Exciting rogue waves, breathers, and solitons in coherent atomic media
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Zhong-Yin Li(黎仲寅), Fei-Feng Li(黎飞凤), Hui-Jun Li(李慧军),Institute of Nonlinear Physics and Department of Physics, Zhejiang Normal University, Jinhua 321004, China
Abstract We propose a scheme that excites rogue waves via electromagnetically induced transparency (EIT), which can also excite breathers and solitons. The system is a resonant Λ-type atomic ensemble. Under EIT conditions, the envelope equation of the probe field can be reduced to several different models, such as the saturable nonlinear Schrödinger equation (SNLSE), and SNLSE with the trapping potential provided by a far-detuned laser field or a magnetic field. In this scheme, rogue waves can be generated by different initial pulses, such as the Gaussian wave with (or without) the uniform background. The scheme can be used to obtain rogue waves, breathers and solitons. We show the existence regions of rogue waves, breathers, and solitons as the function of the amplitude and width of the initial pulse. The novelty of our paper is that, we not only show rogue waves in the integrable system numerically, but also present the method to generate and control rogue waves in the non-integrable system. Keywords：electromagnetically induced transparency;rogue wave;the non-integrable model
PDF (1081KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Zhong-Yin Li(黎仲寅), Fei-Feng Li(黎飞凤), Hui-Jun Li(李慧军). Exciting rogue waves, breathers, and solitons in coherent atomic media. Communications in Theoretical Physics, 2020, 72(7): 075003- doi:10.1088/1572-9494/ab7ed4
1. Introduction
Since the invention of the laser, the rich phenomena and applications of nonlinear optics [1–5] have been of wide concern to researchers. Especially in the field of nonlinear optics, the realization of high resonance media by electromagnetically induced transparency (EIT) [6] has attracted great interest of researchers. On the one hand, EIT has many striking features, including a significant suppression of a large reduction of group velocity [7], optical absorption [8], and giant enhancement of Kerr nonlinearity [9]. On the other hand, the EIT scheme possesses many adjustable parameters, such as atomic density, field strength, and detuning. So, EIT has been selected as an ideal platform to study nonlinear effects. In recent years, there have been many reports on EIT, such as the high-dimensional spatiotemporal optical solitons [10], the high-efficiency four-wave mixing [11], the phase gate [12], the bistable state [13], the optical clock [14], the ultra-low optical transmission and storage [15, 16], etc.
It is well known that rogue waves were discovered firstly in deep oceans [17], which were temporally localized extreme waves [18–20]. In recent years, people pay much attention to the study of rogue waves in nonlinear optics [21–28]. In the EIT system, rogue waves were found through reducing the models into integrable models [29, 30]. However, rogue wave solutions of non-integrable models are not presented in the EIT system. And there are few reports about rogue wave solutions of non-integrable models.
In this paper, we consider a resonant three-level Λ atomic scheme, working in an EIT effect induced by a continuous-wave control field. Taking the different parameters, the envelope equations of the probe field can be reduced into three types of nonlinear Schrödinger equations (NLSE), including the saturable nonlinear Schrödinger equation (SNLSE) with or without the trapping potential. The results of SNLSE have been well reported [31–37]. And its analytical solitons [38], dark solitons [39], and spatial solitons [40] have been reported. However, rogue wave solutions for SNLSE have not been reported.
In this article, we propose a numerical scheme to excite rogue waves, meanwhile it is also used to excite breathers and solitons. Using the numerical propagation method, we obtain rogue wave, breather, and soliton solutions by the initial incident Gaussian pulse. The types of nonlinear modes can be adjusted by the width and amplitude of the initial pulse at will. To the best of our knowledge, the method to excite rogue waves in a non-integrable system has not been reported. So, it will open a way to generate and control nonlinear modes in a non-integrable system.
The paper is arranged as follows: in the next section, we will introduce the model being studied. In section 3, we will excite and control rogue waves, breathers, and solitons. The main results of this paper are summarized in the last section. In the appendix, the results of the NLSE are shown.
