Higher order terms for the quantum evolution of a Wick observable within the Hepp method

Higher order terms for the quantum evolution of a Wick observable within the Hepp method

Sébastien Breteaux IRMAR, UMR-CNRS 6625, Université de Rennes 1, campus de Beaulieu, 35042 Rennes Cedex, France. E-mail: sebastien.breteaux@univ-rennes1.fr
Februar 2011
Abstract

The Hepp method is the coherent state approach to the mean field dynamics for bosons or to the semiclassical propagation. A key point is the asymptotic evolution of Wick observables under the evolution given by a time-dependent quadratic Hamiltonian. This article provides a complete expansion with respect to the small parameter which makes sense within the infinite-dimensional setting and fits with finite-dimensional formulae.

Mathematics subject classification (2000): 81R30, 35Q40, 81S10, 81S30.

Keywords: mean field limit, semiclassical limit, coherent states, squeezed states.

1 Introduction

In this article we derive two expansions with respect to a small parameter of quantum evolved Wick observables under a time-dependent quadratic Hamiltonian.

The Hepp method was introduced in [16] and then extended in [11, 12] in order to study the mean field dynamics of many bosons systems via a (squeezed) coherent states approach. The asymptotic analysis in the mean field limit is done with respect to a small parameter , where the number of particles is of order .

Remember that the mean field dynamics is obtained as a classical Hamiltonian dynamics which governs the evolution of the center  of the Gaussian state (squeezed coherent state). Meanwhile the covariance of this Gaussian as well as the control of the remainder term is determined by the evolution of a quadratic approximate Hamiltonian around .

A key point in this method is the asymptotic analysis of the evolution of a Wick quantized observable according to this quantum time-dependent quadratic Hamiltonian.

Only a few results are clearly written about the remainder terms and some possible expansions in powers of , see the works of Ginibre and Velo [13, 14]. In the finite-dimensional case, entering into the semiclassical theory, accurate results have been given by Combescure, Ralston and Robert in [6]. For the mean field infinite-dimensional setting some results have been proved in [15, 10, 23] with a different approach.

We stick here with the Hepp method with the presentation of [1] which puts the stress on the similarities and differences between the infinite-dimensional bosonic mean field problem and the finite-dimensional semiclassical analysis. Nevertheless, in [1] the authors only considered the main order term although some of their formulae make possible complete expansions. In this article we derive two expansions of the quantum evolved Wick observables which are equal term by term.

Two difficulties have to be solved :

  1. Unlike the time-independent finite-dimensional case, no Mehler type explicit formula (see for example [18] or [8]) is available. A general time-dependent Hamiltonian has no explicit dynamics.

  2. In the infinite-dimensional framework the quantization of a linear symplectic transformation (a Bogoliubov transformation) requires some care. Useful references on this subject are [3] and [2]. Its realization in the Fock space relies on a Hilbert-Schmidt condition on the antilinear part connected with the Shale theorem (see [24] and [21, 7, 5]).

These things are well known but have to be considered accurately while writing complete expansions.

Two different methods, with apparently two different final formulae, will be used. A first one relies on a Dyson expansion approach and provides the successive terms as time-dependent integrals. The second one uses the exact formulae for the finite-dimensional Weyl quantization and after having made explicit the relationship between Wick and Weyl quantizations like in [4] or [1], the proper limit process with respect to the dimension is carried out.

The outline of this article is the following. In Section 2 we recall some facts and definitions about the Fock space and Wick quantization. We then present our main results in Section 3 in Theorems 3.1 and 3.2 and illustrate them by a simple example. Section 4 and Section 5 are devoted to the construction and properties of the classical and quantum evolution associated with a symmetric quadratic Hamiltonian. Section 7 and Section 8 contain the proofs of our two expansion formulae. For the convenience of the reader we recall some facts about real-linear symplectomorphisms and symplectic Fourier transform in the appendices.

2 Wick calculus with polynomial observables

Definitions

We recall some definitions and results about Wick quantization. More details can be found in [1].

In this paper  denotes a separable Hilbert space over , the field of complex numbers. It is also a symplectic space with respect to the symplectic form . We use the physicists convention that all the scalar products over Hilbert spaces are linear with respect to the right variable and antilinear with respect to the left variable. We denote by the symmetrization operator on  (the completion for the natural Hilbert scalar product of the algebraic tensor product ) defined by

where the  are vectors in  and  denotes the set of the permutations of . We will use the notation  for , and  for  when the  terms of this product are equal to . We call monomial of order  a complex-valued application defined on  of the form

with  where  (or ) denotes the Hilbert completion of the -fold symmetric tensor product, and for two Banach spaces  and , the space of continuous linear applications from  to  is denoted by . We then write . The total order of  is the integer . The finite linear combinations of monomials are called polynomials. The set of all polynomials of this type is denoted by . Subsets of particular interest of  are  and , the finite linear combinations of monomials of total order equal to  and not greater than .

