Central limit theorem for Bose gases interacting through singular potentials

We consider a system of N bosons in the limit N→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \rightarrow \infty $$\end{document}, interacting through singular potentials. For initial data exhibiting Bose–Einstein condensation, the many-body time evolution is well approximated through a quadratic fluctuation dynamics around a cubic nonlinear Schrödinger equation of the condensate wave function. We show that these fluctuations satisfy a (multi-variate) central limit theorem.


Introduction
We consider a system of N bosons with Hamilton operator acting on L 2 s (R 3N ), the subspace of L 2 (R 3N ) consisting of functions which are symmetric with respect to permutations. The N -dependent two-body interaction potential is given through Simone Rademacher acknowledges partial support from the NCCR SwissMAP. This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 754411. B Simone Rademacher simone.rademacher@ist.ac.at In the following, we assume V ≥ 0 to be smooth, spherically symmetric, and compactly supported. For β = 0, the Hamiltonian (1.1) describes the mean-field regime characterized by a large number of weak collisions, whereas for β > 1/3 the collisions of the particles are rare but strong. In the Gross-Pitaevskii regime (β = 1), pair correlations play a crucial role. Here, we study intermediate regimes β ∈ (0, 1) in the limit N → ∞ where the particles interact through singular potentials.
The time evolution is governed by the Schrödinger equation For β = 0 (mean-field regime), the solution of (1.2) can be approximated by products of solutions of the Hartree equation with initial data ϕ 0 ∈ L 2 (R 3 ). See, for example, [1][2][3][4][5]10,16,[20][21][22]25,26,34]. For 0 < β ≤ 1, on the other hand, the solution ψ N ,t of (1.2) can be approximated by the nonlinear Schrödinger equation with σ = V (0) if β < 1 and σ = 8π a 0 if β = 1 (Gross-Pitaevskii regime). Hereafter, a 0 denotes the scattering length associated with the potential V defined through the solution of the zero-energy scattering equation with boundary condition f (x) → 1 as |x| → ∞. Then, outside the support of V , the solution f is given through where a 0 is defined as the scattering length of the potential V . In [17][18][19], it is shown that if the one-particle reduced density γ N associated with ψ N satisfies γ N → |ϕ 0 ϕ 0 | in the trace norm topology and then the one-particle reduced density γ N ,t associated with the solution ψ N ,t of (1.2) obeys where ϕ t denotes the solution of (1.3). In fact, in [17] considering the case β < 1, the energy condition (1.6) on the initial data is not needed. For more results in the Gross-Pitaevskii regime, see [7,12,15,30,31]. An overview on the derivation of the nonlinear Schrödinger equation from many-body quantum dynamics is given in [8,23,33].

