Fractional Bloch oscillations in photonic lattices
Giacomo Corrielli, Andrea Crespi, Giuseppe Della Valle, Stefano Longhi, Roberto Osellame
FFractional Bloch oscillations in photonic lattices
Giacomo Corrielli, Andrea Crespi, Giuseppe Della Valle, Stefano Longhi, and Roberto Osellame ∗ Istituto di Fotonica e Nanotecnologie - Consiglio Nazionale delle Ricerche andDipartimento di Fisica - Politecnico di Milano,Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy
Bloch oscillations (BO), i.e., the oscillatory motion of a quantum particle in a periodic potentialdriven by a constant force, constitute one of the most striking and oldest predictions of coherentquantum transport in periodic lattices. In natural crystals, BO have never been observed becauseof dephasing effects. Solely with the advent of semiconductor superlattices and ultracold atoms BOhave been observed for matter waves [1–4]. In their essence, BO are a wave phenomenon. As such,they are found for optical [5–9] and acoustic [10] waves as well. When interactions between particlescompete with their mobility, novel dynamical behaviour can arise where particles form bound states[11] and co-tunnel through the lattice [12]. While particle interaction has been generally associatedto BO damping [13–15], for few strongly-interacting particles it was predicted that bound statesundergo fractional BO at a frequency twice (or multiple) that of single-particle BO [16–18]. Theobservation of fractional BO is challenging in condensed-matter systems and up to now has not beenachieved even in model systems. Here we report on the first observation of fractional BO using aphotonic lattice as a model system of a few-particle extended Bose-Hubbard Hamiltonian. Theseresults pave the way to the visualization of other intriguing phenomena involving few correlatedparticles, such as the interplay between particle interaction and Anderson localization [19, 20],quantum billiards and transition to quantum chaos [21], anyonic BO [22], and correlated barriertunneling and particle dissociation [23].
Ultracold quantum gases in optical lattices have pro-vided in the past decade a powerful laboratory tool tosimulate Hubbard models of condensed-matter physics[24]. For few interacting particles, experiments with ul-tracold atoms have so far demonstrated the existence ofbound particle states and correlated tunneling phenom-ena [11, 12], however fractional BO of atomic pairs havenot been observed yet. A photonic simulator of the Hub-bard model can map the dynamics of two correlated par-ticles hopping on a one dimensional lattice into the mo-tion of a single particle in a two-dimensional lattice withengineered defects [21, 25], thereby offering the possi-bility to easily visualize the effect under simulation, butrequiring a high level of control of the photonic structure.To study correlated BO, we consider an extended Bose-Hubbard (EBH) model [26, 27] describing strongly inter-acting bosons in the lowest Bloch band of a one dimen-sional lattice driven by an external force. This modelis more accurate than the standard Bose-Hubbard (BH)model as it accounts for higher-order processes whosemagnitude is comparable with the one of second-ordertunneling [27]. The Hamiltonian of the system is givenby ˆ H = ˆ H EBH + ˆ H F , where ˆ H F = F d (cid:80) l l ˆ n l describesthe effect of the external constant force F ( d is the latticeperiod) and ˆ H EBH = ˆ H BH + ˆ H + ˆ H + ˆ H is the EBHHamiltonian. ˆ H EBH includes the standard BH Hamilto-nian (with (cid:126) = 1)ˆ H BH = U (cid:88) l ˆ n l (ˆ n l − − J (cid:15) (cid:88) l (cid:16) ˆ a † l ˆ a l +1 + H . c . (cid:17) (1) ∗ [email protected] and three additional terms [26], whose explicit expressionis given in the Supplementary information. In Eq.