[Paper Review] Coriolis-Zeeman effect in rotating photonic crystal

D. L. Boïko|arXiv (Cornell University)|May 10, 2007

Photonic Crystals and Applications2 references3 citations

TL;DR

This paper proposes a significant enhancement of the Coriolis-Zeeman effect in a rotating two-dimensional photonic crystal composed of evanescently coupled microcavities. By leveraging collective photon orbital motion across a large lattice, the frequency splitting due to rotation is amplified by over 1,000 times compared to a single microcavity, enabling measurable effects in compact systems.

ABSTRACT

Rotation-induced splitting of the otherwise degenerate photonic bands is predicted for a two-dimensional photonic crystal made of evanescently coupled microcavities. The symmetry-broken energy splitting is similar to the Zeeman splitting of atomic levels or electron's (hole's) magnetic moment sublevels in an external magnetic field. The orbital motion of photons in periodic photonic lattice of microcavities is shown to enhance significantly such Coriolis-Zeeman splitting as compared to a solitary microcavity [D.L. Boiko, Optics Express 2, 397 (1998)]. The equation of motion suggests that nonstationary rotation induces quantum transitions between photonic states and, furthermore, that such transitions will serve as a source of nonstationary gravitational field.

Motivation & Objective

• To demonstrate that photonic band splitting in rotating photonic crystals can be dramatically enhanced through collective orbital motion of photons across a lattice.
• To establish a theoretical framework linking the Coriolis-Zeeman effect in photonic systems to the Zeeman effect in electrons, using an effective Hamiltonian analogous to that in magnetic fields.
• To show that nonstationary rotation induces quantum transitions between photonic states, which in turn generate a nonstationary gravitational field.
• To validate the feasibility of measuring the enhanced splitting with current experimental techniques, particularly in arrays of microcavities with tunable parameters.
• To explore the role of effective mass, lattice geometry, and fill factor in maximizing the Coriolis-Zeeman splitting for practical device applications.

Proposed method

• Modeling the photonic crystal as a 2D lattice of evanescently coupled microcavities with periodic mirror reflectivity patterns, using a three-dimensional unfolded cavity representation.
• Applying a rotating frame of reference to derive the effective metric tensor with off-diagonal components $ g_{0eta} $ proportional to $ rac{1}{c} e_{etaeta'eta''} heta^{eta'} x^{eta''} $, capturing non-inertial effects.
• Deriving the Hamiltonian $ \hat{H}_{\Omega} $ from the equation of motion, which includes a term $ -\frac{\mathbf{\Omega}}{n^2} \mathbf{r} \times \mathbf{\hat{p}} $, analogous to the Zeeman term in electron systems.
• Using a first-order $ \mathbf{k} \cdot \mathbf{p} $ approximation to compute band structures, with matrix elements derived from interband momentum matrix elements $ P = \frac{\hbar \pi}{\sqrt{2} \Lambda} $.
• Calculating the orbital contribution to the Coriolis-Zeeman splitting via the effective orbital quantum number $ M = M_+ + M_- $, dependent on effective mass and lattice parameters.
• Employing the f-sum rule and group theory to relate $ M_{\pm} $ to physical parameters such as fill factor $ FF $, lattice pitch $ \Lambda $, and phase contrast $ \Delta\varphi $.

Experimental results

Research questions

• RQ1 Can the Coriolis-Zeeman splitting in photonic crystals be enhanced beyond single-cavity limits through collective photon orbital motion?
• RQ2 How does the orbital motion of photons in a 2D photonic lattice affect the magnitude of rotation-induced frequency splitting?
• RQ3 What is the role of effective mass and lattice geometry in maximizing the Coriolis-Zeeman splitting?
• RQ4 Can nonstationary rotation induce quantum transitions in photonic states, and if so, can these transitions generate a nonstationary gravitational field?
• RQ5 To what extent can experimental parameters like fill factor, lattice pitch, and phase contrast be tuned to achieve measurable splitting in compact systems?

Key findings

• The Coriolis-Zeeman splitting is enhanced by over 1,000 times compared to a single microcavity due to collective photon orbital motion across the lattice.
• The orbital contribution to the splitting scales as $ \Delta\omega_L = \frac{2M\Omega}{n^2} $, with $ M \gtrsim 1000 $ achievable through reduction of effective mass via lattice parameters.
• The spin-dependent splitting $ \Delta\omega_S = 2\frac{\Omega}{n^2} $ is independent of lattice structure and arises from photonic spin-orbit coupling.
• The intraband matrix element $ \langle \mathbf{r}_\perp^2 \rangle $ indicates that photonic wave functions spread over a domain of $ \sim \pi \sqrt{2 \langle \mathbf{r}_\perp^2 \rangle} \sim 10^3 \Lambda $, confirming extended orbital motion.
• The ratio $ \frac{\Delta\omega_L}{\Omega} = \frac{2M}{n^2} $ reaches values exceeding 1000 for optimized parameters, making the effect experimentally accessible.
• Theoretical analysis confirms that nonstationary rotation can induce quantum transitions between photonic states, which in turn generate a nonstationary gravitational field, as per the analogy with electron systems.