Vibronic Angular Momentum Transfer

• Vibronic angular momentum transfer is the process of coherently and dissipatively exchanging angular momentum among electronic, vibrational, and occasionally photonic subsystems, driven by spin–orbit coupling and Berry-phase effects.
• Key mechanisms such as spin–orbit coupling, magneto-elastic interactions, and geometric torques enable observable spectroscopic signatures and controlled manipulation in molecular and solid-state systems.
• Experimental techniques including HEOM, DFT, and pump–probe spectroscopy quantitatively reveal angular momentum exchange dynamics, underpinning innovations in quantum magnetism and cavity polaritonics.

Vibronic angular momentum transfer refers to the coherent or dissipative exchange of angular momentum between electronic, vibrational, and, in some cases, photonic subsystems in molecules and solids. Driven by spin–orbit coupling, magneto-elastic interactions, or geometric Berry-phase effects, this phenomenon is now recognized as a central element in molecular junction transport, spin-phonon conversion, quantum magnetism, and the manipulation of chiral phononic and vibronic states. Its ramifications extend to spectroscopic selection rules, topological phonon bands, ultrafast spin-lattice dynamics, and cavity molecular polaritonics.

1. Fundamental Definitions and Operator Formalism

Angular momentum in vibronic systems can reside in electronic ($\hat{\bf S}$, $\hat{\bf L}$, $\hat{\bf J}$), vibrational ($\hat{L}_{\mathrm{vib}}$), and in certain scenarios, photonic ($\hat{L}_{\mathrm{ph}}$) degrees of freedom. The canonical vibrational angular momentum operator for two coupled modes ($x_\nu$, $p_\nu$, $\nu=1,2$) is

$$\hat L_{\rm vib} = x_1 p_2 - x_2 p_1 = i\left(b_1^\dagger b_2 - b_2^\dagger b_1\right)$$

with $b^\dagger_\nu$, $b_\nu$ the bosonic creation/annihilation operators for mode $\nu$ (Rudge et al., 12 Mar 2025). This operator measures net circulation of the vibrational wavepacket in the degenerate mode space.

In polyatomic or solid-state contexts, the quantum-mechanical vibrational angular momentum associated with mode $\nu$ is

$$\bm{\ell}_\nu = -i\hbar\sum_I \mathbf{e}_{I\nu}^* \times \mathbf{e}_{I\nu}$$

where $\mathbf{e}_{I\nu}$ are the complex polarization vectors of normal mode $\nu$ on atom $I$ (Bistoni et al., 2021). Intrinsic $\bm{\ell}_\nu \ne 0$ arise in the absence of external symmetry breaking if the molecular electronic ground state is complex-valued (non-collinear spin, spin-orbit coupling).

The total system angular momentum $\hat{L}_{\rm tot}$ includes all contributions: $$\hat{L}_{\rm tot} = \hat{L}_{\rm el} + \hat{L}_{\rm vib} + \hat{L}_{\rm ph}$$ where conservation laws are enforced microscopically via symmetry, selection rules, and coupling tensors (Pandit et al., 11 Nov 2025).

2. Microscopic Mechanisms of Vibronic Angular Momentum Transfer

2.1 Vibronic Spin–Orbit and Cotunneling Effects

In molecular junctions, spin-polarized charge current imparts angular momentum to vibrations through vibronic spin–orbit coupling. The hierarchical equations of motion (HEOM) approach yields exact dynamics for the reduced density operator $\rho_{\rm mol}(t)$ plus a hierarchy of auxiliary density operators (ADOs): $$\frac{d}{dt}\rho^{(n)}_{\bm j} = -i[H_{\rm mol},\rho^{(n)}_{\bm j}] - \sum_{r=1}^n \kappa_{j_r} \rho^{(n)}_{\bm j} - i\sum_{r=1}^n (-1)^{n-r} \mathcal{C}_{j_r}\rho^{(n-1)}_{\bm j\setminus j_r} - i\sum_{j} \mathcal{A}_{\bar j} \rho^{(n+1)}_{\bm j\cup j}$$ Nonzero $\langle L_{\rm vib} \rangle$ implies angular momentum is explicitly transferred between the electronic and vibrational sectors, with spin-dependent directionality: $\langle L_{\rm vib} \rangle^\uparrow < 0$, $\langle L_{\rm vib} \rangle^\downarrow > 0$ (Rudge et al., 12 Mar 2025). The magnitude is maximized near the onset of the first inelastic transport channel and scales positively with molecule–lead coupling $\Gamma$ and orbital energy detuning $\Delta$.

