Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime

We study dimension reduction for the three-dimensional Gross-Pitaevskii equation with a long-range and anisotropic dipole-dipole interaction modeling dipolar Bose-Einstein condensation in a strong interaction regime. The cases of disk shaped condensates (confinement from dimension three to dimension two) and cigar shaped condensates (confinement to dimension one) are analyzed. In both cases, the analysis combines averaging tools and semiclassical techniques. Asymptotic models are derived, with rates of convergence in terms of two small dimensionless parameters characterizing the strength of the confinement and the strength of the interaction between atoms.


Introduction and main results
In this paper, we study dimension reduction for the three-dimensional Gross-Pitaevskii equation (GPE) with a long-range and anisotropic dipole-dipole interaction (DDI) modeling dipolar Bose-Einstein condensation [11,14]. In contrast with the existing literature on this topic [1], we will not assume that the degenerate dipolar quantum gas is in a weak interaction regime.
Based on the mean field approximation [3,9,13,18,19,20], the dipolar Bose-Einstein condensate is modeled by its wavefunction Ψ := Ψ(t, x) satisfying the GPE with a DDI written in physical variables as ∆Ψ + V (x)Ψ + N g|Ψ| 2 Ψ + N C dip U dip * |Ψ| 2 Ψ, (1.1) where ∆ is the Laplace operator, V (x) denotes the trapping harmonic potential, m > 0 is the mass, is the Planck constant, g = 4π 2 as m describes the contact (local) interaction between atoms in the condensate with the s-wave scattering length a s , N denotes the number of atoms in the condensate, and the dipole-dipole interaction kernel U dip (x) is given as with the dipolar axis n = (n 1 , n 2 , n 3 ) ∈ R 3 satisfying |n| = n 2 1 + n 2 2 + n 3 3 = 1. Here θ is the angle between the polarization axis n and relative position of two atoms (that is, cos θ = n · x/|x|). For magnetic dipoles we have C dip = µ 0 µ 2 dip , where µ 0 is the magnetic vacuum permeability and µ dip the dipole moment, and for electric dipoles we have C dip = p 2 dip /ǫ 0 , where ǫ 0 is vacuum permittivity and p dip the electric dipole moment. The wave function is normalized according to R 3 |Ψ(t, x)| 2 dx = 1. 1 1.1. Nondimensionalization. We assume that the harmonic potential is strongly anisotropic and confines particles from dimension 3 to dimension 3 − d. We shall denote x = (x, z), where x ∈ R 3−d denotes the variable in the confined direction(s) and z ∈ R d denotes the variable in the transversal direction(s). In applications, we will have either d = 1 for disk-shaped condensates, or d = 2 for cigar-shaped condensates. Similarly, we denote n = (n x , n z ) with n x ∈ R 3−d and n z ∈ R d . The harmonic potential reads [2,15,16] V (x) = m 2 ω 2 x |x| 2 + ω 2 z |z| 2 where ω z ≫ ω x . We introduce three dimensionless parameters where the harmonic oscillator length is defined by [2,15,16] a 0 = mω x 1/2 .
The dimensionless parameter λ 0 measures the relative strength of dipolar and swave interactions. Let us rewrite the GPE (1.1) in dimensionless form. For that, we introduce the new variablest,x,z and the associated unknown Ψ defined bỹ The dimensionless GPE equation reads [2,15,16] i∂t where σ = sign a s ∈ {−1, 1}. Define the differential operators ∂ n = n · ∇ and ∂ nn = ∂ n ∂ n . Mathematically speaking, the convolution with U dip in equation (1.1) has to be considered in the distributional sense and we have the following identity (see [3]) We can re-formulate the GPE (1.4) as the following Gross-Pitaevskii-Poisson system (GPPS) [3,7] i∂t Ψ = − 1 2 ∆ Ψ + 1 2 (1.7) Under scaling (1.3), dimension reduction of the above GPPS (1.4) was formally derived from 3D to 2D and 1D in [1,7,17] for any fixed β, λ 0 and n when ε → 0 + . Rigorous mathematical justification was only given in the weak interaction regime, i.e. when β = O(ε) from 3D to 2D and when β = O(ε 2 ) from 3D to 1D [1]. It is an open problem for the case where β is fixed when ε → 0 + . 1.2. New scaling. In order to observe the condensate at the correct space scales, we will now proceed to a rescaling in x and z. Let us denote α = ε 2d/n β −2/n . (1.8) The scaling assumptions are α ≪ 1 and ε ≪ 1.
We define the new variables which means that the typical length scales of the dimensionless variables are ε in the z-direction and α −1/2 in the x-direction. The wavefunction is rescaled as follows: Notice that the L 2 norm of Ψ ε,α is left invariant by this rescaling, so we still have We end up with the following rescaled GPE (for simplicity we omit the primes on the variables): where the transversal Hamiltonian is Let us remark that (1.10) Thanks to identity (1.6), we can remark that U dip is a bounded function of R 3 into [− 1 3 , 2 3 ]. For γ > 0, we denote by V γ dip the tempered distribution whose Fourier transform is Let us note that (1.8) is equivalent to Remark 1.2. The spectrum of H z is the set of integers N. We define (ω k ) k∈N an orthonormal basis of L 2 (R 3 ) made of eigenvectors of H z where ω 0 is the ground state (associated to the eigenvalue 0) In this paper, we study the behavior of the solution of equation (1.12) as ε → 0 and α → 0 independently so that β may be bounded but can also tends to +∞.
Our key mathematical assumption will be that the wavefunction Ψ ε,α at time t = 0 is under the WKB form: Here A 0 is a complex-valued function and S 0 is real-valued. Let us introduce another parameter γ > 0 to get a better understanding of the different phenomena involved during the limiting procedures. In this paper, we will study instead of equation (1.12) the following one : From now on, we denote by Ψ ε,α,γ the solution ψ of equation (1.14). Let us insist on the fact that Ψ ε,α,γ is equal to the solution Ψ ε,α of equation (1.12) if we assume that γ = ε √ α.

