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https://doi.org/10.1209/0295-5075/103/34001http://iopscience.iop.org/article/10.1209/0295-5075/94/54006http://iopscience.iop.org/article/10.1209/0295-5075/94/54006http://iopscience.iop.org/article/10.1088/0034-4885/80/1/016401http://iopscience.iop.org/article/10.1088/0034-4885/80/1/016401http://iopscience.iop.org/article/10.1209/0295-5075/84/10006http://iopscience.iop.org/article/10.1209/0295-5075/84/10006http://dx.doi.org/10.1103/PhysRevA.90.062134http://dx.doi.org/10.1103/PhysRevA.90.062134

August 2013

EPL, 103 (2013) 34001 www.epljournal.orgdoi: 10.1209/0295-5075/103/34001

Realization of the driven nonlinear Schrödinger equationwith stationary light

P. Das1, C. Noh1 and D. G. Angelakis1,2

1 Centre for Quantum Technologies, National University of Singapore - 3 Science Drive 2, Singapore 1175432 Science Department, Technical University of Crete - Chania, Crete, 73100, Greece, EU

received 12 December 2012; accepted in final form 31 July 2013published online 27 August 2013

PACS 42.50.Gy – Effects of atomic coherence on propagation, absorption, and amplificationof light; electromagnetically induced transparency and absorption

PACS 42.65.Pc – Optical bistability, multistability, and switching, including local field effects

Abstract – We introduce a versatile platform for studying nonlinear out-of-equilibrium physics.The platform is based on a slow light setup where an optical waveguide is interfaced with coldatoms to realize the driven nonlinear Schrödinger equation with a potential. We compare theproposed setup with similar setups using Bose-Einstein condensates and investigate the system’sresponse under coherent driving for a lattice potential. The slow light setup provides novel anglesin the study of nonlinear dynamics due to its advantages in introducing and modulating thedriving, the extra tunability over the sign and strength of the available nonlinearities, and thepossibility to electromagnetically carve out the underlying potential on demand.

Copyright c⃝ EPLA, 2013

Introduction. – The nonlinear Schrödinger equation(NLSE) arises naturally in many areas of physics. It de-scribes the propagation of an electromagnetic field in anonlinear medium [1,2] or the interacting ground stateof a Bose-Einstein condensate where it is known as theGross-Pitaevskii equation [3]. The out-of-equilibriumphysics in such systems are of fundamental importance,where the nonlinearity of the system changes transportproperties of a driven system, for example. Previousstudies in this category include the nonlinear transportproperties of Bose-Einstein condensates (BECs) in vari-ous potentials [4–12] and soliton propagation in nonlineardielectrics [13]. Quantum transport of few-quanta pulsesin a strongly nonlinear electromagnetically induced trans-parency (EIT) setup [14] has also been studied recently,aimed at photon switching applications [15,16].

In this letter, we introduce a versatile platform forstudying nonlinear out-of-equilibrium physics, based onthe stationary light system [17,18]. The ability to con-trollably introduce a tunable potential for a trapped-correlated-polariton gas [19], makes it an attractiveplatform to realize and explore quantum nonlinear dynam-ics complementing cold atoms and nonlinear optics setups.As an example of out-of-equilibrium physics, the transportproblem is investigated in the semiclassical limit, where weassume that a coherent input field is continuously drivingthe waveguide. In this case, the resulting dynamics for

the trapped fields are of the driven NLSE nature and thetransmission spectrum in the linear and nonlinear regimescan be studied. In the linear case the setup realizes a text-book transmission problem where the transmission reso-nances and the bandgap due to periodicity are observed.In the nonlinear case, the interplay between the nonlinear-ity and the lattice potential is manifest. We further per-form a linear stability analysis to investigate the feasibilityof observing the transmission spectra. Our analysis revealsthat only the lowest branch of the transmission spectrain the multi-valued region will be populated. We brieflymention a possible method to stabilize other branches.

System. – Figure 1 depicts a schematic diagram ofour proposed setup, an optical waveguide interfaced withan ensemble of cold atoms, and the EIT-based atomiclevel configuration. We note that possible experimentalplatforms can be based on a hollow core fiber filled withultracold atoms [20–24] or a tapered fiber evanescentlycoupled to cold atoms trapped near the surface of thefiber [25,26]. The resulting dynamics is formally simi-lar to that of a BEC in an optical lattice and also sys-tems studied in nonlinear optics. Our system, however,exhibits advantages such as a more efficient preparationof the source (drive), the ability to control the (effective)mass and the (EIT-induced) interaction parameters overa wide range of values. Finally, we note that the proposed

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P. Das et al.

