Dependent random variables in quantum dynamics

Rademacher, Simone

doi:10.1063/5.0086712

We consider the many-body time evolution of weakly interacting bosons in the mean field regime for initial coherent states. We show that bounded k-particle operators, corresponding to dependent random variables, satisfy both a law of large numbers and a central limit theorem.

I. INTRODUCTION AND MAIN RESULTS

We consider N weakly interacting bosons in the mean-field regime described on $L_{s}^{2} (R^{3 N})$ ⁠, the symmetric subspace of $L^{2} (R^{3 N})$ ⁠, by the Hamilton operator

H_{N} = \sum_{i = 1}^{N} (- Δ_{i}) + \frac{1}{N} \sum_{i < j = 1}^{N} v (x_{i} - x_{j}),

(1.1)

with the two-body interaction potential v satisfying

v^{2} \leq C (1 - Δ)

(1.2)

for a positive constant C > 0. The mean-field regime is characterized through weak and long-range interactions of particles. Trapped Bose gases at extremely low temperatures, as prepared in the experiments, are known to relax to the ground state. The ground state $ψ_{N}^{gs}$ of (1.1), if it exists, exhibits Bose–Einstein condensation,¹⁷ i.e., the associated ℓ-particle reduced density

γ_{ψ_{N}^{gs}}^{(ℓ)} ≔ {t r}_{ℓ + 1, \dots, N} | ψ_{N}^{gs} 〉 〈 ψ_{N}^{gs} |

(1.3)

converges in the trace norm to

γ_{ψ_{N}^{gs}}^{(ℓ)} \to | φ 〉 〈 φ |^{\otimes ℓ} as N \to \infty

(1.4)

for all $ℓ \in N$ ⁠, where $φ \in L^{2} (R^{3})$ denotes the condensate wave function, known to be the Hartree minimizer. However, we remark that the factorized state φ^⊗N does not approximate the ground state due to correlations of particles.¹¹

A. Law of large numbers

Turning to the probabilistic picture, the property of Bose–Einstein condensation (1.4) implies a law of large numbers for bounded one-particle operators.³ To be more precise, for $k \in N$ ⁠, we denote with O^(k) a bounded, self-adjoint k-particle operator on $L^{2} (R^{3 k})$ and with ${\underset{̲}{i}}_{k}$ the multi-index

{\underset{̲}{i}}_{k} = (i_{1}, \dots, i_{k}) \in I_{N}^{(k)}, I_{N}^{(k)} ≔ \{(i_{1}, \dots, i_{k}) \in {1, \dots, N}^{k} | i_{j} \neq i_{m} for j \neq m\} .

(1.5)

Then, we define for fixed k ≤ N the N-particle operator

O_{{\underset{̲}{i}}_{k}}^{(k)} with {\underset{̲}{i}}_{k} \in I_{N}^{(k)},

(1.6)

acting as O^(k) on particles i₁, …, i_k and as identity elsewhere. We consider the operator $O_{{\underset{̲}{i}}_{k}}^{(k)}$ as a random variable with probability distribution determined through ψ_N by

P_{ψ_{N}} [O_{{\underset{̲}{i}}_{k}}^{(k)} \in A] = E_{ψ_{N}} [χ_{A} (O_{{\underset{̲}{i}}_{k}}^{(k)})] = 〈 ψ_{N} | χ_{A} (O_{{\underset{̲}{i}}_{k}}^{(k)}) | ψ_{N} 〉,

(1.7)

where χ_A denotes the characteristic function of the set $A \subset R$ ⁠.

For one-particle operators, factorized states correspond to i.i.d. random variables as for any subsets $A_{1}, A_{2} \subset R$ and $i, j \in I_{N}^{(1)}$ with i ≠ j,

\begin{align} P_{φ^{\otimes N}} [O_{i}^{(1)} \in A_{1}, O_{j}^{(1)} \in A_{2}] & = 〈 φ^{\otimes N} | χ_{A_{1}} (O_{i}^{(1)}) χ_{A_{2}} (O_{j}^{(1)}) | φ^{\otimes N} 〉 \\ = 〈 φ | χ_{A_{1}} (O^{(1)}) | φ 〉 〈 φ | χ_{A_{2}} (O^{(1)}) | φ 〉 \\ = 〈 φ^{\otimes N} | χ_{A_{1}} (O_{i}^{(1)}) | φ^{\otimes N} 〉 〈 φ^{\otimes N} | χ_{A_{2}} (O_{j}^{(1)}) | φ^{\otimes N} 〉 \\ = P_{φ^{\otimes N}} [O_{i}^{(1)} \in A_{1}] P_{φ^{\otimes N}} [O_{j}^{(1)} \in A_{2}] . \end{align}

(1.8)

In particular, for factorized states, Chebychef’s inequality implies a law of large numbers for the centered averaged sum,

\frac{1}{N} O_{N}^{(1)} : = \frac{1}{N} \sum_{i = 1}^{N} (O_{i}^{(1)} - 〈 φ | O^{(1)} | φ 〉) .

(1.9)

In contrast to one-particle operators for k-particle operators with k ≥ 2, factorized states do not correspond to i.i.d. random variables. In fact, for k ≥ 2, we have

E_{φ^{\otimes N}} [(O_{{\underset{̲}{i}}_{k}}^{(k)} - 〈 φ^{\otimes k} | O^{(k)} | φ^{\otimes k} 〉) (O_{{\underset{̲}{j}}_{k}}^{(k)} - 〈 φ^{\otimes k} | O^{(k)} | φ^{\otimes k} 〉)] \neq 0

(1.10)

for all ${\underset{̲}{i}}_{k} \neq {\underset{̲}{j}}_{k}$ for which ${\underset{̲}{i}}_{k}$ contains at least one element of ${\underset{̲}{j}}_{k}$ ⁠. We conclude that in this case, the random variables are correlated and, thus, dependent. In contrast, whenever ${\underset{̲}{i}}_{k}$ does not intersect with ${\underset{̲}{j}}_{k}$ ⁠, the random variables $O_{{\underset{̲}{i}}_{k}}^{(k)}$ ⁠, $O_{{\underset{̲}{j}}_{k}}^{(k)}$ are independent [following from arguments similarly to (1.8)]. Consequently, for factorized states, the random variables ${O_{{\underset{̲}{i}}_{k}}^{(k)}}_{{\underset{̲}{i}}_{k} \in I_{N}^{(k)}}$ denote a sequence of m-dependent random variables with $m \in R$ ⁠. Still, as in Theorem 1.1, the centered averaged sum

\frac{1}{(\binom{N}{k})} O_{N}^{(k)} : = \frac{1}{(\binom{N}{k})} \sum_{{\underset{̲}{i}}_{k} \in I_{N}^{(k)}} (O_{{\underset{̲}{i}}_{k}}^{(k)} - 〈 φ^{\otimes k} | O^{(k)} | φ^{\otimes k} 〉)

(1.11)

satisfies a law of large numbers.

