On single-frequency asymptotics for the Maxwell-Bloch equations: pure states

On single-frequenc y asymptotics for the Maxwell–Bloch equations: pure states A.I. K omech 1 and E.A. K opylo v a 1 Institute of Mathematics of BOKU University , Gr e gor Mendel Strasse 33, A-1180, V ienna, A ustria alexander .komech@boku.ac.at, elena.k opylov a@boku.ac.at Abstract W e consider damped driv en Maxwell–Bloch equations for a single-mode Maxwell field coupled to a two-le vel molecule. The equations are used for semiclassical description of the laser action. Our main result is the construction of solutions with single-frequency asymptotics of the Maxwell field in the case of quasiperiodic pumping. The asymptotics hold for solutions with harmonic initial values which are stationary states of a v eraged reduced equations in the interaction picture. W e calculate all harmonic states and analyse their stability . Our calculations rely on the Hopf reduc- tion by the gauge symmetry group U ( 1 ) . The asymptotics follo w by an extension of the averaging theory of Bogolyubo v–Eckhaus–Sanchez-Palencia onto dynamical systems on manifolds. The key role in the application of the av eraging theory is played by a special a priori estimate. MSC classification : 37J40, 58D19, 37J06, 70H33, 34C25, 34C29, 78A40, 78A60, K e ywor ds : Maxwell–Bloch equations; Hamiltonian structure; symmetry gauge group; Hopf fibration; stere- ografic projection; av eraging theory; adiabatic asymptotics; single-frequency asymptotics; asymptotic sta- bility; quantum optics; laser . Contents 1 Introduction 2 2 U ( 1 ) -symmetry and the Hopf fibration 5 3 Resolution of singularity and the reduced dynamics 6 4 A veraging of dynamics in the interaction pictur e 8 5 Stationary states f or av eraged dynamics (harmonic states) 10 5.1 Zero population in v ersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.2 Nonzero population in v ersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.3 Fully in v erted states are not harmonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6 Stability of harmonic states 12 6.1 Zero population in v ersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.2 Nonzero population in v ersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7 Single-frequency asymptotics 13 7.1 KBM vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7.2 Single-frequency asymptotics in the interaction picture . . . . . . . . . . . . . . . . . . . . 14 8 Conflict of interest 15 9 Data a vailability statement 15 A The a priori bounds and well-posedness 15 1 Supported by Austrian Science Fund (FWF) 10.55776/P A T 3476224 B On averaging theory on manif olds 16 C The Maxwell–Bloch equations f or pure states 17 D On possible treatment of the laser action 17 1 Introduction The Maxwell–Bloch equations (MBE) were introduced by Lamb [ 20 ] for semiclassical description of the laser action [ 1 , 6 , 13 , 27 , 28 , 29 , 30 ]. The equations are obtained as the Galerkin approximation of the Maxwell–Schr ¨ odinger system [ 5 , 12 , 15 , 17 , 18 , 19 , 22 , 25 ]. Our main goal is construction of solutions with single-frequency asymptotics of the Maxwell amplitude. The asymptotics provisionally correspond to the laser coherent radiation, which remains a key mystery of laser action since its discov ery around 1960. The damped dri ven MBE for pure states read    ˙ A ( t ) = B ( t ) , ˙ B ( t ) = − Ω 2 A ( t ) − γ B ( t ) + c j ( t ) i ¯ h ˙ C 1 ( t ) = ¯ h ω 1 C 1 ( t ) + ia ( t ) C 2 ( t ) , i ¯ h ˙ C 2 ( t ) = ¯ h ω 2 C 2 ( t ) − ia ( t ) C 1 ( t )       , t ≥ 0 . (1.1) Here A ( t ) , B ( t ) ∈ R , while C 1 ( t ) , C 2 ( t ) ∈ C ; Ω > 0 is the resonance frequency , γ > 0 is the dissipation coef ficient, c is the speed of light, ¯ h is the Planck constant, and ¯ h ω 2 > ¯ h ω 1 are the energy levels of active molecules. The current j ( t ) and the function a ( t ) are gi ven by j ( t ) = 2 κ Im [ C 1 ( t ) C 2 ( t )] , a ( t ) : = κ c [ A ( t ) + A e ( t )] , κ = p ω , ω = ω 2 − ω 1 > 0 , (1.2) where p ∈ R is proportional to the molecular dipole moment. W e suppose that the external field A e ( t ) (the pumping) is a quasiperiodic function: A e ( t ) = Re [ A e e − i Ω t + N ∑ 1 A e k e − i Ω k t ] , where A e , A e k ∈ C , Ω k ∈ R and Ω k  = Ω . (1.3) W e use the Heaviside–Lorentz units, and in Appendix C we comment on the introduction of the equations. For the MBE, the char ge conserv ation holds | C 1 ( t ) | 2 + | C 2 ( t ) | 2 = const. W e will consider solutions with const = 1: | C 1 ( t ) | 2 + | C 2 ( t ) | 2 = 1 , t ≥ 0 . (1.4) Accordingly , the phase space of the system is X = R 2 × S 3 , where S 3 is the unit sphere in C 2 . For any r > 0, solutions X ( t ) = ( A ( t ) , B ( t ) , C 1 ( t ) , C 2 ( t )) to ( 1.1 ) with | p | / γ = r satisfy the following a priori bounds: | X ( t ) | ≤ D r ( | X ( 0 ) | ) , t ≥ 0 , (1.5) which is prov ed in Appendix A . The bound implies the well-posedness of the MBE in the phase space X . Our main goal is asymptotics of the Maxwell amplitudes A ( t ) and B ( t ) for solutions X ( t ) to the MBE as p , γ → 0. Define complex Maxwell amplitudes by M ( t ) = A ( t ) + iB ( t ) / Ω . The first two equations of ( 1.1 ) are equi valent to ˙ M ( t ) = − i Ω M ( t ) − i γ M 2 ( t ) + 2 ic p ω Im [ C 1 ( t ) C 2 ( t )] / Ω , where M 2 ( t ) = Im M ( t ) . (1.6) Note that the parameters | p | , γ are very small for many types of lasers, see Appendix D . For p = γ = 0, all solutions to ( 1.6 ) are single-frequency: M ( t ) = e − i Ω t M ( 0 ) . For small | p | and γ , equation ( 1.6 ) implies that M ( t ) = e − i Ω t M ( 0 ) + Z t 0 R ( s ) d s , where sup s ≥ 0 | R ( s ) | = O ( | p | + γ ) by ( 1.5 ). Hence, M ( t ) = e − i Ω t M ( 0 ) + O ( | p | t ) for p / γ = r with arbitrary fixed r  = 0. In particular , for solutions with any fixed initial states M ( 0 ) , max t ∈ [ 0 , | p | − 1 / 2 ] | M ( t ) − e − i Ω t M ( 0 ) | = O ( | p | 1 / 2 ) , p → 0 , p / γ = r . (1.7) 2 Our main results show that in the resonance case, when Ω = ω , the time scale | p | − 1 / 2 in this asymptotics can be extended to | p | − 1 and e ven more in the case of special harmonic states of the dynamics ( 1.1 ). The harmonic states are defined via the reduction of dynamics ( 1.1 ) by the symmetry gauge group U ( 1 ) with the action g ( θ )( A , B , C 1 , C 2 ) = ( A , B , e i θ C 1 , e i θ C 2 ) , θ ∈ [ 0 , 2 π ] . (1.8) The action turns the sphere S 3 , corresponding to ( 1.4 ), into the Hopf fibration with the base S 2 . W e introduce suitable “global” coordinate Q ∈ C in the base, using the stereographic projection S 2 → C : Q = Z 1 2 − Z 3 , Z 3 = ± r 1 4 − | Z | 2 , Z = C 1 C 2 , (1.9) where | Z | ≤ 1 / 2 by ( 1.4 ), and Z = | C 1 C 2 | e i δ with δ = θ 2 − θ 1 , where θ k = arg C k . The base S 2 is identified with the graph of the two-valued function Q ( Z ) on the disk D : = { Z ∈ C : | Z | ≤ 1 / 2 } . The coordinates Z and Q are in variant with respect to the action ( 1.8 ). The population inv ersion I : = | C 2 | 2 − | C 1 | 2 admits the representation I = | Q | 2 − 1 | Q | 2 + 1 , Q ∈ C . (1.10) Thus, the reduction map Π transforms the system ( 1.1 ) with the phase space X = R 2 × S 3 , into the system of ODEs with the factor-phase space Y = Π X = R 2 × S 2 . Accordingly , the reduced dynamics becomes the system of ODEs for the complex Maxwell amplitude M ( t ) ∈ C and the complex variable Q ( t ) ∈ C which represents the amplitudes ( C 1 ( t ) , C 2 ( t )) ∈ S 3 . The reduced system can be written as ˙ Y ( t ) = F ( Y ( t ) , t ) , t ≥ 0; Y ( t ) = ( M ( t ) , Q ( t )) . (1.11) It is crucially important that the Maxwell amplitudes M ( t ) are not changed by the reduction, so their proper - ties for the Maxwell–Bloch dynamics ( 1.1 ) and for the reduced dynamics ( 1.11 ) are identical. W e consider the interaction picture of the reduced dynamics ( 1.11 ) and the corresponding averag ed equations with the structure ˙ Y ( t ) = p F r ( Y ( t )) , t ≥ 0 , r = p / γ . (1.12) W e define the harmonic states Y ∈ Y of the reduced dynamics ( 1.11 ) as stationary solutions of ( 1.12 ). W e call X ∈ X as harmonic states of the MBE if Y = Π X is the harmonic state of ( 1.11 ). W e show that the harmonic states depend only on the quotient p / γ . Our main results are asymptotics for solutions X ( t ) to equations ( 1.1 ) with a fix ed quotient p / γ = r . Everywhere belo w we consider the case p > 0 only since the extension to p < 0 is obvious. Theorem 1.1. Let Ω = ω and the pumping be quasiperiodic. Then for any r > 0 , the following asymptotics hold. i) Let X ( 0 ) = X r be a harmonic state of ( 1.1 ), and Y r = ( M r , Q r ) : = Π X r be the corresponding stationary solution to ( 1.12 ). Then for solutions ( M ( t ) , Q ( t )) = Π X ( t ) to the r educed system ( 1.11 ), the adiabatic asymptotics holds: max t ∈ [ 0 , p − 1 ] h | M ( t ) − e − i Ω t M r | + | Q ( t ) − e − i Ω t Q r | i = O ( p 1 / 2 ) , p → 0 . (1.13) ii) Let, additionally , Y r = ( M r , Q r ) be an asymptotically stable stationary solution to ( 1.12 ), and D ⊂ C 2 be a bounded domain of attraction. Then for initial states X ( 0 ) with Π X ( 0 ) ∈ D, max t ∈ [ 0 , p − 1 ] h | M ( t ) − e − i Ω t M ( t ) | + | Q ( t ) − e − i Ω t Q ( t ) | i = O ( p 1 / 2 ) , p → 0 , (1.14) wher e M ( t ) and Q ( t ) satisfy ( M ( t ) , A ( t )) → ( M r , Q r ) , t → ∞ ; sup t ≥ 0 [ | ˙ M ( t ) | + | ˙ Q ( t ) | ] = O ( p ) , p → 0 . (1.15) 3 iii) Let, additionally , Y r = ( M r , Q r ) be a linearly stable stationary solution to ( 1.12 ). Then for initial states X ( 0 ) = X r with Π X r = Y r , the uniform in time asymptotics holds, sup t ∈ [ 0 , ∞ ) h | M ( t ) − e − i Ω t M r | + | Q ( t ) − e − i Ω t Q r | i = O ( p ) , p → 0 . (1.16) Mor eover , for initial states X ( 0 ) suf ficiently close to X r , the following asymptotics with attraction holds: for sufficiently small ε > 0 , and some µ > 0 , | M ( t ) − e − i Ω t M r | + | Q ( t ) − e − i Ω t Q r | ≤ C [ p + d 0 e − p µ t ] , t ≥ 0 , p ≤ ε , (1.17) wher e d 0 = | X ( 0 ) − X r | . Remark 1.2. By ( 1.10 ), the asymptotics of type ( 1.13 )–( 1.17 ) hold also for I ( t ) − I r , where I r = ( Q r | 2 − 1 ) / ( | Q r | 2 + 1 ) . Remark 1.3. W ithout the constraint p / γ = r , the a priori bounds ( 1.5 ) and the asymptotics ( 1.13 )–( 1.17 ) do not hold. For example, the bounds must be dif ferent for small and big | p | for a given dissipation γ > 0. The asymptotics do not hold without this restriction since the limiting amplitudes M r depend on r . Let us comment on our approach. T o prov e the asymptotics ( 1.13 ), we sho w that the reduced system ( 1.11 ) admits solutions M ( t ) = e − i Ω t M r ( t ) , Q ( t ) = e − i ω t Q r ( t ) with slowly varying en veloping amplitudes M r ( t ) and Q r ( t ) for small p and γ with p / γ = r . The amplitudes are solutions to the corresponding dynamical system which is the interaction picture (or “rotating frame representation”) of ( 1.1 ). The slow variation of the amplitudes is equi v alent to the fact that the initial state ( M ( 0 ) , Q ( 0 )) = ( M ( 0 ) , Q ( 0 )) is a harmonic state, i.e., a stationary state of ( 1.12 ). W e calculate all harmonic states Y r = ( M r , Q r ) ∈ Y which are stationary solutions to ( 1.12 ). W e show that the states with M r  = 0 exist only in the resonance case Ω = ω and only if A e  = 0. In particular , for single-frequency pumping A e ( t ) = Re [ A p e − i ω p t ] , the asymptotics ( 1.13 )–( 1.17 ) with M r  = 0 hold only in the case A p  = 0 and triple resonance Ω = ω = ω p . (1.18) W e linearise the av eraged dynamics ( 1.12 ) at the harmonic states and calculate the spectrum of the lin- earisations. In particular , we show that in the case cr > | A e | , there exist linearly stable harmonic states Y r = ( M r , Q r ) of ( 1.12 ). For the proof of the asymptotics ( 1.13 )–( 1.17 ), we extend the av eraging theory of Bogolyubov type [ 26 , Theorem 4.3.6] and the a veraging theory of Eckhaus and Sanchez-Palencia [ 26 , Theorem 5.5.1] onto dynamical systems on manifolds. The extension is done in Appendix B . The key role in the application of the av eraging theory is played by the special a priori estimate ( 1.5 ). Remark 1.4. i) The map Π : X → Y reduces one degree of freedom in the MBE. Howe ver , the role of the reduction is more significant. Namely , the single-frequency asymptotics of type ( 1.13 )–( 1.17 ) apparently do not hold for C 1 ( t ) and C 2 ( t ) . This conjecture is suggested by the Rabi perturbati ve solution [ 28 , (5.2.14)– (5.2.16)] since it can contain incommensurable frequencies. ii) Our approach relies on the averaging theory which neglects oscillating terms. So, it gi ves a justification of the “rotating wav e approximation”, which is widely used in Quantum Optics [ 1 , 13 , 27 , 28 , 29 , 30 ]. The asymptotics ( 1.13 )–( 1.17 ) specify the time scale and the error of such approximations. Let us comment on related results. The problem of existence of time-periodic solutions to the MBE has been discussed since 1960s. The first results in this direction were obtained recently in [ 10 ] and [ 31 ] for v arious versions of the MBE. In [ 31 ], the N-th order time-periodic solutions were constructed by perturbation techniques. For the phenomenological model [ 2 , 3 ], time-periodic solutions were constructed in [ 10 ] in the absence of time-periodic pumping for small interaction constants. The solutions are obtained as the result of a bifurcation relying on homotopy in variance of the degree [ 8 ] and developing the av eraging arguments [ 9 ]. The period is determined by the bifurcation. 