A Faithful Communication Hamiltonian in Photonic Lattices
Matthieu Bellec, Georgios M. Nikolopoulos, Stelios Tzortzakis
aa r X i v : . [ qu a n t - ph ] O c t A Faithful Communication Hamiltonian in Photonic Lattices
Matthieu Bellec, Georgios M. Nikolopoulos, , ∗ and Stelios Tzortzakis , Institute of Electronic Structure & Laser, FORTH, P.O.Box 1385, GR-71110 Heraklion, Greece Department of Materials Science and Technology, University of Crete, P.O. Box 2208, GR-71003 Heraklion, Greece ∗ Corresponding author: [email protected]
Compiled November 8, 2018Faithful communication is a necessary precondition for large scale all-optical networking and quantuminformation processing. Related theoretical investigations in different areas of physics have led to variousproposals in which finite discrete lattices are used as channels for short-distance communication tasks. Here, inthe framework of femtosecond-laser-written waveguide arrays, we present the first experimental realization ofsuch a channel with judiciously engineered couplings. c (cid:13)
OCIS codes:
Photonic lattices (PLs) i.e., arrays of evanescently cou-pled waveguides, offer a remarkably simple and versa-tile tool for rigorous and transparent testing of variousmodels associated mainly with tight-binding Hamilto-nians [1–3]. Numerous phenomena encountered in var-ious areas of physics, such as Bloch oscillations [4],Anderson localization [5], Glauber-Fock states displace-ment [6], have been demonstrated and studied exper-imentally in the context of linear PLs. Nonlinear PLsoffer the prospect of creating and controlling optical dis-crete solitons [7] and filaments [8]. Besides their versa-tility, PLs offer scalability and compatibility with thewidespread silica technologies. Hence, they are expectedto play a pivotal role in the route towards all-opticalnetworking [9] and large scale quantum information pro-cessing [10]. The faithful transfer of signals is a necessaryrequirement for further developments in these directions,and has thus attracted considerable interest. Most of theproposed solutions, however, rely on boundary-free sig-nal propagation, which implies that lattice truncationand edge effects will unavoidably lead to a considerabledistortion of the transmission [11]. One way to circum-vent such diffraction problems is to prevent the uncon-trollable spread of the wavepacket, by using discrete soli-tons as information carriers; an approach that requiresnonlinear PLs and large intensities [1, 9]. For linear dis-crete PLs one may resort to the segmentation of appro-priate lattice sites [12], or to the engineering of judiciouscoupling constants between adjacent sites [11,13,14]. Thelatter scenario has been also studied thoroughly in thecontext of quantum networks [15], and various faithful-communication (FC) Hamiltonians have been proposed.Taking advantage of the versatility of PLs, we presentthe first proof-of-principle experiment on the FC Hamil-tonian proposed in [13,16], for a sufficiently large numberof sites so that coupling engineering is necessary. Our ob-servations are compared to theoretical predictions, andthe influence of disorder and losses is discussed.The Hamiltonian describing the dynamics of a singleexcitation in a one-dimensional lattice within the tight-binding and nearest-neighbour (NN) approximations is of the form ( ~ = 1)ˆ H = N − X k =1 C k,k +1 (ˆ a † k ˆ a k +1 + ˆ a † k +1 ˆ a k ) , (1a)where ˆ a † k is the creation operator for an excitation onthe k th site. The N sites are assumed to be on-resonant,and C k,l is the coupling between the sites with indices k and l . As has been shown in [13, 16], when the couplingsin the Hamiltonian (1a) are chosen according to C k,k +1 = C p ( N − k ) k, (1b)the lattice operates as a perfect quantum channel, i.e.,ideally perfect transfer of the excitation from the k th tothe ( N − k + 1)th site of the lattice at time τ = π/ (2 C ). a) b) C o u p l i n g ( c m - ) Exp. fit d d N-1 d z d k = d N-k
Fig. 1. (a) Schematic view of the PL. (b) Measured de-pendence of the coupling constant on the waveguide sep-aration at 633 nm and the exponential fit.The photonic realization of Hamiltonian (1) relies onthe fact that the temporal evolution of a single excita-tion in the Hilbert space spanned by {| k i ≡ ˆ a † k | i} , isisomorphic to the spatial propagation of light in a lineararray of evanescently coupled single-mode waveguides,when one of the waveguides is initially excited. Withinthe coupled-mode theory, the light evolution in such astructure, is described by the set of discrete equations i dψ k dz = C k,k − ψ k − + C k,k +1 ψ k +1 , (2)1here ψ k ( z ) is the normalized modal electric-field ampli-tude for the k th waveguide, and C k,k ± are determinedby the evanescent overlap between the transverse compo-nents of the field modes in adjacent waveguides, whereasthe evolution of the excitation takes place in space ( z ) in-stead of time. In a PL of N waveguides of fixed length L ,faithful (ideally perfect) power transfer from the k th tothe ( N − k + 1)th waveguide can be achieved if { C k,k ± } are chosen according to (1b), with C = π/ (2 L ).In the weak-coupling