On dynamical semigroup for damped driven Jaynes-Cummings equations

On dynamical semigroup 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 addresses the damp ed driven Ja ynes–Cummings for quantised one-mo de Maxwell field coupled to a t w o-level molecule. W e consider a broad class of damping and pumping which are p olynomial in the creation and annihilation op erators. Our main result is the construction of a con traction dynamical semigroup in the Hilbert space of Hermitian Hilbert–Schmidt op erators in the case of a nonpositive dissipation op erator and time-indep enden t pumping. All tra jectories of the semigroup are generalised solutions to the Ja ynes–Cummings equations. As a key example, we pro ve nonpositivity for the basic dissipation op erator of Quantum Optics. MSC classific ation : 81V80, 81S05, 81S08 37K06, 37K40, 37K45, 78A40, 78A60. Keywor ds : Ja ynes–Cummings equations; dynamical semigroup; Hamiltonian op erator; density op- erator; pumping; dissipation op erator; trace; Hilb ert–Sc hmidt op erator; Lumer–Phillips theorem; 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 and of its adjoint 4 4 Semigroup and generalised solutions 5 5 Ac knowledgemen ts 6 6 Conflict of in terest 6 7 Data a v ailabilit y statemen t 6 1 In tro duction The Jaynes–Cummings equations are one of the basic mo dels of Quantum Optics, and it is used for description of v arious asp ects of laser action. The survey of the results on the mo del without damping and pumping can b e found in [ 7 , 10 , 19 ]. V arious v ersions of pumping are considered in [ 5 , 8 , 12 , 18 , 26 ]. Damping was in tro duced for the analysis of quan tum sp on taneous emission [ 1 ]–[ 3 ], [ 6 , 7 , 26 , 27 ]. W e construct global solutions for all initial v alues from the space of Hilb ert–Schmidt op erators in the case of time-indep endent 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 corresp onding annihilation and creation op erators a and a † : a | n ⟩ = √ n | n − 1 ⟩ , a † | n ⟩ = √ n + 1 | n + 1 ⟩ , [ a, a † ] = 1 . (1.1) W e will consider a damped-driven v ersion of the Jaynes–Cummings equations (QRM) ˙ ρ ( t ) = A ρ ( t ) := − i [ H , ρ ( t )] + γ D ρ ( t ) , t ≥ 0 , (1.2) 1 Supp orted partly by Austrian Science F und (FWF) P A T 3476224. where the densit y op erator ρ ( t ) of the coupled field-molecule system is a Hermitian op erator in X . The Hamiltonian H is the sum H = H 0 + pV , where H 0 := ω c a † a + 1 2 ω a σ 3 , V = ( a + a † ) ⊗ σ 1 + A e . (1.3) Here H 0 is the Hamiltonian of the free field and a molecule without in teraction, pV is the in teraction Hamiltonian, ω c > 0 is the ca vit y resonance frequency , ω a > 0 is the molecular frequency , and p ∈ R is proportional to the molecular dipole moment. The pumping is represented by a selfadjoint op erator A e , and σ 1 and σ 3 are the P auli matrices acting on the factor C 2 in F ⊗ C 2 , so [ a, σ k ] = [ a † , σ k ] = 0. Finally , in ( 1.2 ), γ > 0 and D 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 , 26 , 27 ], 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 [ 23 ] ⟨ ρ 1 , ρ 2 ⟩ HS = tr [ ρ 1 ρ 2 ] . (1.5) Definition 1.2. i) | n, s ± ⟩ = | n ⟩ ⊗ s ± form an orthonormal b asis in X , s ± ∈ C 2 and σ 3 s ± = ± s ± . ii) X ∞ is the sp ac e of 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 prov e the well-posedness for the QRM in the Hilb ert space HS in the case of time-indep endent pumping A e . The main