Storage, Splitting, and Routing of Optical Peregrine Solitons in a Coherent Atomic System

We propose a scheme to realize the storage and retrieval of optical Peregrine solitons in a coherent atomic gas via electromagnetically induced transparency (EIT). We show that optical Peregrine solitons with very small propagation loss, ultraslow motional velocity, and extremely low generation power can be created in the system via EIT. We also show that such solitons can be stored, retrieved, split, and routed with high efficiency and fidelity through the manipulation of control laser fields. The results reported here are useful for the active control of optical Peregrine solitons and promising for applications in optical information processing and transmission.


INTRODUCTION
Rogue waves, first observed in ocean surfaces, are highly isolated spatial-temporal wave packets with very large amplitudes when some special conditions are attained [1]. Such waves are ubiquitous in nature and quite intriguing, since they "appear from nowhere and disappear without a trace" and have extremely destructive power [2]. Except for ocean waves, the study on rogue waves has been extended to many other different physical contexts, including atmosphere [3], superfluid helium [4], capillary waves [5], water waves [6], photorefractive ferroelectrics [7], plasmas [8], ferromagnetic materials [9], and so on [10,11].
Peregrine soliton, firstly suggested by D. H. Peregrine in the early 1980s for nonlinear dynamics of deep waters [12], is commonly taken as a prototype of rogue waves [13,14]. Such soliton, i.e., localized rational solution of nonlinear Schrödinger equation, can be taken as a limiting case of the one-parameter family of Kuznetsov-Ma breathers [15] or Akhmediev breathers [16]. There have been considerable interests on Peregrine solitons occurring in a variety of physical systems . Many efforts have also been devoted to the new understanding of Peregrine solitons through the analysis of other types of nonlinear partial differential equations [40][41][42][43][44][45][46][47][48][49][50][51].
Among various rogue waves, optical rogue waves have received much attention due to their interesting properties and promising applications [10,11,[25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][52][53][54]. However, the creation of the optical rogue waves is not an easy task in conventional optical media (such as optical fibers and waveguides). The reason is that the nonlinear optical effect in such media is very weak, and hence a large input optical power is needed to obtain a significant optical nonlinearity required for the formation of rogue waves. Although some resonance mechanisms may be exploited to enhance nonlinear effects, near resonances significant optical absorptions occur, which result in serious attenuation and distortion of optical pulses during propagation.
In recent years, many efforts have been focused on the investigation of electromagnetically induced transparency (EIT), a typical quantum interference effect occurring in threelevel atomic systems, by which the light absorption due to resonance may be largely suppressed and giant Kerr nonlinearity may be obtained simultaneously [55]. By means of EIT, it has been shown that weak-light solitons and their storage and retrieval can be realized [56][57][58]. Recent works [59,60] have demonstrated that it is possible to generate optical Peregrine solitons with low generation power in EIT-base atomic systems.
In this work, we suggest a scheme to realize the memory of optical Peregrine solitons in a Λ-shaped three-level atomic gas via EIT. We show that such solitons may have very small propagation loss, ultraslow motional velocity, and extremely low generation power; they can be stored, retrieved, split, and routed with high efficiency and fidelity through the manipulation of control laser fields. The results reported here are helpful for the active control of optical Peregrine solitons and promising for practical applications in optical information processing and transmission.
The article is arranged as follows. In Section 2, the physical model and ultraslow weak-light Peregrine solitons and their propagation are described. In Section 3, the storage, retrieval, splitting, and routing of such solitons are presented. Finally, Section 4 gives a summary of the main results obtained in this work.

Model
We start to consider a cold three-state atomic gas with Λ-shaped level configuration, interacting with a weak, pulsed probe laser field (center wavenumber k p and center angular frequency ω p ) and a strong, continuous-wave (CW) control laser field (wavenumber k c and angular frequency ω c ). The probe (control) field drives the transition |1〉 ↔ |3〉 (|2〉 ↔ |3〉); see Figure 1A.
The total electric field in the system reads the unit polarization vector (envelope) of the electric field E l . To suppress Doppler effect, both the probe and control fields are assumed to propagate along z direction.
