Mean field approximation of many-body quantum dynamics for Bosons in a discrete numerical model

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Mean field approximation of many-body quantum dynamics for Bosons in a discrete numerical model

Boris Pawilowski

To cite this version:

Boris Pawilowski. Mean field approximation of many-body quantum dynamics for Bosons in a discrete

numerical model. 2015. �hal-01183707�

Mean field approximation of many-body quantum dynamics for Bosons in a discrete numerical model

B. Pawilowski

August 3, 2015

Abstract

The mean field approximation is numerically validated in the bosonic case by considering the time evolution of quantum states and their associated reduced density matrices by many-body Schr¨odinger dynamics. The model phase-space is finite-dimensional. The results are illustrated with numerical simulations of the evolution of quantum states according to the time, the number of the particles, and the dimension of the phase-space.

Mathematics subject classification: 81S30, 81S05, 81T10, 35Q55, 81P40

Keywords: mean field limit, reduced density matrices, Wigner measures, bosons Fock space, second quantization.

1 Introduction

The mean field approximation is known to be a good way to approximate the many-body Schr¨odinger dynamics when the number of particles is large enough (see [2,6,9,10,15,19,20,21,22,24,25,26,32,37, 43, 46,29,30,33,50]).

It consists in looking for the solutions to the non-linear Schr¨odinger equation for one particle called the Hartree equation. We are interested in the density matrix associated with the wave function, this matrix satisfies the quantum Liouville equation dual to the Von Neumann equation.The partial trace operators of this matrix, called the reduced density matrices, satisfy a hierarchy of equations. For instance, by considering the case where the initial state for the Schr¨odinger equation is a Hartree ansatz(a product state) which is suitable for a bosons condensate, the limit, when the number of particles N goes to the infinity of these matrices converge in trace norm to the product of the density matrix associated with the solution to the Hartree equation. And this asymptotic density matrix satisfies the time dependent Hartree equation [8].

When the particles are bosons, the suitable space for the bosons is the symmetric Fock space on the phase- space. Moreover for the sake of numerical computations, a finite-dimensional phase-space will be used instead of an usual phase-space of typeL²(R^d). So here the phase-space will beZ=`²({0,· · ·, K})'C^K whereK is a given integer representing the number of sites. Each particle can live in one of theK sites.

For the numerical implementation, an explicit basis of the N-fold sector of the Fock bosonic spaces is specified. This basis allows the numerical computation of the full N-body quantum problem for N large enough to validate various mean field regimes, in spite of a rapidly increasing complexity.

The resolution of theN-particles Schr¨odinger equation will rely on a splitting method, one part for the free Hamiltonian and the other one for the two particles interaction term.

For the simulations, the considered real bounded potential associated with the interaction term will beV defined onZ/KZbyV(i) =_|i|¹ ifi6= 0 and V(0) = 0.

According to previous results related to the propagation of the Wigner measures [3, 4, 5, 6] knowing the Wigner measure at timet= 0 determines the Wigner measure at timetand all asymptotic reduced density matrices. For many examples, like Hermite states, twin Fock states or states studied in quantum information theory (see [1]) their Wigner measure as well as the order of convergence of reduced density matrices is known explicitly. The evolution of the Wigner measure, and consequently of the asymptotic density matrices, is evaluated after integrating numerically the mean field non linear Hartree time-dependent equation. In order

∗IRMAR, Universit´e de Rennes I, UMR-CNRS 6625, Campus de Beaulieu, 35042 Rennes Cedex, France

†Fak. Mathematik, Univ. Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

to preserve numerically quadratic quantities like the symplectic form, the latter is solved with a symplectic 4^thorder Runge-Kutta method ([35]).

To estimate numerically the error of convergence of the reduced density matrices in the mean field limit, a discretization of a time interval [0, tmax] is considered, in the examples tmax = 1 is chosen, then the quantity max_t∈[0,tmax]

γ^(p)_N (t)−γ∞^(p)(t)

₁ is observed. Here γ_N^(p)(t) denotes the time-evolved p particles reduced density matrix forN bosons, while γ∞^(p)(t) is its theoretical limit whenN goes to the infinity, and k.k1denoting the trace norm.

For the evaluation of the order of convergence, the logarithm of the previous error estimate is drawn according tolog(N). This gives a straight line whose the slope is the order of convergence in 1/N.

These numerical results agree very well and illustrate the theoretical analysis carried out in [1].

By increasingK, we wish to approach a continuous model. The complexity of the computations increase in the same time thatK andN increase because of the dimension of theN-particles bosons Fock space onC^K which is a binomial coefficient.

