Soliton Dimer-soliton scattering in bilayerQuasi-one-dimensional Dipolar Bose-EinsteinCondensates

Gautam Hegde, Pranay Nayak, Ratheejit Ghosh andRejish Nath

Department of Physics, Indian Institute of Science Education and Research, Dr.Homi Bhabha Road, Pune 411008, Maharashtra, India.

Abstract. We discuss scattering between a bright soliton and a soliton dimerin bilayer quasi-one-dimensional dipolar Bose-Einstein condensates. The dimer isformed by each soliton from both layers due to the attractive inter-layer dipole-dipole interaction. The dipoles within each layer repel each other, and a stable,bright soliton is stabilized via attractive contact interactions. In general, thescattering is inelastic, transferring the kinetic energy into internal modes of bothsoliton dimer and single soliton. Our studies reveal rich scattering scenarios,including dimer-soliton repulsion at small initial velocities, exchange of atomsbetween dimer and single soliton and soliton fusion at intermediate velocities.Interestingly, for some particular initial velocities, the dimer-soliton scatteringresults in a state of two dimers. At large initial velocities, the scattering is elasticas expected.

Submitted to: J. Phys. B: At. Mol. Phys.

arX

iv:2

110.

0014

4v1

[co

nd-m

at.q

uant

-gas

] 1

Oct

202

1

CONTENTS 2

Contents

1 Introduction 2

2 Setup and Model 2

3 Soliton-soliton scattering in a single layer 3

4 Dimer-soliton scattering 4

5 Summary 6

6 Acknowledgements 6

1. Introduction

Collisional or scattering dynamics of solitons is ofinterest in both fundamental science and applications[1–3]. True solitons in one dimension, owing to theintegrability of the corresponding non-linear equation,should pass through each other without suffering anychanges in shape and velocity, i.e., undergoes anelastic scattering. In condensates, strictly speaking,we do not deal with solitons but solitary waves dueto both the external confinement and their quasi-one-dimensional (Q1D) nature and thus can experiencein-elastic collisions [4]. The bright solitons typicallyinteract upon contact, with an effective interactiongenerated by the interference of the two wave packets,with its range decays exponentially [5, 6]. The natureof soliton interaction, whether being attractive orrepulsive, depends on their relative phase.

Bose-Einstein condensates (BECs) opened up anew avenue to explore the soliton physics in a morecontrolled manner in particular for bright solitons [3, 7–15]. The collisional dynamics between bright solitonsis studied in both usual condensates with contactinteractions [4, 7, 14, 16–27] and dipolar BECs [28, 30–39], which have been exploited to generate entangledsoliton pairs [17, 18] and design interferometers [19, 22,27, 40, 41]. Contrary to the short-range condensates, indipolar BECs, the long range and anisotropic nature ofdipole-dipole interaction (DDI) lead to novel scenariossuch as the multi-dimensional solitons [42–47] andinterlayer effects [48–53]. Further, interlayer dipolarinteractions can stabilize soliton complexes such asmolecules or dimers, crystals and filaments [31, 51, 54,55].

This paper analyses the scattering dynamics ofthree bright solitons in a bilayer system of dipolar

condensates. Two out of three solitons, one from eachlayer, form a dimer and collide with a single soliton. Toappreciate the new dynamics emerging in the dimer-soliton setup, we first briefly discuss the scatteringof two Q1D solitons in a single tube. In the lattercase, we observe the repulsion between solitons forsmall initial velocities, both inelastic destruction andfusion at intermediate velocities and elastic scatteringat large initial velocities. The elastic scattering iswhat one would expect for true solitons, in which theypass through each other. In the dimer-soliton case,we observe a non-trivial pattern of inelastic scatteringas a function of the initial velocity, especially atintermediate velocities. The latter can be understoodbecause the three-soliton setup possesses more low-lying modes than the two-soliton setup. A completecharacterization of the role of internal modes is notpossible due to its dynamical nature, and we exploreonly a few of the exciting scattering scenarios keepingthe interaction strengths fixed. We show that forsmall initial velocities, the dimer and soliton repel eachother, whereas, for intermediate velocities, an exchangeof atoms between dimer and soliton occurs. Thelatter also leads to soliton fusion in the upper layer,transferring the kinetic energy into internal modes,even destroying the soliton in the bottom layer. Themost striking scenario is the emergence of two solitondimers out of a dimer and soliton collision.