The Hamiltonian of the system in the interaction picture reads is one-(two-) photon detuning; E j is the eigenvalue of the atomic state j〉; Ω p (e p · p 13 )E p /Z (Ω c (e c · p 23 )E c1 /Z) is the half Rabi frequency of the probe (control) field; p ij is the electricdipole matrix element associated with levels |i〉 and j〉. The atomic dynamics is described by a 3 × 3 density matrix σ, obeying the optical Bloch equation where Γ is a relaxation matrix characterizing the spontaneous emission and dephasing [61]. The explicit form of Eq. 1 is presented in Section 1 of the Supplementary Material. The evolution of the probe field E p is governed by the Maxwell equation ∇ 2 E p − (1/c 2 )z 2 E p /zt 2 (1/ε 0 c 2 )z 2 P p /zt 2 , where P p N a {p 13 σ 31 exp[i(k p z − ω p t)] + c.c.} is the electric polarization intensity, with N a the atomic density. Under slowly varying envelope and paraxial approximations, the Maxwell equation is reduced into the form with κ 13 N a ω p p 13 2 /(2ε 0 cZ). Note that we have assumed that the probe field has a large transverse size so that its diffraction effect is negligible. The model described here may be realized, e.g., by a cold 87 Rb atomic gas [62], with the levels selected by Thus we have ω p 2.37 × 10 15 Hz, p 13 2.54 × 10 − 27 C cm. If the atomic density N a 8.8 × 10 11 cm − 3 , κ 13 takes the value of 2.4 × 10 10 cm − 1 s −1 . This set of parameters will be used in the following analysis and calculation.

Ultraslow Weak-Light Peregrine Solitons and Their Propagation
We first investigate the linear propagation of the probe field. When a very weak probe pulse is applied, the system undergoes a linear evolution. In this case, the Maxwell-Bloch (MB) (Eqs. 1 and 2) admit the solution Ω p F exp[i(Kz − ωt)], where F is a constant, is linear dispersion relation, and Γ αβ , and c dep αβ is the dephasing rate associated with the states |α〉 and β〉).
Shown in Figure 1B is the imaginary part Im(K) and the real part Re(K) of K as functions of ω. Due to the quantum interference effect induced by the control field, an EIT transparency window is opened in Im(K) (dashed line), which implies that the probe field can propagate in this resonant atomic gas with a very small absorption. Parameters used for plotting the figure are Δ 2 −2π × 0.64 MHz, Δ 3 −2π × 9.6 MHz, c 21 2π × 1.09 kHz, c 31 2π × 2.5 MHz, and Ω c 2π × 31.8 MHz.
From the MB Eqs. 1 and 2 and using the method of multiplescales [63], we can derive the controlling equation governing the nonlinear evolution of the probe-field envelope F (see Section 2 of the Supplementary Material), which reads where τ t − z/Ṽ g [Ṽ g ≡ (zK/zω) − 1 is the group velocity of the envelope; here and in the following, the quantity with a tilde represents the corresponding real part]; K 2 z 2 K/zω 2 is the coefficient describing group-velocity dispersion; W is the coefficient (describing self-phase modulation) proportional to Kerr nonlinearity. The explicit expression of W is given in Section 2 of the Supplementary Material. If the imaginary parts of K and W are much smaller than their corresponding real parts, Eq. 4 admits the Peregrine soliton solution, which can be expressed by the half Rabi frequency where K 0 ≡ K| ω 0 , U 0 and τ 0 are respectively the characteristic half Rabi frequency and time duration of the probe field, and L Non ≡ 1/(U 2 0 W ) is the characteristic nonlinearity length (which has been assumed to equal the dispersion length defined by L Dis ≡ τ 2 0 / K 2 for simplicity). One sees that the Peregrine soliton consists of a CW background and a bump in its envelope that first grows and then decay rapidly on the background. The physical reason for the formation of such optical Peregrine soliton can be understood as follows. When a plane-wave probe field with a finite amplitude is applied to and propagates in the atomic gas, the Kerr nonlinearity brings a modulational instability and a phase modulation to the probe field; due to the role played by the group-velocity dispersion, the phase modulation is converted into amplitude modulation and peak amplification. Because of the joint phase and amplitude modulations, the probe field reorganizes its spatial distribution and hence the Peregrine soliton is generated in the system.
