Parametric resonant enhancement of motional entanglement under optimal control: an analytical study

TL;DR

This paper addresses the challenge of achieving continuous-variable entanglement between motional modes of two optically trapped nanoparticles in the presence of decoherence. It develops a semi-analytical covariance-matrix framework together with a Mathieu-equation description of the differential mode under parametric modulation $g(t)=g_{0}+2g_{1}\cos(\Omega_c t)$ and optimal feedback control. Key contributions include closed-form expressions for the conditional logarithmic negativity, a compact form for the exact-resonance case showing how parametric gain and decoherence compete to generate entanglement, and an analysis of unconditional negativity via excess-noise dynamics. The results are validated against full numerical simulations, showing good agreement for attractive coupling and outlining the limits in the repulsive regime, with implications for experimentally achieving and optimizing entanglement in levitated-mass systems and potentially other parametrically driven quantum platforms.

Abstract

We study theoretically continuous-variable entanglement between the motional degrees of freedom of optically trapped massive particles coupled via the Coulomb interaction, in the presence of a feedback control scheme. We perform a detailed analysis of the parametric resonance induced by temporal modulation of the coupling strength, based on the system's coupled nonlinear, nonhomogeneous dynamical equations. Our model accurately reproduces the numerical findings and provides closed-form expressions for the entanglement degree. We demonstrate that a stationary nonequilibrium entangled state is realized as a result of the competition between parametric gain and decoherence.

Figures (7)

• Figure 1: Schematic illustration of two interacting optically trapped particles.
• Figure 2: Attractive interaction logarithmic negativity for the modulation frequency $\Omega_{c}=2 \Omega_{-}$ as a function of the efficiency $\eta$ (a) and the modulation depth $g_{1}$ (b) averaged over one period. Solid line with markers denotes the numerical results, and the dashed line describes the value of the analytical expression of the negativity Eq. \ref{['eq: approximate logarithmic negativity']}. The black dashed line indicates the entanglement threshold. System parameters are: $\Omega_{c} = 2\Omega_{-}$, $\Omega_0/2\pi = 29.4\,\text{kHz}$, $g_{0}=0 .2\Omega_{0}$, $\Gamma_{\rm ba} = 1.3\,\text{kHz}$, $\Gamma_{\rm th} = 66.2\,\text{Hz}$, $\gamma = 0.31\,\mu\text{Hz}$, $Q = 1.08\,\mu\text{Hz}$, and $\theta_{\rm EPR} = \pi$.
• Figure 3: (a) Comparison of the ellipse-shaped, standard deviation of the differential mode's Wigner distribution between the numerical and analytical solutions. (b) Decomposition of the ellipse axes into the diverging and decaying modes. Calculated for the attractive case at the end of one modulation period for a frequency of $\Omega_{c} = 2\Omega_{-}$, Parameters match Fig. \ref{['fig: g1-eta logneg']}, except for $g_{1}/g_{0}=0.25$ and $\eta=0.5$.
• Figure 4: Time evolution of the logarithmic negativity over one driving period for attractive interaction with $g_0/\Omega_0 = 0.2$ (a), and for repulsive interaction with $g_0/\Omega_0 = -0.2$ (b). Both approximate (analytical) and numerical solutions are shown for the conditional and unconditional cases. The entanglement threshold is indicated by the black line at zero. System parameters are identical to those of Fig. \ref{['fig: g1-eta logneg']}, where we also set $g_{1}/|g_{0}| = 0.25$ and $\eta = 0.5$.
• Figure 5: Period-averaged logarithmic negativity as a function of the driving modulation frequency $\Omega_c$ and driving amplitude $g_1$. Panels (a) and (b) present the numerical and analytical conditional logarithmic negativity, respectively; panels (c) and (d) show the corresponding unconditional predictions. Black regions in (b) and (d) mark parameter ranges where the model is not applicable. The dashed yellow line signifies the theory-predicted resonance range (Eq. \ref{['eq:range']}) in all of the plots. Results are calculated for attractive interactions with $g_{0}/\Omega_{0}=0.2$. All parameters are the same as in Fig. \ref{['fig: g1-eta logneg']}, with the addition of $g_{1}/g_{0} = 0.25$ and $\eta = 0.5$.