2. Model
We consider an atomic system with a Λ-type energy-level configuration, as shown in figure 1. The system interacts with the probe field, the control field, the large detuned laser field and the magnetic field. In the resonant interaction system, the probe field satisfies the SNLSE with the trapping external potential, contributed by the large detuning laser field and magnetic field.
Figure 1.
New window|Download| PPT slide Figure 1.Excitation scheme of the lifetime broadened three-state atomic system interacting with a weak probe field with the half Rabi frequency Ωp, and a strong continuous-wave control field with the half Rabi frequency Ωc.
According to [40], the dimensionless equation satisfied by the probe field is$ \begin{eqnarray}{\rm{i}}\displaystyle \frac{{\rm{d}}u}{{\rm{d}}s}+\displaystyle \frac{{{\rm{d}}}^{2}u}{{\rm{d}}{\xi }^{2}}+\displaystyle \frac{{c}_{1}u}{1+{c}_{2}| u{| }^{2}}+{c}_{3}| v{| }^{2}u+{c}_{4}{wu}=0,\end{eqnarray}$where $s=z/{L}_{\mathrm{diff}}$, $\xi =x/{R}_{\perp }$, $u={{\rm{\Omega }}}_{{\rm{p}}}/{U}_{0}$, $v={E}_{0}(x)/{V}_{0}$, $w={B}_{0}(x)/{W}_{0}$, with ${L}_{\mathrm{diff}}(\equiv 2{R}_{\perp }^{2}{\omega }_{{\rm{p}}}/c)$, ${R}_{\perp }$, U0, V0, B0(x), and W0 being, respectively, the characteristic diffraction length, beam radius, half Rabi frequency of the probe field, intensity of the far-detuned (Stark) optical lattice field, the weak magnetic field along the transverse direction, and intensity of the magnetic field. The coefficients respectively are ${c}_{1}={\kappa }_{13}{d}_{2}{L}_{\mathrm{diff}}/D$, ${c}_{2}=W{{U}_{0}}^{2}$, ${c}_{3}=\alpha {L}_{\mathrm{diff}}{V}_{0}^{2}$, ${c}_{4}\,=\beta {L}_{\mathrm{diff}}{W}_{0}$, with $\alpha ={\kappa }_{13}\left[{\alpha }_{2}D+{d}_{2}({\alpha }_{2}{d}_{3}+{\alpha }_{3}{d}_{2})\right]/(2{\hslash }{D}^{2})$, $\beta ={\kappa }_{13}\left[{\mu }_{2}D+{d}_{2}({\mu }_{2}{d}_{3}+{\mu }_{3}{d}_{2})\right]/({\hslash }{D}^{2})$, $W=(| {{\rm{\Omega }}}_{{\rm{c}}}{| }^{2}+| {d}_{2}{| }^{2})/| D{| }^{2}$, $D=| {{\rm{\Omega }}}_{{\rm{c}}}{| }^{2}-{d}_{2}{d}_{3}$, ${d}_{{\rm{j}}}={{\rm{\Delta }}}_{{\rm{j}}}+{\rm{i}}{\gamma }_{{\rm{j}}}$, αj is the polarizability of the level $| j\rangle $, ${\mu }_{{\rm{j}}}={\mu }_{{\rm{B}}}{g}_{{\rm{F}}}^{{\rm{j}}}{m}_{{\rm{F}}}^{{\rm{j}}}$, ${g}_{{\rm{F}}}^{{\rm{j}}}$ is the Landé factor, Δj and γj are the detuning and the decay rate of the states $| j\rangle $, ${{\rm{\Omega }}}_{{\rm{p}},{\rm{c}}}\,=({{\boldsymbol{e}}}_{{\rm{x}}}\cdot {{\boldsymbol{p}}}_{\mathrm{13,23}}){{ \mathcal E }}_{{\rm{p}},{\rm{c}}}/{\hslash }$ with ${{\boldsymbol{e}}}_{{\rm{x}}}$, ${{\boldsymbol{p}}}_{13}$, ${{ \mathcal E }}_{{\rm{p}}}$ being the polarization unit vector, the electric dipole matrix element of the transition, and the envelope, ${\kappa }_{13}=N{\omega }_{{\rm{p}}}| {{\boldsymbol{e}}}_{{\rm{x}}}\cdot {{\boldsymbol{p}}}_{13}{| }^{2}/(2{\epsilon }_{0}{\hslash }c)$ with N being the atomic concentration.
