1. Introduction

Rogue waves are unexpectedly large, spatially and temporally localized extreme wave events excited from background states [1–3], firstly discovered in deep oceans [4, 5]. Due to the interest in both fundamental research and practical applications, in recent years the study on rogue waves has received tremendous attention in many branches of science, including fluid dynamics [6–9], nonlinear optics [10–14], plasma physics [15], acoustics [16], Bose-Einstein condensates in ultracold quantum gases [17], and even financial physics [18]. More references about rogue waves can be found in the comprehensive reviews on recent developments in the area of localized structures in optical and matter-wave media [19,20].

The nonlinear Schrödinger (NLS) equation, which governs the evolution of the envelope of weakly nonlinear, dispersive wave packets, has been widely employed as a simple model for describing rogue wave phenomena [21]. However, due to the requirement for modeling a variety of complex systems, it becomes necessary to investigate the rogue wave phenomena beyond the framework of a single-component NLS equation. Recent developments have considered dissipative effects [22–24], higher-order nonlinear terms [25–27], and in particular the interaction between several field components [28–34]. The latter investigations have led to the discovery of multi-component rogue waves, described by coupled NLS equations.

In this work, we suggest a scheme to create vector (or two-component) optical rogue waves, Akhmediev and Kuznetsov-Ma breathers in a coherent atomic system of a M-type five-level configuration interacting resonantly with a probe laser field (with two polarization components) and two control laser fields. By means of electromagnetically induced transparency (EIT), we prove that the propagation velocity of these nonlinear optical excitations can be reduced to 10⁻⁴ c (c is the light speed in vacuum) and the generation power of them can be lowered to the level of microwatt. We also prove that, under the action of a transversal gradient magnetic field, the motion trajectories of the two polarization components of these nonlinear optical excitations may undergo a significant deflection, very similar to the Stern-Gerlach (SG) effect of an atomic beam carrying with a spin angular momentum. We find that, within the propagation distance only several centimeters, the deflection angle of these nonlinear excitations the can reach to 10⁻⁴ radian. At variance with the Stern-Gerlach effect of atomic beams, the deflection angle can be made different for different polarization components and may be actively manipulated in an easy and controllable way.

Before proceeding, we note that the study of the SG deflection for linear optical pulses via EIT was firstly reported by Karpa and Weitz [35]. Later on, some further extensions related to the SG effect of optical pulses in atomic systems were also carried out in [36–38]. However, different from these previous works, the topics we consider here are vector optical rogue waves and breathers, and their SG deflection and active manipulation. The results obtained here not only provide a method for obtaining novel, multi-component optical rogue waves and breathers with coherent atomic gases, but also are promising for practical applications, e.g., for the precise measurement of gradient magnetic fields.

The rest of the article is arranged as follows. In Sec. II, the theoretical model under study is described. In Sec. III, the nonlinear envelope equations governing the evolution of the two polarization components of electromagnetic field are derived by using a method of multiple-scales. In Sec. IV, weak-light vector Peregrine rogue waves, Akhmediev and Kuznetsov-Ma breathers are presented, and their group velocity and generation power are estimated. In Sec. V, the SG effect of the rogue waves and breathers is discussed. Finally, in the last section a summary of main results obtained in this work is presented.

