The 3D Split-Ring Cavity Lattice: A New Metastructure for Engineering Arrays of Coupled Microwave Harmonic Oscillators

A new electromagnetic cavity structure, a lattice of 3D cavities consisting of an array of posts and gaps is presented. The individual cavity elements are based on the cylindrical re-entrant (or Klystron) cavity. We show that these cavities can also can be thought of as 3D split-ring resonators, which is confirmed by applying symmetry transformations, each of which is an electromagnetic resonator with spatially separated magnetic and electric field. The characteristics of the cavity is used to mimic phonon behaviour of a one dimensional chain of atoms. It is demonstrated how magnetic field coupling can lead to phonon-like dispersion curves with acoustical and optical branches. The system is able to reproduce a number of effects typical to one-dimensional lattices exhibiting acoustic vibration, such as band gaps, phonon trapping, and effects of impurities. In addition, quasicrystal emulations predict the results expected from this class of ordered structures. The system is easily scalable to simulate 2D and 3D lattices and shows a new way to engineer arrays of coupled microwave resonators with a variety of possible applications to hybrid quantum systems proposed.


Introduction
Metamaterials are structures made of artificial building blocks with a scale smaller than a working wavelength [1]. Such materials are typically designed to possess properties that cannot be found in natural materials. The classical example of such an application is metamaterials with negative refractive index [2]. Such metamaterials are definitely in need in both science and engineering. Although, there could be another potential application of metamaterials: They can be used to emulate behaviour of systems from one physical realm by making experiments in another one, potentially with some boosted parameters. For example, can the behaviour of a mechanical solid be emulated in an electromagnetic cavity with the speed of sound approaching speed of light? Such system can be used to study solids that cannot be found in nature, for example quasicrystals or phonon trapping lattices.
Although the answer to the question asked above is definitely yes, it is important to propose a practical structure that is simple, scalable, tuneable and with low loss. In this work, we analyse a new metastructure based on the re-entrant cavity, the 3D analog of the 2D split-ring resonator with similar characteristics. 2D split-ring resonators have been a popular choice for metastructures [3,4]. However, one of the weak points of the 2D structure is the inability to reach very high quality factors. Thus, there is an interest to explore metastructures made of 3D cavities where all the benefits could be fully exploited. In this work we take this approach and undertake 3D finite-element electromagnetic modelling of a multiple post 3D cavities and compare it to the phonon behaviour in a one dimensional chain of atoms. These similarities enable the emulation of defects in crystals, as well as some properties of quasicrystals.

System Description
The one dimensional metastructure analysed in this work is based on the reenrant cavity [5,6,7]. The reentrant cavity is a closed (typically cylindrical) 3D microwave resonator built around a central post with a tiny gap between one of the cavity walls and the post tip. It can be demonstrated that a reentrant 3D cavity can be considered as a continuous 2π rotation of a 2D split-ring resonator as it is demonstrated in Fig. 1. To build a solid, a three diminutional structure, one needs to continuously rotate a two diminutional surface, a split-ring resonator, around axis of rotation C C by 2π in the yx plane. As a result such cavity demonstrates a continuous rotational symmetry.
Unlike most other 3D structures the reentrant cavity has spatially separated electrical and magnetic fields. Whereas almost all the electrical field is concentrated in the gap, all the magnetic field is distributed around the post decaying rapidly with the distance from it. Thus, as a resonant system, the cavity can be approximated by an LC model in the vicinity of its main resonance.
The metastructure analysed in this work is constructed from a standard singlepost re-entrant cavity by applying translational symmetry rules and removing all walls  between posts. If this is done along one axis only, e.g. x, one obtains a one-dimensional lattice. Although this could be done in two directions (and maybe three directions if one imbeds posts in a dielectric), in this work we consider only the 1D lattice to simplify the discussion. Side and top views of a rectangular cavity with a chain of multiple posts is shown in Fig. 2. This represents the first realisation of a multiple post reentrant cavity and unequivocally show the existence of higher order post modes, which also represents a new way to engineer arrays of coupled microwave oscillators. The proposed in this work metastructure is a subject of patent [8].
At each moment of time, the vector of the electrical field points out of, or into the post. Accordingly, the magnetic field can take one of two possible directions around the post, Clockwise or Anti-Clockwise. These situations are shown in Fig. 2: Situation "+" describes the case of current flowing up the post and is referred as positive orientation, the opposite orientation of vectors is then understood as the negative direction (or π out of phase) with the current flowing down. In the following sections it is demonstrated that the fields of the different posts can exhibit different orientations at the same time for the given eigenmode.
Whereas modes of a single post cavity has been studied both analytically and experimentally, a system of multiple posts contains extra information. That is the relative orientations of the post currents, "+" or "−", and corresponding eigenmodes. Combinations of these orientations set a pattern of a whole system specific and unique for each system eigenmode. In the following sections, these patterns of orientations are analysed using 3D electro-magnetic field simulation. This phenomenon in a double post cavity has been exploited to focus the magnetic field to demonstrate ultra-strong coupling with magnons [9].
Each post of the chain can be understood as a Harmonic Oscillator (HO) in the frequency range where the transverse effects along the y coordinate could be neglected. Indeed, each post has an associated capacitance, formed by the gap, and inductance, formed by the post itself. In this chain configuration HOs of two posts are coupled only through the magnetic field since the overlap of electrical field components is negligible in the limit of a small gap. As a result the system of identical posts can be formalised in a form of a Hamiltonian as follows: where N is the total number of posts, q i and φ i are corresponding charge and fluxes, G is the coupling between two posts set mostly by the distance between posts. G can be understood as a mutual inverse inductance. The model is based on an approximation of the reentrant cavity as an LC circuit [5] formed by its capacitive and inductive parts. In addition, Hamiltonian (1) considers couplings only between two nearest neighbours.