We select the D1 line transition ${5}^{2}{S}_{1/2}\longrightarrow {5}^{2}{P}_{1/2}$ of the 87Rb atoms. The levels respectively are $| 5{S}_{1/2},F=1,{m}_{{\rm{F}}}\,=-1\rangle $, $| 5{S}_{1/2},F=2,{m}_{{\rm{F}}}=-1\rangle $ and $| 5{S}_{1/2},F=2,{m}_{{\rm{F}}}=-2\rangle $. At the same time, γ1=0, 2γ2=300 s−1, 2γ3=3.6× 107 s−1, ωp=2.37×1015 s−1, ${R}_{\perp }=2.52\times {10}^{-3}\,\mathrm{cm}$, ${{\rm{\Omega }}}_{{\rm{c}}}\,=6.0\times {10}^{7}\,{{\rm{s}}}^{-1}$, ${\kappa }_{13}=1.0\times {10}^{11}$ cm−1 s−1, ${{\rm{\Delta }}}_{1}=0$, ${{\rm{\Delta }}}_{2}\,=3.6\times {10}^{4}{\sigma }_{1}$ s−1, ${{\rm{\Delta }}}_{3}=1.0\times {10}^{9}\,{{\rm{s}}}^{-1}$, ${U}_{0}=6.0\times {10}^{7}\sqrt{{\sigma }_{2}}\,{{\rm{s}}}^{-1}$, V0=380σ3 V cm−1, ${W}_{0}=0.09{\sigma }_{4}$ Gs. Where the size and sign of σ1, σ2, σ3, and σ4 are free parameters that are individually controlled by Δ2, U0, V0, and W0. Substituting these parameters into the formula of coefficients, we can obtain the characteristic diffraction length Ldiff=1.0 cm,$ \begin{eqnarray}{c}_{1}/{\sigma }_{1}=1.0+0.01{\rm{i}},{c}_{2}/{\sigma }_{2}=1.0,\end{eqnarray}$$ \begin{eqnarray}{c}_{3}/{\sigma }_{3}=1.0+0.001{\rm{i}},{c}_{4}/{\sigma }_{4}=1.0+0.001{\rm{i}}.\end{eqnarray}$
The imaginary part of σi(i=1, 2, 3, 4) can be ignored, and equation (1) becomes [40]:$ \begin{eqnarray}{\rm{i}}\displaystyle \frac{{\rm{d}}u}{{\rm{d}}s}+\displaystyle \frac{{{\rm{d}}}^{2}u}{{\rm{d}}{\xi }^{2}}+\displaystyle \frac{{\sigma }_{1}u}{1+{\sigma }_{2}| u{| }^{2}}+{\sigma }_{3}| v{| }^{2}u+{\sigma }_{4}{wu}=0.\end{eqnarray}$Taking the different parameters, we can obtain different σ1, σ2, σ3, and σ4, and get the different models.
The linear dispersion relation of the Λ-type EIT system has been shown in [41], we find our system lies in the transparency window easily. So, the absorption is suppressed, and the imaginary part of the coefficients in equations (2) and (3) is so small that it can be neglected.
