aa r X i v : . [ qu a n t - ph ] F e b Markovian approximation for Pauli Fierz operators
L. Amour, J. NourrigatLaboratoire de Math´ematiques de Reims - UMR CNRS 9008Universit´e de Reims Champagne-Ardenne, France
Abstract
The purpose of this article is to derive a Markovian approximation of the reduced timedynamics of observables for the Pauli-Fierz Hamiltonian with a precise control of the errorterms. In that aim, we define a Lindblad operator associated to the corresponding quan-tum master equation. In a particular case, this allows to study the transition probabilitymatrix.
Keywords:
Pauli-Fierz Hamiltonian, Markovian approximation, Lindblad operator, quantummaster equation, transition probabilities, transition rate matrix, quantum electrodynamics,QED, spin dynamics.
MSC 2010:
Pauli-Fierz operators are used to describe the time evolution of charged particles with spinsinteracting with electric potentials and the quantized electromagnetic field (photons) and pos-sibly with a non quantized external magnetic field [3, 17, 8]. In the Markovian approximation,the interaction between particles and the quantized field is usually assumed to be small andin Pauli-Fierz Hamiltonians, the terms for these interactions are multiplied by a parameter g (coupling constant). We then suppose in the following that g is small in the Pauli-FierzHamiltonian considered here and denoted by H ( g ).The purpose of this work is to give an approximation of a Markov type for the time evolutionof observables for small g . In that aim, we shall omit some terms in the Hamiltonian that weexpect would not modify our results below. The precise definition of the simplified Pauli-FierzHamiltonian H ( g ) is given in Section 2.Let H tot be the Hilbert space of the model. Assuming that the Hamiltonian H ( g ) is acting in H tot , we recall that the evolution X ( t, g ) of an observable X in H tot is usually given for anytime t by: X ( t, g ) = e itH ( g ) Xe − itH ( g ) . In the system studied here we consider moving spinless charged particles together with a spin- fixed particle in the quantized electromagnetic field, with an external electric potential and anexternal constant magnetic field. The Hilbert space H tot is the completed tensor product of1he three Hilbert spaces H ph , H el and H sp respectively corresponding to the quantized elec-tromagnetic field, the moving charged particles and the spin fixed particle. Concerning theobservables, we assume that they act only on the moving charged particle and the spin fixedparticle spaces and not on the photon space, that is, the observables are written as I ⊗ X with I the identity in H ph and X an operator in H el ⊗ H sp . The photon vacuum state is denoted byΨ ∈ H ph .In this article we are concerned with the operator denoted σ S ( t, g ) X mapping from H el ⊗ H sp into itself and defined by: h σ S ( t, g ) Xf, g i H el ⊗H sp = h ( I ⊗ X ) e − itH ( g ) (Ψ ⊗ f ) , e − itH ( g ) (Ψ ⊗ g ) i H tot , for all f and g in H el ⊗ H sp . Thus, we consider time evolutions with initial states in the vacuumof photon. Note that in the case g = 0, one has: σ S ( t, X = e it ( H el + H sp ) Xe − it ( H el + H sp ) where H el and H sp respectively are the Schr¨odinger and the free spin Hamiltonian (see Section2.1). That is, ddt σ S ( t, X = iσ S ( t, H el + H sp ) , X ] . This means that, for g = 0, the observable evolution follows the Heisenberg equation (for theSchr¨odinger and the free spin Hamiltonians).Our goal is to prove that for g small enough, σ S ( t, g ) X follows a differential equation (masterequation) with an explicit term (Lindblad operator) and two negligible terms. The first errorterm is estimated with O ( g ) and the second with O ( g ) but the latter can be negligible for large t . The simplified equation obtained when omitting these two error terms is the quantum masterequation and constitutes the Markovian approximation. Let us be more specific concerning thatresult. We define a mapping L ( g ) (see Proposition 2.3 below) giving for any suitable operator X in H el ⊗ H sp an operator L ( g ) X also in H el ⊗ H sp satisfying for all t > ddt σ S ( t, g ) X = σ S ( t, g ) L ( g ) X + R ( t, g ) X + R ( t, g ) X. The mapping L ( g ) is called Lindblad operator (see [7, 1, 18, 19, 25, 26, 27, 30]). Concerning theerror terms in this approximation, we prove for small g the following estimates for all positivetimes: k R ( t, g ) X k ≤ C g t k X k , k R ( t, g ) X k ≤ Cg ln(1 + t ) k X k and if the ultraviolet cutoff vanishes at the origin: k R ( t, g ) X k ≤ C g t k X k , k R ( t, g ) X k ≤ Cg k X k where the above norm k X k will be specified.This result is precisely stated in Theorem 2.4 below and is the main result of the paper. InSection 2.2, we shall see the precise hypotheses and the definition of the chosen norms in orderto get the control of the error terms.There exists other types of corrections of the Heisenberg equation for the Schr¨odinger operator,for example, the Breit operator (see [6, 5]). There are other reduced dynamics, see e.g. [22, 31],which are different to the one that we consider. Return to equilibrium for Pauli-Fierz are2tudied, e.g., in [4, 12, 14, 15, 16]. For Markovian approximation and weak coupling limitapproximation, see [9, 31] and also [1, 30], and [32, 33, 9, 10, 30, 31] for the control of the error.Let us mention that the main result Theorem 2.4 is proved in view of possible applications inNMR (Nuclear Magnetic Resonance) for a system of several interacting electrons together witha fixed spin- particle subject to a constant external (non quantized) magnetic field. See [2]for the case of several fixed spins but without moving electrons and see [27, 23] in the positivetemperature case.In Section 5, we make explicit the operator L ( g ) in a particular case. Namely, we consider thecase of single electron without constant magnetic field and without the fixed spin particle. Wemake the hypotheses that the Schr¨odinger operator is globally elliptic in the sense of [20]. In thatcase, there exists an Hilbertian basis ( u j ) of L ( R ) with eigenfunctions u j of the Schr¨odingeroperator H el (see Section 2.1 and Section 5). Denoting by π u j the orthogonal projection on the u j , we make explicit for all m and j : M mj = g − h ( L ( g ) π u j ) u m , u m i . The form of this matrix suggests that it is the infinitesimal generator of a Markov semi-group.One can address the question of the existence of this semi-group and whether it defines a goodapproximation for small g of the transition probability matrix, that is, the probabilities foran electron initially in an excited state u m without photon to be at time t in the state u j .Different facts make this realistic. First, the matrix is triangular which means that up to thisapproximation, the probability for the electron to be at time t > The Hilbert space of the model is the completed tensor product of the three Hilbert spaces ofthe elements constituting the system: H tot = H ph ⊗ H el ⊗ H sp and the Hamiltonian of the model is the sum of the Hamiltonians H ph , H el and H sp of the threeelements when they are not interacting, together with the interaction Hamiltonian H int . Photon Hilbert space and Hamiltonian.
