Evading Vacuum Noise: Wigner Projections or Husimi Samples?

The accuracy in determining the quantum state of a system depends on the type of measurement performed. Homodyne and heterodyne detection are the two main schemes in continuous-variable quantum information. The former leads to a direct reconstruction of the Wigner function of the state, whereas the latter samples its Husimi $Q$ function. We experimentally demonstrate that heterodyne detection outperforms homodyne detection for almost all Gaussian states, the details of which depend on the squeezing strength and thermal noise.

Figure 1

Sketch of the experimental setup. Pulses from a shot-noise-limited laser centered at 1560 nm are distributed equally on the principal axes of a 13 m long polarization maintaining fiber. The pulses are individually squeezed due to the Kerr nonlinearity. A birefringent compensator controls the temporal overlap of the emerging pulses and locks them to a relative phase difference of $\pi /2$, hence forming an $S_3$ polarized state. The (a) homodyne and (b) heterodyne schemes are emulated by Stokes measurements.

Figure 2

(a) Illustration of the squeezed $S_3$-polarized state on the Poincaré sphere. (b) Magnification of the polarization state at the pole of the Poincaré sphere. For bright states, the Poincaré sphere has a large radius such that the curvature is locally negligible and the projection in the $S_1\text{−}{S}_2$ dark plane is equivalent to a rescaled canonical $X\text{−}P$ quadrature phase space.

Figure 3

(a) Contours of the reconstructed covariance matrices (shown here for homodyne detection). (b) Color-coded contour plot of the relative accuracy $\gamma$ in terms of the ellipticity $\lambda$ and thermal noise $\mu$. The experimentally tested states are indicated by orange disks.

Figure 4

(a) Experimental results for the accuracy of the covariance matrix reconstruction with heterodyne detection. The Hilbert-Schmidt distance is shown for three different numbers of sampling events. (b) Experimental results for the accuracy of the covariance matrix reconstruction based on homodyne detection. (c) Relative accuracy $\gamma$. The colored lines in the background indicate the theoretical prediction.