2. Model

We start with considering an atomic gas with a M-type five-level configuration. A linearly polarized, pulsed probe laser field (with pulse duration τ₀) E_p = E_p1 + E_p2 = (∊̂₋ℰ_p1 + ∊̂₊ℰ_p2) exp[i(k_p z − ω_pt)] + c.c. drives respectively the transitions |3〉 ↔ |2〉 and |3〉 ↔ |4〉 by its σ⁻ (i.e. left-circular) polarization component E_p1 and σ⁺ (i.e. right-circular) polarization component E_p2; two π-polarized continuous-wave (CW) control laser fields E_c1 = ẑℰ_c1 exp[i(k_c1x − ω_c1t)] + c.c. and E_c2 = ẑℰ_c2 exp[i(k_c2x − ω_c2t)] + c.c. drive respectively the transitions |1〉 ↔ |2〉 and |5〉 ↔ |4〉 [see Fig. 1]. Here c.c. denotes complex conjugate; ${\widehat{∊}}_{-}\equiv (\widehat{\mathbf{x}}-i\widehat{\mathbf{y}})/\sqrt{2}$ and ℰ_p1 [${\widehat{∊}}_{+}\equiv (\widehat{\mathbf{x}}+i\widehat{\mathbf{y}})/\sqrt{2}$ and ℰ_p2] are respectively the unit vector and envelope of the σ⁻ (σ⁺) polarization component of the probe field; x̂, ŷ and ẑ are respectively unit vectors along coordinate axes x, y, and z. The control field envelopes ℰ_c1 and ℰ_c2 are assumed to be strong enough so that they can be taken to be undepleted during the evolution of the probe field. Furthermore, the atoms are assumed to be initially prepared in the ground state |3〉 and trapped in a gas cell with ultracold temperature to cancel Doppler broadening and collisions. The level diagram in Fig. 1 can be taken to be composed by two Λ-type EIT configurations, sharing with the common ground state |3〉.

 

Fig. 1 Possible geometrical arrangement and the coordinate frame chosen for the system. A cold atomic gas with a M-type five-level configuration (right side) are loaded in a gas cell (left side). A pulsed probe field E_p drives respectively the transitions |3〉 ↔ |2〉 and |3〉 ↔ |4〉 by its σ⁻ polarization component and σ⁺ polarization component; Two strong CW control field E_c1 and E_c2 drive respectively the transitions |1〉 ↔ |2〉 and |5〉 ↔ |4〉. δ_cj, δ_p, and δ_p + Δ are detunings. Atoms are assumed to be prepared in the ground-state |3〉, indicated by black dots. B is a magnetic field applied to the atomic gas.

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Under electric-dipole and rotating-wave approximations, the Hamiltonian of the system in interaction picture reads H_int/ħ = (δ_p − δ_c1)|1〉 〈1| + δ_p|2〉 〈2| + (δ_p + Δ)|4〉 〈4| + (δ_p + Δ − δ_c2)|5〉 〈5|+Ω_c1|2〉 〈1|+Ω_p1|2〉 〈3|+Ω_p2|4〉 〈3|+Ω_c2|4〉 〈5|+H.c., where Ω_p1=−(p₂₃ · ∊̂₋)ℰ_p1/ħ and Ω_p2=−(p₄₃ · ∊̂₊)ℰ_p2/ħ [Ω_c1=−(p₂₁ · ẑ)ℰ_c1/ħ and Ω_c2=−(p₄₅ · ẑ)ℰ_c2/ħ] are Rabi frequencies of the two circularly polarized components of the probe field (the two π-polarized control fields), with p_jl being the electric dipole matrix element associated with the transition from |j〉 to |l〉. Detunings are defined as δ_p = ω₂₃ + μ₂₃B₀ − ω_p, δ_c1 = ω₂₁ + μ₂₁B₀ − ω_c1, δ_c2 = ω₄₅ + μ₄₅B₀ − ω_c2, and Δ = μ₄₂B₀, where ${\mu }_{jl}={\mu }_B\left(g_F^j{m}_F^j-g_F^l{m}_F^l\right)/\hslash$ (μ_B, $g_F^j$, and $m_F^j$ are Bohr magneton, gyromagnetic factor, and magnetic quantum number of the level |j〉, respectively) and ω_jl = (E_j − E_l)/ħ, with E_j the eigenenergy of the state |j〉. Here we have applied an homogeneous static magnetic field, B = ẑB₀, to the atomic gas, which results in a Zeeman level shift $\mathrm{\Delta }{E}_{\text{Zeeman}}={\mu }_B{g}_F^j{m}_F^j{B}_0$, and hence removes the degeneracy of ground-state sublevels |j〉 (j = 1, 3, 5) and the excited-state sublevels |j〉 (j = 2, 4). The motion of the atoms is governed by the optical Bloch equation