2.1. SNLSE
When we choose Δ2=−3.46×105 s−1, U0=6.0× 107 s−1, V0=0 V cm−1, and W0=0 Gs, we can obtain σ1=−9.6, σ2=1, σ3=0, and σ4=0. Thus, we can get the SNLSE [42, 43]$ \begin{eqnarray}{\rm{i}}\displaystyle \frac{{\rm{d}}u}{{\rm{d}}s}+\displaystyle \frac{{{\rm{d}}}^{2}u}{{\rm{d}}{\xi }^{2}}-\displaystyle \frac{9.6u}{1+| u{| }^{2}}=0.\end{eqnarray}$
2.2. SNLSE with the trapping potential contributed by far-detuned (Stark) optical lattice field
Choosing Δ2=−3.46×105 s−1, U0=6.0×107 s−1, V0=1.22×103 V cm−1, and W0=0 Gs, we obtain σ1=−9.6, σ2=1, σ3=3.2, and σ4=0. Thus, equation (4) becomes$ \begin{eqnarray}{\rm{i}}\displaystyle \frac{{\rm{d}}u}{{\rm{d}}s}+\displaystyle \frac{{{\rm{d}}}^{2}u}{{\rm{d}}{\xi }^{2}}-\displaystyle \frac{9.6u}{1+| u{| }^{2}}+3.2| v{| }^{2}u=0.\end{eqnarray}$
2.3. SNLSE with the trapping potential contributed by weak magnetic field
When Δ2=−3.46×105 s−1, U0=6.0×107 s−1, V0= 0 V cm−1, and W0=0.36 Gs, we get σ1=−9.6, σ2=1, σ3=0, and σ4=4. Thus, equation (4) becomes$ \begin{eqnarray}{\rm{i}}\displaystyle \frac{{\rm{d}}u}{{\rm{d}}s}+\displaystyle \frac{{{\rm{d}}}^{2}u}{{\rm{d}}{\xi }^{2}}-\displaystyle \frac{9.6u}{1+| u{| }^{2}}+4{wu}=0.\end{eqnarray}$
3. Excitation and control of rogue waves, breathers, and solitons
In integrable models, the rogue wave solution can be obtained by the analytic method and the numerical modulation instability (MI). In non-integrable models, however, the analytic methods are invalid and the MI can just be used to prove the analytic result rather than find the rogue wave solution. Finding a simple and efficient method to generate and control the rogue wave is necessary. Here, we propose a method to excite and control rogue waves through the evolution of the initial pulse. For example,$ \begin{eqnarray}{u}_{0}=C+A{{\rm{e}}}^{\left(-\tfrac{{\xi }^{2}}{{\omega }^{2}}\right)}\end{eqnarray}$is selected as the initial incident pulse, here, C is the height of background, such as the plane wave, A and ω are the amplitude and width of Gaussian pulse, respectively. We can use the split-step Fourier method [44] to get the solution by evolution of the initial pulse.
Since the method can be used to generate a rogue wave, it is also possible to generate the soliton and breather. We take the propagation distance s=20 to distinguish the rogue wave, breather, and soliton. The criterion for the three nonlinear modes are as follows: (1) rogue wave: it is localized spatially and temporally [18–20], and the peak amplitude of the first wave should exceed the significant background C in 2–2.2 times [45] (we choose 2.2 times as a standard) along the propagation direction s, and the wide Gaussian pulse provides the background when C=0. (2) Breather: it is periodic in the s (or ξ) direction and localized in the ξ (s) direction. There exist ten periodic waves along the s direction at least, the error of every peak amplitude is less than 10% of the peak amplitude of the first wave. (3) Soliton: its waveform remains unchanged along the s direction, and the error of the peak amplitude is less than 10% of the initial peak amplitude.
In the appendix, we prove that our method has three advantages by using NLSE. Firstly, it is reliable and simple. Then, it can be used to excite and control rogue waves, breathers, and solitons. Finally, it is also used to the various non-integrable models as following.
3.1. Excitation and control of rogue waves, breathers, and solitons in SNLSE
It is well known that the SNLSE(5) is non-integrable, and there is no report about its rogue wave solution. Now, we use the propagating method to excite its nonlinear solutions.