2 Framework

The bosons Fock space on an Hilbert spaceZis defined as Γ_s(Z) =L

n≥0

Wn

ZwhereWn

Zis the symmetric n-fold Hilbertian tensor product of Z which is the range of the projection defined on the Hilbert tensor productZ^⊗n, by:

S_n(ξ₁⊗ξ₂⊗...⊗ξ_n) = 1 n!

X

σ∈Σn

ξ_σ(1)⊗ξ_σ(2)⊗...⊗ξ_σ(n), whereξi is in Z for eachiin [1, n] and Σn is the set of the permutations ofnelements.

Forz inZandεpositive, theε-scaled annihilation and creation operators are defined for all Φ inZand nin Nby:

a(z)Φ^⊗n =√

εnhz|ΦiΦ^⊗n−1, a^∗(z)Φ^⊗n =p

ε(n+ 1)Sn+1(|zi ⊗Φ^⊗n).

These operators are then extended by linearity and density toWn

Z.

These operators satisfy the canonical commutation relations (CCR):

[a(z₁), a^∗(z₂)] =εhz₁, z₂iId , (1)

[a(z1), a(z2)] = 0, (2)

[a^∗(z1), a^∗(z2)] = 0. (3)

The second quantization of an operator A∈ L(Z) or a self-adjoint operator (A, D(A)) inZ is defined by:

dΓ(A)_|∨n,algD(A)=ε

n

X

i=1

Id^⊗i−1⊗A⊗Id^⊗n−i. The second quantization ofId_Z is the number operator:

N_|∨nZ=εnId_∨ⁿ_Z.

2.1 Orthogonal basis of the N -fold sector

Use the following notations: ZK=Z/KZ.

For α = (α1,· · ·, αK) in N^K, the length of α is written |α| = α1 +· · ·+αK and the factorial of α α! =α1!· · ·αK!.

Let (e1,· · · , eK) be an orthonormal basis ofC^K. Set Z = C^K. Then an orthonormal basis of WN

Z can be built from this basis which is labelled by the

multi-indicesαinN^K such that|α|=N.

With the creation operators, an orthonormal basis can be written as:

eα:= a^∗(e)^α

√

ε^|α|α!|Ωi:= 1

√

ε^|α|α!a^∗(e1)^α1· · ·a^∗(eK)^αK|Ωi, where|Ωi= (1,0,0,0, . . .) is the vacuum of the Fock space.

Then the dimension ofWN

Z iscard({α∈N^K/|α|=N}) =

N+K−1 K−1

. Andcard({α∈N^K/|α| ≤N}) = ^N_K^+K

is the dimension of LN n=0

Wn

Z.

2.2 Hamiltonian

As a binomial number, the dimension of theN particles bosonic sector increases rapidly but not too much, asN increases (see Table6.4 for numerical values).The complexity has to be handled carefully if we want to approach the mean field limit by takingN large or the continuous model by takingK large.

Define ∆K the discrete Laplacian operator onC^K by:

∀z∈C^K , ∀i∈Z/KZ, (∆Kz)i=zi+1+z_i−1. And letH0=dΓ(−∆K) be the free Hamiltonian.

The interaction term denoted byV equals:

V= 1 2

X

(i,j)∈Z²_K

Vija^∗(ei)a^∗(ej)a(ei)a(ej),

whereVij =Vji=V(i−j).

In this framework changing the value ofV(0) add an irrelevant phase factor in the time evolved wave function.

In the sequelV(0) = 0 is assumed.

Or as a Wick quantized operator (14):

V= z^⊗2, 1

2 X

(i,j)∈Z²_K

Vij|ei⊗ejihei⊗ej|

z^{⊗2W ick} .

The considered linear Schr¨odinger equation is:

iε∂tΨ =HεΨ, (4)

where the complete Hamiltonian is defined on the bosonic Fock space by : H_ε=dΓ(−∆_K) +V.

3 Finite dimensional mean field equation

3.1 Energy of the Hamiltonian

The energy of the Hamiltonian corresponds to the symbol of the complete Hamiltonian:

H(z,z) =¯ hz,−∆Kzi+1 2

X

i6=j

Vij|zi|²|zj|², while recalling our convention V_ii =V(0) = 0 for alliinZK.

