Coarsening dynamics of binary Bose condensates

Johannes Hofmann,1, ∗ Stefan S. Natu,1, † and S. Das Sarma1

1Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics,University of Maryland, College Park, Maryland 20742-4111 USA

(Dated: October 4, 2018)

We study the dynamics of domain formation and coarsening in a binary Bose-Einstein condensatethat is quenched across a miscible-immiscible phase transition. The late-time evolution of the systemis universal and governed by scaling laws for the correlation functions. We numerically determinethe scaling forms and extract the critical exponents that describe the growth rate of domain sizeand autocorrelations. Our data is consistent with inviscid hydrodynamic domain growth, whichis governed by a universal dynamical critical exponent of 1/z = 0.68(2). In addition, we analyzethe effect of domain wall configurations which introduce a nonanalytic term in the short-distancestructure of the pair correlation function, leading to a high-momentum “Porod”-tail in the staticstructure factor, which can be measured experimentally.

The thermodynamic ground state of a system that con-sists of multiple species is not always spatially homo-geneous. Indeed, as the thermodynamic state variablesor the interspecies couplings are tuned, often a transi-tion between a miscible and an immiscible ground statetakes place [1]. A system that is quenched across such atransition does not phase-separate instantly but exhibitshighly nontrivial dynamics which generally proceed intwo stages [2–5]: first, domains of one species nucleateover a short time-scale. In the second stage, these do-mains merge and coarsen until in the infinite-time limit,only one large domain of each species remains. In certaincases, the dynamics in the latter stage can be universalin that they do not depend on the microscopic detailsof the system and are only constrained by symmetriesand conservation laws [5, 6]. The time evolution is thenself-similar, i.e., the time dependence of any ensembleaveraged-quantity is captured by a simple rescaling ofunits by a characteristic length scale L(t) (for example,the average domain size). In classical theories of phaseordering kinetics, this scale diverges with time accordingto a characteristic power law L(t) ∼ t1/z. The phaseordering dynamics of different systems can thus be sepa-rated into distinct dynamical universality classes that arecharacterized by the dynamical critical exponent z. Theconcept of scaling applied to the late-time coarsening istruly universal and has found its application to a wide va-riety of distinct problems in physics: originally developedto describe the growth of metallic grain boundaries [7]and the spinoidal decomposition of a binary alloys be-low a critical phase-coexistence temperature [8], scalingconcepts are now used to describe to formation of galax-ies, the domain growth of liquid membranes [9], or evensociopysics [10]. Here, we extend the classical paradigmof phase ordering kinetics to quantum systems by pre-senting an example of a quantum system that exhibitsclassical late-time scaling: we calculate the dynamicalcritical exponent for domain coarsening in a binary su-perfluid in two dimensions, finding a dynamical scalingexponent that is consistent with inviscid hydrodynamic

domain growth. Understanding the late time dynamicsof isolated quantum systems undergoing unitary (energyconserving) time evolution is an active topic of research,our work shows that established paradigms from classicalcoarsening can further our understanding of this funda-mental problem.

Recent experiments in spinor Bose-Einstein conden-sates (BECs) [1, 11–15] (see [16] for a review) have madeit possible to answer long standing questions about theproperties of multicomponent superfluids [17]. Theseultra-cold gases are unique quantum fluids as they arelargely isolated from the environment and their quan-tum dynamics can be probed accurately over long timesboth in-situ [12, 18] and in time-of-flight [1, 11, 14, 15].Furthermore, the unprecedented control offered by quan-tum gas experiments makes them an ideal testbed tostudy the nonequilibrium physics of superfluids follow-ing a parameter quench [19–23]. Indeed, the short-timedynamics of domain formation following a quench arewell understood both theoretically and experimentally[1, 11, 12, 16, 24–27]. However, questions remain aboutthe universal, long time dynamics and the mechanismsgoverning domain growth [19–23, 25, 28–30]. Surpris-ingly, even in the seemingly simple binary Bose systemwe consider, the power law governing long-time domaingrowth at zero temperature was hitherto unknown. Here,we theoretically establish that the dynamical critical ex-ponent to be 1/z = 0.68(2).

We describe the binary BEC at zero temperature bytwo classical fields ψ1 and ψ2, and take into account theinteraction between the particles on a mean field level.In typical ultra-cold gas experiments, these fields corre-spond to two hyperfine states of an atom such as 23Na or87Rb ([1, 11, 14]), or two different atomic species (87Rband 85Rb) as in the experiment of Papp et al. [15]. Thisyields the Gross-Pitaevskii (GP) energy functional:

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0 1000 2000 3000 4000 50000.5

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FIG. 1. (a-d) Snapshot of the time evolution of a system on a 1024 square lattice at times t = 300, 1000, 2000, and 5000.Domains of positive (negative) magnetization are shown in black (white). After the system is quenched across the instability,domains begin to form which then undergo a self-similar coarsening evolution that is governed by universal scaling laws. (e)Average quadratic magnetization 〈m2〉 as a function of time. The magnetization of each domain grows with time.

