Sub-Planck structure quantification in non-Gaussian probability densities

Abstract

Sub-Planck structures in non-Gaussian probability densities of phase space variables are pervasive in bosonic quantum systems. They are almost universally present if the bosonic system evolves via nonlinear dynamics or nonlinear measurements. So far, identification and comparison of such structures remains qualitative. Here we provide a universally applicable and experimentally friendly method to identify, quantify and compare sub-Planck structures from directly measurable or estimated probability densities of single phase space variables. We demonstrate the efficacy of this method on experimental high order Fock states of a single-atom mechanical oscillator, showing provably finer sub-Planck structures as the Fock occupation increases despite the accompanying uncertainty increase in the phonon, position, and momentum bases.

Figures (11)

• Figure 1: Schematic of the computational procedure quantifying the nonclassical sub-Planck structure around the global maximum of the continuous variable distribution, whose distillable squeezing asymptotically approaches that given by Eq. (\ref{['Asym']}). The quantification of sub-Planck structure requires the estimation of the probability distribution $P(x)$ to be distilled, followed by an operationally defined processing of the data. This processing produces an output distribution $Q_N(x)$ with a global maximum displaced to the origin followed by optimised filtration $\mathcal{G}\left(Q_N(x)\right)$. This is illustrated for the discrete variable experimental Fock state $\ket{10}$. The black dashed lines are the ground state and the insets show a direct comparison of the global maximum with the ground state variance. The sub-Planck structure embedded in the relative concavity of $P(x)$ around a global maximum is preserved by the distillation while being promoted from a local feature to a global feature: the variance of the distilled distribution $\mathcal{G}\left(Q_N(x)\right)$. The dashed yellow line shows the Gaussian squeezed state whose variance is calculated from Eq. (\ref{['Asym']}).
• Figure 2: Left: Sub-Planck structure quantification by universal distillation of squeezing from experimental Fock states (empty markers), compared with the pure states (solid markers), as a function of the occupation $n$, where $n\in\{0,2,4,6,8,10\}$. The heavy dashed line indicates the ground state noise, while the upper and lower dashed lines represent 3 dB and 6 dB of squeezing respectively. The numbers above points are the probabilities associated with the experimental Fock state $\ket{n}$. Larger $n$ results in consistently larger distillable squeezing, confirming that larger Fock states possess greater nonclassical sub-Planck structure. The colours indicate the number of copies $M=2^N$ used in the distillation process. Increasing the copies consumed consistently results in an increase in distillable squeezing. Black squares indicate the asymptotic limit of distillable squeezing [Eq. (\ref{['Asym']})] for the experimental (empty squares) and pure (solid squares) Fock states. Right: The sub-Planck structure depth for the experimental Fock states under thermalisation, with $M=16$, and compared with the thermalisation depth of Wigner negativity, the thermalisation depth for $n$-photon QNG criteria calculated from the Fock state distribution, and the thermalisation depth for sub-Poissonian statistics (Fano factor). The coloured numbers connected to the points are the probabilities for the respective Fock state after thermalisation. For the pure states the sub-Planck structure depth is consistently $\mathop{\mathrm{\bar{{\it n}}}}\nolimits\approx0.28$, compared with the Wigner negativity which vanishes at $\mathop{\mathrm{\bar{{\it n}}}}\nolimits=0.5$ for all $n$, and the $n$-photon QNG depth which strictly decreases with $n$ (see SM). For the experimental states, composed of a finite mixture of Fock states, the sub-Planck structure depth advantageously remains constant and similar to that of the pure states.
• Figure 3: Left: The sub-Planck structure depth for the asymptotic distillable squeezing calculated from the simulated thermalisation of the pure Fock states. The depth converges around $\mathop{\mathrm{\bar{{\it n}}}}\nolimits\approx0.28$. Right: The sub-Planck structure depth compared with the QNG hierarchy depth and Wigner negativity depth for the pure Fock states.
• Figure 4: Statistics for the experimental Fock states. Left: The variance increases for higher order Fock states, however the mean values (callout numbers) stay close to $n$. Center: The Fano factor $F=\frac{\text{Var}(n)}{\braket{n}}$, which is a detector of nonclassicality. $F<1$ for all experimental Fock states except the ground state. Right: The signal-to-noise ratio ($\text{SNR}=\frac{\braket{n}}{\sqrt{\text{Var}(n)}}$) of the experimental Fock states. The rising variance (left) does not outcompete the increase in mean value so that the SNR stays above 1, although it falls for higher $n$.
• Figure 5: The effect of phonon noise on the distillable squeezing for the experimental Fock states (empty markers) under thermalisation, with $M=16$ as compared with the theoretical results for the corresponding pure states (solid markers).