3.2 Hartree equation

The mean field equation inZ is written as:

i∂_tz=∂_¯zH(z,z)¯ . For each component inC^K we obtain:

i∂_tz_i=∂_z¯iH=∂_z¯i(X

i⁰

|z_i⁰−z_i⁰₋₁|²−2|z_i|²+1 2

X

i⁰6=j

V_i⁰_j|z_i⁰|²|z_j|²)

=∂z¯i(X

i⁰

( ¯zi⁰−zi⁰¯−1)(zi⁰−zi⁰−1)−2 ¯zizi+1 2

X

i⁰6=j

Vi⁰jz¯i⁰zi⁰z¯jzj)

=zi−zi−1−(zi+1−zi)−2zi+X

j6=i

Vijzi|zj|²

=−(zi+1+z_i−1) + (X

j6=i

Vij|zj|²)zi

˙

zi=−i[−(∆Kz)i+ (X

j6=i

Vij|zj|²)zi] By writingz=q+ ipwhereqandpbelong to R^K, it becomes:

˙

q_i = −(p_i+1+p_i−1) +X

j6=i

V_ij(q_j²+p²_j)p_i

˙

pi = −

−(qi+1+q_i−1) +X

j6=i

Vij(q²_j+p²_j)qi

A 4^th or 6^th order Gauss RK method is used by using the coefficients given in [35] to solve the Hartree equation.

A symplectic method is used to preserve the quadratic part of the energy, the symplectic form and the phase space volume.

3.3 Wigner measures

For f ∈ Z the field operator is defined by Φ(f) = ^√1

2(a^∗(f) +a(f)) which is essentially self-adjoint on Γ_{f in}(Z) =Lalg

n∈N

Wn

Z

The Weyl operator is defined by W(f) =e^iΦ(f).

Let (%_ε)_ε∈E be a family of normal states on Γ_z(Z) withE ⊂(0,+∞), 0∈ E.

A measureµis a Wigner measure for this family,µ∈ M(%_ε, ε∈ E), if there existsE⁰⊂ E, 0∈ E⁰ such that

∀f ∈ Z, lim

ε∈E⁰,ε→0Trh

%εW(√ 2πf)i

= Z

Z

e^2iπRehf,zi dµ(z),

see [42].

The following result valid for separable Hilbert spacesZ, apply to our finite dimensionalZ 'C^K.

Theorem 3.1 [3]. If (%ε)_ε∈E satisfies the uniform estimate Tr[%εN^δ] ≤Cδ < +∞ for someδ > 0 fixed, M(%ε, ε∈ E)is not empty and made of Borel probability measures (Zseparable) such thatR

Z|z|^2δdµ(z)≤Cδ. For eachpinN, the reduced density matrix associated with a state%εis a trace class operator inL¹(Wp

Z) defined by the duality relation:

Trh γ_ε^p˜bi

= Tr

%εb^{W ick}

Tr [%ε(|z|^2p)^{W ick}] (5)

where ˜b∈ L(Wp

Z).

The asymptotic reduced density matrix associated with the Wigner measure µequals:

γ₀^p= R

Z|z^⊗pihz^⊗p|dµ(z) R

Z|z|^2pdµ(z) . (6)

In finite dimension, if the family (%ε)ε∈E satisfiesM(%ε, ε∈ E) ={µ} then the (P I)-condition (see [3]):

∀p∈N, lim

ε∈E,ε→0Tr [%_εN^p] = Z

Z

|z|^2p dµ(z) is always satisfied.

3.4 Reduced density matrices

Theorem 3.2 [5]. If the family(%ε)_ε∈E satisfiesM(%ε, ε∈ E) ={µ} with the(P I)-condition:

∀p∈N, lim

ε∈E,ε→0Tr [%εN^p] = Z

Z

|z|^2p dµ(z), thenTr

%_εb^{W ick}

converges toR

Zb(z)dµ(z)for all polynomial b(z)and

ε∈Elim,ε→0kγ_ε^p−γ^p₀k_L1 = 0 for allp∈N.

Theorem 3.3 [5,6,42]. AssumeM(%ε, ε∈(0,ε) =¯ {µ0}and the condition(P I). ThenM(e^−iεtHε%εe^iεtHε, ε∈ (0,ε)) =¯ {µt}. The measure µt = Φ(t,0)_∗µ0 is the push-forward measure of the initial measure µ0 where Φ(t,0) is the hamiltonian flow associated with the equation

i∂tzk(t) =−∆Kzk(t) +X

j

Vkj|zj|²zk. (7)

After propagation of the Wigner measures, for anyp∈N, the convergence of the reduced densiy matrices is obtained at any timet:

γ_ε^p(t)−

R

Z|z^⊗pihz^⊗p|dµt(z) R

Z|z|^2pdµ0(z)

_L1 −→0.