Fig. 1: (Colour on-line) Schematic diagram of the system con-sisting of an optical waveguide interfaced with an ensemble ofcold atoms. Two counterpropagating control fields, Ω±, drivethe 4-level atoms and create a Bragg grating that traps the in-cident light Ψ+. The modulation in the atomic density inducesan effective periodic potential acting on the trapped light. Thetrapped stationary dark-state polaritons formed in the waveg-uide can be made to obey the nonlinear Schrödinger equationwith optically tunable interaction parameters.

system also allows the study of out-of-equilibrium quan-tum many-body physics, which we plan to investigate inthe future.

As described in [15,16,27,28], the dark-state polari-ton operators describing the trapped fields are Ψ̂± =g√

2πn0ʱ/Ω±, where ʱ are the slowly varying partsof the trapped quantum fields and Ω± = Ω are classicalcontrol fields with ± denoting the directions the fields aretravelling in; n0 is the mean density of atoms coupled tothe waveguide and g is the atom-field coupling strength.Introducing the symmetric and anti-symmetric combina-tions of the polariton operators Ψ = (Ψ+ + Ψ−)/

√2 and

A = (Ψ+ − Ψ−)/√

2, A can be shown to adiabaticallyfollow Ψ, i.e., A = −ik0Lcoh(mR/m)∂zΨ, in the limitof large optical depth. k0 is the wave number of thelattice potential and Lcoh = |∆0|

2+(Γ/2)2

Γ1Dn0|∆0| is the charac-teristic length scale, termed the coherence length [16];m = − Γ1Dn04νg(∆0+iΓ/2) is the complex effective mass and mRis the real part of m; νg ≈ νΩ2/(πg2n0) is the groupvelocity with ν the velocity in an undoped fibre; ∆0 isthe two-photon detuning shown in fig. 1; Γ1D denotesthe sponteneous emission rate into the guided modes; Γis the total spontaneous emission from the excited states|b⟩ and |d⟩.

A polaritonic lattice potential can be introduced bycreating a periodic modulation in the atomic density:n = n0 + n1 cos2(k0z) with n1 ≪ n0 [19]. In this casethe dynamics of the symmetric polariton operator is gov-erned by the NLSE:

i∂tΨ = −1

2m∂2zΨ + 2V cos

2(k0z)Ψ + 2χΨ†Ψ2, (1)

1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

V

b

Fig. 2: (Colour on-line) (a) The real part of the interactionparameter χ̄ as a function of the one-photon detuning ∆p/Γand Rabi frequency Ω/Γ. (b) The effective polaritonic latticedepth as a function of the Rabi frequency Ω/Γ.

where V is an effective lattice depth and χ an effec-tive polariton-polariton interaction strength. SubstitutingΨ±(z, t) = ψ±(z, ϵ)e−iϵt into the above equation yields thefollowing dimensionless coupled mode equations:

∂z̄ψ+ −i

2m

lcohmR(ψ+ − ψ−) −

ilcoh2

[ϵ̄− V̄ − V̄ cos(2z̄)

− χ̄2

(ψ+ + ψ−)†(ψ+ + ψ−)](ψ+ + ψ−) = 0, (2)

∂z̄ψ− −i

2m

lcohmR(ψ+ − ψ−) +

ilcoh2

[ϵ̄− V̄ − V̄ cos(2z̄)

− χ̄2

(ψ+ + ψ−)† (ψ+ + ψ−)]

(ψ+ + ψ−) = 0, (3)

where z and ψ± have been scaled by 1/k0,√

k0 andother effective interaction parameters by the recoil energyER = k20/(2mR), except χ which has been further scaledby 1/k0; lcoh = k0Lcoh quantifies the ratio between the co-herence length and the lattice constant. We note in pass-ing that the above coupled mode equations are similar,but not equivalent, to the nonlinear coupled mode equa-tions arising in the study of nonlinear wave propagationin one-dimensional periodic structures [29].

