Controllable two-photon interference with versatile quantum frequency processor

Quantum information is the next frontier in information science, promising unconditionally secure communications, enhanced channel capacities, and computing capabilities far beyond their classical counterparts. And as quantum information processing devices continue to transition from the lab to the field, the demand for the foundational infrastructure connecting them with each other and their users---the quantum internet---will only increase. Due to the remarkable success of frequency multiplexing and control in the classical internet, quantum information encoding in optical frequency offers an intriguing synergy with state-of-the-art fiber-optic networks. Yet coherent quantum frequency operations prove extremely challenging, due to the difficulties in mixing frequencies efficiently, arbitrarily, in parallel, and with low noise. Here we implement an original approach based on a reconfigurable quantum frequency processor, designed to perform arbitrary manipulations of spectrally encoded qubits. This processor's unique tunability allows us to demonstrate frequency-bin Hong-Ou-Mandel interference with record-high 94% visibility. Furthermore, by incorporating such tunability with our method's natural parallelizability, we synthesize independent quantum frequency gates in the same device, realizing the first high-fidelity flip of spectral correlations on two entangled photons. Compared to quantum frequency mixing approaches based on nonlinear optics, our linear method removes the need for additional pump fields and significantly reduces background noise. Our results demonstrate multiple functionalities in parallel in a single platform, representing a huge step forward for the frequency-multiplexed quantum internet.

Quantum information is the next frontier in information science, promising unconditionally secure communications, enhanced channel capacities, and computing capabilities far beyond their classical counterparts 1,2 . And as quantum information processing devices continue to transition from the lab to the field, the demand for the foundational infrastructure connecting them with each other and their users-the quantum internet 3,4 -will only increase. Due to the remarkable success of frequency multiplexing and control in the classical internet, quantum information encoding in optical frequency offers an intriguing synergy with state-of-the-art fiber-optic networks. Yet coherent quantum frequency operations prove extremely challenging, due to the difficulties in mixing frequencies efficiently, arbitrarily, in parallel, and with low noise. Here we implement an original approach based on a reconfigurable quantum frequency processor, designed to perform arbitrary manipulations of spectrally encoded qubits. This processor's unique tunability allows us to demonstrate frequency-bin Hong-Ou-Mandel interference with record-high 94% visibility. Furthermore, by incorporating such tunability with our method's natural parallelizability, we synthesize independent quantum frequency gates in the same device, realizing the first high-fidelity flip of spectral correlations on two entangled photons. Compared to quantum frequency mixing approaches based on nonlinear optics 5,6 , our linear method removes the need for additional pump fields and significantly reduces background noise. Our results demonstrate multiple functionalities in parallel in a single platform, representing a huge step forward for the frequency-multiplexed quantum internet.
In the classical domain, the ultrabroad bandwidth supported by optical fiber has proven crucial in solidifying fiber optics in the digital communications revolution. Wavelengthdivision multiplexing (WDM)-either in its standard embodiment with independent frequency channels, or more complex versions with channels comprising interleaved bands 7forms an essential component, and will continue to do so even as novel data formats and multiplexing techniques are incorporated 8 . Such success has naturally brought WDM approaches to the forefront for the quantum internet as well. In particular, potentially large amounts of information can be stored in single photons encoded in spectro-temporal modes [9][10][11][12] , and frequency multiplexing is essential for scaling up quantum memories 13 .
To that end, considerable progress has been made in generating multiphoton entanglement across a comb of narrowband frequency modes, or bins, including optical parametric oscillators below threshold 14 , straightforward filtering of broadband parametric downconversion 15 , and, recently, on-chip production of quantum frequency combs using microring resonators [16][17][18][19] . Likewise, an explosion of research in quantum frequency conversion has showcased coherent translation of single-photon states across both wide 20,21 and narrow 22 bandwidths. But the step from viewing frequency as a channel-wherein quantum information is carried by some other parameter, such as time or polarization-to encoding the information itself in frequency is significant, requiring markedly more complex operations: i.e., universal gate sets in frequency modes. As important milestones in that direction, frequency beamsplitters 5,6,23 and quantum pulse gates 10 based on optical nonlinearities have shown coherent interference and mode selection of frequency-encoded photons. Yet the need for powerful optical control fields makes nonlinear approaches challenging, given the potential for extra noise photons from Raman scattering and imperfect isolation.
