Comment on Evolution of the unidirectional electromagnetic pulses in an anisotropic two-level medium

Commen t on ’Ev olution of the unidirectional elect romagnetic pulses in an anisotropic t wo-lev el medium’ N.V. Ustinov ∗ (Dated: October 31, 2018) Recently , Zab olotskii [Phys. Rev. E 77, 036603 (2008)] presented the Lax pair for a version of the reduced Maxw ell–Bloch equations. This ve rsion wa s derived under considering the unidirectional propagation of a t wo-component electromagnetic pulse through an anisotropic tw o-level medium. W e demonstrate that his deriv ation contai n s essen tial omissi on, which led to the wrong version of the reduced Maxw ell–Bloch equations. W e also point out that the Lax pair for correct v ersion of these equations had b een kn o wn in the most general anisotropic case. P ACS num ber s: 41 . 20.-q, 02.30.Ik, 42.50.Md, 42.65.Tg A version o f the reduced Ma xwell–Bloch (RMB) eq ua- tions was der ived in Ref. [1] as follows. The Hamilto nia n of the interaction of t wo-compo nent electromagnetic field with a nisotropic tw o-level medium was taken in the for m ˆ H = ~ ω 0 2 ˆ σ 3 − ( d (1) z x E ′ x + d (1) z y E ′ y ) ˆ σ 11 − ( d (2) z x E ′ x + d (2) z y E ′ y ) ˆ σ 22 − ( d xx E ′ x + d xy E ′ y ) ˆ σ 1 − ( d y x E ′ x + d y y E ′ y ) ˆ σ 2 , (1) where ˆ σ k ( k = 1 , 2 , 3) a re the Pauli matrices, ˆ σ 11 = diag(1 , 0), ˆ σ 22 = diag(0 , 1), E ′ x and E ′ y are the trans - verse compo nents o f the electric field, ω 0 is the frequency of the transition, ~ is the Plank’s constant. E quiv alen t representation of ˆ H is ˆ H = ~ ω 0 2 ˆ σ 3 − ˆ d x E ′ x − ˆ d y E ′ y , (2) where the matrices ˆ d x and ˆ d y corres p o nding to the o p- erators of the dipo le moment comp onents are defined as given ˆ d x = d (1) z x d xx − id y x d xx + id y x d (2) z x ! , ˆ d y = d (1) z y d xy − id y y d xy + id y y d (2) z y ! . Co efficients d ( k ) z s ( k = 1 , 2, s = x, y ), d sp ( s, p = x, y ) of these matrices ar e ass umed to b e rea l consta n ts. New v ar iables were introduced by the transfo r mation E x E y ! = T E ′ x E ′ y ! , (3) where T =      d xx δ x d xy δ x d y x δ y d y y δ y      , ∗ Electronic address: n˙ustino v@mail .ru δ x = q d 2 xx + d 2 xy , δ y = q d 2 y x + d 2 y y . Then, Hamiltonian (1) rea ds as ˆ H = ~ ω 0 2 ˆ σ 3 − ( p (1) x E x + p (1) y E y ) ˆ σ 11 − ( p (2) x E x + p (2) y E y ) ˆ σ 22 − δ x E x ˆ σ 1 − δ y E y ˆ σ 2 . Here p ( k ) x = δ x d ( k ) z x d y y − d ( k ) z y d y x P 0 , p ( k ) y = δ y d ( k ) z y d xx − d ( k ) z x d xy P 0 ( k = 1 , 2), P 0 = d xx d y y − d xy d y x . The dyna mics o f the tw o-le vel medium w a s describ ed by the von Neumann equation on the density matrix ˆ ρ , i ~ ∂ ˆ ρ ∂ t = [ ˆ H , ˆ ρ ] , (4) with ˆ ρ = ρ 11 ρ 12 ρ 21 ρ 22 ! . Eq. (4 ) w as rewritten in the terms of the comp onents of the B lo ch vector S x = ρ 12 + ρ 21 2 , S y = ρ 12 − ρ 21 2 i , S z = ρ 11 − ρ 22 2 and the dimensionless v ariables E x = d 0 E x ~ ω 0 , E y = d 0 E y ~ ω 0 , where d 0 = r 4 δ 2 x + 4 δ 2 y +  p (1) x − p (2) x  2 +  p (1) y − p (2) y  2 , as g iven ∂ S x ∂ τ ′ = (1 − m x E x − m y E y ) S y + µ y E y S z , (5) ∂ S y ∂ τ ′ = ( m x E x + m y E y − 1) S x − µ x E x S z , ( 6 ) ∂ S z ∂ τ ′ = µ x E x S y − µ y E y S x . ( 7 ) 2 Here τ ′ = ω 0 t , µ x = 2 δ x d 0 , m x = p (1) x − p (2) x d 0 , µ y = − 2 δ y d 0 , m y = p (1) y − p (2) y d 0 . An evolution of the electromagnetic field of the pulse has to ob ey the Ma xwell equations