Theorem 1.1

(law of large numbers). For

k \in N

, let O^(k) denote a self-adjoint bounded k-particle operator,

φ \in L^{2} (R^{3})

⁠, and

ψ_{N} \in L_{s}^{2} (R^{3 N})

be a bosonic wave function satisfying

γ_{ψ_{N}}^{(ℓ)} \to | φ 〉 〈 φ |^{\otimes ℓ} as N \to \infty

(1.12)

for all

ℓ \in N

. Then, for any fixed

k \in N

and δ > 0, the averaged sum

O_{N}^{(k)}

defined in (1.11) satisfies

P_{ψ_{N}} [|\frac{1}{(\binom{N}{k})} O_{N}^{(k)}| > δ] \to 0 as N \to \infty .

(1.13)

For factorized states, we have $γ_{φ^{\otimes N}}^{(ℓ)} = | φ 〉〈 φ |^{\otimes ℓ}$ ⁠, and a law of large numbers follows from Theorem 1.1.

In particular, Theorem 1.1 shows that the property of condensation (1.12) implies a law of large numbers for bounded k-particle operators for fixed $k \in N$ ⁠. Thus, Theorem 1.1 generalizes known results from Ref. 3 for bounded one-particle operators to k-particle operators with fixed $k \in N$ ⁠. We recall that the ground state $ψ_{N}^{gs}$ of (1.1) cannot be approximated by a factorized state; nonetheless, the condensation property (1.4) ensures that bounded k-particle operators satisfy a law of large numbers for $ψ_{N}^{gs}$ ⁠, too.

1. Generalization to the Fock space

In order to generalize Theorem 1.1 to any Fock space vector $ψ \in F$ of the bosonic Fock space $F = ⨁_{n \geq 0} L^{2} {(R^{3})}^{\otimes_{s}^{n}}$ ⁠, we introduce some more notation.

For any vector $ψ \in F$ ⁠, we have the following identity for the operator ${\tilde{O}}^{(k)} = O^{(k)} - 〈 φ^{\otimes k} | O^{(k)} | φ^{\otimes k} 〉$ on the N-particle sector:

{(d Γ^{(k)} ({\tilde{O}}^{(k)}) ψ)}^{(N)} = O_{N} ψ^{(N)},

(1.14)

where we introduced the second quantization for any integral operator O^(k) on $L^{2} (R^{3 k})$ ⁠,

d Γ^{(k)} (O^{(k)}) = \int d x_{1} \dots d x_{k} d y_{1} \dots d y_{k} O^{(k)} (x_{1}, \dots, x_{k}; y_{1}, \dots y_{k}) a_{x_{1}}^{*} \dots a_{x_{k}}^{*} a_{y_{1}} \dots a_{y_{k}} .

(1.15)

Note that we can generalize the definition of the probability distribution (1.7) to the Fock space: For any $ψ \in F$ ⁠, integral operator O^(k) on $L^{2} (R^{3 k})$ ⁠, and $A \subset R$ ⁠, we have

P_{ψ} [d Γ^{(k)} (O^{(k)}) \in A] = E_{ψ} [χ_{A} (d Γ^{(k)} (O^{(k)}))] = 〈 ψ | χ_{A} (d Γ^{(k)} (O^{(k)})) | ψ 〉 .

(1.16)

On the Fock space, the k-particle reduced density $γ_{ψ}^{(k)}$ associated with $ψ \in F$ is given by the integral operator with kernel

γ_{ψ}^{(k)} (x_{1}, \dots, x_{k}; y_{1}, \dots y_{k}) ≔ 〈 ψ | a_{y_{1}}^{*} \dots a_{y_{k}}^{*} a_{x_{1}} \dots a_{x_{k}} | ψ 〉 .

(1.17)

It follows from a generalization of Theorem 1.1’s proof in Sec. II that for $ψ \in F$ satisfying (1.12), we have for any δ > 0,

P_{ψ_{N}} [|\frac{1}{(\binom{N}{k})} d Γ^{(k)} ({\tilde{O}}^{(k)})| > δ] \to 0 as N \to \infty .

(1.18)

2. Dynamics

We are interested in the dynamics of initially trapped Bose gases. Removing the trap, the bosons evolve with respect to the Schrödinger equation,

i \partial_{t} ψ_{N, t} = H_{N} ψ_{N, t},

(1.19)

with H_N being the mean-field Hamiltonian given in (1.1). In the following, we consider coherent initial data, i.e., initial data of the form

ψ_{N, 0} = W (\sqrt{N} φ) Ω,

(1.20)

where Ω denotes the vacuum of the bosonic Fock space $F = ⨁_{n \geq 0} L^{2} {(R^{3})}^{\otimes_{s}^{n}}$ equipped with creation and annihilation operators a^*(f), a(f) for $f \in L^{2} (R^{3})$ ⁠, $W (f) = e^{a^{*} (f) - a (f)}$ denotes the Weyl operator, and $f \in H^{1} (R^{3})$ denotes the condensate wave function. Coherent states of the form (1.20) exhibit Bose–Einstein condensation in the quantum state φ, i.e., they satisfy (1.4).

Thus, it follows from Theorem 1.1 that initially a law of large numbers holds true. The property of condensation is preserved along the many-body time evolution (Ref. 4, Theorem 3.1), i.e., the ℓ-particle reduced density $γ_{N, t}^{(ℓ)}$ associated with ψ_N,t satisfies

γ_{N, t}^{(ℓ)} \to | φ_{t} 〉 〈 φ_{t} |^{\otimes ℓ} as N \to \infty for all ℓ \in N,

(1.21)

where $φ_{t} \in H^{1} (R^{3})$ denotes the solution to the Hartree equation,

i \partial_{t} φ_{t} = h_{H} (t) φ_{t} with h_{H} (t) = - Δ + (v * | φ_{t} |^{2}),

(1.22)

with initial data $φ_{0} = φ \in H^{1} (R^{3})$ (for further references, see, e.g., Refs. 1, 2, 7, 9, 10, 15, 22, and 23). Theorem 1.1 and (1.21) show that

\frac{1}{(\binom{N}{k})} d Γ (O_{t}^{(k)}) : = \frac{1}{(\binom{N}{k})} d Γ (O^{(k)} - 〈 φ_{t}^{\otimes k} | O^{(k)} | φ_{t}^{\otimes k} 〉)

(1.23)

satisfies a law of large numbers for positive times t > 0 too, i.e., for any δ > 0,

P_{ψ_{N, t}} [|\frac{1}{(\binom{N}{k})} d Γ (O_{t}^{(k)})| > δ] \to 0 as N \to \infty for all t \in R .

