DQG ...Arindam Ghosh, Khairul Islam, Dipankar Bhattacharyya et al.-Interplay of classical and...

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Journal of Physics B: Atomic, Molecular and Optical Physics PAPER Electromagnetically induced transparency and nonlinear pulse propagation in a combined tripod and Λ atom-light coupling scheme To cite this article: H R Hamedi et al 2017 J. Phys. B: At. Mol. Opt. Phys. 50 185401 View the article online for updates and enhancements. Related content Triplet absorption spectroscopy and electromagnetically induced transparency F Ghafoor and R G Nazmitdinov - Revisiting the four-level inverted-Y system under both Doppler free and Doppler broadened condition: an analytical approach Arindam Ghosh, Khairul Islam, Dipankar Bhattacharyya et al. - Interplay of classical and quantum dynamics in a thermal ensemble of atoms Arif Warsi Laskar, Niharika Singh, Arunabh Mukherjee et al. - This content was downloaded from IP address 140.114.81.132 on 08/11/2017 at 06:16

Transcript of DQG ...Arindam Ghosh, Khairul Islam, Dipankar Bhattacharyya et al.-Interplay of classical and...

Page 1: DQG ...Arindam Ghosh, Khairul Islam, Dipankar Bhattacharyya et al.-Interplay of classical and quantum dynamics in a thermal ensemble of atoms Arif Warsi Laskar, Niharika Singh, Arunabh

Journal of Physics B: Atomic, Molecular and Optical Physics

PAPER

Electromagnetically induced transparency andnonlinear pulse propagation in a combined tripodand Λ atom-light coupling schemeTo cite this article: H R Hamedi et al 2017 J. Phys. B: At. Mol. Opt. Phys. 50 185401

 

View the article online for updates and enhancements.

Related contentTriplet absorption spectroscopy andelectromagnetically induced transparencyF Ghafoor and R G Nazmitdinov

-

Revisiting the four-level inverted-Y systemunder both Doppler free and Dopplerbroadened condition: an analyticalapproachArindam Ghosh, Khairul Islam, DipankarBhattacharyya et al.

-

Interplay of classical and quantumdynamics in a thermal ensemble of atomsArif Warsi Laskar, Niharika Singh, ArunabhMukherjee et al.

-

This content was downloaded from IP address 140.114.81.132 on 08/11/2017 at 06:16

Page 2: DQG ...Arindam Ghosh, Khairul Islam, Dipankar Bhattacharyya et al.-Interplay of classical and quantum dynamics in a thermal ensemble of atoms Arif Warsi Laskar, Niharika Singh, Arunabh

Electromagnetically induced transparencyand nonlinear pulse propagation in acombined tripod and Λ atom-light couplingscheme

H R Hamedi, J Ruseckas and G Juzeliūnas

Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio 3, Vilnius LT-10222,Lithuania

E-mail: [email protected]

Received 1 May 2017, revised 17 July 2017Accepted for publication 8 August 2017Published 24 August 2017

AbstractWe consider propagation of a probe pulse in an atomic medium characterized by a combinedtripod and Lambda (Λ) atom-light coupling scheme. The scheme involves three atomic groundstates coupled to two excited states by five light fields. It is demonstrated that dark states can beformed for such an atom-light coupling. This is essential for formation of the electromagneticallyinduced transparency (EIT) and slow light. In the limiting cases the scheme reduces toconventional Λ- or N-type atom-light couplings providing the EIT or absorption, respectively.Thus, the atomic system can experience a transition from the EIT to the absorption by changingthe amplitudes or phases of control lasers. Subsequently the scheme is employed to analyze thenonlinear pulse propagation using the coupled Maxwell–Bloch equations. It is shown that ageneration of stable slow light optical solitons is possible in such a five-level combined tripodand Λ atomic system.

Keywords: electromagnetically induced transparency, dark state, combined tripod and Lambdaatom-light coupling

(Some figures may appear in colour only in the online journal)

1. Introduction

Electromagnetically induced transparency (EIT) [1–6] plays animportant role in controlling the propagation of light pulses inresonant media. Due to the EIT a weak probe beam of lighttuned to an atomic resonance can propagate slowly and isalmost lossless when the medium is driven by one or severalcontrol beams of light with a higher intensity [1–6]. The EIT isformed because the control and probe beams drive the atoms totheir dark states representing a special superposition of theatomic ground states immune to the atom-light coupling. Theabsorption is suppressed due to a quantum mechanical inter-ference between different excitation pathways of atomic energylevels leading to the EIT. The EIT has various importantapplications in quantum and nonlinear optics, such as slow andstored light [7–12], stationary light [13, 14], multiwave mixing

[15–17], optical solitons [18–23], optical bistability [24, 25] andKerr nonlinearity [26–30]. Using the slow light greatly enhan-ces the light–matter interaction and enables nonlinear opticalprocesses to achieve significant efficiency even at a single-photon level 26, 31–38].

There has been a considerable amount of activities onsingle- [2–4, 8, 11, 12, 39–43] and two-component (spinor)[6, 44–48] slow light in atomic media induced by the EIT. Theformer single-component slow light involves a probe beam oflight and one or several control beams resonantly interactingwith atomic media characterized by three level Lambda (Λ)type [2, 40] or four level tripod type [41, 42, 49–51] atom-lightcoupling schemes. In the later (spinor) case, double-tripod [6,44–48] coupling schemes have been considered to support asimultaneous propagation of two probe beams leading to for-mation of a two-component slow light.

Journal of Physics B: Atomic, Molecular and Optical Physics

J. Phys. B: At. Mol. Opt. Phys. 50 (2017) 185401 (10pp) https://doi.org/10.1088/1361-6455/aa84f6

0953-4075/17/185401+10$33.00 © 2017 IOP Publishing Ltd Printed in the UK1

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In this paper, we propose and analyze a novel five-levelclosed-loop scheme supporting the EIT. Closed-loop quantumconfigurations [30, 45, 52–55] represent a class of atom-lightcoupling schemes in which the driving fields acting on atomsbuild closed paths for the transitions between atomic levels.The interference between different paths makes the systemsensitive to relative phases of the applied fields. In our proposalillustrated in figure 1, the atom-light coupling represents a five-level combined Lambda-tripod scheme, in which three atomicground states are coupled to two excited states by four controland one probe laser fields. In other words, the scheme involvesfour atomic levels coupled between each other by four controlfields and interacting with a ground level through a weak probefield. The existence of dark states, essential for the EIT, isanalytically demonstrated for such an atom-light couplingsetup. It is shown that in some specific limiting cases thisscheme can be equivalent to the conventional Λ- or N-typeatom-light couplings. An advantage of such an atomic systemis a possibility of transitions to the Λ or N-type level schemesjust by changing the amplitudes and phases of the controllasers. The limiting cases are discussed where the schemereduces to atom-light couplings of the Λ- or N-type. By makinga transition between the two limiting cases, one can switchfrom the EIT regime to the absorption for the probe fieldpropagation.

