## Interband resonant transitions in two-dimensional hexagonal lattices: Rabi oscillations, Zener tunnelling, and tunnelling of phase dislocations

Optics Express, Vol. 16, Issue 18, pp. 14076-14094 (2008)

http://dx.doi.org/10.1364/OE.16.014076

Acrobat PDF (1390 KB)

### Abstract

We study, analytically and numerically, the dynamics of interband transitions in two-dimensional hexagonal periodic photonic lattices. We develop an analytical approach employing the Bragg resonances of different types and derive the effective multi-level models of the Landau-Zener-Majorana type. For two-dimensional periodic potentials without a tilt, we demonstrate the possibility of the Rabi oscillations between the resonant Fourier amplitudes. In a biased lattice, i.e., for a two-dimensional periodic potential with an additional linear tilt, we identify three basic types of the interband transitions or Zener tunnelling. First, this is a quasi-one-dimensional tunnelling that involves only *two* Bloch bands and occurs when the Bloch index crosses the Bragg planes away from one of the high-symmetry points. In contrast, at the high-symmetry points (i.e., at the M and Γ points), the Zener tunnelling is essentially two-dimensional, and it involves either *three* or *six* Bloch bands being described by the corresponding multi-level Landau-Zener-Majorana systems. We verify our analytical results by numerical simulations and observe an excellent agreement. Finally, we show that phase dislocations, or optical vortices, can tunnel between the spectral bands preserving their topological charge. Our theory describes the propagation of light beams in fabricated or optically-induced two-dimensional photonic lattices, but it can also be applied to the physics of cold atoms and Bose-Einstein condensates tunnelling in tilted two-dimensional optical potentials and other types of resonant wave propagation in periodic media.

© 2008 Optical Society of America

## 1. Introduction

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7. B. Bourlon, D. C. Glattli, B. Placais, J. M. Berroir, C. Miko, L. Forro, and A. Bachtold, “Geometrical Dependence of High-Bias Current in Multiwalled Carbon Nanotubes,” Phys. Rev. Lett. **92**, 026804 (2004). [CrossRef] [PubMed]

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9. M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener Tunneling in a Periodic Potential,” Phys. Rev. Lett. **91**, 230406 (2003). [CrossRef] [PubMed]

12. V. Agarwal, J. A. del Río., G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, M. Scalbert, M. Vladimirova, and B. Gil, “Photon Bloch Oscillations in Porous Silicon Optical Superlattices,” Phys. Rev. Lett . **92**, 097401 (2004). [CrossRef] [PubMed]

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20. L. Santos, M. A. Baranov, J. I. Cirac, H.-U. Everts, H. Fehrmann, and M. Lewenstein, “Atomic Quantum Gases in Kagomé Lattices,” Phys. Rev. Lett. **93**, 030601 (2004). [CrossRef] [PubMed]

21. A. A. Burkov and E. Demler, “Vortex-Peierls States in Optical Lattices,” Phys. Rev. Lett. **96**, 180406 (2006). [CrossRef] [PubMed]

24. A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, “Resonant Zener tunneling in two-dimensional periodic photonic lattices,” Opt. Lett. **32**, 325–327 (2007). [CrossRef] [PubMed]

27. D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, “Bloch oscillations in two-dimensional lattices,”New J. Phys. **6**, 41 (2004). [CrossRef]

9. M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener Tunneling in a Periodic Potential,” Phys. Rev. Lett. **91**, 230406 (2003). [CrossRef] [PubMed]

30. V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, “Nature of the Intrinsic Relation between Bloch-Band Tunneling and Modulational Instability,” Phys. Rev. Lett. **96**, 150402 (2006). [CrossRef] [PubMed]

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## 2. Shallow-lattice approximation

14. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices,” Phys. Rev. Lett. **96**, 053903 (2006). [CrossRef] [PubMed]

*𝝒*is the normalized nonlinear refractive index, ∇

^{2}=

*∂*

^{2}

_{x}+

*∂*

^{2}

_{y}and

**x**=(

*x*,

*y*),

*I*

*(*

_{g}**x**) describes the optical lattice, and

*I*

*(*

_{m}**x**) is a lattice tilt. We define the lattice potential as

*V*(

**x**)≡-

*𝝒*

*I*

*(*

_{g}**x**) and assume the tilt to be linear,

*𝝒*

*I*

*(*

_{m}**x**)=-

**αx**. Equation (1) in the shallow lattice approximation, |

*V*(

**x**)|≪1 (the main approximation used below), and for a weak linear tilt, |

**α**‖

**d**|/2≪|

*V*(

**d**/2)-

*V*(0)|, reduces to the linear Schrödinger equation

*t*is the propagation distance in the case of an optical beam in a periodic photonic structure and the normalized time in the case of BEC. In the model,

*V*(

*x*) is the periodic lattice potential,

*V*(

**x**+

**d**)=

*V*(

**x**), where

**d**is one of the lattice periods, see Fig. 1(b),

**α**is the acceleration of the lattice in the case of BEC and the steepness of the refractive index tilt in the case of the optical beam propagation. Below, we denote by

**Q**a reciprocal lattice vector.

*z*-direction is assumed to be much stronger than the nonlinearity of BEC, so that the condensate performs the ground-state quantum motion along this direction, while the trap in the (

*x*,

*y*)-plane is assumed to be weak and is neglected, i.e., it contains a large number of the lattice periods. The nonlinearity of BEC can be neglected if the Bloch oscillation period

*t*

*is much less than the characteristic time*

_{B}*t*

_{nonl}of the modulational instability development (see, for instance, [30

30. V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, “Nature of the Intrinsic Relation between Bloch-Band Tunneling and Modulational Instability,” Phys. Rev. Lett. **96**, 150402 (2006). [CrossRef] [PubMed]

*t*

*≪*

_{B}*t*

_{nonl}~1/

*γ*, where

*γ*is proportional to the

*s*-wave scattering length multiplied by the number of BEC atoms per lattice cell [19]. Since

*t*

*~|*

_{B}**Q**|/|

*α*

_{‖}| with

**Q**=

**b**

*(see Fig. 1(a)) and*

_{j}**α**

_{‖}the component of the acceleration parallel to

**Q**, we obtain the condition in the form

*γ*≪|

**α**α‖

**d**| (since |

**b**

*|~1/|*

_{j}**d**|). Under the above conditions Eq. (2) describes the condensate tunneling in a titled two-dimensional optical lattice.

