Manipulation of a weak signal pulse by optical soliton via double electromagnetically induced transparency

We propose a scheme to realize the manipulation of a weak signal pulse by ultraslow optical soliton in a coherent inverted-Y-type atomic system via double electromagnetically induced transparency (EIT). Based on Maxwell-Bloch equations, we derive nonlinear equations governing the spatial-temporal evolution of the probe and signal pulse envelopes. We show the giant enhancement of optical Kerr nonlinearity can be obtained under the condition of the double EIT, which results in the generation of a (2+1)-dimension optical soliton and can realize the manipulation of a weak signal pulse. Applying a far-detuned laser field to the system, we find that a weak signal pulse can be trapped by a (3+1)-dimension light bullet. In particular, the trajectories of the light bullet and trapped signal pulse can be manipulated and controlled by introducing a Stern–Gerlach gradient magnetic field. The results predicted here may not only open a route for the study of weak-light nonlinear optics but also have potential applications in the precision measurements and optical information processing and transmission.


Introduction
Optical trapping is a prospective technique for manipulating material objects [1]. The principle is that a high intensity laser light transfers some of its momentum to small particles and hence a force acts on the particles, which has significant applications in many research fields [2][3][4][5]. In recent years, much attention has been focused on soliton-radiation trapping (i.e., trapping light by light) in many nonlinear optical media through cross-phase modulation (CPM) effect, which have led to the observation of the Raman-induced frequency shifts and supercontinuum generation [6][7][8][9][10][11].
When the group velocities of both the optical soliton and the nonsolitonic pulse are matched, the nonlinearity of the soliton will affect the nonsolitonic pulse via CPM that can balance the dispersion and/or diffraction effects, and thus the phenomenon of trapping light by light occurs [12]. This phenomenon has been intensively investigated due to the rich nonlinear physics and important applications [13][14][15]. However, the mentioned works are utilized passive optical media, in which far-off resonance excitation schemes are employed for avoiding significant optical absorption. Moreover, for obtaining optical pulse trapping, very high light-intensity and ultrashort laser pulses are usually needed to generate nonlinearity strong adequately to balance dispersion and/or diffraction effects; furthermore, an active manipulation on the property of optical pulse trapping is not easy to realize in passive media because of the absence of energy-level structure and selection rules that can be used.
The disadvantages mentioned above can be overcome by the discovery of electromagnetically induced transparency (EIT) in a resonant atomic system at very low light level [16]. Through the giant enhancement of self-Kerr and cross-Kerr nonlinearities induced by EIT, there are many interesting researches paying attention to numerous nonlinear optical progresses, including temporal (spatial) optical solitons [17][18][19], Manakov/ Thirring vector solitons [20][21][22][23][24][25], high-dimensional spatiotemporal optical solitons [26][27][28][29][30], efficient multiwave mixing [31], the bistable state [32], and so on. In addition, different from the mechanism of Manakov/Thirring Figure 1. (a) The life-time broadened inverted-Y atomic level diagram and excitation scheme. Δ 2 and Δ 4 are two-photon detunings and Δ 3 is one-photon detuning. Ω p , Ω s , and Ω c are half Rabi frequencies of the probe, signal, and continuous-wave control fields, respectively. Γ 13 , Γ 23 , and Γ 34 are respectively the spontaneous emission decay rates from ñ |3 to ñ ñ | | 1 , 3 to ñ |2 , and ñ |4 to ñ |3 . (b) The coordinate frame used for calculations and geometrical arrangement of the system. A Stern-Gerlach gradient magnetic field is applied along the z direction with its gradient along the x direction. The (orange) thick arrow denotes the far-detuned optical lattice field used to stabilized LBs. The shaded region represents the atomic gas confined in a cell.
Doppler broadening and collisions, and are distributed in the ground states ñ |1 and ñ |2 . A possible arrangement for LBs trapping a weak signal pulse and their trajectories manipulation is shown in figure 1(b).
For simplicity, we assume that the electric field vector of all the laser fields in the system reads l p s c l l i k z t , , l l , for propagating along the z direction. Here e l ( l ) is the unit polarization vector (envelope) of the lth polarization component, and k l =ω l /c is the wavenumber of the laser field before entering the atomic gas. The continuous-wave control field counterpropagating with the probe and signal fields is used to suppress the Doppler broadening. To investigate the trajectory control of the probe and signal pulses, an SG gradient magnetic field is applied to the system, i.e., where e z is the unit vector in the z direction and B indicates the transverse gradient. Thus a small but spacedependent Zeeman level shift j , and m F j are Bohr magneton, gyromagnetic factor, and magnetic quantum number of the level ñ | j , respectively. We furthermore assume a far-detuned (Stark) laser field is applied to this system which is used to realize stable LB [28,39]. Here( ) E y R a and w a are the envelope of field amplitude and angular frequency, respectively. Because of the presence of Stark field ( ) Here p p s s s 13 23 , and , where p jl is the electric-dipole matrix element related to the states ñ | j and ñ |l .
where σ is a 4×4 density matrix in the interaction picture and Γ is a 4×4 relaxation matrix denoting the spontaneous emission and dephasing of the system. The explicit expression for density matrix elements σ jl is given inappendix A. The equation of motion for probe-field and signal-field Rabi frequencies Ω p and Ω s can be captured by the Maxwell equation under slowly varying envelope approximation [19], which is read as with  a the atomic concentration. In addition, we have assumed the continuous-wave control field is strong enough so that Ω c can be regarded as a constant during the evolution of the probe and signal pulses.

