Bose-Bose mixtures in a weak-disorder potential: Fluctuations and superfluidity
aa r X i v : . [ c ond - m a t . d i s - nn ] A ug Bose-Bose mixtures in a weak disorder potential: Fluctuations and Superfluidity
Abdelˆaali Boudjemˆaa , and Karima Abbas , Department of Physics, Faculty of Exact Sciences and Informatics,Hassiba Benbouali University of Chlef, P.O. Box 78, 02000, Ouled-Fares, Chlef, Algeria. Laboratory of Mechanics and Energy, Hassiba Benbouali University of Chlef,P.O. Box 78, 02000, Ouled-Fares, Chlef, Algeria. ∗ We study the properties of a homogeneous dilute Bose-Bose gas in a weak disorder potentialat zero temperature. By using the perturbation theory, we calculate the disorder corrections tothe condensate density, the equation of state, the compressibility, and the superfluid density as afunction of density, strength of disorder, and miscibility parameter. It is found that the disorderpotential may lead to modifying the miscibility-immiscibility condition and a full miscible phaseturns out to be impossible in the presence of the disorder. We show that the intriguing interplay ofthe disorder and intra- and interspecies interactions may strongly influence the localization of eachcomponent, the quantum fluctuations, and the compressibility, as well as the superfluidity of thesystem.
I. INTRODUCTION
In recent years, degenerate multi-component quantumgases have prompted considerable interest in the com-munity of cold atoms physics both theoretically andexperimentally due to their rich phase diagram. Oneof the most significant characteristics of such multi-component structures is their miscibility-immiscibilitytransition which depends on the ratio of the intra- andinterspecies interactions [1–3], on the condensate num-bers [4], and on thermal fluctuations [5–8]. A mixture oftwo-component Bose-Einstein condensate (BEC) plays acrucial role in various systems, such as solitons (see e.g.[9]), vortices (see e.g. [10]), and bilayer Bose systems(see e.g. [11, 12]). Very recently, it has been found thatthe balance between the mean-field term and the beyond-mean-field quantum fluctuation may lead to the forma-tion of a mixture droplet phase [13–16].On the other hand, the creation of disorder usingspeckle lasers [17, 18] or incommensurate laser beams[19, 20] opens promising new avenues in condensed mat-ter physics and in the ultracold quantum gases field. Thecompetition between disorder and interactions plays anontrivial role in developing a fundamental understand-ing of many aspects of ultracold gases namely: the Boseglass (a gapless compressible insulating state) [21–25],Anderson localization [17–20, 26–31], disordered BEC inoptical lattices [32–35], Bose-Fermi mixtures [36, 37], anddipolar BEC in random potentials [7, 38–45].Until now, there has been little work treating disor-dered ultracold Bose-Bose mixtures. A general mech-anism of random-field-induced order has been analyzedin both lattice [46] and continuum [47] two-componentBEC. Localization of a trapped two-component BEC in aone-dimensional random potential has been numericallyaddressed in Ref.[48]. It has been found in addition thatdisorder plays a crucial role in the dynamics of spin-orbit ∗ [email protected] coupled BEC in a random potential [49].This paper aims to investigate the impacts of a weakdisorder potential on the quantum fluctuations and onthe superfluidity of two-component BEC. To this end,we extend the perturbative theory applicable to the singlecomponent bosonic gas [30, 31, 38, 39, 45, 50] and presenta detailed analysis of weakly interacting homogeneoustwo-component Bose gases subjected to weak disorderpotential with delta-correlation function. The effects ofthe disorder on the miscibility-immiscibility condition arealso deeply investigated. This study not only bridges thegap between superfluidity, interactions and disorder butalso it is important from the viewpoint of elucidating thelocalization phenomenon of