Theorem 3.4 [1]. Let (α(n))n∈N^∗ be a sequence of positive numbers with limα(n) = ∞ and such that (^α(n)_n )_n∈N^∗is bounded. Let(%_n)_n∈N^∗and(γ^(p)_∞)_p∈N^∗be two sequences of density matrices with%_n∈L¹(Wn

Z) andγ∞^(p)∈L¹(Wp

Z)for eachn, p∈N^∗. Assume that there existC0>0,C >2 andγ≥1 such that for all n, p∈N^∗ withn≥γp:

γ_n^(p)−γ_∞^(p)

₁≤C₀ C^p

α(n). (8)

Then for anyT >0 there existsCT >0 such that for allt∈[−T, T] and alln, p∈N^∗ with n≥γp,

γ_n^(p)(t)−γ_∞^(p)(t)

₁≤C_T C^p

α(n), (9)

where

γ_∞^(p)(t) = Z

Z

|z^⊗pihz^⊗p|dµt(z),

withµ_t= (Φ_t)_]µ₀ is the push-forward of the initial measure µ₀ by the well defined and continuous Hartree flowΦ_t onZ.

4 Numerical methods

4.1 Method to solve the Hartree equation

To solve the mean field equation (7), a Runge-Kutta method is used.

Letbi,aij (i, j= 1, . . . , s) be real numbers andci=Ps j=1aij.

An s-stage Runge-Kutta method with a time step hto solve a first-order ordinary equation y⁰ =f(t, y) , y(t0) =y0 is given by:

ki = f(t0+cih, y0+h

s

X

j=1

aijkj), i= 1, . . . , s

y1 = y0+h

s

X

i=1

biki

represented as:

c1 a11 . . . a1s

... ... ... cs as1 . . . ass

b1 . . . bs

.

Here the system is autonomous, and according to [35] the coefficients used for the Gauss RK method are:

0 1/2 1/2

1/2 0 1/2

1 0 0 1

1/6 2/6 2/6 1/6 ,

0

1/3 1/3 2/3 -1/3 1

1 1 -1 1

1/8 3/8 3/8 1/8 or

1/2−√

3/6 1/4 1/4−√

3/6 1/2 +√

3/6 1/4 +√

3/6 1/4

1/2 1/2

.

In our case, the functionf corresponds tof(z) =−i

−∆Kz_k+P

jV_kj|zj|²z_k . As a function inR^2K by replacingz byq+ ip,

f(q, p) =f_q(q, p) + if_p(q, p) fq_i(q, p) =−(pi+1+pi−1) +X

j6=i

Vij(q²_j+p²_j)pi

f_pi(q, p) =q_i+1+q_i−1−X

j6=i

V_ij(q²_j+p²_j)q_i

For an implicit Runge-Kutta method, a Newton method is applied to find the coefficientsk_ifor each step of the RK method to the function gy₀ : (ki)_i=1,···s7→

ki−f(y0+hPs

j=1aijkj)

i=1,···s.

Given the time step h small enough, the starting point of the Newton method is chosen by setting ki=f(y0) for alli.

To apply the Newton’s method the differential off is computed by using the following partial derivatives of f:

∂f_qi

∂qk

= (1−δi,k)2Vikqkpi

∂fp_i

∂q_k =δ_i+1,k+δ_i−1,k+ (δ_i,k−1)2V_ikq_kq_i−δ_i,kX

j6=i

V_ij(q²_j+p²_j)

∂fq_i

∂p_k =−(δ_i+1,k+δ_i−1,k) + (1−δ_i,k)2V_ikp_kp_i+δ_i,kX

j6=i

V_ij(q²_j+p²_j)

∂fp_i

∂p_k = (1−δ_i,k)2V_ikp_kq_i

Then the differential ofgy₀ is:

Dgy₀((ki)i=1,···,s) =



Id2K−hailDf(y0+h

s

X

j=1

aijkj)





i,l=1,···,s

whereg_y0 is considered as a function fromR^2Ks.

4.2 Resolution of the Schr¨ odinger equation in W

Z

4.2.1 Composition method For a given Ψ inWN

Z, the full N-body evolved statee^−itεHεΨ is computed in the basis (eα)_|α|=N. After writing Ψ =P

|α|=NΨαeαa modified splitting method for which the numerical error is carefully controlled (see5), involves only multiplications by the diagonal matrixe^−iεptV and the sparse matrix dΓ(−∆K).

In order to handle the high complexity of the problem (see table 6.4) no matrix, but only vectors or the sparse matricesdΓ(−∆K) and the matrix (V_ij)_i,j are stored.

The complete evolutione^−iεtHε is computed by a composition method based on the Strang splitting method:

e^−itεHε = lim

p→∞ e^{−i2εpt V}e^−iεptH0e^{−i2εpt Vp} .

