Rotational Vacuum Friction of Nonabsorbing Particles

Technical Summary: Rotational Vacuum Friction of Nonabsorbing Particles

Problem Statement
The paper addresses the phenomenon of vacuum friction acting on neutral bodies in motion, specifically focusing on small particles rotating in a vacuum. While vacuum friction is well-established for absorbing particles (where mechanical energy is dissipated via single-photon emission coupled to internal excitations), the behavior of nonabsorbing particles remains less understood. Specifically, the authors investigate the regime where the rotational frequency $\Omega$ lies below the material's optical band gap ($\varepsilon_g$). In this regime, the standard first-order single-photon dissipation channel is energetically forbidden. The central question is whether and how a nonabsorbing particle loses angular momentum and mechanical energy in vacuum, and how the particle's symmetry influences this process.

Methodology
The authors develop a fully quantum-mechanical theory describing the interaction between a rotating particle, the electromagnetic vacuum field, and the particle's internal bosonic excitations.

• Model: The system consists of a particle with moment of inertia $I$ rotating about the $z$-axis with angular velocity $\Omega$ at a vacuum temperature $T_0$. The particle possesses internal excitations with frequencies $\varepsilon_j \gg \Omega$ and $\varepsilon_j \gg k_B T_0/\hbar$.
• Interaction: The coupling is treated within the electric-dipole approximation. The interaction Hamiltonian connects the particle's lab-frame transition dipoles (which rotate with the particle) to the photon creation and annihilation operators.
• Perturbation Theory: Since first-order transitions are forbidden by the large internal energy gap, the authors apply Fermi's Golden Rule to second-order processes. They calculate transition rates for three distinct mechanisms:
  1. Two-Photon Emission (2PE): Spontaneous emission of a photon pair.
  2. Two-Photon Absorption (2PA): Absorption of a photon pair from the thermal bath.
  3. Photon Scattering (PS): Scattering of a single thermal photon.
• Calculation: The torque is derived by summing the angular momentum transfer ($\pm 2\hbar$ or $\pm \hbar$) weighted by the transition rates. The sums over photon modes are converted to integrals over wave vectors and polarizations, expressed in terms of the particle's polarizability tensor components $\alpha_{ab}(\omega)$.

Key Contributions and Results

1. Leading Dissipation Channel for Asymmetric Particles:
   For nonabsorbing particles lacking axial symmetry (anisotropic), the leading-order dissipation channel is the spontaneous emission of correlated photon pairs (2PE).

   • Mechanism: The particle emits two photons with frequencies $\omega$ and $2\Omega - \omega$ (where $0 < \omega < 2\Omega$). This process conserves energy ($\omega_1 + \omega_2 = 2\Omega$) and removes $2\hbar$ of angular momentum from the rotor, leaving the particle's internal state unchanged.
   • Scaling: At zero temperature, the frictional torque $M$ scales as $\Omega^7$. The expression is given by:
     $$M \approx -\frac{128}{315\pi} \frac{\hbar \Omega^7 g_0}{c^6}$$
     where $g_0$ depends on the anisotropy of the polarizability tensor ($|\alpha_{xx} - \alpha_{yy}|^2 + 4|\alpha_{xy}|^2$).
   • Finite Temperature: At finite temperatures, additional contributions arise from 2PA and photon scattering. In the high-temperature limit ($k_B T_0 \gg \hbar \Omega$), the torque becomes linear in $\Omega$ and scales with the sixth power of temperature ($T_0^6$).

2. Protection of Axisymmetric Particles:
   For nonabsorbing particles possessing axial symmetry (e.g., spheres), the authors prove that the frictional torque vanishes identically at zero temperature.

   • Symmetry Argument: The tensor combination $g(\omega)$ governing the 2PE torque becomes zero for axisymmetric particles ($\alpha_{xx} = \alpha_{yy}$ and $\alpha_{xy} = 0$).
   • Higher-Order Cancellation: The authors demonstrate that this cancellation persists to all orders of perturbation theory. Odd-order processes cannot return the particle to its ground state, and even-order processes involve matrix element pairs that cancel out due to rotational invariance in the $x-y$ plane.
   • Finite Temperature Exception: At finite temperatures, a non-zero torque exists for axisymmetric particles, but it arises solely from single-photon absorption of thermal photons with frequencies above the band gap. This torque is linear in $\Omega$ but exponentially suppressed by the factor $e^{-\hbar \varepsilon_g / k_B T_0}$.

3. Quantitative Estimates:
   Applying the theory to a prolate diamond ellipsoid (transparent from microwave to optical frequencies) with semi-axes of 100 nm and 76 nm rotating at 6 GHz:

   • At $T_0 = 300$ K, the torque is estimated at $\sim 2 \times 10^{-33}$ N m.
   • This value is approximately six orders of magnitude below the sensitivity of current state-of-the-art levitated torque sensors.
   • The resulting spin-down time is estimated at $\sim 10^{12}$ seconds, indicating that transparent nanorotors are essentially immune to vacuum friction.

Significance and Claims
The paper establishes a "symmetry-controlled hierarchy" of multiphoton processes governing rotational vacuum friction.

• Fundamental Physics: It identifies a genuine zero-temperature quantum regime where friction is driven by the spontaneous emission of frequency-entangled photon pairs, analogous to degenerate parametric down-conversion driven mechanically.
• Stability: The results demonstrate that axial symmetry provides a robust protection mechanism against vacuum friction for nonabsorbing particles at zero temperature, defining a dynamically stable equilibrium configuration.
• Experimental Relevance: The authors suggest that the predicted $\Omega^7$ scaling and the emission of entangled photon pairs could serve as a background-free experimental signature for quantum rotation tests. They note that parametric implementations in superconducting circuits, where dynamical Casimir emission has been observed, offer a route to emulate and potentially enhance this pair-emission process.
• Quantum Information: Due to the extreme weakness of the friction for transparent particles, the authors conclude that such systems are robust against decoherence from vacuum friction, making them suitable candidates for storing and processing quantum information in levitated optomechanical systems.