Loss tolerant quantum absorption measurement

First, we derive S for a classical absorption measurement (CAM). Figure 1(a) shows a schematic of a CAM. In a CAM, classical light, such as a laser beam, irradiates an absorptive medium as a sample. An absorptivity change in the sample is detected through a change of the transmitted light power. As possible experimental imperfections, we consider the transmittance of the optical path η_A (0 ⩽ η_A ⩽ 1) and the detection efficiency of a photon detector η_D (0 ⩽ η_D ⩽ 1). The path transmittance η_A may be reduced by optical elements such as lenses for focusing the beam on the sample, or filters for cutting unwanted florescence from the sample. Taking into account these possible experimental imperfections, we can represent the count of detected photons as N_CAM ≡ Tη_Aη_Dn for an input of n photons. Because the photon number statistics for coherent light follows a Poisson distribution, the detected photon count has a variance of Δ²N_CAM = N_CAM. Then, the detectable smallest absorptivity change for a CAM is obtained from equation (1) as follows:

$$\begin{align}\hfill \delta {T}_{\text{CAM}}& =\frac{{\Delta}{N}_{\text{CAM}}}{\vert \partial {N}_{\text{CAM}}/\partial T\vert }=\frac{\sqrt{T{\eta }_{\text{A}}{\eta }_{\text{D}}n}}{{\eta }_{\text{A}}{\eta }_{\text{D}}n}=\sqrt{\frac{T}{{\eta }_{\text{A}}{\eta }_{\text{D}}n}}.\hfill \end{align} \tag{ 3 }$$

When there is no experimental imperfection (η_A = 1, η_D = 1), this leads to the SNL of $\sqrt{T/n}$. The sensitivity to absorption for a CAM is derived from equations (2) and (3) as follows:

$$\begin{equation}{S}_{\text{CAM}}=\sqrt{{\eta }_{\text{A}}{\eta }_{\text{D}}}.\end{equation} \tag{ 4 }$$

This shows that the sensitivity to absorption for a CAM never exceeds SNL, namely S_CAM ⩽ 1.

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To exceed the SNL, a quantum enhanced absorption measurement (QAM) has been proposed and demonstrated [10–15]. Figure 1(b) shows the schematic of a QAM. Pump laser light is injected into a nonlinear optical material (NL) and pairs of photons are generated via a nonlinear optical process, such as spontaneous parametric down-conversion or spontaneous four-wave mixing. The absorptive medium is placed in one of the paths (path A). Each photon is detected by a single-photon detector at the output. Similar to a CAM, we introduce experimental imperfections: the transmittances of the paths A and B (η_A, η_B) and the detection efficiencies of the photon detectors. For simplicity, we assume that the two single-photon detectors have the same detection efficiency η_D. Note that the transmittance of path A (η_A) does not include that of the absorptive medium (T). The initial quantum state of the photon pair emitted from NL is approximately described by

$$\begin{align}\hfill \left\vert {{\Psi}}_{\text{i}}\right\rangle & =\sqrt{1-{\alpha }_{1}}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}+\sqrt{{\alpha }_{1}}{\left\vert 1\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}=\left(\sqrt{1-{\alpha }_{1}}+\sqrt{{\alpha }_{1}}{a}_{\text{i}}^{{\dagger}}{b}_{\text{i}}^{{\dagger}}\right)\left\vert 0\right\rangle ,\hfill \end{align} \tag{ 5 }$$

in the low-gain regime where the average photon-pair number per mode α₁ is much less than 1 (α₁ ≪ 1). ${a}_{\text{i}}^{{\dagger}}$ and ${b}_{\text{i}}^{{\dagger}}$ are the creation operators for A and B modes just after the NL, respectively. Before reaching the single-photon detector, one of the photons in the pair passes through the absorptive medium in path A, while the other photon travels through path B. The creation operators are connected by the relations

$$\begin{align*}\hfill {a}_{\text{i}}^{{\dagger}}& =\sqrt{T{\eta }_{\text{A}}{\eta }_{\text{D}}}{a}^{{\dagger}}+\sqrt{1-T{\eta }_{\text{A}}{\eta }_{\text{D}}}{a}_{\text{l}}^{{\dagger}},\hfill \\ \hfill {b}_{\text{i}}^{{\dagger}}& =\sqrt{{\eta }_{\text{B}}{\eta }_{\text{D}}}{b}^{{\dagger}}+\sqrt{1-{\eta }_{\text{B}}{\eta }_{\text{D}}}{b}_{\text{l}}^{{\dagger}},\hfill \end{align*}$$

