Interface dynamics of a two-component Bose-Einstein condensate driven by an external force

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by  D. Kobyakov

Interface dynamics ofatwo-component Bose-Einstein condensate driven by anexternal force
D. Kobyakov,1 V. Bychkov,1 E. Lundh,1 A. Bezett,1 V. Akkerman,2 and M. Marklund1
1Department of Physics, Umea˚ University, 901 87 Umea˚, Sweden
2DepartmentofMechanicalandAerospaceEngineering, PrincetonUniversity,NJ-08544-5263Princeton,USA
The dynamics of an interfaceinatwo-component Bose-Einsteincondensate driven bya spatiallyuniform
time-dependent force is studied. Starting from the Gross-Pitaevskii Lagrangian, the dispersion relation for
linearwavesandinstabilitiesattheinterfaceisderivedbymeansofavariationalapproach.Anumberofdiverse
1 dynamicaleffectsfordifferenttypesofthedrivingforceisdemonstrated, whichincludestheRayleigh-Taylor
1 instabilityforaconstantforce,theRichtmyer-Meshkovinstabilityforapulseforce,dynamicstabilizationofthe
0 Rayleigh-Taylorinstabilityandonsetoftheparametricinstabilityforanoscillatingforce. GaussianMarkovian
2
andnon-Markovianstochasticforcesarealsoconsidered.ItisfoundthattheMarkovianstochasticforcedoesnot

n produceanyaverageeffectonthedynamicsoftheinterface,whilethenon-Markovianforceleadstoexponential
a perturbationgrowth.
J

1
3 I. INTRODUCTION As a particular experimental realization of such a force,
Sasakietal.[12]suggestedasystemoftwophase-segregated,

] interacting BECs with different spins placed in an external
s
a Superfluid dynamics of interfaces between Bose-Einstein magnetic field gradient. This is possible, for example, for
g condensates(BECs) ofdiluteatomic gaseshasgainedactive two magneticsublevelsof the F =1 hyperfinestate of 87Rb
- interest,bothinexperimental[1–5],andtheoreticalworks[6– with F,m = 1, 1 and 1,1 . Starting from the Gross-
nt 13]. These studies started with the experimental realization Pitaev|skiiF(GiP)|act−ionifor th|e twio component BEC system,
a ofamulticomponentBEC[1,2]andthetheoreticalinvestiga- we derive the equations of motion of small perturbations of
u
tionoftheplanarinterfacestructure[6–9]. Atpresent,agood theinterface.
q
deal of the research focuses on the hydrodynamic instabili-
at. ties at the interfaces in BECs, such as the Kelvin-Helmholtz Theotherpurposeofthepresentpaperistodemonstratethe
diversityofhydrodynamiceffectsinasystemoftwoseparated
m instability,theRayleigh-Taylor(RT)instability,theferrofluid
BECs under the action of a time-dependentforce. Recently,
normal-field instability, and the Richtmeyer-Meshkov (RM)
- withinclassicalgasdynamics,therehasbeenincreasedinter-
d instability [12–17]. A common feature of these fundamen-
est in the problems of the modified RT and RM instabilities
n tal instabilities in the classical gas dynamics is the produc-
o subjecttotime-dependentgravity,e.g. see[18–21]. Thecon-
tion of vortices at the unstable interface. In that sense, dy-
c figurationoftime-dependentgravitywasdiscussedinthecon-
namicsofaninterfacebetweentwoquantumfluidshassome
[ textofspeciallydesignedlaboratoryexperiments[22],inthe
uniquefeatureswith quantizedvorticity being, probably,the
scope of the problemof inertial confinedfusion [19, 23, 24]
1 most specific and the most interesting among them. At the
v andinrelationtoflamedynamics[25–28].Inparticular,stabi-
sametime,thereisalsomuchsimilaritybetweentheclassical
8 lizationoftheRTinstabilitybyanoscillatinghigh-frequency
andquantumgasdynamics,whichallows”borrowing”some
9 addition to the gravity acceleration has been proposed both
classical results and applying them directly or with modifi-
9 in the traditional RT configuration [29] and for the inertial
5 cations to the quantum case, see Refs. [12–15]. Such ”bor-
confinedfusion[23, 24]. Developmentof the parametricin-
. rowing”is especially typicalin studiesof the linear instabil-
1 stability at a flame front under the action of acoustic waves
itystages,forwhichthedifferencebetweentheclassicaland
0 andpossiblestabilizationofthehydrodynamicflameinstabil-
1 quantumcases is expectedto be minimal. Still, the problem
ity has been investigated experimentally and theoretically in
1 of rigorous derivation of such effects in BEC systems from
Refs. [30–32]. Here we show that similar effectstake place
: thebasicprinciplesofquantumtheoryremainsopenformost
v inBECs. ApartfromtheRTandRMinstabilitiesconsidered
cases. The derivation is needed not only to justify the ex-
i beforein[12,15],wedemonstratealsothepossibilityofdy-
X trapolationsfromtheclassicaltoquantumgasdynamics,but
namicalstabilization of the RT instability by high-frequency
r also to find limits of such extrapolation as well as to indi-
a oscillations,aswellastriggeringoftheparametricinstability.
cate intrinsic quantumeffectsinvolvedinto the problem. As
Anotherinterestingoption,whichhasnotbeenconsideredyet
an example,we pointoutthestudy ofthe interfacewavesin
evenintheclassicalgas-dynamicscorrespondstoaGaussian
a system of two BECs undertaken in Ref. [8]. Apart from
stochasticforce,whichmaybeMarkovianornon-Markovian
common capillary waves anticipated from the classical gas-
with zero or non-zero time correlations, respectively. In the
dynamicsandrelatedtoin-phaseperturbationsoftwoBECs,
present paper we show that the Markovian stochastic force
Ref.[8]demonstratedalsothepossibilityofquantumcounter-
does not produce any average effect on dynamics of the in-
phaseperturbations,forwhichthelocalinterpenetrationdepth
terface, while the non-Markovian force leads to exponential
of two wave functions for differentBECs oscillates in time.
growthofperturbations.
