Angular Momentum Fluctuations in the Phonon Vacuum of Symmetric Crystals

Abstract

Although time-reversal and inversion symmetry constrain the angular momentum of each phonon mode to vanish, we show that the vacuum state of crystals with such symmetries can nevertheless exhibit finite angular momentum fluctuations, {which persists at finite temperature}. These fluctuations arise from quantum coherence between nondegenerate, orthogonally polarized modes and are encoded in the off-diagonal components of the angular momentum operator. Their origin lies in the noncommutativity between the phonon Hamiltonian and angular momentum, which enables time-dependent rotational dynamics even in symmetric vacua. We provide intuitive insight into the coherence underlying this phenomenon by drawing an analogy with the beating between linearly polarized classical waves. Finally, we show that these angular momentum fluctuations produce distinct spectral signatures that can, in principle, be probed via established techniques sensitive to the polarization content and symmetry of lattice excitations, opening an uncharted avenue for accessing and leveraging rotational vacuum correlations in crystalline systems.

Figures (2)

• Figure 1: A quantum spin with resonance frequency $\omega$, coupled to vacuum angular momentum noise, undergoes spontaneous emission by generating a pair of phonons with orthogonal, linear polarizations which satisfy conservation of energy and linear momentum. The angular momentum fluctuations associated with the phonon pair can be understood through a classical analogy: the superposition of two orthogonal, linearly polarized waves with a frequency mismatch $\delta\omega = \omega_2 - \omega_1$, which results into a finite instantaneous angular momentum $J_z$ that averages to zero over a period $T= 2\pi / \delta\omega$.
• Figure 2: (a) Schematic illustration of a quantum two-level system with resonance frequency $\omega$ undergoing spontaneous emission into a two-dimensional symmetric crystal. As dictated by Eq. \ref{['eq:susceptibility']}, the emitted pair of phonons has opposite linear momenta $\pm k_0$ and orthogonal polarizations, taken here to lie along $\hat{x}$ and $\hat{y}$. Due to the lack of mode degeneracy, the mode with higher group velocity has the higher frequency, i.e., $\omega_y > \omega_x$, with $\omega = \omega_x + \omega_y$. The resulting two-phonon state admits a classical representation as a coherent superposition of orthogonal displacement fields $u_x(t)$ and $u_y(t)$, oscillating at frequencies $\omega_x$ and $\omega_y$ subject to the same constraint. (b) Classical trajectory traced by the displacement field \ref{['superpo']}, plotted as $u_y(t)/u_0$ versus $u_x(t)/u_0$ over half a beating cycle, i.e., $T/2 = \pi/\delta\omega$, for $\omega_x = 1$ and frequency detuning $\delta\omega = \omega_y - \omega_x = 0.25$. The resulting elliptical motion is color-coded by time, with key time instants marked to illustrate the precession of the polarization ellipse. (c) Time evolution of the instantaneous angular momentum \ref{['eq9']}, plotted in dimensionless form as $J_z(t)/(\omega_x u_0^2)$ over one full period $T$. Its oscillations arise from the beating dynamics shown in (b) and exhibit a finite amplitude with vanishing time-averaged angular momentum, mirroring the structured temporal fluctuations revealed by Eq. \ref{['eq6']}.