## Abstract

Realizing a fully connected network of quantum processors requires the ability to distribute quantum entanglement. For distant processing nodes, this can be achieved by generating, routing, and capturing spatially entangled itinerant photons. In this work, we demonstrate the deterministic generation of such photons using superconducting transmon qubits that are directly coupled to a waveguide. In particular, we generate two-photon N00N states and show that the state and spatial entanglement of the emitted photons are tunable via the qubit frequencies. Using quadrature amplitude detection, we reconstruct the moments and correlations of the photonic modes and demonstrate state preparation fidelities of 84%. Our results provide a path toward realizing quantum communication and teleportation protocols using itinerant photons generated by quantum interference within a waveguide quantum electrodynamics architecture.

## INTRODUCTION

Modular architectures of quantum computing hardware have recently been proposed as an approach to realize robust large-scale quantum information processing (*1*–*4*). However, such architectures rely on a means to coherently transfer quantum information between individual, and generally nonlocal, processing nodes. Spatially entangled itinerant photons can be used to achieve this by efficiently distributing entanglement throughout a quantum network. Conventional approaches for generating such photons in optical systems typically use spontaneous parametric down conversion in conjunction with arrays of beamsplitters (*5*) and photodetectors for postselection (*6*, *7*). However, the stochastic nature of these approaches limits their utility in quantum information processing applications.

Recent progress with superconducting circuits has established a path toward realizing a universal quantum node that is capable of storing, communicating, and processing quantum information (*8*–*12*). These works often invoke a cavity quantum electrodynamics (cQED) architecture, where cavities protect qubits from decoherence within a node, enabling the high-fidelity control required to generate arbitrary quantum states. To link distant nodes, this quantum information must propagate along a bus composed of a continuum (or quasi-continuum) of modes. To this end, we strongly couple qubits to a waveguide such that the excitations stored in the qubits are rapidly released as itinerant photons. Such a system is described by waveguide quantum electrodynamics (wQED). Entering the strong coupling regime in wQED enables qubits to serve as high-quality quantum emitters (*13*). More generally, superconducting circuits have been used to produce a wide variety of nonclassical itinerant photons from classical drives (*14*–*18*), such as those with correlations and entanglement in frequency (*18*).

Here, we demonstrate that the indistinguishability and quantum interference between photons directly emitted from multiple sources into a waveguide can deterministically generate spatially entangled itinerant photons. In particular, we generate two-photon N00N states *n*_{L}*n*_{R}〉 denotes the number of photons in the left and right propagating modes of the waveguide, respectively. More generally, we show that our device can generate itinerant photons with states of the form ∣ψ_{ph}〉 = *a*∣20〉 + *b*∣02〉 + *c*∣11〉, where *a*, *b*, and *c* are complex coefficients that are set by the effective qubit spatial separation Δ*x*.

## RESULTS

### Device description and photon generation protocol

The test device consists of three flux-tunable transmon qubits (*19*) that are capacitively coupled to a common 50-ohm transmission line (an electromagnetic coplanar waveguide), as shown in Fig. 1A. The configurations we consider involve two qubits, used as photonic emitters, that are spatially separated by Δ*x* = 3λ/4 and Δ*x* = λ/2. The effective spacing is controlled by the qubit frequencies ω (*20*) via the corresponding wavelengths λ = 2π*v*/ω, where *v* is the speed of light in the waveguide. Setting the transition frequencies of qubits *Q*_{1} and *Q*_{3} to ω/2π = 4.85 GHz corresponds to a spacing of Δ*x* = 3λ/4 between emitters. The frequency of the central qubit *Q*_{2} is detuned hundreds of megahertz such that it can be ignored. In this configuration, the qubits are coupled to the coplanar waveguide with a coupling strength of γ/2π = 0.53 MHz. Alternatively, to realize a spacing of Δ*x* = λ/2 between emitters, the frequencies of *Q*_{1} and *Q*_{2} are set to ω/2π = 6.45 GHz, where the qubit-waveguide coupling strength is γ/2π = 0.95 MHz, while sufficiently detuning *Q*_{3} so that it may be ignored.

The Hamiltonian of the qubit-waveguide system is (*21*)*x _{j}* is the position of the

*j*

^{th}qubit, and

*X*and

*Z*Pauli operators. The coupling strength

*g*(ω) determines the physical qubit-waveguide coupling rate γ(ω

_{j}*) = 4π*

_{j}*g*(ω

_{j}*)*

_{j}^{2}

*D*(ω

*), where*

_{j}*D*(ω) is the density of photonic modes in the waveguide. The qubits couple to the transmission line with equal strength when placed on resonance with each other.

