1State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China2NYU-ECNU Joint Institute of Physics at NYU-Shanghai, Shanghai 200062, China
3Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
(Received 5 November 2018; published 16 April 2019)
We propose a scheme to realize slow-light soliton beam splitters by using a tripod-type four-level atomicsystem. We show that optical solitons, which have ultraslow propagation velocity and ultralow generation power,can be generated in the system via electromagnetically induced transparency and can be stored and retrieved withhigh efficiency and fidelity. In particular, a nonlinear beam splitter that splits one optical soliton into two or moreones can be obtained by switching on and off of two or more control laser fields subsequently. The resultsreported here open a route not only for active manipulation of nonlinear optical pulses in multistate quantumsystems but also for promising applications in optical information processing and transmission.
DOI: 10.1103/PhysRevA.99.043821
I. INTRODUCTION
Optical beam splitters are basic devices that can split anincident optical beam or pulse into two or more beams orpulses in space and/or time domains, which may or maynot have the same light power. Optical beam splitters playsignificant roles in modern physics; especially, they are cru-cial parts of most interferometers, widely exploited in boththought and real-world physical experiments in the areas ofquantum theory, relativity theory, and many other fields ofphysics and engineering [1,2].
In recent years, much attention has been paid to slow-lightbeam splitters and their extension [3–24] via electromagnet-ically induced transparency (EIT) [25] in various multilevelatomic systems that resonantly interact with three or morelaser fields. In addition to the interest for fundamental re-search, the study of slow-light beam splitters has potential ap-plications in both classical and quantum information process-ing and communication networks. For instance, they can beused to design dynamically reconfigurable all-optical routersand to produce highly efficient entanglement for quantumrepeaters [3–27]. However, all works on EIT-based slow-lightbeam splitters reported up to now are only for linear opticalpulses, which may generally experience serious deformationsince these optical pulses interact with atoms in a resonantway and hence significant dispersion is generally unavoid-able. Thus it is desirable to seek techniques to balance thedispersion and hence to obtain optical pulses that have betterperformance for the design of slow-light beam splitters. Onemethod of such techniques is to make the system work in weaknonlinear optics regimes.
In this paper, we present a proposal for realizing slow-lightsoliton beam splitters by exploiting a tripod-type four-levelatomic system coupled with two control laser fields and aweak nonlinear probe laser field. We show that optical solitonswith ultraslow propagating velocity and ultraslow light powercan be created via EIT and they can be stored and retrievedwith high efficiency and fidelity based on an EIT-memory
scheme through two different optical excitation channels. Wealso show that a nonlinear beam splitter that makes one opticalsoliton split into two ones can be obtained by switching on andoff of the two control laser fields separately. In addition, thescheme can also be generalized to cases where one slow-lightsoliton can be split into three or more ones via a multichanneloptical excitation process.
Before proceeding, we note that some studies have beencarried out on soliton-related splitters in systems of photo-voltaic materials [28–30], nonresonant Kerr media [31,32],liquid crystals [33,34], and Bose-Einstein condensates [35].However, our paper is completely different from these studies.First, due to the resonant character of our system, the signif-icant energy transfer via different quantum excitation chan-nels plays a crucial role for the realization of the slow-lightsoliton splitters, which was absent in the systems consideredin Refs. [28–35]. Second, the system suggested in our papercan be actively manipulated, and is easy to extend to caseswith more optical excitation channels. Third, the slow-lightsoliton splitters obtained in our system can work at extremelylow and even single-photon level, which is not viable inthe systems used in Refs. [28–35]. The results reported inour paper open a route not only for active manipulation ofnonlinear optical pulses but also for promising applications inoptical and quantum information processing and transmission.
II. MODEL AND SLOW-LIGHT SOLITONS
A. Model
The system under consideration is a lifetime-broadenedfour-state atomic gas with a tripod-type level configuration,interacting resonantly with a weak pulsed probe laser field(with center wave number kp = ωp/c, center angular fre-quency ωp, and time duration τ0 at the entrance of themedium) and two strong continuous-wave control laser fields(with wave numbers kc1 = ωc1/c and kc2 = ωc2/c, and angu-lar frequencies ωc1 and ωc2, respectively). The probe fielddrives the transition |1〉 ↔ |0〉, and the two control fields
CHONG SHOU AND GUOXIANG HUANG PHYSICAL REVIEW A 99, 043821 (2019)
FIG. 1. (a) Atomic gas (brown dots) couples with a pulsed probe laser field (with center angular frequency ωp and half Rabi frequency�p) and two continuous-wave control laser fields [with angular frequencies ωc1 (half Rabi frequency �c1) and ωc2 (half Rabi frequency �c2),respectively]. Inset: Energy-level diagram and excitation scheme of the atomic gas with a tripod-type level configuration. �10 (�20, �30) is thedecay rate from |0〉 to |1〉 (|0〉 to |2〉, |0〉 to |3〉); �0 (�2 and �3) is one-photon (two-photon) detuning. (b) The imaginary part Im(K ) (K is thelinear dispersion relation of the system) as a function of ω, for �2 �= �3 (solid blue line, which displays two transparency windows), and for�2 = �3 (dashed red line, which displays only one transparency window). The lower half part of the panel is the real part Re(K ), for �2 �= �3
(solid blue line), and for �2 = �3 (dashed red line).
drive, respectively, the transitions |2〉 ↔ |0〉 and |3〉 ↔ |0〉[see Fig. 1(a)]. The total electric field in the system can be ex-pressed as E = Ep + Ec1 + Ec2 = ∑
l=p,c1,c2 elElexp[i(klz −ωl t )] + c.c., where el (El ) is the unit polarization vector(envelope) of the electric field El . For simplicity, we haveassumed that all laser fields propagate along the z direction.
The Hamiltonian of the system in the interaction picturereads
Hint = −h
⎛⎝ 3∑
j=0
� j | j〉〈 j| + �p|0〉〈1| + �c1|0〉〈2|
+�c2|0〉〈3| + H.c.
⎞⎠, (1)
where �1 = 0, �0 = ωp − (E0 − E1)/h (one-photondetuning), �2 = ωp − ωc1 − (E2 − E1)/h, and �3 =ωp − ωc2 − (E3 − E1)/h (two-photon detunings); Ej ( j =0, 1, 2, 3) is the energy eigenvalue of the atomic state| j〉; �p = (ep · p10)Ep/h, �c1 = (ec1 · p20)Ec1/h, and�c2 = (ec2 · p30)Ec2/h are, respectively, the half Rabifrequencies of the probe and the two control fields; pi j
is the electric dipole matrix element associated with levels|i〉 and | j〉, with ep (Ep), ec1 (Ec1), and ec2 (Ec2), respectively,the polarization unit vectors (envelopes) of the probe and twocontrol fields.