Recently, we proposed a general framework 24 for spectrally encoded photonic state control, based on electro-optic phase modulators (EOMs) and Fourier-transform pulse shapers.
Enabling universal quantum information processing in a scalable fashion, our approach is also optically linear, obviating the need for additional pump fields. Figure 1 sketches an example of such a quantum frequency processor, with the particular operations chosen to match the ensuing experiments. In general, an input quantum state consisting of a superposition of photons spread over discrete frequency bins is manipulated by the designed network of EOMs and pulse shapers, which applies various unitary operations to combinations of frequency bins. After each step, some of the frequency bins can be detected, with the newly available bandwidth re-provisioned with freshly encoded photons. Note that, although we draw each frequency bin as a separate "rail" for conceptual purposes, the physical encoding occurs within a single fiber-optic spatial mode, thereby enabling natural phase stability and providing compatibility with current fiber networks. This paradigm has allowed us to experimentally demonstrate frequency beamsplitters and tritters with ultrahigh operation fidelity and parallelizability across a 40-nm optical bandwidth 25 . While then validated with weak coherent states, a quantum frequency network will require multiphoton and nonclassical interference phenomena as well. In this work, we experimentally show that not only does our operation indeed realize unrivaled interference of fully quantum frequency states, performing the desired set of operations. Spheres of a specific color trace the probability amplitudes of a single input photon, so that an ideal measurement will register precisely one click for each color. Frequency superpositions are represented by spheres straddling multiple lines, while entangled states are sums of photon products (visualized by clouds). The specific operations are those we realize experimentally: Hong-Ou-Mandel interference (top) and two-qubit rotation (bottom).
but it also enables independent simultaneous operations in the same device, via its unique tunability and parallelizability.  In the case of HOM interference, we seek to apply such a gate to a pair of photons located in adjacent frequency bins, which we obtain directly by filtering out all but bins 0 and 1 of the source (Fig. 2b); the H gate applied to 0 and 1 should cause both photons to bunch in either bin 0 or 1, with no coincidences between the two bins. To measure the strength of quantum interference, one must scan some parameter which controls the distinguishability of the two-photon probability amplitudes leading to clicks on both output detectors; a visibility exceeding 50% indicates nonclassicality 26 . In the case of frequency mixers, one can introduce a temporal delay between the two modes 5 or scan the photon frequency spacing relative to that of the frequency beamsplitter 23 . In our case, we adjust the mixing probability of the operation itself, analogous to varying the reflectivity R of a spatial beampslitter. To do so, we note a valuable feature of the our spectral beamsplitter: tunability. For by simply changing the depth of the phase shift imparted by the pulse shaper between frequency bins 0 and 1, the spectral reflectivity R can be tuned smoothly from 0 to ∼0.5 and back to 0 (see Methods). Figure 3a plots the theoretically predicted (curves) and experimentally measured (symbols) beamsplitter transmission and reflection coefficients between bins 0 and 1, when scanning the pulse shaper phase. A phase setting of π results in an H gate; 0 and 2π phase shifts yield an identity operation. It is important to note that both EOMs remain fixed throughout the scan, so that the tunability is effected only by adjusting the phase applied by the pulse shaper.
Sending in the photon pair |1 ω 0 A |1 ω 1 B (i.e., one photon in frequency-bin 0, assigned to party A, and one photon in frequency-bin 1, assigned to party B) and scanning the pulse shaper phase, we measure the coincidence counts between output bins 0 and 1 shown in  values measured for frequency-domain HOM interference-namely, 0.71 ± 0.04 5 and 0.68 ± 0.03 23 -and is a consequence of both the reduced optical noise and fine controllability of the operation. We also record the singles counts for bins 0 and 1, as well as the adjacent sidebands (−1 and 2). As shown in Fig. 3c, the two central modes retain nearly constant flux across the full scan, showing that the dip in coincidence counts results from truly quantum HOM interference as opposed to photon loss (see Methods for detailed verification).
