Volker Bach
email: v.bach@tu-bs.de

Sébastien Breteaux
email: sbreteaux@bcamath.org

Sören Petrat
email: soeren.petrat@ist.ac.at

Peter Pickl
email: pickl@math.lmu.de

Tim Tzaneteas
email: timmy.tzaneteas@uni-tuebingen.de

Kinetic Energy Estimates for the Accuracy of the Time-Dependent Hartree-Fock Approximation with Coulomb Interaction

Keywords: 35Q40, 35Q55, 81Q05, 82C10 Hartree-Fock, many-body theory, mean-field limit for fermions

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I Introduction

The Model. In quantum mechanics, the state of a system of N identical particles is described by a wave function Ψ t which evolves in time t ∈ R according to Schrödinger's equation,

i∂ t Ψ t = HΨ t , Ψ t=0 = Ψ 0 . (1)
Given the (Bose-Einstein or Fermi-Dirac) particle statistics and the one-particle Hilbert space h, the wave function Ψ t is a normalized vector in H (N ) b := S (N ) [h ⊗N ], for a system of N bosons, or in H (N ) f := A (N ) [h ⊗N ], for a system of N fermions. Here S (N ) and A (N ) are the orthogonal projections onto the totally symmetric and the totally antisymmetric subspace, respectively, of the N-fold tensor product h ⊗N of the oneparticle Hilbert space h. The dynamics [START_REF] Adami | Rigorous derivation of the cubic NLS in dimension one[END_REF] is generated by the Hamilton operator H which is self-adjointly realized on a suitable dense domain in H In the present article we study a system of N spinless fermions in R 3 , so Ψ t ∈ H (N ) f , and h = L 2 [R 3 ] is the space of square-integrable functions on R 3 . The Hamiltonian is given by

H = ν + N j=1 h (1) j + λ 1≤j<k≤N v(x j -x k ) , (2)
where

• the number ν ∈ R is a constant contribution to the total energy. For example, if we describe a molecule in the Born-Oppenheimer approximation, then ν would account for the nuclear-nuclear repulsion,

• the coupling constant λ > 0 is a small parameter and possibly depends on the particle number N ≥ 1 (while our interest ultimately lies in the description of systems with N ≫ 1, the estimates in this article hold for any N ≥ 1),

• the self-adjoint operator h (1) on h is of the form -a∆ + w(x), where a > 0 and the external potential w is an infinitesimal perturbation of the Laplacian,

• and v(x) := ±|x| -1 is the Coulomb potential, for x ∈ R 3 \ {0}; v(x) = +|x| -1 is the repulsive case, v(x) = -|x| -1 the attractive case.

The Hamiltonian specified in [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF] describes several situations of interest, e.g.:

• Atom. For an atom in the (0 th ) Born-Oppenheimer approximation with a nucleus of charge Z at the origin, we have repulsive interaction and

ν = 0 , h (1) = - ∆ 2 -α Z |x| , λ = α , (3)
where α > 0 is the fine structure constant whose physical value is α ≃ 1/137. Note that our system of units is chosen such that the reduced Planck constant , the electron mass m and the speed of light c are equal to one, and the charge of the electron is -e = -√ α. For more details about this choice of units see [45, p. 21].

• Molecule. More generally, we can consider a molecule with M ∈ N nuclei of charges Z 1 , . . . , Z M > 0 at fixed, distinct positions R 1 , . . . , R M ∈ R 3 in the Born-Oppenheimer approximation. In this case we have

ν = 1≤m<l≤M αZ m Z l |R m -R l | , h (1) = - ∆ 2 - M m=1 αZ m |x -R m | , λ = α . (4)
• Particles in a Trap. For electrons in an external confining potential (realized, e.g., by a laser trap), we have repulsive interaction and ν = 0 , h (1) = -∆ 2 + w(x) , λ = α .

(5)

• Fermion Star. The Hamiltonian also describes systems of gravitating fermions, e.g., neutrons. In this case the interaction is attractive and

ν = 0 , h (1) = - ∆ 2 , λ = G , (6)
where G is Newton's gravitational constant (and recall that we set the mass m = 0). A better description of a fermion star is achieved by replacing the non-relativistic Laplacian by the semi-relativistic operator √ -∆ + ½ .

For these situations the Hartree-Fock description that we are aiming at in this article and that we describe below can only be expected to hold for very short times (short relative to the large particle number N). For times of order 1, we have to choose the coupling constant small in N to see Hartree-Fock behavior ("mean-field scaling"). There are several possibilities to do that:

• Mean-Field Scaling for Large Volume. Let us first note that for systems with large volume proportional to N, the kinetic energy is naturally also of order N. For such a system, the choice ν = 0, h (1) = -∆ 2 + w(x) , λ = 1 N 2/3 [START_REF] Bach | Error bound for the Hartree-Fock energy of atoms and molecules[END_REF] leads to an interaction energy which is of the same order in N as the kinetic energy (see [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF] for a more detailed discussion).

• Mean-Field Scaling for Fixed Volume. For systems with volume independent of N, the mean-field limit is naturally coupled to a semi-classical limit. Note that here the kinetic energy is of order N 5/3 . Then the choice

ν = 0, h (1) = - ∆ 2N 1/3 + w(x) , λ = 1 N 2/3 (8)
leads to an interaction energy of the same order as the kinetic energy and nontrivial mean-field behavior (see in particular [START_REF] Elgart | Nonlinear Hartree equation as the mean field limit of weakly coupled fermions[END_REF][START_REF] Benedikter | Mean-field Evolution of Fermionic Systems[END_REF] for more details).

• λ = N -1 Scaling. Very often, the term "mean-field scaling" is identified with the choice λ = N -1 . However, comparing with [START_REF] Bach | Error bound for the Hartree-Fock energy of atoms and molecules[END_REF] and [START_REF] Bach | Accuracy of Mean Field Approximations for Atoms and Molecules[END_REF], in the two situations considered above, we see that this scaling leads to a subleading interaction.

Theory of the Time-Dependent Hartree-Fock Equation. Although [START_REF] Adami | Rigorous derivation of the cubic NLS in dimension one[END_REF] admits the explicit solution Ψ t = e -itH Ψ 0 , this explicit form is not useful in practice (from the point of view of numerics, for example) because of the large number N ≫ 1 of variables, and it therefore becomes necessary to consider approximations to this equation. One such approximation consists of restricting the wave function Ψ t to a special class of wave functions. For fermion systems, the Hartree-Fock approximation is a natural choice: it restricts Ψ t to the class of Slater determinants, i.e., to those Φ ∈ H (N ) f which assume a determinantal form,

Φ(x 1 , . . . , x N ) = 1 √ N! det    ϕ 1 (x 1 ) • • • ϕ 1 (x N ) . . . . . . . . . ϕ N (x 1 ) • • • ϕ N (x N )    , (9)
where the orbitals ϕ 1 , . . . , ϕ N ∈ h are orthonormal. We express (9) more concisely as

Φ = ϕ 1 ∧ • • • ∧ ϕ N .
In time-independent Hartree-Fock theory, one is interested in determining the minimal energy expectation when varying solely over Slater determinants [START_REF] Bach | Error bound for the Hartree-Fock energy of atoms and molecules[END_REF][START_REF] Bach | Generalized Hartree-Fock theory and the Hubbard model[END_REF][START_REF] Bach | Bogolubov-Hartree-Fock Mean Field Theory for Neutron Stars and other Systems with Attractive Interactions[END_REF][START_REF] Lewin | Geometric methods for nonlinear many-body quantum systems[END_REF][START_REF] Bach | Mathematical Density and Density Matrix Functional Theory (DFT and DMFT)[END_REF], i.e., one is interested in finding

inf Φ, HΦ Φ = ϕ 1 ∧ • • • ∧ ϕ N , ϕ i , ϕ j = δ ij .
One can also study the evolution governed by (1) using Slater determinants, which gives rise to time-dependent Hartree-Fock theory. Here the basic intuition is that, for a system containing a large number of particles, the solution will stay close to a Slater determinant (at least for short times), provided the initial state is close to a Slater determinant. Turning this intuition into mathematics requires the specification of the equation of motion of the approximating Slater determinant, as well as a mathematically rigorous notion of being "close". For the derivation of the former, one assumes that the solution to (1) is of the form [START_REF] Bach | Mathematical Density and Density Matrix Functional Theory (DFT and DMFT)[END_REF]. It is then easy to verify that the orbitals ϕ t,1 , . . . , ϕ t,N necessarily satisfy the time-dependent Hartree-Fock (TDHF) equation, that is the system of N non-linear equations given by i dϕ t,j dt = h (1)

Φ t = ϕ t,1 ∧ • • • ∧ ϕ t,N , as in
ϕ t,j + λ N k=1 [v * |ϕ t,k | 2 ]ϕ t,j -[v * (ϕ t,j φt,k )]ϕ t,k , (10)
for j = 1, . . . , N ( φ is the complex conjugate of ϕ).

The TDHF equation ( 10) can be rewritten in terms of the one-particle density matrix p t = N j=1 |ϕ t,j ϕ t,j | with ϕ t,j ∈ h and ϕ t,j , ϕ t,k = δ jk as

(TDHF) i∂ t p t = [h (1) , p t ] + λTr 2 [v (2) , (p t ⊗ p t )(½ -X)] . ( 11
)
Here X is the linear operator on h ⊗ h such that X(ϕ ⊗ ψ) = ψ ⊗ ϕ and Tr 2 is the partial trace (see [START_REF] Elgart | Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons[END_REF]). Sometimes, we write p

(2) t = (p t ⊗ p t )(½ -X).
In the sequel, when speaking of the TDHF equation, we refer to [START_REF] Bach | Generalized Hartree-Fock theory and the Hubbard model[END_REF]. The term involving ½ is called the direct term, the term involving X the exchange term.

Note that the TDHF equation ( 11) can be written as

i∂ t p t = [h (1)
HF (p t ), p t ], where the effective HF-Hamiltonian h

HF (γ) := h (1) + λTr 2 [v (2) (½ h⊗h -X)(½ h ⊗ γ)] . (12)
Implicitly assuming the existence and regularity of p t , the HF-Hamiltonian h

HF (p t ) is self-adjoint with the same domain as h (1) , and hence the solution to

∂ t U HF,t = -ih (1)
HF (p t )U HF,t , with U HF,0 = ½, is unitary. This has the important consequence that [START_REF] Bach | Generalized Hartree-Fock theory and the Hubbard model[END_REF] preserves the property of the one-particle density matrix p t of being a rank-N orthonormal projection. In other words, if Φ t ∈ H (N ) f evolves according to the TDHF equation and

Φ 0 = ϕ 1 ∧ • • • ∧ ϕ N is a Slater determinant, then so is Φ t , for all t ∈ R.
The TDHF equation for density matrices as in [START_REF] Bach | Generalized Hartree-Fock theory and the Hubbard model[END_REF] has been studied in [START_REF] Bove | An existence proof for the Hartree-Fock timedependent problem with bounded two-body interaction[END_REF] for a bounded two-body interaction. Then the mild solutions of the TDHF equation in the form [START_REF] Bach | Bogolubov-Hartree-Fock Mean Field Theory for Neutron Stars and other Systems with Attractive Interactions[END_REF] have been handled for a Coulomb two-body potential in [START_REF] Chadam | Global existence of solutions to the Cauchy problem for time-dependent Hartree equations[END_REF] for initial data in the Sobolev space H 1 . This result has been extended to the TDHF equation in the form [START_REF] Bach | Generalized Hartree-Fock theory and the Hubbard model[END_REF] in [START_REF] Bove | On the Hartree-Fock time-dependent problem[END_REF][START_REF] Chadam | The time-dependent Hartree-Fock equations with Coulomb twobody interaction[END_REF]. Note that [START_REF] Bove | On the Hartree-Fock time-dependent problem[END_REF] also handles the case of a more general class of two-body potentials and the existence of a classical solution for initial data in a space similar to the Sobolev H 2 space for density matrices. In [START_REF] Zagatti | The Cauchy problem for Hartree-Fock time-dependent equations[END_REF] the existence of mild solutions of the TDHF in the form [START_REF] Bach | Bogolubov-Hartree-Fock Mean Field Theory for Neutron Stars and other Systems with Attractive Interactions[END_REF] was proved for a Coulomb two-body potential with an (infinite sequence of) initial data in L 2 . For the convenience of the reader we state the precise results we use about the theory of the TDHF equation in Appendix A. In [START_REF] Arnold | Self-consistent relaxation-time models in quantum mechanics[END_REF] the existence and uniqueness of strong solutions to the von Neumann-Poisson equation, another nonlinear self-consistent time-evolution equation on density matrices, are proved with the use of a generalization of the Lieb-Thirring inequality. Another direction in which to generalize the Hartree equations is to consider, instead of an exchange term, a dissipative term in the Hartree equation; the existence and uniqueness of a solution for such an equation has been proved in [START_REF] Arnold | Quantum dynamical semigroups for diffusion models with Hartree interaction[END_REF].