The atomic dynamics is described by a 4 × 4 densitymatrix σ , obeying the optical Bloch equation
∂σ
∂t= − i
h[Hint, σ ] − �[σ ], (2)
where � is a relaxation matrix characterizing the spontaneousemission and dephasing in the system [36]. The explicit formof Eq. (2) is presented in Appendix A.
The evolution of the probe field Ep is governedby the Maxwell equation ∇2Ep − (1/c2)∂2Ep/∂t2 =(1/ε0c2)∂2Pp/∂t2, where Pp = Na{p10σ01 exp[i(kpz −ωpt )] + c.c.} is the electric polarization intensity inducedby the probe field, with Na the atomic density. Under slowlyvarying envelope approximation, the Maxwell equation isreduced to
i
(∂
∂z+ 1
c
∂
∂t
)�p + κ10σ01 = 0, (3)
where κ10 = Naωp|p10|2/(2ε0ch) is the coupling coefficientdescribing the interaction between the light field and atoms.
Notice that in the derivation of the above Maxwell-Bloch(MB) Eqs. (2) and (3), we have made the following assump-tions.
(i) The probe pulse has a large transverse size so thatthe diffraction effect [i.e., the term proportional to (∂2/∂x2 +∂2/∂y2)�p] can be neglected.
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(ii) Compared with the probe field, the two control fieldsare strong, so that their amplitudes (Ec1 and Ec2) can beundepleted during the evolution of the probe field. However,when considering the storage and retrieval of the probe field(see the next section), Ec1 and Ec2 (hence �c1 and �c2) will beassumed to be varied adiabatically in time.
(iii) The atomic gas is cold and dilute, thereby the Dopplereffect is negligible and the interaction between atoms can bedescribed by the dephasing parameter γ
depjl (see Appendix A).
Note that the model mentioned above may be re-alized in realistic experiments. One example is an ul-tracold 87Rb atomic gas [37], with the energy lev-els selected to be |1〉 = |5 2S1/2, F = 1, mF = −1〉, |2〉 =|5 2S1/2, F = 2, mF = −1〉, |3〉 = |5 2S1/2, F = 2, mF = 1〉,and |0〉 = |5 2P1/2, F = 1, mF = 0〉, which gives |p10| �|p20| � |p30| = 2.54 × 10−27 C cm. If the atomic density ischosen as Na = 1.1 × 1012 cm−3, κ10 takes the value of3.0 × 1010 cm−1 s−1. This set of parameters will be used inthe analysis and calculation given below.
B. Slow-light solitons
We start to consider the propagation of a linear probepulse in the system. We assume that the two control fields areapplied first and hence all the atoms are prepared in the groundstate |1〉. When a weak probe pulse is present, the systemundergoes a linear evolution. In this case, the MB Eqs. (2) and(3) admit the solution for �p with the form F exp[i(Kz − ωt )][38], where F is a constant and K is the linear dispersionrelation
K (ω) = ω
c− κ01(ω + d21)(ω + d31)
D, (4)
with D = (ω + d01)(ω + d21)(ω + d31) − |�c1|2(ω + d31) −|�c2|2(ω + d21). Shown in Fig. 1(b) is the imaginary part[i.e., Im(K )] and the real part [i.e., Re(K )] of K as func-tions of ω. The solid blue line and dashed red line in thefigure are for (�2, �3) = (−2π × 15.9, −2π × 47.7) and(−2π × 0.16, −2π × 0.16) MHz, respectively. The other pa-rameters are �0 = −2π × 15.9 MHz, γ01 = 2π × 2.8 MHz,γ21 = 2π × 8 Hz, and γ31 = 2π × 15.9 Hz. We see that when�2 �= �3 the system displays a double EIT with two trans-parency windows; however, if �2 = �3, the double EIT isdegenerated into a single EIT with only one transparencywindow. The reason for the occurrence of such degeneration isdue to the symmetry of the tripod level configuration for �2 =�3. With such symmetry, the system is largely simplified,which will be considered in the discussion below.
EIT has been widely used for nearly free propagation oflinear optical pulses in multistate quantum systems [25–27].Yet, linear optical pulses in EIT-based systems generallyexperience serious deformation. The physical reason for thedeformation is that in such systems the optical pulses inter-act with atoms (or other quantum emitters) resonantly andhence a significant dispersion is unavoidable for long-distancepropagation. Thus it is necessary to find a way to suppressthe dispersion and hence to obtain optical pulses that arerobust during propagation. It has been shown in recent yearsthat stable optical pulses are indeed possible if EIT-basedsystems work in a weak nonlinear region [39–41]. We now
demonstrate that stable slow-light solitons are also possible inthe present tripod system.
A weak nonlinear probe pulse in the system still has theform �p = F exp[i(Kz − ωt )], but F is now an envelopefunction modulated slowly in both time and space. By exploit-ing a perturbation expansion with multiple scales developedin Ref. [40], the nonlinear envelope equation describing theevolution of F can be derived based on the MB Eqs. (2) and(3), which reads
i∂
∂zF − 1
2K2
∂2
∂τ 2F + W |F |2Fe−2αz = 0, (5)
where τ = t − z/Vg [Vg ≡ (∂K/∂ω)−1 is the group velocityof the probe pulse], α = Im(K ) is an absorption coefficient,K2 = ∂2K/∂ω2 is a coefficient describing group-velocity dis-persion, and W is a coefficient describing self-phase modu-lation. The detailed derivation of the nonlinear SchrödingerEq. (5) and the explicit expressions of W and solutions upto the third order of the perturbation expansion have beenpresented in Appendix B.
If α is small, and the imaginary parts of K2 and W aresmaller compared with their corresponding real parts, Eq. (5)has the following soliton solution:
�p = 1
τ0
√|K2|W
sech
[1
τ0
(t − z
Vg
)]exp
[i
(K0 − 1
2LD
)z
],
(6)
for sgn(K2) = −1. Here K0 ≡ K|ω=0; the tilde symbols overK0, K2, W , and Vg represent their corresponding real parts;LD ≡ τ 2
0 /|K2| is the typical dispersion length of the system.To check that (6) is indeed a physical solution of the
system, we take the realistic system parameters �c1 = 2π ×27.1 MHz, �c2 = 2π × 27.1 MHz, �0 = −2π × 15.9 MHz,�2 = �3 = −2π × 1.9 MHz, τ0 = 5.0 × 10−8 s, γ21 =2π × 8.0 Hz, γ31 = 2π × 15.9 Hz, γ01 = 2π × 2.5 MHz,ωp = 2.37 × 1015 Hz. Then we obtain α = 0.02, K2 =(−2.4 + 0.3i) × 10−15 cm−1 s2, and W = (3.5 − 0.02i) ×10−16 cm−1 s2. We see that α and the imaginary parts of K2
and W are indeed small. The physical reason is that, underthe EIT condition, the absorption of the system is largelysuppressed by the EIT effect induced by the control fields,which guarantees the validity of the soliton solution given by(6). For more detailed discussion on the slow-light solitonand its stable propagation, see Appendix B.
