Classical analogies for the force acting on an impurity in a Bose-Einstein condensate

Jonas Rønning1, Audun Skaugen2, Emilio Hernández-García3, Cristóbal López3, Luiza Angheluta1

1PoreLab, The Njord Centre, Department of Physics,University of Oslo, P. O. Box 1048, 0316 Oslo, Norway

2Computational Physics Laboratory, Tampere University, P.O. Box 692, FI-33014 Tampere, Finland3IFISC (CSIC-UIB), Instituto de Fisica Interdisciplinar y Sistemas Complejos, 07122 Palma de Mallorca, Spain

(Dated: February 13, 2020)

We study the hydrodynamic forces acting on a finite-size impurity moving in a two-dimensionalBose-Einstein condensate at non-zero temperature. The condensate is modeled by the damped-Gross Pitaevskii (dGPE) equation and the impurity by a Gaussian repulsive potential giving thecoupling to the condensate. The width of the Gaussian potential is equal to the coherence length,thus the impurity can only emit waves. Using linear perturbation analysis, we obtain analyticalexpressions corresponding to different hydrodynamic regimes which are then compared with directnumerical simulations of the dGPE equation and with the corresponding expressions for classicalforces. For a non-steady flow, the impurity experiences a time-dependent force that, for smallcoupling, is dominated by the inertial effects from the condensate and can be expressed in terms ofthe local material derivative of the fluid velocity, in direct correspondence with the Maxey and Rileytheory for the motion of a solid particle in a classical fluid. In the steady-state regime, the force isdominated by a self-induced drag. Unlike at zero temperature, where the drag force vanishes belowa critical velocity, at finite temperatures, the drag force has a net contribution from the energydissipated in the condensate through the thermal drag at all velocities of the impurity. At lowvelocities this term is similar to the Stokes’ drag in classical fluids. There is still a critical velocityabove which the main drag pertains to energy dissipation by acoustic emissions. Above this speed,the drag behaves non-monotonically with impurity speed, reflecting the reorganization of fringesand wake around the particle.

I. INTRODUCTION

The motion of an impurity suspended in a quantumfluid depends on several key factors such as the superfluidnature and flow regime, as well as the size of the impu-rity and its interaction with the surrounding fluid [1–5].Therefore, it is disputable whether the forces acting on animpurity in a quantum fluid should bear any resemblanceto classical hydrodynamic forces. In the case of an impu-rity immersed in superfluid liquid helium, classical equa-tions of motion and hydrodynamic forces are assumed apriori [6], since impurities are typically much larger thanthe coherence length and then quantum hydrodynamiceffects like the quantum pressure can be neglected. ForBose-Einstein condensates (BEC) in dilute atomic gases,impurities can be neutral atoms [7], ion impurities [8, 9]or quasiparticles [10]. The size of an impurity in a BECis typically of the same order of magnitude or smallerthan the coherence length, and quantum hydrodynamiceffects cannot be ignored.

There are several theoretical and computational stud-ies of the interaction force between an impurity and aBEC at zero absolute temperature, using different ap-proaches depending on the nature of the particle andits interaction with the condensate. A microscopic ap-proach is used to analyse the interaction of a rigid par-ticle with a BEC by solving the Gross-Pitaevskii equa-tion (GPE) for the condensate macroscopic wavefunc-tion and using boundary conditions such that the con-densate density vanishes at the particle boundary [11].This methodology allows to study complex phenomenasuch as vortex nucleation and flow instabilities, but it is

more oriented to find the effects of an obstacle on theflow rather than the coupled particle-flow dynamics. Inaddition, the boundary condition introduces severe non-linearities which can only be addressed numerically. Ata more fundamental level of description, the impurity istreated as a quantum particle with its own wavefunctiondescribed by the Schrödinger equation and that is cou-pled with the GPE for the macroscopic wavefunction ofthe BEC [12]. A more versatile model for the interac-tion of impurities with the BEC has been explored inseveral papers [3–5, 13, 14]. Here, an additional repul-sive interaction (a Gaussian or delta-function potential)is added to model scattering of the condensate particleswith the impurity. The hydrodynamic force on the im-purity is determined by this repulsion potential and thesuperfluid density through the Ehrenfest theorem. Thestrong-coupling limit of this repulsive potential wouldbe equivalent to the rigid boundary-condition approach.Within this modeling approach, some works have studiedthe complex motion of particles interacting with vorticesin the flow, and the indirect interactions between themarising from the presence of the fluid [4, 14]. Another lineof research using this type of modeling focused mainlyon the superfluidity criterion of a uniform BEC at zerotemperature regime. Within the Bogoliubov perturba-tion analysis for a small impurity and weak interaction,analytical expressions can be derived for the steady-stateforce exerted by the superfluid as function of the constantvelocity of the impurity [3, 5, 15, 16]. At zero temper-ature, this force vanishes below a critical velocity, thespeed of long-wavelength sound waves, at least when weignore the quantum fluctuations [15], and corresponds

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to the dissipationless motion. Above this velocity, thereis a finite drag force and the motion of the impurity isdamped by acoustic excitations. While this is a form ofdrag, in that the force opposes motion by dissipating en-ergy, it is not the same as the classical Stokes’ drag inviscous fluids. Recent experiments probing superfluidityin a BEC are able to indirectly estimate the drag forceby measuring the local heating rate in the vicinity of themoving laser beam and show that there is still a criticalvelocity even at non-zero temperatures and that the crit-ical velocity is lower for a repulsive potential than for anattractive one [17].

In this paper, we study the forces exerted on an im-purity moving in a two-dimensional BEC at finite tem-perature, using an approach similar to [3–5, 13, 14], inwhich a repulsive Gaussian potential is used to describethe interaction of the particle with the BEC, but us-ing a dissipative version of the GPE to model the fluid.Our aim is to bridge this microscopic approach with thephenomenological descriptions [6] that assume that theforces from the superfluid are the same as those from aclassical fluid in the inviscid and irrotational case. As inthe classical-fluid case, we find that the force is made oftwo contributions: One of them, dominant for very weakfluid-particle interaction, bears a rather complete anal-ogy with the corresponding force in classical fluids (iner-tial or pressure-gradient force), which depends on localfluid acceleration and includes the so-called Faxén correc-tions arising from velocity inhomogeneities close to theparticle position [18]. The difference is that, in a classi-cal fluid, these corrections arise from the finite size of theparticle and vanish when the particle size becomes zero.In the BEC, Faxén-type corrections arise both from theparticle size (modeled by the range of the particle repul-sion potential) and from the BEC coherence length. Asfluid-particle interaction becomes more important, a sec-ond contribution to the force becomes noticeable, whichtakes into account the drag on the particle arising fromthe perturbation of the flow produced by the presence ofthe particle. Thus it can be called a particle self-inducedforce. We are able to obtain explicit formulae for it inthe case of constant-velocity motion of the particle in anotherwise homogeneous and steady BEC. This drag is adissipative (damping) force due to viscous-like drag ofthe perturbed BEC with the thermal cloud. It occurs inaddition to the drag due to acoustic excitations in thecondensate that in the absence of dissipation occurs onlyabove a critical velocity for the particle. Here, it can becompared with the corresponding force in classical flu-ids, namely the viscous Stokes drag. We find that, as theStokes force, the self-induced dissipative drag is linear inthe particle velocity for small velocities, and we obtainan expression for it also at arbitrary velocities.