24. A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, “Resonant Zener tunneling in two-dimensional periodic photonic lattices,” Opt. Lett. **32**, 325–327 (2007). [CrossRef] [PubMed]

14. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices,” Phys. Rev. Lett. **96**, 053903 (2006). [CrossRef] [PubMed]

*an arbitrary lattice*. On the other hand, for a deep lattice, which is the other limiting case, the continuous Eq. (2) is replaced by a set of coupled discrete equations for the amplitudes of Wannier states, which is the so-called tight-binding approximation [26, 40

40. F. Keck and H. J. Korsch, “Infinite-variable Bessel functions in two-dimensional Wannier-Stark systems,” J. Phys. A: Math. Gen. **35**, L105–L116 (2002). [CrossRef]

*x*).

*x*) has a narrow Fourier spectrum,

*D*be much smaller than the size of the Brillouin zone. In other terms, the initial condition is broad enough and covers at least several lattice periods. This means using broad optical beams in photonic crystals (as it was done in the recent experiment [14

**96**, 053903 (2006). [CrossRef] [PubMed]

9. M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J. H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener Tunneling in a Periodic Potential,” Phys. Rev. Lett. **91**, 230406 (2003). [CrossRef] [PubMed]

*C*(

*k*) can be modelled by a sum of the Dirac delta-functions and the Zener tunnelling is described by an LZM model.

*V*=|∑

*E*

*|*

_{l}^{2}. To have a hexagonal lattice the wave vectors should satisfy the corresponding symmetry, i.e., they should transform under the rotation by π/3 as follows:

**k**

_{1}→

**k**

_{2},

**k**

_{2}→

**k**

_{3},

**k**

_{3}→-

**k**

_{1}. The general expression for a hexagonal lattice (not necessarily created by an interference) can be given as the following Fourier series, infinite in general,

**b**

_{1}=

**k**

_{1}-

**k**

_{2},

**b**

_{2}=

**k**

_{1}-

**k**

_{3}, and

**b**

_{3}=

**k**

_{2}-

**k**

_{3}. Note that

**b**

_{3}=

**b**

_{2}-

**b**

_{1}. On the r.h.s. of Eq. (4) we denote by the symbol “…” the omitted higher-order terms in the base vectors

**b**

_{k}, i.e., the cosines of

*n*

**b**

_{l}*x*,

*n*≥3, and the cosines of various sums of

**b**

_{1}

**x**,

**b**

_{2}

**x**, and

**b**

_{3}

**x**, involving more than two terms (all cosines of the same kind have the same amplitude to satisfy the hexagonal rotation symmetry). The hexagonal lattice formed by using three plane waves, for instance, is given just by the first sum on the r.h.s. of Eq. (4):

*V*=

*V*

_{0}∑

^{3}

_{l}=

_{1}cos(

**b**

*l*

**x**).

*ε*

_{1}≠0 the fundamental periods of the lattice (4) can be selected as

*l*=1, 2, where

*b*=|

**b**

*| and the vectors*

_{l}**e**

_{1}and

*e*

_{2}give the reciprocal basis for

**e**

_{l}**n**

*=*

_{m}*δ*

*,*

_{l}*. Note also that*

_{m}*b*=2 (which can be arranged by an appropriate scaling of the spatial co-ordinate). In this case the vectors

**e**

_{1}and

**e**

_{2}connect the Γ-point with the M-points on the border of the Brillouin zone, see Fig. 1(a). We have

**b**

_{1}=(2,0),

**b**

_{2}=(1,√3), and

**b**

_{3}=(-1,√3). Two important special cases of the general hexagonal lattice are considered below. In the first case,

*ε*

_{1}=3/2,

*ε*

_{2}=1/4, and

*ε*

_{3}=1/2 (all other terms are set to zero), while the “triangular” lattice corresponds to

*ε*

_{1}=-1/2,

*ε*

_{2}=-1/4, and

*ε*

_{3}=1/2. Because the latter has the amplitude

*ε*

_{1}three times smaller, the efficiency of interband coupling is much lower in certain cases for the triangular lattices. Therefore, while our analytical theory is applicable to both lattices, we give numerical simulations for the hexagonal cosine lattice (5) only. Its Bloch band structure is given in Fig. 1(d).

## 3. Rabi oscillations

**α**=0 the interband transition can be observed in the form of periodic Rabi oscillations. In the limit of shallow lattice, instead of expanding the solution over the Bloch waves, it is more convenient to use the Fourier version of the Schrödinger equation (2) directly. Setting Θ(

**x**,

*t*)=∫d

**k**

*C*(

**k**,

*t*)

*e*

^{i}**, expanding the lattice potential into the Fourier series,**

^{kx}**k**=

**q**-

**Q**with

**q**lying in the first Brillouin zone, one obtains the equation for the Fourier coefficients

*q*

**q**where

*E*

_{0}(

**q**-

**Q**)=

*E*

_{0}(

**q**). Most of the Fourier amplitudes are nonresonant and can be neglected in the first-order approximation. Then Eq. (7) predicts oscillations between the resonant peaks in Fourier space defined by Bragg condition