One-particle Density Matrix. The notion of proximity of two states we use in this article is defined by expectation values of k-particle observables, where

1 ≤ k ≪ N. More specifically, if Ψ t ∈ H (N ) f
is the (normalized) solution to (1) and Φ HF,t = ϕ t,1 ∧• • •∧ ϕ t,N , where ϕ t,1 , . . . , ϕ t,N are the solutions to [START_REF] Bach | Bogolubov-Hartree-Fock Mean Field Theory for Neutron Stars and other Systems with Attractive Interactions[END_REF], then, for any k-particle operator A (k) (i.e., for any bounded operator A (k) on h ∧k := A[h ⊗k ]), we wish to control the quantity

δ (k) t A (k) := 1 A (k) ∞ Ψ t , (A (k) ⊗ ½ N -k )Ψ t -Φ HF,t , (A (k) ⊗ ½ N -k )Φ HF,t .
Here ½ N -k denotes the identity operator on h ⊗(N -k) and • ∞ denotes the operator norm on

B[h ∧k ].
It is more convenient to reformulate this approach in terms of reduced density matrices. We recall that, given Ψ ∈ H (N ) f , the corresponding reduced k-particle density matrix is the trace-class operator γ

(k) Ψ on H (k)
f whose kernel is given by

γ (k) Ψ (x 1 , . . . , x k ; y 1 , . . . y k ) = N! (N -k)! ˆΨ(x 1 , . . . x k , x k+1 , . . . x N ) Ψ(y 1 , . . . y k , x k+1 , . . . x N ) d 3 x k+1 • • • d 3 x N . (13)
Note that we normalize the reduced density matrices so that Trγ

(k) Ψ = N ! (N -k)! . We may then rewrite δ (k) t (A (k) ) as δ (k) t A (k) = 1 A (k) B(H (k) f ) Tr (γ (k) Ψt -γ (k)
Φ HF,t )A (k) and observe that sup

A (k) ∈B(H (k) f ) δ (k) t A (k) = γ (k) Ψt -γ (k) Φ HF,t L 1 ,
where • L 1 denotes the trace norm. We are thus interested in bounds on γ

(k) Ψt - γ (k)
Φ HF,t L 1 . In the present article we restrict ourselves to the case k = 1.

Derivation of the TDHF Equation. The derivation of the TDHF equation may be seen as part of the quest for a derivation of macroscopic, or mesoscopic, dynamics from the microscopic classical or quantum-mechanical dynamics of many-particle systems as an effective theory. Let us first discuss some generally interesting examples and then come to the case of the TDHF equation for fermions.

In the case of the dynamics of N identical quantum-mechanical particles, the timedependent Hartree equation, that is the TDHF equation [START_REF] Bach | Bogolubov-Hartree-Fock Mean Field Theory for Neutron Stars and other Systems with Attractive Interactions[END_REF] without the exchange term, was first derived rigorously in [START_REF] Spohn | Kinetic equations from Hamiltonian dynamics: Markovian limits[END_REF] for a system of N distinguishable particles in the mean-field limit. For systems of indistinguishable particles, the case of bosons has received considerable attention compared to the case of fermions, and several methods have been developed. The so-called Hepp method has been developed in [START_REF] Hepp | The classical limit for quantum mechanical correlation functions[END_REF][START_REF] Ginibre | The classical field limit of scattering theory for nonrelativistic many-boson systems[END_REF][START_REF] Ginibre | The classical field limit of scattering theory for nonrelativistic many-boson systems[END_REF] in order to study the classical limit of quantum mechanics. It inspired, among others, [START_REF] Fröhlich | On a classical limit of quantum theory and the nonlinear Hartree equation[END_REF], where the convergence to the Hartree equation is proved, [START_REF] Rodnianski | Quantum fluctuations and rate of convergence towards mean field dynamics[END_REF], where the rate of convergence toward mean-field dynamics is studied, and [START_REF] Ammari | Mean field limit for bosons and infinite dimensional phase-space analysis[END_REF][START_REF] Ammari | Mean field limit for bosons and propagation of Wigner measures[END_REF], where the propagation of Wigner measures in the mean-field limit is studied, with special attention to the relationships with microlocal and semiclassical analysis. In this direction, with a stochastic microscopic model, the linear Boltzmann equation was obtained as a weakcoupling limit in [START_REF] Breteaux | A geometric derivation of the linear Boltzmann equation for a particle interacting with a Gaussian random field, using a fock space approach[END_REF] yielding an example for a derivation of an equation with non-local terms using methods of pseudodifferential calculus. The derivation of the linear Boltzmann equation in the earlier work [START_REF] Erdős | Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation[END_REF], along with the series of works following it, used a different method based on series expansions in terms of graphs similar to Feynman diagrams. The result is valid on longer time-scales than in [START_REF] Breteaux | A geometric derivation of the linear Boltzmann equation for a particle interacting with a Gaussian random field, using a fock space approach[END_REF], but with more restrictive initial data. Other limit dynamics have been obtained, a particularly interesting one is the weak-coupling limit for interacting fermions for which a (non-rigorous) derivation of the nonlinear Boltzmann equation has been given in [START_REF] Erdős | On the quantum Boltzmann equation[END_REF]. Series expansion methods and the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy have also proved fruitful in other works, e.g., [START_REF] Spohn | Kinetic equations from Hamiltonian dynamics: Markovian limits[END_REF][START_REF] Bardos | Weak coupling limit of the N-particle Schrödinger equation[END_REF][START_REF] Elgart | Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons[END_REF][START_REF] Adami | Rigorous derivation of the cubic NLS in dimension one[END_REF][START_REF] Erdős | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF][START_REF] Ammari | Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states[END_REF][START_REF] Erdős | Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate[END_REF][START_REF] Erdős | Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential[END_REF]. In [START_REF] Erdős | Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate[END_REF][START_REF] Erdős | Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential[END_REF] the Gross-Pitaevskii equation, which describes the dynamics of a Bose-Einstein condensate has been derived. Also for the Gross-Pitaevskii equation the formation of correlations has been studied in [START_REF] Erdős | Dynamical formation of correlations in a Bose-Einstein condensate[END_REF], providing information on the structure of solutions to the Gross-Pitaevskii equation. The techniques developed in [START_REF] Lukkarinen | Weakly nonlinear Schrödinger equation with random initial data[END_REF] to study the weakly nonlinear Schrödinger equation are used in [START_REF] Lukkarinen | Not to normal order-notes on the kinetic limit for weakly interacting quantum fluids[END_REF] to derive quantum kinetic equations; those techniques resemble the BBGKY hierarchy methods, but they do not impose the normal ordered product of operators when considering expectation with respect to the initial state. The bounds on the rate of convergence in the mean-field limit given in [START_REF] Fröhlich | On a classical limit of quantum theory and the nonlinear Hartree equation[END_REF] have been sharpened in [START_REF] Erdős | Quantum dynamics with mean field interactions: a new approach[END_REF] using a method inspired by Lieb-Robinson inequalities. Another method introduced in [START_REF] Fröhlich | Atomism and quantization[END_REF] shows that the classical time evolution of observables commute with the Wick quantization up to an error term which vanishes in the mean-field limit, yielding an Egorov-type theorem. Recently a new method based on a Grønwall lemma for a well-chosen quantity has been introduced [START_REF] Pickl | A simple derivation of mean field limits for quantum systems[END_REF][START_REF] Knowles | Mean-field dynamics: singular potentials and rate of convergence[END_REF] in the bosonic case, which considerably simplifies the convergence proof for the Hartree equation.

In the fermionic case, the TDHF equation has been derived in [START_REF] Bardos | Mean field dynamics of fermions and the time-dependent Hartree-Fock equation[END_REF] in the λ = N -1 scaling for initial data close to Slater determinants, and with bounded two-body potentials. The same authors give bounds on the accuracy of the TDHF approximation for uncorrelated initial states in [START_REF] Bardos | Accuracy of the timedependent Hartree-Fock approximation for uncorrelated initial states[END_REF], still with a bounded two-body potential. For the same scaling, the TDHF equation has been derived in [START_REF] Fröhlich | A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction[END_REF] for the Coulomb potential for sequences of initial states given by Slater determinants. The semi-classical mean-field scaling from (8) has first been considered in [START_REF] Narnhofer | Vlasov Hydrodynamics of a Quantum Mechanical Model[END_REF] where it is shown that for suitably regular interactions the Schrödinger dynamics is close to the classical Vlasov dynamics. The results have been improved in [START_REF] Spohn | On the Vlasov hierarchy[END_REF]. In [START_REF] Elgart | Nonlinear Hartree equation as the mean field limit of weakly coupled fermions[END_REF], in the semi-classical mean-field scaling, the closeness of the Schrödinger dynamics to the Hartree-Fock dynamics was discussed and bounds for the Husimi function were given, assuming the potential to be realanalytic and thus in particular bounded. Up to that point all the method used to derive the TDHF equation had always been based on BBGKY hierarchies. In [START_REF] Benedikter | Mean-field Evolution of Fermionic Systems[END_REF][START_REF] Benedikter | Mean-field dynamics of fermions with relativistic dispersion[END_REF] estimates of γ N,tp N,t L 1 were given in terms of the number N of electrons and the time t, in the semi-classical mean-field scaling. Their method is based on the Grønwall lemma, similarly to [START_REF] Pickl | A simple derivation of mean field limits for quantum systems[END_REF] in the bosonic case. The second article deals with the semirelativistic case. The authors pointed out that with a bounded potential, in this scaling, the exchange term in the time-dependent Hartree-Fock equation does not play a role so that the time-dependent Hartree-Fock equation reduces to the time-dependent Hartree equation. In [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF], the fermionic Hartree equation in the large volume case is considered by generalizing the method of [START_REF] Pickl | A simple derivation of mean field limits for quantum systems[END_REF]. Interactions of the form |x| -s are considered, with the corresponding λ = N -1+s/3 . Under the condition that the Hartree-Fock kinetic energy per particle is bounded uniformly in time, a derivation of the TDHF equation is given for 0 < s < 3/5, and for Coulomb interaction with either a mild singularity cutoff on a ball with radius N -1/6+ε , for any ε > 0, or for the full Coulomb interaction under certain Sobolev conditions on the solution to the TDHF equation which are not proven in this work. Explicit bounds in terms of N, the Hartree-Fock kinetic energy and t are given. Furthermore, in [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF], the main result of [START_REF] Benedikter | Mean-field Evolution of Fermionic Systems[END_REF] is reproduced with a different method than in [START_REF] Benedikter | Mean-field Evolution of Fermionic Systems[END_REF] and written down for weaker conditions on the closeness of the initial state to a Slater determinant.

Main Estimate of this Article (see Theorem II.1). Given a normalized initial state

Ψ 0 ∈ H (N ) f
and the one-particle density matrix p 0 ≡ γ Φ HF,0 associated with a Slater determinant Φ HF,0 = ϕ 1,0 ∧ • • • ∧ ϕ N,0 , with ϕ i,0 , ϕ j,0 being orthonormal orbitals in H 1 (R 3 ), γ t the one-particle density matrix of the solution Ψ t to (1) and p t the solution to [START_REF] Bach | Generalized Hartree-Fock theory and the Hubbard model[END_REF] obey the trace norm estimate

1 N γ t -p t L 1 ≤ √ 8 N 2/3 1 N γ 0 -p 0 L 1 exp(C λ,N,K t) + N -1/3 exp(C λ,N,K t) -1 , ( 14
) with C λ,N,K = 30λ √ KN 1/6
, where K is a bound on the kinetic energy of p t which is assumed to be uniform in time (see [START_REF] Benedikter | Mean-field dynamics of fermions with relativistic dispersion[END_REF]).

Discussion of the Results. Roughly speaking, the estimate [START_REF] Bardos | Weak coupling limit of the N-particle Schrödinger equation[END_REF] implies that, starting from a state close to a Slater determinant for the N-body Schrödinger equation and from the corresponding one-particle density matrix for the TDHF equation, the Hartree-Fock approximation is justified up to times of order (λ √ KN 1/6 ) -1 , where K is the kinetic energy (which, for repulsive systems, is bounded by the total energy of the system, uniformly in time) and λ the coupling constant. Hence, our assumption on the initial state is given in terms of energy, and not in the form of "increasing" sequences of Slater determinants. This assumption seems more natural to the authors as it is closer to a thermodynamic assumption on the system. In our proof we obtain a rate of convergence of N -1/6 . For the initial data, in order to have convergence, we can allow states with N -1/3 γ 0p 0 L 1 → 0 for N → ∞. This means, e.g., that, for any ε > 0, the initial state can have N 1/3-ε particles outside the condensate, i.e., the Slater determinant structure.

The fact that the estimate ( 14) is relevant when λN 1/6 K 1/2 t is of order one, restricts its applicability to a regime where the kinetic energy dominates the direct and exchange terms. This implies that the evolution is the free evolution to leading order. Estimate ( 14) captures the subleading effect of the direct term on the dynamics and is thus relevant provided that K ≫ N 4/3 . We substantiate this by heuristic arguments in Appendix B. Let us stress that Estimate ( 14) requires no additional assumption on the initial states other than the Hartree-Fock kinetic energy to be finite. Furthermore, Estimate ( 14) applies to the repulsive or attractive Coulomb interaction, which is very relevant for many physical systems.

