On global dynamics for damped driven Jaynes-Cummings equations

On global dynamics for damp ed driv en Ja ynes–Cummings equations A.I. Komec h 1 and E.A. Kop ylov a 1 Institute of Mathematics of BOKU University, Vienna, Austria alexander.k omec h@b oku.ac.at, elena.k op ylo v a@b oku.ac.at Abstract The article concerns damp ed driven Jaynes–Cummings equation whic h describes quantised one-mo de Maxw ell field coupled to a t wo-lev el molecule. W e consider a broad class of damping and pumping which are p olynomial in the creation and annihilation op erators, and their struc- tures corresp ond to the theory of completely positive and trace preserving generators (CPTP) of Lindblad and Kossako wski & al. Our main result is the construction of global generalised solutions with v alues in the Hilbert space of nonnegativ e Hermitian Hilb ert–Sc hmidt op erators in the case of time-dep endent pump- ing. The pro ofs rely on finite-dimensional appro ximations of the annihilation and creation op e- rators. MSC classific ation : 81V80, 81S05, 81S08 37K06, 37K40, 37K45, 78A40, 78A60. Keywor ds : Ja ynes–Cummings equation; dynamical semigroup; Hamiltonian op erator; density op- erator; pumping; dissipation op erator; p ositivity preserv ation; trace; Hilb ert–Sc hmidt op erator; Quan tum Optics; laser. Con ten ts 1 In tro duction 1 2 Notations and main results 3 3 Nonp ositivit y of the dissipation op erator 5 4 Finite-dimensional appro ximations and uniform b ounds 5 5 P assage to limit 7 6 Conflict of in terest 8 7 Data a v ailabilit y statemen t 8 1 In tro duction The Ja ynes–Cummings equation is the basic mo del of Quan tum Optics, and it is used for description of v arious asp ects of the laser action. The mo del without damping and pumping was introduced in [ 16 ] (the surv ey can b e found in [ 19 ]). V arious versions of the pumping are considered in [ 5 , 8 , 12 , 17 , 23 ]. The damping w as introduced analysing quan tum sp ontaneous emission [ 1 ]–[ 3 ], [ 6 , 7 , 23 , 24 ]. W e construct global solutions for all initial v alues from the space of Hilb ert–Schmidt op erators in the case of time-dep enden t pumping. Denote X = F ⊗ C 2 , where F is the single-particle Hilb ert space endow ed with an orthonormal basis | n ⟩ , n = 0 , 1 , . . . , and the corresponding annihilation and creation op erators a and a † : a | n ⟩ = √ n | n − 1 ⟩ , a † | n ⟩ = √ n + 1 | n + 1 ⟩ , [ a, a † ] = 1 . (1.1) 1 The researc h w as funded in whole b y Austrian Science F und (FWF) 10.55776/P A T3476224. W e will consider a damp ed driv en version of the Ja ynes–Cummings equation ˙ ρ ( t ) = A ρ ( t ) := − i [ H ( t ) , ρ ( t )] + γ D ( t ) ρ ( t ) , t ≥ 0 , (1.2) where the density op erator ρ ( t ) of the coupled field-molecule system is a Hermitian op erator in X . The Hamiltonian H ( t ) is the sum H ( t ) = H 0 + pH 1 ( t ) , H 0 := ω c a † a + 1 2 ω a σ 3 , H 1 ( t ) = ( a + a † ) ⊗ σ 1 + A e ( t ) . (1.3) Here H 0 is the Hamiltonian of the free field and atom without interaction, while pH 1 ( t ) is the in teraction Hamiltonian, ω c > 0 is the cavit y resonance frequency , ω a > 0 is the