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Interface dynamics of a two-component Bose-Einstein condensate driven by an external 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, Sweden2Department of Mechanical and Aerospace Engineering, Princeton University, NJ-08544-5263 Princeton, USA

The dynamics of an interface in a two-component Bose-Einstein condensate driven by a spatially uniformtime-dependent force is studied. Starting from the Gross-Pitaevskii Lagrangian, the dispersion relation forlinear waves and instabilities at the interface is derived by means of a variational approach. A number of diversedynamical effects for different types of the driving force is demonstrated, which includes the Rayleigh-Taylorinstability for a constant force, the Richtmyer-Meshkov instability for a pulse force, dynamic stabilization of theRayleigh-Taylor instability and onset of the parametric instability for an oscillating force. Gaussian Markovianand non-Markovian stochastic forces are also considered. It is found that the Markovian stochastic force does notproduce any average effect on the dynamics of the interface,while the non-Markovian force leads to exponentialperturbation growth.

I. INTRODUCTION

Superfluid dynamics of interfaces between Bose-Einsteincondensates (BECs) of dilute atomic gases has gained activeinterest, both in experimental [1–5], and theoretical works [6–13]. These studies started with the experimental realizationof a multicomponent BEC [1, 2] and the theoretical investiga-tion of the planar interface structure [6–9]. At present, a gooddeal of the research focuses on the hydrodynamic instabili-ties at the interfaces in BECs, such as the Kelvin-Helmholtzinstability, the Rayleigh-Taylor (RT) instability, the ferrofluidnormal-field instability, and the Richtmeyer-Meshkov (RM)instability [12–17]. A common feature of these fundamen-tal instabilities in the classical gas dynamics is the produc-tion of vortices at the unstable interface. In that sense, dy-namics of an interface between two quantum fluids has someunique features with quantized vorticity being, probably,themost specific and the most interesting among them. At thesame time, there is also much similarity between the classicaland quantum gas dynamics, which allows ”borrowing” someclassical results and applying them directly or with modifi-cations to the quantum case, see Refs. [12–15]. Such ”bor-rowing” is especially typical in studies of the linear instabil-ity stages, for which the difference between the classical andquantum cases is expected to be minimal. Still, the problemof rigorous derivation of such effects in BEC systems fromthe basic principles of quantum theory remains open for mostcases. The derivation is needed not only to justify the ex-trapolations from the classical to quantum gas dynamics, butalso to find limits of such extrapolation as well as to indi-cate intrinsic quantum effects involved into the problem. Asan example, we point out the study of the interface waves ina system of two BECs undertaken in Ref. [8]. Apart fromcommon capillary waves anticipated from the classical gas-dynamics and related to in-phase perturbations of two BECs,Ref. [8] demonstrated also the possibility of quantum counter-phase perturbations, for which the local interpenetrationdepthof two wave functions for different BECs oscillates in time.Thus, one purpose of the present paper is to provide rigor-ous derivation of the dispersion relations for some fundamen-tal quasi-hydrodynamic instabilities in a two-component BECdriven by a time-dependent force.

As a particular experimental realization of such a force,Sasaki et al. [12] suggested a system of two phase-segregated,interacting BECs with different spins placed in an externalmagnetic field gradient. This is possible, for example, fortwo magnetic sublevels of theF = 1 hyperfine state of87Rbwith |F,mF〉 = |1,−1〉 and |1,1〉. Starting from the Gross-Pitaevskii (GP) action for the two component BEC system,we derive the equations of motion of small perturbations ofthe interface.

The other purpose of the present paper is to demonstrate thediversity of hydrodynamic effects in a system of two separatedBECs under the action of a time-dependent force. Recently,within classical gas dynamics, there has been increased inter-est in the problems of the modified RT and RM instabilitiessubject to time-dependent gravity, e.g. see [18–21]. The con-figuration of time-dependent gravity was discussed in the con-text of specially designed laboratory experiments [22], inthescope of the problem of inertial confined fusion [19, 23, 24]and in relation to flame dynamics [25–28]. In particular, stabi-lization of the RT instability by an oscillating high-frequencyaddition to the gravity acceleration has been proposed bothin the traditional RT configuration [29] and for the inertialconfined fusion [23, 24]. Development of the parametric in-stability at a flame front under the action of acoustic wavesand possible stabilization of the hydrodynamic flame instabil-ity has been investigated experimentally and theoretically inRefs. [30–32]. Here we show that similar effects take placein BECs. Apart from the RT and RM instabilities consideredbefore in [12, 15], we demonstrate also the possibility of dy-namical stabilization of the RT instability by high-frequencyoscillations, as well as triggering of the parametric instability.Another interesting option, which has not been considered yeteven in the classical gas-dynamics corresponds to a Gaussianstochastic force, which may be Markovian or non-Markovianwith zero or non-zero time correlations, respectively. In thepresent paper we show that the Markovian stochastic forcedoes not produce any average effect on dynamics of the in-terface, while the non-Markovian force leads to exponentialgrowth of perturbations.

The paper is organized as follows. In section II we for-mulate the model. In section III we consider a planar (one-dimensional) quantum system with a magnetic gradient push-

2

ing the BECs towards each other, for the case of both constantand time-dependent external forces. In section IV we derivethe equation, which describes the dynamics of small interfaceperturbations. In section V we consider interface dynamicsunder the action of different forces, including a constant force,a harmonic force, a pulse force, and a noisy Gaussian force.The results of the paper are summarized in section VI.

II. THE MODEL

The system of two BECs at zero temperature is describedby two macroscopic wave functionsΨ j(t, r), and the mean-field GP Lagrangian [33]:

L (t) =∫

d3r

[

∑j=1,2

ih2

(

Ψ∗j∂ Ψ j

∂ t− Ψ j

∂ Ψ∗j

∂ t

)

+µ j |Ψ j |2− Vj(t, r)|Ψ j |2−g j j

2|Ψ j |4

− h2

2mj

∣

∣∇Ψ j∣

∣

2

−g12|Ψ1|2|Ψ2|2]

, (1)

whereVj(t, r) are external potential energies of the BEC com-ponents. The tildes in Eq. (1) designate dimensional vari-ables, which will be scaled in the following. Keeping in minda system with two magnetic sublevels of87Rb, we assume thesame particle masses for both componentsm1 = m2 ≡ m andthe same interaction parametersg11= g22≡ g. The interactionparametersg11, g22, g12 are related to the respective s-wavescattering lengthsai j by gi j ≡ 4π h2ai j /m. A dimensionlessinteraction parameter is introduced through the definition

γ ≡ g12/g−1. (2)

The two components are separated in space ifg12>√

g11g22,which impliesγ > 0, see [33]. Throughout this paper we usethe condition of weak separationγ ≪ 1. Modification of allresults of the present paper to the case of different BECs isstraightforward.