Norm approximation
Besides the convergence of the one-particle reduced density γ N ,t associated with ψ N ,t , the norm approximation of ψ N ,t has been studied for different settings of β ∈ (0, 1) in [11,24,28,29]. Our result is based on the norm approximation obtained in [11] covering β < 1 whose ideas we explain in the following.
Truncated Fock space. As first step toward the norm approximation in [11], the contribution of the Bose-Einstein condensate is factored out. This is realized through the unitary U ϕ N ,t : L s R 3N → F ≤N ⊥ϕ N ,t . It maps the N -particle sector of the bosonic Fock space into the truncated Fock space defined over the orthogonal complement L 2 ⊥ϕ N ,t R 3 of the subspace of L 2 (R 3 ) spanned by the condensate wave function ϕ N ,t . This unitary has first been used in [27] in the mean-field regime. Its definition is based on the observation that every ψ N ∈ L 2 s (R 3N ) has a unique decomposition where α (n) ∈ L 2 ⊥ϕ N ,t (R 3 ) ⊗ s n for all n = 1, . . . , N . Then, U ϕ N ,t ψ N = {α (0) , α (1) , . . . , α (n) }.
This unitary satisfies the following properties proven in [27] Here a * ( f ), a( f ) denote the standard creation and annihilation operators on the bosonic Fock space F . On the truncated Fock space, we define modified creation and annihilation operators The modified creation operator b * ( f ) excites one particle from the condensate into its complement, while b( f ) annihilates an excitation into the condensate. We define the vector ξ N ,t := U ϕ N ,t ψ N ,t representing the fluctuation outside the condensate and observe From the truncated Fock space to the bosonic Fock space. We approximate the generator L N ,t acting on the truncated Fock space only with a modified generator L N ,t defined on the whole bosonic Fock space. We consider regimes with a small number of excitations N + (t). For this reason, we realize the approximation of L N ,t through L N ,t by replacing For a precise definition, see [11, eq. (54)].
Correlation structure through Bogoliubov transformation. In the intermediate regime, correlations are important (at least if β > 1/2). For their implementation, we consider for fixed > 0 the ground state of the scattering equation with Neumann boundary conditions on the ball B (0). We fix f N (x) = 1 for all |x| = and extend f N to R 3 by setting f N (x) = 1 for all |x| ≥ .
In [10], the nonlinear Schrödinger equation (1.3) is replaced by the N -dependent Hartree equation (1.12) with initial data ϕ N ,0 = ϕ 0 (the condensate wave function at time t = 0) to approximate the time evolved condensate wave function. The well-posedness of (1.12) is shown in [10,Appendix B]. The correlation structure is implemented through the Bogoliubov transformation 13) where η N ,t denotes the Hilbert-Schmidt operator with integral kernel Here, ω N = 1 − f N and ϕ N ,t are as defined in (1.11) resp. (1.12) and q N , The Bogoliubov transformation acts on the creation and annihilation operators as for all f ∈ L 2 (R 3 ). The operators sinh η N ,t and cosh η N ,t are defined through the absolutely convergent series of products of the operator η N ,t (1. 15) Let G N ,t be the generator given through (1. 16) In fact, the special choice of (1.11) and (1.12) allows crucial cancellations in the generator G N ,t . Note that G N ,t consists of terms which are quadratic in creation and annihilation operators and of terms of higher order. Nevertheless, in [11,Lemma 5], it is shown that G N ,t can be approximated through the generator G 2,N ,t containing quadratic terms only.
Limiting quadratic dynamics. We are interested in the limit N → ∞ of G 2,N ,t defined in (1.16). In order to replace the Bogoliubov transformation T N ,t defined in (1.13) with a limiting one, we define the limiting kernel ω ∞ of ω N through for |x| ≤ and ω ∞ (x) = 0 otherwise. Here, we used the notation b 0 = dx V (x). Furthermore, the solution ϕ N ,t of the modified Hartree equation (1.12) with initial data ϕ 0 ∈ H 4 (R 3 ) can be approximated with the solution ϕ t of (1.3) with σ = V (0) and with initial data ϕ 0 . To be more precise, [10,Proposition B.1] shows that there exists a constant C > 0 (depending on ϕ 0 H 4 ) such that with γ = min{β, 1−β}. Standard arguments (see, for example, [10, Proposition B.1]) imply that there exists a constant C > 0 such that for all n ∈ N. The approximations (1.17) and (1.18) lead to a limiting kernel We define the limiting Bogoliubov transformation where γ = min{β, 1 − β}.
In order to define the limiting dynamics, we introduce some more notation. We use the shorthand notation j x (·) = j(·, x) for any j ∈ L 2 (R 3 × R 3 ). Furthermore, we decompose sh η t = η t + r t , ch η t = 1 + p t and for all x, y ∈ R 3 . A slight modification of the arguments in [10, Appendix C] shows some properties of the kernels. For these, we consider initial data ϕ 0 ∈ H 4 (R 3 ) of (1.3). There exist a constant C > 0 (depending only on ϕ 0 H 4 (R 3 ) and on V ) such that on the one hand ch η t ≤ C, and k t 2 , η t 2 , sh η t 2 , p t 2 , r t 2 , μ t 2 ≤ C, (1.23) where · denotes the operator norm. On the other hand, denoting with ∇ 1 k t and ∇ 2 k t the operator with the kernel ∇ x k t (x; y) (1.24) Furthermore, let Δ 1 r t resp. Δ 2 r t be the operator having the kernel Δ x r t (x; y) resp. Δ y r t (x; y), then for all i = 1, 2 In order to simplify notation, we write in the following sh η t = sh, ch η t = ch resp. r t = r, k t = k, p t = p.