The single photon Hilbert space H ph is: H ph = { f ∈ L ( R , R ) , X j =1 k j f j ( k ) = 0 , k ∈ R } . The photon Hilbert space H ph is the symmetric Fock space F s ( H ph ) over the single photonHilbert space H ph and we use the notations and definitions of [29]. In this Fock space, we shallmainly use two types of unbounded operators. These two unbounded operators are initiallydefined in the subspace H finph of H ph of elements with a finite number of components (see [29]).For each V in the one photon space H ph , we use the Segal field Φ S ( V ) which is an unboundedoperator in the Fock space H ph (see [29]). For each bounded or unbounded operator T in H ph ,3e use the standard operator d Γ( T ) in the Fock space (see [29]). In particular, if M ω is themultiplication in H ph by ω ( k ) = | k | , the operator H ph = d Γ( M ω ) is the usual free photon energyHamiltonian. Electron Hilbert space and Hamiltonian.
The electron Hilbert space is H el = L ( R Q ) where Q is the number of electrons. We assumethat there is a constant external magnetic field B ext = (0 , , B ) where B ∈ R . The correspondingvector potential is denoted by: A ext ( x ) = (cid:16) − B x , B x , (cid:17) . Letting x = ( x (1) , . . . , x ( Q ) ) be the variable in R Q , we set for α = 1 , . . . , Q and j = 1 , , D ( α ) j = 1 i ∂∂x ( α ) j , ∇ ( α ) j = D ( α ) j − A extj ( x ( α ) ) . The Hamiltonian of the system of electrons is the Q -body Schr¨odinger operator with constantmagnetic field ( H el , D ( H el )): H el = Q X α =1 3 X j =1 ( ∇ ( α ) j ) + V ( x ) (2.1)where V is the electric potential supposed to be a real valued function on R Q with polynomialgrowth at infinity and identified with the multiplication operator by this function. This Hamil-tonian operator is initially defined on S ( R Q ). One can also omit the polynomial growth on theelectric potential but replace in the following the Schwartz space by the set of C ∞ functionswith compact support Spin Hilbert space and Hamiltonian.
The spin Hilbert space of the fixed particle is H sp = C . We denote by σ j the standard Paulimatrices (1 ≤ j ≤ , , B ), the energy of the spin- particle corresponds to the Hamiltonian: H sp = Bσ . (2.2) Interaction Hamiltonian.
The interaction between electrons and the spin fixed particle with the quantized electromagneticfield is expressed with two types of operators. One type corresponds to the three componentsof the quantized electromagnetic vector potential and is used to define the interaction betweenthe electron motions and the field. The other type, corresponding to the three components ofthe quantized magnetic field at the position x of the spin- fixed particle, is used to definethe interaction between the field and the spin particle. These operators are the image by theSegal field Φ S of the elements A jx and B jx of H ( x ∈ R , 1 ≤ j ≤
3) that we recall here thedefinitions: A jx ( k ) = ϕ ( | k | ) | k | / e − ik · x (cid:18) e j − ( e j · k ) k | k | (cid:19) , k ∈ R \{ } , (2.3) B j,x ( k ) = iϕ ( | k | ) | k | (2 π ) e − ik · x k × e j | k | , k ∈ R \{ } . (2.4)4n the above equalities, ( e , e , e ) stands for the canonical basis of R . The function ϕ is asmooth ultraviolet cutoff always supposed in the Schwartz space S ( R ). Note that each A jx and B jx ( j and x fixed) maps R into R and satisfy the equalities: k · A jx ( k ) = k · B jx ( k ) = 0 , k ∈ R \{ } . This means that the A j,x and B j,x are elements of the single photon Hilbert espace H ph accordingto the decreasing property of ϕ . The operators corresponding to the three components of thevector potential and of the magnetic field at point x respectively are Φ S ( A jx ) and Φ S ( B jx ).In order to define and use for upcoming computations the operator H int , we temporarily identify H tot and L ( R Q , H ph ⊗ H sp ) and set, for all f ∈ S ( R Q , H ph ⊗ H sp ) and any x ∈ R Q :( H int f )( x ) = Q X α =1 3 X j =1 (cid:16) Φ S ( A j,x [ α ] ) ⊗ I (cid:17) ∇ ( α ) j f ( x ) + X j =1 (cid:16) Φ S ( B j,x [0] ) ⊗ σ j (cid:17) f ( x ) . The tensor product in the above equality refers to H ph ⊗ H sp and I denotes the identity in H sp . It appears to be more convenient in the sequel to use integral expressions with creationoperators a ⋆ ( k ) and annihilation operators a ( k ) ( k ∈ R ) as in [3]. For any V ∈ H ph :Φ S ( V ) = 1 √ Z R (cid:16) a ⋆ ( k ) V ( k ) + a ( k ) V ( k ) (cid:17) dk. Then for each k ∈ R , we define an operator E ( k ) from H el ⊗ H sp now identified with L ( R Q , H sp ) and taking values in ( H el ⊗ H sp ) by:( E ( k ) f )( x ) = Q X α =1 3 X j =1 A j,x [ α ] ( k ) ∇ ( α ) j f ( x ) + X j =1 B j,x [0] ( k ) σ j f ( x ) (2.5)for all f ∈ S ( R Q , H sp ).The operator E ⋆ ( k ) denotes the formal adjoint of E ( k ). Let us emphasize that each E ( k ) takesvalues in ( H el ⊗ H sp ) and verifies: k · E ( k ) = 0 , k ∈ R . Therefore, for each f ∈ S ( R Q , H sp ) and for each x ∈ R Q , the function k → E x ( k ) =( E ( k ) f )( x ) belongs to H ph ⊗ H sp where H ph is the single photon Hilbert space. Thus, ex-tending the Segal field Φ S to the Hilbert space H ph ⊗ H sp as in [11], we have:( H int f )( x ) = Φ S ( E x ) . This operator is initially defined in H regtot = H finph ⊗ S ( R Q ) ⊗ H sp . This definition may be writtenin a somewhat formal way but useful in the following sections as: H int = 1 √ Z R (cid:16) ( a ( k ) ⊗ E ⋆ ( k ) + a ⋆ ( k ) ⊗ E ( k ) (cid:17) dk. (2.6)The above tensor product corresponds to the tensor product of H ph with H el ⊗ H sp . Full Hamiltonian. H regtot : H (0) = H ph + H el + H sp , (2.7) H ( g ) = H ph + H el + H sp + gH int = H (0) + gH int Domain and unitary group.
