Beating the standard quantum limit with single-photon-added coherent states

Abstract

The standard quantum limit (SQL), also known as the shot-noise limit, defines how quantum fluctuations of light constrain measurement precision. In a benchmark experiment using the Mach-Zehnder interferometer (MZI), where a coherent state with the average photon number $\langle n\rangle$ is combined with an ordinary vacuum input, the SQL for the phase uncertainty is given by the well-known relation $Δ\varphi_{\text{SQL}} = 1/\langle n\rangle$. Using a single photon-added coherent state and a weak coherent state as inputs, we report an enhanced phase sensitivity in MZI surpassing the SQL. In stark contrast to other approaches, we focus on the low-photon-number regime, $\langle n\rangle < 10$, and demonstrate that our scheme offers better phase sensitivity compared to the SQL. Beating the SQL at low photon numbers paves the way for the new generation of devices employed in \textquotedblleft photon-starved\textquotedblright quantum sensing, spectroscopy, and metrology.

Figures (4)

• Figure 1: (a) Schematic diagram of an MZI with input modes ($a,b$) and output modes ($e,f$). A phase shift ($\varphi$) is introduced in the upper arm relative to the lower arm and an intensity difference detection scheme ($\langle{n}_{e}\rangle-\langle{n}_{f}\rangle$) at the output was employed to extract the phase shift $\varphi$. (b) Plot of the normalized phase uncertainty $\mathcal{S}_{\text{SQL}}(\varphi)=\Delta\varphi/\Delta\varphi_{\text{SQL}}$ as a function of phase shift ($\varphi$) for $\alpha_{b}$= 0.1 (black line) and $\alpha_{b}$= 1.5 (red line). We see that for both cases $\mathcal{S}_{\text{SQL}}(\varphi) > 1$ i.e. SPACS cannot beat the SQL and $\mathcal{S}_{\text{SQL}}(\varphi) \rightarrow 1$ for $\varphi\rightarrow \pi/2$. (c) Plot of $\mathcal{S}_{\text{SQL}}(\varphi)$ as a function of phase shift ($\varphi$) for $\alpha_{b}$= 1.5 and $m = 1, 2, 3, 4, 5$.
• Figure 2: Density plot of $\mathcal{S}_{\text{SQL}}(\varphi)$ as a function of (a) $|\alpha_{a}|^{2}$ and $|\alpha_{b}|^{2}$ for $\varphi = 0$ and (b) phase shift ($\varphi$) and average photon number $\langle n\rangle$ for $|\alpha_{a}|$= 1.5. The colored region shows the range of the parameters in (a,b) for which $\mathcal{S}_{\text{SQL}}(\varphi) < 1$ i.e. the coherent state $\left|\alpha_{a}\right\rangle$ and the SPACS $|\alpha_{b}, 1\rangle$ as inputs can beat the SQL. (c) Plot of $\mathcal{S}_{\text{SQL}}(\varphi)$ as a function of the average photon number $\langle n\rangle$ for $|\alpha_{a}|$= 1.5 and $m = 1$. We see that there is small range of $\langle n\rangle \sim$$4.03 \leq \langle n\rangle \leq 8.1$ where $\mathcal{S}_{\text{SQL}}(\varphi) < 1$.
• Figure 3: Plot of of $\mathcal{S}_{\text{SQL}}(\varphi)$ as a function of average photon number $\langle n\rangle$ for $|\alpha_{a}|$= 1.5, $\varphi = 0$ for different PACS $|\alpha_{b},m\rangle$ ($m$ = 0, 1, 2, 3, 4). Here, $m=0$ corresponds to no photon addition, i.e., the input modes are in coherent states.
• Figure 4: Phase-space Wigner function profiles for SPACS combined with a coherent state in an MZI. The vertical axis denotes the computed Wigner function value, highlighting the degree of nonclassicality. (a) Combined Wigner function of a SPACS ($m=1$ where $\alpha_a = 1.17$ and $\alpha_b = 1.5$) and a coherent state at the MZI input, showing interference-induced negativity at the origin. (b) Single-mode SPACS Wigner function after propagation through the MZI for $m=1$ and $\varphi=0$, preserving pronounced nonclassical features. (c) Output Wigner function for $m=1$ and $\varphi=5.03$, where the negativity diminishes and the profile becomes more Gaussian-like. (d) Input Wigner function for a higher-order SPACS ($m=4$ where $\alpha_a = 0.819$ and $\alpha_b = 1.5$), exhibiting complex interference fringes. (e) Wigner function at the MZI output for $m=4$ and $\varphi=0.00$, with sharp central negativity retained. (f) Output Wigner function for $m=1$ and $\varphi=3.77$, where nonclassical features are suppressed due to phase-induced redistribution.