atomic frequency , and p ∈ R is prop ortional to the molecular dip ole momen t. The pumping is represen ted b y self- adjoin t op erators A e ( t ), and σ 1 and σ 3 are the Pauli matrices acting on the factor C 2 in F ⊗ C 2 , so [ a, σ k ] = [ a † , σ k ] = 0. Finally , γ > 0, and D ( t ) is a dissipation op erator. W e will consider the op erator D 1 ρ = aρa † − 1 2 a † aρ − 1 2 ρa † a, (1.4) used in [ 1 ]–[ 3 ], [ 6 , 7 , 20 , 23 , 24 ], and also its suitable mo difications. Definition 1.1. HS is the Hilb ert sp ac e of Hermitian Hilb ert–Schmidt op er ators with the inner pr o duct [ 21 ] ⟨ ρ 1 , ρ 2 ⟩ HS = tr [ ρ 1 ρ 2 ] . (1.5) Definition 1.2. i) | n, s ± ⟩ = | n ⟩ ⊗ s ± is the orthonormal b asis in X , s ± ∈ C 2 and σ 3 s ± = ± s ± . ii) X ∞ is the sp ac e of al l finite line ar c ombinations of the ve ctors | n, s ± ⟩ . iii) D ⊂ HS is the subsp ac e of finite r ank Hermitian op er ators ρ = ∞ X n,n ′ =0 X s,s ′ = s ± ρ n,s ; n ′ ,s ′ | n, s ⟩ ⊗ ⟨ n ′ , s ′ | . (1.6) Our main goal is to construct global nonnegative solutions for the equation ( 1.2 ) in the Hilb ert space HS in the case of time-dep endent pumping A e . The main issue is that the op erators a and a † are unbounded by ( 1.1 ), so, the righ t hand side of ( 1.2 ) is not Lipsc hitz contin uous in ρ ( t ). Accordingly , the meaning of the equation ( 1.2 ) must b e adjusted (see Definition 2.3 ). In the case of time-indep endent pumping A e ( t ) = A e , the Hamilton H ( t ) = H is a selfadjoint op erator, so for γ = 0, solutions are given b y ρ ( t ) = e − iH t ρ (0) e iH t , t ≥ 0 . (1.7) In this case, the trace tr ρ ( t ) is conserv ed, and ρ ( t ) ≥ 0 if ρ (0) ≥ 0. How ever, for time-dep enden t pumping or γ > 0, the formula for solutions is missing. W e assume that the pumping A e ( t ) and the dissipation op erator D ( t ) are p olynomials in a and a † . Moreo v er, the structure of D ( t ) corresp onds to the theory of completely p ositiv e and trace preserving generators (CPTP) developed by Lindblad [ 20 ] and Gorini, Kossak owski and Sudarshan [ 14 ]. Examples. A e = a † + a , A e = a † a , and D ( t ) = D 1 satisfy all our assumptions. Our main results are as follows: there exist con tin uous linear op erators U ( t ) : HS → HS with t ≥ 0 such that for each nonnegativ e ρ 0 ∈ HS, the tra jectory ρ ( t ) = U ( t ) ρ 0 is a nonnegativ e generalised solution to ( 1.2 ) with initial condition ρ (0) = ρ 0 , (1.8) and the a priori b ounds hold, ∥ ρ ( t ) ∥ HS ≤ ∥ ρ 0 ∥ HS , t ≥ 0 . (1.9) 2 Our strategy relies on finite-dimensional approximations of the annihilation and creation op erators and the uniform b ounds ( 1.9 ). The b ounds are due to the nonp ositivity of the dissipation op erators D ( t ) under our assumpions. The k ey example of such dissipation operators is D 1 for whic h the nonp ositivit y has b een established in [ 18 ]. Note that the nonnegativity of ρ ( t ) cannot be obtained b y a straightforw ard application of the theory [ 14 , 20 ] of completely p ositiv e and trace preserving generators (CPTP) since the generator A ( t ) is an unbounded operator. How ever, the theory allo ws us