regime, C k,k +1 depends exponen-tially on the separation of the waveguides i.e., C exp k,k +1 = α exp ( − βd k,k +1 ) [2], where α, β are open parameters tobe determined by fitting to related experimental data fora particular setup. For given N and L , the realization ofthe couplings (1b) is achieved for d k,k +1 = h ln( α/C ) − ln( p k ( N − k )) i β − . (3)By means of directional couplers, we measured for oursetup the dependence of the coupling on the separa-tion of the waveguides at λ = 633 nm, obtaining thus α ≃ . − and β ≃ . µ m − (see Fig. 1). Us-ing Eq. (3), we were able to estimate the separationsrequired for the realization of the coupling distribution(1b) in a PL of N = 9 waveguides of length L = 10cm. Subsequently, the PL was inscribed in a fused silicaglass using a standard femtosecond laser writing tech-nique [2, 17]. The waveguides had an elliptical cross sec-tion (4 × µ m ) and the associated refractive indexmodification was estimated to ∼ × − so that at λ = 633 nm a single mode is excited. Light propagationin the PL was monitored using a fluorescence microscopytechnique (FMT) [2]. Following the approach outlinedon pg. 7 of [2], propagation losses in the sample wereestimated to about 0.4dB/cm, and the obtained fluores-cence images, as well as the intensities at the input andthe output of the sample, were normalized accordingly.Figures 2(a) and 2(b) present numerical results on thelight propagation in the structure when one of the twooutermost waveguides and the central waveguide respec-tively, are initially excited. The corresponding experi-mental observations are depicted in Figs. 2(c) and 2(d),and there is a rather good qualitative agreement with thetheoretical predictions. Nevertheless, although propaga-tion losses have been removed by rescaling, yet the powertransfer from the k th to the ( N − k + 1)th waveguide isnot complete, as opposed to the theoretical predictions.As shown in Fig. 2(i), in the case of Fig. 2(c) about 39%of the input intensity is transferred to the 9th waveguide,whereas in the case of Fig. 2(d), this percentage increasesto about 65%. In fact, a small fraction of the input powerseems to have been transferred to waveguides adjacentto the ( N − k + 1)th waveguide; an indication that thediffraction of the signal has been restricted but it hasnot been minimized. Such deviations from the theorycan be attributed to various experimental imperfections.(i) To follow the light propagation in the PL, a FMT wasemployed that requires sufficiently intense writing laser. The optimal energy for waveguides of good quality andintense enough fluorescence signal was 270 nJ, which isquite close to the threshold intensity for damaging thesample ( ∼
300 nJ). So, in view of the low repetitionrate of the writing laser ( ∼ µ m. Our simulations show that such spacing inaccu-racies have a small contribution to the observed devia-tions from theory, since they reduce the power transferby at most 5%; an estimate that can be reduced furtherby increasing the precision of the writing. The imperfec-tion mechanisms and the propagation losses are expectedto be present in the realization of any coupling distribu-tion in PLs (the details may vary though).Propagation losses are expected to be the same in oursetup irrespective of the implemented coupling configu-ration and can be minimized [18]. By contrast, diffrac-tion losses do depend on the realized coupling configu-ration. From Fig. 2(i), the diffraction losses in the im-plemented configuration were about 61% and 35% forFigs. 2(c) and (d), respectively. These losses would besignificantly higher for configurations that are not judi-ciously designed, such as the uniform C k,k +1 = C orthe harmonic one C k,k +1 = C √ k [6]. As depicted inFig. 2, in these configurations the transfer from the 1stto the 9th waveguide does not exceed ideally 83%. Thispercentage drops with increasing N , as opposed to theideally perfect transfer for (1b). Moreover self-imagingis impossible when the central waveguide is initially ex-cited; a distinct feature of configuration (1b) that hasbeen verified experimentally [see Figs. 2(b,d)].The NN approximation in our setup is justified, be-cause the separations between successive waveguides aresufficiently large. According to Eq. (3), the minimumseparations are for the central waveguides, which for N ≫ e − βd min = ǫ for some finite ǫ ≪ d min = d , . The parameter ǫ quantifies the de-viations from the NN Hamiltonian; the larger ǫ is, thelarger deviations we expect. For the implemented PL, d min = (21 . ± . µ m and thus ǫ . . L and α ), we can obtain a theoretical2 P r o p a g a t i o n ( c m ) a) b) c) d)10 10 P r o p a g a t i o n ( c m ) Waveguide number1 5 90510 e) f) g) h)10 10 k = 1k = 5 i) a,bc,de,fg,h
Fraction of input intensity0 0.25 0.5 0.75 1
Fig. 2. (Color online) Light propagation in an array of N = 9 waveguides with coupling distribution given byEq. (1b). (a,b) Numerical results obtained by propaga-tion of Eqs. (2) when the 1st and the 5th waveguide areinitially excited. (c,d) The corresponding fluorescenceimages obtained in our experiment. (e,f) As in (a,b) for C k,k +1 ≈ .
