Critical Rotational Speeds in the GrossPitaevskii Theory on a Disc with Dirichlet Boundary Conditions
Abstract
We study the twodimensional GrossPitaevskii theory of a rotating Bose gas in a discshaped trap with Dirichlet boundary conditions, generalizing and extending previous results that were obtained under Neumann boundary conditions. The focus is on the energy asymptotics, vorticity and qualitative properties of the minimizers in the parameter range where is the rotational velocity and the coupling parameter is written as with . Three critical speeds can be identified. At vortices start to appear and for the vorticity is uniformly distributed over the disc. For the centrifugal forces create a hole around the center with strongly depleted density. For vorticity is still uniformly distributed in an annulus containing the bulk of the density, but at there is a transition to a giant vortex state where the vorticity disappears from the bulk. The energy is then well approximated by a trial function that is an eigenfunction of angular momentum but one of our results is that the true minimizers break rotational symmetry in the whole parameter range, including the giant vortex phase.
MSC: 35Q55,47J30,76M23. PACS: 03.75.Hh, 47.32.y, 47.37.+q.
Keywords: BoseEinstein Condensates, Superfluidity, Vortices, Giant Vortex.
Contents
1 Introduction and Main Results
The GrossPitaevskii (GP) theory is the most commonly used model to describe the behavior of rotating superfluids. Since the nucleation of quantized vortices is a signature of the superfluid behavior it is of great interest to understand that phenomenon in the framework of the GP theory. A fascinating example of superfluid is provided by a cold Bose gas forming a BoseEinstein condensate (BEC). The possibility to nucleate quantized vortices in a rotating BEC has triggered a lot of interest in the last decade, both experimental and theoretical (see the reviews [Co, Fe1] and the monograph [A] for further references).
BoseEinstein condensates are trapped systems: A magnetooptical confinement is imposed on the atoms. When rotating such a system, the strength of the confinement can lead to two different behaviors. If the trapping potential increases quadratically with the distance from the rotation axis (‘harmonic’ trap), there exists a limiting angular velocity that one can impose to the gas. Any larger velocity would result in a centrifugal force stronger than the trapping force. The atoms would then be driven out of the trap. By contrast, a stronger confinement (‘anharmonic’ trap) allows in principle an arbitrary angular velocity. In this paper we focus on the twodimensional GP theory for a BEC with anharmonic confinement.
Theoretical and numerical arguments have been proposed in the physics literature (see, e.g., [FJS, FB, KB]) in favor of the existence of three critical speeds at which important phase transitions are expected to happen:

If the velocity is smaller than the first critical velocity , then there are no vortices in the condensate (‘vortexfree state’);

If is between and , there is a hexagonal lattice of singly quantized vortices (‘vortexlattice state’);

When is taken larger than , the centrifugal force becomes so important that it dips a hole in the center in the condensate. The annulus in which the mass is concentrated still supports a vortex lattice however (‘vortexlatticeplushole state’), until crosses the third threshold ;

If is larger than , all vortices retreat in the central low density hole, resulting in a ‘giant vortex’ state. The central hole acts as a multiply quantized vortex with a large phase circulation.
In [CDY1, CY, CRY, R] we have studied these phase transitions using as model case a BEC in a ‘flat’ trap, that is a constant potential with hard walls. This is the ‘most anharmonic’ confinement one can imagine and serves as an approximation for potentials used in experiments. Mathematically, it has the advantage that the rescaling of spatial variables as and/or is avoided. The GP energy functional in the noninertial rotating frame is defined as
(1.1) 
where we have denoted the physical angular velocity by , is the angular momentum operator and the unit twodimensional disc. We have written the coupling constant as . The subsequent analysis (as well as the papers [CDY1, CY, CRY, R]) is concerned about the ‘ThomasFermi’ (or strongly interacting) limit where .
The simplest way to define the ground state of the system is to minimize the energy functional (1.1) under the mass constraint
with no further conditions. This is the approach that has been considered in the previous papers [CDY1, CY, CRY, R], leading to Neumann boundary conditions on . We will refer to this situation as the ‘flat Neumann problem’ in the sequel.