Compared to [START_REF] Fröhlich | A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction[END_REF], where also the Coulomb potential was considered, our result holds for larger time scales. In [START_REF] Fröhlich | A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction[END_REF], the λ = N -1 -scaling was assumed and, by a rescaling in time and in space, the result also applies to a large neutral atom (i.e., with charge N ≫ 1 and λ = α). With the result of [START_REF] Fröhlich | A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction[END_REF] the Hartree-Fock approximation is then justified up to times of order N -2 . Assuming we have a state with a negative energy, the kinetic energy is controlled by a universal multiple of N 7/3 (see Sect. II for more details), and our estimate allows us to justify the approximation up to much larger times, of order N -4/3 . (Note, however, that our estimate deteriorates if the energy of the state is higher.)

Compared to [START_REF] Benedikter | Mean-field Evolution of Fermionic Systems[END_REF] where the semi-classical scaling ( 8) is considered, our result allows us to control the approximation only up to times of order N -1/3 , whereas the estimates in [START_REF] Benedikter | Mean-field Evolution of Fermionic Systems[END_REF] allow one to control the approximation up to times of order 1 (however, only for bounded two-body potentials). This comes from the fact that we do not assume any semi-classical structure on the initial data. Note that our strategy is similar to the one of [START_REF] Benedikter | Mean-field Evolution of Fermionic Systems[END_REF] since we do not use the BBGKY hierarchy but instead make use of a Grønwall lemma. An important difference lies in the decomposition of the potential: in [START_REF] Benedikter | Mean-field Evolution of Fermionic Systems[END_REF] a Fourier decomposition is used whereas we use the Fefferman-de la Llave formula.

Let us compare our results to [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF] where the mean-field scaling for large volume ( 7) is considered. Note that there the Schrödinger dynamics is compared to the fermionic Hartree equation without exchange term. While in [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF] other interactions are also considered, for Coulomb interaction, essentially two results are proven. First, for regularized Coulomb interaction with singularity cut off on a ball with radius N -1/6+ε for any ε > 0, convergence of the Schrödinger dynamics to the fermionic Hartree dynamics is shown in terms of a counting measure α g , with convergence rate depending on the cutoff. Note, that we use the same measure in our proof, see also Remark III.5, but we formulate our main result only in terms of the trace norm difference of reduced densities. The improvement of our result is that it holds for full Coulomb interaction without any regularization and, in general, with a better convergence rate. For the second result in [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF] a bound on Tr[(-∆) 3+ε p t ] is assumed. Under that condition convergence for full Coulomb interaction in terms of α g and the trace norm difference is shown, with rate N -1/2 in the trace norm sense. This bound on Tr[(-∆) 3+ε p t ] was, however, not proven to hold for t > 0. Compared to that, our result holds for any initial condition with kinetic energy bounded by CN, without further assumptions, but only with a convergence rate of N -1/6 in the trace norm sense.

Sketch of our Derivation of Estimates on the Accuracy of the TDHF Approximation. We derive an estimate on the trace norm of the difference γ tp t between the one-particle density matrix γ t ≡ γ Ψt of the (full) solution Ψ t = e -itH Ψ 0 of (1) and the one-particle density matrix p t solving the TDHF equation [START_REF] Bach | Generalized Hartree-Fock theory and the Hubbard model[END_REF]. Our work is inspired by [START_REF] Pickl | A simple derivation of mean field limits for quantum systems[END_REF], where one of us developed a new method for bosonic systems which was generalized to fermion systems in [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF] by two of us. The method uses a Grönwall estimate for a well-chosen quantity called the number of bad particles in [START_REF] Pickl | A simple derivation of mean field limits for quantum systems[END_REF]. We refer to the quantity we chose to control as the degree of evaporation S g . The subscript g refers to a freedom in the choice of a weight function g which allows us to fine-tune the distance of ρ t (the density matrix of Ψ t ) to p t in a suitable way. For the simplest choice g(x) = x, S g is called the degree of non-condensation in [38, Remark (a) on p. 5], while in [START_REF] Siedentop | The asymptotic behaviour of the ground state energy of the Müller functional for heavy atoms[END_REF] it is called Verdampfungsgrad, which translates to degree of evaporation.

We show that the degree of evaporation S g is directly related to the trace norm γp L 1 . We then calculate the time derivative of S g and split it into three terms that we estimate separately. To obtain the estimates we make use of correlation inequalities which may be seen to be a dynamical version of the correlation estimate presented in [START_REF] Bach | Error bound for the Hartree-Fock energy of atoms and molecules[END_REF]. (See also [START_REF] Graf | A correlation estimate with applications to quantum systems with Coulomb interactions[END_REF] for an alternative proof of that correlation estimate which does not make use of second quantization.) While we estimate two of the terms in a way very similar to [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF], our estimate for the remaining term (here called A; in [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF] called (I)) is very different and allows us to treat the full Coulomb potential. This term is physically the most important, since its smallness is a consequence of cancellations between the Hartree-Fock and the many-body interaction. The bounds on this term are the key estimates of this work. They are obtained by using the Fefferman-de la Llave decomposition formula [START_REF] Fefferman | Relativistic stability of matter[END_REF]. We remark that, in view of the generalization of this decomposition derived in [START_REF] Hainzl | General decomposition of radial functions on R n and applications to N-body quantum systems[END_REF][START_REF] Gneiting | Radial positive definite functions generated by Euclid's hat[END_REF], our result applies to a more general class of twobody interaction potentials. The Lieb-Thirring inequality [START_REF] Lieb | Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter[END_REF] and Hardy's inequality then provide an estimate in terms of kinetic energy. Finally, we note that in many physically relevant cases the estimate in terms of kinetic energy can be stated in terms of an estimate on the initial total energy of the system.

Outline of the Article. In Sect. II we state our main result, along with applications to molecules or the mean-field limit. In Sect. III we introduce the degree of evaporation S g and relate it to the difference between the one-particle density matrix of the solution to our model and the solution to the TDHF equation. We then calculate the time derivative of S g and provide bounds for the different contributions, thus proving our main theorem. In Appendix A we recall some results about the theory of the TDHF equation.

II Main Result and Applications

Our main result is an estimate of the trace norm • L 1 of the difference between the one-particle density matrix of the solution to the many-body Schrödinger equation ( 1) and the solution to the time-dependent Hartree-Fock equation [START_REF] Bach | Generalized Hartree-Fock theory and the Hubbard model[END_REF] in terms of the kinetic energy of the system. As usual, we denote by H 1 (R 3 ) the Sobolev space of weakly differentiable functions with square-integrable derivative.

We henceforth make use of the following notation:

• Let Ψ 0 ∈ H (N ) f
be a normalized initial state, and let γ t := γ Ψt be the one-particle density matrix of the solution Ψ t = e -iHt Ψ 0 to the Schrödinger equation [START_REF] Adami | Rigorous derivation of the cubic NLS in dimension one[END_REF] with Hamiltonian H from (2) (i.e., with Coulomb interaction).

• Let Φ HF,0 = ϕ 1,0 ∧ • • • ∧ ϕ N,0 be a Slater determinant, with ϕ j,0 ∈ H 1 (R 3 ) and ϕ j,0 , ϕ k,0 h = δ jk , for 1 ≤ j, k ≤ N.
Let p 0 := γ Φ HF,0 be the one-particle density matrix of Φ HF,0 and p t be the solution to the time-dependent Hartree-Fock equation ( 11) with initial condition p 0 .

Theorem II.1. Assume that the kinetic energy of p t is uniformly bounded in time,

K := sup t≥0 Tr[-∆p t ] < ∞ . (15)
Under the assumption of (15) the estimate

1 N γ t -p t L 1 ≤ √ 8 N 2/3 1 N γ 0 -p 0 L 1 exp(C λ,N,K t) + N -1/3 exp(C λ,N,K t) -1 (16) holds true with C λ,N,K = 30λ √ KN 1/6 .
The proof of Theorem II.1 is postponed to Sect. III.

Remark II.2. One of the ingredients of our proof is the Fefferman-de la Llave decomposition of the Coulomb potential [31]

1 |x| = ˆ∞ 0 16 π r 5 (1 B(0,r/2) * 1 B(0,r/2) )(x) dr , (17)
an identity that holds for all x ∈ R 3 \ {0}, where 1 B(0,r/2) is the characteristic function of the ball of radius r/2 centered at the origin in R 3 . A generalization of this decomposition to a class of two-body interaction potentials v of the form

v(x) = ˆ∞ 0 g v (r) (1 B(0,r/2) * 1 B(0,r/2) )(x) dr , (18)
with x ∈ R 3 \ {0}, was given in [START_REF] Hainzl | General decomposition of radial functions on R n and applications to N-body quantum systems[END_REF] under Assumption II.3 below, and our proof largely generalizes to those potentials v. More precisely, the assertion of Theorem II.1 holds true and without any change in the constants, if we replace the Coulomb potential by any pair potential v that satisfies Assumption II.4 below, which in particular implies v(x) ≤ |x| -1 . Note that the assumption of semi-boundedness of v is only used to ensure the global existence of a solution to the TDHF equation. One could drop it to study problems up to the time the solution to the TDHF blows up.

Assumption II.3. The function v : R 3 \ {0} → R has the following properties:

• v is a radial function, and there exists a function ṽ ∈ C 3 [(0, ∞); R] such that v(x) = ṽ |x| , for all x ∈ R 3 \ {0},

• r m d m ṽ dr m (r) → 0, as r → ∞, for m = 0, 1, 2,

• lim R→∞ ´R 1 r 3 g v (r) dr exists, with g v (r) := 2 Remark II.5. Note that we actually prove a slightly stronger result in Theorem III.8 in terms of the degree of evaporation S g (t) (with properly chosen g), which is defined in Definition III.3. The way our result is formulated in Theorem III.8 can directly be compared to the results in [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF].

Remark II.6. Note that the two summands in the square root on the right-hand side of ( 16) come from different contributions which we call A t , B t and C t (and which are called (I), (II), (III) in [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF]), see Proposition III.9. It is interesting to note that all three terms contribute to the first summand (which is proportional to γ 0p 0 L 1 ) but only the B t term contributes to the second summand.

Remark II.7. Note that it is sufficient to prove Theorem II.1 with

Ψ 0 in H (N ) f ∩ H 1 (R 3 )
⊗N . A density argument then provides the result for a general Ψ 0 in H (N )

f . Let us discuss some cases when the assumption that the Hartree-Fock kinetic energy is uniformly bounded in time is satisfied. In Propositions II.11 and II.12 we give explicit bounds on the kinetic energy K in terms of the energy expectation value Φ HF,0 , HΦ HF,0 of the initial state Φ HF,0 and the ground state energy for examples presented in Sect. I. In the case of atoms or molecules this follows from known estimates, which we now recall.

To formulate these, we denote the energy expectation value and the kinetic energy expectation value of a normalized wave function

Ψ ∈ H (N ) f ∩ H 1 (R 3 )
⊗N by

E(Ψ) = Ψ, HΨ and K(Ψ) = Ψ, N j=1 -∆ j Ψ .
For atoms and molecules the ground state energy E gs is defined as

E gs = inf E(Ψ) Ψ ∈ H (N ) f ∩ H 1 (R 3 ) ⊗N , Ψ H (N) f = 1 , R 1 , . . . , R M ∈ R 3 , l = m ⇒ R l = R m .
Equipped with this notation, we formulate the coercivity of the energy functional on the Sobolev space of states with finite kinetic energy:

Proposition II.8. Consider a neutral atom or a molecule as in (3) or (4). If E gs ≤ 0 then

K(Ψ) ≤ E(Ψ) -E gs + -E gs 2 ≤ 2E(Ψ) + 4|E gs | .
Proof. See [45, p.132].

Using Proposition II.8 along with the conservation of the total energy for both the Schrödinger equation and the TDHF equation (see Appendix A) we get the following bound on the kinetic energy.

Proposition II.9. Assume that Φ HF,0 = ϕ 1,0 ∧ • • • ∧ ϕ N,0 is a Slater determinant, with ϕ j,0 ∈ H 1 (R 3 ) and ϕ j,0 , ϕ k,0 h = δ jk , for 1 ≤ j, k ≤ N. Then, in the case of atoms or molecules as in (3) or (4),

K := sup t≥0 Tr[-∆p t ] ≤ E(Φ HF,0 ) -E gs + -E gs 2 . ( 19
)
Thus, if E(Φ HF,0 ) ≤ 0 then K ≤ -4E gs . (20)
We also recall a known bound for the ground state energy, see [START_REF] Lieb | Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter[END_REF] or [START_REF] Lieb | The stability of matter in quantum mechanics[END_REF], whose units we use. Proposition II.12 (TDHF equations without external potential and with repulsive interaction). For h (1) = -∆/2 and v(x) = |x| -1 , the Hartree-Fock kinetic energy is bounded by the total Hartree-Fock energy (for any λ > 0), which is preserved in time, i.e., K ≤ E(Φ HF,0 ) .

