⇦Chinese Physics Letters, 2020, Vol. 37, No. 1, Article code 014201 Quantum Scars in Microwave Dielectric Photonic Graphene Billiards * Xiao Wang (王晓)1**, Guo-Dong Wei (魏国东)2 Affiliations 1Department of Computer Science and Technology, Taiyuan University of Science and Technology, Taiyuan 030024 2School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024 Received 31 October 2019, online 23 December 2019 *Supported by the National Natural Science Foundation of China under Grant No. 11847067, the Natural Science Foundation of Shanxi Province under Grant No. 201801D221178, the Science and Technology Innovation Project of Shanxi Higher Education under Grant No. 2019L0648, and Taiyuan University of Science and Technology Scientific Research Initial Funding under Grant No. 20152044.
**Correspondence author. Email: wang.xiao321@163.com Citation Text: Wang X and Wei G D 2020 Chin. Phys. Lett. 37 014201    Abstract In the band structure of graphene, the dispersion relation is linear around a Dirac point at the corners of the Brillouin zone. The closed graphene system has proven to be the ideal model to investigate relativistic quantum chaos phenomena. The electromagnetic material photonic graphene (PG) and electronic graphene not only have the same structural symmetry, but also have the similar band structure. Thus, we consider a stadium shaped resonant cavity filled with PG to demonstrate the relativistic quantum chaos phenomenon by numerical simulation. It is interesting that the relativistic quantum scars not only are identified in the PG cavities, but also appear and disappear repeatedly. The wave vector difference between repetitive scars on the same orbit is analyzed and confirmed to follow the quantization rule. The exploration will not only demonstrate a visual simulation of relativistic quantum scars but also propose a physical system for observing valley-dependent relativistic quantum scars, which is helpful for further understanding of quantum chaos. DOI:10.1088/0256-307X/37/1/014201 PACS:42.70.Qs, 05.45.Pq, 72.10.-d © 2020 Chinese Physics Society Article Text A quantum scar is one of the fundamental phenomena of quantum chaos, which was first noticed by McDonald and Kaufman in 1978[1,2] and later named by Heller in 1984.[3] It is defined as the anomalous condensation of wave functions on unstable periodic orbits.[1–3] Up to now, most of the works in quantum chaos have investigated non-relativistic quantum systems,[4,5] whose energy-momentum relation is parabolic. In the past one and half decades, graphene has attracted significant attention due to its unique optical, electrical, mechanical and thermal properties and its potential use in electronics devices.[6–8] As we know, the band structure of PG possesses Dirac cones at the corners of the Brillouin zone, where the dispersion relation is linear, and the motion of the pseudoparticles satisfies the two-dimensional massless Dirac equation. Thus graphene billiards provide a natural experimental platform to examine the intriguing effects of relativistic quantum chaos. In this regard, relativistic quantum scars in graphene billiards have been not only predicted numerically in Ref. [9], but also observed experimentally in mesoscopic graphene rings via the scanning gate microscopy method.[10] Level spacing statistics, quantum chaotic scattering as well as quantum tunneling in graphene systems have also been investigated extensively.[11–15] Furthermore, the adjacent Dirac points are inequivalent,[16] thus a complete description of the pseudo particles in PG would need a $4\times 4$ matrix representation, i.e., two 2D massless Dirac particles coupled by the boundary. Owing to the same structural symmetry and the similar band structure between PG and electronic graphene, PG has also been used widely to investigate the intriguing phenomena proposed in electronic graphene systems.[17–25] In particular, the microwave PG systems can serve as a convenient platform for examining relativistic quantum chaos experimentally, benefiting from the excellent experimental accessibility of itself. The flat microwave cavities (i.e., the so-called microwave billiards) are analogues to quantum billiards.[26] The microwave billiards consisting of PG inside the cavity are the ideal models to investigate the relativistic quantum chaos,[27–29] which are nominated as the photonic graphene billiard (PGB) in this study. In recent years, many researchers have investigated the statistical fluctuation properties of the energy spectrum in PGB thorough the experimental methods with different boundary shapes.