(1), U is the on-site energy interaction strength ( U > J is related to the single-particle hopping rate between adjacent lattice sites; (cid:15) (cid:28) a † l andˆ a l are the bosonic creation and annihilation operatorsand ˆ n l = ˆ a † l ˆ a l is the particle number operator at lat-tice site l . The standard BH term ˆ H BH describes on-site particle-particle interaction and single-particle tun-neling between adjacent lattice sites [see Figs.1(a) and(b)]. The additional terms ˆ H , ˆ H and ˆ H account fornearest-neighbor hopping of single atoms conditioned bythe on-site occupation number, nearest-neighbor inter-actions, and nearest-neighbor hopping of bosonic pairs,respectively [see Figs.1(c) and (d)].In the absence of particle interaction, each particleundergoes independent BO on the lattice. The single-particle amplitude probabilities A n ( t ) to find the particleat lattice site n evolve according to the coupled equations i dA n dt = − κ ( A n +1 + A n − ) + F dnA n (2)where κ = (cid:15)J/ ω B = F d . For single-siteexcitation, i.e. A n (0) = δ n, , BO appear as a breathing(rather than an oscillatory) motion at the frequency ω B .Photonic demonstrations of such BO breathing dynamicswere previously reported in Refs.[5, 6, 8].Let us now consider the dynamics of two interact-ing particles. The state vector | ψ ( t ) (cid:105) of the sys-tem can be expanded in Fock space as | ψ ( t ) (cid:105) = a r X i v : . [ qu a n t - ph ] M a r FIG. 1.
Dynamics of two interacting bosons in the extended Bose-Hubbard lattice . Two-particle dynamics in theEBH model: (a) Single-particle tunneling; (b) on-site particle interaction; (c) nearest-neighbor-site particle interaction andconditional single-particle tunneling; (d) nearest-neighbor hopping of particle pairs. (e) Fock-space representation of the two-particle dynamics; the upper-right inset represents the two possible pathways for two-particle hopping to the nearest-neighborsite: ( α ) is the second-order pair tunneling, while ( β ) is the direct two-particle tunneling. (1 / √ (cid:80) n,m c n,m ( t )ˆ a † n ˆ a † m | (cid:105) , where c n,m ( t ) is the am-plitude probability to find one particle at site n and theother particle at site m , with c n,m = c m,n for bosons. Theevolution equations for the amplitude probabilities (seethe Supplementary information) show that the dynam-ics of the two correlated bosons on the one-dimensionallattice can be mapped into the motion of a single par-ticle hopping on the two-dimensional square lattice ofFig.1(e). The square lattice shows both energy sitedefects and hopping rate corrections on the three di-agonals m = n, n ±
1. Far from the main diagonal,the square lattice is homogeneous with a uniform hop-ping rate κ = (cid:15)J/
2, which is the single-particle hoppingrate of the original problem [see Fig.1(a)]. The site en-ergy defects U and 2 (cid:15) U on the main ( m = n ) andnearest-neighbor ( m = n ±
1) diagonals account for on-site and nearest-neighbor-site particle repulsion, respec-tively [Figs.1(b) and (c)]. The correction of hoppingrates κ = κ + κ (cid:48) (with κ (cid:48) = − U (cid:15) / ) on the maindiagonal accounts for nearest-neighbor hopping of sin-gle atoms conditioned by the on-site occupation num-ber [Fig.1(c)]. Finally, the cross coupling ρ = − U (cid:15) on the main diagonal n = m describes nearest-neighborhopping of bosonic pairs, i.e. direct two-particle tunnel-ing [Fig.1(d)]. For BO of two strongly-correlated par-ticles ( J ∼ U ), only this last term is important, com-peting with second-order pair tunneling. In fact, let usassume as an initial condition c n,m (0) = A n (0) δ n,m , i.e.that the two particles are initially placed on the samesite, with a probability | A n (0) | to find both particles at site n . Then an asymptotic analysis of the underly-ing equations shows that the two particles form a boundstate, i.e. they co-tunnel along the lattice, undergo-ing BO at the frequency 2 ω B = 2 F d twice the single-particle BO frequency ω B . In Fock space, this meansthat the excitation remains confined along the main di-agonal m = n of the square lattice of Fig.1(e), and pe-riodic revivals at frequency 2 ω B are expected. Precisely,as shown in the Supplementary information, the bound-particle occupation amplitudes A n ( t ) ∼ c n,n ( t ) at thevarious lattice sites evolve according to Eqs.(2), but with F replaced by 2 F and κ replaced by κ eff . The effec-tive hopping rate of the bound particle state is givenby k eff = − κ /U + ρ = − (cid:15) J / (2 U ) − U (cid:15) , whichincludes second-order tunneling ( − κ /U ) and directtwo-boson tunneling ( ρ ) processes. Such two contribu-tions always add with the same sign, for both repulsive( U >