The rest of the paper is structured as follows. In Sect.II, we discuss the general modeling setup and in Sect. IIIa perturbation analysis is used to derive the linearizedequations for the perturbations in the wavefunction re-lated to non-steady condensate flow and the particle re-

pulsive potential. Subsections IIIA and III B derive ana-lytical expressions within perturbation theory for the twocontributions to the force experienced by the particle. InSection IV, we compare our theoretical predictions withnumerical simulations of the dissipative GPE coupled tothe impurity, and the final section summarizes our con-clusions.

II. MODELING APPROACH

We model the interaction between the impurity and atwo-dimensional BEC through a Gaussian repulsive po-tential which can be reduced to a delta-function limitsimilar to previous studies [3, 5]. The BEC itself, whichis at a finite temperature, is described by a macro-scopic wavefunction ψ(r, t) that evolves according to thedamped Gross Pitaevskii equation (dGPE) [19–21]:

i~∂tψ =

(1− iγ)

(− ~2

2m∇2 + g|ψ|2 − µ+ Vext + gpUp

)ψ, (1)

where g is an effective scattering parameter between con-densate atoms. Vext is any external potential used toconfine the condensate atoms or to stir them. The damp-ing coefficient γ > 0 is related to finite-temperature ef-fects due to thermal drag between the condensate andthe stationary thermal reservoir of excited atoms at fixedchemical potential µ. This damping γ is very smallat low temperatures and can be expressed as functionof temperature T , chemical potential µ and the energyof the thermal cloud [22]. The dGPE can be derivedfrom the stochastic projected Gross-Pitaevskii equationin the low-temperature regime [23] and has been usedto study different quantum turbulence regimes and vor-tex dynamics [19, 20, 24, 25] that are also observed inrecent experiments [26]. An hydrodynamic descriptionin terms of density and velocity of the BEC can be de-veloped using the Madelung transformation of the wave-function: ψ = |ψ|eiφ. The macroscopic number den-sity is ρ(r, t) = |ψ(r, t)|2 and the condensate velocity isv(r, t) = (~/m)∇φ(r, t). This velocity can also be ob-tained from the superfluid current J(r, t) as

J =~

2mi(ψ∗∇ψ − ψ∇ψ∗) = ρv . (2)

ψ∗ denotes the complex conjugate of ψ. In addition todamping the BEC velocity, the presence of γ 6= 0 in thedGPE also singles out the value ρh = |ψ|2 = g/µ asthe steady homogeneous density value when the phase isconstant and Vext = 0.

The interaction potential Up(r − rp) between the con-densate and the impurity is modeled by a Gaussian po-tential Up(r − rp) = µ/(2πσ2)e−(r−rp)2/(2σ2). The pa-rameter gp > 0 is the weak coupling constant for repul-sive impurity-condensate interaction, rp = rp(t) denotesthe center-of-mass position of the impurity, and σ its ef-fective size. Here we consider an impurity of size σ of

3

the order the coherence length ξ = ~/√mµ of the con-densate. The impurity is too small to nucleate vorticesin its wake [27]. Instead, the impurity will create acous-tic excitations which at supersonic speeds correspond toshock waves. Similar acoustic fringes in the condensatedensity have been reported numerically in [2] for a differ-ent realization of non-equilibrium Bose-Einstein conden-sates. In the limit of a point-like impurity, the Gaussianinteraction potential converges to a two-body scatteringpotential Up(r − rp, t) = µδ(r − rp(t)) that has beenused in previous analytical studies [3, 4, 13, 14]. Notethat we are modeling only the interaction of the particlewith the BEC, so that the viscous-like drag we will ob-tain arises from the indirect coupling to the thermal bathvia the BEC. Any direct interaction of the particle withthe thermal cloud of normal atoms will lead to additionalforces that we do not consider here.

In order to gain insight into the forces and their rela-tionship with the classical case, we keep the set-up as sim-ple as possible. We consider a strictly two-dimensionalcondensate, which can be obtained by plane optical traps.We assume that the size of the condensate is large enoughso that we can neglect inhomogeneities in the confiningpart of Vext in the region of interest. Also, we considera neutrally buoyant impurity so that effects of gravitycan be neglected. This would imply Vext = 0 except ifan external forcing is introduced to stir the system, inwhich case we assume the support of this external forceis sufficiently far from the impurity.

The impurity and the condensate will exert an inter-action force on each other that is determined by theEhrenfest theorem for the evolution of the center-of-mass momentum of the particle. The potential force−gp∇Up(r − rp) is the force exerted by an impurityon a condensate particle at position r. By space aver-aging over condensate density, we then determine theforce exerted by the impurity on the condensate as−gp

∫dr|ψ(r, t)|2∇Up(r−rp) [14]. Hence, the force act-

ing on the impurity has the opposite sign and is equalto

Fp(t) = +gp

∫d2r|ψ(r, t)|2∇Up(r − rp) (3)

which, through an integration by parts, is equivalent to

Fp(t) = −gp∫d2r Up(r − rp, t)∇|ψ(r, t)|2. (4)

Note that this last expression can also be used, reversingthe sign, to give the force exserted on the BEC by a laserof beam profile given by Up.

At zero temperature, i.e. γ = 0, and neglecting the ef-fect of quantum fluctuations [15, 16], the impurity moveswithout any drag through a uniform condensate belowa critical velocity, which is the low-wavelength speed ofsound c =

õ/m, as determined by the condensate lin-

ear excitation spectrum, in agreement with Landau’s cri-terion of superfluidity [3]. Above the critical speed, theimpurity will create excitations, and depending on the

size of the impurity these excitations range from acousticwaves (Bogoliubov excitation spectrum) to vortex dipolesand to von-Karman street of vortex pairs [27]. Previ-ous studies focused on the theoretical investigations ofthe self-induced drag force and energy dissipation ratein the presence of Bogoliubov excitations emitted by apointwise [3, 15, 16] or finite-size [5] particle, or numer-ical investigations of the drag force due to vortex emis-sions [1, 13, 14]. The energy dissipation rate depends onwhether the impurity is heavier, neutral or lighter withrespect to the mass of the condensate particles [14]. Thedependence on the velocity of the self-induced drag forceabove the critical velocity changes with the spatial di-mensions [3]. This means that the energy dissipation rateis also dependent on the spatial dimensions. If instead ofa single impurity one considers many of them there willbe, besides direct inter-particle interactions, additionalforces between the impurities mediated by the flow, lead-ing to a much more complex many-body dynamics evenin an otherwise uniform condensate, as discussed in [4].Here we neglect all these effects and consider a singleimpurity in a two-dimensional BEC.