Thus, one purpose of the present paper is to provide rigor-
ousderivationofthedispersionrelationsforsomefundamen- The paper is organized as follows. In section II we for-
talquasi-hydrodynamicinstabilitiesinatwo-componentBEC mulate the model. In section III we consider a planar (one-
drivenbyatime-dependentforce. dimensional)quantumsystemwithamagneticgradientpush-
2
ingtheBECstowardseachother,forthecaseofbothconstant
and time-dependentexternalforces. In section IV we derive
theequation,whichdescribesthedynamicsofsmallinterface
perturbations. In section V we consider interface dynamics
undertheactionofdifferentforces,includingaconstantforce,
a harmonicforce, a pulse force, and a noisy Gaussian force.
TheresultsofthepaperaresummarizedinsectionVI.
II. THEMODEL
The system of two BECs at zero temperatureis described
by two macroscopic wave functions Y˜ (t˜,r˜), and the mean-
j
fieldGPLagrangian[33]:
L˜(t˜)= d3r˜ (cid:229) ih¯ Y˜ ¶ Y˜ j Y˜ ¶ Y˜∗j
Z "j=1,2(2 ∗j ¶ t˜ − j ¶ t˜ ! FunIGifo.r1m: Smcahgenmetaitcicfioelfdt.hetwo-component BEC inan external non-
g
+m Y˜ 2 V˜ (t˜,r˜)Y˜ 2 jj Y˜ 4
j j j j j
| | − | | − 2 | |
h¯2 (cid:209) Y˜ 2 g Y˜ 2 Y˜ 2 , (1) particle of j-th condensate, which is madeup of the interac-
j 12 1 2
−2mj (cid:27)− | | | | (cid:21) tionenergyduetos-wavescatteringandofenergyofthepar-
(cid:12) (cid:12) ticleintheexternalpotential. Inordertostudylow-energetic
whereV˜ (t˜,r˜)areexternal(cid:12)poten(cid:12)tialenergiesoftheBECcom-
j hydrodynamicprocessesinBECsweassumethatenergyper
ponents. The tildes in Eq. (1) designate dimensional vari-
particleischangednegligibleduetotheexternalpotential,and
ables,whichwillbescaledinthefollowing.Keepinginmind
fornormalizationonecanusevalueofthechemicalpotential
asystemwithtwomagneticsublevelsof87Rb,weassumethe
forB =0inaninfinitesystem. Suchsystemischaracterized
′
sameparticlemassesforbothcomponentsm =m mand
1 2≡ byauniformdensitysufficientlyfarfromtheinterface(i.e.on
thesameinteractionparametersg =g g.Theinteraction
11 22≡ length scales much larger than the interface thickness); then
parameters g , g , g are related to the respective s-wave
11 22 12 thechemicalpotentialisgivenbyausualresultforauniform
scattering lengths aij by gij≡4p h¯2aij/m. A dimensionless BECwithnumberdensityn˜0, m =gn˜0. However,realBECs
interactionparameterisintroducedthroughthedefinition
arealwaysfinite;ifthelengthscaleofoursystemalongz-axis
g g /g 1. (2) is2Lz (eachcondensateisofsizeLz),thentheconditionthat
12
≡ − theexternalpotentialisasmallperturbation,reads
Thetwocomponentsareseparatedinspaceifg12>√g11g22,
whichimpliesg >0,see[33]. Throughoutthispaperweuse
the condition of weak separation g 1. Modification of all m B(t˜)
≪ L B ′ gn˜ . (4)
results of the present paper to the case of different BECs is z 0
2 ≪
straightforward.
Theunperturbedflatinterfacebetweentwocomponentsof
Dimensionlessunitsareintroducedbyscalingcoordinates
BEC is located in the (x,y) plane with z=0 indicating the byh¯/√mm , time byh¯/m , andatom numberdensity (orcon-
planeofsymmetry,Fig. 1. Thecomponents1and2arechar- centration)bym /g. Thedimensionlessmagneticforceacting
acterizedbytheirprojectionoftheatommagneticmomenton
onthecomponentsis
thex-axism = 1,respectively.
F
±
Using the states F,m = 1, 1 and 1,1 of the atoms
makes it possible t|o apFpily t|he−Steirn-Ge|rlachi force, push- b (t)= b (t)= h¯m BB′(t) b(t).
ing the components in the opposite directions perpendic- 1 − 2 −2m √mm ≡−
ularly to the magnetic moment quantization axis. We
use a magnetic field whose magnitude has the gradient The dimensionless Gross-Pitaevskii Lagrangian for two
(cid:209) B(t) =xˆdBx(t)/dz xˆB′(t),whichisuniforminspacebut macroscopicwavefunctionsY j(t,r)atzerotemperaturereads
m|aybe|time-dependen≡t.Forthecaseof87RbatomstheStern-
GerlachpotentialenergyinEq.(1)reads L(t)= d3r (cid:229) i Y ¶ Y j Y ¶ Y ∗j
V˜ (t˜,r˜)=z˜m BB′(t˜)( 1)j, (3) Z "j=1,2(cid:26)2(cid:18) ∗j ¶ t − j ¶ t (cid:19)
j
2 − 1
+ Y 2 Y(cid:209) 2 zb (t) Y 2
wherem istheBohrmagneton,and1/2istheLande´factor. j −2 j − j j
B
The chemicalpotentials m j in Eq.(1) define normalization 1(cid:12)(cid:12) Y (cid:12)(cid:12) 4 (cid:12)(cid:12)(1+(cid:12)(cid:12)g )Y 2 Y (cid:12)(cid:12) 2(cid:12)(cid:12), (5)
ofstationarywave functionsY j, andare equalto energyper − 2 j − | 1| | 2|
(cid:27) (cid:21)
(cid:12) (cid:12)
(cid:12) (cid:12)
3
andequationsofmotionareobtainedfromtheactionprinciple forz>Z . WiththeseapproximationsEq.(9)becomes:
p
1d2y
d S ¶ d S 01 =[g (2+g )b z]y . (12)
= , (6) 2 dz2 − 0 01
d Y ∗j ¶ td ¶ Y ∗j/¶ t  The type of solution to Eq.(12) changes from exponentially
 (cid:16) (cid:17) decaying to oscillating in space, as z becomes larger than
where the action is S= dtL(t). The result is the coupled Z =g /[(2+g )b ] g /(2b ). However,themagnitudeofthe
o 0 0
GPequations: R spatialoscillations≈dependsonthevaluey 01(Zo):itdecreases
exponentially with increasing Z . Therefore, the spatial os-
¶ Y D o
i ¶ t1 =(cid:20)−2 −1−zb(t)+|Y 1|2+(1+g )|Y 2|2(cid:21)Y 1, (7) cwililtahtiZopn,sobrewchoemnebs0ig=nigficagn/t2w;ihnenthiZsocabseecothmeeTsFcAombepcaorambeles
¶ Y D
i ¶ t2 = −2 −1+zb(t)+|Y 2|2+(1+g )|Y 1|2 Y 2. (8) itniavlaolisdc,ilalantdionEsqms.(a9y),b(e10in)ptemrpursettebdeassoqlvueadnteuxmacitnlyte.rfeTrheencsepoa-f
