aa r X i v : . [ phy s i c s . op ti c s ] D ec Photonic analogue of
Zitterbewegung in binary waveguide arrays
S. Longhi
Dipartimento di Fisica, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy
Compiled July 12, 2018An optical analogue of
Zitterbewegung (ZB), i.e. of the trembling motion of Dirac electrons caused by theinterference between positive and negative energy states, is proposed for spatial beam propagation in binarywaveguide arrays. In this optical system ZB is simply observable as a quiver spatial oscillatory motion of thebeam center of mass around its mean trajectory. c (cid:13)
OCIS codes:
Zitterbewegung (ZB) refers to the rapid trembling mo-tion of a free Dirac electron caused by the interferencebetween positive and negative energy state components[1, 2]. However, the amplitude of ZB oscillations turnsout to be extremely small, of the order of the Comp-ton wavelength, which defines the limit of electron lo-calization. Therefore, its direct observation is unlikely,though observable consequences of ZB have been foundin the response of electrons to external fields [1]. Sim-ilarly to other effects, like the Klein paradox [2], ZBhas been for long time regarded as a relativistic effectrooted in the Dirac equation. However, several authorshave recently shown that ZB is not unique to Dirac elec-trons, rather it is a generic feature of wave packet dy-namics in spinor systems with certain linear dispersionrelations, such as those exhibiting the so-called Diracpoints (DP) that describe massless fermions [3–13]. Incondensed-matter and matter-wave physics, trapped ions[3], graphene [4, 5], and ultracold neutral atoms [6] havebeen proposed as candidate systems for a direct observa-tion of ZB. Similarly, in the optical context it was shownthat DP can occur in two-dimensional photonic crys-tals [7–11] or in negative-zero-positive index metamate-rials [12]. Photonic analogues of ZB have been recentlyproposed in Refs. [11,13], in which the observation of ZBrequires time-resolved measurements of pulse transmis-sion through photonic crystal or metamaterial slabs ofdifferent thickness. In this Letter we propose a simpleroptical system, consisting of a binary waveguide array,which enables an easy visualization in space of photonicZB. The ability of mapping typical ultrafast phenom-ena occurring in the matter as spatial propagation oflight waves in coupled waveguide structures has beensuccessfully demonstrated in several experiments (see,e.g., [14–17]), and our proposal should thus greatly fa-cilitate the way toward a first observation of ZB.Let us consider propagation of monochromatic lightwaves at wavelength λ in a one-dimensional optical lat-tice, which is described by the following scalar equationfor the electric field envelope E ( x, z ) i∂ z E = − [ λ/ (4 πn s )] ∂ x E + (2 π/λ )[ n s − n ( x )] E, (1)where n s is the substrate refractive index and n ( x ) is the n x () x c n + c n c n- a a D n D n w w x q z (a) (c) a a a q w () q - - 00 (b) p a p a p a p a d x Fig. 1. (Color online) (a) Schematic of a one-dimensionalbinary array (left) and refractive index profile (right). (b)Dispersion curves of the first two minibands of a tight-binding binary array (solid curves), and correspond-ing dispersion curves of the Dirac equation (4) (dottedcurves). (c) Broad-beam excitation geometry of the ar-ray at a tilting angle θ .refractive index profile of the lattice. To realize a