*E*

_{0}(

**q**-

**Q**)=

*E*

_{0}(

**q**-

**Q**′) [42

42. V. S. Shchesnovich and S. Chávez-Cerda, “Bragg-resonance-induced Rabi oscillations in photonic lattices,” Opt. Lett. **32**, 1920–1922 (2007). [CrossRef] [PubMed]

**Q**/2 and -

**Q**/2 below) and the oscillations are equivalent to those in the two-level system:

*V*̂

**=**

_{-Q}*V*

^{̂}*

**and introduced the running index**

_{Q}*δ*

**for the points of the two resonant Fourier peaks. Since the Bloch waves are approximated as linear combinations of plane waves, the system (8)–(9) corresponds to the inter band oscillations between two Bloch bands. By analogy, these oscillations can be called Rabi oscillations. Rabi oscillations between several Bloch bands can be realized by placing the wave on one of the high-symmetry points on the boundary of the first Brillouin zone.**

_{q}**Rabi.avi**demonstrate the results of the simulations of Eq. (2) with a Gaussian beam as the initial condition, namely we use exp(-

*x*

^{2}/2

*w*

^{2}+

*i*

**q**

_{0}

**x**) with

*w*=20 and

**q**

_{0}=

**b**

_{1}/

_{2}, i.e., in Fourier space the beam is initially at the right X-point, see Fig. 1(a). We use spilt-step beam propagation method and monitor the dynamics also in the Fourier domain, see Fig. 2(b). The energy is periodically transferred between two X-points (±

**b**

_{1}/2) and, for a quantitative comparison with the predictions of the LZM system (8–9), we integrate the intensities of two interacting beams in the Fourier domain to obtain their normalized powers

*P*

_{1},

_{2}. The results presented in Fig. 2(c) allow to estimate, roughly, the period of these oscillations to be ~28.8. At the same time, the system (8–9) has a solution (

*δ*

**q**=0) in terms of harmonic functions with the period

*T*=

*π*/

*V*̂

**Q**=

*π*/(

*ε*

_{1}

*I*

_{0})≈41.9, for the used parameters of the lattice

*I*

_{0}=0.1 and

*ε*

_{1}=3/2. Most probably, the disagreement is due to the transitional dynamics in Fig. 2 and averaging over longer propagation time provides better comparison with analytical predictions, similar as in the square lattice case [42

42. V. S. Shchesnovich and S. Chávez-Cerda, “Bragg-resonance-induced Rabi oscillations in photonic lattices,” Opt. Lett. **32**, 1920–1922 (2007). [CrossRef] [PubMed]

*δ*

**q**=0 there is the frequency mismatch between the corresponding Fourier amplitudes:

*E*

_{0}(

**Q**/2+

*δ*

**q**)-

*E*

_{0}(-

**Q**/2+

*δ*

_{q})=

**Q**

*δ*

**q**resulting in a higher average Fourier power of the input wave over the Bragg reflected one (see also [42

42. V. S. Shchesnovich and S. Chávez-Cerda, “Bragg-resonance-induced Rabi oscillations in photonic lattices,” Opt. Lett. **32**, 1920–1922 (2007). [CrossRef] [PubMed]

## 4. Zener tunnelling

*x*) with a narrowFourier spectrum, we can use the expansion over the Bloch waves with a definite

*t*-dependent Bloch index

*q*=

**q**(

*t*), similar to Houston’s approach [43

43. W. V. Houston, “Acceleration of Electrons in a Crystal Lattice,” Phys. Rev. **57**, 184–186 (1940). [CrossRef]

*x*,

*t*)=exp{

*i*(

**k**

_{0}-

**α**

*t*)

**x**-

*i*∫

*dτ(*

^{t}**k**

_{0}-

**α**τ)

^{2}/2}, is an alternative to having a partial derivative in the governing equation in Fourier space, which would account for the linear potential according to the rule

**x**→

*i*

*∂*

**. Indeed, we get from Eq. (10)**

_{k}**q**-

**Q**,

*t*)=

*e*

*∫*

^{-i}*dτ*

^{t}*E*

_{0}(

**q**-

**Q**-α

**τ**)

*C*̃(

**q**-

**Q**,

*t*) one sees that any two coefficients

*C*(

**q**-

**Q**) and

*C*(

*-*

**q***′) are effectively coupled on the time interval where the property*

**Q***E*

_{0}(

*-*

**q****-**

*Q***α**

*t*)≈

*E*

_{0}(

**q**-

**Q**′-

*t) is satisfied, otherwise the coupling coefficients are oscillating about zero. Thus the interband transitions take place on the Bragg resonance planes. It is convenient to explicitly account for the Bragg resonance by defining the resonant point*

**α****q**

_{res}by setting

**-**

*q**-*

**Q**

**α***t*=

**q**_{res}-

*-*

**Q***(*

**α***t*-

*t*

_{0}) in the energy

*E*

_{0}. For simplicity, below we set

*t*

_{0}=0 (in this case one obtains the governing LZM-type models in the standard form). There are three types of Zener interband transitions in the hexagonal lattices: (

*i*) the quasi one-dimensional tunnelling (section 4.1), (

*ii*) tunnelling between three Bloch bands at the M-point (section 4.2), and (

*iii*) between the six Bloch bands at the Γ-point (section 4.3).

