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Abstract
We experimentally studied a continuous time evolution of a “plasmonic” walker in a 1-dimensional lattice structure based on long-range surface plasmon polariton waveguides. The plasmonic walker exhibited a typical time evolution of a 1-dimensional quantum walk, which indicates that the plasmonic system is a potential platform to construct quantum walk simulators. By comparing experimental results to numerical simulations, the fidelity of the plasmonic quantum walk simulator is estimated to be > 0.96, which demonstrates that the plasmonic system can be a feasible platform for large-scale and high dimensional quantum walk simulators.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum walk (QW) is of great interest in wide areas of scientific research, i.e. physics, computational science and biology, in the same way as the classical probabilistic random walk [1]. So far, many theoretical and experimental investigations of QW have been performed. The quantum walker exhibits a time evolution completely different from that of a classical random walker. The quantum feature has been already applied to simulations, for instance, to analyse localization phenomena [2–5], topological systems [6,7], and photosynthetic energy transfer [8]. QW can efficiently generate the entanglement which is the indispensable concept for the quantum information science and technology [9,10], and offers the novel approach to implement a search algorithm and build a quantum computer [11,12].
QW has been implemented in several physical systems based on nuclear magnetic resonance samples [13], the optically trapped single atom [14], trapped ions [15], photons in optical circuits using beam splitters [3,16], continuously coupled waveguide arrays [2,17,18], and multicore fiber [5]. Among them, the most promising physical system to realize a large-scale QW is considered to be the photonic one based on on-chip integrated devices which have been developed for a wide range of applications, e.g. the optical interconnect. On-chip photonic integrated devices have scalabilities better than any other physical systems and realize a variety of functionalities and reconfigurability [3,19,20]. Various waveguide platforms have been proposed to implement photonic QWs, e.g. silica (SiO2), silicon (Si), and silicon nitride (SiN) waveguide platforms [2,3,16–20]. After the successful demonstration of 1-dimensional (1D) photonic QWs using planer type of photonic circuits, 2D QWs have recently been demonstrated using 2D continuous-coupled waveguide arrays based on a laser-writing waveguide platform [21]. However, the demonstration of a large-scale 2D QW is still quite challenging, because of difficulties in the fabrication of 3D structured devices.
As an alternative, plasmonic devices can be employed. QWs and physical simulators that utilize unique characteristics of surface plasmon polaritons (SPPs) have been proposed [22–25]. Even if those unique characters are not used, plasmonic circuits are fascinating as the feasible integrated circuit [26,27]. The Bloch oscillation has been experimentally observed in the SPP waveguide array which modelled by the nearest-neighbour coupling with the tight-binding Hamiltonian [23]. The successful demonstration reveals that the quantum dynamics can be simulated using the plasmonic circuit based on the dielectric-loaded SPP (DL-SPP) waveguide. The DL-SPP waveguide allows the electro-magnetic (EM) filed to be confined in the submicron dielectric core. Therefore, the approach using the DL-SPP waveguide is advantageous for down-sizing of the plasmonic circuit which supports a low-loss propagation mode. However, in the viewpoint of the possibility of larger-scale and high-dimensional integrated circuits, another SPP waveguide platform should be considered. One is the plasmonic circuit based on the long-range SPP (LR-SPP) waveguide. It has low-loss and single-polarization features which contribute to the high-fidelity implementation of a large-format integrated circuit [26,28,29]. Moreover, the LR-SPP waveguide platform can offer multi-layered structures with a high reliability. The LR-SPP waveguide can be simply formed by the nanometer-thick metal film embedded in micrometer-thick dielectric, and consequently vertical stacking of metal-dielectric layers may be relatively easy [30].
In this paper, we report on the first observation of a plasmonic quantum walk in a laterally arrayed LR-SPP waveguides. The time evolution of the plasmonic walker was found to be that of the typical continuous quantum walks in a 1D photonic waveguide lattice structure, which indicates that a single LR-SPP is useable as a bosonic qubit [31]. The fidelity of our plasmonic QW device was evaluated via a comparison between experimental results and the corresponding numerical ones. In addition, the numerical simulation demonstrated that the LR-SPP waveguide can be coupled with the nearest neighbour waveguides in the vertical direction, which clearly shows the great potential to realize a 2D plasmonic lattice structure.