to establish the nonnegativit y for sp ecial finite-dimensional appro ximations which k eep the CPTP structure of the generator. Then the nonnegativity of ρ ( t ) follo ws by the limit transition. In con trast, the trace conserv ation do es not follow in the limit, though it is conserv ed for all the approximations. Let us comment on previous related results. In the case of b ounded generators A ( t ), global solu- tions for equations of t yp e ( 1.2 ) ob viously exist. In this case, Lindblad [ 20 ] and Gorini, Kossak o wski, and Sudarshan [ 14 ] found necessary and sufficien t conditions on A ( t ) pro viding the p ositivity and trace preserv ation. F or un b ounded generators, the existence of global solutions to Quantum Dynamical Systems (QDS) is not w ell-dev elop ed, [ 4 , p.110]. In [ 11 ], E.B. Da vies considered autonomous quan tum- mec hanical F okker–Planc k equations (QFP). The existence of the corresp onding p ositiv e con traction semigroup is established in the Banach s pace of self-adjoint trace-class op erators. The uniqueness and trace preserv ation hav e not b een prov ed. The useful sufficient conditions, pro viding the trace preserv ation for QFP equations, were found in [ 9 ]. The detailed c haracterisation of a class of co v ariant QDS with unbounded generators is presented in [ 15 ]. The survey can b e found in [ 10 , 13 ]. In [ 18 ], we ha ve constructed the con traction semigroup for the equation ( 1.2 ) with pumping and dissipation op erator which do not dep end on time, so A ( t ) = A . In this case, the main ingredient in the pro of is the nonp ositivity A ≤ 0 which implies the existence of the semigroup e A t b y the Lumer– Phillips theorem. The nonp ositivit y follows from the same prop ert y of the dissipation op erator D and implies that the semigroup is con tracting, i.e., the b ounds ( 1.9 ) hold. As the k ey example, the nonp ositivit y is prov ed for the op erator ( 1.4 ). Note that the equation ( 1.2 ) is non-autonomous, so, the theory of semigroup is not applicable. Our approac h allo ws us to substitute the theory b y a systematic application of the contraction ( 1.9 ). Up to our knowledge, the w ell-p osedness for nonautonomous damp ed driven Jaynes–Cummings equations has not b een established previously . Ac knowledgemen ts. The authors thank S. Kuksin, M.I. P etelin, A. Shnirelman and H. Spohn for longterm fruitful discussions, and the Institute of Mathematics of BOKU Universit y for the supp ort and hospitality . The research was funded in whole b y Austrian Science F und (FWF) 10.55776/P A T3476224. 2 Notations and main results Ev ery density op erator ρ ∈ HS is defined uniquely b y its matrix entries ρ n,s ; n ′ ,s ′ = ⟨ n, s | ρ | n ′ , s ′ ⟩ , n, n ′ = 0 , 1 , . . . , s, s ′ = s ± . (2.10) The Hilb ert–Schmidt norm, corresponding to the inner pro duct ( 1.5 ), can b e written as ∥ ρ ∥ 2 HS = tr [ ρ 2 ] = ∞ X n,n ′ =0 X s,s ′ = s ± | ρ n,s ; n ′ ,s ′ | 2 < ∞ . (2.11) Note that by ( 1.1 ), the space X ∞ is in v ariant with resp ect to a and a † . Hence, the pro ducts of ρ ∈ D with any p olynomials of a and a † are well-defined