4 In [ 16 ], we ha ve established the existence of solutions with T -periodic Maxwell amplitude for any T - periodic pumping without smallness conditions. Up to our kno wledge, the single-frequency asymptotics for the MBE were not constructed till no w . Let us comment on our e xposition. In Sections 2 and 3 we construct an appropriate model of the reduced dynamics. The averaged equations in the interaction picture are calculated in Section 4 , and all stationary states of these equations are calculated in Section 5 . In Section 6 we analyse the stability of the stationary states. In Section 7 we prove the single-frequency asymptotics ( 1.13 )–( 1.17 ). In Appendix A we establish the bound ( 1.5 ). W e comment on the introduction of the MBE in Appendix C . In Appendix D we discuss possible treatment of the laser threshold and laser amplification relying on our results. Acknowledgements. The authors thank S. K uksin, M.I. Petelin, A. Shnirelman and H. Spohn for longterm fruitful discussions, the revie wer of our paper for useful remarks, and the Institute of Mathematics of BOKU Uni versity for the support and hospitality . The research is supported by Austrian Science Fund (FWF) 10.55776/P A T3476224. 2 U ( 1 ) -symmetry and the Hopf fibration Recall that the phase space of the MBE is X = R 2 × S 3 due to the charge conservation ( 1.4 ). The system is U ( 1 ) -in v ariant with respect to the action ( 1.8 ). This means that the function g ( θ ) X ( t ) is a solution if X ( t ) is. This is ob vious from ( 1.1 ) and ( 1.2 ). Let us denote the factor space Y = X / U ( 1 ) which is the space of all orbits X ∗ = { g ( θ ) X : θ ∈ [ 0 , 2 π ] } with X ∈ X . Thus, the reduction map Π : X → Y is defined by Π : X = ( A , B , C ) 7→ X ∗ = ( A , B , C ∗ ) ∈ Y , X ∈ X , (2.1) where C ∗ = P C , and the map P : C 7→ C ∗ is the Hopf fibration S 3 → S 2 . So, Y = R 2 × S 2 . The MBE induce the corresponding reduced dynamics in the factorspace Y which can be written as ( 1.11 ). Lemma 2.1. The reduced dynamics ( 1.11 ) admits a unique global solution Y ( t ) = ( A ( t ) , B ( t ) , C ∗ ( t )) for every initial state Y ( 0 ) ∈ Y . Pr oof. The uniqueness follows from the smoothness and continuity of F ( · , t ) . T o prove the existence, take any point X 0 ∈ Π − 1 Y ( 0 ) and set Y ( t ) = Π X ( t ) , where X ( t ) is the solution to the MBE with the initial state X 0 . Then Y ( t ) is the solution to ( 1.11 ) by definition of the reduced dynamics, and the initial state Y ( 0 ) = Π X ( 0 ) = Y 0 . The model of the Hopf projection. The Hopf projection P can be represented by the map P : C = ( C 1 , C 2 ) 7→ Z = C 1 C 2 (2.2) since it is constant on the fibers { e − i θ ( C 1 , C 2 ) : θ ∈ [ 0 , 2 π ] } . The orbit populations | C 1 | 2 and | C 2 | 2 can be expressed in Z . Indeed, | C 1 | 2 + | C 2 | 2 = 1 and | Z | = | C 1 || C 2 | . Hence, the populations and the population in v ersion are gi ven by | C 1 | 2 = 1 2 ∓ r 1 4 − | Z | 2 , | C 2 | 2 = 1 2 ± r 1 4 − | Z | 2 , I : = | C 2 | 2 − | C 1 | 2 = ± q 1 − 4 | Z | 2 . (2.3) All functions Z , | C 1 | , | C 2 | and the population inv ersion I = | C 2 | 2 − | C 1 | 2 are constant on the fibers, so they are functions on Y . Hence, the function ( 2.2 ) defines the map S 2 → D onto the disk D : = { Z ∈ C : | Z | ≤ 1 2 } . So, Z is a local coordinate on the base S 2 outside the points with | Z | = 1 2 . On the other hand, the formulas ( 2.3 ) show that the map is not a homeomorphism S 2 → D since the in verse map D → S 2 is two-valued. In particular , Z = 0 for the Hopf projection of ground state ( 1 , 0 ) and for the projection of the excited state ( 0 , 1 ) . Let us introduce an appropriate model S 2 of the base S 2 with a two-folded projection S 2 → D . Denote by B the ball {| Z | ≤ 1 / 2 } ⊂ R 3 . W e identify the base S 2 with the sphere S 2 = { Z ∈ R 3 : | Z | = 1 2 } and D with the equatorial section B ∩ { Z 3 = 0 } . Denote by S 2 ± the hemispheres S 2 ± = { Z ∈ S 2 : Z 3 ≷ 0 } , (2.4) 5 so the orthogonal projections of S 2 ± onto the disc D coincide. Let us identify Z = Z 1 + i Z 2 , see Fig. 1 . T o fix the model, we identify the North Pole ( 0 , 0 , 1 / 2 ) with P ( 0 , 1 ) and the South Pole ( 0 , 0 , − 1 / 2 ) with P ( 1 , 0 ) . Equiv alently , S 2 ± = { Z ∈ S 2 : I ≷ 0 } . (2.5) Remark 2.2. In the ne xt section, we will construct a “global coordinate” on the base S 2 except one point. The r educed dynamics. Let us construct suitable representation for the reduced dynamics ( 1.11 ) in the coordinate Z . Dif ferentiating Z ( t ) = P C ( t ) , we obtain from the MBE that ˙ Z ( t ) = ˙ C 1 C 2 + C 1 ˙ C 2 = ( − i ω 1 C 1 + a ¯ h C 2 ) C 2 + C 1 ( − i ω 2 C 2 − a ¯ h C 1 ) = − i ω Z + a ¯ h ( | C 2 | 2 − | C 1 | 2 ) . (2.6) Using ( 1.2 ) and ( 2.3 ), we re write equation ( 2.6 ) as ˙ Z ( t ) = − i ω Z ± b [ A ( t ) + A e ( t )] q 1 − 4 | Z | 2 , b = κ c ¯ h . (2.7) Remark 2.3. i) The right hand side of the resulting equation ( 2.7 ) is function of Z because of U ( 1 ) -in variance of the system ( 1.1 ). ii) By ( 2.5 ) and ( 2.4 ), we must take the upper sign in ( 2.3 ) and ( 2.7 ) in the case P C ∈ S 2 + and the lower sign in the case P C ∈ S 2 − . 