The unperturbed flat interface between two components ofBEC is located in the(x,y) plane withz= 0 indicating theplane of symmetry, Fig. 1. The components 1 and 2 are char-acterized by their projection of the atom magnetic moment onthex-axismF =±1, respectively.

Using the states|F,mF〉= |1,−1〉 and |1,1〉 of the atomsmakes it possible to apply the Stern-Gerlach force, push-ing the components in the opposite directions perpendic-ularly to the magnetic moment quantization axis. Weuse a magnetic field whose magnitude has the gradient∇|B(t)|= xdBx(t)/dz≡ xB′(t), which is uniform in space butmay be time-dependent. For the case of87Rb atoms the Stern-Gerlach potential energy in Eq.(1) reads

Vj(t, r) = zµBB′(t)

2(−1) j , (3)

whereµB is the Bohr magneton, and 1/2 is the Land ´e factor.The chemical potentialsµ j in Eq.(1) define normalization

of stationary wave functionsΨ j , and are equal to energy per

FIG. 1: Schematic of the two-component BEC in an external non-uniform magnetic field.

particle of j-th condensate, which is made up of the interac-tion energy due to s-wave scattering and of energy of the par-ticle in the external potential. In order to study low-energetichydrodynamic processes in BECs we assume that energy perparticle is changed negligible due to the external potential, andfor normalization one can use value of the chemical potentialfor B′ = 0 in an infinite system. Such system is characterizedby a uniform density sufficiently far from the interface (i.e. onlength scales much larger than the interface thickness); thenthe chemical potential is given by a usual result for a uniformBEC with number density ˜n0, µ = gn0. However, real BECsare always finite; if the length scale of our system alongz-axisis 2Lz (each condensate is of sizeLz), then the condition thatthe external potential is a small perturbation, reads

LzµBB′(t)

2≪ gn0. (4)

Dimensionless units are introduced by scaling coordinatesby h/

√mµ , time by h/µ, and atom number density (or con-

centration) byµ/g. The dimensionless magnetic force actingon the components is

b1(t) =−b2(t) =− hµBB′ (t)2µ√mµ

≡−b(t) .

The dimensionless Gross-Pitaevskii Lagrangian for twomacroscopic wave functionsΨ j(t, r) at zero temperature reads

L (t) =∫

d3r

[

∑j=1,2

i2

(

Ψ∗j∂Ψ j

∂ t−Ψ j

∂Ψ∗j

∂ t

)

+∣

∣Ψ j∣

∣

2− 12

∣

∣∇Ψ j∣

∣

2− zbj(t)∣

∣Ψ j∣

∣

2

− 12

∣

∣Ψ j∣

∣

4

− (1+ γ)|Ψ1|2|Ψ2|2]

, (5)

3

and equations of motion are obtained from the action principle

δSδΨ∗

j=

∂∂ t

δS

δ(

∂Ψ∗j/∂ t

)

, (6)

where the action isS=∫

dtL (t). The result is the coupledGP equations:

i∂Ψ1

∂ t=

[

−∆2−1− zb(t)+ |Ψ1|2+(1+ γ) |Ψ2|2

]

Ψ1, (7)

i∂Ψ2

∂ t=

[

−∆2−1+ zb(t)+ |Ψ2|2+(1+ γ) |Ψ1|2

]

Ψ2. (8)

III. PLANAR DENSITY PROFILES FOR ATWO-COMPONENT BEC

The present work is devoted to multidimensional instabili-ties bending an interface between two BEC components in anexternal magnetic field gradient. As the first step in the study,we have to specify the planar (one-dimensional, 1D) quasista-tionary state, which is influenced by a spatially uniform exter-nal field. In the subsections A and B we consider the cases ofconstant and time-dependent external fields, respectively.

A. Planar profiles in the case of a stationary force

Here we study the 1D hydrostatic state of BECs, when theconstant external forceb0 is balanced by internal forces of thecondensates. Let us assume that such a state was created byadiabatic switching on of the external field, and no perturba-tions bend the interface. The steady state is described by tworeal-valued functionsψ0 j(z), and the system of dimensionlessGP equations reads

0=12

d2ψ01

dz2 +ψ01+ zb0ψ01−ψ301− (1+ γ)ψ2

02ψ01, (9)

0=12

d2ψ02

dz2 +ψ02− zb0ψ02−ψ302− (1+ γ)ψ2

01ψ02. (10)

The time-independent force pushes the BEC components to-wards each other whenb0 > 0. With our choice of equalmasses and scattering lengths for both components, the prob-lem possesses the symmetryψ01(z) = ψ02(−z). In the caseof zero magnetic gradient the density profiles have been foundin [8] within the limit of weak separationγ ≪ 1 as

n01(z) = n02(−z)≈[

1+exp(

−2√

2γz)]−1

. (11)

We calculate the wavefunction of condensate 1 in the bulkof condensate 2. According to Eq. (9), the wave functionspenetrate into each other with the characteristic depthZp =

1/√

2γ, see also [7], which means thatψ201 ≪ 1 for z> Zp.

The density profile of the bulk is described by the Thomas-Fermi approximation (TFA), which means thatψ2

02 ≈ 1− bz

for z> Zp. With these approximations Eq.(9) becomes:

12

d2ψ01

dz2 = [γ − (2+ γ)b0z]ψ01. (12)

The type of solution to Eq.(12) changes from exponentiallydecaying to oscillating in space, asz becomes larger thanZo = γ/[(2+ γ)b0]≈ γ/(2b0). However, the magnitude of thespatial oscillations depends on the valueψ01(Zo): it decreasesexponentially with increasingZo. Therefore, the spatial os-cillations become significant whenZo becomes comparablewith Zp, or whenb0 = γ

√

γ/2; in this case the TFA becomesinvalid, and Eqs.(9),(10) must be solved exactly. The spa-tial oscillations may be interpreted as quantum interference ofmatter waves, caused by interplay between the non-linearityand the dispersion (the kinetic term). This phenomenon is be-yond the scope of the present work, and it will be studied indetail elsewhere. Thus, here we use parameters satisfying thecondition

b0 ≪ b0cr ≡ γ√

γ/2, (13)

or the magnetic gradientB′ satisfying

B′ < B′0cr ≡

√

γ3µ3m/(µBh) , (14)

which means that the spatial oscillations of the hydrostaticprofiles may be neglected. It follows from Eq. (14) thatthe critical magnetic gradientB′

0cr depends on the differ-ence between inter- and intra-species interaction parametersasγ = (g12−g)/g, and on density via the chemical potentialsinceµ ≈ gn0 in the TFA for an infinite system in a suffi-ciently weak external field. Therefore, the effect of quantuminterference becomes stronger at lower densities of the BECs.Equation (13) yields the dimensionless condition of a well-defined interface between two condensates subject to the par-ticular external force. In the system of two magnetic sublevelsof 87Rb we haveγ ≃ 10−2 (see [2]), and the typical density is5 · 1014cm−3. The numerical solution to Eqs. (9), (10) forthese parameters shows that the spatial oscillations are notice-able atb0 ≈ 10−4; so we takeb< 10−4. Numerical solutionsfor the wave functions near the interface are presented in Fig.2 for b0 = 7.5·10−5; 10−4, plots (a) and (b), respectively. Thedashed lines in Fig. 1(a) show the approximate analytical so-lution of Eq.(11) obtained for the case of zero magnetic field.As we can see, the analytical solution Eq.(11) fits the numer-ical results quite well even in the case of non-zero magneticfield.

