THE MEAN-FIELD APPROXIMATION AND THE NON-LINEAR

THE MEAN-FIELD APPROXIMATION AND THE NON-LINEAR ¨ SCHRODINGER FUNCTIONAL FOR TRAPPED BOSE GASES

arXiv:1405.3220v2 [math-ph] 3 Aug 2014

` MATHIEU LEWIN, PHAN THANH NAM, AND NICOLAS ROUGERIE Abstract. We study the ground state of a trapped Bose gas, starting from the full manybody Schr¨ odinger Hamiltonian, and derive the nonlinear Schr¨ odinger energy functional in the limit of large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive nonlinear Schr¨ odinger ground state.

Contents 1. Introduction 2. Main results 2.1. Error bounds for the mean-field approximation 2.2. Error bounds for the Non Linear Schr¨ odinger model 3. Error bounds for Hartree theory 3.1. Quantum de Finetti and localization 3.2. Truncated two-body Hamiltonian 3.3. Bound on the localized energy 3.4. Conclusion: proof of Theorem 2.1 4. The NLS limit 4.1. From Hartree to NLS: proof of Proposition 2.3 4.2. Convergence of the many-body energy: proof of Theorem 2.4 4.3. Convergence of states: proof of Theorem 2.5 References

1 6 6 8 12 12 15 17 18 19 19 21 21 25

1. Introduction The impressive progress of cold atom physics [44, 48] during the last two decades gave a new impetus to the theory of many-bosons systems, in particular the study of Bose-Einstein condensates (BECs). In such a state of matter, many interacting particles (bosons) occupy the same quantum state and may thus be collectively modeled using a one-body nonlinear Schr¨ odinger description. Date: July 30, 2014. 1

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M. LEWIN, P.T. NAM, AND N. ROUGERIE

In this framework, the energy functional of a Bose-Einstein condensate of non-relativistic particles in1 Rd , interacting via a pair potential w, reads as (in appropriate units) Z 1 (1.1) |(∇ + iA) u|2 + V |u|2 + |u|2 (w ∗ |u|2 ). EH [u] := 2 Rd

This so-called Hartree functional includes a trapping potential V : Rd → R confining the particles in a bounded region of space (V (x) → ∞ when |x| → ∞), a feature ubiquitous in experiments with cold atoms. The vector potential A : Rd → Rd can model a (possibly artificial [18]) magnetic field or the Coriolis force due to the rotation of the atoms [2, 14, 23], in which case V has to incorporate a contribution due to the centrifugal force. The ground state of the system is obtained by minimizing (1.1) under the mass constraint Z |u|2 = 1. (1.2) Rd

Samples in which Bose-Einstein condensation is achieved are typically very dilute, and it is therefore relevant to think of the case where the range of the interaction potential w is much smaller than the size of the system. This situation can be modeled by using contact interactions, which amounts to formally set w = aδ0 proportional to the delta function at the origin. One then obtains the nonlinear Schr¨ odinger (or Gross-Pitaevskii) functional Z a |(∇ + iA) u|2 + V |u|2 + |u|4 . (1.3) Enls [u] := 2 Rd

Here a measures the strength of interparticle interactions. An appealing aspect of cold atoms experiments is the possibility to tune the value and even the sign of a, going from repulsive (a > 0) to attractive interactions (a < 0)2. In the latter case the system may collapse, i.e. there might not exist a ground state for (1.3). It is in fact the case if d = 2 and a ≤ −a∗ for some critical value a∗ > 0, and also if d ≥ 3 and a < 0. An important question is that of the relation between the macroscopic, nonlinear description using functionals such as (1.1) and (1.3), and the underlying, linear, physics based on the many-body Schr¨ odinger equation. In the latter model the system is described via a many-body Hamiltonian of the form HN =

N   X
2 − ∇xj + iA(xj ) + V (xj ) + j=1

1 N −1

X

1≤i −1. 

Except for the case of equality, this assumption is again also necessary. (3) d = 1 under no specific assumption on the interaction potential. See Section 2.2.1 for a detailed discussion on these assumptions. Of these three cases, the second is the one that presents the main novelty of the paper: in 2D, we present the first derivation of the attractive NLS ground state in the regime of stability. In 3D on the other hand our conditions on β are much more stringent than those in [38, 35] where the GP limit is covered in the case w ≥ 0. The result is thus not new but our method of proof may still be of interest, in particular because it provides error bounds, and treats on the same footing the case with and without vector potential A. In 1D, because of the Sobolev embedding, one may directly use contact interactions at the level of the many-body Hamiltonian (1.4) as in [34, 54], so the procedure of scaling a regular interaction potential is less relevant. However, we provide a derivation of the attractive 1D ground state with what seems to be an unprecedented precision on the error bound. Our main theorems are stated in the next section, and their proofs occupy the rest of the paper. Acknowledgement: The authors acknowledge financial support from the European Research Council (FP7/2007-2013 Grant Agreement MNIQS 258023) and the ANR (Mathostaq project, ANR-13-JS01-0005-01). PTN and NR have benefited from the hospitality of the Institute for Mathematical Science of the National University of Singapore.

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M. LEWIN, P.T. NAM, AND N. ROUGERIE

2. Main results We do not aim at optimal assumptions on the potentials V and w. We will assume that w is integrable, symmetric and uniformly bounded: We shall denote by

w ∈ L1 (Rd , R),

w(x) = w(−x)

and

|w(x)| ≤ C.

(2.1)

wN (x) := N dβ w(N β x) the scaled interaction potential, and use the same notation wN for the operator acting on the two-particle space H2 ≃ L2 (R2d ) as the multiplication by (x1 , x2 ) 7→ wN (x1 − x2 ). The fact that we consider a trapped system, as appropriate for experiments with cold atomic gases, is materialized by the following assumption on the one-body potential: V ∈ L1loc (Rd , R),

V (x) ≥ c|x|s − C

(2.2)

for some exponent s > 0 and some constants c > 0, C ≥ 0. Our estimates will essentially involve the exponent s. As for the vector potential A, that can model a magnetic field or Coriolis forces acting on rotating particles, it is sufficient to assume A ∈ L2loc (Rd , Rd ).