[30–34] Note that the PGB in these works consists of a basin constructed by cylinders in the form of photonic crystal and a lid made from metal plates. It is intriguing whether relativistic quantum chaos can also occur in PGB with dielectric PG, in which the wave scattering is weaker than that for metal PG. Furthermore, this intriguing phenomenon may not only provide an alternative experimental platform for observing the effects of relativistic quantum chaos but also make the PGB great promising in the fields of device applications. Although the relativistic quantum scar has been observed experimentally in mesoscopic graphene rings, the localization of the wave function on unstable periodic orbits is not obvious. Therefore, investigating relativistic quantum scars in PG billiard systems may provide an effective method to observe relativistic quantum scars experimentally. In this Letter, the numerical simulations are carried out on a system of stadium-shaped dielectric PG billiards, which are confined with armchair or zigzag horizontal boundaries. In addition, the PG billiards in this study can be identified as series of scarring wave functions for two models with different incident directions in an experimentally accessible parameter range. Furthermore, the recurrence rules have been analyzed and confirmed. At the same time, the eigenfrequency is equally spaced for the recurring scars, arising from the linear dispersion relation between the frequency and the wave vector. Thus it is believed that relativistic quantum scars localized on unstable periodic orbits could be observed experimentally in the experimental setup build up according to the model in our study. The property of the Billiard system depends greatly on the geometry of their edges. If the Billiard system is bounded with regular boundary, such as circular, square, or elliptic, then the system can be considered to be an integral one. However, when the Billiard system is bounded with irregular boundary, such as Africa, stadium, or heart-shaped, it can be considered to be chaotic (non-integrable). In this Letter, we investigate the quantum scars in stadium-shaped PGB confined with different horizontal boundaries, where the incident port is placed in the border of the stadium. Strictly speaking, this system is open. Nevertheless, when the incident port is pretty small, part of the characteristics in the closed system can also appear in the so-called open system. In this work, the quantum scars have been researched in the quantum system with entrance and exit ports in a pretty small size. Here the width of the ports is indicated by $a$, which is defined much smaller than the system model.
cpl-37-1-014201-fig1.png
Fig. 1. Band structures for TE modes in an infinite PG. The red triangles show the frequencies at which the scars are mentioned hereinafter.
In this study, the PGB is a resonant cavity consisting of PGs made up of cylindrical alumina rods arranged in a honeycomb lattice. The alumina cylinders with radius 3 mm and dielectric constant 8.35 are squeezed between the two metal surfaces of the cavity. In addition, the distance between two adjacent rods is $a=8.57$ mm. The band structure of this infinite PG is then calculated by plane wave expansion method for the TE modes (electric field parallel to the axial direction of rods),[35] as illustrated in Fig. 1. Furthermore, the Dirac point can be seen at the frequency of 6.39 GHz from Fig. 1 (normalized frequency of $a / \lambda \approx 0.183$, $\lambda$ is the wavelength in vacuum), near which the dispersion relation is linear. Consequently, it is believed that the PGB consisting of PG is just perfect to mimic a relativistic quantum chaotic system.
cpl-37-1-014201-fig2.png
Fig. 2. The structure of the model 1. (a) The plan of the model 1. (b) The enlarged view of Z1 at the structural boundary. (c) The enlarged view of Z2 near the entrance port.
cpl-37-1-014201-fig3.png
Fig. 3. Typical quantum scars (lighter region) for a stadium shaped graphene confinement with armchair horizontal boundaries (model 1). The corresponding frequencies are 6.076 GHz, 5.948 GHz, 5.877 GHz, 5.81 GHz, 5.795 GHz and 5.789 GHz for (a)–(f), respectively. The dashed lines represent classical periodic orbits.
