Abstract

The exact electric polarization operator of the Maxwell–Bloch equations for a three-level Λ-type system is derived without assuming equality of the transition dipole moments. Numerical solution of the resulting solitary wave equations yields a richer spectrum than found earlier for identical dipole moments [ Hioe and Grobe, Phys. Rev. Lett. 73, 2559 (1994) ]. The two components of the new solitary waves each have a zero-pulse area and take the form of modulated oscillations.

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References

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  1. S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
    [Crossref]
  2. M. J. Konopnicki and J. H. Eberly, “Simultaneous propagation of short different-wavelength optical pulses,” Phys. Rev. A 24, 2567-2583 (1981).
    [Crossref]
  3. F. T. Hioe and R. Grobe, “Matched optical solitary waves for three- and five-level systems,” Phys. Rev. Lett. 73, 2559-2562 (1994).
    [Crossref] [PubMed]
  4. Q.-H. Park and H. J. Shin, “Systematic construction of multicomponent optical solitons,” Phys. Rev. E 61, 3093-3106 (2000).
    [Crossref]
  5. H. Eleuch and R. Bennaceur, “An optical soliton pair among absorbing three-level atoms,” J. Opt. A, Pure Appl. Opt. 5, 28-533 (2003).
    [Crossref]
  6. N. A. Ansari, L. N. Towers, Z. Jovanoski, and H. H. Sidhu, “A semi-classical approach to two-frequency solitons in a three-level cascade atomic system,” Opt. Commun. 274, 66-73 (2007).
    [Crossref]
  7. T. Ghannam, “Optical solitons in three-level media,” M.S. thesis (Western Michigan University, 2003), QC 9999.G53x.

2007 (1)

N. A. Ansari, L. N. Towers, Z. Jovanoski, and H. H. Sidhu, “A semi-classical approach to two-frequency solitons in a three-level cascade atomic system,” Opt. Commun. 274, 66-73 (2007).
[Crossref]

2003 (1)

H. Eleuch and R. Bennaceur, “An optical soliton pair among absorbing three-level atoms,” J. Opt. A, Pure Appl. Opt. 5, 28-533 (2003).
[Crossref]

2000 (1)

Q.-H. Park and H. J. Shin, “Systematic construction of multicomponent optical solitons,” Phys. Rev. E 61, 3093-3106 (2000).
[Crossref]

1994 (1)

F. T. Hioe and R. Grobe, “Matched optical solitary waves for three- and five-level systems,” Phys. Rev. Lett. 73, 2559-2562 (1994).
[Crossref] [PubMed]

1981 (1)

M. J. Konopnicki and J. H. Eberly, “Simultaneous propagation of short different-wavelength optical pulses,” Phys. Rev. A 24, 2567-2583 (1981).
[Crossref]

1969 (1)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

H. Eleuch and R. Bennaceur, “An optical soliton pair among absorbing three-level atoms,” J. Opt. A, Pure Appl. Opt. 5, 28-533 (2003).
[Crossref]

Opt. Commun. (1)

N. A. Ansari, L. N. Towers, Z. Jovanoski, and H. H. Sidhu, “A semi-classical approach to two-frequency solitons in a three-level cascade atomic system,” Opt. Commun. 274, 66-73 (2007).
[Crossref]

Phys. Rev. (1)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457-485 (1969).
[Crossref]

Phys. Rev. A (1)

M. J. Konopnicki and J. H. Eberly, “Simultaneous propagation of short different-wavelength optical pulses,” Phys. Rev. A 24, 2567-2583 (1981).
[Crossref]

Phys. Rev. E (1)

Q.-H. Park and H. J. Shin, “Systematic construction of multicomponent optical solitons,” Phys. Rev. E 61, 3093-3106 (2000).
[Crossref]

Phys. Rev. Lett. (1)

F. T. Hioe and R. Grobe, “Matched optical solitary waves for three- and five-level systems,” Phys. Rev. Lett. 73, 2559-2562 (1994).
[Crossref] [PubMed]

Other (1)

T. Ghannam, “Optical solitons in three-level media,” M.S. thesis (Western Michigan University, 2003), QC 9999.G53x.

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Figures (8)

Fig. 1
Fig. 1

Notation for the three-level lambda configuration.

Fig. 2
Fig. 2

Rabi frequency of conventional two-state solitons computed from Eqs. (3, 4a, 4b) with p 32 = 0 for two different ϕ (solid: ϕ = 2 ; dashed: ϕ = 25 ).

Fig. 3
Fig. 3

Pulse area gradient d θ d s for hyperbolic secant-type solitary waves. (A) Single-peaked nonzero-area solitary wave of the MSP-type and (B) double-peaked solitary wave. These pulses can be generated for all values of β. The solid curve is the a-pulse and the dashed curve is the b-pulse.

Fig. 4
Fig. 4

Pulse area gradient versus s for MZA-type solitary waves. Pulses are generated by numerical solution of Eqs. (6a, 6b) for β = 1.75 for different initial phasing of the two pulses. Solid: a-pulse, dashed: b-pulse. (A) Relative phase of the components changes by 90 ° between the two adjacent peaks. (B) Another possible MZA form in which the relative phase is constant. (C) Form in which both pulses show significant modulation.

