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Abstract
Classical optical interferometry requires maintaining live, phase-stable links between telescope stations. This requirement greatly adds to the cost of extending to long baseline separations and limits on baselines will in turn limit the achievable angular resolution. Here we describe a novel type of two-photon interferometer for astrometry, which uses photons from two separate sky sources and does not require an optical link between stations. Such techniques may make large increases in interferometric baselines practical, even by orders of magnitude, with a corresponding improvement in astrometric precision benefiting numerous fields in astrophysics. We tested a benchtop analogue version of the two-source interferometer and unambiguously observe correlated behavior in detections of photon pairs from two thermal light sources, in agreement with theoretical predictions. This work opens new possibilities in future astronomical measurements.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction and basic concepts
Classical optical Michelson interferometers collect photons from a sky source into two or more sub-apertures which are then transported through optical links and brought into interference at a common point. For a pair of sub-apertures separated by a baseline $B$, the measured interference pattern is sensitive to features of the source with angular size $\delta \theta \sim \lambda /B$, where $\lambda$ is the photon wavelength. The optical link between the stations must meet demanding requirements, with lengths remaining stable to within a fraction of a wavelength. This makes the Michelson interferometry expensive and difficult to extend to long baselines, which in turn limits the achievable resolution [1,2].
The use of quantum optics to improve the precision of astronomical measurements is a long-desired goal of both the optical and astronomical communities. In particular, the seminal Gottesman-Jennewein-Croke (GJC) proposal [3] attracted attention as a way to build a quantum-enhanced telescope, i.e. a very-long-baseline interferometer enabled by the use of quantum optical effects. However, the GJC proposal is dependent on the use of quantum repeaters [4–7], a technology which still requires a substantial amount of development, limiting practical implementations of the technique. Alternatively, the proposed Stankus-Nomerotski-Slosar-Vintskevich (SNSV) approach [8] relies solely on two sky sources and the two-photon Hanbury Brown and Twiss (HBT) and Hong-Ou-Mandel (HOM) effects. The SNSV scheme simplifies the requirements for an optical astronomical measurement, providing a practical pathway to achieve the goal of more precise astrometric measurements for astronomical objects, and is based on the same two-photon interference phenomena, while avoiding employment of quantum repeaters. Other theoretical techniques for improved resolution on imaging the starlight were also introduced and some proof-of-principle demonstrations were successfully performed [9–15].
Here we describe a proof-of-principle experiment based on the SNSV scheme [8,16,17], shown schematically in Fig. 1(a). In brief, photons are collected from two sky sources in two observing stations. At each station, light from the two sources is directed into beamsplitters for interference. Correlations of photon detections between the four detectors are sensitive to the relative phase difference of incoming photons from the two sources. From this, the opening angle between the sources can be determined, allowing for longer baselines when compared to classical interferometry, since it entirely removes the need for a physical optical path between the stations. Therefore, such an approach could improve astrometric precision.
Fig. 1. (a) Concept of the proposed Stankus-Nomerotski-Slosar-Vintskevich (SNSV) two-photon interferometer, which interferes and detects photons, shown as plane waves, from two astronomical sources. (b) Equivalent scheme for the tabletop implementation. The argon lamps 1 and 2 indicate the two ports associated with input spatial and polarization modes. Each output detection port $D_{1}, D_{2}, D_{3},$ and $D_{4}$ corresponds to detectors in two observation stations $L$ and $R$ in (a). The phase delays in each arm of the interferometer are labeled as $\delta _{1}, \delta _{2},\delta _{3},\delta _{4}$ and correspond to the detected spatial modes.
The measured coincidence rates of the outputs can be described by a second-order intensity correlation function $\Gamma _{jk}$ which has the simplified form
where $A$ and $B$ are the coefficients dependent on the photon polarization and coherence properties, and $\delta _{j,k}$ are the phases for given modes as described in Fig. (1(b)).We provide a detailed theoretical derivation of the intensity correlation function in quasi-monochromatic approximation in the Supplement focusing on the polarization degrees of freedom, and on the relationship between the mode indistinguishability and interference effects. There, we predict that the output detection ports from Fig. 1 should form pairs such that: detectors 1&3 should be anti-correlated with detector pairs 1&4 and 2&3, however, detectors 1&3 should be correlated with 2&4; the detector pair 1&4 would be correlated with the detector pair 2&3, however, the detector pair 1&4 should be anti-correlated with detector pairs 1&3 and 2&4. We also predict that coincidences of the detector pairs 1&2 and 3&4 would have a stationary rate without oscillations, a non-trivial correlation pattern.
