Abstract

A new scheme of five-level hyper V-type atomic system is proposed with the aim of generating slow temporal vector optical solitons. Two transitions in the five-level atomic medium independently interact with the two orthogonally polarized components of a low intensity linear-polarized pulsed probe field, while two other transitions are driven by control laser fields. We demonstrate that various distortion-free slow temporal vector optical solitons, such as bright-bright, bright-dark, dark-bright and dark-dark vector solitons, can be evolved from the probe field. Besides, we also show that the modified Hubbard model that includes the Manakov system may be realized by adjusting the corresponding self- (cross-) phase modulation and dispersion effects of this system.

©2009 Optical Society of America

1. Introduction

The interaction of radiation with nonlinear medium gives rise to a number of phenomena, including solitons [1–18], bistability [19, 20], entanglement [21–23], four-wave mixing (FWM) [24, 25], and electromagnetically induced transparency (EIT) [26–28]. Formation of vector optical solitons in various nonlinear medium has been extensively investigated in theory and experiment [29–43]. Vector solitons are a particular class of solitons in which each component of the optical field remains almost stable over long propagation distances. They form by the proper balance between dispersion, self-, and cross-modulation (SPM and XPM) in two (or more) components in a nonlinear medium. Compared with the scalar solitons, vector solitons have richer propagation dynamical properties, which make them having promising applications for the design of all-optical switching, logic, and computation. Thus far, most vector optical solitons were produced with intense electromagnetic fields and substantial propagation distance in passive optical media, such as optical fibers [36–43], which lack of distinctive energy levels and strong nonlinear effects. Therefore, far-off resonance excitation schemes are generally employed to avoid unmanageable attenuation and distortion of optical field. As a consequence, vector optical solitons produced in this way generally travel with a propagation speed very close to velocity of light in vacuum.

In recent years, there has been a significant surge of study activities on wave propagations in highly resonant media such as the atomic system [26–28]. Some of the striking features of wave propagation in such a highly resonant medium are the vanishing linear absorption and large nonlinear effects as well as the significant reduction of the propagation velocity of the optical fields [44–49]. Such the attractive properties have been shown to result in several new propagation effects in the field of fundamental physics. In particular, well-characterized and distortion-free slow optical waves, including scalar solitons [4, 5] and vector solitons [31–33], have been testified that they could be formed in several kinds of multi-level atomic systems.

In present paper, we investigate the formation of slow temporal vector optical solitons in a highly resonant nonlinear optical medium, namely, a new scheme of five-level hyper V-type atomic system, illustrated in Fig. 1(a). By analyzing the linear and nonlinear dynamics of the two orthogonally polarized components of a low intensity linear-polarized pulsed probe field, we drive the corresponding nonlinear governing equations, i.e., two coupled nonlinear Schrödinger (NLS) equations, which admit of various distortion-free temporal vector optical solitons, such as bright-bright, bright-dark, dark-bright and dark-dark vector solitons. We show that these results are produced from the proper balance between dispersion, SPM, and XPM. Moreover, we demonstrate that, depending on the the choice of corresponding parameters, the modified Hubbard model which includes the Manakov system may be realized in this five-level hyper V-type atomic system.

 figure: Fig. 1.

Fig. 1. (a) Five-level atomic system in a hyper V-type configuration. The strong cw control field with frequency ω c1 (ω c2) and Rabi frequency Ω c1c2) couples to the ∣1〉 ↔ ∣3〉 (∣2〉 ↔ ∣4〉) transition, and the σ - σ + component of the weak probe field with frequency ωp and Rabi frequency Ωp couples to the ∣0〉 ↔ ∣1〉 (∣0〉 ↔ ∣2)) transition. Δss - Δ) and Δt1t2) are corresponding one- and two-photon detunings with Δ = 2μ 𝓑g𝓑/h̄ being the Zeeman shift of levels ∣1〉 and 2〉. (b) Possible arrangement of experimental apparatus. 𝓑 is an applied magnetic field. Ωc1 and ωc2 represent two control fields. σ - and σ + denote two orthogonally polarized components of a probe field.

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Our paper is organized as follows. In Sec. II, we describe the corresponding model and discuss briefly the linearized dynamics of the two components of the probe field. Sec. III aims to investigate the corresponding nonlinear dynamics to derive two coupled NLS equations and show that various types of slow temporal vector optical solitons can propagate through the atomic system. The modified Hubbard model which includes the Manakov system in the five-level hyper V-type atomic system is also brief discussed in this section. Conclusions with a brief summary are made in Sec. IV.

2. Model and linearized dynamics

We consider a lifetime-broadened five-state atomic system in a hyper V-type configuration (see Fig. 1(a)). The degeneracy of the two atomic sublevels ∣1〉 and ∣2〉 are removed by the applied magnetic field 𝓑, which determine the Zeeman split Δ = 2μ 𝓑g𝓑/h̄, where flag is the Bohr magneton, and g is the gyromagnetic factor. The atom interacts with three laser fields. A strong continuous wave (cw) control field with frequency ω c1 and Rabi frequency Ω c1 couples to the ∣1〉 ↔ ∣3〉 transition, while another one with frequency ω c2 and Rabi frequency Ωc2 couples to the ∣2〉 ↔ ∣4〉 transition. A low intensity linear-polarized pulsed probe field with frequency ωp and Rabi frequency Ωp have two orthogonally polarized components with the σ- (σ +) component coupleing to the ∣0〉 ↔ ∣1〉 (∣0〉 ↔ ∣2〉) transition. Thus the five-level hyper V-type atomic system is composed of two cascade-type configurations, both of them share the ground-state level ∣0〉 [7, 8, 28]. In order to cancel Doppler broadening and reduce interatomic collisions, the atoms are trapped in a cell at enough low temperature. Figure 1(b) is shown as a possible arrangement of experimental apparatus, which is in the collinear Doppler-free geometry.

The electric fields of the probe field and the control fields can be written as Ep = Ep- + Ep+ = [e+ E p- + e⃗-E p+)exp(-pt + i kp · r⃗) + c.c. and Ec1,c2 = ec1,c2 E c1,c2exp(-i ω c1,c2 t + i kc1,c2 · r⃗) + c.c, respectively. Here, e- = (x⃗ - iy⃗)/√2 and e+ = (x + iy)√2 are the unit vectors of the σ - and σ + circular polarization components with the slowly varying envelopes E - and E +, and ec1,c2 are the unit vectors of the control fields with the envelopes E c1,c2 Taking free Hamiltonian Ĥ0/h̄ = ωp∣l〉〈l∣ + ωp∣2〉〈2∣ + ωp + ω c1)∣3〉(3∣ + (ωp + ω c2)∣4〉〈4∣, then under electric-dipole and rotating-wave approximations, we have the Hamiltonian in the interaction picture as follows,

Ĥinth̄=Δs11(ΔsΔ)22Δt133Δt244
(Ωp1eikp·r10+Ωp2eikp·r20+Ωc1eikc1·r31+Ωc2eikc2·r42+H.c.),

where Δs = ωp - ωps - Δ) is the one-photon detuning, and Δt1 = ωp + ω c1 - ω30 (Δt2 = ωp + ω c2 - ω 40) is the two-photon detuning with ωjn denoting the corresponding transition frequencies. Ωc1 = (μ31 · ec1)E c1/h̄, Ωc2 = (μ42 · ec2)Ec2/h̄, Ωp1 = (μ10 · e-)E p-/h̄, and Ωp2 = (μ20 · e+)E p+/h̄ are Rabi frequencies with μij being the dipole moment for the relevant transitions ∣i〉 ↔ ∣j〉.

In order to study the dynamics of this five-level hyper V-type atomic system, we define the atomic state as ∣Ψ〉 = B 0(t)∣0〉 + B 1(t)e ikp·r⃗∣1〉 +B 2(t)e i kp·r∣2〉 + B 3(t)e i(kp+kc1r⃗∣3〉 + B 4(t)e i(kp+kc2r∣4〉. Then, from the Schrödinger and Maxwell’s equations, we readily obtain the atomic equations of motion and the wave equations for the time-dependent two polarization components of the probe field,

B1t=i(Δs+iγ1)B1+iΩc1*B3+iΩp1B0.
B2t=i(ΔsΔ+iγ2)B2+iΩc2*B4+iΩp2B0,
B3t=i(Δt1+iγ3)B3+iΩc1B1,
B4t=i(Δt2+iγ4)B4+iΩc2B2,
B02+B12+B22+B32+B42=1,
Ωp1z+1cΩp1t=iκ10B1B0*,
Ωp2z+1cΩp2t=iκ20B2B0*,

where 2γk (k= 1,2,3,4) is the decay rate of state ∣k〉, and κ 10,20 = pμ10,20 · e-,+2/(2h̄ε0 c) with N and ε 0 being the concentration and vacuum dielectric constant, respectively. In writing Eqs. (7)–(8), we have assumed collinear propagation geometry and applied slowly varying envelope approximation.

 figure: Fig. 2.

Fig. 2. Absorption coefficients α 1 and α 2 versus dimensionless Rabi frequency ∣Ωc1∣/γ 1 and ∣Ωc2∣/γ 2 for several different values of the two-photon detunings Δt1 and Δt2. The other parameters are γ 1γ 2 ≃ 5.6 MHz, γ 3γ 4 ≃ 0.76 MHz, κ 10κ 20 ≃ 6 γ 1/cm, Δs ~ ≃ 25 γ 1, and Δ ~ 0.5 γ 1.

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To provide a clear picture of the interplay between the group-velocity dispersion and nonlinear (SPM and XPM) effects of the atomic system interacting with two cw optical fields and a pulsed probe field, we first investigate the dispersion properties of the system. This requires a perturbation treatment of the system response to the first order of two polarization components Ωp1 and Ωp2 of the probe field while keeping all orders due to control fields Ωc1 and Ωc1. In the next section, we demonstrate SPM and XPM effects that are due to higher-order Ωp1 and Ωp2 that are required for balancing the the group-velocity dispersion effect so that the formation of slow temporal vector optical solitons can occur.