(2.3)

A particularly relevant example is given by A(x) = Ω(−x2 , x1 , 0) for d = 3 and A(x) = Ω(−x2 , x1 ) for d = 2, corresponding to Coriolis forces due to a rotation at speed Ω around the x3 -axis, or a constant magnetic field of strength Ω pointing in this direction. 2.1. Error bounds for the mean-field approximation. Our first main task is to provide quantitative bounds on the discrepancy between the ground state energy per particle corresponding to (1.4) (2.4) E(N ) := inf σHN HN and the nonlinear energy eH := inf EH [u] (2.5) kukL2 (Rd ) =1

given by the minimization of the Hartree functional Z   |(∇ + iA(x)) u(x)|2 + V (x)|u(x)|2 dx EH [u] := Rd ZZ 1 + |u(x)|2 wN (x − y)|u(y)|2 dxdy. (2.6) 2 d d R ×R Note that when β > 0 both EH and eH depend on N and that |eH | ≤ CN dβ . It will be convenient to introduce a slightly modified Hartree energy with the interaction wN replaced by wN − ε|wN | for some ε > 0 (that will later on be taken small enough): ZZ ε EHε [u] := EH [u] − |u(x)|2 |wN (x − y)||u(y)|2 dxdy. (2.7) 2 Rd ×Rd

We denote by

eεH :=

inf

kukL2 (Rd ) =1

EHε [u]

the corresponding ground state energy, which satisfies |eεH | ≤ C(1 + ε)N dβ .

(2.8)

MEAN-FIELD APPROXIMATION FOR TRAPPED BOSE GASES

When β > 0 we also consider the NLS energy functional Z Z   a 2 2 |u(x)|4 dx |(∇ + iA(x)) u(x)| + V (x)|u(x)| dx + Enls [u] = 2 d d R R with ground state energy enls = inf Enls [u] kukL2 (Rd ) =1

7

(2.9) (2.10)

that arises as the limit N → ∞ of (2.6). As announced, a will always be defined as Z w. (2.11) a := Rd

The ground state energy enls is finite in dimensions d ≥ 2 only under appropriate assumptions on a (e.g. a ≥ 0). For β = 0 it is well-known (see [32] and references therein) that

E(N ) = eH . (2.12) N →∞ N For the purpose of this paper we need to provide explicit estimates in the case of a confined system and for β ≥ 0, which is the content of the following lim

Theorem 2.1 (Error bounds for the mean-field approximation). We assume that (2.1), (2.2) and (2.3) hold true. • If d = 1 and β > 0, then we have

1 E(N ) − ≥ eH − CN 4+2/s . N • If d = 1 and β = 0, or if d ≥ 2 and 1 , 0≤β< d(1 + d/s + d/2) then we have E(N ) ε−1−d/2−d/s eH ≥ ≥ eεH − C 1−dβ(1+d/2+d/s) , N N for all 0 < ε ≤ 1.

eH ≥

(2.13)

(2.14)

(2.15)

Without more information on the potential w and when β > 0, it is not obvious that the problem eεH is actually close to eH and we will only be able to go further under some additional stability assumptions on w. We can however make the following Remark 2.2 (Estimates in the mean-field limit). Using the simple estimate eεH ≥ eH − CN dβ ε and optimizing with respect to ε, we can immediately deduce from (2.15) a bound which is valid for a smaller range of β but does not involve eεH anymore. Namely, if 1 0≤β< , (2.16) d(2 + d/s + d/2) then we have 1 E(N ) dβ− 2+d/s+d/2 eH ≥ ≥ eH − CN . (2.17) N

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M. LEWIN, P.T. NAM, AND N. ROUGERIE

If β = 0 then eH is independent of N and we have thus the uniform bound 1 E(N ) − ≥ eH − CN 2+d/2+d/s N which gives a bound on the rate of convergence in (2.12).

eH ≥

(2.18) △

With more information on w, it is clear that (2.18) is not optimal. Indeed, it has been proved in [33, 53, 26] that when the Hartree minimization problem eH has a unique nondegenerate minimizer, then the next order is given by Bogoliubov’s theory and it is of order 1/N . This is for instance the case if3 w b > 0 and the vector potential A is small enough. Without more assumptions on w, we are not aware of any existing quantitative error bound. For instance, in a gas rotating sufficiently fast for vortices to be nucleated (see [15, 16, 17, 52] and references therein), uniqueness is known to fail and our bound (2.18) seems to be the best so far. For illustration, we compute the rate of convergence we obtain in a few physically interesting situations (see Table 1), still in the Hartree case β = 0. We consider space dimensions d = 1, 2, 3, and compare the harmonic oscillator case s = 2 with the case of particles in a box where we set formally s = ∞. In fact our method also applies to the case of particles confined to a bounded domain and we obtain the rates for s = ∞ in this case. d=3 s = 2 N −1/5 s = ∞ N −2/7

d=2 d=1 N −1/4 N −1/3 N −1/3 N −2/5

Table 1. Rates of convergence to Hartree’s energy (β = 0). 2.2. Error bounds for the Non Linear Schr¨ odinger model. If β > 0, then eH and eεH still depend on N in our bounds (2.13) and (2.15). Our next task is to relate these energies to the NLS ground state energy enls . In general, eH will not converge to enls . For this to be true, some stability properties of the interaction potential w are needed, further discussed in the following. 2.2.1. Stability properties of w. An even potential W is called classically stable [51] when X W (xi − xj ) ≥ −CN, ∀x1 , ..., xN ∈ Rd , ∀N ≥ 2. (2.19) 1≤i −1 inf  (2.23) 2 2   u∈H 1 (R2 ) 2 kukL2 (R2 ) k∇ukL2 (R2 )

R which plays the same role as the assumption R2 W > −a∗ in the NLS case. • In dimension d = 1, the NLS nonlinearity is subcritical and the quantum system is always stable thanks to the kinetic energy, whence the absence of specific assumptions in this case. In order to simplify our presentation, from now on we use the word “stable” for a potential W that satisfies (2.23) in dimension d = 2 and (2.20) in dimension d ≥ 3. The importance of these concepts is illustrated in the following. Proposition 2.3 (Convergence of eH towards enls ). Let β > 0. We assume that (2.1), (2.2) and (2.3) hold true, and further suppose that |x|w(x) ∈ L1 (Rd ). • If d = 1, or d = 2, 3 with w stable, then we have |eH − enls | ≤ CN −β .

