Abstract
As an important electron transportation phenomenon, Bloch oscillations have been extensively studied in condensed matter. Due to the similarity in wave properties between electrons and other quantum particles, Bloch oscillations have been observed in atom lattices, photonic lattices, and so on. One of the many distinct advantages for choosing these systems over the regular electronic systems is the versatility in engineering artificial potentials. Here by utilizing dissipative elements in a CMOScompatible photonic platform to create a periodic complex potential and by exploiting the emerging concept of paritytime synthetic photonics, we experimentally realize spatial Bloch oscillations in a nonHermitian photonic system on a chip level. Our demonstration may have significant impact in the field of quantum simulation by following the recent trend of moving complicated tabletop quantum optics experiments onto the fully integrated CMOScompatible silicon platform.
Introduction
Bloch oscillations (BO) effect is a fundamental electron transport phenomenon in condensed matter, which is characterized as the coherent oscillatory motion of electron in a periodic potential driven by an external DC electric field^{1}. The electron is accelerated by the electric field, then reflected when it reaches a momentum satisfying the Bragg reflection condition in the periodic potential and, subsequently, decelerated to the initial state to form a BO cycle. However, because of dephasing effects, it is difficult to observe BO in natural crystals. Owing to the development of semiconductor technology, BO have been experimentally observed in semiconductor superlattices with periodically arranged layers of different semiconductor materials^{2}. Due to the equivalence between the Schrödinger equation in quantum mechanics and the classical wave equation, BO can be also observed for matter waves^{3,4,5,6}, optical waves^{7,8,9,10,11,12,13}, acoustic waves^{14} and surface plasmon polariton waves^{15}. In fact, studying BO and related wave transportation phenomena in some of these waveforms can be very advantageous compared to electrons. For example, coherent optical sources with much longer coherent length are readily available, so the dephasing effect would not become a hurdle. Moreover, advanced nanofabrication technologies allow for the creation of very complicated optical potentials, which has already become a major driving force behind the rapid development of the latest nanophotonic technologies. It is worth pointing out that almost all the experimental studies involving BO are primarily focused on the wave dynamics in Hermitian systems with only realvalued potentials. Recently, a particular class of nonHermitian photonic systems with paritytime (PT) symmetry has drawn great research interest^{16,17,18,19,20,21,22,23,24,25}. It is shown that such systems can possess a spontaneous symmetrybreaking phase transition with an eigen energy transition from real spectra into complex spectra. The phase transition point shows all the characteristics of an exceptional point, that is, having nonHermitian degeneracy, where the real and imaginary parts of eigenvalues are identical. Different from conventional Hermitian degeneracy, not only the eigenvalues but also the eigenvectors coalesce at an exceptional point, giving rise to abundant intriguing physical phenomena. These include unidirectional BO^{26}, unidirectional invisibility^{27,28,29,30} and unidirectional transmission^{31}. In addition, the exceptional point singularity also has broadened the scope of metamaterials^{32,33,34}.
Here we report an experimental study of photonic BO and related wave dynamics in Hermitian and nonHermitian systems realized with complementary metaloxidesemiconductor (CMOS)compatible fabrication processes on a silicononinsulator (SOI) platform. A realvalued periodic potential in a Hermitian photonic lattice in conjunction with a linear gradient potential perturbation acting as a constant acceleration force is created by bending an array of identical SOI waveguides, while a complex PT synthetic potential in a nonHermitian photonic lattice is realized with the same lattice but with an additional optical loss modulation layer, that is, a chrome (Cr) layer on top of every other bent silicon waveguide. Compared with all other platforms for studying BO and its related classic or quantum wave dynamics^{2,3,4,5,6,7,8,9,10,11,12,13,14,15}, our integrated platform demonstrated here shows great advantages in emerging applications requiring ultimate scalability and stability, for instance, quantum optics on a chip^{35,36,37,38,39,40}. In addition, by using scanning nearfield optical microscope (SNOM), we are able to directly visualize the wave dynamics responsible for the BO continuously for both the Hermitian and nonHermitian photonic lattice in spatial domain, which is another key advantage over the largescale discretized PTsymmetric optical networks^{30}, where the wave dynamics is discrete in temporal domain and all waves are needed to be multiplexed with actively controlled optical switching components to construct the BO in temporal domain. Our spatial wave dynamics recorded by the SNOM reveal a prominent secondary emission around the BO recovery point for the nonHermitian system in contrast to the classical picture of BO in a Hermitian system. This feature in such a PT synthetic photonic lattice is related to the spontaneous PT symmetry breaking.
