Solitons in Periodic and Quasiperiodic Optical …...c, the solutions of this equation can...

Introduction BEC Penrose tilling. VA Solution of equations Numerics Recent BEC. Conclusion. Refs

Solitons in Periodic and QuasiperiodicOptical Lattices.

Gennadiy BurlakCentro de Investigacion en Ingenierıa y Ciencias Aplicadas,Universidad Autonoma del Estado de Morelos, Cuernavaca,

Mor. Mexico.

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Outline

1 IntroductionA Bose–Einstein condensate (BEC)

2 BECQuantum theory.Gross-Pitaevskii equation.

3 Penrose tilling.Penrose tilling.

4 VAThe asymmetric variational approximation (VA)

5 Solution of equationsEigenfrequencies

6 NumericsNonlinear case

7 Recent BEC.Super photons.Solitons in 3D.Collapsing 3D soliton

8 Conclusion.Conclusion.

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A Bose–Einstein condensate (BEC)

First let me remind some simple properties of Bose particles. Thenumber of Bose noninteracting particles is:

N =∑k

[exp(

εk − µT

)− 1

]−1, (1)

where εk = ~2k2/2m is energy, and µ is chemical potential. Tocalculate the transition temperature T0, we replace in Eq.(1) thesum with the integral over all momentum states p = ~k ,

k = [2mε]1/2 /~, z = ε/T , that allows to rewrite N as

N

V= C

∫ ∞0

z1/2dz/[exp(z − µ

T)− 1

](2)

where C = C (T ), V is volume. From Eq.(2) µ can be interpretedas the implicit function of T and N/V as µ = µ(T ,N/V ) ≤ 0.For µ = 0 the integral Eq.(2) defines the critical temperatureT = T0 and the particle number corresponding to the negligiblechemical potential conditions µ = 0.

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Simple Einstein’s argument shows that for T ≤ T0 the number ofnoninteracting particles N can be rewritten as N = Nε>0 + Nε=0,

where Nε>0 = (T/T0)3/2N is the particle number with ε > 0,

while Nε=0 =[1− (T/T0)3/2

]N is the number of particles in the

ground state energy ε = 0. Thus, if T → 0 than Nε=0 → N, so allthe Bose particles transit to the ground state with ε = 0(Bose-Einstein condensate - BEC).

This is a simple conclusion of the ideal noninteracting particlesmodel. This state of matter was first predicted by Satyendra NathBose and Albert Einstein in 1924 -25.

What happens in a real situation with the interacting particles?

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A Bose–Einstein condensate (BEC) is a state of matter of a dilutegas of weakly interacting bosons confined in an external potentialand cooled to temperatures very near absolute zero (0 K or -273.15C). Under such conditions, a large fraction of the bosons occupythe lowest quantum state of the external potential, at which pointquantum effects become apparent on a macroscopic scale.

The first gaseous condensate was produced by Eric Cornell andCarl Wieman in 1995 at the University of Colorado at BoulderNIST-JILA lab, using a gas of rubidium atoms cooled to 170nanokelvin (nK) (1.7× 10−7 K). For their achievements Cornell,Wieman, and Wolfgang Ketterle at MIT received the 2001 NobelPrize in Physics.

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Quantum theory.

The state of the BEC can be described by the wavefunction of thecondensate ψ(r), so |ψ(r)|2 is interpreted as the particle density, so

the total number of atoms is N =∫dr |ψ(r)|2 = ‖ψ‖2. Provided

essentially all atoms are in the condensate (that is, have condensedto the ground state), and treating the bosons using mean fieldtheory, the energy E associated with the state ψ(r) is:

E =

∫dr

[~2

2m|5ψ(r)|2 + V (r) |ψ(r)|2 +

1

2G |ψ(r)|4

](3)

Minimising this energy with respect to variations δψ(r), andholding the number of atoms constant, yields the Gross-Pitaevskiiequation (GPE) (also a non-linear Schrodinger equation):

i~∂ψ(r)

∂t=

(−~252

2m+ V (r) + G |ψ(r)|2

)ψ(r) (4)

that normally applied for theoretical analysis.

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Quantum theory.

