🔬 optics

Classical Petermann Factor as a Measure of Quantum Squeezing in Photonic Time Crystals

This paper demonstrates that the classical Petermann factor of the effective Floquet Bogoliubov de Gennes dynamical matrix serves as a quantitative predictor for quantum noise, squeezing dynamics, and photon generation in photonic time crystals, thereby bridging classical mode nonorthogonality measurements with the engineering of quantum resources.

Original authors: Younsung Kim, Kyungmin Lee, Changhun Oh, Young-Sik Ra, Kun Woo Kim, Bumki Min

Published 2026-07-28

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Younsung Kim, Kyungmin Lee, Changhun Oh, Young-Sik Ra, Kun Woo Kim, Bumki Min

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where light doesn't just travel through space, but also dances to a rhythm set by time itself. This is the realm of photonic time crystals, a fascinating corner of physics where materials change their properties not as you move through them, but as time ticks by. To understand the magic happening here, we need two key ideas. First, think of quantum squeezing like a balloon. Normally, a balloon has a certain amount of "wobble" or noise in its shape. Squeezing is the act of pressing that balloon so that it gets thinner in one direction (less noise) but fatter in another (more noise). Scientists love this because it lets them measure things with super-precision, like detecting the faint ripples of gravitational waves. Second, there's a concept called the Petermann factor. In the world of lasers and light cavities, this is a number that tells us how "messy" or "non-orthogonal" the light modes are. Usually, we think of light waves as being perfectly independent, like two people walking on parallel tracks. But in real, imperfect systems, these tracks can cross and tangle. The Petermann factor measures just how tangled they get; a high number means the light modes are leaning heavily on each other, which usually makes the system noisier and less stable.

For decades, physicists have treated these two ideas—squeezing (a quantum trick) and the Petermann factor (a classical messiness)—as if they lived in separate universes. One was about the quantum rules of the very small, and the other was about the classical behavior of light waves. But what if they were actually two sides of the same coin? This is the big question that researchers Youn Sung Kim, Kyungmin Lee, and their team at the Korea Advanced Institute of Science and Technology and Chung-Ang University decided to tackle. They wanted to know if the "messiness" of a light system could actually be used as a ruler to measure how much quantum squeezing was happening. If they could prove this link, it would mean that scientists could predict complex quantum behaviors just by doing simpler, classical measurements.

In their study, the team focused on a special kind of photonic time crystal. They showed that these crystals act like a massive collection of tiny amplifiers, each tuned to a specific momentum (or speed) of light. The researchers discovered a direct, mathematical bridge between the classical and quantum worlds. They found that the Petermann factor, a number calculated from the classical behavior of the light, sets the exact scale for the quantum squeezing. It's as if the amount of "tangling" in the light waves directly dictates how much the quantum balloon gets squeezed.

The team found that this relationship works in two different ways depending on the state of the light. In the "stable" zones, where the light behaves nicely, the Petermann factor tells us exactly how much the vacuum of space is filled with "bare photons" due to the squeezing. It's like the factor measures how much the vacuum is "stirred up." However, in the "unstable" zones (called momentum gaps), where the light amplifies itself rapidly, the Petermann factor acts as a powerful multiplier. It boosts the speed at which new photons are created. The researchers showed that even though the growth rate of the light might slow down near the edge of these gaps, the Petermann factor shoots up to infinity, compensating for the slowdown and ensuring that the creation of photons and the squeezing effect remain incredibly strong.

What makes this finding so exciting is that it turns a classical measurement into a quantum prediction. Instead of needing complex quantum experiments to figure out how much squeezing a system will produce, scientists can now measure the classical non-orthogonality (the "tangling") of the light modes. If the Petermann factor is high, they know the squeezing will be intense. The paper demonstrates that this isn't just a theoretical guess; it's a precise algebraic relationship derived from the equations governing the system. The authors suggest that this link offers a new, compact "knob" for engineers. By designing systems with specific classical properties, they can now intentionally engineer quantum resources, creating the perfect conditions for squeezing and photon generation. This work unifies two previously separate stories of light, showing that the classical geometry of light waves is the hidden architect of quantum noise and precision.

Problem Statement
The paper addresses a historical disconnect in the understanding of quantum noise in optical systems. On one hand, classical non-Hermitian mode theory utilizes the Petermann factor (PF) to quantify mode non-orthogonality, explaining excess noise and linewidth broadening in lasers and unstable cavities. On the other hand, quantum optics describes noise reduction and amplification through the concept of squeezed states within a Hermitian, $SU(1,1)$ framework. While both narratives describe the reshaping and amplification of fluctuations, they have traditionally been expressed in separate languages. The authors identify Photonic Time Crystals (PTCs)—systems with periodically modulated permittivity—as an ideal platform to unify these narratives. In PTCs, a Hermitian Hamiltonian gives rise to a non-Hermitian dynamical matrix, creating momentum gaps (MGs) where Floquet quasienergies become complex. Previous works treated the PF (associated with non-Hermitian mode non-orthogonality) and squeezing (associated with quantum photon generation) separately, failing to establish a direct quantitative link between the classical geometric factor and quantum squeezing parameters.

