A Large Deviation Principle in Many-Body Quantum Dynamics

We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates that are consistent with central limit theorems that have been established in the last years.


Introduction
A system of N bosons in the mean-field regime can be described by the Hamilton operator acting on the Hilbert space L 2 s (R 3N ), the subspace of L 2 (R 3N ) consisting of functions that are symmetric with respect to any permutation of the N particles.
The time evolution of the N particles is governed by the many-body Schrödinger equation (1.1) If the N particles are trapped into a finite region by a confining external potential v ext , the system exhibits, at zero temperature, complete Bose-Einstein condensation in the minimizer of the Hartree energy functional E Hartree (ϕ) = |∇ϕ| 2 + v ext |ϕ| 2 dx + 1 2 v(x − y)|ϕ(x)| 2 |ϕ(y)| 2 dxdy taken over ϕ ∈ L 2 (R 3 ) with ϕ = 1. For this reason, from the point of view of physics, it is interesting to study the solution of (1.1) for an initial sequence ψ N ∈ L 2 s (R 3N ) exhibiting complete Bose-Einstein condensation, in the sense that the one-particle reduced density γ N = tr 2,...,N |ψ N ψ N | associated with ψ N satisfies γ N → |ϕ ϕ| for a normalized one-particle orbital ϕ ∈ L 2 (R 3 ), in the limit N → ∞.
To keep our analysis as simple as possible, we consider solutions of (1.1) for factorized initial data ψ N,0 = ϕ ⊗N (which obviously exhibits condensation, since γ N = |ϕ ϕ|). Notice, however, that our approach could be extended to physically more interesting initial data exhibiting condensation.
Under quite general assumptions on the interaction potential v, one can show that (in contrast with factorization) the property of Bose-Einstein condensation is preserved by the many-body evolution (1.1) and that, for every fixed t ∈ R, the reduced one-particle density γ N,t = tr 2,...,N |ψ N,t ψ N,t | is such that γ N,t → |ϕ t ϕ t |, as N → ∞. Here, ϕ t is the solution of the nonlinear Hartree equation with the initial data ϕ t=0 = ϕ. See for example [1,2,4,[10][11][12][13][14][15]18,24,25]. The convergence γ N,t → |ϕ t ϕ t | of the reduced one-particle density associated with the solution of the Schrödinger equation (1.1) can be interpreted as a law of large numbers. For a self-adjoint operator O on L 2 (R 3 ), let O (j) = 1⊗ · · · ⊗ O ⊗ · · · ⊗ 1 denote the operator on L 2 (R 3N ) acting as O on the j-th particle and as the identity on the other (N −1) particles. The probability that, in the state described by the wave function ψ ∈ L 2 s (R 3N ), the observable O (j) takes values in a set A ⊂ R is determined by , with factorized initial data ψ N,0 = ϕ ⊗N , is not factorized. Nevertheless, the convergence of the reduced density γ N,t → |ϕ t ϕ t | implies that the law of large numbers still holds true, i.e., that for all δ > 0; see, for example, [3].
To go beyond (1.3) and study fluctuations around the limiting Hartree dynamics, it is useful to factor out the condensate.
To reach this goal, we define the bosonic Fock space F = N j=0 L 2 ⊥ϕt (R 3 ) ⊗sj . On F, for any f ∈ L 2 (R 3 ), we introduce the usual creation and annihilation Vol. 22 (2021) A Large Deviation Principle in Many-Body 2597 operators a * (f ), a(f ), satisfying canonical commutation relations. It will also be convenient to use operator-valued distributions a * x , a x , for x ∈ R 3 , so that In terms of a * x , a x , we can express the number of particles operator, defined by (N Ψ) (n) = nΨ (n) , as More generally, for an operator A on the one-particle space L 2 (R 3 ), its second quantization dΓ(A), defined on F so that (dΓ(A)Ψ) (n) = n j=1 A j Ψ (n) , with A j = 1 ⊗ · · · ⊗ A ⊗ · · · ⊗ 1 acting non-trivially on the j-th particle only, can be written as is the integral kernel of A (with this notation N = dΓ (1)). More details on the formalism of second quantization applied to the dynamics of mean-field systems can be found in [5].