Our analysis is based on the 2D Gross-Pitaevskii equation for themean-field wave function, Ψ (x , y , t), written in the dimensionlessform assuming the self-attractive nonlinearity:

i∂Ψ

∂t= −1

2

(∂2

∂x2+

∂2

∂y2

)Ψ− |Ψ|2 Ψ− V (x , y)Ψ, (5)

where the QP lattice potential is taken as[H. Sakaguchi, B. A.Malomed, Phys. Rev. E 74, 026601 (2006); G. Burlak,A.B.Klimov, Phys. Lett. A 369, 510 (2007)]

V = V (x , y) = V0

M∑n=1

cos(k(n)r), (6)

with the set of wave vectorsk(n) = cos(2π(n − 1)k/M, sin(2π(n − 1)k/M and M = 5. Herewe focus on the basic case of the Penrose-tiling potential,corresponding to M = 5.

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Quantum theory.

Nonlinear Schrodinger equation (NSE)

i∂Φ

∂τ= [−O2

⊥ + V (r)+G |Φ|2]Φ, (7)

where G is a real function (the nonlinear coefficient), so that ifG > 0 the nonlinearity is repulsive, whereas for G < 0 thenonlinearity is attractive.

It is well known that for G < 0, and if N = ‖u0‖2 is above athreshold value Nc , the solutions of this equation can self-focusand become singular in a finite time. This phenomenon is calledwave collapse or blowup of the wave amplitude.

When G < 0, there exists only one solution of NSE which is real,positive and radially symmetric and for which ‖u0‖2 has theminimum value and this solution decays exponentially at infinity.This solution is called the ground state or Townes soliton.

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Quantum theory.

010

2030

4050 0

1020

3040

50

0

0.5

1

y

(a)

x

010

2030

4050 0

1020

3040

50

0

0.5

1

y

(b)

x

010

2030

4050 0

1020

3040

50

0

0.5

1

1.5

y

(c)

x

010

2030

4050 0

1020

3040

50

0

1

2

y

(d)

x

Figure : 4. Self-compressing of initial Gaussian package at attractiveinteraction G = −1 and the period of π-phase shift Tt = 8. (a) τ = 15;(b) 20; (c) 25; (d) 30.

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Quantum theory.

From the theory of nonlinear Schrodinger equations it is knownthat the Townes soliton has exactly the critical power for blowupNc = ‖u0‖2 ' 5.85 (for G = −1), therefore, it separates in somesense the region of collapsing and expanding solutions. Moreover,the Townes soliton is unstable, i.e. small perturbations of thissolution lead to either expansion of the initial data or blowup infinite time.

In nonlinear optics a spatial modulation of the nonlinear coefficientof the optical material is used to prevent such a collapse so thatthe optical beam becomes collapsing and expanding in alternatingregions and is stabilized in average. The same idea now has beenused in the field of matter waves to obtain a stable BEC soliton.

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Quantum theory.

Typical optical lattices

0

20

400

50

0

0.5

1

y

(a)

x

0

20

400

50

0

5

10

y

(b)

x

0

20

400

50

−2

0

2

y

(c)

x

0

50 0

50

−2

0

2

y

(d)

x

Figure : 2. (a) Init Gaussian package. (b),(c),(d) two-dimensionaldistribution of the trapping potential V (x , y , 0) induced by opticallattices (OL) for Penrose tiling (a pattern of tiles, which completely coveran infinite plane in an aperiodic manner) for cases (b) M = 4; (c)M = 5; (d) M = 7. 19 / 40


Gross-Pitaevskii equation.

In the normalized form, the respective 2D Gross-Pitaevskiiequation for mean-field wave function Ψ (x , y , t) in cubic-quinticnonlinearity case is

i∂Ψ

∂t= −1

2

(∂2

∂x2+

∂2

∂y2

)Ψ+g |Ψ|2 Ψ+κ |Ψ|4 Ψ−V0T (t)V (x , y)Ψ,

(8)where t is time, (x , y) are coordinates in the 2D plane (scaled so

that the OL period is π), V0 is the depth of the lattice,T (t) = 1 + (ε/2) cos(Ωt), ε and Ω being the amplitude andfrequency of its temporal modulation.