two bosonic species.We derive useful expressions for the condensate fluc-tuations due to the disorder known as glassy fraction ,the equation of state (EoS), the compressibility, and thesuperfluid density. We look at how each species is influ-enced by the disorder and how the interaction betweendisordered bosons influences the coupling and the phasetransition between the two components. Our results re-veal that the localization of each species does not de-pend only on the disorder strength but depends also onthe interspecies interactions and the ratio of intraspeciesinteractions. We show that the disorder effects couldsignificantly enhance chemical potential of each species.The disorder corrections to the superfluid density show asimilar behavior as the glassy fraction of the condensate.Moreover, we obtain disorder corrections to the com-pressibility, and the miscibility condition and accuratelydetermine the critical disorder strength above which atransition from miscible to immiscible phase occurs. Inthe decoupling regime where the interspecies interactiongoes to zero, we find good agreement with the analyti-cal results obtained within the Huang-Meng-Bogoliubovmodel [51] and perturbative theory for a single compo-nent BEC. Experimental evidence of the Huang-Mengtheory for a single BEC has been reported most recentlyin Ref.[52].The rest of this paper is structured as follows. InSec.II we develop the perturbative theoretical descriptionwith respect to disorder which is based on the coupledGross-Piteavskii (GP) equations and discuss its validity.Section III deals with the fluctuations due to the disor-der potential. We focus explicitly on the effects of weakdelta-function correlated disorder and derive an analyti-cal formula for the glassy fraction. Its behavior is deeplyhighlighted as a function of the miscibility parameter andinterspecies interactions. In Sec.IV we calculate the dis-order corrections to the EoS by extending the renormal-ization scheme used in a dirty single BEC [39, 45]. Sec-tion V is dedicated to investigating the compressibilityand to establishing the miscibility condition for a disor-dered homogeneous mixture. We find that a binary Bosemiscible mixture cannot occur in the presence of the dis-order. In Sec.VI we look at how a weak disorder po-tential influences the superfluidity. SectionVII containssome conclusions and outlooks. II. MODEL
Consider weakly interacting binary Bose gases in aweak random potential fulfilling mean-field miscibilitycriterion (see below). The system is described by thecoupled GP equations [6, 7, 16, 53] µ j Φ j = (cid:20) − ¯ h m j ∇ + U j + g j | Φ j | + g | Φ j | (cid:21) Φ j , (1)where Φ j is the wavefunction of each condensate, theindice j is the species label, j = 3 − j , µ j is the chemi-cal potential of each condensate, g j = (4 π ¯ h /m j ) a j and g = g = 2 π ¯ h ( m − + m − ) a with a j and a beingthe intraspecies and the interspecies scattering lengths,respectively. The gas parameter satisfies the condition n j a j ≪
1. The disorder potential U j ( r ) is described byvanishing ensemble averages h U ( r ) i = 0, and a finite cor-relation of the form h U ( r ) U ( r ′ ) i = R ( r − r ′ ).For weak disorder, Eq.(1) can be solved using straight-forward perturbation theory in powers of U using theexpansion [30, 31, 38, 39, 45, 50]Φ j = Φ (0) j + Φ (1) j ( r ) + Φ (2) j ( r ) + · · · , j = 1 , i in the real valued functions Φ ( i ) ( r )signals the i -th order contribution with respect to thedisorder potential. They can be determined by insertingthe perturbation series (2) into Eq.(1) and by collectingthe terms up to U . The zeroth order givesΦ (0) j = vuut µ j − g Φ (0)2 j g j , (3)which is the uniform solution in the absence of a disorderpotential. Combining Eqs.(3), yieldsΦ (0) j = vuut µ j g j − g g j µ j µ j ! ∆∆ − , (4) where ∆ = g j g j /g is the miscibility parameter whichcharacterizes the miscible-immiscible transition. For ∆ >
