Bose-Einstein Condensation on Curved Manifolds
Natália S. Móller, F. Ednilson A. dos Santos, Vanderlei S. Bagnato, Axel Pelster
BBose-Einstein Condensation on Curved Manifolds
Nat´alia S. M´oller
Physics Department and Research Center Optimas, Technische Universit¨atKaiserslautern, 67663 Kaiserslautern, Germany, and Departamento de F´ısica,Universidade Federal de Minas Gerais, 31270-901 Belo Horizonte, MG, BrazilE-mail: [email protected]
F. Ednilson A. dos Santos
Departamento de F´ısica, Universidade Federal de S˜ao Carlos, 13565-905, S˜ao Carlos,SP, Brazil
Vanderlei S. Bagnato
Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo, 13560-550 S˜ao Carlos,SP, Brazil, and The Hagler Institute for Advanced Studies at Texas A&M - USA
Axel Pelster
Physics Department and Research Center Optimas, Technische Universit¨atKaiserslautern, 67663 Kaiserslautern, GermanyAugust 2020
Abstract.
Here we describe a weakly interacting Bose gas on a curved smoothmanifold, which is embedded in the three-dimensional Euclidean space. To this end westart by considering a harmonic trap in the normal direction of the manifold, whichconfines the three-dimensional Bose gas in the vicinity of its surface. Following thenotion of dimensional reduction as outlined in [L. Salasnich et al., Phys. Rev. A ,043614 (2002)], we assume a large enough trap frequency so that the normal degreeof freedom of the condensate wave function can be approximately integrated out. Inthis way we obtain an effective condensate wave function on the quasi-two-dimensionalsurface of the curved manifold, where the thickness of the cloud is determined self-consistently. For the particular case when the manifold is a sphere, our equilibriumresults show how the chemical potential and the thickness of the cloud increase withthe interaction strength. Furthermore, we determine within a linear stability analysisthe low-lying collective excitations together with their eigenfrequencies, which turn outto reveal an instability for attractive interactions.
1. Introduction
Bose-Einstein condensates (BECs) are today among the most explored many-bodyquantum systems. Ranging from more simple thermodynamics [1, 2] to the more a r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug ose-Einstein Condensation on Curved Manifolds ose-Einstein Condensation on Curved Manifolds
2. Differential Geometrical Preliminaries
Due to the fact that in this work we pursue a more general approach to the problem ofBose-Einstein condensation in special geometries, it is necessary to introduce relevantaspects of differential geometry that are used later on. To this end we consider M tobe a smooth manifold [36] to which the Bose gas is confined. The way to describe amanifold is not uniquely defined, i.e., there are a lot of coordinate systems that couldbe used. Furthermore, it is often not possible to describe the whole manifold withonly one coordinate system. In those cases we would have a local coordinate systemfor each portion of the manifold and we should put all these pieces together to havea global description. The way of choosing these portions and their number depend onthe respective manifold and on the chosen coordinate systems. Anyhow, we can assurethat this number can be chosen to be finite for compact manifolds, such as spheresand ellipsoids. For not compact manifolds it could be that this number of portions isinfinite, but it can always be chosen to be countable. For simplicity, we suppose in thefollowing that the manifold is described by only one coordinate system. Even though itis not true for many examples, it will not affect our main result. Furthermore, all thearguments presented here could be straightforwardly reproduced for the case of needingto describe the manifold with more than one piece. ose-Einstein Condensation on Curved Manifolds M , where all the points can be characterizedby two real variables ( x , x ) belonging to an open subset of the real plane IR . So wecan denote the points on this manifold by the vector p ( x , x ). Let v ( x , x ) be a unitnormal vector to the manifold at the point p ( x , x ). Now, we describe the neighbouringpoints, which do not belong necessarily to M , by q ( x , x , x ) = p ( x , x ) + x v ( x , x ) . (1)Note that the points belonging to M are described by x = 0 and that ones, whichdo not belong to M , are described by x (cid:54) = 0. Furthermore, fixing any constant valuefor x , a locally parallel manifold to M is defined, which we denote by M ( x ). Inparticular, M (0) simply coincides with the manifold M .One detail to be pointed out is that equation (1) only represents a local description,so, in principle, x cannot be arbitrarily large. We consider that the Gaussian normalcoordinate system is well-defined in the interval | x | < R/
2, where R is the minimum,over all points p belonging to M , of the smallest curvature radius of each point p . Inorder to be more precise, for any fixed point p we denote by R ( p ) and R ( p ) the twoprincipal curvature radii of the manifold at that point. Choose as R ( p ) the minimumbetween these two radii, i.e., R ( p ) = min { R ( p ) , R ( p ) } . (2)Then, define R as the minimum value of R ( p ) over the whole manifold M , i.e., R = min p ∈M R ( p ) . (3)For the special case of a sphere, its radius coincides with R . Due to global properties, itcould be for some manifold that the Gaussian normal coordinate system describes twicethe same point. We exclude such manifolds in the following, restricting ourselves onlyto manifolds where this situation does not occur.Now we present a heuristic recipe for deriving equation (1). To this end, we supposeto know the manifold equation, which determines the points p belonging to a manifoldportion as a function of x and x , that is p = p ( x , x ). In order to find the respectivetangent vectors we evaluate v ( x , x ) = ∂ p ( x , x ) ∂x , v ( x , x ) = ∂ p ( x , x ) ∂x . (4)Then the unit normal vector to the manifold at p ( x , x ) is defined by the cross productbetween these tangent vectors and a subsequent normalization, i.e., v ( x , x ) = v ( x , x ) × v ( x , x ) | v ( x , x ) × v ( x , x ) | . (5)With this, equation (1) is well-defined and | x | = | ( q − p ) · v | defines the distance ofthe point q from the manifold. ose-Einstein Condensation on Curved Manifolds Figure 1.
Illustration of a portion of the manifold M . The Gaussian normalcoordinate system defines the points belonging to this portion via equation (1) bysetting x = 0. The vectors v and v are two tangent vectors and v ( x , x ) is aunit normal vector to the manifold at the point p ( x , x ). The neighbouring point q ( x , x , x ) does not necessarily belong to the manifold M , and the Gaussian normalcoordinate system defines this point via equation (1) by identifying x with the distancebetween q and the manifold. In the illustration, p represents the point belonging to M , which is the nearest to q . Now, we can define the tangent vectors to the surface M ( x ), which is locallyparallel to M , using an analogous procedure. From formula (1) we calculate thecorresponding tangent vectors as v i ( x , x , x ) = ∂ q ( x , x , x ) /∂x i , for i = 1 ,
2. Thenormal vector to the manifold M ( x ) at the point q ( x , x , x ) is then defined by v ( x , x , x ) = v ( x , x , x ) × v ( x , x , x ) | v ( x , x , x ) × v ( x , x , x ) | (6)and turns out to be v ( x , x , x ) = v ( x , x ). The latter statement can be seen fromthe property v ( x , x ) · v i ( x , x , x ) = 0, which follows due to ∂ v ( x , x ) ∂x i · v ( x , x ) = 12 ∂ [ v ( x , x ) · v ( x , x )] ∂x i = 0 , (7)for i = 1 ,
2. With that, a basis for the 3D space at each point q ( x , x , x ) is given bythe vectors v ( x , x , x ), v ( x , x , x ), and v ( x , x , x ). This allows us to define acovariant metric of the 3D space in the neighbourhood of the manifold M , since eachcomponent of the metric is defined by the scalar product between two respective basisvectors, i.e., G µν ( x , x , x ) = v µ ( x , x , x ) · v ν ( x , x , x ) , (8)where µ and ν range from 0 to 2. From the definition of the normal vector v in equation (5) and from the above discussion, we conclude both G ( x , x , x ) = v ( x , x ) · v ( x , x ) = 1 and G i ( x , x , x ) = G i ( x , x , x ) = v ( x , x ) · v i ( x , x , x ) = 0, for i = 1 ,
2, for all x , x and x where the coordinate system iswell-defined. With that and using the Gaussian normal coordinate system, we obtain ose-Einstein Condensation on Curved Manifolds M canbe represented by the matrix G µν ( x , x , x ) = g ij ( x , x , x ) , (9)for µ and ν ranging from 0 to 2, while i and j range only from 1 to 2. For x fixed, g ij ( x , x , x ) denotes the covariant metric of the manifold M ( x ). Each entry of thismetric represents the scalar product of the respective tangent vectors v ( x , x , x ) and v ( x , x , x ). For the special case x = 0, g ij (0 , x , x ) represents the metric of themanifold M and we use the abbreviated notation g ij ( x , x ).Note that the metric g ij ( x , x , x ) can be Taylor expanded around the metric g ij ( x , x ) and written in terms of other local properties of the manifold M , as issummarized in appendix A. There, it is also shown that calculating the square rootof the determinant of this metric yields (cid:113) det g ij ( x , x , x ) = (cid:113) (det g ) · (cid:34) O (cid:32) x R (cid:33) + O (cid:32) ( x ) R (cid:33) + ... (cid:35) , (10)for | x | < R/
2, where we have introduced the notation det g = det g ij ( x , x ). This resultturns out to be useful in the next sections. Note that we can assume in the followingwithout loss of generality that singular points, where det g vanishes, do not occur. Onepossibility would be the presence of a coordinate singularity, which occurs, for instance,at the north and south poles of a sphere, when a spherical coordinate system is used.But, as we comment explicitly below, such coordinate singularities at the poles of asphere turn out to have no physical consequence. Another possibility would be a non-coordinate system singularity, which includes, for instance, regions similar to the edgeof a cone. But such manifolds with a non-coordinate system singularity are discardedfrom our approach as they would correspond to a not smooth manifold. Basically, sucha real singularity in the metric involves a drastic change of the geometry, which has tobe analysed case by case in more detail within another study.We know that in many-body quantum theory the kinetic energy of a 3D BECis given by the Laplacian of the condensate wave function. A generalization of theLaplacian for the 3D space in a generalized coordinate system is called Laplace-Beltramioperator [39]. It is expressed by∆ LB = 1 √ det G ∂∂x η (cid:32) √ det G G ηκ ∂∂x κ (cid:33) , (11)with η and κ ranging from 0 to 2 and det G denoting the determinant of the covariantmetric. Note that we use the Einstein notation in (11), so that a summation over thesame co- and contravariant indices is implicitly assumed. For a flat 3D space describedby a Cartesian coordinate system, the metric is given by an Euclidean metric withthe Kronecker symbol as its respective components and the above formula recoversthe standard representation of the Laplacian. On the other side, still for the flat 3D ose-Einstein Condensation on Curved Manifolds ∂ ∂x + ∂∂x (cid:18) ln (cid:113) det g (cid:19) ∂∂x + ∆ M ( x ) , (12)with the abbreviation∆ M ( x ) = 1 √ det g ∂∂x i (cid:32)(cid:113) det g g ij ∂∂x j (cid:33) . (13)Note that (13) denotes also a Laplace-Beltrami operator, but this time in the contextof the manifold M ( x ) for each fixed value of x , and the indices i, j range from 1 to 2.In particular, when x = 0, the operator (13) represents the Laplace-Beltrami operatorof the manifold M , denoted in the following simply by ∆ M .