The assumption on the Hamiltonian H el is hypothesis ( H ) below. We use the following nota-tions. For any ( α, J ) with α = ( α , . . . , α m ) and J = ( j , . . . , j m ) where the α µ = 1 , . . . , Q andthe j µ = 1 , ,
3, we set: D α,J = ∇ [ α ] j · · · ∇ [ α m ] j m and set | ( α, J ) | = m . We make the following hypothesis ( H ). Hypothesis (H) . The operator H el defined in (2.1) is essentially self-adjoint and we denoteby H el its self-adjoint extension. We assume that there exists C > satisfying H el + C > and we denote by W elm the domain of ( C + H el ) m/ . For all integers m , W elm is the set of all u ∈ L ( R Q ) satisfying D α,J u ∈ L ( R Q ) if | ( α, J ) | ≤ m with the norm: k u k W elm = k u k + X | ( α,J ) |≤ m k D α,J u k (2.8) where k · k is the L ( R Q ) norm. This hypothesis is fulfilled for some potentials V . In particular, we consider in Section 5 thecase of globally elliptic Schr¨odinger operator ([20]) where this hypothesis is satisfied. Our mainresult could be applied with other potentials.Since we consider H el + H sp with H sp bounded we shall still denote by W elm the spaces associatedto H el + H sp instead of H el in the aim to avoid additional notations such as W el − spm . For example, W el now refers to H el ⊗ H sp . Theorem 2.1.
Under hypothesis ( H ) , the operator H (0) initially defined on H regtot is essentiallyself-adjoint. Let C > be such that H (0) + C > and let W totm be the domain of ( H (0) + C ) m/ .Then, for g small enough, the operator H ( g ) with domain W tot is self-adjoint and the domainof ( H ( g ) + C ) m/ is W totm . Theorem 2.1 is proved in Section 3. By this theorem, the operators e itH (0) and e itH ( g ) makesense for g small enough. See also [21] for self-adjointness of Pauli-Fierz without the condition g small. Full, free and reduced time evolutions of electron-spin observable.
For every operator X in H el ⊗ H sp , the time full evolution of X is defined by: S ( t, g ) X = e itH ( g ) ( I ph ⊗ X ) e − itH ( g ) . For any operator A in H tot , we denote by σ A the operator defined in H el ⊗ H sp by: h σ Af, g i H el ⊗H sp = h A (Ψ ⊗ f ) , (Ψ ⊗ g ) i H tot , (2.9)for all f and g in H el ⊗ H sp .In particular, we have for S ( t, g ) X , t >
0: 6 efinition 2.2.
We call reduced time evolution the mapping X → σ S ( t, g ) X from H el ⊗ H sp into H el ⊗ H sp defined for every t > and any operator X in H el ⊗ H sp by: h σ S ( t, g ) Xf, g i H el ⊗H sp = h ( I ph ⊗ X ) e − itH ( g ) (Ψ ⊗ f ) , e − itH ( g ) (Ψ ⊗ g ) i H tot , for all f and g in H el ⊗ H sp . In rest of the paper, we often omit the Hilbert space as a subscript in k · k and h , i when it is H tot , H ph , H el or H el ⊗ H sp . Similarly we omit the Hilbert subscript in the notation I for theidentity since there is no ambiguity. Also L ( H , H ) denotes the set of bounded linear mapsfrom H to H for any Hilbert spaces H and H .The free evolution of an operator A in H tot is defined by: A free ( t ) = e itH (0) Ae − itH (0) (2.10)and for an operator A in H el ⊗ H sp , we set (using the same notation since no confusion willappear): A free ( t ) = e it ( H el + H sp ) Ae − it ( H el + H sp ) . In particular, with E ( k ) the function used to give an expression of H int in (2.6), we set: E free ( k, t ) = e it ( H el + H sp ) E ( k ) e − it ( H el + H sp ) (2.11)and E free,⋆ ( k, t ) is similarly defined. Definition of the Lindblad operator.
The Lindblad operator is defined by the following Proposition proved in Section 4.Let us recall that the operator E ( k ) defined in (2.5) is an operator from H el ⊗H sp to ( H el ⊗H sp ) .Therefore, we can write E ( k ) = ( E ( k ) , E ( k ) , E ( k )). In this subsection, we agree in orderto avoid running indices in the sums to write E ⋆ ( k ) AE ( k ) instead of P j E ⋆j ( k ) AE j ( k ) for anyoperator A in all the integrals. Proposition 2.3.