The phase transition regions of nonlinear modes as the function of the amplitude and width of the initial pulse are shown in figure 2(a) when C=0, and the phase transition region of rogue wave is shown in figure 2(b) when C=1. We choose some points (black points 1–6 in figure 2(a) and black points 1–2 in figure 2(b)) to exhibit their propagating results. Here, the corresponding nonlinear modes are shown in figures 2(a1)–(a6), and (b1)–(b2). We introduce the full width at half maximum (FWHM) of nonlinear modes W to denote their distribution regions. For the SNLSE, FWHM of figures 2(a1)–(a6) are W=1.02, 1.10, 1.17, 1.33, 1.64, and 0.86, respectively, and the one of figure 2(b1) is W=1.25. According to the above criterions about nonlinear modes, we find the yellow regions located by the black point 1 are for rogue waves, the regions located by the black point 2 are the transition regions of rogue waves and breathers, the blue regions located by the black point 3 are the breathers, the regions located by the black point 4 are the transition regions of breathers and solitons, the red regions located by the black point 5 are the solitons, and there is not the soliton solutions in the regions located by the black point 6. In figure 2(b), the regions located by the black point 1 denote the existence interval of rogue waves as shown in figure 2(b1) when the nonlinear coefficient σ1=−9.6 and −4.8. It is obvious that the existence regions of rogue waves are effected by the nonlinear coefficient σ1.
Figure 2.
New window|Download| PPT slide Figure 2.(a) The phase transition regions of nonlinear modes for SNLSE (5) as the function of the initial amplitude A and width w when taking C=0. (b) The phase transition curves of rogue wave for SNLSE (5) as the function of the initial amplitude A and width w when taking C=1. Here, σ1=−4.8 and −9.6. (a1)–(a6) The evolution results of the initial incident pulse (8), correspond to the black points 1–6 in figure 2(a). Here, A=1.2, ω=4, 3.7, 2.5, 1.6, 1.1, and 0.5, respectively. FWHM of (a1)–(a6) are W=1.02, 1.10, 1.17, 1.33, 1.64, and 0.86, respectively. (b1)–(b2) The evolution results of the initial incident pulse (8), correspond to the black points 1 and 2 in figure 2(b), respectively. Here, A=1.2, ω=1.4 and 0.5. FWHM of (b1)–(b2) is W=1.25 and 2.58.
These results show that the background adjusted by C is more conducive to excite rogue waves. Furthermore, the above scheme also provides a simple way to generate various nonlinear modes experimentally, which can be excited and controlled by the amplitude and width of the initial pulse, in which, one parameter can be adjusted to excite and control rogue waves, breathers, and solitons. Moreover, these results further prove that the propagating method for finding the nonlinear solutions not only adapts to the integrable NLSE as shown in the appendix, but also can be used to generate nonlinear solutions for non-integrable nonlinear models.
3.2. Influence of far-detuned (Stark) optical lattice field on excitation and control rogue waves, breathers, and solitons in SNLSE
When the far-detuned optical lattice field is considered, equation (6) becomes an SNLSE with trapping potential, for instance, dimensionless far-detuned (Stark) optical lattice field $| v{| }^{2}={{\rm{e}}}^{\left(-\tfrac{{\xi }^{2}}{25}\right)}$ is chosen. Taking C=0, we obtain the phase transition regions of nonlinear modes in figure 3(a). Figure 3(a) shows a similar result as shown in figure 2(a). In figure 3(a), the yellow regions located by the point 1, the blue regions located by the point 2, and the red regions located by the point 3 denote the existence interval of rogue waves, breathers, and solitons as shown in figures 3(a1)–(a3), respectively. The corresponding parameters of black points 1, 2, and 3 are chosen as the propagating initial values, the evolution results are shown in figures 3(a1)–(a3). The FWHM of figures 3(a1)–(a3) are W=1.95, 1.18, and 1.64, respectively.
Figure 3.
New window|Download| PPT slide Figure 3.Choosing far-detuned optical lattice field $| v{| }^{2}={{\rm{e}}}^{\left(-\tfrac{{\xi }^{2}}{25}\right)}$ in equation (6). (a) The phase transition regions of nonlinear modes for equation (6) as the function of the initial amplitude A and width w when taking C=0. (b) The phase transition curves of the rogue wave for equation (6) as the function of the initial amplitude A and width w when taking C=1. Here, σ1=−4.8 and −9.6. (c) The phase transition curves of the rogue wave for equation (6) as the function of the initial amplitude A and width w when taking C=1. Here, σ3=0 and 3.2. (d) The phase transition curves of the rogue wave for equation (6) as the function of the initial amplitude A and the background height C with σ3=0 (red solid line) and σ3=3.2 (blue dotted line). Here, ω=1.5. (a1)–(a3) The evolution results of the initial incident pulse (8) with these parameters corresponding to the black points 1–3 in figure 3(a). Here, A=1.2, ω=4, 2.1, and 1.1, respectively. The FWHM of (a1)–(a3) are W=1.95, 1.18, and 1.64, respectively. (b1) The evolution results of the initial incident pulse (8) with these parameters corresponding to the black point 1 in figure 3(b). Here, A=1.2 and ω=1.4. FWHM of (b1) is W=1.21. (c1) The evolution results of the initial incident pulse (8) with these parameters corresponding to the black point 1 in figure 3(c). Here, A=1.2 and ω=1.5. FWHM of (c1) is W=1.25.
