Chromatic interferometry with small frequency differences

By developing a `two-crystal' method for color erasure, we can broaden the scope of chromatic interferometry to include optical photons whose frequency difference falls outside of the 400 nm to 4500 nm wavelength range, which is the passband of a PPLN crystal. We demonstrate this possibility experimentally, by observing interference patterns between sources at 1064.4 nm and 1063.6 nm, corresponding to a frequency difference of about 200 GHz.


INTRODUCTION
Chromatic interferometry refers broadly to experiments which leverage quantum superposition in frequency-space to recover hidden phase information encoded in correlations among photons with different wavelengths [1]. Recently, chromatic interferometry has attracted increasing attention, both for its intrinsic interest and for its possible utility in highresolution imaging and photonic computation [1][2][3][4][5][6][7].
Color erasure is the essential technology enabling chromatic interferometry. Only when the frequency difference between the photons is surpassed by the response of detector can interference be measured [8][9][10]. Information which identifies wavelength (e.g., specifically, energy deposit) is registered in the detection apparatus, even if it is not readily accessible to an experimentalist. Wavelength information is generally harder to erase than polarization or path information, and so color erasure poses an interesting challenge.
The purpose of "color erasure detectors" [1,2] is to erase all wavelength identifying information, thus enabling chromatic interference. The use of such detectors goes beyond previous experiments in chromatic interferometry which implement wavelength conversion either at the light source, or at beamsplitters [3,4,6,[11][12][13]. By contrast, color erasure detectors can recover phase information between different wavelengths of light after interference or phase accumulation has occurred.
Ironically, a significant limitation of existing color erasure detectors is that they can only render photons indistinguishable when their frequency difference is sufficiently large. In order to render reception of two optical photons with frequencies f 1 < f 2 indistinguishable, they employ three-wave mixing with a coherent source at frequency f 3 = f 2 − f 1 . Appropriate crystals or waveguides that implement the mixing are available if f 3 corresponds to a wavelength in the 400 nm to 4500 nm wavelength range, but not otherwise. This consideration significantly restricts the frequencies f 1 and f 2 and thus the scope of applications.
Here we develop a more general method of color erasure, which allows f 2 − f 1 to be very small. We demonstrate its soundness and practicality by performing chromatic intensity (Hanbury Brown-Twiss) interferometry [14,15] between sources with 1064.4 nm and 1063.6 nm photons. Hanbury Brown-Twiss interferometry plays an important role in quantum optics [16] and has wide applications in astronomy and fluorescence microscopy [14,[17][18][19], and so our experiments lay the groundwork for new chromatic generalizations and technologies.
Let us first describe our protocol heuristically, to provide intuition for the mathematics to follow. Consider a superposition of photons with wavelengths f 1 , f 2 . Using a beamsplitter, we can transform this state into a (further) superposition of two distinct spatiotemporal modes. Let us put the photons in the first mode through a PPLN waveguide [20] pumped with a coherent state of many ∆f 31 photons. In this way, we induce upconversions f 1 → f 3 and f 2 → f 2 . Similarly, let us put photons in the second mode through a second PPLN waveguide pumped with a coherent state of many ∆f 21 photons, inducing upconversions f 1 → f 1 and f 2 → f 3 . Then we can filter both beams to allow only photons with frequency f 3 , and finally recombine the two beams using a second beamsplitter. This processing and filtering renders it impossible