pho-tonic analogue of ZB, a two-band model with a dynamicsdescribed by a two-component spinor wave function isneeded. Such a model can be realized by either a singly-periodic lattice with a shallow sinusoidal refractive indexprofile, where the two components of the spinor wavefunctions are related by simple linear transformation tothe amplitudes of counterpropagating waves in the lat-tice (see, for instance, [18]), or to a tight-binding bi-nary superlattice composed by two interleaved sublat-tices A and B [see Fig.1(a)], where the spinor wave func-tions correspond to the occupation amplitudes in thetwo sublattices. Here we study ZB in the latter struc-ture and assume a symmetric intersite coupling, whichcorresponds to the experimental conditions of Ref. [17];the analysis could be extended, if needed, for the moregeneral case of non-symmetric intersite couplings [20]. Inthe tight-binding approximation, light transport in thebinary lattice is described by coupled-mode equationsfor the modal field amplitudes c n in the various waveg-1ides [17, 20, 21] i ( dc n /dz ) = − σ ( c n +1 + c n − ) + ( − n δc n , (2)where 2 δ and σ are the propagation constant mismatchand the coupling rate between two adjacent waveguidesof the array, respectively. The tight-binding model (2)supports two minibands, whose dispersion curves arereadily obtained by the plane-wave Ansatz c n ( q ) ∼ exp( iqna − iωz ) and read [20] ω ( q ) = ± p δ + 4 σ cos ( qa ) (3)[see Fig.1(b)]. Let us assume that the array is excited bya broad beam (e.g. Gaussian shaped) incident onto thearray at an angle close to the Bragg angle θ = λ/ (4 n s a ),i.e. E ( x,
0) = G ( x ) exp(2 πiθn s x/λ ), where a is the spac-ing between adjacent waveguides and G ( x ) varies slowlyon the spatial scale ∼ a [see Fig.1(c)]. At such an inci-dent angle, the modes in adjacent waveguides are excitedwith a nearly equal amplitude but with a phase differ-ence of π/
2. After setting c n ( z ) = ( − n ψ ( n, z ) and c n − = − i ( − n ψ ( n, z ), for broad beam excitation theamplitudes ψ and ψ vary slowly with n , and one canthus write ψ , ( n ± , z ) = ψ , ( n, z ) ± ( ∂ψ , /∂n ) andconsider n ≡ ξ as a continuous variable rather than asan integer index. At the input plane z = 0, the ampli-tudes ψ ( ξ,
0) and ψ ( ξ,
0) are proportional to G (2 na )and G (2 na − a ) ≃ G (2 na ), respectively, so that one canassume ψ ( ξ, ≃ ψ ( ξ,
0) as an initial condition. Undersuch assumptions, from Eqs.(2) it readily follows thatthe two-component spinor ψ ( ξ, z ) = ( ψ , ψ ) T satisfiesthe one-dimensional Dirac equation i∂ z ψ + iσα∂ ξ ψ − δβψ = 0 , (4)where α = (cid:18) (cid:19) , β = (cid:18) − (cid:19) . (5)Note that α and β coincide with the σ x and σ z Paulimatrices, respectively. Assuming the normalization con-dition R dξ ( | ψ | + | ψ | ) = 1, after the formal change σ → c , δ → mc / ¯ h , ξ → x and z → t , Eq.(4) cor-responds to the one-dimensional Dirac equation for anelectron of mass m in absence of external fields [2]. The temporal evolution of the spinor wave function ψ forthe Dirac electron is therefore mapped into the spatial evolution of the modal amplitudes ψ and ψ in thetwo sublattices A and B. The energy-momentum dis-persion relation ¯ hω ( k ) of the Dirac equation (4), ob-tained by making the Ansatz ψ ∼ exp( ikξ − iωz ) inEq.