Below-shot-noise capacity in phase estimation using nonlinear interferometers

TL;DR

This work compares three nonlinear-interferometer configurations (Yurke SU(1,1), Mandel induced coherence, and a tunable hybrid) for phase estimation using only intensity measurements, accounting for loss and high-gain operation. While ideal Yurke setups can exhibit Heisenberg-limited scaling, realistic losses erode this advantage; the Mandel configuration with differential intensity detection provides the most robust, shot-noise-limited performance under loss, outperforming Yurke in high-gain regimes. The study yields practical guidelines: use balanced Yurke operation for modest gains to glimpse Heisenberg scaling, but prefer Mandel differential detection for high photon flux and asymmetric losses. The results inform design choices for bicolor quantum imaging and nonlinear metrology under realistic conditions, and motivate further work on loss-tolerant readouts and multimode extensions.

Abstract

Over the past decade, several schemes for imaging and sensing based on nonlinear interferometers have been proposed and demonstrated experimentally. These interferometers exhibit two main advantages. First, they enable probing a sample at a chosen wavelength while detecting light at a different wavelength with high efficiency (bicolor quantum imaging and sensing with undetected light). Second, they can show quantum-enhanced sensitivities below the shot-noise limit, potentially reaching Heisenberg-limited precision in parameter estimation. Here, we compare three quantum-imaging configurations using only easily accessible intensity-based measurements for phase estimation: a Yurke-type SU(1,1) interferometer, a Mandel-type induced-coherence interferometer, and a hybrid scheme that continuously interpolates between them. While an ideal Yurke interferometer can exhibit Heisenberg scaling, this advantage is known to be fragile under realistic detection constraints and in the presence of loss. We demonstrate that differential intensity detection in the Mandel interferometer provides the highest and most robust phase sensitivity among the considered schemes, reaching but not surpassing the shot-noise limit, even in the presence of loss. Intensity measurements in a Yurke-type configuration can achieve genuine sub-shot-noise sensitivity under balanced losses and moderate gain; however, their performance degrades in realistic high-gain regimes. Consequently, in this regime, the Mandel configuration with differential detection outperforms the Yurke-type setup and constitutes the most robust approach for phase estimation.

Figures (6)

• Figure 1: Sketches of different nonlinear interferometers: In the Yurke setup (a), a nonlinear medium $A$ with vacuum input generates signal and idler fields $j=\text{s},\text{i}$ (blue and red). They may be subject to loss, encoded in a non-unit transmittance $T_j$, before they seed a second nonlinear medium $B$, whose signal output $N_\text{s}$ is detected. In the Mandel setup (b), only the idler seeds medium $B$, while the signal modes of both media are interfered on a 50:50 beam splitter, whose outputs $N_\text{s}$ and $N_\text{s}^\prime$ are detected. In a hybrid configuration (c), only a fraction $1-\varrho$ of the signal mode is seeded into medium $B$, while the remaining part is interfered with the signal output of medium $B$ on a beam splitter, whose transmittance is $1-\varrho/2$. The variable parameter $\varrho$ tunes the setup from a Yurke ($\varrho=0$) to a Mandel configuration ($\varrho=1$).
• Figure 2: Phase uncertainty of the hybrid setup $n\,\sigma^2_\text{H}$ for $n=10$, as a function of the mixing parameter $\varrho$ and the phase $\phi$. The phases minimizing the sensitivity for discrete values of $\varrho$ are represented by white dots. The optimal working point for the Yurke setup ($\varrho=0$) is at $\phi_\text{min}=\pi$, while the sensitivity is minimized at two phases for $\varrho>0$. For the Mandel setup ($\varrho=1$), these phases are $\phi_\text{min} = \pi/2,\,3\pi/2$. The margin shows the minimal phase uncertainties, ranging from the optimal uncertainty of the Yurke setup to the shot-noise limit exhibited by the Mandel configuration.
• Figure 3: Normalized classical Fisher information $F_\text{c}/n= 1/ (n\sigma_\text{Y}^2|_\text{min})$ at the optimal phase setting $\phi_\text{min}$ as a function of $n$ and $T_\text{i}$, assuming $T_\text{s}=0.8$. An increase of the normalized Fisher information with the number of probing photons implies better than shot-noise scaling, while a constant behavior corresponds to a shot-noise limited measurement. A decrease of the normalized Fisher information results in a deterioration of the phase sensitivity with the photon number, therefore worse than shot-noise scaling. We observe a resonance for $T_\text{i}=T_\text{s}$. The cuts along the dashed lines at $T_\text{i}=0.7$ (black) and $T_\text{i}=0.8$ (blue) are displayed in Fig. \ref{['fig:YurkePhases']}.
• Figure 4: Normalized classical Fisher information as a function of $n$. The blue lines correspond to a case with equal losses ($T_\text{s}=T_\text{i}=0.8$). The thick blue solid line denotes the maximal classical Fisher information for each value of $n$, i. e. $F_\text{c}/n$ evaluated at $\phi_\text{min}$. It approaches a constant value for high values of $n$, which corresponds to shot-noise scaling. The thin light blue lines are the normalized classical Fisher information for specific values of the phase, $\phi= 0.97 \pi, 0.95 \pi, 0.9 \pi$ (dotted, dashed, solid). The black lines corresponds to a case with non equal losses ($T_\text{s}=0.8$ and $T_\text{i}=0.7$). The thick black line denotes the maximal classical Fisher information, evaluated at $\phi_\text{min}$, whereas the thin gray lines corresponds to phases $\phi= 0.97 \pi, 0.95 \pi, 0.9 \pi$ (dotted, dashed, solid). The decrease of the normalized classical Fisher information signifies a scaling worse than shot noise, whereas an increase corresponds to sub-shot-noise behavior.
• Figure 5: Minimal phase uncertainty $\sigma_\text{Y}^2|_{\text{min}}$ as a function of $n$. For $T_\text{s}=0.9$ and $T_\text{i}=0.85$ (black), we observe a constant limit (gray line) in the high-gain regime, while for $T_\text{s}=T_\text{i}=0.9$ (blue) we observe shot-noise scaling. For low photon numbers, both cases approach a Heisenberg scaling (dashed, red line). The respective limits and scalings plotted in the figure correspond to the respective terms of Eq. \ref{['eq:high-gain-Yurke']}.