Superiority of heterodyning over homodyning: An assessment with quadrature moments

We examine the moment-reconstruction performance of both the homodyne and heterodyne (double-homodyne) measurement schemes for arbitrary quantum states and introduce moment estimators that optimize the respective schemes for any given data. In the large-data limit, these estimators are as efficient as the maximum-likelihood estimators. We then illustrate the superiority of the heterodyne measurement for the reconstruction of the first and second moments by analyzing Gaussian states and many other significant nonclassical states. Finally, we present an extension of our theories to two-mode sources, which can be straightforwardly generalized to all other multimode sources.

Figure 1

Schema for the (a) homodyne and (b) heterodyne setups. Here, BS denotes the beam splitter and LO denotes the local oscillator.

Figure 2

Plots of ${\gamma }_2$ surfaces for $\varphi =0$ and different displacement magnitudes along the $x$ axis in phase space. The center plot refers to the critical displacement magnitude of $\sqrt{5/32}\approx 0.395$, beyond which ${\gamma }_2<1$ for all $\mu$ and $\lambda$. The surface tip at $\mu =\lambda =1$ for the coherent states is invariant under a displacement rotation. It is clear from these plots that increasing the temperature reduces the value of ${\gamma }_2$, while increasing the squeezing strength counters this reduction.

Figure 3

Plots of ${\gamma }_2$ surfaces against the displacement $r_0$ for $\varphi =0$ and different values of $\mu =\lambda$. In the limit $\mu \to \infty , {\gamma }_2\le 1$ approaches unity at $p_0=0$. The signature peak of ${\gamma }_2=\frac{6}{5}=1.2$ at $x_0=p_0=0$ for $\mu =1$ is consistent with the finding in Refs. [44, 45] for central Gaussian states.

Figure 4

Plot of ${\gamma }_2$ (solid blue squares) against $n$ for Fock states. As $n$ increases, ${\gamma }_2$ decreases monotonically and eventually saturates at a subunit constant of $\frac{2}{5}$ (dashed red line).

Figure 5

Plots of ${\gamma }_2$ for the even (solid blue curve) and odd (dashed red curve) coherent states against the parameter ${\alpha }_0$ that characterizes the even and odd coherent states. For the even coherent states, the unit-${\gamma }_2$ crossover occurs at ${\alpha }_0\approx 0.693$, whereas for the odd coherent states, this happens at ${\alpha }_0\approx 1.128$. Furthermore, for each type of states, ${\gamma }_2$ possesses a stationary global minimum. For the even states, the minimum value of ${\gamma }_{2,\text{min}}=0.77096$ is attained at ${\alpha }_0=1.148\approx 1$. For the odd states, this optimum value is ${\gamma }_{2,\text{min}}=0.86796$ and is achieved with ${\alpha }_0=1.980\approx 2$.

Figure 6

Plots of (a) the optimum (minimum) ${\gamma }_2$ over all ${\alpha }_0$ with $m$ (left) and the minimum point ${\alpha }_0={\tilde{\alpha }}_0$ (right) for the displaced Fock states, as well as those of (b) the photon-added coherent states. For the displaced Fock states in (a), ${\gamma }_{2,\text{min}}$ tends to the limiting value of $6/5-\sqrt{69}/10\approx 0.3693$ (dashed red line), and the brown curve representing the exact expression for ${\tilde{\alpha }}_0$ shows the quadratic behavior for small $m$ and the approximate square-root behavior for large $m$. On the other hand, for the photon-added states in (b), the numerically found ${\gamma }_{2,\text{min}}$ values (solid blue circles) are plotted with the theoretical asymptotic power-law curve (solid dark green curve) to illustrate the accuracy of the latter for $m\gtrsim 10$, both of which approach the limiting value of $\frac{2}{5}$ (dashed red line). The ${\gamma }_{2,\text{min}}$ value for $m=0$ (not plotted) has the analytical value of $3(6-\sqrt{21})/5\approx 0.85$ that occurs at ${\tilde{\alpha }}_0=\sqrt{13+3\sqrt{21}}/4\approx 1.29$. The approximate model [see Eq. (5.30)] for ${\tilde{\alpha }}_0$ (green line) is compared with the numerical minima (solid red circles) as a showcase of its remarkable fit.