Finally, let us note that for attractive Coulomb interaction without external field, we have the bound

K ≤ 2E(Φ HF,0 ) + Cλ 2 N 7/3 , (21)
which follows from the Lieb-Thirring inequality and which we prove in Appendix B. Thus, also for attractive interaction, the bounds K ≤ CN in the mean-field scaling for large volume [START_REF] Bach | Error bound for the Hartree-Fock energy of atoms and molecules[END_REF] and K ≤ CN 5/3 in the semi-classical mean-field scaling (8) hold, if the corresponding bounds hold for the total energy.

III Control of the Degree of Evaporation S g

We first introduce the degree of evaporation S g , which is a function of a state on the Fock space and a one-particle density matrix. We use S g as an indicator of closeness of the Hartree-Fock to the Schrödinger quantum state.

III.1 Definition and Properties of the Degree of Evaporation

For A and more generally B (M ) (M ≤ N) linear operators acting on h and H (M ) f

, respectively, we use the notation

dΓ(A) := N j=1 A j and dΓ (M ) (B (M ) ) := N j 1 ,...,j M =1 j 1 =j 2 ... =j M B (M ) j 1 ...j M , (22)
as operators on H

(N ) f , with A j acting on the j th factor in h ⊗N and B (M ) acting on the j th 1 , . . . , j th M factors in h ⊗N , respectively. Remark III.1. Although we do not use the fermion creation and annihilation operators a * , a, note that ( 22) coincides with the second quantization dΓ in quantum field theory in the sense that dΓ(A) = ˆA(x; y) a * (x)a(y) dxdy , or, more exactly, its restriction to the N-particle sector. Similarly, e.g., dΓ (2) (B (2) ) = ˆB(2) (x 1 , x 2 ; y 1 , y 2 ) a * (x 2 )a * (x 1 )a(y 1 )a(y 2 ) dx 1 dx 2 dy 1 dy 2 .

Let L 1 (H) denote the space of trace class operators on a Hilbert space H. We use the partial trace Tr 2 : L 1 (h ⊗2 ) → L 1 (h) which is defined for B (2) ∈ L 1 (h ⊗2 ) to be the operator Tr 2 [B (2) ] ∈ L 1 (h) such that

Tr Tr 2 (B (2) ) A = Tr B (2) (A ⊗ ½ h ) (23)
holds for all A ∈ B(h).

Definition III.2. For an N-particle density matrix ρ ∈ L 1 + (H

(N )
f ), i.e., a non-negative trace-class operator on H (N ) f of unit trace, the one-(resp. two-)particle density matrix of ρ is denoted by γ ρ (resp. γ

(2) ρ ). It is the operator on h (resp. H (2) f ) such that ∀A ∈ B(h) : Tr H (N) f [ρ dΓ(A)] = Tr h [γ ρ A] , (24)
∀B (2) ∈ B(H

(2) f ) : Tr H (N) f ρ dΓ (2) (B (2) ) = Tr H (2) f γ (2) ρ B (2) . ( 25
)
We note that γ ρ and γ

(2)

ρ satisfy 0 ≤ γ ρ ≤ ½ , Tr h [γ ρ ] = N , 0 ≤ γ (2) ρ ≤ N , Tr H (2) f [γ (2) ρ ] = N(N -1) (26)
(see, e.g., [9, Theorem 5.2]). Further note that we are slightly abusing notation since the one-particle density matrix was defined for wave functions in Eq. ( 13), rather than for density matrices. We thus identify γ Ψ ≡ γ |Ψ Ψ| , for all normalized Ψ ∈ H

(N )
f , whenever this does not lead to confusion. Definition III.3. Let N ∈ N and

S N := η ∈ L 1 (h) 0 ≤ η ≤ ½ , Tr[η] = N ,
and g be a continuous function from R to R. The map S g :

L 1 + (H (N ) f ) × S N → R + 0 defined by S g (ρ, η) := Tr[ρ ĝ] , (27)
where ĝ := g dΓ(½η) , is called the degree of evaporation (of ρ relative to η). The translation of g by k ∈ Z, is denoted by τ k g(y) := g(yk).

Note that when the expression of g is too long to fit under a hat, we write (g) ∧ instead of ĝ.

Remark III.4. ĝ := g dΓ(½η) is defined using the functional calculus [START_REF] Reed | Methods of modern mathematical physics. I. Functional analysis[END_REF][START_REF] Riesz | Functional analysis[END_REF].

Remark III.5. The particular case in which η is a rank-N projector is of importance in the sequel, and we then write p := η and q := ½η .

In this case, only the values of g on {0, . . . , N} are relevant for the definition of ĝ and the continuity assumption on g can be dropped. We can then give an alternative and equivalent viewpoint using the orthogonal projections

P (M ) m := a∈{0,1} M 0 |a|=m M ℓ=1
(1a ℓ )p + a ℓ q = 1 {m} dΓ(q) on h ⊗M , with |a| = a 1 +• • •+a M , for M ∈ N and m ∈ Z. (Note that with this definition P (M ) m = 0 for m / ∈ {0, . . . , M}.) We can then write down the spectral decomposition of dΓ(q), dΓ(q) =

n∈Z n P (N ) n , (28)
i.e., P (N ) n

is the projection on the eigenspace of dΓ(q) associated with the eigenvalue n ∈ Z. It follows that

S g (ρ, p) = N n=0 g(n) Tr ρP (N ) n . (29)
In this form we see directly that for ρ = |Ψ Ψ| we have S g = α g , where α g is the functional used in [50, Definition 2.1] to control the closeness of a Hartree-Fock state to a Schrödinger state. However, note that there is a difference in the choice of normalization. The particular choices of functions, f in [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF] and g in our article, are related through g = N f , such that S g = Nα f . Note also that for g(x) = x we find

S Id R (ρ, p) = Tr[γ ρ (½ -p)].
Remark III.6. For g(x) = x, the functional S g has been used in [START_REF] Bach | Accuracy of Mean Field Approximations for Atoms and Molecules[END_REF][START_REF] Graf | A correlation estimate with applications to quantum systems with Coulomb interactions[END_REF][START_REF] Siedentop | The asymptotic behaviour of the ground state energy of the Müller functional for heavy atoms[END_REF] in the context of mean-field approximations for ground states. In [START_REF] Graf | A correlation estimate with applications to quantum systems with Coulomb interactions[END_REF], S g is called "degree of noncondensation" or "the relative number of particles outside the Fermi sea". For general g, a bosonic variant of S g was introduced for the derivation of mean-field dynamics in [START_REF] Pickl | A simple derivation of mean field limits for quantum systems[END_REF] and for the derivation of the NLS equation in [START_REF] Pickl | Derivation of the Time Dependent Gross-Pitaevskii Equation Without Positivity Condition on the Interaction[END_REF][START_REF] Pickl | Derivation of the Time Dependent Gross Pitaevskii Equation with External Fields[END_REF]. For the derivation of mean-field dynamics for fermions, S g was introduced in [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF]. Note that for g(x) = x and

Ψ 0 ∈ H (N )
f , 2S g (t) coincides with the quantity U N (t; 0)ξ, N U N (t; 0)ξ in [START_REF] Benedikter | Mean-field Evolution of Fermionic Systems[END_REF] in case ξ = R * ν N Ψ 0 . Let us collect some properties of S g (ρ, η) and show how it relates to the distance of γ ρ to η in trace norm. (Note that some of the statements were already proven in [START_REF] Benedikter | Mean-field Evolution of Fermionic Systems[END_REF] and [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF].) We denote the Hilbert-Schmidt norm by

• L 2 .
Proposition III.7. For η ∈ S N and ρ a density matrix with one particle density matrix γ, the degree S g (ρ, η) of evaporation has the properties

inf 0≤x≤N g(x) ≤ S g (ρ, η) ≤ sup 0≤x≤N g(x) , (30)
γ -η 2 L 2 ≤ 2S Id R (ρ, η) , (31) g 1 ≤ g 2 on [0, N] ⇒ S g 1 (ρ, η) ≤ S g 2 (ρ, η) , ( 32
) g → S g (ρ, η) is linear, (33)
for g, g 1 , g 2 functions from R to R. If furthermore p 2 = p is a rank-N orthogonal projection and g(0) = 0, g(x) ≥ x on [0, N], then

1 N γ -p L 1 ≤ 8 N S g (ρ, p) , (34)
S g (ρ, p) ≤ sup 0<x≤N g(x) x γ -p L 1 . ( 35
)
Proof. The spectrum of dΓ(q) (restricted to

H (N ) f ) is included in [0, N], thus inf 0≤x≤N g(x) ≤ ĝ ≤ sup 0≤x≤N g(x) ,
in the sense of quadratic forms. As ρ is a state, Eq. ( 30) follows. Equation ( 31) follows from

γ -η 2 L 2 = Tr (γ -η) 2 = Tr γ 2 + η 2 -2γη = 2S Id R (ρ, η) -Tr[γ -γ 2 ] -Tr[η -η 2 ] ≤ 2S Id R (ρ, η) .
Equations ( 32) and ( 33) follow from the properties of the functional calculus.

For the proof of (34), we first remark that γp has at most N negative eigenvalues (counting multiplicities). This is a well-known consequence of γp ≥ -p and the fact that p is a rank-N orthogonal projection (see, e.g., [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]), but we include its proof for the sake of completeness: Suppose that γp has at least N + 1 negative eigenvalues.

Then there is a subspace W of dimension N + 1 such that ϕ|(γp)ϕ < 0, for all ϕ ∈ W \ {0}. Since γ ≥ 0, this implies that ϕ|pϕ > 0, for all ϕ ∈ W \ {0}. On the other hand, the largest dimension of a subspace with this property is N, by the minmax principle and the fact that p has precisely N negative eigenvalues, which contradicts the existence of W .

Denoting the number of negative eigenvalues (counting multiplicities) of γp by M, we consequently have that M ≤ N. Let λ 1 , . . . , λ M be these M negative eigenvalues of γp, and λ M +1 , λ M +2 , . . . be the non-negative ones. Since Tr[γp] = 0, it follows that

-(λ 1 + • • • + λ M ) = ∞ m=M +1 λ m .
Using the Cauchy-Schwarz inequality and M ≤ N, we obtain

γ -p L 1 = ∞ m=M +1 λ m - M m=1 λ m = -2 M m=1 λ m ≤ 2 √ M M m=1 λ 2 m 1/2 ≤ 2 √ N ∞ m=1 λ 2 m 1/2 = 2 √ N γ -p L 2 ,
and Eq. ( 34) follows from Eq. ( 31) and S Id R (ρ, p) ≤ S g (ρ, p).

To prove Eq. ( 35), we observe that g(x) ≤ sup 0<x≤N g(x) x

x on [0, N] and thus, using positivity preservation and linearity, we have

S g (ρ, p) ≤ sup 0<x≤N g(x)
x x S Id R (ρ, p) .

We conclude with

S Id R (ρ, p) = Tr[γ(1 -p)] = Tr[p(p -γ)p] ≤ γ -p L 1 , using again p = p 2 and Tr[γ] = N = Tr[p].
Let us now state the main result of this section. Recall that we defined K := sup t≥0 Tr[-∆p t ].

Theorem III.8. Assume (15) holds, i.e., the kinetic energy is uniformly bounded, as in Theorem II.1. Then, writing

ρ t = |Ψ t Ψ t |, S g 1/3 (ρ t , p t ) ≤ S g 1/3 (ρ 0 , p 0 ) exp 30λ √ KN 1/6 t + N 2/3 exp 30λ √ KN 1/6 t -1 , (36)
where, for θ > 0, g θ is the function from R to R defined by

∀x ∈ R , g θ (x) := N 1-θ x 1 [0,N θ ] (x) + N 1 (N θ ,∞) (x) . ( 37
)
Note that the function g θ was also used to obtain the results in [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF]. Theorem III.8 will be proved in the following subsections. The strategy is to obtain a bound for dS t /dt in terms of S t and N δ , for some δ < 1, and then integrate it, in the spirit of the Grönwall lemma.

Before we turn to the proof of Theorem III.8, we show how Theorem III.8 and the properties of the degree of evaporation imply Theorem II.1, the main result of this article.

Proof of Theorem II.1. Since g 1/3 ≥ Id R on [0, N], we can apply Eq. ( 34) to Eq. ( 36) which gives

1 N γ t -p t L 1 ≤ 8 N S g 1/3 (ρ 0 , p 0 ) exp(30λ √ KN 1/6 t) + N 2/3 exp(30λ √ KN 1/6 t) -1 .
Equation ( 35) with g 1/3 yields S g 1/3 (ρ, p) ≤ N 2/3 γ-p L 1 which then gives Eq. ( 16).

The rest of this section is devoted to the proof of Theorem III.8.