In terms of bare optical parameters, the dimension-less lattice depth and nonlinearity are given by V̄ =ΛΓ21Dδn0n1|∆0|8Ω2k20(∆

20+Γ2/4)

and χ̄ = Λ2Γ1D∆p

4lcoh(∆2p+Γ2/4)

(1 − i Γ2∆p

), where

Λ = Ω2/(Ω2 − δ∆0/2). ∆p and δ are the single- andtwo-photon detunings shown in fig. 1. For the optical pa-rameters, we assume δ = −0.01, ∆0 = −50, Γ1D = 0.2,n0 = 107m−1, n1 = 106m−1, k0 = 104m−1, whereall the frequencies are in units of the spontaneous emis-sion rate, Γ. With these values, which are within reachof near-future experiments in tapered and hollow corefibers [20–26], a broad range of effective interaction pa-rameters can be obtained as depicted in fig. 2.

Equation (1) is equivalent to the Gross-Pitaevskii equa-tion with an optical lattice potential, showing formalsimilarity between the proposed photonic system and aBEC-based system. While transport problems involvingcold atoms and BECs have been studied extensively [4–12],our proposal utilizes a fundamentally different system

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Realization of the driven nonlinear Schrödinger equation with stationary light

where the tunably interacting fields are hybrid light-matter excitations and therefore offer a complementarysystem with distinct advantages. One important advan-tage is that the non-equilibrium driving conditions arisemuch more naturally in the present system. Preparing acoherent source of an atomic BEC is much more difficultcompared to preparing a coherent source of photons, i.e.,a laser, although there have been a great progress in pro-ducing a guided matter waves [11,30–32]. The proposedsystem also allows efficient measurements of the outputintensities and correlation functions using standard opti-cal methods, which are difficult to perform in experimentsinvolving BECs.

In this letter, we investigate the transport dynamicsin the situation depicted in fig. 1, where an input fieldimpinges on the waveguide from the left. We specifythe natural boundary conditions ψ+(z = 0) = α andψ−(z = d) = 0, where α depends on the field strengthof the driving laser impinging from the left and d = k0Lis the dimensionless length of the waveguide. The trans-mission spectrum, i.e., the transmittivity as a function ofthe probe field detuning ϵ̄, is investigated in both the lin-ear and nonlinear regimes in the semiclassical limit withemphasis on effects of the lattice potential. Finally, a lin-ear stability analysis is performed to show the prospect ofobserving the characteristic features found in the trans-mission spectrum.

Linear regime. – In the linear regime, our systemprovides a realization of a classic transmission problemof a particle moving through a potential. Thus, usingthe periodic optical medium, it is possible to build a tun-able optical experiment to simulate the Schrödinger equa-tion in a controlled environment using stationary photons.Moreover, the shape of the photonic potential can be en-gineered, by carving out the atomic density distributionas shown in [19], to test other classic text-book problems.Note that the regime can be achieved without the 4thlevel in fig. 1, which alleviates experimental difficultiesconsiderably.

Figure 3 shows the transmittivity as a function ofthe input field detuning for two different values of V̄for lcoh = 0.25 with the total loss determined by β =(d/lcoh)(Γ/|∆0|) = 12π/50 ≈ 0.75.

When V̄ = 0, i.e., when Ω/Γ is large enough, thereare transmission peaks at the resonances ϵ̄0 = (nπ)2 forn = 1, 2, 3, . . . and at an additional point ϵ̄0 = 1. Thesetransmission resonances are an artefact of the Bragg trap-ping produced by the interaction of the counterpropagat-ing control fields with the atoms [16]. As Ω/Γ is tuneddown to increase the effective potential depth to 1, thecontinuum shifts towards right by 2V̄ (maximum poten-tial height) as expected, and additional resonances developin ϵ̄ < 2V̄ signifying bound states. The energy of the firstbound state can be readily calculated within the pertur-bation theory as V̄ − V̄ 2/8, which matches the startingpoint of the first resonance in the figure. Furthermore, the

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

e

Transmittivity

a

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

e

Transmittivity

b

Fig. 3: Transmission spectra for lcoh = 0.25, d = 3π, β =12π/50 ≈ 0.75, V̄ = 0 (a), and V̄ = 1.0 (b). The bound stateand the first bandgap are clearly shown in (b).

locations of other resonances can be approximated from ageneralization of the free-field resonance points d

√ϵ̄ = nπ

to d√ϵ̄− V̄ + V̄ 2/8 = nπ, such that only the quasi-kinetic

energy part is used to determine the resonance condition.This approximation works well if the input field detuningis much larger than the effective lattice depth, i.e. ϵ̄ ≫ V̄ ,because in this case the effects of the potential is per-turbative and the solution is expected to be close to thefree-field case.