The 4^thorder composition method is given by:

e^−itεHε = lim

p→∞ e^−ia2εp3tVe^−iaεp3tH0e^−ia2εp3tVe^−ia2εp2tVe^−iaεp2tH0e^−ia2εp2tVe^−ia2εp1tVe^−iaεp1tH0e^{−ia2εp1tVp} ,

where the coefficients of the method are satisfying the two equations (see [35]):

a1+a2+a3= 1 (10)

a³₁+a³₂+a³₃= 0 (11)

and are given by:

a₁=a₃= 1

2−2^1/3 , a₂=− 2^1/3

2−2^1/3 . (12)

4.2.2 Computation of the free evolution e^{−itεdΓ(−∆K)}

The numerical computation of e^{−iεtdΓ(−∆K)}= Γ(e^it∆K), relies on the following two remarks:

• the dimension of theN-fold sectors ^N_K−1^+K−1

prevents the storage of any square matrix.

• the matrix of Γ(e^it∆K) is actually non trivial sparse matrix in the basis (eα).

The matrix of ∆_K is given by:

∆K =

























0 1 0 · · · 0 1 1 . .. . .. . .. 0 0 . .. . .. . .. . .. ... ... . .. . .. . .. . .. 0 0 . .. . .. . .. 1 1 0 · · · 0 1 0























 ,

We are interested in the matrix of the second quantization of the discrete Laplacian on the basis of the bosons space to implement it numerically as a sparse matrix containing only 2K ^N_K−2^+K−2

elements whereas a full matrix contains ^N_K−1^+K−12

.

Then e^{−i∆tεdΓ(−∆K)} will be computed at each time step by a 4^th order Taylor expansion.

This expansion is then replaced in the composition method.

For an operatorA:C^K −→C^K, A= (Ai,j)i,j, dΓ(A)_|^Wn_CK=

K

X

i,j=1

A_i,ja^∗(e_i)a(e_j). This yields:

dΓ(−∆K) = −

K

X

j=1

a^∗(ej+1)a(ej) +a^∗(ej)a(ej+1). (13)

Lemma 4.1 For all multi-indicesγ andαthe following equality holds:

a(e)^γa^∗(e)^α|Ωi=δγ≤αε^|γ| α!

(α−γ)!a^∗(e)^α−γ|Ωi. Proof.

a(e)^γa^∗(e)^α = a(e1)^γ1. . . a(eK)^γKa^∗(e1)^α1. . . a^∗(eK)^αK

=

K

Y

i=1

a(ei)^γia^∗(ei)^αi which is a commutative product because of CCR (1)

=

K

O

i=1

a(e_i)^γia^∗(e_i)^αi

by using the following separation of the variables: Γ(Z) = Γ(Ce1)⊗. . .⊗Γ(CeK). In this space let|Ωibe|Ω1i ⊗. . .⊗ |ΩKi.

Let us considerγi≥1 andαi≥1,

a(e_i)^γia^∗(e_i)^αi|Ωii=a(e_i)^γi−1a(e_i)a^∗(e_i)^αi|Ωii

=a(ei)^γi−1a^∗(ei)^αia(ei)|Ωii+a(ei)^γi−1[a(ei), a^∗(ei)^αi]|Ωii

=εαia(ei)^γi−1a^∗(ei)^αi−1|Ωii.

By induction, we obtain

a(e_i)^γia^∗(e_i)^αi|Ω_ii=α_iε×(α_i−1)ε×. . .×2ε×εa(e_i)^γi−αi|Ω_ii= 0,whenα_i< γ_i

and

a(e_i)^γia^∗(e_i)^αi|Ωii=α_iε×(α_i−1)ε×. . .×(α_i−(γ_i−1))εa^∗(e_i)^αi−γi|Ωii

=ε^γi α_i!

(αi−γi)!a^∗(e_i)^αi−γi|Ω_ii,whenγ_i≤α_i.

The above separation of variables leads under the conditionγ≤αto:

a(e)^γa^∗(e)^α|Ωi=⊗^K_i=1a(ei)^γia^∗(ei)^αi|Ωii

=Y^K

i=1

ε^γiα_i! (αi−γi)!

⊗^K_i=1a^∗(ei)^αi−γi|Ωii

=ε^|γ| α!

(α−γ)!

Y^K

i=1

a^∗(ei)^αi−γi

(|Ω1i ⊗. . .⊗ |ΩKi)

=ε^|γ| α!

(α−γ)!a^∗(e)^α−γ|Ωi.

Proposition 4.2 For all multi-indicesαandβ, the matrix elements of dΓ(−∆K)are given by:

dΓ(−∆K)α,β =−εX

i

(δ_β−e⁺

i,α−ei+1

pβi(βi+1+ 1) +δ_β−e⁺

i+1,α−ei

pβi+1(βi+ 1)), whereδ⁺_α,β=δα,β1_NK(α)forαandβ multi-indices inZ^K.