There are, however, both physical and mathematical reasons for considering also the corresponding problem with a Dirichlet boundary condition, i.e., requiring the wave function to vanish on the boundary of the unit disc. Physically, this corresponds to a hard, repelling wall which is usually a closer approximation to real experimental situations than a ‘sticky’ wall modeled by a Neumann boundary condition. The Dirichlet boundary condition can be formally implemented by replacing the flat trap with a smooth confining potential of the form and taking^{2}^{2}2This limit has to be taken with care, however, because it can not be interchanged with the asymptotic limit we shall consider. This point will be discussed further in [CPRY]. .
Mathematically, the new boundary condition is responsible for some new aspects requiring several modifications of the proofs. For one thing, the density profile is no longer a monotonously increasing function of the radial variable and the position of the density maximum has to be precisely estimated. Furthermore, energy estimates have to be refined to take the boundary effect into account, and a boundary estimate for the GP minimizer, that was an important ingredient in the proof of the giant vortex transition in [CRY], has to be replaced by a different approach.
In addition to these adaptations to the new situation the present paper contains also substantial improvements of results proved previously in the Neumann case. These concern in particular the uniform distribution of vorticity in the bulk (Theorem 1.1) and the rotational symmetry breaking (Theorem 1.6). Besides, the error term in our energy estimate in Theorem 1.4 below is much smaller than the corresponding term in [CRY, Theorem 1.2]. This last improvement is due to the new method for estimating a potential function that we use to avoid the boundary estimate.
From now on the minimization of (1.1) is considered on the domain
(1.2) 
where is the Sobolev space of complex valued functions on with and on . The ground state energy is thus defined as
(1.3) 
and any corresponding minimizer is denoted by . This case will be referred to as the ‘flat Dirichlet problem’. In the following we will often use a different form of the GP functional which can be obtained by introducing a vector potential, i.e.,
(1.4) 
where
(1.5) 
Here are twodimensional polar coordinates and a unit vector in the angular direction.
The GP minimizer is in general not unique because vortices can break the rotational symmetry (see Section 1.3) but any minimizer satisfies in the open ball the variational equation (GP equation)
(1.6) 
with additional Dirichlet conditions at the boundary, i.e.,
(1.7) 
The chemical potential in (1.6) is given by the normalization condition on , i.e.,
(1.8) 
For such a model, variational arguments have been provided in [FB] to support the following conjectures about the three critical speeds:
(1.9)  
(1.10)  
(1.11) 
As for the behavior of the condensate close to , the centrifugal force is not strong enough for the specificity of the anharmonic confinement to be of importance. A consequence is that the analysis developed in [IM1, IM2] (see also [AJR] for recent developments) for harmonic traps applies and leads to the rigorous estimate
(1.12) 
when . In this paper we aim at providing estimates of and and thus will assume that
i.e., we consider angular velocities strictly above . The situation is then very different from that in a harmonic trap because of the onset of strong centrifugal forces when approaches .
Our main results can be summarized as follows. We show that if , the condensate is discshaped, while for the matter density is confined in an annulus along the boundary of . In addition we prove that if
there is a uniform distribution of vorticity in the bulk of the condensate. Although our estimates are not precise enough to show that there is a hexagonal lattice of vortices, these results support the qualitative picture provided in [FB]. We deduce that when
(1.13) 
We refer to Section 1.1 for the detailed statements of these results.
In Section 1.2 we present our results about the third critical speed. We show that if with , then there are no vortices in the bulk of the condensate. This provides an upper bound on the third critical speed
(1.14) 
It should be noted right away that we do believe that this upper bound is optimal. This has been proved in [R] in the flat Neumann case and the adaptation of the adequate tools to the flat Dirichlet case is possible but beyond the scope of this paper. We hope to come back to the regime in the future.
We also remark that the estimates we obtain for the three critical speeds in the limit are the same in the flat Neumann and Dirichlet settings. In the cases of the first and second critical velocities this is plausible because the features that mark the onset of the transition (the first vortices and the appearance of the ’hole’ respectively) occur far from the boundary of the trap. The independence of the third critical velocity of boundary conditions is less obvious but the main reason is that the maximum of the density is to leading order the same for both boundary conditions.
In the regime a very natural question occurs about the distribution of vorticity in the central hole of low matter density: Is the phase of the condensate created by a single multiply quantized vortex at the center of the trap? We show that this is not the case in Section 1.3 and, as a consequence, the rotational symmetry is always broken at the level of the ground state, even when .