if the semiclas si- cal approach is applied. It was c la imed in [1] that the transformed comp onents E x and E y satisfy the ”Maxwell equations” ∂ 2 E x ∂ z 2 − n 2 c 2 ∂ 2 E x ∂ t 2 = 4 π c 2 ∂ 2 P x ∂ t 2 , (8) ∂ 2 E y ∂ z 2 − n 2 c 2 ∂ 2 E y ∂ t 2 = 4 π c 2 ∂ 2 P y ∂ t 2 , (9) where n is the refractive index of the medium, c is the light sp eed in free spa ce. The quantit ies P x and P y in the right hand sides of Eq s. (8) and (9) were interpreted as the comp onents of the medium p ola rization a nd were defined fo r this rea son in the following manner: P x = − N T r ( ˆ ρ ∂ ˆ H ∂ E x ) , (10) P y = − N T r ( ˆ ρ ∂ ˆ H ∂ E y ) , (11) where N is the density of the medium. A t last, the reduced equations ∂ E x ∂ χ ′ = R z E y − µ x S y , (12) ∂ E y ∂ χ ′ = − R z E x + µ y S x , (13) where χ ′ = 2 π N d 2 0 n ~ c  z + c n t  , R z = m x µ y S x + m y µ x S y − µ x µ y S z , were obtained fro m (8) and (9) with the help of the uni- directional propaga tion appr oximation. The misprint s were corr ected in the form ulas presented ab ov e. W e divided the first ter m in the rig ht ha nd side of (1) by t wo, changed a sign in the definition of µ y , m ultiplied v aria ble χ ′ by ω 0 / 2. Also, Eqs. (8)–(11) were written in the terms of the v ariables E x , E y instead o f E x , E y , and the mu ltiplier s d x , d y are omitted in the right hand sides of Eqs. (8), (9). W e found that these correctio ns are necessary fo r the system (5)–(7), (12), (13) to b e obtained. According to Ref. [1 ], an evolution of the unidirectional t wo-comp onent electro magnetic pulses in an anisotropic t wo-level medium is describ ed b y the version (5)–(7), (12), (13) of the RMB equations. W e believe this sta te- men t to b e misleading since the deriv a tion of Eqs. (12), (13) co n ta ins essential o mission. T o ex plain why this statement in [1] is incor rect we consider the Maxwell equations for the comp onents E ′ x and E ′ y of the elec tr ic field. So, we have ∂ 2 E ′ x ∂ z 2 − n 2 c 2 ∂ 2 E ′ x ∂ t 2 = 4 π c 2 ∂ 2 P ′ x ∂ t 2 , (14) ∂ 2 E ′ y ∂ z 2 − n 2 c 2 ∂ 2 E ′ y ∂ t 2 = 4 π c 2 ∂ 2 P ′ y ∂ t 2 , (1 5) where the comp onents of the medium p ola rization a re P ′ x = − N T r ( ˆ ρ ∂ ˆ H ∂ E ′ x ) , (16) P ′ y = − N T r ( ˆ ρ ∂ ˆ H ∂ E ′ y ) (17) (compare with Eqs. (10), (11)). Substitution of (2) in to (16), (17) leads to the s tandard formulas: P ′ x = N T r ( ˆ ρ ˆ d x ) , P ′ y = N T r ( ˆ ρ ˆ d y ) . F rom Eqs. (14), (15) and the tra ns formation (3), one obtains the wa ve equations on transfor med components E x and E y : ∂ 2 E x ∂ z 2 − n 2 c 2 ∂ 2 E x ∂ t 2 = 4 π c 2 ∂ 2 ˜ P x ∂ t 2 , (18) ∂ 2 E y ∂ z 2 − n 2 c 2 ∂ 2 E y ∂ t 2 = 4 π c 2 ∂ 2 ˜ P y ∂ t 2 , (1 9) where q ua nt ities ˜ P x and ˜ P y are defined by the relation ˜ P x ˜ P y ! = T P ′ x P ′ y ! . (20) A co nnection b etw een the comp onents P ′ x , P ′ y and P x , P y exists als o . T a king into account (3), w e deduce from Eqs. (10), (11) and (16), (1 7) that P ′ x P ′ y ! = T T P x P y ! . (21) Define the dimensionles s par ameter ε = d xx d y x + d xy d y y δ x δ y . If the condition ε = 0 (22) holds, then matrix T is orthogonal: T T = T − 1 . It c a n easily be seen from Eqs. (20 ) and (21) that ˜ P x = P x , 3 ˜ P y = P y in this case. F ormula (3) under this condition is nothing but the rotation transforma tion. In the general cas e ( ε 6 = 0), we ha ve fro m (20), (21) that ˜ P x 6 = P x and ˜ P y 6 = P y , i.e. the r ight ha nd sides of