Laser-driven atomic media, on the other hand, can beexploited to exhibit various nonlinear optical properties[15, 16, 18, 20, 24, 26, 28, 31, 56–58]. A particular exampleis formation of optical solitons with applications for opticalbuffers, phase shifters [59], switches [60], routers, transmis-sion lines [61], wavelength converters [62], optical gates [63]and others. Solitons represent a specific type of stable shape-preserving waves propagating through nonlinear media. Theycan be formed due to a balance between dispersive andnonlinear effects leading to an undistorted propagation overlong distance [63–73]. Following a report of ultraslow opticalsolitons in a highly resonant atomic medium by Wu and Deng[67], these solitary waves have received a considerableattention [19, 21–23, 74–82]. Here, the equations of motionthat govern the nonlinear evolution of the probe pulseenvelope are derived for the Lambda-tripod atom-light

coupling by solving the coupled Maxwell–Bloch equations. Itis found that, by properly choosing the parameters of thesystem, the formation and slow propagation of shape-pre-serving optical solitons is feasible.

The paper is organized as follows. In section 2 we pre-sent the proposed setup and investigate propagation of theprobe beam for various configurations of the control fields.Nonlinear propagation of the probe pulse is considered insection 3. Section 4 summarizes our findings.

2. Formulation and theoretical background

2.1. The system

Let us consider a probe pulse described by a Rabi frequencypW . Additional laser fields described by Rabi frequencies ,1W2W , 3W and 4W control propagation of the probe pulse. The

probe and the control fields are assumed to co-propagatealong the z direction. We shall analyze the light–matterinteraction in an ensemble of atoms using a five-levelLambda-tripod scheme shown in figure 1. The atoms arecharacterized by three ground levels añ∣ , cñ∣ and dñ∣ , as well astwo excited states bñ∣ and eñ∣ . Four coherent control fields withthe Rabi frequencies ,1W 2W , 3W and 4W induce dipole-allowedtransitions b cñ ñ∣ ⟷ ∣ , b dñ ñ∣ ⟷ ∣ , e cñ ñ∣ ⟷ ∣ , and e dñ ñ∣ ⟷ ∣ ,respectively. As a result, the control fields couple two excited

states bñ∣ and eñ∣ via two different pathways b c e1 3*

ñ ñ ñW W

∣ ∣ ∣

and b d e2 4*

ñ ñ ñW W

∣ ∣ ∣ making a four level closed-loop coher-ent coupling scheme described by the Hamiltonian ( 1 = )

H

c b d b c e d e h.c.

1

4Levels

1 2 3 4* * * *=-W ñá - W ñá - W ñá - W ñá +∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣( )

Furthermore, the tunable probe field with the Rabi frequencypW induces a dipole-allowed optical transition a bñ ñ∣ ⟷ ∣ . The

total Hamiltonian of the system involving all five atomiclevels of the combined Λ and tripod level scheme is given by

H a b b a H . 2p p5Levels 4Levels*= - W ñá + W ñá +( ∣ ∣ ∣ ∣) ( )

Note that the complex Rabi frequencies of the four controlfields can be written as ej j

i iW = W f∣ ∣ , with j 1, 2, 3, 4= ,where jW∣ ∣ and jf are the amplitude and phase of each appliedfield. As it will be explored below, in this scheme thedestructive interference between different transition pathwaysinduced by the control and probe beams can make the med-ium transparent for the resonant probe beams in a narrowfrequency range due to the EIT. We define 1 2f f f= - -( )

3 4f f-( ) to be a relative phase among the four control fieldsforming a closed-loop coherent coupling. By changing f, onecan substantially modify the transparency and absorptionproperties for the probe field in such a Lambda-tripodscheme.

Figure 1. Schematic diagram of the five-level Lambda-tripodquantum system.

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2.2. Equations of motion

The dynamics of the probe field propagating through theatomic medium is described by the Maxwell–Blochequations. To the first-order of pW the equations have the form

d i i i , 3ba ba ca da p1

11

11

21r r r r= + W + W + W˙ ( )( ) ( ) ( ) ( )

d i i , 4ca ca ba ea1

21

11

31* *r r r r= + W + W˙ ( )( ) ( ) ( ) ( )

d i i , 5da da ba ea1

21

21

41* *r r r r= + W + W˙ ( )( ) ( ) ( ) ( )

d i i , 6ea ea ca da1

31

31

41r r r r= + W + W˙ ( )( ) ( ) ( ) ( )

and

zc

t

N

ci , with

2, 7

p pba

p ba1 12

hr h

w m¶W

¶+

¶W

¶= =- ∣ ∣

( )( )

where u v,1r( ) are the first-order matrix elements of the density

matrix operator u vuvr r= å ñ á∣ ∣. The optical Blochequations (3)–(6) imply the probe field to be much weakerthan the control ones. In that case most atomic population isin the ground state añ∣ , and one can treat the probe field as aperturbation. Therefore, we can apply the perturbationexpansion ,ij k ij

kr r= S ( ) where ijkr( ) represents the kth order

part of ijr in terms of probe field pW . Since p iW Wi 1, 2, 3, 4=( ), the zeroth-order solution is 1aa

0r =( ) , whileother elements being zero ( 0bb cc dd ee

0 0 0 0r r r r= = = =( ) ( ) ( ) ( ) ). Allfast-oscillating exponential factors associated with centralfrequencies and wave vectors have been eliminated from theequations, and only the slowly varying amplitudes areretained.

The wave equation (7) describes propagation of the probefield pW influenced by the atomic medium, where bam is anelectric dipole matrix element corresponding to the transitionb añ ñ∣ ⟷ ∣ , N is the atomic density and pw is the frequency ofthe probe field. The density matrix equations (3)–(6) describethe evolution of the atomic system affected by the control andprobe fields. They follow from the general quantum Liouvilleequation for the density matrix operator [83]

H Li

, , 85Levels

r r= - + r˙ [ ] ( )

where the damping operator Lr describes the decay of thesystem described by parameters d 2 ib p1 = -G + D ,d i p2 2= D - D( ) and d 2 ie p3 3 2= -G + D + D - D( ) inequations (3)–(6). We have defined the detunings as:

,bc bd2D = D = D and ec ed3D = D = D , with bcD =bc1w w- , bd bd2w wD = - , ec ec3w wD = - , ed 4wD = -

edw , and p p baw wD = - , where iw is a central frequency ofthe corresponding control field. Two excited states bñ∣ and eñ∣decay with rates bG and eG , respectively.