1.3.
Heuristics. In this section, we derive formally the limiting behavior of the solution of (1.14) as ε (strong confinement limit), α (semiclassical limit) and γ (limit of the dipole-dipole interaction term) go to 0. Our main result, stated in the next section, will be that in fact these limits commute together: the limit is valid as ε, α and γ converge independently to zero. Thus, this gives us as a by-product the behavior of the solution of equation (1.12) as ε and α converge independently to zero. a) Strong confinement limit : ε → 0. Let us fix α ∈ (0, 1] and γ ∈ [0, 1]. Following [6], in order to analyze the strong partial confinement limit, it is convenient to begin by filtering out the fast oscillations at scale ε 2 induced by the transversal Hamiltonian. To this aim, we introduce the new unknown It satisfies the equation where the nonlinear function is defined by A fundamental remark is that for all fixed Φ, the function θ → F γ (θ, Φ) is 2πperiodic, since the spectrum of H z only contains integers. For any fixed α > 0 and λ 0 = 0, Ben Abdallah et al. [6,5] proved by an averaging argument that we have In our study, we consider the case λ 0 ∈ R and a similar averaging argument should give us the same result Φ ε,α,γ = Φ 0,α,γ + O(ε 2 ). b) Semi-classical limit : α → 0. Let us remark that equation (1.14) is written in the semi-classical regime of "weakly nonlinear geometric optics", which can be studied by a WKB analysis. Here we are only interested in the limiting model, so in the first stage of the WKB expansion. Let us introduce the solution S(t, x) of the eikonal equation and filter out the oscillatory phase of the wavefunction by setting For all fixed ε > 0, we can expect that

Remark 1.4. A key point here in this analysis is that the nonlinearities
and for all t, we have where q > 0 and In this paper, the main difficulty we have to tackle and also the main difference with respect to the previous work of the authors [4] in the case λ 0 = 0, is the study of this limit γ → 0.
d) The simultaneous study of the three limits. We introduce for any (ε, α, γ) which is the solution of the equation We will also consider the solution A ε,0,γ of (1.23) with α = 0, the solution A ε,α,0 of (1.23) with γ = 0 and the solution A 0,α,γ of for all (x, z) ∈ R 3 . As long as the phase S(t, ·) remains smooth, i.e. before the formation of caustics in the eikonal equation (1.18), we expect to have and the solution Ψ ε,α,γ of equation (1.14) is expected to behave as for some q > 0.

1.4.
Main results. In this paper, our main contribution is the rigorous study of the dipole-dipole interaction limits γ → 0 as well as the study of the three simultaneous limits ε → 0, α → 0 and γ → 0 involved in the problem. The techniques used for the study of the limits ε → 0 and α → 0 were developed by the authors in [4]. We will recall and use some of the results proved in this first paper.
1.4.1. Existence, uniqueness and uniform boundedness results. Let us make precise our functional framework. For wavefunctions, we will use the scale of Sobolev spaces adapted to quantum harmonic oscillators: are two algebras. In this paper, we will also make frequent use of the estimate (see [12] and [6] for a more general class of confining potential).