Proof.

According to (13) and to Lemma4.1, we obtain:

a^∗(ei+1)a(ei)a^∗(e)^α|Ωi=δe_i≤αε α!

(α−ei)!a^∗(e)^ei+1+α−ei|Ωi

=δ_1≤αiεα_ia^∗(e)^ei+1+α−ei|Ωi,

and

a^∗(e_i)a(e_i+1)a^∗(e)^α|Ωi=δ_ei+1≤αε α!

(α−ei+1)!a^∗(e)^ei+α−ei+1|Ωi

=δ_1≤αi+1εαi+1a^∗(e)^ei+α−ei+1|Ωi.

Then

heα, dΓ(−∆K)e_βi=−

* e_α,

K

X

i=1

a^∗(e_i+1)a(e_i) +a^∗(e_i)a(e_i+1)

! e_β

+

= −ε

pα!β!ε^2N X

i

a^∗(e)^αΩ| δ_1≤βiβ_ia^∗(e)^β+ei+1−ei+δ_1≤βi+1β_i+1a^∗(e)^β+ei−ei+1 Ω

= −ε

ε^N√ α!β!

X

i

(δ_β−e⁺

i,α−e_i+1β_iε^Np

α!(β−e_i+e_i+1)!

+δ_β−e⁺

i+1,α−eiβ_i+1ε^Np

α!(β−e_i+1+e_i)!)

= −ε

√β!

X

i

(δ_β−e⁺

i,α−ei+1β_ip

(β−e_i+e_i+1)! +δ⁺_β−e

i+1,α−eiβ_i+1p

(β−e_i+1+e_i)!)

=−εX

i

(δ_β−e⁺

i,α−e_i+1β_i s

β_i+1+ 1 βi

+δ_β−e⁺

i+1,α−e_iβ_i+1

sβ_i+ 1 βi+1

) dΓ(−∆_K)_α,β =−εX

i

(δ_β−e⁺

i,α−ei+1

pβ_i(β_i+1+ 1) +δ_β−e⁺

i+1,α−ei

pβ_i+1(β_i+ 1)).

Numerically only the indices of the multi-indices αand β corresponding to the nonzero components of dΓ(−∆K) with their values, are stored in an array.

In the algorithm instead of running over the multi-indicesαorβ with a lengthN, the multi-indicesβ⁰with a lengthN−1 are run over. And for eachiin [1, K], the changes of multi-indicesβ⁰=β−eiorβ⁰=β−ei+1

are used, then the indices of the corresponding multi-indicesαand β with lengthN are looked for.

Therefore an array composed of 2K ^N+K−2_K−2

triplets of elements is numerically stored.

4.2.3 Computation of the interaction factore^−itεV Denote a^#_j =a^#(ej) andNi=a^∗_iai .

By using the relations CCR (1):

a^∗_ia^∗_jaiaj =a^∗_i(aia^∗_j−εδij)aj=a^∗_iaia^∗_jaj−εδija^∗_iaj

=NiNj−εδijNi=Ni(Nj−εδij). ThenV can be rewritten as:

V =1 2

X

(i,j)∈Z²K

VijNi(Nj−εδij).

And sinceNieα=εαieα thenV is diagonal in the basis (eα)α: Veα=



 1 2

X

(i,j)∈Z²_K

Vijεαi(εαj−εδij)



eα

=



 ε²

2 X

(i,j)∈Z²K

Vijαi(αj−δij)



eα

and

e^−itεVe_α = e⁻ⁱ

t ε

ε2 2

P

(i,j)∈Z2 K

V_ijα_i(α_j−δ_ij)

e_α

= e^−itε2

P

i6=jVijαiαj+P

i∈ZKViiαi(αi−1)

eα.

4.3 Numerical computation of the reduced density matrices

Consider b^{W ick}=a^∗(e)^δa(e)^γ with|δ|=|γ|and its associated homogeneous polynomial:

b(z) = ¯z^δz^γ = ¯z₁^δ1. . .z¯_K^δKz₁^γ1. . . z^γ_K^K.

Let us compute the quantity Tr(%εb^{W ick}) when %ε is a normal state. Using an orthonormal basis of the N-fold sectorWN

Z,%εis a linear combination of operators|ΦihΨ|. It suffices to compute Tr(|ΦihΨ|b^{W ick}) = hΨ, b^{W ick}Φi.

Lemma 4.3 Setb^{W ick}=a^∗(e)^δa(e)^γ with|δ|=|γ|and letΦandΨbe in WN

Z then:

hΨ, b^{W ick}Φi=ε^|γ| X

|α⁰|=N−|δ|

Ψ¯α⁰+δΦα⁰+γ

p(α⁰+δ)!(α⁰+γ)!