1, the mixture is miscible while it is immiscible for ∆ < − ¯ h m j ∇ Φ (1) j ( r ) + U j ( r )Φ (0) j + 2 g j Φ (0)2 j Φ (1) j ( r ) (5)+ 2 g Φ (0) j Φ (0) j Φ (1) j ( r ) = 0 , Performing a Fourier transformation, one obtainsΦ (1) j ( k ) = − h U j ( k ) + 2 g Φ (0) j Φ (1) j ( k ) i Φ (0) j E kj + 2 g j Φ (0)2 j , (6)where E kj = ¯ h k / m j .For E kj ≪ g j Φ (0)2 j = µ j (cid:16) − g µ j /g j µ j (cid:17) ∆ / (∆ − n j ( r ) =Φ (0)2 j + n (1) j ( r ), where n (1) j = ∆ (cid:0) n j − g n j /g j (cid:1) / (∆ − n j = ( µ j − V j ) /g j being the decoupled condensatedensity which is nothing else than the standard Thomas-Fermi-like shape. For E k ≫ µ j (cid:16) − g µ j /g j µ j (cid:17) ∆ / (∆ − U have been smoothed out.The second-order term is governed by the followingequation − ¯ h m j ∇ Φ (2) j ( r ) + U j ( r )Φ (1) j + g j (cid:20) (0)2 j Φ (2) j ( r ) (7)+ 3Φ (0) j Φ (1)2 j ( r ) (cid:21) + g (cid:20) (0) j Φ (1) j ( r )Φ (1) j ( r ) + Φ (0) j Φ (1)2 j ( r )+ 2Φ (0) j Φ (0) j Φ (2) j ( r ) (cid:21) = 0 . The solution of this equation in the momentum spacereadsΦ (2) j ( k ) = − Z d k ′ (2 π ) Φ (1) j ( k − k ′ ) h U j ( k ′ ) + 3 g j Φ (0) j Φ (1) j ( k ′ ) i E kj + 2 g j Φ (0)2 j − g (0) j Φ (0) j Φ (2) j ( k ) E kj + 2 g j Φ (0)2 j − g Z d k ′ (2 π ) Φ (1) j ( k − k ′ ) × h (0) j Φ (1) j ( k ′ ) + Φ (1) j ( k ′ )Φ (0) j i E kj + 2 g j Φ (0)2 j . (8)Equation (8) enables us to selfconsistently determine thechemical potential of the system (see below).Finally, the validity of the present perturbation ap-proach requires the condition: U ≪ g j Φ (0)2 j ≃ g j n j ,where Φ (0) j is given in Eq.(4), tells us that the den-sities do not vary much around the homogeneous val-ues. For g = 0, one recovers the well-known condition( U ≪ g Φ (0)2 ) established for a disordered single BEC[50]. Indeed, this simple assumption indicates how local-ization can be destroyed in a regime of weak interactions.However, the perturbation approach is no longer valid inthe regime of strong disorder. III. GLASSY FRACTION
In this section we deal with the mixture fluctuationsdue to the disorder potential. It has been shown that thedisorder contribution to the condensate can be given asthe variance of the wavefunction n Rj = n j − n cj [38, 39],where n j = h Φ j ( r ) i = Φ (0)2 j + h Φ (1)2 j ( r ) i + 2Φ (0) j h Φ (2) j ( r ) i + · · · (9)and n cj = h Φ j ( r ) i = Φ (0)2 j + 2Φ (0) j h Φ (2) j ( r ) i + · · · (10)is the condensed density. Subtracting (10) from (9), oneobtains n Rj = h Φ (1)2 j ( r ) i + · · · , which is in fact analogto the Edwards-Anderson order parameter of a spin glass[39, 54, 55].From now on, we shall consider U = U = U and m = m = m .Employing the Fourier transform of Φ (1) j ( r ) i.e.Eq. (6), and using the fact that h U ( k ′ ) U ( k ′′ ) i =(2 π ) R ( k ′ ) δ ( k ′ + k ′′ ), the glassy fraction, n Rj , can bewritten as: n Rj = n j Z dk (2 π ) R ( k ) " E k + 2 n j (cid:0) g j − g (cid:1) E k , (11)where E k = (cid:0) E k + 2 g j n j (cid:1)(cid:0) E k + 2 g j n j (cid:1) − g n j n j .For analytical tractability, we consider the white noiserandom potential, which assumes a delta distribution R ( r − r ′ ) = R δ ( r − r ′ ) , (12)where R is the disorder strength with dimension(energy) × (length) . The model (12) is valid when thecorrelation length of the correlation function R ( r − r ′ ) issufficiently shorter than the healing length.After some algebra, we get a useful formula for the glassyfraction: n Rj n j = 4 πR ′ j s n j a j π f j (∆) , (13)where R ′ j = R /g j n j is a dimensionless disorder strength and, f j (∆) = (2 β j ) − / q µ j + p β j − (2 β j ) − / q (1 + ¯ µ j ) − p β j ¯ f (∆)(14)+ √ β − j q µ j + p β j + √ β − j q (1 + ¯ µ j ) − p β j ¯ f (∆)where¯ f (∆) = (1 + ¯ µ j ) + 2 α j (1 + ¯ µ j ) − µ j ) (cid:2) µ j (cid:0) ∆ − (cid:1) + α j (cid:3) + 8¯ µ j α j (cid:0) ∆ − (cid:1) ,¯ f (∆) = (1 + ¯ µ j ) + 2 α j (1 + ¯ µ j ) − µ j (cid:0) ∆ − (cid:1) − α j , β j = (1 + ¯ µ j ) − µ j [(∆ − / ∆], α j = ¯ µ j (cid:16) − q g j / (cid:0) g j ∆ (cid:1)(cid:17) , and ¯ µ j = n j g j /n j g j .Equation (13) is appealing since it describes the glassyfraction in terms of the miscibility parameter. The to-tal disorder density is given by n R = n R + n R . For∆ → ∞ (or g →