In order to factor out the condensate, described at time t ∈ R, by the solution ϕ t of (1.2), we observe now that every ψ ∈ L 2 s (R 3N ) can be uniquely written as denotes the orthogonal complement in L 2 (R 3 ) of the condensate wave function ϕ t . This remark allows us to define, for every t ∈ R, a unitary operator , removes the condensate wave function ϕ t and allows us to focus on its orthogonal excitations. It maps the N -particle space L 2 s (R 3N ) into the truncated Fock space F ≤N ⊥ϕt , constructed over the orthogonal complement of ϕ t . The map U t can be used to define the fluctuation dynamics (mapping the orthogonal excitations of the condensate at time t 1 into the orthogonal excitations of the condensate at time t 2 ): (1.4) The fluctuation dynamics satisfies the equation with W N (t 1 ; t 1 ) = 1 for all t 1 ∈ R and with the generator To compute the generator L N (t), we use the rules . We obtain, similarly to [19], the matrix elements Moreover, we introduced the notation N + (t) for the number of particles operator on the space F ≤N ⊥ϕt (N + (t) = dΓ(q t ), with q t = 1 − |ϕ t ϕ t |, if we think of F ≤N ⊥ϕt as a subspace of F) and, for f ∈ L 2 ⊥ϕt (R 3 ), we defined (using the notation introduced in [7 and the corresponding operator-valued distributions b * x , b x , for x ∈ R 3 . In the limit of large N , the fluctuation dynamics W N (t 2 ; t 1 ) can be approximated by a limiting dynamics W ∞ (t 2 ; t 1 ) : with the generator L ∞ (t 2 ), whose matrix elements are given by for all ξ 1 , ξ 2 ∈ F ⊥ϕt 2 ; see [19]. (This line of research started in [17] and was further explored in [11,16,21]; recently, an expansion of the many-body dynamics in powers of N −1 was obtained in [6].) Notice that L ∞ (t 2 ) acts on (a dense subspace of) the Fock space F ⊥ϕt 2 , constructed on the orthogonal complement of ϕ t2 , with no restriction on the number of particles. We have the inclusions F ≤N Observe also that L ∞ (t 2 ) is quadratic in creation and annihilation operators. It follows that the limiting dynamics W ∞ (t 2 ; t 1 ) acts as a time-dependent family of Bogoliubov transformations (in a slightly different setting, this was shown in [3]). In other words, introducing the notation for a two-parameter family of operators Θ(t 2 ; t 1 ) : The convergence towards the limiting Bogoliubov dynamics (1.8) has been used in [3,8] to prove that, beyond the law of large numbers (1.3), the variables O (j) also satisfy the central limit theorem . (The solution of (1.11) is related with the family of Bogoliubov transformations Θ(t 1 ; t 2 ), since Θ(0; t)(f t;t ; Jf t;t ) = (f 0;t ; Jf 0;t ).) For singular interaction potentials, scaling as N 3β v(N β x) for a 0 < β < 1 and converging therefore to a δ-function as N → ∞, the validity of a central limit theorem of the form (1.10) was recently established in [22]; in this case, the correlation structure produced by the interaction affects the variance of the limiting Gaussian distribution. For β = 1 (the Gross-Pitaevskii regime), the validity of a central limit theorem for the ground state was established instead in [23].
In our main theorem, we show, for bounded interactions, a large deviation principle for the fluctuations of the many-body quantum evolution around the limiting Hartree dynamics.
For t ∈ R, let ψ N,t denote the solution of the many-body Schrödinger equation Then, there exists a constant C > 0 (depending only on ϕ H 4 ) such that, denoting by O (j) = 1 ⊗ · · · ⊗ O ⊗ · · · ⊗ 1 the operator O acting only on the j-th particle, and Remark. The result and its proof can be trivially extended to particles moving in d dimensions, for any d ∈ N\{0}.
It follows from (1.12) that with rate function where the infimum is taken over For any fixed t > 0, the infimum is attained at , which is consistent with the central limit theorem (1.10), obtained in [3,8]. This shows, in particular, that the quadratic term on the r.h.s. of (1.12) is optimal.
To prove Theorem 1.1, we first write the expectation on the l.h.s. of (1.12) as (1. 15) in terms of the fluctuation dynamics introduced in (1.4). Here, we used the choice of the initial data to write In the next step, motivated by the bound ±dΓ up to the exponential of a cubic expression in λ, contributing only to the last term on the r.h.s. of (1.12); this is the content of Lemma 3.1. In the next step, Lemma 3.2, we replace the fluctuation dynamics W N (t; 0) by its limit W ∞ (t; 0), as defined in (1.8); as in the first step, also this replacement only produces an error cubic in λ in (1.12). Describing the action of W ∞ through the solution of (1.11), we arrive at the product In the final step, Lemma 3.3, we estimate (1.16), concluding the proof of (1.12). This step makes use of the choice of product initial data (which implies that the expectation is taken in the vacuum); at the expenses of a longer proof, we could have proven Theorem 1.1 to a larger and physically more interesting class of initial data.