Before stating our results more precisely, we want to make a comparison with the 2D GinzburgLandau (GL) theory for superconductors in applied magnetic fields (see [BBH, FH, SS2] for a mathematical presentation). The analogies between GP and GL theories have often been pointed out in the literature, with the external magnetic field playing in GL theory the role of the angular velocity in GP theory. We stress that our results in fact enlighten significant differences between the two theories. Whereas the first critical speed in GP theory can be seen as the equivalent of the first critical field in GL theory, the second and third critical speeds have little to do with the second and third critical fields of the GL theory. The difference can be seen both in the order of magnitudes of these quantities as functions of (which for a superconductor is the inverse of the GL parameter) and in the qualitative properties of the states appearing in the theories. In GP theory there is no equivalent of the normal state and there is no vortexlatticeplushole state in GL theory. The giant vortex state of GP theory could be compared to the surface superconductivity state in GL theory, but the physics governing the onset of these two phases is quite different. The main reason for this different behavior is the combined influence of the centrifugal force and mass constraint in GP theory, two features that have no equivalent in GL theory.
We will now state our results rigorously. The core analysis that we present below is an adaptation of the techniques developed in [CDY1, CY, CRY] for the Neumann case, but the Dirichlet condition leads to important novel aspects that we discuss in the sequel.
1.1 The Regime : Uniform Distribution of Vorticity
Before stating our results we need to introduce some notation. We define the density functional
(1.15) 
for any real function . The minimization is given by
(1.16) 
and is the associated minimizer (see Proposition 2.1). In order to give a precise meaning to the expression ‘bulk of the condensate’, we introduce the following ThomasFermi functional, obtained by dropping the first term in (1.4) or (1.15):
(1.17) 
which is expected to provide the energy associated with the nonuniform density of the condensate. We refer to the Appendix for the properties of its ground state energy and associated minimizer . Let us define
(1.18) 
If , , while if , is an annulus of outer radius and inner radius with . As we shall see below, is close to and thus, if , the mass of is concentrated close to the boundary of .
Our result about the uniform distribution of vorticity in fact holds in a slightly smaller region than , namely the annulus
(1.19) 
where, for a certain quantity such that as (see Section 3.3, Equation (3.35) for its precise definition),
(1.20) 
and is the position of the unique maximum of the density (see Proposition 2.2). It should be noted that is close to and is close to in such a way that
i.e., the domain tends to the support of the TF density as . Also, thanks to the above estimate, we have
(1.21) 
i.e., the mass is concentrated in . We refer to (2.22), (2.23) and (2.32) below for precise estimates of .
We now state our result about the uniform distribution of vorticity. It is the analogue of [CY, Theorem 3.3] but here we prove that the distribution of vorticity is uniform in the whole regime whereas in [CY] this was proved only for .
Theorem 1.1 (Uniform distribution of vorticity).
Let be any GP minimizer and sufficiently small. If , there exists a finite family of disjoint balls^{3}^{3}3Throughout the whole paper the notation stands for a ball of radius centered at , whereas is a ball with radius centered at the origin. such that

, and ,

on for some .
Moreover, denoting by the winding number of on and introducing the measure
(1.22) 
then, for any family of sets such that as ,
(1.23) 
Remark 1.1 (Distribution of vorticity)
The result proven in the above Theorem implies that the vorticity measure converges after a suitable rescaling to the Lebesgue measure, i.e., the vorticity is uniformly distributed. However such a statement is meaningful only for angular velocities at most of order , when the TF support can be bounded independently of . On the opposite if , shrinks and its Lebesgue measure converges to 0 as . To obtain an interesting statement one has therefore to allow the domain to depend on with as .
Remark 1.2 (Conditions on )
We remark that the lower bound on the measure of the set , i.e., , is important, even though not optimal, as it will be clear in the proof: In order to localize the energy bounds to suitable lattice cells, one has to reject a certain number of ‘bad cells’ where nothing can be said about the vorticity of . However since the number of bad cells is much smaller than the total number of cells, this has no effect on the final statement provided the measure of is much larger than the area of a single cell, i.e., . A similar effect occurs in [CY, Theorem 3.3], where the stronger condition is assumed.
