$\mathcal{PT}$-symmetric spectral singularity and negative-frequency resonance

Vacuum consists of a bath of balanced and symmetric positive- and negative-frequency fluctuations. Media in relative motion or accelerated observers can break this symmetry and preferentially amplify negative-frequency modes as in quantum Cherenkov radiation and Unruh radiation. Here, we show the existence of a universal negative-frequency-momentum mirror symmetry in the relativistic Lorentzian transformation for electromagnetic waves. We show the connection of our discovered symmetry to parity-time ($\mathcal{PT}$) symmetry in moving media and the resulting spectral singularity in vacuum fluctuation-related effects. We prove that this spectral singularity can occur in the case of two metallic plates in relative motion interacting through positive- and negative-frequency plasmonic fluctuations (negative-frequency resonance). Our work paves the way for understanding the role of $\mathcal{PT}$-symmetric spectral singularities in amplifying fluctuations and motivates the search for $\mathcal{PT}$ symmetry in novel photonic systems.

Figure 1

(a) The frequency-mirror symmetry condition for ${\beta }_x=0.3$. The equation of the line satisfying frequency-mirror symmetry condition (5) in frame $S^m$ is shown by the solid red line, while the corresponding Lorentz transformed line in the frame of reference $S^m$ is shown by the dashed red line. It can be seen that the frequency ($\omega$) flips its sign, while the momentum $k_x$ is conserved. (b) The momentum symmetry condition (7), where the solid red line is the momentum-mirror symmetry line in the $S$ frame of reference which transforms to the dashed red line the moving frame of reference $S^m$. It can be seen that the momentum ($k_x$) changes its sign, while frequency $\omega$ is conserved.

Figure 2

Imaginary part of reflection coefficient for (a) stationary metallic slab and (b) slab moving with velocity $v_x=0.1c$ in the four quadrants of the $\omega -k_x$ plane. The $p$-polarized plasmonic mode is considered such that $r_p=\frac{{\epsilon }_1{k}_{z1}-{\epsilon }_2{k}_{z2}}{{\epsilon }_1{k}_{z1}+{\epsilon }_2{k}_{z2}}$. Here, $k_{z1}=\sqrt{{{\epsilon }_1{k}_0^2-k_x^2}^{\phantom{1}}}, k_{z2}=\sqrt{{{\epsilon }_2{k}_0^2-k_x^2}^{\phantom{1}}}$, and the dielectric response is governed by the Drude model ${\epsilon }_1\left(\omega \right)=1-\frac{{\omega }_p^2}{{\omega }^2+i\mathrm{\Gamma }\omega }$. It can be seen that the peak in $r_p^{\prime \prime }$ follows the dispersion curve of the surface plasmon polaritons. The $r_p^{\prime \prime }$ is proportional to the normal component of the Poynting vector in the evanescent plasmonic wave and is representative of the loss in the slab. For a lossy medium, $r_p^{\prime \prime }$ is positive for positive frequencies and negative for negative frequencies, as depicted in (a). For a moving slab, in the region where $k_x{v}_{\mathrm{motion}}>\omega$, the negative-frequency mode (represented by the blue peak in $r_p^{\prime \prime }$) is dragged into the positive-frequency quadrant, which exhibits gain at positive frequencies. Therefore, in a moving slab, there are two forward-propagating modes in the positive-frequency region: one is the positive-frequency lossy mode represented by positive peaks in $r_p^{\prime \prime }$, while the other is the gain mode with negative $r_p^{\prime \prime }$ in the positive-frequency region.

Figure 3

Metal slabs separated by a distance $d$ at relative motion interact via near-field evanescent waves. When the velocity of motion is greater than the Cherenkov velocity ($v_{\mathrm{motion}}>\omega /k_x$), the interaction is between positive frequencies of the stationary slab and negative frequencies of the moving slab. This can result in a perfect coupling between positive- and negative-frequency modes, which we call a negative-frequency resonance. When the velocity of motion is exactly twice the Cherenkov limit ($v_{\mathrm{motion}}=2\omega /k_x$), the dielectric response of the two slabs becomes complex-conjugate pairs, resulting in $\mathcal{PT}$-symmetric spectral singularity. The $\mathcal{PT}$ symmetry is achieved as a consequence of the negative-frequency-mirror symmetry.

Figure 4

Dispersion curves for plasmonic MIM waveguide in four quadrants of the $\omega -k_x$ plane for (a) stationary slabs and (b) slabs with relative velocity of $0.2c$. Solid colored lines represent the real part of propagation constant $k_x^{\prime }$, while the dash-dotted lines represent the imaginary part $k_x^{\prime \prime }$ of the respective mode. Black dotted lines represent the light cone. In (b), the dispersion curve lying below the $\mathcal{PT}$-symmetry line is the negative-frequency mode which is dragged into the first quadrant. This is the gain mode. The gain and loss modes merge at the intersection of the $\mathcal{PT}$-symmetry line with the imaginary component of the propagation constant equal to zero at the intersection point. At this point, loss in the stationary metal slab is exactly compensated by gain in the moving metal slab. All points on the dispersion curve below the $\mathcal{PT}$-symmetry line have gain, while all the points above have loss. The inset shows the $H_z$ mode profile at (b1) loss, (b2) gain, and (b3) $\mathcal{PT}$-symmetric points. The magnetic field profile in the lossy region of the dispersion curve (b1) attenuates as it propagates along the $x$ direction, and amplifies in the gain region (b2). On the $\mathcal{PT}$-symmetry point of the dispersion curve, the field profiles propagate without any attenuation or gain. Drude metal with plasma frequency ${\omega }_p={10}^{14}$ Hz; collision frequency $\mathrm{\Gamma }=0.05{\omega }_p$ is considered in the simulation. The separation between the slabs is $d=25$ nm.

Figure 5

(a) Phase velocity along the dispersion curve as a function of frequency when $v_{\mathrm{motion}}=0.2c$. The gain and loss regions of the dispersion curve are separated by the $\mathcal{PT}$-symmetry line with $v_{\mathrm{motion}}/2$. The mode intersects the $\mathcal{PT}$-symmetry line when $v_{\mathrm{motion}}=2v_p$. (b) $Q$ factor of the resonance as a function of the ratio $v_{\mathrm{motion}}/2v_p$. The mode exhibits spectral singularity as $v_{\mathrm{motion}}\to 2v_p$, at the intersection of the $\mathcal{PT}$-symmetry line and dispersion curve.