Fig. 5
Fig. 5

(A) Atomic populations and (B) dipole amplitudes, both near the peak of the pulse in Fig. 4A. In (A), solid is the population of state 2, dashed is the state 1, and dotted is the state 3. (B) Solid is the 21 polarization, dashed is the 23 polarization.

Fig. 6
Fig. 6

Same as Fig. 5A for an MSP-type solitary wave.

Fig. 7
Fig. 7

Pulse area gradient d θ b d s versus distance z for an MSP solitary wave of the first kind ( k = 0.05 , A 1 = 10 , A 2 = 20 , B 1 = 25 , and B 2 = 65 . The propagating b-component of the pulse shape is shown for times 0, 3, and 5 units.

Fig. 8
Fig. 8

Pulse area gradient d θ b d s versus the distance for MZA-type solitary waves. Solitary wave speed is 0.8 c , β = 1.75 . The propagating pulse is shown at times 0, 3.75, 12.5, and 25.0 ns .

Equations (31)

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p 12 = ( e ) e ̂ a 1 r 2 , p 32 = ( e ) e ̂ b 3 r 2
u + = σ 23 + σ 32 , u = i ( σ 23 σ 32 ) , u 3 = σ 22 σ 33 , v + = σ 21 + σ 12 , v = i ( σ 21 σ 12 ) , v 3 = σ 22 σ 11 . t + = σ 13 + σ 31 t = i ( σ 13 σ 31 ) ,
d η d t = B η ,
E a , b = e ̂ a , b A a , b ( r , t ) e ( i k z i ω a , b t ) + c.c. ,
P a , b = e ̂ a , b L a , b ( r , t ) e ( i k z i ω a , b t ) + c.c.
A a ( z , t ) z + 1 c A a ( z , t ) t = i ω a L a 4 ε 0 c = n p 12 ω a 4 ε 0 c X a ,
A b ( z , t ) z + 1 c A b ( z , t ) t = i ω b L b 4 ε 0 c = n p 32 ω b 4 ε 0 c X b ,
L a = n p 12 σ 12 = 1 2 in p 12 X a , L b = n p 32 σ 32 = 1 2 in p 32 X b .
X a = θ a ψ 3 [ θ a 2 sin ( ψ ) + 2 θ b 2 sin ( ψ 2 ) ] ,
X b = 4 θ b θ a 2 ψ 3 sin ( ψ 2 ) sin 2 ( ψ 4 ) .
s = 2 ϕ q [ z v g t ] .
d 2 θ a d s 2 = X a ,
d 2 θ b d s 2 = β X b ,
d θ a d s = p 12 2 ϕ n ω a 2 ε 0 Ω a
Ω a ( s ) = A 1 f 1 ( s ) + A 2 f 2 ( s ) ,
Ω b ( s ) = B 1 f 1 ( s ) + B 2 f 2 ( s ) ,
First kind Second kind f 1 = s n ( s , k ) f 1 = k s n ( s , k ) f 2 = c n ( s , k ) f 2 = d n ( s , k ) .
σ ̇ 11 = γ 11 σ 11 + i 2 [ Ω a * σ 21 Ω a σ 12 ] ,
σ ̇ 22 = γ 22 σ 22 + i 2 [ Ω a σ 12 Ω a * σ 21 + Ω b σ 32 Ω b * σ 23 ] ,
σ ̇ 33 = γ 33 σ 33 + i 2 [ Ω b * σ 23 Ω b σ 32 ] ,
σ ̇ 12 = ( γ 12 + i Δ a ) σ 12 + i 2 [ Ω a * ( σ 22 σ 11 ) Ω b * σ 13 ] ,
σ ̇ 32 = ( γ 32 + i Δ b ) σ 32 + i 2 [ Ω b * ( σ 22 σ 33 ) Ω a * σ 31 ] ,
σ ̇ 13 = ( γ 13 + i Δ R ) σ 13 + i 2 [ Ω a * σ 23 Ω b σ 12 ] .
u + ( t ) = 4 S θ a 2 ψ 3 sin ( ψ 2 ) sin 2 ( ψ 4 ) ,
u ( t ) = 4 R θ a 2 ψ 3 sin ( ψ 2 ) sin 2 ( ψ 4 ) ,
u 3 ( t ) = θ a 2 2 ψ 4 { θ a 2 2 θ b 2 ( θ a 2 + 2 θ b 2 ) cos ( ψ ) + 4 θ b 2 cos ( ψ 2 ) } ,
v + ( t ) = Q ψ 3 { θ a 2 sin ( ψ ) + 2 θ b 2 sin ( ψ 2 ) } ,
v ( t ) = P ψ 3 { θ a 2 sin ( ψ ) + 2 θ b 2 sin ( ψ 2 ) } ,
v 3 ( t ) = 1 2 ψ 4 { θ b 2 ( 2 θ b 2 θ a 2 ) + θ a 2 ( 2 θ a 2 + θ b 2 ) cos ( ψ ) + 4 θ b 2 θ a 2 cos ( ψ 2 ) } ,
t + ( t ) = ( P R + Q S ) ψ 4 { θ a 2 2 θ b 2 + θ a 2 cos ( ψ ) + 2 ( θ b 2 θ a 2 ) cos ( ψ 2 ) } ,
t ( t ) = ( Q R P S ) ψ 4 { θ a 2 2 θ b 2 + θ a 2 cos ( ψ ) + 2 ( θ b 2 θ a 2 ) cos ( ψ 2 ) } .

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