(cid:20) (cid:21)
matter waves, caused by interplay between the non-linearity
andthedispersion(thekineticterm).Thisphenomenonisbe-
III. PLANARDENSITYPROFILESFORA yondthe scope ofthe presentwork, andit willbe studied in
TWO-COMPONENTBEC detailelsewhere. Thus,hereweuseparameterssatisfyingthe
condition
Thepresentworkisdevotedtomultidimensionalinstabili-
b b g g /2, (13)
tiesbendinganinterfacebetweentwoBECcomponentsinan 0≪ 0cr≡
externalmagneticfieldgradient.Asthefirststepinthestudy, orthemagneticgradientB satisfyping
′
wehavetospecifytheplanar(one-dimensional,1D)quasista-
tionarystate,whichisinfluencedbyaspatiallyuniformexter- B <B g 3m 3m/(m h¯), (14)
nalfield. InthesubsectionsAandBweconsiderthecasesof ′ ′0cr≡ B
constantandtime-dependentexternalfields,respectively. which means that the spatipal oscillations of the hydrostatic
profiles may be neglected. It follows from Eq. (14) that
the critical magnetic gradient B depends on the differ-
′0cr
A. Planarprofilesinthecaseofastationaryforce ence between inter- and intra-species interaction parameters
asg =(g g)/g,andondensityviathechemicalpotential
12
since m g−n˜ in the TFA for an infinite system in a suffi-
Herewestudythe1DhydrostaticstateofBECs,whenthe 0
≈
cientlyweakexternalfield. Therefore,theeffectofquantum
constantexternalforceb isbalancedbyinternalforcesofthe
0
interferencebecomesstrongeratlowerdensitiesoftheBECs.
condensates. Let us assume thatsuch a state was createdby
Equation (13) yields the dimensionless condition of a well-
adiabaticswitchingonoftheexternalfield, andnoperturba-
definedinterfacebetweentwocondensatessubjecttothepar-
tionsbendtheinterface. Thesteadystateisdescribedbytwo
real-valuedfunctionsy (z),andthesystemofdimensionless ticularexternalforce.Inthesystemoftwomagneticsublevels
0j of87Rbwehaveg 10 2(see[2]),andthetypicaldensityis
GPequationsreads −
5 1014cm 3. The≃numerical solution to Eqs. (9), (10) for
−
·
1d2y theseparametersshowsthatthespatialoscillationsarenotice-
0= 2 dz201 +y 01+zb0y 01−y 031−(1+g )y 022y 01, (9) ableatb0≈10−4;sowetakeb<10−4. Numericalsolutions
forthewavefunctionsneartheinterfacearepresentedinFig.
0= 12dd2yz202 +y 02−zb0y 02−y 032−(1+g )y 021y 02. (10) 2dafsohrebd0l=ine7s.5in·1F0i−g5.;11(0a)−s4h,opwlottsh(ea)apapnrdo(xbi)m,raetespaencatlivyetilcya.lTshoe-
lutionofEq.(11)obtainedforthecaseofzeromagneticfield.
The time-independentforce pushesthe BEC componentsto-
Aswecansee,theanalyticalsolutionEq.(11)fitsthenumer-
wards each other when b > 0. With our choice of equal
0 ical results quite well even in the case of non-zeromagnetic
massesandscatteringlengthsforbothcomponents,theprob-
field.
lem possesses the symmetry y (z)=y ( z). In the case
01 02
−
ofzeromagneticgradientthedensityprofileshavebeenfound
in[8]withinthelimitofweakseparationg 1as
≪ B. 1DBogoliubovexcitationsofBECduetoaharmonicforce
1
n01(z)=n02( z) 1+exp 2 2g z − . (11) Since we are interested in the development of hydrody-
− ≈ −
h (cid:16) p (cid:17)i namic instabilities for time-dependent external forces, we
Wecalculatethewavefunctionofcondensate1inthebulk need to know how a harmonic force influences the planar
of condensate 2. According to Eq. (9), the wave functions densityprofilesof BECs. Similar to the constantforcestud-
penetrate into each other with the characteristic depth Zp = iedabove,fortime-dependentforceswe also considerrather
1/√2g , see also [7], which means that y 2 1 for z>Z . small magnetic fields, so that the respective 1D Bogoliubov
01 ≪ p
The density profile of the bulk is described by the Thomas- excitationsmaybe interpretedas linear acousticwaves. An-
Fermiapproximation(TFA), which meansthat y 2 1 bz othertypeofatime-dependentforcestudiedinthispaperisa
02 ≈ −
4
the ”wall”, then the wave function and density are specified
onlyforz 0,nohealingofthecondensatewavefunctionis
≥
requirednearthewallinTFA,andequilibriumisspecifiedby
the uniformdensity n =1. In that case the velocity pertur-
0
bationd U(z,t)iszeroatthewalld U(0,t)=0.Theperturba-
tionsofdensityandvelocity,d nandd U,maybeexpressedin
termsoftheperturbationwavefunctiondy
y (z,t)=exp( it)(√n +dy ), (16)
0
−
wheredy (z,t)=u(z)exp(iW t)+v (z)exp( iW t),andu(z),
∗
−
v (z)aretime-independentfunctions.Sincetheperturbations
∗
aresmall,wefind
d n=exp(iW t)(u+v)+c.c.,
¶f 1 ¶
d U = = exp(iW t) (u v) c.c.,
¶ z 2i ¶ z − −
where c.c. denotes complex conjugate. Then the boundary
conditionatthewallreads
¶
(u v) =0. (17)
¶ z −
(cid:12)z=0
(cid:12)
WelinearizetheGPequation(1(cid:12)5),
(cid:12)
¶ 1 ¶ 2
i¶ tdy =−2¶ z2dy +2dy +dy ∗+zbsexp(iW t),
andobtaintheBogoliubovequations
1¶ 2u
FIG.2:Hydrostaticprofilesoftwocomponentspushedtoeachother −W u=−2¶ z2 +u+v+bsz, (18)
byaconstantforce,b=7.510 5;10 4,figures(a)and(b),respec-
− −
·
tively. ThedashedlinesshowtheanalyticalsolutionEq. (14). The
insetinfigure(b)presentstheAiryfunctionofthesecondkind. 1¶ 2v
W v= +v+u, (19)
−2¶ z2
pulseforcecorrespondingtoaweakshockinhydrodynamics. which,inturn,maybereducedto
Still,itisknownfromhydrodynamicsthataweakshockmay
bealsodescribedasalinearacousticwave[34]. Forthisrea- W 2 ¶ (u v)= 1 ¶ 2 +1 2 1 ¶ (u v) W b .
