All-optical switching in a medium of a four-level vee-cascade atomic medium

We proposed a model for all-optical switching in a medium consisting of four-level vee-cascade atomic systems excited by coupling, probe, and signal fields. It is shown that, by changing the intensity or the frequency of the signal field, the medium can be actively switched between either electromagnetically induced transparency or electromagnetically induced absorption, which has behavior of all-optical switching. As a result, a cw probe field is switched into square pulses by modulating the intensity or the frequency of the signal light. Furthermore, width of the square probe pulses can be controlled by tuning the switching period of the signal field. Such a tuneable all-optical switching is useful for finding related applications in optic communications and optical storage devices.


Introduction
All-optical switch is an important component in high-speed optical communication networks and quantum computing (Ishikawa 2008). Over the last few decades, all-optical switching based on optical bistability in two-level atomic systems was studied. However, applications of the two-level atomic system is limited due to strong resonant absorption and only one optical field is employed for both applying and switching, thus lack of control for switching intensity thresholds. The advent of EIT (Imamoǧlu and Harris 1989;Boller et al. 1991;Fleischhauer et al. 2005;Doai et al. 2014;Khoa et al. 2016Khoa et al. , 2017a has created transparent media whose optical properties can be controlled by the external fields. Due to very steep dispersion in transparent spectral region, therefore light pulses can propagate with very small group velocities (Hau et al. 1999;Anh et al. 2018aAnh et al. , 2018b, the medium can achieve giant Kerr nonlinearities (Wang et al. 2001;Khoa et al. 2014;Hamedi et al. 2016;Doai et al. 2019), and optical soliton is also easily achieved with low intensity (Huang et al. 2006;Si et al. 2010;Chen et al. 2014;Khoa et al. 2017b;Dong et al. 2018).

Theoretical model
We consider an atomic medium consists of four-level vee-cascade excitation scheme having a ground state �1⟩ and three excited states �2⟩ , �3⟩ and �4⟩ , as shown in Fig. 1a. An optical probe field with frequency ω p and Rabi frequency Ω p drives the transition �1⟩ ↔ �2⟩ , while the transition �1⟩ ↔ �3⟩ and �2⟩ ↔ �4⟩ are excited by coupling (frequency ω c ) and signal (frequency ω s ) optical fields with Rabi frequency Ω c and Ω s , respectively. The decay rates from the states �2⟩ and �3⟩ to the ground state �1⟩ , and from the excited state �4⟩ to the state �2⟩ denoted by γ 21 , γ 31 and γ 42 , respectively.
Using the rotating-wave and the electric dipole approximations, the total Hamiltonian of the system in the interaction picture can be written as (in units ℏ).
where Δ p = 21 − p , Δ c = 31 − c , and Δ s = 42 − s are frequency detunings of the probe, coupling, and signal fields, respectively. The dynamical evolution of the system can be described by the Liouville equation: where Λρ represents the decaying processes. For the four-level vee-cascade system, the density matrix equations in Eq. (2) are decomposed as: where the matrix elements obey conjugated and normalized conditions, namely ij = * ij (i ≠ j), and 11 + 22 + 33 + 44 = 1 , respectively.

Switching between EIT and EIA
First of all, we consider the influence of the switching signal field on the absorption and dispersion properties of medium for the probe field in the presence of coupling field at the steady regime. The linear susceptibility χ of the atomic medium for the probe light field relates to the matrix element ρ 21 determined by Doai et al. (2014): where the matrix element ρ 21 is solved numerically from set of Eqs. (3a-3j). The imaginary χ′′ and real χ′ parts of the linear susceptibility represent the absorption and dispersion, respectively. In Fig. 2, we plotted the absorption Im(ρ 21 ) and dispersion Re(ρ 21 ) versus the frequency detuning Δ p for the cases of absence (Ω s = 0) and presence (Ω s = 9γ 21 ) signal field. Here we used resonant conditions Δ c = Δ s = 0 and Ω p = 0.01γ 21 , Ω c = 9γ 21 . (3e) As shown in Fig. 2, the absorption and dispersion spectra of the probe field depend sensitively on the ON or OFF mode of the signal field. Indeed, when switching field OFF, Ω s = 0 (dashed lines), the absorption of the probe field can be suppressed thus the medium becomes completely transparent to the probe field at line center, which the corresponding dispersion profile has a positive slope near zero detuning (dashed line Fig. 2b) hence the probe light is slowed down. However, when the signal field is turns on, Ω s = Ω c = 9γ 21 , the medium is switched from transparent to a complete absorbing mode (EIA) at the line center. Furthermore, the absorption profile for the probe field is switched from single transparency window centered at Δ p = 0 to a double-transparency window located at Δ p = ± Ω c . The corresponding dispersion profile has a negative slope near zero detuning (solid line Fig. 2b), thus the medium speeds up group velocity of the probe light.