## 4.1. Quasi-one-dimensional Zener transitions

*I*

_{0}). This is a transition between two Bloch bands at an avoided crossing, which takes place along one of the borders of the irreducible Brillouin zone, i.e. the triangle ΓMX in Fig. 1, to which the Bragg plane is equivalent (after performing rotations by multiples of

*π*/3 and translations by the reciprocal lattice vectors). Let us briefly recall the derivation, which is similar as in the case of the square lattice [23] (see also [28, 29, 44

44. V. S. Shchesnovich and S. B. Cavalcanti, “Finite-dimensional model for the condensate tunnelling in an accelerating optical lattice,” J. Phys. B: At. Mol. Opt. Phys. **39**, 1997–2011 (1997). [CrossRef]

**q**

*=*

_{j}*Q*

*/2. Substituting the expression for the Bloch wave into Eq. (2) and requiring cancellation of the terms linear in*

_{j}*x*, which gives

**q**=-

**α**(the dot denotes derivative with respect to

*t*), we get a system of coupled equations for the incident

*C*

_{1}and Bragg reflected

*C*

_{2}amplitudes of the Bloch wave. By setting

*=*

**q****Q**

*/2-*

_{j}

*α**t*and (

*C*

_{1},

*C*

_{2})=

*e*

^{iϕ(t)}(

*c*

_{1},

*c*

_{2}), with

*ϕ*=(

**q**

^{2}(

*t*)+[

**q**(

*t*)-

**Q***]*

_{j}^{2})/4, the latter system is cast in the form of Zener [2], Landau [4], and Majorana [5

5. E. Majorana, “Orientated atoms in a variable magnetic field,” Nuovo Cimento **9**, 43–50 (1932). [CrossRef]

*E*at an avoided crossing:

*q*≡(

**Q**

_{j}*)*

**α***t*/2 is the running band parameter. All quasi one-dimensional cases are treated in the same way: there is a single resonant reciprocal lattice vector

**Q**which defines a Bragg plane and the probability of tunnelling is given by the LZM formula:

**α**_{⊥}being the component of the bias perpendicular to the respective Bragg plane. Thus, Eq. (14) gives the probability of transition between the two Bloch bands

*E*

_{1},

_{2}(

*q*), defined for the

*j*-th band as

*P*

*=|*

_{j}*c*

_{j}(∞)|

^{2}for the initial condition |

*c*

*(-∞)|=1. Due to the symmetry*

_{j}*t*→-

*t*and

*c*

_{1}→

*c*

^{*}

_{2},

*c*

_{2}→

*c*

^{*}

_{1}the two probabilities coincide. Note also that, in the case of a sufficiently small bias |

*|≪(*

**α***ε*

_{1}

*I*

_{0})

^{2}, Zener tunnelling is negligible when crossing the border of the first Brillouin zone (away from the high-symmetry point M). This means that the wave (beam) is totally Bragg-reflected at the boundary of the first Brillouin zone, i.e., it performs Bloch oscillations. There are three distinct cases corresponding to the three inequivalent Bragg planes, each one being equivalent to one side of the irreducible Brillouin zone. Recall that in the derivation of the LZM model (13) we have subtracted the mean value of the Bloch energy

*E*̄=(

**Q**

*)*

_{j}^{2}/8 (by using the phase transformation) which specifies the two bands involved in the tunnelling transition.

*I. Zener transitions on the XM-border*. In this case the resonant reciprocal lattice vector is

**Q**=

**b**

_{1}(equivalently

*b*

_{2}or

*b*

_{3}). The resonant Fourier components of the lattice are those with the coefficient

*ε*

_{1}in formula (4) and the LZM formula (14) becomes

*II. Zener transitions on the ΓM-border*. The resonant reciprocal lattice vector is

*Q*=2

**b**

_{1}(or its equivalent). The resonant Fourier components of the lattice are those with the coefficient

*ε*

_{2}. The shift by -

**b**_{1}and rotation by -

*π*/3 transforms the resonance point to a point lying on the ΓM-border. The probability of tunnelling reads

*III. Zener transitions on the ΓX-border*. The resonant reciprocal lattice vector is equivalent to

**Q**=

**b**_{1}+

**b**_{2}and the resonant Fourier components are those with the coefficient

*ε*

_{3}. The rotation by

*π*/3 followed by the shift by -

*b*

_{3}transforms the resonance point to a point lying on the ΓX-border. We obtain the probability as follows

*I*. above), similar to Rabi oscillations in Fig. 2. We show in Fig. 3 and the movie

**1D.avi**results of the simulations with the parameters |

**b**

*|=2,*

_{j}*I*

_{0}=0.1,|

*|=0.05, and*

**α***ε*

_{1}=3/2. The LZM result here is

*P*=exp(-9

*π*/80)≃0.7 and it is shown with horizontal dashed line in Fig. 2(c), together with the solution to Eq. (13) with initial condition

*c*

_{1}(-∞)=1 and

*c*

_{2}(-∞)=0. The numerical simulations with a finite beam in Fig. 3(d, solid lines) reproduce the qualitative features of the asymptotic LZM result in Fig. 3(c). To have a quantitative comparison with the numerics one has to match the initial conditions. Instead of the asymptotic initial condition at minus infinity, a better account is given by the LZM system Eq. (13) with the initial condition at

*t*

_{0}=(

**q**

_{X}-

**q**

_{0})/|

*|, i.e.*

**α***c*

_{1}(

*t*

_{0})=1 and

*c*

_{2}(

*t*

_{0})=0. By doing so, we were able to find an excellent agreement between direct numerical results (solid lines in Fig. 2(d)) and the solution to LZM system (dashed lines), including the value of the tunnelling efficiency ~0.77.

