1. Introduction

The generation and propagation of optical solitons is of great interest due to their shaping-preserving wave propagation properties, the variety of the solitons’ interactions, and their potential technological applications [1–3]. The most celebrated soliton is the hyperbolic secant soliton which is formed as the result of interplay between nonlinearity and dispersion properties of the medium under excitations. On the other hand, rogue waves are large, unexpected and suddenly appearing surface waves due to modulation instability, which were first discovered in the ocean [4,5]. In the past decades, rogue waves have been widely identified in many fields [6], including nonlinear optics [7–9], plasma physics [10,11], Bose-Einstein condensatess [12,13], and finance [14]. Among these rogue waves, optical rogue waves have attracted extensive attention because of their potential applications [15], such as photolithography [16], supercontinuum generation [17–19], and high power pulses extraction [20].

Optical rogue waves are statistically rare bursts of white light. Because they are described by so-called L-shaped probability distributions, these extreme light waves are rare, but much more common than expected from Gaussian statistics. It is well known that rogue waves are adequately described by the solutions of NLSE, including Peregrine solitons [21], Kuznetsov-Ma breathers [22] and Ahkmediev breathers [23], where the Peregrine soliton is also a limiting case of the Akhmediev breather and the Kuznetsov-Ma breather, exhibiting properties of localization in both space and time.

Using the giant Kerr nonlinearity and the attenuation of absorption in the electromagnetically induced transparency (EIT) system [24], ultraslow optical solitons, Peregrine solitons, Kuznetsov-Ma breathers, and Ahkmediev breathers have been widely studied [25–28]. However, for a EIT system, a complete elimination of the optical absorption occurs only under the exact resonance condition where the system exhibits no Kerr nonlinearity at all. Thus, to obtain large Kerr nonlinearity and form the optical solitons, the system can NOT work in the resonant region. As a result, the absorption can NOT be suppressed efficiently even under the EIT condition. To overcome the disadvantage of the absorption in EIT system, the active Raman gain (ARG) scheme is proposed and a gain-assisted optical soliton with superluminal group velocity can be obtained [29–31]. However, the gain induced noise and narrow frequency regime for the generation of solitons restrict the applications of this superluminal soliton.

In this paper, we study the generation and propagation of hyperbolic secant solitons, Peregrine solitons, and various breathers in a coherently prepared three-level atomic system. Combining the EIT and ARG characteristics, a large constant phase with almost zero absorption/gain can be achieved over a wide range of the probe field frequency. Based on these features, we show that not only hyperbolic secant solitons but also Peregrine solitons and breathers can be generated with an input probe field at single photon level because a large and flat Kerr nonlinearity can be realized. Due to this flat dispersion relation and near zero absorption/gain of the probe field, the quantum noise of the probe field can also be significantly suppressed. Therefore, the results obtained in this work may pave the way for realizing solitons and breathers at a single photon level. In addition, they hold potential applications in quantum information and communication.

2. Model

To begin with, we consider a three-level $\Lambda$-type atomic system shown in Fig. 1(a), where states $|1\rangle$ and $|3\rangle$ are hyperfine ground states and the state $|2\rangle$ is the excited state [32]. In our system, two lower ground states are coherently prepared prior to the injection of a strong pump field $E_P$ (angular frequency $\omega _{P}$) and a weak probe field $E_p$ (angular frequency $\omega _{p}$). The pump (probe) field couples the $|1\rangle \leftrightarrow |2\rangle$ ($|2\rangle \leftrightarrow |3\rangle$) transition, and $\delta =\omega _P-(\omega _2-\omega _1)$ ($\delta _\textrm {2ph}=\omega _P-\omega _p-(\omega _3-\omega _1)$) is the one-photon (two-photon) detuning with $\hbar \omega _j$ being the energy of state $|j\rangle \ (j=1-3)$. Thus, the system is a combination of EIT and ARG systems, where the ARG process ($|1\rangle \rightarrow |2\rangle \rightarrow |3\rangle$) produces a gain and the EIT process ($|3\rangle \rightarrow |2\rangle \rightarrow |1\rangle$) produces an absorption. Using the rotating wave approximation (RWA) and assuming the one-photon detuning is large enough, the equations of motion for the density matrix elements are given by

 

Fig. 1. (a) A three-level scheme where two lower states are coherently prepared prior to the injection of a strong pump field $E_P$ with angular frequency $\omega _P$ and a weak quantum probe field $\hat {E}_p$ with angular frequency $\omega _p$. In this model, states $|1\rangle$ and $|3\rangle$ are coherently prepared prior to the injection of the pump and probe fields. (b) Probe field phase shift $\phi _0$ (red solid curve) and the linear loss/gain coefficient $\alpha _0$ (blue dashed curve) as a function of the two-photon detuning $\delta _\textrm {2ph}$.