56 cm − . (g,h) As in (a,b) for C k,k +1 = C √ k and C ≈ . − . (i) Fraction of the input intensitytransferred from the k th to the ( N − k + 1)th waveguide.upper bound on N , so that the configuration (3) is im-plementable and couplings beyond NNs do not exceed achosen ǫ . With d min given by Eq. (3) for k = N/ N ≤ Lαǫ/π , which implies that for a fixed α , the designof larger networks requires longer samples. Propagationlosses, however, increase exponentially with L and thus,for large-scale all-optical networking one has to find thefigure of merit between scalability and losses. In general,the presence of couplings beyond NNs does not preclude the existence of FC Hamiltonians [19, 21].In conclusion, we have presented the photonic real-ization of a FC Hamiltonian with engineered couplings.The implemented scheme is capable of minimizing lossesassociated with the diffraction of signals in linear PLs,allowing thus for reliable optical routing and switchingwhen combined with the ideas of [9, 20, 21]. Great effortis needed for minimization of disorder and imperfections,which are expected to affect the signal transmission. Ourresults provide a benchmark case and guide for the plan-ning of future experiments on all-optical networking andshort-distance quantum communication.We acknowledge support by the EU Marie Curie Ex-cellence Grant MULTIRAD Grant No. MEXTCT-2006-042683, and assistance of D. Gray in the experiments. References
1. D. N. Christodoulides, F. Lederer, and Y. Silberberg,Nature , 817 (2003).2. A. Szameit and S. Nolte, J. Phys. B , 163001 (2010).3. S. Longhi, Laser & Photon. Rev. , 243 (2009).4. H. Trompeter, W. Krolikowski, D. Neshev, A. Desy-atnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch,U. Peschel, and F. Lederer, Phys. Rev. Lett. , 053903(2006).5. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Na-ture , 52 (2007).6. R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich,H. Moya-Cessa, S. Nolte, D. N. Christodoulides, andA. Szameit, Phys. Rev. Lett. , 103601 (2011).7. F. Lederer, G. I. Stegeman, D. N. Christodoulides,G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. , 1 (2008).8. M. Bellec, P. Panagiotopoulos, D. G. Papazoglou, N. K.Efremidis, A. Couairon, and S. Tzortzakis, Phys. Rev.Lett. , 113905 (2012).9. R. Keil, M. Heinrich, F. Dreisow, T. Pertsch,A. T¨unnermann, S. Nolte, D. N. Christodoulides, andA. Szameit, Sci. Rep. , 94 (2011).10. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L.O’Brien, Science , 646 (2008).11. S. Longhi, Phys. Rev. B , 041106(R) (2010).12. R. Keil, Y. Lahini, Y. Shechtman, M. Heinrich, R. Pu-gatch, F. Dreisow, A. T¨unnermann, S. Nolte, and A. Sza-meit, Opt. Lett. , 809 (2012).13. R. Gordon, Opt. Lett. , 2752 (2004).14. Y. Joglekar, C. Thompson, and G. Vemuri, Phys. Rev.A , 063817 (2011).15. A. Kay, Int. J. Quantum Inf. , 641 (2010).16. G. M. Nikolopoulos, D. Petrosyan, and P. Lambropou-los, J. Phys.: Cond. Matter , 4991 (2004).17. R. R. Gattass and E. Mazur, Nature Photon. , 219(2008).18. T. Fukuda, S. Ishikawa, T. Fujii, K. Sakuma, andH. Hosoya, Proc. SPIE , 524 (2004).19. V. Kostak, G. Nikolopoulos, and I. Jex, Phys. Rev. A , 042319 (2007).20. G. Nikolopoulos, Phys. Rev. Lett. , 200502 (2008).21. G. M. Nikolopoulos, A. Hoscovec and I. Jex, Phys. Rev.A , 062319 (2012). eferences
