Ultra-precise phase estimation without mode entanglement

Abstract

We explore optical quantum engineering of phase-parameterized continuous-variable (CV) probe states to exploit nonclassical light to solve the problem of precise phase estimation. The optical interferometer consists of a single beam splitter (BS) with tunable transmittance and reflectance, and two single-mode squeezed vacuum states (SMSVs). The reference SMSV state is mixed with a weakly squeezed state carrying an unknown phase at the beam splitter to form an output hybrid entangled state. Then, in the measurement mode, the number of photons is measured to generate the target CV state parameterized by the unknown phase. Using the CV states, we propose a sub-Heisenberg metrology protocol in which the quantum Cramer-Rao (QCR) boundary is saturated by intensity measurement. The advantage of quantum engineering of CV probe states for ultra-precise phase estimation of unknown phase is due solely to the nonclassical photonic properties of the measurement induced CV states of definite parity and is independent of the mode entanglement.

Figures (5)

• Figure 1: Schematic representation of the protocol for quantum estimation of unknown phase shift $\varphi$ using reference and auxiliary $SMSV$ states. The auxiliary $SMSV$ state contains an unknown parameter, and its mixing with the reference $CV$ state on a beam splitter with arbitrary real transmittance $t>0$ and reflectance $r>0$ is the basis for quantum engineering of the $CV$ state of definite parity. After measuring a certain number of photons in the measurement mode using the $PNR$ detector, initial probe $SMSV$ state is parametrized by unknown parameter $\varphi$. In the remaining mode the average number of photons is measured to estimate the unknown parameter $\varphi$ with sub-Heisenberg precision.
• Figure 2: (a-c) Periodic dependences of the $QCR$ boundary $\triangle\varphi_{qcr\,k}^{(02)}$ on the value of the estimated phase $\varphi$ for different values of k with $S_2=0.3 \,\,\,dB$ for any $k=1,\,2,\,3,\,4$ and for such $S$ and B (the $S$ and $B$ are different for different k), which provide the minimum value of the utmost bound at the corresponding points $\varphi_1^{(even)}=\pi,\,\varphi_3^{(even)}=3\pi,\,\varphi_5^{(even)}=5\pi$ and $\varphi_2^{(odd)}=2\pi,\varphi_4^{(odd)}=4\pi$. The graphs in (a) represent the general form of the dependencies, and in (b) the same curves, but on a reduced scale along the vertical axis. In (b) the curves are grouped by parity, so when scaled down along the vertical axis they may appear as one curve rather than two. The corresponding periodic dependencies of the probabilities $P_k^{(02)}$ to generate the $CV$ states \ref{['eq:3']} on the estimated parameter $\varphi$ are presented in (c). The curves $P_3^{(02)}$ and $P_4^{(02)}$ are almost equal to each other and therefore appear as a single horizontal line at the chosen scale along the vertical axis.
• Figure 3: (a-c). Optimized periodic dependencies of the $QCR$ boundary $\triangle\varphi_{qcr\,k}^{(02)}$ on $\varphi$ for $S_2=0.3\,\,\,dB$ on (a) large and (b) small scale along the vertical axis. Optimization is performed based on parameters $S$ and $B$. A periodic dependence is observed with minimum values at the same points as in \ref{['Figure.2']}. Optimization allows to reduce the amplitude of $QCR$ boundary oscillations. Periodic dependencies of the probabilities $P_k^{(02)}$ on $\varphi$, optimized for $S$ and $B$, are shown in (c).
• Figure 4: (a-c). (a) Periodic dependences of the phase uncertainty $\triangle\varphi_{k}^{(02)}$ in equation (14), obtained by direct measurement of the number of photons, on the parameter $\varphi$ for the same values of the parameters $S$, $B$ and $S_2$ that are used in constructing \ref{['Figure.2']}. The minimum values of $\triangle\varphi_{k}^{(02)}$ on a reduced scale in (b) are at the same values of $\varphi$ as the $QCR$ boundary $\triangle\varphi_{qcr\,k}^{(02)}$, so that $\triangle\varphi_{k}^{(02)}-\triangle\varphi_{qcr\,k}^{(02)}$$\ll1$, which may indicate saturation of such a measurement. The downward-pointing peaks of $\triangle\varphi_{k}^{(02)}$ in (b) are anticorrelated with the upward-pointing peaks of the average number of photons $\langle n^{(02)}_{k}\rangle$ in $CV$ states in (c) at the same values of $\varphi_1^{(even)}=\pi$,$\varphi_3^{(even)}=3\pi$,$\varphi_5^{(even)}=5\pi$ and $\varphi_2^{(odd)}=2\pi$,$\varphi_4^{(odd)}=4\pi$. Due to the grouping of even and odd dependencies, they may appear as one curve instead of two for each $\varphi_i^{(even)}$ and $\varphi_j^{(odd)}$ in (b) and (c).
• Figure 5: (a-d). Dependence of (a) the $QCR$ boundary $\triangle\varphi_{qcr\,k}^{(02)}$ and (b) the phase uncertainty $\triangle\varphi_{k}^{(02)}$ on the average number of photons $\langle n^{(02)}_{k}\rangle$ in odd $CV$ states \ref{['eq:3']} with $k=1,\,3$ for phase shift $\varphi=2\pi-\delta$ with $\delta \rightarrow 0$. Similar dependencies of (c) the QCR boundary $\triangle\varphi_{qcr\,k}^{(02)}$ and (d) the phase uncertainty $\triangle\varphi_{k}^{(02)}$ on the average number of photons $\langle n^{(02)}_{k}\rangle$ but for even $CV$ states at $k=2,\,4$ near the phase shift $\varphi_1^{(even)}=\pi$. All four figures also show the dependence of $\langle n^{(02)}_{k}\rangle^{-1}$, which is significantly higher than the corresponding dependences $\triangle\varphi_{qcr\,k}^{(02)}$ and $\triangle\varphi_{k}^{(02)}$, indicating the possibility of achieving sub-Heisenberg precision using the probe state \ref{['eq:3']} encoding information about $\varphi$. The average number of photons $\langle n^{(02)}_{k}\rangle$ of the probe $CV$ state with encoded information $\varphi$ follows from expression \ref{['eq:7']} for (a,b) $\varphi=2\pi-\delta$ and (c,d) $\varphi=\pi-\delta$ with $\delta\rightarrow 0$.