B. 1D Bogoliubov excitations of BEC due to a harmonic force

Since we are interested in the development of hydrody-namic instabilities for time-dependent external forces, weneed to know how a harmonic force influences the planardensity profiles of BECs. Similar to the constant force stud-ied above, for time-dependent forces we also consider rathersmall magnetic fields, so that the respective 1D Bogoliubovexcitations may be interpreted as linear acoustic waves. An-other type of a time-dependent force studied in this paper isa

4

FIG. 2: Hydrostatic profiles of two components pushed to eachotherby a constant force,b= 7.5·10−5; 10−4, figures (a) and (b), respec-tively. The dashed lines show the analytical solution Eq. (14). Theinset in figure (b) presents the Airy function of the second kind.

pulse force corresponding to a weak shock in hydrodynamics.Still, it is known from hydrodynamics that a weak shock maybe also described as a linear acoustic wave [34]. For this rea-son, in the present subsection we study modifications of theBEC density in the 1D geometry due to the harmonic force.

Because of the problem symmetry with respect toz= 0, weconsider a one-component semi-infinite BEC forz< 0 con-fined by a wall atz= 0. The magnitude and frequency of theforce determines the energy of the generated Bogoliubov exci-tations. In this subsection we express density and velocityper-turbations,δn andδU , in terms of magnitude and frequencyof the external force, and find the restrictions on the drivingforce, for which the density perturbations are smallδn≪ n0.

In the case of a harmonic force, the dimensionless externalpotential iszbsexp(iΩt), wherebs is the force amplitude andΩ is the frequency. Our model system is then described by atime-dependent GP equation

i∂ψ∂ t

=−12

∂ 2ψ∂z2 +ψ∗ψ2+ψVtrap(z)+zbsexp(iΩt)ψ . (15)

whereVtrap (z) = +∞ for z< 0 andVtrap (z) = 0 for z≥ 0.Since the trap potentialVtrap (z) experiences discontinuity at

the ”wall”, then the wave function and density are specifiedonly for z≥ 0, no healing of the condensate wave function isrequired near the wall in TFA, and equilibrium is specified bythe uniform densityn0 = 1. In that case the velocity pertur-bationδU(z, t) is zero at the wallδU(0, t) = 0. The perturba-tions of density and velocity,δn andδU , may be expressed interms of the perturbation wave functionδψ

ψ (z, t) = exp(−it) (√

n0+ δψ) , (16)

whereδψ (z, t) = u(z)exp(iΩt)+v∗ (z)exp(−iΩt), andu(z),v∗ (z) are time-independent functions. Since the perturbationsare small, we find

δn= exp(iΩt)(u+ v)+ c.c.,

δU =∂φ∂z

=12i

exp(iΩt)∂∂z

(u− v)− c.c.,

wherec.c. denotes complex conjugate. Then the boundarycondition at the wall reads

∂∂z

(u− v)

∣

∣

∣

∣

z=0= 0. (17)

We linearize the GP equation (15),

i∂∂ t

δψ =−12

∂ 2

∂z2 δψ +2δψ + δψ∗+ zbsexp(iΩt) ,

and obtain the Bogoliubov equations

−Ωu=−12

∂ 2u∂z2 +u+ v+bsz, (18)

Ωv=−12

∂ 2v∂z2 + v+u, (19)

which, in turn, may be reduced to

Ω2 ∂∂z

(u− v) =

[

(

−12

∂ 2

∂z2 +1

)2

−1

]

∂∂z

(u− v)−Ωbs.

(20)The solution to Eq. (20) is a superposition of a runningwave U1exp(iΩt − iqz) and spatially uniform oscillationsU0exp(iΩt),

∂∂z

(u− v) = U0+U1exp(−iqz) , (21)

whereU0,1 are the respective amplitudes. The running wave isthe Bogoliubov elementary excitation (the quantum acousticwave) with the wave numberq(Ω) determined via the Bogoli-ubov spectrum

Ω2 = q4/4+q2. (22)

It follows from the boundary condition at the wall thatU0 =−U1 =−bs/Ω and

δU =bs

Ω[sin(Ωt −qz)− sinΩt] . (23)

5

The perturbations of density follow from the 1D continuityequation

∂∂ t

δn=− ∂∂z

δU

as

δn=bsqΩ2 sin(Ωt −qz) . (24)

Then, the condition of weak perturbationsδn≪ n0 becomesbsq/Ω2 ≪ 1, or in the dimensional form

µBB′q

2mΩ2≪ 1, (25)

where q and Ω are the dimensional wave number and fre-quency of the induced excitations, coupled by the Bogoliubovspectrum (22). In the low energy limitq4 ≪ q2, the Boguli-ubov excitations are sound waves with velocitycs=

√

gn0/m(unity in our dimensionless units). Then the restriction ontheamplitude of the magnetic force reads

µBB′

2m≪ Ω

√

gn0

m. (26)

The criterion (26) may be interpreted as the condition ofincompressible condensates with the speed of sound muchlarger than any other parameter of velocity dimension in-volved in the problem. A more general condition (25) takesinto account the Bogoliubov spectrum for higher energies.

We solved the Bogoliubov equations for a model infinitesystem. The case of a finite system, confined by two walls, issolved analogously, and the boundary condition (17) shouldbe imposed for both walls. Real systems are always finite,and therefore the maximal wavelength is limited by the sizeof the systemLz. In the low-frequency limit condition (25)is equivalent to the energetic condition (4). In this limit,theBogoliubov excitations are sound waves, with maximal wave-length of orderLz; substitutingLz = 2π q−1 to (4), and usingthe sound spectrum, one obtains (26).