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The coefficient g < 0 ( further we normally use g = −1) in frontof the cubic nonlinear term in Eq. (7) implies that the nonlinearityis attractive. Actually, V0 is measured in units of the recoil energyof atoms trapped in the OL.Here, where Ω is the frequency, ε is the amplitude, V0 is the opticalpotential, t is the time, and (x , y) are coordinates in 2D of the OL.The quasiperiodic (QP) lattice potential of depth 2V0 is taken as

V (x , y) = −V0

K∑n=1

cos(k(n)r), (9)

with the set of wave vectorsk(n) = cos (2π(n − 1)/K ) , sin (2π(n − 1)/K ) and K = 5 orK ≥ 7. Here we focus on the basic case of the Penrose-tilingpotential, corresponding to K = 5. The 2D profile of such QPpotential is displayed below

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Fig.1. Quasiperiodic lattice (OL) Fig.2. Bose-Einstein-condensate.

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Penrose tilling.

A Penrose tiling is an example of non-periodic tiling generated byan aperiodic set of prototiles. Penrose tilings are named aftermathematician and physicist Roger Penrose, who investigatedthese sets in the 1970s. The aperiodicity of prototiles implies thata shifted copy of a tiling will never match the original. A Penrosetiling may be constructed so as to exhibit both reflection symmetryand fivefold rotational symmetry, as in the diagram:

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Penrose tilling.

A Penrose tiling has many remarkable properties, most notably: -Itis non-periodic, which means that it lacks any translationalsymmetry. -It is self-similar, so the same patterns occur at largerand larger scales.

Roger Penrose in the foyer of the Mitchell Institute forFundamental Physics and Astronomy, Texas A&M University,standing on a floor with a Penrose tiling.

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The asymmetric variational approximation (VA)

Variational methods (VA) have been quite useful in many problemsof nonlinear optics and BEC. To apply the VA to the presentmodel, we notice that Eq. (7) can be derived from Lagrangian

L =∫ +∞−∞ Ldx , with density

L =i

2(Ψ∗Ψt −ΨΨ∗t )− 1

2

(|Ψx |2 + |Ψy |2

)+

g

2|Ψ|4 +

κ

3|Ψ|6

+ V0V (x , y)T (t) |Ψ|2 , (10)

where the asterisk stands for the complex conjugation. Forisotropic potential the simplest isotropic ansatz and for the solitoncan be used.

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However for asymmetric potential due to insensitive to theparticular orientation of wave vectors k(n), this isotropicapproximation is too coarse and has to be generalized. Suchapproach for asymmetric potential we define as an anisotropicansatz that has to be written as

Ψans (x , y , t) = A(t) exp

(iφ(t) +

i

2(b1(t)x2 + b2(t)y2)

)(11)

· exp(−1

2

(x2

W 21 (t)

+y2

W 21 (t)

)), (12)

where T (t) =[1 + ε

2 cos(ωt)]

and all variables A(t), φ(t), b1,2(t),and W1,2(t) (amplitude, phase, radial chirp, and soliton width,respectively) are real.

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The first Euler-Lagrange equation following from effectiveLagrangian δLeff/δφ = 0 (δ/δφ stands for the variationalderivative), is tantamount to the conservation of the norm of thewave function, which is the single dynamic invariant of Eq. (7).Indeed, the norm of ansatz (11) is∫ ∫

|Ψans(x , y)|2 dxdy = πA2W1W2 ≡ N. (13)

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To study the boundary of stability for asymmetric 2D solitons wesearch for the solution in form

Wi(t) = wi0 + wi1(t)

where wi0 = const and wi1(t) have 1-st order of smallness (itconsidered case with small V0,; general case of V0 is latter studiednumerically).

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In zero order we obtain the equations for w0i

6π2w220 − 3N w10w20π + 4N2=0,

6π2w210 + 4N2 − 3N w10w20π= 0,

with a no-trivial solution

w10 = w20 ≡ w0 = 2N (κ/3π)1/2 /(N − 2π)1/2, and b1,20 = 0.(14)

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Eigenfrequencies

From the latter one can see that such solution (14) exists atN > 2π and κ > 0. Then the solution to linear 1st order equationsfor w1,2(t) can be readily obtained from considered Eqs. bystandard way. To seek a simplicity we write down here suchsolution for a static OL case with ε = 0 in form ( the solutions fortime-managed OL will be studied numerically in next Sec.)

w1(t) = −C4 exp(3 iM√

2π t

4N2κ)+C3 exp(

3 iM3/2√πt4N2κ

) + c .c . (15)

