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Stern-Gerlach effect of multi-component ultraslow optical solitons via electromagnetically induced transparency

We propose a scheme to exhibit a Stern-Gerlach effect of n-component (n>2) high-dimensional ultraslow optical solitons in a coherent atomic system with (n + 1)-pod level configuration via electromagnetically induced transparency (EIT). Based on Maxwell-Bloch equations, we derive coupled (3+1)-dimensional nonlinear Schrodinger equations governing the spatial-temporal evolution of n probe-field envelopes. We show that under EIT condition significant deflections of the n components of coupled ultraslow optical solitons can be achieved by using a Stern-Gerlach gradient magnetic field. The stability of the ultraslow optical solitons can be realized by an optical lattice potential contributed from a far-detuned laser field.


I. INTRODUCTION
In past two decades, (3+1)-dimensional spatiotemporal optical solitons, alias light bullets (LBs) [1], i.e. wave packets localized in three spatial and one time dimensions during propagation, have been intensively investigated due to their rich nonlinear physics and important applications [2]. However, LBs studied up to now are usually produced in passive optical media, in which far-off resonance excitation schemes are used to avoid optical absorption.
Such LBs have several disadvantages in applications. For instances, the generation power of LBs in passive optical media is very high, and it is very hard to realize an active manipulation and control on them. In addition, due to off-resonance character the propagating velocity of such LBs is close to c (i.e. the light speed in vacuum).
However, the disadvantages mentioned above can be overcome by using an active excitation scheme with electromagnetically induced transparency (EIT) [3]. The basic principle of EIT is the use of the quantum interference effect induced by a control laser field to significantly eliminate the absorption of a probe laser field in a resonant atomic system. By using EIT, one can also realize ultraslow group velocity and giant enhancement of Kerr nonlinearity. Based on these intriguing properties, ultraslow LBs have been recently predicted in highly resonant atomic systems via EIT [4].
Particles with nonzero magnetic moments will deflect along different trajectories when passing through a gradient magnetic field. Such phenomenon, called Stern-Gerlach (SG) effect, was first discovered in the early period of quantum mechanics. As one of canonical experiments in modern physics [5], SG effect is not only important for illustrating the basic concepts of quantization, spin, quantum entanglement and measurement [6,7], but also becomes a powerful experimental techniques in the study of molecular radicals [8][9][10], metal clusters [11][12][13][14], and nanoparticles [15], etc.
In a remarkable experiment carried out by Karpa and Weitz [16], a SG deflection of a probe laser beam was observed in a Λ-type three level atomic system via EIT. But the deflection obtained in this experiment can not be simply explained as a standard SG effect since only one "spin" component is involved. In addition, diffraction and dispersion inherent in the resonant atomic system also bring a noticeable distortion of the deflected probe beam.
In a recent work [17], a double EIT scheme with M-type level configuration was proposed to demonstrate a SG effect of vector optical solitons, which has two polarization components (i.e. a quasispin) and allows a stable propagation of probe pulses.
However, the result in Ref. [17] can not be analogous to general case of SG effect in atomic physics, where space quantization of magnetic moments may result in three-and even multicomponent deflection of atomic trajectories if angular-momentum quantum number of the atoms J = 1/2 [18]. Such "SG deflection spectrum" has been widely observed in experiments and nowadays taken to study many physical properties such as magnetic moments and spin relaxation, etc [8,9,[11][12][13][14][15].
In this article, we propose a scheme to exhibit a SG effect of n-component (n > 2) ultraslow LBs in a coherent atomic system with (n + 1)-pod level configuration via EIT.
Based on Maxwell-Bloch (MB) equations, we derive coupled (3+1)-dimensional nonlinear Schrödinger (NLS) equations, which govern the spatial-temporal evolution of n probe-field envelopes. We show that under EIT condition a significant deflection of the n components of the ultraslow LBs can be achieved by using a SG gradient magnetic field. The stability of the ultraslow LBs can be realized by an optical lattice potential contributed from a fardetuned laser field. The results presented here may have potential applications in the study of optical magnetometery, light and quantum information processing, and so on.
The rest of the article is arranged as follows. The next section describes our model and derives the Maxwell-Bloch equations. In Sec. III, we derive nonlinear envelope equations of three (i.e. n = 3) probe pulses using a method of multiple scales, and obtain ultraslow LB solutions. In Sec. IV, the SG effect of the ultraslow LBs is studied. In Sec. V, we investigate SG effect of n component ultraslow LBs. The last section contains a summary of our main results.