III.2 Time-Derivative of the Degree of Evaporation

In this subsection we calculate the time derivative of the degree of evaporation S g (t) := S g (ρ t , p t ) and bring it into a form that can be conveniently estimated. Then most of the following subsections provide bounds on the different contributions to the time derivative. First, recall the Fefferman-de la Llave decomposition, for x = y ∈ R 3 ,

1 |x -y| = ˆR3 d 3 z ˆ∞ 0 dr π r 5 X r,z (x) X r,z (y) , (38)
of the Coulomb potential, where X r,z (x) := 1 |x-z|≤r is the characteristic function of the ball in R 3 of radius r > 0 centered at z ∈ R 3 . This formula can also be written as

v (2) = ˆdµ(ω) X ω ⊗ X ω , (39)
where ω = (r, z) ∈ R + × R 3 and ´dµ(ω) f (ω) := ´R3 d 3 z ´∞ 0 dr π r 5 f (r, z). The form (38) is convenient for the estimates derived below, but we note that it agrees with (18), of course. Note that the terms A t , B t , C t in the following proposition are the same as (I), (II), (III) in [50, Lemma 6.5]. However, an important difference lies in the presentation of the A t term using the decomposition [START_REF] Graf | A correlation estimate with applications to quantum systems with Coulomb interactions[END_REF], which enables us to handle the case of the Coulomb interaction. In the following, ℑ denotes the imaginary part.

Proposition III.9. For all monotonically increasing g : R → R, the time-derivative of S g (t) = S g (ρ t , p t ) (with the notation from Theorems II.1 and III.8) is

dS g (t) dt = λ A t + B t + C t , (40)
where

A t := ˆ2ℑTr dΓ(q t X ω p t ) dΓ(p t X ω p t ) -Tr[X ω p t ] ρ [-1,1] t dµ(ω) , (41)
B t := ℑTr dΓ (2) (q t ⊗ q t )v (2) (p t ⊗ p t ) ρ [-2,2] t , (42)
C t := 2ℑTr dΓ (2)

(q t ⊗ q t )v (2) (p t ⊗ q t ) ρ [-1,1] t , (43)
with q t := ½p t , and

ρ [-j,j] := (τ -j g -g) 1/2∧ ρ(g -τ j g) 1/2∧ . (44)
Before we turn to the proof we note that

q t p t = p t q t = 0 , (45)
since p t is a projection. We further note that, for A, B linear and bounded operators on h, we have that

dΓ(A) dΓ(B) = dΓ (2) (A ⊗ B) + dΓ(AB) . (46)
To prove Proposition III.9 we need several lemmas. We begin with an evolution equation for ĝ.

Lemma III.10. For any function g : R → R, with ĝ = g dΓ(q t ) and h

HF defined in Eq. [START_REF] Bardos | Mean field dynamics of fermions and the time-dependent Hartree-Fock equation[END_REF],

i∂ t ĝ = [dΓ h (1)
HF (p t ) , ĝ] . (47)
Proof. First observe that only the values of g on the spectrum of dΓ(q t ) are used to define ĝ. Hence we could as well consider a polynomial which coincides with g on the spectrum of dΓ(q t ). It is then enough to prove that Eq. ( 47) holds for any monomial dΓ(q t ) n . It indeed holds for n = 1:

i∂ t dΓ(q t ) = dΓ(i∂ t q t ) = dΓ [h (1)
HF (p t ), q t ] = dΓ h

HF (p t ) , dΓ(q t ) .

Then, for any n ∈ N,

i∂ t dΓ(q t ) n = n j=1 dΓ(q t ) j-1 dΓ h (1)
HF (p t ) , dΓ(q t ) dΓ(q t ) n-j

= dΓ h (1)
HF (p t ) , dΓ(q t ) n , as all the terms but two simplify in the sum.

Next, we need a commutation relation (analogous to [50, Lemma 6.4]) involving ĝ that enables us to write the time derivative of S g (t) in terms of a discrete derivative of g. Recall that τ j-k g(x) = g(xj + k).

Lemma III.11. For integers 0 ≤ j, k ≤ M ≤ N, any function g : R → R, with the notations of Definition III.3, and h (M ) ∈ L(h ⊗M ),

dΓ (M ) P (M ) j h (M ) P (M ) k ĝ = τ j-k g dΓ (M ) P (M ) j h (M ) P (M ) k . ( 48
)
Proof. First, note that if

dΓ (M ) P (M ) j h (M ) P (M ) k dΓ(q) = τ j-k Id dΓ (M ) P (M ) j h (M ) P (M ) k
holds, then Eq. ( 48) follows by the same argument as in the proof of Lemma III.10.

Using P (M )

m 1 P (M ) m 2 = δ m 1 m 2 P (M ) m 1 and P (N ) m = d∈Z P (M ) d ⊗ P (N -M ) m-d (recall that P (M ) d
= 0 for d / ∈ {0, . . . , M}), and without loss of generality singling out the first M variables,

P (M ) j h (M ) P (M ) k ⊗ ½ ⊗N -M P (N ) n = P (M ) j h (M ) P (M ) k ⊗ ½ ⊗N -M d∈Z P (M ) d ⊗ P (N -M ) n-d = P (M ) j h (M ) P (M ) k ⊗ P (N -M ) n-k = d∈Z P (M ) d ⊗ P (N -M ) n-k+j-d P (M ) j h (M ) P (M ) k ⊗ ½ ⊗N -M = P (N ) n-k+j P (M ) j h (M ) P (M ) k ⊗ ½ ⊗N -M .
It follows from the spectral decomposition (28) of dΓ(q) that

dΓ (M ) P (M ) j h (M ) P (M ) k dΓ(q) = dΓ (M ) P (M ) j h (M ) P (M ) k n∈Z n P (N ) n = n∈Z n P (N ) n-k+j dΓ (M ) P (M ) j h (M ) P (M ) k = n∈Z (n + k -j) P (N ) n dΓ (M ) P (M ) j h (M ) P (M ) k = τ j-k Id dΓ (M ) P (M ) j h (M ) P (M ) k
, which, as discussed above, implies the result.

Proof of Proposition III.9. Without loss of generality we assume that

Ψ 0 ∈ H (N ) f ∩ H 1 (R 3 )
⊗N (see Remark II.7). Using the evolution equation for ĝ from Lemma III.10, we find

dS g dt (ρ t , p t ) = Tr -i[H, ρ t ]ĝ + ρ t (-i[dΓ(h (1)
HF (p t )), ĝ]) = Tr i[H -dΓ(h (1)
HF (p t )), ĝ]ρ t = λTr i[ 1 2 dΓ (2) (v (2) ) -dΓ(v (1)
HF (p t )), ĝ]ρ t with v (1)
HF (p t ) := Tr 2 [v (2) (1-X)(½⊗p t )]. Now, recall that M m=0 P (M ) m = ½ h ⊗M .
Inserting this identity for M = 1 and M = 2 and using Lemma III.11 gives

dS g dt (ρ t , p t ) = λ 2 Tr i[dΓ (2) ( P (2)
0 + P (2) 1 + P (2) 2 v (2) P (2) 0 + P (2) 1 + P (2) 2 )
-2 dΓ( P

0 + P

HF (p t ) P

(1)

0 + P (1) 1 ), ĝ]ρ t = λ 2
Tr i[dΓ (2) (P

(2) 0 v (2) P (2) 2 + P (2) 2 v (2) P (2) 0 + P (2) 0 v (2) P (2) 1 + P (2) 1 v (2) P (2) 0 + P (2) 1 v (2) P (2) 2 + P (2) 2 v (2) P (2) 1 ) -2 dΓ P (1) 0 v (1)

HF (p t )P

(1)

1 + P (1) 1 v (1) HF (p t )P (1) 0 , ĝ]ρ t
We recall the definition of the direct integral of a family (B(x)) x∈R 3 of operators on h:

ˆ⊕ R 3 B(x ′ 1 )dx ′ 1 ψ (x 1 , x 2 ) := ½ ⊗ B(x 1 ) ψ (x 1 , x 2 ) , (49)
for any ψ ∈ H

f . Lemma III.13. Let A be a bounded non-negative operator on h and (B(x)) x∈R 3 be a family of non-negative trace class operators on h. Then dΓ (2)

( √ A ⊗ ½) ˆ⊕ R 3 B(x 1 )dx 1 ( √ A ⊗ ½) ≤ dΓ √ A Tr[B(x)] √ A . ( 50
)
If A is also trace -class and such that Tr

√ ATr[B(x)] √ A < ∞ then dΓ (2) ( √ A ⊗ ½) ˆ⊕ R 3 B(x 1 )dx 1 ( √ A ⊗ ½) ≤ ˆA(x; x) Tr[B(x)]dx . ( 51
)
where A(x, y) = ∞ i=1 λ i ϕ i (x)ϕ i (y) denotes the integral kernel of A defined in terms of the spectral decomposition of A.

In particular: If B(x) = B does not depend on x,

dΓ (2) (A ⊗ B) ≤ Tr[B] dΓ(A) . (52)
With w : R 3 → R + , w (2) = w(x 1x 2 ) and p = N i=1 |ϕ i ϕ i |, ϕ i |ϕ j = δ ij a rank-N projector on h, we have that dΓ (2) (q ⊗ p)w (2)

(q ⊗ p) ≤ dΓ q(w * f )q ≤ w * f ∞ dΓ(q) , (53)
dΓ (2) (p ⊗ p)w (2)

(p ⊗ p) ≤ dΓ p(w * f )p ≤ (w * f )f 1 , (54)
where f (x) := p(x; x) := N i=1 |ϕ i (x)| 2 are the diagonal values of the integral kernel of p.

Proof. Let Ψ (N ) ∈ H (N ) f . With Ψ(N-1) A,x (x 1 , . . . , x N -1 ) := ½ ⊗N -1 ⊗ √ A Ψ (N ) (x 1 , . . . , x N -1 , x)
and the direct integral representation we get, using Lemma III.12,

Ψ (N ) , dΓ (2) ( √ A ⊗ ½) ˆ⊕ R 3 B(x 1 )dx 1 ( √ A ⊗ ½) Ψ (N ) = N(N -1) ˆ Ψ(N-1) A,x , ½ ⊗N -2 ⊗ B(x) Ψ(N-1) A,x dx = N ˆ Ψ(N-1) A,x , dΓ B(x) Ψ(N-1) A,x dx ≤ N ˆ Ψ(N-1) A,x
, Tr[B(x)] Ψ(N-1)

A,x dx = Ψ (N ) , dΓ √ ATr[B(x)] √ A Ψ (N ) ,
and Eq. ( 50) follows. If Tr[ √ ATr[B(x)] √ A] < ∞, then Eq. ( 51) follows from Lemma III.12.

The case with B(x) independent of x is clear. For the second particular case, observe that the operator (½ ⊗ p)w (2) (½ ⊗ p) can be written as the direct integral

(½ ⊗ p)w (2) (½ ⊗ p) = ˆ⊕ R 3 p(τ x 1 w)p dx 1 with τ x 1 w(x 2 ) = w(x 2 -x 1
) a translation of w. Then with B(x) = p(τ x w)p and A = q we get Eq. ( 53), and with A = p we get Eq. ( 54).

For ρ ∈ L 1 (H

(N )
f ), let us introduce the shorthand notation

ρ [j] := (g -τ j g) 1/2 ∧ ρ (g -τ j g) 1/2 ∧ , (55)
ρ [-j] := (τ -j g -g) 1/2 ∧ ρ (τ -j g -g) 1/2 ∧ , (56)
with j = 1, 2, and let γ [j] and γ [-j] be the corresponding one-particle and γ [j](k) and γ [-j](k) the corresponding k-particle density matrices (see also Definition III.2 extended to non-negative and trace class operators whose trace is not necessarily one). Note that ρ [j] and ρ [-j] are not states because their trace is not one, and thus γ [j] and γ [-j] do not necessarily satisfy Eq. ( 26).

The next lemma shows the advantage we gain from using the function

g θ (x) := N 1-θ x 1 [0,N θ ] (x) + N 1 (N θ ,∞) (x) (57)
in the definition of the degree of evaporation. This lemma is analogous to [50, Lemma 7.1], but note that the use of the functional calculus clarifies the fact that one ultimately uses only inequalities on functions from R to R.

Lemma III.14. For j ∈ {-2, -1, 1, 2}, any (normalized

) state ρ ∈ L 1 H (N ) f
, and the function g θ from (57) (with the notation from (55) and (56)),

Tr ρ [j] ≤ |j| N 1-θ , Tr dΓ(q) ρ [j] ≤ |j| (|j| + 1) S g θ ,
Tr dΓ (2)

(q ⊗ q) ρ [j] ≤ |j| (|j| + 1) 2 N θ S g θ .
Proof of Lemma III.14. The inequalities are a direct consequence of the functional calculus, once we observe that dΓ(q) = Id R , dΓ (2)

(q ⊗ q) = dΓ(q) 2 -dΓ(q) = (Id R • (Id R - 1)) ∧ and τ j g θ -τ k g θ ≤ (k -j) N 1-θ , Id R • (τ j g θ -τ k g θ ) ≤ (k -j)(k -j + 1) g θ , Id R • (Id R -1) • (τ j g θ -τ k g θ ) ≤ (k -j) (k -j + 1) 2 N θ g θ ,
as inequalities of functions from R to R, for -2 ≤ j < k ≤ 2.