where a^† and b^† are the creation operators for the photons detected at the output A and B, respectively, and ${a}_{\text{l}}^{{\dagger}}$ and ${b}_{\text{l}}^{{\dagger}}$ correspond to the loss. Consequently, the photon-pair state changes as follows:

$$\begin{align}\hfill \left\vert {{\Psi}}_{\text{f}}\right\rangle & =\sqrt{1-{\alpha }_{1}}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}+\sqrt{{\alpha }_{1}\left(1-T{\eta }_{\text{A}}{\eta }_{\text{D}}\right)\left(1-{\eta }_{\text{B}}{\eta }_{\text{D}}\right)}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad +\sqrt{{\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{D}}T\left(1-{\eta }_{\text{B}}{\eta }_{\text{D}}\right)}{\left\vert 1\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}+\sqrt{{\alpha }_{1}{\eta }_{\text{B}}{\eta }_{\text{D}}\left(1-{\eta }_{\text{A}}{\eta }_{\text{D}}T\right)}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad +\sqrt{{\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{B}}{\eta }_{\text{D}}^{2}T}{\left\vert 1\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}},\hfill \end{align} \tag{ 6 }$$

where ${\left\vert 1\right\rangle }_{\text{lA}}$ (${\left\vert 0\right\rangle }_{\text{lA}}$) and ${\left\vert 1\right\rangle }_{\text{lB}}$ (${\left\vert 0\right\rangle }_{\text{lB}}$) represent the states which have one (zero) photon in the loss modes corresponding to ${a}_{\text{l}}^{{\dagger}}$ and ${b}_{\text{l}}^{{\dagger}}$, respectively. In a QAM, the output event number for k trials is given by the count difference between the two single-photon detectors:

$$\begin{align}\hfill {N}_{\text{QAM}}& =k\left\langle {{\Psi}}_{\text{f}}\right\vert {b}^{{\dagger}}b-{a}^{{\dagger}}a\left\vert {{\Psi}}_{\text{f}}\right\rangle =k{\alpha }_{1}{\eta }_{\text{D}}\left({\eta }_{\text{B}}-{\eta }_{\text{A}}T\right)=n{\eta }_{\text{D}}\left({\eta }_{\text{B}}-{\eta }_{\text{A}}T\right),\hfill \end{align} \tag{ 7 }$$

with a given number of pairs n = kα₁ created from the photon pair source. This means that n is equal to the number of photons input to the optical path where the absorptive medium is placed. The noise in the measurement is determined by the variance of ΔN_QAM:

$$\begin{align}\hfill {\Delta}{N}_{\text{QAM}}^{2}& =k\left(\langle {\left({b}^{{\dagger}}b-{a}^{{\dagger}}a\right)}^{2}\rangle -{\langle {b}^{{\dagger}}b-{a}^{{\dagger}}a\rangle }^{2}\right)\hfill \\ \hfill & =k\left(\langle {b}^{{\dagger}}b{b}^{{\dagger}}b\rangle +\langle {a}^{{\dagger}}a{a}^{{\dagger}}a\rangle -2\langle {a}^{{\dagger}}a{b}^{{\dagger}}b\rangle -{\langle {b}^{{\dagger}}b\rangle }^{2}-{\langle {a}^{{\dagger}}a\rangle }^{2}+2\langle {a}^{{\dagger}}a\rangle \langle {b}^{{\dagger}}b\rangle \right)\hfill \\ \hfill & =k{\alpha }_{1}{\eta }_{\text{D}}\left({\eta }_{\text{B}}+{\eta }_{\text{A}}T-2{\eta }_{\text{A}}{\eta }_{\text{B}}{\eta }_{\text{D}}T-{\alpha }_{1}{\eta }_{\text{B}}^{2}{\eta }_{\text{D}}-{\alpha }_{1}{\eta }_{\text{A}}^{2}{\eta }_{\text{D}}{T}^{2}+2{\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{B}}{\eta }_{\text{D}}T\right)\hfill \\ \hfill & \approx n{\eta }_{\text{D}}\left({\eta }_{\text{B}}+{\eta }_{\text{A}}T\left(1-2{\eta }_{\text{B}}{\eta }_{\text{D}}\right)\right),\hfill \end{align} \tag{ 8 }$$

where the last expression is an approximation when α₁ ≪ 1. The smallest detectable absorptivity change is obtained from equations (1), (7) and (8).

