Felix Dreisow,1 Gang Wang,2 Matthias Heinrich,1 Robert Keil,1 Andreas Tünnermann,1 ... *Corresponding author: alexander.szameit@uniâjena.de. Received ...
October 15, 2011 / Vol. 36, No. 20 / OPTICS LETTERS
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Observation of anharmonic Bloch oscillations Felix Dreisow,1 Gang Wang,2 Matthias Heinrich,1 Robert Keil,1 Andreas Tünnermann,1 Stefan Nolte,1 and Alexander Szameit1,3,* 1
Institute of Applied Physics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
2
Max Planck Institute for the Physics of Complex Systems, Noethnitzer Str. 38, 01187 Dresden, Germany 3
Solid State Institute and Physics Department, Technion, 32000 Haifa, Israel *Corresponding author: alexander.szameit@uni‑jena.de Received June 17, 2011; accepted September 5, 2011; posted September 7, 2011 (Doc. ID 149453); published October 4, 2011
We report on the experimental observation of Bloch oscillations of an optical wave packet in a lattice with secondorder coupling. To this end, we employ zigzag waveguide arrays, in which the second-order coupling can be precisely tuned. © 2011 Optical Society of America OCIS codes: 190.0190, 190.6135, 190.4390.
Since their prediction 80 years ago, Bloch oscillations (BO) of an evolving wave packet in a periodic potential is a basic concept in physics [1]. The periodic movement of the wave packet is caused by an external potential gradient, imposed on the underlying lattice. Although BO were originally predicted in the context of solids, so far BO could not be observed in this system, due to various experimental difficulties. First, crystalline potentials in solid state physics commonly suffer from time dependence due to the occurrence of phonons. Secondly, it is almost impossible to observe the wave function of an isolated electron, therefore, multibody effects, such as nonlinearities and correlations, will have a significant impact on the experimental result. However, BOs are entirely based on interference of waves. Therefore, this concept is not restricted to quantum systems but applies in principle to any wave-mechanical system. Besides BO in semiconductors [2], ultracold atoms [3,4], and Bose–Einstein condensates [5], their observation in photonic waveguide arrays gathered particular attention in recent years [6–8]. However, in all these equivalent systems, BO were considered only in the special case of nearest-neighbor coupling, where only adjacent lattice sites are allowed to interact. Recently, it was suggested that BO can also occur in lattices with second-order coupling, where also nextnearest lattice sites can interact [9]. This results in a significantly altered band structure beyond the nearestneighbor model, which strongly alters the oscillation trajectory of the evolving wave packet. In this work, we report on the experimental demonstration of such anharmonic BO. To this end, we employ curved zigzag waveguide lattices [10], where the second-order coupling has significant influence [11]. Our setting is sketched in Fig. 1(a), where the firstorder coupling is the coupling between the nearestneighboring waveguides along the diagonal, and the second-order coupling connects the two next-nearest neighbors along the horizontal. The corresponding quantum-mechanical potential, formed by a crystalline lattice of a solid, is shown in Fig. 1(b). Each waveguide in our array forms a potential well, whereas the curved path of the guides is equivalent to a linear potential gradient in quantum-mechanics, caused by an external electric field [12]. The interaction between the individual potential 0146-9592/11/203963-03$15.00/0
wells—corresponding to Fig. 1(a)—is indicated by the arrows. To describe the light evolution in our curved waveguide array, we employ a discretized version of the Schrödinger equation for the amplitudes φm in the mth waveguide: i
d þ mF φm þ ðφmþ1 þ φm−1 Þ dZ þ Δðφmþ2 þ φm−2 Þ ¼ 0:
ð1Þ
The propagation direction Z ¼ c1 z is normalized to the first-order coupling constant c1 and Δ ¼ c2 =c1 describes the relative second-order coupling. Our waveguides, fabricated by femtosecond laser waveguide writing [13] in 100 mm long fused silica samples, follow a parabolic trajectory and therefore induce a linear transverse potential gradient F that is inversely proportional to the curvature radius R [12]. The quantity R is used to fix the Bloch period zB ¼ 95 mm to our sample length of 100 mm, using the condition zB ¼ Rλ=ðn0 dÞ, where λ ¼ 633 nm is the vacuum wavelength of the launched light, n0 ¼ 1:45 is the bulk refractive index, and d ¼ 12 μm is the lattice constant. Note that, due to the elliptic waveguide profile, c1 slightly increases with the increasing lattice angle [14], which is unharmful for our experimental analysis. The relative strength Δ of the second-order coupling c2 is tuned by the lattice angle α, as defined in Fig. 1(a). The extreme value of Δ for 75° is achieved by additionally decreasing the coupling constant c1 and keeping the ( ) α
Fig. 1. (Color online) (a) A sketch of the zigzag waveguide lattice. The interaction between adjacent lattice sites is illustrated by the red solid line, whereas the lattice angle α determines the interaction strength between the next-nearest neighbors (blue dotted line). (b) Corresponding quantum-mechanical potential, formed by the individual potential wells and a linear potential gradient that results from a curvature of the waveguides along the propagation direction. © 2011 Optical Society of America
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Fig. 2. (Color online) (Upper row) Experimentally observed trajectory of the wave packet motion as a function of the lattice angle α and, therefore, the relative second-order coupling Δ. (Second row) Numerical simulations of the light evolution insider our samples, using Eq. (1). (Lower row) Dispersion relation between β and κ of the underlying lattices, as given by Eq. (2).