3 Resolution of singularity and the reduced dynamics Note that the v ector field F ( · , t ) in ( 1.11 ) is smooth on Y . On the other hand, the coef ficients of the equation ( 2.7 ) are singular at | Z | = 1 2 . The singularity is due to the fact that the map ( 2.2 ) is not transversal at the points ( C 1 , C 2 ) ∈ S 3 with | C 1 | 2 = | C 2 | 2 = 1 2 . So, we must construct a transv ersal modification of the map. W e will do it using the stereographic projection in the model S 2 constructed abov e, see ( 2.4 ). The stereographic projection. The equation ( 2.7 ) holds with the corresponding sign on S 2 ± . Let us rewrite the equation in the stereographic projection S from the North Pole ( 0 , 0 , 1 / 2 ) of the sphere S 2 which is the Hopf projection of the excited state ( 0 , 1 ) . W e identify the projection of a point Z ∈ S 2 with the complex number Q = Z 1 2 − Z 3 , Z 3 = ± r 1 4 − | Z | 2 for ± Z 3 > 0; Z = Z 1 + i Z 2 ∈ D . (3.1) Note that Z → Q is the smooth map ˙ S 2 → C , where ˙ S = S 2 \ ( 0 , 0 , 1 / 2 ) . Unlik e Z , the coordinate Q dif ferentiates the hemispheres S 2 ± , see Fig. 1 : by ( 2.4 ), S 2 ± = { Z ∈ S 2 : | Q | ≷ 1 } . (3.2) Hence, by ( 2.5 ), I ≷ 0 ⇐ ⇒ | Q | ≷ 1 . (3.3) T ransf ormation of the MBE. W e will express the equation ( 2.7 ) in the variable Q taking into account dif ferent signs in ( 3.1 ). W e will see that the resulting expressions for both signs are identical. For any complex numbers z 1 , z 2 ∈ C we will denote z 1 · z 2 = Re [ z 1 z 2 ] , z 1 ∧ z 2 = Im [ z 1 z 2 ] (3.4) the inner and vector products of the corresponding real 2D vectors. Dif ferentiating Q ( t ) = Q ( Z ( t )) , we get from ( 3.1 ), ˙ Q ( t ) = ˙ Z 1 2 − Z 3 + Z ( 1 2 − Z 3 ) 2 ˙ Z 3 = ˙ Z 1 2 − Z 3 ∓ Z ( 1 2 − Z 3 ) 2 Z · ˙ Z q 1 4 − | Z | 2 = ˙ Z 1 2 − Z 3 − Q Q · ˙ Z Z 3 . (3.5) 6 Q Figure 1: Stereografic projections Therefore, ( 2.7 ) can be written as ˙ Z = − i ω ( 1 2 − Z 3 ) Q + 2 b [ A ( t ) + A e ( t )] Z 3 . (3.6) Substituting this formula into the right hand side of ( 3.5 ), we obtain ˙ Q = − i ω Q + 2 b [ A ( t ) + A e ( t )] Z 3 1 2 − Z 3 − 2 bQQ 1 [ A ( t ) + A e ( t )] , Q 1 = Re Q , (3.7) since Q · [ iQ ] = 0. Note that Z 2 3 + | Z | 2 = Z 2 3 + | Q | 2 ( 1 2 − Z 3 ) 2 = 1 4 by ( 3.1 ). Hence, Z 3 = 1 2 − 1 ( | Q | 2 + 1 ) , 1 2 − Z 3 = 1 | Q | 2 + 1 , Z 3 1 2 − Z 3 = | Q | 2 − 1 2 . (3.8) Therefore, ( 3.7 ) implies that ˙ Q ( t ) = − i ω Q ( t ) − b [ A ( t ) + A e ( t )]( Q 2 ( t ) + 1 ) . (3.9) Remark 3.1. Due to ( 2.3 ), the population in version reads as ( 1.10 ): I = q 1 − 4 | Z | 2 = 2 Z 3 = | Q | 2 − 1 | Q | 2 + 1 (3.10) since I > 0 for | Q | > 1 by ( 3.3 ). South Pole projection. For any complex number z ∈ C , we will denote z 1 = Re z and z 2 = Im z . The Maxwell amplitudes are gov erned by the first two equations of ( 1.1 ): ˙ A ( t ) = B ( t ) , ˙ B ( t ) = − Ω 2 A ( t ) − γ B ( t ) + c j ( t ) , j ( t ) = 2 κ Z 2 ( t ) = 2 κ ( 1 2 − Z 3 ) Q 2 ( t ) = 2 κ Q 2 ( t ) | Q ( t ) | 2 + 1 . (3.11) The system ( 3.9 ), ( 3.11 ) represents the smooth reduced dynamics ( 1.11 ) in the local coordinates ( A , B , Q ) defined in the chart K + = R 2 × [ S 2 \ ( 0 , 0 , 1 2 )] . Similar smooth representation of the reduced dynamics holds 7 in the chart K − = R 2 × [ S 2 \ ( 0 , 0 , − 1 2 )] with another coordinate Σ ∈ R 2 defined as the stereographic projection from the “South Pole” ( 0 , 0 , − 1 2 ) : now ( 3.1 ) changes to Σ ( Z ) = Z 1 2 + Z 3 , Z 3 = ± r 1 4 − | Z | 2 , ± Z 3 > 0; Z ∈ D . (3.12) Fig. 1 obviously shows that the North and South Pole stereografic projections of any point of the sphere S 2 are related by the in v ersion Q = 1 / Σ . (3.13) It is easy to check that the resulting equations are very similar to ( 3.9 ), ( 3.11 ):      ˙ A ( t ) = B ( t ) , ˙ B ( t ) = − Ω 2 A ( t ) − γ B ( t ) + c j ( t ) , j ( t ) = 2 κ Σ 2 ( t ) | Σ ( t ) | 2 + 1 ˙ Σ ( t ) = i ω Σ ( t ) + b [ A ( t ) + A e ( t )]( Σ 2 ( t ) + 1 )        . (3.14) On repr esentations of the reduced dynamics. The trajectories Y ( t ) = ( A ( t ) , B ( t ) , C ∗ ( t )) of the reduced dynamics satisfy the equations ( 3.9 ), ( 3.11 ) in the local coordinates ( M , Q ) on the phase space Y . The equations hold until C ∗ ( t ) goes through the North Pole. Similarly , the trajectories satisfy the equations ( 3.14 ) in the local coordinates ( M , Σ ) until C ∗ ( t ) goes through the South Pole. 4 A veraging of dynamics in the interaction pictur e Write the system ( 3.9 ), ( 3.11 ) as    ˙ A ( t ) = B ( t ) , ˙ B ( t ) = − Ω 2 A ( t ) − γ B ( t ) + 2 c κ Q 2 ( t ) | Q ( t ) | 2 + 1 ˙ Q ( t ) = − i ω Q ( t ) − b [ A ( t ) + A e ( t )]( Q 2 ( t ) + 1 )       . (4.1) Let us denote M ( t ) = A ( t ) + iB ( t ) / Ω and re write ( 4.1 ) as      ˙ M ( t ) = − i h Ω M ( t ) + γ M 2 ( t ) − ˜ κ Q 2 ( t ) | Q ( t ) | 2 + 1 i ˙ Q ( t ) = − i ω Q ( t ) − b [ A ( t ) + A e ( t )]( Q 2 ( t ) + 1 ) ,        , ˜ κ = 2 c κ Ω . (4.2) For small p , γ > 0, the system is small perturbation of the unperturbed one, ˙ M ( t ) = − i Ω M ( t ) , ˙ Q ( t ) = − i ω Q ( t ) . (4.3) The interaction picture. The unperturbed system ( 4.3 ) admits the single-frequency solutions M ( t ) = e − i Ω t M , Q ( t ) = e − i ω t Q , where M , Q ∈ C . (4.4) Our goal is the construction of similar solutions to the perturbed system ( 4.2 ), M ( t ) = e − i Ω t M ( t ) , Q ( t ) = e − i ω t Q ( t ) (4.5) with slo wly varying en veloping amplitudes: for a wide interval of time [ 0 , T ( p )] , sup t ∈ [ 0 , T ( p )] h | M ( t ) − M ( 0 ) | + | Q ( t ) − Q ( 0 ) | i → 0 , p → 0 , p / γ = r  = 0 . (4.6) Remark 4.1. For such solutions, | Q ( t ) | ∼ const. Therefore, the corresponding population in v ersion also is almost constant by ( 3.10 ). 8 Substituting ( 4.5 ) into ( 4.2 ), we obtain the dynamical equations for the en v eloping amplitudes:      ˙ M ( t ) = − ie i Ω t h γ Im ( e − i Ω t M ( t )) − ˜ κ Im ( e − i ω t Q ( t )) | Q ( t ) | 2 + 1 i ˙ Q ( t ) = − be i ω t [ Re ( e − i Ω t M ( t )) + A e ( t )]( Q 2 ( t ) e − 2 i ω t + 1 )        . (4.7) The equations are called as the interaction picture of ( 4.2 ). The equations can be written as ˙ M ( t ) = p f r ( M ( t ) , Q ( t ) , t ) , ˙ Q ( t ) = p g r ( M ( t ) , Q ( t ) , t ) , (4.8) where the functions f r and g r are gi ven by      f r ( M , Q , t ) = − ie i Ω t h γ 1 ( M 2 cos Ω t − M 1 sin Ω t ) − κ 1 Q 2 cos ω t − Q 1 sin ω t ) | Q | 2 + 1 i g r ( M , Q , t ) = − b 1 e i ω t [ M 1 cos Ω t + M 2 sin Ω t + A e ( t )]( Q 2 e − 2 i ω t + 1 )        , (4.9) and the parameters are γ 1 = γ p = 1 r , κ 1 = ˜ κ p = 2 c ω Ω , b 1 = b p = ω c ¯ h . (4.10) Remark 4.2. It is important that the coefficients γ 1 , κ 1 , b 1 depend only on Ω , ω , r . Hence, for any fixed Ω , ω > 0 and r  = 0, the asymptotics of solutions to the systems ( 4.8 ) as p → 0 and p / γ = r can be