## 4.2. Zener transitions between three Bloch bands

**b**

*. To derive the corresponding LZM model we note that the resonant terms in the lattice potential and in the Bloch wave can be cast as follows:*

_{j}*ν*

_{1}=

*q*_{M}

*α*,

*ν*

_{2}=

**q**

_{M}′

*, and*

**α***ν*

_{3}=

**q**_{M}″

*. The coefficients*

**α***ν*

*satisfy the obvious constraint ν*

_{j}_{1}+ν

_{2}+ν

_{3}=0. The invariance of the system (17)–(19) with respect to the

*π*/3-rotation is evident from the corresponding transformation ν

_{1}→ν

_{2}→ν

_{3}→ν

_{1}. The coefficients

*ν*j can be cast as follows:

*ν*

_{1}=ℓ|

**|cos**

*α**θ*,

*ν*

_{2}=-

*ℓ*|

*|(cos*

**α***θ*-√3sin

*θ*)/2, and

*3=-ℓ*

_{ν}*|*α|(cos

*θ*+√3sin

*θ*)/2, where

*ℓ*=|

*q*_{M}|=

*b*/√3 with

*b*≡|

**b***| (in our case*

_{l}*b*=2) is the length of the hexagon side in Fig. 1 and

*θ*is the polar angle of the bias direction counting from the ΓM-line (the choice of a particular ΓM-line just sets the ordering of the Fourier coefficients

*c*

*). Since the transformation*

_{j}*θ*→-

*θ*, i.e.,

*c*

_{2}→

*c*

_{3}and

*c*

_{3}→

*c*

_{2}, leaves the system (17)–(19) invariant, the polar angle can be restricted to 0≤

*θ*≤

*π*/3.

*q*_{M}|=

*b*/√3. Hence, in the case of the hexagonal lattice (4), two of the three Bloch bands collide at the M-point (in the case of the lattice (5) the first two bands collide, while in the case of the “triangular” lattice the second and third bands collide at the M-point). It is not known whether the

*n*-level LZM system admits an analytical solution for all transition probabilities in the general case (see [35

35. S. Brundobler and V. Elser, “S-matrix for generalized Landau-Zener problem,” J. Phys. A: Math. Gen. **26**, 1211–1227 (1993). [CrossRef]

36. Yu. N. Demkov and V. N. Ostrovsky, “Multipath interference in a multistate Landau-Zener-type model,” Phys. Rev. A. **61**, 032705 (2000). [CrossRef]

*m*→

*m*, where

*m*is the diagonal index of the minimal or maximal value of the coefficient at

*t*in the system (which must be unique). The probabilities are given by the general formula [35

35. S. Brundobler and V. Elser, “S-matrix for generalized Landau-Zener problem,” J. Phys. A: Math. Gen. **26**, 1211–1227 (1993). [CrossRef]

*P*

*→*

_{m}*≡|*

_{m}*c*

*(∞)|*

_{m}^{2}for |

*c*

_{m}(-∞)|=1)

*ν*

_{j}is the coefficient at

*t*in the

*j*-th equation in the system and Δ

*is the coupling coefficient between the*

_{mj}*j*-th and

*m*-th equations. In the generic case, i.e., when 0<

*θ*<

*π*/3, we have

*ν*1>

*ν*

_{2}>

*ν*

_{3}in Eqs. (17)–(19) and the transition probabilities

*P*

_{1}→

_{1}and

*P*

_{3}→

_{3}are

*θ*=0 and

*θ*=

*π*/3, when Eqs. (17)–(19) can be reduced to the LZM system for two amplitudes.

*I. Special case θ=0: symmetric three-fold Zener transitions.*

*b*

_{1}(

*t*)=

*b*

_{1}(0)exp(-

*i*

*νt*^{2}/2+

*i*

*ε*

_{1}

*I*

_{0}

*t*/2), and the system (25)–(26) has an additional integral of motion, |

*b*

_{2}|

^{2}+|

*b*

_{3}|

^{2}≡

*C*=const. We employ the initial conditions

*c*

_{1}(-∞)=1 and

*c*

_{2},

_{3}(-∞)=0, so that

*C*=1. It follows from the initial condition that

*b*

_{1}(

*t*)=0 and, hence,

*c*

_{2}=

*c*

_{3}=

*b*

_{2}/√2 for all

*t*. The conservation of norm can be expressed now as |

*c*

_{2},

_{3}|

^{2}=(1-|

*c*

_{1}|

^{2})/2 and we refer to this case as the symmetric tunnelling. It was analyzed numerically in [24

24. A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, “Resonant Zener tunneling in two-dimensional periodic photonic lattices,” Opt. Lett. **32**, 325–327 (2007). [CrossRef] [PubMed]

*E*

_{1}=(

*ν*

_{1}

*t*-

*ε*

_{1}

*I*

_{0})/2,

*D*(

*t*)=

*ν*

^{2}

_{1}

*t*

^{2}+

*ε*

^{2}

_{1}

*I*

^{2}

_{0}+2

*ε*

_{1}

*ν*

_{1}

*t*/3 (

*D*(

*t*)>0 for

*ε*

_{1}

*I*

_{0}<3, i.e., for a shallow lattice). The tunnelling probability between the Bloch bands corresponding to the system (25)–(26) is still given by the probability

*P*

_{1}→

_{1}of Eq. (22), which in this case becomes the LZM result

**32**, 325–327 (2007). [CrossRef] [PubMed]

**3D0.avi**. It is interesting to note that the three-beam interference in Fig. 4(c) reproduces the wave used to optically induce the lattice itself [45

45. A. S. Desyatnikov, N. Sagemerten, R. Fischer, B. Terhalle, D. Träger, D. N. Neshev, A. Dreischuh, C. Denz, W. Krolikowski, and Yu. S. Kivshar “Two-dimensional self-trapped nonlinear photonic lattices,” Opt. Express **14**, 2851–2863 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-7-2851. [CrossRef] [PubMed]

46. M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Progress in Optics **42**, 219 (2001). [CrossRef]

47. T. J. Alexander, A. S. Desyatnikov, and Yu. S. Kivshar, “Multivortex solitons in triangular photonic lattices,” Opt. Lett. **32**, 1293–1295 (2007). [CrossRef] [PubMed]

48. A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical Vortices and Vortex Solitons,” Progress in Optics **47**, 291–391 (2005). [CrossRef]