The evolution of E_p is controlled by Maxwell equation ∇²E_p − (1/c²)Phys. Lett. ²E_p/Phys. Lett. t² = (1/∊₀c²)Phys. Lett. ²P_p/Phys. Lett. t², where P_p is the electric polarization intensity of the probe field. Under slowly varying envelope approximation, the Maxwell equation reduces to

If the transverse size of the probe pulse is large enough so that the diffraction effect is negligible, the solution for the linear propagation of the probe field can be obtained by taking Ω_pj as small quantities. Then, from the Maxwell-Bloch (MB) equations (1) and (2), one finds that Ω_pj are proportional to exp{i[K_j (ω)z − ωt)]}, with the linear dispersion relation

From Eq. (3) we see that the linear dispersion relation of the system has two independent branches (i.e. the branch K₁ and the branch K₂). Shown in Fig. 2(a) is the imaginary part Im(K₁) and the real part Re(K₁) for the branch K₁, as functions of ω. The red dashed (solid) line in Fig. 2(a) is the profile of Im(K₁) for Ω_c1 = 0 (Ω_c1 = 1.0 × 10⁷ s⁻¹); the black dashed-dotted line is the profile of Re(K₁) for Ω_c1 = 1.0 × 10⁷ s⁻¹. Shown in Fig. 2(b) is the same as Fig. 2(a), but for the branch K₂. When plotting Fig. 2, system parameters are chosen from a laser-cooled ⁸⁷Rb atomic gas with atomic states assigned as |1〉 = |5²S_1/2, F = 2, m_F = −1〉, |2〉 = |5²P_1/2, F = 2, m_F = −1〉, |3〉 = |5²S_1/2, F = 1, m_F = 0〉, |4〉 = |5²P_1/2, F = 2, m_F = 1〉, and |5〉 = |5²S_1/2, F = 2, m_F = 1〉. The atomic state decay rates are given by Γ₂ ≃ Γ₄ ≃ 0.5×10⁷ s⁻¹ and Γ₁ ≃ Γ₃ ≃ Γ₅ ≃ 1.0 × 10⁴ s⁻¹. In addition, we take κ₃₂ ≃ κ₃₄ = 1.0 × 10⁹ cms⁻¹, Ω_c1 = Ω_c2 = 1.0 × 10⁷ s⁻¹, δ_p = δ_c1 = 0, δ_c2 = 2.0 × 10⁶ s⁻¹, and Δ = 2.0 × 10⁶ s⁻¹.

 

Fig. 2 Linear dispersion relation of the system. (a) Im(K₁) and Re(K₁) as functions of ω. The red dashed (solid) line is Im(K₁) for the case Ω_c1 = 0 (Ω_c1 = 1.0 × 10⁷ s⁻¹); the black dashed-dotted line is Re(K₁) for Ω_c1 = 1.0 × 10⁷ s⁻¹. (b) The same as (a) but for Im(K₂) and Re(K₂).

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From Fig. 2(a) and Fig. 2(b) we see that, in the presence of the control fields, a large and deep EIT transparency window is opened in both the absorption spectra of Ω_p1 and Ω_p2 (i.e. the two polarization components of the probe pulse). This is a kind of double EIT phenomenon and it is resulted from the quantum destruction interference contributed by the two control fields. Furthermore, due to the symmetry of the two Λ-type level configurations, the linear dispersion relations for Ω_p1 and Ω_p2 are nearly coincided with each other. Due to the EIT effect and the symmetry of level configuration, the group velocities of the two polarization components [defined by V_gj = Re(∂K_j/∂ω)⁻¹, j = 1, 2] can be very small compared with c for large control fields and can be matched very well [see the expression (9) below]. We shall see that the ultraslow, matched group velocities are very crucial for obtaining significant optical Stern-Gerlach deflections of the vector rogue waves, as shown below.

