is used to tune both the
wave formation and dynamics, the ability to probe such elastic and inelastic scattering properties of the atomic
narrow and hence rapidly varying potentials using these sample to achieve efficient evaporation. Importantly the
wavepacketsalsolendsitselftoanobviousapplicationin resonance at 155 G in collisions between 85Rb atoms in
precisionmeasurement. Atomsclosetoasurfacearesub- the F = 2,mF = −2 state gives control over the s-wave
jecttotheshort-rangeCasimir-PolderandvanderWaals scatteringlengthclosetothezerocrossingof∼40a0/G.
potentialswhichcanbemeasuredusingtheclassicaland The use of a magnetic Feshbach resonance means it
quantum reflection of bright solitary matter-waves [4]. is advantageous to work with a levitated crossed opti-
Our apparatus includes a super-polished Dove prism for cal dipole trap. This is formed from a single 10.1 W,
such studies, see Fig. 1(a). Further in the future, the λ=1064nmlaserbeam(IPG:YLR-15-1064-LP-SF)used
abilitytodeliverandmanipulateultracoldatomsnearto in a bow-tie configuration as shown in Fig. 1(a). The
a solid surface may open up new routes to probe short term ‘levitated’ refers to the use of an additional mag-
range corrections to gravity [28] due to exotic forces be- netic quadrupole field whose vertical gradient is set to
yond the Standard model. just less than that required to support atoms against
5
gravity. This trap allows the magnetic field, and hence ingtheexpansionisnon-linearoverthefullrangeoftimes
scattering length, to be changed independently of the measured, a linear approximation is valid over the range
trapping frequencies. 10 ms<t<100 ms from which we can extract a ‘rate’.
B. Loading the optical waveguide D. Control of the solitary wave velocity
To investigate the creation of solitary waves we begin The position of the magnetic field zero in the axial di-
by forming a BEC containing up to 10,000 atoms at a rection of the waveguide can be displaced by an amount
scattering length of a ≈300 a . The crossed beam trap determined by the magnetic field gradient in this direc-
s 0
in which the BEC is created has a roughly spherically tion, B(cid:48)/2, and a moderate offset field, Boffset, according
symmetric geometry at the point of condensation, with to ∆x = Boffset/(B(cid:48)/2) [31]. In this way the amplitude,
final trap frequencies of ω = 2π × (31,27,25) Hz. and hence velocity, of the solitary wave motion can be
x,y,z
This trap is ill-suited to the observation of bright soli- preciselycontrolledduetothedominanceofthemagnetic
tary matter-waves and thus we transfer the condensate potential over the optical confinement of the waveguide
intoamorequasi-1Dwaveguidecreatedbyanadditional alongtheaxialdirection. Themaximumvelocityisgiven
1064 nm laser beam, focused to a waist of 117 µm and by v = Aωaxial where A is the amplitude of the motion,
intersecting the crossed trap at 45◦ to each beam. This set by the separation between the minimum of the mag-
enters the glass science cell through the back surface of netic potential along the axis of the waveguide and the
an anti-reflection coated fused silica Dove prism (to be release point from the crossed dipole trap. Using this
later used for the study of atom-surface interactions [4]). technique the solitary wave can reach velocities of tens
To load the condensate into the waveguide the scat- ofmms−1 whentravellingthroughthecentreofthehar-
tering length is ramped close to a = 0 in 50 ms thus monic potential or, alternatively, be brought to a near
s
reducing the condensate size and creating a BEC ap- standstill, achieving velocities <0.5 mm s−1.