1. D. N. Christodoulides, F. Lederer, and Y. Silberberg,“Discretizing light behaviour in linear and nonlinearwaveguide lattices,” Nature , 817–823 (2003).2. A. Szameit and S. Nolte, “Discrete optics infemtosecond-laser-written photonic structures,” J. Phys.B , 163001 (2010).3. S. Longhi, “Quantum-optical analogies using photonicstructures,” Laser & Photon. Rev. , 243–261 (2009).4. H. Trompeter, W. Krolikowski, D. Neshev, A. Desy-atnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch,U. Peschel, and F. Lederer, “Bloch Oscillations andZener Tunneling in Two-Dimensional Photonic Lat-tices,” Phys. Rev. Lett. , 053903 (2006).5. T. Schwartz, G. Bartal, S. Fishman, and M. Segev,“Transport and Anderson localization in disorderedtwo-dimensional photonic lattices,” Nature , 52–55(2007).6. R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich,H. Moya-Cessa, S. Nolte, D. N. Christodoulides, andA. Szameit, “Classical Analogue of Displaced FockStates and Quantum Correlations in Glauber-Fock Pho-tonic Lattices,” Phys. Rev. Lett. , 103601 (2011).7. F. Lederer, G. I. Stegeman, D. N. Christodoulides,G. Assanto, M. Segev, and Y. Silberberg, “Discrete soli-tons in optics,” Phys. Rep. , 1–126 (2008).8. M. Bellec, P. Panagiotopoulos, D. G. Papazoglou, N. K.Efremidis, A. Couairon, and S. Tzortzakis, “Observationand optical tailoring of photonic lattice filaments,” Phys.Rev. Lett. , 113905 (2012).9. R. Keil, M. Heinrich, F. Dreisow, T. Pertsch,A. T¨unnermann, S. Nolte, D. N. Christodoulides, andA. Szameit, “All-optical routing and switching for three-dimensional photonic circuitry,” Sci. Rep. , 94 (2011).10. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L.O’Brien, “Silica-on-Silicon Waveguide Quantum Cir-cuits,” Science , 646–649 (2008).11. S. Longhi, “Periodic wave packet reconstruction intruncated tight-binding lattices,” Phys. Rev. B ,041106(R) (2010).12. R. Keil, Y. Lahini, Y. Shechtman, M. Heinrich, R. Pu-gatch, F. Dreisow, A. T¨unnermann, S. Nolte, and A. Sza-meit, “Perfect imaging through a disordered waveguidelattice,” Opt. Lett. , 809 (2012).13. R. Gordon, “Harmonic oscillation in a spatially finitearray waveguide,” Opt. Lett. , 2752–2754 (2004).14. Y. Joglekar, C. Thompson, and G. Vemuri, “Tunablewaveguide lattices with nonuniform parity-symmetrictunneling,” Phys. Rev. A , 063817 (2011).15. A. Kay, “Perfect, efficient, state transfer and its appli-cation as a constructive tool,” Int. J. Quantum Inf. ,641 (2010).16. G. M. Nikolopoulos, D. Petrosyan, and P. Lambropou-los, “Electron wavepacket propagation in a chain of cou-pled quantum dots,” J. Phys.: Cond. Matter , 4991–5002 (2004).17. R. R. Gattass and E. Mazur, “Femtosecond laser micro-machining in transparent materials,” Nature Photon. ,219–225 (2008).18. T. Fukuda, S. Ishikawa, T. Fujii, K. Sakuma, andH. Hosoya, “Low-loss optical waveguides written by fem-tosecond laser pulses for three-dimensional photonic de- vices,” Proc. SPIE , 524–538 (2004).19. V. Kostak, G. Nikolopoulos, and I. Jex, “Perfect statetransfer in networks of arbitrary topology and couplingconfiguration,” Physical Review A , 042319 (2007).20. G. Nikolopoulos, “Directional Coupling for QuantumComputing and Communication,” Phys. Rev. Lett. ,200502 (2008).21. G. M. Nikolopoulos, A. Hoscovec and I. Jex, “Analysisand minimization of bending losses in discrete quantumnetworks,” Phys. Rev. A , 062319 (2012)., 062319 (2012).