Phonon Emulation
Hamiltonian (1) represents an idealised electromagnetic system consisting of identical coupled cavities. The Hamiltonian can be rewritten in the limit of large N as follows: where L −1 = L −1 − G. In terms of eigenmodes this Hamiltonian could be rewritten as: where ω 2 0 = 1 L C and ω k is given by the phonon dispersion relationship: So, the simple model of the post chain predicts the existence of a phonon-like dispersion curve ω k referenced to a bare photon angular frequency ω 0 . Building an analogy between electrical and mechanical systems, Q k plays a role of momentum and Φ k of a coordinate whereas C is an analog of the mass. Moreover, taking into account energy ω 0 that is independent of the wavelength, one may draw an analogy with a relativistic particle. Such that in the long wave length limit, the full dispersion relationship could be written as: where v = G C is an effective "speed of light" and m = C G 2 C L is an effective particle "mass". Accordingly, in the same approximation Hamiltonian (4) could be transformed to the Lagrangian for the one-dimensional Klein-Gordon field. Existence of such a dispersion curve is verified numerically in the subsequent sections.

Chain of Equidistant Identical Posts
As a starting point of the study, a chain of N = 20 identical equidistant posts is considered. Simulation results for this case demonstrate the existence of 20 modes with one variation along all dimensions normal to the chain (first order in width). All these modes could be separated into two groups: 1) in-phase currents through the neighbouring posts, which create time-varying regions of posts with positive and negative directions (Fig. 3), 2) anti-phase currents through the neighbouring posts creating a time-varying analogue of electrical dipole moments (Fig. 4). For each type of mode, one can uniquely identify a mode number n as a number of variations of magnetic and electrical field along the chain, i.e. the number of half-waves. Both Fig. 3 and Fig. 4 demonstrate modes with two half-waves. Resonances of two types of modes form two distinct types of dispersion curves as shown in Fig. 5. The first type gives an ascending curve where frequency increases with the normalised wave vector k = n/N . On the contrary, the second type demonstrates descending behaviour, i.e. decrease of the frequency with n/N . This situations is analogous to acoustic and optical phonons in an one-dimensional lattice. In this interpretation, the role of displacement of an atom is played by the electrical field under the post: Acoustic modes are characterised by the in-phase 'movement' of the neighbouring posts whereas optical modes are characterised by out-of-phase electrical vectors of the nearest neighbours. Although, unlike an optical branch of a mechanical chain of atoms, the acoustical branch in the present system does not demonstrate convergence to zero in the limit of infinitely long wavelengths k −1 . Instead, it asymptotically approaches ω 0 2π (dashed line in Fig. 5) as the minimal photon energy in (4). In the following text, we refer to these branches as acoustic and optical photon dispersion branches which are taken in reference to ω 0 . Fig. 5 shows that all chains with a different number of curves consistently represent the same dispersion curves. The difference of the odd N case is existence of the point where two curves collapse and exhibit the same behaviour. The same frequency correspond to a system with only one post (square in Fig. 5).
Parameters of the dispersion curves for photons in a chain of posts depend on the post gap, radius and spacing between posts. Fig. 5 demonstrate dependence of the curves on the spacing distance. The results demonstrates that the frequencies drop with decreasing density of posts.

