Optical Solitons in an Anisotropic Medium with Arbitrary Dipole Moments

Optical Solitons in an Anisotropic Medium with Arbitrary Dip ole Momen ts N. V. Ustino v Quan tum Field Theory D epartmen t, T omsk State Univ ersit y , 36 Lenin Av en ue, T omsk, 634050, Russia ABSTRA CT W e find the Lax pair for a system of r educed Maxwell–Blo ch equations that descr ibe s the propagation of tw o- comp onent extremely short electromagnetic pulses thro ugh the medium containing tw o- level quan tum particles with ar bitrary dip ole momen ts. Keyw ords: ex tremely short pulse, optical anisotr opy , self-induced transpa rency , solito n 1. INTR ODUCTIO N Generation of extremely shor t pulses 1–5 (ESP) with a dura tio n o f a few p erio ds o f light oscillatio ns ha s offered a strong incentiv e for theore tica l studies of their in ter action with matter (see reviews 6, 7 and refere nces therein). F or ob vious r easons, the slowly v arying en velope approximation commonly exploited in the nonlinear optics of quasi-mono chromatic (or ultras hort) pulses cannot b e applied to in vestigate the propagatio n of ESP . The slo wly v arying en velope approximation was not used in the ca se of the ultra short pulses in Ref. 8 , where an alterna tive approa ch to the theory of self-induced tra nsparency 9, 10 (SIT) was develop e d. This appr o ach w as based on the assumption of lo w de ns it y of the quan tum particles, whic h was consistent with conditions of the SIT exper iments. Since the backscattered wa ve is w e ak in that cas e, the order of deriv atives in the wa ve equation for the pulse field can b e reduced by using the unidirectional pr opagatio n (UP) approximation. 11 The r esulting so-called reduced Maxwell-Blo ch (RMB) equations 8 , as well as the SIT equations 10 , ar e in tegr able by the inv e rse scattering transfo r mation (IST) metho d. 12–14 This metho d is widely reco g nized as one of the most powerful to o ls in studying the nonlinear phenomena. In particular, the pulse dynamics in the integrable mo dels of nonlinear optics is describ ed by the soliton solutions . In the last years, the theoretica l inv estiga tion of coherent nonlinear effects in anisotropic media attrac ts great atten tion. 15–26 This is caused b y the sig nificant dev elo pmen t of the nanotechnologies and the metho ds of pro ducing the low-dimensional quan tum structures. Unlik e the c ase of is otropic media, the parity of the stationary states of the quantum par ticles of the aniso tropic medium is not well defined. That is why the diagona l elements of the matrix of the dip ole moment op era to r and their difference, which is called the per manent dipo le moment (PDM) of the transition, may not v anis h. An o ptical pulse propaga ting in such a medium not only induces tra nsitions b etw een these states, but also shifts the transition frequency v ia dynamic Stark effect. Owing to this, the ultra s hort pulses can propa gate in the medium in the regimes that differ fro m the SIT one. 17, 20 The dynamics of one- comp onent ESP in the anisotropic media was studied in pap ers 15, 16, 18, 19, 21, 23 . It was revealed 16 that the scalar RMB equations with P DM are integrable in the fr a meworks of the IST. The numerical inv estigation o f the pulse formatio n governed b y these equations displayed an existence of solitar y stable bipo lar signal with nonzero time area. 19 Its s o litonic nature w as established in Refs. 21 and 23. The propag ation through anisotro pic media of the tw o-co mpo ne nt E SP was investigated in 22, 24–26 . The case, where o ne of the pulse comp onents causes the quantum tra nsitions, while another o ne shifts the energy lev els , was considered in Ref. 24. Correspo nding tw o-comp onent RMB equations differ b y notations fro m the sy s tem describing the transverse-longitudinal acous tic pulse propa gation in the low-temperature parama g netic crys ta l F urth er author information: E-mai l: n ustino v@mail.ru and are in teg rable. 