2.2 Berry Phase and Geometric Torque in Magnetic Molecules

Non-adiabatic electron–vibron coupling in non-collinear magnets induces a Berry vector potential in ionic coordinate space: $$\mathcal{A}_\lambda(\mathbf{u}) = i\langle \Psi(\mathbf{u}) | \partial_{u_\lambda} \Psi(\mathbf{u}) \rangle$$ This breaks time-reversal symmetry and causes phonon modes to acquire intrinsic angular momentum (Bistoni et al., 2021). In platinum clusters (Pt$_3$, Pt$_5$), first-principles calculations find $\ell_{\nu} \sim 0.02$–$0.05\,\hbar$, comparable to electronic orbital angular momenta. Excitation of such modes transfers angular momentum to the electronic sector, measurable as vibrationally-induced torque or spectroscopic mode splitting.

2.3 Magneto-elastic and Spin-Vibrational Coupling

Spin–vibrational exchange in single-molecule magnets involves higher-rank Stevens operators coupled to chiral vibrational modes (Ullah et al., 16 May 2025). The general Hamiltonian

$$\hat{H}_{s\text{--vib}} = -\frac{1}{4i} \sum_{k,q} \lambda_{k,4}^q\, \hat{O}_k^q(\hat{J}) \otimes \left( \hat{L}_+^{\rm vib} - \hat{L}_-^{\rm vib} \right)$$

permits controlled transfer of vibrational angular momentum (from doubly degenerate $E$-modes) to spin via the selection rule $\Delta m_J = q$. In the presence of external Zeeman fields (breaking $\mathcal{T}$) and molecular inversion ($\mathcal{P}$), the eigenstates acquire Berry phases $\pm \pi$, observable via split circular dichroic absorption lines.

2.4 Relativistic Spin-Lattice (Magnon–Phonon) Interaction

In magnetic crystals, the dominant channel for angular momentum transfer is a Dzyaloshinskii–Moriya-type spin–lattice coupling: $$H_{\rm DM}^{\rm sl} = \sum_{\langle i j\rangle,k,\mu} {\cal D}_{ij,k}^\mu \left( \mathbf{S}_i \times \mathbf{S}_j \right)^\mu u_k^\mu$$ The phonon–spin torque $\boldsymbol{\mathcal T}_{\rm me}$ drives ultrafast ($\sim$10–100 fs) conversion of magnon angular momentum to circularly polarized phonons. In bcc Fe, $\left|{\cal D}_{ij,j}^\mu\right| \sim 0.05$–$0.1$ meV/Å governs these processes (Mankovsky et al., 2022).

2.5 Light–Matter (Structured Light–Molecule) Transfer

Structured light (e.g. Laguerre–Gaussian beams with orbital angular momentum $\ell$) can transfer $\ell+\sigma$ units of angular momentum to molecular rotation or vibration (Maslov et al., 2023). The general interaction Hamiltonian

$$H_{\rm int} = -\sum_{k,q} \nabla^{(k)}_q E(\mathbf{R})\, T^{(k)}_{-q}$$

yields selection rules $\Delta M = \ell + \sigma$ for rotational transitions, allowing access to forbidden lines (e.g., quadrupole $\Delta J = 2$) enabled by the light’s angular momentum content.

3. Selection Rules, Conservation Laws, and Spectroscopic Signatures

Angular momentum transfer processes are tightly constrained by selection rules derived from symmetry, tensor rank, and field configuration.