The paper is structured as follows. In section 2,we introduce the physical setup and the governingequations. In section 3, we briefly discuss thescattering dynamics of two Q1D solitons in a singletube. In section 4, we discuss the scattering of asoliton-dimer and a single soliton. Finally, we concludein section 5.

2. Setup and Model

We consider a bilayer setup of Q1D DBECs in whichthe radial direction is harmonically confined with afrequency ω⊥ and no confinement along the z direction.The two tubes are separated along the y-axis by aseparation of y0. We assume the dipoles are alignedalong the y-axis, and both ω⊥ and y0 are sufficientlylarge to ensure no hopping of particles between thelayers. The dipoles repel along the z-axis, i.e. withineach layer and attract along the y-axis. Due to theattractive inter-layer DDIs (along y-axis), a solitondimer can be formed using one soliton from each layer

CONTENTS 3

Figure 1. Schematic diagram of the scattering setup of bilayerQ1D dipolar BECs. The two solitons on the left side form adimer. Dipoles are oriented along the y-axis and therefore thesolitons in the top layer repel each other.

[51]. Effectively, we have a soliton dimer shared amongtwo layers and a single soliton, which are well separatedalong the z-axis as shown in figure 1. The condensatesare described by coupled non-linear Gross-Pitaevskiiequations (NLGPE)

i~∂tΨj(r) =

[− ~2

2m∇2 + Uj(r) + g |Ψj(r)|2 (1)

+

2∑m=1

∫dr′Vd (r− r′) |Ψm (r′)|2

]Ψj(r, t),

where Ψj is the condensate wavefunction of the jth

layer, U1(r) = mω2⊥[x2 + (y − y0)

2]/2 and U2(r) =

mω2⊥(x2 + y2)/2 are the external trapping potentials

of first and second layers, respectively. Vd(r) = gd(1−3 cos2 θ/|r|3) is the dipole-dipole potential with theinteraction strength gd = Nµ0µ

2/4π, with µ beingthe dipole moment and θ being the angle betweendipole axis y and radial vector r between two dipoles.N is the number of atoms per soliton. Hence theupper layer has 2N atoms, and the lower layer has Natoms. The parameter g = 4πa~2N/m characterizesthe strength of the contact interaction with a beingthe s-wave scattering length. The wavefunctionssatisfy the normalization conditions

∫d3r|Ψ1|2 = 2

and∫d3r|Ψ2|2 = 1, leaving all three solitons with

same number of atoms at t = 0. We keep thecondensates strongly confined in the xy-plane, and thechemical potential is much smaller than ~ω⊥ to ensurethe Q1D nature of the condensates. In the lattercase, we can factorize the BEC wave function at eachlayer as Ψj(x, y, z) = ψj(z)φj(x, y), where φj(x, y) isthe ground-state wave function of the xy harmonicoscillator. Using the Fourier transform of the dipolarpotential, convolution theorem and integrating overthe radial directions, we arrive at coupled 1D NLGPEs[51, 52, 54, 56],

i~∂tψj(z) =

[− ~2

2m∂2z +

g

2πl2⊥|ψj(z)|2

+2gd3

M∑m=1

1

2π

∫dkze

ikzznm (kz)Fmj (kz)

]ψj(z), (2)

where nm(kz) is the Fourier transform of density|ψm(z)|2, and

Fij (kz) =

∫ ∫dkxdky

(3k2y

k2x + k2y + k2z− 1

)× e− 1

2 (k2x+k2

y)l2⊥−ıky(yi−yj). (3)

We use the dimensionless quantities g = g/2π~ω⊥l3⊥and gd = gd/2π~ω⊥l3⊥ with l⊥ =

√~/mω⊥. Also,

we take y0 = 7.5l⊥ throughout the paper. Thestable/unstable regions of both soliton dimer and asingle Q1D bright soliton as a function of g and gdfor the configuration we consider are provided in [51].Based on that, we obtain the ground states of a solitondimer and a single soliton via imaginary time evolutionof the Q1D NLGPEs for g = −0.9, gd = 0.37, which wekept through out the paper.

Before indulging in the scattering dynamics ofsoliton dimer-soliton scattering, we briefly outline themain scattering scenarios in the case of two Q1Dsolitons in a single layer. The scattering of two Q1Dsolitons in disconnected parallel tubes are discussed in[28] and is characterized by inelastic scattering in whichthe kinetic energy is resonantly transferred into theinternal modes of the soliton for an intermediate rangeof initial velocities. Such resonant inelastic scatteringcan even destroy the solitons.