3. Normalization of Condensate Wave Function
Now that we have our mathematical objects well-defined, we introduce two importantphysical quantities, which we need to consider when dealing with a BEC on a manifold:the potential which confines the Bose gas to the manifold and a particular ansatz forthe condensate wave function.We suppose that the Bose gas is confined in the immediate vicinity of the manifold.Such a confinement could be realized, for instance, by a harmonic oscillator potential inthe normal direction to the manifold, which has its minimum on the manifold. In theGaussian normal coordinate system introduced in the previous section, this potentialhas the form V harm ( x ) = 12 M ω ( x ) . (14)Here the particle mass M and the frequency ω define a length scale in terms of theoscillator length σ osc = (cid:113) ¯ h/M ω , which represents the order of magnitude of thethickness of the Bose gas cloud surrounding the manifold. In the following we assumethat the frequency ω is so large that the oscillator length σ osc is much smaller than theminimum R of the respective local curvature radii on the manifold, i.e., σ osc (cid:28) R . Thiscorresponds to the physical situation that the Bose gas forms a thin shell around themanifold.In addition to this harmonic potential, we also allow the Bose gas to be affectedby a potential U depending on the manifold coordinates x and x . Thus, the totalpotential is given by V ( x , x , x ) = V harm ( x ) + U ( x , x ) . (15)Note that V harm ( x ) and U ( x , x ) are 3D potentials, even though each one does notdepend on all three variables.Let us now compute as a physical quantity the number of particles. Denoting thecondensate wave function Ψ( x , x , x ), the number of particles is given by an integralover the whole 3D space of its squared norm, i.e., N = (cid:82) dV | Ψ( x , x , x ) | . But since ose-Einstein Condensation on Curved Manifolds M , the integral becomes naturallyrestricted to the manifold neighbourhood N ( M ), yelding N = (cid:82) N ( M ) dV | Ψ( x , x , x ) | .We consider that this neighbourhood is defined by the points q ( x , x , x ) of the 3D spacedescribed by the Gaussian coordinate system according to equation (1) with | x | < R/ R (cid:29) σ osc . Thus, the number of particles in the gas can be expressed as N = (cid:90) R/ − R/ dx (cid:90) dx dx (cid:113) det g ( x , x , x ) | Ψ( x , x , x ) | . (16)In order to describe the confinement of the Bose gas in a thin shell, one is temptedto follow the arguments of the reference [35] and choose a trial wave function of theform Ψ( x , x , x ) = e − ( x σ ( x ,x √ π (cid:113) σ ( x , x ) · ψ ( x , x ) . (17)We plug this trial function into the above normalization integral (16) and expand theterm (cid:113) det g ( x , x , x ) in a power series using equation (10). After this expansion, weare able to approximately perform the integrals in the limit of large R , which leads toan exponentially small error O ( e − R /σ ) [40, (8.25)]. The integral over x of each termof the power series times a Gaussian is given by (cid:90) ∞−∞ dx ( x ) n e − ( x ) /σ √ πσ = ( n − σ n n/ , (18)for even non-negative integer values of n , while it vanishes for odd non-negative values of n . Therefore, only the even order terms survive and provide results at least of the orderof σ /R . Thus, the number of particles can be calculated through the normalization ofthe 2D function ψ ( x , x ) apart from a polynomial error, i.e., N = (cid:90) dx dx (cid:113) det g (0 , x , x ) | ψ ( x , x ) | + O (cid:32) σ R (cid:33) . (19)But now we argue that a better choice of the trial wave function is provided byΨ( x , x , x ) = e − ( x σ ( x ,x √ π (cid:113) σ ( x , x ) · ψ ( x , x , x ) , (20)where we have defined ψ ( x , x , x ) = φ ( x , x ) (cid:113) det g ( x , x , x ) . (21)For x = 0 we have that ψ (0 , x , x ) represents the 2D wave function of the gas. In orderto calculate the number of particles, we can follow the same procedure as the one above,but now we do not need to perform a Taylor series expansion, since the denominator ofthe term (21) matches with the one from the volume element. To perform the integral itis only necessary to approximately take the limit of large R , leading to an exponentiallysmall error. Thus, the number of particles is given by the integral of the squared normof the 2D wave function apart from only an exponentially small error: N = (cid:90) dx dx (cid:113) det g (0 , x , x ) | ψ (0 , x , x ) | + O ( e − R /σ ) . (22) ose-Einstein Condensation on Curved Manifolds φ ( x , x ) would also vanish at the poles, thus compensating the coordinate singularities.
4. Reducing Dimensionality
In the previous sections we introduced the necessary differential geometrical notationand the physical ideas on how to confine a weakly interacting Bose-Einstein condensatein the neighbourhood of a manifold. Now we proceed with the description of this systemby considering its grand-canonical energy E = (cid:90) dV Ψ ∗ (cid:32) − ¯ h M ∆ + M ω x ) + U ( x , x ) + 12 g int | Ψ | − µ (cid:33) Ψ , (23)where g int = 4 π ¯ h a s /M denotes the interaction strength, determined by the s -wavescattering length a s , and µ is the chemical potential of the system. Since the confinementfrequency ω is supposed to be large enough, we can follow the same procedure as in thelast section and perform this integral only on the manifold neighbourhood N ( M ). Then,the energy (23) is well approximated by E = (cid:90) N ( M ) dx dx dx (cid:113) det g Ψ ∗ (24) · (cid:32) − ¯ h M ∆ + M ω x ) + U ( x , x ) + 12 g int | Ψ | − µ (cid:33) Ψ . Inserting the trial function (20), (21) into the energy functional (24), we expand it intoa Taylor series with respect to x and perform the resulting integral with respect to x approximately in the limit of large R as in section 3.Note that the Laplacian (12) with the trial function (20), (21) reads explicitly∆Ψ = (cid:40)(cid:34) (cid:32) ( x ) σ − σ (cid:33) − (cid:32) ∂ ln √ det g∂x (cid:33) − ∂ ln √ det g∂x (25)+ g ij ( ∂ i σ )( ∂ j σ )2 σ (cid:35) ψ ( x , x , x ) + ∆ M ( x ) ψ ( x , x , x ) (cid:41) e − ( x σ √ π √ σ . Approximately integrating the energy (24) with respect to x , we obtain E = (cid:90) N ( M ) dx dx (cid:113) det g ψ ∗ (cid:40) − ¯ h M ∆ M ( x ) − ¯ h g ij M ( ∂ i σ )( ∂ j σ ) σ − µ (26)+ ¯ h M (cid:34) (cid:32) ∂ ln √ det g∂x (cid:33) + ∂ ln √ det g∂x (cid:35) + ¯ h M σ + M ω σ U + g int | ψ | √ π σ (cid:41) ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 . ose-Einstein Condensation on Curved Manifolds ψ ∗ and σ . In the first case we obtain the time-independent 2D Gross-Pitaevskiiequation µψ = (cid:32) − ¯ h M ∆ M − ¯ h g ij M ( ∂ i σ )( ∂ j σ ) σ + V eff + U + ¯ h M σ + M ω σ g | ψ | (cid:33) ψ, (27)where ψ = ψ ( x = 0 , x , x ) denotes the 2D wave function, while ∆ M is given inequation (13) in the case when x = 0, and V eff ( x , x ) = ¯ h M (cid:34) (cid:32) ∂ ln √ det g∂x (cid:33) + ∂ ln √ det g∂x (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 + ... (28)represents an effective potential due to the non-trivial metric. Moreover, g = g int / ( σ √ π ) turns out to be the 2D interaction parameter, which gets larger for smallervalues of the width σ . It shows that a strong confinement leads to stronger effectivetwo-particle interactions on the manifold.Extremizing instead the energy (26) with respect to the cloud width σ yields¯ h M σ − M ω σ g int | ψ | √ πσ + ¯ h M (∆ M σ ) σ − ¯ h M g ij ( ∂ i σ )( ∂ j σ ) σ + ¯ h M g ij ( ∂ i σ ) σ ( ∂ j | ψ | ) | ψ | = 0 . (29)Thus, in equilibrium, one has to solve both equations (27) and (29) for ψ and σ bytaking into account the particle number in equation (22). Note that our results (27)and (29) for a general manifold contain the corresponding ones for a plane, which werealready treated in reference [35] for a constant width σ .