Nonlinear envelope equations and Giant Kerr effect
3.1. Asymptotic expansion and coupled nonlinear envelope equations Because we are interested in the nonlinear evolution and the possible formation of optical solitons in the system, we employ the standard method of multiple scales [19] to derive nonlinear envelope equations of the probe and signal pulses based on the Maxwell-Bloch(MB) equations (4) and (5) 1 . Here s ( ) jj 0 is the initial population distribution prepared in the state ñ | j ( j=1, 2), which is assumed as 1/2 for simplicity; ò is a dimensionless small parameter characterizing the typical amplitude of the probe pulse. All the quantities on the right-hand sides of the expansion are considered as functions of the multiscale variables x y y z z , , 1 1 , and  = a a t t (α=0, 2).
We furthermore substitute the expansion into MB equations (4) and (5), and compare the coefficients of ò α (α=1, 2, 3, L), thus a set of linear but inhomogeneous equations for s a being a yet to be determined envelope function of the slow variables t 2 , x 1 , y 1 , and z 2 . The linear dispersion relation K p (ω) is given by 2 the envelope function yet to be determined and Note that the property of K s (ω) is similar to K p (ω), thus they have EIT transparency windows in the imaginary parts Im(K p ) and Im(K s ) for W ¹ 0 c , i.e., double EIT [33].
At the third order (α=3), the divergence-free condition yields the nonlinear equation for the envelope F 1 of the probe pulse: At the fourth order (α=4), we obtain the nonlinear equation for the envelope F 2 of the signal pulse: Here we should clearly indicate that is contributed by the far-detuned laser field and the SG gradient magnetic field. The explicit expressions of W 11 , W 21 , M j , N j , and each-order approximation solutions are presented in appendix B. For the convenience of later calculations, we introduce some dimensionless variables , and U 0 are typical diffraction length, pulse duration, and Rabi frequency of the probe pulse, respectively. Thus equations (6) and (7) can be converted into the dimensionless form . We additionally assume the solution with Gaussian envelope propagates in the z direction [19,25] 2 0 2 and ρ 0 being a free real parameter. After integrating over the variable s, equation (8) becomes

Giant Kerr effect and ultraslow matched group velocities
The model mentioned above can be easily realized by selecting realistic physical systems. One of them is the ultracold 87 Rb atoms with the energy levels selected as ñ = [48,49]. Generally speaking, the coefficients in equation (10) are complex which is difficult to support the stable nonlinear localized solutions. Fortunately, when the system keeps in the condition of double EIT, then the real parts of these coefficients can be made much larger than their imaginary parts so that the nonlinear localized solutions are possible.
To this end, we now consider the third-order self-and cross-Kerr nonlinear optical susceptibilities c ( ) 11 3 and c ( ) 21 3 , which are respectively proportional to the SPM coefficient W 11 in equation (6) and CPM coefficient W 21 in equation (7). The relation obeys In the atomic medium, the frequency and wave number of the probe (signal) field are given by w w + , respectively. Thus ω = 0 corresponds to the center frequency of the probe (signal) field.
The system parameters are given by G  1 kHz 2 , which possesses two obvious features. On the one hand, their real parts have the order of magnitude 10 −3 cm 2 V −2 , which is 10 12 times larger than the third-order nonlinear optical susceptibilities found in conventional nonlinear optical media [47]. On the other hand, the real parts are indeed much larger than their imaginary parts, which is due to the quantum destructive interference.
To obtain the nonlinear localized solutions

