Controllable ultraslow optical solitons in a degenerated two-level atomic medium under EIT assisted by a magnetic field

We proposed a simple model for generation of controllable ultraslow optical solitons of a weak probe laser light in a degenerated two-level atomic medium under electromagnetically induced transparency assisted by a magnetic field. It is shown that bright and dark optical solitons can be formed from a probe light with controllable ultraslow group velocities at a few m/s by tuning the strength of a coupling light and/or the magnetic field. In addition to the ultraslow velocity, the advantage of this model is to use a sole laser for delivering both pump and probe lights. Furthermore, one can switch between bright and dark solitons by reversing the direction of the magnetic field. Such controllable ultraslow solitons are interested in finding applications in optical communications and optical data processing.

We proposed a simple model for generation of controllable ultraslow optical solitons of a weak probe laser light in a degenerated two-level atomic medium under electromagnetically induced transparency assisted by a magnetic field. It is shown that bright and dark optical solitons can be formed from a probe light with controllable ultraslow group velocities at a few m/s by tuning the strength of a coupling light and/or the magnetic field. In addition to the ultraslow velocity, the advantage of this model is to use a sole laser for delivering both pump and probe lights. Furthermore, one can switch between bright and dark solitons by reversing the direction of the magnetic field. Such controllable ultraslow solitons are interested in finding applications in optical communications and optical data processing.
Optical soliton formation is a fundamental phenomenon in nonlinear optics that attracts a great attention over the last four decades due to its potential applications [1][2][3] . Recently, gaseous atomic media has been the interesting subject for soliton formation since the advent of EIT [4][5][6] . In addition to a large suppression of optical absorption, EIT medium reduces significantly group velocity of optical lights 7,8 , supports for optical switching and bistability at low-light intensity 9,10 .
In the conventional methods for generating optical solitons, there often need intense electromagnetic fields or ultrashort laser pulses due to small nonlinearity of atomic medium for the far-off-resonance excitation frequencies. Contrastively, in the EIT atomic medium, the frequency of the interacting fields closes to atomic transition, thus the nonlinearity is enhanced significantly [11][12][13] . This is particularly interesting because ultraslow propagation can be achieved at a weak field intensity with controllable group velocity.
Up to date, most of works of slow light propagation and optical soliton formation in the EIT atomic medium focused on the three-level [14][15][16][17][18][19][20][21][22][23][24][25][26] , four-level [27][28][29][30][31][32][33] , and five-level 34,35 systems in which experimental observations of several soliton types were demonstrated [24][25][26] . Despite of extensive proposals in this topic for multi-level atoms in which all interacting fields must be controlled synchronously, the two-level system is particularly interesting because of its simple realization. Furthermore, the previous studies often neglect degeneration of Zeeman levels which should be considered when the atoms placed in external magnetic fields or in the polarized optical fields. In this work, we propose a simple model for manipulating ultraslow-light solitons in a degenerated two-level system under EIT assisted by a static magnetic field. In "Theoretical model" section, we describe the theoretical model based on the Maxwell-Schrödinger equations (MSE) for the evolution of atom-field interaction. "Results and analysis" section discusses the property of ultraslow bright and dark solitons with variation of the magnetic field. A possible experimental realization for the proposed model is presented in "Possible experimental realization" section. Finally, conclusions of the present work are given in "Conclusions" section.