II. THEORETICAL MODEL
We consider a resonant atomic system with (n + 2) levels interacting with (n + 1) laser fields. The excitation scheme constitutes a (n + 1)-pod configuration, where Ω pj is the half Rabi frequency of the jth weak, pulsed probe field E pj = e pj E pj (r, t)e i(k pj ·r−ω pj t) + c.c., Ω c is the half Rabi frequency of a strong continuous-wave control field E c = e c E c e i(kc·r−ωct) + c.c., with e pj and e c (E pj and E c ) are respectively the unit polarization vectors (envelope functions) of the jth probe field and the control field, and ∆ j is the detuning of the jth level; see Fig. 1(a). We assume that initially the atomic population is prepared in the ground states |1 , |2 ,. . .,|n , and cooled to a ultracold temperature in order to eliminate Doppler broadening and collisions. Fig. 1(b) is a possible arrangement of experimental apparatus. The aim of the co-propagating configuration of probe and control fields is also to avoid Doppler shifts.
We further assume a static SG gradient magnetic field is applied to the medium with B 1 ≪ B 0 . Here B 0 contributes to a Zeeman level shift ∆E j,Zeeman = µ B g j F m j F B 0 for level E j ; µ B , g j F and m j F are Bohr magneton, gyromagnetic factor, and magnetic quantum number of the level |j , respectively; B 1 is the transverse gradient of the SG magnetic field, which will lead to SG deflection of the probe fields.
Additionally, we assume a small, far-detuned optical lattice field is also applied into the system, where E 0 , R ⊥ , and ω L are field amplitude, beam radius, and angular frequency, respectively [19]. Because of the existence of E L (x, t), Stark level Here α j is the scalar polarizability of the level |j , Q(x, t) t represents the time average in an oscillation cycle for the quantity Q(x, t), and therefore we have E L (x) = (E 0 / √ 2) cos(x/R ⊥ ). The aim of introducing the far-detuned optical field is to stabilize the LBs [17,20].
Under electric-dipole and rotation wave approximations, the Hamiltonian of the atomic system in interaction picture is given bŷ Here Ω pj = (e pj · p j,n+2 )E pj / and Ω c = (e c · p n+1,n+2 )E c / are Rabi frequencies of the jth probe and control fields, respectively, with p jl being the electric-dipole matrix element related to the states |j and |l . The control field is so strong that it can be considered to be undepleted during the propagation of probe fields.
The equation of motion for the density matrix elements in interaction picture reads [21] ∂σ ∂t where Γ(σ) is a relaxation matrix denoting spontaneous emission and dephasing. The explicit form of Eq. (4) for n = 3 is given in Appendix A.

A. Nonlinear envelope equations
One of our main aim is to get a stable propagation of all probe fields. On the one hand, the probe fields may suffer serious distortion due to the dispersion and diffraction of the system. On the other hand, EIT may result in a giant enhancement of Kerr effect.
It is natural to use the enhanced Kerr effect to balance the dispersion and diffraction for obtaining solitonlike pulses that are shape-preserved during propagation.
To this end, we use the method of multiple scales [22] to derive nonlinear envelop equations of the probe fields. For simplicity, we consider the case of n = 3 (The case for general n will be taken into account in Sec. V). Taking the asymptotic expansion jj is the population distribution prepared in the state |j initially, which is assumed as 1/3 (j = 1, 2, 3) for simplicity; ǫ is a dimensionless small parameter characterizing the typical amplitude of the probe fields. All quantities on the right hand side of the expansions are considered as functions of the multi-scale variables x 1 = ǫx, y 1 = ǫy, z α = ǫ α z and t α = ǫ α t (α = 0, 2). The SG gradient magnetic field and the far-detuned optical lattice field are assumed to be B SG (y 1 ) = B 0 + ǫ 2 B 1 y 1 and 5j ), θ j = K j (ω)z 0 − ωt 0 , with F j being envelope functions depending on slow variables x 1 , y 1 , z 2 and t 2 and K j (ω) being the linear respectively. The parameters used for plotting the figure have been given in the text. dispersion relations given by In most cases K j (ω) can be Taylor expanded around ω pj (which corresponds to ω = 0) [23], i.e. K j (ω) = K j0 + K j1 ω + 1 2 K j2 ω 2 + · · · , with K jl = [∂ l K j (ω)/∂ω l ]| ω=0 (l = 0, 1, 2, · · · ). The coefficients K jl have rather clear physical interpretation, i.e. K j0 = Re(K j0 ) + iIm(K j0 ) gives the phase shift per unit length and absorption coefficient; K j1 determines the group velocity given by j represent the group velocity dispersion which resulting in spreading and attenuation of the probe pulses.
Note that the (blue) solid curve in both panels is a overlapping of three curves, which can not be resolved since they nearly coincide each other due to the symmetry of the system.
The advantage of such symmetry is very useful for obtaining matching group velocities of different probe fields (i.e. Re(K 11 ) ≈ Re(K 21 ) ≈ Re(K 31 ); Fig. 2(b)), which is very essential to obtain significant SG deflection of the probe fields.
At α = 3 order, the giant enhancement of Kerr effect of the system generated by EIT plays an important role. We obtain the coupled NLS equations governing the evolution of (j = 1, 2, 3). Hereā l = ǫ −2 Im(K l0 ); W jl result from the Kerr nonlinearity, which contribute to self-phase modulation (when j = l) and cross-phase modulation (when j = l), with explicit expressions given in Appendix C; M j and N j have the form characterizing the contributions of the SG gradient magnetic field and far-detunned optical lattice field, respectively.