4with W − := − min(0, W ) the negative part of W

(2.24)

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M. LEWIN, P.T. NAM, AND N. ROUGERIE

Furthermore, minimizers for eH converge in the limit N → ∞ to a minimizer for enls , after extraction of a subsequence. • If d = 2 and  ZZ |u(x)|2 |u(y)|2 w(x − y) dx dy   R2 ×R2  < −1, (2.25) inf  2 2   u∈H 1 (R2 ) 2 kukL2 (R2 ) k∇ukL2 (R2 ) or if d ≥ 3 and w is not classically stable (in the sense of (2.20)), then lim eH = −∞.

N →∞

(2.26)

The decay condition |x|w(x) ∈ L1 is a technical assumption which allows us to obtain (2.24) by controling the error induced by replacing the interaction potential wN in the Hartree model with the delta-potential in the NLS model. Without this assumption we still have eH → enls but with no estimate on the convergence rate. Of course (2.26) immediately implies E(N ) = −∞ (2.27) lim N →∞ N since E(N ) ≤ N eH . In the above result, the stability condition on w Ris optimal in dimension d = 3. If w is classically unstable we have (2.27) but it could be that R3 w ≥ 0, and then enls is finite and therefore cannot be related to the limit of the N -body problem. In dimension d = 2 we are only missing the case of equality in (2.23) and the stability condition is therefore nearly optimal. 2.2.2. Convergence of the many-body problem. With Theorem 2.1 and Proposition 2.3 at hand, it is an easy task to deduce error bounds between E(N )/N and the NLS minimum energy. Theorem 2.4 (Derivation of the NLS ground state energy). Let β > 0. We assume that (2.1), (2.2) and (2.3) hold true, and further suppose that |x|w(x) ∈ L1 (Rd ). • If d = 1, then we have CN −β + enls ≥

1 E(N ) − ≥ enls − CN −β − CN 4+2/s N

(2.28)

for all β > 0. • If d = 2 and w is stable (in the sense of (2.23)), or if d = 3 and w − η|w| is stable (in the sense of (2.20)) for some 0 < η < 1, then 1−dβ(1+d/2+d/s) E(N ) − 2+d/s+d/2 , ≥ enls − CN −β − CN N provided that β satisfies (2.14). • Finally, if d = 3 and w is stable (in the sense of (2.20)), then we have

enls + CN −β ≥

enls + CN −β ≥ provided that β satisfies (2.16).

1 E(N ) dβ− 2+d/s+d/2 ≥ enls − CN −β − CN N

(2.29)

(2.30)

MEAN-FIELD APPROXIMATION FOR TRAPPED BOSE GASES

11

We remark that when d = 2, since the stability assumption (2.23) is strict, the stability for w implies the stability for w − η|w| for some 0 < η < 1. However, the same does not hold when d = 3 and the error estimate (2.29) is really better than (2.30) (and similarly for the range of β). The proofs of Theorems 2.1 and 2.4 also imply some estimates on the ground states themselves. In the absence of any assumption on the behavior of the Hartree functional we cannot convert them into convergence rates for the states. In the NLS limit we nevertheless obtain convergence of states and Bose-Einstein condensation: Theorem 2.5 (Convergence of states in the NLS limit). We use the same assumptions as in Theorem 2.4. Denote ΨN a ground state of the manybody Hamiltonian (1.4) and (n)

γN := Trn+1→N |ΨN ihΨN |

its n-body reduced density matrix. We have, modulo a subsequence, Z (n) dµ(u)|u⊗n ihu⊗n | lim γN = N →∞

(2.31)

u∈Mnls

strongly in the trace-class for any n ≥ 1, with µ a Borel probability measure supported on o n (2.32) Mnls = u ∈ L2 (Rd ), kukL2 = 1, Enls [u] = enls .

In particular, when the NLS ground state unls is unique (modulo a constant phase), we have convergence for the whole sequence (n)

⊗n lim γN = |u⊗n nls ihunls |, strongly in trace-class norm.

N →∞

(2.33)

R Uniqueness of unls is ensured when w is either ≥ 0 or small enough in absolute value in dimensions d = 1, 2, and the vector potential A is small enough. Loss of uniqueness does occur if these assumptions are not satisfied, which is intimately linked to symmetry breaking phenomena [3, 5, 16, 17, 27, 52]. When the NLS functional satisfies some stability properties one may obtain error estimates on states, see Remark 4.2. The main virtue of the estimates of Theorems 2.1 and 2.4 is that they do not depend on special properties of the interaction potentials (apart from the necessary stability conditions). They actually also do not depend on the vector potential A and they would be exactly the same if the one-body term in the many-particle Hamiltonian is perturbed by any bounded operator. As we have said, there is however no reason to think that the rates of convergence are optimal. When d ≥ 2, the estimates of Theorem 2.1 deteriorate with increasing β and this results in a decrease of the range of applicability to the NLS limit. For the typical settings described above we obtain the desired result E(N ) → enls when N → ∞ (2.34) N under the conditions listed in Table 2. In 3D we distinguish the case where w − η|w| is stable for some small fixed η where we obtain better estimates than when w is simply stable. In the former case we say that w is η-stable. It remains an open problem to improve the convergence rates we obtain and to generalize the results to larger values of β. In