As an example, we take τ 0 2.36 × 10 − 7 s, U 0 2π × 8.0 MHz, and other system parameters which are the same as those used in Figure 1B. Then we obtain K 0 −1.70+i0.02 cm − 1 ,K 1 zK/zω ≈ (4.5 − i0.05) × 10 − 7 cm − 1 s, K 2 ≈ (−1.5 − i0.1) × 10 − 14 cm − 1 s 2 , and W ≈ (1.05 − 0.004) × 10 − 16 cm − 1 s 2 (estimated at ω 0). We see that the imaginary parts of K j (j 0, 1, 2) and W are much smaller than their corresponding real parts, which is due to the EIT effect that results in the suppression of the optical absorption in the system. Based on these results, we obtain L Non xL Dis x3.8 cm and Thus, the propagation velocity of the optical Peregrine soliton is much slower than the light speed c in vacuum. If the transverse cross-section area of the probe pulse takes the value S 8.0 × 10 − 3 cm 2 , the generation power of the soliton (which can be estimated by using the Poynting vector [56]) reads i.e., very small power needed for creating such soliton. Consequently, the Peregrine solitons given here are different from those obtained in conventional optical systems [25,27,28,31]. We now investigate the propagation of the ultraslow Peregrine soliton by exploiting Runge-Kutta method based on solving the MB Eqs. 1 and 2 numerically. Since solution (5) has an infinite FIGURE 1 | (A) Energy-level diagram and excitation scheme of the Λ-shaped three-level atomic system for realizing EIT. The probe field (with center angular frequency ω p and half Rabi frequency Ω p ) couples the atomic levels |1〉 and |3〉; the CW control field (with angular frequency ω c and half Rabi frequency Ω c ) couples the atomic levels |2〉 and |3〉; Γ 13 (Γ 23 ) is the decay rate from |3〉 to |1〉 (|3〉 to |2〉); Δ 3 (Δ 2 ) is one-photon (two-photon) detuning. (B) The imaginary part Im(K) and real part Re(K) of the linear dispersion relation K as a function of ω (ω 0 corresponds to the central frequency of the probe pulse). Frontiers in Physics | www.frontiersin.org March 2021 | Volume 9 | Article 594680 energy due to the existence of the CW background, it cannot be generated in a real experiment. To avoid this, we assume the probe field at z 0 has the form Here ] is chosen to match the analytical solution (5); the hyperbolic tangent function is used to impose temporal boundaries on both sides of CW background (far from the pump part), which can make the soliton have finite energy and also have a clear illustration on its waveshape (similar to the case for generating dark solitons [64,65]); T The lower part of Figure 2A illustrates the result of a numerical simulation on the propagation of the Peregrine soliton (with Δ 3 −2π × 95.5 MHz, τ 0 1.5 × 10 − 7 s, and other parameters the same as those used in Figure 1B), by taking Ω p τ 0 as a function of t/τ 0 and z. The orange line is the input Peregrine soliton at z 0; the red line denotes the Peregrine soliton propagating to z 4.3 cm; the maximum value ( Ω p τ 0 | max x15.2) of the soliton along the trajectory appears sharply around z 4.3 cm at t 16.9τ 0 . Figure 2B shows the contour map for the propagation of the Peregrine soliton, which can be taken as a projection of Figure 2A onto the t-z plane. One sees that the Peregrine soliton (indicated by the red dashed circle in Figure 2B) appears sharply and disappears suddenly; a secondary peak (soliton) emerges at longer distance, as a result of phase modulation when the first soliton is excited.

STORAGE, RETRIEVAL, SPLITTING, AND ROUTING OF THE OPTICAL PEREGRINE SOLITONS
We now turn to consider the memory of the optical Peregrine solitons and related applications in optical splitting and routing through the manipulation of the control fields.

Storage and Retrieval of the Optical Peregrine Solitons
We first consider the storage and retrieval of optical Peregrine solitons obtained above, which can be implemented by switching off and on the control field described by the following switching function: where Ω c0 is a constant, T c s is the time interval for switching off and switching on the control field (switching time), and T c off (T c on ) is the time when the control field is switched off (on).
As an example, we take Ω c0 2π × 31.8 MHz, T c off 10.0τ 0 , T c on 20.0τ 0 , T c s 3.0τ 0 (τ 0 1.5 × 10 − 7 s), and other system parameters are the same as those used in Figure 2. The upper part of Figure 3A shows the time sequences of the control field (black line) and the probe field (blue line); the red dashed vertical line (black dashed vertical line) represents the time T p off (T c off ). Symbols I, II, and III denote the CW background, the Peregrine soliton, and the low-intensity component of the probe field, respectively. The lower part of the figure shows the result of a numerical simulation on the storage and retrieval of the Peregrine soliton by taking Ω p τ 0 as a function of t/τ 0 and z. Here the orange line is the input Peregrine soliton at z 0; the purple line represents the Peregrine soliton at the storage period; the red line represents the retrieved Peregrine soliton propagating to z 4.1 cm with the maximum intensity Ω p τ 0 | max x15.4 at t 30.3τ 0 . Shown in Figure 3B is the contour map of the storage and retrieval of the Peregrine soliton in the t-z plane 3.0τ 0 . The probe field (the Peregrine soliton) is thus stored in the system (i.e., it is converted into the atomic coherence σ 21 [66,67]).