The Hilbert space

is called the symmetric Fock space associated with , where tensor products and sum completions are made with respect to the natural Hilbert scalar products inherited from . We also consider the dense subspace  of  of states with a finite number of particles

where the tensor products are completed but the sum is algebraic.

The Wick quantization of a monomial  is the operator defined on  by its action on  as an element of ,

where  denotes the identity map on the space  and for , . The Wick quantization is extended by linearity to polynomials.

We have a notion of derivative of a polynomial, first defined on the monomials and then extended by linearity. For  and for any given , the operator

(2.1)

is an element of . We use the “bra” and “ket” notations of the physicists for vectors and forms in Hilbert spaces. Then we can define the Poisson bracket of order  of two polynomials , by

since, for any polynomial , is a -form (on ) and  is a -vector.

Remark 2.1.

The product denoted by a dot in the definition of the Poisson bracket is a -bilinear duality-product between -forms and -vectors. As an example consider the polynomials

The Poisson bracket of order  of  and  is

Some examples of Wick quantizations

Here is a quick review of the notations used for some useful examples of Wick quantization. A vector of  is denoted by , is a bounded operator and  is the variable of the polynomials. In the next table, the first column describes the polynomial and the second the corresponding Wick quantization (as an operator on ).

The operator  is the usual second quantization of an operator restricted to  multiplied by a factor . If  we obtain  the usual number operator multiplied by a factor . The operators  and  are the usual annihilation, creation and field operators of quantum field theory with an additional  factor. The real and imaginary parts of a complex number  are denoted by  and . The field operators  are essentially self-adjoint and this enables us to define the (-dependent) Weyl operators

Calculus

Here are some calculation rules for Wick quantizations of polynomials in . The proofs can be found in [1].

Proposition 2.2.

For every polynomial ,

  • in  for any ,

  • is closable and the domain of the closure contains

    (we still denote by the closure of ),

  • on (where the bar denotes the usual conjugation on complex numbers),

  • for any in , holds on  where .

3 Main results and a simple example

Our two hypotheses are:

H1

Let  be a one parameter family of self-adjoint operators on  defining a strongly continuous dynamical system .

H1’

Assume H1 and additionally that the dynamical system preserves a dense set  such that, for any , belongs to .

H2

Let  be in , ( defines a -antilinear Hilbert-Schmidt operator by ).

With H1’ and H2, the classical flow associated with  of quadratic polynomials is the solution  to the equation

(3.1)

where , written in a weak sense.

Although things are better visualized by writing a differential equation, the hypotheses H1 and H2 suffice to define the dynamical system . Details about this point are given in Section 4. Actually  is a family of symplectomorphisms of  which are naturally decomposed into their -linear and -antilinear parts:

See Appendix A for more details about symplectomorphisms and this decomposition.

Similarly, the quantum flow associated with  is the solution  of

(3.2)

The precise meaning of the solutions to this equation is specified in Section 5.

We are ready to state our two main results dealing with the evolution of a Wick observable , , under the quantum flow, that is to say the quantity . (We use the usual notation .)

Theorem 3.1.

Assume H1 and H2. Let  be a polynomial. Then, for any time , the formula

(3.3)

holds as an equality of continuous operators from  to , where

  • and ,

  • the polynomials  are defined recursively by

    with  for any polynomial c.

Theorem 3.2.

Assume H1 and H2. Let  and  a polynomial. Then introducing

  • the vector  such that for all ,

  • the operator on 

the formula

(3.4)

holds as an equality of continuous operators from  to .

Remark 3.3.

The derivative  is in  and Tr denotes the trace on the subset of trace class operators of .

Remark 3.4.

For  the operators  and  send  into .

Remark 3.5.

The exponential is intended in the sense

for a polynomial  in .

Example 3.6.

To give an idea of the behavior of these formulae we apply them in the simplest (non trivial) possible situation, with  and . As  is time-independent the classical evolution equation is autonomous and thus we can write  and . The solution is . We can then compute both

The first one is easily computed as , and, with ,

and thus

Now we compute the second one. Since  and , we get  and then obtain directly

We thus obtain the same result with the two computations for the term of order 1 in .

Then we can show that

since

because

Remark 3.7.

Since these two formulae will be proven independently and the identification of each term of order  in  in the expansion of the symbol is clear, we carry out a computation only on the formal level for the convenience of the reader to show the link between the two formulae in the general case.

We show (formally) that

Then it is simple to show that

as operators on  once the case  is understood:

In this computation we have used that  as .

We first give  in a more explicit way. As  and  we first get

with  and omitting the time dependence everywhere. Then with  (and thus ) and  we obtain

Then we compute  in several parts. The linear and antilinear parts of the equation give

We now show that ,

And thus .

We then show that

We first observe that . A simple calculation using  shows that  and this immediately gives the result.

