A measurement-based protocol for the generation of delocalised quantum states of a mechanical system

TL;DR

The paper presents a measurement-based protocol to herald delocalized, nonclassical states of a macroscopic mechanical oscillator via Geiger-mode detection of light from a cavity optomechanical system. It analyzes two implementations—a blue-detuned pulsed drive and a continuous-wave scheme with temporal-mode filtering—linking the generation of optomechanical entanglement through two-mode squeezing to conditional mechanical states with negative Wigner functions. The results show that short, low-photon-number pulses yield strong nonclassicality and higher heralding rates, while steady-state operation offers enhanced robustness to higher temperatures, albeit with lower detection probabilities. Together, the work provides a practical, platform-independent route to macroscopic quantum state engineering using realistic detection efficiency and temperature considerations, enabling exploration of macroscopic quantum phenomena and potential table-top tests of foundational physics.

Abstract

Non-Gaussian mechanical states are a key resource for quantum-enhanced sensing and tests of macroscopic quantum physics. We propose a measurement-based protocol to herald delocalized, nonclassical states of a mechanical oscillator in cavity optomechanics by conditioning on Geiger photodetection of the optical output. We analyse under which conditions Stokes-induced optomechanical entanglement give rise to mechanical Wigner Function negativity upon detection. We develop and compare a blue-detuned pulsed scheme and a continuous-wave steady-state scheme employing temporal-mode filtering, and we quantify heralding rates and robustness to finite temperature under realistic detection efficiencies.

Figures (6)

• Figure 1: Logarithmic negativity of output states. (a) In the pulsed scenario, we consider a finesse of $\mathcal{F}=\pi c/L\kappa=5~10^4$ and fixed pulse power$P_\text{p}=3$ mW, ensuring $\kappa/\Omega\simeq 0.3$, $g/\kappa\simeq0.3$. For $T\sim10^{-1}$ K, neglecting thermal decoherence is well justified. The longest duration here considered $\Omega\tau=80$ fixes $N_\text{ph}=1.5~10^{10}$ and one can reach squeezing rates $r\sim1.5$. NotePulsedEntanglement. Entanglement shows slower onset due to higher initial temperatures, reaching values greater then the continuous drive for longer durations. In (b), entanglement of the filtered steady state shows to peak around the Stokes sideband $\nu=-\Omega$. When the mode duration is too short the detected spectral bandwidth consists of a broad range of frequencies, diminishing Stokes contributions. Here the driving is at $\Delta=\Omega$ with $P=30$ mW , with $\mathcal{F}=3.5~10^4$ , so that $\kappa/\Omega\sim0.4$ and $g/\Omega\sim0.4$. The thermal bath is at $T=0.4$ K. The realistic detection efficiency is taken for both plots at $\eta=0.6$, while short detection windows makes dark counts negligible.
• Figure 2: Detection probability and WF negativity for different initial mechanical temperatures. a) Longer pulses generate brighter signals and higher heralding probabilities. Solid line is \ref{['eq:PClickPulse']}, while markers are calculated using the approach developed in \ref{['sec:RobustnessTemp']}, validating the equivalence of the two approaches in the case where thermal decoherence is negligible. b) Higher WF negativity is generated with closer ground-state cooling and shorter pulses NotePulsedEntanglement. Relevant parameters are the same as in \ref{['FIG:Entanglement']}.
• Figure 3: (a): WF of the conditional mechanical state for pulses duration of $\Omega\tau_1=10$ and $\Omega\tau_2=60$, fixed input power at $P=3$ mW and $n_0=0.1$. Both WF shows a negative region around the origin. The heralding probability is $\sim10\%$ and $52\%$ respectively. In the first case, two-modes squeezing interaction excites only the first Fock states, whose WF shows to be more nonclassical than the one resulting from longer pulses, where higher Fock states are excited. Fig (b) illustrate this, showing relative probabilities of detection of each Fock state (blue for $\tau_1$ and orange for $\tau_2$).
• Figure 4: (a)-(b): Detection probability and WF negativity against filtering frequencies, and for different mode durations. The parameters are the same as in Fig. \ref{['FIG:Entanglement']}. Detection on the Anti-Stokes sideband generates no negativity upon measurement due to the absence of entanglement. Maximal negativity is obtained when filtering around the Stokes sideband instead. Mode with shorter duration contains less photons, generating more non-classical states but with smaller detection probability.
• Figure 5: WFs obtained filtering on the Stokes sideband $\nu=-1$, for different modes duration $\Omega\tau=5,10$. The heralding probaiblities are $2.2\%$ and $3.7\%$ respectively. Detection of longer pulses heralds less negative WF due to the higher number of photons. Differently from the pulsed scenario, the length of the detection window is limited sideband resolution, limiting heralding probability. Parameters are the same as in \ref{['FIG:Entanglement']}.