27 An application of the spectr a l o verlap appro ximation to thes e equations g ives one more int e g rable mo del. 25, 26 If b oth components of the ESP excite the quan tum transitions only (i.e., PDM o f the transition is equal to zero), the prop er s ystem o f the tw o-comp onent RMB equations is also in tegr able. 22 W e see that differen t particular cas es of the tw o-co mpo ne nt RMB equations are in teg rable by mea ns of the IST method. The main aims of the presen t study are to join these ca ses and to find more gener al conditions , under which the in teg r ability of the RMB equations takes place. 2. TW O-CO MPONENT SYSTEM OF THE MAXWELL–BLOCH EQ UA TIONS Let us consider the medium con taining tw o -level quantum par ticles. Assume for simplicity that the medium is isotropic, i.e. an a nisotropy is induced b y the quan tum particles. Let the plane ESP propaga te through the medium in the p ositive direction of y ax es o f the C a rtesian co o rdinate system. Then the Max well equations yield the following sys tem for pr o jections E x and E z of the electric field: ∂ 2 E x ∂ y 2 − n 2 c 2 ∂ 2 E x ∂ t 2 = 4 π c 2 ∂ 2 P x ∂ t 2 , (1) ∂ 2 E z ∂ y 2 − n 2 c 2 ∂ 2 E z ∂ t 2 = 4 π c 2 ∂ 2 P z ∂ t 2 , (2) where P x and P z are the components of p ola rization connected with the tw o -level pa rticles; n is the refractive index o f the medium; c is the sp eed of light in free space. T o desc rib e the ev o lution of q ua nt um particles, w e exploit the von Neumann equation for density matrix ˆ ρ : i ¯ h ∂ ˆ ρ ∂ t = [ ˆ H , ˆ ρ ] . (3) Here Hamiltonian H o f the t wo-lev el par ticle is wr itten as follows ˆ H = diag(0 , ¯ hω 0 ) − ˆ d x E x − ˆ d z E z , (4) where ω 0 is the resonant frequency of qua n tum transition; ˆ d x and ˆ d z are the matrices of the pro jection of the dipo le mo ment op erator on x and z axes, res pectively; ¯ h is the Pla nk’s consta n t. The expres s ions for the p olariza tio n c o mpo nent s rea d as P x = N T r ( ˆ ρ ˆ d x ) , (5) P z = N T r ( ˆ ρ ˆ d z ) , (6) where N is the concentration o f the quant um par ticles. W e supp ose that the matrices of the dip ole mo men t ar e defined a s given ˆ d x =  D x d 1 d 1 0  , (7) ˆ d z =  D z δ + id 2 δ − id 2 0  , (8) where d 1 , d 2 , δ , D x and D z are real pa rameters. This repr esentation for the dipo le moment ma trices is ge neral, but we hav e r educed it to a simpler for m. Quantities D x and D z are no thing but the PDM pro jections. Using equatio ns (3)–(8), w e find ∂ W ∂ t = 2 d 2 ¯ h E z U − 2  d 1 ¯ h E x + δ ¯ h E z  V , (9) ∂ U ∂ t = −  ω 0 + D x ¯ h E x + D z ¯ h E z  V − 2 d 2 ¯ h E z W , (10) ∂ V ∂ t =  ω 0 + D x ¯ h E x + D z ¯ h E z  U + 2  d 1 ¯ h E x + δ ¯ h E z  W , (11) where W = ρ 22 − ρ 11 2 , U = ρ 12 + ρ 21 2 , V = ρ 12 − ρ 21 2 i are the comp onents of the Bloch vector; ρ j k ( j, k = 1 , 2) a re the elements o f the dens ity ma tr ix. Let the concentration of the quant um particles b e small: N k ˆ d x, z k 2 / ¯ hω 0 ≪ 1, where k C k is the norm of the matrix C . Then we a re able to r e duce the or der of der iv ativ es in the wa ve equations for the electr ic field comp onents. An applica tion of the UP approximation 11 to equations (1), (2) and exclusion o f the time der iv atives of the elements o f the density matr ix with the help of (9)–(11) give ∂ E x ∂ y + n c ∂ E x ∂ t = 4 π N nc ¯ h h S E z + ¯ hω 0 d 1 V i , (12) ∂ E z ∂ y + n c ∂ E z ∂ t = − 4 π N nc ¯ h h S E x + ¯ hω 0 ( d 2 U − δV ) i , (13) where S = 2 d 1 d 2 W + d 2 D x U + ( d 1 D z − δ D x ) V . The tw o-co mpo nen t system o f RMB equations (9)–(13) describes t he propagatio n o f v ector ESP in the medium containing t wo-lev el quantum par ticles with ar bitrary dip ole moments. It is seen that b oth electric field comp onents, as well a s a n y sup erp osition of them, fulfill tw o different functions in the general case: they excite the quantum transitio ns a nd s hift the energy levels due to P DM. Obviously , the system o btained coincides with the RMB equations for isotropic medium 8 if w e put P DM equa l to zero ( D x = D z = 0) and E z = d 2 = 0 (or E x = d 1 = 0 ). Other integrable cases of equatio ns (9)–(13) were studied in 16, 22, 24 . 