We rewrite the dGPE in dimensionless units by us-ing the characteristic units of space and time in termsof the long-wavelength speed of sound c =

õ/m in

the homogeneous condensate and the coherence lengthξ = ~/(mc) = ~/√mµ. Space is rescaled as r → rξ

and time as t → tξ/c. In addition, the wavefunctionis also rescaled ψ → ψ

√µ/g, where g/µ is the equilib-

rium particle-number density corresponding to the so-lution with constant phase if Vext,Up = 0. The exter-nal potential, Vext = µVext, and the interaction poten-tial, gpUp = µgpUp, are measured in units of the chem-ical potential µ with Up = 1/(2πa2)e−(r−rp)2/(2a2), anda = σ/ξ, gp = gp/(ξ

2µ). Henceforth, the dimensionlessform of the dGPE reads as

∂tψ = (i+ γ)

(1

2∇2 + 1− Vext − gpUp − |ψ|2

)ψ. (5)

We use these dimensionless units and express the force(4) exerted on an impurity as Fp = (µ2ξ/g)Fp, where

Fp(t) = −gp∫d2rUp(r − rp)∇|ψ(r, t)|2. (6)

For the rest of the paper, we will now omit the tildes overthe dimensionless quantities.

In the limit of a point-like particle, Up = δ(r−rp), theforce from Eq. (6) becomes

Fp(t) = −gp∇|ψ(r, t)|2|r=rp(t). (7)

III. PERTURBATION ANALYSIS

For a weakly-interacting impurity, the condensatewavefunction ψ can be decomposed into an unperturbedwavefunction ψ0(r) describing the motion and density of

4

the fluid in the absence of the particle and the pertur-bation δψ1(r) due to the impurity’s repulsive interactionwith the condensate, hence ψ = ψ0 + gpδψ1. The unper-turbed wavefunction ψ0(r, t) can be spatially-dependent,if it is initialized in a nonequilibrium configuration, or ifexternal forces characterized by Vext are at play. Here,we consider deviations with respect to the steady anduniform equilibrium state (which in our dimensionlessunits is ψh = 1). As stated before, we do not considerlarge extended inhomogeneities produced by a trappingpotential, and assume that any stirring force acting onthe BEC is far from the particle. Thus, we treat inho-mogeneities close to the particle as small perturbationsto the uniform state ψh = 1: ψ0(r, t) = 1 + δψ0(r, t).Combining the two types of perturbations, and using therelationships of the wavefunction to the density, velocityand current (Eq. (2), which in dimensionless units readsρv = (ψ∗∇ψ − ψ∇ψ∗)/(2i)) we find

ψ = 1 + δψ0 + gpδψ1 (8)ρ = 1 + δρ0 + gpδρ1, (9)

v = δv(0) + gpδv(1), (10)

whereδρ0 = δψ0 + δψ∗0 , δρ1 = δψ1 + δψ∗1 , (11)

δv(0) =1

2i∇ (δψ0 − δψ∗0) , δv(1) =

1

2i∇(δψ1 − δψ∗1).

(12)Combining Eq. (6) with the expressions for the density

perturbations, we have that the total force can be splitinto the contribution from the density variations in theBEC by causes external to the particle (initial prepara-tion, stirring forces in Vext, ...), and the density perturba-tions due to the presence of the particle Fp = F (0)+F (1):

F (0)(t) = − gp2πa2

∫d2re−

(r−rp(t))2

2a2 ∇δρ0(r, t), (13)

F (1)(t) = −g2p

2πa2

∫d2re−

(r−rp(t))2

2a2 ∇δρ1(r, t). (14)

The perturbative splitting of the force in these two con-tributions is completely analogous to the correspondingclassical-fluid case in the incompressible [18] and in thecompressible [28] situations. The F (0) contribution is theequivalent to the classical inertial or pressure-gradientforce on a test particle, which does not disturb the fluid,in a inhomogeneous and unsteady flow. In the followingit will be called the inertial force. The F (1) contribu-tion takes into account perturbatively the modificationson the flow induced by the presence of the particle, andit will be called the self-induced drag on the particle. Tocomplete the comparison with the classical expressions[18, 28], we need to express Eqs. (13) and (14) in termsof the unperturbed velocity field v(0)(r, t) = δv(0)(r, t)and of the particle speed Vp(t) = rp(t). We are able todo so in a general situation for the inertial force F (0). ForF (1), we obtain analytical expressions in the simple casewhere the impurity is moving with a constant velocity inan otherwise uniform BEC.

The desired relationships between ∇δρ0 and ∇δρ1 inEqs. (13)-(14), and δv(0) and Vp will be obtained fromthe linearization of the dGPE Eq. (5) around the uniformsteady state ψh = 1:

∂tδψ0 = (i+ γ)

(1

2∇2 − 1

)δψ0

− (i+ γ)δψ∗0 , (15)

∂tδψ1 = (i+ γ)

(1

2∇2 − 1

)δψ1

− (i+ γ)δψ∗1 − (i+ γ)Up(r − rp) . (16)

Terms containing Vext are not included in Eq. (15) be-cause of our assumption of sufficient distance betweenpossible stirring sources and the neighborhood of the par-ticle position, the only region that–as we will see– willenter into the calculation of the forces. In the next sec-tions we solve these linearized equations to relate densityperturbations to undisturbed velocity field and particlevelocity.