We consider the situation where all atoms are initially in their ground state, i.e., B 0 (t = 0) = 1, as shown in Fig. 1(a), and the strong pump condition that the control laser is strong enough to make ε = Ωp1,p2c1,c2 be a small parameter [6]. Then, we can make the asymptotic expansion Bj = ∑k B (k) j, where is B (k) j is the kth order part of Bj in terms of ε. Within adiabatic following framework it can be shown that B (0) j = δ j0 and B (1) 0 = 0. Keeping up to the first order of ε and taking Fourier transform of Eqs. (2)–(5) and (7)–(8), we have

(ω+Δs+iγ1)β1(1)+Ωc1*β3(1)=Λp1,
(ω+ΔsΔ+iγ2)β2(1)+Ωc2*β4(1)=Λp2,
(ω+Δt1+iγ3)β3(1)+Ωc1β1(1)=0,
(ω+Δt2+iγ4)β4(1)+Ωc2β2(1)=0,
Λp1ziωcΛp1=iκ10β1(1),
Λp2ziωcΛp2=iκ20β2(1),

where β (1) j and Λp1,p2 are the time Fourier transforms of B (1) j and Ωp1,p2, respectively, and ω is the time Fourier transform variable.

Equations (9)–(14) can be solved analytically, yielding

Λp1(z,ω)=Λp1(0,ω)exp[izK1(ω)],
Λp2(z,ω)=Λp2(0,ω)exp[izK2(ω)],

where K 1(ω) and K 2(ω) are the propagation constants corresponding to σ - and σ + components of the probe field, respectively, and denoted by

K1=ωc+κ10(ω+Δt1+iγ3)Ωc12(ω+Δs+iγ1)(ω+Δt1+iγ3)=K10+K11ω+K12ω2+,
K2=ωc+κ20(ω+Δt2+iγ4)Ωc22(ω+ΔsΔ+iγ2)(ω+Δt2+iγ4)=K20+K21ω+K22ω2+,

where K 10,20 = K 1,2(0), K 11,21 = dK 1,2(ω)ω=0, K 12,22 = 2d 2 K 1,2(ω)/ 2ω=0, which have clear physical signification. K 10,20 = ϕ 1,2 + 1,2/2 describes the phase shift ϕ 1,2 per unit length and absorption coefficient α 1,2 (see Figs. 2(a) and 2(b)) of the σ -(σ +) component of the probe field. K 11,21 = 1/V g1,g2 gives the propagation group velocity, and K 12,22 represents the group-velocity dispersion that contributes to the pulse’s shape change and additional loss of field intensity. Figure 2 illustrates the absorption coefficients α 1 and α 2 of the σ - and σ + components of the probe field versus the dimensionless Rabi frequencies ∣Ωc1∣/γ 1 and ∣Ωc2∣/γ 2 for several different values of two-photon detunings Δt1 and Δt2. These two figures clearly demonstrate that there exist parameter regimes in which the absorptions of the two components of the probe field can be almost simultaneously completely suppressed due to the contribution of the control fields under appropriate conditions in this five-level hyper V-type atomic system. It should be emphasized that the vector optical soliton pairs produced in this way generally travel with, respectively, a group velocity given by V g1 = 1/K 11 and V g2 = 1/K 21, which are nearly matched under appropriate parameter conditions as shown in below.

We stress again that Eqs. (15)–(16) are obtained in the linear regime of the system under the weak-field and adiabatic approximations with ignoring higher-order of Ωp1,p2. In order to preserve the shapes of the two polarization components of the probe field, we need to include the SPM and XPM which may balance the spread effect due to the group velocity dispersion to produce the vector solitons of the probe field. In next section, we will explore the higher-order of e with systematically keeping terms up to ω 2 in Eqs. (17)–(18) for the purpose of demonstrating the formation of slow temporal vector optical solitons in the five-level hyper V-type atomic system.

3. Two coupled NLS equations and vector optical solitons

In this section we will investigate the nonlinear evolution of two polarization components of the probe field. A detailed analysis of the nonlinear coupling between two components of the probe field reveals that the nonlinear Kerr effect due to SPM and XPM may offer an effective remedy to proper balance the rapid increase pulse width in the time domain and lead to the formation of slow temporal vector solitons. We now show that, a reasonable and realistic set of parameters can be found so that the SPM and XPM effects can precisely balance group velocity dispersion in the slow propagation regime and lead to the Eqs. (7) and (8) of describing the propagation of two components of the probe field evolving into two coupled NLS equations, which admit of solutions describing different types of vector solitons.

We now derive the nonlinear envelope equations which govern the propagation of two components of the probe field. In order to balance the interplay between group velocity dispersion and nonlinear effect, we consider the nonlinear polarization on the right-hand sides of Eqs. (7) and (8) and take the trial functions Ωp1,p2(z,t) = Ω1,2(z,t)exp[iz K 10,20] to substitute them into Eqs. (2)–(8), then we obtain the nonlinear wave equations of the slowly varying envelopes Ω1(z,t) and Ω2(z,t),

i(z+1Vg1t)Ω1+K122t2Ω1=𝒩1,
i(z+1Vg2t)Ω2+K222t2Ω2=𝒩2,

where 𝒩1,2 represent the nonlinear terms and are given by 𝒩1,2 = - κ10,20 B (1) 1,2 e -izK10,20[∣B (1) 12 + ∣B (1) 22 + ∣B (1) 32 + ∣ B (1) 42]. B (1) j can be obtain from Eqs. (9)–(12),

B1(1)=Δt1+iγ3Ωc12(Δs+iγ1)(Δt1+iγ3)Ωp1,
B2(1)=Δt2+iγ4Ωc22(ΔsΔ+iγ2)(Δt2+iγ4)Ωp2,
B3(1)=Ωc1Ωc12(Δs+iγ1)(Δt1+iγ3)Ωp1,
B4(1)=Ωc2Ωc22(ΔsΔ+iγ2)(Δt2+iγ4)Ωp2.

We then have the nonlinear evolution equations, i.e., the two coupled NLS equations for Ω1 (z, t) and Ω2(z,t),

i(ξ+δτ)Ω1K122τ2Ω1(W11eα1ξΩ12+W12eα2ξΩ22)Ω1=0,
i(ξδτ)Ω2K222τ2Ω2(W22eα2ξΩ22+W21eα1ξΩ12)Ω2=0,

where we have defined δ = (1/V g1 - 1/V g2)/2, 1/Vg = (1/V g1 + 1/V g2)/2, ξ = z, and t = t - z/Vg. Absorption coefficients α 1,2 = 2Im(K 10,20), SPM coefficients W 11,22, and XPM coefficients W 12,21 of the two components of the probe field are explicitly given by

W11=κ10(Δt1+iγ3)(Ωc12+Δt12+γ32)D1D12],
W12=κ10(Δt1+iγ3)(Ωc22+Δt22+γ42)D1D22],
W22=κ20(Δt2+iγ4)(Ωc22+Δt22+γ42)D2D22],
W21=κ20(Δt2+iγ4)(Ωc12+Δt12+γ32)D2D12],

with D 1 = ∣Ωc1∣ - (Δs + 1)(Δt 1 + 3) and D 2 = ∣Ωc22 - (Δs - Δ + 2)(Δt2 + 4).

Inspection of Eqs. (27)–(30) shows that the two coupled NLS Eqs. (25)–(26) have complex coefficients and generally do not allow vector soliton solutions. However, in the presence of the control fields, the absorption of the probe field can be almost simultaneously completely suppressed under appropriate conditions, which result in exp(α 1 L) ≃ exp(α 2) ≃ 1 (L is the hyper V-type atomic system’s length), just as shown in Fig. 2. Furthermore, as we show below, for the present system practical parameters can be found so that the imaginary parts of these complex coefficients in Eqs. (25)–(26) are much smaller than their corresponding real parts, i.e., K 12,22 = K 12r,22r + i K 12i,22iK 12r,22r, Wlm = Wlmr + iWlmi - Wlmr(l,m =1,2). Therefore, in these parameter regimes, we can neglect the small imaginary parts and make the Eqs. (25)–(26) to be nearly integrable. Besides, we define the characteristic dispersion length Ld, the characteristic nonlinear length Ln and characteristic group velocity mismatch length Ld of the hyper V-type atomic system, respectively, as Ld = τ 2 0/∣K 22r∣, Ln = 1 / (∣W 22rU 2 0) and Lδ = τ 0/∣ δ∣ with τ 0 and U 0 being, respectively, the characteristic pulse length and the typical Rabi frequency of the probe field. With the aim of searching for the formation of shape-preserving vector optical solitons, we set Ld = Ln, which means the balance between the group-velocity dispersion and nonlinearity effects in our system and thus we have U 0 = 1/τ 0K 22r/W 22r1/2. If we define s = ξ/Ld, σ = τ/τ 0, u 1,2 = Ω1,2/U 0, Ωδ = sgn(δ)Ld/Lδ, Q 1,2 = K 12r,22r/∣K 22r∣ and Qlm = Wlmr/∣W 22r∣, then Eqs. (25)–(26) can be written in the dimensionless forms,

iu1s+iQδu1σQ12u1σ2(Q11u12+Q12u22)u1=0,
iu2siQδu2σQ22u2σ2(Q22u22+Q21u12)u2=0,

which admit of solutions describing various types of vector solitons [1, 31–33, 50–55], such as bright-bright, bright-dark, dark-bright, dark-dark vector solitons, depending on the choice of corresponding parameters.

Based on the analysis above, we find that Eqs. (31)–(32) are nearly integrable, the SPM and XPM coefficients in Eqs. (25)–(26) defined by Eqs. (27)–(30) obviously satisfy the relation W 11 W 22 = W 12 W21, i.e., Q 11 Q 22 = Q 12 Q 21, which represents the balance between the group-velocity dispersion and nonlinearity, namely, SPM and XPM effects. Therefore, various types of temporal vector optical solitons including bright-bright, bright-dark, dark-bright and dark-dark vector solitons [31, 32] are possible that can propagate for an extended distance without significant deformation in this system. If the parameters fulfill the condition Q 1 Q 22 = Q 2 Q 12, we can easily obtain bright-bright, bright-dark, dark-bright, and dark-dark vector soliton solutions [50–55] of Eqs. (31)–(32) as shown below.