to determine whether the triggering photons had frequency f 1 or f 2 . Now let us treat this mathematically. Let |Ω be the vacuum state, and let a † γ create a γ photon in some fixed spatiotemporal mode. Then, for instance, a † γ a † γ would create two photons γ and γ in the same fixed spatiotemporal mode. For simplicity, consider the initial state where |α| 2 + |β| 2 = 1. This state corresponds to a superposition of a γ 1 photon and a γ 2 photon in a single spatiotemporal mode. Consider a second spatiotemporal mode, with corresponding creation operators given by b † γ . A 50-50 beamsplitter between the first and second spatiotemporal modes corresponds to for all γ. Applying such a 50-50 beamsplitter to |Ψ 0 , we obtain Evolution of the first (second) mode, propagating through a PPLN waveguide pumped with a coherent state of a large N number of ∆f 31 (∆f 32 ) photons, is described by the Hamiltonian H 31 (H 32 ) where The ξ parameters control the speed of up-and downconversion, and the φ parameters dictate the phases accumulated by the converted photons during the process. These effective Hamiltonians, which cause the f 1 and f 2 photons to become entangled with the large N coherent state of the pump, were derived in [1] using a systematic 1/N expansion. As emphasized in [1], large N coherent states are physically essential, since we want to 'lose track' of the loss or gain of single photons. In our setup, we consider the combined Hamiltonian and evolve (3) by e −iHT . We apply a second 50-50 beamsplitter to both spatiotemporal modes, and finally filter to γ 3 photons in the first outputted spatiotemporal mode. This corresponds to projecting onto a † γ3 |Ω . The resulting state is where θ ij ≡ T ξ ij . These angular θ ij parameters control the amount of up-and down-conversion that have occurred between the photons with frequencies f i and f j . By tuning φ 31 = φ 32 = 0, and say θ 31 = π/2 and θ 32 = 2π, we get Putting everything together, we have as was desired. What we have effectively done is mapped a † γ1 → 1 √ 2 a † γ3 + · · · and a † γ2 → 1 √ 2 a † γ3 + · · · , and then postselected onto the outcome of receiving a γ 3 photon.
Our arrangement in its entirety embodies a single color erasure detector. Equation (9) summarizes the manner in which the detector decoheres a state [21,22]. Color erasure is achieved through an entangling measurement, as described above.
We conclude this section by seeing how color erasure detectors allow us to perform Hanbury Brown-Twiss interferometry with sources having distinct wavelength. Here we will be schematic, but full details can be found in [1,2]. Suppose we consider the standard Hanbury Brown-Twiss experiment with two sources of the same wavelength. Let a † γ and b † γ denote creation operators for γ photons at the locations of two detectors A and B, respectively. Suppose each source emits a single photon at some moment in time. Then once the photons have reached the detectors, we will have a state which allows us to extract |α+β| 2 = |α| 2 +αβ * +α * β +|β| 2 corresponding to the probability that each detector received exactly one of the two photons (i.e., a coincidence count). This probability crucially contains an interference term αβ * + α * β, which encodes desired phase information in the Hanbury Brown-Twiss setup.
By contrast, if the first source emits photons of wavelength γ 1 and the second source emits photons of wavelength γ 2 , then the analog of Eqn. (10) is Since a † γ1 b † γ2 |Ω and a † γ2 b † γ1 |Ω are orthogonal, we can only extract |α| 2 and |β| 2 via measurement, and so we do not have access to the interference term αβ * + α * β. To gain access to this interference term, we can let detectors A and B be color erasure detectors, taking |Ω as per (9). Accordingly, (11) becomes from which we can extract |α + β| 2 and the desirable interference term αβ * + α * β by determining the frequency of coincidence counts of γ 3 photons at the color erasure detectors A and B.