$$\begin{align}\hfill \delta {T}_{\text{QAM}}& =\frac{{\Delta}{N}_{\text{QAM}}}{\vert \partial {N}_{\text{QAM}}/\partial T\vert }=\frac{\sqrt{n{\eta }_{\text{D}}\left({\eta }_{\text{B}}+{\eta }_{\text{A}}T\left(1-2{\eta }_{\text{B}}{\eta }_{\text{D}}\right)\right)}}{n{\eta }_{\text{D}}{\eta }_{\text{A}}}=\frac{\sqrt{{\eta }_{\text{B}}+{\eta }_{\text{A}}T\left(1-2{\eta }_{\text{B}}{\eta }_{\text{D}}\right)}}{\sqrt{n{\eta }_{\text{D}}}{\eta }_{\text{A}}}\hfill \end{align} \tag{ 9 }$$

From equation (2) together with δT_QAM, the sensitivity for a QAM is given as follows:

$$\begin{align}\hfill {S}_{\text{QAM}}& =\frac{\sqrt{T}}{\delta {T}_{\text{QAM}}\sqrt{n}}=\frac{\sqrt{T{\eta }_{\text{D}}}{\eta }_{\text{A}}}{\sqrt{{\eta }_{\text{B}}+{\eta }_{\text{A}}T\left(1-2{\eta }_{\text{B}}{\eta }_{\text{D}}\right)}}.\hfill \end{align} \tag{ 10 }$$

In an ideal case (η_A = η_B = η_D = 1), S_QAM becomes $\sqrt{T}/\sqrt{1-T}$, indicating that the sensitivity to absorption for a QAM increases with T and diverges to infinity at T = 1. This means that the sensitivity for a QAM becomes larger for smaller absorptivity. In other words, a QAM is suited for samples with small absorptivity. In fact, when T ≈ 1, a QAM in the ideal case reaches the quantum Fisher information that determines the smallest uncertainty [32]. However, the performance of a QAM will be significantly degraded due to unwanted photon losses. Since it may be difficult to achieve η_D ≈ 1, while it is relatively easy to improve η_A and η_B closer to 1, let us consider a case where η_D < 1, η_A = η_B = 1 and the sample has a small absorptivity (T ≈ 1). In this case, equation (10) becomes ${S}_{\text{QAM}}\approx \frac{\sqrt{{\eta }_{\text{D}}}}{\sqrt{2-2{\eta }_{\text{D}}}}$. From this, the condition where the SNL is exceeded (S_QAM > 1) is given as η_D > 2/3 (figure 2). Therefore, a QAM requires η_D = 66.7% to reach the SNL. Additionally, even with a very high detection efficiency, the enhancement is moderate. For instance, to achieve S_QAM = 3, a QAM requires η_D = 94.7%, which is barely reachable using current state of the art photon detectors [33]. S_QAM for η_A = η_B ∼ 1 is plotted as a blue curve in figure 2.

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To suppress the performance degradation due to photon loss at the detection stage in a QAM, we propose a loss-tolerant quantum absorption measurement (LTQAM). Figure 1(c) shows a schematic of an LTQAM. Pairs of photons are generated by a nonlinear optical process in the same way as in a QAM. Then, pump light and the generated photon pairs are injected to the second nonlinear optical material (NL2). The probability amplitude for a photon pair from NL1 interferes with that from NL2. The phase ϕ between the pump light and the photon pairs is tuned to be at the destructive interference condition where the photon pairs vanish. Consequently, when there is no absorptive medium, no photons are found at the output in the condition in which the quantum destructive interference is perfect and there are no optical losses in paths A and B. When the absorptive medium is placed in path A and absorbs one of the photons in a pair, destructive interference does not occur and the other photon will be detected at the output mode. Thus, the photon counts at output B increase with the number of the absorbed photons in the absorptive medium and an absorptivity change is thus detected.

Next, we derive the sensitivity to absorption for an LTQAM. A pair of photons generated from NL1 is represented by equation (5). Since it undergoes a phase-shift ϕ in path B together with losses in paths A and B, as shown in figure 1(c), the quantum state just before NL2 is given by

$$\begin{align}\hfill \left\vert {{\Psi}}_{1}\right\rangle & =\sqrt{1-{\alpha }_{1}}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}+\sqrt{{\alpha }_{1}\left(1-T{\eta }_{\text{A}}\right)\left(1-{\eta }_{\text{B}}\right)}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad +\sqrt{{\alpha }_{1}{\eta }_{\text{A}}T\left(1-{\eta }_{\text{B}}\right)}{\left\vert 1\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}+{\text{e}}^{\text{i}\phi }\enspace \sqrt{{\alpha }_{1}{\eta }_{\text{B}}\left(1-{\eta }_{\text{A}}T\right)}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad +{\text{e}}^{\text{i}\phi }\enspace \sqrt{{\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{B}}T}{\left\vert 1\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}.\hfill \end{align} \tag{ 11 }$$

