Gaussian-state interferometry with passive and active elements

We address the precision of optical interferometers fed by quantum and semiclassical Gaussian states involving passive and/or active elements, such as beam splitters, photodetectors, and optical parametric amplifiers. We first address the ultimate bounds to precision by discussing the behavior of the quantum Fisher information. We then consider photodetection at the output and calculate the sensitivity of the interferometers taking into account the nonunit quantum efficiency of the detectors. Our results show that in the ideal case of photon number detectors with unit quantum efficiency the best configuration is the symmetric one, namely, a passive (active) interferometer with a passive (active) detection stage: in this case one may achieve Heisenberg scaling of sensitivity by suitably optimizing over Gaussian states at the input. On the other hand, in the realistic case of detectors with nonunit quantum efficiency, the performances of the passive scheme are unavoidably degraded, whereas detectors involving optical parametric amplifiers allow us to fully compensate for the presence of loss in the detection stage, thus restoring the Heisenberg scaling.

Figure 1

Top: The general scheme of an interferometric setup. A factorized pure state $\left|\psi \right.〉\left|\phi \right.〉$ of the two modes is injected in the device. The two modes are then coupled by the unitary operator $U$, and an unknown phase shift $V\left(\varphi \right)$ is applied to one of the beams. At the output of the interferometer, an observable described by the POVM $E$ is measured. Bottom left: A schematic diagram of the passive measurement stage. Before being measured, the two beams go through a beam splitter. The difference photocurrent $D_{-}$ is then measured. Bottom right: Scheme of the active measurement stage. The beam splitter is here replaced by an OPA. The measured observable is the sum photocurrent $D_{+}$.

Figure 2

Top left: $H_{\varphi }$ as a function of $\theta$ (rad), where $N_{\mathrm{tot}}=1$ and $\delta =1/4$ (orange lines), $\delta =1/2$ (blue lines), and $\delta =1$ (green lines). Solid lines show ${\beta }_{\mathrm{tot}}=3/4$ and $\beta =2/3$, and dashed lines show ${\beta }_{\mathrm{tot}}=1/3$ and $\beta =1/6$. Top right: $H_{\varphi }$ as a function of $\delta$, where $N_{\mathrm{tot}}=1$ and $\theta =0$. Solid lines show ${\beta }_{\mathrm{tot}}=3/4$, and dashed lines show ${\beta }_{\mathrm{tot}}=1/3$; orange lines are for $\beta =0$, blue lines are for $\beta ={\beta }_{\mathrm{tot}}/4$, and green lines are for $\beta ={\beta }_{\mathrm{tot}}/2$. Bottom left: The values of ${\beta }_{\mathrm{tot}}$ (blue line) and $\beta$ (orange line) maximizing the auantum Fisher information $H_{\varphi }$ as a function of $N_{\mathrm{tot}}$. Bottom right: The standard deviation ${\sigma }_{\varphi }$ (blue solid line) as a function of $N_{\mathrm{tot}}$. The shot-noise limit $1/\sqrt{N_{\mathrm{tot}}}$ is shown by the orange dashed line, and the Heisenberg limit $1/N_{\mathrm{tot}}$ is shown by the green dotted line.

Figure 3

Top left: The QFI $H_{\varphi }$ for an active interferometer as a function of $\theta$ (rad) and $\delta$ for fixed values of $\beta =0.7$ and $N_{\mathrm{tot}}=1$. Top right: The minimum detectable fluctuation ${\sigma }_{\varphi }$ (blue solid line) as a function of $N_{\mathrm{tot}}$, with the shot-noise limit (orange dashed line) and the Heisenberg limit (green dotted line). Bottom: The parameters ${\beta }_{\max }$ and ${\delta }_{\max }$, maximizing $H_{\varphi }$, as a function of $N_{\mathrm{tot}}$.

