We present physical onsets of novel integrable generalizations of - PDF document

Self-Induced Transparency (SIT) in a Dispersive Medium A. A. Zabolotskii ∗ Institute of Automation & Electrometry of Siberian Branch of the RAS, Academic Koptug ave.1, 690090 Novosibirsk, Russian Federation (Dated:) Abstract We present physical onsets of novel integrable generalizations of the Maxwell-Bloch equa- tions describing electromagnetic field interac- tion with a two-level systems (TLS). ∗ Electronic address: zabolotskii@iae.nsk.su 1

Self-induced-transparency (SIT) soliton phenomenon in two- level atomic systems is one of the most well-known coherent pulse propagation phenomena. S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18 , 908 (1967). 2. Integrable Maxwell-Bloch equations (MBE) in a two-level system (TLS) with slow varying envelope approximation. G. L. Lamb. Jr, Rev. Mod. Phys. 43 , 99 (1971). 3. Integrable generalizations of the MBEs in TLS. A. A. Zabolotskii, Phys. Lett. A 124 , 500 (1987).(Nonlinear Stark shift). M. Agrotis, N.M. Ercolani, S.A. Glasgow, J.V. Moloney, Physica D, 138 134 (2000).(Permanent dipole momentum) A.A. Zabolotskii, JETP Lett. 77 , 464 (2003).(Two polarizations) H. Steudel, A.A. Zabolotskii, R. Meinel, Phys. Rev. E 72 , 056608 (2005). A.A. Zabolotskii, Phys. Rev. E 77 , 036603 (2008).(Two polariza- tions + permanent dipole). + N-level systems, + unified models (Nonlinear Schr odinder + Maxwell-Bloch). 2

I. DISPERSIVE HOST MEDIUM The Maxwell equation is ∂ 2 E ∂ 2 E ∂ 2 P nl ∂x 2 − 1 ∂t 2 = 4 πN ∂t 2 , (1) c 2 c 2 here N is a density of the TLS. Nonlinear part of the polarizability is � t ǫ ( t − t ′ ) P TLS ( x, t ′ ) dt ′ P nl ( x, t ) = (2) 0 The susceptibility ǫ ( t ) describes the retarded reaction. For a TLS P TLS = ( d 21 ρ 12 + d 12 ρ 21 ), where d 12 = d ∗ 21 are the elements of the dipole matrix d ij . ρ ij , i, j = 1 , 2 is the density matrix of the TLS. Dielectric medium � � ∆ ǫ p ω 2 p ǫ Lorentz ( ω ) = ǫ 0 ǫ ∞ + , (3) p − ω 2 − 2 i Γ p ω ) ( ω 2 Present the electromagnetic field amplitude as � E ( x, t ) e i ( k 0 x − ω 0 t ) + E ∗ ( x, t ) e − i ( k 0 x − ω 0 t ) � E ( x, t ) = 1 , (4) 2 where ± ω 0 and ± k 0 are the carrying frequencies and the wave vectors, respectively, ω 0 = ck 0 . E ( x, t ) is the slow envelope: � � � � � ∂ 2 E � � � ∂ E � � � � � ≪ ω 0 � , (5) � � ∂t 2 ∂t � � � � � ∂ 2 E � � � ∂ E � � � � � ≪ k 0 � . (6) � � ∂x 2 ∂x 3

Let � S ( x, t ) e i ( k 0 x − ω 0 t ) + S ( x, t ) ∗ e − i ( k 0 x − ω 0 t ) � P TLS ( x, t ) = d 12 , (7) 2 S ( x, t ) is the slow amplitude of off-diagonal elements of the density matrix ρ 12 ( x, t ): ρ 12 ( x, t ) = S ( x, t ) e i ( k 0 x − ω 0 t ) , ρ 21 ( x, t ) = S ∗ ( x, t ) e − i ( k 0 x − ω 0 t ) . (8) Neglect the terms ( | ∂ t S | /ω 0 ) n , n = 1 , 2, ( | ∂ t ǫ ( t ) | /ω 0 ) n and ( | ∂ t S ( t ) | /ω 0 ) n , n = 1 , 2. Let � � � ∂ 2 ǫ � ǫ ( ω ) = ǫ ( ω 0 )+ ∂ǫ ( ω − ω 0 )+ 1 � � ( ω − ω 0 ) 2 + · · · . (9) � � ∂ω 2 ∂ω 2 ω = ω 0 ω = ω 0 Then � � � k P nl ( x, t ) = e i ( k 0 x − ω 0 t ) � ∂ k ǫ � 1 i ∂ � d 21 S ( x, t )( x, t ) � ∂ω k k ! ∂t ω = ω 0 k � � � k +e − ( k 0 x − ω 0 t ) � ∂ k ǫ � 1 i ∂ � d 21 S ∗ ( x, t )( x, t ) . (10) � ∂ω k k ! ∂t ω = − ω 0 k From Maxwell equations we get � 1 � = − 2 πNω 2 ∂ E ∂t + ∂ E 0 d 12 ie i ( k 0 x − ω 0 t ) c 2 c ∂x � � � ∂ � k e i ( k 0 x − ω 0 t ) � i k q k × S ( x, t ) , (11) ∂t k where � ∂ k ǫ ( ω ) � q k = 1 � . (12) � ∂ω k k ! ω = ω 0 4