proximately in the harmonic oscillator ground state of
the crossed trap. The BEC is then held for 10 ms to al-
low the magnetic field to stabilise before simultaneously E. Classical reflection from a Gaussian barrier
switching the waveguide beam on, the crossed beams off
and jumping the quadrupole gradient in the vertical di- To produce the repulsive potential barrier we use a
rectionfrom B(cid:48) =21.5G cm−1 to26 Gcm−1. Although 532nmGaussianlaserbeam(derivedfromaLaserQuan-
it is advantageous in terms of the evaporation to be un- tum Finesse laser), focussed to a waist of 131 µm hori-
der levitated during the condensation phase, we must zontally and 495 µm vertically, with a power of up to
increasethegradientoncewewishtotransfertheatoms. 2 W. The barrier is aligned to cross the waveguide in
Thisensuresatruerlevitationoftheatomsinthewaveg- the horizontal plane at an angle of ∼45◦ and is offset by
uide trap, thus maximising the trap depth of the beam. 455µmfromwheretheBECisreleasedfromthecrossed
Inaddition,thepresenceofthequadrupolegradientpro- dipole trap, see Fig. 3(a). This angle is restricted by the
vides much of the, albeit weak, axial trapping along the available optical access close to the trap centre.
(cid:112)
beam,ω =1/2 µB(cid:48)2/mB ≈2π×1Hz[30]. Here,µ
axial 0
isthemagneticmomentofatomswithmassmandB is
0
the magnetic bias field. The waveguide beam itself con- F. Theoretical modelling
tributes < 0.1 Hz to the axial trapping, hence the mag-
neticconfinementdominatesinthisdirection. Atabeam
The release of the BEC into the waveguide poten-
powerof0.17Wthewaveguideandquadrupolepotential
tial,anditssubsequentexpansion,wasmodelledatzero-
produce a trap of ω = 2π×(1,27,27) Hz. Here the
x,y,z temperature by solving the Gross-Pitaevskii equation in
radial trap frequency (ω ) approximately matches that
y,z 3D using a cylindrically-symmetric Fourier pseudospec-
of the crossed beam trap at the point of condensation.
tral method. In all cases the initial non-interacting
ground state of a harmonic trap with axial (radial) fre-
quency 30 (27) Hz (corresponding to the crossed dipole
C. Propagation in the waveguide trappotential)wasreleasedinstantaneouslyintoanother
harmonictrapwithaxial(radial)frequency1(27)Hzand
A small offset (2.6 mm) between the crossed dipole offsetby2.6mmalongtheaxialdirection(corresponding
trap, i.e. the waveguide loading position, and the to the waveguide potential). The scattering length was
quadrupole centre means that once loaded into the instantaneously changed to the appropriate value of a
s
waveguide, the BEC propagates freely towards the mag- at the time of release.
netic field minimum along the direction of the waveg- In cases where the barrier was present this was mod-
uide, undergoing harmonic motion. As the BEC propa- elled as a Gaussian ‘light-sheet’ potential centered on a
gates its rate of expansion in the axial direction is deter- plane perpendicular to the axial direction, offset from
mined by thescattering length. Although strictlyspeak- the initial harmonic trap by 2.145 mm, and with height
6
√
760 nK and width 131 2 µm. Compared to the ex- linear least-squares.
perimental barrier beam this model neglects the vertical
width of the beam which is large compared to the radial
extent of the BEC in the waveguide, and includes the
√
geometric factor 2 to account for the 45 degree angle
Acknowledgements
of the beam.
Expansionrateswerecalculatedfromthefull-widthat
halfmaximumoftheBECaxialdensityprofilepredicted We thank the Durham soliton theory group for many
bytheGPE(obtainedbyintegratingovertheradialcoor- useful discussions. We acknowledge financial support
dinate)after10msand100msofexpansion. Inallcases, fromtheUKEngineeringandPhysicalSciencesResearch
the change in radius over this time interval was approxi- Council (EPSRC grant EP/F002068/1) and the Euro-
matelylinear. ForthesimulationsinFig.3(e),thewidth pean Science Foundation within the EUROCORES Pro-
was calculated by convolving the BEC axial density pro- gramme EuroQUASAR (EPSRC grant EP/G026602/1).
file predicted by the GPE with a 10 µm width Gaussian TPB was supported by The Marsden Fund of New
(to account for finite imaging resolution), and fitting a Zealand (UOO162), and The Royal Society of New
Gaussian distribution to the resulting profile using non- Zealand (UOO004).
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