5. Equilibrium on a Sphere
The simplest case to be studied is the ground state of a sphere with radius R . Forsimplicity, we suppose that we do not have any external potential, i.e., U ( x , x ) = 0.Furthermore, from (28) we conclude that the effective potential V eff vanishes for such asphere. Due to the rotational symmetry of the sphere, the gas in the ground state isdescribed by a uniform distribution, thus the 2D wave function ψ satisfies ∆ M ψ = 0.Moreover, from the normalization (22) we obtain ψ = N πR . (30)With this the time-independent 2D Gross-Pitaevskii equation (27) reduces to analgebraic equation for the chemical potential µ , i.e., the equation of state µ ¯ hω = 14 (cid:32) σ σ + σ σ (cid:33) + P σ osc σ , (31)where P = a s σ osc N √ π R , (32) ose-Einstein Condensation on Curved Manifolds P can be tuned by changingthe particle number N , the s -wave scattering length a s , as well as by changing theoscillator length σ osc or the radius R .Correspondingly, for a sphere also the ground state thickness σ is uniform, so (29)reduces to σ σ = 1 + P σ σ osc . (33)In figure 2 we plot the results for the dimensionless Gaussian width σ /σ osc and thedimensionless chemical potential µ/ ¯ hω as functions of the dimensionless interactionstrength P .The width of the Gaussian is positive for any value of P . It coincides with theharmonic oscillator length σ osc for vanishing interactions and increases for repulsiveinteraction strengths ( P > σ /σ osc = √ P . For attractive interaction strengths ( P < | P | increases, tending to zero with an asymptotic behaviour of the form σ /σ osc = − /P .The dimensionless chemical potential coincides with 1 / P . Its asymptotic behaviour for large positive valuesof P is µ/ ¯ hω = (5 / P / . It reaches zero at about P ≈ − .
44 and turns to be negativefor smaller values of P . Its asymptotic behaviour for large negative values of P is givenby µ/ ¯ hω = − P / N = 10 rubidium atoms on a sphere with a radius about R = 10 µ m and a harmonictrap such that the harmonic oscillator length is of the order of σ osc = 1 µ m [41]. As the s -wave scattering length a s is about 100 times the Bohr radius, we obtain a dimensionlessinteraction of about P = 2 .
1. For later figures we always use those experimentallyrealistic parameters.Note that our equilibrium results recover the corresponding ones for an infiniteplane in the limit of an infinite curvature radius, i.e., R → ∞ . To this end we just haveto redefine equations (30) and (32) as ψ = ρ and P = 2 √ πa s σ osc ρ by introducing theparticle density ρ of the plane.
6. Dynamics on a Curved Manifold
In the last section, we calculated equilibrium results for a gas confined on a sphere,finding that the cloud has a positive width for all interactions strengths. In order todetermine the stability of the system, we now embark upon a linear stability analysis.To this end, we extend the equilibrium consideration of section 4 and treat the Bose gasdynamically. ose-Einstein Condensation on Curved Manifolds Figure 2.
Dimensionless width σ /σ osc and chemical potential µ/ ¯ hω as functions ofdimensionless interaction strength P . We begin a dynamical analysis deriving the temporal evolution of the 2D wavefunction and of the cloud width. To do that, instead of the energy, we use thecorresponding action S = (cid:90) dt (cid:90) dV Ψ ∗ (cid:32) i ¯ h∂ t + ¯ h M ∆ − M ω x ) − U ( x , x ) − g int | Ψ | (cid:33) Ψ , (34)and a more general ansatz than that of equation (20), which includes an imaginarywidth term in the Gaussian exponent, according to [42, 43]Ψ( x , x , x , t ) = exp (cid:104) − ( x ) (cid:16) σ ( x ,x ,t ) + iB ( x , x , t ) (cid:17)(cid:105) √ π (cid:113) σ ( x , x , t ) · ψ ( x , x , x , t ) , (35)with ψ ( x , x , x , t ) = φ ( x , x , t ) (cid:113) det g ( x , x , x ) . (36)Here B represents the variational parameter conjugated to the cloud width, which isnecessary to be included in order to properly describe the dynamics of the system.Using the same procedure as in the last section, we insert this ansatz intoequation (34) and integrate approximately the variable x . With this we get thefollowing expression for the action: S = (cid:90) N ( M ) dtdx dx √ gψ ∗ (cid:34) i ¯ h∂ t ψ + ¯ h M (∆ M ( x ) ψ ) + ¯ hσ ∂ t B ) ψ − ¯ h M σ B ψ − ¯ h M g ij (cid:32) ( ∂ i σ )( ∂ j σ )2 σ ψ + 3 σ
16 ( ∂ i B )( ∂ j B ) ψ + iσ ∂ i σ )( ∂ j B ) ψ + iσ ∂ i B )( ∂ j ψ ) (cid:33) − i ¯ h σ M (∆ M B ) ψ − V eff ψ − ¯ h M σ ψ − M ω σ ψ − g int | ψ | √ πσ ψ (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 . (37)Following the standard approach, we extremize the action (37) with respect to ψ ∗ , σ and B . In this way, we obtain at first the evolution equation for ψi ¯ h∂ t ψ = − ¯ h M (∆ M ψ ) − ¯ hσ ∂ t B ) ψ + ¯ h M σ B ψ + i ¯ h σ M (∆ M B ) ψ ose-Einstein Condensation on Curved Manifolds
13+ ¯ h M g ij (cid:32) ( ∂ i σ )( ∂ j σ )2 σ ψ + 3 σ
16 ( ∂ i B )( ∂ j B ) ψ + iσ ∂ i σ )( ∂ j B ) ψ + iσ ∂ i B )( ∂ j ψ ) (cid:33) + V eff ψ + ¯ h M σ ψ + M ω σ ψ + g int | ψ | √ πσ ψ, (38)as well as the corresponding equation for σM ω ¯ h σ = 1 + g int M σ | ψ | ¯ h √ π + σ ∆ M σ + M σ ¯ h ∂ t B − σ B − g ij ( ∂ i σ )( ∂ j σ ) + g ij σ ( ∂ i σ ) ∂ j | ψ | | ψ | (39) − σ g ij ( ∂ i B )( ∂ j B ) + i σ g ij ( ∂ i B )( ψ∂ j ψ ∗ − ψ ∗ ∂ j ψ ) . A subsequent extremization of the action with respect to B yields B = − M ¯ h ∂ t σσ − M h ∂ t | ψ | | ψ | + 3 σ M B + i (cid:32) ∆ M ψψ − ∆ M ψ ∗ ψ ∗ (cid:33) (40)+ i g ij ∂ i σσ (cid:32) ∂ j ψψ − ∂ j ψ ∗ ψ ∗ (cid:33) + 3 σ g ij ( ∂ i B ) ∂ j | ψ | | ψ | + 3 σ g ij ∂ i σσ ∂ j B. We remark that in the above three equations (38)–(40), the value of the first variableis fixed x = 0, such that ψ stands for ψ (0 , x , x , t ). These equations describe thedynamics of a Bose gas on a curved manifold by determining the evolution of the 2Dwave function, as well as the real and imaginary cloud width self-consistently.