Based on the parameters given above, we obtain the prop-agation velocity of the soliton:
Vg � 6.14 × 10−5c, (7)
which is much smaller than the light speed c in vacuum,thereby (6) is a slow-light soliton. The maximum averagepower density Pmax of the slow-light soliton can be estimatedby using Poynting’s vector [40]. For the transverse radius ofthe probe pulse R = 300 μm, we obtain
Pmax � 7.06 × 10−9 W, (8)
which is very low. From results (7) and (8), we see thatthe optical soliton obtained in the present tripod system hasultraslow propagation velocity and ultralow generation power,
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CHONG SHOU AND GUOXIANG HUANG PHYSICAL REVIEW A 99, 043821 (2019)
FIG. 2. Storage and retrieval of the slow-light pulses in the tripod system, for �2 = �3 = −2π × 1.9 MHz and �c1 = �c2. (a) Evolutionof |�pτ0| in the soliton region as a function of t/τ0 and z, with �p(0, t )τ0 = 3.3 sech(t/τ0). Lines 1, 2, and 3 are, respectively, soliton profiles atz = 0, 0.6, and 1.2 cm (before storage); lines 4, 5, and 6 are, respectively, soliton profiles at z = 1.8, 2.4, and 3.0 cm (after the storage). The solidblack line shows the switching off and on of the two control fields (they are overlapped completely) simultaneously. (b) Atomic coherences|σ21|, |σ31| (|σ21| ≈ |σ31|) in the soliton region [corresponding to (a)] before, during, and after the soliton storage. (c) |�pτ0| in the dispersion-dominant region, with �p(0, t )τ0 = 0.05 sech(t/τ0 ). (d) |�pτ0| in the nonlinearity-dominant region, with �p(0, t )τ0 = 5.0 sech(t/τ0 ).
which is very different from that obtained by using conven-tional optical media (e.g., fibers).
III. SLOW-LIGHT SOLITON BEAM SPLITTERS
EIT is one of the important methods for the storage andretrieval of optical pulses [25–27,42,43]. Recently, it has beenshown that the EIT-based linear light memory can be extendedto weak nonlinear optics regimes [44–46]. In the following,we show that the slow-light solitons of the tripod systemobtained above can be used to build slow-light soliton beamsplitters through the manipulation of the control fields in thesystem.
A. Storage and retrieval of slow-light solitons
Before the presentation on how to realize slow-light solitonbeam splitters, we give a simple description for the storageand retrieval of optical solitons in the tripod system for thesimple case where the two control fields are switched on andoff simultaneously. The simultaneous switching on and offof the two control fields can be described by the followingswitching function:
�c j = �c j0
[1 − 1
2tanh
(t − T j
off
Ts
)+ 1
2tanh
(t − T j
on
Ts
)],
(9)
where �c j0 ( j = 1, 2) is constant, Ts is the switching time,and T j
off (T jon) is the time when switching off (on) �c j .
We take �c10 = �c20 = 2π × 27.1 MHz, T 1off/τ0 = T 2
off/τ0 =19.0, T 1
on/τ0 = T 2on/τ0 = 30.0, and Ts/τ0 = 0.2 (with τ0 =
5.0 × 10−8 s−1), and exploit Runge-Kutta method to solveEqs. (2) and (3).
Shown in Fig. 2(a) is the result of a numerical simulationon the storage and retrieval of the slow-light soliton for �2 =�3 = −2π × 1.9 MHz, by taking |�pτ0| as a function of t/τ0
and z. Lines 1, 2, and 3 are, respectively, soliton profiles atthe positions z = 0, 0.6, and 1.2 cm (before storage); lines 4,5, and 6 are, respectively, soliton profiles at positions 1.8, 2.4,and 3.0 cm (after the storage). The solid black line in the figureshows the simultaneous switching off and on of the two con-trol fields (i.e., �c1 = �c2). The initial condition used in thesimulation is �p(0, t )τ0 = 3.3 sech(t/τ0); system parametersare �0 = −2π × 15.9 MHz, γ21 = 2π × 8.0 Hz, γ31 = 2π ×15.9 Hz, γ01 = 2π × 2.5 MHz. From the figure, we see thatthe soliton is very stable before the storage, and the retrievedsoliton (after the storage) is also quite stable and has nearlythe same wave shape as that before the storage. The physicalreason of the shape preservation during the soliton generationprocess is due to the balance between the dispersion and theKerr nonlinearity of the system. The approximated expressionof the probe soliton during the storage and retrieval is pre-sented in Appendix C1.
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SLOW-LIGHT SOLITON BEAM SPLITTERS PHYSICAL REVIEW A 99, 043821 (2019)
Figure 2(b) shows the result of atomic coherences |σ21|and |σ31| (|σ21| ≈ |σ31|), as functions of t/τ0 and z during theprocess of the storage and retrieval of the soliton. One seesthat |σ21| and |σ31| are nonzero during the period when bothcontrol fields are switched off. Such result is a manifestationof the information transfer from the probe field to the atomicinternal states during the storage, which can be understoodthrough the stable propagation of the dark-state polaritons inthe system [44–46].
The efficiency of the slow-light soliton memory can becharacterized by [44–46]
η =∫ +∞−∞
∣∣Eoutp (t )
∣∣2dt∫ +∞
−∞∣∣E in
p (t )∣∣2
dt, (10)
where E inp (t ) = E in
p (z, t )|z=0 (the input soliton pulse) andEout
p (t ) = Eoutp (z, t )|z=Lz [the output (i.e., retrieved) soliton
pulse], with Lz (=3.0 cm) the medium length. From Fig. 2(a)we obtain η = 0.92. The soliton memory efficiency will beincreased if its storage time is shortened (or Lz is reduced).