son, in the present subsection we study modifications of the ¶ z − "(cid:18)−2¶ z2 (cid:19) − #¶ z − − s
BECdensityinthe1Dgeometryduetotheharmonicforce. (20)
Becauseoftheproblemsymmetrywithrespecttoz=0,we The solution to Eq. (20) is a superposition of a running
consider a one-componentsemi-infinite BEC for z<0 con- wave U exp(iW t iqz) and spatially uniform oscillations
1
finedbyawallatz=0. Themagnitudeandfrequencyofthe U exp(iW t), −
0
forcedeterminestheenergyofthegeneratedBogoliubovexci-
¶
tations.Inthissubsectionweexpressdensityandvelocityper- (u v)=U˜ +U˜ exp( iqz), (21)
turbations,d nandd U, intermsofmagnitudeandfrequency ¶ z − 0 1 −
of the externalforce, and find the restrictions on the driving
force,forwhichthedensityperturbationsaresmalld n n . whereU˜0,1aretherespectiveamplitudes.Therunningwaveis
0
≪ the Bogoliubov elementary excitation (the quantum acoustic
Inthecaseofaharmonicforce,thedimensionlessexternal
potentialiszb exp(iW t), whereb is theforceamplitudeand wave)withthewavenumberq(W )determinedviatheBogoli-
s s
W isthefrequency. Ourmodelsystemisthendescribedbya ubovspectrum
time-dependentGPequation W 2=q4/4+q2. (22)
i¶y¶ t =−12¶¶2zy2 +y ∗y 2+y Vtrap(z)+zbsexp(iW t)y . (15) ItUf˜oll=owsbfr/oWmatnhdeboundaryconditionatthe wallthatU˜0 =
1 s
− −
SwihnecreetVhteratpra(pz)p=ote+n¥tiafloVrtrzap<(z0) eaxnpdeVriteranpc(ezs)d=isc0onfotirnuzi≥ty0at. d U = bW s[sin(W t−qz)−sinW t]. (23)
5
The perturbations of density follow from the 1D continuity For the variational calculation, we employ an ansatz that is
equation able to describe bending and compression of the interface
between the two condensates. We are interested in pertur-
¶ ¶
d n= d U bation wavelengthsmuch largerthan the interfacethickness,
¶ t −¶ z kZ =k/√2g 1, so that the density deformationsmay be
p
≪
treatedasabendingoftheunperturbedprofileasawhole,
as
d n= bsqsin(W t qz). (24) nj(z,x,t)=nj0[z−d zj(x,t)], (28)
W 2 −
wheren arethestationarydensityprofilesfortheBECs. The
j0
Then,theconditionofweakperturbationsd n n becomes variational functions are taken to be of the form of Fourier
0
b q/W 2 1,orinthedimensionalform ≪ modeswithseparationofspaceandtimedependence,
s
≪
m BB′q˜ 1, (25) d zj(x,t)=z j(t)cos(kx), (29)
2mW˜2 ≪
f (z,x,t)=x (t)f (z)cos(kx). (30)
where q˜ and W˜ are the dimensional wave number and fre- j j j
quencyoftheinducedexcitations,coupledbytheBogoliubov Inclassicalhydrodynamics,perturbationsofbothcomponents
spectrum(22). In the low energylimit q4 q2, the Boguli- of an inert interface are related by the continuity condition
≪
ubovexcitationsaresoundwaveswithvelocitycs= gn˜0/m z 1 =z 2. In the quantum gas-dynamics of two BECs this is
(unityinourdimensionlessunits). Thentherestrictiononthe notnecessarily the case, since we may have interpenetration
p
amplitudeofthemagneticforcereads ofthewavefunctions.
TheLagrangeequationswithrespecttothevariablesx are
m B gn˜ j
B ′ W˜ 0. (26)
2m ≪ r m z˙ dzn f =x dz f d n f +k2f2n . (31)
j ′j j j jdz j j′ j j
The criterion (26) may be interpreted as the condition of Z Z (cid:20) (cid:21)
(cid:0) (cid:1)
incompressible condensates with the speed of sound much TheseLagrangeequationsaresatisfiediftherespectiveconti-
larger than any other parameter of velocity dimension in- nuityequationsare
volved in the problem. A more generalcondition(25) takes
intWoaecscooluvnetdththeeBBogoogloiulibuobvovspeeqcutrautmionfsorfohrigahemroendeerlgiinefis.nite z˙jn′j=x j ddz njfj′ +k2fjnj . (32)
(cid:20) (cid:21)
system. Thecaseofafinitesystem,confinedbytwowalls,is (cid:0) (cid:1)
First, the time-dependent terms on each side have to be
solved analogously, and the boundary condition (17) should
equated,which,uptoconstantfactors,leadsto
be imposed for both walls. Real systems are always finite,
and therefore the maximal wavelength is limited by the size
z˙ (t)=x (t). (33)
of the system L . In the low-frequency limit condition (25) j j
z
is equivalentto the energeticcondition(4). In this limit, the Thisreflectsthefactthatx andz arecollectivecanonically
j j
Bogoliubovexcitationsaresoundwaves,withmaximalwave-
conjugate variables. For the functions f , Ref.[8] gave the
lengthoforderL ; substitutingL =2p q˜ 1 to (4), andusing j
z z − approximatesolution
thesoundspectrum,oneobtains(26).