47. T. J. Alexander, A. S. Desyatnikov, and Yu. S. Kivshar, “Multivortex solitons in triangular photonic lattices,” Opt. Lett. **32**, 1293–1295 (2007). [CrossRef] [PubMed]

*II. Special case θ=π/3: Zener tunnelling of Rabi oscillations*. We have in this case

*ν*

_{1}=

*ν*

_{2}=-

*ν*

_{3}/2=-

*ν*(note that now

*ν*=-

*l*|

*|/2). Introducing the amplitudes*

_{α}*b*

_{1}=(

*c*

_{1}-

*c*

_{2})/√2,

*b*

_{2}=(

*c*

_{1}+

*c*

_{2})/√2, and

*b*

_{3}=

*c*

_{3}we obtain formally the same system (24)–(26) (in fact, complex conjugate to it with inverted time

*t*→-

*t*, if one takes into account change of sign of

*ν*). The initial conditions now read

*b*

_{1}(0)=

*b*

_{2}(0)=1/√2 and

*b*

_{3}(0)=0, so that |

*b*

_{2}|

^{2}+|

*b*

_{3}|

^{2}=1/2. It follows from the solution to Eq. (24) that

*c*

_{1}(

*t*)=

*c*

_{2}(

*t*)+exp(-

*iνt*

^{2}/2+

*i*

*ε*

_{1}

*I*

_{0}

*t*/2).

*c*

_{1}and

*c*

_{2}form a Rabi oscillating state, described by the two-level system (17)–(18) (equivalent up to a phase transformation to the system (8)–(9) with

**Q**=

**b**

_{1}/2 and

*V*̂

**=**

_{Q}*ε*

_{1}

*I*

_{0}/2) before and after the three-fold resonance crossing at

*t*~0, where they couple to the third Fourier amplitude

*c*

_{3}(away from the resonance the coupling is non-resonant and can be neglected, i.e.,

*c*

_{3}rapidly varies with respect to

*c*

_{1}and

*c*

_{2}). In the equivalent Bloch band population interpretation the Rabi oscillations are between the first and the second Bloch bands in Fig. 1(d) (along the XM-line) before the resonance crossing and between the second and third bands (now along the ΓM-line) after the crossing. Before the resonance, the first two band population amplitudes are, in fact, given by

*b*

_{1}and

*b*

_{2}(and the third by

*b*

_{3}=

*c*

_{3}). After the resonance crossing

*b*

_{1}and

*b*

_{2}now give the second and third band populations, while

*b*

_{3}is the population of the first band. As the result of the resonance, a part of the Rabi energy (given by |

*b*

_{3}|

^{2}) is left in the first band and the amplitude of oscillations is reduced (recall the conservation law |

*b*

_{2}|

^{2}=1/2-|

*b*

_{3}|

^{2}). Thus the tunnelling of the oscillating state, described by the LZM system (25)–(26), can be thought of as Zener tunnelling of Rabi oscillations.

**3D3.avi**. Although the short time which our main moving beam spends near M-point is not sufficient to observe Rabi oscillations, we could successfully match these results with corresponding solution to the LZM system in Fig. 5(e). Note that after the tunnelling starts to develop, the LZM system shows the onset of oscillations between

*c*

_{1}and

*c*

_{2}. In contrast, the powers

*P*

_{1}and

*P*

_{2}in (e) remain close because these two beams remain no longer in the vicinity of resonant M-points. Nevertheless, the efficiency of tunnelling to the level P3~0.277 is in perfect agreement with LZM model.

## 4.3. Zener transitions between six Bloch bands

*b*_{1},±

*b*_{2},±

*b*_{3}} if taken from the center of the Brillouin zone would point at one of the six nearest equivalent Γ-points, see Fig. 6(a), and the Bragg vectors giving the translations between these Γ-points correspond to non-zero Fourier amplitudes of the lattice (4) if all

*ε*

*≠0.*

_{j}*t*=0 and the Bloch index is

**q**(0)=

**q**

_{Γ1}=ΓΓ

_{1}, i.e.,

**q**=

**q**

_{Γ}

_{1}-

*α*

*t*. Then the corresponding resonant terms of the Bloch wave can be cast as follows

**q**

_{Γ}

*is the reciprocal lattice vector connecting the center of the Brillouin zone with the Γ*

_{l}*-point. Setting*

_{l}*C*

*=*

_{j}*e*

^{iϕ}^{(t)}

*c*

*, with*

_{j}*ϕ*=(

*q*^{2}

_{Γ}

_{1}+

*α*

^{2}

*t*

^{2})/2, and following the same procedure as above we get the system of equations for the Fourier amplitudes

*c*

*of the Bloch wave. Defining*

_{j}*C*=(

*c*

_{1}, …,

*c*

_{6})

*we have*

^{T}*Λ*=diag(-

*λ*

_{1},-

*λ*

_{2},-

*λ*

_{3},

*λ*

_{1},

*λ*2,

*λ*

_{3}), where

*λ*

*=*

_{j}

*b**α,*

_{j}*=1,2, 3, and*

_{j}**H**is a circulant, i.e., it is a Toeplitz matrix,

*H*

*,*

_{i}*=*

_{j}*h*(

*i*-

_{j}), with the property

*h*(

*l*±6)=

*h*(

*l*). This, together with the transformation

*λ*

_{1}→

*λ*

_{2}→

*λ*

_{3}→-

*λ*

_{1}under the rotation by

*π*/3, guarantees the invariance of the system (28) under the rotations multiple of

*π*/3. Due to

**b**

_{3}=

**b**

_{2}-

**b**

_{1}, the coefficients

*λ*

*satisfy the identity*

_{j}*λ*

_{3}=

*λ*

_{2}-

*λ*

_{1}. They can be given as follows:

*λ*

_{1}=

*b*|

**|cos**

*α**φ*,

*λ*

_{2}=

*b*|

**|(cos**

*α**φ*+√3sin

*φ*)/2, and

*λ*

_{3}=

*b*|

**|(-cos**

*α**φ*+√3sin

*φ*)/2, with

*φ*being the polar angle of the bias direction counting from the ΓΓ

_{1}-line. The angle

*φ*can be restricted to 0≤

*φ*<

*π*/3 by the rotation symmetry.