Then, the pair of photons is injected into NL2 with the pump laser light. After NL2, the photon pair state emitted from NL2 is coherently added to that from NL1 (equation (11)) as follows [18]:

$$\begin{align}\hfill \left\vert {{\Psi}}_{2}\right\rangle & ={\alpha }_{0}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}+{\alpha }_{{0}^{\prime }}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad +\sqrt{{\alpha }_{1}{\eta }_{\text{A}}T\left(1-{\eta }_{\text{B}}\right)}{\left\vert 1\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}+{\text{e}}^{\text{i}\phi }\enspace \sqrt{{\alpha }_{1}{\eta }_{\text{B}}\left(1-{\eta }_{\text{A}}T\right)}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad +{\text{e}}^{\text{i}\phi }\enspace \sqrt{{\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{B}}T}{\left\vert 1\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}+\sqrt{{\alpha }_{2}}\left(\sqrt{\gamma }{\left\vert 1\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}+\sqrt{1-\gamma }{\left\vert 1\right\rangle }_{{\text{A}}^{\prime }}{\left\vert 1\right\rangle }_{{\text{B}}^{\prime }}\right){\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}},\hfill \\ \hfill & ={\alpha }_{0}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}+{\alpha }_{{0}^{\prime }}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad +\sqrt{{\alpha }_{1}{\eta }_{\text{A}}T\left(1-{\eta }_{\text{B}}\right)}{\left\vert 1\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}+{\text{e}}^{\text{i}\phi }\enspace \sqrt{{\alpha }_{1}{\eta }_{\text{B}}\left(1-{\eta }_{\text{A}}T\right)}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad +\left({\text{e}}^{\text{i}\phi }\enspace \sqrt{{\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{B}}T}+\sqrt{{\alpha }_{2}\gamma }\right){\left\vert 1\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}+\sqrt{{\alpha }_{2}\left(1-\gamma \right)}{\left\vert 1\right\rangle }_{{\text{A}}^{\prime }}{\left\vert 1\right\rangle }_{{\text{B}}^{\prime }}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}.\hfill \end{align} \tag{ 12 }$$

where α₀ and α_0' are introduced to represent the probability amplitudes of the first two terms which have no photons in the both modes A and B, and α₂ is the average number of photon pairs generated from NL2. The mode matching parameter γ determines the ratio of the photon pairs generated from NL2 in the modes A' and B', which do not contribute to the quantum interference but photons in these modes are also detected by the detectors at the outputs. Consequently, γ represents the quality of the interference.

For quantum destructive interference (ϕ = π), equation (12) becomes

$$\begin{align}\hfill \left\vert {{\Psi}}_{2}\right\rangle & ={\alpha }_{0}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}+{\alpha }_{{0}^{\prime }}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad +\sqrt{{\alpha }_{1}{\eta }_{\text{A}}T\left(1-{\eta }_{\text{B}}\right)}{\left\vert 1\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}-\sqrt{{\alpha }_{1}{\eta }_{\text{B}}\left(1-{\eta }_{\text{A}}T\right)}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad +\left(-\sqrt{{\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{B}}T}+\sqrt{{\alpha }_{2}\gamma }\right){\left\vert 1\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}+\sqrt{{\alpha }_{2}\left(1-\gamma \right)}{\left\vert 1\right\rangle }_{{\text{A}}^{\prime }}{\left\vert 1\right\rangle }_{{\text{B}}^{\prime }}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}.\hfill \end{align} \tag{ 13 }$$

The optimal condition for destructive interference is α₂ = α₁η_Aη_BTγ, which minimizes the sum of the possibilities of the last two terms in equation (13). This condition is achievable by tuning the power of the pump laser light incident on NL2. In determining α₂, we can obtain α₁, η_A, η_B and γ by independent measurements in advance. Although the exact value of T is unknown in advance, T can be approximately set to be 1 for measuring very small absorptivity, and thus it is possible to substantially determine α₂. After optimizing α₂, we obtain the following state:

$$\begin{align}\hfill \left\vert {{\Psi}}_{2}\right\rangle & ={\alpha }_{0}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}+{\alpha }_{{0}^{\prime }}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad +\sqrt{{\alpha }_{1}{\eta }_{\text{A}}T\left(1-{\eta }_{\text{B}}\right)}{\left\vert 1\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}-\sqrt{{\alpha }_{1}{\eta }_{\text{B}}\left(1-{\eta }_{\text{A}}T\right)}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad -\sqrt{{\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{B}}T}\left(1-\gamma \right){\left\vert 1\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}+\sqrt{{\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{B}}T\gamma \left(1-\gamma \right)}{\left\vert 1\right\rangle }_{{\text{A}}^{\prime }}{\left\vert 1\right\rangle }_{{\text{B}}^{\prime }}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}.\hfill \end{align} \tag{ 14 }$$