7. Collective Modes of BEC on a Sphere
Collective modes of a condensate are of great value for experimental studies, since theyallow a quantitative characterization of the underlying system, even when an opticalabsorption projection is made in the data collection. In addition, collective modes areassociated with the equilibrium state around which they occur. Being able to analysethe collective modes of a confined condensate creates the possibility of understandingthe influence of various system parameters on the hydrodynamics of the system.In order to determine the low-lying collective modes, we now study smallperturbations of the ground state for a Bose gas confined on the surface of a sphereof radius R . To this end, we perform a linear stability analysis of the evolutionequations derived in the previous section. Note that the effective potential V eff ( x , x )in equation (38) vanishes for the case of a sphere. We suppose a small perturbation ofthe ground state in the form ψ = ( ψ + δψ ) e − iµt/ ¯ h (41) σ = σ + δσ (42) B = δB, (43)where ψ is given in equation (30), the chemical potential µ follows from equation (31),and σ is defined via equation (33). ose-Einstein Condensation on Curved Manifolds δψ , δσ and δB , we obtain i ¯ h ( ∂ t δψ ) + ¯ hσ ∂ t δB ) = 12 M R ( L δψ ) + g ψ ( δψ + δψ ∗ ) − iσ hM R ( L δB ) (44)+ (cid:32) − ¯ h M σ + M ω σ − g ψ σ (cid:33) ψ δσ,M σ ¯ h ∂ t δB = − g M σ ψ ¯ h ( δψ + δψ ∗ ) + (cid:32) σ ¯ h R L + 4 M ω σ ¯ h − g M σ ψ ¯ h (cid:33) δσ, (45) M h ∂ t ( δψ + δψ ∗ ) ψ + M ¯ h ∂ t δσσ = − i h R ψ L ( δψ − δψ ∗ ) − (cid:32) σ h R L + 1 (cid:33) δB. (46)Here we have used that for a sphere the Laplace-Beltrami operator is proportional tothe square of the angular momentum operator L via∆ M = − L ¯ h R . (47)Note that the eigenvalues of L are given by ¯ h l ( l +1), for l = 0 , , , ... being the angularmomentum quantum numbers and its eigenfunctions are proportional to the sphericalharmonics Y lm , with m = 0 , ± , ..., ± l .The technical details on how to solve equations (44)–(46) are relegated toappendix B. There it is shown that these equations can be straight-forwardly solvedby decomposing the functions δψ , δσ , δB for all l = 0 , , ... in terms of ¯ Y lm = Y lm + Y ∗ lm for m = 0 , ..., l and proportional to ¯ Y lm = − i ( Y lm − Y ∗ lm ) for m = − l, ..., −
1. Here werestrict ourselves to summarize and discuss the respective results.For l = 0, the collective oscillation mode frequency is given byΩ = ω (cid:115) σ σ , (48)which is of the order of the transversal confinement frequency ω . It coincides with theformula obtained in reference [25], where the mode associated to this frequency wascalled accordion mode .For l ≥
1, irrespective of the sign of the dimensionless interaction strength P wefind two branches of collective oscillation mode frequencies, a larger one and a lowerone, but degenerate with respect to the magnetic quantum number m . The branch withlarger oscillation frequencies is approximately given byΩ l = Ω + ω Ω (cid:32) σ σ + 7 σ σ + 7 σ σ (cid:33) δ l , (49)while the branch with smaller oscillation frequencies reads approximatelyΛ l = ω Ω (cid:34) P σ osc σ (cid:32) σ σ (cid:33) δ l (50)+ ω Ω (cid:32) − σ σ + 454 + 11 σ σ + 7 σ σ + 3 σ σ (cid:33) δ l (cid:35) / . ose-Einstein Condensation on Curved Manifolds (a) (b)(c) (d) Figure 3.
Frequencies (48)–(50) as functions of dimensionless interaction strength P .(a) Higher branch frequency Ω l /ω for l = 0 , , ,
3. (b) The same graphic as in (a),but plotted for an enlarged scale. (c) Lower branch frequency Λ l /ω for l = 1 , ,
3. (d)The same graphic as in (c), but plotted around P = 0. Here δ l = σ R l ( l + 1) (51)represent smallness parameters, since the width σ osc is supposed to be much smallerthan the radius R of the sphere. As discussed at the end of section 5, we consider σ /R = 0 .
01 to be realistic for a bubble trap in microgravity.Note that also our dynamical results recover the corresponding ones for an infiniteplane in the limit of an infinite curvature radius, i.e. R → ∞ . To this end thesmallness parameter (51) has just to be redefined via σ ( k x + k y ), where k x , k y denotethe components of a 2D wave vector. This means that the collective frequencies (49)and (50) still hold but represent each a continuous spectrum above a ground frequency.The latter is given in case of the upper branch (49) by the minimal value Ω given inequation (48), whereas it vanishes for the lower branch (50).A plot of the frequencies Ω l and Λ l as functions of the dimensionless interactionstrength P for the lowest values of l can be found in figure 3. These graphics are madefor the realistic range of P values, according to the parameters given at the end ofsection 5.From equations (33) and (48) we read off that Ω /ω equals to 2 for vanishing ose-Einstein Condensation on Curved Manifolds √ P , as can be seen in panels 3(a) and (b). For l = 1 , ,
3, the frequencies Ω l /ω have a similar behaviour as Ω /ω , but they turn out to be larger than Ω /ω . Fromthe plots we can also see that frequencies Ω l /ω increase with the angular momentumquantum number l .The dimensionless frequencies Λ l /ω are positive for vanishing interaction, eventhough they are much smaller than 1. The frequencies Λ l /ω reach zero for somenegative value of P and monotonically increase with P , see figure 3(c) and (d). Fromequation (50) we see that these frequencies decrease for a smaller value of the parameter δ l . On the other hand, for some negative value of the dimensionless interaction strength P the lower frequencies could become imaginary, such that the corresponding solutionexhibit an exponential behaviour. We stress that only for quite small negative valuesof P we still have a stable solution, as can be seen in figure 3(d), and they turn out tobe unstable as soon as P is decreased even to relatively moderate values. Note that for l = 0, the lower frequency is not defined due to the conservation of the particle number,as is discussed in more details in appendix B.
8. Analysis of Modes
Within a linear stability analysis of small perturbations on a sphere, we have derivedanalytic expressions for two types of collective oscillation mode frequencies Ω l and Λ l ,as well as understood their dependences on the angular momentum quantum number l and on the dimensionless interaction strength P . Now, we analyse the respectivedensity profiles of these oscillations on the sphere, whose calculations are relegatedto appendix B. We first discuss the accordion mode, which occurs for the angularmomentum quantum number l = 0, and afterwards we analyse the modes for largerangular momentum quantum numbers l , and also illustrate some examples. Finally,we discuss in detail the direction of the oscillations, which can be in the confinementdirection, along the surface of the sphere, or even have a mixed behaviour.For l = 0 only the mode with frequency (48) appears. In this case, the temporalevolution of each component of the wave function associated with the frequency Ω turnsout to be δψ ( t ) = iC ψ ω Ω (cid:32) P σ σ + σ σ osc (cid:33) sin(Ω t ) ¯ Y ,δσ ( t ) = C σ osc cos(Ω t ) ¯ Y , (52) δB ( t ) = C Ω σ osc σ ω sin(Ω t ) ¯ Y , where C is a proportionality constant defined by the intensity of the perturbation,which has to be small. Note that we have Re δψ = 0 in order to satisfy the conservationof the particle number, see appendix B for further details.To illustrate the accordion mode, the evolution of its density profile given by thesquared norm | Ψ ( r, θ, ϕ, t ) | of the 3D wave function (35) is pictured in figure 4(a)–(c) ose-Einstein Condensation on Curved Manifolds (a) (b) (c) (d) Figure 4.
Accordion mode oscillation: density profile | Ψ( r, θ, ϕ, t ) | at (a) t = 0, (b) t = π/ and (c) t = π/ Ω , in x – z plane. (d) Radial density profiles of (a) (in orangedashed), (b) (in blue continuous), and (c) (in green dotted), for fixed ϕ = 0 and θ = 0.Here r , θ and ϕ denote the spatial variables in terms of the spherical coordinates. at different times in x – z plane, for the chosen parameters at the end of section 5. Theproportionality constant is chosen to be C = 0 .