H =

∫d2r

(∑i=1,2

{− ψ†i

∇2

2miψi − µi|ψi|2 +

gii2|ψi|4

}+ g12|ψ1|2|ψ2|2

), (1)

where mi is the particle mass, µi the chemical poten-tial, gii is the interaction between like particles of typei, and g12 describes the scattering of atoms of spin 1and 2. In the absence of long-range dipolar interactions[12], the interaction Hamiltonian does not contain spin-flip terms and conserves the total density of each species.Note that the GP equation is strictly applicable onlyat zero temperature and does not account for quantumfluctuations. More complex models describing the time-evolution of the superfluid mixture, such as the “ModelF” in the Hohenberg-Halperin classification [6], reduceto the GP equation in the low-temperature limit [28].Here, we model the GP equation directly and discuss thelimitations of our approach later.

In typical spinor BECs such as the hyperfine states of87Rb or 23Na, the scattering length is nearly identical inall channels [31], and we choose g11 = g22 = g > 0 inthe following. If the intra-species interaction dominates,g12 <

√g11g22, the two condensates can coexist. How-

ever, if g12 >√g11g22, the ground state is no longer ho-

mogeneous and the system phase-separates [11, 32–34].We investigate how the system evolves when suddenlyquenched from the miscible phase with g12 < g to the im-miscible phase with g12 > g. Experimentally, this can bedone either using a Feshbach resonance in systems suchas 85Rb-87Rb mixtures [35], or by preparing a miscibleinitial state in an otherwise immisicible mixture (suchas the hyperfine states of 87Rb or 23Na [1, 11]) using atransverse magnetic field and observing the subsequentdynamics [14]. We do not expect that our conclusionswill be modified if g11 6= g22, as the precise mechanismfor coarsening does not depend on the specific choice ofinteraction parameters.

We consider the time-evolution of the polarizationm(r) = (n1(r)−n2(r))/(n1(r)+n2(r)) as an order param-eter. The spin texture of a spinor gas, and thus the mag-netization, can be measured directly using spin-sensitivephase contrast imaging [13, 14]. At early times, domainsof opposite spin form due to a spin-wave instability [24].Taking into account only the most unstable mode, theinitial domain size is of order L0 ≈ ξs, where we de-fine the spin healing length ξs =

√1/2mn(g12 − g), and

n = |ψ1|2 + |ψ2|2 is the total density. When the domainsize becomes much larger than the spin healing length,the dynamics are universal. Expressed in units of thecharacteristic length scale L(t), all correlation functionsof the order parameter m have no explicit time depen-dence, and collapse to a single, universal scaling function.The pair correlation function, which describes the corre-lation of the magnetization at two points separated by adistance r, can be written as follows:

g(r, t) =1

V

∫d2R 〈m(R)m(R+ r)〉 = f(rL−1(t)). (2)

The bracket 〈. . .〉 denotes an ensemble-average. The cor-relation function is normalized such that g(0, t) = 1. Sim-ilarly, the static structure factor assumes the scaling form

S(q) =

∫d2r eiq·rg(r, t) = L2f(qL(t)). (3)

It should be emphasized that the scaling is a conjecturewhich must be proven on a case-to-case basis [5].

The equations of motion corresponding to the Hamil-tonian (1) are the well-known GP equations:

i∂tψi =

{− ∇

2

2m1+ g|ψi|2 + g12|ψj |2

}ψi (4)

where i, j = {1, 2} and j 6= i. In the following, we con-sider two species of equal mass mi = m and equal chemi-cal potential µi = µ. We simulate the time evolution (4)on a square lattice of dimension l = 1024 and spac-ing d using a split-step spectral method as introduced

3

in Ref. [36]. Although we only present results for sizel = 1024, we have performed numerical simulations overa wide range of system sizes l = 64, 128, 512, finding thesame dynamical critical exponent in each case. We ini-tialize the system in the ground state ψi =

õ/(g + g12)

and add a small Gaussian noise to seed the instabilityafter a sudden quench. The results are averaged overseveral noise realizations. The chemical potential is cho-sen such that the initial domain size (L0 ∼ ξs) is muchlarger than the lattice spacing but is still much smallerthan the system size.