The fidelity of the soliton memory can be described by theparameter ηJ2, where J2 describes the degree of coincidenceof the wave shapes for the input and output solitons, definedby the overlap integral
J2 =∣∣ ∫ +∞
−∞ Eoutp (t )E in
p (t + �T )dt∣∣2
∫ +∞−∞
∣∣Eoutp (t )
∣∣2dt
∫ +∞−∞
∣∣E inp (t + �T )
∣∣2dt
, (11)
where �T is the time interval between the peak of the inputsoliton pulse E in
p and the peak of the output soliton pulse Eoutp .
From Fig. 2(a) we obtain J2 = 0.98 and hence ηJ2 = 0.90.Thus, the memory of the slow-light soliton in the tripodsystem has a nice quality due to the balance between thedispersion and nonlinearity in the system.
Plotted in Fig. 2(c) is the result for the optical pulsememory in a dispersion-dominant (i.e., linear) region, with�p(0, t )τ0 = 0.05 sech(t/τ0). In this case, the dispersionlength of the system LD = 1.05 cm is much smaller than thenonlinearity length LNL = 719.0 cm. From the figure we seethat the optical pulse is broadened significantly before andafter the storage, with the memory efficiency and fidelitygiven by (η, ηJ2) = (0.93, 0.75). Figure 2(d) shows the pulsememory in a nonlinearity-dominant (i.e., strong nonlinear)region, with �p(0, t )τ0 = 5.0 sech(t/τ0). In this situation,the nonlinearity length LNL = 0.36 cm is much smaller thanthe dispersion length LD = 1.05 cm. We see that the opticalpulse has a very large distortion before and after the storage,with (η, ηJ2) = (0.79, 0.60). Consequently, the optical pulsememory in these two regions has lower quality compared withthat of the soliton region.
B. Slow-light soliton beam splitters through two opticalexcitation channels
We now turn to describe how to realize a slow-light solitonbeam splitter in the system. Similar to linear beam splitters[22], the slow-light soliton beam splitter can be realizedthrough an adiabatical switching off and on of the two controllaser fields subsequently. The switching off and on of the two
control fields can be described by the following switchingfunctions:
�c1 = �c10
[1
2− 1
2
2∑i=1
tanh
(t − T 1
off i
Ts
)+ 1
2tanh
(t − T 1
on1
Ts
)],
(12a)
�c2 = �c20
[1 − 1
2tanh
(t − T 2
off1
Ts
)+ 1
2tanh
(t − T 2
on1
Ts
)],
(12b)
where �c j0 ( j = 1, 2) is constant, Ts is the switching time, andT j
off i(T j
oni ) is the ith switching off (switching on) of the controlfield �c j . The timing sequence of the switching off and on of�c j for obtaining a slow-light soliton beam splitter is shownin Fig. 3(a), with T 1
off1= T 2
off1< T 1
on1< T 1
off2< T 2
on1.
Figure 3(b) shows the result of a numerical simulation onhow to obtain a slow-light soliton beam splitter by taking|�pτ0| as a function of t/τ0 and z (with τ0 = 5.0 × 10−8 s).The operation steps can be described as follows.
(1) A weak nonlinear probe pulse is incident at z = 0with the waveform �p(0, t )τ0 = 3.3 sech(t/τ0). A slow-lightsoliton forms via the balance between dispersion and Kerrnonlinearity, and propagates stably to z = 0.6 and 1.2 cm(corresponding times are 5.3τ0 and 11.0τ0), respectively. Thisis the soliton (the leftmost red pulses) before the storage.
(2) Both �c1 and �c2 are switched off at t = T 1off1
= T 2off1
=15τ0; the (input) probe soliton is stored in the two atomiccoherences σ21 and σ31 (which are not shown in the figurefor saving space) simultaneously.
(3) By switching on �c1 at t = T 1on1
= 30τ0 but keeping�c2 switched off, the atomic coherence σ21 is converted backinto the probe pulse, and hence a new soliton (called soliton1) is retrieved. The figure (the middle blue pulses) showsthe propagation of the soliton to z = 2.4, 2.7, and 3.0 cm(corresponding times are t = 44.9τ0, 50.4τ0, and 56.1τ0),respectively.
(4) At t = T 1off2
= 80τ0, �c1 is switched off; after time20τ0, �c2 is switched on at t = T 2
on1= 100τ0. The atomic
coherence σ31 is converted back into the probe pulse, and thusanother new soliton (called soliton 2) is retrieved. The figure(the rightmost yellow pulses) shows the propagation of thesoliton 2 to z = 2.4, 2.7, and 3.0 cm (corresponding times aret = 114.9τ0, 120.4τ0, and 126.0τ0), respectively.
In the simulation, the other system parameters arechosen to be �0 = −2π × 15.9 MHz, �2 = �3 = −2π ×1.9 MHz, γ21 = 2π × 8.0 Hz, γ31 = 2π × 15.9 Hz, γ01 =2π × 2.5 MHz, and �c10 = �c20 = 2π × 28.6 MHz, Ts =0.2τ0. From the figure, we see that the input probe solitoncan indeed be stored and partially retrieved by switching onboth control fields subsequently, behaving like a soliton beamsplitter. The retrieved soliton 1 and soliton 2 are very similar tothe stored one due to the balance between the dispersion andKerr nonlinearity, except for a smaller amplitude and a widertemporal width compared with the input soliton. Note that theamplitude of each retrieved pulse has the same magnitude oforder as that of the stored soliton, thereby two retrieved pulsesare still solitons of the system. The approximated expression
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CHONG SHOU AND GUOXIANG HUANG PHYSICAL REVIEW A 99, 043821 (2019)
FIG. 3. Slow-light soliton beam splitters through two optical excitation channels. (a) Timing sequence of the switching on and off of thecontrol fields �c1 and �c2 for obtaining the slow-light soliton beam splitter, with T j
off i(T j
oni) the ith time of switching off and on of the control
field �c j ( j = 1, 2). (b) Soliton splitting by taking |�pτ0| as a function of t/τ0 (τ0 = 5.0 × 10−8 s) and z. The leftmost three red pulses are forthe probe soliton before the storage when it propagates to z = 0.6 and 1.2 cm (corresponding times are t = 5.3τ0 and 11.0τ0), respectively.The middle three blue pulses are for the retrieved new soliton (soliton 1) when it propagates to z = 2.4, 2.7, and 3.0 cm (corresponding timesare t = 44.9τ0, 50.4τ0, and 56.1τ0), respectively. The rightmost three yellow pulses are for another retrieved new soliton (soliton 2) when itpropagates to z = 2.4, 2.7, and 3.0 cm (corresponding times are t = 114.9τ0, 120.4τ0, and 126.0τ0), respectively. For more details, see thetext. (c) Similar to panel (b) but for the pulse splitting in a dispersion-dominant region. (d) Similar to panel (b) but for the pulse splitting in anonlinearity-dominant region.
of the probe pulse during the soliton splitting is presented inAppendix C2.
