Bridging Particle and Wave Sensitivity in a Configurable Detector of Positive Operator-Valued Measures

We report an optical detector with tunable positive operator-valued measures. The device is based on a combination of weak-field homodyne techniques and photon-number-resolving detection. The resulting positive operator-valued measures can be continuously tuned from Fock-state projectors to a variety of phase-dependent quantum-state measurements by adjusting different system parameters such as local oscillator coupling, amplitude, and phase, allowing thus not only detection but also preparation of exotic quantum states. Experimental tomographic reconstructions of classical benchmark states are presented as a demonstration of the detector capabilities.

Figure 1

(a) Proposed scheme for the POVM configurable detector. $D_a$ and $D_b$ are PNR detectors, $\alpha$ is a weak coherent state. (b),(c) POVM elements ${\widehat{\Pi }}_{\beta \gamma }$ corresponding to click events (b) ${\beta }_1=(1,3)$ and (c) ${\beta }_2=(1,1)$ for LO settings $\gamma =(|\alpha |=1.5,\theta =0)$, $R=50\%$ and 90% efficient PNR detectors. ${\widehat{\Pi }}_{\beta \gamma }$ projects onto a single-mode (b) mesoscopic quantum superposition state and (c) squeezed state, with high probabilities.

Figure 2

Experimental set up for the photon-number-resolving homodyne detector (PNRHD). The output of a Ti:sapphire laser is split into two arms (by a PBS), which eventually correspond to signal $\widehat{\rho }$ and LO beams. A piezoelectric moves a mirror to set the phase $\theta$ of the LO. Both arms are then recombined and sent through the configurable homodyne detector.

Figure 3

Wigner function and corresponding contour plot (inset) of the experimentally reconstructed phase-averaged weak coherent states, with average photon numbers $⟨n^{\mathrm{est}}⟩$ equal to (a) $0.29\pm 0.03$, (b) $0.61\pm 0.06$, and (c) $0.71\pm 0.07$. (d) One extreme POVM element used in the error estimation ${\widehat{\Pi }}_{\beta \gamma }^{\max }$, (e) difference between two extreme POVM elements ${\widehat{\Pi }}_{\beta \gamma }^{\mathrm{diff}}={\widehat{\Pi }}_{\beta \gamma }^{\max }-{\widehat{\Pi }}_{\beta \gamma }^{\min }$.

Figure 4

Wigner function (left) and amplitude of the density matrix in the photon-number basis $|{\widehat{\rho }}_{nm}|$ (right) for experimentally reconstructed weak coherent states. Here $(x,p)$ labels the quadratures and $(n,m)$ labels the photon numbers. Insets show corresponding contour plot (left) and diagonal matrix elements (right). Rows correspond to an average photon number $⟨n^{\mathrm{est}}⟩$ of (a) $0.29\pm 0.03$, (b) $0.59\pm 0.06$, and (c) $1.53\pm 0.1$.