• Lastly, the control field is switched on again at t T c on 20.0τ 0 . The atomic coherence σ 21 is converted back to the probe field, and hence the probe pulse is retrieved. Particularly, at time tx30.3τ 0 , the retrieved probe field manifests as a Peregrine soliton with the maximum intensity Ω p τ 0 max x15.4 at the position zx4.1 cm.
The efficiency of the Peregrine soliton memory can be characterized by the parameter η dt [57,67], where E in p (t) E in p (0, t) (i.e., the input Peregrine soliton) and E Pere p (t) E Pere p (L Pere , t) (i.e., the retrieved Peregrine soliton), with L Pere (x4.1 cm) as the position where the Peregrine soliton is retrieved. Based on the result of Figure 3A, we obtain η 85.9%. The fidelity of the Peregrine soliton memory can be described by the parameter ηJ 2 , where J 2 describes the degree of coincidence of the wave shapes for the input and retrieved solitons, defined by the overlap integral J 2 [57,67], where T 1 is a coefficient related to the temporal width of the Peregrine soliton (i.e., corresponding to symbol II in Figure 2A), and ΔT is the time interval between the peak of the input soliton pulse E in p and the peak of the retrieved soliton pulse E Pere p . Here we take ΔT 30.3τ 0 and T 1 10τ 0 . We obtain ηJ 2 84.3%. We see that the efficiency and fidelity of the storage and retrieval of the Peregrine soliton are quite high.
The numerical result shown in Figure 3C is similar to that of Figure 3A but for T p off > T c off . In this case, the storage and retrieval of the Peregrine soliton can also be implemented; however, compared with Figure 3A (which is for T p off < T c off ), the retrieved waveshape is little more distorted. Figure 3D illustrates the contour map of the Peregrine soliton in the t-z plane with T p off > T c off . One sees that the probe field has a nonzero value in the region indicated by the dashed white circle, which means that some parts of the probe field are not stored when the control field is switched off. We obtain the efficiency and fidelity of the Peregrine soliton memory for T p off > T c off are η 77% and ηJ 2 65%, respectively. Based on these results, we conclude that The energy-level diagram and excitation scheme of the double-Λ-type four-level atomic system. In this configuration, two probe laser fields Ω p1 and Ω p2 drive the transitions |1〉↔|3〉 and |1〉↔|4〉, respectively; two CW control laser fields Ω c1 and Ω c2 drive the transitions |2〉↔|3〉 and |2〉↔|4〉, respectively; Δ 3 and Δ 4 are one-photon detunings, and Δ 2 is two-photon detuning. (B) Time sequences of the two control fields Ω c1 , Ω c2 and the probe field Ω p1 for realizing the soliton routing. The vertical red dashed line denotes the time t T p off ; the vertical black dashed lines denote the times when switching off and on the control fields. (C) Contour maps for the routing process of the Peregrine soliton, where the upper panel is for Ω p1 and the lower panel is for Ω p2 . The Peregrine soliton is input as the probe field Ω p1 , but it is retrieved as the probe field Ω p2 (i.e., "retrieved PS" indicated by the red dashed circle in the lower panel of the figure). Here, a probe laser field Ω p drives the transition |1〉 ↔ |3〉; the two CW control fields Ω c1 and Ω c2 drive the transitions |2〉 ↔ |3〉 and |4〉 ↔ |3〉, respectively; Γ j3 is the decay rate from |3〉 to |j〉 (j 1, 2, 4); Δ 3 and Δ l (l 2, 4) are respectively one-photon and two-photon detunings. in order to get a high memory quality, the choice of T p off < T c off is better than that of T p off > T c off .