III.4 Bound for A t

Let us first estimate the integrand A t (X ω ) of (41), i.e., A t (X) := 2ℑTr dΓ(q t Xp t ) dΓ(p t Xp t ) -

Tr[Xp t ] ρ [-1,1] t ,
where X is an operator on h such that 0 ≤ X ≤ ½.

Proposition III.15. Let X be an operator on h such that 0 ≤ X ≤ ½ and set γ

[-1] t ⊥ := Tr[ρ [-1] t ] -γ [-1] t . Then A t (X) ≤ Tr[p t X] Tr[X(2q t γ [1] t q t + p t γ [-1] t ⊥ p t )] . (58)
Proof. Using the Cauchy-Schwarz inequality and 2ab ≤ a 2 + b 2 , and then Eq. ( 46) and Lemma III.12, we get

A t (X) = 2ℑTr dΓ(q t Xp t ) dΓ(p t Xp t ) -Tr[Xp t ] ρ [-1,1] t ≤ Tr dΓ(q t Xp t ) dΓ(p t Xq t ) ρ [1] t + Tr Tr[p t X] -dΓ(p t Xp t ) 2 ρ [-1] t ≤ Tr dΓ (2) (q t Xp t ⊗ p t Xq t ) ρ [1] t + Tr dΓ(q t Xp 2 t Xq t ) ρ [1] t + Tr[p t X] Tr Tr[p t X] -dΓ(p t Xp t ) ρ [-1] t . (59)
For the first term on the right-hand side of (59), we apply the Cauchy-Schwarz inequality again and obtain

Tr dΓ (2) (q t Xp t ⊗ p t Xq t ) ρ

[1] t = Tr q t √ X ⊗ p t √ X √ Xp t ⊗ √ Xq t γ [1](2) t ≤ Tr (q t Xq t ⊗ p t Xp t ) γ [1](2) t
Tr (p t Xp t ⊗ q t Xq t ) γ [START_REF] Adami | Rigorous derivation of the cubic NLS in dimension one[END_REF](2) t = Tr (q t Xq t ⊗ p t Xp t ) γ [START_REF] Adami | Rigorous derivation of the cubic NLS in dimension one[END_REF](2) t = Tr dΓ (2) (q t Xq t ⊗ p t Xp t ) ρ

[1] t ] . (60)
Using Lemma III.13 yields in turn Tr dΓ (2) (q t Xq t ⊗ p t Xp t ) ρ [START_REF] Adami | Rigorous derivation of the cubic NLS in dimension one[END_REF] t ] ≤ Tr p 2 t X] Tr dΓ(q t Xq t ) ρ

[1] t = Tr[p t X] Tr Xq t γ [1] t q t . (61)
For the second term on the right-hand side of (59), we observe that

Tr[dΓ(q t Xp 2 t Xq t ) ρ [1] t ] = Tr[Xp 2 t Xq t γ [1] t q t ] ≤ Tr[p t X] Tr[Xq t γ [1] t q t ] ,
and for the third term on the right-hand side of (59),

Tr[p t X] Tr Tr[p t X] -dΓ(p t Xp t ) ρ [-1] t = Tr[p t X] Tr[(Tr[ρ [-1] t ]p t -p t γ [-1] t p t ) X] ,
which yields [START_REF] Segal | Non-linear semi-groups[END_REF].

We now give a bound on the integral ´At (X ω )dµ(ω) using the estimate from Proposition III.15 on A t (X). To get good estimates we take g to be g θ as in Eq. [START_REF] Gneiting | Radial positive definite functions generated by Euclid's hat[END_REF]. We use the notation

f HF (x) := p t (x; x) ≥ 0 , (62)
where p t = N i=1 |ϕ i,t ϕ i,t |, with ϕ i,t |ϕ j,t = δ ij and p t (x; y) := N i=1 ϕ i,t (x)ϕ i,t (y), which allows us to rewrite the traces as integrals. For example,

Tr[p t X r,z ] = ˆ|x-z|≤r f HF (x) d 3 x .
Observe that ´fHF = N and that the quantity ´f 5/3 HF appearing in Proposition III.17 is controlled by the Lieb-Thirring inequality, as is discussed in Sect. III.7.

Before we give the bound for A t , let us prove an auxiliary lemma. Let A c denote the complement of a set A and recall that B(0, R) denotes the ball of radius R centered at 0 in R 3 .

Lemma III.16.

For 1 p 1 + 1 p 2 + 1 s = 2, with 1 ≤ p j , s ≤ ∞, measurable functions χ, f 1 , f 2 : R 3 → R and v(x) = |x| -1 , ˆ(χ v)(x -y) f 1 (x) f 2 (y) d 3 x d 3 y ≤ f 1 p 1 f 2 p 2 χ v s . Additionally, for s < 3, 1 B(0,R) v s = 4π 3 -s 1/s R 3/s-1 ,
and, for s > 3,

1 B(0,R) c v s = 4π s -3 1/s R 3/s-1 , with the convention that 4π ∞-3 1/∞ := 1.
Proof. The first relation follows directly from applying Hölder's and Young's inequalities. The second and third relations follow directly from integration. With the ingredients above we can give a bound on A t .

Proposition III.17. The estimate

A t ≤ 5 -5/6 72 π 1/3 N 1/6 f HF 5/6 5/3 S g θ (63) holds.
Proof. By Proposition III.15,

A t ≤ 2 ˆTr[p t X ω ] Tr[q t γ [1] t q t X ω ] dµ(ω) + ˆTr[p t X ω ] Tr[p t γ [-1] t ⊥ p t X ω ] dµ(ω) . (64)
We now explicitly use the Fefferman-de la Llave decomposition of the Coulomb potential. Then, we find that for any non-negative trace-class operator h, using ´h(y; y)dy = Tr[h], Hölder's inequality and Lemma III.16 in the end (where we distinguish between the short-range and the long-range part of the potential) gives us that

ˆTr[p t X ω ] Tr[hX ω ] dµ(ω) = 1 π ˆTr[p t X r,z ] Tr[hX r,z ] dr r 5 d 3 z = 1 π ˆ ˆ|x-z|≤r f HF (x) d 3 x ˆ|y-z|≤r h(y; y) d 3 y dr r 5 d 3 z = ˆ1 |x -y| f HF (x) h(y; y) d 3 x d 3 y ≤ 1 B(0,R) v 5/2 f HF 5/3 + 1 B(0,R) c v ∞ f HF 1 Tr[h] ≤ (8π) 2/5 R 1/5 f HF 5/3 + R -1 f HF 1 Tr[h] .
Optimizing with respect to R > 0 yields R = (8π) -1/3 5 5/6 f HF We now apply this inequality to (64), i.e., with h = q t γ [START_REF] Adami | Rigorous derivation of the cubic NLS in dimension one[END_REF] t q t and with h = p t γ

t q t ] = Tr[q t γ [1] t ] = Tr[dΓ(q t )ρ [1] t ] ≤ 2S g θ , (65)
Tr[p t γ [-1] t ⊥ p t ] = Tr[p t ]Tr[ρ [-1] t ] -Tr dΓ(p t )ρ [-1] t = Tr dΓ(q t )ρ [-1] t ≤ 2S g θ . (66)
This proves [START_REF] Zagatti | The Cauchy problem for Hartree-Fock time-dependent equations[END_REF].

III.5 Bound for B t

We estimate B t in the same fashion as in [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF]Lemma 7.3]. Note, that for this term and the C t term it is not necessary to use the Fefferman-de la Llave decomposition.

Proposition III.18. The estimate

B t ≤ 2 1/3 π 2/3 f HF 5/6 5/3 N 1/6 6S g θ + N 1-θ (67) holds.
Proof. We estimate the B t term by using the Cauchy-Schwarz inequality to arrive at a three-particle term.

Recall that ρ [-2,2] t is a rank one operator, i.e., ρ

[-2,2] t = Ψ [-2]
Ψ [2] . For a linear operator A on h, j ∈ {-2, 2} and almost every x ∈ R 3 , we define the vectors Ψ

[j] A,x ∈ H (N -1) f by Ψ[j] A,x (x 1 , . . . , x N -1 ) := (½ ⊗N -1 ⊗ A)Ψ [j] (x 1 , . . . , x N -1 , x) .
Inserting the form of v (2) as a direct integral into the expression of B t in Eq. ( 42) yields

B t = ℑ Ψ [-2] , dΓ (2) (q t ⊗ ½) ˆ⊕ R 3 q t (τ x 1 v)p t dx 1 (p t ⊗ ½) Ψ [2] = Nℑ ˆR3 Ψ[-2] qt,x , dΓ q t (τ x v)p t Ψ[2] pt,x dx .
Taking the modulus of both sides and using the Cauchy-Schwarz inequality we obtain

B t ≤ N ˆR3 Ψ[-2] qt,x 2 dx 1/2 N ˆR3 dΓ(q t (τ x v)p t ) Ψ[2] pt,x 2 dx 1/2 = Ψ [-2] , dΓ(q t )Ψ [-2] 1/2 N ˆR3 Ψ[2] pt,x , dΓ(p t (τ x v)q t ) dΓ(q t (τ x v)p t ) Ψ[2] pt,x dx 1/2 .
From Lemma III.14 we deduce that Ψ [-2] , dΓ(q t ) Ψ [-2] ≤ 6S g θ . We estimate the remaining integral using Eq. ( 46), the Cauchy-Schwarz inequality, Lemma III.13 and Lemma III.14:

N ˆR3 Ψ[2] pt,x , dΓ (2) (p t (τ x v)q t ) ⊗ (q t (τ x v)p t ) + dΓ p t (τ x v)q t (τ x v)p t Ψ[2] pt,x dx ≤ N ˆR3 Ψ[2] pt,x , dΓ (2) (p t (τ x v) 2 p t ) ⊗ q t + dΓ p t (τ x v) 2 p t Ψ[2]
pt,x dx = Ψ [2] , dΓ (2)

p ⊗2 t (v 2 ) (2) p ⊗2 t dΓ(q t ) + dΓ (2) p ⊗2 t (v 2 ) (2) p ⊗2 t Ψ [2]
≤ Ψ [2] , dΓ(q t )Ψ [2] + Ψ [2] , Ψ [2] ˆR3

(f HF * v 2 )f HF ≤ (6S g θ + 2N 1-θ ) ˆR3 (f HF * v 2 )f HF .
By the Hardy-Littlewood-Sobolev inequality (see, e.g., [START_REF] Lieb | Analysis[END_REF]Theorem 4.3]) we find

(v 2 * f )f 1 ≤ 4 1/3 π 4/3 f 2 3/2 . (68)
We then apply Hölder's inequality with

1 = 3 4 + 1 4 to obtain f 2 3/2 = f 5/4 f 1/4 4/3 1 ≤ f 5/4 4/3 4/3 f 1/4 4/3 4 = f 5/3 5/3 f 1/3 1 . (69)
Applying this to f HF and ab ≤ (a 2 + b 2 )/2 to the bound we obtained on B t yields the result.

III.6 Bound for C t

Our estimate for C t is analogous to [START_REF] Petrat | A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics[END_REF]Lemma 7.3]. Note that for this estimate our choice of the function g θ is crucial, while in the bounds for A t and B t we could have used the identity function to obtain the desired estimate. By using g θ with appropriate θ < 1 we obtain the desired N-dependence in the estimate for C t .

• If, moreover, T 0 ∈ H A 2,1 (E) then T ∈ C 1 ([0, t 0 ); H A 1 (E)) and i dT dt (t) = A, T (t) + W(T (t)) , T (t) T (0) = T 0 .

Such a function T is called a classical solution of the TDHF equation.

• Any mild solution to the TDHF equation satisfies Tr T 0 W(T 0 ) .

• If ∃k 1 ∈ R such that * T ∈ H A 1 (E) , 0 ≤ T ≤ ½ ⇒ W(T ) ≥ k 1 ,
and T 0 ∈ H A 1 (E), 0 ≤ T 0 ≤ ½, then T can be extended to the entire positive real axis. Moreover if T 0 ∈ H 2,1

A (E), then T is the unique global classical solution.

Remark A.2. In [START_REF] Bove | On the Hartree-Fock time-dependent problem[END_REF] the space H A 2,1 (E) is not used. They use a space larger than H A 2,1 (E) which is more natural, but less explicit. As it is enough for us to use classical solutions for initial data in H A 2,1 (E) and then use a density result, we restrict ourselves to this framework. We now quote a result which, although not explicitly stated in [START_REF] Bove | On the Hartree-Fock time-dependent problem[END_REF], is a direct consequence of [START_REF] Bove | On the Hartree-Fock time-dependent problem[END_REF] along with [START_REF] Segal | Non-linear semi-groups[END_REF].

Proposition A.3. The map

H A 1 (E) × [0, ∞) → H A 1 (E) (T 0 , t) → T (t) ,
where T (t) is the (mild) solution to the TDHF equation with initial data T 0 , is jointly continuous in T 0 and t.

Indeed the proof of existence and uniqueness in [START_REF] Bove | On the Hartree-Fock time-dependent problem[END_REF] is based on the results in [START_REF] Segal | Non-linear semi-groups[END_REF] which also ensure the continuity with respect to the initial data (see [START_REF] Segal | Non-linear semi-groups[END_REF]Corollary 1.5,p. 350]).

