Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state

Phase precision in optimal two-channel quantum interferometry is studied in the limit of large photon number $N\gg 1$, for losses occurring in either one or both channels. For losses in one channel an optimal state undergoes an intriguing sequence of local bifurcations as the number of photons (or losses) increase. The optimal state has a continuous form in the Fock state basis for large $N$. The loss parameter limits any precision improvement over classical light to at most a constant factor independent of $N$. We determine a crossover value of photon number $N_c$ beyond which supraclassical precision is progressively lost.

Figure 1

Probability weights ${\rho }_{\ell }$, represented as stacked histograms (top) and positions $x_{\ell }$ (bottom), as a function of $r$. Different components are indicated using color. Black solid line on the bottom figure is the rescaled Fisher information $\tilde{\mathcal{F}}(r)$ and the blacked dashed is its asymptote $4/r$; convergence takes place for much larger values of $r$. Thresholds $r_2,r_3,r_4$ correspond to appearances of new components at the origin. Components separate from the origin at critical values $r_1^{\prime },r_2^{\prime },r_3^{\prime },r_4^{\prime }$. The data for $20⩽r⩽25$ are magnified (top figure, lower right corner) showing components with very small weight.

Figure 2

Optimal 20-photon states for 95% loss ($r=380$) in one (left) and two (right) modes. Red bars represent amplitudes ${\varphi }_n$ obtained by numerical optimization. Black lines represent analytical approximation with an Airy function and a Gaussian. These optimal states offer a precision improvement (square root of Fisher information) over coherent light of just 6% (single-mode losses) and 0.4% (symmetric losses), owing to high loss amount. (For single-mode losses, using coherent light, the precision used in calculation is for the optimal reflectivity of the input beamsplitter in Mach-Zehnder interferometer.)

Figure 3

Left: Fisher information for symmetric ($R^{(1)}=R^{(2)}=R$) and single-mode ($R^{(1)}=R$, $R^{(2)}=0$) losses for optimal $N$-photon states as a fraction of the linear upper bound [$N(1-R)/R$ and $4N(1-R)/R$, respectively]. The curves must tend to $1$ for large $N$, but the convergence is faster for losses in both arms. For symmetric losses we use exact Fisher information, not the approximate upper bound. Right: Collapse of data when replotted as a function of $r=NR/(1-R)$ [see Eq. (3)].