0) and attractive ( U <
0) interactions, and gen-erally are of the same order of magnitude. The boundparticle state thus behaves like a single particle hoppingon the one-dimensional lattice, but with a driving forcewhich is doubled as compared to that of a single particle[16, 17] and with a modified hopping rate κ eff . Notethat the effect of direct two-atom tunneling on the BOdynamics, not considered in previous works [17, 18, 25],is to increase the amplitude of the breathing motion (ow-ing to the correction of κ eff ), whereas the Wannier-Starkenergy spectrum is not altered, i.e. ω B is not affected.In our experiments, single-particle hopping dynamicsin the square lattice of Fig.1(e) has been simulated by R Y d 100 µm a z b FIG. 2.
The photonic simulator of correlated BO. (a) Sketch of the waveguide array structure; red-coloured waveguideshave a different refractive index change with respect to the others to implement the on-site particles-interaction defect U . (b)Section of the fabricated array, imaged with an optical microscope. discretized spatial light transport in an engineered two-dimensional square lattice of evanescently-coupled opti-cal waveguides [28]. This lattice has been fabricated ina fused silica substrate by direct waveguide writing withfemtosecond lasers [29–33], taking advantage of the three-dimensional capabilities of this technology. The spatial propagation of the light intensity in the generic ( m, n )-th lattice site maps the temporal evolution of the two-particle probability distribution | c m,n ( t ) | in Fock space[28, 32]. In the photonic simulator of ˆ H EBH , the roleof the hopping coefficients κ and ρ is played by theevanescent coupling constants between first- and second-neighboring waveguides (¯ κ and ¯ ρ ), respectively. The for-mer is in general greater than the latter and the relativeweight can be tuned by properly engineering the latticeparameter d (see Supplementary Information). The on-site particle-interaction defect U is realized by fabricat-ing the main diagonal waveguides with a different refrac-tive index change (this is achieved by slightly modifyingthe writing speed of the waveguides) and thus with thepropagation constant β (cid:48) slightly detuned with respect tothat of all the others ( β ). The positive or negative signof the relative detuning ∆ β = β − β (cid:48) determines therepulsive or attractive nature of the interaction, respec-tively. Lastly, we did not add any special feature in thenearest-neighboring diagonals, because, as already dis-cussed above, the nearest-site effects described in Fig.1(c)are negligible with respect to the other terms of the EBHmodel. The implementation of the driving force term ˆ H F is obtained by a constant bending of the waveguide axes(radius of curvature R , F ∝ /R ) in the plane deter-mined by the lattice main diagonal direction Y and thelight propagation direction z [8]. In Fig.2(a) the wholelattice structure is depicted. Note that the array cross-section is rotated in such a way that the bending plane ishorizontal in the laboratory reference frame [45 o rotationwith respect to Fig.1(e)]. This simplifies the fabricationprocess and the imaging of the light propagating in thewaveguides; in this way, although being a 3D structure, each waveguide remains on a specific horizontal plane.In a first experiment we checked that our photonicsimulator can reliably mimic the dynamics governed byˆ H EBH alone, namely without the action of the exter-nal constant force. For this purpose we fabricated 225straight waveguides ( R = ∞ ) arranged in a 15x15 squarelattice, with spacing d = 19 µm and a length L = 2 . cm .For such a lattice spacing the coupling coefficients are¯ κ = 0 . cm − and ¯ ρ = 0 . cm − . To simulate particleinteraction, we fabricated the main diagonal waveguideswith a propagation constant detuning ∆ β = − cm − ,therefore representing an attractive interaction betweenthe particles. A microscope image of the array section isreported in Fig.2(b). We coupled a 633 nm He-Ne laserradiation into the central waveguide, belonging to themain diagonal subset and thus corresponding to an ini-tial condition where the two particles occupy the samelattice site. As previously discussed light is expected