The basis of the proposed technique relies on two phenomena: the Hanbury Brown-Twiss (HBT) [18] and Hong-Ou-Mandel (HOM) [19] effects. We expect the two photons to bunch within their coherence time as in the HBT effect, resulting in the HBT peaks in the time difference distributions. In addition to this, we predict that the population of the HBT peak would depend on the relative phase difference between the two photons. The HOM effect would play a role when correlations between two outputs of the same beamsplitter are considered, leading to two indistinguishable photons coalescing at a beamsplitter output. Note that the presented scheme has similarities with the Noh-Fougères-Mandel experiments [20–23] but, in contrast to these, the present work employs a second-order correlation analysis, which describes coincidences between pairs of detectors. The optical effects discussed above are a manifestation of the interference of two indistinguishable photons. To be indistinguishable the photons must have similar frequencies and arrive at the beamsplitters at the same time. These arguments establish requirements on the timing and spectral resolution for the interferometer instrumentation [16,24–26]. We also note that while appropriate for the proof-of-principle measurements, polarization analysis is difficult in actual telescopes.
An important clarification to make is about the role of the HBT effect. The SNSV proposal can be considered as a variation of the HBT effect. An important distinction is that the original HBT effect applies to two photons from the same source, in that case from the same star, while the SNSV scheme, also a two-photon technique, relies on two photons from two different stars. Another difference is that the original HBT effect requires thermal light, while in the SNSV proposal thermal light is not a requirement. The SNSV scheme would also work for coherent light (e.g. a laser) and single-photon light (e.g. a SPDC source). The reason we are using thermal light in these measurements is that we plan to use the SNSV scheme for astronomical measurements with stars, which are natural thermal light sources.
The SNSV technique has parallels with the GJC approach [3], which employs a path-entangled source of single photons distributed between two observing stations, and its variations [27–33]. In the SNSV proposal, photons provided by the second thermal light source would replace photons from a single photon source, typically a spontaneous parametric down-conversion (SPDC) source [34,35]. While an on-sky light source is inherently thermal, we note that this has important practical advantages, compared to the above-mentioned quantum source, such as the complete absence of the direct optical link between the stations and uniformity of instrumentation for observations, when all stations have the same detection systems. Assuming the spectral binning that would be fast spectrometers with single photon sensitivity.
The two-photon interferometer presented here allows for precision improvement, by orders of magnitude, in principle, which could benefit numerous areas of cosmology and astrophysics. There are many scientific opportunities that would benefit from substantial improvements in astrometric precision such as testing theories of gravity by direct imaging of black hole accretion discs, precision parallax for the cosmic distance ladder, mapping microlensing events, peculiar motions, and dark matter; see Ref. [8] for a more comprehensive discussion.
As a numerical example, it was determined in Ref. [8] that a nominal precision on the order of 10 $\mu$as on the opening angle between two bright stars of magnitude 2 in a single night’s observation could be reached for baselines of 100 m and wavelength of 1 $\mu$m. We note that the estimate is based on fundamental limitations due to the photon statistics and it does not take into account possible systematic uncertainties of a real instrument. For astronomical measurements under real conditions, a number of practical issues must be taken into consideration, such as atmospheric fluctuations. Also, as with any two-photon intensity interferometry scheme, requiring a coincidence results in low count rates compared to the single-photon Michelson scheme. A comprehensive comparison of precision for the amplitude versus intensity interferometry for astronomy was presented recently in [36].
In the following, we start with discussing the benchtop experimental setup of the interferometer in Section 2. We then focus on the methods and results in Section 3. Section 4 provides a discussion of the results. Lastly, we present the conclusions and the future outlook in Section 5. Detailed theoretical considerations and extended experimental setup description are presented in the Supplement 1.
2. Experimental setup
The experimental setup utilized in the measurements is shown in Fig. 1(b). Two argon lamps with isolated 794.82 nm spectral lines were used as sources of thermal photons. The photons were directed to four 50:50 non-polarizing beamsplitters to arrange interference as in the original scheme in Fig. 1(a). The photon phases remained stable to environmental disturbances for extended periods of time and could be adjusted deterministically with specialized phase shifters to study the phase dependence. The shifters were implemented as small angle glass wedges, which can be moved laterally in fine steps. The four outputs of the interferometer were instrumented with fast single-photon detectors. Two types of detectors were used, Single Photon Avalanche Diodes (SPAD) [37–39], and Superconducting Nanowire Single-Photon Detectors (SNSPD) [40–42], both with a temporal resolution of the order of 100 ps. The SPAD/SNSPD digital output signals were then sent to a time-to-digital converter (TDC) module for time stamping. Typical single photon rates for the argon lamps operating at full power were about 200-400 k counts per second per detector depending on the configuration. A detailed description of the setup is provided in the Supplement 1.