issue is that the op erators a and a † are unbounded b y ( 1.1 ), so the generator A in QRM is also unbounded. Accordingly , the meaning of the QRM must b e adjusted (see Definition 2.3 ). The Hamilton op erator H is selfadjoint, so, in the case γ = 0, solutions are given by 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 γ > 0 the formula for solutions is missing. W e assume that • The pumping A e and the dissipation op erator D are p olynomials ( 2.14 ) in a and a † . • The dissipation op erator D and its adjoint D † are nonp ositiv e on the dense domain D : ⟨ ρ, D ρ ⟩ HS ≤ 0 , ⟨ ρ, D † ρ ⟩ HS ≤ 0 , ρ ∈ D . (1.8) F or example, A e = a † + a , A e = a † a , and D = D 1 fulfill the first assumption. As a k ey example, we pro v e in Theorem 2.2 that the dissipation op erator D 1 satisfies ( 1.8 ). Our main results are as follo ws. I. The generator A admits a closure A in HS from the domain D . This closure is a generator of a strongly con tin uous con traction semigroup in HS. I I. F or all ρ (0) ∈ HS, the tra jectories ρ ( t ) = e A t ρ (0) ∈ C (0 , ∞ ; HS) are generalised solutions to the QRM (Definition 2.3 ). Our strategy is as follo ws. W e rewrite the QRM as ˙ ρ ( t ) = A ρ ( t ) := K ρ ( t ) + γ D ρ ( t ) , where K ρ = − i [ H , ρ ] , ρ ∈ HS . (1.9) W e sho w that the op erator K is antisymmetric on D : ⟨ K ρ 1 , ρ 2 ⟩ HS = −⟨ ρ 1 , K ρ 2 ⟩ HS , ρ 1 , ρ 2 ∈ D , (1.10) 2 and that its quadratic form v anishes: ⟨ ρ, K ρ ⟩ HS = 0 , ρ ∈ D . (1.11) Then b oth op erators A and A † are nonp ositive on D by ( 1.8 ). Hence, b oth op erators A and A † are dissipativ e, so the existence of the semigroup follows from the Lumer–Phillips theorem [ 21 ]. The crucial role in the pro ofs is play ed b y the p olynomial structure ( 2.14 ). Remark 1.3. The zero quadratic form ( 1.11 ) means that the v ector field K ρ is orthogonal to ρ in the space HS. Thus, the first term on the righ t hand side of ( 1.9 ) can b e in terpreted as a rotations in HS. By ( 1.8 ), the second vector field γ D ρ is the generator of con tractions of HS whic h correspond to quan tum sp on taneous emission. Let us comment on previous results in the field. In the case of b ounded generators A , semigroups for equations of type ( 1.2 ) obviously exist. In this case, Lindblad [ 20 ] and Gorini, Kossak o wski, and Sudarshan [ 16 ] found necessary and sufficient conditions on A providing the p ositivity and trace preserv ation. F or unbounded generators, the existence of dynamical semigroup for Quan tum Dynamical Sys- tems (QDS) is not well-dev elop ed, [ 4 , p.110]. In [ 11 ], E.B. Davies considered quantum-mec hanical F okk er–Planck equations (QFP). The existence of the corresp onding p ositive con traction semigroup is established in the Banach space of self-adjoint trace-class op erators. The uniqueness and trace preserv ation hav e not b een pro ved. Sufficien t conditions, pro viding the trace preserv ation for QFP equations, were found in [ 9 ]. The detailed c haracterisation of a class of co v arian t QDS with un- b ounded generators is presen ted in [ 17 ]. The surv ey can b e found in [ 10 , 15 ]. The theory [ 16 , 20 ] implies that for a class of dissipation op erators, ( 1.2 ) can b e represented in the form of the QFP equation if A is a b ounded generator of a completely p ositive semigroup (see Theorem 4.2 of [ 10 , Chapter 9]). F or unbounded generators, suc h a representation holds under appropriate conditions [ 14 ]. W e construct strongly con tin uous contraction semigroups for equation ( 1.2 ) in the Hilb ert space of the Hilb ert–Schmidt op erators which, in particular, contains all trace class op erators. This framew ork allows us to emplo y the well-dev elop ed theory of con traction semigroups in the Hilb ert space by Lumer and Phillips and others. As a result, w e prov e that the semigroups exist under simple and easily verifiable conditions on the pumping and dissipation op erators. How ever, the questions of p ositivit y and trace preserv ation remain op en; the needed developmen t will b e addressed elsewhere. 2 Notations and main results A densit y op erator ρ ∈ HS is defined uniquely by its matrix entries ρ n,s ; n ′ ,s ′ = ⟨ n, s | ρ | n ′ , s ′ ⟩ , n, n ′ = 0 , 1 , . . . , s, s ′ = s ± . (2.12) The Hilb ert–Sc hmidt norm, corresp onding 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 < ∞ . (2.13) Note that by ( 1.1 ), the space X ∞ is in v arian t with resp ect to a and a † . Hence, the pro ducts of ρ ∈ D with an y p olynomials of a and a † are w ell-defined as op erators in X ∞ . Denote M 2 = C 2 ⊗ C 2 the space of 2 × 2-matrices. W e assume that the pumping and the dissipation op erator satisfy the follo wing conditions. H1. A e and D ρ are p olynomials in the creation and annihilation op erators: A e = P ( a, a † ) , Dρ = X j Q j ρR j , (2.14) where P , Q k , R k are p olynomials in a and a † with co efficien ts from M 2 . H2. The op erators A e and D ρ with ρ ∈ D are symmetric in X ∞ . H3. D and D † are nonp ositiv e in D (that is, ( 1.8 ) holds). 3 Remark 2.1. i) H1–H2 imply that A ρ ∈ D for ρ ∈ D . ii) H3 holds if Q j = R † j for all j . Our first result is as follo ws. Lemma 2.2. Al l c onditions H1–H3 hold for the dissip ation op er ator D = D 1 . T o form ulate other results, w e need to giv e a meaning to the QRM. The issue is that the op erator A is not w ell-defined on HS. Note that the equation admits the treatment via the matrix en tries ( 2.12 ) as the system ˙ ρ n,s ; n ′ ,s ′ ( t ) = [ A ρ ( t )] n,s ; n ′ ,s ′ , t ≥ 0 , n, n ′ ≥ 0 , s, s ′ = s ± , (2.15) since the righ t hand side is well-defined for all ρ ( t ) ∈ HS. Indeed, by H1 and ( 1.1 ), the generator A is w ell-defined on the domain D , and [ A ρ ] n,s ; n ′ ,s ′ = X | k − n | + | k ′ − n ′ | ≤ N r, r ′ = s ± A k,r ; k ′ ,r ′ n,s ; n ′ ,s ′ ρ k,r ; k ′ ,r ′ , n, n ′ ≥ 0 , s, s ′ = s ± , ρ ∈ D , (2.16) where N = max(2 , deg P , max j [deg Q j + deg R j ]). Finally , since the summation in ( 2.16 ) is finite, the matrix entries [ A ρ ] n,s ; n ′ ,s ′ admit a unique extension b y contin uit y from ρ ∈ D to all ρ ∈ HS. Hence, the op erator A admits a closed extension A from the domain D to D ( A ) ⊃ D . The structure ( 2.16 ) means that the matrix of the generator in the basis | n, s ⟩ is almost diagonal. Definition 2.3. We say that a tr aje ctory ρ ( t ) ∈ C (0 , ∞ ; HS) is a gener alise d solution to the QRM if it satisfies the system ( 2.15 ) in the sense of distributions; that is, ρ n,s ; n ′ ,s ′ ( t ) − ρ n,s ; n ′ ,s ′ (0) = Z t 0 [ A ( τ ) ρ ( τ )] n,s ; n ′ ,s ′ dτ , t ≥ 0 , ∀ n, n ′ , s, s ′ . (2.17) Our main result is the follo wing theorem. Theorem 2.4. L et c onditions H1–H3 hold. Then i) A is a gener ator of a str ongly c ontinuous c ontr action semigr oup U ( t ) = e A t in HS , and for ρ (0) ∈ D ( A ) , the tr aje ctories ρ ( t ) = U ( t ) ρ (0) ar e solutions to the e quation ˙ ρ ( t ) = A ρ ( t ) , t ≥ 0 , (2.18) ii) F or al l ρ (0) ∈ HS , the tr aje ctories ρ ( t ) = U ( t ) ρ (0) ∈ C (0 , ∞ ; HS) ar e gener alise d solutions to the QRM. 3 Nonp ositivit y of the dissipation op erator and of its adjoin t Here we prov e Lemma 2.2 . Conditions H1 and H2 obviously hold. It remains to chec k ( 1.8 ). First, let us calculate the adjoin t op erator D † 1 : for ρ 1 , ρ 2 ∈ D , tr [ ρ 1 ( D 1 ρ 2 )] = tr  ρ 1 ( aρ 2 a † − 1 2 a † aρ 2 − 1 2 ρ 2 a † a )  = tr  ρ 1 aρ 2 a † − 1 2 ρ 1 a † aρ 2 − 1 2 ρ 1 ρ 2 a † a  = tr  a † ρ 1 aρ 2 − 1 2 ρ 1 a † aρ 2 − 1 2 a † aρ 1 ρ 2  = tr  ( a † ρ 1 a − 1 2 ρ 1 a † a − 1 2 a † aρ 1 ) ρ 2  = tr [( D † 1 ρ 1 ) ρ 2 ] . Hence, the adjoin t op erator D † 1 differs from D 1 b y sw apping a and a † : D † 1 ρ = a † ρa − 1 2 ρa † a − 1 2 a † aρ, ρ ∈ D . (3.1) 4 Second, let us pro v e the nonp ositivity ( 1.8 ) for D = D 1 . F or ρ ∈ D , ⟨ ρ, D 1 ρ ⟩ HS = tr  ρD 1 ρ  = tr  ρ  aρa † − 1 2 a † aρ − 1 2 ρa † a   = tr  ρaρa † − ρa † aρ  = tr  ρaρa † − a † aρ 2  . (3.2) No w w e use the fact that ρ is a finite rank Hermitian operator ( 1.6 ). Then ( 2.14 ) implies that the op erators ρaρa † and a † aρ 2 ha v e only finite num b er of nonzero entries ( 2.12 ), so their traces are w ell-defined. Moreo v er, ρ admits a finite sp ectral resolution in the orthonormal basis of its eigen v ectors e i ∈ X ∞ : ρ = ν X i =1 ρ i e i ⊗ e i . (3.3) In this basis, the entries ρ ij = ρ i δ ij , and the en tries a j k = ⟨ e j , ae k ⟩ and a † kl = ⟨ e k , a † e l ⟩ of the op erators a and a † are w ell-defined. Hence, ( 3.2 ) implies, with summation in rep eated indices, ⟨ ρ, D 1 ρ ⟩ HS = ρ i δ ij a j k ρ k δ kl a † li − a † kl a lj ρ 2 j δ j k = ρ i a ik ρ k a † ki − a † kl a lk ρ 2 k = ρ i a ik ρ k a † ki − a † ki a ik ρ 2 k = a ik a † ki ( ρ i ρ k − ρ 2 k ) = 1 2  a ik a † ki ( ρ i ρ k − ρ 2 k ) + a ki a † ik ( ρ k ρ i − ρ 2 i )) = − 1 2 | a ik | 2 ( ρ i − ρ k ) 2 ≤ 0 (3.4) since a † ik = a ki . Hence, the nonp ositivity is pro ved for D 1 . F or D † 1 the pro of is the same with the sw apping a and a † . Remark 3.1. The pro of of the nonp ositivity essen tially dep ends on the symmetry of ρ . 4 Semigroup and generalised solutions Lemma 4.1. Both op er ators A and A † ar e wel l-define d and nonp ositive on the domain D . Pr o of. i) F or ρ 1 , ρ 2 ∈ D , w e ha v e in the notation ( 1.9 ): ⟨ ρ 1 , K ρ 2 ⟩ HS = − i tr ( ρ 1 [ H , ρ 2 ]) = − i tr ( ρ 1 ( H ρ 2 − ρ 2 H )) = − i tr ( ρ 1 H ρ 2 − ρ 2 H ρ 1 ) . (4.1) where all the terms are w ell-defined b y H1 . Hence, ( 1.11 ) holds. Therefore, A = K + γ D is nonp ositiv e b y H3 . Similarly , w e obtain ( 1.10 ): ⟨ K ρ 1 , ρ 2 ⟩ HS = i tr ([ H , ρ 1 ] † ρ 2 ) = − i tr (( H ρ 1 − ρ 1 H ) ρ 2 ) = − i tr ( ρ 1 ρ 2 H − ρ 1 H ρ 2 ) = i tr ( ρ 1 [ H , ρ 2 ]) = − tr ( ρ 1 ( K ρ 2 )) = −⟨ ρ 1 , K ρ 2 ⟩ HS . (4.2) Hence A † | D = − K + γ D † , so A † is also w ell-defined on D and nonp ositive by H3 . Pro of of Theorem 2.4 . i) Lemma 4.1 together with Proposition [ 13 , I I.3.23] imply that b oth op erators A and A † are dissipativ e. Moreov er, the op erator A is densely defined and admits a closed extension A b y ( 2.16 ). Now Theorem 2.4 i) follows from [ 13 , Corollary II.3.17] which is a corollary of the Lumer–Phillips theorem [ 21 , I I.3.15]. ii) The con traction means that for ρ (0) ∈ HS, w e ha v e: ∥ ρ ( t ) ∥ HS ≤ ∥ ρ (0) ∥ HS , t ≥ 0 . (4.3) T o pro ve the iden tit y ( 2.17 ), let us tak e ρ k (0) ∈ D ( A ) and ρ k (0) HS − − − → ρ (0). Then also ρ k ( t ) ∈ D ( A ) and ρ k ( t ) HS − − − → ρ ( t ) uniformly for t ≥ 0 by ( 4.3 ). The equation ( 2.18 ) for ρ k ( t ), implies the corresp onding equations ( 2.15 ) since the formulas ( 2.15 ) hold for ρ ∈ D ( A ). Hence, ( 2.17 ) for ρ ( t ) follo ws from similar integral identities for ρ k ( t ) as k → ∞ since the summations are finite for every n, s, n ′ , s ′ . 