calculated by methods of the av eraging theory [ 7 , 26 ]. The a veraging. A veraging equations ( 4.8 ), we obtain the equation ( 1.12 ) in the form ˙ M ( t ) = p f r ( M ( t ) , Q ( t )) , ˙ Q ( t ) = p g r ( M ( t ) , Q ( t )) , (4.11) where f r ( M , Q ) = ⟨ f r ( M , Q , · ) ⟩ = lim T → ∞ 1 T Z T 0 f r ( M , Q , t ) d t , g r ( M , Q ) = ⟨ g r ( M , Q , · ) ⟩ . (4.12) Let us calculate the averages ( 4.12 ). The results differ drastically for the resonance case Ω = ω and non- resonance Ω  = ω . In notation ( 1.3 ), A e = A e 1 + i A e 2 , where A e = 2 ⟨ A e ( t ) e i Ω t ⟩ , A e 1 = 2 ⟨ A e ( t ) cos Ω t ⟩ , A e 2 = 2 ⟨ A e ( t ) sin Ω t ⟩ . (4.13) Resonance case Ω = ω . Using the expressions ( 4.9 ), we obtain f r ( M , Q ) = − i γ 1 2 ( M 2 − i M 1 ) + i κ 1 2 ( | Q | 2 + 1 ) ( Q 2 − i Q 1 ) = − γ 1 2 M + κ 1 2 Q | Q | 2 + 1 . (4.14) Similarly , g r ( M , Q ) = − b 1 2 h ( M 1 − i M 2 + A e 1 − i A e 2 )( Q 2 1 − Q 2 2 + 2 i Q 1 Q 2 ) + M + A e i = b 1 2 h ( M 1 + A e 1 )( | Q | 2 − 2 Q 1 ( Q 1 + i Q 2 )) + i ( M 2 + A e 2 )( | Q | 2 + 2 Q 2 ( i Q 1 − Q 2 )) − M − A e i = b 1 2 h ( M + A e )( | Q | 2 − 1 ) − 2 ( M 1 + A e 1 ) Q 1 ( Q 1 + i Q 2 ) − 2 ( M 2 + A e 2 ) Q 2 ( Q 1 + i Q 2 )] = b 1 2 h M ( | Q | 2 − 1 ) − 2 Q M · Q − 2 Q A e · Q + A e ( | Q | 2 − 1 ) i = b 1 2 h [ M + A e ]( | Q | 2 − 1 ) − 2 Q [ M + A e ] · Q i . (4.15) Non-resonance case Ω  = ω . In this case the calculations ( 4.14 ) simplify to f r ( M , Q ) = − γ 1 2 M , (4.16) 9 which implies nonexistence of stationary states with nonzero Maxwell field for the system ( 4.11 ), and also the decay M ( t ) = M ( 0 ) e − γ 2 t . In contrast, the resonance equation ( 4.14 ) includes the interaction term which can pre vent the decay of the Maxwell amplitude, that is e xpected physically . The a veraged system. By ( 4.14 ) and ( 4.15 ), the averaged equations ( 4.11 ) (or ( 1.12 )) in the resonance case read as        ˙ M ( t ) = p h − γ 1 2 M + κ 1 2 Q | Q | 2 + 1 i ˙ Q ( t ) = pb 1 2 h [ M + A e ]( | Q | 2 − 1 ) − 2 Q [ M + A e ] · Q i         . (4.17) 5 Stationary states for a veraged dynamics (harmonic states) In this section, we calculate all harmonic states ( M , Q ) , i.e., stationary states for the av eraged reduced equations ( 4.17 ), in the resonance case Ω = ω . The states satisfy the system 0 = − γ 1 M + κ 1 Q | Q | 2 + 1 , 0 = [ M + A e ]( | Q | 2 − 1 ) − 2 Q [ M + A e ] · Q . (5.1) It is important that the stationary equations depend on r but do not depend on p . The first equation of ( 5.1 ) gi ves M = α Q where α = κ 1 γ 1 ( | Q | 2 + 1 ) = 2 cr ( | Q | 2 + 1 ) > 0 (5.2) according to ( 4.10 ) with Ω = ω . Now the second equation of ( 5.1 ) becomes − α Q ( | Q | 2 + 1 ) − 2 Q A e · Q + A e ( | Q | 2 − 1 ) = 0 . (5.3) Note that in the case A e = 0, we hav e Q = M = 0, so we consider below A e  = 0 only . T aking the vector product of ( 5.3 ) with Q , we get A e ∧ Q ( | Q | 2 − 1 ) = 0 . Hence, it suf fices to consider the cases 1) | Q | = 1, and 2) | Q |  = 1 while A e ∧ Q = 0. 5.1 Zero population in version In the case | Q | = 1, we hav e | Z | = 1 / 2, so by ( 2.3 ), the population inv ersion v anishes: I = 0. Lemma 5.1. Let A e = e i ϕ | A e |  = 0 . Then solutions ( M , Q ) to the system ( 5.1 ) with | Q | = 1 exist if f cr ≤ | A e | . (5.4) The solutions r ead as Q ± = e i ( ϕ ± θ ) , M ± = cre i ( ϕ ± θ ) , where θ = arccos − cr | A e | . (5.5) F or cr < | A e | , ther e ar e two solutions. F or cr = | A e | , the unique solution is given by Q = − A e / | A e | , M = − A e . (5.6) Pr oof. For | Q | = 1, equation ( 5.3 ) simplifies to A e · Q = − κ 1 2 γ 1 = − cr (5.7) according to ( 4.10 ) with Ω = ω . Hence, | A e | cos \ ( A e , Q ) = − cr (5.8) Evidently , the equation has solutions if f ( 5.4 ) holds. Finally , ( 5.5 ) and ( 5.6 ) follow from ( 5.8 ) and ( 5.2 ). 10 5.2 Nonzero population in version Here we calculate stationary states of the system ( 5.1 ) in the case | Q |  = 1, A e ∧ Q = 0. Lemma 5.2. Let A e  = 0 . Then nonzer o solutions ( M , Q ) to ( 5.1 ) with | Q |  = 1 and A e ∧ Q = 0 e xist iff cr > | A e | . (5.9) The solutions ar e given by Q ± = h − cr ± q c 2 r 2 − | A e | 2 i A e | A e | 2 , M ± = α ± Q ± , where α ± = 2 cr | Q ± | 2 + 1 . (5.10) The following r elations hold: | Q + | < 1 and | Q − | > 1 . (5.11) Pr oof. Assume first that A e 1  = 0. Then A e ∧ Q = 0 implies Q 2 = A e 2 A e 1 Q 1 . Hence A e · Q = A e 1 Q 1 + A p 2 A e 2 A e 1 Q 1 = Q 1 | A e | 2 A e 1 , | Q | 2 = Q 2 1  1 + | A e 2 | 2 | A e 1 | 2  = Q 2 1 | A e | 2 | A e 1 | 2 . Substituting into ( 5.3 ), we obtain for the first component, − κ 1 γ 1 Q 1 − 2 Q 2 1 | A e | 2 A e 1 + Q 2 1 | A e | 2 A e 1 − A e 1 = 0 . Simplifying, we get γ 1 | A e | 2 A e 1 Q 2 1 + κ 1 Q 1 + γ 1 A e 1 = 0 . (5.12) Hence we get the solutions Q ± = − κ 1 ± q κ 2 1 − 4 γ 2 1 | A e | 2 2 γ 1 A e | A e | 2 , M ± = α ± Q ± , where α ± = κ 1 γ 1 ( | Q ± | 2 + 1 ) . (5.13) The formula also holds in the case when A e 1 = 0 and A e 2  = 0. At last, the solution can be rewritten as ( 5.10 ) since κ 1 / γ 1 = 2 cr by ( 4.10 ) with Ω = ω . Finally , ( 5.11 ) holds since | Q + | · | Q − | = 1 by ( 5.10 ). 5.3 Fully in verted states are not harmonic Our analysis above does not include the North Pole which corresponds to Q = ∞ . T o extend the analysis, we can use the equations ( 3.14 ) corresponding to the stereographic projection from the South Pole ( 3.12 ). Slightly modifying calculations of Section 4 , it is easy to check that for A e  = 0 and any r  = 0, the subset { Σ = 0 , M ∈ C } ⊂ Y = X / U ( 1 ) does not contain stationary states of the corresponding av eraged equations in the interaction picture. The population in version at these states is gi ven by ( 3.10 ) with Q = ∞ , i.e., I = 1. So, all fully in v erted states are not harmonic. 5.4 Summary The results of this section imply the follo wing corollary . Corollary 5.3. i) In the non-r esonance case, Ω  = ω , the avera ged system ( 4.11 ) does not have stationary states with nonzer o Maxwell field. ii) In the resonance case, Ω = ω , the system ( 4.11 ) admits two stationary states given by ( 5.5 ) for cr < | A e | and by ( 5.10 ) for cr > | A e | . In the case cr = | A e | , the states coincide and ar e given by ( 5.6 ). Remark 5.4. Let ( M ( t ) , Q ( t )) be a solution to the averaged equations ( 4.17 ). Then e i φ ( M ( t ) , Q ( t )) with φ ∈ R is the solution to the same equations with A e replaced by e i φ A e . Hence, this correspondence also holds for solutions to stationary equations ( 5.1 ) that is obvious from the formulas ( 5.5 ) and ( 5.10 ). 