We employed the following experimental procedure. Firstly, one of the phase shifters was moved in small steps by a distance of about 0.45 mm over the duration of 15 minutes, which corresponded to a shift by five wavelengths, followed by a pause of two minutes. Then, the second phase shifter was moved in the same manner followed by another two-minute pause. The total duration of the undisturbed measurement was about 35 minutes. The photon time-stamps of the four detectors were continuously logged on disk for post-processing. A variety of runs were performed with two different detector types and varying polarizer configurations. These configurations included experiments with unpolarized photons, when the polarizers were removed from the beam paths, and experiments with different relative polarizations for the two lamps.
3. Methods and results
The main goal of the analysis was to determine the dependence of two-photon correlations on the relative phase of the photons. Algorithms were developed to condition the raw data by removing the afterpulses, then to find coincidences of photon pairs in different detectors to identify the HBT peaks, and to determine the dependence of the peak population on the photon phase.
3.1 HBT peaks
The HBT peaks appear in the time difference distributions of detector pairs due to the two-photon interference yielding photon bunching [18]. The characteristic shape of the distribution is determined by the convolution of the corresponding photon coherence time due to the spectral width of the argon line, and timing resolution [16]. We study this effect by analyzing the distribution of photon detection time differences by combining various detector channels. The analysis algorithm searches for pairs of single photons detected within 20 ns of each other.
In previous work [16], we also determined that timing resolution is the predominant contribution to the HBT peak temporal width, so we model the HBT peaks with a normalized $g^{(2)}$ autocorrelation function of this form:
As expected, prominent peaks are seen for most of the six combinations of detectors, as shown in Fig. 2. SubFigs. 2(a) and 2(b) show the HBT peaks with the vertical-vertical (VV) and vertical-horizontal (VH) input polarization orientations, respectively. Time offsets in the peak positions are due to varying path lengths in different detectors including small differences in the length of optical fibers. Note that for the VH case, the interference is happening for two photons from the same source; VH photon pairs from different sources do not contribute. We discuss properties of the HBT peak distributions in Section 4.
Fig. 2. Normalized coincidence count rates of two-photon detections as a function of the time difference between them, $\Delta t$, for different detector combinations and different input polarizations, as labeled. The peak in each case indicates the enhanced correlation between two photons, calibrating where simultaneous pairs will appear. (a): results with both polarizers aligned vertically, called the VV configuration. (b): results with one polarizer aligned vertically and the other polarizer aligned horizontally, called the VH configuration. The parameters $t_0$ and $\sigma$ are the time offset and the width of the Gaussian fitting, respectively, according to Eq. (2). The peaks are signatures of the Hanbury Brown-Twiss (HBT) effect, appearing due to the photon bunching.
3.2 Coincidence rates
As the first step, the HBT peaks of the entire 35-minute dataset were fit with a Gaussian function to determine the peak width $\sigma$ and the central value. Then the coincidence rate for detector pairs was determined using two techniques. In the first, simple approach, a window of $\pm 1.5 \sigma$ around the HBT peak central value was selected, where $\sigma$ was taken from the corresponding fit of the peak. Then the number of entries in this window in a predefined time bin, typically 10-30 sec, was determined and plotted as a function of time as shown in Fig. 3 for all six detector combinations for the SNSPD data set with polarizers. SubFigs. 3(a) and 3(b) show, respectively, the number of coincidences in the 20 sec time bins with VV and VH polarizer orientation at the interferometer inputs. It is expected that these time trending plots will have oscillatory behavior in the VV case due to the advance of the photon phase caused by the phase shifters as predicted in Eq. (1). We indeed observe this behavior, but we defer a detailed discussion of the main features of these measurements to Section 4. We presented more examples of the raw coincidence rate information in [17].
Fig. 3. Number of two-photon detections within a $\pm 1.5\sigma$ window about $t_0$ of the HBT peak in 20 sec time bins plotted versus time for the SNSPD data set with polarizers. The time accounts for the real time during which the phase shifters were slowly moving a preset distance corresponding to five wavelengths, then paused for two minutes, then moved again for five wavelengths. The graphs show results for six combinations of detector pairing, 1&2, 1&3, 1&4, 2&3, 2&4 and 3&4. (a): Results with VV configuration of polarizers. (b): Results with VH configuration of polarizers. See the text for discussion.