Let X be a bounded operator in H el ⊗ H sp and bounded from W el into itself.Then, for every g small enough and all t > , the following operator is well defined from W el into W el = H el ⊗ H sp : L ( t, g ) X = i [ H el + H sp , X ]+ g Z R × (0 ,t ) (cid:0) e − is | k | [ E ⋆ ( k ) , X ] E free ( k, − s ) − e is | k | E free ( k, − s ) ⋆ [ E ( k ) , X ] (cid:1) dkds. (2.12) Moreover, for small g , the limit as t → + ∞ of L ( t, g ) X exists in L ( W el , W el ) and this limit isdenoted by L ( g ) X . We also have for some C > : k L ( t, g ) X − L ( g ) X k L ( W el ,W el ) ≤ Cg t k X k L ( W el ,W el ) , for all t > . If the ultraviolet cutoff function ϕ in (2.3)(2.4) vanishes at the origin then thisinequality can be replaced by: k L ( t, g ) X − L ( g ) X k L ( W el ,W el ) ≤ Cg t k X k L ( W el ,W el ) . The operator L ( g ) : X → L ( g ) X is called Lindblad operator. Main result. heorem 2.4. Let X be an operator in W el = H el ⊗ H sp bounded in W elj for all ≤ j ≤ .We have: ddt σ S ( t, g ) X = σ S ( t, g ) L ( g ) X + R ( t, g ) X + R ( t, g ) X (2.13) where R ( t, g ) X is an operator from W el into W el and R ( t ) X is an operator from W el into W el satisfying: k R ( t, g ) X k L ( W el ,W el ) ≤ C g t k X k L ( W el ,W el ) k R ( t, g ) X k L ( W el ,W el ) ≤ Cg ln(1 + t ) sup m ≤ k X k L ( W elm ,W elm ) . If the ultraviolet cutoff ϕ in (2.3)(2.4) vanishes at the origin then these inequalities can beimproved by: k R ( t, g ) X k L ( W el ,W el ) ≤ C g t k X k L ( W el ,W el ) k R ( t, g ) X k L ( W el ,W el ) ≤ Cg sup m ≤ k X k L ( W elm ,W elm ) . Theorem 2.4 is proved in Section 4.
Theorem 2.1 follows from Proposition 3.2 and Proposition 3.6.
We are first concerned with studying the action of the operator E ( k ) defined in (2.5) in theSobolev spaces W elm . Spherical coordinates for k , namely k = ρω ( ρ > ω ∈ S the unitsphere centered at the origin) are used in that purpose. Proposition 3.1.
Under hypothesis ( H ) the operator E ( k ) defined in (2.5) for fixed k ∈ R maps W elm into ( W elm − ) . Moreover, for any α , all m and p , there is C αmp > such that: k ∂ αρ (cid:0) ρ / E ( ρω ) (cid:1) k L ( W elm , ( W elm − ) ) ≤ C αmp (1 + ρ ) − p . (3.1) In particular, there exists C mp satisfying: k E ( k ) k L ( W elm , ( W elm − ) ) ≤ C mp (1 + | k | ) − p . The analogous estimates holds for E ⋆ ( k ) .Proof. We use hypothesis (2.8) noticing according to the expressions (2.3) and (2.4) for thefunctions A jx and B jx that the commutator:[ ∇ [ α ] j , E ( k )] = ik j E ( k ) . Since the smooth ultraviolet cutoff ϕ in (2.3)(2.4) is supposed to belong to S ( R ), we see thatfor each integer p : k ∂ αρ (cid:0) ρ / E ( ρω ) (cid:1) k L ( W el , ( W el ) ) ≤ C αmp (1 + ρ ) − p . The proposition follows. (cid:3) .2 Free Hamiltonian. We first have the following property ([28]).
Proposition 3.2.
The operator H (0) defined in H regtot by (2.7) is essentially self-adjoint. Wealso denote by H (0) its self-adjoint extension. If C > satisfies H (0) + C > then W totm standsfor the domain of ( C + H (0)) m/ . For all integers m , we have: k f k W tot m = X p + q ≤ m k H pph H qel f k . (3.2)We then deduce the Proposition below. Proposition 3.3.
We have the two following properties.(i) If the operator A maps W elm into W elp then I ⊗ A maps W totm into W totp .(ii) If the operator B maps W totm into W totp then σ ( B ) maps W elm into W elp .Proof. Let us prove the point ( ii ). We see that ( C + H el ) p σ ( B ) = σ (( I ⊗ ( C + H el ) p ) B ).Therefore, for each f in W elm : k σ ( B ) f k W elp ≤ k ( I ⊗ ( C + H el ) p ) B (Ψ ⊗ f ) k ≤ k B (Ψ ⊗ f ) k W totp ≤ K k Ψ ⊗ f k W totm = K k Ψ ⊗ H mel f k = k f k W elm , for some K >
0. We have used above that H ph Ψ = 0. (cid:3) We shall also use the fact that e itH (0) is bounded from W totm into itself for all m . We now study the action of the operator H int in the Sobolev spaces defined in Section 3.2. Proposition 3.4.
The operator H int is bounded from W totm to W totm − . For the proof, we need the following Lemma which is Lemma I.6 of [3].
Lemma 3.5.
Let k → F ( k ) be a function on R taking values in L ( H el ⊗ H sp , ( H el ⊗ H sp ) ) and satisfying k · F ( k ) = 0 for all k ∈ R . We define two operators T F and U F from W tot to W tot by: T F f = Z R ( a ( k ) ⊗ F ⋆ ( k )) f dk,U F f = Z R ( a ⋆ ( k ) ⊗ F ( k )) f dk (with the same notation convention as for E ( k )) . Then: k T F f k ≤ k ( H / ph ⊗ I ) f k Z R k F ⋆ ( k ) k dk | k | , k U F f k ≤ k ( H / ph ⊗ I ) f k Z R k F ( k ) k dk | k | + k f k Z R k F ( k ) k dk. The norms of F ( k ) and F ⋆ ( k ) above are the norm of L ( H el ⊗ H sp , ( H el ⊗ H sp ) ) . roof of Lemma 3.5. For the convenience of the reader we recall the proof in [3]. We have, forall f and g : |h T F f, g i| ≤ Z R | k |k ( a ( k ) ⊗ I ) f k dk Z R k ( I ⊗ F ⋆ ( k )) g k dk | k | . We know that: Z R | k |k ( a ( k ) ⊗ I ) f k dk ≤ k ( H / ph ⊗ I ) f k . We also have: k U F f k = Z R h ( a ( p ) a ⋆ ( k ) ⊗ F ( k )) f, ( I ⊗ F ( p ) f i dkdp = Z R k ( I ⊗ F ( k )) f k dk + Z R h ( a ⋆ ( k ) a ( p ) ⊗ F ( k )) f, ( I ⊗ F ( p )) f i dkdp ≤ Z R k ( I ⊗ F ( k )) f k dk + Z R | p || k | k ( a ( p ) ⊗ F ( k )) f k dkdp ≤ Z R k ( I ⊗ F ( k )) f k dk + Z R | k | k ( H / ph ⊗ F ( k )) f k dk. The Lemma follows. (cid:3)
Proof of Proposition 3.4.