Splitting of the Optical Peregrine Solitons
To realize an optical splitting [67] of the Peregrine soliton, we generalize the system into a four-level one with a tripod-type level configuration. Here a probe field Ω p drives the transition |1〉↔|3〉; two CW control fields Ω c1 and Ω c2 drive respectively the transitions |2〉↔|3〉 and |4〉↔|3〉; Γ j3 is the decay rate from |3〉 to j〉 (j 1, 2, 4), Δ 3 and Δ l (l 2, 4) are respectively onephoton and two-photon detunings (see Figure 4A). The Hamiltonian of the system and the optical Bloch equations controlling the dynamics of the atoms have been presented in Section 5 of the Supplementary Material. The timing sequences of the switching-off and -on of Ω cj (t) for obtaining a Peregrine soliton splitter are shown in Figure 4B, with T p off < T c1  Figure 4C is the numerical result on the simulation for obtaining the Peregrine soliton splitter by taking Ω p τ 0 as a function of t/τ 0 and z (with τ 0 1.5 × 10 − 7 s). The operation steps can be described as follows: 1) Firstly, the two control fields Ω c1 and Ω c2 are applied and a probe field with the waveform Ω p0 (0,t) 6.67 .0]} is incident to the system. 2) Then, both control fields are simultaneously switched off at time t T c1 off 1 T c2 off 1 6.0τ 0 . Thus the probe field is stored in the two atomic coherences σ 21 and σ 41 simultaneously 67. 3) Later on, switching on Ω c1 at t T c1 on 15.0τ 0 (but Ω c2 is remained to be switched off), the atomic coherence σ 21 is converted back into the probe field, and hence a new probe pulse is retrieved. At time tx23.0τ 0 , this retrieved probe pulse turns into a Peregrine soliton (i.e., "1st retrieved PS", indicated by a red circle in Figure 4B) with the maximum intensity Ω p τ 0 max x15.4 at the position z 3.7 cm. 4) By switching off Ω c1 at t T c1 off 2 35.0τ 0 and switching on Ω c2 at t T c2 on 45.0τ 0 , the atomic coherence σ 41 converts back into the probe field; this retrieved probe field turns into another Peregrine soliton (i.e., "2nd retrieved PS", indicated by another red circle in Figure 4B) with the maximum intensity Ω p τ 0 max x14.0 at the position z 5.5 cm at tx61.5τ 0 .
In the simulation, we have taken Δ 2 Δ 4 −2π × 0.64 MHz, c 21 c 41 2π × 1.09 kHz, with the other parameters the same as those used in Figure 3A. The reason for taking Δ 2 Δ 4 and c 21 c 41 is to keep the symmetry of the tripod level configuration, which gives two nearly degenerated EITs in the system; for details, see [67]. The splitting efficiency and fidelity of the first (second) Peregrine soliton are η 1 89.8% and η 1 J 2 1 85.4% (η 2 89.3%, η 2 J 2 2 84.9%), respectively.
The Hamiltonian of the system and the MB equations governing the dynamics of the atoms and light fields have been given in Section 6 of the Supplementary Material.
For simplicity, here we consider a frequency routing process, i.e., the probe field Ω p1 is converted into the Ω p2 (which has different frequency from Ω p1 ). The time sequence of the switching off and on of Ω cj for obtaining routing of Peregrine soliton is shown in Figure 5B, with T p off < T c1 off < T c2 on , where T c1 off is the switching-off time of Ω c1 and T c2 on is the switching-on time of Ω c2 . The corresponding switching functions have been given in Section 5 of the Supplementary Material. Without loss of generality, the system parameters are set to be Ω c1 Ω c2 2π × 31.8 MHz, T c1  Figure 5C as a function of propagation distance t/τ 0 and z. One sees that a trajectory of the soliton shows up before its storage. Second, by switching off Ω c1 at time t 10τ 0 , the probe field Ω p1 is stored in the atomic coherence σ 21 . Third, by switching on the control field Ω c2 at t 25τ 0 , another probe pulse Ω p2 appears from the atomic coherence σ 21 , i.e., "retrieved PS" in the lower panel of Figure 5C. We stress that during this routing process, the Peregrine soliton in the probe field Ω p1 is annihilated and a new Peregrine soliton in the probe field Ω p2 (which has no input) is created. Since the frequency of Ω p2 is different from that of Ω p1 , the system performs as a frequency router of the Peregrine soliton.

CONCLUSION
We have proposed a scheme for realizing the storage and retrieval of optical Peregrine solitons in a coherent atomic gas via EIT. We have shown that the optical Peregrine solitons with very small propagation loss, ultraslow motional velocity, and extremely low generation power can be generated in the system via EIT. We have demonstrated that such solitons can be stored, retrieved, split, and routed with high efficiency and fidelity through the manipulation of control laser fields.
The scheme can also be generalized to cases with more optical output channels through the use of more control fields, and hence the two-channel splitting and routing processes can be generalized to multiple channel ones. Furthermore, the storage and retrieval of the optical Peregrine solitons can be extended to solid materials, like on-chip optical resonator systems [69]. The research results reported here may be useful for the active control of optical Peregrine solitons and promising for potential applications in optical information processing and transmission.