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
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.
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 E⃗p = E⃗p- + E⃗p+ = [e⃗+ E p- + e⃗-E p+)exp(-iωpt + i k⃗p · r⃗) + c.c. and E⃗c1,c2 = e⃗c1,c2 E c1,c2exp(-i ω c1,c2 t + i k⃗c1,c2 · r⃗) + c.c, respectively. Here, e⃗- = (x⃗ - iy⃗)/√2 and e⃗+ = (x + iy)√2 are the unit vectors of the σ - and σ + circular polarization components with the slowly varying envelopes E - and E +, and e⃗c1,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,
where Δs = ωp - ωp (Δs - 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 = (μ⃗31 · e⃗c1)E c1/h̄, Ωc2 = (μ⃗42 · e⃗c2)E⃗c2/h̄, Ωp1 = (μ⃗10 · e⃗-)E p-/h̄, and Ωp2 = (μ⃗20 · e⃗+)E p+/h̄ are Rabi frequencies with μ⃗ij 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 ik⃗p·r⃗∣1〉 +B 2(t)e i k⃗p·r∣2〉 + B 3(t)e i(k⃗p+k⃗c1)·r⃗∣3〉 + B 4(t)e i(k⃗p+k⃗c2)·r∣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,
where 2γk (k= 1,2,3,4) is the decay rate of state ∣k〉, and κ 10,20 = Nωp∣μ⃗10,20 · e⃗-,+∣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.
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,p2/Ωc1,c2 be a small parameter . 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
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.
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
where K 10,20 = K 1,2(0), K 11,21 = dK 1,2(ω)dω∣ω=0, K 12,22 = 2d 2 K 1,2(ω)/dω 2∣ω=0, which have clear physical signification. K 10,20 = ϕ 1,2 + iα 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),
We then have the nonlinear evolution equations, i.e., the two coupled NLS equations for Ω1 (z, t) and Ω2(z,t),
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
with D 1 = ∣Ωc1∣ - (Δs + iγ 1)(Δt 1 + iγ 3) and D 2 = ∣Ωc2∣2 - (Δs - Δ + iγ 2)(Δt2 + iγ 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,22i≃K 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 22r∣U 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 0∣K 22r/W 22r∣1/2. 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,
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.
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.
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
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.
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 : 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 11 ≃ W 22 ≃ W 12 ≈ W 21 ≃ (-1.347 + 0.023i) × 10-18 s2cm-1, V g1/c≃ V 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.,
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 -iσ + sech(σ - 2.0)eiσ for Fig. 4(a) and u 1(z = 0) = u 2(z = 0) = sech(σ + 2.0)e -iσ(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.
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 11 ≃ K 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 11 ≃ W 22 ≃ W 12 ≃ W 21 ≃ (-2.615 + 0.026i) × 10-19 s2cm-1, V g1/c ≃ V g2/c ≃ 0.152, and α 1 ≃ α 2 ≃ 0.00093 cm-1. Thus, we have the dimensionless coefficients Qδ ≃ -0.00043, Q 1 ≃ 0.9999, Q 2 = 1, and Q 11 ≃ Q 12 ≃ Q 21 ≃ Q 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.
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.
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.
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