3. Soliton-soliton scattering in a single layer

Here, we briefly discuss the scattering of two Q1Dsolitons in a single tube along the z-axis. As mentionedbefore, we take g = −0.9 (attractive short-rangeinteraction) and gd = 0.37. Since the dipoles arealong the y-axis, the solitons repel each other alongthe z-axis. Note that the contact interactions arekept attractive to stabilize the initial solitons. Wetake the initial positions of the solitons are at z(t =0) = ±100l⊥, which is sufficiently far enough toneglect the DDI between them. Then, each of thesolitons is provided with equal initial velocities butin opposite directions. As a function of the initialvelocity or equivalently initial momentum ki, weobserve four scattering scenarios. For sufficiently smallinitial velocities, the kinetic energy of the solitonsis insufficient to overcome the dipolar repulsion, andthe solitons repel away from each other, as shownin figure 2(a). As the initial velocity increases,the solitons come closer and closer, and above acritical initial speed, they come in contact. For theintermediate velocities, we observe both soliton fusionand soliton destruction. In the case of soliton fusion[see figure 2(b)], the resultant final soliton remains at

CONTENTS 4

Figure 2. Different scenarios in two soliton scattering dynamicsfor g = −0.9, gd = 0.37. (a) depicts the repulsion between thesolitons for a small initial momentum of kil⊥ = 0.0015. (b)shows the solitons fusion at kil⊥ = 0.0025 whereas (c) showsthe soliton destruction at kil⊥ = 0.078. (d) shows the elasticscattering at a large initial momentum of kil⊥ = 0.5.

0.02 0.04 0.06 0.08 0.10 0.12 0.14

kil⊥

1

3

5

7

wm/w

i

−0.3

−0.1

0.1

0.3

∆M

Figure 3. The ratio of maximum width wm and the initialwidth wi of the soliton in the second layer as a function of theinitial momentum ki. Here, the value of ki is restricted to theintermediate values for which inelastic scattering occurs. Wealso show the mass imbalance ∆M of the first layer calculate atsufficiently long time after the scattering.

rest at the point of contact, with a partial excitationof its breathing mode. In the destructive case [seefigure 2(c)], the kinetic energy is completely transferredto the internal modes, and the solitons disperse afteremerging out of the collision. At sufficiently largevelocities, they undergo elastic scattering similar tothat of an ideal soliton in which the solitons passthrough each other. The latter scenario is alsocharacterized by an interference pattern in the contactregion, as seen in figure. 2(d). The initial velocitiesat which the four scattering scenarios occur dependscritically on the interaction strengths g and gd.

4. Dimer-soliton scattering

In this section, we study the scattering between asoliton dimer and a single soliton for g = −0.9, gd =0.37. At t = 0, the ground state solutions of solitondimer and single soliton are placed at z = ±100l⊥.Then they are subjected to collisions providing thesame initial velocities to both the dimer and the singlesoliton. The setup is such that we have two solitonsin the first layer and a single soliton in the secondlayer. Before the collision, the soliton in the secondlayer forms the dimer with a soliton in the first layer.In general, the soliton collision is sensitive to thephase difference between them, and thus, the initialseparation can play an important role. By the timethe scattering becomes prominent, the long-range DDIcan induce different phases in three solitons.

As we show below, the ratio wm/wi together withthe mass imbalance in the first layer can provide uswith a hint at the nature of the scattering, where wm

is the maximum width attained by the lone solitonin the second layer during the scattering and wi isits initial width. A large wm indicates a significanttransfer of kinetic energy into the internal modes anda resonant transfer can even blow up the soliton (wm →∞). The mass imbalance is defined as the normalizeddifference in probabilities or masses in the left and

right halves of the upper layer: Ml =∫ L/2

0|ψ1|2dz and

Mr =∫ L

L/2|ψ1|2dz at a sufficiently long time after the

scattering, where L is the total box size, i.e.,

∆M =Mr −Ml

Mr +Ml, (4)

where Ml + Mr = 2 in our case. ∆M < 0 (∆M > 0)implies there is more fraction of condensates in the left(right) half of the cylindrical tube after the scattering.The behaviour of wm as a function of ki for kil⊥ ≥ 0.02is shown in figure 3. The most interesting scatteringscenarios are shown in figure 4 and the correspondingki are marked by dashed vertical lines in figure 3.