A. Inertial force

To convert Eq. (13) for the inertial force into an ex-pression suitable for comparison for the correspondingterm in classical fluids, we need to express ∇δρ0 in termsof the undisturbed velocity field v(0)(r, t) = δv(0)(r, t).To this end, we substract Eq. (15) from its complex con-jugate, obtaining:(

∇2 − 4)∇δρ0 = 4

(∂t −

γ

2∇2)δv(0), (17)

where we have used Eqs. (11) and (12). Since the forceformulae require to obtain the condensate density in aneighborhood of the particle position, it is convenient tomove to a coordinate frame with center always at the(possibly moving) particle location r = rp(t). Thus wechange variables from (r, t) to (z, t), with z = r − rp(t),and the velocity field will be now referred to the particlevelocity Vp(t) = rp(t): δw(0)(z, t) = δv(0)(r, t) − Vp(t).Equation (17) becomes:(

∇2z − 4

)∇zδρ0 =

4(∂t − Vp · ∇z −

γ

2∇2z

)δw(0) + Vp(t), (18)

which has the corresponding equation for its Green’sfunction given by(

∇2z − 4

)G(z) = δ(z) (19)

with the boundary condition G(|z| → ∞) → 0 (corre-sponding to vanishing ∇zδρ0(r) at |r| = ∞). The solu-tion is given by the zeroth order modified Bessel functionG(z) = −K0(2|z|)/(2π). Hence, the gradient of the den-sity perturbation can be written as the convolution withthe Green’s function:

5

∇zδρ0(z, t) = − 2

π

∫dz′K0(2|z − z′|)

[(∂t − Vp · ∇z′ −

γ

2∇2

z′

)δw(0)(z′, t) + Vp(t)

], (20)

and the expression for the force (13), using the comoving variables (z, t), becomes:

F (0)(t) = − gpπ2a2

∫dze−

z2

2a2

∫dz′K0(2|z − z′|)

[(∂t − Vp · ∇z′ −

γ

2∇2

z′

)δw(0)(z′, t) + Vp(t)

]. (21)

The above expression is a weighted average of contri-butions from properties of the fluid velocity in a neigh-borhood of the impurity center-of-mass position (z = 0in the comoving frame). The size of this neighborhoodis given by the combination of the range of the Besselfunction kernel, which in dimensional units would be thecorrelation length ξ, and the range of the Gaussian po-tential, a, giving an effective particle size. In classicalfluids, the analogous force on a spherical particle involvesthe average of properties of the undisturbed velocity fieldwithin the sphere size [28], and there is no equivalent tothe role of ξ.

As in the classical case [18, 28], if fluid velocity varia-tions are weak at scales below a and ξ, we can approxi-mate the condensate velocity by a Taylor expansion nearthe impurity, i.e.:

δw(0)i (z′, t) ≈ δw(0)

i (t) +∑j

eij(t)z′j

+1

2

∑jk

eijk(t)z′jz′k + . . . , (22)

where the indices i, j, k = x, y denote the coordinatecomponents. eij(t) = ∂jδw

(0)i (z, t)|z=0 and eijk(t) =

∂j∂kδw(0)i (z, t)|z=0 are gradients of the unperturbed

condensate relative velocity. Inserting this expansioninto Eq. (21), and performing the integrals of theGaussian and of the Bessel function (using for ex-ample

∫K0(2|z|)dz = π/2 and

∫zizjK0(2|z|)dz =

(δij/2)∫∞

02πz3K0(2z)dz = δijπ/4), we obtain:

F (0)(t) ≈ gpVp(t) + gp

[∂t − Vp(t) · ∇z +

a2

2∂t∇2

z

− γ

2∇2z +

1

4∂t∇2

z

]δw(0)(z, t)|z=0 . (23)

The terms containing Laplacians are analogous to theFaxén corrections in classical fluids [18] which arise forparticles with finite size. Here, they arise from a combi-nation of the finite effective size of the particle, a, andof the quantum coherence length, ξ = 1. This last effectremains even in the limit of vanishing particle size a→ 0.Interestingly, one of the two terms in these quantum cor-rections depend on γ hence indirectly on the presence ofthe thermal cloud.

As in the classical case, if flow inhomogeneities areunimportant below the scales a and ξ, we can neglect theLaplacian terms in Eq. (23). Returning to the variables(r, t) in the lab frame of reference, the terms containing

Vp cancel out, showing that the inertial force is mainlygiven by the local fluid acceleration:

F (0)(t) = gp∂tδv(0)(r, t)

∣∣r=rp(t)

. (24)

We have assumed a small non-uniform unperturbed ve-locity field v(0)(r, t) = δv(0)(r, t). To leading order inthis small velocity, the partial derivative ∂tδv(0) and thematerial derivative Dδv(0)/Dt = ∂tδv

(0) + δv(0) · ∇δv(0)

are identical. In classical fluids the same ambiguity oc-curs and it has been established, on physical groundsand by going beyond linearization, that using the ma-terial derivative is more correct [18]. After all, usingthis material derivative in the equation of motion sim-ply means that, under the above approximations and inplaces where stirring and other external forces are ab-sent, the local acceleration on the impurity arises fromthe corresponding acceleration of the condensate. Sincefor a→ 0 the condensate-impurity interaction has a sim-ilar scattering potential (delta function) as that for theinteraction between condensate particles, similar accel-erations would be experienced by a condensate particleand by the impurity, just modulated by a different cou-pling constant. Thus, replacing ∂t by D/Dt in (24) theapproximate inertial force becomes:

F (0)(t) = gpDv(0)

Dt

∣∣∣∣r=rp(t)

, (25)

or, if we return back to dimensional variables:

F (0)(t) =gpgmDv(0)

Dt

∣∣∣∣r=rp(t)

. (26)

This is equivalent to the equation for the inertial forcein classical fluids [18] except that the coefficient of thematerial derivative in the classical case is the mass of thefluid fitting in the size of the impurity. In the comovingframe, replacement of the partial by the material deriva-tive amounts to replace (∂t − Vp · ∇z′)δω

(0) in Eq. (21)by Dδω(0)/Dt. Eq. (25) is expected to be valid for smallvalues of gp and in regions where fluid velocity and den-sity inhomogeneities are both small and weakly varying.At this level of approximation neither compressibility nordissipation effects appear explicitly in the inertial force,in analogy with classical compressible fluids [28]. Butthese effects are indirectly present by determining thestructure of the field v(0)(r, t).

6

B. Self-induced drag force

In classical fluids, consideration of the self-inducedforce on a particle moving at arbitrary time-dependentvelocity, coming from the perturbation in produces inthe flow, leads to different terms, namely [18, 28] the vis-cous (Stokes) drag, the unsteady-inviscid term that in theincompressible case becomes the added-mass force, andthe unsteady-viscous term that in the incompressible casebecomes the Basset history force. They are expressed interms of the undisturbed velocity flow v(0) and the parti-cle velocity Vp(t). Here, for the BEC case, we are able toobtain the self-induced force only for a particle moving atconstant speed on the condensate. For the classical fluidcase, in this situation the only non-vanishing force is theStokes drag, so that this is the force we have to compareour quantum result with. We note that the condensateitself in the absence of the particle perturbation can be inany state of (weak) motion since in our perturbative ap-proach summarized in Eqs (15)-(16), the inhomogeneityδψ0 and the gp-perturbation δψ1 are uncoupled.