For comparison, Fig. 3(c) [Fig. 3(d)] shows the result ofnumerical simulations on the pulse splitting in a dispersion-dominant (nonlinearity-dominant) region, with the inputpulse given by �p(0, t )τ0 = 0.3 sech(t/τ0) [�p(0, t )τ0 =4.5 sech(t/τ0)] and all other parameters the same as thoseused in panel (b) but τ0 = 2.5 × 10−8 s (τ0 = 5.0 × 10−8 s).We see that, though one can get pulse storage and splitting,the retrieved pulses have large deformation compared withthe input one. The reason is that in both the dispersion- andnonlinearity-dominant regions there is no balance between thedispersion and Kerr nonlinearity, and thereby a spreading or
distortion of the pulse is unavoidable in the processes of thestorage and splitting.
We now give a simple explanation for the memory andsplitting of the slow-light soliton described above, based onthe Bloch Eq. (2) (its explicit expression is given in Ap-pendix A). Since for a weak probe pulse d21σ21, d31σ31, �pσ02,and �pσ03 are small, from Eqs. (A2a) and (A2b) one has
σ01 = −i1
|�c1|2 + |�c2|2(
�c1∂σ21
∂t+ �c2
∂σ31
∂t
). (13)
Substituting Eq. (13) into Eq. (A2d) and using σ11 ≈ 1 andσ00 ≈ 0, we obtain
�c1σ21 + �c2σ31 = −�p − 1
|�c1|2 + |�c2|2(
∂
∂t+ id01
)(�c1
∂σ21
∂t+ �c2
∂σ31
∂t
)� −�p. (14)
Equation (14) can be rewritten as the form σ21 +(�c2/�c1)σ31 � −(�p/�c1). Thereby, σ21 and σ31 may ac-quire nonzero values if the rations �c2/�c1 and �p/�c1
remain finite constant values, though during the storage �c1
and �c2 are switched off and �p approaches zero; in theretrieval process, the probe pulse can be recovered when �c1
and �c2 are switched on again. This can also be understoodby the existence of the two dark-state polaritons in the system,which read [6,8].
� = cos θ�p − g√Na sin θ [cos φ exp(iχc1)σ21
+ sin φ exp(iχc2)σ31], (15a)
Z = sin φeiχc1σ21 − cos φeiχc2σ31, (15b)
where g2 = |p01|2ωp/(2ε0h), cos φ = |�c1|/|�c|, sin φ =|�c2|/|�c|, sin θ = g
SLOW-LIGHT SOLITON BEAM SPLITTERS PHYSICAL REVIEW A 99, 043821 (2019)
FIG. 4. Slow-light soliton beam splitter via multiple optical excitation channels. (a) Excitation scheme of the N-pod system. (b) Timingsequence for adiabatically manipulating the three control fields �c1, �c2, and �c3 in a quadripod system, for obtaining a slow-light solitonbeam splitter via three optical excitation channels. (c) Numerical result on the splitting of the slow-light soliton in the quadripod system, bytaking |�pτ0| as a function of t/τ0 and z. (d) When the system works in a dispersion-dominant region, the retrieved (linear) optical pulsedeforms significantly. See the text for more details.
with |�c| =√
|�c1|2 + |�c2|2. Since the nonlinear effectunder consideration is weak, and during the soliton memoryprocess both �c1 and �c2 are changed adiabatically, it canbe shown that the dark-state polariton functions (15a) and(15b) satisfy equations ∂�/∂t + c cos2 θ∂�/∂z ≈ 0 and∂Z/∂t ≈ 0. This means that during the soliton memoryprocess the information of the probe soliton �p is transferredinto the two atomic coherences σ21 and σ31, and vice versa,as shown in Fig. 2(b). This explanation applies also forthe storage and retrieval processes in the soliton splittingillustrated in Fig. 3. For expressions of the probe pulse duringthe soliton splitting, see Appendix C2.
C. Slow-light soliton beam splitters through multiple opticalexcitation channels
The scheme proposed above can be generalized to systemswith more optical excitation channels. Shown in Fig. 4(a) arethe level diagram and excitation scheme of a N-pod system,where one probe field (with half Rabi frequency �p) drivestransition |1〉 to |0〉 and N − 1 control fields (with half Rabifrequencies �c j ) drive transitions | j〉 to |0〉 ( j = 2, · · · , N).� j0 is the decay rate from |0〉 to | j〉 and � j ( j �= 1) is thetwo-photon detuning ( j = 1, · · · , N). To realize a slow-lightsoliton beam splitter in such system, the control fields �c j
( j = 1, · · · , N − 1) must be adiabatically manipulated in asuitable way. For simplicity, here we consider only the caseof a quadripod system, with the timing sequence of the threecontrol fields �c1, �c2, and �c3 shown in Fig. 4(b). Plottedin Fig. 4(c) is the result of a numerical simulation for howto obtain the slow-light soliton beam splitter via three opticalexcitation channels, with operation steps described as follows.
(1) Initially (before the storage), the three control fieldsare switched on simultaneously, so a probe soliton formsin the system. The leftmost three red pulses in Fig. 4(c)are the soliton profile when it propagates to z = 0, 0.6, and1.2 cm (corresponding times are t = 0, 4.5τ0, and 9.1τ0),respectively.
(2) At the time T 1off1
, the three control fields are switchedoff simultaneously. When �c1 is switched on again at the timeslot T 1
on1< t < T 1
off2(but �c2 and �c3 remain switched off),
a new soliton (called soliton 1) is retrieved from the atomiccoherence σ21. Blue pulses in Fig. 4(c) are for the retrievedsoliton 1 when it propagates to z = 2.4, 2.7, and 3.0 cm(corresponding times are t = 41.0τ0, 48.2τ0, and 55.5τ0),respectively.
(3) At the time T 1off2
, we switch off �c1 and hence thesoliton 1 disappears. Then, at the time T 2
on1, �c2 is switched
on again in the time slot T 2on1
< t < T 2off2
(but �c1 and �c3
remain switched off), and another new soliton (called soliton2) is retrieved from the atomic coherence σ31. Yellow pulsesin Fig. 4(c) are for the retrieved soliton 2 when it propagatesto z = 2.4, 2.7, and 3.0 cm (corresponding times are t =111.0τ0, 118.3τ0, and 125.6τ0), respectively.