Figure 4

The passive-passive interferometer with ideal photodetection. Top left: The sensitivity $S_1$ as a function of $\varphi$ (rad) for $N_{\mathrm{tot}}=2$ and $\xi =0.5$ and $r=0.7$ (blue line) or $\xi =0.7$ and $r=0.5$ (green line) and for different values of $\theta$ (rad): $\theta =0$ (dashed lines), $\pi /4$ (dotted lines), $\pi /2$ (solid lines), or $3/4\pi$ (dot-dashed lines). Top right: The sensitivity $S_1$ as a function of $\theta$ (rad) for $\varphi =\pi /2,N_{\mathrm{tot}}=2$ and for $\xi =0.5$ and $r=0.7$ (orange line) or $\xi =0.7$ and $r=0.5$ (blue line) or $\xi =0.5$ and $r=0.5$ (green line). Bottom left: The ideal sensitivity $S_{1\min }$, minimized with respect to $\beta$, as a function of ${\beta }_{\mathrm{tot}}$ for $N_{\mathrm{tot}}={10}^{-1}$ (dashed line), $N_{\mathrm{tot}}=1$ (dotted line), $N_{\mathrm{tot}}=10$ (solid line), or $N_{\mathrm{tot}}={10}^2$ (dot-dashed line). Bottom right: The optimized sensitivity $S_{1\min }$ as a function of $N_{\mathrm{tot}}$ (solid blue line). Also the shot-noise limit $1/\sqrt{N_{\mathrm{tot}}}$ (dashed orange line) is shown.

Figure 5

The passive-passive interferometer with realistic photodetection. Top left: The sensitivity $S_{\eta }$ as a function of $\varphi$ (rad) for $N_{\mathrm{tot}}=1,{\beta }_{\mathrm{tot}}=0.8$, and $\beta =0.6$. The blue lines are for $\eta =0.9$ and $\theta =\pi /4$ (dashed) or $\theta =3/4\pi$ (solid). The orange lines are for $\eta =0.3$, with $\theta =\pi /2$ (dotted) or $\theta =\pi$ (dot-dashed). The green lines are for $\eta =0.1$ and $\theta =\pi /4$ (dashed) or $\theta =3/4\pi$ (solid). Top right: The sensitivity $S_{\eta }$ as a function of $\theta$ (rad) for $\varphi =\pi /2,N_{\mathrm{tot}}=10$, and ${\beta }_{\mathrm{tot}}=0.9$. Solid lines are for $\eta =0.8$, and dashed lines show $\eta =0.2$ for $\beta =0.1$ (orange), $\beta =0.3$ (green), or $\beta =0.6$ (blue). Bottom left: The fraction ${\beta }_{\min }$, minimizing the sensitivity $S_{\eta }$ for $N_{\mathrm{tot}}\ll 1$, as a function of $\eta$. Bottom right: The optimized sensitivity (solid lines) for $N_{\mathrm{tot}}\gg 1$ as a function of $N_{\mathrm{tot}}$ for $\eta =0.9$ (blue line), $\eta =0.4$ (orange line), or $\eta =0.1$ (green line). The shot-noise (red dashed line) and the Heisenberg limit (magenta dotted line) are also shown.

Figure 6

The passive-active interferometer with ideal photodetection. Top left: The minimized sensitivity $S_{1\min }$ as a function of the parameter $r_1$ for $N_{\mathrm{tot}}=1$ (orange line), $N_{\mathrm{tot}}=10$ (green line), and $N_{\mathrm{tot}}=100$ (blue line). Top right: The parameter $\beta$ minimizing the sensitivity as a function of $N_{\mathrm{tot}}$. Bottom left: The optimal sensitivity $S_{1\min }$ as a function of the total number of photons $N_{\mathrm{tot}}$, together with the shot-noise limit (orange dashed line) and the Heisenberg limit (green dotted line). Bottom right: The behavior of the ratio $R=S_{1\min }/S_{\mathrm{HL}}$, where $S_{\mathrm{HL}}=1/N_{\mathrm{tot}}$ is the Heisenberg scaling, as a function of $N_{\mathrm{tot}}$. The asymptotic value of $R$ is $1+1/\sqrt{2}$.

Figure 7

The passive-active interferometer with realistic photodetection. The plot shows the optimized sensitivity $S_{\eta \min }$ as a function of $r_1$ for $N_{\mathrm{tot}}=1$ (solid lines), $N_{\mathrm{tot}}=10$ (dashed lines), and $N_{\mathrm{tot}}=100$ (dotted lines) and for $\eta =1$ (orange lines), $\eta =0.6$ (blue lines), and $\eta =0.2$ (green lines).