It is convenient to transform the problem to the frameof reference moving with the particle (r, t)→ (z, t) withz = r − rp(t), so that Eq. (16) becomes

∂tδψ1 − Vp · ∇δψ1 = (i+ γ)

(1

2∇2 − 1

)δψ1

− (i+ γ)δψ∗1 − (i+ γ)U(r) . (27)

Note that such Galilean transformations of the GPE us-ing a constant Vp are often accompanied by a multipli-cation of the transformed wavefunction by a phase fac-tor exp(iVp · z + i

2V2p t), in order to transform the con-

densate velocity (see below) to the new frame of refer-ence, and account for the shift in kinetic energy. In-deed, such a combined transformation leaves the GPEunchanged at γ = 0 [29] (but not for γ > 0). The

density perturbation δρ1 is already given correctly byδψ1 + δψ∗1 , where δψ1(z, t) is the solution of (27), with-out the need of any additional phase factor. The velocityin the comoving frame would need to be corrected asδω(1)(z, t) = δv(1) − Vp, with δv(1) given by expression(12) in terms of he solution of (27).

Eq. (27) in the steady-state can be solved by using theFourier transform δψ1(z) = 1/(2π)2

∫d2keik·zδψ1(k). It

follows that the linear system of equations for δψ1(k) andδψ∗1(−k) is given by[

−2ik · Vp + (i+ γ)(k2 + 2)]δψ1+2(i+ γ)δψ∗1 =

− 2(i+ γ)e−a2k2

2 ,[−2ik · Vp + (−i+ γ)(k2 + 2)

]δψ∗1+2(−i+ γ)δψ1 =

− 2(−i+ γ)e−a2k2

2 .

(28)

By solving these equations, we find δψ1(k) and δψ∗1(−k),and the Fourier transform of the density perturbationδρ1 = δψ∗1 + δψ1 then follows as

δρ1 =e−

k2a2

2 (4k2(1 + γ2)− 8iγk · Vp)4k · Vp(Vp · k + iγk2 + 2iγ)− k2(4 + k2)(1 + γ2)

.

(29)Using the convolution theorem, we can express the self-induced force (14) (in the co-moving frame, i.e. withrp = 0) in terms of δρ1 as

F (1) = −g2p

(2π)2

∫d2kikδρ1(k)e−

k2a2

2 . (30)

This force can be decomposed into the normal and tan-gential components relative to the particle velocity Vp:F (1) = F‖e‖ + F⊥e⊥. Due to symmetry, the normalcomponent vanishes upon polar integration, and we areleft with the tangential, or drag, force

F‖ = −g2p

(2π)2

∫ ∞0

dk

∫ 2π

0

dθe−k2a2 ik2 cos(θ)

[4k2(1 + γ2)− 8iγkVp cos(θ)

]4kVp cos(θ)(kVp cos(θ) + iγk2 + 2iγ)− k2(4 + k2)(1 + γ2)

. (31)

Vp is the modulus of Vp. At zero temperature, i.e. whenγ = 0, the drag force reduces to the one that has alsobeen calculated for a point particle in Refs. [3] and in [5]for a finite-a particle:

F‖ = −g2p

π2

∫ ∞0

dk

∫ 2π

0

dθik2 cos θe−k

2a2

4V 2p cos2 θ − (4 + k2)

, (32)

which is zero for particle speed smaller than the criticalvalue given by the long-wavelength sound speed, Vp <c = 1. Above the critical speed, the integral has poles

and acquires a non-zero value given by

F‖ = −g2pk

2max

4Vpe−

a2k2max2

[I0

(a2k2

max

2

)− I1

(a2k2

max

2

)](33)

in terms of the modified Bessel functions of the first kindIn(x) and where kmax = 2

√V 2p − 1. For vanishing a

the dominant term is proportional to (V 2p − 1)/Vp [3].

This drag is pertaining to energy dissipation by radiatingsound waves in the condensate away from the impurity.We stress again that we assume a small enough such thatemission of other excitations, such as vortex pairs, does

7

not occur. It is important to note [3, 5] that in order toobtain a real value for the force in Eq. (33) one has toconsider that it has been obtained from the limit γ →0+ in (31), which implies that an infinitesimal positiveimaginary part needs to be considered in the denominatorto properly deal with the poles in the integral.

In general, for a non-zero γ, Eq. (31) simplifies uponan expansion in powers of Vp to the leading order. For thelinear term in Vp, we can perform the polar integrationand arrive at

F‖ = − 2

πVp

γ

1 + γ2g2p

∫k3e−a

2k2

(4 + k2)2dk . (34)

Substituting u = a2(k2 + 4), we find

F‖ = −Vpγ

1 + γ2g2p

1

π

[e4a2E1(4a2)(1 + 4a2)− 1

],

(35)

where E1(x) denotes the positive exponential integral.When a → 0, the expression inside the bracket divergesas −γE−1− ln(4a2) with γE begin the Euler-Mascheroniconstant. It is therefore necessary to keep a finite size a.

This drag force is analogous to the viscous Stokes dragforce in classical fluids since it is due to an effective inter-action of the impurity with the normal fluid through thethermal drag on the BEC. The effective drag coefficientdepends on the thermal drag such that it vanishes at zerotemperature. But it also depends non-trivially on the sizeof the impurity and it diverges in the limit of point-likeparticle. Faxén corrections involving derivatives of theunperturbed flow are not present here because of the de-coupling between δψ0 and δψ1 arising in the perturbativeapproach leading to (15)-(16).

IV. NUMERICAL RESULTS

To test the analytical predictions of the inertial forceand the self-induced drag deduced above from the totalforce expression Eq. (6), we performed numerical simu-lations of the dGPE. Actually, our simulations are donein the co-moving frame of the impurity moving at con-stant velocity Vp, so that the equation we solve is (seenumerical details in the Appendix):

∂tψ − Vp · ∇ψ = (i+ γ)[

12∇

2ψ +(1− gpUp − |ψ|2

)ψ],

(36)

where the impurity is described by the Gaussian potentialof intensity gp = 0.01 and effective size a = ξ = 1, and issituated in the middle of the domain with the coordinatesx/ξ = 128 and y/ξ = 64. As an initial condition, we startwith the condensate being at rest and in equilibrium withthe impurity. This is done by imaginary time integrationof Eq. (36) for Vp = 0 and γ = 0. Then, at t = 0, we solvethe full Eq. (36), and as a consequence, sound waves areemitted from the neighborhood of the impurity (the size

of the impurity is below the critical size for vortex nu-cleation). Their speed is determined by the dispersionrelation ω(k) giving the frequency as a function of thewavenumber and can be obtained by looking for plane-wave solutions to Eq. (15). If γ = 0, ω(k) is given by theBogoliubov dispersion relation [30] ω(k) = k

√1 + k2/4

(with c = ξ = 1). Note that the smallest velocity,c = 1, is that of long waves, and that waves of smallerwavelength travel faster. For γ > 0, the planar wavesare dampened out and the dispersion relation becomesω(k) = −iγ(k2/2 + 1) +

√k2 + k4/4− γ2. The damp-

ing rate is determined by γ and increases quadraticallywith the wavenumber. Also, in this case all waves have agroup velocity faster than a minimum one that for smallγ is close to c = 1.