We use the identity (3.2) for W totm together with the integral expression(2.5) for H int . We know that: H ph a ( k ) = a ( k )( H ph − | k | ) . Therefore, for all f in W tot m , for all integers p and q such that p + q ≤ m − H pph ⊗ H qel )( a ( k ) ⊗ E ⋆ ( k )) f = p X j =0 ( − j C jp | k | p − j ( a ( k ) H jph ⊗ H qel E ( k )) f. We apply Lemma 3.5 with: F j ( k ) = | k | p − j H qel E ( k )( C + H el ) − q − , g j = ( H jph ⊗ ( C + H el ) q +1 ) f. Thus: ( H pph ⊗ H qel )( a ( k ) ⊗ E ⋆ ( k )) f = p X j =0 ( − j C jp T F j g j . By Proposition 3.1: Z R k F j ( k ) k dk | k | < ∞ . The norm for k F j ( k ) k in the above integral is the norm of L ( H el ⊗ H sp , ( H el ⊗ H sp ) ). By theLemma: k ( H pph ⊗ H qel )( a ( k ) ⊗ E ⋆ ( k )) f k ≤ C p X j =0 k ( H / ph ⊗ I ) g j k≤ C p X j =0 k ( H j +1 / ph ⊗ ( C + H el ) q +1 ) f k ≤ C k f k W tot m . Proposition 3.4 then follows for m even for other m by interpolation. (cid:3) roposition 3.6. For g small enough, H ( g ) is essentially self-adjoint on its initial domain.The domain of its self-adjoint extension is W tot . The domain of H ( g ) m is W tot m for any integer m . The operator e itH ( g ) is bounded in W totm uniformly in t .Proof. By Proposition 3.4, we have for g small enough: k H ( g ) m f − H (0) m f k ≤ Cg k f k W tot m . Since W tot m is the domain of the self-adjoint extension of H (0) m , the Proposition follows fromKato Rellich Theorem. The last point of the Proposition is deduced from the first point foreven m since e itH ( g ) maps D ( H ( g ) m/ ) into itself and by interpolation for the other m . (cid:3) We write A ∼ B if σ ( A − B ) = 0 for two given operators A and B in H tot where σ is theoperator defined in (2.9).We begin this subsection with formal equalities and the norm estimates making sense foroperators will be studied in the next subsections.For each t >
0, set: E ( t ) = { ( s , s ) ∈ R , s + s < t } . (4.1) Theorem 4.1.
For each operator X in H el ⊗ H sp , we have: ddt e itH ( g ) ( I ⊗ X ) e − itH ( g ) ∼ e itH ( g ) ( I ⊗ L ( t, g ) X ) e − itH ( g ) + R ( t, g ) X + R ( t, g ) X where L ( t, g ) X is defined in (2.3) and: R ( t, g ) X = i g Z R × E ( t ) e i ( t − s ) | k | e itH ( g ) (cid:16) I ⊗ [ E ⋆ ( k ) , X ] (cid:17) e − is H ( g ) (4.2)[ H int , I ⊗ E ( k, s + s − t )] e i ( s − t ) H ( g ) dkds ds ,R ( t, g ) X = − i g Z R × E ( t ) e i ( s − t ) | k | e i ( t − s ) H ( g ) [ H int , I ⊗ E free ( k, s + s − t ) ⋆ ] (4.3) e is H ( g ) ( I ⊗ [ E ( k ) , X ]) e − itH ( g ) dkds ds . Concerning the dependance on the parameter g in the notations, we make it explicit in impor-tant terms such as H ( g ) , S ( t, g ) , L ( t, g ) , L ( g ) , R ( t, g ) , R ( t, g ) , R ( , g ) , R ( t, g ) and not for theterms used in the computations such as f ( t ) , I ( t ) , I ( t ) , L ( t ) , L ( t ) , Φ( s, X ).The first three steps of the proof correspond to the three Propositions below. Proposition 4.2.
For all operators X in H el ⊗ H sp , we have: ddt e itH ( g ) ( I ⊗ X ) e − itH ( g ) = ie itH ( g ) (cid:16) I ⊗ [( H el + H sp ) , X ] + g [ H int , I ⊗ X ] (cid:17) e − itH ( g ) ie itH ( g ) ( I ⊗ [( H el + H sp ) , X ]) e itH ( g ) + ig √ Z R e itH ( g ) (cid:16) a ( k ) ⊗ [ E ⋆ ( k ) , X ] + a ⋆ ( k ) ⊗ [ E ( k ) , X ] (cid:17) e − itH ( g ) dk. The proof of this Proposition is only a combination of the Heisenberg equation for the Pauli-Fierz Hamiltonian with the integral expression (2.6) of H int . Proposition 4.3.
For each k in R , we have: e itH ( g ) ( a ( k ) ⊗ I ) e − itH ( g ) = e − it | k | ( a ( k ) ⊗ I ) − ig √ Z t e i ( s − t ) | k | e isH ( g ) ( I ⊗ E ( k )) e − isH ( g ) ds,e itH ( g ) ( a ⋆ ( k ) ⊗ I ) e − itH ( g ) = e it | k | ( a ⋆ ( k ) ⊗ I ) + ig √ Z t e i ( t − s ) | k | e isH ( g ) ( I ⊗ E ⋆ ( k )) e − isH ( g ) ds. Proof.