as op erators in X ∞ . W e assume that the pumping and the dissipation op erator satisfy the following conditions. Denote by M 2 = C 2 ⊗ C 2 the space of 2 × 2-matrices. H1. The pumping A e ( t ) is a symmetric op erator in X ∞ , and it is a p olynomial in the creation and annihilation op erators: A e ( t ) = X A e l,l ′ ( t ) a † l a l ′ , A e l,l ′ ∈ C (0 , ∞ ; M 2 ) , (2.12) 3 H2. The dissipation op erator D ( t ) ρ admits a “double” structure of the CPTP theory of completely p ositiv e trace preserving operators [ 14 , 20 , 22 ]: D ( t ) ρ = [ V ( t ) ρ, V † ( t )] + [ V ( t ) , ρV † ( t )] , (2.13) where V ( t ) is also a p olynomial in the creation and annihilation op erators: V ( t ) = X V l,l ′ ( t ) a † l a l ′ , V l,l ′ ∈ C (0 , ∞ ; M 2 ) , (2.14) Remark 2.1. i) H1–H2 imply that A ( t ) ρ ∈ D for ρ ∈ D and t ≥ 0. ii) The op erator ( 1.4 ) admits the structure ( 2.14 ). Remark 2.2. In [ 17 ], the pumping A e ( t ) = σ + e − iω p t + σ − e iω p t , σ + =  0 1 0 0  , σ − =  0 0 1 0  (2.15) has b een applied to the study of the collapse and reviv al phenomenon (see [ 17 , (2.1)]). T o formulate our results, w e need give a meaning to the equation ( 1.2 ). The issue is that the op erators A ( t ) are not w ell-defined on HS. The equation admits the treatment via the matrix entries ( 2.10 ) as the system ˙ ρ n,s ; n ′ ,s ′ ( t ) = [ A ( t ) ρ ( t )] n,s ; n ′ ,s ′ , t ≥ 0 , n, n ′ ≥ 0 , s, s ′ = s ± , (2.16) since the right hand side is well-defined for all ρ ( t ) ∈ HS. Indeed, by H1 and ( 1.1 ), the op erators A ( t ) are well-defined on the domain D , and [ A ( t ) ρ ] n,s ; n ′ ,s ′ = X | k − n | + | k ′ − n ′ | ≤ N r, r ′ = s ± A k,r ; k ′ ,r ′ n,s ; n ′ ,s ′ ( t ) ρ k,r ; k ′ ,r ′ , n, n ′ ≥ 0 , s, s ′ = s ± , ρ ∈ D , (2.17) where N = max(2 , deg A e , deg V ), and A k,r ; k ′ ,r ′ n,s ; n ′ ,s ′ ( · ) ∈ C (0 , ∞ ) , ∀ n, s, n ′ , s ′ , k , r , k ′ , r ′ . (2.18) Finally , since the summation in ( 2.17 ) is finite, the matrix en tries admit a unique extension by con tin uit y from ρ ∈ D to all ρ ∈ HS. The structure ( 2.17 ) means that the matrix of the generator in the basis | n, s ⟩ is almost diagonal. Let us denote by HS w the Hilb ert space HS endo w ed with the weak top ology , and the space of tra jectories T = L ∞ (0 , ∞ ; HS) ∩ C (0 , ∞ ; HS w ) . (2.19) Definition 2.3. A tr aje ctory ρ ( t ) ∈ T is a gener alise d solution to ( 1.2 ) if it satisfies the system ( 2.16 ) in the sense of distributions, i.e., ρ n,s ; n ′ ,s ′ ( t ) − ρ n,s ; n ′ ,s ′ (0) = Z t 0 [ A ( τ ) ρ ( τ )] n,s ; n ′ ,s ′ dτ . (2.20) Our main result is the follo wing theorem. Denote by HS + the set of nonnegative ρ ∈ HS. Theorem 2.4. L et c onditions H1–H2 hold. Then ther e exist c ontinuous line ar op er ators U ( t ) : HS → HS with t ≥ 0 such that i) F or e ach ρ 0 ∈ HS , the tr aje ctory ρ ( t ) = U ( t ) ρ 0 ∈ T is the gener alise d solution to ( 1.2 ) with initial c ondition ( 1.8 ); ii) ρ ( t ) ∈ HS + for t ≥ 0 if ρ 0 ∈ HS + , and the b ounds ( 1.9 ) hold. The pro of relies on finite-dimensional F aedo–Galerkin appro ximations defined via finite-dimen- sional approximations