toremain on the detuned diagonal, representing the factthat the correlated particles will hop together to the nextsite and will never separate. To study the evolution ofthe system we monitored the light propagation along thearray by imaging from above the fluorescence light at650 nm emitted from the waveguides and arising fromcolour centres created in the femtosecond laser writingprocess. Since the colour centres are formed exclusivelyinside the waveguides, this technique yields a high signal-to-noise ratio and permits a direct imaging of the lightpropagation in the waveguide arrays [32, 33]. The resultobtained in this first experiment is visible in Fig.3(a).The measured distribution of light in the photonic modelimplementing both contributions to the bound state hop-ping rate, i.e. two-particle tunneling and second-orderpair tunneling, is compared with the corresponding nu-merical simulation [Fig.3(b)]. The excellent agreementdemonstrates that our photonic model can indeed takeinto account both contributions and thus correctly im-plements the EBH model. The role of the direct two-particle tunneling [Fig.1(d)], neglected in the standard a bc ExperimentTheoryTheory
FIG. 3.
Photonic simulation of the delocalization dy-namics of two interacting particles in a crystal. (a)Experimental simulation in a waveguide lattice implementingdirect two-particle tunneling and second-order pair tunnelingwith a similar weight. (b) Numerical simulation includingboth contributions. (c) Numerical simulation including onlysecond-order pair tunneling.
BH model, can be quite relevant as shown by the differentlight distributions obtained from numerical simulationsof the same structure but including only the second-ordertunneling [Fig.3(c)].In a second experiment, we proceeded to the observa-tion of fractional BO. To this purpose we fabricated an-other 15x15 square waveguide lattice, with the same lat-tice spacing and detuning on the diagonal as in the previ-ous structure. To mimic the external force F , the waveg-uides are now circularly-bent with a radius R = 400 cm ,as depicted in Fig.2(a). The array length is L = 8 . cm .As in the previous experiment, we injected the probe lightinto the central waveguide and we imaged its propagationalong the waveguide lattice diagonal (Y), where light isconfined. Figure 4(a) shows the oscillatory behaviour ofthe light dynamics along the waveguide array. The accu-racy of the fabricated photonic simulator is confirmed bythe very good agreement with the numerical simulationsof light propagation in the designed structure [Fig.4(a)].Initially, light spreads into several waveguides, until itreaches a maximum of the breathing amplitude (around L max = 3 . cm ). Then light refocuses into the cen-tral waveguide ( L foc = 6 . cm ) and the breathing startsagain. This periodic behavior represents the correlatedBO of two interacting particles, watched in Fock space.To better analyze this result we cut the sample at L max and L foc and we imaged onto a CCD camera the lightdistribution exiting the output facet of the array (Fig.5).Figure 5(a) clearly shows that, even in correspondence of ba ExperimentExperimentTheoryTheory L max L foc FIG. 4.
Photonic simulation of Bloch oscillations forcorrelated and single particles. a) Experimental and nu-merical simulation of the light distribution in the waveguidelattice diagonal, representing in Fock space the BO for twointeracting particles . (b) Experimental and numerical simu-lation of the light distribution corresponding to single particleBO; the position where the BO amplitude is at maximum cor-responds to the refocusing condition for the two-particle BO L foc = 6 . cm . the breath maximum amplitude, light remains confinedin the waveguide lattice detuned diagonal. This fact is afurther demonstration that two interacting particles, ini-tially in the same lattice site, form a bound state and hoptogether to neighboring sites. Figure 5(b) shows that atthe refocusing distance, all the light is indeed confinedin the central waveguide, which was the one originallyexcited (indicated by a white arrow in both panels).The ’fractional’ nature of BO for correlated particles,namely frequency doubling with respect to the case of a b
50 µm
FIG. 5.
