1. Introduction

The breathers on a finite background have attracted increasing attention in many fields including hydrodynamics and optics [1,2], due to their intimate connection to the formation of extreme rogue wave events [3,4]. One type of such breathers is the Kuznetsov-Ma (KM) soliton [5,6], which behaves like an ordinary soliton, localized yet oscillatory as it propagates. The other one is the Ahkmediev breather (AB) [7] who is localized in the longitudinal dimension while oscillating along the transversal direction. Interestingly, as the oscillating period becomes infinity, both types of breathers can reduce to the Peregrine soliton [8], which is localized in both space and time. To date, because of their fundamental interests and practical implications, both breathers, including their reduced Peregrine soliton state, have become a hot topic, either theoretical [9–13] or experimental [14,15], at the cutting edge of nonlinear optics. Generally, the generation of optical breathers uses far-off-resonant mechanisms and high-intensity optical fields [2,15] which, as a result, cause a group velocity close to the light speed in vacuum.

In the past decades, there have been significant achievements concerning the propagation of electromagnetic waves in highly resonant media [16–22]. Among them, an important quantum interference effect known as electromagnetically induced transparency (EIT) was uncovered [23,24], by which the light pulses are allowed to propagate through an otherwise opaque atomic medium. Due to this EIT effect, the light-matter interaction process may exhibit some unique properties such as reducing the group velocity of light pulses [25,26], enhancing the Kerr nonlinearity [27,28], and lasing without population inversion [29]. In addition, ultraslow ordinary optical solitons were predicted to exist in the three-level [30,31] and four-level [32,33] atomic media. Also, the weak-light KM solitons and ABs were found to occur in such highly resonant media, within the framework of the nonlinear Schrödinger (NLS) equation [34–36].

In this article, we will investigate the formation of KM solitons and ABs in a cold three-level atomic medium exposed to a probe field of nanosecond pulse duration. Within the integrable Hirota equation framework, we obtain for the first time the exact KM soliton and AB solutions. We then confirm by numerical simulations that both types of breathers are stable against non-integrable perturbations caused by the nonvanishing decay rates of atomic states and reveal that these breathers thus generated can propagate at a quite low group velocity.

2. Model, multiscale method, and generalized NLS equation

For our study, let us start with a lifetime broadened three-state atomic system that interacts with a weak pulsed probe field $E_p(\mathrm {\mathbf {r}},t)=(\hbar /D_0)\Omega _p\exp [i\omega _p(z/c-t)]+c.c.$ (central frequency $\omega _{p}$, Rabi frequency $2\Omega _p$) tuned to the transition $|1\rangle \rightarrow |3\rangle$ and a strong continuous-wave (cw) control field (angular frequency $\omega _c$, Rabi frequency $2\Omega _c$) tuned to the transition $|2\rangle \rightarrow |3\rangle$ (see Fig. 1). Here $\hbar$ is the Planck constant divided by $2\pi$, $D_{0}$ the dipole moment for the transition $|1\rangle \rightarrow |3\rangle$, and $c.c.$ is the shorthand of the complex conjugate terms. As a cold atomic medium is adopted here, we preclude the atomic collisions and Doppler broadening effect. Therefore, in the interaction picture and under the slowly varying envelope and rotating-wave approximations, the atomic equations for motion and the wave equation for the time-dependent probe field, often termed Maxwell–Bloch (MB) equations, can be expressed as [31]

 

Fig. 1. The excitation scheme of a $\Lambda$-type lifetime broadened three-state atomic system interacting with a pulsed probe field (central frequency $\omega _{p}$, Rabi frequency $2\Omega _{p}$) and a strong cw control field (angular frequency $\omega _{c}$, Rabi frequency $2\Omega _{c})$.