2.3. Transition to a new basis

The Hamiltonian for the atomic four-level subsystem (1) canbe represented as

H D b B b B e h.c., 9e e e4Levels b a= - ñá - ñá - W ñá +∣ ∣ ∣ ∣ ∣ ∣ ( )where

D c d1

, 10e 4 3ñ =W

W ñ - W ñ∣ ( ∣ ∣ ) ( )

B c d1

11e 3 4* *ñ =W

W ñ + W ñ∣ ( ∣ ∣ ) ( )

are the internal dark and bright states for the Λ- scheme madeof the two ground states states cñ∣ and dñ∣ , as well as an excitedstates eñ∣ . One can also introduce another set of dark andbright states corresponding to the Λ-scheme made of the samepair of ground states states cñ∣ and dñ∣ , yet a different excitedstate bñ∣ :

D c d1

, 12b 2 1ñ =W

W ñ - W ñ∣ ( ∣ ∣ ) ( )

B c d1

. 13b 1 2* *ñ =W

W ñ + W ñ∣ ( ∣ ∣ ) ( )

In writing equation (9), the coefficient

D B D B1

14b e e d 1 4 2 3* * * *b = á ñ = á ñ =W

W W - W W∣ ∣ ( ) ( )

represents the quantum inteference between the four controlfields playing the main role in tuning dispersion andabsorption properties in the combined tripod and Λ scheme.In addition, we define

D D B B1

15b e e d 1 3 2 4* *a = á ñ = á ñ =W

W W + W W∣ ∣ ( ) ( )

and the total Rabi frequency

. 1632

42W = W + W∣ ∣ ∣ ∣ ( )

Figure 2. Five-level quantum system in the transformed basis for0b ¹ and 0a ¹ .

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By changing the quantum interference coefficient β and thecoefficient α one arrives at three different situations. For each

of them we shall plot the level schemes in the basis involvingthe transformed states Beñ∣ and Deñ∣ .

2.3.1. Situation (a): β ≠ 0 and α ≠ 0. In the case when bothcoefficients α and β are nonzero, the five-level tripod andΛ scheme shown in figure 2 looks similar to the originalscheme (figure 1) in the transformed basis, but the couplingbetween the states Deñ∣ and eñ∣ is missing. The coefficient β isnonzero when 1 4 2 3W W ¹ W W . The condition 0b ¹ is validprovided 1 4 2 3W W ¹ W W∣ ∣∣ ∣ ∣ ∣∣ ∣ and phase f is arbitrary, or

1 4 2 3W W = W W∣ ∣∣ ∣ ∣ ∣∣ ∣ with 0f ¹ .When both α and β are nonzero, one can define a global

dark state Dñ∣ for the whole atom-light coupling scheme

D a D . 17p ebñ = ñ - W ñ∣ ∣ ∣ ( )

The dark state is an eigenstate of the full atom-lightHamiltonian with a zero eigen-energy: H D 05Levels ñ =∣ . Thestate Dñ∣ has no contribution by the ground state superpositionBeñ∣ , as well as no contribution by the bare excited states eñ∣and bñ∣ . As a result, there is no transition from the state Dñ∣ tothe excited states eñ∣ and bñ∣ , making the five-level closed-loop scheme transparent to the electromagnetic field. This is anew mechanism for the EIT compared with the Λ [2, 8],tripod [41, 42], or double tripod schemes [45, 46].

It is known that the real and imaginary parts of ba1r( )

correspond to the probe dispersion and absorption,

respectively. A steady-state solution to the density matrixelement bar reads under the resonance condition 2D =

03D =

where the interference term β is involved.A denominator of equation (18) represents the fourth

order polynomial which contains zero points at four differentdetunings pD of the probe field from the EIT resonance. Thisprovides four maxima in the absorption profile of the system,as one can see in figure 5(a). Furthermore the EIT window isformed for zero detuning. In appendix A we have presentedeigenstates and the corresponding eigenvalues of theHamiltonian(9) describing the four-level subsystem. Onecan see that all four eigenstates niñ∣ characterized by the

eigenvalues S Y1,2 2l = - and S Y

3,4 2l = + contain con-

tributions due to the excited state bñ∣ (note that S and Y can befound in appendix A). This results in four peaks in absorptionprofile of the system.

2.3.2. Situation (b): β ¼ 0 and α ≠ 0. The condition 0b = isfulfilled if 1 4 2 3W W = W W , or equivalently 2 3 1 4W W = W W∣ ∣∣ ∣ ∣ ∣∣ ∣and 0f = . In that case the state Deñ∣ is not involved, so theinteraction Hamiltonian(9) for the four-level subsystem canbe rewritten as

H B b B e h.c. 19e e4Levels a= - ñá - W ñá +∣ ∣ ∣ ∣ ( )

Consequently, the five-level tripod and Λ scheme becomes

Figure 3. The level scheme in the transformed basis for 0b = and0a ¹ . The ground state superposition Deñ∣ (now shown in the figure)

is decoupled from the remaining four states.

Figure 4. Schematic diagram of the five-level quantum system for0b ¹ and 0a = .

i

i i

i i i i i, 18ba p

p p p

p p p p p p

13

24

22

2 22 1

22

22 3

24

22

e

e b e

rb

= WD W + W + D - + D

W + D - + D W + W + D - + D W + W + D - + D

G

G G G

( )( )

( )( ) ( ) ( )

∣ ∣ ∣ ∣

∣ ∣ (∣ ∣ ∣ ∣ ) ∣ ∣ ∣ ∣( )( )

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equivalent to a conventional N-type atomic system [29, 84]shown in figure 3.

Since the quantum interference term β vanishes,equation (18) simplifies to

The denominator of ba1r( ) in equation (20) is now a cubic

polynomial, providing three absorption maxima. The eigen-states and eigenvalues corresponding to this situation arepresented in appendix B. The eigenvector n2ñ∣ characterizedby a zero eigen-energy coincides with the dark state Deñ∣ andis decoupled from the radiation fields. Only the remainingthree eigenvectors n1ñ∣ , n3ñ∣ and n4ñ∣ contain the contributiondue to an excited state bñ∣ . This leads to three absorption peaksdisplayed in figure 5(b). In this way, the absence of quantuminterference term β between the control fields destroys one ofthe peaks in the absorption profile leading to three absorptionmaxima.

2.3.3. Situation (c): β ≠ 0 and α ¼ 0. When the coefficient βis nonzero but the coefficient α is zero, the interactionHamiltonian(9) can be represented as

H D b B e h.c. 21e e4Levels b= - ñá - W ñá +∣ ∣ ∣ ∣ ( )

As illustrated in figure 4, the five-level tripod and Λ scheme isthen equivalent to a conventional Λ-type atomic system[2, 40] which is decoupled from the two-level systeminvolving the states Beñ∣ and eñ∣ .