Considering the photon losses at the photon detectors, this quantum state changes as follows:

$$\begin{align}\hfill \left\vert {{\Psi}}_{\text{f}}\right\rangle & =\left({\alpha }_{0}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}{\left\vert 0\right\rangle }_{{\text{lA}}^{\prime }}{\left\vert 0\right\rangle }_{{\text{lB}}^{\prime }}+{\alpha }_{{0}^{\prime }}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}{\left\vert 0\right\rangle }_{{\text{lA}}^{\prime }}{\left\vert 0\right\rangle }_{{\text{lB}}^{\prime }}+{\alpha }_{{0}^{{\prime\prime}}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}{\left\vert 1\right\rangle }_{{\text{lA}}^{\prime }}{\left\vert 0\right\rangle }_{{\text{lB}}^{\prime }}\right.\hfill \\ \hfill & \left.\quad +{\alpha }_{{0}^{\prime \prime \prime }}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}{\left\vert 0\right\rangle }_{{\text{lA}}^{\prime }}{\left\vert 1\right\rangle }_{{\text{lB}}^{\prime }}+{\alpha }_{{0}^{\prime \prime \prime \prime }}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}{\left\vert 1\right\rangle }_{{\text{lA}}^{\prime }}{\left\vert 1\right\rangle }_{{\text{lB}}^{\prime }}\right){\left\vert 0\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}\hfill \\ \hfill & \quad +\left(\sqrt{{\eta }_{\text{D}}{\alpha }_{1}{\eta }_{\text{A}}T\left(1-{\eta }_{\text{B}}\right)}{\left\vert 1\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 1\right\rangle }_{\text{lB}}-\sqrt{{\eta }_{\text{D}}{\alpha }_{1}{\eta }_{\text{B}}\left(1-{\eta }_{\text{A}}T\right)}{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}\right){\left\vert 0\right\rangle }_{{\text{lA}}^{\prime }}{\left\vert 0\right\rangle }_{{\text{lB}}^{\prime }}\hfill \\ \hfill & \quad -\sqrt{{\eta }_{\text{D}}\left(1-{\eta }_{\text{D}}\right){\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{B}}T}\left(1-\gamma \right)\left({\left\vert 1\right\rangle }_{\text{A}}{\left\vert 0\right\rangle }_{\text{B}}{\left\vert 0\right\rangle }_{{\text{lA}}^{\prime }}{\left\vert 1\right\rangle }_{{\text{lB}}^{\prime }}+{\left\vert 0\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}{\left\vert 1\right\rangle }_{{\text{lA}}^{\prime }}{\left\vert 0\right\rangle }_{{\text{lB}}^{\prime }}\right){\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}\hfill \\ \hfill & \quad +\sqrt{{\eta }_{\text{D}}\left(1-{\eta }_{\text{D}}\right){\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{B}}T\gamma \left(1-\gamma \right)}\left({\left\vert 1\right\rangle }_{{\text{A}}^{\prime }}{\left\vert 0\right\rangle }_{{\text{B}}^{\prime }}{\left\vert 0\right\rangle }_{{\text{lA}}^{\prime }}{\left\vert 1\right\rangle }_{{\text{lB}}^{\prime }}+{\left\vert 0\right\rangle }_{{\text{A}}^{\prime }}{\left\vert 1\right\rangle }_{{\text{B}}^{\prime }}{\left\vert 1\right\rangle }_{{\text{lA}}^{\prime }}{\left\vert 0\right\rangle }_{{\text{lB}}^{\prime }}\right){\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}},\hfill \\ \hfill & \quad -\left({\eta }_{\text{D}}\sqrt{{\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{B}}T}\left(1-\gamma \right){\left\vert 1\right\rangle }_{\text{A}}{\left\vert 1\right\rangle }_{\text{B}}+{\eta }_{\text{D}}\sqrt{{\alpha }_{1}{\eta }_{\text{A}}{\eta }_{\text{B}}T\gamma \left(1-\gamma \right)}{\left\vert 1\right\rangle }_{{\text{A}}^{\prime }}{\left\vert 1\right\rangle }_{{\text{B}}^{\prime }}\right){\left\vert 0\right\rangle }_{\text{lA}}{\left\vert 0\right\rangle }_{\text{lB}}{\left\vert 0\right\rangle }_{{\text{lA}}^{\prime }}{\left\vert 0\right\rangle }_{{\text{lB}}^{\prime }},\hfill \end{align} \tag{ 15 }$$