Moreover, the small reduction in singles counts around π-accompanied by the increase in singles counts for bins −1 and 2-also qualitatively matches expectations, given the fact that the full H gate scatters 2.61% of the input photons out of the computational space into adjacent sidebands. We note that even this scattering could be removed by driving the EOMs with more complicated waveforms 24 .
The quantum frequency processor's tunability, invoked in the above realization of HOM interference, relies only on modifying the spectral phase, which suggests the ability to perform independent operations by setting different phase shifts on appropriate subbands in the pulse shaper's bandwidth. Accordingly, this form of parallelizability is even stronger than previously shown, where the same operation was replicated across the bandwidth 25 .
To demonstrate this, we set the BFC shaper to pass modes {−4, −3, 4, 5} (cf. Fig. 2b), Here φ 0 is an offset with no physical significance, while α = π for the ideal Hadamard. Yet α can be tuned as well; doing so actually permits tunable reflectivity. Specifically, if we write out the 2 × 2 transformation matrix on modes 0 and 1 as a function of this phase, we can define the variable reflectivities (i.e., mode-hopping probabilities) and transmissivities (probabilities of preserving frequency) as where J k (Θ) is the Bessel function of the first kind. We note that, when α = π, the elements {V 00 , V 01 , V 10 } are all real and positive, while V 11 is real and negative-in accord with the ideal Hadamard and leading to destructive HOM interference between the reflect/reflect and transmit/transmit two-photon probability amplitudes. Additionally, these expressions satisfy R 0→1 = R 1→0 ≡ R and T 0→0 = T 1→1 ≡ T . As α is tuned over 0 → π → 2π, R follows from 0 to a peak of 0.4781 and back to 0, while T starts at 1, drops to 0.4979, and returns to 1. The sum R + T defines the gate success probability, which drops slightly at α = π due to the use of single-frequency electro-optic modulation. These particular values are confirmed experimentally in Fig. 2a with coherent state measurements 25 .
Hong-Ou-Mandel interference. The generated biphoton frequency comb can be described as a state of the form or in terms of bosonic mode operators, whereâ n (â † n ) annihilates (creates) one photon in the frequency bin centered at ω n . The A and B nomenclature defines the modes held by each of two parties: A consists all ω n such that n ≤ 0, B everything with n ≥ 1. We favor this notation over the more traditional "signal" and "idler" classification because (i) our frequency operations can move photons between A and B mode sets-and indeed does in the case of HOM-and (ii) there are no other distinguishing degrees of freedom to label the photons.
Our quantum frequency processor transforms these bins into outputsb m (at frequencies The matrix V describes the entire operation over all modes. Then at the output we measure the spectrally resolved coincidences between bins n A and n B , i.e., as well as the singles In the case of HOM interference, we filter out all photon pairs except c 1 [Eq. (1)], so the input state is |Ψ = |1 ω 0 A |1 ω 1 B , which gives C 01 = |V 00 V 11 +V 01 V 10 | 2 and S n = |V n0 | 2 +|V n1 | 2 .
In light of the previous discussion on beamsplitter tunability, we thus predict: where the nonunity success probability [R(π) + T (π) = 0.976] results in some photons scattering into bins −1 and 2. (Scattering beyond these modes is not observable in experiment, consistent with the theoretical prediction of only ∼10 −4 probability to leave the center four bins.) Invoking the theoretically predicted values for R and T , we use weighted least-squares to fit the function f (α) = K 0 + K 1 C 01 (α) to the data in Fig. 3b and extract the visibility = 0.94 ± 0.01. Now, because the singles S 0 and S 1 drop slightly at α = π (cf. Fig. 3b)-which is not the case in a traditional HOM experiment-we also look at the visibility of the normalized cross-correlation function, g 01 = C 01 S 0 S 1 . For in the most pathological case, a reduction in the unnormalized coincidences C 01 could in principle be due to dropping singles S 0 or S 1 , which would not be surprising from a classical view: if one detector rarely clicks, of course its coincidences with another detector will drop as well. On the other hand, the normalized g does not suffer from this issue, by accounting for singles counts directly. Accordingly, we repeat the least-squares fit using the theoretically predicted g (2) 01 (α), along with the measured coincidences (Fig. 3b) divided by the product of mode 0 and 1 single counts (Fig. 3c). In this more conservative case, we still retrieve V = 0.94 ± 0.01, fully confirming the nonclassicality of our HOM interference.