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M. LEWIN, P.T. NAM, AND N. ROUGERIE

particular, we expect that one should obtain the same NLS limit for all 0 < β < 1 when d ≥ 2, see [38, 39] for the repulsive case. s=2 s=∞

d = 3, w stable d = 3, w η-stable β < 1/15 β < 1/12 β < 2/21 β < 2/15

d=2 β < 1/8 β < 1/6

Table 2. Maximal value of β in the NLS limit (β > 0). The rest of the paper is organized as follows: the core of the analysis is in Section 3 where we prove Theorem 2.1. Next we turn to the proof of Proposition 2.3 in Section 4 where we also conclude the proofs of Theorems 2.4 and 2.5. 3. Error bounds for Hartree theory In this section, we prove Theorem 2.1. 3.1. Quantum de Finetti and localization. Let γN be an arbitrary (mixed) state in N the bosonic Hilbert space HN = N s H, i.e. a positive operator satisfying Tr γN = 1. For (k) every k = 1, 2, ..., N , the k-particle density matrix γN is obtained by taking the partial trace over all but the first k variables: (k)

γN = Trk+1→N [γN ] One of the main advantages of the reduced density matrices is that we can write i h 1 1 E(N ) (2) = hΨN , HN ΨN i = TrH2 H2 γN N N 2 where γN = |ΨN ihΨN | and

(3.1)

H2 = − (∇x1 + iA(x1 ))2 + V (x1 ) − (∇x2 + iA(x2 ))2 + V (x2 ) + N dβ w(N β (x1 − x2 )).

When β = 0, H2 is independent of N and the limit E(N )/N → eH essentially comes (2) from the structure of γN in the large N limit. If one does not need an error estimate, this convergence follows easily from a compactness argument and the quantum de Finetti theorem [57, 29]. This is explained in [32, Section 3]. When β > 0, we have to deal with a N -dependent interaction potential and the compactness argument in [32] is not sufficient. Our strategy in this paper is to use a localization method to reduce the problem to a finite dimensional setting. We may then employ the following quantitative version of the quantum de Finetti theorem, originally proved in [12] (see [11, 28, 31] for variants of the proof and [22] for an earlier result in this direction): Theorem 3.1 (Quantitative quantum de Finetti in finite dimension). N Let K be a finite dimensional Hilbert space. For every state GN on KN := N s K and for every k = 1, 2, ..., N we have Z 4k dim K (k) (3.2) |u⊗k ihu⊗k |dµGN (u) ≤ TrKn GN − N SK where

(n)

GN := Trn+1→N [GN ]

MEAN-FIELD APPROXIMATION FOR TRAPPED BOSE GASES

13

and

 dµGN (u) := dim KN u⊗N , GN u⊗N du with du being the normalized uniform (Haar) measure on the unit sphere SK.

(3.3)

Remark 3.2. The measure µGN is a probability measure thanks to Schur’s formula Z N |u⊗k ihu⊗k |du. 1KN = dim K SK

We shall not need the explicit expression (3.3). If one could find a different construction giving a better error estimate, the convergence rates of our main theorems would be improved using the method we describe below. △ We will apply Theorem 3.1 to the low-lying energy subspace of the (magnetic) Schr¨ odinger 2 2 d operator H1 = − (∇ + iA) +V acting on H = L (R ). We denote by P− and P+ the spectral projectors above and below the energy cut-off L: P− = 1(−∞,L) (H1 ) , P+ = 1H − P− = P−⊥ .

(3.4)

NL := dim(P− H) = number of eigenvalues of H1 below L

(3.5)

Thanks to our assumption (2.2) and (2.3), the dimension of the low-lying subspace is finite. Moreover it is controlled by a semi-classical inequality “`a la Cwikel-Lieb-Rosenblum”, stated in the next lemma. We refer to [36, Chapter 4] for a thorough discussion of related inequalities. Lemma 3.3 (Low-lying bound states of the one-body Hamiltonian). Let V and A satisfy (2.2) and (2.3), respectively. Then for L large enough we have NL ≤ CLd/s+d/2 .

(3.6)

Proof. The number of eigenvalues of (−i∇ + A)2 + V below L can be estimated by    ZZ |p|2 +V −L 1 (−i∇ + A)2 + V − L e− L ≤ dx dp, NL ≤ TrL2 (Rd ) exp − d L (2π) Rd ×Rd

using [13, Thm. 2.1] and [55, Thm 15.8]. Using our assumption (2.2) that V (x) ≥ c|x|s − C and changing variables gives the result.  We will combine the de Finetti theorem and the localization method in Fock space, which provides the correct way of restricting a quantum N -body state to a subspace of H. Let us quickly recall this procedure, following the notation of [30]. Let γN be an arbitrary N -body (mixed) state. With the given projections P− , P+ , there + are localized states G− N , GN in the Fock space of the form

F(H) = C ⊕ H ⊕ H2 ⊕ · · · ± ± ± G± N = GN,0 ⊕ GN,1 ⊕ · · · ⊕ GN,N ⊕ 0 ⊕ · · ·

with the crucial property that their reduced density matrices satisfy  −1 X N   h i
N k ± (n) ⊗n (n) ⊗n P± γN P± = GN = Trn+1→k G± N,k n n k=n

(3.7)

(3.8)

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M. LEWIN, P.T. NAM, AND N. ROUGERIE

for any 0 ≤ n ≤ N . Here we use the convention that (n)

γN := Trn+1→N [γN ], which differs from the convention of [30], whence the different numerical factors in (3.8). + The relations (3.8) determine the localized states G− N , GN uniquely and they ensure that + G− N and GN are (mixed) states on the Fock spaces F(P− H) and F(P+ H), respectively: N X k=0

Tr

h

G− N,k

i

=

N X k=0

h i Tr G+ N,k = 1.

(3.9)

Due to (3.1), we are mainly interested in the two-particle density matrices. Applying the quantitative de Finetti Theorem 3.1 to the localized state G− N , we obtain the following Lemma 3.4 (Quantitative quantum de Finetti for the localized state.). Let γN be an arbitrary N -body (mixed) state and for every L > 0, we have Z ⊗2 (2) ⊗2 8NL ⊗2 ⊗2 |u ihu |dµN (u) ≤ TrH2 P− γN P− − N SP− H

where

dµN (u) =

N X k(k − 1) dµN,k (u), N (N − 1) k=2

E D ⊗k du. (3.10) u dµN,k (u) = dim(P− H)ks u⊗k , G− N,k

Proof. Applying the quantitative de Finetti Theorem 3.1 we have i h i Z h NL − ⊗2 ⊗2 TrHk G− TrH2 Tr3→k GN,k − |u ihu |dµN,k (u) ≤ 8 N,k k SP− H

⊗k where dµN,k (u) = dim(P− H)ks hu⊗k , G− N,k u idu. Combining this and (3.8) we get Z N  −1   i h X ⊗2 (2) ⊗2 N k 8NL ⊗2 ⊗2 |u ihu |dµN (u) ≤ TrH2 P− γN P− − TrHk G− N,k 2 2 k SP− H k=2

N i 8N h 8NL X k − 1 L = ≤ TrHk G− , N,k N N −1 N k=2

since

PN

− k=0 TrHk [GN,k ]

= 1.