0, equivalently), we find fromEq.(13) that f ( ∞ ) = f ( ∞ ) = 1 /
2. Therefore, weshould reproduce the famous Huang and Meng result [51], n R /n = 2 πR ′ p na /π for the single component disorderfraction. The intriguing interplay between the strong in-tercomponent coupling and the disorder effects in theregime ∆ − ≪ f j (∆). Near the phase separation i.e. ∆ → g → √ g g , equivalently), the functions f j (∆) arediverging. They are complex for ∆ < f j have the following asymp-totic behavior for small a f j ( a ) = 12 − n j n j a j (cid:16) q n j a j n j a j (cid:17) a + · · · , and for large a f j ( a ) = (cid:16)q a j /a j − (cid:17) (cid:16) n j a j n j a j + 1 (cid:17) / r(cid:16) a j a j /a (cid:17) − · · · . It is straightforward to check that these asymptotic re-sults perfectly agree with the solutions shown in Fig. 1(a) in the asymptotic regime.As an illustration of our theoretical formalism, weconsider a two-component Bose condensate of rubid-ium atoms in two different internal states Rb- Rb.We have taken the intra-component scattering lengths: a = 100 . a and a = 95 . a ( a is the Bohr ra-dius) [56], and the densities: n = 1 . × m − , and n = 10 m − . Thus, the parameter n j a j is as small as ∼ − .Figure 1 (a) shows that for a /a ≤ .
89, the func-tions f j are decreasing with the interspecies interaction H a L a (cid:144) a f j H b L a (cid:144) a f j FIG. 1. (Color online) (a) Behavior of the disorder functions f j as a function of the interspecies interaction strength a for Rb- Rb mixture. (b) Behavior of the disorder functions f j as a function of the ratio a /a for a = 90 a . Blue dottedlines: f . Red dashed lines: f . Here a can be adjusted viaFeshbach resonance. giving rise to the delocalization of both species. In thevicinity of the transition between the miscible and immis-cible phases i.e. a /a = 97 .
89, the functions f j exhibitan anomalous behavior where they develop a small mini-mum. Then they start to increase for a /a > .
89. Insuch a regime, both species are srongly localized in thelocal wells of the random potential.The situation is quite different for fixed a and varyingthe interactions ratio a /a . The disorder functions f and f decrease/increase with the ratio a /a as is shownin Fig.1 (b). The function f develops a minimum at a ≃ a . For a /a > ∼ f is very small and thus, thefirst component becomes almost superfluid due to thesuppression of the localization, while the second BECremains localized regardless of the value of a . One canconclude that the localization of one component does nottrigger the localization of the second component due tothe interplay of the intra- and interspecies interactionsand the disorder potential. IV. EQUATION OF STATE
The EoS can be calculated by substituting Eqs.(3)-(8)into Eq.(9) and solving the equation h Φ j ( µ bj ) i = n ( µ bj ),where µ bj represents the bare chemical potential. It di-verges for uncorrelated disorder [39, 45]. We then obtain µ bj ( n j , n j ) = g j n j + g n j − Z d k (2 π ) R ( k )( g j g j − g ) E k ( ( g j g j − g )[ E k − n j ( g − g j )] + g g j [ E k − n j ( g − g j )] − g g j g j n j [ E k − n j ( g − g j )] − g j g j n j [ E k − n j ( g − g j )] E k − g n j ( g j g j − g )[ E k − n j ( g − g j )] × [ E k − n j ( g − g j )] E k − g g j g j n / j [ E k − n j ( g − g j )] ( g j g j − g ) E k ) . (15)To overcome this unphysical ultraviolet divergence, werenormalize the chemical potential. The renormalized chemical potential is defined as: µ j ( n j , n j ) = µ bj ( n j , n j ) − µ bj (0) , (16)where µ bj (0) = − Z d k (2 π ) R ( k ) " E k + g j g ( g j g j − g ) E k . (17)Omitting higher order in g , we obtain, in second-orderof