α⁰! in the orthonormal basis_a√∗(e)^α

ε^Nα!|Ωi

|α|=N of WN

Z.

Proof. Given Ψ and Φ in the N-particles bosons space and the formula4.1 a^∗(e)^δa(e)^γa^∗(e)^α|Ωi=δ_γ≤αε^|γ| α!

(α−γ)!a^∗(e)^δ+α−γ|Ωi, we can write :

hΨ, b^{W ick}Φi=h X

|α|=N

Ψ_αa^∗(e)^α

√

ε^Nα!Ω, ε^|γ| X

|β|=N,β≥γ

Φ_β

√β!

ε^N/2(β−γ)!a^∗(e)^δ+β−γ|Ωi

=ε^|γ|

ε^N X

|α|=N

Ψ¯_α X

|β|=N,β≥γ

Φ_β

√β!

√

α!(β−γ)!ha^∗(e)^αΩ, a^∗(e)^δ+β−γ|Ωi

=ε^|γ| X

|α|=N,α≥δ

Ψ¯_αΦ_α+γ−δ

p(α+γ−δ)!

√

α!(α−δ)! α!

=ε^|γ| X

|α|=N,α≥δ

Ψ¯_αΦ_α+γ−δ

pα!(α+γ−δ)!

(α−δ)!

=ε^|γ| X

|α⁰|=N−|δ|

Ψ¯_α0+δΦ_α0+γ

p(α⁰+δ)!(α⁰+γ)!

α⁰! .

because

ha^∗(e)^αΩ, a^∗(e)^δ+β−γ|Ωi 6= 0 if and only ifα=δ+β−γ soβ =α+γ−δandβ ≥γ meansα−δ≥0.

The last line is obtained by a change of multi-indices by setting for eachα, α⁰=α−δbecauseα≥δ, and

then|α⁰|=|α| − |δ|=N− |δ|.

Numerically, all multi-indices ofN^K with length not larger than a givenN_max are stored in the lexico- graphic order.

For our algorithms, we pay attention to preserve this lexicographic order (or reverse).

For a givenN ≤ N_max, the list of relevant multi-indices (with length N) is extracted and handled in the lexicographic order.

For a givenδ, numerically the above summation is performed over multi-indicesα⁰such that|α⁰|=N−|δ|

in the lexicographic order.

Then for eachα⁰, the multi-indicesαof lengthN written asα=α⁰+δare looked for. Theseαare exactly the multi-indices such thatα≥δ and|α|=N.

Note in particular that the mappingα⁰ 7→α⁰+δpreserves the lexicographic order.

First let us compute the matrix elements ofγ_ε^p in the orthonormal basis (eα)α. The matrix element corresponding to the lineβ and columnαis:

Da^∗(e)^β

√ε^pβ!Ω

γ_ε^pa^∗(e)^α

√ε^pα!

ΩE

= Tr γ_ε^p(t)

a^∗(e)^α

√ε^pα!ΩEDa^∗(e)^β

√ε^pβ!Ω

= Tr(%ε(t)b^{W ick}) ε^pN(N−1). . .(N−p+ 1) according to the duality relation (5) of the reduced density matrices with ˜b=

a^∗(e)^α

√ε^pα!ΩED_a∗(e)^β

√ε^pβ!Ω , and b(z) =hz^⊗p,˜bz^⊗pi ∈ Pp,p.

If ˜b=

a^∗(e)^α

√ε^pα!ΩED_a∗(e)^β

√ε^pβ!Ω

, its Wick quantized is : b^{W ick}= p!

√α!β!a^∗(e)^αa(e)^β.

And then

γ_ε^p(t)(β, α) = p!

√α!β!

Tr(%ε(t)a^∗(e)^αa(e)^β) ε^pN(N−1). . .(N−p+ 1) .

Then owing to Lemma4.3, all the elements of the matrices γ^p_ε can be numerically computed.

In the case where the initial state is a Hermite state%ε=|z^⊗Nihz^⊗N|,z^⊗N needs to be expanded in the orthonormal basis (e_α) which is given by the following lemma.

Lemma 4.4 For allp∈N, andz∈ Z, we obtain in the basis(eα)α: z^⊗p= X

|α|=p

rp!

α!z^αeα.

Proof.

a^∗(z)^p|Ωi=a^∗(z1e1+. . .+zKeK)^p|Ωi= (z1a^∗(e1) +. . .+zKa^∗(eK))^p|Ωi

= X

|α|=p

p!

α!z₁^α1. . . z^α_K^Ka^∗(e1)^α1. . . a^∗(eK)^αK|Ωi

= X

|α|=p

p!