(1.24)

B. Central limit theorem

While the law of large numbers characterizes the mean of the probability distribution, fluctuations around the mean are governed through the central limit theorem. Before stating our result on a central limit theorem for fluctuations of order O(N^k−1/2), we introduce some notations. For a bounded k-particle integral operator O^(k) and $φ \in L^{2} (R^{3})$ ⁠, we define

\begin{align} {({\bar{φ}}_{t}^{\otimes (k - 1)} O^{(k)} φ_{t}^{\otimes k})}_{j} (x) \\ ≔ \int d x_{1} \dots d x_{j - 1} d x_{j + 1} \dots d x_{k} d y_{1} \dots d y_{k} O^{(k)} (x_{1}, \dots, x_{i - 1}, x, x_{i + 1}, \dots, x_{k}; y_{1}, \dots y_{k}) \prod_{\begin{array}{c} i = 1 \\ i \neq j \end{array}}^{k} {\bar{φ}}_{t} (x_{i}) \prod_{m = 1}^{k} φ_{t} (y_{m}), \end{align}

(1.25)

and furthermore, for $t \in R$ ⁠, 0 ≤ s ≤ t, and j ∈ {1, …, k}, the function $f_{s; t}^{(j)}$ is given by

i \partial_{s} f_{s; t}^{(j)} = (h_{H} (s) + K_{1, s} - K_{2, s} J) f_{s; t}^{(j)} with f_{t; t}^{(j)} = q_{t} {({\bar{φ}}_{t}^{\otimes (k - 1)} O^{(k)} φ_{t}^{\otimes k})}_{j},

(1.26)

with the anti-linear operator $J f = \bar{f}$ for any $f \in L^{2} (R^{3})$ ⁠, q_t = 1 − |φ_t⟩⟨φ_t|, the Hartree Hamiltonian h_H defined in (1.22), and the operators

K_{1, t} (x; y) = φ_{t} (x) v (x - y) {\bar{φ}}_{t} (y), K_{2, t} (x; y) = φ_{t} (x) v (x - y) φ_{t} (y) .

(1.27)

Theorem 1.2

(central limit theorem). For $k, N \in N$ with k ≤ N, let O^(k) be a self-adjoint, bounded k-particle integral operator and φ_t be the solution to the Hartree equation (1.22) with initial datum $φ_{0} = φ \in H^{1} (R^{3})$ . Let $ψ_{N, t} \in L_{s}^{2} (R^{3 N})$ denote the solution to the Schrödinger equation (1.19) with the initial datum of the form $ψ_{N, 0} = W (\sqrt{N} φ) Ω$ .

Let

a, b \in R

with a < b; then, there exists a constant C_a,b,k > 0 such that the centered averaged sum

d Γ (O_{t}^{(k)})

defined in (1.23) satisfies

|P_{ψ_{N, t}} [N^{- k + 1 / 2} d Γ (O_{t}^{(k)}) \in [a, b]] - P [G_{t} \in [a, b]]| \leq C_{a, b, k} e^{C | t |} N^{- 1 / 12},

(1.28)

where

G_{t}

denotes the centered Gaussian random variable with variance given by

σ_{t}^{2} = \sum_{i, j = 1}^{k} 〈 f_{0; t}^{(i)} | | f_{0; t}^{(j)} 〉 .

(1.29)

We remark that for a factorized state, we can explicitly compute the variance

\begin{align} σ_{N}^{2} & = E_{φ^{\otimes N}} [{(O_{N}^{(k)})}^{2}] - E_{φ^{\otimes N}} {[O_{N}^{(k)}]}^{2} \\ = \sum_{{\underset{̲}{i}}_{k}, {\underset{̲}{j}}_{k} \in I_{N}^{(k)}} E_{φ^{\otimes N}} [{\tilde{O}}_{{\underset{̲}{i}}_{k}}^{(k)} {\tilde{O}}_{{\underset{̲}{j}}_{k}}^{(k)}] - {(\sum_{{\underset{̲}{i}}_{k} \in I_{N}^{(k)}} E_{φ^{\otimes N}} [{\tilde{O}}_{{\underset{̲}{i}}_{k}}^{(k)}])}^{2}, \end{align}

(1.30)

where we introduced the centered k-particle operator

{\tilde{O}}^{(k)} = O^{(k)} - 〈 φ^{\otimes k} | O^{(k)} | φ^{\otimes k} 〉 .

(1.31)

The last sum of the rhs of (1.30) vanishes. Furthermore, the first sum vanishes whenever ${\underset{̲}{j}}_{k}$ does not intersect with ${\underset{̲}{i}}_{k}$ ⁠, and for the remaining terms, we find

σ_{N}^{2} = \sum_{i, j = 1}^{k} \frac{N \dots (N - 2 k + 1)}{k! (k - 1)!} M_{φ^{\otimes N}} (i, j) + O (N^{2 k - 2})

(1.32)

using the definition

\begin{align} M_{φ^{\otimes N}} (i, j) & = {⟨({({\bar{φ}}^{\otimes (k - 1)} {\tilde{O}}^{(k)} φ^{\otimes k})}_{i}| {({\bar{φ}}^{\otimes (k - 1)} {\tilde{O}}^{(k)} φ^{\otimes k})}_{j}⟩}_{L^{2} (R^{3})} \\ = {⟨(q {({\bar{φ}}^{\otimes (k - 1)} O^{(k)} φ^{\otimes k})}_{i}| q {({\bar{φ}}^{\otimes (k - 1)} O^{(k)} φ^{\otimes k})}_{j}⟩}_{L^{2} (R^{3})} \end{align}

(1.33)

with q = 1 − |φ⟩⟨φ| and (1.25). In particular, we observe that the variance scales as $σ_{N}^{2} = O (N^{2 k - 1})$ ⁠, and thus, we expect fluctuations to be O(N^k−1/2).

We observe that Theorem 1.2 shows that the fluctuations of the many-body dynamics scale similarly to the fluctuations of a factorized state. Moreover, for t = 0, the variance $σ_{0}^{2}$ of the many-body dynamics defined in (1.29) agrees with the covariance matrix $M_{φ^{\otimes N}} (i, j)$ in (1.33) of a factorized state.

