Analogs of spontaneous emission and lasing in photonic time crystals

TL;DR

The paper demonstrates dynamic control of the electromagnetic vacuum by implementing photonic time crystals (PTCs) through a time-periodically modulated array of LC resonators, analyzed with non-Hermitian Floquet theory. By measuring broadband-noise–driven radiated power, the authors map the spectrally resolved LDOS and reveal a cusp at the momentum-gap frequency, accompanied by a decomposition into absorptive and dispersive Lorentzians that reflects non-orthogonal in-gap Floquet modes and exceptional points. A two-mode Floquet model captures the gap-mode behavior, including the Petermann-factor–driven enhancement and the modulation-depth threshold $\delta_c$ for a transition to a parametric lasing (PTC-laser) regime. These results establish dynamic Purcell engineering and nonequilibrium photonics with time-periodic LDOS shaping, enabling potential applications in emission control, dynamical Casimir analogs, and emitter–PTC hybrids. The work highlights how temporal modulation can sculpt light–matter interactions in ways complementary to static photonic crystals.

Abstract

We report the first direct mapping of the frequency-resolved local density of states (LDOS) in a photonic time crystal (PTC) implemented as an array of time-periodically modulated LC resonators at microwave frequencies. Broadband white noise probes the system and yields an LDOS lineshape near the momentum gap that can be decomposed into absorptive and dispersive Lorentzian components. The finite LDOS peak at the gap frequency, which grows with modulation strength, implies that the spontaneous emission rate of an emitter coupled to the PTC would be maximized at that frequency. The measured spectra are in good agreement with classical non-Hermitian Floquet theory. As the modulation-induced gain exceeds intrinsic losses, the system undergoes a transition to a narrow-band self-oscillation (lasing) regime. These results open a route to nonequilibrium photonics and bring time-periodic LDOS engineering closer to practical realization.

Figures (5)

• Figure 1: (a) Schematic representation of the equivalent circuit for the photonic time crystal. The system consists of an array of LC resonators side‑coupled to a transmission line. In each unit cell, the resonator capacitors are sinusoidally modulated at angular frequency $\Omega$, thereby producing a time‑periodic modulation of the effective permittivity of the structure. (b) Map of the momentum-resolved density of states (kDOS) together with band structures (main panel), LDOS (left panel), and imaginary parts of the quasi-eigenfrequencies (bottom panel) for $\Omega \approx 2\omega_{k}$ at $\delta = 0.023$.
• Figure 2: (a) Real (solid lines) and imaginary (dashed lines) parts of the normalized field intensity $I_\pm(k)$. Inside the momentum gap (green shaded region), a finite $\Im[I_\pm(k)]$ generates the antisymmetric Lorentzian component in the kDOS. Outside the gap, $\Im[I_\pm(k)] \approx 0$, resulting in a purely symmetric Lorentzian profile. (b) Square root of the Petermann factor, $\sqrt{\mathrm{PF}}=\sqrt{\mathrm{PF}_+}=\sqrt{\mathrm{PF_-}}$ (black), and source-mode overlap magnitude $\Pi_\alpha$ for the two modes ($+$ in blue and $-$ in red)
• Figure 3: (a) Experimentally measured radiated-power map at the momentum gap frequency as functions of modulation frequency and power. Blue (linear) and yellow (oscillation/lasing) regions are separated by the oscillation threshold; colored markers indicate operating points used in the panels b and g. (b) Sub-threshold spectra measured for the gap-center modes. Increasing the modulation power from -17 dBm to -16 dBm increases the peak amplitude and narrows the symmetric Lorentzian profile. (c) The calculated LDOS for modulation depths in the range $0.023 \le \delta \le 0.027$ reproduces the observed peak growth and linewidth narrowing. (d) The imaginary part of the eigenfrequencies as a function of wavenumber $k$. The two gap modes dominate the LDOS at the momentum gap frequency. (e) Real and imaginary parts of $I_\alpha$ for the gap modes as a function of modulation frequency. The imaginary parts of $I_\alpha$ decrease as the mode approaches the center of the momentum gap, reducing the antisymmetric weight in the LDOS. (f) Extracted linewidth $\Gamma_-=\Im[\omega_-]$ of the lower-loss gap mode. The linewidth decreases toward the gap center, accompanied by a narrowing of the LDOS peak. (g) When the modulation frequency is detuned so the modes move off-center in the momentum gap, the measured spectrum becomes skewed. Black curve: total fit obtained from the two-mode model of Eq. \ref{['eqn:mode decomposition']}. Blue and orange curves: individual contributions of the $\alpha=+$ and $-$ gap modes, respectively. (h) Theoretical LDOS for this detuned case reproduces the asymmetric profile. (i) Corresponding imaginary part of the eigenfrequencies versus wavenumber $k$.
• Figure 4: (a) Measured radiated power at the momentum gap frequency and the corresponding linewidth as functions of modulation power at a fixed modulation frequency $\Omega/2\pi=2.998 \,\mathrm{GHz}$. As the modulation power exceeds the threshold, the system undergoes a sharp transition into an oscillating regime where the radiated power saturates due to nonlinearities. (b) Spectral profile at $-13 \,\mathrm{dBm}$, deep in the lasing regime, showing a sharp, narrow-band oscillation.
• Figure S1: (a) Schematic of the one-dimensional array of 12 coupled LC resonators embedded in a waveguide. (b) Schematic layout and photograph of the fabricated unit cell, incorporating a varactor diode for time-periodic capacitance modulation via a DC-biased AC voltage. The unit cell is designed with the following geometrical parameters: $\mathrm{h}_1=40\ \mathrm{mm}$, $\mathrm{h}_2=1\ \mathrm{mm}$, $\mathrm{h}_3=3\ \mathrm{mm}$, $\mathrm{h}_4=10\ \mathrm{mm}$, $\mathrm{w}_1=105\ \mathrm{mm}$ and $\mathrm{w}_2=27.5\ \mathrm{mm}$. (c) Schematic of the experimental setup.