theoretical model
We consider a degenerated two-level atomic system consists of an upper non-degenerated level (corresponds to hyperfine state F = 0 with magnetic quantum number m F = 0) and a lower degenerated level (correspond to hyperfine state F = 1 with m F = ± 1), as shown in Fig. 1a (Fig. 1b). All the atoms assumed to be optically pumped to the states |1〉 and |2〉 with the same populations, i.e. ρ 11 = ρ 22 = 1/2. A weak probe laser field E p with the left-circularly polarized component σ − (with frequency ω p and one-half Rabi-frequency � p = µ 21 E p /2 ) drives the transition |1〉 to |3〉. At the same time, a strong coupling laser field E c with the right-circularly polarized component σ + (with frequency ω c and one-half Rabi-frequency � c = µ 23 E c /2 ) is introduced to couple the transition between the states |2〉 and |3〉. The decay rate from the states |3〉 to |1〉 and |2〉 is given by γ. The relaxation rates of coherence between the ground states |1〉 and |2〉 by collisions are neglected.
Using the rotating-wave and the electric dipole approximations, the interaction Hamiltonian of system in the interaction picture can be written as (in the units of ): and p = c = , are detuning of the probe field and coupling field from the atomic transition frequencies, respectively.
In the interaction picture, by using the time-dependent Schrödinger equations, the probability amplitudes equations for the relevant states are given by where A n (n = 1, 2 3) represents amplitude of atomic wave function for each state, γ is decaying rate of the states |3〉.
Under the slowly varying envelope and rotating-wave approximations, evolution of the probe field is represented by the following wave Eq. 9 : here κ 13 = 2Nω p |µ 31 | 2 /( c) is the propagation constant, with N, μ 13 , c, and ε 0 , are the atomic density, dipole moment between levels |1〉 and |3〉, vacuum speed of light, and vacuum dielectric constant, respectively.

Results and analysis
In this section, we focus on interplay between the dispersion and nonlinear effects in the atomic system which can form solitons 19,29 . Firstly, we consider the dispersion properties of the atomic system by using perturbation treatment to the first order of weak probe field Ω p while keeping all orders due to control field Ω c . To attain this aim, the perturbation approach is applied to the atomic part in terms of the expansion A n = k A (k) n , where A (k) is the k-th order part of A n in the probe field Ω p . To the first-order of the probe field Ω p , we assume that the atomic is initially in the ground states |1〉 and |2〉 with A  2) and (3) and keeping up to the first order of Ω p , we obtained n and Ω p respectively, and ω is the Fourier variable. By solving Eq. (4c) with a substitution form Eqs. (4a) and (4b), we obtain solution for the probe field where β(ω) is the propagation constants denoted by with where β 0 (0), β 1 (0), and β 2 (0) relates to physics quantities. Indeed,β 0 (0) = φ + iα/2 describes the phase shift ϕ per unit length with absorption coefficient α (see Figs. 2a, 3a) of the probe field; V g = Re[1/β 1 (0)] denotes the propagation group velocity of optical solitons; and β 2 (0) represents the group-velocity dispersion (GVD) that leads to change in probe pulse's shape and loss of probe field intensity.
To study formation of optical solitons, there should balance the interplay between group velocity dispersion and nonlinear effects. We consider the nonlinear polarization on the right-hand sides of Eq. (4c) and take a trial function � p (z, t) = � p (z, t) exp [iβ 0 (0)z] for Eq. (3), we obtain the nonlinear wave equations for the slowly varying envelope � p (z, t).
where NLT is a nonlinear term given by NLT = −κ 13 A It is convenient to transform Eq. (8) into a moving frame by changing ξ = z and τ = t − z/V g , we obtain the following equation for Ω p : where absorption coefficient α = 2Im[β 0 (0)] and Since the coefficients in NLS Eq. (10) are therefore complex, thus Eq. (10) generally does not have soliton solutions. However, in the presence of the coupling field, the absorption for the probe field can be suppressed under EIT conditions, where the probe field intensity relates to exp (−αl) ≃ 1 (l is length of the atomic medium). Furthermore, as we can see below, for the practical parameters one may find conditions so that the imaginary part of the complex coefficient in Eq. (11) much smaller than their corresponding real part, i.e., β 2 (0) = β 2r (0) + iβ 2i (0) ≃ β 2r (0) , and W = W r + iW i ≃ W r . Under the regime of these parameters, we can neglect the imaginary parts and make the Eq. (11) to be integrable, then Eq. (10) can be reduced to the standard nonlinear Schrödinger equation: www.nature.com/scientificreports/ which admits the solutions describing various types of solitons 2,19,28,29 , such as the right ( β 2r W r > 0 ) and dark ( β 2r W r < 0 ) solitons, depending on choosing the parameters. The fundamental bright soliton is given by: where sech(τ/τ 0 ) is the hyperbolic secant function, amplitude Ω p0 and width τ 0 subject only to the constraint � p0 τ 0 2 = 2β 2r (0)/W r . Note that the condition � p0 τ 0 2 ≪ |� c τ 0 | 2 used to derive Eqs. (10) and (12) is fulfilled for weak probe field. Therefore width τ 0 should be chosen to meet 2β 2r (0)/W r ≪ |� c τ 0 | 2 . The fundamental dark soliton of Eq. (12) with β 2r W r < 0 is given by where the envelope of the probe pulse is chosed as � p (ξ = 0, τ ) = � p0 tanh(τ/τ 0 ) 37 . We now consider practical parameters to show the existence of bright and dark solitons in the degenerated two-level atomic system. For this purpose, we plot the cases of bright and dark solitons in Figs. 2 and 3, respectively.
In Fig. 2, the absorption coefficient α and the ratios β 2i /β 2r and W i /W r are plotted versus the dimensionless Rabi frequency Ω c /γ with parameters κ 13 = 1 × 10 9 cm −1 s −1 , Δ = 3 × 10 8 s −1 , and γ = 6 × 10 6 s −1,19,28 , Δ B = − 2 × 10 6 s −1 = − 0.33γ, which corresponds to B = − 0.33γ c (we note that when the Zeeman shift Δ B is scaled by γ, then the magnetic strength B should be in unit of the combined constant γ c = µ −1 B g −1 F γ ). The figure clearly demonstrates that there exists a region of the parameters in which absorption for the probe field can be www.nature.com/scientificreports/ almost suppressed under presence of the magnetic field 9 . In this region we see that β 2r .W r > 0, thus, the bright soliton can be formed. Using the same parameters as used in Fig. 2 except for Δ B = 2 × 10 6 s −1 , (corresponds to B = 0.33γ c ), we plot the absorption coefficient α and the ratios β 2i /β 2r and W i /W r versus the dimensionless Rabi frequency Ω c /γ, as shown in Fig. 3. In this parameter regions, β 2r .W r < 0, thus, dark solitons can be admitted. From both Figs. 2 and 3 one can see a possible switching between a bright and dark soliton by reversing direction of the external magnetic field.
Finally, we consider influences of the coupling light and magnetic strength on the group velocity of the probe light by plotting v g /c versus Rabi frequency Ω c and versus magnetic strength, as shown in Figs. 6 and 7,