1.4.2.
Study of the limits α → 0, ε → 0 and γ → 0. We are now able to study the behavior of A ε,α,γ as α → 0, ε → 0 and γ → 0. (1.30) The constants C and C q do not depend on α, ε and γ but C q does depend on q. The estimates related to the original equation (1.12) can be summarized in the following diagram: Remark 1.8. The case λ 0 = 0 has already been studied by the authors in [4] where we got estimates that are similar to (1.27) and (1.28).
Remark 1.9. Assume that either n x = 0 or n z = 0. Then, for all (ε, α) ∈ (0, 1] 2 , for all q such that we get the same conclusion as in Theorem 1.7. The following immediate corollary gives a more accurate approximation of A ε,α,ε √ α than A 0,0,0 . This result can be useful for numerical simulations and has to be related to the ones of Ben Abdallah et al. [5]. Corollary 1.10. Assume the hypothesis of Theorem 1.6 true. Then, for all (ε, α) ∈ [0, 1] 2 , we have the following bound: The following proposition concerns the special case of an initial data polarized on one mode of H z . It generalizes the case studied by Bao, Ben Abdallah and Cai [1, Theorems 5.1 and 5.5] where the initial data was taken on the ground state of H z . Proposition 1.11. Let k ∈ N. Assume the hypothesis of Theorem 1.6 true. Assume also that A 0 (x, z) = a 0 (x)ω k (z), (x, z) ∈ R 3 where ω k is defined in Remark 1.2. Then, the function A 0,α,γ stays polarized on the mode ω k i.e.
Here, B α,γ is the solution of we have moreover the following bound for all α ∈ [0, 1] : and for α ∈ (0, 1] fixed The paper is organized as follows. In Section 2, we study some properties of the dipolar term that are needed in the proofs of Theorems 1.6, 1.7 and 1.11 given in Section 3.

Study of the dipolar term
Let us define for θ ∈ R, γ ∈ [0, 1] and Φ ∈ L 2 (R 3 ) so that F γ = F 1 + F γ 2 and F γ av = F 1,av + F γ 2,av . In order to prove the uniform well-posedness of the nonlinear equations (1.23) and (1.24), we will need Lipschitz estimates for F γ (θ, ·) defined by (1.15) and F γ av (·) defined by (1.17). We only study here the dipolar terms F γ 2 and F γ 2,av since the cubic ones F 1 (θ, Φ) and F 1,av (Φ) have already been studied in [ by (1.11). Then, we get for all u ∈ H m (R 3 ) The following lemma gives Lipschitz estimates for the dipolar terms.
To begin, assume that θ = 0, then we get that Lemma 2.1 and Remark 1.5 ensure that For the second term, we get

This gives us
2.2. The limit γ → 0. Let us study now the behavior of F γ 2 and F γ 2,av as γ → 0.

Case of a function which is polarized on one mode of
where k ∈ N and ω k is defined in Remark 1.2. The constant C m,q depends neither on u nor on γ.

Proofs of our main Theorems
This section is devoted to the proofs of Theorems 1.6 and 1.7 and Proposition 1.11 which are inspired by the ones of [4, Theorem 1.3. and 1. 4.]. To do so, we recall without any proof some of the results the authors obtained in this paper for the sake of readability.
We give then in Proposition 3.2 the local in time well-posedness of the eikonal equation [4,Proposition 2.2.].
The following lemma is related to the non-homogeneous linear equation (3.1) (see [4,Lemma 2.6.]). The crucial bound (3.2) is obtained by energy estimate.
Moreover for all t ∈ [0, T ], a satisfies the estimates where C is a generic constant which depends only on m and on  [4] where the case λ 0 = 0 was treated.
Let us now prove Theorem 1.7.
The semi-classical limit α → 0: proof of (1.28). The proof of the error estimate (1.28) follows exactly the same arguments as the ones of [4, Theorem 1.4.] since for γ fixed, the new dipolar term can be treated exactly as the cubic term.
Proof of Proposition 1.11. In this case, we remark that the solutions remain polarized on a single mode of H z as time evolves. Hence, we can apply Lemma 2.4 instead of Lemma 2.3 and Proposition 1.11 follows from the arguments used in the proof of the estimate (1.29) in Theorem 1.7