2. Experiments
2.1 Plasmonic lattice device based on gold strip waveguides
In order to implement the continuous plasmonic QW, we fabricated arrayed plasmonic waveguides supporting SPPs. SPPs are the quantized EM field propagating along a dielectric-metal interface [32]. The EM field is coupled to the plasma oscillation of free electrons on the metal surface. A symmetrical structure of a thin metal film embedded in dielectric supports the two identical mode, namely the symmetrical and asymmetrical modes (with respect to the electric field orientation). The symmetrical mode which is the so-called LR-SPP has extremely low propagation loss and its mode size is close to that of the standard single-mode optical fiber [33,34]. These features are beneficial to the application to integrated quantum circuits.
The LR-SPP waveguide array are illustrated in Fig. 1. Each waveguide is composed of the 20-nm-thin and 8-µm-wide gold (Au) strip embedded in dielectric (upper and lower cladding layers with a thickness of 22 µm) [28,29]. We used a UV-curable fluorinated resin (ZPU12-RI, ChemOptics) as the dielectric material. Waveguides are designed for an efficient LR-SPP excitation by a laser (photons) at 1550 nm via the end-fire coupling method [28,29,31,35] using a polarization maintaining fiber (PMF), and excited LR-SPPs can propagate the dielectric-gold-dielectric interface with a low propagation loss. The coupling efficiency and the propagation loss were −3 dB and −1.5 dB/mm, respectively. The coupling loss would be reduced to the theoretical value of −0.1 dB, using a mode-shaping structure and reducing the roughness of the input facet of the device [34]. The 1D lattice structure is composed of 50 Au-strips. The separation of adjacent Au strips was set to 2 µm.
Fig. 1. 1D lattice of long-range surface plasmon polariton waveguides. (a) the cross-section and (b) perspective illustrations of the plasmonic lattice device.
Since the LR-SPP waveguide we fabricated is composed of 8-µm-wide Au strip, only the fundamental LR mode can be supported, and consequently the single-mode propagation is realized (higher-order modes must be cut off with an Au-strip width of lower than ∼20 µm [33]). Then, the coupling between nearest-neighboring LR-SPP waveguides can be modeled using the orthogonal symmetric and asymmetric coupler supermodes [33,34]. The modeling have been widely used for the analysis of the coupling between general parallel waveguides based on dielectric structure (the coupling between a dielectric waveguide and the LR-SPP waveguide was also studied by the same way in Ref. [36]). Thus, the coupling coefficient $C\; $ for the nearest-neighboring LR-SPP waveguides is simply given by $C = \pi ({\textrm {Re}} \{ {n^ + }\} - {\textrm{Re}} \{ {n^ - }\} )/\lambda $, where ${n^ + }$ and ${n^ - }$ are the effective refractive indices of the symmetric and asymmetric coupler supermodes, respectively, $\lambda $ is the free-space wavelength. The fact that the general coupling model has been adopted indicates that the dynamics in the 1D array of the LR-SPP waveguides can also be modeled using the nearest-neighbor-coupling with the tight-binding Hamiltonian [18,23,37]. The excitation of a single LR-SPP in $k\textrm{ - th}$ waveguide is governed by the Heisenberg equation for the LR-SPP (bosonic) creation operator ${a^\dagger }$,
2.2 Experimental setup for the plasmonic quantum walk
Figure 2(a) shows the experimental setup for 1D plasmonic QWs. Optical pulses with a pulse width of 1 ps at 1550 nm from the Er-doped-fiber mode-locked laser were led to the 1D plasmonic lattice device by PMFs. Here, the intensity and polarization of the laser pulses were adjusted in advance by the variable attenuator (V-ATT) and the half wave plate (HWP), respectively. Using the 5-axis (x, y, z, θ, ϕ) fiber positioner, the PMF was aligned to the front facet of the center Au-strip waveguide of the 1D lattice device. Since substantial mode matching between wave vectors in a photonic mode of the PMF and a LR-SPP mode of the Au-strip waveguide was achieved, the LR-SPPs were efficiently excited by the end-fire (butt) coupling (Fig. 2(b)) [28,29,31,35]. Here, the orientation of the slow axis of the PMF was fixed at the TM polarization direction of the LR-SPP waveguide (see Fig. 2(a)). The probabilities that LR-SPPs exit at end-facets of each Au-strip waveguide correspond to the intensity distribution of the cross-section near-field image measured by the infrared (IR) vidicon tube camera. Figure 2(c) shows imaging results when the lattice device with a length L of 3 mm was used. When the polarization of the excitation laser pulse was oriented to the slow axis of the PMF, many bright spots were observed on the horizontal line. On the other hand, when the polarization of the excitation laser pulse was oriented to the fast axis of the PMF, bright spots disappeared and a speckle-like and very weak intensity distribution was observed. The excitation of the LR-SPPs is actually allowed only when the laser pulses have TM polarization. Therefore, when the excitation laser pulses have TE polarization, photonic guiding modes of the slab structure formed by dielectric cladding layers were dominant.
