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# Mean field limit of bosonic systems in partially factorized states and their linear combinations.

Marco Falconi Dipartimento di Matematica, Università di Bologna
Piazza di Porta San Donato 5 - 40126 Bologna, Italy
August 17, 2020
###### Abstract

We study the mean field limit of one-particle reduced density matrices, for a bosonic system in an initial state with a fixed number of particles, only a fraction of which occupies the same state, and for linear combinations of such states. In the mean field limit, the time-evolved reduced density matrix is proved to converge: in trace norm, towards a rank one projection (on the state solution of Hartree equation) for a single state; in Hilbert-Schmidt norm towards a mixed state, combination of projections on different solutions (corresponding to each initial datum), for states that are a linear superposition.

## I Introduction.

The mathematics of mean field limit of quantum systems has been widely investigated in the late 35 years. The first to put on a sound mathematical basis the concept of mean field limit of many-boson systems were Ginibre and Velo [1979], developing an idea by Hepp [1974]. Actually, in their work they performed the classical limit , and used the formalism of Fock space: they showed that, in the limit, bounded functions of annihilation and creation operators converge in some sense to bounded functions of the solution of classical equation corresponding to the system (Hartree equation); also, the quantum evolution between -dependent coherent states converges when to the evolution of quantum fluctuations around the classical solution. Their result was extended by Rodnianski and Schlein [2009]: they studied the convergence of reduced density matrices of many-bosons systems in the mean field limit, and also provided a bound on the rate of convergence. To do that they proved the convergence of normal ordered products of time evolved creation and annihilation operators (averaged on initial states that depend suitably on the number of particles). Further improvements were made by Chen and Lee [2011] and Chen et al. [2011]. The Fock space method has also been used to study the mean field limit of other bosonic systems, such as the Nelson model with cut off, or with a different scaling of the potential [see Ginibre et al., 2006, Falconi, 2013, Benedikter et al., 2012, Lee, 2013, for further details].

Another method has been widely used to study mean field limit of quantum systems, and it is based on a hierarchy of equations called BBGKY. This method has been very successful but has some limitations: in particular, due to its abstract argument, it does not give information on the rate of convergence of reduced density matrices [see Golse, 2013, for a review of BBGKY methods, and detailed references]. Recently, yet another simple method has been developed by Pickl [Knowles and Pickl, 2010, Pickl, 2011]: it avoids the technicalities of BBGKY hierarchies and to introduce the formalism of Fock spaces (in particular the use of Weyl operators).

To sum up, we review briefly the results of the works cited above. Let be a one-particle normalized state; then is a state with particles, all in the same state , and ( is the Weyl operator, the vacuum of Fock space) a coherent state with an average number of particles . Now let and be the corresponding one-particle reduced density matrices evolved in time by quantum dynamics. In the mean field limit , the following convergences in trace norm are proved, for suitable bosonic systems (Pickl [2011] proves convergence in operator norm):

 (1) Tr∣∣Tr1ρφ⊗n(t)−|φt⟩⟨φt|∣∣n→∞⟶0,Tr∣∣Tr1ρC(√nφ)Ω(t)−|φt⟩⟨φt|∣∣n→∞⟶0

where is the solution of the classical equation with initial datum . A bound on the rate of convergence of order [e.g. Rodnianski and Schlein, 2009, Knowles and Pickl, 2010, Benedikter et al., 2012] is given; for some systems, using quantum fluctuations, it can be improved to order [e.g. Chen et al., 2011, Falconi, 2013].

### i.1 Basic notions on symmetric Fock spaces.

We would like to extend the convergence results above to other types of states (that include as a particular case). We will use the Fock space method, so we recall some basic concepts of Fock spaces.

Let be a separable Hilbert space, with scalar product

 ⟨f,g⟩H=∫dx¯f(x)g(x).

From , construct the symmetric Fock space as follows. Define:

 H0=C,Hn:=Sn(H⊗⋯⊗H) with Sn orthogonal symmetrizer on the n-th fold tensor product of H, i.e. Hn={ϕn(x1,…,xn)∣∣ ϕn is invariant for any permutation of variables}. Then Fs(H):=∞⨁n=0Hn.

Let be a vector of . Then , endowed with the norm

 ∥∥ϕ∥∥:=(∞∑n=0∥∥ϕn∥∥2Hn)1/2

is a (separable) Hilbert space. The vector plays a special role in Fock spaces and it is often called the vacuum.