Remark 1.3 (Vortex balls)
The balls contained in the family are not necessarily vortex cores in the sense that each one might contain a large number of vortices. However the conditions stated at point 1 of the above Theorem 1.1 have important consequences on the properties of the family. For instance, if , the last one, i.e., , guarantees that the area covered by balls is smaller than the area of the annulus where the bulk of the condensate is contained. At the same time the other two conditions imply that the radius of any ball in the family is at most and their number can not be too large: Assuming that for each ball , the second condition would yield a number of balls of order at most , which is expected to be close to the total winding number of any GP minimizer.
An important difference between the flat Neumann and the flat Dirichlet problems can be seen directly from the energy asymptotics. Indeed, in the flat Neumann case (see [CY, Theorem 3.2]) the energy is composed of the contribution of the TF profile (leading order) and the contribution of a regular vortex lattice (subleading order). In the flat Dirichlet case the radial kinetic energy arising from the vanishing of the GP minimizer on might be larger (see Remark 1.4 below) than the contribution of the vortex lattice. As a result the functional (1.16) that includes this radial kinetic energy plays a key role in the energy asymptotics of the problem:
Theorem 1.2 (Ground state energy asymptotics).
As ,
(1.24) 
if , and
(1.25) 
if .
Remark 1.4 (Composition of the energy)
The leading order term in the GP energy asymptotics is given by the energy which contains the kinetic contribution of the density profile (see (1.15)), i.e., one can decompose as , where the first remainder is the most relevant in the regime and the second becomes dominant for angular velocities much larger than .
The kinetic energy of the density profile can in turn be decomposed into the energy associated with Dirichlet conditions and the one due to the inhomogeneity of the profile , which is (see Remark 2.1). The first contribution dominates for any angular velocity and this is why it is the only one appearing in (1.24) and (1.25).
Note also that the kinetic energy due to Dirichlet boundary conditions is, in general, much larger than the vortex energy contribution, i.e., the second term in (1.24) and (1.25), except in the narrow regime
where the latter becomes predominant.
An important consequence of the above energy asymptotics is that we always have (see Proposition 3.1)
(1.26) 
which allows to deduce
(1.27) 
This implies that if , the mass of is concentrated in an annulus, marking the transition to the vortexlatticeplushole state. We thus have
(1.28) 
Note that we actually prove stronger results than (1.21) and (1.27). If , any GP minimizer is in fact exponentially small in the central hole, minus possibly a very thin layer close to (see Proposition 3.2).
1.2 The Regime : Emergence of the Giant Vortex
When the angular velocity reaches the asymptotic regime a transition in the GP ground state takes place above a certain threshold: Vortices are expelled from the essential support of any GP minimizer . The density is concentrated in a shrinking annulus where such a wave function is vortex free. Anticipating this transition we shall throughout this section assume that
(1.29) 
for some constant .
The bulk of the condensate has to be defined differently in this regime: We set
(1.30) 
where
(1.31) 
The main result in this regime is contained in the following
Theorem 1.3 (Absence of vortices in the bulk).
If the angular velocity is given by (1.29) with , then no GP minimizer has a zero inside if is small enough.
More precisely, for any ,
(1.32) 
Remark 1.5 (Bulk of the condensate)
As the notation indicates, the domain contains the bulk of the condensate: Using the explicit expression (A.1) of , one can easily verify that
(1.33) 
which implies by (1.32) that the same estimate holds true also for .
A consequence of this result is the estimate
(1.34) 
As already noted, we believe that this upper bound is optimal, i.e., we actually have
The proof of this conjecture could use the tools of [R] but we leave this aside for the present.
The theorem above is based on a comparison of a minimizer with a giant vortex wave function of the form
where is some additional phase. Therefore we introduce a density functional stands for the integer part and
(1.35) 
where is realvalued and
(1.36) 
We also set
(1.37) 
By simply testing the GP functional on a trial function of the form above, one immediately obtains the upper bound
(1.38) 
In the following Theorem we prove that the r.h.s. of the expression above gives precisely the leading order term in the asymptotic expansion of as and we state an estimate of the phase optimizing .
Theorem 1.4 (Ground state energy asymptotics and optimal phase).