1. The Gaussian width of the statein (b) coincides with the one of the equilibrium state, while the width of the statesin (a) and in (c) are, respectively, larger and smaller than the one of the equilibriumstate. From the initial state in (a) which has the largest width, the system evolves tothe state shown in (b) and finally reaches the state with the smallest width in (c). Thenit returns to the state in (b), to the state in (a), and so on. This oscillation happenswith frequency Ω . The radial density profile of these three stages for fixed ϕ = 0 and θ = 0 are plotted in figure 4(d). The thinnest stage in green shows a higher peak andthe thickest in orange shows a smaller one. This happens since the number of particles N is conserved.For the cases where l ≥
1, there are two types of oscillation frequencies Ω l and Λ l ,given in equations (49) and (50), respectively. For Ω l , the temporal evolution of theassociated components of the wave function are given by δψ lm ( t (cid:48) ) = C lm ψ (cid:40) − P σ ω σ Ω δ l cos(Ω l t )+ iω Ω (cid:34) σ σ osc + P σ σ + P ω (cid:32) − σ σ − σ σ (cid:33) δ l (cid:35) sin(Ω l t ) (cid:41) ¯ Y lm ,δσ lm ( t (cid:48) ) = C lm σ osc cos(Ω l t ) ¯ Y lm , (53) δB lm ( t (cid:48) ) = C lm Ω σ osc σ ω (cid:34) − P ω Ω (cid:32) σ σ + 5 σ σ (cid:33) δ l (cid:35) sin(Ω l t ) ¯ Y lm . where C lm is a proportionality constant. For Λ l the solutions are δψ lm ( t ) = C lm ψ (cid:32)
12 cos(Λ l t ) − i Λ l ωδ l (cid:32) − P σ osc ω σ Ω δ l (cid:33) sin(Λ l t ) (cid:33) ¯ Y lm ,δσ lm ( t ) = C lm P σ ω σ Ω (cid:40) (cid:34) − σ ω σ Ω + P σ osc ω σ Ω (cid:32) σ σ (cid:33)(cid:35) δ l (cid:41) cos(Λ l t ) ¯ Y lm , (54) δB lm ( t (cid:48) ) = C lm P σ osc Λ l ωσ Ω (cid:40) (cid:34) − σ ω σ Ω + P σ osc ω σ Ω (cid:32) σ σ (cid:33)(cid:35) δ l (cid:41) sin(Λ l t ) ¯ Y lm , with C lm also being a proportionality constant. ose-Einstein Condensation on Curved Manifolds (a) (b) (c) (d) Figure 5.
Illustration of condensate density | Ψ( r, θ, ϕ, t ) | for l = 1 and l = 2, with m = 0, at t = 0 (top row) and t = π/ Ω l (bottom row). (a) Density profile for l = 1in x – z plane. (b) Density on surface of the sphere for l = 1, i.e, | Ψ( R, θ, ϕ, t ) | . (c)Profile for l = 2 in x – z plane. (d) Density on surface of the sphere for l = 2. We illustrate the density profiles of these modes in figure 5 for l = 1 and l = 2,with m = 0. Figures 5(a) and (c) show the density profile | Ψ( r, θ, ϕ, t ) | in the x – z plane. Figures 5(b) and (d) show the condensate density on the surface of the sphere,i.e., | Ψ( R, θ, ϕ, t ) | . The proportionality constants are chosen to be C = C = 0 . σ lm ( t ) = σ + δσ lm ( t ) and the density of the 2D wavefunction | ψ lm ( t ) | = | ψ | + 2 | ψ | Re δψ lm ( t ) turn out to oscillate in time.From panels 5(a) and (c) we read off that the regions on the sphere where thedensity maxima are located have the minimal width. The oscillations of δσ changethe shape of the Gaussian, similarly to what happens in figure 4(d). The differenceis that for the accordion mode this happens in a spherically symmetric way, while forlarger values of l there is an angular dependence, as is illustrated in figures 5(a) and(c). Oscillations of Re δψ do not change the width of the Gaussian nor its shape, andonly the amplitude of the wave function is changed. For l = 0 such oscillations do notoccur, because they would change the number of particles in the system. For l ≥ l given in equation (53), the amplitude of δσ lm is much larger than the amplitude of the real part of δψ lm , since the latter one isproportional to δ l , which is small. In this sense, we say that the density oscillationspredominantly occur in the direction perpendicular to the sphere. For the modescorresponding to Λ l given in equation (54), we see that the oscillations of Re δψ lm havea fixed amplitude, while the amplitude oscillations of δσ lm depend on the interaction ose-Einstein Condensation on Curved Manifolds Figure 6.
Normalized vectors in the direction of ( | Re δψ lm | /ψ , δσ lm /σ osc ) for variousvalues of l and P , for the modes corresponding to the frequencies Ω l (in black) and Λ l (in blue). (a) Fixed l = 1 and P = 0 , ...,
5. Note that all black arrows practically lieon y axis, and that the blue arrow corresponding to P = 0 lies on the x axis. The bluevectors increase their angle with the horizontal for increasing P . (b) Fixed P = 2 . l = 0 , ...,
10 for the larger frequencies Ω l and l = 1 , ...,
10 for the lower frequenciesΛ l . Both black and blue vectors increase their angle with the vertical for increasing l . strength P . This dependence occurs explicitly and implicitly, since both σ and Ω depend on P according to (33) and (48). If P is zero, the amplitude oscillations of δσ lm vanish, so we can say that the mode is in a parallel direction to the sphere. If P increases, the amplitude oscillations of δσ lm increase and it can even be comparableto the amplitude oscillations of Re δψ lm . In this sense, we say that the direction ofoscillations turns out to have a mixed behaviour with both the parallel and perpendicularcomponents, which we call a diagonal oscillation.In order to illustrate the above statements, in figure 6 we plot the normalizedvectors in the direction of the vectors ( | Re δψ lm | /ψ , δσ lm /σ osc ) for various values of l and P in order to illustrate the difference in the contributions of Re δψ lm and δσ lm in themodes associated to the frequencies Ω l and Λ l , respectively. Note that the amplitudeof the collective modes have a degeneracy on the values of m for l fixed. Moreover, thedirections of these vectors coincide with the notion given above, of oscillations being ina direction perpendicular, parallel or diagonal to the sphere.In panel 6(a) the angular momentum quantum number l = 1 is fixed and wevary the dimensionless interaction strength P from 0 to 5. We see that the amplitudeoscillations for the modes with larger frequency almost do not change, while theamplitude oscillations for the modes corresponding to the smaller frequency are quitesensitive to the values of P . The amplitude oscillation that is parallel to the sphere for P = 0 turns to be diagonal as P increases.In panel 6(b) the dimensionless interaction strength P = 2 . l . For the modes corresponding to Ω l , l isvaried from 0 to 10, while for the modes corresponding to Λ l , l is varied from 1 to 10, ose-Einstein Condensation on Curved Manifolds l = 0. We see that for both branches the vectors increasetheir angle with the vertical for increasing l , but this is more pronounced for the modesassociated with the frequencies Λ l .From figure 6 we see that the modes corresponding to the Ω l branch predominantlyoscillate in the perpendicular direction, with a negligible dependence on P and a smalldependence on l . On the other hand, the modes from the branch corresponding toΛ l depend on both P and l , but have a dominant parallel component. With this,we conclude that the oscillations with higher frequencies are in the direction of theconfinement trap, while the oscillations with smaller frequencies occur along the sphere,i.e., in the not confined direction.
9. Conclusions
Motivated by recent experimental advances in the field, we have studied both the staticand the dynamic properties of a Bose-Einstein condensate confined on the surface of acurved manifold. To this end, we provided a general formulation of the problem andderived a self-consistent set of equations for the 2D condensate wave function on themanifold and its width. In particular, we found an effective potential in the resulting 2DGross-Pitaevskii equation, which depends on the metric of the manifold in a non-trivialway but vanishes for a sphere. For the latter special case we determined in equilibriumhow both the width of the condensate and its chemical potential increase with therepulsive interaction strength. Moreover, we found via a linear stability analysis twobranches of collective excitations, with distinctly different frequencies. The larger branchfrequencies turned out to be of the order of the harmonic confinement frequency and,thus, correspond to oscillations predominantly in the direction of the confinement, i.e, inthe perpendicular direction to the sphere. The lower branch frequencies are much smallerand represent oscillations predominantly on the sphere. Our results represent concretepredictions, which presumably could be confirmed in upcoming bubble trap experimentsin the NASA Cold Atom Laboratory at the International Space Station [24]. But inorder to become experimentally more realistic it would be necessary to deal with a Bosegas confined on an ellipsoid as it represents a better approximation to the bubble trap,than the sphere [17, 23].Note that the collective modes analysed in this paper differ from the ones identifiedin reference [33]. This is due to the fact that our ansatz (35), (36) for the wave functionneglects the possibility that the condensate levitates below or above the minimum ofthe confinement potential.