of the annihilation and creation op erators, and on uniform a priori b ounds in the Hilb ert–Schmidt norm. Remark 2.5. i) Theorem 2.4 also holds for dissipation op erators D ( t ) = P j D j ( t ) where each D j ( t ) admits the structure ( 2.13 )–( 2.14 ) with the corresp onding p olyniomial V j ( t ). The pro of is unc hanged. ii) The trace conserv ation holds for every finite-dimensional approximation, ho w ev er, we cannot conclude the conserv ation in the limit. 4 3 Nonp ositivit y of the dissipation op erator In this section we pro ve the nonp ositivity of the dissipation op erator D ( t ) satisfying H2 . Lemma 3.1. The op er ators D ( t ) with the structur e ( 2.13 ), ( 2.14 ) ar e nonp ositive in D : ⟨ ρ, D ( t ) ρ ⟩ HS ≤ 0 , ρ ∈ D , t ≥ 0 . (3.1) Pr o of. W e will omit the dep endence on t . F or ρ ∈ D , ⟨ ρ, D ρ ⟩ HS = tr  ρD ρ  = tr  ρ  V ρV † − 1 2 V † V ρ − 1 2 ρV † V   = tr  ρV ρV † − ρV † V ρ  = tr  ρV ρV † − V † V ρ 2  . (3.2) No w w e use the fact that ρ is a finite rank Hermitian op erator ( 1.6 ). Then ( 2.13 ) and ( 2.14 ) imply that the operators ρV ρV † and V † V ρ 2 ha v e only finite n um b er of nonzero entries ( 2.10 ), so their traces are well-defined. Moreov er, ρ admits a finite spectral resolution in the orthonormal basis of its eigenv ectors e i ∈ X ∞ : ρ = ν X i =1 ρ i e i ⊗ e i . (3.3) In this basis, the en tries ρ ij = ρ i δ ij , and the en tries V j k = ⟨ e j , V e k ⟩ and V † kl = ⟨ e k , V † e l ⟩ of the op erators V and V † are well-defined. Hence, ( 3.2 ) implies, with summation in rep eated indices, ⟨ ρ, D ρ ⟩ HS = ρ i δ ij V j k ρ k δ kl V † li − V † kl V lj ρ 2 j δ j k = ρ i V ik ρ k V † ki − V † kl V lk ρ 2 k = ρ i V ik ρ k V † ki − V † ki V ik ρ 2 k = V ik V † ki ( ρ i ρ k − ρ 2 k ) = 1 2  V ik V † ki ( ρ i ρ k − ρ 2 k ) + V ki V † ik ( ρ k ρ i − ρ 2 i )) = − 1 2 | V ik | 2 ( ρ i − ρ k ) 2 ≤ 0 (3.4) since V † ik = V ki . Hence, ( 3.1 ) is prov ed. Remark 3.2. The pro of of the nonp ositivity essentially dep ends on the symmetry of ρ . 4 Finite-dimensional approximations and uniform b ounds Replace the Hilb ert space X b y its finite-dimensional subspace X ν = ( | n ⟩ ⊗ s ± : n ≤ ν ) , ν = 1 , 2 , . . . . (4.1) The subspace is inv arian t under action of the annihilation op erator a , and w e define by a ν its restriction to X ν : a ν [ | n ⟩ ⊗ s ] = [ a | n ⟩ ] ⊗ s, 0 ≤ n ≤ ν. (4.2) On the other hand, X ν is not inv ariant under action of the creation op erator a † , so w e define a † ν as the adjoint to a ν : a † ν [ | n ⟩ ⊗ s ] =  [ a † | n ⟩ ] ⊗ s, 0 ≤ n < ν 0 , n = ν     . (4.3) Remark 4.1. The definitions ( 4.2 ), ( 4.3 ) are equiv alent to kno wn matrix represen tations of a n,n ′ and a † n,n ′ restricted to n, n ′ ≤ ν . 