For small initial velocities (corresponds to kil⊥ <0.02), both dimer and single soliton reflect each otheras shown in figure 4(a) for kil⊥ = 0.008, which isalso identical to that of two-soliton scattering discussedin section 3. The latter occurs due to the repulsiveinteraction between the solitons in the upper layer. Infigure 4(a), the upper plot shows the trajectories of twosolitons in the first layer, and the lower plot shows thetrajectory of the soliton in the second layer. For thescattering scenario in figure 4(a), we have ∆M = 0(not shown in figure 3) since there is no transfer ofatoms between the solitons in the upper layer.

Upon increasing the initial speeds further, enoughto overcome the long-range repulsion, we observe non-trivial collisional dynamics as a function of the initialvelocity that is entirely different from simple two

CONTENTS 5

Figure 4. Different scenarios in the scattering between soliton dimer and a single soliton for g = −0.9, gd = 0.37. (a) depicts therepulsion between the solitons for a small initial momentum of kil⊥ = 0.008. (b)-(e) show the inelastic scattering at the intermediatevelocities. The initial velocities are (b) kil⊥ = 0.022, (c) kil⊥ = 0.0247, (d) kil⊥ = 0.06 and (e) kil⊥ = 0.063. (e) depicts the solitonfusion in the upper layer and (f) is for large initial velocity of kil⊥ = 0.5 for which the elastic scattering occurs.

soliton scattering. Due to more degrees of freedom,the dimer-soliton setup possesses more low-lying modesthan the two-soliton system, which intuitively explainswhy one would expect more complicated scatteringdynamics in the former. That is also supported bythe series of inelastic peaks in the wm/wi vs ki plotshown in figure 3. In both two soliton and threesoliton cases, the modes dynamically change as thesolitons approach each other. The latter prevents acomprehensive analysis of the role of these elementarymodes on the soliton scattering.

In figure 4(b) for kil⊥ = 0.022, we see that afterthe scattering, the soliton dimer passes through the

single soliton, but at the cost of the destruction ofthe single soliton which it collided with [see the upperpanel in figure 4(b)]. The lower panel in figure 4(b)indicates that the breathing mode of the lone solitonin the second layer is excited by the collision event.This signifies the importance of inter-layer effects indisconnected dipolar condensates. In addition, for thiscase, we see that ∆M > 0 (figure 3) implying thatthere are more atoms on the right half of the first layerafter the collision. That means there was a transfer ofatoms among the solitons in the first layer, leaving abigger soliton in the dimer, making it asymmetric afterthe collision. All the blue shaded regions in figure 3

REFERENCES 6

−20 0 20 40 60z/l⊥

−10

0

10

20y/l⊥

Blah

0.000

0.004

0.008

0.012

0.016

0.020

0.024

0.028

Blahs

Figure 5. The density of the condensates in both layers afterthe scattering for kil⊥ = 0.02671 at ω⊥t = 4680, for which wesee the emergence of two dimers. The values of other parametersare the same as that of figure 4.

correspond to the same dynamics as that for kil⊥ =0.022, indicating the non-monotonous dependence ofscattering dynamics on the initial velocity. Also, thedensity plot in figure 4(b) reveals that the velocityof the dimer increased slightly after the destructivecollision with the single soliton.

At the intermediate initial velocities, a smallchange in ki can lead to drastically different dynamicsas shown in figure 4(c) for which kil⊥ = 0.0247. Afterthe scattering, a dimer does survive but moves in theopposite direction. The remaining soliton in the firstlayer is destroyed by the impact of the scattering.As one can see, the soliton in the second layer isalso significantly affected by the scattering due to theinterlayer DDI. We also see that figure 4(c) is almosta mirror image of what is depicted in figure 4(b).Figure 4(c) is characterized by ∆M < 0 since in upperlayer significant population is moving towards the left.As ki increases further [see figure 4(d)], we see thatall three solitons survive after the scattering, leavinga new dimer and a single soliton. Two features arevisible in this case: (i) in the first layer the final singlesoliton carries more atoms (hence higher density) thanthe initial single soliton and (ii) the vibrational modesof the dimer are also weakly excited after the collision.