One knows that: e itH (0) ( a ( k ) ⊗ I ) e − itH (0) = e − it | k | ( a ( k ) ⊗ I ) . Let: f ( t ) = e it | k | e itH ( g ) ( a ( k ) ⊗ I ) e − itH ( g ) = e itH ( g ) e − itH (0) ( a ( k ) ⊗ I ) e itH (0) e − itH ( g ) . We see that: f ′ ( t ) = ige it | k | e itH ( g ) [ H int , ( a ( k ) ⊗ I )] e itH ( g ) . According to the integral expression (2.6) of H int , since the operators a ( k ′ ) and a ( k ) are com-muting, and using [ a ( k ) , a ⋆ ( k ′ )] = δ ( k − k ′ ), we get:[ H int , ( a ( k ) ⊗ I )] = − √ I ⊗ E ( k )) . Consequently: f ′ ( t ) = − ig √ e it | k | e itH ( g ) ( I ⊗ E ( k )) e − itH ( g ) . The Proposition then follows. (cid:3)
The next Proposition is of the type of Proposition 4.2.
Proposition 4.4.
For all operators Y in H el ⊗ H sp , we have: e − itH ( g ) ( I ⊗ Y ) e itH ( g ) = I ⊗ Y free ( − t ) − ig Z t e − isH ( g ) [ H int , ( I ⊗ Y free ( s − t ))] e isH ( g ) ds. Proof.
For any operator Z , set: f ( t ) = e − itH ( g ) e itH (0) ( I ⊗ Z ) e − itH (0) e itH ( g ) . We have: f ′ ( t ) = − ige − itH ( g ) [ H int , ( I ⊗ Z free ( t ))] e itH ( g ) . Thus: e − itH ( g ) ( I ⊗ Z free ( t )) e itH ( g ) = ( I ⊗ Z ) − ig Z t e − isH ( g ) [ H int , ( I ⊗ Z free ( s ))] e isH ( g ) ds. Z = Y free ( − t ). (cid:3) End of the proof of Theorem 4.1.
By Proposition 4.2, we have: ddt e itH ( g ) ( I ⊗ X ) e − itH ( g ) = ie itH ( g ) (cid:16) I ⊗ [( H el + H sp ) , X ] (cid:17) e − itH ( g ) + I ( t ) X + I ( t ) X, with: I ( t ) X = ig √ Z R e itH ( g ) ( a ( k ) ⊗ [ E ⋆ ( k ) , X ]) e − itH ( g ) dk,I ( t ) X = ig √ Z R e itH ( g ) (cid:16) a ⋆ ( k ) ⊗ [ E ( k ) , X ] (cid:17) e − itH ( g ) dk. Proposition 4.3 is used to rewrite I ( t ) X . This gives: I ( t ) X = ig √ Z R e itH ( g ) ( I ⊗ [ E ⋆ ( k ) , X ])( a ( k ) ⊗ I ) e − itH ( g ) = I ′ ( t ) X + I ′′ ( t ) X, with: I ′ ( t ) X = ig √ Z R e itH ( g ) ( I ⊗ [ E ⋆ ( k ) , X ]) e − itH ( g ) e it | k | ( a ( k ) ⊗ I ) dk,I ′′ ( t ) X = g Z R × (0 ,t ) e itH ( g ) ( I ⊗ [ E ⋆ ( k ) , X ]) e i ( s − t ) | k | e i ( s − t ) H ( g ) ( I ⊗ E ( k )) e − isH ( g ) dkds. Notice that I ′ ( t ) X ∼
0. To rewrite I ′′ ( t ) X , we use Proposition 4.4 with t remplaced by t − s and Y by E ( k ). Thus, we obtain: e i ( s − t ) H ( g ) ( I ⊗ E ( k )) e i ( t − s ) H ( g ) = I ⊗ E free ( k, s − t ) − ig Z t − s e − is H ( g ) h H int , ( I ⊗ E free ( k, s + s − t )) i e is H ( g ) ds . Consequently: I ( t ) X ∼ I ′′ ( t ) X = e itH ( g ) ( I ⊗ L ( t ) X ) e − itH ( g ) + R ( t, g ) X where R ( t, g ) X is defined in (4.2) and: L ( t ) X = g Z R × (0 ,t ) e i ( t − s ) | k | [ E ⋆ ( k ) , X ] E f ree ( k, s − t ) dkds. Next, we similarly consider I ( t ) X and obtain: I ( t ) X ∼ e itH ( g ) ( I ⊗ L ( t ) X ) e − itH ( g ) + R ( t, g ) X,L ( t ) X = − g Z R × (0 ,t ) e is | k | E free ( k, − s ) ⋆ [ E ( k ) , X ] dkds where R ( t, g ) X is defined in (4.3). We see that the operator L ( t, g ) X defined in (2.3) satisfies: L ( t, g ) X = i [( H el + H sp ) , X ] + L ( t ) X + L ( t ) X. The proof is completed. (cid:3) .2 Norm estimates of the Lindblad operator. We study here the main term L ( t, g ) X and its limit as t → + ∞ . To this end, we need the nextProposition. Proposition 4.5.