the disorder strength, the following renormalized EoS µ j = g j n j + g n j + Z d k (2 π ) R ( k )( g j g j − g ) E k E k (18) × ( g j g j n j ( E k + g j n j ) ( E k + 2 g j n j ) + 4 g j g j g n j (cid:20) ( E k + g j n j )( E k + 2 g j n j ) + E k ( E k + 2 g j n j ) (cid:21)(cid:27) . This equation allows us to calculate the sound velocityand the inverse compressibility.For delta-correlated disorder (12), the EoS reads µ j = g j n j + g n j + 16 πg j n j R ′ j s n j a j π h j (∆) , (19)where h j (∆) = 1(2 β j ) / ∆∆ − (cid:20) ¯ h (∆) + n j g n j g j ¯ h (∆) (cid:21) , (20)and¯ h (∆) = q µ j + p β j − q (1 + ¯ µ j ) − p β j H (∆)+ p β j q µ j + p β j + p β j q (1 + ¯ µ j ) − p β j H (∆) , ¯ h (∆) = q µ j + p β j − q (1 + ¯ µ j ) − p β j H (∆)+ p β j q µ j + p β j + p β j q (1 + ¯ µ j ) − p β j H (∆) , where H (∆) = (1 + ¯ µ j ) + (1 + ¯ µ j ) h + 2¯ µ j − ¯ µ j (cid:16) ∆∆ − (cid:17)i − µ j ) (cid:2) ¯ µ j + ¯ µ j (cid:0) (cid:0) ∆ − (cid:1)(cid:1)(cid:3) +2¯ µ j (cid:0) ∆ − (cid:1) (1+4¯ µ j )+6¯ µ j , H (∆) = (1 + ¯ µ j ) + (1 + ¯ µ j ) h µ j (cid:16) ∆∆ − (cid:17)i − µ j (cid:0) ∆ − (cid:1) , H (∆) = 2(1 + ¯ µ j ) + (1 + ¯ µ j ) h − ¯ µ j − (cid:16) ∆∆ − (cid:17)i +2¯ µ j (cid:8) (cid:0) ∆ − (cid:1) [ − µ j ) + 3¯ µ j + 4] (cid:9) , and H (∆) = 2(1 + ¯ µ j ) + (1 + ¯ µ j ) h (cid:16) ∆∆ − (cid:17) + 3¯ µ j i − µ j (cid:16) ∆∆ − (cid:17) .The last term in Eq.(19) accounts for the disorder correc-tions to the EoS. For ∆ → ∞ (or g →
0, equivalently),one has h j ( ∞ ) = 3 / µ = gn (1 + 12 πR ′ p na /π ), found in Refs.[55, 57, 58]using the Huang-Meng-Bogoliubov theory. H a L a (cid:144) a h j H b L a (cid:144) a h j FIG. 2. (Color online) (a) Behavior of the disorder functions h j as a function of a for Rb- Rb mixture. (b) Behaviorof the disorder functions h j as a function of the ratio a /a for a = 90 a . Blue dotted lines: h . Red dashed lines: h . Figure 2 (a) depicts that the functions h j grow with a and diverge at a → √ a a results in an enhancementof the total chemical potential. In this case, the quantumfluctuations arising from interactions are viewed as beingpredominated by disorder effects.Moreover, we see from Fig.2 (b) that the disorder func-tions h j behave differently with the interactions ratio a /a . Both functions diverge for a /a → a /a = 5. The chemical potential associated with thefirst component µ enhances when h rises, while µ de-cays for lowering h . This reveals that the competitionof the intraspecies interactions and the disorder poten-tial may perceptibly alter the behavior of the EoS of thewhole mixture. V. MISCIBILITY CONDITIONS
We now discuss a possible energetic instability, associ-ated with the presence of the disorder and the occurenceof miscible-immiscible phase transition. For a homo-geneous mixture to be stable, the following conditionsshould be fulfilled [59]: ∂µ j ∂n j > , (21a) (cid:18) ∂µ j ∂n j (cid:19) ∂µ j ∂n j ! > ∂µ j ∂n j ! . (21b)These conditions are derived from the variation of theenergy with respect to the densities. For the EoS (19),we obtain ∂µ j ∂n j = g j πR ′ j s n j a j π (cid:18) h j + 2 n j ∂h j (∆) ∂n j (cid:19) . (22)The second term in the r.h.s of Eq.(22) constitutes thedisorder corrections to the inverse compressibility κ − j = n j ∂µ j /∂n j .Figure 3 (a) shows that the disorder functions n j ∂h j /∂n j possess identical behavior over almost theentire range of the interspecies interactions. They van-ish for a = 0 where the two components are spatiallyseparated and remain negligibly small in the domain0 ≤ a /a < ∼
65, indicating that the disorder effect ismarginally relevant in this regime. For a /a > ∼ n j ∂h j /∂n j decrease and display a negative divergence at a → √ a a , leading to appreciably reduce the com-pressibility of the system.We observe from Fig.3 (b) that the disorder functions n ∂h /∂n and n ∂h /∂n vary in the opposite way withthe ratio a /a . They diverge for a /a →
0, and haveminimum/maximum at a /a ≃ .