When the propagation time for photons between the qubits is small relative to the time scale γ^{−1} of the qubit emission, the system can be simulated for arbitrary initial conditions and spacings by integrating a master equation derived from the Hamiltonian in Eq. 1 and applying input-output theory (*21*). The input-output relations that provide the dynamics of the photons emitted into the left- and right-propagating modes are*t* whose corresponding fields are taken to be in the vacuum state.

The two resonant qubits in each spacing configuration are initialized to their excited states, while the detuned qubit is left in the ground state. Under these conditions, the final (unnormalized) state of the photons emitted by the excited qubits is given by*j* is multiplied over the two active qubits that are prepared in the state∣*ee*〉 (*Q*_{1}, *Q*_{3} for Δ*x* = 3λ/4 and *Q*_{1}, *Q*_{2} for Δ*x* = λ/2). From Eq. 3, we may verify that∣ψ_{ph}〉 is a two-photon N00N state when the spatial separation between qubits is Δ*x* = λ/4, 3λ/4, …, (2*n* + 1)λ/4, where *n* is an integer. This can be understood by considering the interference between the four possible coherent emission pathways for two excitations to leave the system (shown in Fig. 1B). The emission pathways containing a single photon in both the left and right propagating modes destructively interfere, resulting in the entangled state*x* = 0, λ/2, …, *n*λ/2, depicted in Fig. 1C, the destructive interference no longer occurs, resulting in a standard (equal) partitioning of the photons into the left and right propagating modes. For this latter configuration, the decay of the qubits from ∣*ee*〉 to ∣*gg*〉 is determined by super-radiant emission (*20*).

### Measurement techniques and protocols

Figure 2A shows the control and measurement schematic. First, we measure the scattering of coherent microwave fields to extract qubit parameters and calibrate the absolute power of photons at the qubit (see the Supplementary Materials). Next, we independently prepare the qubits by detuning them from each other and then applying resonant microwave pulses to the transmission line. The qubits can be individually prepared anywhere on the Bloch sphere α∣*g*〉 + β∣*e*〉, where α and β are complex coefficients determined by the amplitude and phase of the pulse. We then verify the state of the photons that are emitted by the qubits using quadrature amplitude detection of the left and right outputs of the transmission line. These photons are amplified and downconverted to an intermediate frequency *f*_{d} using in-phase and quadrature (IQ) mixing. For example, we can prepare a single detuned qubit in the state *e*〉, as will be required for the N00N-state generation protocol. In this case, the emitted photon has no coherence relative to the vacuum state∣00〉, and thus, the voltage averages to zero as shown in Fig. 2C.

To uniquely identify the state and correlations of the photons emitted from two qubits, it is necessary to measure higher-order moments of the fields. To do this, time-independent values for the field quadratures of both the left *S*_{L} = *X*_{L} + *iP*_{L} and right *S*_{R} = *X*_{R} + *iP*_{R} emission signals are obtained through digital demodulation and integration of individual records of *S*_{L} and *S*_{R}

We account for the noise added by the amplifiers in the measurement chain by using the input-output relations for a phase-insensitive amplifier *22*–*24*), where *G*_{L(R)} is the gain of the left (right) amplification chain. The moments of the noise channels *S*_{L} and *S*_{R} while leaving the qubits in the ground state. We account for residual thermal photons with an effective temperature ≈46 mK in

Before generating the photonic states of interest, we first obtain the properties of the measurement chains. We are able to calibrate the net amplification gain by preparing a single qubit in an equal superposition of its ground and excited states (*22*), as done in Fig. 2B. For this case, provided the qubits are prepared with a sufficiently high fidelity, the state of the emitted photon will approximately be *G*_{L(R)}, the gain can be calibrated by finding the value for which *n*_{noise} is the average number of photons added by the noise, we can find the detection efficiency of our measurement chains η = (1 + *n*_{noise})^{−1} by performing a maximum likelihood estimation on the measured moments of *n*_{noise} that best describes the measurements and find the detection efficiencies to be η_{L(R)} ≈ 10.4 % (12.1 %). Finally, we alternate between initializing the qubits in the fully excited (∣*ee*〉) and ground (∣*gg*〉) states while measuring

### Photon state verification via state tomography

We first initialize the qubits to∣ψ_{qb}〉 = ∣*ege*〉 with *Q*_{1} and *Q*_{3} separated by a distance of Δ*x* = 3λ/4 along the waveguide. In doing so, we generate the two-photon N00N state*25*, *26*). We are able to validate the state of the emitted photons through the moments and correlations between the left and right output modes shown in Fig. 3A. We observe

