Three-dimensional squeezing of optically levitated nanospheres

Abstract

We propose a protocol to measure impulses beyond the standard quantum limit. The protocol reduces noise in all three spatial dimensions and consists of squeezing a mechanical system's state via a series of jumps in the frequency of the harmonic potential. We quantify how decoherence in a realistic system of an optically levitated, dielectric nanoparticle limits the ultimate sensitivity. We predict that $\sim$10 dB of squeezing is achievable with current technology, enabling quantum-enhanced detection of weak impulses.

Figures (4)

• Figure 1: (a) An example of the optical potential as a function of time for the squeezing protocol. The particle is cooled close to the ground state of the trap at frequency $\omega$, and at $t=0$ the squeezing begins. After two cycles of squeezing (i.e. two time segments in the smaller potential $\omega'$), the trap is switched off at $t=t_\mathrm{off}$. The particle evolves freely until $t=t_\mathrm{on}$, at which point the a phase rotation begins. At $t=t_\mathrm{end}$, the particle's position is measured. (b) Projection of the particle's Wigner function at different times when an impulse acts on the particle at $t = t_\mathrm{kick}$. We sketch the Wigner functions just before the kick (at $t = t^-_\mathrm{kick}$) as well as just after (at $t = t^+_\mathrm{kick}$).
• Figure 2: The squeezing of the momentum variance, measured in decibels, as a function of the decoherence rate. We plot the amount of squeezing for a different number of squeezing cycles $n_\mathrm{cycles}$ for measurement efficiency $\eta = 0.2$ and $\omega'/\omega=0.5$.
• Figure 3: (a) The squeezing of the momentum variance as a function of the frequency ratio $\omega'/\omega$ for a varying number of squeezing cycles $n_\mathrm{cycles}$. We distinguish between longitudinal modes (parallel to the laser direction) and those that are perpendicular due to their different decoherence rates. We note that the lines for the longitudinal mode lines lie largely on top of one another since the squeezing quickly reaches its asymptotic value. (b) The width of the state in momentum space in absolute units, as a function of the frequency ratio.
• Figure 4: The ratio of the transverse to longitudinal frequency for a circularly polarized laser. The gray and green dashed lines correspond to the two different harmonic matching conditions of \ref{['eqn:harmonicCondition']} with $k_z = 0$.