It was shown in [START_REF] Bove | On the Hartree-Fock time-dependent problem[END_REF] that those results apply to the case

E = h = L 2 (R 3 ), A = -∆, W(γ) = Tr 2 v (2) (½ -X)(½ ⊗ γ) ,
and v (2) = |x -y| -1 . The proof then extends to the case A = h (1) with h (1) = -C∆ + w(x), where the external potential w is an infinitesimal perturbation of the Laplacian.

B Some Estimates of the Direct Term and the Kinetic Energy

The dynamics is the free dynamics to leading order in the λK 1/2 N 1/6 t ∼ 1 regime. We now substantiate by heuristic argument that, in the particular case of the Coulomb interaction potential, if λK 1/2 N 1/6 t is assumed to be of order one, which is the regime where our estimates are relevant, then the evolution is the free evolution to leading order. Note that the exchange term is expected to be subleading with respect to the direct term; we thus neglect the exchange term in the following computation.

We now estimate the effect of the direct term on the time derivative of the average momentum per particle. We denote the Hartree-Fock density at time t by f HF,t = N j=1 |ϕ t,j | 2 ; thus the direct term is the convolution λv * f HF,t . For the absolute value of the time derivative of the expectation value of the momentum per particle we find, using |∇v| = 3v 2 and (68) with (69),

N -1 ∂ t Tr p t (-i∇) = N -1 Tr p t [h HF , ∇] ≤ λN -1 Tr p t (∇v * f HF,t ) ≤ 3λN -1 (v 2 * f HF,t )f HF,t 1 ≤ CλN -2/3 K. ( 74
)
After a time t the effect of the direct term on the momentum is thus expected to be of order λN -2/3 Kt. Since λK 1/2 N 1/6 t is assumed to be of order one, the average change in momentum is of order K 1/2 N -5/6 . Since this is much smaller than the average momentum of a particle (K/N) 1/2 , we conclude that the dynamics is, to leading order, free.

The estimate [START_REF] Bardos | Weak coupling limit of the N-particle Schrödinger equation[END_REF] allows to distinguish the free dynamics and the Hartree-Fock dynamics to the next order. Again using heuristic arguments we substantiate that, for large enough kinetic energy K ≫ N 4/3 , estimate [START_REF] Bardos | Weak coupling limit of the N-particle Schrödinger equation[END_REF] allows to distinguish the effect of the direct term on the evolution, i.e., our main result shows that the Hartree-Fock equation gives a better approximation to the Schrödinger equation than the free equation. This is because our convergence rate is N -1/6 (let us assume γ 0 = p 0 ), i.e., for λK 1/2 N 1/6 t of order 1, the error between Schrödinger and Hartree-Fock dynamics is for any bounded observable of order N -1/6 . For K ≫ N 4/3 this rate is much smaller than the average change in momentum estimated above, i.e., N -1/6 ≪ K 1/2 N -5/6 .

Estimate on the Kinetic Energy. We prove the estimate [START_REF] Chadam | Global existence of solutions to the Cauchy problem for time-dependent Hartree equations[END_REF]. Recall that f HF,t = N j=1 |ϕ t,j | 2 and that the direct term is the convolution λv * f HF,t . First, we use the conservation of the total Hartree-Fock energy and the fact that the exchange energy is bounded by the direct energy (which follows directly from applying the Cauchy-Schwarz inequality). Then, by Hölder's inequality, we find Tr[-∆p t ] ≤ E(Φ HF,0 ) + λ 2 ˆ(v * f HF,t )(x)f HF,t (x)d 3 x ≤ E(Φ HF,0 ) + λ 2 v * f HF,t ∞ N .

By splitting the area of integration in space in the convolution product v * f HF,t into a short-range and a long-range part, using Hölder's inequality, Lemma III. [START_REF] Benedikter | Mean-field Evolution of Fermionic Systems[END_REF]

dr 2

 2 (r) . Note that g |•| -1 (r) = 16 π r -5 in case of the Coulomb potential which is prototypical for the following further assumption. Assumption II.4. (With the same notation as in Assumption II.3.) The function v : R 3 → R satisfies Assumption II.3, |g v (r)| ≤ 16 π r -5 and, for some µ ∈ R, v(x) ≥ µ for all x.

Proposition II. 10 (1/3 2 .

 102 Ground state energy of a molecule). For a molecule with nuclei of charges Z 1 , . . . , Z M > 0 at pairwise distinct positions R 1 , . . . , R M ∈ R 3 , with λ = α, ν = m<l αZ m Z l /|R m -R l | as in (4), and Z = max{Z 1 , . . . , Z M }, the ground state energy satisfies the bound 0 < -E gs ≤ (0.231)α 2 N 1 + 2.16 Z M N Proposition II.11 (Neutral atom). In case of an atom with N = Z the ground state energy satisfies 0 < -E gs ≤ (2.31)α 2 N 7/3 .

6 5/ 3 , 3 HF 1 3 N 1 / 6

 6331316 so that (recall f HF 1 = N) ˆTr[p t X ω ] Tr[hX ω ] dµ(ω) ≤ 5 -5/6 6(8π) 1/3 ˆf 5/Tr[h] .

  ∀t ∈ [0, t 0 ) , Tr MT (t)M + 1 2 Tr T (t) W(T (t)) = Tr MT 0 M + 1 2

  and the Lieb-Thirring inequality, we findv * f HF,t ∞ ≤ f HF,t 5/3 1 B(0,R) v 5/2 + f HF,t 1 1 B(0,R) c v ∞ ≤ CK 3/5 R 1/5 + NR -1 . (75)Optimizing with respect to R gives the boundv * f HF,t ∞ ≤ CK 1/2 N 1/6 .(76)Then, by Eq. (76) and ab ≤ a 2 2 + b 2 2 , we findTr[-∆p t ] ≤ E(Φ HF,0 ) + Cλ (Tr[-∆p t ]) 1/2 N 1/6 N

	≤ E(Φ HF,0 ) +	1 2	Tr[-∆p

t ] + Cλ 2 N 7/3 , which proves

[START_REF] Chadam | Global existence of solutions to the Cauchy problem for time-dependent Hartree equations[END_REF]

.

* There was a typographical error in Assumption iv) in[START_REF] Bove | On the Hartree-Fock time-dependent problem[END_REF], namely, W(T )T ≥ k 1 shall be read W(T ) ≥ k 1 .

Acknowledgments

The authors are indebted to N. Benedikter, M. Porta and B. Schlein for helpful discussions and sharing their results prior to publication. S. B.'s research is supported by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323. S. P.'s research has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n°291734. T. T. is supported by the DFG Graduiertenkolleg 1838, and for part of this work was supported by DFG Grant No. Ba-1477/5-1 and also by the European Research Council under the European Community's Seventh Framework Program (FP7/2007-2013)/ERC grant agreement 20285.

= λℑTr dΓ (2) (P

HF (p t )P

(1) 0 ρ [-1,1] t + λℑTr dΓ (2) (P

+ λℑTr dΓ (2) (P

We then insert the Fefferman-de la Llave decomposition and Eq. [START_REF] Lieb | Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter[END_REF] in the first term to get ℑTr[(dΓ (2) (P (1)

where we used P (1) 0

= q t ⊗ p t + p t ⊗ q t .

III.3 Auxiliary Lemmas

We prove here three lemmas that we frequently need for estimating the terms A t , B t and C t from Proposition III.9.

For fermionic systems the following bound on dΓ(A) is well-known. Note that this is the only point at which the Fermi statistics enter our paper.

Lemma III.12. Let A be a trace-class and self-adjoint operator on a separable Hilbert space h. Then, as quadratic forms on H

Proof. We use the spectral decomposition A = j λ j |ϕ j ϕ j | with λ j ∈ R, j |λ j | < ∞, for some orthonormal basis (ϕ j ) ∞ j=1 , and we write any vector

where

which yields the result.

Proposition III.19. The estimate

holds.

Proof. Using the Cauchy-Schwarz inequality, Lemma III.13 and Lemma III.14 we find

Tr dΓ (2)

which is the result.

III.7 Kinetic Energy Estimates and Proof of Theorem III.8

In order to estimate f HF Proposition III.20 (Lieb-Thirring Inequality, see [START_REF] Lieb | Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter[END_REF] or [45, p.73] ). Let γ ∈ L 1 (h) be a one-particle density matrix of finite kinetic energy, i.e., 0 ≤ γ ≤ ½ and Tr[-∆γ] < ∞.

with C LT = 9 5 (2π) 2/3 and where f (x) := γ(x; x) is the corresponding one-particle density.

Proposition III.21 (Hardy's Inequality, see [START_REF] Hardy | Note on a Theorem of Hilbert[END_REF] or, e.g., [START_REF] Yafaev | Sharp constants in the Hardy-Rellich inequalities[END_REF]). Let γ ∈ L 1 (h) be a one-particle density matrix of finite kinetic energy, i.e., 0 ≤ γ ≤ ½ and Tr[-∆γ] < ∞. Then ˆf (x)

where f (x) := γ(x; x) is the corresponding one-particle density.

We now combine the results of Sects. III.2 to III.6 to prove Theorem III.8.

Proof of Theorem III.8. We choose θ = 1 3 so that our bound for C t is good enough. Collecting the estimates for the A t , B t and C t terms from Propositions III.17, III.18 and III.19 and using the kinetic energy inequalities from Propositions III.20 and III.21, we can continue the estimate for the time derivative of S g 1/3 ,t from Proposition III.9 and find (recall that K := sup t≥0 Tr[-∆p t ])

Integrating this inequality (Grönwall lemma) yields Theorem III.8.

A Some Results about the Theory of the Time-Dependent Hartree-Fock Equation

In this appendix we recall some known facts about the theory of the TDHF equation. We begin by stating a theorem summarizing those results proved in [START_REF] Bove | On the Hartree-Fock time-dependent problem[END_REF] which we use.

equipped with the norm T k,p,A = M k T M k p where X p = Tr[|X| p ] 1/p for 1 ≤ p < ∞ or X B(E) for p = ∞ (we write L ∞ (E) for B(E)). We adopt the special notations H(E) := H A 0,∞ (E) for the space of bounded self-adjoint operators on E and

Then

• For any T 0 ∈ H A 1 (E) there exists t 0 > 0 and T ∈ C([0, t 0 ); H A 1 (E)) such that, ∀t ∈ [0, t 0 ), T (t) = e -itA T 0 e itAi ˆt 0 e -i(t-s)A W(T (s)), T (s) e i(t-s)A ds .

Such a function T is called a local mild solution of the TDHF equation and, provided its interval of definition is maximal, it is unique.


Konstantin Mishchik

Yannick Petit
email: yannick.petit@u-bordeaux.fr

Etienne Brasselet

Arnaud Royon

Thierry Cardinal

Lionel Canioni

Patterning linear and nonlinear optical properties of photosensitive glasses by femtosecond structured light

Keywords:

Patterning linear and nonlinear optical properties of photosensitive glasses by femtosecond structured light

During the last decade, the versatile and cost-limited technique of 3D femtosecond (fs) direct laser writing (DLW) has led to huge progress [1]. Indeed, new laserinduced structures and functionalities have been obtained, especially because of two complementary fields of research, namely the engineering of complex light [2] and the development of tailored materials [3].

DLW with tightly focused Gaussian TEM 00 beams can result in locally isotropic material modifications, but also in highly anisotropic structures such as nanogratings [4]. However, the bell-shaped nature of the Gaussian beam limits the range of accessible topologies. Such limitation was first partially overcome by considering structured light, referring to electromagnetic fields endowed with phase or polarization singularities (also called vortices). This appeared to be a powerful tool to carry out original vortex-induced processing of optical materials, by enabling surface ablations with donut-shaped topologies [5,6] or the retrieval of the polarization distributions of complex light [7]. Moreover, vortex-assisted DLW, which derives from the stimulated emission depletion (STED) approach [8], has also led to super-resolution DLW [9,10]. Since structured light can be engineered on demand with spatial light modulators, with setups based on anisotropic crystals, or with specially designed plates engineered by DLW [11,12], it opens a large range of accessible shapes. However, vortex-inducted DLW has only led to the pattering of linear optical properties.

Concerning developments of optical materials, the photosensitizing with silver ions appears promising [3]. Indeed, the silver-doping of gallo-phosphate glasses highly improves the manufacturing quality of fs laserinduced nanogratings [13]. With noncommercial silverdoped phosphate glasses, it has been shown that the interaction of a high repetition rate fs pulse train gives rise to an original spatial distribution of photo-induced silver clusters below the diffraction limit [14], leading to fluorescent patterns [15], as well as second- [16][17][18] and third-order [19] nonlinear optical properties. Thus, such developments in silver-doped glasses appear relevant for new patterning with structured light, of both linear and nonlinear optical properties, the latter remaining an open challenge.