Another interesting scenario is the soliton fusionin the first layer, which is shown in figure 4(e) forkil⊥ = 0.063. In this case, after the collision, almosta complete transfer of kinetic energy into the internalmodes leaves the two solitons in the first layer fusedtogether. It results in the destruction of the solitonin the lower layer. It is clear from the top panelof figure 4(e) that the final soliton in the first layerexhibits strong breathing mode oscillations and also ajet of atoms keep emitting from it. The destructionof the soliton in the bottom layer is featured by twostructureless clouds of atoms [see the bottom panelin figure 4(e)]. At high initial velocities, shown infigure 4(f), the dimer and the soliton pass through eachother after the collision indicating an elastic scattering.

The foremost exciting scenario we observed is

the emergence of an additional dimer (effectivelyfour solitons) from the dimer-soliton (three solitons)scattering. An example of it is shown in figure 5, inwhich we see two soliton dimers are moving away fromeach other after the collision. This indicates that thesoliton in the bottom layer breaks up into two solitonsupon collision. Due to the asymmetry of the setup,it is hard to obtain the case in which the soliton inthe lower layer breaks up into two identical solitons.We would also like to stress that scattering dynamicsin multilayer dipolar condensates generally show highsensitivity to initial separations and velocities apartfrom their dependence on interaction strengths.

5. Summary

In conclusion, we studied the scattering dynamicsbetween a soliton dimer and a single soliton in asetup of bilayer dipolar condensates. The long-rangeand anisotropic nature leads to interesting inelasticscattering dynamics, and in particular, the effectof interlayer DDI becomes manifest in all scatteringscenarios. The intriguing nature of scatteringdynamics is attributed to the internal modes of thesoliton and dimer. Our studies also open up manypossibilities to explore shortly, for instance, the roleof the orientation of dipoles in bilayer systems andthe potential to engineer soliton interferometers usingdipolar condensates. We would also expect morecomplex scattering dynamics in the case of four dipolarQ1D solitons.

6. Acknowledgements

R.N. acknowledges DST-SERB for Swarnajayanti fel-lowship File No. SB/SJF/2020-21/19. We acknowl-edge National Supercomputing Mission (NSM) for pro-viding computing resources of ’PARAM Brahma’ atIISER Pune, which is implemented by C-DAC and sup-ported by the Ministry of Electronics and InformationTechnology (MeitY) and Department of Science andTechnology (DST), Government of India. G.H. andP.N. acknowledge the funding from DST India throughan INSPIRE scholarship.

References

[1] Hasegawa A and Kodama Y 1995 Solitonsin Optical Communications Clarendon Press,Oxford, UK

[2] Mollenauer L F and Gordon J P 2006, Solitonsin Optical Fibers: Fundamentals and ApplicationsAcademic Press, San Diego, CA

[3] Kevrekidis P G, Frantzeskakis D J, and Carretero-Gonzalez R 2007 Emergent Nonlinear Phenomena

REFERENCES 7

in Bose-Einstein Condensates: Theory andExperiment (Berlin: Springer)

[4] Martin A D, Adams C S and Gardiner S A 2008Phys. Rev. A 77 013620

[5] Gordon J P 1983 Opt. Lett. 8 596

[6] Mitschke F M and Mollenauer L F 1987 Opt. Lett.12 355

[7] Nguyen J H V, Dyke P, Luo D, Malomed B A andHulet R G 2014 Nat.Phys. 10 918

[8] Khaykovich L, Schreck F, Ferrai G, Bourdel T,Cubizolles J, Carr LD, Castin Y and Salomon C2002 Science 296 1290

[9] Strecker K E, Partridge G B, Truscott A G andHulet R G 2002 Nature 417 150

[10] Marchant A L, Billam T P, Wiles T P, Yu M M H,Gardiner S A and Cornish S L 2013 Nat. Commun.4 1865

[11] Medley P, Minar M A, Cizek N C, BerryrieserD and Kasevich M A 2014 Phys. Rev. Lett. 112060401

[12] Khawaja U A, Stoof H T C, Hulet R G, StreckerK E and Partridge G B 2002 Phys. Rev. Lett. 89200404

[13] Cornish S L, Thompson S T and Wieman C E2006 Phys. Rev. Lett. 96 170401

[14] Nguyen J H V, LuoD and Hulet R G 2017 Science356 422

[15] Everitt P J et al 2017 Phys. Rev. A 96 041601(R)