Definition 1
We define the limiting dynamics U 2 (t; s) satisfying where G 2,t is given by (1.30) Here, we used the notation K = dx a * x (−Δ x ) a x and K 1,t = q t K 1,t q t and K 2,t = (q t ⊗ q t ) K 2,t where K 1,t is the operator with integral kernel and K 2,t is the function given through for all N sufficiently large and all t ∈ R.

Bogoliubov transformation
The limiting dynamics U 2 (t; s) defined in (1.26) is quadratic in creation and annihilation operators. As the following proposition shows, it gives rise to a Bogoliubov transformation defined in the following. For this, we first define On the one hand, On the other hand, the commutation relations imply for (1.34) It turns out that a Bogoliubov transformation ν is of the form The following proposition is proven in Sect. 2.2.

Proposition 1
Let U 2 (t; s) be the dynamics defined in (1.27). For every t, s ∈ R, there exists a bounded linear map (1.36)

Central limit theorem
From a probabilistic point of view, (1.7) implies a law of large numbers, in the sense that for a one-particle self-adjoint operator O on L 2 (R 3 ) and for every δ > 0 Here O ( j) denotes the operator on L 2 (R 3N ) acting as O on the jth particle and as identity elsewhere. The proof of (1.37) follows from Markov's inequality (see [13]). As a next step, we are interested in a central limit theorem. For this, we consider the rescaled random variable where ϕ N ,t denotes the solution of (1.12) with initial data ϕ N ,0 = ϕ 0 . We consider initial data ψ N ,0 of the form ψ N ,0 = U * ϕ 0 1 ≤N T * N ,0 Ω exhibiting Bose-Einstein condensation [11,Theorem 3]. As a consequence, such a initial data satisfy a law of large numbers in the sense of (1.37). Moreover, such initial data obeys a central limit theorem in the sense that Here, G 0 denotes the centered Gaussian random variable with variance σ 0 2 2 , where following from Theorem 1 for time t = 0. Note that initial data of the form ψ N ,0 = U * ϕ 0 1 ≤N T * N ,0 Ω describe approximate ground states of trapped systems [9]. In experiments, such initial data are prepared by trapping particles through external fields and by cooling them down to extremely low temperatures so that the system essentially relaxes to its ground state.
The validity of a central limit theorem for the ground state of trapped systems has already been addressed in [32]. To be more precise, [32] considers the ground state of (1.1) for β = 1, i.e. in the Gross-Pitaevskii regime. The ground state is known to exhibit Bose-Einstein condensation. It is proven that the ground state satisfies a central limit theorem. The arguments of the proof can be adapted to the intermediate regime β < 1 using the norm approximation for the ground state obtained in [9]. Now, we consider the time evolution of the initial data ψ N ,0 = U * ϕ 0 1 ≤N T * N ,0 Ω with respect to the Schrödinger equation (1.2) and show the validity of a (multi-variate) central limit theorem.
Theorem 1 Let β ∈ (0, 1) and assume V to be radially symmetric, smooth, compactly supported and point-wise nonnegative. Furthermore, fix > 0 (independent of N ). Let ϕ t denote the solution of (1.3) and ϕ N ,t the solution of (1.12) both with initial data ϕ 0 ∈ H 4 (R 3 ). Moreover, we denote by ψ N ,t the solution of the Schrödinger equation Assume t ∈ C k×k , given through A similar result has been established in [6,13] for the mean-field regime characterized through weak interaction of the particles. It is shown that fluctuations around the nonlinear Hartree equation of bounded self-adjoint one-particle operators satisfy a (multi-variate) central limit theorem. We show that this result is true in the intermediate regime, where the interaction is singular, too. In particular, the correlation structure which becomes of importance in the intermediate regime does not affect the validity of a central limit theorem. However, it affects the covariance matrix (1.41) through the Bogolioubiv transform T t .
Similarly as in [13,Corollary 1.3 ], Theorem 1 implies a Berry-Esséen-type central limit theorem. To be more precise, we consider a bounded self-adjoint operator O on L 2 (R 3 ) and the random variable For every α < min{β/2, where G t is the centered Gaussian random variable with variance σ t Moreover, note that the covariance matrix (1.41) resp. the variance (1.43) are completely determined by the Bogoliubov transform T t defined in (1.21) and the quadratic fluctuation dynamics U 2 (t; 0) defined in (1.27). Theorem 1 resp. the properties (1.36) of the operators U (t; 0), V (t; 0) show that the solution of the Schrödinger equation (1.2) modulo the extraction of the condensate is approximately a quasi-free state for quasi-free initial data. This observation coincides with results in [24,28,29].