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We now use a standard multiscale approach [37] to derive the nonlinear envelope equation describing the evolution of the weak probe optical field, under the assumption of a small population depletion of the ground state. To this end, one can perform the asymptotic expansions $A_{j}=\sum ^{\infty }_{m=0}\varepsilon ^{m}a_{j}^{(m)}$ and $\Omega _{p}=\sum ^{\infty }_{n=1}\varepsilon ^{n}\Omega _{p}^{(n)}$, and let all quantities therein be the functions of $z_{\iota }=\varepsilon ^{\iota }z~(\iota =0,\ldots ,3)$, $t_{1}=\varepsilon t$, $x_{1}=\varepsilon x$, and $y_{1}=\varepsilon y$. It should be noted that there is a slight difference between our multiscale variables and those used in [31,34]. Then, inserting these expanded formulas into Eqs. (1)–(4), followed by equating the coefficients of $\varepsilon ^{m}$ to zero, one can obtain a succession of linear but nonhomogeneous equations involving $a_{j}^{(m)}$ and $\Omega _{p}^{(n)}$. The preceding low-order equations can be written as

Obviously, the system of linear Eqs. (5)–(7) can be solved in an iterative yet exact manner, with the help of the conservation relation (4). After some algebra, we find $\Omega _{p}^{(1)}=F(z_1,z_2,z_3,t_1,x_1,y_1)\exp (iK_0z_0)$ and $\Omega _{p}^{(n)}=0$ for $n\geqslant 2$, where $F$ is the complex amplitude and $K_0=\kappa d_2/D$ is the complex wave number, with $D=|\Omega _c|^2-d_2d_3$ and $d_{j}=i\gamma _{j}+\Delta _j$ ($j=2,3$). In terms of the function $F$ and its derivatives, the probability amplitudes $a_{j}^{(m)}$ can also be exactly defined (we do not present their explicit expressions here, for the sake of brevity). Recalling that $\Omega _p=\varepsilon \Omega _p^{(1)}=U\exp [i\mathrm {Re}(K_0)z_0]$, where $U=\epsilon F \exp (-\alpha z_0)$, with $\alpha =\mathrm {Im}(K_0)$ and $\mathrm {Im}$ standing for the imaginary part, is the envelope amplitude, we find, after returning to the original variables ($z,t,x,y$), that the nonlinear envelope equation for $U$ can read as

We should point out that our Eq. (8) obtained here is a direct result of Eqs. (5)–(7) without any further approximations; it seems to be similar in form to Eq. (18) derived in [31], but involves different equation coefficients defined by Eqs. (9)–(14). It is apparent that these equation coefficients are all complex (because of the introduction of the complex $d_2$ and $d_3$) and can be separated into the real and imaginary parts, namely, $K_j=K_j^{\prime }+iK_j^{\prime \prime }$ ($j=1,2,3$), $\gamma =\gamma ^\prime +i\gamma ^{\prime \prime }$, and $\beta _{1,2}=\beta _{1,2}^{\prime }+i\beta _{1,2}^{\prime \prime }$. Usually, $V_g=1/K_1^{\prime }$ gives the group velocity of the probe field. In this regard, Eq. (8) becomes an analog of the (3+1)D complex Ginzburg-Landau equation [38], comprising the spectral filtering denoted by $K_2^{\prime \prime }$, the third-order spectral correction by $K_3^{\prime \prime }$, the linear loss by $\alpha$, and the nonlinearity gain by $\gamma ^{\prime \prime }$. However, in a very short distance and considering that $\Delta _2\gg \gamma _2$ and $\Delta _3\gg \gamma _3$, which is accessible by typical alkali atoms, these extra effects resulting from the imaginary parts of the coefficients will become insignificant and thus can be neglected. Under the circumstances, Eq. (8) can be reduced to the generalized NLS equation, whose coefficients are still given by Eqs. (9)–(14), but with $d_j$ being replaced by $\Delta _j$. For this case, one can obtain the group velocity $V_g$ of the weak probe field:

3. Dynamics of KM solitons and ABs and numerical simulations

More interestingly, if we neglect the spatial diffraction effect and the linear loss, which holds true for a propagation distance of a few dispersion lengths $L_D=\tau _0^2/|K_2|$, and let further

The Hirota Eq. (17) is integrable and can be solved by many analytical methods [39,40]. It admits plenty of exact solutions, such as solitons, rogue waves, and breathers [41,42]. Of special interest are the KM and AB solutions, either of which arises from the modulation instability of the unstable background field [2]. By means of the bilinear Hirota approach [39] and starting from the seeding plane-wave solution