_{1}=

*ε*

_{2}-

*ε*

_{1}-

*ε*

_{3}, Δ

_{2}=

*ε*

_{1}-

*ε*

_{2}-

*ε*

_{3}, Δ

_{3}=2(

*ε*

_{3}-

*ε*

_{1})-

*ε*

_{2}, and Δ

_{4}=2(

*ε*

_{3}+

*ε*

_{1})+

*ε*

_{2}. For the lattice (5) we get (we use

*b*=2, compare with Fig. 1(d))

*λ*

*two tunnelling probabilities corresponding to the system (28) can be given in the analytic form by using formula (21). Noticing that*

_{j}*λ*

_{1}>

*λ*

_{3}and λ

_{2}≥-λ

_{3}(the equality sign for

*φ*=0) we have the two possible orderings of the coefficients:

*π*/6<φ<

*π*/3. Under the rotation by

*π*/3, due to the transformation

*λ*

_{1}→

*λ*

_{2}→

*λ*

_{3}→-

*λ*

_{1}, the probabilities in Eq. (30) go into those of Eq. (31), as it should be.

*φ*=0 (the tilt is along the ΓΓ

_{1}-line, i.e., in the direction of

**b**

_{1}) the system decouples into a two-level system and a four-level system, whereas in the case

*φ*=

*π*/6 (the tilt is in the direction of

**b**

_{1}+

**b**

_{2}) the general system decouples into two three-level systems. Since the special cases could be relevant for future experiments, we consider them in detail.

*Special case φ*=0. In this case we haveλ

_{1}=

*b*|

**α**|,

*λ*

_{2}=λ

_{1}/2 andλ

_{3}=-λ

_{1}/2. Introducing new amplitudes

*b*

*by an orthogonal transformation,*

_{l}*b*

_{1}=

*c*

_{1},

*b*

_{2}=(

*c*

_{2}+

*c*

_{6})/√2,

*b*

_{3}=(

*c*

_{3}+

*c*

_{5})/√2,

*b*

_{4}=

*c*

_{4},

*b*

_{5}=(

*c*

_{2}-

*c*

_{6})/√2 and

*b*

_{6}=(

*c*

_{3}-

*c*

_{5})/√2, we get for

**B**

_{0}=(

*b*

_{1},

*b*

_{2},

*b*

_{3},

*b*

_{4})

*a four-level system*

^{T}_{0}=diag(-

*λ*

_{1},-

*λ*

_{1}/2,

*λ*

_{1}/2,

*λ*

_{1}). The rest two amplitudes,

*b*

_{5}and

*b*

_{6}, are decoupled from the system (32), they evolve according to the LZM system:

*b*

*. In the case of tunnelling between the two bands the probability is given by the LZM result*

_{j}*c*

_{1}(0)=

*b*

_{1}(0)=1, so that, solving Eqs. (33)–(34), we obtain

*b*

_{5},

_{6}=0 and

*c*

_{2}=

*c*

_{6},

*c*

_{3}=

*c*

_{5}. The probability of the transition

*P*

_{1}→

_{1}=

*P*

_{4}→

_{4}is known, and, in fact, follows from Eq. (30) which becomes

**6D.avi**. It is clearly seen that tunnelling is indeed symmetric with respect to reflection in

*x*-axis (i.e.,

*c*

_{2}=

*c*

_{6},

*c*

_{3}=

*c*

_{5};

*x*-axis corresponds to the

**b**

_{1}-direction).

*Special case φ*=

*π*/6. We have

*λ*

_{3}=0. Therefore, in this case system (28) decouples into two three-level systems for the amplitudes given by the following orthogonal transformation

*b*

_{1}=(

*c*

_{1}+

*c*

_{2})/√2,

*b*

_{2}=(

*c*

_{3}+

*c*

_{6})/√2,

*b*

_{3}=(

*c*

_{4}+

*c*

_{5})/√2,

*b*

_{4}= (

*c*

_{1}-

*c*

_{2})/√2,

*b*

_{5}=(

*c*

_{3}-

*c*

_{6})/√2,

*b*

_{6}=(

*c*

_{4}-

*c*

_{5})/√2. We get:

**B**_{1}=(

*b*

_{1},

*b*

_{2},

*b*

_{3})

*,*

^{T}

**B**_{2}=(

*b*

_{4},

*b*

_{5},

*b*

_{6})

*,*

^{T}*Λ*

_{1}=

*Λ*

_{2}=diag(-

*λ*

_{1},0,λ

_{1}), and

*P*

^{(j)}≡

*P*

^{(j)}

_{1}→

_{1}=

*P*

^{(j)}

_{3}→

_{3}is known, it reads

*j*=1 and the minus for

*j*=2. For the initial condition

*c*

_{1}(-∞)=1 the tunnelling involves all six Bloch bands since in this case we have

*b*

_{1}(-∞)=

*b*

_{4}(-∞)=1, see Fig. 6(c,d).

## 5. Vortex tunnelling

46. M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Progress in Optics **42**, 219 (2001). [CrossRef]

*n*,

**k**=

*k*

**n**

*+*

_{x}*λ*

**n**

*. In the absence of the lattice, but with the linear potential alone,*

_{y}*V*

_{lin}=

**α**

**x**, the wave (37) expands (diffracts), σ

^{2}(

*t*)=σ

^{2}+

*it*, acquiring a chirp and an overall phase according to the law

**x**

_{0}(

*t*)=

**x**

_{0}+∫

*t*

_{0}

**k**

_{0}(

*t*)d

*t*,

**k**

_{0}(

*t*)=

**k**

_{0}-

**α**

*t*. Note that a vortex initially placed close to the beam center

**x**=

**x**

_{0}remains close to

**x**

_{0}(

*t*) for all times.