B. Light bullet solutions
For convenience, we introduce the dimensionless variables s = z/L Diff , τ = t/τ 0 , v gj = V gj τ 0 /L Diff , and u j = (Ω pj /U 0 ) exp[−iRe(K j0 )z]. Here L Diff = ω p R 2 ⊥ /c, τ 0 , and U 0 are typical diffraction length, probe-field pulse duration, and half Rabi frequency, respectively. Then Eq. (8) can be written into the dimensionless form where g jl = W jl /|W 12 | characterize nonlinear effect; and A j = Im(K j0 )L Diff (j =1, 2, 3) are absorption coefficients. The potential functions in Eq. (10) read with M j = L Diff M j R ⊥ B 1 and N j = L Diff N j E 2 0 (j = 1, 2, 3). Note that in the derivation of Eq. (10) we have assumed τ 0 is large so that group-velocity dispersion term (i.e., the term proportional to ∂ 2 u j /∂τ 2 ) can be neglected, which can be easily realized experimentally.
Furthermore, the absorption can also be negligible by choosing suitable system parameters under the condition of EIT. In fact, when we take τ 0 = 9.0×10 −7 s, U 0 = 3.38×10 6 s −1 , ∆ 2 = 1.2×10 4 s −1 , ∆ 3 = 1.5×10 4 s −1 , ∆ 4 = 2.0×10 5 s −1 , ∆ 5 = 9.0×10 6 s −1 , Ω c = 2.0×10 7 s −1 , and R ⊥ = 36 µm with other parameters the same as in Fig. 2, we obtain the typical diffraction length L Diff = 1.0 cm, which is approximately equal to the typical nonlinearity length However, the typical linear absorption length L A j = 1/Im(K j0 ) is around 924.0 cm and typical second-order dispersion length L Disp j = τ 2 0 /Re(K j2 ) (j = 1, 2, 3) is 20.5 cm, both of them are much larger than L Diff and L N . Base on the results, we thus have the ratio coefficients d j = L Diff /L Dispj ≈ 0.0495, and A j = L Diff /L A j ≈ 0.0011, which describe the significance of the various characteristic interaction lengths relative to the diffraction effect. Therefore we can safely neglect the corresponding terms because they are much smaller than 1.
With the above parameters we obtain group velocities of three probe field envelopes (i.e. We see that three group velocities of probe field envelopes are very small comparing with c (c is light speed in vacuum), and nearly matched each other. The ultraslow and matched group velocities are essential to obtain significant SG deflection of the probe fields, as will be shown below.
We now seek approximated analytical solutions of the Eq. (10) with the form [25] u j (ρ j , τ, ξ, η) = f j (ρ j )v j (τ, ξ, η), where f j (ρ j ) are normalized Gaussian functions, that is, Integrating out the variable ρ j , Eq. (10) becomes To obtain the LB solutions of Eq. (15), we consider several reasonable approximations: (i)In the presence of the SG gradient magnetic field, the three probe-field envelopes will separate each other after propagating some distance. In such situation, the interaction between different envelopes becomes weak and hence the cross-phase-modulation terms can be neglected; (ii)The potential wells of the optical lattice are assumed to be deep enough, so that the probe-field envelopes are almost trapped in the wells in x direction. Hence V j (ξ, η) given in Eq. (11) can be approximated by M j η + N j − N j ξ 2 . Thus Eq. (15) can be approximated as Assuming v j (τ, ξ, η) = w j (τ, η)φ j (ξ) exp(iN j v gj τ ), where φ j (ξ) is a normalized ground state solution satisfying the eigenvalue problem (∂ 2 /∂ξ 2 − 2N j ξ 2 )φ j = 2E ξ φ j with E ξ = − N j /2, and integrating over the variable ξ, one obtains Eq. (17) is a (1+1)-dimensional NLS equation with a linear potential, which admits the exact single-soliton solutions [26] w j = A j e iϕ j sechΘ j , where A j = (2 5/4 N 1/4 j πρ 0 /|g jj |) 1/2 , ϕ j = M j v gj τ (η − M j v 2 gj τ 2 /6), and Θ j = (2N j ) 1/4 (η − M j v 2 gj τ 2 /2). Finally, we obtain the solution of Eq. (15) which is a nonlinear solution localized in three space and one time dimensions, i.e. the (3+1)-dimensional LB solution of the system.