For any and small enough
(1.39) 
Moreover with satisfying
(1.40) 
Remark 1.6 (Composition of the energy)
We refer to [CRY, Remark 1.4] for details on the energy (denoted in that paper). Let us just emphasize that in this setting the Dirichlet boundary condition is responsible for a radial kinetic energy contribution that was not present in the flat Neumann case and gives the leading order correction to in the asymptotic expansion of .
A consequence of Theorem 1.3 is that the degree of is well defined on any circle of radius centered at the origin, as long as
We are able to estimate this degree, proving that it is in agreement with that of the optimal giant vortex trial function (1.40):
Theorem 1.5 (Degree of a GP minimizer).
If and is small enough,
(1.41) 
for any .
We note that because of the Dirichlet condition there is a small region close to where the density goes to zero. We have basically no information on the GP minimizer in this layer that could a priori contain vortices. The existence of this layer is the main difference between the flat Dirichlet case and the flat Neumann case considered in [CRY]. In particular the lack of a priori estimates on the phase circulation of on requires new ideas in the proof.
1.3 Rotational Symmetry Breaking
As anticipated above, a very natural question arising from the results in Section 1.2 is that of the repartition of vortices in the central hole of low matter density. In particular, does one have
modulo a constant phase factor, which would imply that all the vorticity is contained in a central multiply quantized vortex?
We show below that this can not be the case: the GP functional is rotationally symmetric but if the angular velocity exceeds a certain threshold this symmetry is broken at the level of the ground state. No minimizer of the GP energy functional is an eigenfunction of the angular momentum, i.e. a function of the form with real and an integer. A straightforward consequence is that there is not a unique minimizer but for any given minimizing function one can obtain infinitely many others by simply rotating the function by an arbitrary angle. In other words as soon as the rotational symmetry is broken, the ground state is degenerate and its degeneracy is infinite.
In [CDY1, Proposition 2.2] we have proven that the symmetry breaking phenomenon occurs in the case of a bounded trap with Neumann boundary conditions when , for some given constant . We are now going to show that such a result admits an extension to angular velocities much larger than , i.e., the rotational symmetry is still broken even for very large angular velocities. Such an extension is far from obvious in view of the main result about the emergence of a giant vortex state discussed above: Since vortices are expelled from the essential support of the GP minimizer, there might a priori be a restoration of the rotational symmetry but the behavior of any GP minimizer inside the hole remains unknown.
Theorem 1.6 (Rotational symmetry breaking).
If is small enough and large enough, no minimizer of the GP energy functional (1.1) is an eigenfunction of the angular momentum.
1.4 Organization of the Paper
The paper is organized as follows. Section 2 is devoted to general estimates that will be used throughout the paper. We then prove our results about the regime in Section 3. The analysis of the energy functional (1.15) is the main new ingredient with respect to the method of [CY]. We adapt the techniques developed in that paper for the evaluation of the energy of a trial function containing a regular lattice of vortices. The corresponding lower bound is proved via a localization method allowing to appeal to results from GL theory [SS1, SS2]. The inhomogeneity of the density profile is dealt with using a Riemann sum approximation.
Section 4 is devoted to the giant vortex regime. Our main tools are the techniques of vortex ball construction and jacobian estimates, originating in the papers [Sa, J, JS] (see also [SS2]). We implement this approach using a cell decomposition as in [CRY]. New ideas are necessary to control the behavior of GP minimizers on .
The symmetry breaking result is proved in Section 5. Following [Seir], given a candidate rotationally symmetric minimizer, we explicitly construct a wave function giving a lower energy. Finally the Appendix gathers important but technical results about the TF functional and the third critical speed.
2 Preliminary Estimates: The Density Profile with Dirichlet Boundary Conditions
This section is devoted to the proof of estimates which will prove to be very useful in the rest of the paper but are independent of the main results. We mainly investigate the properties of the density profile which captures the main traits of the modulus of the GP minimizer : More precisely we study in details the minimization of the density functional (1.15) and prove bounds on its ground state energy (1.16) and associated minimizers .
The leading order term in the ground state energy is given by the infimum of the TF functional (1.17), i.e.,
(2.1) 
We postpone the discussion of the properties of as well as the corresponding minimizer to the Appendix.
Proposition 2.1 (Minimization of ).
If as ,
(2.2) 
Moreover there exists a minimizer that is unique up to a sign, radial and can be chosen to be positive away from the boundary . It solves inside the variational equation
(2.3) 
with boundary condition and .