In the following we consider a symmetric case where1 2W = W∣ ∣ ∣ ∣, 3 4W = W∣ ∣ ∣ ∣ and f p= . In such a situation the

conditions 0b ¹ and 0a = are fulfilled, with 2212b = W∣ ∣.

equation (18) for ba1r( ) then simplifies considerably, giving

i i. 22ba p

p

p p

1

22b

rb

= WD

+ D - + DG( )∣ ∣( )( )

Obviously, the polynominal in denominator becomes quad-ratic in pD resulting in two absorption peaks or a single EITwindow, which is a characteristic feature of the Λ scheme.Furthermore, the probe absorption and dispersion do notdependent on Ω, only β contributes to the optical properties.This is because the Λ-type scheme is now decoupled from the

transition B eeñ ñW

∣ ∣ , and the system behaves as a three- levelΛ-type scheme containing a ,ñ∣ bñ∣ , and Deñ∣ .

In the following we summarize our results for thebehavior of real and imaginary parts of ba

1r( ) corresponding tothe probe dispersion and absorption for different situations(a)–(c) described above. Without a loss of generality, in allthe simulations we take e b gG = G = . Other frequencies arescaled by γ which should be in the order of MHz for cesium(Cs) atoms. Figure 5 shows that the our model provides ahigh control of dispersive-absorptive optical properties ofthe probe field. The absorption profile has four, three andtwo peaks featured in figures 5(a)–(c), respectively. There isthe resulting change in the sign of the slope of thedispersion at 0pD = for different situations (a)–(c). Thisgives rise to switching in the group velocity of the probepulse from subluminal to superluminal or visa versa. Inparticular, for the choice of parameters satisfying situation(a) and (c), there is the subluminality accompanied by EITat line center. On the other hand, the superluminality

0.015

0.01

0.005

0

-0.005

-0.01-3 -2 -1 0 1 2 3

0.015

0.01

0.005

0

-0.005

-0.01-3 -2 -1 0 1 2 3

-0.01-3 -2 -1 0 1 2 3

0.02

0.015

0.01

0.005

0

-0.005

∆p/γ ∆p

/γ ∆p/γ

Figure 5. Probe absorption (Im ba1r( ( ))) versus pD for (a) 0.91 gW = , 0.72 gW = , 0.43 gW = , 0.84 gW = , and 0f = corresponding to the first

situation, (b) 0.51 2 gW = W = , 0.73 4 gW = W = and 0f = corresponding to the second situation, and (c) 0.21 2 gW = W = , 3 4W = W =0.1g and f p= corresponding to the third situation. Other parameters are e b gG = G = , 02 3D = D = , and 0.01p gW = . Note that allfrequencies are scaled by γ which should be of the order of MHz, like for cesium (Cs) atoms.

i i

i i i i i i. 20ba p

p p

p p p p

13

24

22

2 12

22

2 32

42

2

e

e b e

r = WW + W + D - + D

- + D W + W + - + D W + W + D - + D

G

G G G

( )( )

( )( ) ( ) ( )

∣ ∣ ∣ ∣

(∣ ∣ ∣ ∣ ) ∣ ∣ ∣ ∣( )( )

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accompanied by a considerable absorption is observed forthe parametric condition satisfying the situation (b).

3. Linear and nonlinear pulse propagation incombined tripod and Λ scheme

In this section we consider propagation of the probe pulse inthe proposed tripod and Λ scheme. Performing the timeFourier transform of equations (3)–(7) one can obtain

t F F F 0, 23ba ca da p1 1 2w + W + W + L =( ) ( )

t F F F 0, 24ca ba ea2 1 3* *w + W + W =( ) ( )

t F F F 0, 25da ba ea2 2 4* *w + W + W =( ) ( )t F F F 0, 26ea ca da3 3 4w + W + W =( ) ( )

and

z cFi i , 27

pp ba

wh

¶L

¶- L = ( )

where t i 2b p1 w w= + G + D( ) , t p2 2w w= + D - D( ) , andt i 2e p3 2 3w w= + G + D - D + D( ) . Note that Fij and pLrepresent the Fourier transforms of ij

1r( ) and pW , respectively,where ω is a deviation from the central frequency.

A solution of equation (27) is a plane wave of the form

z, 0, e , 28p pziw wL = L k w( ) ( ) ( )( )

where

c

S

Q, 291k

w h ww

= -( )

( )( )

describes the linear dispersion relation of the system.Expanding κ in power series around the center frequency ofthe probe pulse ( 0w = ) and taking only the first three terms,we get

, 300 1 22k k k w k w= + + ( )

where the detailed expressions for the coefficients 0k , 1k and2k are given in appendix C, while S1 w( ) and Q w( ) can be

found in appendix D. In equation (30), id

d 0j

jk = k ww w=∣( ) with

j 0, 1, 2= are the dispersion coefficients in different orders.In general, the real part of i 20k c= ¡ + defines the phaseshift ϒ per unit length, while the imaginary part indicates thelinear absorption χ of the probe pulse. The group velocity vgis given by 1 1k , whereas the quadratic term 2k is associatedwith the group velocity dispersion which causes the pulsedistortion.

In the linear regime, we take an incoming probe pulse to beof the Gaussian shape, t0, ep p

t0 02W = W t-( ) ( ) , with a duration

0t . The subsequent time evolution is obtained from equation (28)by carrying out an inverse Fourier transform [67]

z tL

zt z

L, exp i , 31p

p

z z

0

01

2

02

kkt

W =W

--⎡

⎣⎢⎤⎦⎥( ) ( ) ( )

with L s z s ziz 1 2= -( ) ( ), s z z1 4 Re1 2 02k t= +( ) ( ) and

s z z4 Im2 2 02k t=( ) ( ) . In this way, even if there is no

absorption due to EIT (Im 0ik =( ) , i 0, 1, 2= ), the dispersion

effects can contribute to the pulse attenuation and spreadingduring propagation.

Our goal is to obtain shape-preserving optical pulseswhich can propagate without significant distortion and loss inour medium. The idea is to include the optical Kerr non-linearity of the probe laser field into the light propagation, andshow that the Kerr nonlinear effect can compensate the dis-persion effects and result in shape-preserving optical pulses.To balance the dispersion effects and optical nonlinearity, inthe following a theoretical model is employed based on thecoupled Maxwell–Bloch equations for the nonlinear pulsepropagation. Following [67], we take a trial function

e . 32p pzi 0L = L k˜ ( )

Substituting equation (32) into the wave equation (27) andusing equations (29), (30) and (D.1) we obtain

ze i e , 33

p zp

zi1 2

2 i0 0k w k w¶L

¶= + Lk k( ) ( )

where we have replaced pL̃ with pL for the sake of con-venience. Here we only keep terms up to the order 2w inexpanding the dispersion relation jk .