5 Denote by D ν the space of Hermitian op erators ( 1.6 ) with n, n ′ ≤ ν , and define the finite- dimensional approximations of the JC dynamics ( 1.2 ) by ˙ ρ ν ( t ) = A ν ( t ) ρ ν ( t ) := − i [ H ν ( t ) , ρ ν ( t )] + γ D ν ( t ) ρ ν ( t ) , t ≥ 0 , (4.4) where ρ ν ( t ) is a linear op erator in X ν , and H ν ( t ) and D ν ( t ) are defined as H ( t ) and D ( t ) with a and a † replaced by a ν and a † ν resp ectiv ely . By the conditions H1–H2 and ( 4.2 )–( 4.3 ), H ν ( t ) and D ν ( t ) ρ ν are Hermitian op erators in X ν . Hence, A ν ( t ) ρ ν is also Hermitian, so the linear equation ( 4.4 ) admits a unique global Hermitian solution ρ ν ( t ) ∈ C (0 , ∞ ; D ν ) for any Hermitian initial state ρ ν (0) ∈ D ν . Since ρ 0 ∈ HS, ρ 0 = X n,n ′ X s,s ′ ρ 0 n,s ; n ′ ,s ′ | n, s ⟩⟨ n ′ , s ′ | , ∥ ρ 0 ∥ HS = X n,n ′ X s,s ′ | ρ 0 n,s ; n ′ ,s ′ | 2 < ∞ . (4.5) Define the approximate initial state b y ρ 0 ν = X n,n ′ ≤ ν X s,s ′ ρ 0 n,s ; n ′ ,s ′ | n, s ⟩⟨ n ′ , s ′ | . (4.6) It is the Hermitian op erator since ρ 0 is, and ∥ ρ 0 ν ∥ HS ≤ ∥ ρ 0 ∥ HS . (4.7) Ob viously , ρ 0 ν ≥ 0 if ρ 0 ≥ 0. Denote by ρ ν ( t ) the solution to ( 4.4 ) with the initial state ρ 0 ν . Lemma 4.2. i) The a priori b ounds hold ∥ ρ ν ( t ) ∥ HS ≤ ∥ ρ 0 ν ∥ HS , t ≥ t 0 , ν = 0 , 1 , . . . . (4.8) ii) ρ ν ( t ) ≥ 0 for t ≥ 0 if ρ 0 ν ≥ 0 . Pr o of. ad i) It suffices to pro ve that the generator A ν ( t ) is nonp ositive in X ν : ⟨ ρ, A ν ( t ) ρ ⟩ HS ≤ 0 , ρ ∈ X ν . (4.9) The generator can b e written as A ν ( t ) ρ ν = K ν ( t ) ρ ν + γ D ν ( t ) ρ ν , (4.10) where K ν ( t ) ρ ν = − i [ H ν ( t ) , ρ ν ] , D ν ( t ) ρ ν = [ V ν ρ ν , V † ν ] + [ V ν , ρ ν V † ν ] . (4.11) Here V ν = V ν ( t ) is obtained from the p olynomial V ( t ) by replacement of a and a † b y a ν and a † ν , resp ectiv ely , while V † ν = ( V ν ( t )) † b y ( 4.3 ). F or ρ ν ∈ D ν , we hav e ⟨ ρ ν , K ν ( t ) ρ ν ⟩ HS = − i tr ( ρ † ν [ H ν ( t ) , ρ ν ]) = − i tr ( ρ ν [ H ν ( t ) ρ ν − ρ ν H ( t )]) = 0 . (4.12) On the other hand, ⟨ ρ ν , D ν ( t ) ρ ν ⟩ HS ≤ 0 , ρ ν ∈ D ν (4.13) b y Lemma 3.1 applied to the op erator D = D ν ( t ). Therefore, ( 4.9 ) is prov ed. ad ii) The nonnegativit y preserv ation holds since, by H1–H2 , the generator A ν ( t ) with ev ery t ≥ 0 admits the structure of the CPTP theory [ 14 , 20 ]. Note that the theory addresses the case when H ν ( t ) and D ν ( t ) do not depend on t . Ho w ev er, the con tin uit y of the coefficients in ( 2.12 ) and ( 2.14 ), allo ws us to approximate the time-dependent coefficients b y pice-wise constant ones on the in terv als εn < t < ε ( n + 1), for which the nonnegativity preserv ation holds. Finally , the Gronw all estimate implies that the discrepancy con verges to zero as ε → 0. Remark 4.3. The zero quadratic form ( 4.12 ) means that the v ector field K ν ( t ) ρ ν on the righ t hand side of ( 4.10 ) is orthogonal to ρ ν in the space HS. Thus, this term in ( 4.10 ) corresp onds to rotations in HS. By ( 4.13 ), the second v ector field γ D ν ( t ) ρ ν in ( 4.10 ) is the generator of con tractions of HS whic h corresp ond to the quan tum sp ontaneous emission. 