where ${\alpha }_{{0}^{{\prime\prime}}}$, ${\alpha }_{{0}^{\prime \prime \prime }}$ and ${\alpha }_{{0}^{\prime \prime \prime \prime }}$ are introduced to represent the probability amplitudes of the terms which have no photons in the both modes A and B, and ${\left\vert 1\right\rangle }_{{\text{lA}}^{\prime }}$ (${\left\vert 0\right\rangle }_{{\text{lA}}^{\prime }}$) and ${\left\vert 1\right\rangle }_{{\text{lB}}^{\prime }}$ (${\left\vert 0\right\rangle }_{{\text{lB}}^{\prime }}$) represent the states which have one (zero) photon in the loss modes of the output A and B, respectively. In an LTQAM, we measure the number of photons at the output B. For k trials, the average number of detected photons is given by

$$\begin{align}\hfill {N}_{\text{LTQAM}}& =k{p}_{\text{B}}\hfill \\ \hfill & =k\left\langle {{\Psi}}_{\text{f}}\right\vert \left({b}^{{\dagger}}b+{b}^{\prime {\dagger}}{b}^{\prime }\right)\left\vert {{\Psi}}_{\text{f}}\right\rangle \hfill \\ \hfill & =k{\alpha }_{1}{\eta }_{\text{B}}{\eta }_{\text{D}}\left(1-\gamma {\eta }_{\text{A}}T\right)\hfill \\ \hfill & =n{\eta }_{\text{B}}{\eta }_{\text{D}}\left(1-\gamma {\eta }_{\text{A}}T\right),\hfill \end{align} \tag{ 16 }$$

where p_B is the probability that a photon is detected at output B and b' and b'^† are the annihilation and creation operators for mode B'. The noise in this measurement is given by ${\Delta}{N}_{\mathrm{B}}^{2}=k{p}_{\text{B}}\left(1-{p}_{\text{B}}\right)$ and the smallest detectable absorptivity change is obtained from equations (1) and (16) as

$$\begin{align}\hfill \delta {T}_{\text{LTQAM}}& =\frac{{\Delta}{N}_{\text{LTQAM}}}{\vert \partial {N}_{\text{LTQAM}}/\partial T\vert }=\frac{\sqrt{\left(1-\gamma {\eta }_{\text{A}}T\right)\left(1-{\alpha }_{1}{\eta }_{\text{B}}{\eta }_{\text{D}}\left(1-\gamma {\eta }_{\text{A}}T\right)\right)}}{\vert -\gamma {\eta }_{\text{A}}\sqrt{n{\eta }_{\text{B}}{\eta }_{\text{D}}\vert }}\hfill \\ \hfill & \approx \frac{\sqrt{1-\gamma {\eta }_{\text{A}}T}}{\gamma {\eta }_{\text{A}}\sqrt{n{\eta }_{\text{B}}{\eta }_{\text{D}}}},\hfill \end{align} \tag{ 17 }$$

where the second equation is approximated to the last under the assumption α₁ ≪ 1. From equations (2) and (17), the sensitivity to absorption for an LTQAM is

$$\begin{align}\hfill {S}_{\text{LTQAM}}& \equiv \frac{\sqrt{T}}{\delta {T}_{\text{LTQAM}}\sqrt{n}}\hfill \end{align} \tag{ 18 }$$

$$\begin{align}\hfill & =\frac{\gamma {\eta }_{\text{A}}\sqrt{{\eta }_{\text{B}}{\eta }_{\text{D}}T}}{\sqrt{1-\gamma {\eta }_{\text{A}}T}}.\hfill \end{align} \tag{ 19 }$$

In the simplest case where η_A = η_B = γ = 1 and η_D < 1, S_LTQAM is always higher than S_QAM because ${S}_{\text{LTQAM}}=\frac{\sqrt{{\eta }_{\text{D}}T}}{\sqrt{1-T}}$ and ${S}_{\text{QAM}}=\frac{\sqrt{{\eta }_{\text{D}}T}}{\sqrt{1+T\left(1-2{\eta }_{\text{D}}\right)}}$, demonstrating the loss-tolerance of an LTQAM. In the situation of interest, the absorption is so small that T ≈ 1. In this case, S_LTQAM can be much higher than 1 even when the QAM is inferior to the SNL (S_QAM < 1).