α!z^αa^∗(e)^α|Ωi. And then

z^⊗p=a^∗(z)^p

√ε^pp!|Ωi= X

|α|=p

rp!

α!z^αeα.

In the case where the initial state is a twin state, the following lemma is used to obtain an expansion in the basis (e_α).

Lemma 4.5 Let φ , ψ be in Z andz the state

a^∗(φ)ⁿa^∗(ψ)^m

√ε^n+mn!m! |Ωi such thatn+m=N.

Then we obtain

z=√

n!m! X

|γ|=N

X

|α|=n,α≤γ

√γ!

α!(γ−α)!φ^αψ^γ−αa^∗(e)^γ pε^Nγ!|Ωi. Proof.

a^∗(φ)ⁿ= X

|α|=n

n!

α!φ^αa^∗(e)^α a^∗(ψ)^m= X

|β|=m

m!

β!ψ^βa^∗(e)^β a^∗(φ)ⁿa^∗(ψ)^m= X

|α|=n,|β|=m

n!m!

α!β!φ^αψ^βa^∗(e)^αa^∗(e)^β= X

|α|=n,|β|=m

n!m!

α!β!φ^αψ^βa^∗(e)^α+β because of the CCR relations (1).

a^∗(φ)ⁿa^∗(ψ)^m

√

n!m! |Ωi= X

|α|=n,|β|=m

√ n!m!

p(α+β)!

α!β! φ^αψ^β a^∗(e)^α+β p(α+β)!|Ωi

=

√

n!m! X

|γ|=N,|α|=n,α≤γ

√γ!

α!(γ−α)!φ^αψ^γ−αa^∗(e)^γ

√γ! |Ωi.

By settingγ=α+β,

z=√

n!m! X

|γ|=N

X

|α|=n,α≤γ

√γ!

α!(γ−α)!φ^αψ^γ−αa^∗(e)^γ pε^Nγ!|Ωi.

Further the limit reduced density matrices (6) have to be computed numerically. In order to do this, the integration overZ of the Wigner measure is discretized. The problem is then reduced to the computation of the matrix elements of|z^⊗pihz^⊗p| in the basis (eα)_|α|=p.

Compute the matrix elements of|z^⊗pihz^⊗p|, according to Lemma4.4:

hz^⊗p|a^∗(e)^α

√

ε^pα!|Ωi= rp!

α!¯z^α Then

(|z^⊗pihz^⊗p|)eα= rp!

α!z¯^α X

|β|=p

s p!

β!z^βeβ

= X

|β|=p

√p!

α!β!z¯^αz^βeβ. For the computation of the integralR

Z|z^⊗pihz^⊗p|dµ(z), the Wigner measure is approximated by a convex combination of gauge invariant delta functionsδ_z^S1, where δ_z^S1= _2π¹ R2π

0 δ_eiθzdθ.

For the Wigner measure associated with the Hermite states,µ=δ^S_z¹, and the discretization is trivial and exact. it is not needed to be approximated because of the gauge invariance.

In the case of the twin states given by ΨN =^a∗√^{(ψ1)n1a∗(ψ2)n2}

εⁿ1 +n2n₁!n₂! |Ωi, whereψ1, ψ2∈ Z,kψ1k=kψ2k= 1, and n₁=n₂=^N₂, the Wigner measure isµ₀=_2π¹ R2π

0 δ_ψ^S1

φdφaccording to [6], with:

ψφ= cos(φ)ψ0+ sin(φ)ψ^π

2 ,

ψ0=

√ 2

2 (ψ1+ψ2), ψ^π

2 = i

√ 2

2 (ψ1−ψ2). Numerically the interval [0,2π] is discretized and µ0is approximated by _m¹ Pm

k=1δ^S_z¹

k. The Wigner measureµtpropagated at the time tof the twin states is given by :

µt= 1 2π

Z 2π 0

δ^S_ψ¹_φ(t)dφ ,

whereψφ(t) is solution to the Hartree equation at the timet with initial conditionψφ. Numerically it is now approximated by _m¹ Pm

k=1δ_z^S1

k(t)wherez_k(t) solves the Hartree equation (7).

Thus the matrix elements ofR

Z|z^⊗pihz^⊗p|dµ_t(z) are given by the formula:

1 m

m

X

k=1

√p!

α!β!z¯k(t)^αzk(t)^β. And the approximation of the scalarR

Z|z|^2pdµ0(z) is given by the formula _m¹ Pm

k=1|zk|^2p. The matrix of γ_ε^p(t)−

R

Z|z^⊗pihz^⊗p|dµt(z) R

Z|z|^2pdµ₀(z) can then be computed at any time t numerically with a good approximation.