Let X be a bounded operator in H el ⊗ H sp which is also bounded from W el to itself. Then for all t > , the following operators are well defined from W el into W el : Φ( s, X ) = Z R e is | k | [ E ( k ) ⋆ , X ] E free ( k, − s ) dk Φ ⋆ ( s, X ) = Z R e − is | k | E free ( k, − s ) ⋆ [ X, E ( k )] dk. Moreover, under the above hypotheses, there exists
C > such that: k Φ( s, X ) k L ( W el ,W el ) + k Φ ⋆ ( s, X ) k L ( W el ,W el ) ≤ C s k X k L ( W el ,W el ) . If in addition X is bounded from W el to W el then we have: k Φ( s, X ) k L ( W el ,W el ) + k Φ ⋆ ( s, X ) k L ( W el ,W el ) ≤ C s k X k L ( W el ,W el ) . If the function ϕ in (2.3)(2.4) is vanishing at the origin then the first inequality can be replacedby: k Φ( s, X ) k L ( W el ,W el ) + k Φ ⋆ ( s, X ) k L ( W el ,W el ) ≤ C s k X k L ( W el ,W el ) and similarly for the second inequality.Proof. We use spherical coordinates for k , k = ρω , ρ > ω ∈ S . We have:Φ( s, X ) = X j =1 Z R + × S e isρ [ F j ( ρ, ω ) , X ] G j ( ρ, ω, s ) ρdρdω, with: F j ( ρ, ω ) = ρ / E j ( ρω ) ⋆ , G j ( ρ, ω, s ) = ρ / E freej ( ρω, − s ) . According to Proposition 3.1, we get: k ∂ αρ F j ( ρ, ω ) k L ( W elp ,W elp − ) ≤ C (1 + ρ ) − . Using the hypotheses on the operator X , we obtain: k ∂ αρ [ F j ( ρ, ω ) , X ] k L ( W elp ,W elp − ) ≤ C (1 + ρ ) − sup m ≤ p k X k L ( W elm ,W elm ) . By Proposition 3.1 and since e it ( H el + H sp ) is uniformly bounded from W elk into itself for each k ,we get: k ∂ αρ G j ( ρ, ω, s ) k L ( W el ,W el ) ≤ C (1 + ρ ) − . We can integrate twice by parts with the variable ρ in the expression of Φ( s, X ). We deduce: s Φ( s, X ) = X j =1 Z R + × S e isρ ∂ ρ (cid:16) ρ [ F j ( ρ, ω ) , X ] G j ( ρ, ω, s ) (cid:17) dρdω.
14o prove the last points of the Theorem, we note that if the function ϕ vanishes at the originthen we can integrate by parts three times instead of two.The proof of the Proposition is completed. (cid:3) We then deduce the next result.
Proposition 4.6.
Let X be a bounded operator in H el ⊗ H sp which is also bounded from W el toitself. Then the operator L ( t, g ) X defined in (2.3) is bounded uniformly in t from W el to W el .Moreover, as t tends to + ∞ , this operator tends in L ( W el , W el ) to a limit L ( g ) X satisfying: k L ( t, g ) X − L ( g ) X k L ( W el ,W el ) ≤ Cg t k X k L ( W el ,W el ) and the above factor / (1 + t ) is improved to / (1 + t ) if the ultraviolet cutoff ϕ vanishes atthe origin. Propostion 2.3 is then proved.
We now need to control the norms in some spaces of the operators R ( t, g ) X and R ( t, g ) X given by (4.2) and (4.3). Proposition 4.7.
For each integer m , there exists C m > such that the operator R ( t, g ) defined in (4.2) satisfies: k R ( t, g ) X k L ( W totm ,W totm − ) ≤ C m g ln(1 + t ) sup j ≤ m k X k L ( W elj ,W elj ) . We also have: k σ R ( t, g ) X k L ( W el ,W el ) ≤ C m g ln(1 + t ) sup j ≤ k X k L ( W elj ,W elj ) . If the smooth cutoff ϕ in (2.3) and (2.4) vanishes at the origin then the factor ln(1 + t ) can beomitted in the above inequalities.The same estimates holds true for R ( t, g ) instead of R ( t, g ) .Proof. In the definition (4.2) of R ( t, g ) X , we use spherical coordinates for k , namely k = ρω with ρ > ω ∈ S the unit sphere centered at the origin. Thus, we can write: R ( t, g ) X = g Z E ( t ) Φ( t, s , s ) ds ds with: Φ( t, s , s ) = Z R + × S e i ( t − s ) ρ e itH ( g ) ( I ⊗ [ F ( ρ, ω ) , X ]) e − is H ( g ) [ H int , I ⊗ G ( ρ, ω, s , s , t )] e i ( s − t ) H ( g ) ρdρdω (4.4)where: F ( ρ, ω ) = ρ / E ⋆ ( ρω ) G ( ρ, ω, s , s , t ) = ρ / E ( ρω, s + s − t ) . F takes its values in L ( W elj , W elj − ) and we have: k ∂ αρ F ( ρ, ω ) k L ( W elj ,W elj − ) ≤ C j (1 + ρ ) − . Under the hypothesis of Theorem 2.4, we see: k ∂ αρ [ X, F ( ρ, ω )] k L ( W elj ,W elj − ) ≤ C j (1 + ρ ) − sup k ≤ j k X k L ( W elk ,W elk ) . By Proposition 3.3: k ∂ αρ ( I ⊗ [ X, F ( ρ, ω )]) k L ( W totj ,W totj − ) ≤ C j (1 + ρ ) − sup k ≤ j k X k L ( W elk ,W elk ) . In the same way and since e it ( H el + H sp ) is uniformly bounded from W elk into itself for all k , wededuce: k ∂ αρ G ( ρ, ω, s , s , t ) k L ( W elj ,W elj − ) ≤ C j (1 + ρ ) − where C j is idependent on t, s , s . Therefore, by Proposition 3.3: k ∂ αρ ( I ⊗ G ( ρ, ω, s , s , t ) k L ( W totj ,W totj − ) ≤ C j (1 + ρ ) − . By Proposition 3.4, H int is bounded from W totj to W totj − . Thus: k ∂ αρ [ H int , I ⊗ G ( ρ, ω, s , s , t ) k L ( W totj ,W totj − ) ≤ C j (1 + ρ ) − . Finally, we know that e itH ( g ) , e − is H ( g ) and e i ( s − t ) H ( g ) are uniformly bounded from W totj intoitself for all j . We can integrate twice by parts with the variable ρ in (4.3). In view of theabove estimates, we get:(1 + | t − s | ) k Φ( t, s , s ) k L ( W totj ,W totj − ) ≤ C j sup k ≤ j k X k L ( W elk ,W elk ) . If the cutoff ϕ vanishes at 0, we have F (0 , ω ) = 0 and G (0 , ω, s , s , t ) = 0. In equality (4.3),we can integrate three times by parts and obtain:(1 + | t − s | ) k Φ( t, s , s ) k L ( W totj ,W totj − ) ≤ C j sup k ≤ j k X k L ( W elk ,W elk ) . If E ( t ) is the set defined in (4.1), we see that: Z E ( t ) ds ds | t − s | ≤ C ln(1 + t ) , Z E ( t ) ds ds | t − s | ≤ C. The first point in the Theorem is then deduced. The second one follows by Proposition 3.3. (cid:3)
End of the proof of Theorem 2.4.