(1+kz) 1, if z<0,
f1(z)=−f2(−z) −k−1exp(− kz), if z>0. (34)
IV. LINEARANALYSISOFTHEINTERFACEDYNAMICS (cid:26) − −
The same solution holds in the presence of a small external
Wenowconsiderthedynamicsoftheinterfacebetweentwo force under the conditions specified in Sec. III. We take Eq.
infiniteBECsinthepresenceofanexternalforce,whichmay (34)asouransatzfor fj.Moreelaborateexpressionsfor f1(z),
be time-dependent, b(t). Similarly to Refs. [8, 33, 35] we and f2(z)mayprovideasmoothtransitionintheinterfacere-
use a variationalapproach. We representthe wave functions gion; however, the ansatz Eq. (34) yields reliable analytical
intermsofdensityn andvelocitypotentialf asy (t,x,z)= results, as we will demonstrate below by comparison to the
j j j
√njexp(if j),sothattheactionforthesystemoftwoBECsis numericalsolutiontotheproblem.
We now derive the Lagrange equation for z (t). No time
j
S= dt d3r (cid:229) n ¶f j +1n ((cid:209) f )2 + edqeuriavtaiotinverseaidnsz j appear in the Lagrangian, so the Lagrange
j ¶ t 2 j j
Z Z (j=1,2(cid:20)
d S
12 (cid:209) √nj 2−nj +zb(t)(n2−n1)+ 0= dz j =I1(j)+I3(j)+I5(j)+I7(j), (35)
(cid:21)
(cid:0) 1(cid:1) where we introduce I to label the nonvanishing terms of
n2+n2 +(1+g )n n . (27) l(j)
2 1 2 1 2 d S/dz j; thenumberingl inIl(j) refertotherespectiveterms
(cid:27)
(cid:0) (cid:1)
6
in Eq. (27). Forconvenience,we assume a finite size of the L d Lz+d z2
system,Lz,Lx,thoughthefinalresultdoesnotdependonthe 2xb(t)dz dz′ z′n02 z′ −d z2n02 z′ .
sizerestrictions.Wehave 2 Z¥
− (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
d ¶f
I = d2rn j = Theintegralscanbecalculatedas
1 dz j ¶ t
jZ L d 1 L
dzdxn′0j(z+d zj)fjx˙jcos2(kx)= I5(1)=− 2xb(t)dz 1(cid:18)−2d z21+d z21(cid:19)=− 2xb(t)z 1, (40)
Z
L
2xx˙jZ dzn′j(z)fj(z). (36) I5(2)= L2xb(t)dzd 12d z22−d z22 =−L2xb(t)z 2. (41)
The next nonvanishingterm comes from the zero-point mo- 2(cid:18) (cid:19)
tionenergy, Thenexttermofd S/dz doesnotcontributetotheequations
j
d 1 ofmotion,whilethelast(seventh)termI7isimportant.Anal-
I3≡ dz j Z d2r 2 (cid:209) √nj 2= yarseisgoofvethrneetderbmystwcaolccuoluapteleddsoosfcailrlasthoorweqsuthaatitotnhse,avnadriathbelecsozu-j
21dzd jZ d2r"(cid:18)¶¶x√(cid:0)nj(cid:19)2(cid:1)+(cid:18)¶¶z√nj(cid:19)2#= aparnleidngicdhoeaunpntiptceearnl-,pshtbhaeescean,uozsremo=aflI7m.zoSdi.nescFemirthsatey,elbqeeutauitnsi-opcnhasalfcsouerl,azzt1e1aI=ndfzzo22r,
1 2 7
1 d ¶ 2 ¶ 2 the case of z =z , whic−h we denote by I . In this case
2dz jZ d2r(cid:18)¶ z√nj(cid:19) "1+(cid:18)¶ xd zj(cid:19) #. (37) d z1=d z2≡1d z,an2dwefind 7(in)
d
The first term within the parentheses yields a constant inde- I = d2r(1+g )n n =
7(in) dz 1 2
pendentofz ,andthesecondtermequals jZ
+¥
I3= 21dzd j Z d2r(cid:18)¶¶z√nj(cid:19)2(cid:18)¶¶xd zj(cid:19)2= L2xdzd j Z¥ dz(1+g )n01(z+d z)n02(z+d z)=0. (42)
−
L2x12dzd jZ dz(cid:18)ddz√n0j(cid:19)2(z jk)2= Nd ze1x=tw−edczo2n≡siddezrathnedcza1s=e o−fzc2ou≡ntze,r-wphhiacsheloesacdisllatotionswith
L2xz jk2Z+¥ ¥ dz(cid:18)ddz√n0j(cid:19)2. (38) I7(co)= L2xdzd +¥ dz′(1+g )n01 z′ n02 z′−2d z .
− Z¥
− (cid:0) (cid:1) (cid:0) (cid:1)
Thenextnon-vanishingtermcorrespondstothe perturbation
To calculate d /dz in the last expression, it is easiest to as-
ofthepotentialenergyofthesystem,
sumed zsmallandTaylorexpandn (z 2d z)tosecondor-
02 ′
d der,whichresultsin −
I = d2rzb(t)(n n )=
5(j) dz j Z 2− 1 L +¥ d2n
L2x(−1)jb(t)dzd jZ dzznj= I7(co)= 2x4(1+g )z −Z¥ dzn01 dz202.