3. Asymptotic expansion and nonlinear envelope equations

One of the main goals of the present work is to obtain vector rogue waves and breathers in the system described above. To this end, we employ the method of multiple-scales [37, 38] to derive nonlinear envelope equations of the envelopes of the two probe-field components. We take the following asymptotic expansion ${\rho }_{mm}={\sum }_{l=1}{∊}^l{\rho }_{mm}^{(l)}$ (m = 1–5), ${\rho }_{33}=1+{\sum }_{l=1}{∊}^l{\rho }_{33}^{(l)}$, ${\rho }_{mn}={\sum }_{l=1}{∊}^l{\rho }_{mn}^{(l)}$ (m, n = 1–5; m ≠ n), and ${\mathrm{\Omega }}_{pj}={\sum }_{l=1}{∊}^l{\mathrm{\Omega }}_{pj}^{(l)}$ (j = 1, 2). Here ∊ is a small parameter characterizing the small population depletion of the ground state |3〉, and all quantities on the right hand side of the asymptotic expansion are considered to be functions of the multi-scale variables z_l=∊^l z (l = 0, 1, 2), x₁= ∊x, and t_l=∊^lt (l = 0, 1).

Substituting the above expansion into the optical MB equations (1) and (2), we obtain a chain of linear, but inhomogeneous equations, which can be solved order by order. At the first order (l = 1), we obtain the solution for the half Rabi frequencies of the probe field ${\mathrm{\Omega }}_{pj}^{(1)}=F_j\text{exp}\left\{i\left[K_j(\omega )z_0-\omega t_0\right]\right\}$ (j = 1, 2), where F_j are yet to be determined envelope functions of slow variables z₂ and t₂, and K_j (ω) is given by the linear dispersion relation (3). The solution for the density-matrix elements at this order reads ${\rho }_{13}^{(1)}=-{\mathrm{\Omega }}_{c1}^{*}{\mathrm{\Omega }}_{p1}^{(1)}/D_1$, ${\rho }_{23}^{(1)}=-(\omega -d_1){\mathrm{\Omega }}_{p1}^{(1)}/D_1$, ${\rho }_{34}^{(1)}=-\left(\omega -d_5^{*}\right){{\mathrm{\Omega }}_{p2}^{*}}^{(1)}/D_2^{*}$, and ${\rho }_{35}^{(1)}=-{\mathrm{\Omega }}_{c2}{{\mathrm{\Omega }}_{p2}^{*}}^{(1)}/D_2^{*}$, with the other density-matrix elements being zero.

At the second order (l = 2), one obtains the solution for ${\rho }_{mm}^{(1)}$ (m = 1–5), ${\rho }_{12}^{(1)}$, ${\rho }_{14}^{(1)}$, ${\rho }_{15}^{(1)}$ ${\rho }_{24}^{(1)}$, ${\rho }_{25}^{(1)}$, and ${\rho }_{45}^{(1)}$, which are lengthy and hence omitted here. The other second-order density-matrix elements are zero. A solvability condition for ${\mathrm{\Omega }}_{pj}^{(2)}$ requires

At the third order (l = 3), a solvability condition for ${\mathrm{\Omega }}_{pj}^{(3)}$ yields

For the convenience of following discussions, we introduce δ = (1/V_g1 − 1/V_g2)/2, V_g = 2V_g1V_g2/(V_g1 + V_g2), and τ = t − z/V_g. Then Eq. (6) combined with Eq. (4) can be written into the dimensionless form