Remark 3.5 (De Finetti measure for the n-body reduced density matrix). We will later apply the same idea to any n-body reduced density matrix of γN with n ≥ 2. The de Finetti measures obtained in this way, namely N  −1   N X X N k k(k − 1)...(k − n + 1) n dµN,k (u), dµN (u) = dµN,k (u) = N (N − 1)...(N − n + 1) n n k=n

k=n

G− N,k

= 0 for any k = 0 . . . N − 1, namely unless all the a priori depend on n, unless particles are P− localized. However, in some situations (e.g. in our case in Section 4.3), we can show that most particles are P− localized, and thus all these n-dependent measures actually converge to the same limit. △

MEAN-FIELD APPROXIMATION FOR TRAPPED BOSE GASES

15

We shall prove Theorem 2.1 in the following subsections. The core of the proof is in Section 3.3, where Lemma 3.4 plays an essential role, with some preparation in Section 3.2 and some final computations in Section 3.4. 3.2. Truncated two-body Hamiltonian. Recall from (3.1) that 1 E(N ) (2) = Tr[H2 γN ] N 2 with γN = |ΨN ihΨN |. In order to reduce to a finite dimensional setting, we will replace the Hamiltonian H2 by the localized operator P− ⊗ P− H2 P− ⊗ P− , up to a small modification of the interaction potential wN . The crucial fact is that if the energy cut-off L is large enough (in comparison with kwN kL∞ ), then the kinetic energy of the P+ localized particles can be used to control both the localization error and the number of P+ localized particles. For convenience we introduce a two-particle operator with modified interaction: H2ε = −(∇x1 − iA(x1 ))2 + V (x1 ) − (∇x2 − iA(x2 ))2 + V (x2 ) An important tool is then the following.

+ wN (x1 − x2 ) − ε|wN (x1 − x2 )|. (3.11)

Lemma 3.6 (Truncated two-body Hamiltonian). Assuming that 0 < ε ≤ 1 and L ≥ CN dβ ε−1 (resp. L ≥ Cε−2 if d = 1 and β > 0) for a large enough constant C, we have L (3.12) H2 ≥ P−⊗2 H2ε P−⊗2 + (P+ H1 P+ ⊗ 1 + 1 ⊗ P+ H1 P+ ). 2 The estimates in the d = 1 case use a simple lemma that we prove at the end of this subsection: Lemma 3.7 (Sobolev-type inequality in dimension d = 1). For any even potential W ∈ L1 (R), we have in dimension d = 1 Z 2 − W − C. H1 ⊗ 1 + 1 ⊗ H1 + W (x − y) ≥ −C

(3.13)

R

Proof of Lemma 3.6. For the two-body non-interacting Hamiltonian H1 ⊗ 1 + 1 ⊗ H1 = −(∇x1 − iA(x1 ))2 + V (x1 ) − (∇x2 − iA(x2 ))2 + V (x2 ),

we use that

H1 = P− H1 P− + P+ H1 P+ and obtain H1 ⊗ 1 + 1 ⊗ H1 = P− H1 P− ⊗ 1 + 1 ⊗ P− H1 P− + P+ H1 P+ ⊗ 1 + 1 ⊗ P+ H1 P+   = (P− )⊗2 H1 ⊗ 1 + 1 ⊗ H1 (P− )⊗2 + P− H1 P− ⊗ P+ + P+ ⊗ P− H1 P− + P+ H1 P+ ⊗ 1 + 1 ⊗ P+ H1 P+ .

Since H1 is bounded from below, we have P− H1 P− ≥ −CP− ≥ −C and we obtain   H1 ⊗1+1⊗H1 ≥ (P− )⊗2 H1 ⊗1+1⊗H1 (P− )⊗2 +P+ (H1 −C)P+ ⊗1+1⊗P+ (H1 −C)P+ . (3.14)

16

M. LEWIN, P.T. NAM, AND N. ROUGERIE

We now consider the interaction term and use the shorthand notation wN for the twobody multiplication operator (x1 , x2 ) 7→ wN (x1 − x2 ). We write again wN = (P− + P+ )⊗2 wN (P− + P+ )⊗2

= (P− )⊗2 wN (P− )⊗2 + (P− )⊗2 wN Π + ΠwN (P− )⊗2 + ΠwN Π,

(3.15)

with the orthogonal projection Π := P− ⊗ P+ + P+ ⊗ P− + P+ ⊗ P+ . To bound the error terms we use the inequality P AQ + QAP ≥ −εP |A|P − ε−1 Q|A|Q,

valid for any self-adjoint operator A, and any orthogonal projectors P, Q. If A is positive this follows by writing5     ε1/2 P ± ε−1/2 Q A ε1/2 P ± ε−1/2 Q ≥ 0

and the general case is obtained by using the same bound applied to A+ and A− separately. We thereby deduce that (P− )⊗2 wN Π + ΠwN (P− )⊗2 ≥ −ε(P− )⊗2 |wN |(P− )⊗2 − ε−1 Π|wN |Π

and, therefore,

wN ≥ (P− )⊗2 (wN − ε|wN |)(P− )⊗2 − (1 + ε−1 )Π|wN |Π.

