Resolution and sensitivity of a Fabry-Perot interferometer with a photon-number-resolving detector

With photon-number resolving detectors, we show compression of interference fringes with increasing photon numbers for a Fabry-Perot interferometer. This feature provides a higher precision in determining the position of the interference maxima compared to a classical detection strategy. We also theoretically show supersensitivity if $N$-photon states are sent into the interferometer and a photon-number resolving measurement is performed.

Figure 1

(Color online) Fabry-Perot cavity with complex amplitudes $R$ and $T$ for reflected and transmitted modes, respectively, and incident intensity $I$. Each mode can be assigned a mode operator marked by the hat to quantize the respective mode, where the subscripts $u$ and $d$ stand for up and down. We assume, for simplicity, that both mirrors have identical complex reflection and transmission coefficients denoted with $r$ and $t$.

Figure 2

(Color online) Effective beam-splitter for the Fabry-Perot cavity with complex amplitudes $R$ and $T$ for reflected and transmitted modes, respectively, which satisfy the conditions ${\left|T\right|}^2+{\left|R\right|}^2=1$ as well as $T{R}^{\ast }+R{T}^{\ast }=0$.

Figure 3

(Color online) (a) shows the classical normalized transmission function as a result of Eq. (5) displayed for a reflectivity of ${\left|r\right|}^2=70\%$ (LHS) and ${\left|r\right|}^2=90\%$ (RHS) of the FPI-mirrors. Note that the transmission peaks become narrower and the maxima slightly shifted to the left as the reflectivity increases. Also the minima go to zero for 90% reflectivity whereas they do not reach zero for lower reflectivities. (b) shows the probabilities of detecting $k$ photons, $p_k^{\text{coh}}$ for a single-mode coherent input state $|\alpha ⟩$ with mean photon number $\overline{n}={\left|\alpha \right|}^2=4$ incident on the FPI and a photon-number resolving measurement displaying $k=1,\mathrm{...},4$ (solid lines). The reflectivity ${\left|r\right|}^2$ of the mirrors is 70%. The dashed lines shows the transmission probabilities $p_k^{\text{F}}$ for a single-mode photon-number state $|4⟩$. We observe that the Fock states (dashed lines) show transmission peaks that are sharper in general. This effect is equivalent to operating at a larger reflectivity of the mirrors, which is demonstrated in (a) for the classical curves. A major difference appears for $k=4$ where the transmission maxima reach one for the Fock state, whereas the coherent state stays low. Here and in the following, we choose a reflectivity of 70% as the important features are more pronounced than for larger reflectivities.

Figure 4

(Color online) Transmission probabilities for a single-mode photon-number state and a photon-number resolving measurement, for photon numbers $n=1,2,3$. The free spectral range $\Delta L/\lambda$ is the distance between two adjacent maxima. Transmission probabilities are calculated for 70% reflectivity of the mirrors.

Figure 5

(Color online) Dimensionless uncertainty $\delta L_k^{\text{coh}}/\lambda$ (where $L$ is the length change of the FPI and $\lambda$ is the wavelength of the coherent laser beam) for a mean intensity measurement with $\overline{n}=4$ compared to photon number resolving measurements $k=1$ and $k=4$. The solid line also represents the shot-noise limit. The reflectivity ${\left|r\right|}^2$ of the mirrors is 70%.

Figure 6

(Color online) Transmission experiment with an $N$-photon state incident on a Fabry-Perot interferometer and photon-number resolving detection of the output.

Figure 7

(Color online) Dimensionless uncertainty $\delta L_k^{\text{F}}/\lambda$ (where $L$ is the length change of the FPI and $\lambda$ is the wavelength of the light) for a mean intensity measurement with $\overline{n}=4$ compared to photon number resolving measurements $k=1$ and $k=4$. The solid line also represents the shot-noise limit. The reflectivity ${\left|r\right|}^2$ of the mirrors is 70%. The shot-noise limit (solid line) is beaten by the $k=4$ curve.

Figure 8

(Color online) Comparison of the sensitivity $\delta L/\lambda$ [Eq. (12)] for a coherent state (solid line) and mean intensity detection (shot-noise limit) as a function of mean number of detected photons versus a $n$-photon state $|n⟩$ with $n$-photon resolving measurement [Eq. (15)] as a function of the photon number, where we take $n=\overline{n}$, slightly away from the transmission maximum. The reflectivity ${\left|r\right|}^2$ of the mirrors is 70%. The parameter of the phase is chosen for each $n$, $\overline{n}$, respectively, so that the sensitivity is at its minimum.

Figure 9

(Color online) Transmission experiment with a weak coherent laser beam incident on a length tunable FPI and a fiber coupled TES. The light is collected and coupled into a single-mode fiber and transmitted to the detector.

Figure 10

Sample histogram of the output pulse integrals of the photon-number resolving TES used in our experiment. The histogram indicates the probabilities of detecting each photon number, as well as the probability no photons were detected (labeled “0”). The vertical lines show the thresholds between the individual photon-number peaks.

Figure 11

(Color online) Transmission probabilities for a single-mode coherent state and a photon-number resolving measurement, for photon numbers $k=1,\mathrm{..},7$. The blue solid lines show the individual theoretical fits. The peaks become narrower as $k$ increases. The fit yields the mean photon number $\overline{n}\approx 3.9$ and the reflectivity of the mirrors $\approx 91\%$. The graph at the bottom right is the reconstructed classical signal from the photon-number resolved curves with $p^{\text{coh}}={\sum }_{k=1}^{7}k{p}_k^{\text{coh}}$. The black solid line is the theoretical fit. From the fit the mean photon-number is determined as ${\overline{n}}^{\prime }=2.6$, which is not in good agreement with the mean photon-number obtained from the photon-number resolved curves of $\overline{n}\approx 3.9$ and reflects insufficient stability of the used FPI. We also only have access to photon-number counts in the range $1\le k\le 7$ which amounts to reconstructing the classical signal with an amplitude 15% too low.

Figure 12

(Color online) Figure of the standard deviations as a function of $k$ presented in Table .