It follows easily that the KM soliton solution (20) oscillates along the propagation direction $z\chi +\tau =0$, with the amplitude of all peaks as high as $[1+2\cosh (\phi )]\sqrt {P}~(\geqslant 3\sqrt {P})$, but located on $(\tau _j,z_j)=(-2j\pi \chi /g, 2j\pi /g)$, $(j=0, \pm 1,\pm 2,\ldots )$. By contrast, the AB solution (21) is found to oscillate along the transversal (i.e., $z=0$) direction, with its peaks being located on $(\tau _j,z_j)=(2j\pi /q,0)$, $(j=0, \pm 1,\pm 2,\ldots )$ and as high as $[1+2\cos (\phi )]\sqrt {P}~(\leqslant 3\sqrt {P})$. On the other hand, the peak power associated to the incident pulsed probe beam of cross-section area $S_0$ can be calculated by $P_{\mathrm {peak}}=2\epsilon _0c n_pS_0(\hbar /D_0)^2P$, where $\epsilon _0$ is the permittivity of free space and $n_p=1+cK_0/\omega _p$ is the phase refractive index experienced by the probe field. Albeit $P$ is a free parameter in Eqs. (20) and (21), there is a practical requirement on the input peak power $P_{\mathrm {peak}}$ for forming stable KM solitons and ABs within a limited-size cell of alkali atom gases. Besides, we need to point out that although both the KM and AB solutions discussed above bear a similarity to the corresponding ones of the standard NLS equation [35,36,42], they can describe more accurately the propagation of the probe pulses of duration down to the picosecond order, due to the inclusion of the parameters $\eta$ and $\chi$, which are related to the third-order dispersion ($K_3$) and self-steepening ($\gamma K_3/K_2$) effects.

We now consider some numerical examples to demonstrate these two types of breathers in a typical alkali (Rubidium 87 $\mathrm {D}_2$ transition, say) atomic system, where the coupling strength is $\kappa =1.0\times 10^9 \mathrm {cm^{-1}s^{-1}}$ and the decay rates are $\Gamma _2=2\gamma _2=10\mathrm {kHz}$ and $\Gamma _3=2\gamma _3=5\mathrm {MHz}$ [31]. When a weak probe pulse of 0.12 cm beam diameter (central wavelength $\lambda _p=2\pi c/\omega _p=780~\mathrm {nm}$, pulse duration $\tau _0=30~\mathrm {ns}$, peak power $P_{\mathrm {peak}}=700~\mathrm {mW}$ or $P=6.25\times 10^{15}~\mathrm {s^{-2}}$) was launched into the medium (let $D_0=2.1\times 10^{-27}~\mathrm {cm~C}$), we tuned the detunings by $\Delta _3=50\mathrm {MHz}$ and $\Delta _2=3\Delta _3=150\mathrm {MHz}$, which are much larger than the decay rates, and adjusted the Rabi frequency of the control field by $2\Omega _c=6\sqrt {3}\Delta _3=519.6\mathrm {MHz}$, which will be significantly greater than that of the probe field, $2\sqrt {P}=158~\mathrm {MHz}$. In fact, for a typical nanosecond pulse laser of 100 kHz repetition rate, the above peak power value corresponds to $2.1\mathrm {mW}$ average output power, which is also much less than the nominal average power of most current commercial 780nm cw lasers, thereby justifying the choice of the above adopted parameters. With these characteristic values, one can calculate the equation coefficients as $K_1=2.5\times 10^{-6}~\mathrm {sm^{-1}}$, $K_2 = 2.5\times 10^{-14}~\mathrm {s^2m^{-1}}$, $K_3 = 5.0\times 10^{-22}~\mathrm {s^3m^{-1}}$, $\gamma = 6.25\times 10^{-15}~\mathrm {s^2m^{-1}}$, $\beta _1= 0$, and $\beta _2= 1.25\times 10^{-22}~\mathrm {s^3m^{-1}}$, where the insignificant imaginary parts are dropped. Then the dispersion length is found to be $L_D=3.6~\mathrm {cm}$, which has been made to equal to the nonlinearity length $L_{\mathrm {NL}}=1/(\gamma P)$ in order to favor the formation of fundamental soliton. The analytical dynamics of the KM soliton solution (20) and the AB solution (21) are shown in Figs. 2(a) and 2(b), respectively, with $\phi$ and $\omega$ being specified in the caption. It is shown that the KM soliton oscillates periodically along the $z$ axis, whereas the AB oscillates periodically along the $\tau$ axis, as stated above, both propagating with a group velocity of $V_{g}=c/751$. When $\phi =0$, the recurring peaks of both breathers disappear, left with only one single peak and two side troughs, namely, a Peregrine soliton structure that has been shown in Fig. 2(c). As indicated by the red cross-section profiles of the framed patterns at $z=0$, the peak amplitude of the KM is always larger than that of the Peregrine soliton, but the latter is always larger than that of the AB, no matter what parameters are used.