Bandgap
In a chain of vibrating atoms, a band gap is created by a periodic repetition of atoms with different masses. Similarly, in the present system a band gap could be created by the variation of post parameters, e.g. its radius or gap. As a result the L and C of each post in the Hamiltonian (1) will depend on its number. The dispersion curves shown in Fig. 6 demonstrate the dispersion characteristic for a chain with periodically changing post radii. As it is clearly seen from the figure, the splitting between two branches increases for increasing ratio between post radii as it is expected from a chain of vibrating atoms. Fig. 7 shows a family of curves for a chain with periodic change of post gaps. Although, a band gap is also observed, it shows no further increase after a certain value of ratios. Instead the modes loose their identity, since the reentrant type of resonance becomes lost.

Potential Well
A variation of spacing between the posts could be used to change the energy distribution along the chain. For example, photons can be concentrated at the centre of the chain by increasing gaps between posts in both directions of photon propagation. In this section we consider the case of a potential well, where the post coordinate x depends on the post site number n as follows: x n = sign(n)h/2 + hn + gn 2 , n ∈ [ − N/2, N/2] where h and g are some constants. Such a law implies quadratic growth of spacing between posts G i,i+1 as one moves from the chain centre. Correspondingly, the coupling energy between the two nearest neighbouring posts decreases. It can be shown that the Hamiltonian density in the limit N → ∞ and k → 0 becomes where G(x) grows with x as prescribed by (7). As a result, energy of the quasiparticle is minimised at the origin. The increased concentration of the field is clearly seen in Fig. 8, compared to the uniform post distribution shown in Fig. 4. The result could be understood as photon trapping similar to phonon trapping techniques used in high-Q acoustical cavities [10,11,12]. The variable post separation results in the creation of an effective potential well along the chain, which distorts the dispersion curves (Fig. 9). It can be inferred from this figure that in the extreme case of g → ∞, all the modes collapse to   the same frequency where all the energy is concentrated between the posts closest to the origin. Similar results can be obtained if the post parameters are quadratically varied along the chain.

Interstitial Defect
Real solids are always subject to various defects, imperfections of the crystal structure. An example of such a defect is an interstitial impurity. It arises when an alien ion or atom exists in intetratomic space of the crystal. In natural crystal, this situation is very common for quartz where interstitial H + , Li + , N a + exists as charge compensators near substitutional Al 3+ . In case of the simple chain of atoms, the interstitial impurity may be modelled as an extra post between regular posts of a chain. Simulated dispersion curves for such a system are shown in Fig. 10. Various curves are shown for different relative size of the impurity post. Fig. 10 demonstrates an effect of separation of the chain into two separate chains. In the large impurity limit, the impurity works as a wall (large inertia) making two separate resonant systems. For this reason, two adjacent points on the dispersion curve tend to have the same angular frequency. One of these frequencies correspond to the left part of the chain and another to the right half-chain.
In the case of infitimesly small the additional post, the system is unperturbed. Thus, in this limit the curves converge to the dispersion curve of the ideal chain.

Substitutional Defect
Another common type of impurities in crystals is substitutional impurity. The situations arises when an alien atom substitutes a normal atom in a crystal. This situation also could be simulated with a chain of posts. The result is shown in Fig. 11.
The effect of a large substitutional impurity is similar to that of a large interstitial impurity. The substitutional post acts as a wall separating the lattice of resonators into two sub-chains. So, in the limit of an infinitely large impurity, two subsystems are completely separated and uncoupled giving two separate dispersion curved. But unlike in the case of infinitesimally small limit of the substitutional impurity, an infinitesimally small substitutional defect effectively acts as a gap. This gap is also a kind of separator that decreases coupling between two ideal sub-chains. Thus, the similar effects of doubling the dispersion curves are also observed in this limit (Fig. 11).

Quasicrystal
Quasicrystals [13] are an ordered form of matter that do not have translational symmetry.
The analysed system of posts can be used to study both 1D and 2D quasicrystal structures. One of the simplest examples of quasicrystals is a Fibonacci chain formed by the rule F n = F n−1 + F n−2 . Such a chain does not have translational symmetry unlike the regular chain of two posts studied in Section 4. For example, for a chain of N = 21 posts the structure is LSLLSLSLLSLLSLSLLSLSL, where L and S represent posts of two different types or two different distances between two neighbouring posts. Fig. 12 compares dispersion characteristics of such a megastructure with the regular one for N = 33. Quasicrystalline megastructure demonstrates effects significantly different from that of regular chains. In particular, multiple discontinuities of spectra are observed.