After taking C=1, the phase transition curves of rogue waves are plotted in figure 3(b) with a different nonlinear coefficient σ1, which can be realized by changing the detuning Δ2. With the increasing of the nonlinear coefficient $| {\sigma }_{1}| $, the existing regions for the rogue wave were broadened. We choose a set of parameters remarked by black point 1 to obtain the evolution result shown in figure 3(b1). We also give the phase transition curves of the rogue wave with the different trapping potential σ3 in figure 3(c). With the increasing of σ3, the existing region of the rogue wave is changed. The corresponding rogue wave marked by the black point 1 are shown in figure 3(c1). The FWHM of figures 3(b1) and (c1) are W=1.21 and 1.25. When ω=1.5, the phase transition curves of the rogue wave as the function of the initial amplitude A and the background height C is shown in figure 3(d), where the red solid line and black dotted line are σ3=0 and σ3=3.2, respectively. The region denoted by b in figure 3(d) is the existing region of the rogue wave. When the initial pulse amplitude is the same, it needs a higher background to excite the rogue wave under the far-detuned (Stark) optical lattice field. From these results, we conclude that the propagating method is suitable to generate the nonlinear modes, and the trapping potential is advantageous for the rogue wave.
3.3. Influence of weak magnetic field on excitation and control rogue waves in SNLSE
After introducing the weak magnetic field, the model(7) is still an SNLSE with the trapping potential. But due to the Zeeman effect of weak magnetic field, the potential is the linear function of the weak magnetic field. We can choose the dimensionless weak magnetic field w=1−0.5ρ(ξ) in equation (7), and ρ are random numbers which can be realized by adding a demagnetized neodymium-iron-boron ferromagnet [46]. When C=1, we obtain the phase transition curves of the rogue wave in figure 4(a) without (by the blue solid line) or with (by the red dashed line) the random weak magnetic field. The corresponding parameters of the black point 1 are chosen as the propagating initial value, the evolution results are shown in figure 4(a1). The FWHM of figure 4(a1) is W=1.33. When ω=1.5, the phase transition curves of the rogue wave as the function of the initial amplitude A and the background height C is shown in figure 4(b), where the red solid line and blue dotted line denote σ4=0 and σ4=4, respectively. The region denoted by b in figure 4(b) is the existing region of the rogue wave, respectively. When C<1, the excitation of rogue waves in a weak magnetic field requires a larger background of the initial pulse. When C≥1, the excitation of rogue waves in a weak magnetic field requires a smaller background for the initial pulse. These results show that the random weak magnetic field is not conducive to exciting the rogue wave.
Figure 4.
New window|Download| PPT slide Figure 4.(a) The phase transition curves of the rogue wave for equation (7) as the function of the initial amplitude A and width w with σ4=0 and 4, when C=1. Here, the dimensionless weak magnetic field w=1−0.5ρ(ξ). (b) The phase transition curves of the rogue wave for equation (7) as the function of the initial amplitude A and the background height C with σ4=0 (red solid line) and σ4=4 (blue dotted line). Here, ω=1.5, w is a random number. (c) The phase transition curves of the rogue wave for equation (7) as the function of the initial amplitude A and width w when taking C=1. Here, the dimensionless weak magnetic field $w={{\rm{e}}}^{\left(-\tfrac{{\xi }^{4}}{{5}^{4}}\right)}$, σ4=0 and σ4=4, respectively. (a1) The evolution results of the initial incident pulse (8) with these parameters corresponding to the black point 1 in figure 4(a). Here, A=1.2 and ω=1.3. The FWHM of (a1) is W=1.33. (c1) The evolution results of the initial incident pulse (8) by taking these parameters corresponding to the black point 1 in figure 4(c). Here, A=1.2 and ω=1.3. The FWHM of (c1) is W=1.25.