11 6 Stability of harmonic states In this section we calculate the spectra of the linearization of the av eraged equations ( 4.11 ) at the harmonic states calculated abov e in the resonance case Ω = ω . As the result, we will obtain that the states ( 5.5 ) and ( 5.6 ) with | Q | = 1 are not linearly stable, the states Q + in ( 5.10 ) is linearly stable, while Q − is unstable . Recall the resonance formulas ( 4.14 ), ( 4.15 ): omitting the index r ,      f ( M , Q ) = − γ 1 2 M + κ 1 2 Q | Q | 2 + 1 , g ( M , Q ) = b 1 2 h M ( | Q | 2 − 1 ) − 2 Q M · Q − 2 Q A e · Q + A e ( | Q | 2 − 1 ) i . Recall that z 1 and z 2 denote the real and imaginary parts for any comple x number z ∈ C . Differentiating, we get for i , j = 1 , 2,            ∂ f i ∂ M j = − γ 1 2 δ i j , ∂ f i ∂ Q j = κ 1 2  δ i j | Q | 2 + 1 − 2 Q i Q j ( | Q | 2 + 1 ) 2  , ∂ g i ∂ M j = b 1 2 δ i j ( | Q | 2 − 1 ) − b 1 Q i Q j , ∂ g i ∂ Q j = b 1 [ M i Q j − δ i j M · Q − Q i M j − δ i j A e · Q − Q i A e j + A e i Q j ] . T aking into account ( 5.2 ), we obtain the Jacobian J =      − γ 1 2 δ i j κ 1 2  δ i j | Q | 2 + 1 − 2 Q i Q j ( | Q | 2 + 1 ) 2  b 1 2 ( | Q | 2 − 1 ) δ i j − b 1 Q i Q j − b 1 [ δ i j M · Q + δ i j A e · Q + Q i A e j − A e i Q j ]      (6.1) W e can choose real coordinates in C = R 2 such that Q = ( | Q | , 0 ) . Then the Jacobian ( 6.1 ) becomes J =        − γ 1 2 0 κ 1 2 1 −| Q | 2 ( | Q | 2 + 1 ) 2 0 0 − γ 1 2 0 κ 1 2 1 | Q | 2 + 1 − b 1 2 ( | Q | 2 + 1 ) 0 − b 1 ( s + κ 1 γ 1 | Q | 2 | Q | 2 + 1 ) − b 1 w 0 b 1 2 ( | Q | 2 − 1 ) b 1 w − b 1 ( s + κ 1 γ 1 | Q | 2 | Q | 2 + 1 )        , (6.2) where s = A e · Q , w = A e ∧ Q . 6.1 Zero population in version. Denote s ± = A e · Q ± and w ± = A e ∧ Q ± . For the states ( 5.5 ) with | Q ± | = 1, the matrix ( 6.2 ) becomes J ± =     − γ 1 2 0 0 0 0 − γ 1 2 0 κ 1 4 − b 1 0 0 b 1 w ± 0 0 − b 1 w ± 0     (6.3) since s ± + κ 1 γ 1 | Q | 2 | Q | 2 + 1 = A e · Q + κ 1 2 γ 1 = 0 by ( 5.7 ). The eigen v alues of J ± equal λ 1 , 2 = − γ 1 / 2 , λ 3 , 4 = ± ib 1 w ± . (6.4) Note that the spectrum of the linearised system ( 4.11 ) at the stationary states ( 5.5 ), ( 5.6 ) consists of p λ 1 , 2 and p λ 3 , 4 , where p ∈ R . Thus, we ha ve pro ved the follo wing lemma. Lemma 6.1. F or cr ≤ | A e | and all p ∈ R , stationary states ( 5.5 ) of the system ( 4.11 ) ar e not linearly stable. 12 6.2 Nonzero population in version Here we consider the stationary states ( 5.10 ) corresponding to | cr | > | A e | . F or a moment, let us write Q instead of Q ± . By ( 5.10 ), A e = ( −| A e | , 0 ) in the case Q = ( | Q | , 0 ) . Hence, equation ( 5.12 ) becomes − γ 1 | A e || Q | 2 + κ 1 | Q | − γ 1 | A e | = 0 . This implies that κ 1 γ 1 | Q | | Q | 2 + 1 = | A e | , and therefore s + κ 1 γ 1 | Q | 2 | Q | 2 + 1 = A e · Q + κ 1 γ 1 | Q | 2 | Q | 2 + 1 = −| A e || Q | + κ 1 γ 1 | Q | 2 | Q | 2 + 1 = | Q | h κ 1 γ 1 | Q | | Q | 2 + 1 − | A e | i = 0 . No w w = Q ∧ A e = 0, and the Jacobian ( 6.2 ) becomes J =      − γ 1 2 0 κ 1 2 1 −| Q | 2 ( | Q | 2 + 1 ) 2 0 0 − γ 1 2 0 κ 1 2 1 | Q | 2 + 1 − b 1 2 ( | Q | 2 + 1 ) 0 0 0 0 b 1 2 ( | Q | 2 − 1 ) 0 0      . (6.5) The characteristic equation det ( 2 J − λ ) = 0 reads as − ( γ 1 + λ )   − ( γ 1 + λ ) 0 κ 1 | Q | 2 + 1 0 − λ 0 b 1 ( | Q | 2 − 1 ) 0 − λ   − κ 1 ( | Q | 2 − 1 ) ( | Q | 2 + 1 ) 2   0 − ( γ 1 + λ ) κ 1 | Q | 2 + 1 − b 1 ( | Q | 2 + 1 ) 0 0 0 b 1 ( | Q | 2 − 1 ) − λ   = 0 . Ev aluating, we obtain ( γ 1 + λ ) 2 λ 2 − ( γ 1 + λ ) λ b 1 κ 1 ( | Q | 2 − 1 ) | Q | 2 + 1 + κ 1 ( | Q | 2 − 1 ) ( | Q | 2 + 1 ) 2  b 2 1 κ 1 ( | Q | 2 − 1 ) − b 1 ( | Q | 2 + 1 )( γ 1 + λ ) λ  = 0 . Equi valently , ( γ 1 + λ ) 2 λ 2 − 2 ( γ 1 + λ ) λ b 1 κ 1 ( | Q | 2 − 1 ) | Q | 2 + 1 + b 2 1 κ 1 2 ( | Q | 2 − 1 ) 2 ( | Q | 2 + 1 ) 2 = h ( γ 1 + λ ) λ − b 1 κ 1 I i 2 = 0 by ( 3.10 ). Hence, we hav e the follo wing roots of multiplicity two: λ 1 , 2 = λ 1 , 2 ( Q ) = 1 2 h − γ 1 ± q γ 2 1 + 4 b 1 κ 1 I ( Q ) i , I ( Q ) = | Q | 2 − 1 | Q | 2 + 1 . (6.6) By ( 4.10 ), the spectrum of the linearised system ( 4.11 ) at the stationary states ( 5.10 ) consists of p λ 1 , 2 = 1 2 h − γ ± q γ 2 + 4 p 2 b 1 κ 1 I ( Q ) i . Note that κ 1 , b 1 > 0 by ( 4.10 ), so for p > 0, Re [ λ 1 , 2 ( Q )] = − ν < 0 (6.7) if f I ( Q ) < 0 . Recall that I ( Q + ) < 0 and I ( Q − ) > 0 by ( 5.11 ). Thus, we have pro v ed the following lemma. Lemma 6.2. In the case cr > | A e | , stationary states ( M + , Q + ) r esp. ( M − , Q − ) of the system ( 4.11 ), which ar e given by ( 5.10 ), ar e linearly stable resp. unstable. 7 Single-frequency asymptotics In this section we prove asymptotics ( 1.13 )–( 1.17 ) in the resonance case Ω = ω . W e obtain the asymptotics applying the results of the av eraging theory [ 26 ]. T o justify the application, we are going to check suitable properties of the system ( 4.8 ). 13 7.1 KBM vector field Let us denote v r ( M , Q , t ) = ( f r ( M , Q , t ) , g r ( M , Q , t )) the vector field ( 4.9 ) of the system ( 4.8 ). It is easy to check that in the case of almost periodic pumping A e ( t ) the asymptotics holds sup ( M , Q ) ∈ D 1 T    Z T 0 [ v r ( M , Q , t ) − v r ( M , Q )] d t    → 0 , T → ∞ (7.1) for each r  = 0 and any bounded region D ⊂ C 2 . Hence, according to [ 26 , Definition 4.2.4], v r is a KBM (Krylov–Bogolyubo v–Mitropolsky) vector field on any bounded region D ⊂ C 2 . Furthermore, for the quasiperiodic pumping ( 1.3 ), formulas ( 4.9 ) imply that δ D ( p ) : = p sup ( M , Q ) ∈ D sup T ∈ [ 0 , p − 1 ]    Z T 0 [ v r ( M , Q , t ) − v r ( M , Q )] d t    = O ( p ) , p → 0 , (7.2) where δ D ( p ) is the corresponding or der function defined in [ 26 , Lemma 4.6.4]. Remark 7.1. For more general almost periodic pumping A e ( t ) , the order function can be different [ 26 , Section 4.6]. 