Possible Applications
The proposed type of the megastructure may be used for various applications in physics. Its main advantage is high tunability [14] and potentially high quality factors. High tunability of the order of 1 GHz/µm for µm-size gaps is achieved due to the concentration of the electrical field in the tiny gaps [6]. Such high tunability has been utilised for transducers in gravitational wave antennae [15]. High quality factors Q > 10 8 have been demonstrated for superconducting reentrant-type resonators [16]. These 3D closed cavities outperform all known 2D structures popular in Circuit Quantum Electrodynamics (cQED) [17] with the reported value only bettered by cryogenic dielectric resonators at low temperatures [18] and the best acoustical systems [11,12].
The functionality of the linear reentrant metastructure can be extended by  additional nonlinear or parametric elements: For example, the reentrant cavity could be used for cQED experiments by incorporating Josephson Junctions. In this case, a Josephson Junction can be formed by the dielectric interface between the post and the cavity bottom, a pin contact [19,20]: where E J is the Josephson energy. A chain of such posts could serve as an alternative realisation of the supersolid matter state [21]. High sensitivity to mechanical motion of the gap makes the reentrant cavity a good transducer of mechanical motion. In principle, the chain in this case may be thought as an array of intercoupled harmonic oscillators controlled by cavity photon modes: where ω m , a † i (a i ) are the angular frequency and the creation (annihilation) operators of the mechanical resonators under each post, ω, b † i (b i ) are the angular frequency and the creation (annihilation) operators of the electromagnetic resonance associated with each post. Concentration of the magnetic field around each post allows the coupling of the photon modes to spins states of the crystal impurities that can be put around the posts in order to achieve photon-spin strong coupling regimes [22]. The ultra-stroncg coupling regimes between magnons and double post cavity phonons has been demonstrated [9].
Applications for the presented metastructure include, but are not limited to [8]: • Spectroscopy of small samples of magnetic materials, which requires maximization of the magnetic filling factor of the material under study [9]. Advantages are that the cavity is highly tunable by changing the electrical length by the gap by either mechanical or dielectric methods, and that the magnetic field is focused via the bright mode into the small sample. Also frequencies much smaller than the natural resonant frequencies of a small sample may be realized. The same set up can be used to characterize the magnetic properties of a material.
• Implementation of arrays which magnetically couple to more than one sample simultaneously. Samples could be spins or arrays of qubits. This would allow complex qubit/spin circuitry, with engineered couplings to specified samples for quantum computing and engineering applications. For example, if each post is associated with a Josephson Junction and cooled to low enough temperatures, the multi-post re-entrant structure provides a new way of arrangement of quantum information, including 2D and 3D arrangement of qubits. The different modes of the structure are then used to address and probe different subsystems.
• The study of electrical response of dielectric materials (materials characterization); some applications could require either concentration of the electric field or its gradient, this could be achieved by putting a sample (film or wafer) under different posts, thus different modes will apply different electric field patterns to the sample.
• Multi-mode tunable filters realisable due existence of several modes of the multipost re-entrant cavity can be utilized to create multi-mode filter system. Such systems have application in multi-band antenna applications for instance. Unlike previous inventions with multiple posts, we propose to utilize all the fundamental resonant modes of the structure, which equals to the number of posts.
• Space-resolving displacement detectors are the extension of one post re-entrant cavity that is a highly sensitive displacement detector. The multi-post structure has a few resonant frequency, each of them has a unique sensitivity distribution in gap plane. Thus such structure will allow to resolve the displacement patterns by identifying frequency shifts of all resonant frequencies.
• Multi-mode gas/fluid sensor is an extension of a single post cavity allowing multi-frequency analysis. Due to existence of several mode of the multi-post re-entrant cavity is utilized to probe gas/fluid at different frequencies. In this application, the cavity extends the functionality of the single-post cavities buy allowing to simultaneously probe gas/fluid at distinct frequencies which leads to better sensitivity and ability to probe different resolve different components of the gas/fluid.
• Band gap filters and isolators; the multi post metastructure built of posts of different size/gaps/spacings leads to creation of wide band gaps use for isolation purposes.
Existence of the such gaps is demonstrated in the present paper. In addition to that, the proposed type of a megastructure is a convenient tool to study novel structures of solids such as quasicrystals in one and two dimensions which is not possible in real matter.