In the present study, the quantum scars are researched in an open system, as shown in Fig. 2(a). There are two simulation models with different horizontal boundaries in this study. Model 1 consists of 14756 alumina cylinders with armchair horizontal boundaries illustrated in Fig. 2(a), while the number of rods is 14978 for model 2 with zigzag horizontal boundaries. Figures 2(b) and 2(c) stand for enlarged views of the two zones in Fig. 2(a). The typical quantum scars in the stadium-shaped PGB are simulated by Computer Simulation Technology (CST) Microwave Studio. All surfaces of stadium are set to be perfect-electric-conductors. It is concluded from Fig. 3 that the typical scars of model 1 will emerge as an incident beam impinging on the armchair interface. In other words, the beam is along the $\mathit{\Gamma} K$ direction. At the same time, the dashed line segments are for eye guidance. Figure 4 demonstrates the typical scars for model 2 when the incident beam impinges on the zigzag interface ($\mathit{\Gamma} M$ direction).
cpl-37-1-014201-fig4.png
Fig. 4. Typical quantum scars (lighter region) for a stadium shaped graphene confinement with zigzag horizontal boundaries (model 2). The corresponding frequencies are 6.131 GHz, 6.019 GHz, 5.979 GHz, 5.938 GHz, 5.818 GHz and 5.801 GHz for (a)–(f), respectively. The dashed lines represent classical periodic orbits.
The quantum scars in Fig. 3 are markedly different from those in Fig. 4, owing to that these two models of PGB are confined with different horizontal boundaries. Our previous work has concluded that the incident beam near Dirac point will split into two beams for the zigzag interface, while the incident beam is self-collimated for the armchair interface.[36] The reason is that the trigonal warping distortion in energy band lifts the degeneracy of two inequivalent valleys. In view of this analysis, the quantum scars of the PGB with different boundaries may be valley-dependent quantum scars. As the wave vector or frequency varies, scars will appear and disappear repeatedly. The differences between the frequency of the recurring adjacent scars are pretty much the same, which will bring about an interesting phenomenon that the wave vector is equally spaced for recurring scars. According to the semi classical theory,[37] for graphene confinement $E=\hslash v_{\rm F} k$ at low energy, $v_{\rm F}$ refers to the Fermi velocity, and $k$ stands for the magnitude of the wave vector from the Dirac point. In particular, the energy interval for scars focusing on a given periodic orbit is determined by $\Delta E=2\pi {\hslash v_{\rm F} } / L$. That is to say, the orbit dimensions are quantized by $k={2\pi } / L$. We have identified series of recurring patterns for the scars shown in Fig. 3(a) and those in Fig. 4(c). For the scars similar to Fig. 3(a), the wave vector interval between consecutive scars is fixed at $\Delta k=1.5$, at the same time the length of the orbit is fixed at $L=469a$, leading to ${2\pi } / {L=1.56}$. Correspondingly, for the scars in Fig. 3(c), the wave vector interval is denoted by $\Delta k=1.04$, while the length of the orbit is replaced by $L=730a$, yielding ${2\pi } / {L=1.004}$. In all, the results aforementioned demonstrate that the orbit dimensions are quantized. In addition, the recurrence of quantum scars in the systems verify that there does exist a linear relation between the energy and wave vector $k$ near the Dirac point. As illustrated in Fig. 5, the recurrence law of quantum scars in the PG system follows an approximately linear relationship, where the red circle and the black triangle represent the scars in Figs. 3(a) and 4(c), respectively. As shown in Fig. 5, the horizontal axis represents the wave vector for recurring scars ($k_{\rm D}$ is the wave vector at Dirac point), and the vertical axis shows the index $n$. Furthermore, the symbol $n$ represents the order of scars appearing, which refers to relative values essentially. According to Fig. 5, it can be concluded that the most wave vectors are equally spaced for recurring scars, except that individual wave vectors have considerable gaps, which is mainly attributed to its semicircular boundary. In addition, considering the fact that the systems in this study are not the strictly closed systems, the number of the typical quantum scars is less than those in completely closed systems.
cpl-37-1-014201-fig5.png
Fig. 5. Recurrence of quantum scars in the PG system. The scars are shown in Figs. 3(a) and 4(c).