Output light distribution at two propagationlengths . Measured light distribution in the waveguide lat-tice section [corresponding to Fig.4(a)] at positions (a) L max ,where the fractional BO amplitude is at maximum and (b) L foc where light refocuses to the initial waveguide (indicatedby a white arrow in both panels). a single particle hopping in the same lattice under theaction of the same force F , is shown in Fig.4(b). Wefabricated a planar 1D array of 23 identical waveguides,equally spaced of d (cid:48) = d = 19 µm , and uniformly bent inthe array plane with a constant radius of curvature R (cid:48) .Such a structure reliably simulates the dynamics of sin-gle particle BO [8]. In order to implement the same forceon the single particle we have to project the curvatureon the diagonal for the two-particles structure on one ofthe main axis of the square lattice: this yields a curva-ture radius for the linear array R (cid:48) = R √ √ cm .We imaged the light propagation in this waveguide ar-ray for a length L (cid:48) = L = 8 . cm [Fig.4(b)]. Also inthis case we observe a breathing propagation mode cor-responding to the single particle BO, but the position ofthe maximum breath amplitude L (cid:48) max corresponds to therefocusing position L foc for two-particles BO [Fig.4(a)].This demonstrates that the frequency of a two-particlesBO is twice that for a single-particle. It may be worthnoting that the breathing amplitude of a single particleBO is larger than that for the two-particle BO. This isdue to a higher hopping rate in the former case, k > k eff ,and to the double force experienced by the two stronglycorrelated particles with respect to the single one.In conclusion, we have experimentally observed thephotonic quantum-analogy of fractional Bloch Oscilla-tions of two strongly correlated particles. We have shownthat our photonic simulator is capable of implementingtwo-particle tunneling in addition to second-order pairtunneling, thus enabling the simulation of the extendedBose-Hubbard model. The simple visualization of thephenomenon under study and the high control on thefabricated structure, and thus of the implemented model,make this approach a very powerful tool to investigateother exotic phenomena associated to the formation ofparticle bound states which are of difficult access in thematter. ACKNOWLEDGMENTS
This work was supported by the European Union through the project FP7-ICT-2011-9-600838 (
QWAD - QuantumWaveguides Application and Development ). AUTHOR CONTRIBUTIONS
All authors conceived the experiment. G.C., A.C., R.O.designed and fabricated the photonic device and performedthe measurements. G.D. and S.L. developed the theory un-derlying the experiment. All authors discussed the resultsand participated in the manuscript preparation.
ADDITIONAL INFORMATION
Supplementary information is available in the online ver-sion of the paper.
COMPETING FINANCIAL INTERESTS
The authors declare no competing financial interests.
METHODSFabrication of the waveguide arrays.
Waveguide ar-rays have been fabricated by femtosecond laser waveguidewriting in fused silica samples. The second harmonic of aHighQLaser femtoREGEN Yb-based amplified laser systemhas been used, consisting in 400-fs pulses at 520-nm wave-length with repetition rates up to 960 kHz. Actual writingconditions consisted in 300-nJ pulses delivered at a repetitionrate of 20 kHz, reduced using the internal pulse-picker. Opti-mal range for the writing speed was identified as 8 − mm/s ,yielding propagation losses of about 0 . dB/cm . In order toimplement the detuning in the lattice diagonal we modulatedthe writing speed, therefore the waveguides on the diagonalwere fabricated at a speed of 9 mm/s , while all the others werewritten at 14 mm/s . The variation in writing speed modifiesthe propagation constants of the guided modes, producing thedesigned detuning ∆ β = − cm − , but has a negligible effecton the coupling rate between the waveguides. End-faces ofthe substrates were polished after writing to improve lightlaunching and output imaging. Characterization of the light distribution in the ar-rays.