(4) At the time T 2off2
, we switch off �c2 and hence thesoliton 2 also disappears. Then, at the time T 3
on1, �c3 is
switched on again (but �c1 and �c2 remain switched off),and a third new soliton (called soliton 3) is retrieved fromthe atomic coherence σ41. Purple pulses in the figure are forthe retrieved soliton 3 when it propagates to z = 2.4, 2.7, and3.0 cm (corresponding times are t = 181.1τ0, 188.3τ0, and195.6τ0), respectively.
In the simulation, we use the parameters τ0 = 5.0 ×10−8 s, T 1
off1= 25τ0, T 1
on1= 30τ0, T 1
off2= 80τ0, T 2
on1= 100τ0,
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CHONG SHOU AND GUOXIANG HUANG PHYSICAL REVIEW A 99, 043821 (2019)
T 2off2
= 150τ0, and T 3on1
= 170τ0. The probe pulse at the en-trance of the medium is �p(0, t )τ0 = 3.5 sech(t/τ0). We seethat the slow-light soliton beam splitter can be indeed realizedthrough three optical excitation channels through adiabati-cally manipulating the three control fields. We stress that theretrieved pulses are indeed solitons because there is a balancebetween the dispersion and nonlinearity in the system, therebythese retrieved pulses are very stable during propagation. Forcomparison, panel (d) shows a retrieved linear pulse withτ0 = 2.5 × 10−8 s (which has been stored in the system firstby switching off the three control fields simultaneously) when�c1 is switched on again. Here the probe pulse at the entranceof the medium is �p(0, t )τ0 = 0.1 sech(t/τ0). One sees thatthe retrieved pulse (green profiles) is deformed significantlyduring propagation since in this case the system works in adispersion-dominant region.
IV. SUMMARY
Note that a retrieval of multiple optical pulses stored in athree-level �-type atomic system was considered in an inter-esting work by Reim et al. [48]. We stress that our work is dif-ferent from that reported in Ref. [48]. First, the storage and re-trieval of the optical pulses in our work are based on a schemeof EIT memory, while Ref. [48] is based on a Raman memory.Second, what we have investigated here is for the storage andretrieval of nonlinear (soliton) optical pulses, while Ref. [48]is for those of linear optical pulses. Third, the slow-lightsoliton beam splitter is hard to realize by using the three-level�-type atomic system with the control field consisting ofseveral narrow subpulses in the retrieval period, proposed inRef. [48]. To demonstrate this, based on a three-level �-typememory scheme we have carried out a numerical simulationon the single-pulse storage and multiple-pulse retrieval byusing the retrieval technique described in Ref. [48]. We foundthat, though a beam splitting for the input narrow probe pulsemay be obtained and the retrieved probe subpulses can beshaped by engineering the control field, a slow-light solitonbeam splitter is not possible to realize with such scheme. Thephysical reason is that with such scheme the retrieved probesubpulses are unstable during propagation due to the loss andthe unbalanced nonlinearity and (large) dispersion that cannotbe avoided in the multiple-pulse retrieval process. In contrast,in our four-level tripod-type scheme with multiple excitationchannels illustrated above, during the retrieval process the lossis small and the dispersion (also small) is well balanced bythe weak nonlinearity of the system. Thus the optical solitonsobtained in our scheme are quite stable during the processes ofthe propagation, storage, and retrieval, which makes the highperformance realization of the soliton beam splitters possible.
In conclusion, in this paper we have proposed a schemefor realizing slow-light soliton beam splitters with a tripod-type four-state atomic system. By using the method of mul-tiple scales, we have derived the nonlinear envelope equationgoverning the evolution of the probe pulse. We have shownthat optical solitons with ultraslow propagation velocity andultralow generation power can be produced in the system viaEIT. We have also shown that these slow-light solitons can bestored and retrieved with high efficiency and fidelity throughtwo different photonic channels. Based on this, a slow-light
soliton beam splitter that splits one optical soliton into twoones can be implemented through the switching off and onof the two control laser fields subsequently. Furthermore, thepossibility of the soliton beam splitter that splits one opticalsoliton into three or more ones via multiphotonic channels hasalso been illustrated.
The results reported here can be generalized to high-dimensional nonlinear optical pulses with orbital angularmomenta, surface polaritons in metal-metamaterial interfaces,excitons in semiconductor quantum wells and dots, etc. Henceour paper opens a route not only for active manipulation ofnonlinear optical pulses in multistate quantum systems butalso for potential applications in optical information process-ing and transmission.
ACKNOWLEDGMENTS
This work was supported by the National Natural ScienceFoundation of China under Grants No. 11475063 and No.11474099.
APPENDIX A: BLOCH EQUATION FOR THE DENSITYMATRIX
The explicit form of the Bloch Eq. (2) reads [36]
i∂
∂tσ11 − i�10σ00 + �∗
pσ01 − �pσ∗01 = 0, (A1a)
i∂
∂tσ22 − i�20σ00 + �∗
c1σ02 − �c1σ∗02 = 0, (A1b)
i∂
∂tσ33 − i�30σ00 + �∗
c2σ03 − �c2σ∗03 = 0, (A1c)
i∂
∂tσ00 + i�0σ00 − �∗
pσ01 + �pσ∗01 − �∗
c1σ02 + �c1σ∗02
−�∗c2σ03 + �c2σ
∗03 = 0, (A1d)
for diagonal matrix elements, and(i∂
∂t+ d21
)σ21 − �pσ
∗02 + �∗
c1σ01 = 0, (A2a)
(i∂
∂t+ d31
)σ31 − �pσ
∗03 + �∗
c2σ01 = 0, (A2b)
(i∂
∂t+ d32
)σ32 − �c1σ
∗03 + �∗
c2σ02 = 0, (A2c)
(i∂
∂t+ d01
)σ01 − �p(σ00 − σ11) + �c1σ21 + �c2σ31 = 0,
(A2d)(i∂
∂t+ d02
)σ02 − �c1(σ00 − σ22) + �pσ
∗21 + �c2σ32 = 0,
(A2e)(i∂
∂t+ d03
)σ03 − �c2(σ00 − σ33) + �pσ
∗31 + �c1σ
∗32 = 0,
(A2f)
for nondiagonal matrix elements, where djl = � j − �l +iγ jl . Here γ jl = (� j + �l )/2 + γ
depjl is the decay rate of the
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SLOW-LIGHT SOLITON BEAM SPLITTERS PHYSICAL REVIEW A 99, 043821 (2019)
atomic coherence σ jl , with � j = ∑El <Ej
�l j the spontaneous-
emission rate of the state | j〉 and γdepjl the dephasing rate
representing the loss of phase coherence between | j〉 and|l〉 contributed by atomic collisions and other incoherentprocesses [36].