10. Acknowledgments
The authors thank Antun Balaˇz for valuable suggestions at all the stages of the project,as well as Arnol Garcia, Aristeu Lima, H´el`ene Perrin, Milan Radonji´c, Luca Salasnich,Enrico Stein, and Andrea Tononi for useful discussions. ose-Einstein Condensation on Curved Manifolds [1] Pitaevskii L and Stringari S 2016
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Amer. J. Math. Appendix A. The Fundamental Forms of a Manifold and the GaussianNormal Coordinate System
There are two fundamental forms [44] which are used to characterize a 2D manifold M embedded in the 3D space. The first fundamental form is simply the metric g ij of themanifold M , as discussed in section 2. This is a natural instrument constructed to treatlengths of curves, areas of regions as well as other metric quantities, and its expressionis given by g ij = v i · v j , (A.1)where v and v are defined in equation (4). Note that this is a symmetric form.The second fundamental form is constructed to deal with changes of the normalvector along a path on the surface, providing information about its curvature, and isdefined by s ij = v i · ∂ j v , (A.2)where v is the normal vector given by equation (5). Note that s ij is also a symmetricform, meaning that s ij = s ji . In order to see that, we start from the orthogonalityrelations v i · v = 0 and v j · v = 0, i.e., ∂ i p · v = 0 and ∂ j p · v = 0, for i, j = 1 , x j and the second with respect to x i we obtain ∂ ij p · v + v i · ∂ j v = 0 , (A.3) ∂ ji p · v + v j · ∂ i v = 0 . (A.4)From the symmetry of the second derivatives and (A.3), (A.4), we then conclude that v i · ∂ j v = v j · ∂ i v . Thus, equation (A.2) can be rewritten as s ij = ( v i · ∂ j v + v j · ∂ i v ),proving the above statement that s ij is symmetric. ose-Einstein Condensation on Curved Manifolds s · g − , which is known in the literature as the Gaussmap. The directions of its eigenvectors e and e are called principal directions , and thesedirections correspond to the minimal and maximal curvatures. The matrix representing s · g − is given in this basis by s · g − = (cid:32) κ κ (cid:33) , (A.5)where κ and κ denote the curvatures of the manifold in the direction of e and e ,respectively. The mean and the Gaussian curvatures of this manifold are defined by H = Tr( s · g − )2 = κ + κ , K = det( s · g − ) = κ κ . (A.6)Furthermore, there is a third fundamental form [45] defined as h ij = ∂ i v · ∂ j v . (A.7)It can be shown that this form turns out to be a combination of the previous two, viathe relation [45] h ij = − Kg ij + 2 Hs ij , (A.8)and the matrix which represents h · g − in the principal directions is given by h · g − = (cid:32) κ κ (cid:33) . (A.9)Now, let us consider a manifold M ( x ) parallel to the manifold M , as explained insection 2. The metric, i.e., the first fundamental form of this manifold M ( x ), for | x | < R/
2, is defined by g ij ( x , x , x ) = ∂ i q · ∂ j q (A.10)and from expression (1) we conclude g ij ( x , x , x ) = ( v i + x ∂ i v ) · ( v j + x ∂ j v ) . (A.11)Combining this equation with the definitions (A.1), (A.2), and (A.7), and taking intoaccount that the fundamental forms are symmetric, we read off that this metric ofthe manifold M ( x ) can be expressed in terms of the three fundamental forms of themanifold M as follows: g ij ( x , x , x ) = g ij + 2 x s ij + ( x ) h ij . (A.12)From the corresponding matrix forms (A.5) and (A.9), it is then possible to deducedet g ( x ) = det g (cid:104) x H + ( x ) (4 H + 2 K ) + 4( x ) HK + ( x ) K (cid:105) . (A.13)Thus, we obtaindet g ( x ) ≤ det g (cid:34) x R + 6( x ) R + 4( x ) R + ( x ) R (cid:35) , (A.14)where R is defined as the minimum mean radius over all points p belonging to M according to equation (3). The square root of this formula can be expanded in a Taylorseries, yielding equation (10) from the main text. ose-Einstein Condensation on Curved Manifolds Appendix B. Determining the Collective Modes
In this section we present the detailed calculations for the respective results of sections 7and 8. For the sake of simplicity, we will use a dimensionless form for our equations.To this end, we define the following dimensionless variables σ (cid:48) = σσ osc , B (cid:48) = σ B , ψ (cid:48) = ψψ , t (cid:48) = ωt, (B.1)and the dimensionless derivative operators ∂ t (cid:48) = ∂ t ω , L (cid:48) = L ¯ h . (B.2)With that, the dimensionless form of the linearized equations (44)–(46) reads i∂ t (cid:48) δψ (cid:48) + σ (cid:48) ∂ t (cid:48) δB (cid:48) = σ R L (cid:48) δψ (cid:48) + (cid:32) σ (cid:48) − σ (cid:48) (cid:33) ( δψ (cid:48) + δψ (cid:48)∗ ) (B.3)+ 12 (cid:32) σ (cid:48) − σ (cid:48) (cid:33) δσ (cid:48) − iσ (cid:48) σ R L (cid:48) δB (cid:48) ,∂ t (cid:48) δB (cid:48) = (cid:32) σ (cid:48) − (cid:33) ( δψ (cid:48) + δψ (cid:48)∗ ) + (cid:32) σ (cid:48) (cid:33) δσ (cid:48) σ (cid:48) + 1 σ (cid:48) σ R L (cid:48) δσ (cid:48) , (B.4)12 ∂ t (cid:48) ( δψ (cid:48) + δψ (cid:48)∗ ) + ∂ t (cid:48) δσ (cid:48) σ (cid:48) = − i σ R L (cid:48) ( δψ (cid:48) − δψ (cid:48)∗ ) − δB (cid:48) − σ (cid:48) σ R L (cid:48) δB (cid:48) . (B.5)The solution of these coupled equations can be written in terms of the decompositions δψ (cid:48) = ∞ (cid:88) l =0 l (cid:88) m = − l δψ (cid:48) lm , δσ (cid:48) = ∞ (cid:88) l =0 l (cid:88) m = − l δσ (cid:48) lm , δB (cid:48) = ∞ (cid:88) l =0 l (cid:88) m = − l δB (cid:48) lm . (B.6)At this point we could have chosen the fundamental solutions δψ (cid:48) lm , δσ (cid:48) lm and δB (cid:48) lm tobe proportional to Y lm . But if we had done that, the respective expansion coefficientswould be coupled. Instead, we define for all l = 0 , , , .. the function ¯ Y lm = Y lm + Y ∗ lm for m = 0 , ..., l , and ¯ Y lm = − i ( Y lm − Y ∗ lm ) for m = − , ..., − l and choose the followingfundamental solutions δψ (cid:48) lm = k lm ( t (cid:48) ) + ih lm ( t (cid:48) )2 ¯ Y lm , δσ (cid:48) lm = s lm ( t (cid:48) ) ¯ Y lm , δB (cid:48) lm = b lm ( t (cid:48) ) ¯ Y lm , (B.7)where the coefficients k lm , h lm , s lm and b lm are real functions of t (cid:48) for all l = 0 , , , ... and m = 0 , ± , ... ± l . With this, it is possible to decouple the equations of differentindices l and m , since ¯ Y ∗ lm = ¯ Y lm . Thus, the following equations turn out to be muchsimpler than they would have been with the standard decomposition.Now we insert the ansatz (B.7) into equations (B.3)–(B.5), by taking into accountthat L (cid:48) Y lm = l ( l + 1) Y lm . With this, we obtain that the real part of equation (B.3) isgiven by − ∂ t (cid:48) h lm σ (cid:48) ∂ t (cid:48) b lm = (cid:32) σ (cid:48) − σ (cid:48) + δ l (cid:33) k lm + 12 (cid:32) σ (cid:48) − σ (cid:48) (cid:33) s lm , (B.8) ose-Einstein Condensation on Curved Manifolds ∂ t (cid:48) k lm δ l h lm − σ (cid:48) δ l b lm . (B.9)Correspondingly, equation (B.4) reduces to ∂ t (cid:48) b lm = (cid:32) σ (cid:48) − (cid:33) k lm + (cid:32) σ (cid:48) + 3 σ (cid:48) + δ l σ (cid:48) (cid:33) s lm , (B.10)whereas equation (B.5) becomes ∂ t (cid:48) k lm ∂ t (cid:48) s lm σ (cid:48) = δ l h lm − (cid:32) σ (cid:48) δ l (cid:33) b lm . (B.11)Rearranging these equations they turn out to be of the form of a system of lineardifferential equations: ∂ t (cid:48) X lm ( t (cid:48) ) = Q l X lm ( t (cid:48) ) , (B.12)where we introduce the vector X lm ( t (cid:48) ) T = ( k lm ( t (cid:48) ) , h lm ( t (cid:48) ) , s lm ( t (cid:48) ) , b lm ( t (cid:48) )) , (B.13)and the matrix Q l = Q + δQ l with Q = − σ (cid:48) σ (cid:48) − σ (cid:48) + 5 σ (cid:48) − σ (cid:48) σ (cid:48) − σ (cid:48) + 3 σ (cid:48) (B.14)and δQ l = δ l − σ (cid:48) δ l − δ l δ l σ (cid:48)