L d
x( 1)jb(t) dzzn (z+d z ). (39) Thelastresultmaybetransformedtoasymmetricformsimi-
2 − dz jZ 0j j larto[8]:
It is more convenient to calculate these terms for j=1 and +¥
L dn dn
fzo+rdj=zj2wseempaordaitfeylyt.hUeslainstgetxhperceososirodninaastetransformationz′= I7(co)=− 2x4(1+g )z Z¥ dz dz01 dz02. (43)
−
L d +¥ Taking into account all the terms obtained above we write
I5(1)=− 2xb(t)dz dzzn01(z+d z1)= equations of motion for z j. The integrals are calculated us-
1 Z ingthedensityprofile(11)
Lz
−
−L2xb(t)dzd 1−LzZ++d¥ z1 dz′(cid:2)z′n01(cid:0)z′(cid:1)−d z1n01(cid:0)z′(cid:1)(cid:3), −Z+¥ ¥ dzddnz0j fj=1/k, (44)
L d Lz +¥ d 2
I5(2)= 2xb(t)dz dzzn02(z+d z2)= dz dz√n0j = g /8, (45)
2 Z¥ Z¥ (cid:18) (cid:19)
− − p
7
+¥
dn dn
dz 01 02 = 8g /3. (46)
dz dz −
Z¥
− p
Wedesignatetheinterfaceperturbationsofthein-phaseoscil-
lationsbyz =z z andobtainthefinalequation
1 2 in
≡
¶ 2z
in +z k3 g /8 b(t)k =0. (47)
¶ t2 in −
h p i
In the opposite case of the counter-phaseperturbationsz =
1
z z wefind
2 co
− ≡
¶ 2z g 4
co +z k2 + (1+g ) 8g b(t) k=0. (48)
¶ t2 co 8 3 −
(cid:20) r (cid:21)
p
Inthelimitingcaseofb=0ourtheoryreproducestheresults
of[8]. Withinthelimitofaweakfield,b b =g √2g ,and
m
long wavelength perturbations k/√2g ≪1, the terms in the FIG.3: LowestBogoliubov modes(theRTdispersionrelation)for
≪ differentvaluesofthedrivingforce.Theanalyticalresultisobtained
parenthesisof Eq. (48) are of differentordersof magnitude,
withtheterm4(1+g )√8g /3dominating.Forthisreason,to fromEq. (49),andpresentedbylines. Dashedlinesshowrealparts
oftheBogoliuboveigenvalues(waves),andsolidlinescorrespondto
the leadingterms, the counter-phasemodeis notaffectedby
the imaginary parts (instability). The markers show respective nu-
theexternalpotential,andwedonotconsideritinthepresent
mericalsolutiontothelinearizedGPequations; filledmarkerscor-
work.Inthefollowing,weworkonlywiththein-phasemode respond to theinstabilityregime and empty markers correspond to
omittingthesubscript”in”asz (t) z in(t),anduseEq. (47) waves.
≡
asthestartingpointforthesubsequentanalysis.
iszero,wefindcapillarywaveswithfrequencyrelatedtothe
V. DYNAMICSUNDERDIFFERENTTYPESOF wavenumberas
TIME-DEPENDENTDRIVINGFORCE
W 2=k3 g /8. (51)
c
In this section we demonstrate the diversity of hydrody- Inthe case of positivegradientpofthe magneticfield the dis-
namic effects in a system of two separated BECs under the persionrelationEq. (49)yieldsw 2<0forperturbationswith
actionof a time-dependentforce. In particular,we are inter-
sufficiently long wavelengths, which corresponds to the RT
estedintheRTinstabilitydevelopingduetoaconstantforce, instabilityz (t)(cid:181) exp(a t)withthegrowthratedependingon
0
in the RM instability for a pulse force, in the parametric in-
thewavelengthas
stabilityforanoscillatingforce,andintheinterfacedynamics
undertheactionofastochasticGaussianforce. a 2=b k(1 k2/k2) (52)
0 0 − c
andwiththecut-offwavenumber
A. TheRTinstabilitydrivenbyaconstantforce
k =(8/g )1/4 b . (53)
c 0
Hereweconsideraconstantforceb(t)=b0=const,which AccordingtoEq. (49),themaximaplgrowthratecorresponds
drivesthe RT instability and capillary-gravitationalwaves in
tok =k /√3. Figure3comparestheanalyticalresultsof
the unstable or stable configurations, respectively. Constant max c
Eq. (49), (50) tothe numericalsolutionto thelinearizedGP
coefficients of Eq. (47) lead to harmonic oscillations of the
equations. The numerical solution is obtained by diagonal-
interfacez (t)(cid:181) exp(iw t)withthedispersionrelation
izationofthelinearizedGPequations,approximatedbyfinite
g difference method on a grid of 3335 points on the interval
√
w 2=k3 kb0, (49) [ 240;240]inthedimensionlesscoordinates.Figure3shows
2√2− −
good agreement of the analytical theory with the numerical
whichmaybewrittenindimensionalformas results. TheinstabilitygrowthrateEq. (52)agreesalsowith
theformulasemployedin[12].
h¯2n˜1/2 m B Thoughthepresentworkconsidersonlythelinearstageof
w˜2=k˜3 2m02 2p (a12−a)−k˜ 2Bm′, (50) theRTinstability,forillustrativepurposes,itisworthpresent-
ing the results obtained for the nonlinear stage. For exam-
p
wherew˜ andk˜arethedimensionalperturbationfrequencyand ple,Fig. 4showsthedevelopmentoftheRTinstabilityatthe
wave number. Note that Eq. (49) has the same form as the nonlinear stage in a confined system of two BECs obtained
dispersion relation for capillary-gravitational waves in clas- numericallyusing the methodsof [15]. In the numericalso-
sical hydrodynamics[34]. When the externalmagnetic field lution, we begin with a system of 3.2 107 atoms of 87Rb,
·
8
FIG. 4: Development of the RT instability in a trapped system of
two-componentBECpresentedforoneofthecomponents.