Remark 2.1 (Composition of the energy )
The remainders appearing on the r.h.s. of (2.2) can be interpreted as the kinetic energy due to Dirichlet boundary conditions: The bending of the TF density close to in order to fulfill the boundary condition produces some kinetic energy which is not negligible and can be estimated by means of the trial function used in the proof of the above proposition, i.e., as long as , and for larger angular velocities. Note indeed that the second correction becomes relevant only if .
The orders of those corrections can be explained as follows: If the TF density goes from its maximum of order 1 to 0 in a layer of thickness (because of the nonlinear term), yielding a gradient and thus a kinetic energy of order . If the thickness of the annulus where varies from to 0 becomes of order and the associated kinetic energy is .
Note that in both cases the kinetic energy associated with the boundary conditions is much larger than the radial kinetic energy of the profile which is in the first case and in the second one (see [CY, Section 4]): The condition is precisely due to the comparison of such energies for large angular velocities.
Finally we point out that, if , the correction of order due to Dirichlet boundary conditions can become much larger than two terms of order and contained inside (see the explicit expression (A.3) in the Appendix), so that the upper bound could be stated in that case .
Proof of Proposition 2.1.
The lower bound is trivial since it is sufficient to neglect the positive kinetic energy to get .
The upper bound is obtained by evaluating on a trial function of the form
(2.4) 
where is the normalization constant and a cutoff function equal to 1 everywhere except in the radial layer , , where it goes smoothly to 0, so that satisfies Dirichlet boundary conditions. The density coincides with if is below the threshold and is given by a regularization of above it, i.e., if , we set as in [CY, Eq. (4.9)]
(2.5) 
Notice that differs from only inside the interval and
(2.6) 
In order to estimate the normalization constant we use the bound , which implies
(2.7) 
The kinetic energy of is bounded as follows:
(2.8) 
where we refer to [CY, Eqs. (4.14) and (4.15)] for the estimate of the kinetic energy of .
The interaction term can be easily estimated as
(2.9) 
To evaluate the centrifugal term we act as in [CY, Eqs. (4.44) – (4.46)]: With analogous notation
(2.10) 
where we have integrated by parts twice and used (2.6), (2.7) and the normalization of . Hence one finally obtains
(2.11) 
It only remains to optimize w.r.t. , which yields , if , and otherwise, and thus the result. ∎
A crucial property of the density is stated in the following
Proposition 2.2 (Behavior of ).
The density admits a unique maximum at some point .
Proof.
The method is very similar to what is used in [CRY, Lemma 2.1], although in that case one considers the Neumann problem. After a variable transformation the functional becomes
(2.12) 
and the normalization condition
(2.13) 
We first observe that the Dirichlet boundary condition implies that cannot be constant, otherwise we would have everywhere, contradicting the mass constraint.
Suppose now that has more than one local maximum. Then it has a local minimum at some point with , on the right side of a local maximum at the position , i.e., . For , we consider the set : Since is continuous, the function
(2.14) 
is strictly positive and as . Likewise, for any , we set , so that
(2.15) 
has the same properties as .
Hence, by the continuity of , there always exist , such that and . Note that this implies that and are disjoint.
We now define a new normalized function by
(2.16) 
The gradient of vanishes in the intervals and and equals the gradient of everywhere else, so that the kinetic energy of is smaller or equal to the one of . The centrifugal term is lowered by , because is strictly decreasing and the value of on is larger than on . Finally since mass is rearranged from to , where the density is lower, .
Therefore the functional evaluated on is strictly smaller than , which contradicts the assumption that is a minimizer. Hence has only one maximum.
∎
The energy asymptotics (2.2) implies that the density is close to the TF minimizer :
Proposition 2.3 (Preliminary estimates of ).
If as ,
(2.17) 
(2.18) 
Proof.
Next proposition is going to be crucial in the proof of the main results since it allows to replace the density with the TF density : On the one hand, using the fact that the latter is explicit, this result will be used to obtain a suitable lower bound on in some region far from the boundary and, on the other hand, it implies that the boundary layer where goes to 0 is very small.
Proposition 2.4 (Pointwise estimate of ).
If with as ,
(2.20) 
for any .
On the other hand^{4}^{4}4This second estimate applies also if , in which case has to be set equal to 0. if ,
(2.21) 
for any .