IV. LINEAR ANALYSIS OF THE INTERFACE DYNAMICS

We now consider the dynamics of the interface between twoinfinite BECs in the presence of an external force, which maybe time-dependent,b(t). Similarly to Refs. [8, 33, 35] weuse a variational approach. We represent the wave functionsin terms of densityn j and velocity potentialφ j asψ j (t,x,z) =√

n j exp(iφ j), so that the action for the system of two BECs is

S=

∫

dt∫

d3r

∑j=1,2

[

n j∂φ j

∂ t+

12

n j (∇φ j )2 +

12

(

∇√n j)2−n j

]

+ zb(t)(n2−n1)+

12

(

n21+n2

2

)

+(1+ γ)n1n2

. (27)

For the variational calculation, we employ an ansatz that isable to describe bending and compression of the interfacebetween the two condensates. We are interested in pertur-bation wavelengths much larger than the interface thickness,kZp = k/

√2γ ≪ 1, so that the density deformations may be

treated as a bending of the unperturbed profile as a whole,

n j(z,x, t) = n j0 [z− δzj(x, t)] , (28)

wheren j0 are the stationary density profiles for the BECs. Thevariational functions are taken to be of the form of Fouriermodes with separation of space and time dependence,

δzj (x, t) = ζ j(t)cos(kx), (29)

φ j(z,x, t) = ξ j(t) f j (z)cos(kx). (30)

In classical hydrodynamics, perturbations of both componentsof an inert interface are related by the continuity conditionζ1 = ζ2. In the quantum gas-dynamics of two BECs this isnot necessarily the case, since we may have interpenetrationof the wave functions.

The Lagrange equations with respect to the variablesξ j are

ζ j

∫

dzn′j f j = ξ j

∫

dz

[

f jddz

(

n j f ′j)

+ k2 f 2j n j

]

. (31)

These Lagrange equations are satisfied if the respective conti-nuity equations are

ζ jn′j = ξ j

[

ddz

(

n j f ′j)

+ k2 f jn j

]

. (32)

First, the time-dependent terms on each side have to beequated, which, up to constant factors, leads to

ζ j(t) = ξ j(t). (33)

This reflects the fact thatξ j andζ j are collective canonicallyconjugate variables. For the functionsf j , Ref.[8] gave theapproximate solution

f1(z) =− f2(−z)

−(1+ kz)−1, if z< 0,−k−1exp(−kz), if z> 0.

(34)

The same solution holds in the presence of a small externalforce under the conditions specified in Sec. III. We take Eq.(34) as our ansatz forf j . More elaborate expressions forf1(z),and f2(z) may provide a smooth transition in the interface re-gion; however, the ansatz Eq. (34) yields reliable analyticalresults, as we will demonstrate below by comparison to thenumerical solution to the problem.

We now derive the Lagrange equation forζ j(t). No timederivatives inζ j appear in the Lagrangian, so the Lagrangeequation reads

0=δSδζ j

= I1( j)+ I3( j)+ I5( j)+ I7( j), (35)

where we introduceIl( j) to label the nonvanishing terms ofδS/δζ j ; the numberingl in Il( j) refer to the respective terms

6

in Eq. (27). For convenience, we assume a finite size of thesystem,Lz, Lx, though the final result does not depend on thesize restrictions. We have

I1 =δ

δζ j

∫

d2rn j∂φ j

∂ t=

∫

dzdxn′0 j(z+ δzj) f j ξ j cos2(kx) =

Lx

2ξ j

∫

dzn′j(z) f j (z). (36)

The next nonvanishing term comes from the zero-point mo-tion energy,

I3 ≡δ

δζ j

∫

d2r12

(

∇√n j)2

=

12

δδζ j

∫

d2r

[

(

∂∂x

√n j

)2

+

(

∂∂z

√n j

)2]

=

12

δδζ j

∫

d2r(

∂∂z

√n j

)2[

1+

(

∂∂x

δzj

)2]

. (37)

The first term within the parentheses yields a constant inde-pendent ofζ , and the second term equals

I3 =12

δδζ j

∫

d2r(

∂∂z

√n j

)2( ∂∂x

δzj

)2

=

Lx

212

δδζ j

∫

dz

(

ddz

√n0 j

)2

(ζ jk)2 =

Lx

2ζ jk

2

+∞∫

−∞

dz

(

ddz

√n0 j

)2

. (38)

The next non-vanishing term corresponds to the perturbationof the potential energy of the system,

I5( j) =δ

δζ j

∫

d2rzb(t)(n2−n1) =

Lx

2(−1) jb(t)

δδζ j

∫

dzznj =

Lx

2(−1) jb(t)

δδζ j

∫

dzzn0 j (z+ δzj) . (39)

It is more convenient to calculate these terms forj = 1 andfor j = 2 separately. Using the coordinate transformationz′ =z+ δzj we modify the last expression as

I5(1) =−Lx

2b(t)

δδζ1

+∞∫

−Lz

dzzn01(z+ δz1) =

−Lx

2b(t)

δδζ1

+∞∫

−Lz+δz1

dz′[

z′n01(

z′)

− δz1n01(

z′)]

,

I5(2) =Lx

2b(t)

δδζ2

Lz∫

−∞

dzzn02(z+ δz2) =

Lx

2b(t)

δδζ2

Lz+δz2∫

−∞

dz′[

z′n02(

z′)

− δz2n02(

z′)]

.

The integrals can be calculated as

I5(1) =−Lx

2b(t)

δδζ1

(

−12

δz21+ δz2

1

)

=−Lx

2b(t)ζ1, (40)

I5(2) =Lx

2b(t)

δδζ2

(

12

δz22− δz2

2

)

=−Lx

2b(t)ζ2. (41)

The next term ofδS/δζ j does not contribute to the equationsof motion, while the last (seventh) termI7 is important. Anal-ysis of the terms calculated so far shows that the variablesζ jare governed by two coupled oscillator equations, and the cou-pling happens because ofI7. Since the equations forζ1 andζ2are identical, the normal modes may be in-phase,ζ1 = ζ2,and counter-phase,ζ1 = −ζ2. First, let us calculateI7 forthe case ofζ1 = ζ2, which we denote byI7(in). In this caseδz1 = δz2 ≡ δz, and we find

I7(in) =δ

δζ j

∫

d2r (1+ γ)n1n2 =

Lx

2δ

δζ j

+∞∫

−∞

dz(1+ γ)n01(z+ δz)n02(z+ δz) = 0. (42)

Next we consider the case of counter-phase oscillations withδz1 =−δz2 ≡ δzandζ1 =−ζ2 ≡ ζ , which leads to

I7(co) =Lx

2δ

δζ

+∞∫

−∞

dz′ (1+ γ)n01(

z′)

n02(

z′−2δz)

.