Manipulation of a signal pulse by (2+1)D soliton
We primarily investigate the manipulation of a signal pulse by (2+1)D probe soliton. To neglect the diffraction in the y direction, we assume the transverse radii of the two pulses satisfying the relation  R R x y . In addition, Stark laser field and SG gradient magnetic field are not applied to the system, hence equation (10) can be reduced to which is similar to [14,33] but in the different physical model. We see that the solution v 1 plays a role of 'external potential' for v 2 in equation (12b). It is obvious that equation (12a) can support a single-soliton solution [50], which is written as s e c h , 1 3 with ς 1 , η 0 , ξ 0 , and ϑ 0 being free real parameters. Thus the trapped solution of v 2 of equation (12b) is determined by the solution (13). For convenience of the following calculations, we set η 0 =ξ 0 =ϑ 0 =0 for the soliton solution (13) . In this case, the solution form of v 2 can be described as with ς 2 and β being the amplitude and propagation constants [14]. Therefore, we can obtain a eigenvalue equation , and −β are respectively the potential well, the eigenfunction, and the corresponding eigenvalue. One can analytically calculate two localized modes (i.e., even and odd modes) of the trapped signal pulse for this system. The even mode solution is read as . Based on the results in section 3.2, we can obtain n 0 =1 and n 1 =0, thus the spatial profiles (i.e., eigenfunction) of the even and odd modes are described as h 0 (ξ)=sech(pξ) and x x = ( ) ( ) h p tanh 1 , respectively. We see that the trapped signal pulse displays a bright soliton or a dark soliton under the interaction of the probe soliton pulse in the coherent atomic system as long as V V  2 1 . Shown in figure 2 is the numerical result of the manipulation of a weak signal pulse by probe soliton pulse based onequations (12a) and (12b) by taking | | v 1 (for probe pulse) and | | v 2 (for signal pulse) as functions of τ/τ 0 and x/R ⊥ . We obviously see that the launched signal pulse with the even mode (figure 2(a)) or odd mode ( figure 2(b)) expands rapidly during propagation when the probe pulse is absent. As a result, no trapping of the signal pulse occurs. When a probe soliton pulse is launched into the system (shown in figure 2(c)) and its group velocity matches well with that of the signal pulse, the CPM effect induced by the probe soliton pulse thus plays a significant role for manipulating the weak signal pulse. In this case, the diffraction-induced broadening of the signal pulse is compensated and both the probe and the localized signal pulses propagate together stably; as shown in figures 2(d) and (e). In the numerical simulation, the amplitudes of the probe and signal pulses we consider are ς 1 =1.0 and ς 2 =0.15, respectively. The trapping phenomenon predicted here maybe have potential applications in quantum information processing, i.e., designing an all-optical switcher at very low light level [33]. . In this case, equation (10) becomes

Manipulation of a signal pulse by (3+1)D LB and their trajectory control
For simplicity, we firstly investigate the manipulation of a weak signal pulse by (3+1)D probe LB without SG gradient magnetic field (i.e.,   = = 0 1 2 ). Here we assume the solution of equation (14) has the form is a normalized ground state solution for the eigenvalue problem being the eigenvalue [28]. Note that ψ 1 (η) and ψ 2 (η) have the same forms because of   » 1 2 . After integrating over the variable η, one obtains f t 2). We obviously see that equation (15a) has the same form with equation (12). Therefore, one can obtain the single-soliton solution f 1 , i.e.,  (10) is s e c h , 1 6 . We see the nonlinear solution (16) is localized in three space and one time dimensions, i.e., the (3+1)D LB solution of the system. Here we have set η 0 =ξ 0 =ϑ 0 =0 for simplicity.
, where ς 2 and β are the amplitude and propagation constants. Thus h(ξ) has two localized modes: (1) the even mode solution 11 1 . As a result, the trapped signal pulse has the form = for odd mode. The consequence indicates that the manipulation of a signal pulse by (3+1)D probe LB can also be realized. Shown in figure 3 is the numerical result of the manipulation of the weak signal pulse by a (3+1)D probe LB without the SG gradient magnetic field (i.e., B=0) based on equation (14), by taking | | u 1 and | | u 2 as functions of (x, y)/R ⊥ and z/L Diff . For simplicity, we consider only the even mode (i.e., bright soliton) of the weak signal pulse. Figures 3(a)-(c) show the light intensity | | u 1 of the signal pulse under the diffraction-induced broadening effect solely (i.e., without probe pulse) when pulse travels to = = z L z L 2 , 4 Diff Diff , and = z L 6 Diff , respectively. In this case, there is no trapping effect occurs and the signal pulse spreads rapidly during propagation. When the probe pulse is present, the trapping of the signal pulse happens; see figures 3(d)-(f), the left spot and right spot in each panel respectively denoteing the probe LB and the trapped signal pulse. Physically, thanks to the matched group velocities of signal and probe pulses, the CPM induced by probe LB is acted on the signal pulse, which leads to a localized pulse. As a result, the profiles of the signal and probe pulses keep nearly unchanged for Diff Diff , and = z L 6 Diff , respectively. Note that for a better visualization the intensity of the signal pulse plotted in figures 3(d)-(f) has been amplified four times, and the central positions of the both pulses have been separated a small distance artificially (the same treatment is also used in figure 4). Here, the amplitudes of the probe and signal pulses are ς 1 =1.0 and ς 2 =0.15, respectively, and the amplitude of the Stark field is =É 2.79 10 V 0 3 cm −1 . By using Poynting's vector, we can make an estimation on the threshold of the optical power densityP probe (the power of the probe pulse) andP signal (the power of the signal pulse) for realizing the trapping phenomenon. Based on the parameters used above, it is easy to obtain = P 4.24 probe nW and = P 0.1 signal nW. We see that, to realize the manipulation of a signal pulse by (3+1)D probe LB, a very low input light power is needed, which is due to the resonance character and the double EIT effect in the system. This is very different from conventional optical media, such as glass-based optical fibers, where picosecond or femtosecond laser pulses are usually needed to reach a high power to produce a sufficiently nonlinear effect needed for the formation of an optical soliton [6][7][8][9][10][11][12][13][14].