Figure 1 shows the domain structure at various timest = md2t = 300, 1000, 2000, and 5000 for a quench withfinal couplings g = 1 and g12 = 1.1. In typical exper-iments with 87Rb or 23Na, the coupling strengths arealmost identical with (g12 − g)/g ≈ 10−3, however thesecan be tuned using a Feshbach resonance [15, 37–39]. Thechoice of couplings determines the initial domain size af-ter the quench and the onset of the universal late-timescaling regime, but does not affect the scaling behavioritself. Regions with positive (negative) polarization areshown in black (white). Shortly after the quench, dis-tinct domains form (Fig. 1(a)) the size of which growswith time (Figs. 1(b)-(c)). In the following, we computethe universal scaling exponent that governs this time evo-lution. Note that even at late times, the domains arenot fully polarized but contain small patches of oppositepolarization, which correspond to vortex or skyrmion de-fects [19]. For non-dissipative evolution, these defectsare long-lived [19], but they decay if dissipative effectsare included into the GP equations [24, 27, 29].

In Figure 2(a), we show the scaling function f of thepair correlation function (g(r, t)) at various times. Theinset shows the unrescaled pair correlation function. Allresults are averaged over 25 initial conditions. For smallseparation r, the correlation is positive, indicating thattwo nearby points have predominantly the same polariza-tion. This changes at larger distances, where two pointsbecome anticorrelated. We take the position of the firstzero in the pair correlation function as a measure forthe average domain size L(t). For sufficiently large do-mains, the results obtained from this method are consis-tent with those obtained by extracting the domain sizedirectly from the simulations (Fig. 1). Higher oscillationsare visible but very weak, and the correlation betweentwo points is lost at large distances. As can be seen fromFig. 2(a), when expressed in terms of the rescaled coor-dinate r/L(t), the correlation functions at different timescollapse onto a single scaling function. Indeed, the timeevolution is consistent with the scaling hypothesis.

The scaling behavior is also visible in the structure fac-tor S(q) shown in Fig. 2(b). Moreover, we see that thestructure factor decays as a power law ∼ 1/q3 at largemomentum. This feature is a consequence of the pres-ence of domain walls, which determine the short-distancestructure of the pair correlations. It can be understood

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FIG. 2. (Color Online) (Top) Scaling function of the paircorrelations function at different times. Inset: Pair corre-lation function. (Bottom) Universal scaling function of thestructure factor at different times t = 300, 1000, 1500, 2000,and 5000. The structure factor collapses to a single functionwhen expressed in units of the domain size. The presence ofdomain walls induces a high-momentum tail S(q) ∼ q−3.

as follows: two points with separation r are positivelycorrelated if they are in domains of the same type andnegatively otherwise. At short distances, this correlationis determined by the probability to cross a single domainwall, which is proportional to r/L(t) [5]. This gives riseto a nonanalytic linear term in the pair correlation func-tion g(r) = 1 + f ′(0) r

L(t) + O(r2). The linear slope is

clearly present in our numerical data, Fig. 2(a). This im-plies a universal high-momentum “Porod” tail [5] in thestructure factor S(q) = C

Lq3 + O(1/q4). Crucially, thishigh-momentum tail is not present in the initial stage ofthe coarsening, where the structure factor is Gaussian.The structure factor changes from its initial Gaussianshape to the form in Fig. 2(b) only after the domains haveformed and the system has entered the second coarsen-ing stage. At late times when the density of domain wallsdecreases, the magnitude of the tail vanishes as 1/L(t).

We proceed to extract the scaling exponent from themeasured domain sizes. The scaling argument is strictlyvalid only in the limit of infinitely large domain sizes, andthe exponent has a finite size scaling correction. Huse ar-gued that for a conserved order parameter, this correctionis of order 1/L(t) [40]. We determine the time-dependentscaling exponent z(t) by taking the logarithmic differen-tial quotient of the domain size at different times t′ > t:

4

1/z(t) = log(L′/L)/ log(t′/t). As is apparent from the re-sult in Fig. 3, the scaling exponent displays a weak drifttowards smaller values at later times. Extrapolating tothe limit of infinite domain size (red line in Fig. 3), weobtain 1/z(∞) = 0.68(2). The same dynamical exponentis found in classical binary fluids where 1/z = 2/3 [41].