− 2

135κ5/2 (−2N I15 + N + 2N I25 + 4π I15 + 8π − 4π I25)

V0

√3N5

√πM3√M

exp(−πN2κ

12M), (16)

and

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Eigenfrequencies

w2(t) = C4 exp(3 iM√

2π t

4N2κ) + C2 exp(

3 iM3/2√πt4N2κ

)+c .c . (17)

+2

135κ5/2 (−2N I15 + N + 2N I25 − 12π + 4π I15 − 4π I25)

V0

√3N5

√πM3√M

exp(−πN2κ

12M), (18)

where M = N − 2π > 0 and C4,2 are complex-valued amplitudes,so in the follows we will use only Re(w1,2(t)) .

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Eigenfrequencies

From equations (15) and (17) we observe that consideredVA-system is a dynamic system having for κ > 0 two nontrivialeigenfrequencies ω1,2 which depending on the number of particlesN is

ω1 = 3 (N − 2π)√

2π /4N2κ, ω2 = 3 (N − 2π)3/2√π/4N2κ (19)

Such ”elastic” eigenfrequencies ω1,2 exist only at κ > 0 due to acompeting form of cubic-quintic nonlinearities. The dependence ω1and ω2 as function of N for case κ = 1 is shown in the followingfigures

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Eigenfrequencies

Eigenfrequencies plots:

Fig.2. The dependence of theeigenfrequencies ω1,2 2D soliton vsnumber of particles N.

Fig.3. Steady-state width of 2D solitonw10 = w20.

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Nonlinear case

The analytical results of preceding Section can be extended beyondof the smallness approximation only with the use of the numericalmethods applied directly to nonlinear equations for W1,2(t). Inwhat follows we study numerically the dynamics of soliton structure(widths) W1,2(t) for typical values N = 25 and ε = 1.96.

Such dynamics is shown for static OL (Ω = 0) in figures (timedependencies) and (phase plane) for values W1(0) = 3.77 andW2(0) = 3.76 which are close to the stationary values w1,20.

From figures we observe that for the solitons keeps the symmetricshape (with W1(t) 'W2(t)) with very small regular variations.The chirps keep their zero values as well.

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Nonlinear case

Fig.4. The dynamics of soliton widthW1,2(t).

Fig.5. Phase plane W2 vs W1.

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Nonlinear case

Fig.6. Dynamical chaos. Fig.7. Dynamical chaos in a phaseplane.

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Nonlinear case

Fig.8. Resonance to an eigenfrequency. Fig.9. Phase plane for a resonantdynamics.

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Nonlinear case

Fig.10. Fig.11.

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Nonlinear case

Fig.12. Fig.13.

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Super photons.

Recent BEC - (super-photons)!!!

Jan Klaers, Julian Schmitt, Frank Vewinger, Martin Weitz, Institutfur Angewandte Physik, Bonn, GermanyBose-Einstein condensation of photons in an opticalmicrocavity; Nature 468, 545-548 (November 2010);

Upon increasing the photon density, authors observed the followingBEC signatures: the photon energies have a Bose Einsteindistribution with a massively populated ground-state mode.

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Solitons in 3D.

In 3D case only the numerical methods (and powerful computers)have to be applied to study the dynamics of solitons

Figure : Numerical study of 3D solitons. For visualization of 3D space itis used the isosurfaces and direct color mapping at each level.

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Collapsing 3D soliton

Figure : Collapse 3D solitons.

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Conclusion.

1 We studied the properties and stability boundary of 2Dsolitons in asymmetric potentials in materials with competingcubic-quintic nonlinearity. In such potentials become possiblethe spatially asymmetric instabilities which however cannot bedescribed by radial symmetric solutions that normally areanalyzed by well-known variant of Variation Approximation(VA).

2 The VA is generalized for solutions with possible ellipticstructure. We defined the soliton profile for which theradial-symmetric shape is stable. Small deviations from such ashape lead to oscillatory solutions having two independenteigenfrequencies.

3 The behavior of such solutions in a potential with largeamplitude is studied numerically. It is considered the resonantcase when the frequency of the time variation (time managed)potential is close to one of the eigenfrequencies. In thissituation even stable (at static potential) solution leads to aweak time decay of the soliton state.

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Thank you for attention!

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