When Vp < 1 all the waves escape the neighborhoodof the impurity (see an example in Fig. 1(a)) and aredissipated in a boundary buffer region that has large γ(see numerical details in the Appendix and SupplementalMaterial [31]). After a transient the condensate achievesa steady state when seen in the frame comoving with theimpurity. Fig. 1(b) shows a steady spatial configurationfor γ = 0 and Vp = 0.9. Figs. 1(d-e) show differentprofiles of the condensate density along the x directionacross the impurity position for Vp. The condensate den-sity is depleted near the impurity due to the repulsiveinteraction, and its general shape depends on the speedVp and thermal drag γ. If γ = 0 and Vp ≤ 1 the den-sity of this steady state has a rear-front symmetry withrespect to the particle position (see specially Fig. 1(d)),so that under integration in Eq. (6) the net force is zero.The presence of dissipation (γ > 0) breaks this symmetryeven if Vp < 1 so that a net drag will appear in agreementwith the calculation of Sect. III B. When Vp > 1 thereare waves that can not escape from the neighborhood ofthe impurity, forming fringes in front of it and a wakebehind it. (see Fig. 1(c) and 1(f) and Supplemental ma-terial [31]). Similar modulations in the condensate den-sity around an obstacle in supersonic flows has also beenobserved experimentally [32]. As a consequence of thisrear-front asymmetry, drag is nonvanishing even whenγ = 0.

Movies showing the transient and long-time density be-havior for several values of Vp and at γ = 0 are includedas Supplemental Material [31]. They are presented inthe comoving impurity frame, so that the impurity ap-pears static. The fluid suddenly stars to move towardsthe negative x direction, and density approaches a steadystate after the transient. Note that during all the dynam-ics, the density deviation with respect to the equilibriumvalue ρ = 1 is very small, justifying the perturbative ap-proach of Sect. III for this situation. The time evolutionfor γ > 0 is qualitatively similar to the γ = 0 one shownin the movies, except that the waves become damped andthat there is a front-rear asymmetry in the steady state.

Our numerical setup is well suited to measure the forceproduced by the perturbation of the impurity on thefluid, i.e. the self-induced drag. Nevertheless, in the ab-

8

0 50 100 150 200 250x/

0

20

40

60

80

100

120

y/

0.9985 0.9990 0.9995 1.0000

(a)

0 50 100 150 200 250x/

0

20

40

60

80

100

120

y/

0.9985 0.9990 0.9995 1.0000

(b)

0 50 100 150 200 250x/

0

20

40

60

80

100

120

y/

0.9990 0.9995 1.0000 1.0005 1.0010

(c)

100 110 120 130 140 150x/

0.9970

0.9975

0.9980

0.9985

0.9990

0.9995

1.0000

1.0005 = 0

Vp = 0.1Vp = 0.6Vp = 0.9Vp = 1 (d)

120 130 140 150 160x/

0.9980

0.9985

0.9990

0.9995

1.0000

Vp = 0.9

= 0= 0.01= 0.05= 0.1 (e)

120 130 140 150 160x/

0.9990

0.9995

1.0000

1.0005

Vp = 1.6

= 0= 0.01= 0.05= 0.1 (f)

FIG. 1. Panels (a)-(c) show 2D snapshots of the condensate density for γ = 0. In these snapshots, the impurity is at x/ξ = 128and y/ξ = 64. (a) is for Vp = 0.9 and corresponds to time t = 200, with transient waves still in the system. (b) is for the sameVp = 0.9 and corresponds to time t = 2000, when the final steady state (in the comoving frame) has been reached. Panel (c)is for Vp = 1.6 > 1, for which some waves remain attached to the impurity as front fringes and a rear wake. Panels (d-f) showcross-section profiles along the x direction of the steady-state condensate density around the impurity. Panel (d) stresses thefront-rear symmetry of the steady profiles when Vp ≤ 1 and γ = 0. An asymmetry develops (panel (e)) for γ > 0, which relatesto the net viscous-like drag. Panel (f) displays density profiles for Vp = 1.6 > 1 and different values of γ. The asymmetricdensity profile corresponds to waves trapped in front of the moving particle. With increasing γ, these waves are damped out.

sence of the impurity the unperturbed state is the trivialψ = 1, so that δψ0 = 0 and the inertial force is identicallyzero. In order to test the accuracy of our expressions forthe inertial force without the need of additional simula-tions under a different set-up, we still use the computedcondensate density and velocity dynamics, produced bythe impurity introduced in the system at t = 0, but weevaluate the inertial force exerted by this flow on anothertest particle located at a different position. In fact, thereis no need to think on the flow as being produced by animpurity: it can be produced by a moving laser beamthat can modeled by an external potential Vext and theonly impurity present in the system is the test particle onwhich the force is evaluated. In the following we evaluatethe inertial and the self-induced drag forces on the differ-ent particles from the general expressions Eqs. (13)-(14)and from the approximate expressions of Sects. III A andIII B.

A. Numerical evaluation of the inertial force

We consider a test particle traveling at the same speedVp as the impurity or laser beam producing the flow, butlocated at a distance of 10 coherence lengths in front ofit, and 20 coherence lengths in the y direction apart from

it. This distance is sufficient to avoid inclusion of Up orVext in Eq. (15) for the neighborhood of the test parti-cle. Condensate and test particle interact via a couplingconstant g′p sufficiently small so that the full force on thelater, Eq. (6), is well approximated by the inertial partEq. (13), being the perturbation the particle induces onthe flow, and thus the force (14) completely negligible.

Figure (2) shows, for different values of Vp = 0.1, 0.8, 1at γ = 0, the x component of the time-dependent forceproduced by the transient flow inhomogeneities hittingthe test particle in the form of sound waves. The sizeof the test particle, taking several values, is called a′

to distinguish it from the size a of the particle pro-ducing the flow perturbation. Blue lines are computedfrom the exact Eq. (6) or equivalently from Eq. (13)to which it reduces for sufficiently small g′p. Because ofthe rather explicit appearance of the interaction poten-tial in this formula, we label the blue lines in Fig. (2)as ‘potential force’. High frequency waves arrive beforelow-frequency ones, because its larger sound speed. Wealso see how the frequencies become Doppler-shifted forincreasing Vp. We have derived in Sect. III A several ap-proximate expressions for the inertial force. First, Eq.(21) is obtained with the sole assumption (besides gpsufficiently small) of smallness of the unsteady and/orinhomogeneous part δψ0 of the wavefunction, which al-

9

0 20 40 60 80 100t

0.60.40.20.00.20.40.60.81.0

F x/g

′ p×10 6 a′ = 0.25, Vp = 0.1

Potential forceInertial force

(a)