Figure 8

The active-passive interferometer with ideal photodetection. Top left: The sensitivity $S_1$ as a function of $\theta$ (rad), when $N_{\mathrm{tot}}=10,\delta =1/2$, and $\varphi =\pi /2$. The blue line refers to $\beta =0.01$, the orange line refers to $\beta =1/3$, and the green one refers to $\beta =2/3$. Top right: The sensitivity $S_1$ as a function of $\beta$, when $\delta =1/2,\varphi =\pi /2$, and $\theta =\pi$. The orange line is taken for $N_{\mathrm{tot}}=0.1$, the green one is for $N_{\mathrm{tot}}=1$, and the blue one is for $N_{\mathrm{tot}}=10$. Bottom left: The value of $\beta$ minimizing $S_{1\min }$ as a function of $N_{\mathrm{tot}}$. Bottom right: The optimized $S_1$ as a function of $N_{\mathrm{tot}}$ (blue solid line), with the shot-noise limit (orange dashed line) and the Heisenberg limit (green dotted line).

Figure 9

The active-passive interferometer with realistic photodetection. Top left: The sensitivity $S_{\eta }$ as a function of $\beta$ for $\delta =1/2,\varphi =\pi /2$, and $\theta =\pi$. The solid lines are for $\eta =0.8$, and the dashed lines are for $\eta =0.2$. The orange lines are for $N_{\mathrm{tot}}=0.1$, green lines are for $N_{\mathrm{tot}}=1$, and blue lines are for $N_{\mathrm{tot}}=10$. Top right: The value of $\beta$ minimizing $S_{\eta }$ as a function of $N_{\mathrm{tot}}$. The blue line refers to $\eta =1$, the orange one refers to $\eta =0.6$, and the green one refers to $\eta =0.2$. Bottom left: The optimized sensitivity $S_{\eta \min }$ (solid lines) as a function of $N_{\mathrm{tot}}$ for $\eta =1$ (blue line), $\eta =0.8$ (orange line), and $\eta =0.2$ (green line). The shot-noise limit (red dashed line) and the Heisenberg limit (magenta dotted line) are also shown. Bottom right: The exponent $\varepsilon \left(\eta \right)$ as a function of the quantum efficiency.

Figure 10

The active-active interferometer with ideal photodetection. Top left: The minimum ideal sensitivity $S_{1\min }$ as a function of $r_2$ for $N_{\mathrm{tot}}$ equal to 1 (orange line), 10 (green line), and 100 (blue line). Top right: The value of $\beta$ minimizing $S_1$ as a function of $r_2$ for the same values of $N_{\mathrm{tot}}$ as in the top left plot. Bottom left: The sensitivity $S_{1\min }$ as a function of $\varphi$ (rad) for $N_{\mathrm{tot}}$ equal to $0.1$ (blue line), 1 (orange line), and 10 (green one). Bottom right: The sensitivity (solid line) $S_{1\min }$ as a function of $N_{\mathrm{tot}}$. The shot noise (orange dashed line) and the Heisenberg limit (green dotted line) are also shown for comparison.

Figure 11

The active-active interferometer with realistic photodetection. The plot shows the sensitivity $S_{\eta \min }$ as a function of $r_2$ for values of $N_{\mathrm{tot}}$ equal to 1 (solid lines) and 100 (dashed lines) and values of $\eta$ equal to 1 (orange lines), $0.6$ (blue lines), and $0.2$ (green lines).

Figure 12

Sum and difference photocurrents $D_{+}\left(\eta \right)$ and $D_{-}\left(\eta \right)$ with realistic photodetectors. Here, we use a beam splitter with transmission coefficient $\eta$ to model quantum efficiency of detectors. Since a beam splitter has two input modes, one is filled with the radiation signal (modes $a$ and $b$ in the scheme), while in the other (that is, modes $c$ and $d$) we put a vacuum state $\left|0\right.〉$.