The numerical value of the scaling exponent z indicatesthat the dominant mechanism driving domain growth isinertial hydrodynamic transport of superfluid from low-density to high-density regions [5]. Heuristically, this canbe understood as follows: in a binary fluid, the changein a domain wall’s velocity dv/dt is equal to the gradientof the pressure −∇P , where the pressure P is propor-tional to the energy density of the domain walls. Theinertial term scales as dv/dt ∼ L/t2, and the pressuregradient scales as −∇P ∼ σ/L2, where σ is the sur-face tension [42]. Equating inertial and gradient termsgives the dynamical exponent 1/z = 2/3 [5]. This ex-ponent has not been measured in classical binary liquidsprimarily because coarsening dynamics is complicated byadvective and viscous terms which give rise to other scal-ing exponents, and can even lead to a breakdown of scaleinvariance [43, 44]. However binary BECs may be theideal testbed for observing this type of scaling behavior.

Our previous analysis was restricted to the equal-timecorrelation function. It turns out that a single dynamicalexponent is not sufficient to describe the scaling behaviorof correlators at different times [5]. Consider the dynam-ical pair correlation function, which in the scaling limitdepends on two length scales:

g(r, t; r′, t′) =〈m(r, t)m(r′, t′)〉

〈m2(r, t)〉1/2〈m2(r′, t′)〉1/2

= f

(r

L(t),r′

L(t′)

). (5)

In the limit of t � t′, the dependence on one lengthscale separates, and the correlation function becomes a

t`

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s

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0.55

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0.65

0.70

Ξs�LHtL

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tL

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10

100

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ççt`'=300

á

á

á

á

áá

áá

áá

áá

áá

áá

áá

áá

áá

áá

áá

áá

áá

áááááááááááááááá

ááááááááááá

áát`'=400

í

í

í

íí

íí

íí

íí

íí

íí

íí

íí

íí

íí

íííííííííííí

íííííííííííííííííí

íít`'=500

óót`'=600

1.0 2.0 3.01.5

0.02

0.05

0.10

0.20

0.50

1.00

LHtL�LHt'L

gHt,

t'L

FIG. 4. (color online) Autocorrelation function g(t; t′) as afunction of L(t)/L(t′) for different values of t′ = 300, 400, 500,and 600 (left to right). A linear fit (red line) gives the scalingexponent λ = 3.90(2).

homogeneous function of the ratio L(t)/L′(t):

g(r, t; r′, t′) =

(L(t)

L(t′)

)−λf(r/L(t)). (6)

In general, the exponent λ differs from z. We show thebehavior of the zero-range value of the pair correlationfunction g(0, t; 0, t′) as a function of time in Fig. 4. Weassume that the correction arising from the finite domainsize is small and perform a direct fit to the data overthe whole measurement interval. This procedure gives adynamical exponent of λ = 3.90(2).

We now discuss the significance of our results to non-equilibrium experiments in binary Bose condensates. Inthe experiments to date, the spin healing length is oforder ζ ∼ 10µm [1, 11, 12], which corresponds to a timeof t ∼ 1s beyond which scaling should be observed. Onthese timescales, the dynamics of the physical system iscomplicated by particle losses and heating in the trap inpresent-day experiments [14], which however should notpreclude the observation of universal late-time scaling infuture, higher-precision experiments. Another promisingapproach for realizing an effective strongly interactingspin-1/2 system is to use lattice modulation [18]. Whilestronger interactions imply smaller spin healing lengths,inelastic collisions, three-body losses and heating maycomplicate the observation of universal scaling.

Finally, even at temperatures T � µ, where the GPapproach is valid, on long timescales of order tcoll ∼(~an/mlz

√(na3/lz))

−1 � mξ−2s , (where lz is the ax-ial confinement length), inelastic scattering processes willstart to become important, in which case the systemshould be modeled using a quantum kinetic equationrather than the GP ansatz. A proper modeling of in-elastic effects requires more sophisticated methods suchas the c-field techniques currently being developed byseveral groups [45, 46]. While coarsening may still beobserved on times t < tcoll, it remains an exciting fu-

5

ture problem to investigate the transition between theGross-Pitaevskii and kinetic limit.

In conclusion, we provide numerical evidence that thecoarsening dynamics of a binary Bose-Einstein conden-sate quenched across a miscible-immiscible phase bound-ary obeys universal scaling laws. Equal-time correla-tion functions depend implicitly on time through a singlecharacteristic length scale which grows according to apower law with time with an exponent 1/z = 0.68(2),consistent with inertial hydrodynamic growth. Our find-ings can be verified experimentally either via direct imag-ing of magnetic domain growth [1, 11, 12, 14, 18], or bymeasuring the structure factor [47].

Acknowledgements.— This work is supported by JQI-NSF-PFC, AFOSR-MURI, and ARO-MURI.

∗ hofmann@umd.edu† snatu@umd.edu

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