2, where the secondcomponent is extremely dilute compared to the first com-ponent, then they increase/decrease for a /a > . ∂h j /∂n j are negative in the whole range of interactions. The stability conditions (21) turn out to be given as g j πR ′ j s n j a j π (cid:18) h j + 2 n j ∂h j (∆) ∂n j (cid:19) > , (23a)and∆ πR ′ j s n j a j π (cid:18) h j + 2 n j ∂h j (∆) ∂n j (cid:19) × πR ′ j s n j a j π h j + 2 n j ∂h j (∆) ∂n j ! > πR ′ j s n j a j π a j a n j ∂h j (∆) ∂n j . (23b)Expressions (23) clearly show that the miscibility condi-tion for a mixture of two interacting BEC is significantlyaffected by the disorder potential. This gives rise to aphase transition to an immiscible phase even though thecleaned mixture is miscible. For relatively large disorderstrength, the mixture may drive a transition to an im-miscible phase with complete spatial separation betweenthe two BEC. For R ′ j = 0, the conditions (23) reduce tothose of the cleaned binary BEC mentioned above.The critical disorder strength above which a quantummiscible-immiscible phase transition occurs can be di-rectly determined from (23b) as R ′ cj = − A j − q A j − B j (∆ − / ∆16 π q n j a j /πB j , (24)where A j = ( h j + 2 n j ∂h j /∂n j ) + q g j n j /g j n j ( h j +2 n j ∂h j /∂n j ) − − ( n j g j /g )( ∂h j /∂n j ), and B j =( h j + 2 n j ∂h j /∂n j )( h j + 2 n j ∂h j /∂n j ) q g j n j /g j n j − − ( n j g j /g ) ( ∂h j /∂n j ) with R ′ j = R ′ j ( g j n j /g j n j ).In the case of Rb- Rb mixture with parameters : a = 100 . a , n = 1 . × m − and a = 95 . a , n = 10 m − , and a = 90 a , the miscible-immisciblephase transition arises for disorder strengths R ′ c = 0 . R ′ c = 1 . VI. SUPERFLUID FRACTION
Let us consider a Bose mixture superfluid movingwith velocity v sj = ¯ h k sj /m , where k sj is a wavevec-tor corresponding to the velocity of superfluid, subjectedto a moving weak random potential with the velocity v u = ¯ h k u /m , where k u is a wavevector corresponding tothe velocity of disorder. At finite temperatures, the Bosefluid is separated into a superfluid density n sj and a nor-mal density n nj that moves with the disorder component H a L - - - - - - a (cid:144) a n j ¶ h j ‘ ¶ n j H b L - - - - - - - - - a (cid:144) a n j ¶ h j ‘ ¶ n j FIG. 3. (Color online) (a) Behavior of the disorder functions ∂h j /∂n j in units of n j as a function of a for Rb- Rbmixture. (b) Behavior of the disorder functions ∂h j /∂n j inunits of n j as a function of the ratio a /a for a = 90 a .Blue dotted lines: n ∂h /∂n . Red dashed lines: n ∂h /∂n . n Rj . Then the coupled time-dependent GP equationsread i ¯ h ∂ Φ j ( r , t ) ∂t = (cid:18) − ¯ h m ∇ + U ( r − v u t ) (25)+ g j | Φ j ( r , t ) | + g | Φ j ( r , t ) | (cid:19) Φ j ( r , t ) . We treat the solution of Eq.(25) perturbatively by intro-ducing the functionΦ j ( r , t ) = (cid:2) Φ (0) j + Φ (1) j ( r , t ) + Φ (2) j ( r , t ) + · · · (cid:3) (26) × e i k sj . r e − i ¯ h (cid:0) ¯ h k sj m + µ j (cid:1) t , which corresponds to the clean-case solution [39, 45, 57].After inserting the expansion (26) into Eq.(25), and usingthe transformation r ′ = r + v u t , one obtains (cid:18) − ¯ h m ∇ − i ¯ h m K j . ∇ + U ( r ′ ) − µ j + g j | Φ j ( r ′ ) | (27)+ g | Φ j ( r ′ ) | (cid:19) Φ j ( r ′ ) = 0 , where K j = k sj − k u .In the two-fluid model, the total momentum P ( r ) of themoving system is defined as: P j = − i ¯ h h Φ j | i k sj + ∇| Φ j i = ¯ h k sj n j − i ¯ h h Φ ∗ j ∇ Φ j i . (28)We neglect higher than linear terms in k sj and keepingin mind that in zeroth order P j does not depend on k sj .This yields P j = ¯ h k sj n j − i ¯ h h Φ ∗ (1) j ∇ Φ (1) j i + · · · , (29)where the first-order correction to the wavefunction