Testing Nonclassicality and Non-Gaussianity in Phase Space

We theoretically propose and experimentally demonstrate a nonclassicality test of a single-mode field in phase space, which has an analogy with the nonlocality test proposed by Banaszek and Wódkiewicz [Phys. Rev. Lett. 82, 2009 (1999)]. Our approach to deriving the classical bound draws on the fact that the Wigner function of a coherent state is a product of two independent distributions as if the orthogonal quadratures (position and momentum) in phase space behave as local realistic variables. Our method detects every pure nonclassical Gaussian state, which can also be extended to mixed states. Furthermore, it sets a bound for all Gaussian states and their mixtures, thereby providing a criterion to detect a genuine quantum non-Gaussian state. Remarkably, our phase-space approach with invariance under Gaussian unitary operations leads to an optimized test for a given non-Gaussian state. We experimentally show how this enhanced method can manifest quantum non-Gaussianity of a state by simply choosing phase-space points appropriately, which is essentially equivalent to implementing a squeezing operation on a given state.

Figure 1

(a) Four phase-space points chosen at the vertices of a rectangle for a nonclassicality test (blue solid) and of a parallelogram for a non-Gaussianity test (red solid). (b) An optimal choice of the rectangle for testing a squeezed state occurs at $\theta -\varphi =(\pi /4)$ ($\varphi$: squeezing axis).

Figure 2

(a) Optimal $\mathcal{J}$ for a pure Gaussian state ${\sigma }_G$ against squeezing $\xi =2\tanh 2r$ up to ${\mathcal{B}}_G=(8/3^{9/8})\approx 2.32$ (gray dashed line). (b) Detection of nonclassicality for Gaussian states against purity $\mu$ and squeezing parameter $\xi$. Our test adopting four (three) phase-space points detects nonclassicality with purity $\mu >(3^{9/8}/4)\approx 0.86$ ($\mu >0.5$) represented by a purple (red) shaded region. (c) Maximal $\mathcal{J}$ for $f|0⟩⟨0|+(1-f)|2⟩⟨2|$ and (d) $e^{-{\gamma }^2}(\sinh {\gamma }^2|0⟩⟨0|+\cosh {\gamma }^2|{\psi }_{\gamma }⟩⟨{\psi }_{\gamma }|)$, using a rectangle (blue solid line) and a parallelogram (red dashed line) test by optimizing phase-space points for each state. $\mathcal{J}>8/3^{9/8}\approx 2.32$ and $\mathcal{J}>2$ represent quantum non-Gaussianity and nonclassicality, respectively. The Wigner function becomes positive for $\frac{1}{2}<f\le 1$ and $0<\gamma$ in (c) and (d), respectively.

Figure 3

(a) $\mathcal{J}$ for a Fock state $|2⟩$ by using a parallelogram test [Fig. 1] that essentially implements squeezing ($r$: strength) on a given state. Theoretical predictions (black solid line) are in good agreement with the experimental data (filled circles). Genuine non-Gaussianity is manifested above ${\mathcal{B}}_G=8/3^{9/8}$ (blue dashed line). (b) $\mathcal{J}$ for $f|0⟩⟨0|+(1-f)|2⟩⟨2|$ measured in experiment using the parallelogram test with the fixed set of four phase-space points, ($-0.110$, $-0.110$), (0.121, 0.100), (0.100, 0.121), and (0.331, 0.331). Red circles (i), (ii), and (iii) are results for states $|2⟩⟨2|$, $(|0⟩⟨0|+|2⟩⟨2|)/2$, and $|0⟩⟨0|$, respectively, with their experimentally constructed Wigner functions shown below.