The basic operators of Fock spaces are the annihilation and creation operators. They are the adjoint of one another and are defined as following. Let , ; then

 a(f) =∫dxf(x)a(x) a∗(f) =∫dxf(x)a∗(x);

with

 (a(x)ϕ)n(x1,…,xn) =√n+1ϕn+1(x,x1,…,xn) (a∗(x)ϕ)n(x1,…,xn) =1√nn∑i=1δ(x−xj)ϕn−1(x1,…,^xj,…,xn),

where means such variable is omitted. The satisfy the following commutation properties:

 [a(x),a∗(x′)]=δ(x−x′),[a(x),a(x′)]=[a∗(x),a∗(x′)]=0.

From annihilation and creation operators we can construct an unitary operator, called the Weyl operator that plays a crucial role in our treatment of mean field limits. Let ; then the Weyl operator, denoted by , is defined as follows:

 C(α)=exp(a∗(α)−a(¯α)).

For all , is unitary on . In addition to its unitarity, we will use the following properties:

1. , and therefore ;

2. ;

3. .

Another useful class of operators are the ones defined by the so-called second quantization [see e.g. Reed and Simon, 1975, Chapter X.7]. Let be a self-adjoint operator on , with domain of essential self-adjointness . We define the second quantization of , denoted by , as the following operator of : let , then

 (dΓ(A)ϕ)n(x1,…,xn)=n∑i=1A(xi)ϕn(x1,…,xn),

where denotes the operator acting on the subspace of corresponding to the -th variable. is essentially self-adjoint on the domain

 DA={ϕ∈Fs(H)∣∣∃~n,∀n≥~nϕn=0;∀m∈Nϕm∈D⊗⋯⊗D}.

The last notion we introduce is that of number operator . For all , every is an eigenvector of with eigenvalue . Precisely, let , then is defined as

 (Nϕn)n(x1,…,xn)=nϕn(x1,…,xn).

is a self-adjoint operator with domain

 D(N)={ϕ∈Fs(H),∞∑n=0n2∥∥ϕn∥∥2<∞};

it satisfies the following properties:

1. .

2. Let ; then for all :

 ∥∥a#(f)ϕ∥∥≤∥∥f∥∥H∥∥(N+1)1/2ϕ∥∥.

Furthermore, let be the spectral family of the operator . Then for all -measurable operator valued function , we have

 g(N)a(x)=a(x)g(N−1) g(N)a∗(x)=a∗(x)g(N+1);

on suitable domains.

3. Let ; then for all :

 ∥∥∫dxdyf(x,y)a#(x)a#(y)ϕ∥∥ ≤∥∥f∥∥H⊗H∥∥(N+1)ϕ∥∥ ∥∥∫dxdyf(x,y)a∗(x)a(y)ϕ∥∥ ≤∥∥f∥∥H⊗H∥∥Nϕ∥∥.

### i.2 Main results.

The quantum system we would like to study describes non-relativistic interacting bosons; its dynamics is dictated by the following Hamiltonian of : let be a real and symmetric function of (other assumptions on the potential will be specified later), then

 (2) Hn=H0n+Vn=n∑j=1−Δxj+1nn∑i

This operator can be written in the language of second quantization on as:

 (3) H=∫dx(∇a)∗(x)∇a(x)+12n∫dxdyV(x−y)a∗(x)a∗(y)a(x)a(y);

and agree on . Let now with ; we denote by the time evolution of and by

 ρϕ(t)=|ϕ(t)⟩⟨ϕ(t)|

the corresponding density matrix, with integral kernel

 ρϕ(t,x1,…,xn;y1,…,yn)=ϕ(t,x1,…,xn)¯ϕ(t,y1,…,yn).

Also, we denote by the one-particle reduced density matrix, with integral kernel

 Tr1ρϕ(t,x;y)=∫dx1⋯dxn−1ϕ(t,x,x1,…,xn−1)¯ϕ(t,y,x1,…,xn−1).

In the mean field limit , using suitable initial states, the one-particle reduced density matrix is expected to converge in some sense to the solution of Hartree equation:

 (4)

As discussed above, such convergence has already been proved for factorized () and coherent () states. In this paper we prove that mean field limit convergence can be obtained for a wider class of states. First of all we consider states with particles, only a fraction of which is factorized in the same state [see also Lewin et al., 2012, where these states are used to construct an isomorphism between and the truncated Fock space orthogonal to ]. The precise definition is the following:

###### Definition 1 (θn,m).

Let be a Hilbert space; the corresponding symmetric Fock space. We also denote by a subspace of .

Now let such that and such that and

 (5) ⟨φ,ψm⟩H1=∫dx¯φ(x)ψm(x,x1,…,xm−1)=0.

We define

 θn,m:=cn,mSn(φ⊗n−m⊗ψm)∈Hn;

where:

 φ⊗j=φ⊗⋯⊗φj, Sj:H⊗⋯⊗Hj⟶Hj orthogonal projector (% symmetrizer), cn,m=(nm)1/2 (such that ∥∥θn,m∥∥=1).