The scarring states have been obtained so far, as proposed in Ref. [12]. In this study, the $\eta$ is used to characterize the wave vector difference between the repetitive scars on the same orbit, which is defined as $\eta ={(k-k_{0})} / {\Delta k-}[{(k-k_{0})} / {\Delta k}]$,[38] where [$x$] denotes the largest integer less than $x$, $k_{0}$ indicates the wavevector for a scar setting as the reference point, $k$ stands for the wavevector for the repetitive scars on the same orbit, and the difference of wave vector $\Delta k={2\pi } / L$ with the orbital length $L$. Figure 6 plots the corresponding $\eta$ values of scars from Figs. 3(a) and 4(c), from which it can be seen that most of the $\eta$ values is either close to 0 or 1. Considering that the scars in Figs. 3(a) and 4(c) are the even orbits, the $\eta$ values in this study agree well with the semi classical theory proposed by Berry and Mondragon.[39]
cpl-37-1-014201-fig6.png
Fig. 6. The corresponding $\eta$ values of scars shown in Figs. 3(a) and 4(c). The black triangles are for the scar in Fig. 3(a), the red circles are for the scar in Fig. 4(c).
In conclusion, the relativistic quantum scars have been mimicked under dielectric PG systems. The typical quantum scars in PG confined with different horizontal boundaries are markedly different, where the two kinds of horizontal boundaries means $\mathit{\Gamma}K$ and $\mathit{\Gamma}M$ incident directions respectively. For the $\mathit{\Gamma}K$ and $\mathit{\Gamma}M$ directions, there are different valley-dependent beam transportations, which may represent a valley-decided quantum scars in PGB. In addition, the scars appear and disappear repeatedly. What is more, owing to the linear energy-wave vector relation, the wave vector interval for recurring scars is approximately equal. Thus, the exploration will not only demonstrate a visual microwave simulation of relativistic quantum chaos, but also broaden the knowledge of researchers in quantum chaos, which will be greatly beneficial to design of optical devices based on PG chaos characteristic in both experimental and theoretical basics. References Spectrum and Eigenfunctions for a Hamiltonian with Stochastic TrajectoriesWave chaos in the stadium: Statistical properties of short-wave solutions of the Helmholtz equationBound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic OrbitsSmoothed wave functions of chaotic quantum systemsQuantum Scars of Classical Closed Orbits in Phase SpaceThe electronic properties of grapheneThe rise of grapheneColloquium : Andreev reflection and Klein tunneling in grapheneRelativistic Quantum ScarsRecurrent Quantum Scars in a Mesoscopic Graphene RingScarring of Dirac fermions in chaotic billiardsChiral Scars in Chaotic Dirac Fermion SystemsQuantum chaotic tunneling in graphene systems with electron-electron interactionsModulating quantum transport by transient chaosHarnessing quantum transport by transient chaosThe Band Theory of GraphiteObservation of a Dirac point in microwave experiments with a photonic crystal modeling grapheneDirac point and edge states in a microwave realization of tight-binding graphene-like structuresTopological Transition of Dirac Points in a Microwave ExperimentExtremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiardAnomalous transmission of disordered photonic graphenes at the Dirac pointTransmission properties near Dirac-like point in two-dimensional dielectric photonic crystalsTransport properties of disordered photonic crystals around a Dirac-like pointExperimental Observation of Strong Edge Effects on the Pseudodiffusive Transport of Light in Photonic GrapheneObservation of unconventional edge states in ‘photonic graphene’Relativistic quantum chaos—An emergent interdisciplinary fieldChaotic Dirac Billiard in Graphene Quantum DotsPhase-Coherent Transport in Graphene Quantum BilliardsRelativistic quantum chaosSpectral properties of superconducting microwave photonic crystals modeling Dirac billiardsQuantum and wave dynamical chaos in superconducting microwave billiardsSpectral Properties of Dirac Billiards at the van Hove SingularitiesFrom graphene to fullerene: experiments with microwave photonic crystalsQuantization of massive Dirac billiards and unification of nonrelativistic and relativistic chiral quantum scarsComment on “Band gaps structure and semi-Dirac point of two-dimensional function photonic crystals” by Si-Qi Zhang et al .Observation of valley-dependent beams in photonic graphenePeriodic Orbits and Classical Quantization ConditionsScars in Dirac fermion systems: the influence of an Aharonov–Bohm fluxNeutrino Billiards: Time-Reversal Symmetry-Breaking Without Magnetic Fields
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