The solutions of Eqs. (31)–(32) that describe vector solitons for bright-bright vector solitons have the form

u1=C1sech(σ)exp[i(F11σ+F12s)],
u2=C2sech(σ)exp[i(F21σ+F22s)],

where sech(σ) is the hyperbolic secant function. We have defined F 11 = Qδ/(2Q 1), F 12 = -Q 1 - Qδ /(4Q 1), F 21 = -Qδ/(2Q 2), F 22 = -Q 2 - G 2δ /(4Q 2), and C 2 = [(2(Q 1 - Q 11 C 12)/Q 12]1/2 with C 1 being a free parameter.

The bright-dark vector soliton solutions of Eqs. (31)–(32) have the form

u1=C1sech(σ)exp[i(F11σ+F12s)],
u2=C2tanh(σ)exp[i(F21σ+F22s)],

where tanh(σ) is the hyperbolic tangent function. We also have defined F 11 = Qδ/(2Q 1), F 12 = -F 11 Qδ - Q 1(1 - F 112) - Q 12 C 22, F 21 = -Qδ/(2Q 2), F 22 = F 21 Q δ + Q 2 F 212 - Q 22 C 22, and C 2 = [(Q 11 C 12 - 2Q 1)/Q 12]1/2 with C 1 being a free parameter.

The dark-bright vector soliton solutions are

u1=C1tanh(σ)exp[i(F11σ+F12s)],
u2=C2sech(σ)exp[i(F21σ+F22s)],

where F 11 = Qδ/(2Q 1), F 12 = -F 11 Qδ + Q 1 F 2 11 - Q 11 C 2 1, F 21 = -Qδ/(2Q 2), F 22 = F 21 Qδ + Q 2(1 - F 2 21) - Q 21 C 12, and C 2 = [(Q 11 C 2 1 + 2Q 1)/Q 12)1/2 with C 1 being a free parameter.

We also have the dark-dark vector soliton solutions of Eqs. (31)–(32)

u1=C1tanh(σ)exp[i(F11σ+F12s)],
u2=C2tanh(σ)exp[i(F21σ+F22s)],

where F 11 = Qδ/ (2Q 1), F 12 = -F 11 Qδ + Q 1(2 + F 11), F 21 = -Qδ/(2Q 2), F 22 = F 21 Q δ + Q 2(2 + F 2 12), and C 2 = [-(Q 11 C 12 + 2Q 1)/Q 12)1/2 with C 1 being a free parameter. It is worth while pointing out that all four types of temporal vector optical solitons pairs described by Eqs. (33)–(40) are allowed in our system and travel with slow group velocity Vg.

In our five-level hyper V-type atomic system, the modified Hubbard model that includes the Manakov system [55,56] may be realized by adjusting the corresponding parameters. Below we will give a practical example to show that a realistic atomic system can be found to support these viewpoints. An realistic candidate for the proposed system can be found in 87Rb atoms with the designated states chosen as follows [49]: 52 S 1/2, F = 1,MF = 0 as ∣0〉, 52 P 1/2, F = 52, MF = - 1 as ∣1〉, 52 P 1/2, F = 2, MF = 1 as ∣2〉, 52 D 3/2, F = 2, MF = -1 as ∣3〉, and 52 D3/2, F = 2, MF = 1 as ∣4〉. Thus, the decay rates of excited states ∣1〉, ∣2〉, ∣3〉, and ∣4〉 are, respectively, Γ1 = 2γ 1 ≃ Γ2 = 2γ 2 - 11.2 MHz, Γ3 = 2γ 3 - Γ4 = 2γ 4 ~ 1.52 MHz.

We now give a practical parameters to show the existence of bright-bright vector optical solitons in the five-level hyper V-type atomic system. For this purpose, we take κ 10 - κ 20 - 33.6 cm-1MHz, Ωc1 ≃ Ωc2 - 112 MHz, Δt1 ≃ Δt2 - 84 MHz, Δs ≃ 560 MHz, and Δ ≃ 5.6 × 104 s-1, then we have K 11 ≃ (5.860 -0.251i) × 10-10 cm-1s, K 21 - (5.861 - 0.251i) × 10-10cm-1s, K 12 - (-7.937 + 0.533i) × 10-18s2cm-1, K 22 - (-7.940 + 0.533i) × 10-18 s2cm-1, W 11W 22W 12W 21 ≃ (-1.347 + 0.023i) × 10-18 s2cm-1, V g1/cV g2/c≃ 0.057, and α 1 ≃ 0,2 - 0.0028 cm-1. Notice that the imaginary parts of these quantities are indeed much smaller than their relevant real parts. With these quantities, we have Ld ≃ 12.59 cm and Lδ ≃ 6.64 × 104 cm with τ 0 = 1.0 × 10-8 s and U 0 ≃ 2.43 × 108 s-1 and the dimensionless coefficients read Qδ ≃ -0.00019, Q 1 ≃ -0.9997, Q 2 = -1, and Q 11 ~ Q 12 - Q 21 - Q 22 = -1. Then the two coupled NLS Eqs. (25) and (26) in the dimensionless form (31) and (32) are well characterized and hence we have demonstrated the existence of slow bright-bright vector solitons which are evolved from two polarization components of the probe field in the system. More importantly, with these dimensionless quantities obtained above, the two coupled NLS Eqs. (31) and (32) can be written as the standard integrable Manakov equations, i.e.,

 figure: Fig. 3.

Fig. 3. (a) and (c) are, respectively, the σ - and σ + polarization components of the probe field, obtained numerically from Eqs. (31) and (32), versus dimensionless time τ/τ 0 and distance ξ/Ld with τ 0 = 1.0 × 10-8 s and Ld ≃ 12.59 cm for bright-bright vector optical solitons formation. (b) shows the Manakov bright-bright vector solitons given in Eqs. (43) and (44) for the same parameters. Other parameters are explained in the main text.

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isu1+2σ2u1+(u12+u22)u1=0,
isu2+2σ2u2+(u22+u12)u2=0,

which admit of exact N-soliton solutions [52] and is one of the modified Hubbard model [55, 56]. The bright-bright vector soliton solutions of Eqs. (41) and (42) are given by

u1=2cos(θ)sech(σ)eis,
u2=2sin(θ)sech(σ)eis,

where θ is a free parameter. Obviously, we have θ = π/4 because of the fact that two orthogonally polarization components of the injected linear-polarized probe field have equal amplitudes. Thus, we have demonstrated the existence of the Manakov temporal vector solitons in the five-level hyper V-type atomic system.

In Fig. 3, we have plot the evolutions of the σ - and σ + polarization components of the probe field versus dimensionless time τ/τ 0 and distance ξ/Ld for τ 0 = 1.0 × 10-8 s and Ld ≃ 12.59 cm with the parameters given above. (a) and (c) show the bright-bright vector soliton solutions obtained directly from Eqs. (31) and (32), whereas (b) is the result obtained from the standard integrable Manakov Eqs. (41) and (42). It is remarkable that this set of parameters has generated the vector soliton wave form that agrees excellently with that of the Manakov bright-bright vector solitons Eqs. (43) and (44) for a propagation distance of z = t ≃ 1.6Ld. This is a remarkable propagation effect in such a highly resonant system.

To test the stability of the bright-bright vector optical solitons formed in the five-level hyper V-type atomic system, we have investigated the collision of two solitons evolved from the σ - polarization component of the probe field under different initial conditions by using numerical simulations as shown in Fig. 4. The similar figures of two-soliton collisions for the σ + polarization component can be also obtained under the initial conditions, which are omitted here. In the simulation, we choose the initial condition u 1 (z = 0) = u 2(z = 0) = sech(σ + 2.0)e - + sech(σ - 2.0)e for Fig. 4(a) and u 1(z = 0) = u 2(z = 0) = sech(σ + 2.0)e -(σ + sech(σ - 2.0)e i(σ+π) for Fig. 4(b). The Fig. 4(a) shows that the collision between two solitons presents attraction each other if two solitons are in phases and have the same amplitudes. We see that the solitons pass through each other and then recover their initial waveforms after the interaction due to the particle property of the solitons. When the solitons are out of phase and also have the same amplitudes, then the interaction between the two solitons is repulsive, which is shown in Fig. 4(b). From the figures, we find that as time goes on the solitons collide and then depart each other with recovering their initial waveforms.

 figure: Fig. 4.

Fig. 4. The wave shape of the σ - polarization component for two-soliton collisions under different initial conditions: (a), u 1(z = 0) = u 2(z = 0) = sech(σ + 2.0)e - + sech(σ-2.0)e; (b), u 1(z = 0) = u 2(z = 0) = sech(σ + 2.0)e - + sech(σ - 2.0)ei (σ+π). All the parameters are present in the main text.

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We further note that it is possible, by choosing different control field Rabi frequencies and detunings, to obtain other vector solitons. For instance, adjusting Δt1 ≃ Δt2 ≃ -84 MHz with all other parameters given above unchanged, we obtain K 11K 21 ≃ (2.188 - 0.015i) × 10-10cm-1s, K 12 ≃ (6.868 + 0.152i) × 10-19 s2cm-1, K 22 ≃ (6.689 + 0.533i) × 10-19 s2cm-1, W 11W 22W 12W 21 ≃ (-2.615 + 0.026i) × 10-19 s2cm-1, V g1/cV g2/c ≃ 0.152, and α 1α 2 ≃ 0.00093 cm-1. Thus, we have the dimensionless coefficients Qδ ≃ -0.00043, Q 1 ≃ 0.9999, Q 2 = 1, and Q 11Q 12Q 21Q 22 = -1.0. Then the two coupled NLS Eqs. (25) and (26) in the dimensionless form (31) and (32) are well characterized and have described the dark-dark vector optical solitons [50–52]. Hence, we have demonstrated the formation of slow dark-dark vector optical solitons in a cold five-level hyper V-type atomic system.

