First Experimental Realization of the Dirac Oscillator

We present the first experimental microwave realization of the one-dimensional Dirac oscillator, a paradigm in exactly solvable relativistic systems. The experiment relies on a relation of the Dirac oscillator to a corresponding tight-binding system. This tight-binding system is implemented as a microwave system by a chain of coupled dielectric disks, where the coupling is evanescent and can be adjusted appropriately. The resonances of the finite microwave system yield the spectrum of the one-dimensional Dirac oscillator with and without a mass term. The flexibility of the experimental setup allows the implementation of other one-dimensional Dirac-type equations.

Figure 1

A chain of resonators in dimeric configuration, with atoms of types $A$ and $B$. The index $n$ stands for the dimer number, $\Delta$ is the coupling between elements of the same dimer (intradimer coupling, kept constant throughout the array), and ${\Delta }_n$ is the coupling between the right end of dimer $n$ and the left end of dimer $n+1$ (interdimer coupling). Have in mind that here we present the couplings only, later on $\Delta \le {\Delta }_n$, which means that in the experiment, the intradimer distance is the largest distance in the chain.

Figure 2

Disks with a high index of refraction are placed between two metallic plates as a chain with center to center distances $d_n$ or $d$. The microwaves are induced by a vector network analyzer (VNA) via a microwave cable and a kink antenna.

Figure 3

Reflection spectra of a Dirac oscillator without mass for a 15 dimer chain with minimal dimer distance of $d_{\min }=13\mathrm{mm}$. The upper spectrum (up-shifted) is measured at the 15th disk, whereas the lower is measured at the 3rd disk. The vertical lines indicate the predicted resonance positions from Eq. (12) with $\Delta =0.023\mathrm{GHz}$.

Figure 4

Dirac oscillator without mass for an intradimer distance of 13 mm. The continuous curve corresponds to the analytical prediction [Eq. (12)]. The symbols correspond to different numbers of dimers: 6 dimers (circles), 9 dimers (squares), 12 dimers (diamonds), and 15 dimers (triangles).

Figure 5

Reflection spectrum of a Dirac oscillator with mass for a 15 dimer chain, where the intradimer distance of 10 mm and the interdimer distance of $d_{\min }=10.81\mathrm{mm}$. The vertical lines indicate the predicted resonance positions from Eq. (12) with $\mu =1.066\mathrm{GHz}$ and $\Delta =0.028\mathrm{GHz}$.

Figure 6

Dirac oscillator with mass for a different number of dimers, where the last dimers are removed. The intradimer distance is $d=10\mathrm{mm}$. Displayed are 6 ($d_{\min }\approx 12.32\mathrm{mm}$, circles), 9 ($d_{\min }\approx 11.61\mathrm{mm}$, squares), 12 ($d_{\min }\approx 11.15\mathrm{mm}$, diamonds), and 15 ($d_{\min }\approx 10.81\mathrm{mm}$, triangles). Additionally, the corresponding theoretical curves resulting from Eq. (9) are plotted as solid lines.