Large Quantum Delocalization of a Levitated Nanoparticle Using Optimal Control: Applications for Force Sensing and Entangling via Weak Forces

We propose to optimally control the harmonic potential of a levitated nanoparticle to quantum delocalize its center-of-mass motional state to a length scale orders of magnitude larger than the quantum zero-point motion. Using a bang-bang control of the harmonic potential, including the possibility of inverting it, the initial ground-state-cooled levitated nanoparticle coherently expands to large scales and then contracts to the initial state in a time-optimal way. We show that this fast loop protocol can be used to enhance force sensing as well as to dramatically boost the entangling rate of two weakly interacting nanoparticles. We parameterize the performance of the protocol, and therefore the macroscopic quantum regime that could be explored, as a function of displacement and frequency noise in the nanoparticle’s center-of-mass motion. This noise analysis accounts for the sources of decoherence relevant to current experiments.

Figure 1

Applications of the proposed loop protocol. (a) coherent expansion to large scales, (b) static force sensing, and (c) entangling two particles via weak forces. (d) Dimensionless variance $v_x\equiv ⟨{\widehat{X}}^2⟩/x_0^2$ (solid black line, left axis in logarithmic scale) and ${\omega }^2/{\omega }_0^2$ of the harmonic potential (solid blue line, right axis in linear scale) as a function of time in the optimal bang-bang loop.

Figure 2

Coherent expansion. The purity at the end of the protocol as a function of the total protocol time $T$ and the decoherence rate (a) ${\mathrm{\Gamma }}_1$ [(b) ${\mathrm{\Gamma }}_2$]. The lower panel in (a) relates the decoherence rate ${\mathrm{\Gamma }}_1$ to the displacement noise spectrum $S_1$ for ${\omega }_0=2\pi \times {10}^5\mathrm{Hz}$ and several typical particle masses $m$. The lower panel in (b) relates the decoherence rate ${\mathrm{\Gamma }}_2$ to the frequency noise spectrum $S_2$ for two different frequencies (there is no mass dependence for frequency noise). As the initial state, we consider the ground state $\overline{n}=0$ with $\mathcal{P}=1$. We considered a silica nanoparticle of $R=100\mathrm{nm}$ and mass density $\varrho =2201\mathrm{kg}/{\mathrm{m}}^{-3}$.

Figure 3

Force sensing. Minimal detectable force as a function of the total time $T$. Results are shown in the complete absence of decoherence (black lines) and in color for various values of (a) ${\mathrm{\Gamma }}_1$ with ${\mathrm{\Gamma }}_2=0$ [(b) ${\mathrm{\Gamma }}_2$ with ${\mathrm{\Gamma }}_1=0$]. Solid lines show the performance of the loop protocol. Dashed (dash-dotted) lines show the performance of a particle that is evolving freely (in an inverted potential) for a time $T$. The right (top) axis shows the values in Newton (microseconds) for a silica nanoparticle of $R=100\mathrm{nm}$ and ${\omega }_0=2\pi \times {10}^5\mathrm{Hz}$.

Figure 4

Entanglement via weak forces. (a) The purity at the end of the protocol as a function of the protocol time $T$ and coupling strength $g$. In the absence of decoherence, the decrease of purity signals the creation of entanglement (starting from the ground state $\overline{n}=0$ with initial $\mathcal{P}=1$). (b) Coupling $g$ as a function of the particle separation $d/R$ for Coulomb, Casimir, and gravitational interaction for a silica nanoparticle of $R=100\mathrm{nm}$ and ${\tilde{\omega }}_0=2\pi \times {10}^5\mathrm{Hz}$. In the case of Coulomb interaction, each particle is assumed to carry a single electron charge. See [28] for further details.