Methodology
The authors develop a Bogoliubov–de Gennes (BdG) effective-Hamiltonian framework for a one-dimensional PTC. They consider a homogeneous medium with a time-periodic permittivity $\varepsilon(t)$ with period $T$ .

Classical-Quantum Correspondence: They demonstrate that the quantum BdG evolution and the classical wave dynamics are governed by formally equivalent monodromy matrices. Specifically, they show that the quantum and classical monodromy matrices share the same eigenvalues and, crucially, identical Petermann factors due to unitary equivalence between the classical and quantum representations.
Regime Classification: The system is analyzed in two regimes:
- Stable Bands: Where Floquet quasienergies are real ( $\epsilon_k$ ).
- Momentum Gaps (MG): Where quasienergies become complex, characterized by a real growth rate $\kappa_k$ .
Derivation of Relations:
- In the stable band, they derive the effective Hamiltonian and relate the Bogoliubov mixing parameter $r_k$ (the squeezing parameter) directly to the PF.
- In the momentum gap, they parameterize the effective dynamical matrix using parameters $g_k$ and $\Delta_k$ to derive the PF in terms of the growth rate and the photon-number prefactor.
Loss Analysis: The authors extend the analysis to include symmetric Markovian damping using Heisenberg-Langevin equations to test the robustness of the derived relations against intrinsic loss.

Key Contributions and Results
The primary contribution is the establishment of the Petermann factor as a direct, quantitative measure of quantum squeezing and photon generation in PTCs.

In Stable Bands: The authors prove that the Petermann factor $K_k$ is solely governed by the Bogoliubov mixing parameter $r_k$ via the relation $K_k = \cosh^2(2r_k)$ . Consequently, $K_k$ fixes the "mismatch occupation" ( $n_k^{mis}$ ), which represents the mean bare-photon occupation of the Floquet vacuum (or vice versa). This provides a quantum interpretation of classical mode non-orthogonality: the divergence of $K_k$ near the momentum-gap edge corresponds to extreme Bogoliubov mixing.
In Momentum Gaps: The paper derives that the average photon number generated from the vacuum state, $\langle \hat{n}_k(t) \rangle$ , is given by $K_k \sinh^2(\kappa_k t)$ . Here, the Floquet growth rate $\kappa_k$ (the imaginary part of the quasienergy) sets the time scale, while the Petermann factor $K_k$ acts as a multiplicative "gain factor" or prefactor that enhances the initial squeezing rate. Specifically, the initial squeezing rate is enhanced by a factor of $\sqrt{K_k}$ relative to a reference system with $K_k=1$ .
Critical Point Behavior: As the system approaches the momentum-gap edge, the growth rate $\kappa_k$ vanishes, but the diverging $K_k$ compensates for this. The photon number exhibits a universal quadratic scaling $\langle \hat{n}_k(t) \rangle \approx |\Delta_k|^2 t^2$ , consistent with the spectral response of a second-order exceptional point.
Robustness to Loss: The analysis shows that intrinsic loss produces only subleading corrections (third-order in time) to the initial squeezing rate. The relation between the initial squeezing rate and the Petermann factor persists even in the presence of symmetric Markovian damping.

Significance
The paper claims to close the gap between classical non-Hermitian mode theory and quantum squeezing dynamics. By demonstrating that the classical Petermann factor, extracted from the BdG dynamical matrix, sets the scale for quantum noise and squeezing, the authors provide a "compact design parameter" for engineering quantum resources.

The significance lies in the unification of two previously separate narratives:

Quantitative Prediction: Classical measurements of mode non-orthogonality (via the PF) can now be used to make quantitative predictions for squeezing and photon generation without needing a full separate quantum derivation.
Design Knob: The biorthogonal mode geometry, quantified by the PF, is identified not as a nuisance but as a practical resource. It serves as a control parameter for parametric quantum noise in a broad class of two-mode $SU(1,1)$ bosonic systems.
PTC Realization: Photonic Time Crystals are presented as a particularly transparent and experimentally accessible realization of this link, where the momentum gap directly corresponds to the region of enhanced squeezing and photon-pair creation.

The work suggests that for any quadratic bosonic platform admitting an effective two-mode $SU(1,1)$ BdG reduction, the biorthogonality of the modes is the fundamental geometric property that dictates the resulting quantum noise and squeezing yield.

More like this