We contrast the case of Δ*x* = 3λ/4 with Δ*x* = λ/2 to demonstrate the tunability of ∣ψ_{ph}〉. Here, we use *Q*_{1} and *Q*_{2} and initialize the qubits to∣ψ_{qb}〉 = ∣*eeg*〉. Constructive quantum interference of∣11〉 leads to the output state_{ph}〉 are now consistent with the standard partitioning of two classical particles, with each being independently and equally likely to appear in one of the two modes. The moments for this case are shown in Fig. 3B and once again verify the predicted outcome. We obtain

To further characterize the state of the emitted photons, we obtain the density matrix *x* = 3λ/4 is evident in Fig. 4A with a trace overlap fidelity of *x* = λ/2 is shown in Fig. 4B with a state preparation fidelity of 87%. In both cases, we attribute the majority of the infidelity to waveguide-induced *T*_{1} decay of the qubits during the state initialization, as evidenced by a finite population of 0.09 and 0.11 in the∣00〉 state of *27*, *28*), where qubit-waveguide couplings can be tuned in situ such that the qubits are not subject to waveguide-induced decoherence during state preparation. Furthermore, giant atoms can also be used to engineer tailored qubit-waveguide coupling, waveguide-mediated qubit-qubit coupling, and correlated decay spectra (*28*) with the desired properties for a given interference condition.

## DISCUSSION

Our results demonstrate that a wQED architecture supports the high-fidelity generation of spatially entangled microwave photons. Our approach is extensible to higher-order photonic states through the addition of qubits, such that more photons are emitted, and with the appropriate choices of Δ*x* to obtain the desired quantum interference. These types of photonic states are also known to be useful for high-precision phase measurements in quantum metrology (*29*). Although current limitations in detector efficiency hinder the ability to measure higher-order moments, and thus verify the resulting higher-order photonic states, recent proposals for number-resolved microwave photon detectors (*30*, *31*) can address this issue. Finally, devices of the type studied in this work can be further generalized with the addition of direct qubit-qubit coupling, which can be used to dynamically select the direction in which photons are emitted or absorbed (*32*). We envision an architecture where quantum information and entanglement are routed and spread throughout a quantum network via the quantum interference between the photons emitted by qubits that are coupled to a waveguide. Generating itinerant photons using the principles and techniques outlined in this work can then be applied toward realizing interconnected quantum networks for both quantum communication and distributed quantum computation.

## MATERIALS AND METHODS

### Moment inversion

We describe an efficient procedure for determining the moments of the field before amplification, *n*, *m*, *k*, *l* ∈ {0, *N*} are integers up to a desired moment order *N*. In our experiment, we consider moments of order up to *N* = 2. The standard input-output relationship for phase-insensitive amplifiers is given by*G*_{L(R)} is the amplification of the left (right) amplification chain (*22*–*24*). When the gain of the amplifiers are large (*G*_{L(R)} ≫ 1), as is the case in our setup, Eq. 4 can be simplified to

Furthermore, we assume that modes of interest

As described in the measurement techniques and protocols section, we use heterodyne detection on the output of the measurement chain to form a 4D probability distribution,

To obtain the moments of the noise added by the amplifiers *k*_{B}*T* ≪ ℏω, then the state of these photonic modes can be approximated as vacuum. Under these conditions, we have

Equation 6 is then significantly reduced such that moments of the noise channels can be directly obtained from the measured moments of _{ph}〉 = ∣00〉

After determining both *N* + 1)^{4} and takes the form

We can then relate *H*, such that *N* + 1)^{4} × (*N* + 1)^{4} and be of the form

The moments in the *H* is lower triangular, and thus, the system can be solved efficiently using back substitution.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/41/eabb8780/DC1

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**We thank J. Braumüller and A. Vepsäläinen for the valuable discussions.

**Funding:**This research was funded in part by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under contract no. DE-AC02-05-CH11231 within the High-Coherence Multilayer Superconducting Structures for Large Scale Qubit Integration and Photonic Transduction program (QISLBNL), and by the Department of Defense via MIT Lincoln Laboratory under U.S. Air Force contract no. FA8721-05-C-0002. B.K. gratefully acknowledges support from the National Defense Science and Engineering Graduate Fellowship program. M.K. gratefully acknowledges support from the Carlsberg Foundation during a portion of this work.

**Author contributions:**B.K., D.L.C., S.G., and W.D.O. conceived and designed the experiment. D.L.C. designed the devices. B.K. and D.L.C. conducted the measurements, and B.K., D.L.C., and F.V. analyzed the data. D.K.K., A.M., B.M.N., and J.L.Y. performed sample fabrication. B.K. and D.LC. wrote the manuscript. M.K., P.K., and R.W. assisted with the experimental setup. T.P.O., S.G., and W.D.O. supervised the project. All authors discussed the results and commented on the manuscript.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**The data and code that support the findings of this study are available from the corresponding author upon reasonable request. All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

- Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).