*n*-vortex in a Gaussian beam, i.e., the

*n*-order Laguerre-Gauss beam. The Bragg resonance will result in several identical Gaussian peaks in the Fourier space with the indices shifted by the resonant reciprocal lattice vectors. For instance, near the resonance at

*t*=0, for the output Fourier amplitude with the index

**q**-

**Q**we have

*C*(

**q**,0) is the Fourier image of the input wave (in this case,

**q**=

**k**0, it has the Fourier index inside the first Brillouin zone). Therefore, from Eq. (11) and Eq. (39) we conclude that every output peak in the Fourier space tagged by

**q**-

**Q**will have an

*n*-vortex placed on it at its center index

**k**=

**k**

_{0}(

*t*)-

**Q**. It is important to note that the position of the output beam

*C*(

**q**-

**Q**),

**x**

_{0}(

*t*)=

**x**

_{0}+(

**k**

_{0}-

**Q**)

**t**-

**α**

*t*

^{2}/2, is

*carrier-dependent through*

**Q**. Returning to the real space, we obtain an n-vortex sitting on each of the output waves (whose velocities are different). Thus the phase dislocations resulting from the tunnelling are identical and preserved in the evolution between the resonances. The vortices can be replaced by anti-vortices without any affect on the distribution of power in the Fourier space.

**32**, 325–327 (2007). [CrossRef] [PubMed]

*t*→-

*t*.

## 6. Conclusions

**26**, 1211–1227 (1993). [CrossRef]

36. Yu. N. Demkov and V. N. Ostrovsky, “Multipath interference in a multistate Landau-Zener-type model,” Phys. Rev. A. **61**, 032705 (2000). [CrossRef]

**26**, 1211–1227 (1993). [CrossRef]

30. V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, “Nature of the Intrinsic Relation between Bloch-Band Tunneling and Modulational Instability,” Phys. Rev. Lett. **96**, 150402 (2006). [CrossRef] [PubMed]

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28. | B. Wu and Q. Niu, “Nonlinear Landau-Zener tunneling,” Phys. Rev. |

29. | O. Zobay and B. M. Garraway, “Time-dependent tunneling of Bose-Einstein condensates,” Phys. Rev. |

30. | V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, “Nature of the Intrinsic Relation between Bloch-Band Tunneling and Modulational Instability,” Phys. Rev. Lett. |

31. | V. V. Konotop, P. G. Kevrekidis, and M. Salerno, “Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice,” Phys. Rev. |

32. | V. A. Brazhnyi, V. V. Konotop, V. Kuzmiak, and V. S. Shchesnovich, “Nonlinear tunneling in two-dimensional lattices,” Phys. Rev. |

33. | V. S. Shchesnovich and V. V. Konotop, “Nonlinear tunneling of Bose-Einstein condensates in an optical lattice: Signatures of quantum collapse and revival,” Phys. Rev. |

34. | V. S. Shchesnovich and V. V. Konotop, “Nonlinear intraband tunneling of a Bose-Einstein condensate in a cubic three-dimensional lattice,” Phys. Rev. |

35. | S. Brundobler and V. Elser, “S-matrix for generalized Landau-Zener problem,” J. Phys. A: Math. Gen. |

36. | Yu. N. Demkov and V. N. Ostrovsky, “Multipath interference in a multistate Landau-Zener-type model,” Phys. Rev. A. |

37. | A. V. Shytov, “Landau-Zener transitions in a multilevel system: An exact result,” Phys. Rev. |

38. | Yu. S. Kivshar and G. P. Agrawal, |

39. | V. V. Konotop and M. Salerno, “Modulational instability in Bose-Einstein condensates in optical lattices,” Phys. Rev. |

40. | F. Keck and H. J. Korsch, “Infinite-variable Bessel functions in two-dimensional Wannier-Stark systems,” J. Phys. A: Math. Gen. |

41. | G. A. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, “Wannier functions analysis of the nonlinear Schrdinger equation with a periodic potential,” Phys. Rev. |

42. | V. S. Shchesnovich and S. Chávez-Cerda, “Bragg-resonance-induced Rabi oscillations in photonic lattices,” Opt. Lett. |

43. | W. V. Houston, “Acceleration of Electrons in a Crystal Lattice,” Phys. Rev. |

44. | V. S. Shchesnovich and S. B. Cavalcanti, “Finite-dimensional model for the condensate tunnelling in an accelerating optical lattice,” J. Phys. B: At. Mol. Opt. Phys. |

45. | A. S. Desyatnikov, N. Sagemerten, R. Fischer, B. Terhalle, D. Träger, D. N. Neshev, A. Dreischuh, C. Denz, W. Krolikowski, and Yu. S. Kivshar “Two-dimensional self-trapped nonlinear photonic lattices,” Opt. Express |

46. | M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Progress in Optics |

47. | T. J. Alexander, A. S. Desyatnikov, and Yu. S. Kivshar, “Multivortex solitons in triangular photonic lattices,” Opt. Lett. |

48. | A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical Vortices and Vortex Solitons,” Progress in Optics |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: May 19, 2008

Revised Manuscript: August 19, 2008

Manuscript Accepted: August 20, 2008

Published: August 26, 2008

**Citation**

Valery S. Shchesnovich, Anton S. Desyatnikov, and Yuri S. Kivshar, "Interband resonant transitions in
two-dimensional hexagonal lattices:
Rabi oscillations, Zener tunnelling, and
tunnelling of phase dislocations," Opt. Express **16**, 14076-14094 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14076

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