To calculateδ/δζ in the last expression, it is easiest to as-sumeδz small and Taylor expandn02(z′−2δz) to second or-der, which results in

I7(co) =Lx

24(1+ γ)ζ

+∞∫

−∞

dzn01d2n02

dz2 .

The last result may be transformed to a symmetric form simi-lar to [8]:

I7(co) =−Lx

24(1+ γ)ζ

+∞∫

−∞

dzdn01

dzdn02

dz. (43)

Taking into account all the terms obtained above we writeequations of motion forζ j . The integrals are calculated us-ing the density profile (11)

+∞∫

−∞

dzdn0 j

dzf j = 1/k, (44)

+∞∫

−∞

dz

(

ddz

√n0 j

)2

=√

γ/8, (45)

7

+∞∫

−∞

dzdn01

dzdn02

dz=−

√

8γ/3. (46)

We designate the interface perturbations of the in-phase oscil-lations byζ1 = ζ2 ≡ ζin and obtain the final equation

∂ 2ζin

∂ t2 + ζin

[

k3√

γ/8−b(t)k]

= 0. (47)

In the opposite case of the counter-phase perturbationsζ1 =−ζ2 ≡ ζco we find

∂ 2ζco

∂ t2 + ζco

[

k2

√

γ8+

43(1+ γ)

√

8γ −b(t)

]

k= 0. (48)

In the limiting case ofb= 0 our theory reproduces the resultsof [8]. Within the limit of a weak field,b≪ bm = γ

√2γ, and

long wavelength perturbationsk/√

2γ ≪ 1, the terms in theparenthesis of Eq. (48) are of different orders of magnitude,with the term 4(1+ γ)

√8γ/3 dominating. For this reason, to

the leading terms, the counter-phase mode is not affected bythe external potential, and we do not consider it in the presentwork. In the following, we work only with the in-phase modeomitting the subscript ”in” asζ (t)≡ ζin(t), and use Eq. (47)as the starting point for the subsequent analysis.

V. DYNAMICS UNDER DIFFERENT TYPES OFTIME-DEPENDENT DRIVING FORCE

In this section we demonstrate the diversity of hydrody-namic effects in a system of two separated BECs under theaction of a time-dependent force. In particular, we are inter-ested in the RT instability developing due to a constant force,in the RM instability for a pulse force, in the parametric in-stability for an oscillating force, and in the interface dynamicsunder the action of a stochastic Gaussian force.

A. The RT instability driven by a constant force

Here we consider a constant forceb(t) = b0 = const, whichdrives the RT instability and capillary-gravitational waves inthe unstable or stable configurations, respectively. Constantcoefficients of Eq. (47) lead to harmonic oscillations of theinterfaceζ (t) ∝ exp(iωt) with the dispersion relation

ω2 = k3√γ2√

2− kb0, (49)

which may be written in dimensional form as

ω2 = k3 h2n1/20

2m2

√

2π(a12−a)− kµBB′

2m, (50)

whereω andk are the dimensional perturbation frequency andwave number. Note that Eq. (49) has the same form as thedispersion relation for capillary-gravitational waves inclas-sical hydrodynamics [34]. When the external magnetic field

FIG. 3: Lowest Bogoliubov modes (the RT dispersion relation) fordifferent values of the driving force. The analytical result is obtainedfrom Eq. (49), and presented by lines. Dashed lines show realpartsof the Bogoliubov eigenvalues (waves), and solid lines correspond tothe imaginary parts (instability). The markers show respective nu-merical solution to the linearized GP equations; filled markers cor-respond to the instability regime and empty markers correspond towaves.

is zero, we find capillary waves with frequency related to thewave number as

Ω2c = k3

√

γ/8. (51)

In the case of positive gradient of the magnetic field the dis-persion relation Eq. (49) yieldsω2 < 0 for perturbations withsufficiently long wavelengths, which corresponds to the RTinstabilityζ (t) ∝ exp(α0t) with the growth rate depending onthe wavelength as

α20 = b0k(1− k2/k2

c) (52)

and with the cut-off wave number

kc = (8/γ)1/4√

b0. (53)

According to Eq. (49), the maximal growth rate correspondsto kmax= kc/

√3. Figure 3 compares the analytical results of

Eq. (49), (50) to the numerical solution to the linearized GPequations. The numerical solution is obtained by diagonal-ization of the linearized GP equations, approximated by finitedifference method on a grid of 3335 points on the interval[−240;240] in the dimensionless coordinates. Figure 3 showsgood agreement of the analytical theory with the numericalresults. The instability growth rate Eq. (52) agrees also withthe formulas employed in [12].

Though the present work considers only the linear stage ofthe RT instability, for illustrative purposes, it is worth present-ing the results obtained for the nonlinear stage. For exam-ple, Fig. 4 shows the development of the RT instability at thenonlinear stage in a confined system of two BECs obtainednumerically using the methods of [15]. In the numerical so-lution, we begin with a system of 3.2 · 107 atoms of87Rb,

8

FIG. 4: Development of the RT instability in a trapped systemoftwo-component BEC presented for one of the components.

equally split between the two states, with|F,mF〉 = |1,−1〉and |1,1〉. We use the ”pancake” trap geometry withωx =ωz = 2π · 100Hz andωy = 2π · 5kHz. We solve the coupledGP equations numerically to find the ground state of the sys-tem. The two components occupy the regions withz> 0 andz< 0, respectively. A perturbation of size 9µm is used, andthe field of view for all images is 55µm square. The mag-netic field of magnitudeB′ = 1.78G/m is added to the sys-tem, directed so that the two components are pushed towardeach other. The system is shown at times 15.9, 19.08, 22.26,25.44 (ms) after the magnetic gradient is added. The nonlin-ear stage may be described qualitatively as development of theinitial sinusoidal perturbations into a mushroom structure withsubsequent generation of quantum vortices. More detailed in-vestigation of the nonlinear stage of the RT instability willbe presented elsewhere. Similar results may be found also in[12].

B. High-frequency stabilization of the RT instability due to anoscillating force and parametric instability

In this subsection we consider an oscillating forceb(t) =b0 + bscos(Ωt), whereb0 and bs are some constant ampli-tudes andΩ is the frequency of the force. We show that anoscillating force leads to two interesting effects: 1) High-frequency oscillations may stabilize the RT instability pro-duced by the constant background forceb0; 2) The oscillat-ing force may trigger a parametric instability even in the ab-sence of any constant acceleration, e.g. forb0 = 0. We startwith the first effect, which is similar to the mechanical phe-nomenon known as the Kapitsa pendulum [36]. We assumethat the frequencyΩ of the external force is much larger thanthe RT instability growth rateα0 =

√

b0k(1− k2/k2c), that is,

Ω ≫ α0. Then we separate the slow averaged terms (labeled”a”) and fast oscillating terms (labeled ”f”) in the solution toEq. (47) asζ = ζa (t)+ζ f (t)cos(Ωt), whereζa (t) andζ f (t)change slowly on the time scale of high-frequency oscillations2π/Ω. Following the calculation method of [30], we take intoaccount the relationζ ≈ ζa −Ω2ζ f cos(Ωt) and modify Eq.(47) to

ζa−Ω2ζ f cos(Ωt) = b0k

(

1− k2

k2c

)

[

ζa+ ζ f cos(Ωt)]

FIG. 5: Stabilization of the RT instability by a high-frequency forcefor the amplitude ratiosbs/b0 = 0;7;15;20, forb0 = 2.5 ·10−5, k=kmax andΩ = 20α0.