We remark that for k = 1, i.e., considering bounded one-particle observables, Theorem 1.2 generalizes known results^3,6 to more general one-particle observables. This generalization is due to a different strategy of the Proof of Theorem 1.3 than in Refs. 3 and 6. We follow the ideas of Ref. 6; however, we directly use as a first step in Lemma 4.1 the norm approximation (4.1) of the many-body time evolution (for more details, see Sec. IV B). Furthermore, the authors of Ref. 6 proved a multivariate central limit theorem: it is shown that the expectation value of products of functions f₁, …, f_k of bounded, self-adjoint, and centered one-particle operators O₁, …, O_k [i.e., operators of the special form (1.9)] can be approximated with the integral of f₁, …, f_k against a complex-valued Gaussian density.

Recently, for one-particle operators, the probability distribution’s tails were characterized through large deviation estimates,^14,21 showing that

\lim_{N \to \infty} \frac{1}{N} \log P_{ψ_{N, t}} [\frac{1}{N} d Γ (O_{t}^{(1)}) > x] = - \frac{x^{2}}{‖ {\tilde{f}}_{0; t}^{(1)} ‖_{2}^{2}} + O (x^{5 / 2})

(1.34)

for sufficiently small $x \leq C e^{- e^{C | t |}}$ ⁠, where ${\tilde{f}}_{t, 0}^{(1)}$ is defined similarly to (1.26), but using the projected kernels ${\tilde{K}}_{j, s} (x, y) = q_{s} K_{j, s} (x, y) q_{s}$ ⁠.

Furthermore, for one-particle operators, a central limit theorem is proven for stronger particles’ interactions in the intermediate regime,¹⁹ interpolating between the mean-field and the Gross–Pitaevski regime. In the Gross–Pitaevski regime of singular particles’ interaction, a central limit theorem is proven for quantum fluctuations in the ground state,²⁰ too.

Theorem 1.2 follows from an approximation of the random variable’s characteristic function given in the following.

Theorem 1.3.

Under the same assumptions as in Theorem 1.2, we have

|E_{ψ_{N, t}} [e^{i N^{- k + 1 / 2} d Γ (O_{t}^{(k)})}] - e^{- σ_{t}^{2} / 2}| \leq C_{k} e^{C | t |} {‖ O^{(k)} ‖}_{op} \sum_{ℓ = 1}^{2 k - 1} N^{- ℓ / 2} {(1 + \sum_{i, j} {‖ O^{(k)} ‖}_{op}^{2})}^{(ℓ + 1) / 2} .

(1.35)

In the following, we will now first turn to the Proof of Theorem 1.1 in Sec. II, then prove Theorem 1.2 from Theorem 1.3 in Sec. III, and finally prove Theorem 1.3 in Sec. IV.

II. PROOF OF THEOREM 1.1

We generalize ideas from Ref. 3 on a law of large numbers for bounded one-particle observables to the case of k-particle operators.

Proof.

By Chebycheff’s inequality, we have

P_{ψ_{N}} [|\frac{1}{(\binom{N}{k})} O_{N}^{(k)}| > δ] \leq \frac{1}{{(\binom{N}{k})}^{2} δ^{2}} E_{ψ_{N}} [{|\sum_{{\underset{̲}{i}}_{k} \in I_{N}^{(k)}} {\tilde{O}}_{{\underset{̲}{i}}_{k}}^{(k)}|}^{2}],

(2.1)

where we used the notation

{\tilde{O}}^{(k)}

defined in (). Furthermore, we denote with

♯ {{\underset{̲}{i}}_{k}, {\underset{̲}{j}}_{k}}

the number of elements of

{\underset{̲}{i}}_{k}

agreeing with

{\underset{̲}{j}}_{k}

⁠. Then, we can write

{|\sum_{{\underset{̲}{i}}_{k} \in I_{N}^{(k)}} {\tilde{O}}_{{\underset{̲}{i}}_{k}}^{(k)}|}^{2} = \sum_{ℓ = 0}^{k} \sum_{\begin{array}{c} {\underset{̲}{i}}_{k}, {\underset{̲}{j}}_{k} \in I_{N}^{(k)} \\ ♯ {{\underset{̲}{i}}_{k}, {\underset{̲}{j}}_{k}} = ℓ \end{array}} {\tilde{O}}_{{\underset{̲}{i}}_{k}}^{(k)} {\tilde{O}}_{{\underset{̲}{j}}_{k}}^{(k)} .

(2.2)

We can express the rhs of () in terms of j-particle reduced density matrices defined in () and find

E_{ψ_{N}} [{|\sum_{i_{1}, \dots, i_{k} \in I_{N}} {\tilde{O}}_{{i_{1}, \dots, i_{k}}}^{(k)}|}^{2}] = \sum_{ℓ = 0}^{k} \sum_{\begin{array}{c} {\underset{̲}{i}}_{k}, {\underset{̲}{j}}_{k} \in I_{N}^{(k)} \\ ♯ {{\underset{̲}{i}}_{k}, {\underset{̲}{j}}_{k}} = ℓ \end{array}} t r γ_{ψ_{N}}^{(2 k - ℓ)} {\tilde{O}}_{{\underset{̲}{i}}_{k}}^{(k)} {\tilde{O}}_{{\underset{̲}{j}}_{k}}^{(k)} .

(2.3)

Plugging () into the rhs of (), we find

P_{ψ_{N}} [|O_{N}^{(k)}| > δ] \leq \frac{1}{{(\binom{N}{k})}^{2} δ^{2}} \sum_{ℓ = 0}^{k} \sum_{\begin{array}{c} {\underset{̲}{i}}_{k}, {\underset{̲}{j}}_{k} \in I_{N}^{(k)} \\ ♯ {{\underset{̲}{i}}_{k}, {\underset{̲}{j}}_{k}} = ℓ \end{array}} t r γ_{ψ_{N}}^{(2 k - ℓ)} ({\tilde{O}}_{{\underset{̲}{i}}_{k}}^{(k)} {\tilde{O}}_{{\underset{̲}{j}}_{k}}^{(k)}) .

(2.4)

For ℓ = 0, the term of the sum of the rhs of () is given by

\frac{1}{{(\binom{N}{k})}^{2} δ^{2}} \sum_{\begin{array}{c} {\underset{̲}{i}}_{k}, {\underset{̲}{j}}_{k} \in I_{N}^{(k)} \\ ♯ {{\underset{̲}{i}}_{k}, {\underset{̲}{j}}_{k}} = 0 \end{array}} t r γ_{ψ_{N}}^{(2 k)} {\tilde{O}}_{{\underset{̲}{i}}_{k}}^{(k)} {\tilde{O}}_{{\underset{̲}{j}}_{k}}^{(k)} = \frac{(\binom{N}{2 k})}{{(\binom{N}{k})}^{2} δ^{2}} t r γ_{ψ_{N}}^{(2 k)} ({\tilde{O}}^{(k)} \otimes {\tilde{O}}^{(k)}) \leq C_{k} t r γ_{ψ_{N}}^{(2 k)} ({\tilde{O}}^{(k)} \otimes {\tilde{O}}^{(k)}) .