In deriving the linearized wave equation (7), the non-linear polarization due to the optical Kerr nonlinearity of theprobe field has been neglected. Now we turn to investigate thenonlinear propagation of light due to the Kerr effect. Toincorporate the nonlinear optical terms in the pulse propaga-tion, the right-hand side of wave equation (7) must berewritten as i i Kerrba

1hr h-( ) . The Kerr nonlinear term has anopposite sign than the linear term i ba

1hr( ) and the probeabsorption and dispersion are proportional to imaginary andreal parts of ba

1r( ), respectively. Consequently, a large opticalnonlinearity can cancel the dispersion and suppress theabsorption of probe field, effectively. A derivation of the Kerrnonlinear coefficient is provided in appendix E.

Performing the inverse Fourier transform of equation (33),using the expression (E.11) for Kerr and introducing newcoordinates z,z = and t z v ,gh = - we arrive at the non-linear wave equation for the slowly varing envelope pW

i e , 34p p p p2

2

22

zk

h¶¶

W -¶¶

W = Q W Wcz- ∣ ∣ ( )

where KerrhQ = - , and 2 Im 2 Im S

Q01c k h= = -( )( ) .

Equation (34) contains generally complex coefficients.However, for suitable set of system parameters, the absorp-tion coefficient χ may be very small, i.e., 0c , and ima-ginary parts of coefficients Θ and 2k may be made very smallin comparison to their real parts, i.e., ir i r2 2 2 2k k k k= + » ,and ir i rQ = Q + Q » Q . In this case, equation (34) can bewritten as

i . 35p r p r p p2

2

22

zk

h¶¶

W -¶¶

W = Q W W∣ ∣ ( )

This represents the conventional nonlinear Schrödinger (NLS)equation which describes the nonlinear evolution of the probepulse and allows bright and dark soliton solutions. The nature

6

J. Phys. B: At. Mol. Opt. Phys. 50 (2017) 185401 H R Hamedi et al

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of the soliton solution is determined by a sign of the productr r2k Q . A bright soliton is obtained for 0r r2k Q > , and the

solution is then given by

sech exp i 2 . 36p p r p02h t zW = W - Q W( ) ( ∣ ∣ ) ( )

For 0r r2k Q < , one obtains the dark soliton solution of theform

tanh exp i 2 . 37p p r p02h t zW = W - Q W( ) ( ∣ ∣ ) ( )

Here 1p r r0 2t kW = Q( ) ∣ ∣ represents an amplitude of theprobe field and τ is the typical pulse duration (soliton width).

In the following, we explore a possibility for the formationof the shape preserving optical solitons in this combined tripodand Λ scheme for a realistic atomic system and present numericcalculations. The proposed scheme involving the five-levelcombined tripod and Λ structure can be experimentallyimplemented using the cesium (Cs) atom vapor. In our proposal,the levels cñ∣ , dñ∣ and eñ∣ can correspond to S F6 , 3 ,1 2 =∣M 1F = + ñ, S F M6 , 3, 3F1 2 = = + ñ∣ and P F6 , 2 ,3 2 =∣M 2F = + ñ, respectively. In addition, the levels bñ∣ and añ∣ cancorrespond to P F6 , 43 2 = ñ∣ and S F6 , 41 2 = ñ∣ , respectively.The two excited states are assumed to decay with therates 2 5.2 MHze b g pG = G = = ´ .

Assuming the parameteric situation (a) described in theprevious section, we take 1.97 10 s1 2

9 1W = W = ´ -∣ ∣ ∣ ∣ ,2.3 10 s3

9 1W = ´ -∣ ∣ , 16.4 10 s47 1W = ´ -∣ ∣ , 6.42D = ´

10 s ,9 1- 5.9 10 sp9 1D = ´ - and 82 10 s3

7 1D = ´ - . Conse-quently, we obtain i3.9 0.008 cm0

1k » + -( ) , 4.71k » -(i2.1 10 10 cm s2 9 1´ - - -) , i8.06 3.6 102

2k » - - ´ -( )10 cm s17 1 2- - , and i3.6 1.12 10 10 cm s2 19 1 2Q » - ´ - - -( ) .In this case, the standard NLS equation (35) with 0r r2k Q < iswell characterized, leading to the formation of dark solitons inthe proposed system. With this set of parameters, the funda-mental soliton has a width and amplitude satisfying

4.7p r r0 2t kW = Q ∣ ∣ . As shown in figure 6, the darksoliton of this type remains fairly stable during propagation,which is due to the balance between the group-velocitydispersion and Kerr-type optical nonlinearity. According toequation (C.2) in appendix C, the group velocity vg has a general

form v c g Q S g Qg1 1

1 1 22h= + - +- - ( ), with all coefficients

given in appendix C. With the above system parameters, one canfind v c7 10 ,g

3» ´ - indicating that the soliton propagates witha slow velocity.

The formation and propagation of such a slow lightoptical soliton in the system is due to the EIT conditiondescribed in the situation (a) of the previous section, i.e.,

0b ¹ and 0a ¹ , or equivalently 1 4 2 3W W ¹ W W . Due to theEIT, the absorption of the probe field becomes negligiblysmall. In this case, an enhanced Kerr nonlinearity can com-pensate the dispersion effects in such a highly resonantmedium resulting in shape preserving slow light opticalsolitons.

4. Concluding remarks

In conclusion we have demonstrated the existence of darkstates which are essential for appearance of EIT for a situationwhere the atom- light interaction represents a five-levelcombined tripod and Λ configuration. The EIT is possible inthe combined tripod and Λ scheme when the Rabi frequenciesof the control fields obey the condition 0b ¹ , 0a ¹ , whereβ and α given by equations (14) and (15) characterize therelative amplitudes and phases of the four control fields.Under this condition, the medium supports the lossless pro-pagation of slow light. It is analytically demonstrated thatcombined tripod and Λ scheme can reduce to simpler atom-light coupling configurations under various quantum inter-ference situations. In particular, this scheme is equivalent to afour-level N-type scheme when 0b = and 0a ¹ . On theother hand, for 0b ¹ but 0a = , a three level Λ-type atom-light coupling scheme can be established. As a result, bychanging the Rabi frequencies of control fields, it is possibleto make a transition from one limiting case to the another one.This can lead to switching from subluminality accompaniedby EIT to superluminality along with absorption and visaversa. Based on the coupled Maxwell–Bloch equations,a nonlinear equation governing the evolution of the probepulse envelope is then obtained. This leads to formation ofstable optical solitons with a slow propagating velocity due tothe balance between dispersion and Kerr nonlinearity of thesystem.