(4), is composed by the two branches ω ( k ) = ± ǫ ( k ),corresponding to positive and negative energy states ofthe relativistic free electron, where ǫ ( k ) = √ δ + σ k .Such branches reproduces the two minibands of the bi-nary array [see Eq.(3)] near the boundary of the Bril-louin zone [see Fig.1(b)], where k = 2 aq − π . ZB ofthe Dirac electron corresponds to a rapid oscillationof the average position h ξ i ( z ) = R dξ ξ ( | ψ | + | ψ | ) around the classical trajectory. The usual method of un-derstanding ZB in the framework of the Dirac equa-tion is to derive equations of motion for the Heisen-berg operators, and show that they oscillate in time[1, 2]. We instead work directly in the Schr¨odinger pic-ture and consider wave packet evolution in momentumspace, namely we set ψ , ( ξ, z ) = R dk ˆ ψ , ( k, z ) exp( ikξ )and calculate the spectra ˆ ψ , ( k, z ) by solving Eq.(4)in momentum space. One then obtains ˆ ψ , ( k, z ) =ˆ G ( k )[cos( ǫz ) ∓ i ( ± σk + δ ) sin( ǫz ) /ǫ ], where ˆ G ( k ) =(1 / π ) R dξG (2 aξ ) exp( − ikξ ) is the Fourier spectrum ofthe exciting broad input beam. The average position h ξ i is then calculated as h ξ i ( z ) = 2 πi R dk [ ˆ ψ ∗ ( k ) ∂ k ˆ ψ ( k ) +ˆ ψ ∗ ( k ) ∂ k ˆ ψ ( k )], which yields after some algebra h ξ i ( z ) = h ξ i (0) + 4 πσ z Z dk ( k/ǫ ) | ˆ G ( k ) | ++ 2 πσδ Z dk (1 /ǫ ) sin(2 ǫz ) | ˆ G ( k ) | (6)The last oscillatory term in Eq.(6) is ZB, superimposedto the straight trajectory defined by the first two termson the right hand side of Eq.(6). For ˆ G ( k ) spectrallynarrow at around k = 0, the frequency of ZB is equalto 2 ǫ ( k = 0) = 2 δ ; spectral broadening of ˆ G ( k ) is re-sponsible for damping of ZB. It should be noted that h ξ i , calculated as the average position for the Diracequation (4), basically reproduces the evolution of thebeam center of mass h n i in real space, defined as h n i =( P n n | c n | ) / P n | c n | . In fact, it is straightforward toshow that h n i = 2 h ξ i + 12 − πδσ Z dk ( k/ǫ ) | ˆ G ( k ) | sin ( ǫz ) . (7)The last term in Eq.(7) is usually negligible for a spec-trum ˆ G ( k ) narrow at around k = 0 (see the examples tobe discussed below), so that one has h n i ≃ h ξ i + 1 / | c n ( z ) | [Figs.2(a) and (c)], and correspond-ing behavior of beam center of mass h n i ( z ) [Figs.2(b)and (d), solid curves], for σ = 1 and for two values ofdetuning δ . In both cases, the array has been excitedby a broad Gaussian beam G ( x ) = exp[ − ( x/w ) ] with w /a = 12, and the results have been obtained by numer-ical integration of coupled-mode equations (2) with ini-tial conditions c n (0) = i n exp[ − ( na/w ) ]. A clear trem-bling motion of the beam, corresponding to ZB, is ob-served, with an oscillation frequency (amplitude) whichincreases (decreases) as δ increases, according to Eq.(6)[compare Fig.2(b) and Fig.2(d)]. A similar trembling mo-tion would be observed for a binary lattice with asym-metric intesrite coupling [20]. Damping of ZB oscilla-tions, which arises from the broadening of the spectrumˆ G ( k ), is also visible in Figs.2(b) and (d); note also thatthe averaged beam path has a nonvanishing drift veloc-ity which arises from the second term on the right handside of Eq.(6).We finally checked the correctness of the analysis, based2
10 20 30 40-202460 10 20 30 40-60060 z w a v egu i de nu m be r n bea m c en t e r o f m a ss n > > (a) (b)(d) w a v egu i de nu m be r n z (c) Fig. 2. (Color online) (a) Evolution of | c n ( z ) | in a bi-nary array excited by a broad Gaussian beam for σ = 1and δ = 0 .
