Bicircularly Polarized Light

• BCL is an engineered electromagnetic field created by superposing circularly polarized waves with different frequencies and phases to yield complex polarization trajectories such as rose patterns.
• Its tunable parameters enable Floquet engineering to drive controlled symmetry breaking, induce valley-selective transport, and trigger topological phase transitions in various materials.
• BCL leads to nonlinear optical responses, generating tunable photocurrents and opening avenues for ultrafast manipulation of quantum and topological material properties.

Bicircularly polarized light (BCL) describes an electromagnetic field comprised of two superposed circularly polarized components with different frequencies, often possessing opposite helicities and a tunable relative phase. This field produces a complex time-dependent polarization trajectory (frequently a rose or Lissajous shape) in the polarization plane. BCL, as addressed in contemporary research, serves as a versatile tool for symmetry engineering, nonlinear optical response, Floquet topological transitions, and controlled electronic transport phenomena in both centrosymmetric and noncentrosymmetric materials.

1. Physical Definition and Mathematical Description

BCL is engineered by superposing two circularly polarized fields, typically of frequencies $\omega$ and $\eta\omega$, with amplitude ratio $r$ and phase difference $\alpha$. The general vector potential is

$$A(t) = \sqrt{2} \mathcal{A}_0 \, \mathrm{Re}\left\{ r e^{-i(\eta \omega t - \alpha)} \epsilon_R + e^{-i\omega t} \epsilon_L \right\},$$

where $\epsilon_{R/L}$ are right/left circularly polarized basis vectors (Trevisan et al., 2021, Ganguli et al., 7 Oct 2024).

For the plane electric field components,

$$A_x(t) = \mathcal{A}_0 [r \cos(\eta \omega t - \alpha) + \cos(\omega t)], \quad A_y(t) = \mathcal{A}_0 [-r \sin(\eta \omega t - \alpha) + \sin(\omega t)].$$

The field’s polarization trajectory can then display $C_n$-fold symmetry depending on the relative frequency ratio and phase, generating rose patterns, trefoils, and star-like structures in the polarization plane (Ogawa et al., 2023).

2. Floquet Engineering and Light-Induced Symmetry Breaking

Illumination with BCL imparts strong nonequilibrium modifications to material band structures via Floquet engineering. The high-frequency expansion for a time-periodic Hamiltonian $H(t)$ yields a static effective Hamiltonian,

$$H_\text{eff} = H_0 + \frac{1}{\omega} \sum_n \frac{1}{n} [H_n, H_{-n}] + O(1/\omega^2),$$

where $H_n$ are the Fourier components with respect to the BCL cycle (Trevisan et al., 2021, Ganguli et al., 7 Oct 2024, Zhan et al., 5 Nov 2024).

Application of BCL can dynamically break both time-reversal (TRS) and inversion symmetry (IS). For example, in graphene irradiated with BCL of even frequency ratio $\beta$ (i.e., $\beta = 2$), a staggered sublattice potential $\Delta_\text{BCL} \neq 0$ and a complex next-nearest-neighbor hopping are Floquet-induced, leading to a valley-selective Hall effect (Arakawa et al., 5 Jun 2024): $$A_x(t) + iA_y(t) = A_0 e^{i\Omega t} + A_0 e^{-i(\beta \Omega t - \theta)},$$ where $\theta$ is the phase difference. For even values of $\beta$, inversion symmetry is broken, producing valley degeneracy lifting and a tunable Hall response; for odd $\beta$, inversion symmetry remains unbroken.

In Dirac and line-node semimetals, BCL Floquet engineering yields topological transitions, opening band gaps, shifting Dirac points, splitting nodal lines into pairs of Weyl nodes, or maintaining node protection depending on the amplitude and frequency ratio tuning ($r^2 = \eta$ preserves nodal points) (Ganguli et al., 7 Oct 2024, Zhan et al., 5 Nov 2024).

3. Nonlinear and Geometric Optical Responses

BCL enables nonlinear photocurrent generation in centrosymmetric systems where conventional second-order processes (shift current) are symmetry-forbidden. The third-order nonlinear conductivity under BCL excitation is characterized by

$$J^{(u)}_\text{BCL} \sim \operatorname{Re}\left[\sigma^{u\alpha\beta\gamma}_\text{BCL} E^{(\omega)}_\alpha E^{(\omega)}_\beta E^{(-2\omega)}_\gamma\right],$$

where $\sigma^{u\alpha\beta\gamma}_\text{BCL}$ encapsulates the quantum geometric ingredients—gauge-invariant shift vectors, quantum geometric tensors (QGT), and triple-phase products (Guo et al., 5 Mar 2025). Crucially, the real part of the QGT and the imaginary part of the triple-phase product remain nonzero under inversion symmetry, activating multi-band geometric current mechanisms (beyond ordinary Berry curvature) even in centrosymmetric materials.