isgiven in Fourier space byΦ (1) j ( k ) = − U ( k )Φ (0) j ( E k − ¯ h m k . K j ) h − E k − E k Φ (0)2 j ( g j − g ) + ( ¯ h m k . K j ) i (0)2 j Φ (0)2 j g E k − (cid:16) E k + 2 E k g j Φ (0)2 j − ( ¯ h m k . K j ) (cid:17) (cid:16) E k + 2 E k g j Φ (0)2 j − ( ¯ h m k . K j ) (cid:17) . (30)For small K j , the normal density reads n nj = n j − h ∂P j ∂K j (cid:12)(cid:12)(cid:12) K j =0 . (31)In the case of delta-correlated random potential (12), weget for the normal fraction n nj = 16 π R ′ j s n j a j π f j (∆) = 43 n Rj . (32) We see that Eq.(32) well recovers the result of Huang-Meng for a single component BEC with contact interac-tion [51]. The fact that n nj is larger than n Rj is dueto the localization of bosons in the respective minimaof the random potential which leads to reduction of thesuperfluid density. Obviously, the interplay of the dis-order potential, interspecies interaction and the ratio ofintraspecies interactions may strongly affect the super-fluid fraction n sj = 1 − (4 / n Rj . VII. CONCLUSIONS
We investigated the impact of a weak disorder poten-tial with a delta-correlated function of a homogeneous bi-nary BEC at zero temperature. Within the realm of theperturbative theory, we derived analytical expressions forthe physical quantities of interest such as the condensatedepletion due to the disorder, the EoS, the compress-ibility, and the superfluid density in terms of density,strength of disorder and the miscibility parameter. Ourresults revealed that the intriguing interplay of the dis-order and intra- and interspecies coupling may stronglyinfluence both the quantum fluctuations and the super-fluidity yielding a variety of interesting situations for rel-evant experimental parameters. In particular we foundeither both species are localized, or only one species islocalized and the second species remains extended. Weshowed in addition that the localization of one compo-nent does not necessarily trigger the localization of theother species. Interestingly, we found that the disorderpotential leads to a dramatic phase separation betweenthe two species, changing the miscibility criterion of themixture. We expect that the introduction of the Lee-Huang-Yang (LHY) corrections that stem from quantumfluctuations in the EoS [60] may stabilize the misciblestate analogously to the quantum mechanical stabiliza-tion of the droplet phase [13]. The same scenario takesplace in a disordered dipolar BEC with the LHY quan-tum corrections [45]. Furthermore, as in the disordered single BEC, the disorder corrections to the normal partof each Bose fluid have been found to be greater thanthe disorder condensate depletion in each species becausethe bosons scattered by the disorder environment providerandomly distributed obstacles for the motion of the su-perfluid. The results obtained by Huang and Meng andthe perturbation theory in a single BEC for the fluctua-tions of the condensate and of the superfluid density dueto the disorder have been well-recovered.Strictly speaking, in the regime of a strong disorder,each component fragments into a number of low-energy,localized single-particle states with no gauge symmetrybreaking forming the so-called Bose glass phase. Theexploration of such a regime would need either a non-perturbative approach or Quantum Monte Carlo simula-tions. We believe that the findings of this work add extrarichness to the diversity of disordered ultracold atoms.They open up a new avenue for controlling phase sep-aration of Bose-Bose mixtures. Finally, an importantextension of this work would be to analyze effects of aweak disorder in a mixture droplet state.
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