APPENDIX B: DERIVATION OF THE NONLINEARENVELOPE EQUATION AND ITS SOLITON SOLUTION
To obtain the envelope equation governing the nonlinearevolution of the probe field based on the MB Eqs. (2) and (3),we take the asymptotic expansions [40,47]
σ jl = σ(0)jl + εσ
(1)jl + ε2σ
(2)jl + ε3σ
(3)jl + · · · , (B1a)
�p = ε�(1)p + ε2�(2)
p + ε3�(3)p + · · · , (B1b)
with σ(0)jl = δ j0δl0 the steady-state solution when the probe
field is absent and ε the dimensionless small parameter char-acterizing the typical amplitude of the probe field. To have adivergence-free expansion that is valid for the nonlinear evolu-tion of the system, all the components on the right-hand side
of the expansion (B1) for σ jl and �p must be considered tobe functions of the multiscale variables zl = εl z (l = 0, 1, 2),tl = εl t (l = 0, 1) [40,47]. Substituting such expansion to theMB Eqs. (2) and (3), we obtain a set of expansion equationswhich can be solved order by order.
At the first order of the expansion, we obtain the solution
�(1)p = F exp (iθ ), (B2a)
σ(1)21 = [�∗
c1(ω + d31)/D]F exp (iθ ), (B2b)
σ(1)31 = [�∗
c2(ω + d21)/D]F exp (iθ ), (B2c)
σ(1)01 = −[(ω + d21)(ω + d31)/D]F exp(iθ ), (B2d)
with all other σ(1)jl zero. Here θ = K (ω)z0 − ωt0 [38]; F is the
envelope function of the slow variables z1, z2, and t1; K (ω) isthe linear dispersion relation [given by Eq. (4)]. The explicitexpression of the quantity D can be found just below Eq. (4).
At the second order of the expansion, we obtain the enve-lope equation
i
(∂F
∂z1+ 1
Vg
∂F
∂t1
)= 0, (B3)
where Vg ≡ (∂K/∂ω)−1 is the group velocity of the probe pulse. Explicit expressions of the solution at this order are given byσ
(2)j1 = a(2)
j1 i ∂∂t1
F exp(iθ ) ( j = 0, 2, 3), σ(2)02(03,32) = a(2)
02(03,32)|F |2 exp(−2αz2), and σ(2)j j = a(2)
j j |F |2 exp (−2αz2) ( j = 1, 2, 3, 0),with
a(2)01 = 1
κ01
(1
Vg− 1
c
), (B4a)
a(2)21 = −�∗
c1
D2(ω + d31)2(2ω + d01 + d21) + �∗
c1
D2|�c2|2(d31 − d21), (B4b)
a(2)31 = −�∗
c2
D2(ω + d21)2(2ω + d01 + d31) + �∗
c2
D2|�c1|2(d21 − d31), (B4c)
a(2)22 = −�30X + �10(B − B∗)
�10(A − A∗), (B4d)
a(2)33 = Q + i�10|�c1|2(N∗ − N )a(2)
22
i�10P, (B4e)
a(2)11 = X − i�10
(a(2)
22 + a(2)33
)i�10
, (B4f)
a(2)02 = M
[�∗
c2
(1
i�10X + a(2)
33
) + a(1)31
] − a∗(1)21 − �c1
(1
i�10X + a(2)
22
)N
, (B4g)
a(2)03 = − M∗d∗
32
�c1�c2
[�c2
(1
i�10X + a(2)
33
)+ a∗(1)
31 − �c1�c2
d∗32
a∗(2)02
], (B4h)
a(2)32 = 1
d32
(�c1a∗(2)
03 − �∗c2a(2)
02
), (B4i)
where
A = −d∗32M∗
�∗c1
{ |�c1|2�c2
P(N∗ − N ) − �c1�c2
d∗32N∗
[M∗|�c1|2�c2
P(N∗ − N ) − �∗
c1
]}, (B5a)
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CHONG SHOU AND GUOXIANG HUANG PHYSICAL REVIEW A 99, 043821 (2019)
B = d∗32M∗
�∗c1
{�c2
1
i�10X + �c2Q
i�10P+ a∗(1)
31 − �c1�c2
d∗32N∗
[M∗�c2
(1
i�10X+ Q
i�10P
)+ M∗a∗(1)
31 − a(1)21 − �∗
c11
i�10X
]}, (B5b)
M = �c1�c2
d32
( |�c1|2d32
+ d∗03
)−1
, (B5c)
N = d02 + �∗c1�
∗c2M − |�c1|2
d32, (B5d)
P = N∗M�∗c1�
∗c2 − NM∗�c1�c2, (B5e)
Q = [−i|N |2�20 − P + |�c1|2(N∗ − N )]X − i�10[�∗
c1N∗(Ma(1)31 − a∗(1)
21
) − c.c.], (B5f)
X = (ω + d21)(ω + d31)
D− (ω + d∗
21)(ω + d∗31)
D∗ . (B5g)
With the solutions obtained above, we proceed to the thirdorder of the expansion and obtain the envelope equation by adivergence-free condition
i∂
∂z2F − 1
2K2
∂2
∂t21
F + W |F |2Fe−2αz2 = 0, (B6)
where α = ε−2α [α ≡ Im(K )] is the absorption coefficient,K2 = ∂2K/∂ω2 is the coefficient of group-velocity dispersion,and W is the coefficient of self-phase modulation, whichreads
W = −κ01S + �c1(ω + d31)a∗(2)
02 + �c2(ω + d21)a∗(2)03
D,
(B7)
where S = (ω + d21)(ω + d31)(2a(2)11 + a(2)
22 + a(2)33 ). Combin-
ing the above two envelope Eqs. (B3) and (B6), we obtainEq. (5) in the main text when returning to the original vari-ables.