All-optical switching
Now we extend our consideration from the steady to a dynamical regime. Under the slowly-varying envelope and rotating-wave approximations, volution of the probe field is represented by the following wave equation (Khoa et al. 2017b): is the propagation constant. For a convenience, we represent Rabi frequency of the probe field by Ω p (z, t) = Ω p0 f (z, t) , where Ω p0 is a real constant indicating the maximal value of the Rabi frequency at the entrance (i.e., at z = 0), and f (z, t) is a dimensionless spatiotemporal pulse-shaped function. In a moving frame with ξ = z and = t − z∕c , the optical Bloch matrix Eqs. (3a-3j) for the density matrix elements ij ( , ) and Maxwell's wave Eq. (5) for the probe field f ( , ) can be rewritten by: In the following, we solve numerically the set of Eqs. (6a-k) on a space-time grid by using a combination of the four-order Runge-Kutta and finite difference methods which were developed from our previous work (Khoa et al. 2017b). We assumed the initial condition is all atoms in the ground state �1⟩ whereas the boundary condition is the probe pulse having a Gaussian-type shape f ( = 0, ) = exp[−(ln2)( − 30) 2 ∕ 2 0 ] , with 0 = 6∕ 21 is the temporal width of the pulse at the entrance of the medium.
In Fig. 3 we plotted spatiotemporal evolution of intensity (square of magnitude) of the probe field at different intensities of the signal field when Ω c = 9γ 21 , Δ c = Δ s = Δ p = 0. It is shown that the OFF or ON mode of the signal field affects sensitively on the probe pulse absorption. When the signal field OFF (i.e., Ω s = 0), the atomic medium is transparent to the probe pulse, namely, the probe pulse remains unchanged over a long traveling distance (Fig. 3a). As the signal field ON at Ω s = Ω c = 9γ 21 , the probe pulse can be absorbed completely, even in a noticeably short propagation distance (see Fig. 3b). This strong absorption associates with EIA phenomenon, as indicated in Fig. 2a.  Consequently, by switching intensity of the signal field the medium can be switched to either EIT or EIA for the probe pulse. In Fig. 4, we plotted the normalized probe field intensity for different values of the signal detuning Δ s while keeping other parameters at Δ p = 0, Δ c = 9γ 21 , Ω 0p = 0.01γ 21 , Ω c = 8γ 21 , and Ω s = 9γ 21 . When Δ s = 0, the medium is almost transparent for probe pulse (Fig. 4a) whereas it absorbs completely probe pulses as Δ s increasing to a value Δ s = 10γ 21 (see Fig. 4b). From this feature one can conclude that the medium can be switched from EIT to EIA by tuning frequency of the signal field. Next, we consider a possible way to realize all-optical switching for the probe field by tuning parameters of the signal field, as shown in Fig. 5. Here, the probe field (solid lines) is assumed to be a continuous wave (cw) whereas the switching signal field (dashed lines) to be nearly-square pulses with smooth rising and falling edges. The signal field is switched by the following rules: here the intensity of the signal field in Eqs. (7) or (8) is switched with approximate period 50/γ 21 and 100/γ 21 , respectively. In Fig. 5 we plotted the switching field when the amplitude of the signal field is normalized by its peak value Ω s0 = 9γ 21 . Figure 5 shows that the switching periods of both probe and signal fields are the same. Furthermore, the probe transmission is switched to the ON or OFF mode when the intensity of the signal field is OFF or ON, respectively. On the other hand, oscillations at the front edge of the probe pulse can be extinguished when increasing width of the signal pulse (Fig. 5b). In Fig. 6, we consider the dynamics of the cw probe field (solid lines) under modulating frequency of the signal field (dashed lines). Here, frequency detuning of the probe field is chosen as a nearly-square pulse with smooth rising and falling edges as follows: here the frequency detuning of the signal field in Eqs. (9) or (10) is switched with approximate period 50/γ 21 and 100/γ 21 , respectively. In both cases, the frequency detuning is normalized by its peak value as Δ s0 = 10γ 21 . We can see that the probe transmission is switched to either ON or OFF when the frequency detuning of the signal field is switched OFF or ON, respectively. As indicated in the Sect. 3.1, the ON or OFF mode for probe field can be attributed from EIT or EIA mode of the medium, respectively.
Finally, we consider the behavior of optical switching according as intensity (Fig. 7) and the frequency (Fig. 8) of the signal field in strong probe regime. Figures 7 and 8 show the cases of Figs. 5b and 6b, respectively, while the probe intensity is increased to Ω p0 = 0.5γ 21 (a) and Ω p0 = 1γ 21 (b). From these figures we can see that when intensity of the probe field is of the same order with the coupling field, the switching efficiency decreases. To explain (10)  switched at approximate period 100/γ 21 , and the probe field intensity Ω p0 = 0.5γ 21 (a) and Ω p0 = 1γ 21 (b). Other parameters given as same as those in Fig. 6 the phenomenon, we plot spatiotemporal evolution of intensity of the probe pulse at Ω s = Ω c = 9γ 21 , Δ c = Δ s = Δ p = 0 and Ω p0 = 0.5γ 21 (a) and Ω p0 = 1γ 21 (b) as shown in Fig. 9. Due to strong intensity of the probe field, its intensity will not be absorbed completely, then it may transfer to the next period, which results higher background (Figs. 7b and 8b), namely lower switching efficiency. Moreover, the strong probe pulse corresponding to selfphase modulation is formed and the switching process can be broken down gradually (Rajitha et al. 2015).

Conclusion
We have proposed a model of four-level vee-cascade atomic medium to realize all-optical switching for a probe light in weak and strong probe regimes. Due to quantum interferences among atomic transition paths induced by coupling, probe and signal optical fields, the medium can be switched to either EIT or EIA regime by switching the signal field to OFF or ON, respectively. As a result, an all-optical switching can be generated in which the cw probe field is switched into pulses by modulating intensity or frequency of the signal light. Furthermore, width of the probe pulses can be controlled by tuning switching period of the signal field. Such a tuneable all-optical switching is useful for finding related applications in optics communications and optical storage devices.