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Under the slow varying envelope approximation (SVEA), the Maxwell equation for the weak probe field is given by

Solving Eqs. (1) and Eq. (2) by assuming $\Omega _{23}=F\exp {(i\theta )}$ with $\theta =K(\omega )z-\omega t$ and $F$ being the envelope of the probe field, the linear dispersion of the system is then given by

In Fig. 1(b), we plot the phase shift $\phi _0=\textrm {Re}(K_0)L$ and the loss/gain coefficient $\alpha _0=\textrm {Im}(K_0)L$ of the probe field as a function of the two-photon detuning $\delta _{2ph}$. Here, we choose the system parameters as $\rho _{11}=\rho _{33}=0.5$, $\kappa _{32}=3\times 10^{10}\ \textrm {s}^{-1}\textrm {cm}^{-1}$, $\gamma _{21}/2\pi =6$ MHz, $\gamma _{23}/2\pi =6$ MHz, $\gamma _{13}/2\pi =2.5$ KHz, $\delta /2\pi =-1.08$ GHz, $\Omega _{21}/2\pi =30$ MHz and $L=1$ cm. Obviously, there exists a wide range of two-photon detuning (corresponding to the probe field frequency) to realize a constant phase shift with zero gain or loss in the grey shaded area. We should stress that such flat dispersion with zero gain and loss contributes to extremely low quantum noise fluctuation for the probe field at single photon level [32].

To show the physical mechanism more clearly, we also plot the phase shift $\phi _0$ and the loss/gain coefficient $\alpha _0$ with different population in ground states $\rho _{11}$ and two photon detuning $\delta _\textrm {2ph}$. As shown in Fig. 2, the zero loss/gain and constant phase shift can be realized around $\rho _{11}=0.5$, where the coherence between two ground states reaches its maximum. However, for $\rho _{11}\rightarrow 0$ or $1$, these features disappear since the system goes back to the ARG or EIT scheme.

 

Fig. 2. (a) The loss/gain coefficient $\alpha _0$ and (b) the phase shift $\phi _0$ of the probe field as functions of the populations in ground states $\rho _{11}$ and two-photon detuning $\delta _\textrm {2ph}$.

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3. Asymptotic expansion and the nonlinear Schrödinger equation

Next, we are going to search for the formation and propagation of a shape-preserving probe pulse in the coherently prepared atomic system. To get a quantitative description of the formation and dynamics of optical solitons, rogue waves, and breathers in the system, as a first step, we derive a nonlinear envelope equation that describes the evolution of the probe field envelope by employing a standard weak nonlinear perturbation theory. Firstly, we take the following asymptotic expansion [33]: $\rho _{ij}=\rho _{ij}^{(0)}+\epsilon \rho _{ij}^{(1)}+\epsilon ^2\rho _{ij}^{(2)}+\epsilon ^3\rho _{ij}^{(3)}$ ($i,\,j=1-3$), $\Omega _{23}=\epsilon \Omega _{23}^{(1)}$ and $\Omega _{21}=\Omega _{21}^{(0)}$ with $\rho _{22}^{(0)}\approx 0$, where $\epsilon$ is a small parameter characterizing the amplitude of the probe field. Substituting these expansions into Eqs. (1), one obtains a set of linear but inhomogeneous equations of $\rho _{ij}^{(n)}$, which can be solved order by order. For the leading-order solution, we have

For the second order solution (i.e., $n=2$), we can obtain

Likewise, the third-order perturbation solution is given by

Defining $\xi =z$ and $\eta =t-z/V_g$, we then obtain the nonlinear evolution equation for $U(\xi ,\eta )$ with complex coefficients of the form

To satisfy the conditions described above, we choose $\rho _{11}=0.6$ and other system parameters are the same as those used in Fig. 2. In Fig. 3(a), we plot $\eta _{\alpha }=|{\textrm {Re}(\alpha )/ \textrm {Im}(\alpha )}|$ ($\alpha =K_{1},K_{2},W$) and loss/gain coefficient as a function of the two-photon detuning $\delta _{2ph}$. It is clear to see that $\eta _{\alpha }\gg 1$ (solid curves) and near zero loss/gain (indicated by the dashed curve) can be realized over a wide range of the two photon detuning. In addition, it is noted that real parts of these parameters change slowly with the two-photon detuning [see Fig. 3(b)], which allows a wide range of probe field frequency to achieve soliton propagation in this system.

