Astrometry in two-photon interferometry using an Earth rotation fringe scan

Optical interferometers may not require a phase-stable optical link between the stations if instead sources of quantum-mechanically entangled pairs could be provided to them, enabling long baselines. We developed a new variation of this idea, proposing that photons from two different astronomical sources could be interfered at two decoupled stations. Interference products can then be calculated in post-processing or requiring only a slow, classical connection between stations. In this work, we investigated practical feasibility of this approach. We developed a Bayesian analysis method for the earth rotation fringe scanning technique and showed that in the limit of high signal-to-noise ratio it reproduced the results from a simple Fisher matrix analysis. We identify candidate stair pairs in the northern hemisphere, where this technique could be applied. With two telescopes with an effective collecting area of $\sim 2{\mathrm{m}}^2$, we could detect fringing and measure the astrometric separation of the sources at $\sim 100\mathrm{\mu }\mathrm{as}$ precision in a few hours of observations, in agreement with previous estimates.

Figure 1

Schematic picture of the two-photon amplitude interferometer. We assume that there are two sources which can be observed simultaneously from two stations, $\mathbf{L}$ and $\mathbf{R}$, with single spatial mode inputs $a$, $b$ and $e$, $f$. Both sources send out photons in the form of plane wave, the path length difference between the two stations yielding phase delays ${\delta }_1$ and ${\delta }_2$ between the photons observed at channels $a$, $e$ from source 1 and $b$, $f$ from source 2, respectively. If the two detected photons are close enough in frequency and arrival time, then the pattern of coincidences measured at the outputs $c$, $d$ and $g$, $h$ will be sensitive to the difference of the phase differences, $({\delta }_1-{\delta }_2)$, after interference at the symmetric beam splitter in each station.

Figure 2

Schematic drawing of the experiment geometry considering an arbitrary baseline and star pair. Two tangent planes are defined by a unit vector point from the center of Earth to the midpoint between the baseline, B, and the separation vector between the two sources. Two coordinates, $d_N$ and $d_E$, are introduced to characterize the position of the star pair where the origin is defined to be the midpoint of separation vector, which is the black dot between the two stars. $d_E$ points in the azimuth direction or into the page, and $d_N$ points up to the north pole.

Figure 3

Schematic explanation on how simulated coincidences are generated. (i) We first determine the period of each fringe cycle denoted by $\mathrm{\Delta }t$’s, and then the number of event occurrences in each fringe cycle (denoted by the orange points) is determined by Poisson distribution with expected number of coincidences equal to $\overline{n}\mathrm{\Delta }t$. Note that in practice there will be at most one event for each fringe cycle for a realistic $\overline{n}$ of a typical star pair; (ii) Next we determine the phase of each event in the cycle through inverse transform sampling from Eq. (6); (iii) Finally, we calculate the corresponding timestamp of each event based on their phase.

Figure 4

Comparison between the error estimation from the MCMC simulation to the error estimation using the Fisher’s analysis from our previous work [8] for different pair count multiplier. The vertical axis shows the ratio between the separation error of the two sources in the East-West direction from calculating the square root of the covariance matrix of the MCMC simulation and using Equation (13). The horizontal axis shows the coincidence pair rate multiplier used for boosting the sample data size. As the coincidence pair multiplier increases, the ratio gradually approaches 1 as expected.

Figure 5

Right: theoretical fringe patterns for the 15 arcsec separation star pair with East-West and North-South baseline orientation. Left: frequency of the signal changes over time. We observe that the oscillation frequency is higher for the East-West baseline compared to the North-South baseline as expected.

Figure 6

These triangle correlation plots are generated using the corner package [36]. Top left: 200 m North-South baseline with a telescope effective collecting area of $1.4m^2$ and star pair of 1 arcsec separation. Top right: 200 m East-West baseline with a effective collecting area of $1.4m^2$ and star pair of 1 arcsec separation. Bottom left: 200 m North-South baseline with a telescope effective collecting area of $3.9m^2$ and star pair of 15 arcsec separation. Bottom right: 200 m East-West baseline with a telescope effective collecting area of $3.9m^2$ and star pair of 15 arcsec separation. The 4 parameters used for the MCMC simulation are visibility, $\mathrm{\Delta }{d}_E$, $\mathrm{\Delta }{d}_N$, and a constant offset phase, $\psi$. The vertical dashed lines represent 2.3%, 16%, 50%, 84%, and 99.4% quantiles of the Gaussian. $\mathrm{\Delta }{d}_E$ and $\mathrm{\Delta }{d}_N$ are defined to be the offset from the 50% quantile of the Gaussian measured in miliarcseconds. The orange points indicate the true value of each parameter. The contour plots characterizing the correlation between visibility and the two separation parameters have an overall spread behavior as visibility approaches 0. This is expected since likelihood approaches constant as visibility goes to 0.