4 Classical evolution of a Wick polynomial under a quadratic evolution

The adjoint of a -antilinear operator is defined in Appendix A.

Definition 4.1.

A -antilinear operator  on  is said of Hilbert-Schmidt class if  is finite, where  is the usual trace norm for -linear operators. The set of Hilbert-Schmidt antilinear operators is denoted by .

Let  with norm

for , where  and  are respectively -linear and -antilinear. The space  is a Banach algebra.

Remark 4.2.

The norm  is well defined as the decomposition  is unique ( and ).

Construction of the classical flow without the  term

Let  and . Observe that  and so  is a continous one parameter family of , so that the theory of ordinary differential equations in Banach algebras (see for example [17]) asserts that there exists a unique two parameters family  of elements of  such that

with  of  class in both parameters such that for all  and ,

The classical flow  is a symplectomorphism with respect to the symplectic form . It can be checked deriving

with respect to .

The strongly continuous dynamical system associated with 

We first state a proposition which is a direct consequence of Theorem X.70 in [22] in the unitary case. This proposition provides a set of assumptions ensuring the existence of a strongly continuous dynamical system associated with a family  of self-adjoint operators. Other more general situations can be considered as in [19, 20] for example.

Proposition 4.3.

Let  be a family of self-adjoint operators on the Hilbert space  satisfying the following conditions.

  1. The  have a common domain  (from which it follows by the closed graph theorem that  is bounded).

  2. For each , is uniformly strongly continuous and uniformly bounded in  and  for  lying in any fixed compact interval.

  3. For each , exists uniformly for  in each compact interval and  is bounded and strongly continuous in .

The approximate propagator  is defined by if  and .

Then for all , in a compact interval and any ,

exists uniformly in  and . Further, if , then  is in  for all  and satisfies

Construction of the classical flow with the  term

Assume H1 and H2. Let  be the solution of

with , . What we call here the solution of

(4.1)

with  is

Depending on the assumptions on  it will be possible to precise if  sloves Equation (4.1) in a usual sense (strongly, weakly, on some dense subset…).

With the particular set of assumptions of Theorem 4.3 we get that for all  and ,

Composition of a Wick polynomial with the classical evolution

The composition of a polynomial with the classical flow defines a time-dependent polynomial.

Definition 4.4.

We define a norm on  by

where  is a polynomial with  and  is a shorthand for . For a polynomial  in , we will sometimes write .

Proposition 4.5.

Let  be a polynomial, and . Then  and we have the estimate

Proof.

The proof is essentially the same as in Proposition 2.12 of [1]. ∎

5 Quantum evolution of a Wick polynomial

Without the  term

Definition 5.1.

Let  and . A family  of unitary operators on  defined for  real is a solution of

(5.1)

if

  1. is strongly continuous in with respect to  with ,

  2. , ,

  3. exists for almost every  (depending on ) and is equal to ,

  4. , , .

This definition is made to fit the general framework of Theorems 4.1 and 5.1 of [19]. More precisely we may check the following theorem.

Theorem 5.2.

Let  and .

Then the quantum flow equation (5.1) associated to the family  has a unique solution. This solution preserves the sets  for .

To establish this theorem we will use the following estimates.

Lemma 5.3.

Let  and . Then, on , and for , satisfies the estimates

(5.2)

and

(5.3)

The second estimate is in the sense of quadratic forms, for all ,

Proof.

The first estimate is a consequence of  associated to

For the second estimate, consider . The first term of this commutator is

Then we deduce easily the second term and a reindexation gives the following form for the whole commutator:

Newton’s binomial formula and the inequalities  and yield

Using also  to control  we obtain

Cauchy-Schwarz’s inequality gives the claimed estimate. ∎

Lemma 5.4.

Let  and . Then  is essentially self-adjoint on  and its closure is essentially self-adjoint on any other core for . Inequalities (5.2) and (5.3) still hold on .

We still denote by  this self-adjoint extension.

Proof.

We apply the commutators Theorem X.37 of [22] with the estimates of Lemma 5.3 for .∎

Lemma 5.5.

If a solution of the quantum flow equation (5.1) exists then it leaves  invariant for any integer .

In the time-independent case the estimate

holds.

Proof.

From Lemma 5.4, for any , . We can adapt the proof of Theorem 2 of [9] to the case of the quantization of a continuous one parameter family of quadratic polynomials with the estimates of Lemma 5.3. ∎

Proof of theorem 5.2.

We use Theorems 4.1 and 5.1 of [19] with the family of operators  (here we directly consider the self-adjoint extension of ). We set .

  1. This family is stable in the sense that  (we actually have an equality here).

  2. The space  is admissible for this family in the sense that for each , leaves  invariant and

    for .

    This is true because, as we have seen in Lemma 5.5, leaves  invariant and, thanks to the estimate of the same lemma, we can apply the resolvent formula