Because under the EIT condition the absorption of theprobe pulse is largely suppressed, the imaginary parts of K2
and W2 are thus much smaller than their corresponding realparts. Neglecting these small imaginary parts in K2 and W2,we get the dimensionless nonlinear equation
i∂U
∂s− sd
∂2U
∂σ 2+ 2|U |2U = ig0 U, (B8)
with U = εF exp(−iαz)/U0, s = z/(2LD), σ = (t −z/Vg)/τ0, LD = τ 2
0 /|K2| (the tilde symbol denotes takingthe real part), sd = sgn(K2), and g0 = 2LD/L0. HereL0 = 1/Im(K ) is typical absorption length; LD is typicaldispersion length, which has been assumed to be equalto typical nonlinearity length LNL ≡ 1/(WU 2
0 ), and henceU0 = (1/τ0)
√|K2|/W . Note that W has been assumed to
be positive, which is valid if both two-photon detunings�2 and �3 are negative, the case we consider here. ForU0 = 2π × 8.3 MHz and the other system parameters thesame as given in the main text, we have LD = 1.1 cm,L0 = 46.3 cm, and hence g0 = 0.05. Since sd = −1 and ig0Uis very small, one can get the approximated single bright
soliton solution of Eq. (B8), i.e.,
U = β sech[β(σ − c0s − σ0)]
× exp
[ic0
2σ + i
(β2 − c2
0
4
)s − iφ0
]. (B9)
Here β, c0, σ0, and φ0 are free parameters which determinethe amplitude (temporal width), velocity, initial position, andinitial phase of the soliton, respectively. When taking β = 1,c0 = 0, σ0 = 0, and φ0 = 0, the half Rabi frequency of theprobe pulse corresponding to the solution (B9) is just thatgiven by Eq. (6) after returning to the original variables.
Figure 5 shows the time evolution of the slow-light solitonby taking U as a function of t/τ0 and z/(2LD). The boundarycondition at z = 0 is chosen to be U = 0.8 sech(0.8t/τ0) withτ0 = 1.0 × 10−7 s. We see that the soliton is very stable duringpropagation due to the balance between the dispersion and theKerr nonlinearity in the system.
APPENDIX C: EXPRESSIONS OF PROBE PULSES FORSOLITON MEMORY AND SPLITTING
For the problems of soliton memory and soliton splitting,it is not possible to get exact solutions of the MB Eqs. (2)and (3), because these equations not only are nonlinearones (due to the coupling between light and atoms) butalso have variable coefficients (due to the time-dependentcontrol fields needed to implement the manipulation of the
FIG. 5. Evolution of the slow-light soliton by taking U as func-tion of t/τ0 and z/(2LD).
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SLOW-LIGHT SOLITON BEAM SPLITTERS PHYSICAL REVIEW A 99, 043821 (2019)
probe solitons). However, it is possible to provide approxi-mated analytical solutions of the probe field in different timeslots.
1. The case of soliton memory
In the case of the storage and retrieval (Sec. II.B.1 in themain text), the two control fields �c1 and �c2 are (or recoverto) constant values in the time slots before the storage andafter the retrieval of the slow-light soliton. In these time slots,the description by the envelope Eq. (5) in the main text isstill valid. Thus one has the following solution for the probepulse:
�p ≈
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
A0τ0
√|K2|W
sech[
1τ0
(t − z
Vg
)]ei[K0−1/(2LD )]z,
for t < Toff ,
0, for Toff < t < Ton,
B0τ0
√|K2|W
sech[
1τ0
(t − z
Vg
)]ei[K0−1/(2LD )]z+iφ0 ,
for t > Ton,
where A0 and B0 are constants, φ0 is a constant phase factor,Toff ≡ T 1
off = T 2off , and Ton ≡ T 1
on = T 2on.
2. The case of soliton splitting
The realization of the soliton beam splitter requires thatinitially the two control fields are switched on simultaneously,and then switched off and on in subsequent time slots [seeFig. 3(a) in the main text]. In the time slot t < T 1
off1, the two
control fields are constants and the envelope of the probepulse is described by Eq. (5), so it is a slow-light solitongiven by Eq. (6). In the time slot T 1
on1< t < T 1
off2, the control
field �c1 is switched on again but �c2 remains switched off,thus the four-level tripod system is reduced into a three-level�-type system (with �p and �c1 coupling the levels |0〉, |1〉,and |2〉). In this case, due to the existence of the constant�c1, the envelope of the probe pulse is still described byEq. (5) but with �c2 = 0. In the time slot T 1
off2< t < T 2
on1,
the control field �c1 is switched off but �c2 is switched onagain to a constant value; the four-level tripod system is thusconverted into another three-level �-type system (with �p
and �c2 coupling the levels |0〉, |1〉, and |3〉). In this case,due to the constant �c2, the envelope of the probe pulse isstill described by Eq. (5) but with �c1 = 0. That is to say,in the time slots of t < T 1
off1, T 1
on1< t < T 1
off2, and t > T 2
on1,
the description by the envelope Eq. (5) in the main text isvalid but with different control fields. Consequently, duringthe soliton splitting, the probe pulse can be written as the
form
�p ≈
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
A0τ0
√|K2|W
sech[
1τ0
(t − z
Vg
)]ei[K0−1/2LD]z,
for t < T 1off1
,
0, for T 1off1
< t < T 1on1
,
B′0
τ0
√|K ′
2|W ′ sech
[1τ0
(t − z
V ′g
)]ei[K ′
0−1/2L′D]z+iφ′
0 ,
for T 1on1
< t < T 1off2
,
0, for T 1off2
< t < T 2on1
,
B′′0
τ0
√|K ′′
2 |W ′′ sech
[1τ0
(t − z
V ′′g
)]ei[K
′′0 −1/2L′′
D]z+iφ′′0 ,
for t > T 2on1
,
where A0, B′0, B
′′0, φ
′0, and φ
′′0 are constants. In the above
expression,
K ′(ω) = ω/c + κ10(ω + d21)/D′, (C1a)
K ′0 = K ′(ω)|ω=0, (C1b)
V ′g = (∂K ′/∂ω)−1, (C1c)
K ′2 = ∂2K ′/∂ω2, (C1d)
W ′ = κ01[�c1a′(2)∗
32 + (ω + d21)(2a′(2)
11 + a′(2)22
)]/D′,
(C1e)
L′D = τ ′2
0 /|K ′2|, (C1f)
D′ = |�c1|2 − (ω + d21)(ω + d01), (C1g)
and
K ′′(ω) = ω/c + κ10(ω + d31)/D′′, (C2a)
K ′′0 = K ′′(ω)|ω=0, (C2b)
V ′′g = (∂K ′′/∂ω)−1, (C2c)
K ′′2 = ∂2K ′′/∂ω2, (C2d)
W ′′ = κ01[�c2a′′(2)∗
32 + (ω + d31)(2a′′(2)
11 + a′′(2)22
)]/D′′,
(C2e)
L′′D = τ ′′2
0 /|K ′′2 |, (C2f)
D′′ = |�c2|2 − (ω + d31)(ω + d01). (C2g)
Expressions of a′(2)jl (a′′(2)
jl ) can be obtained from that of a(2)jl
in Sec. IV described above by taking �c2 = 0 (�c1 = 0).
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