Preliminaries
The proof of Theorem 1 is based on the norm approximation (1.31) from [11]. In the following, we collect useful properties of the unitaries used therein.
To this end, we define the more general quadratic dynamics U gen (t; s).

Definition 3
Let U gen (t; s) be the dynamics satisfying i∂ t U gen (t; s) = G gen,t U gen (t; s), where the generator G gen,t is of the form for constants c, C > 0.
In the following, we prove the results for the dynamics U gen (t; s). As the next Lemma shows, the results then apply to U 2 (t; s), too.

Lemma 1
The dynamics U 2 (t; s) defined in Definition 1 is of the form of U gen (t; s) defined in Definition 3.
Proof By the definition (1.27) of G 2,t , we split and consider each of the summands separately. First, we consider G V 2,t defined in (1.28), which is again split into four terms. The first one, G V , 1 2,t of the r.h.s. of (1.28), satisfies assumption (2.3) since on the one hand (1.19) and (1.23). For the same reasons, the second term G V , 2 2,t of the r.h.s. of (1.28) satisfies assumption (2.3), too. For the third term G V , 3 2,t , the definition of K 2,t implies 19) and (1.23). The fourth term G V , 4 2,t satisfies the assumption (2.2) due to (1.19).
Furthermore, a transformation of variables shows Therefore, G λ 2,t is of form (2.2). Moreover, for the term G K 2,t − K , we observe with (1.23) and (1.25) The remaining bounds follow in the same way. Note that (1.24) implies the bound Moreover, by definition (1.17) of the limiting kernel, ω ∞ ∈ L p (R 3 ) for all p < 3. Hence, the remaining terms of G K 2,t satisfy the assumptions, too. We are left with the first term of the r.h.s. of (2.4). We write T t = e −B(η t ) . The properties (1.14) of the Bogoliubov transformation lead to Since η t 2 ≤ Ce C|t| from (1.24), these terms satisfy assumption (2.3), too.
As proven in [11,Proposition 8], any moments of the number of particles operator are approximately preserved with respect to conjugation with the Bogoliubov transformation T N ,t . To be more precise for every fixed k ∈ N and δ > 0, there exists C > 0 such that As the following Lemma shows, the moments of number of particles operator are propagated in time with respect to the quadratic U gen (t; 0).

Lemma 2 Let U gen (t; s) be as defined in Definition 3 and ψ ∈ F . For every k ∈ N,
there exists a constant C > 0 such that for all t ∈ R ψ, U gen (t; s) * (N + 1) k U gen (t; s)ψ ≤C exp(C exp(C|t − s|)) ψ, (N + 1) k ψ .
Proof We compute the derivative Using the commutation relations and the definition (2.2), we find (2.6) For the first term of the right-hand side, the commutation relations yield dxdy H where C depends on k ∈ N. The second of the r.h.s. of (2.6) follows in the same way. Hence, there exists C > 0 such that Hence, the Gronwall inequality implies ψ, U * gen (t; s)(N + 1) k U gen (t; s)ψ ≤ C exp (C exp (C|t − s|)) ψ, (N + 1) k ψ .
In [13,Proposition 3.4], it is shown that for every k ∈ N and δ ∈ R, there exists a constant C > 0 such that for all ψ ∈ F and α ≥ 1. Hereafter, we denote dΓ (H ) = N j=1 H ( j) for a bounded operator H on L 2 (R 3 ). A similar estimate holds true for when replacing the creation and annihilation operators a( f ), In fact, as proven in [32, Lemma 3.2], for every k ∈ N, there exists a constant C > 0 such that for all ξ ∈ F ≤N + (t) and α ≥ 1.