00 0 0 − σ (cid:48) δ l δ l σ (cid:48) . (B.15)The general solution of this system is given by e Q l t (cid:48) X lm (0), where X lm (0) denotes theinitial condition.The problem of computing the exponential matrix e Q l t (cid:48) applied to some vector isstandard when the matrix Q l has four different eigenvectors. Let X i , for i = 1 , , , ose-Einstein Condensation on Curved Manifolds λ i , and X lm (0) to be written in theeigenbasis, i.e., X lm (0) = (cid:80) i =1 v i X i . Then the solution of (B.12) with initial condition X lm (0) is given by X lm ( t (cid:48) ) = (cid:80) i =1 v i e λ i t (cid:48) X i . This procedure is going to be used forsolving equation (B.12) when l ≥ l = 0 and P = 0 in thenext subsection.But when l = 0 and P (cid:54) = 0, it turns out that the matrix Q has only threeeigenvectors, meaning that the eigenbasis is incomplete. In this case one has to considergeneralized eigenvectors, as is shown in detail in the following subsection. Appendix B.1. Solving Differential Equation for l = 0To find the solution of (B.12) when l = 0, we have to compute e Q t (cid:48) X (0). Theeigenvalues of Q are λ ± = ± i Ω (cid:48) with multiplicity one and λ = 0 with multiplicitytwo. Here, Ω (cid:48) = Ω /ω and Ω is given in equation (48) of the main text. We discussnow separately the cases P = 0 and P (cid:54) = 0.When P = 0 the respective eigenvectors are X ± = ∓ i ∓ i , X = , X − = , (B.16)where X ± are associated to the eigenvalues ± i Ω (cid:48) and X ± correspond to the eigenvalue0. Then the initial condition is decomposed according to X (0) = v X + v X − + v X + v X − , (B.17)where v i , for i = 1 , , , X ( t (cid:48) ) = e Q t (cid:48) X (0)yields the solution X ( t (cid:48) ) = v e i Ω (cid:48) t (cid:48) X + v e − i Ω (cid:48) t (cid:48) X − + v X + v X − . (B.18)Let us consider now P (cid:54) = 0. In this case it turns out that the matrix Q has only threeeigenvectors. In order to have a basis of the four-dimensional vectorial space, we choosethe vectors X ± = ∓ i Ω (cid:48) (cid:32) P σ (cid:48) + 2 σ (cid:48) (cid:33) ∓ i Ω (cid:48) σ (cid:48) , X = , ¯ X = − σ (cid:48) P (cid:16) σ (cid:48) (cid:17) Ω (cid:48) σ (cid:48) Pσ (cid:48) , (B.19)where X ± are the eigenvectors associated to the eigenvalues ± i Ω (cid:48) , X is theeigenvector associated to the eigenvalue 0 and ¯ X is a generalized eigenvector with ose-Einstein Condensation on Curved Manifolds Q ¯ X = X . Remember that σ (cid:48) = 1 and Ω (cid:48) = 2 when P = 0, meaningthat the vectors X ± of equation (B.19) reduce to the vectors X ± of equation (B.16),when P tends to 0. Because of this relation, we treat formally the two different regimes, P (cid:54) = 0 and P = 0, from now on as the same case when we study the solutions associatedto the vectors X ± .The Jordan normal form of the matrix Q in the basis (B.19) is represented by thematrix Q J = i Ω (cid:48) − i Ω (cid:48) , (B.20)while Q = T Q J T − , where the matrix T = ( X X − X ¯ X ) consists of thevectors (B.19) as column vectors. With that the exponential of Q t (cid:48) is given by e Q t (cid:48) = T e i Ω (cid:48) t (cid:48) e − i Ω (cid:48) t (cid:48) t (cid:48) T − . (B.21)With this, for an initial condition given by the vector X (0) = v X + v X − + v X + v ¯ X , (B.22)the solution of (B.12) reads X ( t (cid:48) ) = v e i Ω t (cid:48) X + v e − i Ω t (cid:48) X − + v X + v ( ¯ X + t (cid:48) X ) . (B.23)Thus, there is a secular term in the last term of the right-hand side, which leads tolinear growth in time. In the following we show that this solution is eliminated due tothe conservation of the number of particles, which is described by N = (cid:90) | ψ | dA = (cid:90) | ψ ( t ) | dA. (B.24)The last integral in (B.24) can be computed with the decomposition (41)–(43) byconsidering only the first order of the perturbation: (cid:90) | ψ ( t ) | dA = (cid:90) [ | ψ | + | ψ | ( δψ (cid:48) + δψ (cid:48)∗ )] dA. (B.25)Comparing equations (B.24) and (B.25), we then conclude that (cid:90) ( δψ (cid:48) + δψ (cid:48)∗ ) dA = 0 . (B.26)Using the decomposition (B.6) for δψ (cid:48) we obtain at first4 πR ( δψ (cid:48) + δψ (cid:48)∗ ) + ∞ (cid:88) l =1 l (cid:88) m = − l (cid:90) ( δψ (cid:48) lm + δψ (cid:48)∗ lm ) dA = 0 . (B.27)Due to the orthogonality relations of the spherical harmonics, for all l ≥ m = − l, ..., l , we have that (cid:90) δψ (cid:48) lm dA = (cid:90) δψ (cid:48)∗ lm dA = 0 . (B.28) ose-Einstein Condensation on Curved Manifolds δψ (cid:48) + δψ (cid:48)∗ = 0 , (B.29)which means that δψ ( t (cid:48) ) is a purely imaginary number. Thus, from equation (B.7) weconclude that k ( t (cid:48) ) = 0. Due to (B.13), (B.16), (B.17), (B.19) and (B.22), this fixesthe coefficient v = 0 in solutions (B.18) and (B.23) for P = 0 and P (cid:54) = 0, respectively.This is important for the case of equation (B.23), because it removes the secular term.Moreover, note that the solutions in (B.18) and (B.23) associated to the eigenvector X are temporally constants. With this, we find that only the eigenvectors X ± describe oscillating solutions in time. However, we can see that the vectors X ± arecomplex, so the physical meaning is not evident here. On the other hand, we remarkthat the first and third entries of both vectors in (B.16) and (B.19) are real numbers,while the second and fourth entries are purely imaginary numbers. Then the linearcombinations X + X − i ( X − X − )2 (B.30)are real vectors. With this we obtain X ( t (cid:48) ) = e Q t (cid:48) X + X − e λ t (cid:48) X + e − λ t (cid:48) X −
2= cos(Ω (cid:48) t (cid:48) ) X + X − i sin(Ω (cid:48) t (cid:48) ) ( X − X − )2 , (B.31)which is a real solution. Analogously, we could compute the solution e Q t (cid:48) i X − X − , butit turns out to lead to the same dynamics as above apart from a phase. So, we willconsider only the solution (B.31) which can be written as X ( t (cid:48) ) T = (cid:34) , C Ω (cid:48) (cid:32) P σ (cid:48) + 2 σ (cid:48) (cid:33) sin (Ω (cid:48) t (cid:48) ) , C cos (Ω (cid:48) t (cid:48) ) , C Ω (cid:48) σ (cid:48) sin (Ω (cid:48) t (cid:48) ) (cid:35) (B.32)with the initial condition X (0) T = (0 , , C , , (B.33)where C is a proportionality constant. This solution means that, when one performsat t = 0 a small perturbation of the ground state given by δσ (cid:48) (0) = C ¯ Y , (B.34)then the temporal evolution of this perturbation is given by δψ (cid:48) ( t (cid:48) ) = i C Ω (cid:48) (cid:32) P σ (cid:48) + σ (cid:48) (cid:33) sin(Ω (cid:48) t (cid:48) ) ¯ Y ,δσ (cid:48) ( t (cid:48) ) = C cos(Ω (cid:48) t (cid:48) ) ¯ Y , (B.35) δB (cid:48) ( t (cid:48) ) = C Ω (cid:48) σ (cid:48) sin(Ω (cid:48) t (cid:48) ) ¯ Y . These equations are the dimensionless form of equations (52) in the main text. ose-Einstein Condensation on Curved Manifolds Appendix B.2. Solving Differential Equations for l ≥ l ≥
1. The solution of this system is givenby X lm ( t (cid:48) ) = e Q l t (cid:48) X lm (0) , (B.36)where X lm (0) denotes the initial state. In order to evaluate the matrix exponential e Q l t (cid:48) we have to determine at first the eigenvalues of the matrix Q l . They turn out to be theroots of the polynomial λ + (Ω (cid:48) + a l ) λ + c l = 0 , (B.37)where we consider for the coefficients a l = (cid:32) σ (cid:48) σ (cid:48) (cid:33) δ l + δ l c l = P σ (cid:48) (cid:32) σ (cid:48) (cid:33) δ l + (cid:32)