• In vibronic spin–orbit coupling: Transfer is linear in difference $$\langle L_{\rm vib} \rangle^\downarrow - \langle L_{\rm vib} \rangle^\uparrow$$, manifesting as spin-polarized current $P_{\rm spin}$ (Rudge et al., 12 Mar 2025).
• Magneto-elastic channels: Coupling to quadrupolar Stevens operators $Q(\Gamma)$ allows only $\Delta J_z = \pm1$ transitions in the presence of degenerate phonons (Pai et al., 2022).
• Berry-phase-induced mode splitting in magnets and single-molecule systems requires broken inversion and time-reversal symmetry to manifest split chiral states and dichroic absorption features (Ullah et al., 16 May 2025).
• In structured-light experiments: Laguerre–Gaussian OAM beams unlock ro-vibrational transitions with $\Delta M = \ell + \sigma$, detectable via high-resolution IR spectroscopy (Maslov et al., 2023).

4. Collective Effects in Polaritonic and Cavity-Coupled Systems

When multiple vibronically active molecules couple coherently to cavity photons, collective angular momentum phenomena emerge.

• In Fabry–Perot cavities hosting $N$ Jahn-Teller molecules, the Hamiltonian supports a cascade of vibronic eigenstates with increasing angular momentum (Pandit et al., 11 Nov 2025). Light–matter interaction conserves $L_{\rm tot}$ yet enables efficient population transfer across vibrational angular momentum states.
• Participation ratio analysis reveals that eigenstates distribute over several vibronic angular momentum sectors, especially as $N$ increases.
• Observable consequences include broadening and fine structure of upper polariton spectral lines, rapid loss of photon polarization memory ($C_{\sigma,\sigma'}(t)$ decay), and the ability to harness high-$L$ vibronic states for cavity photochemistry.

5. Experimental Realizations and Quantitative Metrics

First-principles calculations, transport experiments, pump–probe spectroscopy, and Raman/IR absorption provide direct access to vibronic angular momentum transfer observables.

System/Technique	Key Observable	Quantitative Value
Molecular junction (HEOM)	Spin-polarized current $P_{\rm spin}$	$8\%$ at $\Gamma=20$ meV
Pt$_3$/Pt$_5$ clusters (DFT/LDA)	Intrinsic vibrational angular momentum $\ell_\nu$	$0.02$–$0.05\,\hbar$ per mode
Single-molecule magnet (Ce) (DFT/ab initio)	Chiral splitting, Berry phase	$\Delta E\simeq 0.005$ cm$^{-1}$
bcc Fe (KKR-relativistic)	DMI spin–lattice coupling	$0.05$–$0.1$ meV/Å
Cavity-coupled JT molecules	Polariton spectral broadening	$\sim30$ meV $\to$ $60$ meV
LG beam–molecule (IR absorption)	Quadrupole-enhanced/fobidden lines	$\sim$ few$\%$ absorbance

6. Consequences, Applications, and Outlook

Vibronic angular momentum transfer underpins several contemporary research directions:

• Spintronic devices: Vibronically-assisted spin-polarization in molecular junctions (Rudge et al., 12 Mar 2025).
• Quantum magnetism and spin control: Einstein–de Haas torque and magnon–phonon conversion (Mankovsky et al., 2022).
• Topological phononics: Chiral phonon bands, engineered Berry curvature, and topological edge modes (Pai et al., 2022).
• Molecular polaritonics and cavity quantum chemistry: Collective vibronic cascades, dynamic polarization control (Pandit et al., 11 Nov 2025).
• Advanced spectroscopy: OAM-resolved transitions for rotational state manipulation, chiral discrimination, and optomechanical control (Maslov et al., 2023).

A plausible implication is that by tuning parameters—molecule–lead coupling, bias, field symmetry, cavity configuration—researchers can selectively route angular momentum between electronic, vibrational, and photonic reservoirs. This enables precise examination and utilization of quantum‐mechanical angular momentum transfer processes for information transport, energy conversion, and the engineering of novel quantum matter states.