Heralded quantum non-Gaussian states in pulsed levitating optomechanics

TL;DR

This work presents a practical route to herald and verify quantum non-Gaussian states in levitated optomechanics using pulsed interactions and nonlinear photon detection. By combining cavity-based and cavityless schemes, the authors show how single-phonon addition/subtraction, and potentially multiphonon states, can be generated in a levitated nanoparticle near-ground-state cooling and read out via optical pulses. Verification relies on QNG criteria based on Fock-space statistics and robust Wigner-function characterizations that tolerate loss, recoil heating, and detector inefficiencies. The approach enables enhanced phase-sensitive force sensing and provides a framework for exploring macroscopic quantum phenomena and quantum thermodynamics with levitated systems.

Abstract

Optomechanics with levitated nanoparticles is a promising way to combine very different types of quantum non-Gaussian aspects induced by continuous dynamics in a nonlinear or time-varying potential with the ones coming from discrete quantum elements in dynamics or measurement. First, it is necessary to prepare quantum non-Gaussian states using both methods. The nonlinear and time-varying potentials have been widely analyzed for this purpose. However, feasible preparation of provably quantum non-Gaussian states in a single mechanical mode using discrete photon detection has not been proposed yet for optical levitation. We explore pulsed optomechanical interactions combined with non-linear photon detection techniques to approach mechanical Fock states and confirm their quantum non-Gaussianity. We also predict the conditions under which the optomechanical interaction can induce multiple-phonon addition processes, which are relevant for $n$-phonon quantum non-Gaussianity. The practical applicability of quantum non-Gaussian states for sensing phase-randomized displacements is shown. Besides such applications, generating quantum non-Gaussian states of levitated nanoparticles can help to study fundamental questions of quantum thermodynamics, and macroscopic quantum effects.

Figures (9)

• Figure 1: (a) Illustration of the proposed heralded generation of QNG states for an optomechanical system with a levitated NP trapped within an optical cavity by an optical tweezer. The motion of the NP is coupled to the cavity field via coherent scattering. (b) Experimental implementation of single-phonon addition or subtraction with known experimental parameters and technical imperfections. (c) The protocol and theory model for generating and verifying QNG mechanical states in a pulsed optomechanical system. The system comprises a cooled mechanical oscillator (denoted by annihilation operator $b$) coupled to an optical cavity mode (denoted by $a$). The coupling is enabled by pulsed laser light and lasts for a duration $\tau_1$ with Hamiltonian $H_1$ (beam splitter or parametric squeezing interaction). Upon a photon detection at the array of APDs 1, the mechanical state is projected onto a non-Gaussian quantum state. The generated mechanical state is then transferred to the optical mode via the beam splitter interaction $H_2$ for a duration $\tau_2$ and verified at the array of APDs 2.
• Figure 2: Single-phonon-subtracted squeezed thermal mechanical state (a) and single-phonon-added thermal mechanical state (b). The probability of finding $n$ phonons in the mechanical mode conditioned on detecting a photon at the output versus initial mechanical occupation with initial squeezing with $(r,\phi_0)=(1,\pi/2)$ and (inset) versus squeezing and fixed value of initial mechanical occupation $n_0=0.1$. Dotted, dashed, and solid lines correspond to a fixed heating rate $\gamma \bar{n}= 0$, $\gamma \bar{n}= 0.01 \kappa$ and $\gamma \bar{n} =0.06 \kappa$, respectively. The initial state of the mechanics is a squeezed thermal state with mean phonon number $n_0$ and squeezing parameters $(r,\phi_0)$ (\ref{['eq:sqz:definitions']}) in (a) and a thermal state with mean occupation $n_0$ in (b). Other parameters are $g/\kappa=0.02$ and $\kappa \tau=2$. The dashed gray line is the first-order absolute quantum non-Gaussianity threshold ${Q_1^G} \approx 0.48$ (\ref{['Eq:NG']}). States marked by the dots are further investigated in Figs. \ref{['Fig4']} and \ref{['Fig5']}.
• Figure 3: Quantum non-Gaussianity after heralding without a cavity. The probability of finding $n$ phonons in the mechanical mode conditioning on detecting photon at the output (a) versus initial mechanical occupation with $\omega \tau=1$ and $\Gamma/\omega_m=0.0082$militaru_ponderomotive_2022. (b) Phonon probabilities as a function of the pulse duration assuming $n_0=0.1$. Dashed, and solid lines correspond to a fixed heating rate $\gamma \bar{n} = 10^{-1} H_0$ and $\gamma \bar{n} = H_0$ respectively, where $H_0 = 0.054 \omega_m$. The dashed gray line is the first-order absolute quantum non-Gaussianity threshold ${Q_1^G} \approx 0.48$ (\ref{['Eq:NG']}). States marked by the dots are further investigated in Figs. \ref{['Fig4']} and \ref{['Fig5']}.
• Figure 4: Multiple phonon addition. (a) Illustration of the proposed method for generating higher-order QNG mechanical states. The probability of finding $n$ phonons in the mechanical mode conditioning on detecting photons at the output versus optomechanical thermal occupation for pulses with blue detuning (b) two pulses $j=2$ and (c) three pulses $j=3$. Here $r=0$, $g/\kappa=0.02$ and $\kappa \tau=2$. The mechanical mode is initially in equilibrium with the environment $\bar{n}$. Dotted, dashed and solid lines correspond to a fixed heating rate $\gamma \bar{n}=0$, $\gamma \bar{n}= 0.01 \kappa$ and $\gamma \bar{n} =0.06 \kappa$. The dotted black line is the $j$th-order non-Gaussianity threshold.
• Figure 5: Non-Gaussian depth as a function of initial thermal occupation of the mechanical oscillator $n_0$. (a) Phonon-added (blue lines) and subtracted (red lines) and (b) cavityless system. Each curve corresponds to a different mechanical state shown with markers in Fig. \ref{['Fig2']} and Fig. \ref{['Fig3']}. The upper bound corresponds to a perfect single phonon state shown by the dashed gray line.