If we choose the weak magnetic field as the point defect, for instance the dimensionless weak magnetic field $w={{\rm{e}}}^{\left(-\tfrac{{\xi }^{4}}{{5}^{4}}\right)}$ in equation (7). When C=1, we obtain the phase transition curves of the rogue wave in figure 4(c) without (by the blue solid line) or with (by the red dashed line) the weak magnetic field. The corresponding parameters of black point 1 are chosen as the propagating initial value, the evolution results are shown in figure 4(c1). The FWHM of figure 4(c1) is W=1.25. These results show that the weak magnetic field can be used to change the parameter regions or to control the generation of the rogue wave.
4. Conclusion
In this work, we have proposed a resonant Λ-type atomic ensemble as an ideal experimental platform to generate and control nonlinear modes via EIT. Under the different parameter conditions, we illustrated the envelope equation of probe field became the integrable NLSE or the non-integrable SNLSE. A simple scheme was mainly designed to excite and control rogue waves. And it was also used to excite breathers and solitons. Using the scheme, these nonlinear modes could be obtained by controlling the width or amplitude of the initial incident Gaussian pulse. Besides, the background of the initial pulse was more conducive to excite rogue waves, rather than others modes. Our numerical scheme would be helpful in exciting and controlling different nonlinear modes in other physical models. In addition, we also found the weak magnetic field was not conducive to exciting rogue waves, but the far-detuned (Stark) optical lattice field was conducive to the generation of rogue waves.
Acknowledgments
This work was supported by the NSF-China under Grant Nos.11835011, No.11574274, No.11675146.
Appendix. Excitation and control of rogue waves, breathers, and solitons in NLSE
Taking Δ2=−1.44×105 s−1, U0≈4.24×107s−1, V0= 0V cm−1, and W0=0Gs, we can obtain σ1=−4, σ2=0.5, σ3=0, and σ4=0. Then equation (4) becomes$ \begin{eqnarray}{\rm{i}}\displaystyle \frac{{\rm{d}}u}{{\rm{d}}s}+\displaystyle \frac{{{\rm{d}}}^{2}u}{{\rm{d}}{\xi }^{2}}+2| u{| }^{2}u=0\end{eqnarray}$by Taylor series expansion and the phase transformation.
It is well known that the first order rogue wave solution for NLSE(9) can be written as [47]$ \begin{eqnarray}u(s,\xi )=\left[1-\displaystyle \frac{4(1+4{\rm{i}}s)}{1+4{\xi }^{2}+16{s}^{2}}\right]{{\rm{e}}}^{2{\rm{i}}s}.\end{eqnarray}$This solution can be obtain by Darboux transformation or others’ analytical methods.
In figure A1, the profile for the analytical solution (10) is shown in panel (a). Taking u(s=0, ξ) of the analytical solution (10) as the initial value, the evolution results are shown in panel (b). Panel (c) is the propagating results of taking (8) as the initial value, here, A=0.3, ω=0.6, and C=1. Panel (d) shows the results of numerical MI. For NLSE, the FWHM of figures A2(a)–(d) are W=0.78, 0.78, 0.76, and 0.80, respectively. Due to NLSE(9) with translational invariance, the cross sections of panels (a)–(d) are shown in the special s located at the peak value of their profiles. From figure A1, we know the results obtained by the propagating method are reliable. Using our method, there are several advantages: (1) it is feasible and correct; (2) it is not dependent on the integrability of model; (3) the realization of the rogue wave can be controlled by the height of the background, the amplitude and width of initial incident pulse at will. It will avoid the uncontrollability of MI.
Figure A1.