Results
BO in a Hermitian photonic lattice
As depicted in Fig. 1a, our Hermitian photonic lattice consists of an array of straight equally spaced identical SOI stripe waveguides, which support only a fundamental transverse electric mode at the wavelength of 1,550 nm. In the tightbinding approximation, the propagation of optical wave in this Hermitian photonic lattice is described by coupledmode equations
where Α_{n} is the propagating amplitude in the nth waveguide, and κ is the coupling coefficient between adjacent waveguides. The corresponding dispersion relation is determined (see Methods) to be
where d is the centre distance between adjacent waveguides and β_{H} and q_{H} are the longitudinal propagation constant and the transverse Bloch momentum of the Hermitian photonic lattice, respectively. A plot of this dispersion relation or band diagram is shown in Fig. 1b, where the blue and red curves represent the real and imaginary parts of the band, respectively. As expected, in this Hermitian photonic lattice the imaginary part of the band is always equal to zero regardless of the Bloch momentum. To obtain Hermitian BO, the Hermitian photonic lattice is bent to modify its spatial distribution of optical potential, as sketched in Fig. 2a. The curvature is seen as inertial force acting on the optical wave, and the linear position dependence of curvature in the radial direction thus can result in a linear potential gradient, which ultimately gives rise to an equivalent DC field required for BO. The corresponding dynamic evolution of this Hermitian bent photonic lattice becomes
where Α_{n} is the mode envelop for the nth waveguide, and is the equivalent DC field corresponding to the equivalent index gradient in this particular kind of photonic lattice systems (Supplementary Note 1). In the presence of this DC field, the eigenmode can be represented as , where , m=0, ±1, ±2, … is WannierStark ladder and Ψ_{m}(n) are the corresponding WannierStark eigenstates in the Wannier representation (Supplementary Note 2). We can see that the equivalent DC field F acts as a constant WannierStark ladder spacing, which induces field recovery with Bloch period , and WannierStark eigenstate is a localized optical mode caused by total internal reflection on the lowindex side and Bragg reflection on the opposite highindex side.
To explore the wave dynamics of this photonic lattice, a single waveguide is selectively excited at the input plane. The corresponding evolution of the mode in this waveguide resembles a breathing mode^{9,10}. To verify this characteristic, we simulate this photonic lattice by means of 3D finitedifference timedomain (FDTD) simulations where the electric field distribution is shown in Fig. 3a. The optical wave initially couples from the centre waveguide into the parallel waveguides towards the transverse and forward direction, and then after propagating pass a certain curvature point (approximately half of Bloch Period) it starts to converge back to a recovering point, which forms a Bloch period , thus finishing a breathing cycle or a BO cycle. The Bloch period can also be conveniently expressed as an inplane polar angle . As seen from Fig. 3a, the analytical calculated value θ_{B}≈84° agrees well with the simulation. In experiment, this Hermitian bent photonic lattice is fabricated using CMOScompatible nanofabrication processes (see Methods). A scanning electron micrograph of the fabricated device is shown in Fig. 2b,c. The input waveguide used as the single waveguide excitation is located at the centre of the bent photonic lattice. The electric field distribution of the optical wave recorded by a SNOM (see Methods) in the area of 30 μm × 15 μm within the vicinity of the first BO recovery point is shown in Fig. 3c. As would be expected for a Hermitian BO system, the experimental result indicates that the dynamics of the optical wave has a breathing motion and possesses a very distinct recovery point. This experimental observation in fact agrees very well with the simulated electric field distribution shown in Fig. 3b.