(2.5)

Since ψ_N exhibits Bose–Einstein condensation, it follows by assumption (),

t r γ_{ψ_{N}}^{(2 k)} {({\tilde{O}}^{(k)} \otimes {\tilde{O}}^{(k)})}^{(2 k)} \to t r | φ 〉 〈 φ |^{\otimes (2 k)} {({\tilde{O}}^{(k)} \otimes {\tilde{O}}^{(k)})}^{(2 k)} as N \to \infty,

(2.6)

and by definition () of

{\tilde{O}}^{(k)}

⁠, we arrive at

t r | φ 〉 〈 φ |^{\otimes (2 k)} {({\tilde{O}}^{(k)} \otimes {\tilde{O}}^{(k)})}^{(2 k)} = 0 .

(2.7)

For ℓ ≥ 1, the terms of the sum of the rhs of () consist of (2k − ℓ)-particle operators whose expectation values are computed with (2k − ℓ)-particle operators. In particular, we find

\frac{1}{{(\binom{N}{k})}^{2} δ^{2}} \sum_{ℓ = 1}^{k} \sum_{\begin{array}{c} {\underset{̲}{i}}_{k}, {\underset{̲}{j}}_{k} \in I_{N}^{(k)} \\ ♯ {{\underset{̲}{i}}_{k}, {\underset{̲}{j}}_{k}} = ℓ \end{array}} t r γ_{ψ_{N}}^{(2 k - ℓ)} ({\tilde{O}}_{{\underset{̲}{i}}_{k}}^{(k)} {\tilde{O}}_{{\underset{̲}{j}}_{k}}^{(k)}) \leq C_{k} ‖ O^{(k)} ‖_{op}^{2} \sum_{ℓ = 1}^{k} \frac{(\binom{N}{2 k - ℓ})}{{(\binom{N}{k})}^{2}} \leq C_{k} ‖ O^{(k)} ‖_{op}^{2} \sum_{ℓ = 1}^{k} \frac{1}{N^{ℓ}} \to 0

(2.8)

as N → ∞.

We conclude with (), (), and () by

P_{ψ_{N}} [|\frac{1}{(\binom{N}{k})} O_{N}^{(k)}| > δ] \to 0 as N \to \infty .

(2.9)

□

III. PROOF OF THEOREM 1.2

We use standard arguments from probability theory to prove Theorem 1.2 from Theorem 1.3. We follow the arguments from Ref. 6, Corollary 1.2.

Proof.

We consider the difference

\begin{align} P_{ψ_{N, t}} & [N^{- k + 1 / 2} d Γ (O_{t}^{(k)}) \in [a, b]] - P [G_{t} \in [a, b]] \\ = 〈 ψ_{N, t} | χ_{[a, b]} (N^{- k + 1 / 2} d Γ (O_{t}^{(k)})) | ψ_{N, t} 〉 - \frac{1}{\sqrt{2 π} σ_{t}} \int d x e^{- \frac{x^{2}}{2 σ_{t}^{2}}} χ_{[a, b]} (x) \\ = E_{ψ_{N, t}} [χ_{[a, b]} (N^{- k + 1 / 2} d Γ (O_{t}^{(k)}))] - E [χ_{[a, b]} (G_{t})], \end{align}

(3.1)

where χ_[a,b] denotes the characteristic function of the set [a, b]. We observe that for

g \in L^{1} (R)

with Fourier transform

\hat{g} \in L^{1} (R, (1 + s^{2 k}) d s)

⁠, we have, on the one hand,

\begin{align} E_{ψ_{N, t}} [g (N^{- k + 1 / 2} d Γ (O_{t}^{(k)}))] & = 〈 ψ_{N, t} | g (N^{- k + 1 / 2} d Γ (O_{t}^{(k)})) | ψ_{N, t} 〉 \\ = \int d τ \hat{g} (τ) 〈 ψ_{N, t} | e^{i τ N^{- k + 1 / 2} d Γ (O_{t}^{(k)})} | ψ_{N, t} 〉 \end{align}

(3.2)

and, on the other hand,

E [g (G_{t})] = \frac{1}{\sqrt{2 π σ^{2}}} \int d x g (x) e^{- \frac{x^{2}}{2 σ_{t}^{2}}} = \int d τ \hat{g} (τ) e^{- \frac{τ^{2} σ_{t}^{2}}{2}},

(3.3)

and, in particular, by Theorem 1.3,

\begin{align} |E_{ψ_{N, t}}, [g (N^{- k + 1 / 2} d Γ (O_{t}^{(k)}))] - E [g (G_{t})]| \\ \leq & C_{k} e^{C | t |} {‖ O^{(k)} ‖}_{op} \int d τ | \hat{g} (τ) | \sum_{ℓ = 1}^{2 k - 1} N^{- ℓ / 2} {(1 + τ^{2} {‖ O^{(k)} ‖}_{op}^{2})}^{(ℓ + 1) / 2} . \end{align}

(3.4)

Thus, in order to find an estimate for (), we shall find an approximation from above f_+,ɛ and from below f_−,ɛ of the characteristic function χ_[a,b], which satisfy

f_{-, ε}, f_{+, ε} \in L^{1} (R^{3})

and

{\hat{f}}_{-, ε}, {\hat{f}}_{+, ε} \in L^{1} (R, (1 + s^{2 k}) d s)

⁠. For this, let

η \in C_{0}^{\infty} (R)

with η ≥ 0, η(s) = 0 for all |s| ≥ 1 and ∫ds η(s) = 1. Furthermore, for ɛ > 0, let η_ɛ(s) = ɛ⁻¹η(s/ɛ). Then, for any ɛ > 0, we define

f_{-, ε} ≔ χ_{[a + ε, b - ε]} * η_{ε} and f_{+, ε} ≔ χ_{[a - ε, b + ε]} * η_{ε},

(3.5)

which satisfy

f_{-, ε} \leq χ_{[a, b]} \leq f_{+, ε} .

(3.6)

Moreover, the Fourier transform is given by

{\hat{f}}_{-, ε} (τ) = - i τ^{- 1} (e^{i τ (b - ε)} - e^{i τ (a + ε)}) \hat{η} (ε τ) .