BULLETS
When returning to original variables, the LB solution (19) has the form We see that the LB travels in z direction with ultraslow group velocityṼ gj . In addition, it has an acceleration M j R ⊥Ṽ 2 gj /L 2 Diff in y direction, which results in the SG deflection. Note that M j is proportional to the parameter B 1 , i.e. the deflection comes from the SG gradient magnetic field given by Eq. (1). Fig. 3 is the result of the SG deflection spectrum of the 3-component LB by Through the information of Fig. 3, we can see that an obvious deflection track of LBs occurs due to the SG gradient magnetic field existing. The phenomenon is similar to the SG deflection for atoms.

Shown in
We now determine the SG deflection angles of each LB components. From the solution (20) we get the propagating velocity of the jth LB component at time t Assume the medium length in the z direction is L. The running time in z direction is thus L/Ṽ gj . At the exit of the medium the velocity of the jth LB component will be As a result, we have the deflection angle after passing through the medium where r = R ⊥ /L Diff , p j = k pj is photon momentum, and µ sol j = L Diff M jṼgj k pj is effective magnetic moment [27]. With the data given in Fig. 3 From the formula (22) we see that the deflection angle θ j is inversely proportional toṼ gj . So than that for linear polariton obtained in [16].
The SG effect of the LBs demonstrated above may show many intriguing applications.
For instance, through measuring the deflection angles of LBs components, one can obtain the gradient magnetic field B 1 . Fig. 4(b) shows the deflection angles as functions of B 1 for L = 4L Diff . The results of θ 1 , θ 2 and θ 3 are respectively labeled by solid, dashed and dashed-dotted lines. We see that the larger the magnetic field gradient, the larger the SG deflection angles. We expect that the significant SG deflection obtained here may have potential applications in optical magnetometery and quantum information processing, etc.

V. SG EFFECT OF n-COMPONENT ULTRASLOW LIGHT BULLETS
We now investigate the SG deflection of multi-component ultraslow LBs in a (n + 2)-level system via EIT. For the (n + 1)-pod level configuration (n > 3) shown in Fig. 1(a), the theoretical approach is a direct generation of that developed in last two sections. Using the weak nonlinear perturbation theory [22], we can obtain the following coupled NLS equations (j = 1, 2, . . . , n). Here F j is the envelope of the jth probe field; W jl are coefficients of self-phase (for j = l) and cross-phase (for j = l) modulations; M j (N j ) characterize the amplitude of the SG gradient magnetic field (the far-detuned optical lattice field). Explicit expressions of these coefficients are lengthy and omitted here.
A similar approach as in the last two sections can also be done.  (16), and LB solution with the same form of (20).
In this case, we also numerically simulate Eq.  potential contributed from a far-detuned laser field. We expect that the results predicted here may have potential applications in the research fields of optical magnetometery, light information processing, and so on.  The Kerr coefficients W jl in Eq. (8) are given by , (j = 1, 2 , 3) (C1a) where the superscript of (0) in d