+bsk

[

ζacos(Ωt)+12

ζ f +12

ζ f cos(2Ωt)

]

.

Separating the terms oscillating with different frequencies wefind the relation between the amplitudes of the slow and fastterms

ζ2 f =− bskΩ2+b0k(1− k2/k2

c)ζa ≈−bsk

Ω2 ζa

and reduce Eq. (47) to

ζa = b0k

(

1− k2

k2c− b2

sk2b0Ω2

)

ζa. (54)

Equation (54) describes stabilization of the RT instability byhigh-frequency oscillations for

1− k2

k2c− b2

sk2b0Ω2 < 0.

Thus, stabilization of the RT perturbations of wave numberk by the high-frequency term is expected for the oscillationamplitude

bs

b0>√

2Ω√b0k

√

1− k2

k2c, (55)

which may be written in dimensional units as

B′s

B′0>

√

4mΩ2

µBB′0k

(

1− k2

k2c

)

. (56)

The stabilization of the RT instability by high-frequency os-cillations is illustrated in Figure 5, which presents the resultsof direct numerical simulation of Eq. (47) for the amplituderatiosbs/b0 = 0;7;15;20, for the stationary component of themagnetic fieldb0 = 2.5 ·10−5, the wave number correspond-ing to the maximal growth ratek= kmax and frequency of theoscillating magnetic field componentΩ= 20α0. In agreementwith the theoretical predictions, unstable exponential growth

9

FIG. 6: Stability limits (white line) in the parameter spaceof thescaled wave numberk/kc and the oscillation amplitudebs/b0, whereb0 = 2.5 · 10−5, Ω = 20max(α0). Shading indicates value of theinstability growth rate in the unstable region, Eq. (54).

FIG. 7: The RT instability growth rate, Eq. (54), in presenceofexternal high-frequency field withΩ = 20max(α0) vs perturbationwave number for different values ofb1/b0 = 0;7;15;20 andb0 =2.5·10−5.

of perturbations (in average) forbs/b0 = 0;7 is replaced bywaves at the interface for sufficiently large amplitudes of theoscillating part of the magnetic field,bs/b0 = 15;20. Figure6 shows stability limits in the parameter space of the scaledwave numberk/kc and the oscillation amplitudebs/b0; theshading indicates the absolute value of the instability growthrate in the unstable region. Figure 7 shows the RT instabil-ity growth rate in the presence of the external high-frequencymagnetic field plotted versus the perturbation wave numberfor different values of the external field.

The second effect related to the oscillating external fieldis the parametric instability, which is known from mechanics[36] and which has been encountered in gas dynamics, e.g.,of combustion systems [30–32]. The constant part of the ”ac-celeration” is not required to obtain the parametric instabilityand it may be omitted in the calculations by takingb0 = 0.

Similar to [30, 36] we look for a solution to Eq. (47) in theform ζ = [ξ1cos(Ωt/2)+ ξ2sin(Ωt/2)]exp(αt). Mark thatthe growth of the parametric instability is accompanied by os-cillations of the perturbation with frequencyΩ/2, i.e. half ofthe driving force frequency. Substitutingζ into Eq. (47) weobtain(

α02− Ω2

4+Ω2

c

)[

ξ1cos

(

Ωt2

)

+ ξ2sin

(

Ωt2

)]

+α0Ω[

−ξ1sin

(

Ωt2

)

+ ξ2cos

(

Ωt2

)]

= bskcos

(

Ωt2

)[

ξ1cos

(

Ωt2

)

+ ξ2sin

(

Ωt2

)]

.

Separating the terms with cos(Ωt/2) and sin(Ωt/2) and omit-ting higher-frequency terms, we reduce the equation to the lin-ear algebraic system for the amplitudesξ1 andξ2

(

α02− Ω2

4+Ω2

c−bsk2

)

ξ1 =−αΩξ2,

(

α02− Ω2

4+Ω2

c+bsk2

)

ξ2 = αΩξ1.

Solving the system we obtain the dispersion relation

(

α2− Ω2

4+Ω2

c

)2

− b2sk2

4+α2Ω2 = 0. (57)

Substitutingα = 0 into Eq. (57) we find the limiting oscilla-tion amplitudebs required to induce the parametric instability

bs >2k

∣

∣

∣

∣

Ω2c −

Ω2

4

∣

∣

∣

∣

. (58)

In dimensional variables the last relation reads as

µBB′s

m> k−1

∣

∣4Ω2c − Ω2

∣

∣ , (59)

where the dimensional frequency of capillary waves is

Ωc(

k)

= k3 h2n1/20

2m2

√

2π (a12−a) .

Minimal (infinitely small) critical amplitude of the magneticfield corresponds to the perturbations with the wave numberdetermined by the equationΩc(k) = Ω/2. Therefore, whenapplying an external magnetic forceb(t) = bscos(Ωt) to thesystem of two BECs, we should expect resonance pumping ofthe capillary waves with frequencyΩ/2 and the wave numberk = Ω2/3(2γ)1/6. If capillary waves of another non-resonantwave number dominate initially, then pumping of the wavesrequires finite (not necessarily small) amplitude of the drivingforce. The instability domain is shown in Fig. 8 in the param-eter space of the force amplitude and the wave number withthe shading indicating the perturbation growth rate. Figure 9presents the dispersion relation for the parametric instability(the growth rate versus the wave number) for different ampli-tude values of the oscillating magnetic field.

10

FIG. 8: Stability limits (white line) in the parameter spaceof thewave numberk and amplitudebs of the external high-frequency forcewith Ω= 3.6·10−3. Shading indicates value of the instability growthrate in the unstable region, obtained from Eq. (57).

FIG. 9: Growth rate of the parametric instability, Eq. (57),asfunction of k for different amplitudes of the external oscillatingmagnetic forcebs = 0.25·10−4;1.75·10−4;3.5 ·10−4;5 ·10−4 withΩ = 3.6·10−3.