(3.7)

Thus, it follows from () and () that

\begin{align} P_{ψ_{N, t}} & [N^{- k + 1 / 2} d Γ (O_{t}^{(k)}) \in [a, b]] - P [G_{t} \in [a, b]] \\ \geq - |E [f_{-, ε} (G_{t})] - E_{ψ_{N, t}} [f_{-, ε} (N^{- k + 1 / 2} d Γ (O_{t}^{(k)}))]| - |E [f_{-, ε} (G_{t})] - E [χ_{[a, b]} (G_{t})]|, \end{align}

(3.8)

and with () and (), we arrive at

\begin{align} P_{ψ_{N, t}} & [N^{- k + 1 / 2} d Γ (O_{t}^{(k)}) \in [a, b]] - P [G_{t} \in [a, b]] \\ \geq - C_{k} e^{C | t |} \sum_{ℓ = 1}^{2 k - 1} N^{- ℓ} {(| a - b | ε^{- 1} + ε^{- 2})}^{((ℓ + 1) / 2)} - C ε . \end{align}

(3.9)

Similarly, using f_+,ɛ, we have

\begin{align} P_{ψ_{N, t}} & [N^{- k + 1 / 2} d Γ (O_{t}^{(k)}) \in [a, b]] - P [G_{t} \in [a, b]] \\ \leq C_{k} e^{C | t |} \sum_{ℓ = 1}^{2 k - 1} N^{- ℓ} {(| a - b | ε^{- 1} + ε^{- 2})}^{((ℓ + 1) / 2)} + C ε . \end{align}

(3.10)

Now, we optimize with respect to ɛ > 0 and arrive at ().□

IV. PROOF OF THEOREM 1.3

A. Fluctuations around the Hartree dynamics

In the following, we consider the bosonic N-particle wave function ψ_N,t as an element of the bosonic Fock space $F = ⨁_{n \geq 0} L^{2} {(R^{3})}^{\otimes_{s}^{n}}$ with creation and annihilation operators a^*(f), a(f) for $f \in L^{2} (R^{3})$ ⁠. Theorem 1.3 characterizes the fluctuations around the Hartree dynamics, which are well described by the approximation of the many-body time evolution (Ref. 4, Theorem 4.1, and Ref. 6, Proposition 3.3) in the $L^{2} (R^{3 N})$ -norm,

‖ ψ_{N, t} - W (\sqrt{N} φ_{t}) U_{\infty} (t; 0) Ω ‖ \leq C | t | N^{- 1 / 2},

(4.1)

where the limiting dynamics $U_{\infty} (t; 0)$ is given by

i \partial_{t} U_{\infty} (t; 0) = L (t) U_{\infty} (t; 0), U_{\infty} (0; 0) = 1,

(4.2)

with the generator

L (t) = d Γ (h_{H} (t) + K_{1, t}) + \int d x d y (K_{2, t} (x; y) a_{x}^{*} a_{y}^{*} + {\bar{K}}_{2, t} (x; y) a_{x} a_{y}) .

(4.3)

Here, $d Γ (A) = \int d x d y A (x; y) a_{x}^{*} a_{y}$ denotes the second quantization of an operator A on $L^{2} (R^{3})$ ⁠, h_H(t) denotes the Hartree Hamiltonian defined in (1.22), and K_j,t denote the operators defined in (1.27). For further references, see also Refs. 8, 12, 13, 16, and 18. The generator $L_{\infty} (t)$ is quadratic in creation and annihilation operators and thus (Ref. 3, Theorem 2.2) (see also Refs. 5 and 19) gives rise to a Bogoliubov dynamics, i.e., there exist bounded operators U(t; 0), V(t; 0) on $L^{2} (R^{3})$ such that for $f, g \in L^{2} (R^{3})$ and the operator $A (f, g) = a (f) + a^{*} (\bar{g})$ ⁠, we have

U_{\infty}^{*} (t; 0) A (f, g) U_{\infty} (t; 0) = A (Θ (t; 0) (f, g)) with Θ (t; 0) = (\begin{matrix} U (t; 0) & J V (t; 0) J \\ V (t; 0) & J U (t; 0) J \end{matrix}),

(4.4)

where $J f = \bar{f}$ for any $f \in L^{2} (R^{3})$ ⁠. In particular, for the operator

ϕ (f) = a (f) + a^{*} (f) for f \in L^{2} (R^{3}),

(4.5)

from (4.4), we have

U_{\infty}^{*} (t; 0) ϕ (f) U_{\infty} (t; 0) = ϕ ((U (t; 0) + J V (t; 0)) f),

(4.6)

and it follows from Ref. 3, Theorem 2.2, and the subsequent remark that

i \partial_{s} (U (t; s) + J V (t; s)) f = (h_{H} (s) + K_{1, s} - K_{2, s} J) (U (t; s) + J V (t; s)) f .

(4.7)

Compared with (1.26), we note that the variance σ_t defined in (1.29) is determined by the limiting Bogoliubov dynamics (4.2), i.e., the fluctuations’ quasi-free approximation.

B. Proof of Theorem 1.3

Proof 1.3 is split into three steps covered by Lemmas 4.1–4.3.

For the first step, Lemma 4.1, we use a strategy similar to the strategy in Refs. 19 and 20, directly the norm approximation (4.1). This allows us to consider more general k (respectively, one) particle operators than in Refs. 3 and 6 where the difference of the limiting fluctuation dynamics $U_{\infty} (t; 0)$ defined in (4.2) to the full many-body dynamics was estimated in (4.10) by Duhamel’s formula and a Gronwall estimate. The remaining steps use the same ideas as in Refs. 3 and 6.

Lemma 4.1.

Under the same assumptions as in Theorem 1.2, let

ξ_{N, t} = W (\sqrt{N} φ_{t}) U_{\infty} (t; 0) Ω .

(4.8)

Then, there exists C > 0 such that

|E_{ψ_{N, t}} [e^{i N^{- k + 1 / 2} d Γ (O_{t}^{(k)})}] - E_{ξ_{N, t}} [e^{i N^{- k + 1 / 2} d Γ (O_{t}^{(k)})}]| \leq C | t | N^{- 1 / 2} .

(4.9)

Proof.

We have

\begin{align} E_{ψ_{N, t}} & [e^{i N^{- k + 1 / 2} d Γ (O_{t}^{(k)})}] - E_{ξ_{N, t}} [e^{i N^{- k + 1 / 2} d Γ (O_{t}^{(k)})}] \\ = | 〈 ψ_{N, t} | e^{i τ N^{- k + 1 / 2} d Γ (O_{t}^{(k)})} | ψ_{N, t} - ξ_{N, t} 〉 | + | 〈 ψ_{N, t} - ξ_{N, t} | e^{i τ N^{- k + 1 / 2} d Γ ((O_{t}^{(k)}))} | ξ_{N, t} 〉 | . \end{align}

(4.10)

The operator O^(k) is a self-adjoint operator; thus,

‖ e^{i N^{- k + 1 / 2} O_{N}^{k}} ‖_{op} \leq 1

⁠, and we find with () and (),

|E_{ψ_{N, t}} [e^{i N^{- k + 1 / 2} d Γ (O_{t}^{(k)})}] - E_{ξ_{N, t}} [e^{i N^{- k + 1 / 2} d Γ (O_{t}^{(k)})}]| \leq C | t | N^{- 1 / 2} .