BO in a nonHermitian photonic lattice
PT synthetic photonic structures exhibit intriguing and unusual wave dynamics due to the presence of symmetry breaking, resulting in phase transition from a Hermitian system to a nonHermitian system. To obtain strict photonic PT symmetry, it requires precise engineering of the spatial distribution of complex optical potential by using optical gain and loss media. However, gain media is not straightforward to be incorporated into the CMOScompatible fabrication process; therefore, instead of gain, a class of dissipative PT symmetry systems^{19,29} with an average background loss have been widely adopted in the literature to study PTrelated physics. Here the dissipative element consists of an array of lossy metal stripes deposited on top of every other silicon waveguide of the photonic lattice, as shown in Fig. 4a. By referencing to the background loss, the optical wave in silicon waveguides without (with) metal strips experiences an equivalent gain (loss). The dynamic evolution of the straight photonic lattice, initially Hermitian, is drastically altered by the presence of the synthetic dissipative PT symmetric potential. This change is reflected in the coupledmode equations governing the dynamics of the mode amplitude Α_{n′}:
where there exists an alternating mode attenuation (0↔γ/2) in adjacent waveguides in addition to the mode coupling κ between them. The dispersion relation governing the wave dynamics of this PT synthetic photonic lattice is determined (see Methods) as
which is plotted in Fig. 4b. This class of dissipative PT symmetric systems possesses two exceptional point singularities at as highlighted by the black points, where both the real and imaginary parts of the band diagram are degenerate. Between these two exceptional point singularities, the state of the system falls into a symmetrybroken phase with a coalesced real part of the band but bifurcated imaginary part. Since for any γ>0 there will also be complex eigenvalues near the Brillouin Zone edge (q_{NH}=π/(2d)), the PT symmetry breaking is spontaneous or in other words thresholdless. However, it is worth noting that the thresholdless PT symmetry breaking in the straight photonic lattice discussed here is local, where PT symmetry breaking only happens for particular kvectors along the transverse direction^{20}, which is strongly related to optical excitation conditions.
Similarly to the realization of BO in the Hermitian photonic lattice, a constant DC field is also mimicked by a circular bending of the lattice, as sketched in Fig. 5a. In the presence of this DC field the spectra have two interleaved WannierStark ladders , m=0, ±1, ±2, …, where is a complex function of and (Supplementary Fig. 1 and Supplementary Note 3). In other words, the eigen energy of a classical BO is split into two in a complex energy plane to reflect the emergence of two localized modes, one with pure optical loss and the other with equivalent optical gain biased by the background loss. The splitting of eigen energy level is a direct outcome of the PT symmetry breaking near the Brillouin boundary. Note that the PT symmetry breaking phenomenon for a particular Bloch mode of the previously described nonHermitian straight photonic lattice occurs when γ>4κcos(q_{NH}d) as shown in Fig. 4b. In this bent configuration the PT symmetry breaking can always be experienced by the optical wave under any excitation condition, because the transverse wave vector in the initial excitation will change under the acceleration force caused by the bending and it will eventually approache the edge of the Brillouin zone, where the thresholdless PT symmetry breaking takes place. To elucidate this, the optical wave dynamics in the nonHermitian bent photonic lattice is simulated and shown in Fig. 6a. The dynamics of BO in this case still possesses a periodic feature, but with decreasing intensity due to the dissipative background of this system. The BO here, however, is accompanied by a noticeable secondary emission at the BO recovery point as indicated in the zoomin view of Fig. 6b. In experiment, the designed nonHermitian bent photonic lattice is fabricated with an additional Cr metal layer on top of the Hermitian bent photonic lattice (see Methods). An example of a fabricated device is shown in Fig. 5b,c, where the waveguide bending radius and the input/output waveguide follow the same design as its Hermitian counterpart. The nearfield electric field distribution of the optical wave in this nonHermitian system is recorded by a SNOM and is shown in Fig. 6c. The overall electric field distribution obtained experimentally near the BO recovery point (indicated as red dots) shows good agreement with the theoretical results in Fig. 6b. Although some of the finest features are not well resolved due to the instrumental limitation, the secondary emission (highlighted in while dotted circles) is clearly visible. Such a emission is originated from the presence of two WannierStark ladders as a result of the spontaneous PT symmetry breaking experienced by the optical wave as it propagates down the photonic lattice with its transverse wave vector evolving to touch the Brillouin zone edge (q_{NH}=π/(2d)) under the gradient force F as shown in Fig. 4b.
Discussion
By engineering the spatial distribution of the complex potential for photonic lattices on a SOI photonic platform, we have experimentally demonstrated photonic BO in both a Hermitian and a nonHermitian system. Apart from the classic oscillatory evolution of the optical wave in Hermitian BO, the dynamics of nonHermitian BO exhibits an intriguing feature characterized by a secondary emission formed near the BO recovery point. This unique feature is closely related to the spontaneous/thresholdless PT symmetrybreaking phase transition for the BO in our dissipative PT synthetic photonic lattice. The fully integrated nonHermitian photonic lattice provides a major step forward in expanding the application scope of onchip photonics in quantum simulation^{35,36,37,38,39,40,41,42}. In fact, the thresholdless PT symmetrybreaking characteristic in our passive system makes it a perfect candidate for studying the quantum version of nonHermitian BO as long as the loss modulation is kept low enough to allow for a meaningful highfidelity quantum measurement, which, however, is difficult to conduct in a gainloss balanced nonHermitian system because of the degradation of the initial quantum input state by the inherent quantum noise inside the gain medium. Moreover, thanks to stateoftheart nanofabrication technology, which allows for the deliberately designed complex potential distribution to be created, this CMOScompatible photonic platform with great flexibility opens the door towards the exploration of unprecedented nonHermitian transport phenomena on a chip^{43,44,45,46,47,48}.