6, and (b) corresponding behavior of beamtrajectory h n i ( z ) (solid curve) as obtained by numeri-cal analysis of Eqs.(2). The dotted curve in (b), almostoverlapped with the solid curve, reproduces the behaviorof 2 h ξ i ( z ) + 1 /
2, where h ξ i ( z ) is calculated according toEq.(5). (c) and (d): same as (a) and (b), but for δ = 1. bea m c en t e r ( m ) m (b) -290 10 z propagation distance (cm) z propagation distance (cm) x ( m ) m (a) Fig. 3. (Color online) (a) Beam propagation (snapshotof | E ( x, z ) | ) in a binary array, as obtained by numericalanalysis of Eq.(1), for broad Gaussian beam excitationtilted at the Bragg angle; parameter values are given inthe text. (b) Corresponding behavior of h x i ( z ).on the tight-binding model (2), and the feasibility ofan experimental observation of ZB by numerical sim-ulations of the wave equation (1) for parameter valuesthat typically apply to binary arrays realized in fusedsilica by femtosecond laser writing [17]. A typical nu-merical result is shown in Fig.3 for parameter values λ = 633 nm, n s = 1 . a = 10 µ m, w = w = 3 . µ m,∆ n = 0 . n = 0 . E ( x,
0) of spot size w = 80 µ m, tilted at the Braggangle θ ≃ . o . The trembling motion of the beam as itpropagates along the 10-cm-long array is clearly visible,and should be easily observed by microscope fluorescenceimaging [17]. The beam trajectory in Fig.3(b) has beencomputed as h x i ( z ) = R dx x | E | / R dx | E | , where theintegration interval is limited by the size of the numeri-cal domain. Note that the first oscillations of h x i internalto the shaded area of Fig.3(b) do not correspond to ZB, rather they arise because of an initial beam break upand appearance of higher-order beams [as indicated bythe arrows in Fig.3(a)] belonging to higher-order bandsof the array. Such higher-order beams, however, refractat large angles and, after few centimeter propagation,they are no more overlapped with the main beam un-dergoing ZB.In conclusion, a photonic analogue of the trembling mo-tion of Dirac electrons has been proposed for spatialbeam propagation in binary waveguide arrays. As com-pared to previous proposals [11, 13], the easy experi-mental visualization of beam dynamics in waveguide ar-rays [17] should greatly facilitate the way toward the firstobservation of ZB.Author E-mail address: longhi@fisi.polimi.it References
1. K. Huang, Am. Phys. J. , 479 (1952).2. W. Greiner, Relativistic Quantum Mechanics (Springer-Verlag, Berlin, 1990).3. L. Lamata, J. Le´on, T. Sch¨atz, and E. Solano, Phys.Rev. Lett. , 172305 (2006).5. T. M. Rusin and W. Zawadzki, Phys. Rev. B , 195439(2007).6. J.Y. Vaishnav and C.W. Clark, Phys. Rev. Lett. ,153002 (2008).7. F.D.M. Haldane and S. Raghu, Phys.Rev.Lett. ,013904 (2008).8. R.A. Sepkhanov, Ya. B. Bazaliy, and C.W.J. Beenakker,Phys. Rev. A , 063813 (2007).9. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev,and D.N. Christodoulides, Phys. Rev. Lett. , 103901(2007).10. O. Bahat-Treidel, O. Peleg, and M. Segev, Opt. Lett. , 2251 (2008).11. X. Zhang, Phys. Rev. Lett. , 113903 (2008).12. L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu,Opt. Lett. , 1510 (2009).13. L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, EPL , 47008(2009).14. H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U.Streppel, A. Br¨auer, and U. Peschel, Phys. Rev. Lett. , 023901 (2006).15. S. Longhi, Laser & Photon. Rev. , 243261 (2009).16. G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V.Foglietti, M. Finazzi, L. Duo, and S. Longhi, Phys. Rev.Lett. , 180402 (2009).17. F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S.Nolte, A. T¨unnermann, and S. Longhi, Phys. Rev. Lett. , 076802 (2009).18. J. Feng, Opt. Lett. , 1302 (1993).19. A.A. Sukhorukov and Y.S. Kivshar, Phys. Rev. Lett. ,113902 (2003).20. A.A. Sukhorukov and Y.S. Kivshar, Opt. Lett. , 2112(2002).21. R. Morandotti, D. Mandelik, Y. Silberberg, J.S. Aitchi-son, M. Sorel, D.N. Christodoulides, A.A. Sukhorukov,and Y.S. Kivshar, Opt. Lett. , 2890 (2004)., 2890 (2004).