[16] Martin A D, Adams C S and Gardiner S A 2007Phys. Rev. Lett. 98 020402

[17] Gertjerenken B, Billam T P, Blackley C L, SueurC R L, Khaykovich L, Cornish S L and Weiss C2013 Phys. Rev. Lett. 111 100406

[18] Billam T P, Blackley C L, Gertjerenken B, CornishS L and Weiss C 2013 J. Phys.: Conf. Ser. 497012033

[19] McDonald G D et al 2014 Phys. Rev. Lett. 113013002

[20] Lepoutre S et al 2016 Phys. Rev. A 94 053626

[21] Meznarsic T et al 2019 Phys. Rev. A 99 033625

[22] Wales O J et al 2020 Commun. Phys. 3 51

[23] Parker N G, Martin A M, Cornish S L and AdamsC S 2008 J. Phys. B At. Mol. Opt. Phys. 41045303

[24] Billam T P, Cornish S L and Gardiner S A 2011Phys. Rev. A 83 041602

[25] Choudhury S, Sreedharan A, Mukherjee R,Streltsov A and Wuster S 2020 Phys. Rev. A 101,043604

[26] Khawaja U Al and Stoof H T C 2011 New J. Phys.13 085003

[27] Martin A D and Ruostekoski J 2012 New J. Phys.14 043040

[28] Nath R, Pedri P and Santos L 2007 Phys. Rev. A76 013606

[29] Nath R, Pedri P and Santos L 2009 Phys. Rev.Lett. 102 050401

[30] Abdullaev F K and Brazhnyi V A 2012 J. Phys.B: At. Mol. Opt. Phys. 45, 085301

[31] Baizakov B B, Al-Marzoug S M and Bahlouli H2015 Phys. Rev. A 92 033605

[32] Chiquillo E 2014 Laser Phys. 24 085502

[33] Cuevas J, Malomed B A, Kevrekidis P G, andFrantzeskakis D J 2009 Phys. Rev. A 79 053608

[34] Eichler R, Zajec D, Koberle P, Main J and WunnerG 2012 Phys. Rev. A 86 053611

[35] Gao P et al 2021 J. Phys. B: At. Mol. Opt. Phys.54 135301

[36] Abdullaev F Kh et al 2014 J. Phys. B: At. Mol.Opt. Phys. 47 075301

[37] Young-S L E et al 2011 J. Phys. B: At. Mol. Opt.Phys. 44 101001

[38] Edmonds M J, Bland T, Doran R and Parker NG 2017 New J. Phys. 19 023019

[39] Abdullaev F Kh, Gammal A, Malomed B A andTomio L 2013 Phys. Rev. A 87 063621

[40] Polo J and Ahufinger V 2013 Phys. Rev. A 88053628

[41] Helm J L, Cornish S L and Gardiner S A 2015Phys. Rev. Lett. 114 134101

[42] Raghunandan M, Mishra C, Lakomy K, Pedri P,Santos L and Nath R 2015 Phys. Rev. A 92 013637

[43] Nath R, Pedri P and Santos L 2008 Phys. Rev.Lett. 101 210402

[44] Mishra C and Nath R Phys. Rev. A 94 033633

[45] Pedri P and Santos L 2005 Phys. Rev. Lett. 95200404

[46] Tikhonenkov I, Malomed B A, and Vardi A 2008Phys. Rev. Lett. 100 090406

[47] Chen X Y, Chuang Y L, Lin C Y, Wu C M, Li YY, Malomed B A and Lee R K 2017 Phys. Rev. A96 043631

[48] Huang C-C and Wu W-C 2010 Phys. Rev. A 82053612

[49] Klawunn M and Santos L 2009 Phys. Rev. A 80013611

[50] Koberle P and Wunner G 2009 Phys. Rev. A 80063601

REFERENCES 8

[51] Lakomy K, Nath R and Santos L 2012 Phys. Rev.A 86 013610

[52] Lakomy K, Nath R and Santos L 2012 Phys. Rev.A 86 023620

[53] Muller S, Billy J, Henn E A L, Kadau H,Griesmaier A, Jona-Lasinio M, Santos L and PfauT 2011 Phys. Rev. A 84 053601

[54] Lakomy K, Nath R and Santos L 2012 Phys. Rev.A 85 033618

[55] Elhadj K M et al 2019 Phys. Scr. 94 085402

[56] Cai Y, Rosenkranz M, Lei Z and Bao W 2010Phys. Rev. A 82 043623