4. Conclusions

In conclusion, we have discussed the possibility of generating temporal vector optical solitons with slow group velocities in a cold five-level hyper V-type atomic system. We have shown that, in the presence of two strong cw control field, the dispersion of the two orthogonally polarized components of a low intensity linear-polarized pulsed probe field can be proper balanced by the SPM and XPM which lead to the formation of various distortion-free slow temporal vector optical solitons, such as bright-bright, bright-dark, dark-bright and dark-dark vector solitons, in the atomic system. Besides, as one of the modified Hubbard models, the Manakov system may be realized in our system by adjusting the corresponding parameters. Due to their robust propagation nature the vector optical solitons suggested in present work may provide the possibility of the promising applications for the design of new types of all-optical switches and logic gates.

Acknowledgments

We would like to acknowledge Professor Y. Wu for his enlightening sugesstions and encouragement. The work is supported in part by the NSF of China (Grant Nos. 60478029, 10575040, 10634060, and 90503010), the National Fundamental Research Program of China (Grant No. 2005CB724508), and the Foundation from the Ministry of the National Education of China (Grant No.200804870051). The authors also express their sincere appreciation to the reviewer for his/her valuable advice and comments.

References and links

1. G. P. Agrawal, Nonlinear Fiber Optics, (Academic, New York, 2001).

2. B. A. Malomed, Soliton management in periodic systems, (Springer, 2006).

3. H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996). [CrossRef]  

4. Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett. 93, 143904 (2004). [CrossRef]   [PubMed]  

5. Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett. 29, 2064–2066 (2004). [CrossRef]   [PubMed]  

6. Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71, 053820 (2005). [CrossRef]  

7. X. Yang and Y. Wu, “Ultra-slow Bright and Dark Optical Solitons in Cold Media,” Commun. Theor. Phys. 45, 335–342 (2006). [CrossRef]  

8. W.-X. Yang and R.-K. Lee, “Slow optical solitons via intersubband transitions in a semiconductor quantum well,” Europhys. Lett. 83, 14002 (2008) [CrossRef]  

9. W.-X. Yang, J.-M. Hou, and R.-K. Lee, “Ultraslow bright and dark solitons in semiconductor quantum wells,” Phys. Rev. A 77, 033838 (2008) [CrossRef]  

10. X.-T. Xie, W.-B. Li, and X. Yang, “Bright, dark, bistable bright, and vortex spatial-optical solitons in a cold three-state medium,” J. Opt. Soc. Am. B 23, 1609–1614 (2006). [CrossRef]  

11. X.-T. Xie, W. Li, J. Li, W.-X. Yang, A. Yuan, and X. Yang, “Transverse acoustic wave in molecular magnets via electromagnetically induced transparency,” Phys. Rev. B 75, 184423 (2007). [CrossRef]  

12. J.-B. Liu, X.-Y. Lü, N. Liu, M. Wang, and T.-K. Liu, “Microwave solitons in molecular magnets via electromagnetically induced transparency,” Phys. Lett. A 373, 413–417 (2008). [CrossRef]  

13. B. Wu, J. Liu, and Q. Niu, “Controlled Generation of Dark Solitons with Phase Imprinting,” Phys. Rev. Lett. 88, 034101 (2002). [CrossRef]   [PubMed]  

14. X.-J. Liu, H. Jing, and M.-L. Ge, “Solitons formed by dark-state polaritons in an electromagnetic induced transparency,” Phys. Rev. A 70, 055802 (2004). [CrossRef]  

15. H. E. Nistazakis, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and R. Carretero-González, “Bright-dark soliton complexes in spinor Bose-Einstein condensates,” Phys. Rev. A 77, 033612 (2008). [CrossRef]  

16. L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-Λ system,” Phys. Rev. E 72, 055601(R) (2005). [CrossRef]  

17. D. V. Skryabin, A. V. Yulin, and A. I. Maimistov, “Localized Polaritons and Second-Harmonic Generation in a Resonant Medium with Quadratic Nonlinearity,” Phys. Rev. Lett. 96, 163904 (2006). [CrossRef]   [PubMed]  

18. G. T. Adamashvili, C. Weber, and A. Knorr, “Optical nonlinear waves in semiconductor quantum dots: Solitons and breathers,” Phys. Rev. A 75, 063808 (2007). [CrossRef]  

19. Y. Wu and R. Côtù, “Bistability and quantum fluctuations in coherent photoassociation of a Bose-Einstein condensate,” Phys. Rev. A 65, 053603 (2002). [CrossRef]  

20. J.-H. Li, X.-Y. Lü, J.-M. Luo, and Q.-J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006). [CrossRef]  

21. Y. Wu, M. G. Payne, E. W. Hagley, and L. Deng, “Ultraviolet single-photons on demand and entanglement of photons with a large frequency difference,” Phys. Rev. A 70, 063812 (2004). [CrossRef]  

22. Y. Wu and L. Deng, “Achieving multifrequency mode entanglement with ultraslow multiwave mixing,” Opt. Lett. 29, 1144–1146 (2004). [CrossRef]   [PubMed]  

23. X.-Y. Lü, J.-B. Liu, C.-L. Ding, and J.-H. Li, “Dispersive atom-field interaction scheme for three-dimensional entanglement between two spatially separated atoms,” Phys. Rev. A 78, 032305 (2008). [CrossRef]  

24. Y. Wu and X. Yang, “Highly efficient four-wave mixing in double-Λ system in ultraslow propagation regime,” Phys. Rev. A 70, 053818 (2004). [CrossRef]  

25. Y. Wu and X. Yang, “Four-wave mixing in molecular magnets via electromagnetically induced transparency,” Phys. Rev. B 76, 054425 (2007). [CrossRef]  

26. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997). [CrossRef]  

27. S. E. Harris and L. V. Hau, “Nonlinear Optics at Low Light Levels,” Phys. Rev. Lett. 82, 4611–4614 (1999). [CrossRef]  

28. Y. Wu and X. Yang, “Electromagnetically induced transparency in V-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005). [CrossRef]  

29. S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Observation of Polarization-Locked Vector Solitons in an Optical Fiber,” Phys. Rev. Lett. 82, 3988–3991 (1999). [CrossRef]  

30. G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Λ system,” Phys. Rev. E 73, 056606 (2006). [CrossRef]  

31. C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008). [CrossRef]  

32. L.-G. Si, J.-B. Liu, X.-Y. Lü, and X. Yang, “Ultraslow temporal vector optical solitons in a cold five-state atomic medium under Raman excitation,” J. Phys. B 41, 215504 (2008). [CrossRef]  

33. L.-G. Si, W.-X. Yang, and X. Yang, “Ultraslow temporal vector optical solitons in a cold four-level tripod atomic system,” J. Opt. Soc. Am. B 26, 478–486 (2009). [CrossRef]  

34. D. V. Skryabin, F. Biancalana, D. M. Bird, and F. Benabid, “Effective Kerr Nonlinearity and Two-Color Solitons in Photonic Band-Gap Fibers Filled with a Raman Active Gas,” Phys. Rev. Lett. 93, 143907 (2004). [CrossRef]   [PubMed]  

35. D. Y. Tang, H. Zhang, L. M. Zhao, and X. Wu, “Observation of High-Order Polarization-Locked Vector Solitons in a Fiber Laser,” Phys. Rev. Lett. 101, 153904 (2008). [CrossRef]   [PubMed]  

36. A. E. Korolev, V. N. Nazarov, D. A. Nolan, and C. M. Truesdale, “Experimental observation of orthogonally polarized time-delayed optical soliton trapping in birefringent fibers,” Opt. Lett. 30, 132–134 (2005). [CrossRef]   [PubMed]  

37. Y. Barad and Y. Silberberg, “Polarization Evolution and Polarization Instability of Solitons in a Birefringent Optical Fiber,” Phys. Rev. Lett. 78, 3290–3293 (1997). [CrossRef]  

38. D. Rand, I. Glesk, C.-S. Brès, D. A. Nolan, X. Chen, J. Koh, J. W. Fleischer, K. Steiglitz, and P. R. Prucnal, “Observation of Temporal Vector Soliton Propagation and Collision in Birefringent Fiber,” Phys. Rev. Lett. 98, 053902 (2007). [CrossRef]   [PubMed]  

39. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994). [CrossRef]   [PubMed]  

40. Z. Chen, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Observation of incoherently coupled photorefrac-tive spatial soliton pairs,” Opt. Lett. 21, 1436–1438 (1996). [CrossRef]   [PubMed]  

41. J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, “Observation of Manakov spatial solitons in AlGaAs planar waveguides,” Phys. Rev. Lett. 76, 3699–3702 (1996). [CrossRef]   [PubMed]  

42. C. Anastassiou, J. W. Fleischer, T. Carmon, M. Segev, and K. Steiglitz, “Information transfer via cascaded collisions of vector solitons,” Opt. Lett. 26, 1498–1500 (2001). [CrossRef]  

43. M. Delquè, T. Sylvestre, H. Maillotte, C. Cambournac, P. Kockaert, and M. Haelterman, “Experimental observation of the elliptically polarized fundamental vector soliton of isotropic Kerr media,” Opt. Lett. 30,, 3383–3385 (2005). [CrossRef]  

44. H. Schmidt and Imamoǧlu A., “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996). [CrossRef]   [PubMed]  

45. M. D. Lukin and A. Imamoǧlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000) [CrossRef]   [PubMed]  

46. M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett. 82, 5229–5232 (1999). [CrossRef]  

47. L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, “Optical-wave group-velocity reduction without electro-magnetically induced transparency,” Phys. Rev. A 65, 051805(R) (2002). [CrossRef]  

48. L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, “Opening Optical Four-Wave Mixing Channels with Giant Enhancement Using Ultraslow Pump Waves,” Phys. Rev. Lett. 88, 143902 (2002). [CrossRef]   [PubMed]  

49. Y. Wu, L. Wen, and Y. Zhu, “Efficient hyper-Raman scattering in resonant coherent media,” Opt. Lett. 28, 631–633 (2003). [CrossRef]   [PubMed]  

50. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Optical solitary waves induced by cross-phase modulation,” Opt. Lett. 13, 871–873 (1988). [CrossRef]   [PubMed]  

51. V. V. Afanasyev, Y. S. Kivshar, V. V. Konotop, and V. N. Serkin, “Dynamics of coupled dark and bright optical solitons,” Opt. Lett. 14, 805–807 (1989). [CrossRef]   [PubMed]  

52. J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics,” Phys. Rev. E 59, 2393–2405 2393 (1999). [CrossRef]  

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

54. T. Kanna, M. Vijayajayanthi, and M. Lakshmanan, “Periodic energy switching of bright solitons in mixed coupled nonlinear Schrödinger equations with linear self-coupling and cross-coupling terms,” Phys. Rev. A 76, 013808 (2007). [CrossRef]  

55. E. Yomba, “Generalized hyperbolic functions to find soliton-like solutions for a system of coupled nonlinear Schrödinger equations,” Phys. Lett. A 372, 1612–1618 (2008). [CrossRef]  

56. T. Kanna, M. Lakshmanan, P. Tchofo Dinda, and N. Akhmediev, “Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations,” Phys. Rev. E 73, 026604 (2006). [CrossRef]  

References

  • View by:

  1. G. P. Agrawal, Nonlinear Fiber Optics, (Academic, New York, 2001).
  2. B. A. Malomed, Soliton management in periodic systems, (Springer, 2006).
  3. H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
    [Crossref]
  4. Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett. 93, 143904 (2004).
    [Crossref] [PubMed]
  5. Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett. 29, 2064–2066 (2004).
    [Crossref] [PubMed]
  6. Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71, 053820 (2005).
    [Crossref]
  7. X. Yang and Y. Wu, “Ultra-slow Bright and Dark Optical Solitons in Cold Media,” Commun. Theor. Phys. 45, 335–342 (2006).
    [Crossref]
  8. W.-X. Yang and R.-K. Lee, “Slow optical solitons via intersubband transitions in a semiconductor quantum well,” Europhys. Lett. 83, 14002 (2008)
    [Crossref]
  9. W.-X. Yang, J.-M. Hou, and R.-K. Lee, “Ultraslow bright and dark solitons in semiconductor quantum wells,” Phys. Rev. A 77, 033838 (2008)
    [Crossref]
  10. X.-T. Xie, W.-B. Li, and X. Yang, “Bright, dark, bistable bright, and vortex spatial-optical solitons in a cold three-state medium,” J. Opt. Soc. Am. B 23, 1609–1614 (2006).
    [Crossref]
  11. X.-T. Xie, W. Li, J. Li, W.-X. Yang, A. Yuan, and X. Yang, “Transverse acoustic wave in molecular magnets via electromagnetically induced transparency,” Phys. Rev. B 75, 184423 (2007).
    [Crossref]
  12. J.-B. Liu, X.-Y. Lü, N. Liu, M. Wang, and T.-K. Liu, “Microwave solitons in molecular magnets via electromagnetically induced transparency,” Phys. Lett. A 373, 413–417 (2008).
    [Crossref]
  13. B. Wu, J. Liu, and Q. Niu, “Controlled Generation of Dark Solitons with Phase Imprinting,” Phys. Rev. Lett. 88, 034101 (2002).
    [Crossref] [PubMed]
  14. X.-J. Liu, H. Jing, and M.-L. Ge, “Solitons formed by dark-state polaritons in an electromagnetic induced transparency,” Phys. Rev. A 70, 055802 (2004).
    [Crossref]
  15. H. E. Nistazakis, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and R. Carretero-González, “Bright-dark soliton complexes in spinor Bose-Einstein condensates,” Phys. Rev. A 77, 033612 (2008).
    [Crossref]
  16. L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-Λ system,” Phys. Rev. E 72, 055601(R) (2005).
    [Crossref]
  17. D. V. Skryabin, A. V. Yulin, and A. I. Maimistov, “Localized Polaritons and Second-Harmonic Generation in a Resonant Medium with Quadratic Nonlinearity,” Phys. Rev. Lett. 96, 163904 (2006).
    [Crossref] [PubMed]
  18. G. T. Adamashvili, C. Weber, and A. Knorr, “Optical nonlinear waves in semiconductor quantum dots: Solitons and breathers,” Phys. Rev. A 75, 063808 (2007).
    [Crossref]
  19. Y. Wu and R. Côtù, “Bistability and quantum fluctuations in coherent photoassociation of a Bose-Einstein condensate,” Phys. Rev. A 65, 053603 (2002).
    [Crossref]
  20. J.-H. Li, X.-Y. Lü, J.-M. Luo, and Q.-J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
    [Crossref]
  21. Y. Wu, M. G. Payne, E. W. Hagley, and L. Deng, “Ultraviolet single-photons on demand and entanglement of photons with a large frequency difference,” Phys. Rev. A 70, 063812 (2004).
    [Crossref]
  22. Y. Wu and L. Deng, “Achieving multifrequency mode entanglement with ultraslow multiwave mixing,” Opt. Lett. 29, 1144–1146 (2004).
    [Crossref] [PubMed]
  23. X.-Y. Lü, J.-B. Liu, C.-L. Ding, and J.-H. Li, “Dispersive atom-field interaction scheme for three-dimensional entanglement between two spatially separated atoms,” Phys. Rev. A 78, 032305 (2008).
    [Crossref]
  24. Y. Wu and X. Yang, “Highly efficient four-wave mixing in double-Λ system in ultraslow propagation regime,” Phys. Rev. A 70, 053818 (2004).
    [Crossref]
  25. Y. Wu and X. Yang, “Four-wave mixing in molecular magnets via electromagnetically induced transparency,” Phys. Rev. B 76, 054425 (2007).
    [Crossref]
  26. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
    [Crossref]
  27. S. E. Harris and L. V. Hau, “Nonlinear Optics at Low Light Levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
    [Crossref]
  28. Y. Wu and X. Yang, “Electromagnetically induced transparency in V-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
    [Crossref]
  29. S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Observation of Polarization-Locked Vector Solitons in an Optical Fiber,” Phys. Rev. Lett. 82, 3988–3991 (1999).
    [Crossref]
  30. G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Λ system,” Phys. Rev. E 73, 056606 (2006).
    [Crossref]
  31. C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
    [Crossref]
  32. L.-G. Si, J.-B. Liu, X.-Y. Lü, and X. Yang, “Ultraslow temporal vector optical solitons in a cold five-state atomic medium under Raman excitation,” J. Phys. B 41, 215504 (2008).
    [Crossref]
  33. L.-G. Si, W.-X. Yang, and X. Yang, “Ultraslow temporal vector optical solitons in a cold four-level tripod atomic system,” J. Opt. Soc. Am. B 26, 478–486 (2009).
    [Crossref]
  34. D. V. Skryabin, F. Biancalana, D. M. Bird, and F. Benabid, “Effective Kerr Nonlinearity and Two-Color Solitons in Photonic Band-Gap Fibers Filled with a Raman Active Gas,” Phys. Rev. Lett. 93, 143907 (2004).
    [Crossref] [PubMed]
  35. D. Y. Tang, H. Zhang, L. M. Zhao, and X. Wu, “Observation of High-Order Polarization-Locked Vector Solitons in a Fiber Laser,” Phys. Rev. Lett. 101, 153904 (2008).
    [Crossref] [PubMed]
  36. A. E. Korolev, V. N. Nazarov, D. A. Nolan, and C. M. Truesdale, “Experimental observation of orthogonally polarized time-delayed optical soliton trapping in birefringent fibers,” Opt. Lett. 30, 132–134 (2005).
    [Crossref] [PubMed]
  37. Y. Barad and Y. Silberberg, “Polarization Evolution and Polarization Instability of Solitons in a Birefringent Optical Fiber,” Phys. Rev. Lett. 78, 3290–3293 (1997).
    [Crossref]
  38. D. Rand, I. Glesk, C.-S. Brès, D. A. Nolan, X. Chen, J. Koh, J. W. Fleischer, K. Steiglitz, and P. R. Prucnal, “Observation of Temporal Vector Soliton Propagation and Collision in Birefringent Fiber,” Phys. Rev. Lett. 98, 053902 (2007).
    [Crossref] [PubMed]
  39. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
    [Crossref] [PubMed]
  40. Z. Chen, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Observation of incoherently coupled photorefrac-tive spatial soliton pairs,” Opt. Lett. 21, 1436–1438 (1996).
    [Crossref] [PubMed]
  41. J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, “Observation of Manakov spatial solitons in AlGaAs planar waveguides,” Phys. Rev. Lett. 76, 3699–3702 (1996).
    [Crossref] [PubMed]
  42. C. Anastassiou, J. W. Fleischer, T. Carmon, M. Segev, and K. Steiglitz, “Information transfer via cascaded collisions of vector solitons,” Opt. Lett. 26, 1498–1500 (2001).
    [Crossref]
  43. M. Delquè, T. Sylvestre, H. Maillotte, C. Cambournac, P. Kockaert, and M. Haelterman, “Experimental observation of the elliptically polarized fundamental vector soliton of isotropic Kerr media,” Opt. Lett. 30,, 3383–3385 (2005).
    [Crossref]
  44. H. Schmidt and Imamoǧlu A., “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996).
    [Crossref] [PubMed]
  45. M. D. Lukin and A. Imamoǧlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000)
    [Crossref] [PubMed]
  46. M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
    [Crossref]
  47. L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, “Optical-wave group-velocity reduction without electro-magnetically induced transparency,” Phys. Rev. A 65, 051805(R) (2002).
    [Crossref]
  48. L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, “Opening Optical Four-Wave Mixing Channels with Giant Enhancement Using Ultraslow Pump Waves,” Phys. Rev. Lett. 88, 143902 (2002).
    [Crossref] [PubMed]
  49. Y. Wu, L. Wen, and Y. Zhu, “Efficient hyper-Raman scattering in resonant coherent media,” Opt. Lett. 28, 631–633 (2003).
    [Crossref] [PubMed]
  50. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Optical solitary waves induced by cross-phase modulation,” Opt. Lett. 13, 871–873 (1988).
    [Crossref] [PubMed]
  51. V. V. Afanasyev, Y. S. Kivshar, V. V. Konotop, and V. N. Serkin, “Dynamics of coupled dark and bright optical solitons,” Opt. Lett. 14, 805–807 (1989).
    [Crossref] [PubMed]
  52. J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics,” Phys. Rev. E 59, 2393–2405 2393 (1999).
    [Crossref]
  53. Q. Park and H. J. Shin, “Systematic construction of multicomponent optical solitons,” Phys. Rev. E 61, 3093–3106 (2000).
    [Crossref]
  54. T. Kanna, M. Vijayajayanthi, and M. Lakshmanan, “Periodic energy switching of bright solitons in mixed coupled nonlinear Schrödinger equations with linear self-coupling and cross-coupling terms,” Phys. Rev. A 76, 013808 (2007).
    [Crossref]
  55. E. Yomba, “Generalized hyperbolic functions to find soliton-like solutions for a system of coupled nonlinear Schrödinger equations,” Phys. Lett. A 372, 1612–1618 (2008).
    [Crossref]
  56. T. Kanna, M. Lakshmanan, P. Tchofo Dinda, and N. Akhmediev, “Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations,” Phys. Rev. E 73, 026604 (2006).
    [Crossref]