Non-linear regime. – The nonlinear transport prob-lem in the presence of a potential has been studied in detailin various physical systems. As we have explained earlier,however, the present system offers unique advantages overthe previous works: tunability of the parameters, largenonlinearity and readily available coherent sources. Wefocus on the transmission spectrum as an example of thetransport problem and study the interplay between thenonlinearity and periodicity.

As we tune the single-photon detuning ∆p from a posi-tive to a negative value, the self-nonlinearity changes andthe magnitude and the direction of the nonlieanr shiftchanges. This is depicted in fig. 4(a) where the trans-mission spectra for χ̄ = −0.02, 0, 0.02 are shown by thethin (red), dashed (blue) and thick (black) curves, respec-tively. Higher magnitudes of nonlinearity, correspondingto smaller ∆p, result in lower transmission peaks due tohigher nonlinear losses. The sign and magnitude of χ̄ canbe tuned by varying, for example, the single-photon de-tuning ∆p as shown in fig. 2(a).

Fixing χ̄ and increasing the input field strength α hassimilar effects on the transmission spectrum to fixing αand increasing χ̄. This is expected because the effectivephotonic nonlinearity inducing the nonlinear shift is deter-mined by χ̄|ψ̄|2, where the overbar denotes averaging. |ψ̄|2of course depends on the value of the driving coherent fieldamplitude α, but because of the nonlinearity, the exactnature of this dependence is not obvious. To better un-derstand the physics, we have investigated the amount ofshift of the first transmission peak, with respect to the lin-ear case, as a function of the effective nonlinearity, χ̄|α|2.The result is depicted in fig. 4(b), where the solid line(black) represents the shift for α = 0.5 and the triangular

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P. Das et al.

0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

e

Transmittivity

a

0.004 0.008 0.012 0.016

0.1

0.15

0.2

0.25

0.3

c a2

Nonlinearshift

b

Fig. 4: (Colour on-line) (a) Transmission spectra for V̄ = 0.5,lcoh = 0.25, α = 0.5, d = 3π, β ≈ 0.75, and three differentvalues of |χ̄|. For repulsive nonlinearity (χ̄ = 0.02, ∆p > 0),one can see the shift of the transmission peaks to the right(thick, black curve). For attractive nonlinearity (χ̄ = −0.02,∆p < 0), the shift is in the opposite direction (thin, red curve).(b) Nonlinear shift as a function of the effective nonlinearity,χ̄|α|2, for different values of |α|. The solid line (black) andthe triangular points (red) correspond to α = 0.5 and 0.25,respectively.

0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

e

Transmittivity

a

0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

e

b

Fig. 5: Transmission spectra for lcoh = 0.25, α = 1.0, χ̄ = 0.02,d = 3π, β ≈ 0.75, and V̄ = 0 (a) and 1 (b).

points (red) correspond to α = 0.25. The nonlinear shiftdepends only on the effective nonlinearity to a very goodapproximation and furthermore, the dependence seems tobe almost linear for larger values of the effective nonlin-earity. Therefore, one should be able to observe similareffects for even lower values of nonlinearity.

Next, we investigate the effects of the polaritonic lat-tice on the transmission properties of the photons in therepulsive case, ∆p > 0. As we decrease Ω, the entire spec-trum shifts to the right as shown in fig. 5. At the sametime, a gap develops where the transmission is stronglysuppressed. The result of the shift and the formation ofthe bandgap is that the resonance peaks get pushed to-wards each other as the polaritonic lattice deepens. NearV̄ = 1.0, the second and the third peaks merge and a pecu-liar feature is observed between ϵ̄ = 1.5 and ϵ̄ = 2.0: a hor-izontal peak near the bandgap region appears, as seen infig. 5(b). As far as we are aware, this feature has not beenobserved before and is a unique feature arising due to theinterplay between the nonlinear-shift and the bandgap. Itis easier to see the origin of this feature with more extremeparameters as shown in fig. 6. A “duck head”-like featureis observed, which arises due to stronger suppression ofthe transmission at the tip of the transmission resonancethat falls on the edge of the bandgap. As lcoh is increased,

Fig. 6: Appearance of a “duck head”-like feature for smallcoherence length. The parameter values are: lcoh = 0.1, α =0.1, χ̄ = 0.1, d = 3π, β ≈ 0.94, and V̄ = 1.

the resonance peaks broaden and part of the “duck head”structure gets absorbed, resulting in the horizontal peakshown in fig. 5(b).