0 20 40 60 80 100t

4

2

0

2

4

F x/g

′ p

×10 5 a′ = 0.25, Vp = 0.8Potential forceInertial force

(b)

0 20 40 60 80 100t

1.5

1.0

0.5

0.0

0.5

1.0

F x/g

′ p

×10 4 a′ = 0.25, Vp = 1Potential forceInertial force

(c)

0 20 40 60 80 100t

0.60.40.20.00.20.40.60.81.0

F x/g

′ p

×10 6 a′ = 0.5, Vp = 0.1Potential forceInertial force

(d)

0 20 40 60 80 100t

4

2

0

2

4

F x/g

′ p

×10 5 a′ = 0.5, Vp = 0.8Potential forceInertial force

(e)

0 20 40 60 80 100t

1.5

1.0

0.5

0.0

0.5

1.0

F x/g

′ p

×10 4 a′ = 0.5, Vp = 1Potential forceInertial force

(f)

0 20 40 60 80 100t

0.60.40.20.00.20.40.60.81.0

F x/g

′ p

×10 6 a′ = 1, Vp = 0.1Potential forceInertial force

(g)

0 20 40 60 80 100t

4

2

0

2

4

F x/g

′ p

×10 5 a′ = 1, Vp = 0.8Potential forceInertial force

(h)

0 20 40 60 80 100t

1.5

1.0

0.5

0.0

0.5

1.0

F x/g

′ p

×10 4 a′ = 1, Vp = 1Potential forceInertial force

(i)

FIG. 2. x component of the time-dependent force Fx/g′p, using direct numerical simulations of the dGPE Eq. (36), on a test

particle of size a′ = 0.25, 0.5, 1 at a relative position (∆x,∆y) = (10, 20) with respect to the position of the particle producingthe flow perturbation. Speed of both particles is Vp = 0.1, 0.8, 1, and γ = 0. Cyan continuous lines correspond to the fullforce (x component) from the exact expression Eq. (6). They are labeled as ‘potential force’ because of the rather explicitappearance of the interaction potential in this formula. Black dotted lines are the predictions for the inertial force from theapproximation Eq. (25) (computed in the comoving frame as explained in the text).

lows linearization. Eq. (23) assumes in addition weakinhomogeneities below scales a and ξ, and finally Eqs.(25) and (26) (equivalent under the previous lineariza-tion approximation) completely neglects such inhomo-geneities (or equivalently, they correspond to a, ξ → 0).We show as black lines in Fig. (2) the prediction of thislast approximation, similar to the most standard classicalexpressions. Since we have computed the wavefunctionψ = 1 + δψ0 in the comoving frame from Eq. (36), weactually use expression (23) without the Faxén Lapla-cian terms, with δω(0) = ∇(δψ0 − δψ∗0)/(2i) − Vp, andVp = 0. Fig. (36) shows that the full force computedfrom Eq. (6) is well-captured by the approximate ex-pression of the inertial force for small test-particle sizea′. Accuracy progressively deteriorates for increasing a′,and also for increasing Vp, but this classical expressionremains a reasonable approximation until a′ ≈ 1. Theaccuracy can be improved by considering higher-orderFaxén corrections, Eq. (23), or even better, by consid-ering the integral form in Eq. (21). We have explicitly

checked that keeping the full Gaussian integration in Eq.(21) but approximating the integrand in the Bessel inte-gral by its value at the particle position gives a very goodapproximation to the exact force even for a′ = 1.

B. Numerical evaluation of the drag force

We now return to the situation in which there is asingle impurity in the system, with size a = ξ = 1 andgp = 0.01. It moves in the positive x direction with speedVp producing a perturbation on the uniform and steadycondensate state ψ = 1. We compute it in the comovingframe, in which the particle is at rest and fluid moveswith speed −Vp, by using Eq. (36). Since in the absenceof the impurity there is no inhomogeneity nor time de-pendence, δψ0 = 0 and the exact force on the impurity,Eq. (6), is also given by the self-induced drag expressiongiven by Eq. (8). After a transient, that in analogy withthe results for compressible classical fluids [28, 33] we ex-

10

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6Vp

0.0

0.2

0.4

0.6

0.8

1.0

|Fd|

×10 5

= 0= 0.01= 0.05= 0.1

0.0 0.1 0.20

1

2

×10 8

FIG. 3. Plot of the drag force in the steady-state regime asfunction of the constant speed Vp. Dashed lines are the an-alytical predictions based on Eq. (33) (for γ = 0) and Eq.(31) (for γ > 0). The symbols correspond to the numericallycomputed force from Eq. (6) based on direct simulations ofthe dGPE Eq. (36). The inset figure shows the small Vp be-havior, with solid straight lines giving the linear dependenceof the drag force on the speed for γ > 0, in the small-Vp ap-proximation given by Eq. (35). We use a = 1 and gp = 0.01.

pect to be of the order of the time needed by the soundwaves to cross a region of size a or ξ, the condensatedensity near the particle achieves a steady state in thecomoving frame, and we then measure the steady dragon the particle. Figure (3) shows this force, for severalvalues of Vp and γ, as dots. The approximate value ofthe drag force that is obtained under the assumption ofsmall perturbation (small gp) that allows linearization isshown as dashed lines. It is computed from Eq. (33)for γ = 0 and Eq. (31) for γ > 0. the agreement isexcellent. As shown in the inset figure, in the regime ofsmall velocities, the self-induced drag is indeed linearlydependent on the speed with an effective drag coefficientthat is well captured by Eq. (35). This Stokes-like dragat small speeds is due to the thermal drag of the conden-sate by the normal fluid, quantified by γ. We notice thatthe dependence of the drag force on Vp is consistent withhaving a critical velocity for superfluidity even at γ > 0,in the sense that there is still a relatively abrupt changein the force (sharper for smaller γ) around a particularimpurity speed. The superfluidity of BECs at finite tem-perature is still an open question. Recent experiments[34, 35] report superfluid below a critical velocity whichis related to the onset of fringes [36]. In the dGPE, thesteady state drag is always nonzero. Nonetheless, there isa critical velocity above which acoustic emission significa-tively increases that can be associated to the breakdownof superfluidity. This is the regime where the drag force isdominated by the interaction of the impurity with the su-personic shock waves that are reminiscent of the Kelvin

wake in classical fluids. This is seen in Fig. 1(c) andobserved experimentally [32]. The maximum drag forceoccurs near the velocity for which the cusp lines formingthe wake still retain an angle close to π. With increasingspeed, this angle becomes more acute (Fig. 1(f)), andthis lowers the density gradient around the impurity.