Fig. 2. Experimental 1D plasmonic quantum walk. (a) Experimental setup. PMF, polarization maintaining fiber; HWP, half wave plate; V-ATT, variable attenuator. (b) End-fire coupling between the PMF and the plasmonic lattice device. (c) Near-field images at the end facets of the 3-mm-long lattice device when the polarization of the excitation light was vertical (upper panel) and horizontal (lower panel). (d) and (e) show photon-counting-based imaging results by means of the DMD-SPI and the raster scanning, respectively.
As described above, to observe 1D QW behaviours, the single-photon source or the single-LR-SPP excitation is not necessary [2,37]. This is because the outcomes are the same, when the coherent optical (classical) pulses are used. This is a natural conclusion that the second and the higher-order interferences do not appear in the present system. Therefore, experiments were carried out in the classical regime using coherent pulses for the LR-SPP excitation. Figure 3 shows the 1D intensity distribution extracted from an imaging result with the 1.0-mm-long plasmonic lattice device. The fitted curve is simultaneously plotted in the same figure. It was calculated assuming that the intensity distributions across each Au-strip waveguide are given by Gaussian with a mode field diameter (MFD) of 14 µm. As shown in the figure, the intensity distributions of adjacent Au-strip waveguides are overlapped each other. Therefore, the measured intensities do not represent the accurate time evolution of the 1D plasmonic QW. However, the intensities at vertexes of each Au-strip output are not affected by the outputs of adjacent waveguides. Thus, we were adopted the intensity values at each Au-strip center to reconstruct the intensity distribution representing the time evolution of the plasmonic QW.
Fig. 3. 1D intensity distribution obtained with the 1-mm-long plasmonic lattice device. Red circles show measured intensities that were extracted from an image taken by means of the vidicon tube camera. The blue line shows the fitted curve. The intensity distributions across individual waveguides are assumed to be Gaussian with a MFD of 14 µm.
3. Results
Figure 4 shows the time evolution of measured output intensity distribution with 1.0-, 1.5-, 2.0-, 2.5-, 3.0-mm-long plasmonic lattice devices. The time evolution exhibits the so-called ballistic diffusion which is a typical behavior of the continuous QWs [2,15,16]. In order to evaluate the fidelity of the experimental QWs, we numerically simulated the plasmonic QW. In general, numerical simulations of SPPs propagating along the waveguides require a very high computational cost, compared to that required for the photonic system, e.g. silica-based planer light wave circuits. This is because the plasmonic system includes nano-scaled structures, namely 20-nm-thin Au films, while the typical mode field diameter is ∼10 µm. Thus, we employed a time-division parallel method [38] to numerically simulate the SPPs propagating and interfering in the Au-strip waveguide array. In this method, the EM field in the frequency domain is calculated by the finite-difference complex-frequency-domain (FDCFD) method [39], and then the calculated EM field is transformed into the time domain by using the fast inverse Laplace transform (FILT) [40–42]. The parallel method can solve the time evolution of the EM field in the plasmonic structure much more efficiently than the other time-domain solvers, such as the conventional finite-difference time-domain (FDTD) method [43] and the Constrained Interpolation Profile (CIP) method [44]. The computational cost is however too large to numerically simulate the time evolution of the EM field in the 1D lattice of laterally coupled LR-SPP waveguides shown in Fig. 1. Therefore, we calculated the time evolution of the EM field coupling between neighboring waveguides in the vertical direction. Thanks to the stronger coupling coefficient and the shorter Au-strip separation than those of the lateral coupling structure [30], the calculation area can be drastically reduced. The numerical results show the LR-SPP distributions at the output facets that are identical with the output photon distributions of the arrayed photonic waveguides except for the coupling constant. In other word, the numerical results can be used as a generalized intensity (probability) distributions exhibiting outcomes of the 1D continuous QW. Hence, the experimental results were fitted by the numerical results by adjusting the coupling constant.