(4.11)

□

Lemma 4.2.

Under the same assumptions as in Theorem 1.2, let ϕ(f) be defined as in (4.5) and

h_{t} = \sum_{j = 1}^{k} h_{j, t} \in L^{2} (R^{3})

be defined with (1.25) by

h_{j, t} = {({\bar{φ}}_{t}^{\otimes (k - 1)} O^{(k)} φ_{t}^{\otimes k})}_{j} .

(4.12)

Then, there exists C > 0 such that

|E_{ξ_{N, t}} [e^{i N^{- k + 1 / 2} d Γ (O_{t}^{(k)})}] - E_{U_{\infty} (t; 0) Ω} [e^{i ϕ (h_{t})}]| \leq C_{k} e^{C | t |} {‖ O^{(k)} ‖}_{op} \sum_{j = 1}^{2 k - 1} N^{- j / 2} {(1 + {‖ O^{(k)} ‖}_{op}^{2})}^{(j + 1) / 2} .

(4.13)

Proof.

Recalling (), we shall estimate the expectation value

\begin{align} E_{ξ_{N, t}} [e^{N^{- k + 1 / 2} d Γ (O_{t}^{(k)})}] & = ⟨W^{*} (\sqrt{N} φ_{t}) e^{i τ N^{- k + 1 / 2} d Γ^{(k)} ({\tilde{O}}^{(k)})} W (\sqrt{N} φ_{t})⟩ U_{\infty} (t; 0) Ω \\ = E_{U_{\infty} (t; 0) Ω} [W^{*} (\sqrt{N} φ_{t}) e^{i τ N^{- k + 1 / 2} d Γ^{(k)} ({\tilde{O}}^{(k)})} W (\sqrt{N} φ_{t})] . \end{align}

(4.14)

In order to compute the operator

O_{N, t} = N^{- k + 1 / 2} W^{*} (\sqrt{N} φ_{t}) d Γ^{(k)} ({\tilde{O}}_{t}^{(k)}) W (\sqrt{N} φ_{t}),

(4.15)

we use the Weyl operators’ shifting properties on creation and annihilation operators, i.e.,

W^{*} (\sqrt{N} φ_{t}) a_{x} W (\sqrt{N} φ_{t}) = a_{x} + \sqrt{N} φ_{t} (x), W^{*} (\sqrt{N} φ_{t}) a_{x}^{*} W (\sqrt{N} φ_{t}) = a_{x}^{*} + \sqrt{N} {\bar{φ}}_{t} (x) .

(4.16)

With

O_{t}^{(k)} ≔ O^{(k)} - 〈 φ_{t}^{\otimes k} | O^{(k)} | φ_{t}^{\otimes k} 〉

⁠, we find

\begin{align} O_{N, t} & = N^{- k + 1 / 2} \int d x_{1} \dots d x_{k} d y_{1} \dots d y_{k} O_{t}^{(k)} (x_{1}, \dots, x_{k}; y_{1}, \dots y_{k}) \\ \times (a_{x_{1}}^{*} + \sqrt{N} {\bar{φ}}_{t} (x_{1})) \dots (a_{x_{k}}^{*} + \sqrt{N} {\bar{φ}}_{t} (x_{k})) (a_{y_{1}} + \sqrt{N} φ_{t} (y_{1})) \dots (a_{y_{k}} + \sqrt{N} φ_{t} (y_{k})) . \end{align}

(4.17)

We observe that the leading order term O(N^k) vanishes by the definition of

{\tilde{O}}^{(k)}

in (). Thus, the first non-vanishing leading order term is O(N^k−1/2), and in particular, we have with () and (),

\begin{align} O_{N, t} & = \int d x_{1} \dots d x_{k} d y_{1} \dots d y_{k} {\tilde{O}}^{(k)} (x_{1}, \dots, x_{k}; y_{1}, \dots y_{k}) \\ \times (\prod_{m = 1}^{k} φ_{t} (y_{m}) \sum_{j = 1}^{k} a_{x_{j}}^{*} \prod_{\begin{array}{c} i = 1 \\ i \neq j \end{array}}^{k} {\bar{φ}}_{t} (x_{i}) + \prod_{m = 1}^{k} {\bar{φ}}_{t} (x_{m}) \sum_{j = 1}^{k} a_{y_{j}} \prod_{\begin{array}{c} i = 1 \\ i \neq j \end{array}}^{k} φ_{t} (y_{i})) + R_{N} \\ = ϕ (h_{t}) + R_{N} . \end{align}

(4.18)

The remainder

R_{N} = O_{N, t} - ϕ (h_{t})

(4.19)

is the sum of (2^k − 2k) terms. The estimates

‖ a (f) ξ ‖ \leq ‖ f ‖_{2} ‖ N^{1 / 2} ξ ‖, ‖ a^{*} (f) ξ ‖ \leq ‖ f ‖_{2} ‖ {(N + 1)}^{1 / 2} ξ ‖

(4.20)

for any

f \in L^{2} (R^{3})

and any Fock space vector

ξ \in F

together with () yield the upper bound

‖ R_{N} ξ ‖ \leq C_{k} ‖ O^{(k)} ‖_{op} \sum_{j = 1}^{2 k} N^{- j / 2} ‖ {(N + 1)}^{(j + 1) / 2} ξ ‖

(4.21)

for any

ξ \in F

⁠. We use the fundamental theorem of calculus to write

\begin{align} E_{ξ_{N, t}} & [e^{i N^{- k + 1 / 2} d Γ (O_{t}^{(k)})}] - E_{U_{\infty} (t; 0) Ω} [e^{i ϕ (h_{t})}] \\ = ⟨ e^{i τ N^{- k + 1 / 2} O_{N, t}} ⟩ U_{\infty} (t; 0) Ω - 〈 U_{\infty} (t; 0) Ω | e^{i τ ϕ (h_{t})} | U_{\infty} (t; 0) Ω 〉 \\ = - \int_{0}^{1} d s \frac{d}{d s} 〈 U_{\infty} (t; 0) Ω | e^{i (1 - s) O_{N, t}} e^{i s ϕ (h_{t})} | U_{\infty} (t; 0) Ω 〉 \\ = \int_{0}^{1} d s 〈 U_{\infty} (t; 0) Ω | e^{i (1 - s) O_{N, t}} R_{N} e^{i s ϕ (h_{t})} | U_{\infty} (t; 0) Ω 〉 . \end{align}