2009 (1)

2008 (9)

D. Y. Tang, H. Zhang, L. M. Zhao, and X. Wu, “Observation of High-Order Polarization-Locked Vector Solitons in a Fiber Laser,” Phys. Rev. Lett. 101, 153904 (2008).
[Crossref] [PubMed]

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

L.-G. Si, J.-B. Liu, X.-Y. Lü, and X. Yang, “Ultraslow temporal vector optical solitons in a cold five-state atomic medium under Raman excitation,” J. Phys. B 41, 215504 (2008).
[Crossref]

X.-Y. Lü, J.-B. Liu, C.-L. Ding, and J.-H. Li, “Dispersive atom-field interaction scheme for three-dimensional entanglement between two spatially separated atoms,” Phys. Rev. A 78, 032305 (2008).
[Crossref]

W.-X. Yang and R.-K. Lee, “Slow optical solitons via intersubband transitions in a semiconductor quantum well,” Europhys. Lett. 83, 14002 (2008)
[Crossref]

W.-X. Yang, J.-M. Hou, and R.-K. Lee, “Ultraslow bright and dark solitons in semiconductor quantum wells,” Phys. Rev. A 77, 033838 (2008)
[Crossref]

J.-B. Liu, X.-Y. Lü, N. Liu, M. Wang, and T.-K. Liu, “Microwave solitons in molecular magnets via electromagnetically induced transparency,” Phys. Lett. A 373, 413–417 (2008).
[Crossref]

H. E. Nistazakis, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and R. Carretero-González, “Bright-dark soliton complexes in spinor Bose-Einstein condensates,” Phys. Rev. A 77, 033612 (2008).
[Crossref]

E. Yomba, “Generalized hyperbolic functions to find soliton-like solutions for a system of coupled nonlinear Schrödinger equations,” Phys. Lett. A 372, 1612–1618 (2008).
[Crossref]

2007 (5)

T. Kanna, M. Vijayajayanthi, and M. Lakshmanan, “Periodic energy switching of bright solitons in mixed coupled nonlinear Schrödinger equations with linear self-coupling and cross-coupling terms,” Phys. Rev. A 76, 013808 (2007).
[Crossref]

D. Rand, I. Glesk, C.-S. Brès, D. A. Nolan, X. Chen, J. Koh, J. W. Fleischer, K. Steiglitz, and P. R. Prucnal, “Observation of Temporal Vector Soliton Propagation and Collision in Birefringent Fiber,” Phys. Rev. Lett. 98, 053902 (2007).
[Crossref] [PubMed]

G. T. Adamashvili, C. Weber, and A. Knorr, “Optical nonlinear waves in semiconductor quantum dots: Solitons and breathers,” Phys. Rev. A 75, 063808 (2007).
[Crossref]

X.-T. Xie, W. Li, J. Li, W.-X. Yang, A. Yuan, and X. Yang, “Transverse acoustic wave in molecular magnets via electromagnetically induced transparency,” Phys. Rev. B 75, 184423 (2007).
[Crossref]

Y. Wu and X. Yang, “Four-wave mixing in molecular magnets via electromagnetically induced transparency,” Phys. Rev. B 76, 054425 (2007).
[Crossref]

2006 (6)

J.-H. Li, X.-Y. Lü, J.-M. Luo, and Q.-J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[Crossref]

G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Λ system,” Phys. Rev. E 73, 056606 (2006).
[Crossref]

X. Yang and Y. Wu, “Ultra-slow Bright and Dark Optical Solitons in Cold Media,” Commun. Theor. Phys. 45, 335–342 (2006).
[Crossref]

X.-T. Xie, W.-B. Li, and X. Yang, “Bright, dark, bistable bright, and vortex spatial-optical solitons in a cold three-state medium,” J. Opt. Soc. Am. B 23, 1609–1614 (2006).
[Crossref]

D. V. Skryabin, A. V. Yulin, and A. I. Maimistov, “Localized Polaritons and Second-Harmonic Generation in a Resonant Medium with Quadratic Nonlinearity,” Phys. Rev. Lett. 96, 163904 (2006).
[Crossref] [PubMed]

T. Kanna, M. Lakshmanan, P. Tchofo Dinda, and N. Akhmediev, “Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations,” Phys. Rev. E 73, 026604 (2006).
[Crossref]

2005 (5)

M. Delquè, T. Sylvestre, H. Maillotte, C. Cambournac, P. Kockaert, and M. Haelterman, “Experimental observation of the elliptically polarized fundamental vector soliton of isotropic Kerr media,” Opt. Lett. 30,, 3383–3385 (2005).
[Crossref]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-Λ system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71, 053820 (2005).
[Crossref]

A. E. Korolev, V. N. Nazarov, D. A. Nolan, and C. M. Truesdale, “Experimental observation of orthogonally polarized time-delayed optical soliton trapping in birefringent fibers,” Opt. Lett. 30, 132–134 (2005).
[Crossref] [PubMed]

Y. Wu and X. Yang, “Electromagnetically induced transparency in V-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
[Crossref]

2004 (7)

Y. Wu, M. G. Payne, E. W. Hagley, and L. Deng, “Ultraviolet single-photons on demand and entanglement of photons with a large frequency difference,” Phys. Rev. A 70, 063812 (2004).
[Crossref]

Y. Wu and L. Deng, “Achieving multifrequency mode entanglement with ultraslow multiwave mixing,” Opt. Lett. 29, 1144–1146 (2004).
[Crossref] [PubMed]

Y. Wu and X. Yang, “Highly efficient four-wave mixing in double-Λ system in ultraslow propagation regime,” Phys. Rev. A 70, 053818 (2004).
[Crossref]

D. V. Skryabin, F. Biancalana, D. M. Bird, and F. Benabid, “Effective Kerr Nonlinearity and Two-Color Solitons in Photonic Band-Gap Fibers Filled with a Raman Active Gas,” Phys. Rev. Lett. 93, 143907 (2004).
[Crossref] [PubMed]

Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett. 29, 2064–2066 (2004).
[Crossref] [PubMed]

X.-J. Liu, H. Jing, and M.-L. Ge, “Solitons formed by dark-state polaritons in an electromagnetic induced transparency,” Phys. Rev. A 70, 055802 (2004).
[Crossref]

2003 (1)

2002 (4)

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, “Optical-wave group-velocity reduction without electro-magnetically induced transparency,” Phys. Rev. A 65, 051805(R) (2002).
[Crossref]

L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, “Opening Optical Four-Wave Mixing Channels with Giant Enhancement Using Ultraslow Pump Waves,” Phys. Rev. Lett. 88, 143902 (2002).
[Crossref] [PubMed]

Y. Wu and R. Côtù, “Bistability and quantum fluctuations in coherent photoassociation of a Bose-Einstein condensate,” Phys. Rev. A 65, 053603 (2002).
[Crossref]

B. Wu, J. Liu, and Q. Niu, “Controlled Generation of Dark Solitons with Phase Imprinting,” Phys. Rev. Lett. 88, 034101 (2002).
[Crossref] [PubMed]

2001 (1)

2000 (2)

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

M. D. Lukin and A. Imamoǧlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000)
[Crossref] [PubMed]

1999 (4)

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[Crossref]

J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics,” Phys. Rev. E 59, 2393–2405 2393 (1999).
[Crossref]

S. E. Harris and L. V. Hau, “Nonlinear Optics at Low Light Levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[Crossref]

S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Observation of Polarization-Locked Vector Solitons in an Optical Fiber,” Phys. Rev. Lett. 82, 3988–3991 (1999).
[Crossref]

1997 (2)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[Crossref]

Y. Barad and Y. Silberberg, “Polarization Evolution and Polarization Instability of Solitons in a Birefringent Optical Fiber,” Phys. Rev. Lett. 78, 3290–3293 (1997).
[Crossref]

1996 (4)

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[Crossref]

H. Schmidt and Imamoǧlu A., “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996).
[Crossref] [PubMed]

Z. Chen, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Observation of incoherently coupled photorefrac-tive spatial soliton pairs,” Opt. Lett. 21, 1436–1438 (1996).
[Crossref] [PubMed]

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, “Observation of Manakov spatial solitons in AlGaAs planar waveguides,” Phys. Rev. Lett. 76, 3699–3702 (1996).
[Crossref] [PubMed]

1994 (1)

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

1989 (1)

1988 (1)

A., Imamoglu

Adamashvili, G. T.

G. T. Adamashvili, C. Weber, and A. Knorr, “Optical nonlinear waves in semiconductor quantum dots: Solitons and breathers,” Phys. Rev. A 75, 063808 (2007).
[Crossref]

Afanasyev, V. V.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, (Academic, New York, 2001).

Aitchison, J. S.