The convergence result about -vectors is formulated in the following theorem:

###### Theorem 1.

Suppose there exists such that the operator inequality

 V2(x)≤D(1−Δx)

is satisfied. Let satisfy definition 1 with . Also, let be the solution of Hartree equation with initial datum . Then :

where the , , are positive and depend only on and .

###### Remark.

For technical reasons (see lemma 2 below), the result above holds only if . However since we are considering the limit , a stricter bound for has to be imposed if we want convergence to hold. In particular,

 Tr−limn→∞Tr1ρθn,m(t)=|φt⟩⟨φt|

whenever exists such that, for large ,

 m∼alnn.

Also, we study the mean field limit for superpositions of , or states. Such linear combinations have to satisfy the following definition:

###### Definition 2 (Φ, Θ, Ψ).

Let be a Hilbert space; the corresponding symmetric Fock space. We also denote by a subspace of . Furthermore, assume:

1. .

2. such that , and . Also we ask either

1. , and linearly independent for all ; or

2. for all .

3. To each vector of satisfying iia we associate and satisfying definition 1, such that if , for all . To each vector of satisfying iib we associate the Weyl operator .

Then define

 Φ=∑i∈Nαi(n)(φ(i))⊗n,Θ=∑i∈Nβi(n)θ(i)n,miΨ=∑i∈Nγi(n)C(√nφ(i))Ω.

The suites , and are chosen such that .

###### Remark 1.

In particular we have

 αi(n) =αi(∑i,j∈N¯αiαj⟨φ(i),φ(j)⟩n)−1/2, βi(n) =βi(∑i,j∈N¯βiβj⟨θ(i)n,mi,θ(j)n,mj⟩)−1/2, γi(n) =γi(∑i,j∈N¯γiγj⟨C(√nφ(i))Ω,C(√nφ(j))Ω⟩)−1/2.

For all , , and are convergent when , and their absolute value is uniformly bounded in respectively by , and , where the constants depend only on the -norm of the respective suite. Further details are discussed in section IV.2.

###### Remark 2.

Since states do not belong to a fixed particle subspace, we define the integral kernel of the reduced density matrix to be

 Tr1ρΨ(t,x;y)=1⟨Ψ(t),NΨ(t)⟩⟨Ψ(t),a∗(y)a(x)Ψ(t)⟩.

With a linear superposition of states as initial condition, the mean field limit is not a pure state. The reduced density matrix converges to a linear combination of projections on the solution of Hartree equation corresponding to each initial datum ; in the topology induced by the Hilbert-Schmidt norm ():

###### Theorem 2.

Suppose there exists such that the operator inequality

 V2(x)≤D(1−Δx)

is satisfied. Let , and satisfy definition 2 with . Also, for all , let be the solution of Hartree equation with initial datum . Then , if there is such that, for large , :

 HS-limn→∞ HS-limn→∞ HS-limn→∞

In particular, the following bounds hold:

 (6) ∥∥Tr1ρΦ(t)−∥∥(αi)i∈N∥∥−2l2∑i∈N|αi|2|φ(i)t⟩⟨φ(i)t|∥∥HS≤K1∑i

where the , , are positive and depend on , and .

###### Remark 3.

The first term on the right hand side of equation (6) converges to zero when because, by definition 2 and Riesz’s lemma, (and ). The first term on the right hand side of equation (7) also converges to zero if . This is because

furthermore since we can exchange summation with the limit .

###### Remark 4.

In this paper we focused attention only on one-particle reduced density matrices. The same method can be used to calculate the limit of -particle reduced density matrices (with ).

The rest of the paper is organized as follows. In section II we analyze the dynamics of classical (Hartree) and quantum system. In section III the combinatorial properties of states are studied. Finally in section IV we consider the limit , and prove theorems 1 and 2.

## Ii Classical and quantum dynamics.

In this section we review some properties of quantum and classical evolution needed to perform the classical limit.

The first proposition concerns the existence and unicity of solutions of the Hartree Cauchy problem [see Rodnianski and Schlein, 2009, Remark 1.3]:

 (9) {i∂tφt=−Δφt+(V∗|φt|2)φtφ(t0)=φ0.
###### Proposition 1.

Under the assumption , (9) has a unique solution for all . Furthermore, using also the conservation in time of and energy

 E(φ)=∫dx|∇φ(x)|2+12∫dxdyV(x−y)∥∥φ(x)∥∥2∥∥φ(y)∥∥2,

we obtain the following bound for :

 ∥∥φt∥∥2H1≤c(∥∥φ0∥∥2H1(1+∥∥φ0∥∥2L2)+∥∥φ0∥∥4L2+∥∥φ0∥∥2L2),

for some positive that depends only on .

We can formulate also the following proposition concerning quantum evolution [see Ginibre and Velo, 1979, Rodnianski and Schlein, 2009].