Maxwell equation, neglecting all terms with k > 2, � � ∂x = 2 πω 2 ∂ 2 S 1 ∂ E ∂t + ∂ E 0 Nd 21 ∂S iq 0 S − q 1 ∂t − iq 2 . (13) k 0 c 2 ∂t 2 c The Bloch equations are: ∂ t ρ 12 = − iω 12 ρ 12 − i ( ρ 11 − ρ 22 ) d 12 � E, (14) ∂ t ρ 11 = id 12 � E ( ρ 21 − ρ 12 ) , (15) ∂ t ρ 22 = id 12 � E ( ρ 12 − ρ 21 ) , (16) here ω 12 is a frequency of the two-level transition. Novel integrable dispersive Maxwell-Bloch equations (DMBEs): ∂S ∂τ = iνS − i U S z , (17) ∂S z ∂τ = i ( U S ∗ − U ∗ S ) , (18) ∂ 2 S ∂ U ∂S ∂χ = ir 0 S + r 1 ∂τ + ir 2 (19) ∂τ 2 , here τ = ( t − x/c ) ω R , ν = ( ω 0 − ω 12 ) /ω R , S z = ρ 11 − ρ 22 , and U = d 12 E , (20) � ω R r 0 = q 0 , r 1 = − q 1 ω R , r 2 = − q 2 ω 2 R , (21) ∂ c � ω R ∂ ∂χ = ∂x. (22) 2 πd 2 12 ω 0 N 5

II. A ZERO CURVATURE PRESENTATION r 0 , r 1 , r 2 , ν ∈ R , r 2 � = 0 , r 0 − νr 1 − ν 2 r 2 � = 0.   − iλ m 0 ( λ + b − ) U  Φ ,  ∂ τ Φ = (23) m 0 ( λ + b + ) U ∗ iλ   iα 0 S z µ ( α 1 S + ir 2 ∂ τ S )  Φ ,  ∂ χ Φ = A Φ ≡ (24) µ ( α 1 S ∗ − ir 2 ∂ τ S ∗ ) − iα 0 S z � µ = m 0 ( λ + b − ) , � µ = m 0 ( λ + b + ) , (25) � r 2 b ∓ = r 1 1 + 4 r 0 r 2 ∓ , (26) 4 r 2 4 r 2 2 r 2 m 2 0 = , (27) r 0 − νr 1 − ν 2 r 2 α 0 = r 0 − 2 r 1 λ − 4 r 2 λ 2 α 1 = r 0 + νr 1 + 2 νr 2 λ , . (28) 2( ν + 2 λ ) ν + 2 λ Spectral problem is generalization of the Wadati-Konno-Ichikawa (WKI) problem. Inverse transform technique for ( b ± = 0) by K. Konno et al, (1981). Reduction is : ν = r 1 = 0, Im ( i U ) = 0 ⇒ ∂ 2 θ ∂ 2 ∂τ∂χ = sin θ + r 2 ∂τ 2 sin θ, (29) i U = ∂ τ θ . A. Fokas (1995). Eq. (29) transforms to Rabelo equations (R. Beals, M. Rabelo (1989)). 6

There are 4 sets of symmetries determines by constants. I. Abnormal dispersion (subindex below 1): r 2 < 0 , m 2 0 < 0 , r 2 1 − 4 r 0 | r 2 | < 0 . (30) � 4 r 0 | r 2 | − r 2 1 b ∓ = β 0 ± iβ 1 , β 1 = (31) 4 | r 2 | II. Normal dispersion (subindex 2): r 2 > 0 , m 2 0 > 0 , r 2 1 + 4 r 0 r 2 > 0 . (32) � 4 r 0 r 2 + r 2 1 b ∓ = β 0 ± β 2 , β 2 = , (33) 4 r 2 β 0 = r 1 . (34) 4 r 2 λ → λ − β 0 + gauge transform:    e iβ 0 τ 0  Φ 1 , 2 , Φ 1 , 2 = (35) e − iβ 0 τ 0   − iλ ( λ + iβ 1 ) W 1  Φ 1 ,  ∂ τ Φ 1 = (36) − ( λ − iβ 1 ) W ∗ iλ 1 where W 1 = i | m 0 | U e − 2 iβ 0 τ , (37) 7

