Quantum limit of precision for phase estimation in squeezing-enhanced interferometry with a single-mode readout

Abstract

We consider an optical interferometer with coherent light in one input and a squeezed vacuum in another. Such an interferometer is known to beat the standard quantum limit of sensitivity to the difference of phase shifts in its arms. We find the ultimate limit of precision for such an interferometer by calculating quantum Fisher information of the mixed quantum state in one of the interferometer's outputs. We show that this information is asymptotically close to quantum Fisher information for the two-mode readout. We conclude that the single-mode readout is optimal for phase estimation in squeezing-enhanced interferometry.

Figures (3)

• Figure 1: Mach-Zehnder interferometer created by beam splitters BS1 and BS2. The interferometer has a coherent (mode $a$) and a squeezed (mode $b$) fields at two its inputs. Unknown phase shifts $\theta_a$ and $\theta_b$ are introduced inside the interferometer to modes $a$ and $b$ respectively. The task is to infer the difference phase $\theta=\theta_a-\theta_b$ from the measurement data of mode $b_\text{out}$ alone.
• Figure 2: Normalized $N$-precision for measuring the difference phase $\theta$ by detecting the number of photons in mode $b$ in the strong-field regime (the mode $a$ has $|\alpha|^2=10^6$ photons at the input). The standard quantum limit is shown by the blue dashed line. The solid green line corresponds to the routinely obtained squeezing $20r\lg e=12.5$ dB and shows that the normalized precision drops to zero at the black fringe ($\theta=0$) but raises to about 15 in its vicinity. The maximal value of $e^{2r}$ corresponds to the limit $|\alpha|^2\to\infty$ (black dotted line).
• Figure 3: Normalized $N$-precision and normalized QFI $Q^{\theta\theta}_\theta/|\alpha|^2$ in the weak-field regime, where the mode $a$ has $|\alpha|^2=10^3$ photons in the input. The standard quantum limit is obtained when no squeezing is used (blue dashed line). The QFI (thin magenta line) is much higher than the $N$-precision (thick green line) in the vicinity of the black fringe and even exceeds the limiting value of the latter $e^{2r}$ (black dotted line).