the equations (8), (9) and (18), (19) on tra nsformed comp o- nent s E x , E y are different. This shows that it is incor rect to determine the quantities P x and P y in Eqs . (8), (9) by means of the fo r mulas (1 0) and (11) if ε 6 = 0 . These for- m ula s are v alid in the pa rticular case ε = 0 when trans- formation (3) is the rotation transformation. Thus, the version (5)–(7), (1 2), (13) o f the RMB equations can b e applied only if the co nditio n (22) is impo sed o n the e le- men ts of the matrices ˆ d x and ˆ d y . It is wrong to exploit this version in the genera l case. An applicatio n of the unidirectional pro pagation ap- proximation to E qs. (18) and (19) gives ∂ E x ∂ χ ′ = R z E y − µ x S y − ε ( R z E x − µ y S x ) , (23) ∂ E y ∂ χ ′ = − R z E x + µ y S x + ε ( R z E y − µ x S y ) . (24) The wrong version (5)– (7), (12), (13) of the RMB equa- tions contains four par ameters µ x , µ y , m x and m y con- nected b y the relation µ 2 x + µ 2 y + m 2 x + m 2 y = 1 . (25) It was claimed in [1] that the Lax pair exists for this version. The corr e ct version (5)–(7), (23), (24) of the RMB equations contains fiv e parameter s µ x , µ y , m x , m y and ε . The r e lation (25) is fulfilled a ls o. The system of the RMB equations equiv alent to the correct version w as consid- ered in Ref. [2 ]. It was s hown that this system p oss e sses the Lax pa ir in the mo s t gener al anisotropic case when all the elements of the ma trices ˆ d x and ˆ d y are arbitrary . The system of the RMB equations a nd its Lax pair were written in [2] in the terms o f physical v ariables and parameters . Having rewr itten these systems in the ter ms of the v a riables τ ′ , χ ′ and E x , E y , we o bta in equations (5)–(7), (23 ), (2 4) and their Lax pair: ∂ ψ ∂ τ ′ = L ( λ ) ψ , (26) ∂ ψ ∂ χ ′ = A ( λ ) ψ . (27) Here ψ = ψ ( τ ′ , χ ′ , λ ) is a solution of the Lax pair, λ is the sp ectral par ameter, 2 × 2 matr ic es L ( λ ) and A ( λ ) a re defined a s given L ( λ ) = 1 2      i  λ 2 − b λ 2  λE ∗ + δ 2 δ 1 E λ λδ 1 E + δ ∗ 2 E ∗ λ − i  λ 2 − b λ 2       , (28) A ( λ ) = r      − i  λ 2 − b λ 2  R z µ x µ y δ 1 R µ x µ y R ∗ i  λ 2 − b λ 2  R z      , (29) E = E x − s ∗ E y √ 1 − ε 2 + δ 3 δ 1 , R = λQ ∗ + δ 2 δ 1 Q λ , Q = δ 3 S z + δ 4 S x + δ 5 S y , δ 1 = − p 1 − ε 2 µ 2 x µ 2 y + µ 2 x m 2 y + µ 2 y m 2 x µ x µ y , δ 2 = s 2 ( µ 2 x + m 2 x ) + µ 2 y + m 2 y + 2 sm x m y 4 , δ 3 = µ 2 y m x − s ∗ µ 2 x m y µ x µ y , δ 4 = µ 2 y + m 2 y + s ∗ m x m y µ y , δ 5 = − s ∗ ( µ 2 x + m 2 x ) + m x m y µ x , b = | δ 2 | 2 δ 2 1 , s = ε + i p 1 − ε 2 , r = − 1 2 √ 1 − ε 2 λ 2 + b λ 2 + 1 + 2 εm x m y 2 δ 1 . It can b e chec ked immediately that the ov erdetermined system (26), (27) is the Lax pa ir o f the correct version of the RMB equations (5)–(7), (23), (24 ). Indeed, the compatibility co ndition of Eqs . (26), (27) is ∂ L ( λ ) ∂ χ ′ − ∂ A ( λ ) ∂ τ ′ + [ L ( λ ) , A ( λ )] = 0 . (30) A substitution o f (28) and (29) int o (30) yields the s ystem of the RMB equations (5)–(7), (2 3), (24). The system eq uiv alen t to Eqs. (5)–(7), (2 3), (24) is obtained by a pplying the unidirectiona l pro pa gation ap- proximation to the Maxwell equations (14), (15). The co efficients of the Lax pair found in [2] for this system were e xpressed directly through the e le ments of the ma- trices ˆ d x and ˆ d y . [1] A. A. Zab olotskii, Phys. Rev. E 77 , 036603 (2008). [2] N. V. Ustinov, Pro c. SPIE 6725 , 67250F (2007); nlin.SI/0705.2833.