In summary, color erasure detectors allow us to perform Hanbury Brown-Twiss interferometry using the standard procedure, even when the sources have distinct wavelength. We will experimentally implement this color erasure version of Hanbury Brown-Twiss interferometry in the next section. EXPERIMENT We have implemented the theoretical proposal given above and used the resulting detectors to perform chromatic intensity interferometry. As shown in Figure 1, 1064.4 nm photons and 1063.6 nm photons, prepared in weak coherent states, meet at a 50-50 beamsplitter labeled BS1. The linewidth of the photons is about 1 kHz. The 1064.4 nm photons and 1063.6 nm photons will not mutually interfere because their frequency difference is about 200 GHz. To recover chromatic interference, we build up two color erasure detectors, each having a traditional Si single photon detector, two beamsplitters, a special-made PPLN waveguide [20], a pump laser, and a filter. Figure 1 shows a diagram of the setup. After the initial BS1, the superposed mixture of 1064.4 nm photons and 1063.6 nm photons is further split and superposed by BS2 and BS3. A delay controller is inserted before BS2 to control the phase of the photons. The two paths emanating from BS2 go into two separate PPLN waveguides (denoted by PPLN WG 1 and 2), and similarly the two paths emanating from BS3 go into separate waveguides (labeled PPLN WG 3 and 4). We use a 1548.7 nm laser with about 100 kHz linewidth to pump PPLN WG 1 and 3, and a 1550. When SPD A or SPD B receives a 630.8 nm photon, it in principle cannot tell if the photon was originally 1064.4 nm or 1063.6 nm. Every photon arrival time at SPD A and SPD B is recorded by a time-to-digital converter (TDC). To observe chromatic Hanbury Brown-Twiss interferometry, we measure the g (2) correlation. Our calculation of g (2) (τ ) amounts to where n coincidence is the number of coincidence counts between SPD A and SPD B, n bin is the number of time bins in our trial, n A is the number of counts at SPD A, and n B is the number of counts at SPD B. Also, τ is the delay applied on the signal of detector B. The second order correlation g (2) (τ = 0) of two lasers is (see the supplemental materials of [1], as well as [2], for a detailed derivation) where ε is the visibility of the interferometry. Above, ∆φ 1AB is the phase difference between the paths from the first source to A and the first source to B, whereas ∆φ 2AB is the phase difference between the paths from the second source to A and the second source to B. In our experimental color erasure setting, the phases from the sources to detector B are fixed. By adjusting the reflector in the delay controller, we can increase the optical path by ∆L, and thus t delay = ∆L c and ∆φ = 2π∆f 21 t delay . Then we can write g (2) (τ = 0) more explicitly as [1,2] g (2) (τ = 0) = 1 + ε 2 cos(φ 0 + 2π∆f 21 t delay ) .
We can also write a more explicit expression for ε. Let n 1A be the number of photons from source 1 which arrive at detector A, and similarly define n 2A , n 1B , n 2B . We also let n dA and n dB denote the unwanted photon counts, including dark counts, environment light, and counts from unfiltered signal and pump light. Then we have As shown in Figure 2, g (2) (τ = 0) oscillates as we change the phase of the interferometer. If the photons were still distinguishable upon measurement, we would have g (2) (τ = 0) = 1. Instead, since the color erasure detectors render the photons indistinguishable, g (2) (τ = 0) need not be near one.
As we see in Figure 2, by changing the length of the optical path from the output of BS1 to detector A, the photons can both bunch and anti-bunch when they arrive at the detectors [23]. Performing a least squares fitting to (15), we find This is consistent with our experimental parameters since the frequency difference between 1064.4 nm and 1063.6 nm corresponds to ≈ 212 GHz (with some systematic uncertainty corresponding to drifting of the sources by up to several GHZ around 212 GHz).
The frequencies of the pump lasers are carefully tuned so that the received photons are indistinguishable to the Si APD. Although ideally we are engineering the processes f 1 → f 3 and f 2 → f 3 , in reality we have 3 . This is okay, so long as f are close enough to be rendered indistinguishable due to the time resolution of the receiving detectors. The difference f 3 appears in the theoretical formula for g (2) (τ ), namely where γ is the spectral linewidth. Indeed, as shown in Figure 3, g (2) (τ ) oscillates as we apply different τ by post-processing. The speed of the oscillations encodes the original frequency difference of the color-  The fitted value of ε for g (2) (τ ) is necessarily similar to the fitted value for ε for g (2) (τ = 0) in (17), and the fitted value of the spectral linewidth γ is consistent with known experimental parameters.

DISCUSSION
We have presented a new methodology for color erasure detectors which enables chromatic interferometry of photons with small frequency differences. This more general method can also be used for large frequency differences, as an alternative to the procedure in [1].
Multi-photon interference enables higher phase sensitivity to light sources, and better resolution of their geometries. However, if the source or sources in question emit photons with distinct wavelengths, then interference between their emitted photons will not occur and the desired phases cannot be extracted. But color erasure detectors allow one to gain access to the desired phase information by retroactively recovering interference (akin to a quantum eraser [24,25]) between the photons emitted from the sources.
In several circumstances, including stars or exoplanets [17] having very different temperatures or differentially fluorescent structures [18,19,26,27], chromatic interferometry promises to be a natural tool for achieving high resolution. We are actively pursuing these directions.