A possible realistic experimental realization of the pro-posed combined tripod and Λ setup can be implemented forthe Cs atoms. The lower levels añ∣ , cñ∣ and dñ∣ can then beassigned to S F6 , 41 2 = ñ∣ , S F M6 , 3, 1F1 2 = = + ñ∣ and

S F M6 , 3, 3F1 2 = = + ñ∣ , respectively. Two excited states bñ∣and eñ∣ can be attributed to the Cs states P F6 , 43 2 = ñ∣ and

P F M6 , 2, 2F3 2 = = + ñ∣ , respectively.

Acknowledgments

The presented work has been supported by the LithuanianResearch Council (No. VP1-3.1-ŠMM-01-V-03-001).

Figure 6. Propagation dynamics of an ultraslow optical soliton with10 s7t = - , l 1 cm= , and 1.0 10 cm s10 1 1h = ´ - - and the para-

meters given in the main text.

7

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Appendix A. Eigenstates and eigenvalues forsituation (a)

The expressions for the eigenstates and their correspondingeigenvalues for situation (a) are:

nS Y

X Yb

S YB

S Y

X YD e

2

2, A.1

e

e

1*

*

a

a b

ñ =-

W -ñ -

-W

ñ

--

W -ñ + ñ

∣ ( )( )

∣ ∣

( )( )

∣ ∣ ( )

nS Y

X Yb

S YB

S Y

X YD e

2

2, A.2

e

e

2*

*

a

a b

ñ =-

W -ñ +

-W

ñ

+-

W -ñ + ñ

∣ ( )( )

∣ ∣

( )( )

∣ ∣ ( )

nS Y

X Yb

S YB

S Y

X YD e

2

2, A.3

e

e

3*

*

a

a b

ñ =+

W +ñ -

+W

ñ

-+

W +ñ + ñ

∣ ( )( )

∣ ∣

( )( )

∣ ∣ ( )

nS Y

X Yb

S YB

S Y

X YD e

2

2, A.4

e

e

4*

*

a

a b

ñ =+

W +ñ +

+W

ñ

++

W +ñ + ñ

∣ ( )( )

∣ ∣

( )( )

∣ ∣ ( )

with eigenvalues

S Y

2, A.51l = -

- ( )

S Y

2, A.62l =

- ( )

S Y

2, A.73l = -

+ ( )

S Y

2, A.84l =

+ ( )

where

S , A.92 2 2a b= + + W∣ ∣ ∣ ∣ ( )

X , A.102 2 2a b= - + W∣ ∣ ∣ ∣ ( )

Y S 4 . A.112 2 2b= - W∣ ∣ ( )

Appendix B. Eigenstates and eigenvalues forsituation (b)

The expressions for the eigenstates and their correspondingeigenvalues for situation (b) are:

n b e , B.11a

ñ = -W

ñ + ñ∣ ∣ ∣ ( )

n D , B.2e2ñ = ñ∣ ∣ ( )

n b B e , B.3e3

2 2*a añ =

Wñ -

+ WW

ñ + ñ∣ ∣∣ ∣

∣ ∣ ( )

n b B e , B.4e4

2 2*a añ =

Wñ +

+ WW

ñ + ñ∣ ∣∣ ∣

∣ ∣ ( )

with eigenvalues

0, B.51l = ( )0, B.62l = ( )

, B.732 2l a= - + W∣ ∣ ( )

. B.842 2l a= + W∣ ∣ ( )

Appendix C. Explicit expressions for κ0, 1 vg�

and κ2

Expressions for 0k , v1 g and 2k read

S

Q, C.10

1k h= ( )

v c

g

Q

S g

Q

1 1, C.2

g

1 1 22

h= + - +⎛⎝⎜

⎞⎠⎟ ( )

g

Q Qg g S g

S g

Q

12 , C.32

32 1 2 1 4

1 22

3k h= - + - -

⎛⎝⎜⎜

⎞⎠⎟⎟( ) ( )

with

g t t t2 , C.41 22

2 3 32

42= + - W + W(∣ ∣ ∣ ∣ ) ( )

g t t

t t t t t t t t t2 ,

C.5

2 2 3 12

22

1 2 32

42

1 22

3 22

1 2 3

= + W + W

+ + W + W - - -

( )(∣ ∣ ∣ ∣ )

( )(∣ ∣ ∣ ∣ )( )

g 6 2 6 i , C.6p e3 2 3= - D + D + D + G ( )

g

t t t t t t t

2

2 2 2 , C.74 1

22

23

24

2

1 2 1 3 2 3 22

= - W + W + W + W

+ + + +

(∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ )( ) ( )

where t t t, ,1 2 3, S1 and Q can be obtained by substituting0w = in coefficients t1 w( ), t2 w( ), t3 w( ), S1 w( ) and Q w( ),

respectively.

Appendix D. Explicit expressions of Fba, Fca, Fda andFea

After some algebraic calculations, the solutions ofequations (23)–(26) can be obtained as

FS

Q, D.1ba

p 1 ww

=-L ( )

( )( )

FS

Q, D.2ca

p 2 ww

=L ( )

( )( )

FS

Q, D.3da

p 3 ww

=L ( )

( )( )

FS

Q, D.4ea

p 4 ww

=L ( )

( )( )

8

J. Phys. B: At. Mol. Opt. Phys. 50 (2017) 185401 H R Hamedi et al

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where

S t t t , D.51 22

3 2 32

42w w w w= - W + W( ) ( ) ( ) ( )(∣ ∣ ∣ ∣ ) ( )

S t te ,D.6

2 1 42

2 3 4i

1 2 3w w w= W W - W W W - Wf-( ) ∣ ∣∣ ∣ ∣ ∣∣ ∣∣ ∣ ∣ ∣ ( ) ( )( )

S t te ,D.7

3 2 32

1 3 4i

2 2 3w w w= W W - W W W - Wf( ) ∣ ∣∣ ∣ ∣ ∣∣ ∣∣ ∣ ∣ ∣ ( ) ( )( )

S t e , D.84 2 2 4 1 3iw w= W W + W W f( ) ( )(∣ ∣∣ ∣ ∣ ∣∣ ∣ ) ( )

Q t t

t t

t t t

2 cos . D.9

2 3 12

22

1 2 32

42

1 22

3 22

32

12

42

1 2 3 4

w w ww ww w w

f

= W + W+ W + W

- - W W - W W+ W W W W

( ) ( ) ( )(∣ ∣ ∣ ∣ )( ) ( )(∣ ∣ ∣ ∣ )( ) ( ) ( ) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣ ∣∣ ∣∣ ∣∣ ∣ ( ) ( )