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, “Observation of Manakov spatial solitons in AlGaAs planar waveguides,” Phys. Rev. Lett. 76, 3699–3702 (1996).
[Crossref] [PubMed]

Akhmediev, N.

T. Kanna, M. Lakshmanan, P. Tchofo Dinda, and N. Akhmediev, “Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations,” Phys. Rev. E 73, 026604 (2006).
[Crossref]

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, “Observation of Manakov spatial solitons in AlGaAs planar waveguides,” Phys. Rev. Lett. 76, 3699–3702 (1996).
[Crossref] [PubMed]

Akhmediev, N. N.

S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Observation of Polarization-Locked Vector Solitons in an Optical Fiber,” Phys. Rev. Lett. 82, 3988–3991 (1999).
[Crossref]

Anastassiou, C.

Barad, Y.

Y. Barad and Y. Silberberg, “Polarization Evolution and Polarization Instability of Solitons in a Birefringent Optical Fiber,” Phys. Rev. Lett. 78, 3290–3293 (1997).
[Crossref]

Benabid, F.

D. V. Skryabin, F. Biancalana, D. M. Bird, and F. Benabid, “Effective Kerr Nonlinearity and Two-Color Solitons in Photonic Band-Gap Fibers Filled with a Raman Active Gas,” Phys. Rev. Lett. 93, 143907 (2004).
[Crossref] [PubMed]

Bergman, K.

S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Observation of Polarization-Locked Vector Solitons in an Optical Fiber,” Phys. Rev. Lett. 82, 3988–3991 (1999).
[Crossref]

Biancalana, F.

D. V. Skryabin, F. Biancalana, D. M. Bird, and F. Benabid, “Effective Kerr Nonlinearity and Two-Color Solitons in Photonic Band-Gap Fibers Filled with a Raman Active Gas,” Phys. Rev. Lett. 93, 143907 (2004).
[Crossref] [PubMed]

Bird, D. M.

D. V. Skryabin, F. Biancalana, D. M. Bird, and F. Benabid, “Effective Kerr Nonlinearity and Two-Color Solitons in Photonic Band-Gap Fibers Filled with a Raman Active Gas,” Phys. Rev. Lett. 93, 143907 (2004).
[Crossref] [PubMed]

Brès, C.-S.

D. Rand, I. Glesk, C.-S. Brès, D. A. Nolan, X. Chen, J. Koh, J. W. Fleischer, K. Steiglitz, and P. R. Prucnal, “Observation of Temporal Vector Soliton Propagation and Collision in Birefringent Fiber,” Phys. Rev. Lett. 98, 053902 (2007).
[Crossref] [PubMed]

Cambournac, C.

Carmon, T.

Carretero-González, R.

H. E. Nistazakis, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and R. Carretero-González, “Bright-dark soliton complexes in spinor Bose-Einstein condensates,” Phys. Rev. A 77, 033612 (2008).
[Crossref]

Chen, X.

D. Rand, I. Glesk, C.-S. Brès, D. A. Nolan, X. Chen, J. Koh, J. W. Fleischer, K. Steiglitz, and P. R. Prucnal, “Observation of Temporal Vector Soliton Propagation and Collision in Birefringent Fiber,” Phys. Rev. Lett. 98, 053902 (2007).
[Crossref] [PubMed]

Chen, Z.

Christodoulides, D. N.

Collings, B. C.

S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Observation of Polarization-Locked Vector Solitons in an Optical Fiber,” Phys. Rev. Lett. 82, 3988–3991 (1999).
[Crossref]

Coskun, T. H.

Côtù, R.

Y. Wu and R. Côtù, “Bistability and quantum fluctuations in coherent photoassociation of a Bose-Einstein condensate,” Phys. Rev. A 65, 053603 (2002).
[Crossref]

Crosignani, B.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

Cundiff, S. T.

S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Observation of Polarization-Locked Vector Solitons in an Optical Fiber,” Phys. Rev. Lett. 82, 3988–3991 (1999).
[Crossref]

Delquè, M.

Deng, L.

G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Λ system,” Phys. Rev. E 73, 056606 (2006).
[Crossref]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-Λ system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

Y. Wu, M. G. Payne, E. W. Hagley, and L. Deng, “Ultraviolet single-photons on demand and entanglement of photons with a large frequency difference,” Phys. Rev. A 70, 063812 (2004).
[Crossref]

Y. Wu and L. Deng, “Achieving multifrequency mode entanglement with ultraslow multiwave mixing,” Opt. Lett. 29, 1144–1146 (2004).
[Crossref] [PubMed]

Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett. 29, 2064–2066 (2004).
[Crossref] [PubMed]

L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, “Opening Optical Four-Wave Mixing Channels with Giant Enhancement Using Ultraslow Pump Waves,” Phys. Rev. Lett. 88, 143902 (2002).
[Crossref] [PubMed]

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, “Optical-wave group-velocity reduction without electro-magnetically induced transparency,” Phys. Rev. A 65, 051805(R) (2002).
[Crossref]

Dinda, P. Tchofo

T. Kanna, M. Lakshmanan, P. Tchofo Dinda, and N. Akhmediev, “Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations,” Phys. Rev. E 73, 026604 (2006).
[Crossref]

Ding, C.-L.

X.-Y. Lü, J.-B. Liu, C.-L. Ding, and J.-H. Li, “Dispersive atom-field interaction scheme for three-dimensional entanglement between two spatially separated atoms,” Phys. Rev. A 78, 032305 (2008).
[Crossref]

DiPorto, P.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994).
[Crossref] [PubMed]

Fleischer, J. W.

D. Rand, I. Glesk, C.-S. Brès, D. A. Nolan, X. Chen, J. Koh, J. W. Fleischer, K. Steiglitz, and P. R. Prucnal, “Observation of Temporal Vector Soliton Propagation and Collision in Birefringent Fiber,” Phys. Rev. Lett. 98, 053902 (2007).
[Crossref] [PubMed]

C. Anastassiou, J. W. Fleischer, T. Carmon, M. Segev, and K. Steiglitz, “Information transfer via cascaded collisions of vector solitons,” Opt. Lett. 26, 1498–1500 (2001).
[Crossref]

Frantzeskakis, D. J.

H. E. Nistazakis, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and R. Carretero-González, “Bright-dark soliton complexes in spinor Bose-Einstein condensates,” Phys. Rev. A 77, 033612 (2008).
[Crossref]

Fry, E. S.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[Crossref]

Ge, M.-L.

X.-J. Liu, H. Jing, and M.-L. Ge, “Solitons formed by dark-state polaritons in an electromagnetic induced transparency,” Phys. Rev. A 70, 055802 (2004).
[Crossref]

Glesk, I.

D. Rand, I. Glesk, C.-S. Brès, D. A. Nolan, X. Chen, J. Koh, J. W. Fleischer, K. Steiglitz, and P. R. Prucnal, “Observation of Temporal Vector Soliton Propagation and Collision in Birefringent Fiber,” Phys. Rev. Lett. 98, 053902 (2007).
[Crossref] [PubMed]

Haelterman, M.

Hagley, E. W.

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-Λ system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

Y. Wu, M. G. Payne, E. W. Hagley, and L. Deng, “Ultraviolet single-photons on demand and entanglement of photons with a large frequency difference,” Phys. Rev. A 70, 063812 (2004).
[Crossref]

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, “Optical-wave group-velocity reduction without electro-magnetically induced transparency,” Phys. Rev. A 65, 051805(R) (2002).
[Crossref]

L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, “Opening Optical Four-Wave Mixing Channels with Giant Enhancement Using Ultraslow Pump Waves,” Phys. Rev. Lett. 88, 143902 (2002).
[Crossref] [PubMed]

Hang, C.

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

Harris, S. E.

S. E. Harris and L. V. Hau, “Nonlinear Optics at Low Light Levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[Crossref]

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[Crossref]

Hau, L. V.

S. E. Harris and L. V. Hau, “Nonlinear Optics at Low Light Levels,” Phys. Rev. Lett. 82, 4611–4614 (1999).
[Crossref]

Haus, H. A.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[Crossref]

Hollberg, L.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[Crossref]

Hou, J.-M.

W.-X. Yang, J.-M. Hou, and R.-K. Lee, “Ultraslow bright and dark solitons in semiconductor quantum wells,” Phys. Rev. A 77, 033838 (2008)
[Crossref]

Huang, G.

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Λ system,” Phys. Rev. E 73, 056606 (2006).
[Crossref]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-Λ system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

Huang, Q.-J.

J.-H. Li, X.-Y. Lü, J.-M. Luo, and Q.-J. Huang, “Optical bistability and multistability via atomic coherence in an N-type atomic medium,” Phys. Rev. A 74, 035801 (2006).
[Crossref]

Imamoglu, A.

M. D. Lukin and A. Imamoǧlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000)
[Crossref] [PubMed]

Jiang, K.

G. Huang, K. Jiang, M. G. Payne, and L. Deng, “Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Λ system,” Phys. Rev. E 73, 056606 (2006).
[Crossref]

Jing, H.

X.-J. Liu, H. Jing, and M.-L. Ge, “Solitons formed by dark-state polaritons in an electromagnetic induced transparency,” Phys. Rev. A 70, 055802 (2004).
[Crossref]

Kang, J. U.

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, “Observation of Manakov spatial solitons in AlGaAs planar waveguides,” Phys. Rev. Lett. 76, 3699–3702 (1996).
[Crossref] [PubMed]

Kanna, T.

T. Kanna, M. Vijayajayanthi, and M. Lakshmanan, “Periodic energy switching of bright solitons in mixed coupled nonlinear Schrödinger equations with linear self-coupling and cross-coupling terms,” Phys. Rev. A 76, 013808 (2007).
[Crossref]

T. Kanna, M. Lakshmanan, P. Tchofo Dinda, and N. Akhmediev, “Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations,” Phys. Rev. E 73, 026604 (2006).
[Crossref]

Kash, M. M.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[Crossref]

Kevrekidis, P. G.

H. E. Nistazakis, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and R. Carretero-González, “Bright-dark soliton complexes in spinor Bose-Einstein condensates,” Phys. Rev. A 77, 033612 (2008).
[Crossref]

Kivshar, Y. S.