By taking realistic parameter set Ω_c1 = Ω_c2 = 1.6 × 10⁸ s⁻¹, δ_p = 1.0 × 10⁸ s⁻¹, δ_c1 = 0, δ_c2 = 3.0 × 10⁶ s⁻¹, and Δ = 2.0 × 10⁶ s⁻¹, with other parameters the same as those given in Sec. II, we obtain K₁₂ = (−4.56 + i0.25) × 10⁻¹⁵ cm⁻¹s², K₂₂ = (−4.64 + i0.26) × 10⁻¹⁵ cm⁻¹s², W₁₁ ≈ (−9.37 + i0.15) × 10⁻¹⁶ cm⁻¹s², W₁₂ ≈ (−9.44 + i0.15) × 10⁻¹⁶ cm⁻¹s², W₂₁ ≈ (−9.34 + i0.15) × 10⁻¹⁶ cm⁻¹s², W₂₂ ≈ (−9.40 + i0.15) × 10⁻¹⁶ cm⁻¹s². We see that the imaginary parts of these quantities are much smaller than their relevant real parts. The physical reason for such small imaginary parts is due to the EIT effect induced by two CW control fields. With these parameters, the numerical values of the group velocities of the two polarization components of the probe field (5) read

Since g_δ and a_1,2 are small, one can treat the terms ig_δ∂u_j/∂σ and −ia_ju_j as a small perturbation. Then Eq. (8) after neglecting the small perturbation reduces to

4. Ultraslow weak-light vector rogue waves, breathers and their interactions

To present a simple solution of Eq. (10), we introduce the transform [34] $(u_1,u_2)=\left(\text{cos}\alpha +\text{sin}\alpha ,\text{cos}\alpha -\text{sin}\alpha \right)\mathrm{\Psi }/\sqrt{2}$. Here α is a free, real constant and Ψ obeys the standard (1+1)-dimensional NLS equation i∂Ψ/∂s + ∂²Ψ/∂σ² + |Ψ|²Ψ = 0, which admits the Peregrine soliton solution with the form Ψ = [1−4(1+2is)/(1+4s²+2σ²)]e^is. Then, we have the two-component Peregrine soliton (vector rogue wave) solution with the form

The Manakov equation (10) also admits solutions of vector Kuznetsov-Ma breather and Ahkmediev breather. In our system, the (dimensional) vector Kuznetsov-Ma breather solution reads

Shown in Fig. 3 are numerical results on the propagation of the vector optical rogue wave and breathers as functions of $\sqrt{2}\tau /{\tau }_0$ and z/L_D by using split-step Fourier method for α = π/3 and $\mathrm{\Omega }=\sqrt{2}$. The left and right panels in the figure are the intensity distributions for the first (i.e. σ⁻) and the second (i.e. σ⁺) polarization components of the probe field, respectively. The upper panels [(a) and (b) ] are the intensity distributions for the first and second polarization components of the vector Kuznetsov-Ma breather, i.e. |Ω_pKMB1/U₀|² (left panel) and |Ω_pKMB2/U₀|² (right panel), obtained by using the vector Kuznetsov-Ma breather (13) as an initial condition; the middle panels [(c) and (d) ] are the intensity distributions for the vector Peregrine soliton, i.e. |Ω_p1/U₀|² (left panel) and |Ω_p2/U₀|² (right panel), obtained by using the vector Kuznetsov-Ma breather (11) as an initial condition; the lower panels [(e) and (f) ] are the intensity distributions for the vector Ahkmediev breathers, i.e. |Ω_pAB1/U₀|² (left panel) and |Ω_pAB2/U₀|² (right panel), obtained by using the vector Kuznetsov-Ma breather (14) as an initial condition. From the figure we see that waveshapes of the two probe-field components are matched well, but the intensity of the first (left panel) polarization component is different from that of the second (right panel) polarization component.