Appendix E. Kerr nonlinear coefficient

One may write the Maxwell equations under the slowlyvarying envelope approximation as

zc

ti , E.1

p pb a

1 *h¶W

¶+

¶W

¶= F F- ( )

where b a ba* rF F = , as well as aF and bF (together with cF , dFand eF ) represent the amplitudes of atomic wavefunctions foreach atomic state and satisfy the relation

1. E.2a b c d e2 2 2 2 2F + F + F + F + F =∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( )

Initially all atoms are assumed to be in the ground state añ∣ . Asthe Rabi frequency of the probe field is much weaker than thatof the control fields, one can neglect the depletion of groundlevel añ∣ , and one has 1aF » . Adopting a perturbation treat-ment of the system response to the first order of probe field,we can take L L L

kF = S F( ) L a b c d e, , , ,=( ). Here LkF( ) is

the kth order part of LF in terms of pW , where0b c d e

0 0 0 0F = F = F = F =( ) ( ) ( ) ( ) , and 1a0F =( ) , while 0a

1F =( ) .Thus, to the first order in pW we may write

, E.3a a0F = F ( )( )

, E.4b a b a b a b ba1 0 1 0 1 1* * rF F = F F = F F = F = ( )( ) ( ) ( ) ( ) ( ) ( )

, E.5c a c a c a c ca1 0 1 0 1 1* * rF F = F F = F F = F = ( )( ) ( ) ( ) ( ) ( ) ( )

, E.6d a d a d a d da1 0 1 0 1 1* * rF F = F F = F F = F = ( )( ) ( ) ( ) ( ) ( ) ( )

. E.7e a e a e a e ea1 0 1 0 1 1* * rF F = F F = F F = F = ( )( ) ( ) ( ) ( ) ( ) ( )

In this limit equation (E.2) reduces to

1. E.8a b c d e0 2 1 2 1 2 1 2 1 2F + F + F + F + F =∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( )( ) ( ) ( ) ( ) ( )

Using equations (E.3)–(E.8), the right-hand side of waveequation (E.1) can be represented as

i i

i 1

i i .

E.9

b a b a

b b c d e

ba ba ba ca da ea

1 0 2

1 1 2 1 2 1 2 1 2

1 1 1 2 1 2 1 2 1 2

*h h

h

hr hr r r r r

F F = F F

= F - F + F + F + F

= - + + +

∣ ∣[ (∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ )]

(∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ )( )

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

The first term i ba1hr( ) shows the linear part of the right-hand side

of wave equation (E.1) which was featured in equation (7). Inaddition, i Kerr i ba ba ca da ea

1 1 2 1 2 1 2 1 2h hr r r r r- = - + + +(∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ )( ) ( ) ( ) ( ) ( )

represents the nonlinear part of the right-hand side of waveequation (E.1). From this expression, the explicit form of thenonlinear coefficient Kerr can be readily derived as

Kerr . E.10ba ba ca da ea1 1 2 1 2 1 2 1 2r r r r r= + + +(∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ) ( )( ) ( ) ( ) ( ) ( )

As Fij is the Fourier transform of ij1r( ), the coefficients for ij

1r( )

can be obtained by taking 0w = in the coefficients given inappendix D. Replacing the coefficients for ij

1r( ) in this way intoequation (E.10) yields

S

Q QS S S SKerr . E.111

2 12

22

32

42=

-+ + +

∣ ∣(∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ) ( )

ORCID iDs

J Ruseckas https://orcid.org/0000-0002-7151-6368

References

[1] Arimondo E 1996 Progress in Optics (Elsevier: Amsterdam)[2] Harris S E 1997 Phys. Today 50 36[3] Lukin M D 2003 Rev. Mod. Phys. 75 457–72[4] Fleischhauer M, Imamoglu A and Marangos J P 2005 Rev.

Mod. Phys. 77 633–73[5] Wu Y and Yang X 2005 Phys. Rev. A 71 053806[6] Fleischhauer M and Juzeliūnas G 2016 Slow, stored and

stationary light Optics in Our Time ed M D Al-Amri et al(Cham: Springer) pp 359–83

[7] Hau L V, Harris S E, Dutton Z and Behroozi C H 1999 Nature397 594

[8] Fleischhauer M and Lukin M D 2000 Phys. Rev. Lett. 845094–7

[9] Liu C, Dutton Z, Behroozi C H and Hau L V 2001 Nature409 490

[10] Phillips D F, Fleischhauer A, Mair A, Walsworth R L andLukin M D 2001 Phys. Rev. Lett. 86 783

[11] Fleischhauer M and Lukin M D 2002 Phys. Rev. A 65 022314[12] Juzeliūnas G and Carmichael H J 2002 Phys. Rev. A 65 021601[13] Bajcsy M, Zibrov A S and Lukin M D 2003 Nature 426 638[14] Lin Y W, Liao W T, Peters T, Chou H C, Wang J S, Cho H W,

Kuan P C and Yu I A 2009 Phys. Rev. Lett. 102 213601[15] Wu Y, Saldana J and Zhu Y 2003 Phys. Rev. A 67 013811[16] Zhang Y, Anderson B and Xiao M 2008 Phys. Rev. A 77

061801[17] Zhang Y, Khadka U, Anderson B and Xiao M 2009 Phys. Rev.

Lett. 102 013601[18] Wu Y and Deng L 2004 Phys. Rev. Lett. 93 143904[19] Huang G, Deng L and Payne M G 2005 Phys. Rev. E 72

016617[20] Li L and Huang G 2010 Phys. Rev. A 82 023809[21] Si L G, Yang W X, Lü X Y, Hao X and Yang X 2010 Phys.

Rev. A 82 013836[22] Yang W X, Chen A X, Lee R K and Wu Y 2011 Phys. Rev. A

84 013835[23] Chen Y, Bai Z and Huang G 2014 Phys. Rev. A 89 023835[24] Joshi A, Brown A, Wang H and Xiao M 2003 Phys. Rev. A 67

041801

9

J. Phys. B: At. Mol. Opt. Phys. 50 (2017) 185401 H R Hamedi et al

Page 11: DQG ...Arindam Ghosh, Khairul Islam, Dipankar Bhattacharyya et al.-Interplay of classical and quantum dynamics in a thermal ensemble of atoms Arif Warsi Laskar, Niharika Singh, Arunabh

[25] Li J H, Lü X Y, Luo J M and Huang Q J 2006 Phys. Rev. A 74035801

[26] Schmidt H and Imamoglu A 1996 Opt. Lett. 21 1936–8[27] Wang H, Goorskey D and Xiao M 2002 Opt. Lett. 27 258–60[28] Niu Y and Gong S 2006 Phys. Rev. A 73 053811[29] Sheng J, Yang X, Wu H and Xiao M 2011 Phys. Rev. A 84

053820[30] Hamedi H R and Juzeliūnas G 2015 Phys. Rev. A 91 053823[31] Harris S E and Yamamoto Y 1998 Phys. Rev. Lett. 81 3611–4[32] Lukin M D and Imamoğlu A 2000 Phys. Rev. Lett. 84 1419–22[33] Wang Z B, Marzlin K P and Sanders B C 2006 Phys. Rev. Lett.