Femtosecond laser writing in fused silica creates colorcenters that provide fluorescent emission at about 650 nmwhen light at 633 nm is propagated in the waveguide. Top-view imaging of the fluorescence signal is employed to visual-ize and quantitatively estimate the light distribution along thewaveguide array, rejecting the background light by a notch fil-ter at 633 nm. In order to achieve a high resolution in imagingthe light in the waveguide array all along its length, severalimages have been acquired and then stitched together. Prop-agation losses in the waveguides have been compensated byrenormalizing the intensity levels in the acquired images. [1] Waschke, C., Roskos, H.G., Schwedler, R., Leo, K., Kurz,H. & K¨ohler, K. Coherent submillimeter-wave emissionfrom Bloch oscillations in a semiconductor superlattice.
Phys. Rev. Lett. , 3319-3322 (1993).[2] Ben Dahan, M., Peik, E., Reichel, J., Castin, Y. & Sa-lomon, C. Bloch Oscillations of Atoms in an Optical Po-tential. Phys. Rev. Lett. , 4508-4511 (1996).[3] Wilkinson, S.R., Bharucha, C.F., Madison K.W., Niu Q.& Raizen, M.G. Observation of Atomic Wannier-StarkLadders in an Accelerating Optical Potential. Phys. Rev.Lett. , 4512-4515 (1996).[4] Anderson, B.P. & Kasevich, M.A. Macroscopic QuantumInterference from Atomic Tunnel Arrays. Science ,1686-1689 (1998).[5] Morandotti, R., Peschel, U., Aitchison, J.S., Eisenberg,H.S. & Silberberg, Y. Experimental Observation of Lin-ear and Nonlinear Optical Bloch Oscillations.
Phys. Rev.Lett. , 4756-4759 (1999).[6] Pertsch, T., Dannberg, P., Elflein, W., Br¨auer, A. &Lederer, F. Optical Bloch Oscillations in TemperatureTuned Waveguide Arrays. Phys. Rev. Lett. , 4752-4755(1999).[7] Sapienza, R., Costantino, P., Wiersma, D., Ghulinyan,M., Oton, C.J. & Pavesi, L. Optical Analogue of Elec-tronic Bloch Oscillations. Phys. Rev. Lett. , 263902(2003).[8] Chiodo, N., Della Valle, G., Osellame, R., Longhi, S.,Cerullo, G., Ramponi, R., Laporta, P. & U. Morgner,U. Imaging of Bloch oscillations in erbium-doped curvedwaveguide arrays. Opt. Lett. , 1651-1653 (2006).[9] Trompeter, H., Krolikowski, W., Neshev, D.N., Desyat-nikov, A.S., Sukhorukov, A.A., Kivshar, Yu. S., Pertsch,T., Peschel, U. & Lederer F. Bloch Oscillations and ZenerTunneling in Two-Dimensional Photonic Lattices. Phys.Rev. Lett. , 053903 (2006).[10] Sanchis-Alepuz, H., Kosevich, Y.A. & Sanchez-Dehesa, J.Acoustic Analogue of Electronic Bloch Oscillations andResonant Zener Tunneling in Ultrasonic Superlattices. Phys. Rev. Lett. , 134301 (2007).[11] Winkler, K., Thalhammer, G., Lang, F., Grimm, R.,Hecker Denschlag, J., Daley, A.J., Kantian, A., B¨uchler,H.P. & Zoller, P. Repulsively bound atom pairs in anoptical lattice. Nature , 853-856 (2006).[12] F¨olling, S., Trotzky, S., Cheinet, P., Feld, M., Saers, R.,Widera, A., M¨uller, T. & Bloch, I. Direct observationof second-order atom tunnelling.
Nature , 1029-1032(2007).[13] Buchleitner, A. & Kolovsky, A. R. Interaction-InducedDecoherence of Atomic Bloch Oscillations.