second-order coupling the same as for 60°. The value α ¼ 0° corresponds to the trivial lattice with Δ ¼ 0. The effective lattice constant between the nearest-neighboring guides is therefore given by d ¼ 12 μm cosðαÞ, which requires a correction of the curvature radius with respect to the lattice angle to achieve the same Bloch period for all samples. In the first part of our experimental analysis, we focused on the evolution of broad wave packets that cover approximately 10 lattice sites. We launched the light at λ ¼ 633 nm into our sample, using a microscope objective and directly observed the movements of the wave
packet inside our sample by employing waveguide fluorescence microscopy [15]. In Fig. 2, in the upper row of panels, the experimentally observed wave packet trajectories are shown, where as in the second row the numerically calculated trajectories, obtained from Eq. (1), are depicted for comparison. One can clearly see that for increasing lattice angle α and therefore increasing relative second-order coupling Δ, the wavepacket follows a more complex trajectory, where multiple reflection occurs as shown in Fig. 2. The reason for this behavior originates from the unique dispersion relation of our lattice that is given by [10,11]
Fig. 3. (Color online) (Upper row) Experimentally observed breathing of the excited wave packet when only a single lattice site is excited as a function of the lattice angle α and, therefore, the relative second-order coupling Δ. (Second row) Numerical simulations of the light evolution insider our samples, using Eq. (1).
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β ¼ 2c1 cosðκÞ þ 2c2 cosð2κÞ;
ð2Þ
where β is the longitudinal and κ is the transverse wavenumber. A linear potential gradient in real space forces an excitation to scan along the dispersion relation in reciprocal space with a constant velocity. Hence, the observed trajectory of the wave packet in Fig. 2 matches the shape of the respective dispersion relation of the underlying lattices, which are depicted in the lower row of Fig. 2. Anharmonic BO in lattices with higherorder coupling can therefore be utilized to perform bandstructure spectroscopy, revealing the dispersion relation of the system in great detail. In a second set of experiments, we focus on a narrow excitation with Gaussian profile covering only a single waveguide. Such a narrow excitation in real space corresponds to a broad excitation on reciprocal space. In this case, the BO drastically change in their appearance. For such narrow excitations, light cannot distinguish between left and right anymore. In more mathematical terms, if φm is a solution of Eq. (1), then φ0m ¼ ð−1Þm φ−m is also a solution with the same intensity profile but a phase jump of π between adjacent lattice sites [6]. One solution tends to move toward increasing and the other one toward decreasing wavenumbers. When only a single waveguides is excited, the field is evenly decomposed into both solutions, so that the intensity distribution stays approximately symmetric and shows a breathing oscillation during propagation. Our experimental results are summarized in Fig. 3 (upper row). One clearly sees the transition of the trivial single-site BO at α ¼ 0° with a period of z ¼ zB to oscillations that exhibit at α ¼ 75° almost complete refocusing already after z ¼ zB =2. Our experimental findings are confirmed by numerically solving Eq. (1), with propagation patterns that are shown for comparison in Fig. 3 (lower row). In conclusion, we experimentally demonstrated BO under the influence of second-order coupling in curved
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waveguide arrays with zigzag geometry. For broad excitations, the wave packet moves on a complex trajectory, which is directly determined by the band structure of the lattice. The authors acknowledge support by the German Academy of Science Leopoldina (grant LPDS 2009-13) and the Leibniz program of the Deutsche Forschungsgemeinschaft (DFG). G. Wang acknowledges the support from the Fundamental Research Funds for the Central Universities (grant 2012QNA38). References 1. F. Bloch, Z. Phys. 52, 555 (1928). 2. C. Waschke, H. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Koehler, Phys. Rev. Lett. 70, 3319 (1993). 3. M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996). 4. S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, Phys. Rev. Lett. 76, 4512 (1996). 5. B. P. Anderson and M. A. Kasevich, Science 282, 1686 (1998). 6. U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998). 7. T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999). 8. R. Morandotti, U. Peschel, J. Aitchinson, H. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999). 9. G. Wang, J. Ping Huang, and K. W. Yu, Opt. Lett. 35, 1908 (2010). 10. N. Efremidis and D. Christodoulides, Phys. Rev. E 65, 056607 (2002). 11. F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Lett. 33, 2689 (2008). 12. G. Lenz, I. Talanina, and M. de Sterke, Phys. Rev. Lett. 83, 963 (1999). 13. A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010). 14. A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Express 15, 1579 (2007). 15. A. Szameit, F. Dreisow, H. Hartung, S. Nolte, and A. Tünnermann, Appl. Phys. Lett. 90, 241113 (2007).