FIG.5: StabilizationoftheRTinstabilitybyahigh-frequencyforce
equally split between the two states, with F,mF = 1, 1 fortheamplituderatiosbs/b0=0;7;15;20,forb0=2.5 10−5,k=
and 1,1 . We use the ”pancake” trap geo|metryiwith| w˜−x =i kmaxandW =20a 0. ·
w˜ =|2p i100Hz and w˜ =2p 5kHz. We solve the coupled
z y
· ·
GPequationsnumericallytofindthegroundstateofthesys-
1 1
tem. Thetwocomponentsoccupytheregionswithz>0and +b k z cos(W t)+ z + z cos(2W t) .
s a f f
z<0, respectively. A perturbationof size 9m m is used, and 2 2
(cid:20) (cid:21)
the field of view for all images is 55m m square. The mag-
Separatingthetermsoscillatingwithdifferentfrequencieswe
netic field of magnitude B =1.78G/m is added to the sys-
′ findthe relationbetweenthe amplitudesof theslow andfast
tem, directed so that the two componentsare pushed toward
terms
eachother. Thesystemisshownattimes15.9,19.08,22.26,
25.44(ms)afterthemagneticgradientisadded. Thenonlin- b k b k
z = s z s z
earstagemaybedescribedqualitativelyasdevelopmentofthe 2f −W 2+b k(1 k2/k2) a≈−W 2 a
initialsinusoidalperturbationsintoamushroomstructurewith 0 − c
subsequentgenerationofquantumvortices.Moredetailedin- andreduceEq. (47)to
vestigation of the nonlinear stage of the RT instability will
bepresentedelsewhere. Similarresultsmaybefoundalsoin z¨ =b k 1 k2 b2sk z . (54)
[12]. a 0 −k2 −2b W 2 a
(cid:18) c 0 (cid:19)
Equation(54) describesstabilizationofthe RT instabilityby
B. High-frequencystabilizationoftheRTinstabilityduetoan high-frequencyoscillationsfor
oscillatingforceandparametricinstability
k2 b2k
1 s <0.
In this subsection we consider an oscillating force b(t)= −kc2 −2b0W 2
b +b cos(W t), where b and b are some constant ampli-
0 s 0 s
tudes and W is the frequencyof the force. We show that an Thus, stabilization of the RT perturbations of wave number
k by the high-frequency term is expected for the oscillation
oscillating force leads to two interesting effects: 1) High-
amplitude
frequency oscillations may stabilize the RT instability pro-
duced by the constant backgroundforce b ; 2) The oscillat-
ingforcemaytriggera parametricinstabili0tyevenin theab- bs >√2 W 1 k2, (55)
sence ofanyconstantacceleration, e.g. for b0=0. We start b0 √b0ks −kc2
with the first effect, which is similar to the mechanical phe-
nomenon known as the Kapitsa pendulum [36]. We assume whichmaybewrittenindimensionalunitsas
thatthefrequencyW oftheexternalforceismuchlargerthan
tWhe≫RTa 0in.sTtahbeinlitwyegrsoewpatrharteattehae0sl=owpabv0ekra(g1e−dkte2r/mkc2s),(ltahbaetleisd, BB′0′s >sm4mBBW ′02k˜ (cid:18)1−kk˜˜c22(cid:19). (56)
”a”)andfastoscillatingterms(labeled”f”)inthesolutionto
Eq.(47)asz =z a(t)+z f(t)cos(W t),wherez a(t)andz f(t) ThestabilizationoftheRTinstabilitybyhigh-frequencyos-
changeslowlyonthetimescaleofhigh-frequencyoscillations cillationsisillustratedinFigure5,whichpresentstheresults
2p /W .Followingthecalculationmethodof[30],wetakeinto of directnumericalsimulation of Eq. (47) forthe amplitude
account the relation z¨ z¨a W 2z fcos(W t) and modify Eq. ratiosbs/b0=0;7;15;20,forthestationarycomponentofthe
(47)to ≈ − magneticfieldb0=2.5 10−5, thewavenumbercorrespond-
·
ingtothemaximalgrowthratek=k andfrequencyofthe
max
z¨ W 2z cos(W t)=b k 1 k2 z +z cos(W t) oscillatingmagneticfieldcomponentW =20a 0.Inagreement
a− f 0 −k2 a f with the theoreticalpredictions, unstable exponentialgrowth
(cid:18) c(cid:19)
(cid:2) (cid:3)
9
Similar to [30, 36] we look for a solution to Eq. (47) in the
form z =[x cos(W t/2)+x sin(W t/2)]exp(a t). Mark that
1 2
thegrowthoftheparametricinstabilityisaccompaniedbyos-
cillationsoftheperturbationwithfrequencyW /2,i.e. halfof
the drivingforcefrequency. Substitutingz intoEq. (47) we
obtain
W 2 W t W t
a 2 +W 2 x cos +x sin
0 − 4 c 1 2 2 2
(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
W t W t
+a W x sin +x cos
0 1 2
− 2 2
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
W t W t W t
=b kcos x cos +x sin .
s 1 2
2 2 2
(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
Separatingthetermswithcos(W t/2)andsin(W t/2)andomit-
tinghigher-frequencyterms,wereducetheequationtothelin-
FIG. 6: Stability limits (white line) in the parameter space of the earalgebraicsystemfortheamplitudesx andx
scaledwavenumberk/kcandtheoscillationamplitudebs/b0,where 1 2
ibn0st=ab2il.i5ty·g1r0o−w5t,hWra=tei2n0tmheaxu(nas0ta)b.leSrheagdioinng,Einqd.i(c5a4te).s value of the a 2 W 2 +W 2 bsk x = a W x ,
0 − 4 c− 2 1 − 2
(cid:18) (cid:19)
W 2 b k
a 2 +W 2+ s x =a W x .