 

Fig. 2. Contour plots of the normalized field amplitude of (a) a KM soliton, (b) an AB, and (c) a Peregrine soliton versus dimensionless distance $z/L_D$ and time $\tau /\tau _0$, obtained with different $\phi$ and $\omega$ values: (a) $\phi = 1/2$, $\omega = 1.29\times 10^8~\mathrm {s^{-1}}$, (b) $\phi = 1/2$, $\omega = 1.22\times 10^8 ~\mathrm {s^{-1}}$, and (c) $\phi = 0$, $\omega = 1.25\times 10^8~\mathrm {s^{-1}}$.The red curves show the cross-section profiles $|U(z=0,\tau )|/\sqrt {P}$ of the patterns framed by the white box.

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Lastly, one may wonder whether both the KM soliton and the AB shown in Fig. 2 are stable against perturbations, even when the imaginary parts of equation coefficients are taken into account. To clarify this concern, we numerically integrate Eq. (17) to unwrap the KM soliton and AB dynamics from an initial profile defined by the analytical solution (20) at $z=-3 L_D$ and by the solution (21) at $z=-2 L_D$, respectively, using a split-step Fourier method [9,10]. Here we particularly remark that one may simulate the original MB Eqs. (1)–(3) for more accurate results, but basically, for a very short propagation distance, an efficient simulation of Eq. (17) will suffice for our present purpose. Typical simulation results are shown in Fig. 3, where we have considered the non-integrable perturbations, that is, violating the integrability of Eq. (17) by use of a set of complex coefficients that is caused by the nonvanishing decay rates of atomic states. It is exhibited that our KM soliton and AB solutions are still robust against such non-integrable perturbations, within a distance of a few dispersion lengths. More interestingly, in Fig. 3(a), there also appears a partial AB dynamics at around $z=2L_D$, which agrees well with its analytical solution shown in the middle of picture (see the patterns framed by the white box). This excited AB tends to interfere with the original KM soliton and eventually distorts it.

 

Fig. 3. Numerical excitations of (a) the KM soliton and (b) the AB that have been shown in Fig. 2(a) and 2(b), respectively, by integrating Eq. (17) with complex coefficients. The middle contour plot between (a) and (b) gives the analytical solution of AB, using the same $\phi$ and $\omega$ as in (a).

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4. Conclusion

In conclusion, we investigated the formation of the KM solitons and ABs in a cold three-level atomic medium, within the framework of the Hirota equation that has comprised the third-order dispersion and self-steepening effects seen by nanosecond probe pulses. We revealed the unique amplitude and oscillatory characteristics associated with these breathers and confirmed numerically their stability against non-integrable perturbations. It was shown that such types of breathers can propagate at a quite low group velocity. However, for probe pulses of shorter (picosecond, say) duration, our results developed here still apply. In the latter situation, the bandwidth of incident pulses will fall outside the EIT transparency window seriously, leading to a light-matter interaction in the proximity of the off-resonant regime and consequently making the group velocity of probe pulses closer to the light speed $c$. We envision that these results may shed some light on how extreme rogue waves form in highly resonant media.