 

Fig. 3 Vector optical rogue waves and breathers as functions of $\sqrt{2}\tau /{\tau }_0$ and z/L_D. The left and right panels are the intensity distributions for the first (i.e. σ⁻) and the second (i.e. σ⁺) polarization components of the probe field, respectively. (a) [(b)] |Ω_pKMB1/U₀|² [|Ω_pKMB2/U₀|²] obtained by using the vector Kuznetsov-Ma breather (13) as an initial condition; (c) [(d)] |Ω_p1/U₀|² (left) [|Ω_p2/U₀|²] obtained by using the vector Peregrine soliton (11) as an initial condition; (e) [(f)] |Ω_pAB1/U₀|² [|Ω_pAB2/U₀|²] obtained by using the vector Ahkmediev breathers (14) as an initial condition.

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By using the relation V_g = 2V_g1V_g2/(V_g1 + V_g2) and the result (9), we obtain

The threshold of the optical power P_th for generating the vector rogue waves and breathers in the present system can be estimated by using the Poynting’s vector. Assuming that the beam diameter ≈ 0.1 mm, we have

To explore the interaction property of the optical rogue waves and breathers obtained above, a numerical simulation on two-soliton collisions is carried out by taking the initial condition u′_1,2(s = 0) = u_1,2(σ + d)e^i0.5σ + ρu_1,2(σ − d)e^{−i0.5σ+iφ} with ρ = 1 and d = 5. The parameter φ is the initial relative phase between the two solitons. Fig. 4(a) [Fig. 4(b)] shows motion trajectories of two Peregrine solitons during their collision with φ = 0 (φ = π/4). We see that for φ = 0 (φ = π/4) the interaction between the two solitons is attractive (repulsive). The physical reason is that for φ = 0 (φ = π/4) there is an increase (a decrease) of the optical refractive index when the two solitons approach each other, resulting in an attraction (a repulsion) of more photons into (from) the central region of the collision. In the figure, only the first polarization component is plotted; similar result for the second polarization component is also obtained but omitted here. Shown in Fig. 4(c) and Fig. 4(d) are respectively collisions between two Kuznetsov-Ma breathers for attractive (φ = 0) and repulsive (φ = π/4) interactions.

 

Fig. 4 Collisions between two vector optical rogue waves and breathers. (a) ((b)) Motion trajectories of two Peregrine solitons with attractive (repulsive) interaction during collision; (c) ((d)) Motion trajectories of two Kuznetsov-Ma breathers with attractive (repulsive) interaction during collision. In panels (a) and (c), the initial phase between the two solitons φ = 0; in panels (b) and (d), φ = π/4.

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5. Stern-Gerlach effect of the Kuznetsov-Ma breathers

One of the main advantages of the present atomic system is that it is possible to realize an active control over the vector optical rogue waves and breathers by manipulating system parameters. As an example, we consider the inhomogeneous static magnetic field

The nonlinear envelope equation in the presence of the gradient magnetic field can also be derived by the use of the multiple-scale perturbation method, as done in the last section. But here we are interested in the case in which the probe-field envelope depends on the slow variables x₁, z₂, and t₂ (i.e. the dispersion effect of the system is negligible, valid for the probe pulse with a large time duration). Carrying out the perturbation calculation to the third order, we obtain the following envelope equations:

Equations (18) can be written into the dimensionless form

Due to the presence of the gradient magnetic field, the two polarization components of the probe pulse, u₁ and u₂, will separate each other after propagating some distance, and hence the cross-phase-modulation terms can be neglected. In this case, Eq. (19) reduces to

In this way, we can get various solutions of vector rogue waves, Kuznetsov-Ma and Ahkmediev breathers of Eq. (20) in the presence of the SG gradient magnetic field. In the following, we only discuss the solution of vector Kuznetsov-Ma breather, which reads