6 5 P assage to limit The following lemma implies Theorem 2.4 . Lemma 5.1. Ther e exists a subse quenc e ν ′ → ∞ , such that i) for al l ρ 0 ∈ HS , ρ ν ′ ( · ) → ρ ( · ) , (5.1) wher e the c onver genc e holds in C (0 , ∞ ; HS w ) with limiting functions ρ ( · ) ∈ L ∞ (0 , ∞ ; HS) ∩ C (0 , ∞ ; HS w ); ii) the limiting functions ar e gener alise d solutions to ( 1.2 ) with the initial c ondition ( 1.8 ); iii) the b ounds ( 1.9 ) hold; iv) the maps U ( t ) : ρ (0) 7→ ρ ( t ) ar e line ar in HS ; v) ρ ( t ) ≥ 0 for t ≥ 0 if ρ (0) ≥ 0 . Pr o of. Using ( 2.17 ), the system ( 4.4 ) can b e written as ˙ ρ ν ; n,s ; n ′ ,s ′ ( t ) = X | n − k | + | n ′ − k ′ | ≤ N k, k ′ ≤ ν, r, r ′ = s ± ˜ A k,r ; k ′ ,r ′ n,s ; n ′ ,s ′ ( t ) ρ ν ; k,r ; k ′ ,r ′ ( t ) , t ≥ 0 , n, n ′ ≤ ν, s, s ′ = s ± , (5.2) where by ( 4.3 ), either ˜ A k,r ; k ′ ,r ′ n,s ; n ′ ,s ′ ( t ) = A k,r ; k ′ ,r ′ n,s ; n ′ ,s ′ ( t ) or ˜ A k,r ; k ′ ,r ′ n,s ; n ′ ,s ′ ( t ) = 0 . (5.3) By ( 4.8 ) and ( 4.7 ), X n,s ; n ′ ,s ′ | ρ ν ; n,s ; n ′ ,s ′ ( t ) | 2 ≤ ∥ ρ 0 ∥ 2 HS , t ≥ 0 . (5.4) Hence, sup ν ≥ max( n,n ′ ) , t ≥ 0 | ρ ν ; n,s ; n ′ ,s ′ ( t ) | ≤ ∥ ρ 0 ∥ HS , ∀ n, s, n ′ , s ′ . (5.5) Therefore, ( 5.2 ) and ( 2.18 ) imply that for any T > 0, sup ν ≥ max( n, n ′ ) t ∈ [0 , T ] | ˙ ρ ν ; n,s ; n ′ ,s ′ ( t ) | ≤ C n,n ′ ( ∥ ρ 0 ∥ HS ) < ∞ , ∀ n, s, n ′ , s ′ . (5.6) No w let us prov e Lemma 5.1 for each fixed ρ 0 ∈ HS. First, by the Arzela–Ascoli theorem, for an y fixed n, s, n ′ , s ′ , the sequence { ρ ν ; n,s ; n ′ ,s ′ ( t ) : ν ≥ 0 } , is compact in C (0 , ∞ ). Hence, for a subsequence ν ′ → ∞ and all n, s, n ′ , s ′ , ρ ν ′ ; n,s ; n ′ ,s ′ ( t ) C (0 , ∞ ) − − − − → ρ n,s ; n ′ ,s ′ ( t ) , t ≥ 0 , ν ′ → ∞ . (5.7) By the F atou lemma, the conv ergence and the b ounds ( 5.4 ), ( 5.6 ) imply that the limiting entries ρ n,s ; n ′ ,s ′ ( t ) represent a tra jectory ρ ( · ) ∈ T = L ∞ (0 , ∞ ; HS) ∩ C (0 , ∞ ; HS w ). The b ounds ( 5.4 ) imply ( 1.9 ): ∥ U ( t ) ρ (0) ∥ HS ≤ ∥ ρ 0 ∥ HS , (5.8) where U ( t ) : ρ 0 7→ ρ ( t ). Second, ( 5.3 ) implies that the limiting entries ρ n,s ; n ′ ,s ′ ( t ) satisfy the system ( 2.16 ) in the sense of distributions since the n umber of summands in ( 2.17 ) is finite. At last, let us v erify the initial condition ( 1.8 ). It suffices to chec k that ρ n,s ; n ′ ,s ′ ( t ) → ρ 0 n,s ; n ′ ,s ′ , t → +0 , ∀ n, s, n ′ , s ′ . (5.9) 7 This con vergence holds since ρ 0 ν ; n,s ; n ′ ,s ′ → ρ 0 n,s ; n ′ ,s ′ b y ( 4.7 ), and b ounds ( 5.6 ) are uniform in ν ≥ max( n, n ′ ). Finally , ρ ( t ) ≥ 0 if ρ (0) ≥ 0 since ρ ν ( t ) ≥ 0 by Lemma 4.2 ii). It remains to sho w that the subsequence ν ′ in ( 5.7 ) can b e ch osen the same for all initial states ρ 0 ∈ HS, and the maps U ( t ) : HS → HS for all t ≥ 0 are linear and preserving the nonnegativity . First, the subsequence can b e chosen the same for all ρ 0 ∈ D ∞ := ∪ ν ≥ 0 D ν with rational matrix en tries. Then the limit m aps U ( t ) ρ 0 7→ ρ ( t ) with all t ≥ 0 are linear maps D ∞ → HS ov er the field of rational num b ers, and preserving the nonnegativity . Second, the linearit y and bounds ( 5.8 ) imply , that the maps U ( t ) : D ∞ → HS with all t ≥ 0 are globally Lipsc hitz in the metric of HS with the Lipschitz constan t one. Hence, by the density argumen ts, the conv ergence ( 5.7 ) holds for all ρ 0 ∈ HS. Therefore, all the maps U ( t ) are linear ov er C and preserving the nonnegativit y . 6 Conflict of in terest W e ha ve no conflict of interest. 