Nonequilibrium lateral force and torque by thermally excited nonreciprocal surface electromagnetic waves

We show that an isotropic dipolar particle in the vicinity of a substrate made of nonreciprocal plasmonic materials can experience a lateral thermal-fluctuations-induced force and torque when the particle's temperature differs from that of the slab and the environment. We connect the existence of the lateral force to the asymmetric dispersion of nonreciprocal surface polaritons and the existence of the lateral torque to the spin-momentum locking of such surface waves. Using the formalism of fluctuational electrodynamics, we show that the features of lateral force and torque should be experimentally observable using a substrate of doped indium antimonide (InSb) placed in an external magnetic field, and for a variety of dielectric particles. Interestingly, we also find that the directions of the lateral force and the torque depend on the constituent materials of the particles, which suggests a sorting mechanism based on nonequilibrium fluctuational electrodynamics.

Figure 1

We illustrate nonequilibrium lateral force and torque on an isotropic dipolar nanoparticle in the near field of a doped InSb slab placed in magnetic field parallel to its surface. We assume a static magnetic field is applied along $\mathbf{x}$ axis of the geometry and clarify the asymmetric interaction of the particle with nonreciprocal SPPs of the slab along $\mathbf{y}$ axis (Voigt configuration). The asymmetric dispersion shown in the inset reveals the unequal magnitude of linear momentum for forward and backward SPPs of equal frequency. The SPPs also carry transverse spin angular momentum locked to their momentum as displayed. When the particle's temperature $T_p$ is different from the temperature $T_e$ of the rest of the system, the net exchange of linear and spin angular momentum via fluctuations-generated photons is asymmetric with respect to forward and backward SPPs causing a lateral force transverse to applied field ($F_y$) and a lateral torque parallel to applied field ($M_x$).

Figure 2

(a) The frequency-dependent local permittivity of InSb slab in magnetic field $B=1\mathrm{T}$ reveals the approximate locations of surface polaritons. (b) Dispersion $\omega \left(k_{\parallel }\right)$ for SPPs propagating at an in-plane angle $\varphi$ with $B$ field shows redshift for $\varphi \in [0,\pi )$ ($k_y>0$) and blueshift for $\varphi \in [\pi ,2\pi )$ ($k_y<0$). Dispersion is plotted using both a nonlocal model (solid lines) and a local model (dashed lines). (c) The force spectral density calculated using a nonlocal model for a particle at a distance $d=0.3\mu \mathrm{m}$ from the slab and under operating temperatures $T_p=305\mathrm{K}$, $T_e=300\mathrm{K}$ is shown by black line. Its characteristic shape follows from the collective contributions of red-shifted ($k_y>0$) forward polaritons and blue-shifted ($k_y<0$) backward polaritons, which are plotted separately as dashed lines. (d) and (e) respectively demonstrate the power and torque spectral densities (black lines). The separately plotted collective contributions of red-shifted and blue-shifted polaritons (dashed lines) clarify the asymmetric flow of energy and angular momenta respectively. (f) shows the force spectral density obtained using local and nonlocal models.

Figure 3

(a) We consider nanoparticles of radius $R=200\mathrm{nm}$ and made of different materials such that they exhibit dipolar resonances over the frequency range of interest. [(b),(c)] demonstrate the dependence of force and torque on surface-to-surface distance $d_s=d-R$ for AgBr (red) and NaCl (blue) nanoparticles for two different operating temperatures $T_p=310\mathrm{K}$ (solid lines) and $T_p=500\mathrm{K}$ (dashed lines) assuming $T_e=300\mathrm{K}$. (d) We consider a potential experiment where the same AgBr and NaCl particles are trapped near InSb slab in $B=1T\mathbf{x}$ inside a vacuum chamber of pressure ${10}^{-5}\mathrm{torr}$. The particles reach a steady state angular velocity at which the lateral torque is balanced by the rotational drag of imperfect vacuum. The figure plots the distance dependence of steady-state rotational velocities for operating temperatures $T_p=310\mathrm{K}$ and $T_e=300\mathrm{K}$.

Figure 4

Figure shows the angular velocity dependence of the lateral nonequilibrium torque on AgBr (red) and NaCl (blue) nanoparticle of $R=200\mathrm{nm}$ at a distance $d_s=100\mathrm{nm}$ from the surface of InSb half-space in $B=1T$ magnetic field along $\mathbf{x}$. The operating temperatures are $T_p=400\mathrm{K}$ and $T_e=300\mathrm{K}$. The magnitude of the torque is affected appreciably at large angular velocities of $\mathrm{\Omega }\sim \mathcal{O}\left({10}^{11}\right)\mathrm{rad}$/s because the rotational Doppler effect on thermal emission dominant at THz frequencies ($\omega \gtrsim {10}^{12}\mathrm{rad}$/s) is non-negligible.