Knorr, A.

G. T. Adamashvili, C. Weber, and A. Knorr, “Optical nonlinear waves in semiconductor quantum dots: Solitons and breathers,” Phys. Rev. A 75, 063808 (2007).
[Crossref]

Knox, W. H.

S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Observation of Polarization-Locked Vector Solitons in an Optical Fiber,” Phys. Rev. Lett. 82, 3988–3991 (1999).
[Crossref]

Kockaert, P.

Koh, J.

D. Rand, I. Glesk, C.-S. Brès, D. A. Nolan, X. Chen, J. Koh, J. W. Fleischer, K. Steiglitz, and P. R. Prucnal, “Observation of Temporal Vector Soliton Propagation and Collision in Birefringent Fiber,” Phys. Rev. Lett. 98, 053902 (2007).
[Crossref] [PubMed]

Konotop, V. V.

Korolev, A. E.

Kozuma, M.

L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, “Opening Optical Four-Wave Mixing Channels with Giant Enhancement Using Ultraslow Pump Waves,” Phys. Rev. Lett. 88, 143902 (2002).
[Crossref] [PubMed]

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, “Optical-wave group-velocity reduction without electro-magnetically induced transparency,” Phys. Rev. A 65, 051805(R) (2002).
[Crossref]

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X.-T. Xie, W. Li, J. Li, W.-X. Yang, A. Yuan, and X. Yang, “Transverse acoustic wave in molecular magnets via electromagnetically induced transparency,” Phys. Rev. B 75, 184423 (2007).
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D. V. Skryabin, A. V. Yulin, and A. I. Maimistov, “Localized Polaritons and Second-Harmonic Generation in a Resonant Medium with Quadratic Nonlinearity,” Phys. Rev. Lett. 96, 163904 (2006).
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D. Y. Tang, H. Zhang, L. M. Zhao, and X. Wu, “Observation of High-Order Polarization-Locked Vector Solitons in a Fiber Laser,” Phys. Rev. Lett. 101, 153904 (2008).
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D. Y. Tang, H. Zhang, L. M. Zhao, and X. Wu, “Observation of High-Order Polarization-Locked Vector Solitons in a Fiber Laser,” Phys. Rev. Lett. 101, 153904 (2008).
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Commun. Theor. Phys. (1)

X. Yang and Y. Wu, “Ultra-slow Bright and Dark Optical Solitons in Cold Media,” Commun. Theor. Phys. 45, 335–342 (2006).
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Europhys. Lett. (1)

W.-X. Yang and R.-K. Lee, “Slow optical solitons via intersubband transitions in a semiconductor quantum well,” Europhys. Lett. 83, 14002 (2008)
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J. Phys. B (1)

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Phys. Lett. A (2)

E. Yomba, “Generalized hyperbolic functions to find soliton-like solutions for a system of coupled nonlinear Schrödinger equations,” Phys. Lett. A 372, 1612–1618 (2008).
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J.-B. Liu, X.-Y. Lü, N. Liu, M. Wang, and T.-K. Liu, “Microwave solitons in molecular magnets via electromagnetically induced transparency,” Phys. Lett. A 373, 413–417 (2008).
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Phys. Rev. A (14)

X.-J. Liu, H. Jing, and M.-L. Ge, “Solitons formed by dark-state polaritons in an electromagnetic induced transparency,” Phys. Rev. A 70, 055802 (2004).
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H. E. Nistazakis, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and R. Carretero-González, “Bright-dark soliton complexes in spinor Bose-Einstein condensates,” Phys. Rev. A 77, 033612 (2008).
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Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71, 053820 (2005).
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Figures (4)

Fig. 1.
Fig. 1. (a) Five-level atomic system in a hyper V-type configuration. The strong cw control field with frequency ω c1 (ω c2) and Rabi frequency Ω c1 c2) couples to the ∣1〉 ↔ ∣3〉 (∣2〉 ↔ ∣4〉) transition, and the σ - σ + component of the weak probe field with frequency ωp and Rabi frequency Ω p couples to the ∣0〉 ↔ ∣1〉 (∣0〉 ↔ ∣2)) transition. Δ s s - Δ) and Δ t1 t2) are corresponding one- and two-photon detunings with Δ = 2μ 𝓑g 𝓑/h̄ being the Zeeman shift of levels ∣1〉 and 2〉. (b) Possible arrangement of experimental apparatus. 𝓑 is an applied magnetic field. Ω c1 and ω c2 represent two control fields. σ - and σ + denote two orthogonally polarized components of a probe field.
Fig. 2.
Fig. 2. Absorption coefficients α 1 and α 2 versus dimensionless Rabi frequency ∣Ω c1∣/γ 1 and ∣Ω c2∣/γ 2 for several different values of the two-photon detunings Δ t1 and Δ t2. The other parameters are γ 1γ 2 ≃ 5.6 MHz, γ 3γ 4 ≃ 0.76 MHz, κ 10κ 20 ≃ 6 γ 1/cm, Δ s ~ ≃ 25 γ 1, and Δ ~ 0.5 γ 1.
Fig. 3.
Fig. 3. (a) and (c) are, respectively, the σ - and σ + polarization components of the probe field, obtained numerically from Eqs. (31) and (32), versus dimensionless time τ/τ 0 and distance ξ/Ld with τ 0 = 1.0 × 10-8 s and Ld ≃ 12.59 cm for bright-bright vector optical solitons formation. (b) shows the Manakov bright-bright vector solitons given in Eqs. (43) and (44) for the same parameters. Other parameters are explained in the main text.
Fig. 4.
Fig. 4. The wave shape of the σ - polarization component for two-soliton collisions under different initial conditions: (a), u 1(z = 0) = u 2(z = 0) = sech(σ + 2.0)e - + sech(σ-2.0)e ; (b), u 1(z = 0) = u 2(z = 0) = sech(σ + 2.0)e - + sech(σ - 2.0)ei (σ+π). All the parameters are present in the main text.

Equations (45)

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Ĥinth̄=Δs 11(ΔsΔ)22Δt133Δt24 4
(Ωp1eikp·r10+Ωp2eikp·r20+Ωc1eikc1·r31+Ωc2eikc2·r42+H.c.) ,
B1t=i(Δs+iγ1) B1+iΩc1*B3+iΩp1B0 .
B2t=i(ΔsΔ+iγ2) B2+iΩc2*B4+iΩp2B0 ,
B3t=i(Δt1+iγ3)B3+iΩc1B1,
B4t=i(Δt2+iγ4)B4+iΩc2B2,
B02+B12+B22+B32+B42=1,
Ωp1z+1cΩp1t=iκ10B1B0* ,
Ωp2z+1cΩp2t=iκ20B2B0* ,
(ω+Δs+iγ1)β1(1)+Ωc1*β3(1)=Λp1,
(ω+ΔsΔ+iγ2)β2(1)+Ωc2* β4(1) = Λp2,
(ω+Δt1+iγ3)β3(1)+Ωc1β1(1)=0,
(ω+Δt2+iγ4)β4(1)+Ωc2β2(1)=0,
Λp1ziωcΛp1=iκ10β1(1),
Λp2ziωcΛp2=iκ20β2(1),
Λp1(z,ω)=Λp1 (0,ω) exp [izK1(ω)] ,
Λp2(z,ω)=Λp2 (0,ω) exp [izK2(ω)] ,
K1=ωc+κ10(ω+Δt1+iγ3)Ωc12(ω+Δs+iγ1)(ω+Δt1+iγ3) =K10+K11ω+K12ω2+,
K2=ωc+κ20(ω+Δt2+iγ4)Ωc22(ω+ΔsΔ+iγ2)(ω+Δt2+iγ4) =K20+K21ω+K22ω2+,
i(z+1Vg1t)Ω1+K122t2Ω1=𝒩1,
i(z+1Vg2t)Ω2+K222t2Ω2=𝒩2,
B1(1)=Δt1+iγ3Ωc12(Δs+iγ1)(Δt1+iγ3)Ωp1,
B2(1)=Δt2+iγ4Ωc22(ΔsΔ+iγ2)(Δt2+iγ4)Ωp2,
B3(1)=Ωc1Ωc12(Δs+iγ1)(Δt1+iγ3)Ωp1,
B4(1)=Ωc2Ωc22(ΔsΔ+iγ2)(Δt2+iγ4)Ωp2.
i(ξ+δτ)Ω1K122τ2Ω1(W11eα1ξΩ12+W12eα2ξΩ22)Ω1=0,
i(ξδτ)Ω2K222τ2Ω2(W22eα2ξΩ22+W21eα1ξΩ12)Ω2=0,
W11=κ10(Δt1+iγ3)(Ωc12+Δt12+γ32)D1D12],
W12=κ10(Δt1+iγ3)(Ωc22+Δt22+γ42)D1D22],
W22=κ20(Δt2+iγ4)(Ωc22+Δt22+γ42)D2D22],
W21=κ20(Δt2+iγ4)(Ωc12+Δt12+γ32)D2D12],
iu1s+iQδ u1σQ12u1σ2(Q11u12+Q12u22) u1 =0,
iu2siQδ u2σQ22u2σ2(Q22u22+Q21u12) u2 =0,
u1=C1sech(σ)exp[i(F11σ+F12s)] ,
u2=C2sech(σ)exp[i(F21σ+F22s)] ,
u1=C1sech(σ)exp[i(F11σ+F12s)] ,
u2=C2tanh(σ)exp[i(F21σ+F22s)] ,
u1=C1tanh(σ)exp[i(F11σ+F12s)] ,
u2=C2sech(σ)exp[i(F21σ+F22s)] ,
u1=C1tanh(σ)exp[i(F11σ+F12s)] ,
u2=C2tanh(σ)exp[i(F21σ+F22s)] ,
isu1+2σ2u1+(u12+u22)u1=0,
isu2+2σ2u2+(u22+u12)u2=0,
u1=2 cos(θ) sech(σ)eis,
u2=2 sin(θ) sech(σ)eis,

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