7 Data av ailabilit y statemen t The manuscript has no asso ciated data. References [1] G.S. Agarw al, Op en quantum Marko vian systems and the microreversibilit y , Z. Physik 258 (1973), 409–422. [2] G.S. Agarw al, Master equation metho ds in quantum optics, pp 3–90 in: Progr. in Optics, vol. 11, North-Holland, Amsterdam, 1973. [3] G.S. Agarwal, Quantum statistical theories of sp ontaneous emission and their relation to other approac h, Springer T racts in Mo dern Ph ysics, vol. 70, Springer, Berlin, 1974. [4] R. Alic ki, K. Lendi, Quantum Dynamical Semigroups and Applications, Springer, Berlin, 2007. [5] P . Alsing, D.-S. Guo, H.J. Carmic hael, Dynamic Stark effect for the Jaynes-Cummings system, Phys. R ev. A 45 (1992), 5135–5143. [6] S.M. Barnett, P .M. Radmore, Metho ds in Theoretical Quan tum Optics, Oxford Universit y Press, Oxford, 1997. [7] S.M. Barnett, J. Jeffers, The damp ed Jaynes–Cummings mo del, Journal of Mo dern Optics 54 (2007), no. 13-15, 2033–2048. [8] I. A. Bo canegra-Garay , L. Hern´ andez-S´ anc hez, I. Ramos-Prieto, F. Soto-Eguibar, H. Mo y a- Cessa, Inv arian t approac h to the driven Jaynes-Cummings mo del SciPost Phys. 16 (2024), 007. [9] A.M. Cheb otarev, F. F agnola, Sufficien t conditions for conserv ativity of quantum dynamical semigroups, J. F unct. Anal. 153 (1998), 382. 994), 1993–1998. [10] E.B. Da vies, Quantum Theory of Op en Systems, AcademicPress, 1976. [11] E.B. Davies, Quantum dynamical semigroups and the neutron diffusion equation, R ep. Math. Phys. 11 (1977), 169. [12] S.M. Dutra, P .L. Knight, H. Mo y a-Cessa, Large-scale fluctuations in the driven Jaynes- Cummings mo del, Phys. R ev. A 49 (1 8 [13] F. F agnola, Quan tum Marko v semigroups and quan tum flo ws, Pr oye c ciones 18 (1999), 1–144. [14] V. Gorini, A. Kossako wski, E. C. G. Sudarshan, Completely p ositiv e dynamical semigroups of N-lev el systems, J. Math. Phys. 17 (1976), 821–825. [15] A.S. Holev o, Cov arian t quantum Marko vian evolutions, J. Math. Phys. 37 (1996), 1812. [16] E.T. Jaynes, F.W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the b eam maser, Pr o c. IEEE 51 (1963), no. 1, 89–109. [17] I.V. Jy otsna, G.S. Agarw al, The Ja ynes–Cummings mo del with contin uous external pumping, Optics Communic ations 99 (1993), 344–349. [18] A. Komec h, E. Kopylo v a, On dynamical semigroup for autonomous damp ed driven Jaynes– Cummings equations, submitted to J. Math. A nal. Appl. , 2025. [19] J. Larson, T. Mavrogordatos, The Jaynes–Cummings Mo del and Its Descendan ts. Mo dern researc h directions, IOP Publishing, Bristol, UK, 2021. [20] G. Lindblad, On the Generators of Quan tum Dynamical Semigroups, Comm. Math. Physics 48 (1976), 119–130. [21] M. Reed, B. Simon, Metho ds of Mo dern Mathematical Physics I: F unctional Analysis, Aca- demic Press, New Y ork, 1980. [22] H. Sp ohn, Kinetic equations from Hamiltonian dynamics: Marko vian limits, R ev. Mo d. Phys. 52 (1980), 569–615. [23] S. Sw ain, Z. Ficek, The damp ed and coherently-driv en Ja ynes–Cummings mo del, Journal of Optics B: Quantum and Semiclassic al Optics 4 (2002), no. 4, 328–336. [24] W. V ogel, D.-G. W elsc h, Quantum Optics, Wiley , New Y ork, 2006. 9