###### Proposition 2.

Let be a real symmetric function such that for some the operator inequality is satisfied. Then:

1. is a self-adjoint operator with domain , defined through the direct sum decomposition

 H=∞⨁n=1Hn.

is the self-adjoint operator (2) on (defined as a Kato sum).

2. Let , the associated solution of (9). Define also

 U(t)=exp(−itH), W(t,t0)=C∗(√nφt)U(t−t0)C(√nφ0).

Then for all , the following inequality holds:

 (10) ∥∥(N+1)δW(t,t0)ϕ∥∥≤K1(δ)eK2(δ)|t−t0|∥∥(N+1)max{2,2δ+1}ϕ∥∥;

where , , are positive and depend only on and .

## Iii Properties of θn,m states.

In this section we study the partially factorized states defined in section I. The technical result formulated in lemma 2 is crucial to perform the mean field limit . Its proof mimics the one performed in the case of completely condensed states, found in [Chen and Lee, 2011, Lemma 6.3].

###### Lemma 1.

The following identities hold:

 θn,m =1√m!∫dy1⋯dymψm(y1,…,ym)a∗(y1)⋯a∗(ym)φ⊗n−m =1√(n−m)!m!∫dy1⋯dymψm(y1,…,ym)(a∗(φ))n−ma∗(y1)⋯a∗(ym)Ω =dn,m√m!∫dy1⋯dymψm(y1,…,ym)PnC(√nφ)a∗(y1)⋯a∗(ym)Ω;

where is the projector on and

 dn,m=√(n−m)!exp(−n/2)n(n−m)/2 (dn,m∼(n−m)1/4em/2 for large n).
###### Proof.

First two equalities are easy to prove. To prove the last one, recall the formula

 φ⊗n−m=dn,mPn−mC(√nφ)Ω.

We then obtain:

 θn,m=dn,m√m!∫dy1⋯dymψm(y1,…,ym)Pna∗(y1)⋯a∗(ym)Pn−mC(√nφ)Ω=dn,m√m!∫dy1⋯dymψm(y1,…,ym)PnC(√nφ)(a∗(y1)+√n¯φ(y1))⋯(a∗(ym)+√n¯φ(ym))Ω;

now all the terms containing at least a vanish by (5). ∎

###### Lemma 2.

Let satisfy definition 1, with . Then :

for some positive that depends only on .

###### Proof.

Consider ; by definition of Weyl operators, we obtain :

By lemma 1 (second equality) we write

 (a∗(φ))i(a(¯φ))n−j+iθn,m=1√(n−m)!m!∫dy1⋯dymψm(y1,…,ym)(a∗(φ))i(a(¯φ))n−j+i(a∗(φ))n−ma∗(y1)⋯a∗(ym)Ω;

so if , i.e. , then either acts on some , or on , yielding zero (by (5), and definition of ). Therefore we must have , and consequently . In such case we obtain

 (a∗(φ))i(a(¯φ))n−j+iθn,m=√(n−m)!(j−m)!(j−m−i)!θj−m,m;

 (C∗(√nφ)θn,m)j=⎧⎪ ⎪⎨⎪ ⎪⎩0 if j

So if we set , for all :

 (C∗(√nφ)θn,m)k+m=e−n/2(√n)n−m−k√(n−m)!k!(k∑i=0(−1)inik!i!(k−i)!(n−m−k+i)!)θk,m.

Now define

 Ak:=∥∥(C∗(√nφ)θn,m)k+m∥∥Hk+m,

and the Laguerre polynomial

 (11) L(α)k(x):=k∑i=0(−1)ixi(k+α)!i!(k−i)!(α+i)!.

Then we have

 Ak=e−n/2(√n)n−m−k√k!(n−m)!∣∣L(n−m−k)k(n)∣∣ for all 0≤k≤n−m.

Actually, an explicit calculation shows that , . For , we use a sharp estimate for Laguerre polynomials obtained by Krasikov [2005]: let , and ; then for , and

 (12) ∣∣L(α)k(x)∣∣<√(k+α)!k!√x(s2−q2)r(x)ex/2x−(α+1)/2.

In the case of we obtain: , and . The condition implies that . Therefore we obtain:

 Ak< ⎷4√k(n−m+1)(2√k(n−m+1)+k−m+1)(2√k(n−m+1)−k+m−1).

Then, since :

 Ak<(√2/2−1/2)−1/2(k+1)−1/4(n−m+1)−1/4.

Let , and consider

since for all . Using the bound above, we obtain:

furthermore, since

 ∞∑k=n−m+1|Ak|2≤∥∥C∗(√nφ)θn,m∥∥2=1,

we have

 ⟨C∗(√nφ)θn,m,(N+1)−δC∗(√nφ)θn
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