(4.22)

Then, it follows from () that

\begin{align} |E_{ξ_{N, t}}) & ([e^{i N^{- k + 1 / 2} d Γ (O_{t}^{(k)})}] - E_{U_{\infty} (t; 0) Ω} [e^{i ϕ (h_{t})}]| \\ \leq \int_{0}^{1} d s ‖ R_{N} e^{i s ϕ (h_{t})} U_{\infty} (t; 0) Ω ‖ \\ \leq C_{k} ‖ O^{(k)} ‖_{op} \int_{0}^{1} d s \sum_{j = 1}^{2 k - 1} N^{- j / 2} ‖ {(N + 1)}^{(j + 1) / 2} e^{i s ϕ (h_{t})} U_{\infty} (t; 0) Ω ‖ . \end{align}

(4.23)

With Ref. 6, Proposition 3.4 and

{‖ h_{t} ‖}_{2}^{2} \leq {‖ O^{(k)} ‖}_{op}^{2}

by definition (), we have

‖ {(N + 1)}^{(j + 1) / 2} e^{i s ϕ (h_{t})} U_{\infty} (t; 0) Ω ‖ \leq C ‖ {(N + 1 + s^{2} {‖ O^{(k)} ‖}_{op}^{2})}^{(j + 1) / 2} U_{\infty} (t; 0) Ω ‖,

(4.24)

and furthermore, with Ref. 6, Lemma 3.2,

\begin{align} ‖ & {(N + 1)}^{(j + 1) / 2} e^{i s ϕ (h_{t})} U_{\infty} (t; 0) Ω ‖ \\ \leq C e^{C | t |} ‖ {(N + 1 + s^{2} {‖ O^{(k)} ‖}_{op}^{2})}^{(j + 1) / 2} Ω ‖ \leq C e^{C | t |} {(1 + s^{2} {‖ O^{(k)} ‖}_{op}^{2})}^{(j + 1) / 2} . \end{align}

(4.25)

We use now estimate () for () and arrive at

\begin{align} |E_{ξ_{N, t}}) & ([e^{i N^{- k + 1 / 2} d Γ (O_{t}^{(k)})}] - E_{U_{\infty} (t; 0) Ω} [e^{i ϕ (h_{t})}]| \\ \leq C_{k} e^{C | t |} ‖ O^{(k)} ‖_{op} \sum_{j = 1}^{2 k - 1} N^{- j / 2} \int_{0}^{1} d s {(1 + s^{2} {‖ O^{(k)} ‖}_{op}^{2})}^{(j + 1) / 2} \\ \leq C_{k} e^{C | t |} ‖ O^{(k)} ‖_{op} \sum_{j = 1}^{2 k - 1} N^{- j / 2} {(1 + {‖ O^{(k)} ‖}_{op}^{2})}^{(j + 1) / 2}, \end{align}

(4.26)

which proves the lemma.□

Lemma 4.3.

Under the same assumptions as in Theorem 1.2, let

f_{t; s} = \sum_{i = 1}^{k} f_{t; s}^{(i)} \in L^{2} (R^{3})

be given by (1.26). Then, we have

E_{U_{\infty} (t; 0) Ω} [e^{i ϕ (h_{t})}] = e^{- ‖ f_{t; 0} ‖_{2}^{2} / 2} .

(4.27)

Proof.

We need to compute the expectation value

E_{U_{\infty} (t; 0) Ω} [e^{i ϕ (h_{t})}] = 〈 U_{\infty} (t; 0) Ω | e^{i ϕ (h_{t})} | U_{\infty} (t; 0) Ω 〉 .

(4.28)

We recall that the limiting dynamics

U_{\infty} (t; 0)

defined in () acts as a Bogoliubov transform. In particular, in follows from () and () and the notations introduced therein that

U_{\infty}^{*} (t; 0) ϕ (h_{t}) U_{\infty} (t; 0) = ϕ ([U (t; 0) + J V (t; 0)] h_{t}) = ϕ (f_{0; t}),

(4.29)

with f_0;t defined in (). Hence, we have

E_{U_{\infty} (t; 0) Ω} [e^{i ϕ (h_{t})}] = 〈 Ω | e^{i U_{\infty} {(t; 0)}^{*} ϕ (h_{t}) U_{\infty} (t; 0)} | Ω 〉 = 〈 Ω | e^{i ϕ (f_{0; t})} | Ω 〉 .

(4.30)

With the Baker–Campbell–Hausdorff formulas, we split the sum in the exponential and arrive at

E_{U_{\infty} (t; 0) Ω} [e^{i ϕ (h_{t})}] = e^{- ‖ f_{0; t} ‖_{2}^{2} / 2} 〈 Ω | e^{i a^{*} (f_{0; t})} e^{i a (f_{0; t})} | Ω 〉 = e^{- ‖ f_{0; t} ‖_{2}^{2} / 2} .

(4.31)

□

Proof of Theorem 1.3.

Combining now Lemmas 4.1–4.3, we arrive at Theorem 1.3.□

ACKNOWLEDGMENTS

S.R. would like to thank Robert Seiringer and Benedikt Stufler for helpful discussions. Funding from the European Union’s Horizon 2020 Research and Innovation Program under the ERC grant (Grant Agreement No. 694227) and under the Marie Skłodowska-Curie grant (Agreement No. 754411) is acknowledged.

AUTHOR DECLARATIONS

Conflict of Interest

The author has no conflicts to disclose.

DATA AVAILABILITY

The data that support the findings of this study are available within the article.

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2022

Author(s)

Dependent random variables in quantum dynamics

I. INTRODUCTION AND MAIN RESULTS

A. Law of large numbers

1. Generalization to the Fock space

2. Dynamics

B. Central limit theorem

II. PROOF OF THEOREM 1.1

III. PROOF OF THEOREM 1.2

IV. PROOF OF THEOREM 1.3

A. Fluctuations around the Hartree dynamics

B. Proof of Theorem 1.3

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

REFERENCES

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Dependent random variables in quantum dynamics

I. INTRODUCTION AND MAIN RESULTS

A. Law of large numbers

1. Generalization to the Fock space

2. Dynamics

B. Central limit theorem

II. PROOF OF THEOREM 1.1

III. PROOF OF THEOREM 1.2

IV. PROOF OF THEOREM 1.3

A. Fluctuations around the Hartree dynamics

B. Proof of Theorem 1.3

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

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