(3.16)

Using now |wN | ≤ CN dβ and collecting our estimates we find

H2 ≥ (P− )⊗2 H2ε (P− )⊗2 + P+ (H1 − C)P+ ⊗ 1 + 1 ⊗ P+ (H1 − C)P+ − C(1 + ε−1 )N dβ Π   L 1 −1 ⊗2 ε ⊗2 − C − C(1 + ε N dβ )Π ≥ (P− ) H2 (P− ) + (P+ H1 P+ ⊗ 1 + 1 ⊗ P+ H1 P+ ) + 2 2 and the result follows in dimensions d ≥ 2 or when β = 0 and d = 1. When d = 1 and β > 0, we may come back to (3.16) and use Lemma 3.7 to obtain 1 H2 ≥ (P− )⊗2 H2ε (P− )⊗2 + (P+ H1 P+ ⊗ 1 + 1 ⊗ P+ H1 P+ ) 2  
+ Π L/4 − C + (H1 ⊗ 1 + 1 ⊗ H1 )/4 − (1 + ε−1 |wN | Π


1 ≥ (P− )⊗2 H2ε (P− )⊗2 + (P+ H1 P+ ⊗ 1 + 1 ⊗ P+ H1 P+ ) + Π L/4 − C − Cε−2 Π. 2

Proof of Lemma 3.7. In 1D we can bound  Z Z

2 |W | kuk2L∞ (R) ≤ CkW kL1 (R) kukL2 (R) u′ L2 (R) |W (x)||u(x)| ≤ R



(3.17)

R

for every u ∈ H 1 (R). We conclude that

2 −d2 /dx2 + W ≥ −d2 /dx2 − W − ≥ −C W − L1 (R) − C.

Using the pointwise diamagnetic inequality |(∇ + iA)u| ≥ |∇|u| | and removing the center of mass, the estimate is similar for the two-particle operator.  5We copy the proof that the diagonal part of a positive hermitian matrix controls the off-diagonal part.

MEAN-FIELD APPROXIMATION FOR TRAPPED BOSE GASES

17

3.3. Bound on the localized energy. Now we turn to the main step of the proof of Theorem 2.1: we compare the energy corresponding to the localized operator in the right side of (3.12) with the Hartree energy eH . Let γN = |ΨN ihΨN | be the density matrix of a ground state ΨN ∈ HN of the N -body Hamiltonian HN (known to exist in our setting of trapped systems). We have Lemma 3.8 (Lower bound to the localized energy). For 0 < ε ≤ 1, L ≥ CN dβ ε−1 (resp. L ≥ Cε−2 if d = 1 and β > 0) and N large enough, we have i CL1+d/s+d/2 1 h ⊗2 ε ⊗2 (2) i L h (1) Tr P− H2 P− γN + Tr P+ γN ≥ eεH − . (3.18) 2 4 N

Proof. We will employ the same notations for localized states G± N obtained from the N -body state γN and the projections P± as in Subsection 3.1. By Lemma 3.4, we have Z 8NL ⊗2 (2) ⊗2 ⊗2 ⊗2 |u ihu |dµN (u) ≤ TrH2 P− γN P− − N SP− H

with dµN defined as in (3.10). On the other hand,

⊗2 ε ⊗2

P H P ≤ 2L + (1 + ε)kwN kL∞ ≤ CL − 2 −

(3.19)

in operator norm because we have truncated the high energy spectrum of the one-body part and the two-body potential is bounded by kwN kL∞ ≤ CN dβ ≤ CL. In dimension d = 1, the estimate is the same but we use that |wN | ≤ H1 ⊗ 1 + 1 ⊗ H1 + C, by Lemma 3.7. Therefore, Z   CLNL 1 h ⊗2 ε ⊗2 (2) i 1 Tr P− H2 P− γN ≥ TrH2 H2ε |u⊗2 ihu⊗2 | dµN − 2 2 SP− H N Z CL1+d/s+d/2 EHε [u]dµN − ≥ N SP− H Z
CL1+d/s+d/2 EHε [u] − eεH dµN + eεH µN (SP− H) − ≥ (3.20) N SP− H where we have used the estimate NLR≤ CLd/s+d/2 of Lemma 3.3. Applying the variational principle EHε [u] ≥ eεH and computing dµN explicitly using (3.10), we get i CL1+d/s+d/2 h X k(k − 1) 1 h ⊗2 ε ⊗2 (2) i Tr P− H2 P− γN ≥ eεH TrHk G− . N,k − 2 N (N − 1) N N

(3.21)

k=2

If eεH ≤ 0 then we simply write

N N h i X h i X k(k − 1) − ≤ G TrHk G− Tr k H N,k N,k ≤ 1 N (N − 1) k=2 k=2 R ε and we are done. If eH > 0, then we need to prove that dµN ≃ 1 and for this we use the positive term i i  h i h h (1) (1) (2) . Tr (P+ ⊗ 1 + 1 ⊗ P+ )γN = 2 Tr P+ γN = 2 1 − Tr P− γN

18

M. LEWIN, P.T. NAM, AND N. ROUGERIE

First, recall that by (3.8) Tr

h

(1) P− γN

i

N h i X k TrHk G− = N,k . N k=1

Then, using Jensen’s inequality, we have

N N i i Tr P γ (1) h h X N X k2 k(k − 1) − N − = G TrHk G− Tr k N,k N,k − N − 1 N (N − 1) N −1 N2 H k=0 k=2 ! N i 2 Tr P γ (1) h X k N − N − − Tr k GN,k ≥ N −1 N H N −1 k=0

= Therefore, denoting by

 Tr P− γN N (1) 2 − Tr P− γN . N −1 N −1 

(1)

(1)