and   − iλ ( λ + β 2 ) W 2   Φ 2 , ∂ τ Φ 2 = (38) ( λ − β 2 ) W ∗ iλ 2 where W 2 = | m 0 | U e − i 2 β 0 τ . (39) Introduce the new variables T, Θ and the new functions F 1 , 2 ( χ, τ ), G 1 , 2 ( χ, τ ) as � τ � τ G − 1 G − 1 1 χ, τ ′ ) dτ ′ , Θ = 2 ( χ, τ ′ ) dτ ′ , T = (40) 0 0 W 1 , 2 1 F 1 , 2 = � 1 ± | W 1 , 2 | 2 , G 1 , 2 = � 1 ± | W 1 , 2 | 2 . (41) Then   − iλG 1 ( λ + iβ 1 ) F 1  Φ 1 ,  ∂ T Φ 1 = (42) − ( λ − iβ 1 ) F ∗ iλG 1 1 and   − iλG 2 ( λ + β 2 ) F 2   Φ 1 , ∂ Θ Φ 2 = (43) ( λ − β 2 ) F ∗ iλG 2 2 F 1 , 2 = 0 , S z = − 1 , S = 0 , T, Θ → ±∞ . (44) The ISTM applications by means of solution of the Marchenko equations or Riemann-Hilbert problem give 8

solitons in implicit form. Abnormal dispersion ( r 2 < 0) F 1 ( τ, χ ) = 2 ζ 0 e − ic 1 − iψ 1 cosh( ψ ) , (45) cosh( ψ ) 2 + ζ 2 0 G 1 ( τ, χ ) = cosh( ψ ) 2 − ζ 2 0 , (46) cosh( ψ ) 2 + ζ 2 0 � � 0 − 4 | r 2 | η 2 T − r ′ ψ = 2 η ν ′ 2 + 4 η 2 χ + c 0 , (47) 0 − 4 | r 2 | η 2 ψ 1 = ν ′ r ′ ν ′ 2 + 4 η 2 χ, (48) η ζ 0 = η + | β 1 | . (49) Time dependence is obtained by integration of ∂ τ T = G − 1 1 ( τ ). Soliton solution � � 2 ζ 0 ζ 0 ψ − � arctanh � tanh[ ψ ] = 2 η [ τ − τ 0 ( χ )] , (50) ζ 2 ζ 2 0 + 1 0 + 1 τ 0 ( χ ) = − ψ ( τ = 0) / (2 η ). G 1 ( τ, χ ) = − 2 iζ 0 e 2 iβ 0 τ − ic 1 − iψ 1 cosh( ψ ) W = F 1 ( τ, χ ) � � , (51) cosh 2 ( ψ ) − ζ 2 | m 0 | 0 9

Figure 1: η = 1 . 0 . Modulus of the soliton amplitude U a vs τ is shown by the dashed line for | r 2 | = 0 . 0001, by pointed line for | r 2 | = 0 . 02, and by solid line for | r 2 | = 2 . 0 . III. SOLITON. NORMAL DISPERSION ( r 2 > 0 ) C 2 ( χ ) = b 2 ( χ ; iη ) /∂ λ a 2 ( χ ; λ ) | λ = iη � = 0 , (52) where a 2 ( χ ; iη ) = 0. 10

Denote | C | 2 ω 2 e − 2 i ( λ ∗ − λ )(Θ − V 1 χ ) = e − 2 θ , (53) β 2 − λ ∗ β 2 + λ = e − 2 iδ , (54) Im λ κ = | β 2 + λ | , (55) C (0) = e iδ 1 | C (0) | . (56) Then the soliton is D 2 ( τ ) = | cosh( θ + iδ ) | 2 + κ 2 (57) | cosh( θ + iδ ) | 2 − κ 2 F 2 ( τ ) = 2 κ cosh ( θ + iδ ) e 2 iV 2 χ − iδ − iδ 1 . (58) | cosh( θ + iδ ) | 2 − κ 2 For Im η = 0 , r 1 = 0 , ν = 0 we have V 2 = 0, δ = 0 , β 2 = � 1 / 2 r 0 /r 2 , 2 η √ r 2 κ = � , (59) 4 η 2 r 2 + r 0 � � Θ − r 0 + 4 r 2 η 2 θ = 2 η χ . (60) 4 η 2 The modulus of the soliton � 1 + 4 η 2 r 2 cosh( θ ) 2 η U n ( τ, χ ) = , (61) (1 + 4 η 2 r 2 ) cosh 2 ( θ ) + 4 η 2 r 2 θ is found by integration of D − 1 = ∂ τ Θ. 2 11

Figure 2: η = 2 . 0 . Modulus of the soliton amplitude U n vs τ is shown by the pointed line for r 2 = 0 . 001, by dashed line for r 2 = 0 . 05, and by solid line for r 2 = 0 . 2 (Topless soliton). 12