Trajectory control of the trapped signal pulse and (3+1)D probe LB
We then turn to investigate the trajectories of the trapped signal pulse and (3+1)D probe LB by an SG gradient magnetic field (i.e., ¹ B 0). Thus equation (15) contains an external potential proportional to ξ written as  The single-soliton solution can be read as s e c h , 1 9 where ς 1 , η 0 , ξ 0 , and ϑ 0 are free real parameters. However, equation (17b) is transferred into another but complicated form due to  = 0 2 . Finally we can obtain the solution and η 0 =ξ 0 =ϑ 0 =0. One can see that (3+1)D probe LB moves along a parabolic trajectory. Because the analytical solution u 2 is not available, we just make a numerical calculation.
Shown in figure 4 represent the deflection of the probe and signal pulses by numerically simulating equation (14) with the SG gradient magnetic field B=50.3 mG cm −1 , by taking | | u 1 and | | u 2 as functions of (x, y)/R ⊥ and z/L Diff . Figures 4(a)-(d) indicate the deflection of the probe LB and trapped signal pulses when pulses Diff Diff Diff , and = z L 8 Diff , respectively. The left spot and right spot in each panel respectively denote the probe LB pulse and the trapped signal pulse. We can see the following: (i) the weak signal pulse can be perfectly trapped by the LB because of the CPM effect induced by the probe pulse; (ii) the probe LB and the trapped signal pulses undergo a deflection in the negative x direction by the contribution of the SG gradient magnetic field. Obviously, the trajectory of the trapped signal pulse and (3+1)D probe LB can be steered by manipulating the SG gradient magnetic field. Such manipulation is useful for optical information processing, e.g., for the control of the behavior of all-optical switching.
In addition, we can determine the deflection angles of the (3+1)D probe LB and trapped signal pulse. Due to  = 0 2 , the trapped signal pulse deflect along the trajectory of (3+1)D probe LB. It is easy to obtain the propagating velocity of the (3+1)D probe LB by solution u 1 , which is The expected deflection angle of the output probe LB is defined as the ratio 2 Diff 2 and = V V z g 1 are the propagating velocities respectively along the x axis and the z axis. Here we assume the probe LB passes through the atomic medium with length L along the z direction, thus the traveling time in the z direction is = t L V g1 . As a result, the deflection angle of probe LB is From the figure 4, we obtain deflection angle q q q q = -´- Diff corresponding to figures 4(a)-(d), which is two orders of magnitude larger than that for linear polariton obtained in [51]. On the contrary, one can precisely measure the micro magnetic fields through the detection of the deflection angle of the LB. We expect that the significant deflection obtained here may have potential applications in optical magnetometery, quantum information processing, i.e., designing optical beam splitters.

Summary
In this work, we have proposed a physical scheme for manipulating a weak signal pulse by (2+1)D optical soliton and (3+1)D LB in a coherent inverted-Y four-level atomic system via double EIT. Based on MB equations, we have derived nonlinear equations governing the spatial-temporal evolution of the probe and signal pulse envelopes. We have shown that the optical Kerr nonlinearity of the system can be enhanced dramatically under the condition of double EIT. We have found that the probe pulse can induce an enough strong SPM effect to balance the diffraction effect, while the weak signal pulse relies on the CPM effect of the probe pulse to form a localized wave packet with matched group velocities; and thus the trapping light by light can be obtained. Furthermore, we have investigated the manipulation of a weak signal pulse by (3+1)D LB by introducing a fardetuned laser field into the system to stabilize the LB. In particular, the trajectories of the LB and the trapped signal pulse can be manipulated when an SG gradient magnetic field is applied to the system. The research results predicted here may not only open a route for the study of weak-light nonlinear optics but also have potential applications in the precision measurements and optical information processing and transmission (e.g., design of all-optical switching at very low light level).