7.2 Single-frequency asymptotics in the interaction pictur e Here we pro ve Theorem 1.1 . W e denote by Y ( t ) = ( M ( t ) , Q ( t )) and Y ( t ) = ( M ( t ) , Q ( t )) the solutions to the systems ( 4.8 ) and ( 4.17 ), respecti vely , with the same initial v alue Y ( 0 ) = Y ( 0 ) = Π X ( 0 ) . i) Let us prov e the asymptotics max t ∈ [ 0 , p − 1 ] h | M ( t ) − M r | + | Q ( t ) − Q r | i = O ( p 1 / 2 ) , p → 0 , p / γ = r . (7.3) In other words, max t ∈ [ 0 , p − 1 ] | Y ( t ) − Y r | = O ( p 1 / 2 ) , p → 0 , p / γ = r , (7.4) where Y r = ( M r , Q r ) is a stationary state for the a v eraged system ( 1.12 ), or equi valently , to ( 4.17 ). Note that Y r : = Π X r = ( M ( 0 ) , Q ( 0 )) since X ( 0 ) = X r . Hence, ( 4.5 ) implies that Y ( 0 ) = ( M ( 0 ) , Q ( 0 )) = Y r . Recall that v r is the KBM vector field in each bounded region D ⊂ C . Hence, ( 7.4 ) would follow from Theorem 4.3.6 of [ 26 ] for the case when all trajectories Y ( t ) are uniformly bounded for small p > 0. The Maxwell amplitudes M ( t ) are uniformly bounded by the a priori estimate ( A.6 ) which is uniform in small p > 0 and γ = pr for every fixed r > 0. On the other hand, the trajectories Q ( t ) can be unbounded if Z ( t ) , defined by ( 3.1 ), approaches the North pole ( 0 , 0 , 1 / 2 ) of the sphere S 2 . This is why the Theorem 4.3.6 of [ 26 ] cannot be applied since the vector field v r ( M , Q ) is not Lipschitz continuous in Q ∈ C . This suggests to consider ( 4.8 ) as the dynamical system for ( M ( t ) , Z ( t )) ∈ C × S 2 with the corresponding vector field v r ( M , Z ) on the manifold C × S 2 . The key observation is that the vector field v r ( M , Z ) is Lipschitz continuous on ev ery compact subset of the manifold C × S 2 in the sense ( B.5 ). This follo ws from the representations ( 4.9 ) in the coordinates M , Q which holds outside the North pole, and similar representations for the system corresponding to ( 3.14 ) outside the South pole. Hence, the bounds ( 7.4 ) hold by Theorem B.2 which extend the Theorem 4.3.6 of [ 26 ] onto dynamical systems on compact manifolds. As the result, ( 7.3 ) follows, so ( 1.13 ) holds by the relations ( 4.5 ). ii) By the same Theorem B.2 , max t ∈ [ 0 , p − 1 ] | Y ( t ) − Y ( t ) | = O ( p 1 / 2 ) , p → 0 , p / γ = r . (7.5) In other words, max t ∈ [ 0 , p − 1 ] h | M ( t ) − M ( t ) | + | Q ( t ) − Q ( t ) | i = O ( p 1 / 2 ) , p → 0 , p / γ = r . (7.6) 14 Hence, asymptotics ( 1.14 ) follows by ( 4.5 ). Finally , ( 1.15 ) holds since Π X ( 0 ) ∈ D and Y ( t ) = ( M ( t ) , Q ( t )) is a solution to ( 4.17 ). iii) Let us prov e the asymptotics | M ( t ) − M r | + | Q ( t ) − Q r | ≤ C [ p + d 0 e − p µ t ] , t ≥ 0 , p ≤ ε , p / γ = r . (7.7) By the smoothness of the projection Π , | Y ( 0 ) − Y r | = | Π ( X ( 0 ) − X r ) | ≤ C d 0 for small d 0 = | X ( 0 ) − X r | . Moreov er , Y r is a linearly stable stationary state of the av eraged system ( 4.17 ) by our assumptions. There- fore, extending Theorem 5.5.1 of [ 26 ] onto the system ( 4.8 ) on the phase space C × S 2 for Z ( t ) ∈ S 2 and M ( t ) ∈ C , we obtain that for small d 0 , sup t ∈ [ 0 , ∞ ) | Y ( t ) − Y ( t ) | = O ( p ) , p → 0 , p / γ = r . (7.8) On the other hand, for small d 0 > 0 and µ ∈ ( 0 , ν ) , ( 6.7 ) implies that | Y ( t ) − Y r | ≤ C d 0 e − p µ t , t > 0 . (7.9) Hence, ( 7.7 ) follo ws. Now ( 1.17 ) holds by ( 4.5 ). Finally , ( 1.16 ) is a particular case of ( 1.17 ) when d 0 = 0. Corollary 7.2. i) Asymptotics ( 1.13 ) hold for solutions X ( t ) to the MBE with Π X ( 0 ) = ( M ± , Q ± ) given by ( 5.5 ) and ( 5.10 ). ii) Asymptotics ( 1.14 ) hold for solutions with Π X ( 0 ) = ( M ± , Q ± ) , given by ( 5.5 ), if the states are asymptot- ically stable for the dynamics ( 4.17 ). iii) Asymptotics ( 1.16 ) and ( 1.17 ) hold for solutions with Π X ( 0 ) = ( M + , Q + ) given by ( 5.10 ). Remark 7.3. i) The asymptotic stability of the harmonic states ( 5.5 ) remains an open problem. ii) For a suitable class of almost periodic pumpings A e ( t ) , asymptotics similar to ( 1.13 )–( 1.17 ) hold if the corresponding order function δ D ( p ) = O ( p ν ) with ν ∈ ( 0 , 1 ) , see [ 26 , Section 4.6]. 8 Conflict of interest W e have no conflict of interest. 9 Data av ailability statement The manuscript has no associated data. A The a priori bounds and well-posedness In this section, we pro ve the a priori bounds ( 1.5 ) assuming that A e ( t ) ∈ C [ 0 , ∞ ) . The bounds imply the well-posedness of the MBE in the phase space X = R 2 × S 3 . The Schr ¨ odinger amplitudes C 1 ( t ) , C 2 ( t ) are bounded by the charge conservation ( 1.4 ). Hence, it remains to prov e the bounds for the Maxwell amplitudes ( A ( t ) , B ( t )) . The follo wing lemma and its proof refine the estimate (2.1) from [ 16 ]. Lemma A.1. Ther e exists a L yapunov function V ( A , B ) such that a 1 [ A 2 + B 2 ] ≤ V ( A , B ) ≤ a 2 [ A 2 + B 2 ] where a 1 , a 2 > 0 , (A.1) and for solutions to ( 1.1 ), the function V ( t ) = V ( A ( t ) , B ( t )) satisfies the inequality ˙ V ( t ) ≤ − γ b V ( t ) + d p 2 γ , t > 0; b , d > 0 . (A.2) 15 Pr oof. Denote by E ( A , B ) = 1 2 ( Ω 2 A 2 + B 2 ) “the energy” of the Maxwell field. The first two equations of ( 1.1 ) imply d d t E ( A ( t ) , B ( t )) = Ω 2 A ( t ) B ( t ) + B ( t )[ − Ω 2 A ( t ) − γ B ( t ) + c j ( t )] = − γ B 2 ( t ) + cB ( t ) j ( t ) . (A.3) W e construct the L yapunov function following the standard approach to dissipative perturbations of the Hamiltonian systems [ 4 , 14 ]: V ( A , B ) = E ( A , B ) + ε AB . Then ( A.1 ) holds for small ε > 0. It remains to demonstrate ( A.2 ) for suf ficiently small ε > 0. Differentiating, we obtain ˙ V ( t ) = ˙ E ( t ) + ε ˙ A ( t ) B ( t ) + ε A ( t ) ˙ B ( t ) = − γ B 2 ( t ) + cB ( t ) j ( t ) + ε B 2 ( t ) + ε A ( t )[ − Ω 2 A ( t ) − γ B ( t ) + c j ( t )] = − ( γ − ε ) B 2 ( t ) − ε Ω 2 A 2 ( t ) − ε γ A ( t ) B ( t ) + c j ( t )( ε A ( t ) + B ( t )) . Note that | ε γ AB | ≤ γ 2 B 2 + 1 2 γ ε 2 A 2 . Hence, for small ε > 0, ˙ V ( t ) ≤ − [ γ 2 − ε ] B 2 ( t ) − ε 2 Ω 2 A 2 ( t ) + c j ( t )( ε A ( t ) + B ( t )) , t > 0 . (A.4) Moreov er , | j ( t ) | ≤ κ = p ω by ( 1.2 ) and ( 1.4 ). Hence, for any N > 0, c | j ( t )( ε A ( t ) + B ( t )) | ≤ γ N c 2 ( ε A ( t ) + B ( t )) 2 + ω 2 N p 2 γ . Therefore, ( A.4 ) with small ε < γ / 2 and suf ficiently large N > 0 implies ( A.2 ). Corollary A.2. Solving the inequality ( A.2 ), we obtain: V ( t ) ≤ V ( 0 ) + d b r 2 , t ≥ 0 , r = p / γ . (A.5) Hence, for solutions to ( 1.1 ) with p / γ = r , the following bounds hold: A 2 ( t ) + B 2 ( t ) ≤ D r ( A 2 ( 0 ) + B 2 ( 0 )) , t ≥ 0 . (A.6) Now ( 1.5 ) follows. B On av eraging theory on manif olds The a veraging theory is usually considered for dynamical systems on linear spaces, [ 26 ]. On the other hand, definition of the KBM vector field ( 7.1 ) and of the order