5 5 Ac kno wledgements The authors thank S. Kuksin, M.I. P etelin, A. Shnirelman, and H. Sp ohn for long-term fruitful discussions, and the Institute of Mathematics of BOKU Univ ersit y for the supp ort and hospitalit y . The researc h is supp orted by Austrian Science F und (FWF) P A T 3476224. 6 Conflict of in terest W e ha ve no conflict of interest. 7 Data av ailabilit y statemen t The man uscript has no asso ciated data. References [1] G.S. Agarwal, 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 quan tum optics, pp 3–90 in: Progr. in Optics, v ol. 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, v ol. 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. Carmichael, 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 Quantum 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 ya- Cessa, Inv arian t approac h to the driven Ja ynes-Cummings mo del SciPost Phys. 16 (2024), 007. [9] A.M. Cheb otarev, F. F agnola, Sufficient conditions for conserv ativity of quantum dynamical semigroups, J. F unct. A nal. 153 (1998), 382. [10] E.B. Da vies, Quan tum 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. Knigh t, H. Moy a-Cessa, Large-scale fluctuations in the driven Ja ynes- Cummings mo del, Phys. R ev. A 49 (1994), 1993–1998. [13] K.J. Engel, R. Nagel, One-parameter Semigroups for Linear Ev olution Equations, Springer, New Y ork, 2000. [14] D.E. Ev ans, J.T. Lewis, Dilations of Dynamical Semi-Groups, Commun. math. Phys 50 (1976), 219–227. [15] F. F agnola, Quantum Mark ov semigroups and quantum flo ws, Pr oye c ciones 18 (1999), 1–144. 6 [16] V. Gorini, A. Kossako wski, E. C. G. Sudarshan, Completely p ositive dynamical semigroups of N-lev el systems, J. Math. Phys. 17 (1976), 821–825. [17] A.S. Holev o, Co v ariant quan tum Marko vian evolutions, J. Math. Phys. 37 (1996), 1812. [18] I.V. Jy otsna, G.S. Agarwal, The Jaynes–Cummings mo del with con tin uous external pumping, Optics Communic ations 99 (1993), 344–349. [19] J. Larson, T. Mavrogordatos, The Jaynes–Cummings mo del and its descendants. Mo dern re- searc h directions, IOP Publishing, Bristol, UK, 2021. [20] G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Physics 48 (1976), 119–130. [21] G. Lumer, R.S. Phillips, Dissipative operators in a Banac h space, Pacific J. Math. 11 (1961), 679–698. [22] H. Nussenzv eig, In tro duction to Quan tum Optics, Gordon and Breach, London, 1973. [23] M. Reed, B. Simon, Methods of Mo dern Mathematical Physics I: F unctional Analysis, Aca- demic Press, New Y ork, 1980. [24] M. Reed, B. Simon, Metho ds of Mo dern Mathematical Physics I I: F ourier Analysis, Self- Adjoin tness, Academic Press, New Y ork, 1978. [25] H. Sp ohn, Kinetic equations from Hamiltonian dynamics: Marko vian limits, R ev. Mo d. Phys. 52 (1980), 569–615. [26] S. Swain, Z. Ficek, The damp ed and coherently-driv en Jaynes–Cummings mo del, Journal of Optics B: Quantum and Semiclassic al Optics 4 (2002), no. 4, 328–336. [27] W. V ogel, D.-G. W elsch, Quan tum Optics, Wiley , New Y ork, 2006. 7