Controllable Kerr Nonlinearity in a Degenerate V-Type Inhomogeneously Broadening Atomic Medium Aided By a Magnetic Field


 We propose a degenerate V-type inhomogeneously broadening atomic model for Kerr nonlinear control under the support of an external magnetic field. The magnetic field is used as a control knob to convert the medium from transparency to absorption at the resonance center. It is shown that the magnitude and sign of the Kerr nonlinearity index are manipulated by the magnetic field. Especially, we can easily convert the Kerr nonlinearity index from the positive peak into a negative peak, and from zero value can be transferred to enhanced positive or negative peaks simply only by changing the direction or strength of the magnetic field. In addition, the Kerr nonlinearity index under the influence of the temperature, the intensity, and detuning of the control field is also considered. The proposed model can be useful for applications in optical switching and nonlinear optical devices as well as possible experimental implementations.

Kerr nonlinear enhancement based on the EIT effect has been widely studied both experimentally and theoretically [23][24][25][26][27][28][29][30][31][32][33][34]. One of the pioneering works in this field was proposed by Schmidt and Imamoglu in 1996 [23]. It is shown that can be achieved a giant increase of Kerr nonlinearity in a four-level N-type atomic system under the EIT regime by introducing an additional off-resonant level. Then, this scheme was proved experimentally in cold Rb atoms by Kang and Zhu [24]. Furthermore, for the first time in the experiment, Wang et al. [26] measured the Kerr nonlinear index of the lambda-type three-level atomic medium at room temperature inside an optical ring cavity. This work has shown that the Kerr nonlinear index can be modified and significantly enhances according to the coherence parameters of the coupling laser fields. At the same time, the sign of the nonlinear index can be modified by changing the sign of the control beam frequency detuning. Sheng et al. [28] demonstrated that adjusting the switching laser power makes it possible to enhance the Kerr nonlinearity in the N-type four-level system in the resonant neighborhood. Besides, using the double-dark resonant interaction in the four-level tripod-type atomic system such as large Kerr nonlinearity through cross-phase modulation [29,30], store a light pulse in two channels [31] and obtain a sub-Doppler and subnatural narrowing of an absorption line [32], and via the incoherent pumping [34]. In addition, a series of studies to enhance Kerr nonlinearity under the influence of spontaneously generated coherence and relative phase between coupling laser fields have also been performed [36][37][38][39][40][41][42], such as colossal Kerr nonlinearity can be obtained in a V-type system under the influence of SGC [36], or the relative phase of the coupling fields [41].
Recently, the external magnetic field used as a control parameter has also attracted many studies such as the ultraslow light-dark soliton transition [43], pulse propagation and optical switching [44[, controlling the period and threshold intensity of the bistability [45][46][47][48].
However, using the external magnetic field to control Kerr nonlinearity in a V-type inhomogeneously broadening medium has not been studied. In this paper, the Kerr nonlinear index in a degenerated V-type atomic medium under the influence of Doppler broadening can be manipulated by a magnetic field. To facilitate this, we derive an analytic solution expression for the Kerr nonlinear index n2 under Doppler broadening and the magnetic field. From there, we control the amplitude and sign of the Kerr nonlinear index under the influence of the magnetic field, intensity, and frequency detuning of the control field in the presence of Doppler broadening. The results obtained are useful for applications in magneto-optic switching and logic gates devices. Furthermore, the model is easy to realize in an experiment because the system can be performed in different frequency regimes using only a single laser for both coupling and probe fields.

Model and basic equations
The model of a degenerate V-type atomic system under the effect of an external magnetic field, and two probe and control laser fields are depicted in Fig. 1 [44][45][46], respectively. The decay rates from the states |3 and |2 to |1 are given by γ31 and γ21, respectively. The total Hamiltonian in the interaction picture of the atomic and the laser field for this system is given by [14,44]: here assume h =1, and notation The dynamic evolution of the system is described by the Liouville equation: ( The density matrix equations for this system can be written as: 33 31 33 13 31 ,   21 31    , and ij is the decay rate between levels |i and |j, respectively. To obtain linear and nonlinear susceptibilities, we need to give the steady-state solution of the density matrix equations (3). To attain this aim, the perturbation approach is used to the density matrix elements are expressed as: . We assume that the coupling field is much stronger than the probe field and the zeroth-order solution of the population   0 11 1   , while the other elements are equal to zero. In the weak-probe field approximation, we obtain the first-order and third-order solutions of the matrix element 21  from equations (3d) and (3f): and To explicitly derive the element   Substituting Eq. (7) into Eq. (5), we obtain the element   3 21  as follows: Therefore, the first-and third-order susceptibilities are determined as follows: where the total susceptibility     where, 2/ B u k T m  is the root mean square atomic velocity and N0 is the total atomic density of the vapour. In this way the Doppler effect can be included in the susceptibilities as: The first-and third-order susceptibility obtained by integrating equations (12) and (13)

Results and discussions
We apply the investigate model for the 87 Rb atomic medium with the levels |2, |3, and |1 chosened corresponding to states: 5P1/2 (F = 1, mF = -1 ), 5P1/2 (F = 1, mF = +1), and 5S1/2 (F = 1, mF = 0 ). The parameters are selected [26,44]: γ31 = γ21 = 2π here can be understood that under Doppler expansion (such as at room temperature), the |2 and |3 sublevels are overlapping. Therefore, to overcome this the magnetic field needs to be much stronger so that the Zeeman shift can be able to overcome this overlap. It is clear as described in Fig 3, where the coupling field is chosen Ωc = 80γ21. From Fig 3(a), we can see the EIT window is shifted to the left (or right) by an amount p = 12021 for B = 60c, corresponds to B = 6021 (or B = -60c, corresponds to B = -6021), respectivelly. Therefore, the medium converting from transparency to fully absorbent at the resonance center and corresponding to the normal dispersion domain is also transferred to the anomalous dispersion domain and vice versa. = -15γc. This behavior can also be seen clearly as described in Fig. 4.    and Δp = Δc = 0 for (b). Other parameters are the same as in Fig. 4.
Finally, we consider the influence of the Doppler broadening effect on the amplitude and shape of the Kerr nonlinear index curve n2 as shown in Fig. 8. Fig. 8, shows the amplitude of the Kerr nonlinear index n2 is degraded as the absolute temperature increases [27]. At the same time, the envelope of the nonlinear dispersion curve also gradually expands. The Kerr nonlinear index with controllable amplitude and sign has created many interesting applications in pulse propagation and all-optical switching technique.

Conclusions
We have investigated the influence of external magnetic field on the Kerr nonlinearity index in a degenerate V-type atomic medium under the Doppler broadening and EIT effects.
We have shown that the amplitude and sign of the Kerr nonlinear index are controlled by the external magnetic field, intensity, and frequency detuning of the control field. The result has been shown that by using the magnetic field as a control knob, we can convert the positive peak of the Kerr nonlinear index into the negative peak and vice versa. Moreover, it is possible to switch from the zero-nonlinear index into the largest value at the resonant center simply only by changing the direction and amplitude of the magnetic field. Manipulating the behavior of the Kerr nonlinear index is a favorable condition for generating steady propagating pulses (solitons) that have potential applications in all-optical switching and communications devices.

Declaration of Interest:
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.