Preliminaries
To begin with, we introduce some notation and we recall some basic facts. For a given normalized ϕ ∈ L 2 (R 3 ), we consider the Hilbert space For g 1 , g 2 , g, h ∈ L 2 ⊥ϕ (R 3 ), we find the commutation relations Ann. Henri Poincaré More generally, for any self-adjoint operators H. We also recall the bounds for every bounded operator H on L 2 ⊥ϕ (R 3 ). For more details, we refer to [7, Section 2].
Furthermore, we introduce the notation ad for all n ≥ 0 and for all n ≥ 1.

Proposition 2.2.
Let g, h ∈ L 2 ⊥ϕ (R 3 ). With the shorthand notation γ s = cosh s and σ s = sinh s, we have  In particular, it follows from (2.10) that, for (2.12) We will also need a formula for e Using (2.12) (and its Hermitian conjugate) and then integrating over s ∈ [0; 1], we arrive at (2.13) Integrating (2.13) against the integral kernel of a self-adjoint operator, we can also get a formula for e   Writing b(h) = (φ + (h) + iφ − (h))/2 and b * (h) = (φ + (h) − iφ − (h))/2, we arrive at (2.15). Hence, if t → A t is an operator-valued functions, differentiable in t, we find Dividing by h and letting h → 0, we find In particular, Integrating over τ , we arrive at (2.16).

Proof of main theorem
To prove Theorem 1.1, we start from (1.15), writing Proof. For s ∈ [0; 1] and a fixed κ > 0, we define Note that ξ s ∈ F ≤N ⊥ϕt for all s ∈ [0; 1]. Then, we have To compare ξ 1 2 with ξ 0 2 , we compute the derivative We have ∂ s ξ s = M s ξ s , with With Proposition 2.3 we find, With Proposition 2.4, we obtain Using the bounds (2.6), (2.7), and the fact that Choosing κ = c O (which also implies that λκ ≤ c), we conclude that By Gronwall, we obtain (3.1).

Lemma 3.2. For a bounded self-adjoint operator
Proof. For s ∈ [0; t] and with κ s as in (3.2), we define and that To compare ξ t (0) 2 with ξ t (t) 2 , we are going to compute the derivative with respect to s. Since the two norms are taken on different spaces, it is convenient to embed first the s-dependent space F ≤N ⊥ϕs into the full, s-independent, Fock space F = n≥0 L 2 (R 3n ) ⊗sn . To this end, we observe that Remark that only the antisymmetric part of J N,t (s) contributes to the growth of the norm. Next, we compute J N,t (s), focusing in particular on its antisymmetric component. We recall the definition (1.6) of the generator L N (s). We introduce the notation h s;t = λf s;t /2 ∈ L 2 ⊥ϕs (R 3 ). From (2.14), we find, on vectors in F ≤N ⊥ϕs (since we consider matrix elements on vectors in F ≤N ⊥ϕs , we can replace the operator h H (s) + K 1,s , which does not leave L 2 ⊥ϕs (R 3 ) invariant, with its restriction to L 2 ⊥ϕs (R 3 ); this is the reason why we can apply Prop. 2.3) With Prop. 2.4, we obtain, again in the sense of forms on F ≤N ⊥ϕs , Removing symmetric terms (which do not contribute to (3.3)) and focusing on terms that are at most quadratic in λ (recall that h s;t = λf s;t /2), we arrive at where S 1 = S * 1 does not contribute to the antisymmetric part of J N,t (s) and for all λ > 0 with Nφ + (f s;t )/2 1 2  We consider now Conjugating separately b * x and a * y a x (or a * x a y and b x in the second term), we arrive, using (2.12) (and its Hermitian conjugate), (2.13) Finally, we consider the term Conjugating separately a * x a x and a * y a y (and also the operator N + (s), From (1.11) and (3.4), we conclude that for all s ∈ [0; t] and all λ > 0 with λ O ≤ 1 and λκ s ≤ 1 for all s ∈ [0; t].
With the choice (3.2), we find Inserting in (3.3), we obtain that By Gronwall, we arrive at

Lemma 3.3.
Let κ t be defined as in (3.2). Then, there exists a constant C > 0 such that Remark. The lemma could be extended to bound the expectation on the l.h.s. of (3.10) for a larger class of states, including quasi-free states, rather than only in the vacuum. This would allow us to consider more general initial data in Theorem 1.1. To keep the focus on the main novelty of our paper (the possibility of proving a large deviation principle for many-body quantum dynamics), we restricted our attention on the simplest case of factorized initial data (leading to the vacuum in (3.10).