Optically Hyperpolarized Materials for Levitated Optomechanics

TL;DR

This work introduces optically hyperpolarized, non-permanent electron spins embedded in levitated naphthalene as a platform for multi-spin matter-wave interferometry and advanced NMR, leveraging long nuclear spin lifetimes and homogeneous spin distributions. The proposed approach combines diamagnetic levitation with magic-angle spinning to suppress spin-spin decoherence, and uses a multi-spin Stern-Gerlach-like interferometer to probe quantum mechanics at mesoscopic mass scales while testing CSL-type collapse models. A modified, pulse-rich protocol amplifies sensitivity to CSL decoherence, enabling stronger bounds on CSL parameters than prior experiments, while providing measurement schemes that translate spin polarization into observable center-of-mass displacements. Beyond foundational tests, the scheme offers practical routes to unprecedented nuclear-spin coherence times and new NMR capabilities, with potential extensions to tailor materials for specific quantum-technological applications.

Abstract

We explore the potential of levitating solids embedded with non-permanent, optically controllable electron spins, which can be used to hyperpolarize their nuclear spin environment with exceptionally long lifetimes. For example, pentacene-doped naphthalene, which will also serve as our prime example, can achieve bulk polarization exceeding $80\,\%$ at cryogenic temperatures with polarization lifetimes extending over weeks. These materials make a compelling case for applications such as matter-wave interferometry and novel uses of established NMR techniques. In that spirit, we design a multi-spin Stern-Gerlach-type interferometry protocol which, thanks to the homogeneous spin distribution and the absence of a preferential nuclear-spin quantization axis in such materials, avoids many of the limitations associated with solid state crystals hosting electronic spin defects, such as nanodiamonds containing NV centers. We assess the potential of our interferometer to enhance existing bounds on the free parameters of objective collapse models. Beyond matter-wave interferometry, we analyze the prospects for implementing magic angle spinning at frequencies surpassing the current standard in NMR, capitalizing on the exceptional rotational capabilities offered by levitation. Additionally, we outline a novel protocol for measuring spin ensemble polarization via the position of the nanoparticle and conduct an analysis of dominant noise sources, benchmarking the required isolation levels for various applications.

Figures (12)

• Figure 1: (a) and (b): The structure of an individual naphthalene molecule and naphthalene in its crystalline form. (c) Structure of pentacene. The carbon atoms are colored green while the hydrogen atoms are colored gray. The hydrogen atoms carry a nuclear spin of $1/2$. Naturally, the $^{13}$C isotope of carbon, which carries a spin, constitutes only $1.1\,\%$ of all carbon atoms, whereas the $^{12}$C isotope does not possess a spin. To ensure that only the hydrogen atoms carry a spin, $^{12}$C enriched naphthalene can be used.
• Figure 2: Here, the time evolution of the $\kappa$ wave packets in a magnetic field gradient is displayed. The $\kappa^\mathrm{th}$ wave packet oscillates around the equilibrium position $x_\mathrm{eq,\kappa}$, as depicted on the left. After a full period of $2\pi/\Omega$, all trajectories meet again in the center of the trap. The equilibrium positions $x_\mathrm{eq,\kappa}=(2\kappa-N)\chi$ are spaced by $\chi$. If the initial spin state is given by \ref{['eq:InitialState']}, the wave packets corresponding to $\kappa$ are in a superposition, as depicted on the right. The Gaussians are displaced by $x_\mathrm{eq,\kappa}[1-\cos(\Omega t)]$ and enclosed in an envelope that follows a binomial distribution. The width of the envelope is given by \ref{['eq:PositionVariance']}.
• Figure 3: Protocol for rapid expansion and recombination of the wave function of the center-of-mass motion in naphthalene. A hyperpolarized particle is placed in a magnetic gradient, forming a harmonic trap. Then a $\pi/2$ pulse is applied. The spin state is now a superposition of the $N+1$ possible Dicke state. Depending on the spin state, the particle travels along another trajectory and the wave function expands for a time $t_1$. By applying two well-timed $\pi$ pulses spaced by $t_2$, all trajectories recombine after another time $t_1$. The trajectories are denoted by $\kappa\in[0,N]$. To better illustrate the intrinsic dynamics of the protocol, we consider an initial particle position at $\kappa = N/2$ rather than at the equilibrium position $\kappa = N$. In this case, the center of the wave packet remains stationary.
• Figure 4: $\expval{\prod_{n=1}^N \hat{\sigma}_x^{(n)}}$ for an arbitrary set of system parameters and protocol duration. $\expval{\prod_{n=1}^N \hat{\sigma}_x^{(n)}}$ is greater for even values of $N$ and smaller for odd ones. This discrepancy arises from the fact that for even $N$, there exists a trajectory $\tilde{\kappa}=N/2$ for which $\tilde{\kappa}=N-\tilde{\kappa}$ and thus $\Lambda_{N/2,N/2}=0$, a condition not satisfied for odd $N$. Moreover, this term carries the greatest weight in the sum of \ref{['eq:AlternativeObservable']}. For large $N$, the difference between odd and even $N$ diminishes as expected and $\expval{\prod_{n=1}^N \hat{\sigma}_x^{(n)}}$ approaches the bound given by $\exp(-\xi T_\mathrm{tot})$, which is depicted in red.
• Figure 5: Example for the protocol to test the CSL model with two spins. A $\pi/2$ is applied in the beginning, bringing all spins into a superposition of up and down. Depending on the spin state, the particle starts oscillating around $d_\kappa$, the wave function expands in space. Here, the trajectory the particle takes if all spins are pointing up, in this case $\kappa=2$, is highlighted. After half a period at $t_1=\pi/\Omega$, a second $\pi/2$ pulse is applied and each trajectory again splits up into $N+1$ trajectories. Each of the trajectories is denoted by $\alpha$. For the $\kappa=2$ branch, the different spin states of the $\alpha^\mathrm{th}$ trajectory are color-coded. The times are given by $t_1=\pi/\Omega-2t_{2,1}-t_{2,2}$, and $t_{2,2}$ as in \ref{['eq:t2']}. In the unitary case, the particle continues traveling on the $\kappa^\mathrm{th}$ trajectory and all paths meet after another time $t_1$ in the origin with zero momentum. In the presence of noise, the coherence between the $\alpha^\mathrm{th}$ trajectories breaks down. This decoherence is mapped on the spin state when the $\alpha^\mathrm{th}$ trajectories meet. Therefore, there are also small contributions to Dicke states different from $\kappa$ after the $3\pi/2$ pulse is applied. The $N+1$ trajectories on the $\kappa\mathrm{th}$ branch after the $3\pi/2$ pulse are denoted with $\beta$.