Figure 2 shows the sensitivities while changing η_D when the transmittances of the paths A and B are η_A = η_B = 1 × 10⁻⁴, which is achievable with current state of the art technology [34]. We assume that the absorptive medium has a very small absorptivity of 10⁻⁶ (namely T = 1 × 10⁻⁶). Note that this value itself is unimportant in the following analysis because the condition 1 − T ≪ 1 − η_A is satisfied, and in this case, the change in T negligibly affects the sensitivity. The red curve in figure 2 shows the sensitivity for an LTQAM when the quantum destructive interference is perfect (γ = 1). While the sensitivity to absorption for a QAM (blue curve in figure 2) rapidly decreases with η_D and is lower than that for a CAM (green curve in figure 2) with η_D below 0.5, an LTQAM maintains an advantage even for lower η_D.

In figures 3(a) and (b), we show the sensitivity to absorption when η_A and η_B are varied, respectively, while η_B and η_A are fixed to 1 × 10⁻⁴ in order to check how η_A and η_B affect the sensitivity to absorption. We adopt a detection efficiency η_D of 0.80, which is achievable with commercial single-photon detectors. The two single-photon detectors in the LTQAM and QAM are assumed to have the same detection efficiency. Because the sensitivity for an LTQAM diverges to infinity at η_A = 1.0 when γ = 1.0, we set the plot range for η_A to be from 0 to 1 × 10⁻⁴ in figure 3(a). A comparison of figures 3(a) and (b) reveals that the sensitivities for an LTQAM have a stronger dependence on η_A than on η_B, and thus increasing η_A is more effective for improving the sensitivity. In figure 3(a), the LTQAM is inferior to the CAM for η_A less than around 0.5. This can be explained by equation (19), which indicates that η_A affects the sensitivity S_LTQAM in exactly the same way as γ.

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To analyze the effect of imperfection in the destructive quantum interference, figure 4 shows the sensitivities while changing the mode matching parameter γ when η_A = η_B = 1 × 10⁻⁴. The detection efficiency η_D is set to 0.80 and 0.67 for figures 4(a) and (b), respectively. The sensitivity to absorption for an LTQAM (red curve in figures 4(a) and (b)) rapidly decreases with γ. When η_D = 0.80, the thresholds of γ where an LTQAM surpasses a QAM (blue curve in figure 4(a)) and a CAM (green curve in figure 4(a)) are 0.77 and 0.62, respectively. Figure 4(b) shows the case for η_D = 0.67 where the sensitivity to absorption for a QAM is equal to the SNL (S_QAM = 1). In this case, while a QAM does not show a quantum advantage, an LTQAM can exceed the SNL for γ larger than 0.69.

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In LTQAM, photons are not found at output A when there is no experimental imperfection. However, if the quantum destructive interference is imperfect and/or the transmittance of path B is not unity, photons leak to output A. In this case, the information about absorptivity which the leaked photons have is discarded because the leaked photons are not detected in an LTQAM. To solve this problem, we also propose a hybrid approach (HybQAM) in which an additional single-photon detector at output A detects the leaked photons, as shown in figure 1(d).

Next, we derive the sensitivity to absorption for HybQAM. We consider a linear combination of the detected photon counts as follows:

$$\begin{equation}{N}_{\text{HybQ}}\equiv -m{N}_{\text{A}}+l{N}_{\text{B}},\end{equation} \tag{ 20 }$$

where N_A and N_B are the detected photon counts at the output A and B, respectively, and m and l are the weights of N_A and N_B, respectively. From equations (15) and (20), N_HybQ becomes

$$\begin{align}\hfill {N}_{\text{HybQ}}& =k\left\langle {{\Psi}}_{\text{f}}\right\vert l\left({b}^{{\dagger}}b+{b}^{\prime {\dagger}}{b}^{\prime }\right)-m\left({a}^{{\dagger}}a+{a}^{\prime {\dagger}}{a}^{\prime }\right)\left\vert {{\Psi}}_{\text{f}}\right\rangle .\hfill \\ \hfill & =n{\eta }_{\text{D}}\left(l{\eta }_{\text{B}}\left(1-{\eta }_{\text{A}}T\gamma \right)-m{\eta }_{\text{A}}T\left(1-{\eta }_{\text{B}}\gamma \right)\right)\hfill \end{align} \tag{ 21 }$$