32 + 5 σ (cid:48) − σ (cid:48) (cid:33) δ l . (B.39)both first and second orders of the smallness parameters δ l defined in equation (51),even though for the final result of the frequencies we restrict ourselves to the first ordercorrections.The roots of equation (B.37) are given by λ l = ± (cid:118)(cid:117)(cid:117)(cid:116) − Ω (cid:48) − a l ± (cid:113) (Ω (cid:48) + a l ) − c l , (B.40)which can be expanded by taking into account only first and second orders of a l and c l ,yielding λ ± l = ± i Ω (cid:48) (cid:32) a l (cid:48) − c l (cid:48) (cid:33) , λ ± l = ± i (cid:118)(cid:117)(cid:117)(cid:116) c l Ω (cid:48) + c l Ω (cid:48) − a l c l Ω (cid:48) . (B.41)The roots λ ± l are always imaginary, so they correspond to oscillating solutions. Instead,the roots λ ± l are imaginary for positive values of the radicand, leading also to oscillatingsolutions, but they are real when the radicand is negative, leading to exponentiallyincreasing and decreasing solutions. The radicand is positive for positive and smallnegative values P , but it turns out to be negative for most negative values of P . Thus,we can conclude that the system is stable for positive interactions and small enoughnegative interaction strengths, while it becomes unstable for most negative interactionstrengths.The frequencies of oscillation are given by Ω (cid:48) l = Ω l /ω = | Re( iλ ± l ) | and Λ (cid:48) l = Λ l /ω = | Re( iλ ± l ) | , and read up to the first order of δ l Ω (cid:48) l = Ω (cid:48) + 1Ω (cid:48) (cid:32) σ (cid:48) σ (cid:48) + 78 σ (cid:48) (cid:33) δ l (B.42) ose-Einstein Condensation on Curved Manifolds (cid:48) l = 1Ω (cid:48) (cid:118)(cid:117)(cid:117)(cid:116) P σ (cid:48) (cid:32) σ (cid:48) (cid:33) δ l + 1Ω (cid:48) (cid:32) − σ (cid:48) σ (cid:48) + 74 σ (cid:48) + 32 σ (cid:48) (cid:33) δ l . (B.43)These equations are the dimensionless counterpart of equations (49) and (50) from themain text. The matrix Q l has 4 different eigenvalues λ ± l = ± i Ω (cid:48) l , λ ± l = ± i Λ (cid:48) l , where Ω (cid:48) l and Λ (cid:48) l are positive real numbers if P is positive or for small negative values of it. Theassociated eigenvectors read X ± l = − P σ (cid:48) Ω (cid:48) δ l ∓ i Ω (cid:48) (cid:32) σ (cid:48) + P σ (cid:48) + P (cid:48) [5 σ (cid:48) − σ (cid:48) − σ (cid:48) δ l (cid:33) ∓ i Ω (cid:48) σ (cid:48) (cid:32) − P Ω (cid:48) [7 σ (cid:48) + 5] σ (cid:48) δ l (cid:33) (B.44)and X ± l = ± i Λ (cid:48) l δ l (cid:32) − P σ (cid:48) Ω (cid:48) δ l (cid:33) Pσ (cid:48) Ω (cid:48) (cid:40) (cid:34) − σ (cid:48) Ω (cid:48) + P σ (cid:48) Ω (cid:48) (cid:32) σ (cid:48) (cid:33)(cid:35) δ l (cid:41) ∓ i P Λ (cid:48) l σ (cid:48) Ω (cid:48) (cid:40) (cid:34) − σ (cid:48) Ω (cid:48) + P σ (cid:48) Ω (cid:48) (cid:32) σ (cid:48) (cid:33)(cid:35) δ l (cid:41) . (B.45)The solution of the system (B.12) is thus given by (B.36), so we have the following timedependences e Q l t (cid:48) X ± l = e λ ± l t (cid:48) X ± l , e Q l t (cid:48) X ± l = e λ ± l t (cid:48) X ± l , (B.46)Note that the eigenvectors (B.44), (B.45) are not pure real vectors, thus in principle,these vectors and the solution (B.46) have no physical meaning. But the first and thirdentries of both vectors (B.44), (B.45) are real numbers, while the second and fourthentries are purely imaginary numbers. Then the vectors X l + X − l , i ( X l − X − l )2 , X l + X − l i ( X l − X − l )2 (B.47)are real vectors. We point out that the dynamics obtained with the combinations i ( X n + l − X n − l ) /
2, for n = 1 ,
2, turn out to be the same of that obtained with( X n + l + X n − l ) /
2, for n = 1 ,
2, apart from a phase. Therefore, we consider from nowonly the latter ones. ose-Einstein Condensation on Curved Manifolds X l ( t (cid:48) ) = e Qt (cid:48) X l + X − l e λ l t (cid:48) X l + e − λ l t (cid:48) X − l (cid:48) l t (cid:48) ) X l + X − l i sin(Ω (cid:48) l t (cid:48) ) ( X l − X − l )2 , which reads explicitly X l ( t (cid:48) ) = − P σ (cid:48) Ω (cid:48) δ l cos(Ω (cid:48) l t (cid:48) )1Ω (cid:48) (cid:32) σ (cid:48) + P σ (cid:48) + P (cid:48) [5 σ (cid:48) − σ (cid:48) − σ (cid:48) δ l (cid:33) sin(Ω (cid:48) l t (cid:48) )cos(Ω (cid:48) l t (cid:48) )Ω (cid:48) σ (cid:48) (cid:32) − P Ω (cid:48) [7 σ (cid:48) + 5] σ (cid:48) δ l (cid:33) sin(Ω (cid:48) l t (cid:48) ) (B.49)with the initial condition X l (0) = − P σ (cid:48) Ω (cid:48) δ l . (B.50)Analogously, we also obtain X l ( t (cid:48) ) = cos(Λ (cid:48) l t (cid:48) ) − (cid:48) l δ l (cid:32) − P σ (cid:48) Ω (cid:48) δ l (cid:33) sin(Λ (cid:48) l t (cid:48) ) Pσ (cid:48) Ω (cid:48) (cid:40) (cid:34) − σ (cid:48) Ω (cid:48) + P σ (cid:48) Ω (cid:48) (cid:32) σ (cid:48) (cid:33)(cid:35) δ l (cid:41) cos(Λ (cid:48) l t (cid:48) ) P Λ (cid:48) l σ (cid:48) Ω (cid:48) (cid:40) (cid:34) − σ (cid:48) Ω (cid:48) + P σ (cid:48) Ω (cid:48) (cid:32) σ (cid:48) (cid:33)(cid:35) δ l (cid:41) sin(Λ (cid:48) l t (cid:48) ) (B.51) ose-Einstein Condensation on Curved Manifolds X l (0) = Pσ (cid:48) Ω (cid:48) (cid:40) (cid:34) − σ (cid:48) Ω (cid:48) + P σ (cid:48) Ω (cid:48) (cid:32) σ (cid:48) (cid:33)(cid:35) δ l (cid:41) . (B.52)Thus, performing at t (cid:48) = 0 a small perturbation given by δψ (cid:48) lm (0) = − C lm P σ (cid:48) Ω (cid:48) δ l ¯ Y lm , δσ (cid:48) lm (0) = C lm ¯ Y lm , (B.53)where C lm is a proportionality constant, the evolution of this system is given by δψ (cid:48) lm ( t (cid:48) ) = C lm (cid:40) − P σ (cid:48) Ω (cid:48) δ l cos(Ω (cid:48) l t (cid:48) )+ i Ω (cid:48) (cid:32) σ (cid:48) + P σ (cid:48) + P (cid:48) [5 σ (cid:48) − σ (cid:48) − σ (cid:48) δ l (cid:33) sin(Ω (cid:48) l t (cid:48) ) (cid:41) ¯ Y lm ,δσ (cid:48) lm ( t (cid:48) ) = C lm cos(Ω (cid:48) l t (cid:48) ) ¯ Y lm , (B.54) δB (cid:48) lm ( t (cid:48) ) = C lm Ω (cid:48) σ (cid:48) (cid:32) − P Ω (cid:48) [7 σ (cid:48) + 5] σ (cid:48) δ l (cid:33) sin(Ω (cid:48) l t (cid:48) ) ¯ Y lm . These equations are the dimensionless form of equations (53) in the main text. But asmall perturbation at t (cid:48) = 0 given by δψ (cid:48) lm (0) = C lm Y lm , (B.55) δσ (cid:48) lm (0) = C lm Pσ (cid:48) Ω (cid:48) (cid:40) (cid:34) − σ (cid:48) Ω (cid:48) + P σ (cid:48) Ω (cid:48) (cid:32) σ (cid:48) (cid:33)(cid:35) δ l (cid:41) ¯ Y lm , where C lm is also a proportionality constant, leads to the solution δψ (cid:48) lm ( t (cid:48) ) = C lm (cid:32)
12 cos(Λ (cid:48) l t (cid:48) ) − i Λ (cid:48) l δ l (cid:32) − P σ (cid:48) Ω (cid:48) δ l (cid:33) sin(Λ (cid:48) l t (cid:48) ) (cid:33) ¯ Y lm ,δσ (cid:48) lm ( t (cid:48) ) = C lm Pσ (cid:48) Ω (cid:48) (cid:40) (cid:34) − σ (cid:48) Ω (cid:48) + P σ (cid:48) Ω (cid:48) (cid:32) σ (cid:48) (cid:33)(cid:35) δ l (cid:41) cos(Λ (cid:48) l t (cid:48) ) ¯ Y lm , (B.56) δB (cid:48) lm ( t (cid:48) ) = C lm P Λ (cid:48) l σ (cid:48) Ω (cid:48) (cid:40) (cid:34) − σ (cid:48) Ω (cid:48) + P σ (cid:48) Ω (cid:48) (cid:32) σ (cid:48) (cid:33)(cid:35) δ l (cid:41) sin(Λ (cid:48) l t (cid:48) ) ¯ Y lm , which are the dimensionless form of equations (54) in the main text.A small perturbation of δσ (cid:48) lm and of the real part of δψ (cid:48) lmlm