Methods
Sample fabrication
The Hermitian photonic lattice was fabricated by means of a single step of electronbeam (ebeam) lithography to define the silicon waveguide layer, which was followed by resist development and inductively coupled plasma etching using a mixture of SF_{6} and C_{4}F_{8}. The nonHermitian photonic lattice was fabricated using two steps of aligned ebeam lithography. First, the Cr layer was defined by using a positive ebeam resist and developed, followed by 4 nm of Cr deposition by means of ebeam deposition and liftoff. The silicon waveguides were defined by means of a second step of aligned ebeam lithography using a negative resist, followed by development and finally inductively coupled plasma dry etching using a mixture of etchant aforementioned to define the waveguides. Both photonic lattices went through additional final processing steps before optical testing. These include (1) a photolithographic process to fabricate SU8 polymer waveguidebased mode converters surrounding the silicon input/output waveguides and (2) dicing and facet polishing.
Dispersion calculation
Coupled mode equations describing the wave dynamics in both the Hermitian and nonHermitian photonic lattices were solved to obtain the dispersion spectrum or band diagram. This was done by inserting a plane wave ansatz Α_{n}∝exp(iβz−inqd) for the nth waveguide’s mode amplitude into the coupled mode equation, that is, equations (1) and (4), and by enforcing the nontrivial solutions for Α_{n}. The corresponding transverse Bloch momentum qdependent dispersion β in a complex wavevector domain represents the underlying dispersion spectrum or band diagram.
Measurement
In measurements, a nearinfrared optical wave at the wavelength of 1,550 nm from a tunable external cavity laser (Photonetics, 1,500–1,640 nm) was launched into the SU8 mode converter through a lensed tapered singlemode fibre. The testing chip and the lensed fibre are bonded onto a glass slide so that the relative position of the fibre to the waveguide is fixing when the testing chip is translated on a moving stage under a SNOM (SNOM100 Nanonics). The SNOM is equipped with a metalcoated fibre tip with a 50 nm radius to map the intensity distribution. A schematic of our measurement setup is shown in Supplementary Fig. 2 and details of this setup are included in Supplementary Note 4.
Additional information
How to cite this article: Xu, Y.L. et al. Experimental realization of Bloch oscillations in a paritytime synthetic silicon photonic lattice. Nat. Commun. 7:11319 doi: 10.1038/ncomms11319 (2016).
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Acknowledgements
This work was jointly supported by the National Basic Research Program of China (Grants 2012CB921503, 2013CB632700 and 2015CB659400) and the National Nature Science Foundation of China (Grants 11134006, 11474158 and 61378009). M.H.L. also acknowledges the support of Natural Science Foundation of Jiangsu Province (BK20140019) and the support from Academic Program Development of Jiangsu Higher Education (PAPD). W.S.F. and A.S. acknowledge Boeing for their support under their SRDMA program and also thank the NSF CIAN ERC (Grant EEC0812072).
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Y.L.X., W.S.F., M.H.L. and X.P.L. conceived the idea and designed the devices. Y.L.X. performed the FDTD simulations, W.S.F. designed the chip layout and fabricated the devices, and L.G. carried out the SNOM measurements. All the authors contributed to discussion of the project. A.S., Z.Y.L. and Y.F.C. guided the project. Y.L.X., X.P.L. and M.H.L. wrote the manuscript with revisions from other authors.
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Supplementary Figures 1 & 2, Supplementary Notes 14 and Supplementary References. (PDF 102 kb)
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Xu, YL., Fegadolli, W., Gan, L. et al. Experimental realization of Bloch oscillations in a paritytime synthetic silicon photonic lattice. Nat Commun 7, 11319 (2016). https://doi.org/10.1038/ncomms11319
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DOI: https://doi.org/10.1038/ncomms11319
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