The noise of the measurement ΔN_HybQ is obtained in a similar way to equation (8). The smallest detectable absorptivity change is derived as follows:

$$\begin{align}\hfill \delta {T}_{\text{HybQ}}& =\frac{{\Delta}{N}_{\text{HybQ}}}{\vert \partial {N}_{\text{HybQ}}/\partial T\vert }\hfill \\ \hfill & \approx \frac{\sqrt{{m}^{2}{\eta }_{\text{A}}T\left(1-\gamma {\eta }_{\text{B}}\right)+{l}^{2}{\eta }_{\text{B}}\left(1-\gamma {\eta }_{\text{A}}T\right)-2lm{\eta }_{\text{A}}{\eta }_{\text{B}}{\eta }_{\text{D}}T\left(1-\gamma \right)}}{\vert \sqrt{n{\eta }_{\text{D}}}{\eta }_{\text{A}}\left(m-\gamma {\eta }_{\text{B}}\left(m-l\right)\right)\left.\right)\vert },\hfill \end{align} \tag{ 22 }$$

which is approximated using the assumption α₁ ≪ 1. From the definition of the sensitivity shown in equation (2), the sensitivity of HybQAM is given by

$$\begin{align}\hfill {S}_{\text{HybQ}}& =\frac{\sqrt{T}}{\delta {T}_{\text{HybQ}}\sqrt{n}}=\frac{\vert \sqrt{{\eta }_{\text{D}}}{\eta }_{\text{A}}\left(m-\gamma {\eta }_{\text{B}}\left(m-l\right)\right)\vert }{\sqrt{{m}^{2}{\eta }_{\text{A}}\left(1-\gamma {\eta }_{\text{B}}\right)+{l}^{2}{\eta }_{\text{B}}\left(1/T-\gamma {\eta }_{\text{A}}\right)-2ml{\eta }_{\text{A}}{\eta }_{\text{B}}{\eta }_{\text{D}}\left(1-\gamma \right)}}.\hfill \end{align} \tag{ 23 }$$

S_HybQ will be maximized when m and l satisfy the following relations:

$$\begin{align}\hfill m& =-1+\gamma {\eta }_{\text{A}}T\left(1-{\eta }_{\text{B}}{\eta }_{\text{D}}\frac{1-\gamma }{1-\gamma {\eta }_{\text{B}}}\right)\hfill \end{align} \tag{ 24 }$$

$$\begin{equation}l=-{\eta }_{\text{A}}T\left(\gamma \left(1-{\eta }_{\text{D}}\right)+{\eta }_{\text{D}}\right).\end{equation} \tag{ 25 }$$

MATHEMATICA was used to find this condition. For an extreme example where γ = 0 and η_A = η_B = 1, ${S}_{\text{HybQ}}=\frac{\sqrt{{\eta }_{\text{D}}T}}{\sqrt{1-T{\eta }_{\text{D}}^{2}}}{\geqslant}{S}_{\text{QAM}}=\frac{\sqrt{{\eta }_{\text{D}}T}}{\sqrt{1+T\left(1-2{\eta }_{\text{D}}\right)}}$, whereas S_LTQAM = 0, which shows that the HybQAM is resistant to decreasing γ. Note that in an ideal case for LTQAM where η_A = η_B = γ = 1, S_HybQ becomes the same as S_LTQAM: ${S}_{\text{HybQ}}={S}_{\text{LTQAM}}=\frac{\sqrt{{\eta }_{\text{D}}T}}{\sqrt{1-T}}$.

Figure 5 shows the sensitivity to absorption while changing γ. As for figure 4, the detection efficiency η_D is set to be 0.80 and 0.67 for figureS 5(a) and (b), respectively. The other parameters η_A and η_B are set to be the same as those used in figure 4, namely 1 × 10⁻⁴. The red curve indicates the sensitivity for HybQAM. Compared with the LTQAM shown in figure 4 (red curve), the HybQAM maintains a sensitivity higher than both the QAM (blue curve) and the CAM (green curve) for any γ. Figure 5(b) shows the case for η_D = 0.67 where the sensitivity to absorption for the QAM is equal to the SNL (S_QAM = 1). We can see that the HybQAM exceeds the SNL for all γ even when the QAM does not show a quantum advantage.

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The above results show that improving γ is important for achieving a high sensitivity for both LTQAM and HybQAM. Although there have been no reported values of γ, it is possible to estimate it from the fringe visibility of a nonlinear interferometer. For instance, from the fringe visibility of 0.965 reported in reference [35], we can obtain a lower bound of γ of 0.931. Note that γ for this experiment would be higher when the loss in the setup is compensated. We believe that the recent intensive investigations of nonlinear interferometers will meant that nearly perfect mode-matching in nonlinear interferometer can be achieved in the near future.