function ( 7.2 ) are inv ariant. Namely , let M be a smooth finite-dimensional manifold, and consider the dynamical system ˙ Y ( t ) = ε V ( Y ( t ) , t ) , t ≥ 0 , (B.1) where V ( Y ) is a vector field on M . The corresponding averaged equation is defined by ˙ Y ( t ) = ε V ( Y ( t )) , t ≥ 0; V ( Y ) = lim T → ∞ 1 T Z T 0 V ( Y , t ) d t , Y ∈ M . (B.2) Definition B.1. i) ([ 26 , Definition 4.2.4]) V ( Y ) is a KBM vector field if for any compact set K ⊂ M , sup Y ∈ K 1 T    Z T 0 [ V ( Y , t ) − V ( Y )] d t    → 0 , T → ∞ . (B.3) ii) The or der function is defined as in [ 26 , Lemma 4.3.1]: δ K ( ε ) : = ε sup Y ∈ K sup T ∈ [ 0 , ε − 1 ]    Z T 0 [ V ( Y , t ) − V ( Y )] d t    . (B.4) 16 Accordingly , the classical Theorem 4.3.6 of [ 26 ] admits the following extension. Let the vector field V ( Y , t ) be Lipschitz continuous uniformly in time: in each local chart on ev ery subset K ⊂ M , | V ( Y 1 , t ) − V ( Y 2 , t ) | ≤ L ( K ) | Y 1 − Y 2 | , Y 1 , Y 2 ∈ K , t ≥ 0 , (B.5) where L ( K ) < ∞ can depend also on the choice of the local coordinates. Note that ( B.5 ) implies ( B.3 ). Theorem B.2. Let M be a compact manifold, ( B.5 ) hold, Y ( t ) = Y ε ( t ) be solutions to ( B.1 ) with ε ∈ ( 0 , 1 ] and the same initial state Y ε ( 0 ) = Y 0 , and Y ( t ) = Y ε ( t ) be solutions to ( B.2 ) with Y ε ( 0 ) = Y 0 . Then max t ∈ [ 0 , ε − 1 ] ρ ( Y ε ( t ) , Y ( t )) = O ( δ 1 / 2 ) , ε → 0 , (B.6) wher e ρ is an arbitrary distance defined on M . Pr oof. The compact manifold M is diffeomorphic to a smooth submanifold of R N with a suitable N ≥ 1. Now the bounds ( B.6 ) follow by the same arguments as Theorem 4.3.6 of [ 26 ] with the interpretation of points Y ∈ M and tangent vectors V ( Y , t ) ∈ T Y M as the corresponding vectors from R N . W ith this interpretation, the vector field V ( Y , t ) on the compact submanifold M is Lipschitz continuous by ( B.5 ). C The Maxwell–Bloch equations for pur e states In [ 16 ], the MBE is obtained as the Galerkin approximation of the coupled Maxwell–Schr ¨ odinger system. The approximation consists of a single-mode Maxwell field coupled to two-lev el molecule in a bounded cavity V ⊂ R 3 : A ( x , t ) = A ( t ) X ( x ) , ψ ( x , t ) = C 1 ( t ) ψ 1 ( x ) + C 2 ( t ) ψ 2 ( x ) , x ∈ V . (C.1) Here A ( x , t ) denotes the v ector potential of the Maxwell field, and X ( x ) is a normalised eigenfunction of the Laplace operator in V under suitable boundary v alue conditions with an eigen value − Ω 2 / c 2 . By ψ l we denote some normalised eigenfunctions of the Schr ¨ odinger operator H : = − ¯ h 2 2m ∆ + e Φ ( x ) with the corresponding eigen v alues ¯ h ω 1 < ¯ h ω 2 , where Φ ( x ) is the molecular (ion’ s) potential. The MBE read as the Hamiltonian system with a dissipation and an external source: 1 c 2 ˙ A ( t ) = ∂ B H , 1 c 2 ˙ B ( t ) = − ∂ A H − γ c 2 B ; i ¯ h ˙ C l ( t ) = ∂ C l H , l = 1 , 2 . (C.2) The Hamiltonian H is defined as H ( A , B , C 1 , C 2 , t ) = H ( A X , B X , C 1 ϕ 1 + C 2 ϕ 2 , t ) , where H is the Hamiltonian of the coupled Maxwell–Schr ¨ odinger equations with pumping [ 5 , 12 , 15 , 17 , 18 , 19 , 22 , 25 ]. Neglecting the spin and scalar potential (which can be easily added), the Hamiltonian H , in the traditional dipole appr oximation , reads as [ 16 , (A.5)]: H ( A , B , C 1 , C 2 , t ) = 1 2 c 2 [ B 2 + Ω 2 A 2 ] + ¯ h ω 1 | C 1 | 2 + ¯ h ω 2 | C 2 | 2 − 2 κ c [ A + A e ( t )] Im [ C 1 C 2 ] . (C.3) No w the Hamilton equations ( C.2 ) become ( 1.1 ). The char ge conservation | C 1 ( t ) | 2 + | C 2 ( t ) | 2 = const follows by dif ferentiation from the last two equations of ( 1.1 ) since the function a ( t ) is real-valued. W e consider solutions with const = 1, see ( 1.4 ). D On possible treatment of the laser action Here we discuss possible treatment of the laser action relying on the obtained results. On the smallness of the parameters. Note that the dipole moment p and dissipation coefficient γ are very small for many types of lasers. In particular, the dissipation coefficient for the Ruby laser is the electrical conduction of corundum which is γ ∼ 10 − 14 in the Hea viside–Lorentz units [ 11 , 29 , 30 ]. For the dipole 17 moment typically | p | ∼ 10 − 18 according to [ 21 ] that agrees with the classical dipole moment ed / 2, where d ∼ 10 − 8 cm is the molecular diameter , and e ∼ 10 − 10 is the elementary charge in the same units. On the laser thr eshold. The asymptotics ( 1.13 ) and ( 1.16 ) hold for solutions with the harmonic initial states ( M r ± , S r ± ) . On the other hand, ( 1.14 ) and ( 1.17 ) hold for solutions with initial states from an open domain of attraction in the phase space. Hence, the asymptotics ( 1.14 ) and ( 1.17 ) appear with a “nonzero probability” in contrast to ( 1.13 ) and ( 1.16 ). This fact, provisionally , shed new light on the “laser threshold” which is necessary to ignite the laser action: the intensity of random pumping must be sufficiently large to bring the solution to the domain of attraction, and then the solution is captured in the domain with the single-frequency asymptotics. On the laser amplification. The equations ( 1.1 ) describe one molecule coupled to the Maxwell field. The limiting amplitudes of the Maxwell field in all asymptotics ( 1.13 )–( 1.16 ) do not depend on non-resonance harmonics in the pumping ( 1.3 ) with the frequencies Ω k  = ω . This means that the dynamics ( 1.1 ) acts as a filter , selecting only the resonant harmonics, that itself cannot explain the amplification of the Maxwell field in laser devices. The amplification could be explained by a large number of activ e molecules, typically N ∼ 10 20 , under the traditional assumption that the molecules interact with the Maxwell field but do not interact with each other [ 23 ]. In this case, the amplitude of the total Maxwell field is multiplied by √ N ∼ 10 10 by the Law of Large Numbers. The amplification is possible only due to the single-frequency asymptotics of indi vidual contributions. The amplification also depends significantly on the fact that the phases of the amplitudes M r in asymp- totics ( 1.13 )–( 1.17 ) for all molecules are uniformly distributed. The uniformity takes place if it holds for phases of A e . This follows from the asymptotics and Remark 5.4 since e i φ e − i Ω t M differs from e − i Ω t M by a shift of time. 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