Abstract
When two indistinguishable single photons impinge at the two inputs of a beam splitter they coalesce into a pair of photons appearing in either one of its two outputs. This effect is due to the bosonic nature of photons and was first experimentally observed by Hong, Ou and Mandel^{1}. Here, we present the observation of the Hong–Ou–Mandel effect with two independent singlephoton sources in the microwave frequency domain. We probe the indistinguishability of single photons, created with a controllable delay, in timeresolved secondorder cross and autocorrelation function measurements. Using quadrature amplitude detection we are able to resolve different photon numbers and detect coherence in and between the output arms. This scheme allows us to fully characterize the twomode entanglement of the spatially separated beamsplitter output modes. Our experiments constitute a first step towards using twophoton interference at microwave frequencies for quantum communication and information processing^{2,3,4,5}.
Main
So far, Hong–Ou–Mandel (HOM) twophoton interference has been demonstrated exclusively using photons at optical or telecom wavelengths. Experiments were performed with photons emitted from a single source using parametric downconversion^{1}, trapped ions^{3}, atoms^{6}, quantum dots^{7} and single molecules^{8}. The HOM effect has also been observed with two independent sources^{9,10,11,12,13,14,15} realizing indistinguishable singlephoton states, which are required as a resource in quantum networks or linearoptics quantum computation. Such experiments have also been performed using donor impurities as sources^{16} including nitrogen–vacancy centres in diamond (see ref. 17 and references therein). Furthermore, the HOM effect has been employed to create entanglement between ions^{18} in spatially separated traps, and to realize a controlledNOT gate in a smallscale photonic network^{19}. Similar physics is also actively explored with ballistic electrons in solids (see ref. 20 and references therein).
Here, we demonstrate the HOM interference of two indistinguishable microwave photons emitted from independent triggered sources realized in superconducting circuits (see Methods). The photons are prepared in two separate microwave resonators A (B) using transmontype qubits and decay exponentially at rates κ/2π = 4.1 (4.6) MHz through their strongly coupled output ports into the input modes and of the beam splitter (see Fig. 1). The two photons then interfere at the beam splitter and are emitted into the output modes and (see Fig. 1a). Using the dispersive interaction between qubit and resonator, we tune the emission frequencies of the two sources to an identical value of ν_{r} = 7.2506 GHz. For our experiments, we sequentially create 20single photons in each source at a rate 1/t_{r} = 1/512 ns∼1.95 MHz in a sequence repeated every 12.5 μs.
To probe the photon statistics in the beamsplitter output modes and we use two spatially separated heterodyne detection channels^{21,22} (see dashed rectangle in Fig. 1b). Each channel consists of a set of semiconductor linear amplifiers, and a microwave frequency mixer for downconversion to 25 MHz, followed by an analogtodigital converter. After further digital downconversion to d.c. and filtering, we extract the two quadrature amplitudes X_{a/b}(t) and P_{a/b}(t) corresponding to the complex envelope of the two timedependent signals^{23}S_{a/b}(t) = X_{a/b}(t)+i P_{a/b}(t). In contrast to many other HOM experiments in which photons in the beamsplitter outputs are detected by singlephoton counters, our measurement of the complex envelope is intrinsically photonnumber resolving for averaged measurements and allows us to measure coherences of the electromagnetic field. The complex envelopes S_{a/b}(t) are used to compute the statistics required to extract all relevant quantum correlations measured here^{21,23}. To analyse the digital data, signal processing and statistical analysis are performed in realtime using fieldprogrammable gate arraybased electronics^{24}. The noise added by the detection chain is fully characterized by measuring its statistical properties when the output modes and of the beam splitter are left in the vacuum state with both sources idle^{21,23}. To verify the singlephoton character of each source individually, we have measured their secondorder crosscorrelation functions G_{a b}^{(2)}(τ). These exhibit clear antibunching^{24} (see Fig. 2a,b).
To investigate twophoton quantum interference, we simultaneously generate two indistinguishable photons ideally realizing a twomode entangled state at the beamsplitter outputs. The measured crosscorrelation of the beamsplitter output powers is observed to vanish G_{a b}^{(2)}(τ)≈0 for all τ between −t_{r}/2 and t_{r}/2 (see coloured region in Fig. 2c). Therefore, we conclude that both microwavefrequency photons coalesce at the beam splitter. This is the HOM effect with microwave photons. In our experiments the spatiotemporal coherence of the singlephoton states is governed by resonator decay alone, and shows no significant additional dephasing. This is in stark contrast to many other experiments in which decoherence resulting in random frequency differences between interfering photons causes finite correlations at τ∼0 (refs 6, 11, 13). At τ = n t_{r} (n = ±1,±2,,±10), the peak at integer nonzero multiples of the photon generation period reflects the product of the power in each output , which we have normalizedto one.