possible experimental realization
In this section, we discuss a possible experimental realization for the case of 87 Rb atoms on the 5S 1/2 ↔ 5P 3/2 transitions. Here, the states |1〉, |2〉, and |3〉 are given as 5S 1/2 (F = 1, m F = − 1), 5S 1/2 (F = 1, m F = 1), and 5P 3/2 (F = 0, m F = 0), respectively (see Fig. 1). Both the probe and coupling fields can be delivered by a sole laser working at 780 nm. The atomic medium can be produced in a vapor cell placed inside a solenoid tube connected with a variation DC current source via interchangeable anode-cathode switch. This configuration can deliver controllable magnetic field in both magnitude and direction. In order to ensure selection rules for the excitation configuration, the coupling and probe beams are directed to quarter-wave plates to produce circularly polarized beams (see Fig. 1) where both of which propagate in opposite directions. The generation of dark solitons can be used the trapezoidal optical integrator whereas detection of the probe pulse can be used by the first-derivative optical differentiator and the first-order Butterworth detector 37 .

conclusions
We have proposed a simple model for generation of tunable ultraslow optical solitons of a weak probe laser pulse in a degenerated two-level atomic medium under an external magnetic field. The system can generate and switch between bright and dark solitons by reversing the direction of the external magnetic field. Furthermore, the solitons can be controlled to propagate with ultraslow group velocity by tuning the strength of the coupling light and/or the magnetic field. In addition to the ultraslow velocity, the advantage of this model is to use a sole laser for delivering both pump and probe lights. Such tunable optical solitons are interesting for finding applications in optical information processing and logic gates.  www.nature.com/scientificreports/

Methods
Using a method of multiple scales with amplitude variable approach we derive the nonlinear Schrödinger equation that governs the time evolution of probe pulse envelope. The formation, evolution and dynamics of the ultraslow optical soliton by using a standard soliton perturbation theory. All analytical predicts are checked by numerical simulations in the MATLAB.