C. RM instability produced by a magnetic pulse

Within the context of quantum gases, the RM instabilityproduced by a magnetic pulseb(t) = bdδ (t) has been dis-cussed recently in Ref. [15], which indicated considerabledifferences in the instability development in BECs in compar-ison with the traditional configuration of classical gases sep-arated by an inert interface. Here we analyze the linear inter-face dynamics by a more general (matrix) method of solvingEq. (49) applicable to a wide class of time-dependent forces.Particularly, the matrix method will be extended to the caseof stochastic forces in the next subsection. We introduce thevector functionζv =

(

ζ ζ)T and rewrite Eq. (49) as

ddt

ζv =

(

0 1−Ω2

c + kb(t) 0

)

ζv, (60)

where the matrix may be split into a constant and time-dependent parts. First, Eq. (60) is solved for the caseb= 0,

which yields

ζv(t) = exp

[

t

(

0 1−Ω2

c 0

)]

ζv(0)

=

(

cos(Ωct) Ω−1c sin(Ωct)

−Ωcsin(Ωct) cos(Ωct)

)

ζv(0). (61)

For convenience, we introduce the designations

V0 ≡(

0 01 0

)

, (62)

G0(t)≡(

cos(Ωct) Ω−1c sin(Ωct)

−Ωcsin(Ωct) cos(Ωct)

)

. (63)

We look for a general solution to Eq. (60) in the formζv (t) =G0(t)ν (t) and reduce Eq. (60) to

ν (t) = kb(t)V(t)ν(t), (64)

where

V(t) = G−10 (t)V0G0 (t)

=

(

−(2Ωc)−1sin(2Ωct) −Ω−2

c sin2 (Ωct)cos2 (Ωct) (2Ωc)

−1sin(2Ωct)

)

.

The solution to Eq.(64) is the time-ordered exponentialF(τ),e.g. see [37],

exp

t∫

0

dτF(τ)

≡ 1+

+∞

∑n=1

t∫

0

dt1

t1∫

0

dt2 · · ·tn−1∫

0

dtnF(t1)F(t2) · · ·F(tn). (65)

We solve Eq. (64), go back to the original variable and obtainthe general solution to Eq. (60)

ζv (t) = G0 (t)exp

t∫

0

dskb(s)V(s)

ζv(0). (66)

On the basis of the general solution, we may recover the re-sults of Ref. [15] on the RM instability triggered by a pulseb(t) = bdδ (t). In this case all terms of time-ordered exponen-tial commute with each other and we obtain

exp

t∫

0

dskbdδ (s)V(s)

= exp

(

0 0kbd 0

)

=

(

1 0kbd 1

)

,

11

FIG. 10: Development of the RM instability in a trapped system oftwo-component BEC presented for one of the components.

which yields the final solution

ζ (t) =

√

ζ 2 (0)+1

Ω2c

[

kbdζ (0)+ ζ (0)]2

sin(Ωct +ϕd)

(67)with the new oscillation phase determined by

tanϕd =Ωcζ (0)

ζ (0)+ kbdζ (0). (68)

Thus, the general solution Eq. (66) reproduces the analyticalresults of Ref. [15] given by Eq. (67). Particularly, Eq. (67)describes the jump in the amplitude and phase of a standingcapillary wave because of the pulse for a non-zero bending ofthe interface,ζ (0) 6= 0. In the opposite case ofζ (0) = 0 andζ (0) 6= 0 the pulse does not influence the interface dynamics.Eq. (67) shows that the transfer of magnetic energy of thepulse to superfluid oscillations is the most efficient when thedeviation of the interface just before the pulse is maximal.

The linear analysis works rigorously as long as the pertur-bation amplitude after the pulse remains sufficiently small.Once it becomes large enough, nonlinear effects dominate,and the interface evolves into a mushroom pattern with de-tachment of droplets as discussed in [15]. Figure 10 illus-trates the development of the RM instability in BECs at thenonlinear stage obtained using the numerical methods similarto [15]. We use the same system of BECs as in Fig. 4. A mag-netic pulse of durationτ = 0.02ms and size

√πµBB′τ/2m=

1.05·10−3m/s is applied to the initial capillary wave seen inFig. 10 (a). The following graphs correspond to the time in-stants 2.39; 4.77; and 7.16 ms after the pulse (figures b, c, d,respectively). Development of the RM instability at the initialstages (e.g. Fig. 10 b) resembles the RT instability shown inFig. 4. However, the difference between two instabilities isnoticeable at later stages shown in Fig. 10 (c, d). In particular,we observe the detachment of droplets in the RM instabilityin agreement with the model presented in Ref. [15].

D. Gaussian noise

Finally, we consider the interface dynamics under the actionof a Gaussian random forcebr (t) with mean zero〈br (t)〉= 0,where the brackets imply averaging over a statistical ensemble

of realizations. The force is switched on att = 0. Because ofthe Gaussian property, the force is characterized only by vari-anceQ and by the correlation timeτc. The case of zeroτc isknown as the Markovian case, and in this case averaging maybe performed exactly. On the contrary, the case of non-zeroτc is not solvable in general, but an analytical approximate so-lution may be obtained in the limit of small correlation timeτcΩc ≪ 1.

In order to obtain an averaged form of Eq. (47) we em-ploy the general solution (66). Since Eq. (66) is a time-ordered exponential with non-commutative kernel at differentmoments of time, we use the cumulant method for averaging[37]. Within this method, ifG(τ) is a stochastic Gaussianoperator with mean zero〈G(t)〉 = 0, then the time-orderedexponential ofG(τ) is averaged by the formula

⟨

exp

t∫

0

dτG(τ)

⟩

= exp

t∫

0

dττ∫

0

ds〈G(τ)G(s)〉

.

(69)In the Markovian case this formula is exact, but in the non-Markovian case this is an approximation valid whenτc issmall compared to the intrinsic time scale of the system,τcΩc≪ 1, and the noise force is sufficiently weak to be treatedas a perturbation [37]. The kernel of the right-hand side ofEq.(69) is the cumulant series terminated at the first non-vanishing term (the first-order term vanishes because〈G(t)〉=0). We note that all cumulants in Eq. (69) vanish identicallyfor τc = 0 because of the Gaussian property, but they are non-zero for finiteτc due to non-commutativity of the kernel in Eq.(66).

We start with the Markovian case when the autocorrelationfunction is given by the delta-function

〈br(t)br(s)〉= Qδ (t − s). (70)

Using Eqs. (66), (69), we find that the averaged solution toEq. (47) reads

〈ζ (t)〉= ζ (0)cos(Ωct)+ζ (0)Ωc

sin(Ωct) , (71)

i.e. the Markovian stochastic force does not influence aver-aged dynamics of the interface.