In this Letter, we report on structured light-induced fs DLW of silver-containing phosphate glasses. Complex fluorescent patterns that cannot be obtained using Gaussian beams are created. In particular, we demonstrate the creation of (i) nested fluorescent rings using Laguerre-Gauss-like beams, and (ii) multi-ring fluorescent structures using Hermite-Gauss-like beams. Noteworthy, the latter case gives access to inner spatial structuring with estimated dimensions down to 210 30 nm. Moreover, effective second harmonic generation (SHG) is reported, highlighting the relationship between the laser-induced linear and nonlinear optical properties. Finally, we discuss our results according to mechanisms at play during DLW in these tailored glasses [3,17].

Our tailored glass is a silver-containing zinc phosphate glass with a 5% Ag 2 O concentration (mol. %), prepared by a standard glass melting method, cut to 1 mm thick sample and polished to optical quality [20]. DLW was performed with a Ti:Sa regenerative laser amplifier (RegA 9000 from Coherent, up to 1 W, 250 kHz, 60 fs at 800 nm). The irradiation duration and transmitted irradiance were controlled by an acousto-optic modulator, enabling the accumulation of N 10 5 -10 6 pulses with energies from 50 to 150 nJ. The positioning and displacement of the sample were performed with a high-precision 3D translation stage (XMS-50 stages, Micro-Contrôle). Irradiations were carried out by focusing laser pulses with a microscope objective (Mitutoyo, APO PLAN VIS, 50 × NA 0.55). Pulse duration (FHWM) at the focus in the sample was 145 fs (Gaussian beams) and 200 fs (structured beams).

To perform structured light-induced DLW, the spatial engineering of the writing light field was set both in phase and amplitude [2]. We used a c-cut 8 mm-thick uniaxial calcite crystal slab (optical axis oriented along the propagation direction, n o 1.649, n e 1.482 at 800 nm), two quarter-wave plates, and a polarization beam splitter (Fig. 1). In a first configuration, the incident linearly polarized Gaussian field E 0 [Fig. 2(a)] was transformed into a circularly polarized light field by the use of a λ∕4 plate, then focused with the lens L1 (f 50 mm) through the calcite crystal, and re-collimated with the second lens L2 (f 50 mm). After the second λ∕4 plate, the inhomogeneously polarized light field was separated into two orthogonal linearly polarized components, thanks to a polarizing beam splitter (PBS). With our settings (Fig. 1), the component E ⊥ led to a charge-2 optical vortex beam [21], whose intensity distribution is shown in Fig. 2(b). In a second configuration, both λ∕4 plates were removed. The incident Gaussian beam led to a four-lobe intensity pattern for the component E ⊥ [22], as shown in Fig. 2(c).

The resulting linear and nonlinear optical responses were observed with fluorescence and SHG microscopy. Fluorescence imaging of the photo-modified areas was performed with a confocal microscope with an excitation wavelength at 405 nm, for an emission in the 430-480 nm range (Leica, SP8, objective 63×, NA 1.3, and lateral resolution of 250-290 nm). Nonlinear SHG imaging was performed with a homemade microscope setup (typical lateral resolution of 300-350 nm), as detailed elsewhere [18].

DLW has been carried out with Gaussian, charge-2 optical vortex and four-lobe laser beams, whose intensity profiles before the focusing objective and in the laserwriting focal plane are shown in Figs. 2(a -First, DLW activates four-photon ionization [14], generating mobile photoelectrons, mostly at the center of the focused spot where the intensity is higher. This is followed by the photoelectron diffusion toward the periphery of the intensity profile of the laser beam. Such multi-pulse DLW may also promote band gap excitation of photoelectrons from laser-induced species [23].

-Second, such diffusion of charged species followed by their trapping at the periphery of the focused laser beam leads to a space charge separation and, consequently, to the creation of an intense buried electric field (estimated to up to 8 × 10 8 V • m -1 [18]). Such an electric field is remarkably stable under thermal constraint, as long as the temperature remains below the glass transition temperature (T g ∼ 385°C), while it completely vanishes above T g [18]. Moreover, it couples with the third-order nonlinear response of the glass, giving rise to an electric field induced second-harmonic generation (EFISHG) with χ 2 eff ∼ 0.6 pm • V -1 [18], corresponding to efficient direct laser poling. From fluorescence and SHG correlative microscopy, both the silver cluster stabilization and the creation of a space charge separation had been monitored. In particular, the EFISHG response was proved to appear prior to the fluorescence emission, meaning that the creation and stabilization of the silver cluster is a consequence of the creation of the static electric field [17].

-Third, fluorescence and SHG correlative microscopy studies had evidenced that the silver clusters get stabilized where the electric potential modifications (resulting from the frozen space charge separation) locally lead to more favorable reduction-oxidation conditions.

For Gaussian beams [Fig. 2(d)], the single-wall hollow pipe shape is independent of the polarization of the incident laser, suggesting that such shape is driven only by the laser intensity profile. In contrast, DLW using structured light creates original fluorescent patterns. Indeed, a charge-2 vortex beam [Fig. 2

(e)] induces a fluorescent pattern that corresponds to a double-wall hollow pipe [Figs. 2(h) and 2(k)], whereas a four-lobe beam [Fig. 2(f)] creates four single-wall hollow pipes [Figs. 2(i) and 2(l)

]. These observations corroborate an intensitydriven mechanism, where phase plays no significant role in the multiphoton ionization step that releases the initial free electrons. Moreover, DLW is independent of the sign of the topological charge of the charge-2 vortex. Regarding the four-lobe beam, the dark cross is associated to π-phase steps, though the four bright spots produced identical fluorescent hollow pipes, showing again the independence of DLW with respect to phase. Still, the spatial distribution of the phase of the writing beam plays a crucial role, as it allows controlling the intensity profile of the incident light field, thereby the lightinduced patterns.

The dependence of structured light-induced patterns on the irradiation parameters has also been studied. For 10 5 pulses, DLW thresholds of fluorescence are well below the typical self-focusing power P cr ∼ 6 MW, and the related energies are estimated to be E p 23, 100, and 67 nJ for Gaussian, charge-2 vortex, and four-lobe beams, respectively, locally corresponding to 3.8, 4.3, and 3.7 TW∕cm 2 . The increase of the number of pulses or of the incident irradiance mainly leads to patterns with larger dimensions and brighter fluorescence emission [Fig. 3(a)]. However, it also reveals that significant spatial distortions may occur. The four-ring pattern obtained with the four-lobe beam at low intensity turns into a four-leaf clover pattern as the irradiance is increased [Fig. 3(a), E p 98 nJ, 10 6 pulses]. In such a distorted case, the four single-wall hollow pipes cannot be considered as independent anymore because their diameters become large enough to repel each other. The photoelectrons' diffusion of each single-wall hollow pipe is balanced by the Coulomb repulsion from the diffusing photoelectrons of the neighboring pipes, resulting in such a distorted pattern. This shows that the involved diffusion processes mostly correspond to free charged particles, namely the induced photoelectrons. This frustrated diffusion is also depicted in Fig. 3(b) by plotting the cross sections of the fluorescent patterns of Fig. 3(a) along the dashed line across the diameters of two neighboring pipes, for different pulse energies. This evidences the decrease of the inter distance Δl between two close fluorescent walls from two neighboring pipes, depending on the incident laser irradiance. The 3D distribution of the pattern obtained with E p 52 nJ has been deconvolved (AutoQuant algorithm), permitting us to estimate the thickness of the fluorescent walls to about 210 30 nm, which appears relevant for photonic applications. The evolution of the inter distance Δl versus pulse energy is plotted [Fig. 3(d)]. We observe that the minimal interdistance between two fluorescent walls saturates to a minimal value, corresponding to a plateau behavior around 500 nm [Fig. 3(d)]. A quantitative assessment of this behavior is beyond the scope of this work, and remains an open issue.

Finally, fluorescence and SHG correlative microscopy has been performed on a charge-2 vortex-induced pattern. Fluorescence shows a nested double-ring pattern [Fig. 4(a), also depicted in Fig. 2 the inner ring being more intense than the outer ring. These two rings are located rather close to the edges of the inner fluorescent ring [Fig. 4(c)]. However, the modeling of the relative fluorescence and SHG distributions could not be provided here with vortex-induced DLW, unlike in previous works with Gaussian beams [17]. Still, the incident linearly polarized probe beam (vertical polarization in Fig. 4(b), related to the angle θ 0°) leads to a radial intensity distribution of the frequencydoubled beam [Fig. 4(d)] with the typical 9∶1 ratio between the directions θ 0°and 90°, respectively. This is in excellent agreement with the theoretical model 9 cos 2 θ sin 2 θ, proving the radial distribution of the buried electric field and the EFISHG nature of the frequency-doubling nonlinear process [18].

DLW with structured light opens new patterning abilities. First, we notice that the double-wall and four singlewall hollow pipes most probably cannot be obtained by means of successive DLW with a Gaussian beam. Actually, overwriting effects would lead to the erasure by the center of intense spots [14]. Indeed, the edges of the intense spots write fluorescent clusters while their more intense central parts erase them. Second, beyond the complex pattern issue, structured light-induced DLW offers the ability for parallelized optical processing [24], as with the four-lobe beam, by simultaneously writing multiple identical structures. Third, 3D patterns with sub-diffraction features are obtained, since we estimated the thickness of some fluorescence walls down to 210 30 nm [Fig. 3(c)], which is well below the diffraction limit [14].

More generally, our observations suggest that any structured light, such as Laguerre-Gauss LG pl and Hermite-Gauss HG nm light modes, or laser interference patterns, could generate complex photo-induced linear and nonlinear patterns, with periodic, quasi-periodic, or aperiodic features. As an example, LG pl beams (corresponding to p 1 concentric annular intensity distributions) are expected to produce 2p 1 embedded nested fluorescent rings, while HG mn should produce n 1 × m 1 rings. The resulting shaping of the nonlinear optical properties would be worth exploring. Additionally, these nontrivial patterns obtained with structured light could also apply for original spatial modifications of various properties for laser-induced composite materials, such as the local precipitation of metallic silver nanoparticles in these phosphate glasses [25], potentially leading to bulk optical devices with engineered dispersion.

To conclude, in this Letter, we performed original DLW using structured light, which has led to nontrivial fluorescence patterns that cannot be obtained with successive steps of DLW with a Gaussian beam. Moreover, we have discussed the principle of single-beam parallelized DLW by using appropriately shaped light fields. We have also observed sub-diffraction features. Finally, correlative microscopy has confirmed both fluorescent and nonlinear SHG responses, with correlated but distinct spatial distributions of the photo-induced linear and nonlinear optical properties. Such bi-functionality could be exploited for innovative three-dimensional photonic architectures.

  )-2(c) and in Figs.2(d)-2(f), respectively. Gaussian-beam based DLW leads to the creation of a fluorescent hollow pipe [Figs. 2(g) and 2(j)], composed of fluorescent silver clusters, as reported elsewhere[14]. Previous works had proposed the following interpretation of the mechanisms at play:

Fig. 1 .

 1 Fig. 1. Structured light generator from an incident linearly polarized Gaussian beam. Two settings, with or without the two λ∕4 plates, are used. Structured light-induced DLW is performed with the E ⊥ polarized component.

Fig. 2 .

 2 Fig. 2. Intensity profiles of the incident and focused beams, respectively: (a), (d) Gaussian; (b), (e) charge-2 vortex; and (c), (f) four-lobe. Related fluorescence patterns under top view and 3D view, respectively: (g), (j) single-wall hollow pipe; (h), (k) double-wall hollow pipe; and (i), (l) four single-wall hollow pipes.

  (h)] with radial symmetry. The SHG pattern, created by direct laser poling, also shows ring patterns, but with distinct diameters and angular dependence [Fig. 4(b)]. The cross sections along the vertical dashed lines give a better insight into the fluorescence and SHG profiles. The latter profile mostly shows two main concentric rings [Fig. 4(c)],

Fig. 3 .

 3 Fig. 3. Four single-ring fluorescent pattern versus incident irradiance. Nonmerging individual structures at high irradiances, underlying the electrically charged nature of the diffusive species, and highlighting their repulsion behavior: (a), (b) 2D top views and related horizontal profiles versus irradiance; (c) profile along the white dashed line for E p 52 nJ pulse energy and N 10 6 pulses, with four fluorescent rings with an estimated thickness down to 210 nm; and (d) inter-distance Δl between two fluorescent rings, showing a plateau around 500 nm, which we attribute to the balance between diffusion and repulsion forces.

Fig. 4 .

 4 Fig. 4. Fluorescence and SHG correlative microscopy: 2D top views from (a) confocal fluorescence imaging and (b) SHG imaging. (c) Fluorescence and SHG radial profiles along SHG probe beam polarization (dashed line at θ 0°). (d) Polarization dependence of the EFISHG signal with the typical 9:1 ratio in agreement with the expected angular dependence 9 cos 2 θ sin 2 θ.

This work was funded by French Agency of National Research [FELINS 2010 ANR BLAN 946 03], French Aquitaine Region [FELINS REGION], group of scientific interest AMA LasINOF, Cluster of Excellence LAPHIA, and Photonic Imaging platform of Bordeaux Imaging Center (UMS 420 CNRS).