By Theorem 4.1, we have (2.13) with: R ( t, g ) X = L ( t, g ) X − L ( g ) X, R ( t, g ) X = R ( t, g ) X + R ( t, g ) X. The norm estimate of R ( t, g ) X comes from Proposition 4.6 and the norm estimate of R ( t, g ) X and R ( t, g ) X from Proposition 4.7. (cid:3) Transition probabilities.
We shall make explicit in this section the matrix of the operator L ( g ) in a suitable basis, and ina particular case. In this Section, there is one (spinless) electron ( Q = 1), no fixed spin particleand no external magnetic field ( A ext = 0). Thus: H tot = H ph ⊗ H el , H el = L ( R )Therefore H el is the usual Schr¨odinger operator: H el = − ∆ + V. We make the following assumptions.( H ′ ) The potential V is a non negative real valued C ∞ function on R , and there exists a real M > and a constant C α > for each multi-index α , such that: | ∂ α V ( x ) | ≤ C α (1 + | x | ) M −| α | . ( H ′ ) There exists γ > such that: V ( x ) ≥ γ | x | M . The following result is proved in [20].
Proposition 5.1.
Under hypotheses ( H ′ ) and ( H ′ ) the operator H el initially defined on S ( R ) has a unique self-adjoint extension. The spectrum of H el is discrete. It is the set of finitemultiplicity eigenvalues µ j going to + ∞ . There exists an Hilbertian basis ( u j ) of H el satisfying: H el u j = µ j u j . Each u j belongs to the intersection of the spaces W elm . We have for all m > : D ( H mel ) = { u ∈ H m ( R ) , V ( x ) m u ∈ L ( R ) } . Consequently, Hypothesis ( H ) of Section 3 is satisfied if the above assumptions ( H ′ )( H ′ ) hold.We can rearrange the eigenfunctions in such a way that the sequence ( µ j ) is non decreasing.We next define the transition probabilities. Let π u j be the orthogonal projection in H el on thevector space spanned by u j . Definition 5.2.
For all unitary eigenfunctions u j and u m of H el , and for each t > , thetransition probability of u m to u j is defined by: P mj ( t ) = h ( σ S ( t, g ) π u j ) u m , u m i H el = h ( I ⊗ π u j ) e − itH ( g ) (Ψ ⊗ u m ) , e − itH ( g ) (Ψ ⊗ u m ) i H tot = h σ S ( t, g ) π u j , π u m i HS . where h , i HS is the scalar product of two Hilbert-Schmidt operators in H el . These identities together with our main result leads us to make explicit the following matrix: M mj = g − h ( L ( g ) π u j ) , π u m i HS = g − h ( L ( g ) π u j ) u m , u m i H el . (5.1)We now omit the subscript H el in the scalar product of H el .We believe that this matrix is the generator of a semi-group of Markov matrices which perhapswill be a good approximation of the matrix P mj ( t ).17 heorem 5.3. The M mj defined by (5.1) satisfy: M mj = π X α =1 Z | k | = µ m − µ j |h E α ( k ) u m , u j i| R dσ ( k ) if µ j < µ m M mj = 0 if µ j > µ m M jj = − X k = j M kj M mj = 0 if µ j = µ m and j = m. Therefore M mj ≥ µ j < µ m , M mj ≤ µ j = µ m and M mj = 0 if µ j > µ m . We have, foreach j : X m M mj = 0 . We remark that these properties are supposed to be verified by the infinitesimal generator ofa Markov semi-group. We do not know if the matrix M mj is indeed the infinitesimal generatorof a semi-group neither if we get a good approximation of the P mj ( t ). If this holds true then,up to this approximation, the transition probability of an initial electronic state to a higherelectronic energy level state is zero. Proof.
By the definition (2.3) of L ( g ), since [ H el , π u j ] = 0, we have: L ( g ) π u j = g t → + ∞ Z R × (0 ,t ) (cid:16) e − is | k | [ E ⋆ ( k ) , π u j ] E free ( k, − s ) − e is | k | E free ( k, − s ) ⋆ [ E ( k ) , π u j ] (cid:17) dkds. We have, for all integers j and m : h L ( g ) π u j , π u m i HS = Tr (cid:0) π u m L ( g ) π u j (cid:1) . By the definition (2.11), we have for the traces:Tr (cid:0) [ E ⋆ ( k ) , π u j ] E free ( k, − s ) π u m (cid:1) = e is ( µ m − µ j ) |h E ( k ) u m , u j i| − e isµ j δ jm h e − isH el E ( k ) u j E ( k ) u j i and Tr (cid:0) E free ( k, − s ) ⋆ [ E ( k ) , π u j ] π u m (cid:1) = e − isµ j δ jm h e isH el E ( k ) u j E ( k ) u j i− e is ( µ j − µ m ) |h E ( k ) u m , u j i| . Therefore: g A mj = h L ( g ) π u j , π u m i HS = g ( A mj + δ jm λ j )with: A mj = lim t → + ∞ Z R × (0 ,t ) cos( s ( | k | + µ j − µ m )) |h E ( k ) u m , u j i| dkds. We know that, for each suitable function F , we have, if λ < t → + ∞ Z R × (0 ,t ) cos( s ( | k | + λ )) F ( k ) dkdt = π Z | k | = − λ F ( k ) dσ ( k ) . This limit is zero if λ ≥
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