Phys. Rev.Lett. , 253002 (2003).[14] Gustavsson, M., Haller, E., Mark, M.J., Danzl,J.G., Rojas-Kopeinig, G. & N¨agerl, H.-C. Control ofInteraction-Induced Dephasing of Bloch Oscillations, Phys. Rev. Lett. , 080404 (2008).[15] Eckstein, M. & Werner, P. Damping of Bloch Oscilla-tions in the Hubbard Model.
Phys. Rev. Lett. , 186406(2011).[16] Claro, F., Weisz, J.F. & Curilef, S. Interaction-induced oscillations in correlated electron transport.
Phys. Rev.B , 193101 (2003).[17] Dias, W.S., Nascimento, E.M., Lyra, M.L. & de Moura,F. A. B. F. Frequency doubling of Bloch oscillations forinteracting electrons in a static electric field. Phys. Rev.B , 155124 (2007).[18] Khomeriki, R., Krimer, Do.O., Haque, M. & Flach, S.Interaction-induced fractional Bloch and tunneling oscil-lations. Phys. Rev. A , 065601 (2010).[19] Krimer, D.O., Khomeriki, R. & Flach, S. Two interactingparticles in a random potential. JEPT Lett. , 406-412(2011).[20] Albrecht C. & Wimberger S. Induced delocalization bycorrelation and interaction in the one-dimensional An-derson model. Phys. Rev. B , 045107 (2012).[21] Krimer, D.O. & Khomeriki, R. Realization of discretequantum billiards in a two-dimensional optical lattice. Phys. Rev. A , 041807(R) (2011).[22] Longhi, S. & Della Valle, G. Anyonic Bloch oscillations. Phys. Rev. B , 165144 (2012).[23] Kolovsky, A. R., Link, J. & Wimberger, S. Energeticallyconstrained co-tunneling of cold atoms. New J. Phys. ,075002 (2012).[24] Bloch, I., Dalibard, J. & Nascimbene, S. Quantum sim-ulations with ultracold quantum gases. Nature Phys. ,267-276 (2012).[25] Longhi, S. Photonic Bloch oscillations of correlated par-ticles. Opt. Lett. , 3248-3250 (2011).[26] Mazzarella, G., Giampaolo, S. M. & Illuminati, F. Ex-tended Bose-Hubbard model of interacting bosonic atomsin optical lattices: From superfluidity to density waves. Phys. Rev. A , 013625 (2006).[27] Trotzky,S., Cheinet, P., F¨olling, S., Feld, M., Schnor-rberger, U., Rey A.M., Polkovnikov, A., Demler, E.A.,Lukin, M.D. & Bloch I. Time-Resolved Observation andControl of Superexchange Interactions with UltracoldAtoms in Optical Lattices Science , 295-299 (2008).[28] Christodoulides, D., Lederer, F. & Silberberg, Y. Dis-cretizing light behavious in linear and nonlinear waveg-uide lattices.
Nature , 817-823 (2003).[29]
Femtosecond Laser Micromachining: Photonic and Mi-crofluidic Devices in Transparent Materials , Topics inApplied Physics Vol. , eds. Osellame, R., Cerullo,G., Ramponi, R. (Springer-Verlag Berlin, Berlin, 2012).[30] Gattass, R. R. & Mazur, E. Femtosecond laser micro-machining in transparent materials.
Nature Photon. ,219-225 (2008).[31] Crespi, A., Longhi, S. & Osellame, R. Photonic realiza-tion of the quantum Rabi model. Phys. Rev. Lett. ,163601 (2012).[32] Szameit, A. & Nolte, S. Discrete optics in femtosecond-laser-written photonic structures.
J. Phys. B: At. Mol.Opt. Phys. , 163001 (2010).[33] Szameit, A., Garanovich, I. L., Heinrich, M., Sukho-rukov, A. A., Dreisow, F., Pertsch, T., Nolte, S., Tnner-mann, A. & Kivshar, Y. S. Polychromatic dynamic lo-calization in curved photonic lattices. Nature Phys.5