Quantum state manipulation. For the state rotation experiments, we filter out all modes except four, leaving the entangled qubits [n = 4, 5 in Eq. (1)]: Ideally, parametric downconversion and filtering should produce this relative phase automatically; but in order to compensate any residual dispersion, we also fine-tune the phase with the BFC pulse shaper, experimentally maximizing spectral correlations in the H A ⊗ H B measurement case (see below). We have six initially empty modes between those populated in A and B, allowing us to apply combinations of Hadamard operations and the identity to each pair of modes-{−4, −3} and {4, 5}-without any fear of the photon in A jumping over to B's modes, and vice versa (cf. guardband discussion in ref. 25). Accordingly, after the frequency-bin transformation V (chosen to apply the desired joint operation), the coincidence probability for any (n A ≤ 0, n B ≥ 1) is given by This expression accounts for all aspects of the potentially nonideal mode transformation.
Focusing on the qubit modes (n A ∈ {−4, 3}, n B ∈ {4, 5}), we have the ideal coincidences under all four cases of Fig. 4 as: To calculate the conditional entropies corresponding to the measurements in Fig. 4, we employ Bayesian mean estimation (BME) on the raw count data 34,35 . State reconstruction. To estimate the complete two-qubit density matrix, we again employ BME 34,35 but now with the assumption of a single quantum state underlying all four measurements in Fig. 4. As noted above, these four combinations are equivalent to joint measurements of the two-qubit observables  Our likelihood function, P (D|α), is then a multinomial distribution over the aforementioned probabilities and outcomes, where α = {ρ, η A , η B , N } is the underlying parameter set. The idealized probabilities {p AB , p A0 , p 0B , p 00 } are all functions of the density matrixρ, which we limit to physically allowable states 35 .
Up to this point, we have focused on a specific choice of POVMs, Λ (A) ⊗ Λ (B) . To account for all 16 POVM combinations (basis pairs and frequency-bin pairs) in the two-qubit space of Fig. 4, we form the product over all settings, leaving the complete posterior distribution where the bolded D represents the union of the respective results D j from each particular setting (j = 1, 2, ..., 16). Our prior P (α) is taken to be uniform in a Haar-invariant sense, and the marginal P (D) is found by integrating the numerator in Eq. (4). With this posterior distribution, we can estimate any parameter of interest via integration, such as the mean density matrixρ BME = dα P (α|D)ρ.
Due to the complexity of integrals of this form, we employ numerical slice sampling for their evaluation 35,36 . The density matrix estimate is plotted in Extended Data Fig. 1. To highlight BME's natural treatment of error, we include the standard deviation for each matrix element as well. The error is extremely low for the real elements, due to our complete coverage of the X and Z bases, whereas several of the imaginary components (particularly the center two of the antidiagonal) have significantly higher error; this is a natural consequence of our lack of any measurements in the Pauli Y basis. Nevertheless, physical requirements do bound this error, such that our Bayesian estimate of the fidelity, F = Ψ|ρ|Ψ -with |Ψ as defined in Eq. (2)-has extremely low uncertainty: F = 0.92 ± 0.01. Such high fidelity provides positive corroboration of our frequency-bin control, and is fairly conservative, given that: (i) dark counts are not removed, and thus can degrade the state; and (ii) we lump any imperfections of our H gate rotation onto the state itself, so that impurities in either the input state or quantum frequency processor will contribute to lower F.