3. LAX P AIR An int e g rability of the nonlinear equations given by means of the IST metho d implies an opp ortunity to represe n t them as the co mpatibilit y c ondition of the overdetermined system o f linea r equations (Lax pair). Consider the following Lax pair ∂ ψ ∂ t = L ( λ ) ψ ( λ ) , ∂ ψ ∂ y = A ( λ ) ψ ( λ ) , (14) where λ is so-called sp ectral para meter; L ( λ ) a nd A ( λ ) are matrices; ψ = ψ ( y , t , λ ) is a solution of the o verde- termined s ystem. Its co mpatibilit y condition reads as ∂ L ( λ ) ∂ y − ∂ A ( λ ) ∂ t + [ L ( λ ) , A ( λ )] = 0 . (15) Using the o verdetermined linear systems for the cases discussed in 16, 22 , w e can offer p os sible for m of the Lax pair for (par ticular case of ) the equations (9)–(13). This form contains coefficie nts in matrices L ( λ ) and A ( λ ), which are a sub ject of subseq uen t definition, and may b e v alid only under impos ing some constra in ts o n the elements o f matr ices (7) and (8) of the dip ole moment pro jections. H aving written down the compatibility condition and having excluded the deriv atives with the help of (9)–(1 3), we obta in the overdetermined system o f algebraic equa tio ns on the entered co efficients. The num b er of the co e fficien ts should b e great eno ugh to include int o a consideratio n b oth the cases we start with. F or this reaso n, the sy stem of the algebra ic equations is strongly overdetermined. Nevertheless, we hav e b een able to solve this system a fter straig ht forward, but tedio us calculations. Moreov er , it has b een done without imp osing additional constraints on the dip ole momen ts o f the transition. W e hav e found the following express ions for ma tr ices L ( λ ) and A ( λ ) of sys tem (14): L ( λ ) =      i 2  λ 2 − b λ 2  1 2 √ 2¯ h  λE ∗ + δ 2 δ 1 E λ  δ 1 2 √ 2¯ h  λE + δ ∗ 2 δ 1 E ∗ λ  − i 2  λ 2 − b λ 2       , (16) A ( λ ) = 2 π N nc 1 λ 2 + b λ 2 + B      i ¯ h  λ 2 − b λ 2  S √ 2 d 1 d 2 ¯ h 2 δ 1  λQ ∗ + δ 2 δ 1 Q λ  √ 2 d 1 d 2 ¯ h 2  λQ + δ ∗ 2 δ 1 Q ∗ λ  − i ¯ h  λ 2 − b λ 2  S      − n c L ( λ ) , (17) where E = E x + iE z + δ 3 δ 1 , Q = δ 3 W + δ 4 U + δ 5 V , δ 1 = 2 ω 0 d 1 d 2 h (4 d 2 1 + D 2 x ) d 2 2 + ( d 1 D z − δ D x ) 2 i , δ 2 = 2 d 2 2 + 2( δ + id 1 ) 2 + ( D z + iD x ) 2 2 , δ 3 = 2¯ h d 1 d 2 h d 2 2 D x + ( δ − id 1 )( δ D x − d 1 D z ) i , δ 4 = ¯ h d 1 d 2 h δ D x D z − d 1 (4 d 2 2 + D 2 z ) + i ( d 1 D z − δ D x ) D x i , δ 5 = ¯ h d 1 h 4 d 1 δ + D x D z − i (4 d 2 1 + D 2 x ) i , b = | δ 2 | 2 δ 2 1 , B = 1 δ 1 h 4( d 2 1 + d 2 2 + δ 2 ) + D 2 x + D 2 z i . Substituting (16) and (17 ) in to (15), we see that the equalit y is fulfilled only if equa tions (9)–(13) take pla ce. Thu s, the tw o-co mpo ne nt RMB equations (9)–(13) belo ng to the cla ss of nonlinear models in tegrable b y means of the IST metho d at any v alues o f parameter s d 1 , d 2 , δ , D x and D z . Equations (15), (1 6 ) and (17) with δ = D x = D z = 0 give the Lax pair presen ted in Ref. 22. A connection of the Lax pair we found with the pair obtained in Ref. 1 6 is not so o b v ious since d 1 d 2 = 0 in the last ca se. 4. CONCLUSION W e have established the integrability in the fra meworks of the IST metho d o f the system o f tw o-comp onent RMB equations for anis o tropic medium in the most ge ner al ca se. This implies that these equa tions have multi-soliton solutions, Darb oux and B¨ acklund transformations , infinite hiera rchies of the c o nserv ation laws and infinitesimal symmetries and o ther attributes of the in tegra ble models. 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