λ := Tr P− γN ≤ 1 the fraction of particles localized on the low energy states, we have the lower bound N λ2 − λ ε L CL1+d/s+d/2 1 h ⊗2 ε ⊗2 (2) i L (1) Tr P− H2 P− γN + Tr P+ γN ≥ eH + (1 − λ) − . 2 4 N −1 4 N When L > 8N eεH /(N − 1), the minimum of the right side is attained at λ = 1. In dimension d ≥ 2, we have eεH ≤ CN dβ and L ≥ CN dβ /ε, and hence the condition L > 8N eεH /(N − 1) is always fulfilled for N large enough. Similarly, if d = 1 and β > 0, we use that |eεH | ≤ C and L ≥ Cε−2 . In all cases we get i 1 h ⊗2 ε ⊗2 (2) i L h CL1+d/s+d/2 (1) Tr P− H2 P− γN + Tr P+ γN ≥ eεH − (3.22) 2 4 N which is the desired inequality.  3.4. Conclusion: proof of Theorem 2.1. The upper bound in (2.15) is trivial, taking a factorized trial state for the N -body energy. To show the lower bound, let us consider a ground state γN = |ΨN ihΨN | of HN . From Lemma 3.6 and Lemma 3.8, we have the lower bound i L1+d/s+d/2 1 h E(N ) (1) (3.23) ≥ eεH − C + Tr P+ H1 γN N N 4 which gives the result for L = CN dβ ε−1 in dimensions d ≥ 2 and, if β = 0, in dimension d = 1. The last term in (3.23) is positive and will be useful later. For now we can just drop it from the lower bound. If d = 1 and β > 0, we take L = Cε−2 . By Lemma 3.7 we have |eH | ≤ C and H2ε = (1 − ε)H2 + ε (H1 ⊗ 1 + 1 ⊗ H1 − (wN )− ) ≥ (1 − ε)H2 − εC,

which implies eεH ≥ eH − Cε. Thus we find

i ε−3−2/s 1 h E(N ) (1) ≥ eH − Cε − C + Tr P+ H1 γN . N N 4 The estimate (2.13) then follows after optimizing with respect to ε.



MEAN-FIELD APPROXIMATION FOR TRAPPED BOSE GASES

19

4. The NLS limit Now we explain how to go from the Hartree model to the NLS model and prove Proposition 2.3 and Theorems 2.4 and 2.5. 4.1. From Hartree to NLS: proof of Proposition 2.3. The key observations are summarized in the following lemma. Lemma 4.1 (Stability of the effective one-body functionals). Let d ≤ 3. We assume that (2.1), (2.2) and (2.3) hold true, and that |x|w(x) ∈ L1 (Rd ). When d = 2, 3 we also assume that w is stable (in the sense of (2.23) when d = 2 and (2.20) when d = 3). Then the set of minimizers for Enls is non-empty and compact in the quadratic form domain of H1 . Also, for any normalized function u ∈ L2 (Rd ) we have Z |∇|u|(x)|2 dx ≤ C(EH [u] + C) (4.1) Rd

and

|EH [u] − Enls [u]| ≤ CN

−β



1+

Z

2

Rd

|∇|u|(x)| dx

2

.

(4.2)

Proof. We start by proving (4.1). When d = 3, since w satisfies (2.20) the nonlinear term is positive, and therefore the inequality (4.1) follows immediately from the diamagnetic inequality |(∇ + iA)u| ≥ |∇|u| | and the assumption that V ≥ −C. If d = 2, we use the stability assumption on w to obtain, for η small enough, ZZ Z 1 2 |u(x)|2 |u(y)|2 w(x − y) dx dy |∇|u|| + C + EH (u) ≥ 2 2 2 2 R ×R R  Z ZZ 1 1−η 2 2 2 ≥ |∇|u|| + |u(x)| |u(y)| w(x − y) dx dy 1−η 2 R2 R2 ×R2 Z η + |∇|u||2 1 − η R2 Z η |∇|u||2 . ≥ 1 − η R2 Finally, in dimension d = 1, we use Lemma 3.7 and obtain Z 2 Z  1 ⊗2 1 1 − ⊗2 EH [u] = u , H2 u w + hu, H1 ui ≥ |u′ |2 − C. ≥ −C − C 2 2 2 R R

Next, we prove (4.2). We change variables to write Z  Z ZZ 1 1 |u|4 |u(x)|2 wN (x − y)|u(y)|2 dxdy − EH [u] − Enls [u] = w 2 2 d d d R Z ZR ×R   1 |u(x)|2 w(z) |u(x + zN −β )|2 − |u(x)|2 dxdz = 2 Rd ×Rd ZZ Z 1   1 2 = |u(x)| w(z) ∇|u|2 (x + tzN −β ) · zN −β dt dxdz 2 Rd ×Rd 0

20

M. LEWIN, P.T. NAM, AND N. ROUGERIE

and hence EH [u] − Enls [u] ≤ N −β

Z

Rd



|z||w(z)| dz |u|2 ∗ ∇|u|2 L∞ (Rd ) .

In dimensions d ≤ 3, we can write by the Young and Sobolev inequalities

2

|u| ∗ |∇|u|2 | ∞ d ≤ kuk3 6 k∇|u|k 2 d ≤ C k|u|k4 1 d , L (R ) L H (R ) L (R )

which concludes the proof of (4.2).



With Lemma 4.1 at hand, it is now easy to prove Proposition 2.3. Proof of Proposition 2.3. We first prove |eH − enls | ≤ CN −β .

(4.3)

Let v be a minimizer for enls . Then |v| ∈ H 1 (Rd ) by (4.1). Consequently,

eH ≤ EH (v) ≤ Enls (v) + CN −β k|v|k4H 1 (Rd ) = enls + CN −β .

Similarly, pick for every N a minimizer uN for eH . By (4.1), |uN | is uniformly bounded in H 1 (Rd ). Hence enls ≤ Enls (uN ) ≤ EH (uN ) + CN −β k|uN |k4H 1 (Rd ) ≤ eH + CN −β , which concludes the proof of (4.3). Next we prove that eH → −∞ as N → ∞ when w is not stable in dimensions d ≥ 2. Let u ∈ Cc∞ (Rd ) with support in the unit ball and kukL2 = 1. Using as trial state vN (x) = N dβ/2 u(N β x), our assumption (2.2) on V and the pointwise estimate |∇vN + iAvN |2 ≤ (1 + ε)|∇vN |2 + (1 + ε−1 )|A|2 |vN |2 , we find that eH ≤ C + (1 + ε)N + = (1 + ε)N 2β

Z

2β

Z

2

Rd

N dβ 2

Rd

−1

|∇u| + (1 + ε

ZZ

Rd ×Rd

|∇u|2 +

)N

dβ

kukL∞

Z

2

|x|N β ≤1

|A| + CN

−Sβ

Z

Rd

|x|S |u|2

|u(x)|2 w(x − y)|u(y)|2 dxdy

N dβ 2

ZZ

Rd ×Rd

|u(x)|2 w(x − y)|u(y)|2 dxdy + o(N dβ ).