To illustrate clearly the feature of the trajectory deflection of the vector Kuznetsov-Ma breather, a numerical simulation is carried out on Eq. (20) with a nonzero B₁. System parameters are chosen as Ω_c1 = Ω_c2 = 1.6 × 10⁸ s⁻¹, δ_p = 1.0 × 10⁸ s⁻¹, δ_c1 = 0, δ_c2 = 3.0 × 10⁶ s⁻¹, Δ = 1.8 × 10⁶ s⁻¹, R_⊥ = 57 μm, U₀ = 1.72 × 10⁷ s⁻¹, and B₁ = 2.2 × 10⁻³ G cm⁻¹. We get L_Diff ≈ 2.6 cm, ℳ₁ = 0.01, and ℳ₂ = −0.01. Shown in Fig. 5(a) and Fig. 5(c) are respectively the deflection trajectories of the first and second polarization components (i.e. |Ω_p1/U₀|² and |Ω_p2/U₀|²) as functions of x/R_⊥ and z/L_Diff. Their corresponding contour maps (i.e. the motion trajectories in the x–z plane) are plotted in Fig. 5(b) and Fig. 5(d). In the simulation, the initial condition is chosen as the Kuznetsov-Ma breather (24), with α = π/3 and $\mathrm{\Omega }=\sqrt{2}$. From the figure, we see that the motion trajectories of the two polarization components of the breather are parabolic and symmetric with respect to the x axis, which can be regarded as a SG effect of optical rogue wave, very similar to the SG effect of an atomic beam. In fact, a close analog exists between the well-known SG effect for atomic beam described in textbook [41] and the SG effect for the vector Kuznetsov-Ma breather described here. An atom in the lowest S-state has a spin angular momentum and can display a SG effect when passing through a gradient magnetic field. The vector Kuznetsov-Ma breather considering here has also a spin-like angular momentum (i.e. the two polarization components) and hence may undergo a SG-like effect in the presence of the gradient magnetic field.

 

Fig. 5 Stern-Gerlach deflection of the vector Kuznetsov-Ma breather in the presence of a gradient magnetic field. (a) Intensity distribution of the first polarization component of the probe field, i.e. |Ω_pKMB1/U₀|², as a function of t/τ₀ and z/L_Diff, with (b) the corresponding contour map (motion trajectory in the x–z plane); (c) Intensity distribution of the second polarization component of the probe field, i.e. |Ω_pKMB2/U₀|², with (d) the corresponding contour map (motion trajectory in the x–z plane). The initial condition is chosen to be the Kuznetsov-Ma breather (24), with α = π/3 and $\mathrm{\Omega }=\sqrt{2}$.

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However, different from the atomic SG effect where the motion trajectories of atoms are always symmetric for two different spin components [41], in the SG deflection of the vector Kuznetsov-Ma breather the motion trajectories of the two polarization components can be asymmetric. The reason is that the two polarization components of the breather can propagate with different velocities. To show this we repeat the simulation by taking Ω_c2 = 1.54 × 10⁸ without changing other system parameters. Then the motion velocities of the two polarization components become (V_g1, V_g2) = (2.3, 1.9) × 10⁻⁴ c. In this situation the first polarization component of the breather keeps the same waveshape and trajectory as that shown in Fig. 5(a) and Fig. 5(b) [see Fig. 6(a) and 6(b)], whereas the second component displays a trajectory different from that shown in Fig. 5(c) and Fig. 5(d) [see Fig. 6(c) and Fig. 6(d)].

 

Fig. 6 Stern-Gerlach deflection of the Kuznetsov-Ma breather in the presence of a gradient magnetic field. (a) Intensity distribution of the first polarization component of probe field, i.e. |Ω_pKMB1/U₀|² versus $\sqrt{2}x/R_{\perp }$ and z/L_Diff, with (b) the corresponding contour map; (c)Intensity distribution of the second polarization component of probe field, i.e. |Ω_pKMB2/U₀|², with (d) the corresponding contour map. Due to the different velocities, the deflections of the two polarization components are asymmetric.