All measured secondorder correlation functions are normalized by a common scaling factor and a relative gain between the two amplification channels. A remaining small offset subtracted from the reconstructed correlation functions is indicated together with its standard deviation (extracted from multiple measurements) in the upper right corner of each panel in Fig. 2. This offset results from the finite statistical uncertainty of measured firstorder correlation terms^{23} and is expected to vanish in the limit of infinite averaging. The measured secondorder correlation functions are in good agreement with analytical calculations^{25} (solid lines in Fig. 2), taking into account the cavity decay rates κ extracted from independent measurements and a fixed detection bandwidth of 20 MHz chosen to reject experimental noise outside the desired band.
To explore the level of indistinguishability between the two interfering photons we introduce a time delay δ τ on the order of the photon decay time 1/κ∼37 ns. For δ τ = 50 ns the correlation function G_{a b}^{(2)}(τ) remains close to zero at τ = 0, indicating the coalescence of those photons detected with a vanishingly small time difference^{6,25,26} (see Fig. 2d). Typically this effect is difficult to observe with detectors of insufficient bandwidth or sources with significant dephasing rates^{13}. The small positive correlations observed at τ∼δ τ are due to the decreased temporal overlap of the singlephoton spatiotemporal mode functions at the beam splitter, that is, the increased distinguishability of the two photons. At τ∼n t_{r} we observe broadened, lower amplitude correlations. All features are in agreement with theory^{25} (solid lines).
For δ τ = 150 ns (and δ τ = 100 ns, not shown) the envelopes of the two singlephoton mode functions barely overlap at the beam splitter, resulting in fully distinguishable single photons. At τ = ±δ τ and τ = n t_{r}±δ τ (see Fig. 2e) we observe positive correlations with an amplitude 1/4, which originate from single photons impinging on the beam splitter at different times and at different input arms (compare Fig. 2ab). At τ = n t_{r} the correlations between photons in the same beamsplitter input arm sum up to 1/2, as expected.
To clearly distinguish between singlephoton antibunching and twophoton coalescence in timeresolved correlation function measurements, we have also measured the secondorder autocorrelation function G_{a a}^{(2)}(τ) of mode . When operating only one singlephoton source G_{a a}^{(2)}(0) is expected to vanish. However, in the HOM configuration with δ τ = 0 we find G_{a a}^{(2)}(0) = G_{a a}^{(2)}(n t_{r}) = G_{a b}^{(2)}(n t_{r}) = 1 (see Fig. 2f) as there is a 50% probability of detecting two photons in mode . All measurements of G_{a a}^{(2)}(τ) are in good agreement with calculations^{25}, both for δ τ = 0 (solid lines) and for δ τ = 100 ns (not shown). Here, it is interesting to note that a secondorder autocorrelation function is rarely directly measured, because of the the lack of sufficiently fast singlephoton detectors^{27}. In contrast, using our heterodyne detection scheme we are capable of measuring G_{a a}^{(2)}(τ) for multiphoton states.
To distinguish between an equal mixture of the states 20〉 and 02〉, compatible with the observed correlations, and their coherent superposition , we fully characterize the twophoton states created in our HOM experiment by quantum state tomography. This allows us to probe the entanglement generated between the coalescing twophoton states in the two output ports of the beam splitter. The created states are also referred to as NOON states. In contrast to the NOON states of propagating photons investigated here, NOON states have also been investigated in superconducting circuits with photons localized in resonators^{28,29}.
To perform full quantum state tomography on propagating photons, we record fourdimensional histograms of the measured field quadratures X_{a}, P_{a}, X_{b} and P_{b}. This is an extension of the scheme discussed in ref. 21 to two spatially separated modes. From these measurements we extract all moments of the twomode field, that is, expectation values of the form , with n,m,k,l∈{0,1,2}. The total gain of the detection chain is calibrated by preparing an equal superposition state in only one mode, for which we expect half a photon in mode . In the analysis we take a residual thermal steadystate population of 0.03 in modes and into account.
We observe that the firstorder moments and all other oddorder moments are zero because all singlephoton Fock states are characterized by a fixed photon number and consequently a fully random phase (see Fig. 3a). As each mode carries exactly one photon on average, and are close to unity, whereas all other secondorder moments vanish. The fourthorder moment is observed to be zero, whereas and are unity, consistent with the coalescence of the two photons into either output. The above observations are consistent with those based on the correlation function measurements in Fig. 2. Most importantly, the twomode entanglement is indicated by the moment , which is close to 1, as expected. All measured moments of the twomode entangled state created in our HOM experiment are in good agreement with the predicted ones (see wire frames in Fig. 3a). Note that moments of order five and higher are all close to zero within their statistical errors.