V. CONCLUSIONS

We have studied, from analytic and numerical analy-sis of the dGPE, the hydrodynamic forces acting on asmall moving impurity suspended in a 2D BEC at finitetemperature. In the regime of small coupling constant gpand thermal drag γ, the force arising from the gradient ofthe condensate density can be decomposed onto the in-ertial force that is produced by the inhomogeneities andtime-dependence of the condensate in the absence of theparticle, and the self-induced force which is determinedby the perturbation produced by the impurity on thecondensate. When the unperturbed flow can be consid-ered homogeneous on scales below the particle size andthe condensate coherence length, the classical Maxey andRiley expression [18], giving the inertial force in terms ofthe local or material fluid acceleration, is a good descrip-tion of the force. When inhomogeneities become relevantbelow these scales, Faxén-type corrections arise, similarto the classical ones in the presence of a finite-size par-ticle, but here the coherence length plays a role similarto the particle size. In addition, the condensate ther-mal drag enters into these expressions, at difference withthe classical viscous case. We also determined the self-induced force in the steady-state regime and shown thatit is non-zero at any velocity Vp of the moving impurityif γ > 0. For small Vp, this force is given as a Stokes dragwhich is linearly proportional to Vp with a drag coeffi-cient dependent on the thermal drag γ. With increasingvelocities, there are corrections to the linear drag andabove a critical speed of the order of Vp = c = 1, theself-induced drag is dominated by the interactions of theimpurity with the emitted shock waves. Following therecent experimental progress on testing the superfluidityin BEC at finite temperature [17], it would be interestingto test experimentally our prediction of the linear dragon the impurity due to the condensate thermal drag atsmall velocities by using measurements of the local heat-ing rate.

We have checked our analytical expressions with nu-merical simulations in the situation in which the impu-rity moves at constant velocity, possibly driven by exter-nal forces different from the hydrodynamic ones analyzedhere. When the coupling constant gp is sufficiently smallso that only the inertial force is relevant, the equationof motion of the impurity under the sole action of theinertial force would be mpdVp(t)/dt = F (0)(t), with mp

the mass of the particle and F (0)(t) one of the suitableapproximations to the inertial force given in Sect. IIIA.For larger gp, when the condensate becomes distorted by

11

the impurity, we have computed the self-induced dragonly in the steady case, so that can not write a generalequation of motion for the impurity in interaction with atime-dependent perturbed flow. In analogy with classicalcompressible flows [28, 33], we expect history-dependentforces in this unsteady situation. The dependence on thethermal drag, however, would be quite different from thatof viscous classical fluids, because of the lack of viscousboundary layers in the BEC case.

In this study, we have focused on a small impurity thatcan only shed acoustic waves. Another interesting exten-sion of this would be to further investigate the drag andinertial forces for larger impurity sizes, which can emitvortices, and study the effect of vortex-impurity interac-tions on the hydrodynamics forces.

ACKNOWLEDGEMENTS

We are thankful to Vidar Skogvoll, Kristian Olsen, Za-karias Laberg Hejlesen and Per Arne Rikvold for stimu-lating discussions. This work was partly supported bythe Research Council of Norway through its centers ofExcellence funding scheme, Project No. 262644, and bySpanish MINECO/AEI/FEDER through the María deMaeztu Program for Units of Excellence in R&D (MDM-2017-0711).

APPENDIX: NUMERICAL INTEGRATION OFDGPE

0 50 100 150 200 250x/

0

20

40

60

80

100

120

y/

0.9990 0.9995 1.0000 1.0005 1.0010

γ =1 + γ

γ =γ

Vp

0

0

FIG. 4. Simulation domain showing the buffer region, outsidethe main simulation region, in which thermal drag is greatlyenhanced to eliminate the emitted waves sufficiently far fromthe moving particle (which is at x/ξ = 128, y/ξ = 64). Thedensity shown is the steady state (in the comoving frame,hence the direction of the arrows indicating the flow velocityin this frame) for Vp = 1.6 and γ = 0.

Numerical simulations of dGPE Eq. (36) are run for asystem size of 128× 256 (in units of ξ) corresponding to

the grid size dx = 0.25ξ, and dt = 0.01ξ/c. To simulatean infinite domain where the density variations emittedby the impurity do not recirculate under periodic bound-ary conditions, we use the fringe method from [27]. Thismeans that we define buffer (fringe) regions around theouter rim of the computational domain (see Fig. 4) wherethe thermal drag γ is much larger than its value insidethe domain, such that any density perturbation far fromthe impurity is quickly damped out and a steady inflowis maintained. The thermal drag becomes thus spatially-dependent and given by γ(r) = max[γ(x), γ(y)], where

γ(x) = 12

(2 + tanh [(x− xp − wx)/d]

− tanh [(x− xp + wx)/d])

+ γ0, (37)

and similarly for γ(y). Here rp = (xp, yp) = (128ξ, 64ξ)is the position of the impurity and γ0 is the constantthermal drag inside the buffer regions (bulk region). Weset the fringe domain as wx = 100ξ, wy = 50ξ and d = 7ξas illustrated in Figure 4.

By separating the linear and non-linear terms in Eq.(36), we can write the dGPE formally as [37]

∂tψ = ω(−i∇)ψ +N(r, t), (38)

where ω(−i∇) = i[ 12∇

2 +1]+Vp ·∇ is the linear differen-tial operator and N(r, t) = −(i+ γ)(Up + |ψ|2)ψ + γψ +12γ∇

2ψ is the nonlinear function including the spatially-dependent γ and Up. Taking the Fourier transform, weobtain ordinary differential equations for Fourier coeffi-cients ψ(k, t) as

∂tψ(k, t) = ω(k)ψ(k, t) + N(k, t), (39)

which can be solved by an operator-splitting andexponential-time differentiating method [38]. It meansthat we exploit the fact that the linear part of Eq. (39)can be solved exactly by multiplying with the integratingfactor e−ω(k)t. This leads to

∂t

(ψ(k, t)e−ω(k)t

)= e−ω(k)tN(k, t). (40)

The nonlinear term N(k, t) is linearly approximated intime for a small time-interval (t, t+ ∆t), i.e

N(k, t+ τ) = N0 +N1

∆tτ (41)

where N0 = N(t) and N1 = N(t + ∆t) − N0. Insertingthis into Eq. (40) and integrating from t to t+∆t we get

ψ(k, t+ ∆t) = ψ(k, t)eω(k)∆t + N0

ω(k)

(eω(k)∆t − 1

)+ N1

ω(k)

[1

ω(k)∆t (eω(k)∆t − 1)− 1

]. (42)

Since computing the value of N1 requires knowledge ofthe state at t+ ∆t before we have computed it, we startby setting it to zero and find a value for the state att + ∆t given that N(t) is constant in the interval. Wethen use this state to calculate N1, and add correctionsto the value we got when assuming N1 = 0.

12

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13

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