Fig. 4. Measured output intensity distributions with numerical ones for the various length of 1D plasmonic lattice devices. Red bars show experimental values and solid diamonds show numerical values. The numerical values are calculated by taking the experimental condition that LR-SPPs are excited not only in the center waveguide 0 but also in the two neighboring waveguides (−1 and 1) into account
The numerical values are plotted together with the experimental values in Fig. 4. The measured intensity distributions are in good agreement with the corresponding numerical ones. Indeed, the plasmonic lattice devices have defects and nonuniformity, but their effects on the results seem to be quite small because the intensity distributions show a bi-lateral symmetry centered at waveguide 0. We must note that the numerical results shown in Fig. 4 are not calculated assuming the ideal LR-SPP excitation in which only the central waveguide is excited. The core diameter of the PMF is 9 µm, and the mode field diameter (MFD) at the front facet of the plasmonic device is expanded to ∼17 µm (the gap between the PMF and the device’s facet is several tens of micrometers). On the other hand, the Au strips have a width of 8 µm with a 2-µm in-plane separation between the nearest neighbors. Thus, LR-SPPs will be excited not only in the central Au-strip waveguide 0 but also in the nearest neighboring waveguides −1 and 1. Figure 5(a) and (b) show the numerical results with the single-waveguide and the simultaneous three-waveguide excitation, respectively. The intensities of the ballistically diffusing components indicated by the white arrows in Fig. 5(a) are much higher than those in Fig. 5(b). It means that the simultaneous three-waveguide excitation makes the ballistic behavior somewhat weak. The experimental results are fit in best with the numerical ones when we assume that the 40% of the total excitation energy is coupled equally to the nearest neighboring waveguide −1 and 1 (see Fig. 4). The 20% of the coupling efficiency to one nearest neighboring waveguide can be roughly estimated from the overlapping ratio of the plasmonic field to the optical excitation field expressed by,
Fig. 5. Numerical intensity distributions. (a) All of the excitation energy is coupled to the central waveguide (waveguide number 0). (b) The 60% of the total excitation energy is coupled to the central waveguide and the rest is coupled to the nearest neighboring waveguides (20% for each waveguide). (I) and (IV), (II) and (V), and (III) and (VI) are cross-section electric-field intensities at device lengths of 1.0, 2.0, and 3.0 mm, respectively. White arrows indicate the fastest diffusion components.
