Fluctuation-induced Hall-like lateral forces in a chiral-gain environment

TL;DR

The paper shows that vacuum fluctuations can produce a Hall-like lateral force on a small two-level particle placed near a chiral-gain substrate, where a Berry curvature dipole induces polarization-dependent gain and nonreciprocal field correlations. By formulating field quantization in a gain medium and deriving a Lindblad-type reduced dynamics, the authors connect gain-induced asymmetries in surface-plasmon propagation to directional photon emission/absorption and lateral momentum transfer. The fluctuation-induced force has no x-component and a negative y-component, is enhanced near the surface-plasmon resonance, scales as $1/z_q^4$, and is proportional to the cyclotron-type frequency $ oldsymbol{ abla}_0$ set by the Berry-curvature dipole $D$, providing a tunable mechanism for nanoscale optical manipulation. This work links quantum geometry to fluctuation electrodynamics and opens possibilities for steering nanoscale motion through engineered non-Hermitian environments.

Abstract

Here, we demonstrate that vacuum fluctuations can induce lateral forces on a small particle positioned near a translation-invariant uniform non-Hermitian substrate with chiral gain. This type of non-Hermitian response can be engineered by biasing a low-symmetry conductor with a static electric field and is rooted in the quantum geometry of the material through the Berry curvature dipole. The chiral-gain material acts as an active medium for a particular circular polarisation handedness, while serving as a passive, dissipative medium for the other polarisation handedness. Owing to the nonreciprocity and gain characteristics, momentum is continuously exchanged in a preferred direction parallel to the surface between the test particle and the surrounding electromagnetic field, giving rise to lateral forces. Interestingly, the force can be viewed as a fluctuation-induced drag analogous to a Hall force. Indeed, although the gain is driven by an electric current, the resulting force acts perpendicular to the bias -- unlike conventional current-drag effects. This effect stems from the skewed propagation characteristics of surface modes and gain-momentum locking. Our theory reveals a Hall-like asymmetry in the field correlations and establishes a novel link between quantum geometry and fluctuation-induced phenomena, offering new possibilities for nanoscale control via tailored electromagnetic environments.

Figures (6)

• Figure 1: Schematic illustration of the setup under study. A point particle is placed at a distance $z _ \mathrm{q}$ above a low-symmetry material exhibiting a chiral-gain response. The electrostatic bias is applied along the $x$ direction. The particle is modelled as a two-level system.
• Figure 2: Two eigenvalues $\epsilon" _ {\ell,\pm}$ of the non-Hermitian part of the dielectric response $\tensor\epsilon" _ \ell$ in the Drude and electro-optic channels ($\ell = \mathrm{D}, \mathrm{EO}$) as a function of frequency. The dashed curve represents the response $\epsilon _ {\mathrm{D},\pm}"$ from the Drude channel, and the red and blue curves represent the one $\epsilon _ \mathrm{EO,\pm}"$ from the electro-optic channel. In the Drude channel ($\ell = \mathrm{D}$), both the right-handed polarisation (RCP) and left-handed polarisation (LCP), $\vb{u} _ +$ and $\vb{u} _ -$, experience dissipation ($\epsilon _ {\mathrm{D},\pm}" > 0$). On the other hand, in the electro-optic channel ($\ell = \mathrm{EO}$), the right-handed (left-handed) polarisation is subject to dissipation (gain) [$\epsilon _ \mathrm{EO,+}" > 0$ ($\epsilon _ \mathrm{EO,-}" < 0$)]. The following parameters were used to generate the plot: $\gamma/\omega _ \mathrm{p} = 0.5$ and $\omega _ 0/\omega _ \mathrm{p} = 0.1$.
• Figure 3: Amplitude of the emission rate spectrum $\norm{\tensor{\gamma} _ \mathrm{L}}$, characterising the directionality of the photon emission for various frequencies. The simulation parameters are: $\gamma/\omega _ \mathrm{p} = 0.5$, $\omega _ 0/\omega _ \mathrm{p} = 0.1$, and $\omega _ \mathrm{p} z/c = 0.0001$. Note that panel (a) has an individual colour bar; panels (b–d) share a colour bar located in the top right corner; and the colour bar for panels (e–h) is shown in the bottom right corner.
• Figure 4: Amplitude of the absorption rate spectrum $\norm{\tensor{\gamma} _ \mathrm{G}}$, characterising the photon absorption for various frequencies. The simulation parameters are as in Fig. \ref{['fig:E-E+']}. Note that panel (a) has an individual colour bar; panels (b–d) share a colour bar located in the top right corner; and the colour bar for panels (e–h) is shown in the bottom right corner.
• Figure 5: Fluctuation-induced lateral force $F _ y = \vb{u} _ {y} \cdot \vb{F} _ \parallel$ as a function of the transition frequency of the two-level system. The green curve represents the total lateral force. The orange (blue) curve corresponds to the contribution from the excited (ground) state, corresponding to Eqs. \ref{['eq:F _ L']} and \ref{['eq:F _ G']}, respectively. The force is exerted along the negative $y$ direction. The following parameters were used to generate the plots: $\vb{d} _ \mathrm{e}/\abs{\vb{d} _ \mathrm{e}} = \vb{u} _ {z}$, $\abs{\vb{d} _ \mathrm{e}} = 100\,\mathrm{D}$, $\omega _ 0/\omega _ \mathrm{p} = 0.1$, $\gamma/\omega _ \mathrm{p} = 0.5$, $\omega _ \mathrm{p} z _ \mathrm{q}/c = 0.0001$, and $\omega _ \mathrm{p}/(2\pi) = 1.0\,\mathrm{THz}$. The dashed line represents the surface plasmon resonance frequency in the absence of chiral gain ($\omega \approx \omega _ \mathrm{p}/\sqrt{2}$).