Linear stability analysis. – So far, we have focusedon the presence of multiple-valued transmission peaks,since it is a common phenomenon in externally drivennonlinear systems. At certain parameter regime we havefound that a “duck head” feature appears, which arisesdue to the interplay between the nonlinearity and theperiodic lattice potential. More generally, three trans-mission points may appear in the transmission spectrafor sufficiently strong nonlinearity, which we denote asthe upper (U), middle (M) and lower (L) branch. Aprevious time-dependent–driving study in a similar BECsetting suggests that only the lower branch is stable [6,33].We performed a linear stability analysis to check thestability of all three branches in the present system bythe following method. Imagine applying a small pertur-bation (δψ ≪ 1) to the obtained stationary solutionsψI0 : ΨI(z, t) = (ψI0(z) + δψ(z, t))e−iϵt, where I = U ,M and L. Now, substituting this in eq. (1) and neglectinghigher-order terms, one obtains the equation in terms ofthe perturbation δψ,

i∂δψ

∂t= − 1

2m∂2δψ

∂z2+ (2V cos2(z) − ϵ)δψ

+ 4χ|ψI0|2δψ + 2χψ2I0δψ. (4)

Figure 7 illustrates the stationary solutions of eqs. (2)and (3) and the perturbed solutions found from eq. (4)for the three branches. In our numerical calculation, wehave applied a small Gaussian profile as an initial per-turbation and considered the following boundary condi-tions: δψ(0, t) − i lcohmRm ∂zδψ(0, t) = 0 and δψ(d, t) +i lcohmRm ∂zδψ(d, t) = 0. It can be clearly seen fromfigs. 7(a) and (b) that the solutions of the upper and mid-dle branches are not linearly stable. We have further cal-culated the quantity,

Ξ =ln{Re[δψ(z, t + δt)]} − ln{Re[δψ(z, t)]}

δt, (5)

34001-p4

Realization of the driven nonlinear Schrödinger equation with stationary light

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

z

ΨU z ΨU0 z

ΨU0 z

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

z

ΨM zΨM0 z

ΨM0 z

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

z

ΨL zΨL0 z

ΨL0 z

(a) (b) (c)

Fig. 7: (Colour on-line) The solutions of the (a) upper, (b) middle, and (c) lower branches along with the perturbation areshown as a function of space. The blue (solid) lines represent the solutions of eqs. (2) and (3), whereas the red (dotted) linesshow the behavior of the solutions along with the perturbation. Only the lower branch is stable.

which in the limit of δt → 0 and t → ∞, approachesthe largest eigenvalue of the perturbation mode evolvingunder eq. (4) [34,35]. Explicit calculations show that theeigenvalues for the upper and middle branches are positivein the steady state and therefore unstable.

We therefore conclude that the resonant transport willbe suppressed during the propagation in the presence ofa strong nonlinearity. This is in good agreement with thefindings reported in [6,33]. It may, however, be possibleto stabilize the upper branch to experimentally access itby applying a temporal modulation in the potential, asshown in [6] for a double-well potential. We leave this fora future work.

Summary and conclusion. – We have introduced aslow light system exhibiting EIT nonlinearities that pro-vides an interesting platform for investigating linear andnonlinear out-of-equilibrium physics under the influence ofa potential. We have shown that through varying certainoptical parameters, such as single-photon detunings, onecan probe linear and nonlinear regimes with or withoutthe lattice potential and thereby observe interesting phe-nomena in the transmission spectrum. Moreover, we haveobserved that a “duck head”-like feature appears at theedge of the bandgap that results from the interplay be-tween the lattice potential and nonlinearity. We have alsoperformed a linear stability analysis in order to show whichbranches of the transmission spectra can be populated.

∗ ∗ ∗

We would like to acknowledge the financial support pro-vided by the National Research Foundation and Ministryof Education, Singapore. We thank Dr B. M. Rodŕıguez-Lara for many helpful discussions, especially at the earlystage of the work. PD thanks Dr J. Bandyopadhyay formany useful discussions.

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