If d ≥ 3 and w is not classically stable, we choose u to have ZZ |u(x)|2 w(x − y)|u(y)|2 dxdy < 0, Rd ×Rd

and we conclude R that limN →∞ eH = limN →∞ E(N )/N = −∞ since E(N )/N ≤ eH . If d = 2, we take ε2 = |x|≤N −β |A|2 and obtain  Z ZZ 1 eH 2 2 2 |∇u| + |u(x)| w(x − y)|u(y)| dxdy . inf lim sup 2β ≤ 2 R2 ×R2 u∈H 1 (R2 ) N →∞ N R2 kukL2 (R2 ) =1

The right side is strictly negative by assumption.



MEAN-FIELD APPROXIMATION FOR TRAPPED BOSE GASES

21

4.2. Convergence of the many-body energy: proof of Theorem 2.4. In the cases d = 1, d = 2 with w stable, the result immediately follows from Theorem 2.1, Remark 2.2 and Theorem 2.3. If w − η|w| is stable for some 0 < η < 1, so is w − ε|w| for all 0 ≤ ε ≤ η. Then we can apply Proposition 2.3 with w replaced by w − ε|w| for some ε < 1 to be tuned later on and get |eεnls − eεH | ≤ CN −β (4.4) R R R ε where enls is the NLS minimization problem, with a = w replaced by w−ε |w|. Arguing as in the proof of Lemma 4.1, it is not difficult to see that |eεnls − enls | ≤ Cε. Therefore Theorem 2.1 provides the bound ε−1−d/2−d/s E(N ) ≥ enls − Cε − CN −β − C 1−dβ(1+d/2+d/s) . N N Optimizing over ε gives (2.29). enls + CN −β ≥



4.3. Convergence of states: proof of Theorem 2.5. We split the proof in four steps for clarity. (n) Step 1, strong compactness of density matrices. We first note that γN is by definition bounded in the trace-class, so that we can extract a subsequence along which (n)

γN ⇀∗ γ (n)

(4.5)

as N → ∞. Modulo a diagonal extraction argument, one can assume that the convergence is along the same subsequence for any n. We now argue that the convergence is actually strong. We start by proving that i h h
2
(1) i (1) (4.6) Tr H1 γN = Tr − ∇ + iA + V γN ≤ C,

independently of N . To this end, pick some α > 0 and define HN,α =

N   2  1+α X − ∇ + iA(xj ) + V (xj ) + N −1 j=1

Noticing for instance that

X

1≤i 2, Z (n) |u⊗n ihu⊗n |dµ(u), γN → Mnls

strongly in the trace-class when N → ∞. In view of the definition of µ this follows from the estimate Z (n) ⊗n ⊗n Tr γN − |u ihu |dµN (u) → 0. (4.9) SP− H

(n)

To see that (4.9) holds we first define a measure approximating γN , as indicated in Remark 3.5. Arguing as in the proof of Lemma 3.4 we have Z nNL (n) |u⊗n ihu⊗n |dµnN (u) ≤ C TrHn P−⊗n γN P−⊗n − (4.10) N SP− H

where

dµnN (u)

=

N  −1   X N k

k=n

n

n

dµN,k (u)

(4.11)

and dµN,k is the same measure as in (3.10). An estimate similar to (4.7) next shows that Z (n) ⊗n ⊗n n |u ihu |dµN (u) → 0. TrHn γN − SP− H

Using the easy bound (see [32, Section 2])  −1    n N k k = + O(N −1 ) n n N

together with the triangle inequality and Schur’s formula (see Remark 3.2) we next deduce from (4.10) that  2  n ! Z N i h X (n) k k ⊗n ⊗n − TrHk G− |u ihu |dµN (u) ≤ Tr γN − N,k N N SP− H k=0 n−1 i h X  k n + TrHk G− N,k N k=0  2  h i X k 2 (4.12) TrHk G− + N,k + o(1). N k=0

24

M. LEWIN, P.T. NAM, AND N. ROUGERIE

Finally, combining our previous estimates gives  N  X k 2

i h TrHk G− N,k → 1

 N  X k 2

i h TrHk G− N,k → 1.

k=2

N

and using (3.9) we in fact have

k=0

N

Then by Jensen’s inequality 1≥

 N  X k n k=0

N

TrHk

h

G− N,k

i

≥

 N  X k 2 k=0

N

TrHk

h

G− N,k

i

!n/2

→ 1.

Inserting this and (3.9) in (4.12) concludes the proof of (4.9) and thus that of the theorem.  We finally note that our method can give quantitative estimates on reduced density matrices in some special cases: Remark 4.2 (The case of a stable NLS functional). If a stability estimate of the form Enls [u] ≥ enls + c inf

v∈Mnls

ku − vk2

(4.13)

holds at the level of the NLS functional, for some norm k . k (say the L2 norm), it is easy to see that the previous method leads to quantitative estimates on density matrices. In particular, if the NLS minimizer is unique (up to a constant phase), non-degenerate, and (4.13) holds, one may obtain quantitative bounds on the depletion of the condensate. Indeed, it immediately follows from the above considerations and (4.13) that Z

|ui hu| − |unls i hunls | 2 2 dµN (u) ≤ rN S u∈SP− H

where S2 denotes the Hilbert-Schmidt class. Then, using Jensen’s inequality together with the fact that |unls ihunls | is a rank-one projection we have

Z



≤ C √rN |uihu|dµ (u) − |u ihu | N nls nls

u∈SP− H

S1

and thus, in view of our previous bounds,

(n) ⊗n  ⊗n

γN − unls unls

S1 (L2 (R3 ))

√ ≤ Cn rN ,

where Cn depends only on n. Since we do not want to rely on assumptions such as (4.13) we do not pursue in this direction. One may consult e.g. [9] or [24, Section 6] for discussions of estimates of the form (4.13). △

MEAN-FIELD APPROXIMATION FOR TRAPPED BOSE GASES

25

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