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Equation (25) tells us that the larger the gradient magnetic field B₁ (and hence the larger the value of ℳ_1,2) is, the more apparently the motion trajectory of the breather bends. To confirm this prediction, in Fig. 7(a) we show the numerical result on the symmetrical and asymmetrical SG deflections of the Kuznetsov-Ma breather, where the SG deflection distance x/R_⊥ is taken as a function of z/L_Diff. The blue lines are for B₁ = 2.2 × 10⁻³ G cm⁻¹ and the red lines are for B₁ = 4.4 × 10⁻³ G cm⁻¹. The upbent lines are for the first (i.e. σ⁻) polarization component, and downbent lines are for the second (i.e. σ⁺) polarization component. The solid lines (dashed lines) are for the symmetrical (asymmetrical) SG deflections, realized by taking Ω_c2 to be 1.6 × 10⁸ s⁻¹ (1.54 × 10⁸ s⁻¹). From the figure, we can see clearly that the SG deflection distance of the Kuznetsov-Ma breather becomes larger when the magnetic field gradient B₁ increases.

 

Fig. 7 Symmetrical and asymmetrical SG deflections of the Kuznetsov-Ma breather. (a) SG deflection distance x/R_⊥ as a function of z/L_Diff for B₁ = 2.2 × 10⁻³ G cm⁻¹ (blue lines) and 4.4 × 10⁻³ G cm⁻¹ (red lines). The upbent lines are for the first (i.e. σ⁻) polarization component, and downbent lines are for the second (i.e. σ⁺) polarization component. The solid lines (dashed lines) are for the symmetrical (asymmetrical) SG deflections, obtained by taking Ω_c2 to be 1.6 × 10⁸ s⁻¹ (1.54 × 10⁸ s⁻¹). (b) SG deflection angles θ as a function of the magnetic field gradient B₁ when the breather propagating to z = 4L_Diff ≈ 10.4 cm. The blue (red) line is for σ₋ (σ₊) polarization component.

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One can make an estimation on the SG deflection angle. The expected deflection angle of the output jth polarization component of the Kuznetsov-Ma breather is determined by the ratio between the propagating velocity along the x axis (i.e. $V_j=R_{\perp }{ℳ}_j{\lambda }_j^{2}t/{\tau }_0^2$) and that along the z axis (i.e. V_gj). Thus, after passing through the medium with length L the deflection angle of the output jth polarization component of the breather is given by

The SG effect of the vector Kuznetsov-Ma breathers is one of examples illustrating how to actively manipulating the vector optical rogue waves and breathers. The reason for such active manipulation possible is due to the resonant characters of the multi-level atomic systems coupling with laser fields, in which many system parameters can be selected and adjusted in a controllable way. Besides the interest in fundamental research, such active manipulation maybe find promising practical applications. For instance, one can utilize the SG effect of the vector Kuznetsov-Ma breathers (or other rouge wave and breathers) to precisely detect the gradient magnetic field by measuring their SG deflection angles.

6. Conclusion

In this work, we have presented a scheme for creating vector optical rogue waves and breathers in a M-type five-level atomic system coupled with a probe laser field of two polarization components and two control laser fields. By means of EIT we have proved that the propagation velocity of these nonlinear optical excitations may be reduced to 10⁻⁴ c and their generation power may be lowered to the level of microwatt. We have also proved that, by the use of a transversal gradient magnetic field, the motion trajectories of the two polarization components of these nonlinear excitations can display a large deflection, a phenomenon quite similar to the SG effect of an atomic beam carrying with a spin angular momentum. We have found that within the propagation distance only several centimeters the deflection angle of these nonlinear excitations can reach to the value of 10⁻⁴ radian. At variance with the SG effect of atomic beams, the deflection angle can be made different for different polarization components and may be actively manipulated in an easy and controllable way. The results obtained here have not only provided a method for generating novel, multi-component optical rogue waves and breathers with coherent atomic gases at weak light level, but also opened an avenue for their active manipulation and possible practical applications in future.