0 − 4 c 2 2 1
(cid:18) (cid:19)
Solvingthesystemweobtainthedispersionrelation
W 2 2 b2k2
a 2 +W 2 s +a 2W 2=0. (57)
− 4 c − 4
(cid:18) (cid:19)
Substitutinga =0intoEq. (57)wefindthelimitingoscilla-
tionamplitudeb requiredtoinducetheparametricinstability
s
2 W 2
b > W 2 . (58)
s k c− 4
(cid:12) (cid:12)
(cid:12) (cid:12)
Indimensionalvariablesthe(cid:12)lastrelatio(cid:12)nreadsas
(cid:12) (cid:12)
m B
B ′s >k 1 4W˜2 W˜2 , (59)
FIG. 7: The RT instability growth rate, Eq. (54), in presence of m − c−
external high-frequency fieldwithW =20max(a 0) vs perturbation (cid:12) (cid:12)
wave number for different values of b1/b0 =0;7;15;20 and b0 = wherethedimensionalfrequen(cid:12)cyofcapil(cid:12)larywavesis
2.5 10 5.
−
· h¯2n˜1/2
W˜ k˜ =k˜3 0 2p (a a).
c 2m2 12−
of perturbations (in average) for b /b =0;7 is replaced by
s 0 (cid:0) (cid:1) p
wavesattheinterfaceforsufficientlylargeamplitudesofthe Minimal(infinitelysmall)criticalamplitudeofthemagnetic
oscillatingpartof the magneticfield, b /b =15;20. Figure field correspondsto the perturbationswith the wave number
s 0
6 shows stability limits in the parameter space of the scaled determined by the equation W c(k)=W /2. Therefore, when
wave number k/kc and the oscillation amplitude bs/b0; the applyinganexternalmagneticforceb(t)=bscos(W t)tothe
shadingindicatestheabsolutevalueoftheinstabilitygrowth systemoftwoBECs,weshouldexpectresonancepumpingof
rate in the unstable region. Figure 7 shows the RT instabil- thecapillarywaveswithfrequencyW /2andthewavenumber
itygrowthrateinthepresenceoftheexternalhigh-frequency k=W 2/3(2g )1/6. If capillary wavesof anothernon-resonant
magnetic field plotted versus the perturbation wave number wave number dominate initially, then pumping of the waves
fordifferentvaluesoftheexternalfield. requiresfinite(notnecessarilysmall)amplitudeofthedriving
The second effect related to the oscillating external field force.TheinstabilitydomainisshowninFig. 8intheparam-
istheparametricinstability,whichisknownfrommechanics eter space of the force amplitude and the wave number with
[36] and which has been encountered in gas dynamics, e.g., theshadingindicatingtheperturbationgrowthrate. Figure9
ofcombustionsystems[30–32]. Theconstantpartofthe”ac- presents the dispersion relation for the parametric instability
celeration”isnotrequiredtoobtaintheparametricinstability (thegrowthrateversusthewavenumber)fordifferentampli-
and it may be omitted in the calculations by taking b =0. tudevaluesoftheoscillatingmagneticfield.
0
10
whichyields
0 1
z (t)=exp t z (0)
v W 2 0 v
(cid:20) (cid:18)− c (cid:19)(cid:21)
cos(W t) W 1sin(W t)
= c −c c z (0). (61)
W sin(W t) cos(W t) v
c c c
(cid:18)− (cid:19)
Forconvenience,weintroducethedesignations
0 0
V , (62)
0≡ 1 0
(cid:18) (cid:19)
cos(W t) W 1sin(W t)
G (t) c −c c . (63)
0 ≡ W csin(W ct) cos(W ct)
(cid:18)− (cid:19)
FIG. 8: Stability limits (white line) in the parameter space of the
wavenumberkandamplitudebsoftheexternalhigh-frequencyforce WelookforageneralsolutiontoEq. (60)intheformz v(t)=
withW =3.6 10−3.Shadingindicatesvalueoftheinstabilitygrowth G0(t)n (t)andreduceEq.(60)to
·
rateintheunstableregion,obtainedfromEq.(57).
n˙(t)=kb(t)V(t)n (t), (64)
where
V(t)=G 1(t)V G (t)
−0 0 0
= −(2W c)−1sin(2W ct) −W −c2sin2(W ct) .
(cid:18) cos2(W ct) (2W c)−1sin(2W ct)(cid:19)
ThesolutiontoEq.(64)isthetime-orderedexponentialF(t ),
e.g.see[37],
t
exp dt F(t ) 1+
 ≡
FIG. 9: Growth rate of the parametric instability, Eq. (57), as Z
0
fmuangcntieotnicofofrckefbosr=di0ff.2er5en1t0a−m4;p1li.t7u5de1s0−of4;t3h.e5e1x0te−r4n;a5l o1s0c−il4lawtiinthg +(cid:229) ¥ t t1 tn−1  
W =3.6 10 3. · · · · dt1 dt2 dtnF(t1)F(t2) F(tn). (65)
· − n=1Z Z ··· Z ···
0 0 0
C. RMinstabilityproducedbyamagneticpulse WesolveEq. (64),gobacktotheoriginalvariableandobtain
thegeneralsolutiontoEq.(60)
Within the context of quantum gases, the RM instability
produced by a magnetic pulse b(t)= b d (t) has been dis- t
d
cussed recently in Ref. [15], which indicated considerable z v(t)=G0(t)exp dskb(s)V(s) z v(0). (66)
 
differencesintheinstabilitydevelopmentinBECsincompar- Z0 
ison with thetraditionalconfigurationof classicalgasessep-
On the basis of the generalsolution, we may recoverthe re-
aratedbyaninertinterface. Hereweanalyzethelinearinter-
facedynamicsbyamoregeneral(matrix)methodofsolving sults of Ref. [15] on the RM instability triggeredby a pulse
Eq. (49)applicabletoawideclassoftime-dependentforces. b(t)=bdd (t). Inthiscasealltermsoftime-orderedexponen-
Particularly, the matrix method will be extended to the case tialcommutewitheachotherandweobtain
ofstochastic forcesin thenextsubsection. We introducethe
vectorfunctionz = z z˙ T andrewriteEq. (49)as t
v exp dskb d (s)V(s)
d
 
d z =(cid:0) (cid:1) 0 1 z , (60) Z0 
dt v W 2+kb(t) 0 v
(cid:18)− c (cid:19)  
where the matrix may be split into a constant and time- 0 0 1 0
=exp = ,
dependentparts. First, Eq. (60) issolvedforthecase b=0, kb 0 kb 1
d d
(cid:18) (cid:19) (cid:18) (cid:19)