In a more sophisticated approach, the HBT peak in each predefined, sequential time bin was fit with a Gaussian profile, only allowing the peak amplitude and background detection level to vary. The peak center $t_0$ and standard deviation $\sigma$ of these fits were held constant at the values determined from the initial fit of the entire dataset’s overall HBT peak. Then the area under each of these Gaussian peaks within $\pm 1.5 \sigma$ of the peak center was determined, effectively subtracting the background detection rate, and plotted as a function of time. We show an example of the resultant data in Fig. 4.
Fig. 4. Two-photon coincidences count rates for the oscillations for detectors 1&3 in the SNSPD data set with VV polarization fit to Eq. (3). The coincidences rates were determined by fitting a Gaussian peak in each 10-second time bin. Data points are presented together with one standard deviation error bars.
Mathematically this result can be described as the convolution of the second-order correlation function with a filtering function, see Eqs. (S13) and (S14) in Supplement 1. All fits in this method were made using LMFIT [43], which was also used to explicitly calculate uncertainties on the parameter best-fit values. This technique should give better statistical accuracy since it uses more statistics for the flat background and can also account for a slow drift in the background coincidence count level.
To determine the visibility and relative phase, the oscillatory behavior of trending plots due to the phase evolution was fit with a cosine function:
4. Analysis and discussion
Oscillations in the trending plots were fit using a cosine function as explained above, from which the main parameters were extracted. These parameters included the oscillation period, visibility, and phase. Out of these, the latter two, visibility and relative phase, are critical parameters, unambiguously predicted by the theory.
4.1 Oscillations in coincidence rate
We show the coincidence rates for all detector combinations in Fig. 3. As previously shown in Eq. (1), the detector pairs on the opposite interferometer arms (pairs 1&3, 1&4, 2&3, and 2&4) are expected to oscillate in or out of phase, while detector pairs on the same interferometer arm (pairs 1&2, 3&4) are not expected to oscillate. Since for the HOM effect only indistinguishable photons are supposed to interfere, we do not expect any oscillations in the VH configuration of polarization.
Only the first five oscillation periods were used for the cosine fits in these results extracting the relative phase, which was calculated with respect to the detector pair 1&4 phase, and the visibility, calculated using Eq. (4). This analysis was performed for several datasets: VV polarized and unpolarized datasets with SNSPD detectors, and an unpolarized dataset with SPAD detectors. We summarize the measured visibility for the cross-station detector combinations and for the datasets in Fig. 5(a) and the relative phases, in radians, in Fig. 5(b). The measured parameters all behave as expected. As can be seen, the visibility was higher for the polarized dataset, as expected, by approximately a factor of two compared to the unpolarized dataset. In the unpolarized case, the visibility of oscillations for the SPAD detectors were consistently smaller compared to the visibility of the peaks for the SNSPD detectors. This can be explained by the inferior timing resolution of SPAD detectors.
Fig. 5. (a) Visibility of oscillations for the two-photon coincidences count rates. (b) Relative phases, in radians, were calculated with respect to the detector pair 1&4 phase. Colors correspond to different datasets. Pol and noPol are the VV polarized and unpolarized datasets, respectively. Data points are presented together with one standard deviation error bars.
We also checked the behavior of the interferometer for varying polarizations of the two beams. Figure 6 shows the visibilities of two-photon detections as a function of polarization angle for detector pairs 1&3, 1&4, 2&3, and 2&4. Here, one input polarization angle was varied in $22.5^\circ$ increments from $0^\circ$ to $180^\circ$ starting from either VV ($\theta = 0$) or VH ($\theta = \pi /2$) configuration, while the other polarizer was not moved. The dependence on the relative angle for the two polarizations was changing in anti-phase for these two cases, as expected. We compare the experimental data to the theoretical predictions, as in Eq. (5), derived in Supplement 1, and find a good agreement.
Here. $r_{ij}$ are the normalized rates and the visibilities $V_{V\theta }$ are defined in Supplement 1.
Fig. 6. Visibility of two-photon detections as a function of polarization angle, for detector pairs 1&3, 1&4, 2&3, and 2&4. Here, one input polarization angle was varied in 22.5$^\circ$ increments from 0$^\circ$ to 180$^\circ$ starting from either VV or VH configuration, while the other polarizer was not moved. Different colors refer to different configurations as indicated. Data points are presented together with one standard deviation error bars. The measured experimental data is fitted by Eq. (5). To fit the results we added to the fit function a small constant offset, which described the accidental coincidence counts.
