Adaptive phase measurements for narrowband squeezed beams

We have previously [Phys. Rev. A 65, 043803 (2002)] analyzed adaptive measurements for estimating the continuously varying phase of a coherent beam, and a broadband squeezed beam. A real squeezed beam must have finite photon flux $\mathcal{N}$ and hence can be significantly squeezed only over a limited frequency range. In this paper we analyze adaptive phase measurements of this type for a realistic model of a squeezed beam. We show that, provided it is possible to suitably choose the parameters of the beam, a mean-square phase uncertainty scaling as ${(\mathcal{N}∕\kappa )}^{-5∕8}$ is possible, where $\kappa$ is the linewidth of the beam resulting from the fluctuating phase. This is an improvement over the ${(\mathcal{N}∕\kappa )}^{-1∕2}$ scaling found previously for coherent beams. In the experimentally realistic case where there is a limit on the maximum squeezing possible, the variance will be reduced below that for coherent beams, though the scaling is unchanged.

Figure 1

The model of the experiment. The phase shift $\theta$ is imposed on the continuous beam, and a phase shift $\Phi$ is imposed on the local oscillator. These beams are incident on a 50:50 beam splitter, and the difference photocurrent $I\left(t\right)$ is determined. The processor then adjusts $\Phi$ based on $I\left(t\right)$. The simplest case is when the signal beam is in a coherent state, but for more accurate phase estimation a squeezed beam, produced by the apparatus to the left of the dashed vertical line, is used. This consists of a parametric down-converter characterized by intensity damping rate $\gamma$ and ${\chi }^{\left(2\right)}$ nonlinearity parameterized by $r$.

Figure 2

The maximum of $g(r,\Phi -\Theta )$ divided by $e^{3r}$ as a function of $r$.

Figure 3

The optimal values of various quantities as well as the variance for adaptive measurements with arbitrary squeezing. The values of ${\sigma }^2∕{(\kappa ∕\mathcal{N})}^{5∕8}$ are shown as the solid line, $e^r∕{(\mathcal{N}∕\kappa )}^{1∕8}$ is shown as the dotted line, $(\gamma ∕\kappa )∕{(\mathcal{N}∕\kappa )}^{3∕4}$ is shown as the dashed line, $(\chi ∕\kappa )∕{(\mathcal{N}∕\kappa )}^{5∕8}$ is shown as the dash-dotted line and $\delta ∕{(\kappa ∕\mathcal{N})}^{1∕4}$ is shown as the crosses.

Figure 4

The ratio of the value of $\kappa ∕{\sigma }^2$ to $\chi$ (continuous line), and the ratio of $\gamma ∕e^r$ to $\chi$ (dashed line).

Figure 5

The phase variance multiplied by $\sqrt{\mathcal{N}∕\kappa }$. The results for adaptive measurements on arbitrarily squeezed states, states limited to $e^{2r}⩽2$, and coherent states are shown as the solid, dotted, and dashed lines, respectively. The results for heterodyne measurements on arbitrarily squeezed states, states limited to $e^{2r}⩽2$, and coherent states are shown as the crosses, plusses, and asterisks, respectively.

Figure 6

The range of values of $\gamma ∕\kappa$ for which it is possible to obtain phase variances within 10% of the minimum values with limited squeezing $(e^{2r}⩽2)$ and adaptive measurements. The ranges are shown by the vertical solid lines between the crosses, and the predicted upper and lower bounds $2\mathcal{N}∕\left(\kappa {\sinh }^{2}r\right)$ and $e^{2r}\sqrt{\mathcal{N}∕\kappa }$ from Eq. (2.14) are shown by the diagonal solid lines.

Figure 7

The ratio of the phase variance obtained via $\arg \left(C_t\right)$ to the phase variance obtained via the Bayesian method (solid line). The estimated ratio, based on the difference being due to the phase information from the noise and using the maximum value of $g(r,\Phi -\Theta )$, is shown as the dashed line. The mean-square difference between the two phase estimates, as a ratio to the phase variance for Bayesian estimates, is shown as the dotted line.