97 063901[34] Shiau B W, Wu M C, Lin C C and Chen Y C 2011 Phys. Rev.

Lett. 106 193006[35] Chen Y H, Lee M J, Hung W, Chen Y C, Chen Y F and Yu I A

2012 Phys. Rev. Lett. 108 173603[36] Vivek Venkataraman K S and Gaeta A L 2013 Nat. Photon. 7

138–41[37] Maxwell D, Szwer D J, Paredes-Barato D, Busche H,

Pritchard J D, Gauguet A, Weatherill K J, Jones M P A andAdams C S 2013 Phys. Rev. Lett. 110 103001

[38] Baur S, Tiarks D, Rempe G and Dürr S 2014 Phys. Rev. Lett.112 073901

[39] Boller K J, Imamoğlu A and Harris S E 1991 Phys. Rev. Lett.66 2593–6

[40] Ruseckas J, Juzeliūnas G, Öhberg P and Barnett S M 2007Phys. Rev. A 76 053822

[41] Ruseckas J, Mekys A and Juzeliūnas G 2011 Phys. Rev. A 83023812

[42] Paspalakis E and Knight P L 2002 J. Opt. B: QuantumSemiclass. Opt. 4 S372

[43] Schnorrberger U, Thompson J D, Trotzky S, Pugatch R,Davidson N, Kuhr S and Bloch I 2009 Phys. Rev. Lett. 103033003

[44] Unanyan R G, Otterbach J, Fleischhauer M, Ruseckas J,Kudriašov V and Juzeliūnas G 2010 Phys. Rev. Lett. 105173603

[45] Ruseckas J, Kudriašov V, Yu I A and Juzeliūnas G 2013 Phys.Rev. A 87 053840

[46] Lee M J, Ruseckas J, Lee C Y, Kudriasov V, Chang K F,Cho H W, Juzeliūnas G and Yu I A 2014 Nat. Commun.5 5542

[47] Ruseckas J, Kudriašov V, Juzeliūnas G, Unanyan R G,Otterbach J and Fleischhauer M 2011 Phys. Rev. A 83063811

[48] Bao Q Q, Zhang X H, Gao J Y, Zhang Y, Cui C L and Wu J H2011 Phys. Rev. A 84 063812

[49] Raczynski A, Rzepecka M, Zaremba J andZielinska-Kaniasty S 2006 Opt. Commun. 260 73

[50] Raczynski A, Zaremba J and Zielinska-Kaniasty S 2007 Phys.Rev. A 75 013810

[51] Beck S and Mazets I E 2017 Phys. Rev. A 95 013818[52] Payne M G and Deng L 2002 Phys. Rev. A 65 063806[53] Korsunsky E A and Kosachiov D V 1999 Phys. Rev. A 60

4996–5009[54] Shpaisman H, Wilson-Gordon A D and Friedmann H 2005

Phys. Rev. A 71 043812[55] Fleischhaker R and Evers J 2008 Phys. Rev. A 77 043805[56] Dey V R K, Evers T N and Kiffner M J 2015 Phys. Rev. A 92

023840[57] Braje D A, Balić V, Goda S, Yin G Y and Harris S E 2004

Phys. Rev. Lett. 93 183601[58] Dey T N and Agarwal G S 2007 Phys. Rev. A 76 015802[59] Kang H and Zhu Y 2003 Phys. Rev. Lett. 91 093601[60] Rodrigo A, Vicencio M I M and Kivshar Y S 2003 Opt. Lett.

28 1942–4[61] Heebner E, Boyd J and Park Q H R W 2002 Phys. Rev. E 65

036619[62] Andrea Melloni F M and Martinelli M 2003 Opt. Photonics

News 14 44–8[63] Liu X J, Jing H and Ge M L 2004 Phys. Rev. A 70 055802[64] Agrawal G P 2001 Nonlinear Fiber Optics 3rd edn (New York:

Academic)[65] Hasegawa A and Matsumoto M 2003 Optical Solitons in

Fibers (Berlin: Springer)[66] Xie X T, Li W B and Yang W X 2006 J. Phys. B: At. Mol. Opt.

Phys. 39 401[67] Wu Y and Deng L 2004 Phys. Rev. Lett. 93 143904[68] Burger S, Bongs K, Dettmer S, Ertmer W, Sengstock K,

Sanpera A, Shlyapnikov G V and Lewenstein M 1999 Phys.Rev. Lett. 83 5198–201

[69] Denschlag J et al 2000 Science 287 97[70] Huang G, Szeftel J and Zhu S 2002 Phys. Rev. A 65 053605[71] Kivshar Y S and Luther-Davies B 1998 Phys. Rep. 298 81[72] Lin Y and Lee R K 2007 Opt. Express 15 8781[73] Xie X T, Li W, Li J, Yang W X, Yuan A and Yang X 2007

Phys. Rev. B 75 184423[74] Yang W X, Hou J M, Lin Y and Lee R K 2009 Phys. Rev. A 79

033825[75] Wu Y and Deng L 2004 Opt. Lett. 29 2064[76] Hang C, Huang G and Deng L 2006 Phys. Rev. E 74 046601[77] Si L G, Yang W X, Liu J B, Li J and Yang X 2009 Opt.

Express 17 7771[78] Hang C and Huang G 2010 Opt. Express 18 2952[79] Zhu C and Huang G 2011 Opt. Express 19 1963[80] Liu J B, Liu N, Shan C J, Liu T K and Huang Y X 2010 Phys.

Rev. E 81 036607[81] Li L and Huang G 2010 Phys. Rev. A 82 023809[82] Chen Y, Chen Z and Huang G 2015 Phys. Rev. A 91 023820[83] Scully M O and Zubairy M S 1997 Quantum Optics

(Cambridge: Cambridge University Press)[84] Harris S E and Hau L V 1999 Phys. Rev. Lett. 82 4611–4

10

J. Phys. B: At. Mol. Opt. Phys. 50 (2017) 185401 H R Hamedi et al