4.2 HBT peak visibility, cancellation of HOM and HBT effects
As already discussed in Section 3 the HBT peaks were fit to Gaussian peaks. The resulting timing resolution ($\sigma$) was found to be equal to 280 ps and 140 ps, respectively for SPAD and SNSPD detectors. This is in agreement with the expected timing resolution of those detectors assuming a time difference measurement for two independent photons. A graph summarizing the visibilities, introduced in Eq. (2), for all combinations of detector pairs in the main datasets is shown in Fig. 7. Similar to the oscillation visibility, the HBT visibility was significantly higher for the polarized case, decreasing for the unpolarized case and for the SPAD detectors, which have worse timing resolution.
Fig. 7. Visibility of the HBT peaks presented in Fig. 2 for various datasets. Colors correspond to different datasets. Pol and noPol are the VV polarized and unpolarized datasets, respectively. Data points are presented together with one standard deviation error bars.
We also note that the HBT peak visibility is almost zero for the 1&2 and 3&4 pair combinations in the VV configuration. These particular detectors evaluate coincidences of two photons exiting the two opposite sides of the same beamsplitter so their coincidence rate will have a dip due to the Hong-Ou-Mandel (HOM) effect. This will cancel out the photon bunching peak due to the HBT effect. However, the results in Figs. 2 and 7 show it’s not an exact cancellation, which is likely due to the nonideal equalization of the interferometer arms, also a valid explanation for the difference in visibilities in Fig. 5(a) and in the middle four detector combinations in Fig. 7.
5. Conclusions and outlook
In this work we implemented a proof-of-principle demonstration of the SNSV proposal for precision astrometry [8] and described bench-top experiments with a two-photon interferometer. The results, in particular the observed phase dependence, confirm the predicted functionality of the proposed instrument, suggesting that it is a viable experimental approach that can improve astrometric precision.
The next steps in the exploration of this approach are observations of on-sky light sources. We note that the phase shifting performed in the above experiments and corresponding oscillations in the coincidence rates are directly analogous to the Earth rotation fringe shifting [44]. Due to Earth’s rotation, the effective baseline between the two stations change, which induces changes in the interference patterns, in particular, in the frequency of corresponding fringe oscillations. The frequency value is predicted in [8] to be proportional to the opening angle between the stars. The uncertainty of the fringe rate has a more favorable scaling with the observation period than simple photon statistic methods, so it can be considered a promising observable for the quantum astrometry approach. However, it would be instructive for applications to review and compare the capabilities of the presented method with already existing methods such as super-resolution techniques with spatial mode demultiplexing for a single telescope, known as SPADE technique [10,45] and, also, the BLESS technique, based on the target beam modulation and examination of shot statistics [46]).
Star spectra are typically broadband, so another obvious extension of the technique is spectroscopic binning, which would allow us to estimate the observables in numerous spectral bins [47]. Each bin would act as an independent experiment, so the sensitivity of the interferometer would improve as $\sqrt {N}$, where $N$ is the number of spectral bins. The sensitivity also improves with a larger number of stations and better timing resolution, as discussed in Ref. [8].
Though we operate here with thermal states of light, which have classical Gaussian quasiprobability distribution, we consider a quantum description of these phenomena, as in this work, to be very instructive. It emphasizes the role of mode (or path) indistinguishability in the quantum interference phenomena. Moreover, this description can be extended further by employing the quantum Continuous Variable (CV) formalism [48]. The CV formalism will play an important role in expanding the presented technique to multiple observing stations assisted by auxiliary ground-based quantum states such as squeezed states and non-trivial quantum channels [49–54], providing novel opportunities in astronomy for extraction of valuable information. At the same time, the complexity of analytical descriptions and data processing would grow tremendously with an increased number of stations and auxiliary quantum states involved in the measurements, including entangled ones [3,27,28,55], so an important future research goal is to provide a theoretical description of such expansions based on both entities, the Gaussian states and quantum channels, powered by machine-learning methods [56,57].
In summary, we have built and characterized a tabletop prototype of a two-photon interferometer, which could improve the astrometric precision by orders of magnitude, by means of enabling extra long baselines between observing stations. The implemented interferometer allowed us to test the important features of the SNSV proposal. In particular, it allowed us to demonstrate that the relative phase of two photons from two independent thermal sources has a direct effect on their bunching due to the HBT effect. The approach demonstrated here is technically feasible with existing technologies of single photon detection [16,44] and allows us to move towards measurements with on-sky sources. This work represents a major step towards quantum telescopes.
Funding
Office of Science (SULI, FWP PO202).
Acknowledgments
This work was supported by the U.S. Department of Energy QuantISED award and BNL LDRD grants 19-30 and 22-22. A.M. acknowledges support under the Lourie Fellowship from the Stony Brook University Department of Physics and Astronomy. We are grateful to Jonathan Schiff and Rom Simovitch for the software development. S.V. is grateful to Rene Reimann and Konstantin Katamadze for discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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