The fidelity of the plasmonic QW was evaluated taking the simultaneous three-waveguide excitation into account. The fidelity $F(p,q)$ can be calculated by [45],
where ${p_k}$ and ${q_k}$ are numerical and experimental probabilities that the LR-SPP exits out of the $k\textrm{ - th}$ waveguide. ${q_k}$ is approximately given by ${q_k} = {I_k}/{I_{ent}}$, where ${I_k}$ and ${I_{ent}}$ are the measured output intensity of waveguide i and the entire intensity of all of the waveguides, respectively. According to the numerical and experimental results shown in Fig. 4, fidelities $F(p,q)$ for the lattice devices with lengths of 1.0, 1.5, 2.0, 2.5, 3.0 mm are evaluated to be 0.98, 0.99, 0.98, 0.97, and 0.96, respectively, which indicates that the plasmonic lattice structure implements high-fidelity QWs.Our experiments described above were performed in the classical regime. The QWs in the quantum regime can be implemented using a single-photon source or entangled photon pairs to excite LR-SPPs, and single-photon detectors to detect them. Single-photon-sensitive image sensors which realize “on-line” detection of visible single-photons in each waveguide are recently developed. However, since our plasmonic devices must be operated with near infrared (NIR) wavelengths especially at 1550 nm to reduce the propagation losses, such an on-line detection is not possible so far. As an alternative, we can use imaging techniques based on a single-pixel-type of single-photon detector, i.e. the raster scanning and the digital-micromirror-device-based single-pixel imaging (DMD-SPI) [46]. As preliminary experiments, we carried out the probability distribution measurements using these techniques with a weak coherent pulse (average photon number per pulse ∼ 0.01). Figure 2(d) and (e) show the single-photon imaging results with the DMD-SPI and the raster scanning with the 3-mm-long lattice device, respectively. In both experiments, we used a sinusoidally gated InGaAs/InP single-photon avalanche diode (SG-SPAD) [47]. These successful demonstrations clearly show the feasibility of the perfect implementation of QW with our plasmonic device if the single-photon source, e.g. a heralded single-photon source [48,49] and an entangled photon source [50] at 1550 nm, is used for the LR-SPP excitation. Furthermore, these photon-counting-based imaging systems are very beneficial to implement the higher dimensional QW.
4. Discussions and conclusions
Finally, we discuss the feasibility of a higher dimensional plasmonic QW device. As mentioned above, the present numerical simulation is performed for the plasmonic QW in a vertically coupled Au-strip waveguide array, which indicates that not only lateral coupling but also vertical one can be employed to construct a waveguide network. Thus, 2D plasmonic QW would be feasible. The waveguide structure is composed of the very thin Au film (several tens of nanometers), while the dielectric layers are very thick (several to tens of micrometers). Hence, the laterally arrayed Au-strip waveguide structure would be stacked vertically (multi-layered) without additional fabrication processes, e.g. precise polishing and etching for each layer. The laser-written waveguide platform has realized a large-format 2D continuous quantum walk [21]. In the platform, only the modulation of the waveguide separation has been demonstrated for the control of the coupling constant between waveguides. In the plasmonic platform, we can utilize the Au-strip-width modulation (diagonal-disorder [51]) for the control of the coupling (off-diagonal-disorder [3]) as well, which allows us to implement a large variety of continuous QW with those disorders.
For the experimental convenience, the end-fire coupling using the PMF was employed to excite LR-SPPs, which resulted in the simultaneous excitation of neighboring waveguides. It is obviously undesired that multiple plasmonic quantum walkers are excited beyond control. One of the solutions is to adopt the end-fire free-space coupling. Our device was designed to match the wavenumber of the LR-SPP to $1.45{k_0}$, where ${k_0}$ is the wavenumber of light in vacuum. The refractive indices of dielectric layers of our device are ${\sim} 1.45$. Therefore, if front facets of Au strips are completely embedded in the dielectric, the wavenumber of light focused at the Au-strip’s front facet can be matched to that of the LR-SPP. In this way, the efficient access to the individual LR-SPP waveguide can be realized. The free-space coupling can be extended to allow the simultaneous excitations of LR-SPPs in the arbitrary waveguides, which is necessary to implement QWs using multiple or entangled LR-SPPs as quantum walkers.
In conclusions, we experimentally demonstrated plasmonic QW in an in-plane array of Au-strip waveguides. The propagation of LR-SPPs along the waveguides exhibits the same behavior as the continuous photonic quantum walk with a high fidelity, which indicates that the plasmonic circuit is one of the potential platforms to implement a large-format and high-dimensional continuous QW.
Funding
Japan Society for the Promotion of Science (18H04292).
Acknowledgments
The devices were partly fabricated in the clean room for analog-digital superconductivity (CRAVITY) in National Institute of Advanced Industrial Science and Technology (AIST). A part of this work was conducted at the AIST Nano-Processing Facility supported by “Nanotechnology Platform Program” of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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