Proof of Proposition 1
It follows from Lemma 1 that it is enough to prove Proposition 1 with respect the dynamics U gen (t; s). First, we prove that for f ∈ L 2 (R 3 ) the Fock space vectors U * gen (t; s)a * ( f )U gen (t; s)Ω and U * gen (t; s)a( f )U gen (t; s)Ω are elements of the one-particle sector. The following Lemma is a generalization of [14, Lemma 8.1].

Lemma 3 Let U gen (t; s) be the dynamics defined Definition 3.
Then for all f ∈ L 2 (R), where either a ( f ) = a( f ) or a ( f ) = a * ( f ) and where P 1 denotes the projection onto the one-particle sector of the Fock space F .

Proof
The proof follows the arguments of the proof of [14, Lemma 8.1]. For m ∈ N, m = 1, we define for arbitrary m-particle wave function ψ ∈ F with ψ = 1 the function Since m = 1, we observe that F(s) = 0 and furthermore e iK t a( f )e −iK t = a(e −iΔt f ) = a( f t ), using the notation f t = e −itΔ f . Since e −iΔt is a unitary operator, we find Then, and the definition of G gen,t in (2.2) leads to The assumption (2.3) implies on the one hand and on the other hand Hence, and analogously, Note that these bounds are independent of f ∈ L 2 (R 3 ). Thus, Using the bounds a ( f )ψ ≤ f 2 (N + 1) 1/2 ψ , we obtain Here, we used Lemma 2 for the last estimate. Since F(s) = 0, the Gronwall inequality implies F(t) = 0 for all t ∈ R.

Proof of Proposition 1
We prove the Proposition with respect to the dynamics U gen (t; s) defined in Definition 3. Then, Proposition 1 follows from Lemma 1. The proof follows the arguments of the proof of [6, Theorem 2.2]. Let P k denote the projection onto the k-particle sector F k of the Fock space. It follows from Lemma that 3 for all f ∈ L 2 (R 3 ) and k = 1. Thus, there exist linear operators U (t; s), V (t; s) : The operators U (t; s) and V (t; s) are bounded in L 2 (R 3 ). This follows from Lemma 2, since We define the bounded operator Θ on N ), we define furthermore Here, a , a are either creation or annihilation operators. Since e −iK t a ( f )e iK t = a (e itΔ f ) and e itΔ f 2 = f 2 for all f ∈ L 2 (R 3 ), we can write The commutation relations imply that F(s) = 0. Furthermore, using the notation f t = e −iΔt f . Analogous calculations as in the proof of Lemma 3 show that The assumption (2.3) implies h i,t 2 ≤ Ce C|t| f 2 for i = 1, 2. Thus, for all f ∈ L 2 (R 3 ) and therefore Since F(s) = 0, the Gronwall inequality implies F(t) = 0 for all t ∈ R. Hence, for all ψ ∈ D (K + N ) with ψ = 1. Combining (2.9) with (2.11), we find where S is defined in 1.34. It follows that 1 , h 1 )) .
On the one hand, (2.9) shows that RΩ = 0, and on the other hand, it follows from (2.12), that R commutes with any creation and annihilation operator. Since states of the form a * ( f 1 ) . . . a * ( f n )Ω build a basis of the Fock space F , we conclude . Now, we are left with proving (2.11). For this, note that (2.10) implies where P ψ resp. P Ω denote the projection on the subspace of F spanned by ψ resp. Ω. Therefore, on the one hand and on the other hand, Assuming that ψ, Ω = 0, claim (2.11) follows. If ψ, Ω = 0, we repeat the same arguments with ψ = 1 √ 2 (ψ + Ω). This leads to (2.11).
It remains to prove the properties (1.35). Since for all f , g ∈ L 2 (R 3 ) the first property follows. Furthermore, from we deduce the second property.