New window|Download| PPT slide Figure A1.(a) The profile of the analytical rogue wave solution (10). (b) The evolution result by taking the analytical solution (10) as the initial value of propagation. (c) The propagating results by taking (8) as the initial value, here, A=0.3, ω=0.6, and C=1. (d) The profile of the rogue wave by the numerical MI. The inset shows the sea of the rogue wave generated by MI. The FWHM of (a)–(d) are W=0.78, 0.78, 0.76, and 0.80, respectively. (e) The cross sections of the profiles in panels (a)–(d) by taking a special s located at the peak value of rogue wave profiles. The blue solid line is the cross section of panel (a), the red dashed line for panel (b), the green dashed–dotted line for panel (c), and the black dotted line for panel (d).
Figure A2.
New window|Download| PPT slide Figure A2.(a) The phase transition regions of nonlinear modes for NLSE (9) as the function of the initial amplitude A and width w when taking C=0. (b) The phase transition curve of the rogue wave for NLSE (9) as the function of the initial amplitude A and width w when taking C=1. (c) The phase transition curve of the rogue wave as the function of the initial amplitude A and the background height C. (a1)–(a3) The evolution results of the initial incident pulse (8) with these parameters corresponding to the black points 1–3 in figure A2(a), respectively. Here, A=1.2, ω=6.5, 3, and 1.45, respectively. The FWHM of (a1)–(a3) are W=0.59, 0.86, and 1.88, respectively. (a11)–(a31) The peak amplitudes of the rogue wave, breather and soliton as the function of the propagation distance, respectively. (a12)–(a32) The profiles of nonlinear modes corresponding to points a–c of figure A2(a11)–(a31) denoted by the black solid line, red dashed line, and blue dashed–dotted line, respectively. (b1)–(b2) The evolution results of the initial incident pulse (8) with these parameters corresponding to the black points 1 and 2 in figure A2(b), respectively. Here, A=1.2, ω=0.6, and 0.2, respectively. The FWHM of figures A2(b1) are W=0.64 and 0.42.
In figure A2(a), we show the phase transition regions of various nonlinear modes for NLSE (9). Here, we take C=0. The existing regions of these nonlinear modes as the function of the amplitude A and width ω are shown. The yellow regions located by the black point 1 are rogue waves. The blue regions located by the black point 2 are the breathers. The red regions located by the black point 3 are solitons. After fixing A=1.2, we choose one point in every region remarked by the number 1–3 (here, ω=6.5, 3, and 1.45, respectively) in figure A2(a) to show the evolution results as shown in figures A2(a1)–(a3). The FWHM of figures A2(a1)–(a3) are W=0.59, 0.86, and 1.88, respectively. The excitation process of the rogue waves, breathers, and solitons can be clearly observed in the two-dimensional graphs. Figures A2(a11)–(a31) are the peak amplitudes of the rogue waves, breathers and solitons as the function of the propagation distance, respectively. And the profiles corresponding to points a–c are denoted by the black solid line, red dashed line, and blue dashed–dotted line in figures A2(a12)–(a32), respectively. With the increasing of the propagating distance, the black solid lines denote the initial pulses, the red dashed lines denote the nonlinear modes excited, the blue dashed–dotted lines show these modes maintain or disappear.
When C=1, the phase transition curve of the rogue wave is shown in figure A2(b). The yellow regions located by the black point 1 denote the existence interval of rogue waves. There is no rogue wave in the regions located by the black point 2. Taking A=1.2, we take black points 1 (ω=0.6) and 2 (ω=0.2) as the example as shown in figures A2(b1) and A2(b2). The FWHM of figures A2(b1)–(b2) are W=0.64 and 0.42. The phase transition curve of the rogue wave as the function of the initial amplitude A and background height C are shown in figure A2(c). The region denoted by b in figure A2(c) is the existing yellow region of the rogue wave. To excite the rogue wave, the amplitude A of the initial incident pulse for NLSE(9) will decrease with the increase of background height C.
From the above results, it is obvious that rogue waves, breaths, and solitons can be excited and controlled by adjusting the amplitude A and width ω of the initial incident pulse. Moreover, the background height adjusted by C is more conducive to excite rogue waves. The method will provide a simple way to excite various nonlinear modes experimentally.
,Institute of Nonlinear Physics and Department of Physics, 
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