In addition, we have determined the most likely density matrix ρ characterizing the created twomode entangled propagating photon state from the measured moments and their respective standard deviation following ref. 21. We have restricted the evaluation of moments to less than three photons per output mode, because we create no more than two single photons with our sources. Note that a related analysis has recently been performed in ref. 30. The real part of ρ is shown in Fig. 3b; all elements of the imaginary part of ρ are smaller than 0.02 (not shown). We extract a fidelity of the NOONtype state of F = 〈ψρψ〉 = 84%and a negativity^{31} . Inefficiencies arise predominantly from residual thermal populations of qubits and resonators, which we estimate to be approximately 3%, from finite qubit coherence times and the associated imperfect cavity state preparations, as well as slight differences in the linewidth of the two resonators.
Finally, to explore the interplay between single and twophoton interference, we have performed experiments with modes and prepared in superpositions of 0 and 1photon Fock states, that is, and with variable phase ϕ, ideally creating the state at the beamsplitter output. With these input states, we have first measured the power in the beamsplitter output modes (blue line) and (red line) when operating only source B and keeping source A idle (see Fig. 3d). In this case, we observe a power level corresponding to 1/4 of a single photon independent of the phase angle ϕ, as expected for an equal superposition of 0〉 and 1〉 impinging on a balanced beam splitter. Operating both sources A and B, we observe a sinusoidal interference with phase ϕ of the two superposition states in the beamsplitter output power of mode (blue dots) and (red dots), respectively. The sinusoidal oscillation is a result of the interference between one photon in either output port whereas the offset in power of 1/4 is the result of twophoton coalescence.
For the phase angle ϕ≈0 we have also performed full quantum state tomography (see Fig. 3c). As for the twophoton NOON state we observe antibunching, coalescence and entanglement in the moments , and , and , respectively with close to expected amplitudes (wire frames). The interference of the superposition states is revealed not only in the power and but also in the coherences with an unbalanced number of creation and annihilation operators. The corresponding state has a fidelity of 84% with respect to the ideal one.
Our results suggest that multiple ondemand singlephoton sources emitting indistinguishable single photons could be used for creating nonlocal entanglement in quantum repeater or quantum communication applications based on microwave photons.
Methods
The photon generation is realized similarly as discussed in ref. 24. Each photon source consists of a transmontype qubit (E_{c}/h = 416 (419) MHz) strongly coupled with rate g/2π = 169 (177) MHz to a transmission line resonator (see Fig. 1a). By adjusting the qubit transition frequencies ν_{a} to 8.575 (8.970) GHz with a static magnetic flux applied to the superconducting quantum interference device loop of each qubit, both microwave resonators have the same frequency ν_{r} = 7.2506 GHz. To create a single photon using one of the sources, we coherently excite the qubit (lifetime T_{1} = 1.0 (0.9) μs and dephasing time T_{2}^{*} = 0.4 (0.6) μs) into a state αg〉+βe〉 using a resonant 20 ns microwave pulse with a Gaussian envelope (σ = 5 ns). Then, we swap the qubit state into the resonator mode by tuning the qubit transition frequency with a dynamical magnetic flux pulse into resonance with the resonator for half a vacuumRabi period. The flux pulse has a rise and fall time of about 1 ns and a total duration of 3 ns≈π/g. This creates the state α0〉+β1〉in the resonator, which for β = 1 corresponds to a singlephoton Fock state. The photon is then emitted with a Lorentzian spectrum of linewidth κ/2π = 4.1 (4.6) MHz.
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Acknowledgements
This work was supported by the European Research Council (ERC) through a Starting Grant and by ETHZ. L.S. was supported by EU IP SOLID. A.B. and M.J.W. were supported by NSERC, CIFAR and the Alfred P. Sloan Foundation.
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C.L. and C.E. performed the experiments and analysed the data. C.L. developed the FPGA firmware. M.J.W. and A.B. contributed to its theoretical interpretation. C.L., L.S. and J.M.F. designed and fabricated the sample. C.L., C.E. and L.S. contributed to constructing and maintaining the measurement setup. C.L. and A.W. cowrote the manuscript and all authors commented on it. A.W. supervised the project. All experiments were carried out at ETH Zurich.
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Lang, C., Eichler, C., Steffen, L. et al. Correlations, indistinguishability and entanglement in Hong–Ou–Mandel experiments at microwave frequencies. Nature Phys 9, 345–348 (2013). https://doi.org/10.1038/nphys2612
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