Under BCL, both injection and shift currents stem from composite geometric objects:

• Gauge-invariant shift vectors $R^{u,\alpha}_{ab}$ from Berry connections,
• Quantum metric contributions $\operatorname{Re} Q^{\alpha\beta}_{ab}$,
• Triple-phase products $T^{\alpha\beta\gamma}_{abc}$ encapsulate higher-order interband coherence.

The photocurrent direction and amplitude exhibit tunable dependence on BCL’s phase difference and symmetry order (Ikeda et al., 2023, Guo et al., 5 Mar 2025).

4. Topological Phase Transitions and Control

BCL illumination offers a unique degree of tunability for electronic topology in materials:

• In multifold fermion semimetals, tuning $r^2$ relative to $\eta$ maintains or shifts band touching points or opens gaps, enabling transitions between gapless and gapped phases (Ganguli et al., 7 Oct 2024).
• In line-node semimetals, the presence of BCL can either preserve nodal lines ($r^2 = \eta$) or break them into point nodes or gapped phases.
• In altermagnets with Rashba spin-orbit coupling, BCL not only controls the band topology and Hall conductivity but also enables direct manipulation of spin textures and Fermi surfaces. A cascade of topological phase transitions is observed as band gap closings occur, reflected in discrete jumps of the Chern number and Berry curvature sign (Ganguli et al., 8 Sep 2025).

BCL’s phase parameter provides an additional knob for symmetry manipulation. Effective Floquet Hamiltonians feature renormalized hopping terms and momentum shifts whose symmetry can be controlled by the relative phase, with the shift vector $k_0$ and form factors $f_0^\alpha$ directly varying with $\alpha$ (Ganguli et al., 8 Sep 2025).

5. Experimental Realizations and Technological Advancements

Recent efforts demonstrate programmable generation of phase-stable BCL pulses in both mid-infrared and multi-terahertz ranges, using spectral broadening, pulse shaping via spatial light modulators (SLM), and intra-pulse difference frequency generation (DFG) within nonlinear crystals (e.g., GaSe). The SLM allows independent control of the phase and ellipticity of spectral components, realizing customizable Lissajous trajectories, rotational symmetries (Cn), and helicity (Ogawa et al., 2023).

Phase stabilization—addressed via in-line DFG—and the absence of suitable polarization elements in the multi-terahertz regime are overcome through pulse shaping and inline conversion. Measured azimuthal fluctuations of 14.7 mrad and temporal jitter of 0.23 fs reflect excellent phase stability, essential for exploiting BCL in ultrafast photonics and valleytronics.

6. Applications in Topological and Quantum Materials

BCL enables control over:

• Valley-selective transport in graphene and other honeycomb materials by breaking valley degeneracy and tuning the Hall response (with conductivities two orders of magnitude larger than conventional cases) (Arakawa et al., 5 Jun 2024).
• Magnetic and topological phase transitions (e.g., Dirac to Weyl semimetal, magnetic topological crystalline insulator) in Dirac semimetals with strain (Trevisan et al., 2021).
• Gyrotropic magnetic current generation in materials such as compressed black phosphorus, through light-induced separation of Weyl nodes in momentum and energy, with first-principles calculations supporting large achievable gyropropic currents under realistic conditions (Zhan et al., 5 Nov 2024).

Programmable BCL is poised to manipulate ultrafast electron dynamics, valley degrees of freedom, topological states, and coherent control protocols for imaging, optoelectronic device development, and nonlinear spectroscopy (Ogawa et al., 2023).

7. Fundamental Limits, Symmetry Constraints, and Future Directions

Symmetry arguments dictate when BCL-induced effects are allowed and their magnitude:

• For even frequency ratios (e.g., $\beta = 2$), BCL breaks inversion symmetry in a manner inaccessible to monochromatic fields, yielding valley polarization and tunable Berry curvature.
• Multi-band quantum geometric effects (quantum metric and triple-phase products) are robust against inversion symmetry and thus enable geometric currents in a broad class of materials (Guo et al., 5 Mar 2025).

Continued development is expected in optimization of BCL pulse generation, exploring higher symmetry orders, and integration into ultrafast pump-probe setups. Applications may extend to all-optical control of Fermi surfaces, spin-orbit textures, and dynamic observation of symmetry-protected surface states in next-generation quantum materials.

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In summary, bicircularly polarized light forms a structurally rich electromagnetic field for active Floquet engineering and quantum geometric manipulation of solids, providing precise and programmable control over material symmetries, band topology, transport, and optical response. Its theoretical and experimental examination reveal both robust and tunable light-induced phenomena, establishing BCL as an essential instrument in contemporary condensed matter, photonics, and quantum materials research.