Proof of Theorem 1
The proof uses ideas introduced in [32]. We consider the expectation value The norm approximation (1.31) from [11] implies that for every α < min{β/2, (1 − β)/2} there exists C > 0 such that We are hence left with computing the expectation value We split this computation in several Lemmata.
Lemma 4 (Action of the unitary U ϕ N ,t ) Let T N ,t and U 2 (t; 0) be as defined in (1.13) resp. (1.26). Moreover, let ξ N ,t = T * N ,t U 2 (t; 0)Ω. Then, using the same notations as in Theorem 1, there exists C > 0 such that In order to show Lemma 4, we define for j ∈ {1, . . . , k} We observe that where Hence, We compute Using the fundamental theorem of calculus, we can write the difference as an integral It follows from (2.5) and Lemma 2 that for a constant C > 0 uniform in N . Hence, Lemma 5 ( Replace modified creation and annihilation operators with standard ones) Let T N ,t and U 2 (t; 0) be as defined in (1.13) resp. (1.26). Moreover, let ξ N ,t = T * N ,t U 2 (t; 0)Ω. Then, with the same notations as in Theorem 1, there exists C > 0 such that with the standard creation and annihilation operators a * ( f ), a( f ), while with the modified creation and annihilation operators defined in (1.9). To this end, we compute By definition of the modified creation and annihilation operators (1.9), we obtain Since Now, Lemma 2 together with (2.7) and (2.5) implies Proof By linearity of the operator φ a ( f ), we compute As We conclude again with Lemma (2.7), Lemma (2.5), and Lemma 2 Lemma 7 (Action of T N ,t ) Let T N ,t and U 2 (t; 0) be as defined in (1.13) resp. (1.26). Moreover, let ξ N ,t = T * N ,t U 2 (t; 0)Ω and ξ t = T N ,t ξ N ,t = U 2,t Ω. Then, using the same notations as in Theorem 1, there exists C > 0 such that with h j,t = cosh(η t )q t Oϕ t + sinh(η t )q t O j ϕ t and η t as defined in (1.20).
Proof We compute using the properties (1.14) of the Bogoliubov transformation with η N ,t as defined in (1.20). In the following, we denote h j,N ,t = cosh(η N ,t )q t Oϕ t + sinh(η N ,t )q t O j ϕ t . Since a (h1,N,t ) . . . e is k φ a (hk,N,t ) ξ t , we need to consider ξ t , e is 1 φ a (h1,N,t ) . . . e is k φ a (hk,N,t ) ξ t − ξ t , e is 1 φ a (h1,t ) . . . e is k φ a (hk,t ) ξ t .
Note that the Baker-Campbell-Hausdorff formula implies on the one hand e iφ a ( f ) e iφ a (g) = e iφ a ( f +g) e −iIm f ,g for f , g ∈ L 2 (R 3 ), i.e., k j=1 e is j φ a (ν j,t ) = e iφ a (ν t ) k i< j e −is i s j Im ν i,t ,ν j,t with ν t = k j=1 ν j,t . On the other hand, the Baker-Campbell-Hausdorff formula applied to the creation and annihilation operator implies k j=1 e is j φ a (ν j,t ) = e − ν t 2 2 /2 e a * (ν t ) e a(ν t ) k i< j e −is i s j Im ν i,t ,ν j,t .
Hence, we write the expectation value (2.14) as ξ t , e iφ a (h 1,t ) . . . e is k φ a (h k,t ) ξ t = e − ν t 2 2 k i< j e −is i s j Im ν i,t ,ν j,t Ω, e a * (ν t ) e a(ν t ) Ω = e − ν t 2 2 k i< j e −is i s j Im ν i,t ,ν j,t .