 

Fig. 3. (a) Parameter $\eta _{\alpha }=|{\textrm {Re}(\alpha )/ \textrm {Im}(\alpha )}|$, $\alpha =K_{1},K_{2},W$ (solid curves) and the loss/gain coefficient (dashed curve) as functions of the two-photon detuning $\delta _\textrm {2ph}$. (b) Coefficient $|\rm {Re}(\alpha )|$ versus the two-photon detuning $\delta _\textrm {2ph}$. We choose $\rho _{11}=0.6,\rho _{33}=0.4$ and other system parameters are the same as in Fig. 2.

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4. Hyperbolic secant solitons and rogue waves

Now, we study the generation of hyperbolic secant solitons and rogue waves in this system. Neglecting the imaginary part of the coefficients in Eq. (6) and introducing the dimensionless variables $s=\xi /(2L_D)$, $\sigma =\eta /\tau _0$ and $u=U/U_0$, one can easily obtain the typical NLSE, i.e.,

To verify the above analysis, we carry out the numerical simulation by solving Eqs. (1) and (2) directly. The initial condition is given by Eq. (8). As shown in Fig. 4(a), the shape of the wave packet preserves well after a long distance propagation (i.e., propagate for the distance $z=6L_{D}=55$ cm). In Fig. 4(b), we also show the collision of two solitons. It is clear to see that wave packets remain unchange after their collision.

 

Fig. 4. (a) Probe field intensity $|\Omega _{23}/U_{0}|^{2}$ versus normalized parameters $\tau /\tau _{0}$ and $z/2L_{D}$. The initial condition is given by Eq. (8). (b) Collision between two solitons traveling in opposite directions.

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Apart from the NLSE (7), Eq. (6) can also be reduced as the following dimensionless form, i.e.,

Equation (9) yields the solutions of rogue waves, such as the Peregrine solitons, the Ahkmediev breathers and Kuznetsov-Ma breathers. The Peregrine soliton is considered as the limit case of Ahkmediev breathers and Kuznetsov-Ma breathers, and its solution having the rational form $u(s,\sigma )=[1-4(1+2is)/(1+4s^2+4\sigma ^2)]e^{is}$, which is an algebraically decaying wave packet in the $z$ direction and time, with the maximum intensity $|u|_{max}^2=9$ at $(s,\sigma )=(0,0)$. After returning to the original variables, the solution of Peregrine solitons can be expressed as [35]

If $q>1/2$, the solution of Eq. (9) is Kuznetsov-Ma breathers, which is given by

In Fig. 5, we show the probe field intensity $|\Omega _{23}/U_{0}|^2$ as functions of normalized parameters $\tau /\tau _{0}$ and $z/L_{D}$ by integrating Eqs. (1) and (2). For panels (a) and (b), the initial condition is chosen as Peregrine soliton [i.e., Eq. (10)] at $z/L_D=-5$, and the generation of the Peregrine soliton is stimulated by a very weak pulse with an input power $P_\textrm {in}\approx 0.13$ nw, which is three orders of magnitude smaller than that used in nonlinear fiber [36]. It is clear to see that there exists a sharp peak around $(z,\tau )=(0,0)$. Besides, by using the exact Peregrine solution, one can also use the pure phase engineering technique for producing rogue waves [see Ref. [12]]. In panels (c) and (d), the Ahkmediev breathers are chosen as the initial condition with modulation parameters $\Omega =\sqrt {2}$ and $q=1/4$. Obviously, they are locally distributed on the $z-$axis but periodically distributed in the $\tau -$axis. Panels (e) and (f) show the numerical results by choosing the Kuznetsov-Ma breathers as the initial condition with modulation parameters $\Omega =\sqrt {2}$ and $q=3/4$. As expected, they are locally distributed on the time axis but periodically distributed in the space axis.

 

Fig. 5. Probe field intensity $|\Omega _{23}/U_{0}|^2$ versus normalized parameters $\tau /\tau _{0}$ and $z/L_{D}$. The initial conditions are chosen as Peregrine solitons [panels (a,b)], Ahkmediev breathers [panels (c,d)] and Kuznetsov-Ma breathers [panels (e,f)], respectively.

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5. Conclusion

In conclusion, we study the generation and propagation of hyperbolic secant solitons, Peregrine solitons and breathers in a coherently prepared three-level $\Lambda -$type atomic medium. Due to coherence between two ground states, the probe field undergoes an almost lossless propagation and its profile remains unchanged since the balance between the dispersion effect and the nonlinear effect can be achieved in a wide range of probe field frequencies, rather than a specific frequency. Therefore, we show the generation of stable hyperbolic secant solitons and rogue waves with a probe field intensity at single photon level. These features demonstrated in this work are conducive to the realization of quantum solitons and promote the research of nonlinear quantum optics.