Now we consider a non-Markovian exponentially corre-lated noise with autocorrelation function given by

〈br(t)br(s)〉=Q

2τcexp(−|t − s|/τc) , (72)

which reduces to the Markovian case in the limit of zero cor-relation timeτc → 0. We average Eq. (47) using the Novikovtheorem [38]:

d2

dt2〈ζ (t)〉+Ω2

c〈ζ (t)〉+ k

t∫

0

ds〈br(t)br(s)〉⟨

δζ (t)δ br(s)

⟩

= 0.

(73)

12

We calculate the functional derivative in Eq.(73) using thegeneral solution Eq.(66)

δζv(t)δbr (s)

=δ

δbr (s)G0 (t)exp

t∫

0

dskb(s)V(s)

ζv (0)

= θ (t − s)θ (s)G0(t)G−10 (s)kV0G0(s)

×exp

t∫

0

dskb(s)V(s)

ζv(0)+∆ [br ] (t,s)ζv(0)

= θ (t − s)θ (s)G0(t − s)kV0G−10 (t − s)ζv(t)

+∆ [br ] (t,s)ζv(0), (74)

whereθ (t) is the Heaviside step function, and∆ [br ] (t,s) isa functional arising from the non-commutativity of the kernelof the exponential in Eq.(74). The functional∆ [br ] (t,s) maybe dropped in the present limit of weak noise. We obtain fromEq. (74)

⟨

δζ (t)δbr (s)

⟩

≃ θ (t − s)θ (s)k[V(t − s)]11〈ζ (t)〉+

[V(t − s)]12ddt

〈ζ (t)〉

. (75)

Substituting the operatorsG0 andV0 to Eq. (75), and usingEq. (73) we find

d2

dt2〈ζ (t)〉+Ω2

c 〈ζ (t)〉+t∫

0

dsk2

2Ωc〈br(t)br(s)〉sin2Ωc(t − s)〈ζ (t)〉−

t∫

0

dsk2

Ω2c〈br(t)br(s)〉sin2Ωc(t − s)

ddt

〈ζ (t)〉= 0.

(76)

Calculating the integrals on time scales much larger than thecorrelation time of the noiset ≫ τc, we arrive to an equationdescribing the averaged interface dynamics

d2

dt2〈ζ (t)〉−2κ

ddt

〈ζ (t)〉+(

Ω2c +ν2

0

)

〈ζ (t)〉= 0, (77)

where

κ =12

k2Qτ2c

1+(2Ωcτc)2 (78)

is the average growth rate and

ν20 =

12

k2Qτc

1+(2Ωcτc)2 (79)

modifies the frequency due to noise. The solution to Eq. (77)grows exponentially in time as

ζ ∝ exp

[

t

(

κ ± i√

Ω2c +ν2

0 −κ2

)]

. (80)

FIG. 11: (a) Single realization of the external stochastic GaussianMarkovian force withΩ = 5 ·10−9, τc = 50 andΩcτc = 1.5 ·10−2.(b) The respective numerical solution to Eq. (47). The dashed lineshows amplitude of the averaged analytical solution, Eq. (80).

FIG. 12: (a) Single realization of the external stochastic GaussianMarkovian force withΩ = 2 ·10−7 andτc = 1200 andΩcτc = 0.38.(b) The respective numerical solution to Eq. (47) (two realizations).The dashed line shows amplitude of the averaged analytical solution,Eq. (80).

Thus, a stochastic non-Markovian force leads to unstablegrowth of the average perturbation amplitude with modifiedfrequency of the capillary oscillations. The analytical theoryfor interface dynamics under the action of a stochastic forceis compared in Figs. 11, 12 to the numerical solution to Eq.(47) calculated fork = 0.0154. We note that real stochas-tic processes always have finite frequency band width, thoughit may be very large. Therefore, any real stochastic processis characterized by some non-zero correlation time leadingto exponential growth of amplitude of the oscillator Eq.(47),which means that the Markovian case may be recovered onlyas a limit ofΩcτc → 0. Figure 11 shows the numerical so-lution to Eq. (47) for the stochastic force withQ = 5 ·10−9

and Ωcτc = 1.5 · 10−2 ≪ 1, which may be considered as agood approximation for the Markovian case. The dashed linerepresents the envelope function of the averaged amplitudeofthe interface oscillations predicted by the theory, Eqs. (78),(80). In the Markovian case, the averaged amplitude does notchange, which is supported by the numerical results shown inFig. 11. Figure 12 corresponds to a non-Markovian case with

13

Q= 2·10−7, τc = 1200 andΩcτc = 0.38. The numerical solu-tion to Eq. (47) is shown by the solid lines for two realizationsof the noise, while the dashed line represents envelope of theaveraged amplitude increasing exponentially in time accord-ing to the analytical theory Eqs. (78), (80). In agreement withthe theoretical predictions, the numerical solution also demon-strates exponential increase of the perturbation amplitude.

VI. SUMMARY

We have studied the dynamics of an interface in a two-component BEC driven by a spatially uniform time-dependentforce. Experimentally, such a force may be produced by amagnetic field gradient acting on the system of two interact-ing BECs with different spins [12]. By applying the varia-tional principle to the GP Lagrangian, we have derived the dis-persion relation (or the respective ordinary differentialequa-tions) for linear waves and instabilities at the interface.Wehave considered different time-dependent forces leading to adiverse collection of dynamical effects: (i) A constant forcepushing the components towards each other generates the RTinstability for a certain domain of perturbation wave numbers.Because of the RT instability, small perturbations grow expo-nentially in time without oscillations. (ii) A sinusoidal forcewith frequencyΩ leads to the parametric instability at the in-

terface. The critical strength of the force required to drivethe instability depends on the perturbation wave number. Thisstrength is infinitely small for the intrinsic capillary waves atthe interface with frequency equal toΩ/2. Because of theparametric instability, small perturbations not only growexpo-nentially in time, but also oscillate with frequencyΩ/2. (iii)A pulse force leads to the quantum counterpart of the RM in-stability. Within the linear approximation, the force producesa discontinuous jump in the amplitude and phase of the cap-illary waves at the interface. (iv) A non-Markovian stochas-tic external field (with non-zero correlation time) drives theinstability at the interface accompanied by oscillations.Thegrowth rate of the instability is determined by the varianceofthe driving force and by the correlation time. The oscillationfrequency is shifted in comparison to the intrinsic frequencyof the capillary waves. A Markovian force on average doesnot lead to any effect.

Acknowledgments

This research was supported partly by the Swedish Re-search Council (VR) and by the Kempe Foundation. Calcu-lations have been conducted using the resources of High Per-formance Computing Center North (HPC2N).

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