Decay of semiclassical massless Dirac fermions from integrable and chaotic cavities

Conventional microlasing of electromagnetic waves requires (1) a high-$Q$ cavity and (2) a mechanism for directional emission. Previous theoretical and experimental work demonstrated that the two requirements can be met with deformed dielectric cavities that generate chaotic ray dynamics. Is it possible for a massless Dirac spinor wave in graphene or its photonic counterpart to exhibit a similar behavior? Intuitively, because of the absence of backscattering of associated massless spin-1/2 particles and Klein tunneling, confining the wave in a cavity for a long time seems not feasible. Deforming the cavity to generate classical chaos would make confinement even more difficult. Investigating the decay of a spin-1/2 wave from a scalar potential barrier-defined cavity characterized by an effective refractive index $n$ that depends on the applied potential and the particle energy, we uncover the striking existence of an interval of the refractive index in which the average lifetime of the massless spin-1/2 wave in the cavity can be as high as that of the electromagnetic wave for both integrable and chaotic cavities. We also find scaling laws for the ratio between the mean escape time associated with electromagnetic waves and that with massless spin-1/2 particles versus the index outside of this interval. The scaling laws hold regardless of the nature of the classical dynamics. All the results are verified numerically. The findings provide insight into the emergent field of Dirac electron optics and have potential applications in developing unconventional electronics using two-dimensional Dirac materials.

Figure 1

Approach of Dirac electron optics to solving the fermion decay problem: four distinct intervals of the effective refractive index in the domain of electrical potential confinement. (a)–(d) The transmission coefficient versus the incident angle in the polar representation for $n\in (-\infty ,-1],n\in [-1,0],n\in [0,1]$, and $n\in [1,\infty )$, respectively. (e)–(h) The corresponding Dirac cone structures outside and inside of the confinement region. Note that, for (a) and (b) [or (e) and (f)], inside the potential region, the directions of the wave vector and the velocity are opposite to each other because of the negative refractive index in these two cases.

Figure 2

Exponential decay of the survival probability for the integrable (circular) cavity for $-1<n<1$. (a) Two examples of exponential decay of $P\left(t\right)$ ($n=-0.5$ and $n=0.5$). (b) The exponential decay exponent $\gamma$ versus $n$. Insofar as the value of $n$ is not close to one, the whispering gallery mode decays most slowly. For $n\to 1$, the cavity becomes transparent, so $\gamma \to \infty$.

Figure 3

Algebraic decay of the survival probability for the integrable (circular) cavity for $\left|n\right|>1$. Shown are decay behaviors of $P\left(t\right)$ for (a) $n=-2$ and $n=-10$ and (b) $n=2$ and $n=10$. The solid and dashed lines represent numerical and theoretical results, respectively.

Figure 4

Decay of spin-1/2 and electromagnetic waves from a ring cavity. (a) For four different ring configurations ($r_1=0$—the original circular cavity, $r_1=0.5$, 0.75, and 0.9), exponential decay behavior of the survival probability $P\left(t\right)$ and (b) the corresponding decay exponent $\gamma$ versus $n$ for $0<n<1$. As the long-time decay behavior is dominated by the whispering gallery modes that concentrate on the larger circumstance, the central forbidden region has little effect on the overall exponential decaying behavior. (c) and (d) The corresponding results for a TM electromagnetic wave where the decay rates for the three actual ring configurations are larger than that of the original circular cavity due to the blockage of the slowest decaying modes along the diameter. (e) and (f) Results for a TE electromagnetic wave where the inaccessibility of the central region has an even more significant impact on the decay.

Figure 5

Exponential decay of a spin-1/2 fermion from a chaotic stadium cavity. (a) The survival probability $P\left(t\right)$ for $n=0$ and $n=0.5$ (on a semilogarithmic scale) where the dashed lines are theoretical predictions. (b) Exponential decay of $P\left(t\right)$ for $n=-10$ and $n=10$. (c) The exponential decay exponent $\gamma$ versus $n$ for $-1<n<1$. (d) The exponent $\gamma v$ versus $\left|n\right|$ for $1<\left|n\right|<100$. In (c) and (d), the dashed line is the analytic prediction. The parameters of the chaotic stadium are $A=\pi ,r_0/l=1/2$, and $L=2\pi r_0+2l$, where $l$ is the length of the straight segment.

Figure 6

Scaling with the relative refractive index of the ratio between the mean escape time for an electromagnetic wave and that of a spin-1/2 wave: (a) the ratio ${\tau }_{\mathrm{TM}}/{\tau }_{\mathrm{S}}$ and (b) the ratio ${\tau }_{\mathrm{TE}}/{\tau }_{\mathrm{S}}$ for three different types of cavity shapes (two with integrable and one with chaotic ray dynamics). For $n\ll 1$, the ratio is proportional to $n^{-1}$, whereas it is proportional to $n$ for $n\gg 1$, regardless of whether the electromagnetic wave in comparison is TM or TE. The scaling laws hold regardless of whether the classical ray dynamics are integrable or chaotic. The surprising result is the existence of an interval in $n$ in which the spin-1/2 wave can have a high-$Q$ value and chaos-enabled directional emission as for the electromagnetic wave.