Modeling Kleinian cosmology with electronic metamaterials

This paper deals with the propagation of Klein-Gordon particles in flat background spacetime exhibiting discontinuous metric changes from a Lorentzian signature $(-,+,+,+)$ to a Kleinian signature $(-,+,+,-)$. A formal analogy with the propagation of electrons at a junction between an anisotropic semiconductor and an electronic metamaterial is presented. From that analogy, we study the dynamics of these particles falling onto planar boundary interfaces between these two families of media and show a mirror-like behavior for the particle flux. Finally, the case of a double junction of finite thickness is examined and the possibility of tunneling through it is discussed. A physical link between the metamaterial and the Kleinian slabs is found by calculating the time of flight of the respective traversing particles.

Figure 1

Top: Tunneling of the wave function $\varphi$ (real part) through a Kleinian slab of length $l$: the blue line represents the incident and transmitted waves while the orange line denotes the reflected one. Bottom: Graph showing the total wave function for $z<0$ ($\text{incident}+\text{reflected}$). The probability current is conserved since in Eq. (11) we have ${|r|}^2+{|t|}^2=1$. In both graphs, $z$ is given in units of $k_z^{-1}$.

Figure 2

Top: A hyperboloid of two sheets for a constant frequency $\omega$ in Eq. (19). Dispersion relations like these are features of hyperbolic metamaterials. Bottom: Ellipsoid of revolution featuring the dispersion relation (22) for ballistic electrons.

Figure 3

Behavior of the wave functions $\mathrm{\Psi }$ and $\varphi$ (real part) through an effective mass (signature) change at $z=0$. The incident wave (blue line) is completely reflected (orange line) with a $3\pi /2$ phase shift. $z$ is given in units of $k_z^{-1}$.

Figure 4

Top: Tunneling of the wave functions $\mathrm{\Psi }$ and $\varphi$ (real part) through a metamaterial (Kleinian spacetime) sandwich of length $l$: comparison with Fig. 1 shows a phase shift of $\pi$ between the two reflected waves (orange lines). Bottom: Graph of the total wave function with a sharp peak at $z=0$. In both graphs, $z$ is given in units of $k_z^{-1}$.

Figure 5

Graph showing the transmission and reflection coefficients as functions of the length $l$ of the slab (in units of $k_z^{-1}$). As we can see, the probability current is conserved besides the minor changes in the wave function $\mathrm{\Psi }$.

Figure 6

Time of flight of the ballistic electrons (KG particle) as a function of the slab length (Kleinian spacetime) $l$. The blue line shows the function given by Eqs. (68) and (70). The orange one is for the case where the slab (Kleinian spacetime) is absent (where $v_g$ is given by $\hslash p/|m_2|$ for ballistic electrons and $p{c}^2/\omega$ for KG particles). In this graph, $l$ is given in units of $k_z^{-1}$ and $\mathrm{\Delta }t$ in units of $|m_2|{\hslash }^{-1}{k}_z^{-2}$.