Abstract
We report a highpurity EinsteinPodolskyRosen (EPR) state between light modes with the wavelengths separated by more than 200 nm. We demonstrate highly efficient EPRsteering between the modes with the product of conditional variances \({{{{{{{{\mathcal{E}}}}}}}}}^{2}=0.11\pm 0.01\ll 1\). The modes display − 7.7 ± 0.5 dB of twomode squeezing and an overall state purity of 0.63 ± 0.16. EPRsteering is observed over five octaves of sideband frequencies from RF down to audioband. The demonstrated combination of high state purity, strong quantum correlations, and extended frequency range enables new matterlight quantum protocols.
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Introduction
Entanglement is the backbone of quantum information science and its applications^{1}. Entangled states of light are necessary for distributed quantum protocols, quantum metrology^{2} and quantum internet^{3}. A distributed quantum network requires entanglement between light modes of different colours optimized for interaction with the nodes as well as for communication between them. This enables quantum protocols such as teleportation and steering between disparate quantum systems^{4,5}, quantum sensing^{2,6} and quantumenhanced gravitational wave detection^{7,8}.
In photonics the nonlinear optics toolbox is the primary resource for generating quantum correlations in continuous variables (CV). The χ^{(2)} nonlinearity can produce a variety of singlemode squeezed states^{9,10}.The interference of singlemode squeezed states has been used to demonstrate strongly correlated EPR states on sideband modes^{11,12} and ultralarge timebin cluster states^{13} enabling complex state topologies necessary for measurementbased quantum computing^{14}. However, this interference method is applicable only to monochromatic states. Multimode light can be used to generate frequency combs with quantum correlations among different nearby frequencies^{15,16}, but the constrains on the frequency span limit the applications in hybrid quantum networks.
Alternatively, secondharmonic generation (SHG)^{17} and the process of nondegenerate parametric downconversion^{18,19} allow for the generation of correlations between modes with a large difference in wavelengths. The nondegenerate parametric process produces nonclassical correlation via annihilation of a photon with the frequency ω_{0} (pump) generating twin photons pairs with frequencies ω_{1} (signal) and ω_{2} (idler); satisfying ω_{0} = ω_{1} + ω_{2} and having the squeezed state production as the degenerate case where ω_{1} = ω_{2}. Previous approaches to generation of multicolour CV quantum correlations with frequency nondegenerate optical parametric oscillators (NOPO) utilized operation above the oscillation threshold and resulted in modest levels of entanglement^{20,21,22}. Noteworthy, to the best of our knowledge, lowfrequency (<500 kHz) twocolour entanglement relevant for sensing applications has never been demonstrated.
Here we demonstrate a highquality, tunable and versatile twocolour EPR state source enabled by a novel experimental scheme (see Fig. 1a). Two coherent laser sources are upconverted via the sumfrequency generation (SFG) ω_{1} + ω_{2} = ω_{SFG}, and the output is used as a pump beam for the NOPO below threshold. The NOPO generates a large number of entangled output modes, Ω_{1,i} and Ω_{2,i}. Combining temperature phasematching with the dualwavelength resonance of the NOPO we broadly tune it to generate quantum correlated modes around two desirable disparate colours of the lasers, ω_{1}, ω_{2}, respectively. The two beams with the EPR mode sets are separated by a dichroic mirror (see Supplementary Figure 1 for details) and are superimposed with the strong coherent states at ω_{1} and ω_{2} in independent homodyne measurements. Optoelectronic control of the doubleresonance NOPO and wide tunability of the relative phases of the four quadrature operators allow for generation and measurement of a robust twocolour EPR state. The particular choice of the wavelengths of the entangled modes at 852 nm and 1064 nm has been motivated by the envisioned application for quantumenhanced gravitational wave detection^{7,8}, but it can be readily applied towards a quantum channel between telecom wavelengths and atomic quantum memories^{23,24}.
In the early days of quantum mechanics the EPR paradox triggered the discussion on quantum nonlocality^{25}. Recent advances in quantum information theory make possible the characterization of different quantum correlations and the specific nonclassical tasks they allow^{26}. Quantum steering has emerged as an intermediate effect between quantum entanglement and Bell nonlocality^{27}. Given a system composed of two quantum objects A and B, described by the state ρ_{AB}, quantum steering implies that by performing measurements on the system A one can influence the quantum state of the system B. Steering is an asymmetric phenomena, measurements on the system B may or may not steer the quantum state of A^{28,29}. This quantum correlation stronger than entanglement leads to applications to onesided device independent quantum key distribution^{30,31,32} and quantum metrology^{33}.
To demonstrate the quantum correlations, we apply the EPRparadox framework^{5,25} of Reid’s EPR criterion^{34}. In this context, we reproduce the original EPRparadox situation if by measurements on one of the subsystems one can infer the expected values of variables in the other subsystem in such a way as to obtain an apparent violation of the Heisenberg uncertainty principle. Consider noncommuting variables associated with the signal (1) and idler (2) field quadratures, [x_{j}, y_{j}] = 2i, j ∈ {1, 2}. We take the violation of the inequality defined in ref. 34, 35.
as a witness of EPRsteering, where the conditional variance is defined as \({V}_{{{{{{{{{\mathcal{O}}}}}}}}}_{12}}={\min }_{{w}_{{{{{{{{\mathcal{O}}}}}}}}}}{{{{{{{\rm{Var}}}}}}}}\left[{{{{{{{{\mathcal{O}}}}}}}}}_{1}{w}_{{{{{{{{\mathcal{O}}}}}}}}}{{{{{{{{\mathcal{O}}}}}}}}}_{2}\right]\), with the parameter \({w}_{{{{{{{{\mathcal{O}}}}}}}}}\in {\mathbb{R}}\). \({{{{{{{{\mathcal{E}}}}}}}}}_{12}^{2} < 1\) is an EPRsteering criterion sufficient for Gaussian states and homodyne measurements^{35}, ruling out the local hidden state description of the system (1) or (2) if the indices are swapped. Moreover, for the symmetric case, the product of conditional variances can be used as a quantifier for the degree of EPR entanglement in a system^{5}.
The theory comprising the dynamics of the EPR variables from a NOPO can be found in^{19,36}. In a nutshell, the generalized quadrature operator \({x}_{1}(\theta )\equiv {e}^{i\theta }{a}_{1}^{{{{\dagger}}} }+{e}^{i\theta }{a}_{1}\) is correlated with x_{2}( − θ), while y_{1}(θ) ≡ x_{1}(θ + π/2) is anticorrelated with y_{2}( − θ), here the pump phase is taken as a reference. The variances of the twomode operators \({X}^{\pm }(\theta )=[{x}_{1}(\theta )\pm {x}_{2}(\theta )]/\sqrt{2}\), Y^{±}(θ) ≡ X^{±}(θ + π/2) in case of symmetric losses are given by the wellknown expressions^{36}.
where σ = P/P_{th} is the pump power (P) normalized by the threshold power (P_{th}), \(\tilde{{{\Omega }}}={{\Omega }}/\delta \nu\) is the measured noise sideband frequency (Ω) normalized by the cavity bandwidth (δν), and η^{tot} is the total efficiency^{19,36}. Thus the sum and the difference of the quadratures behave as two independent singlemode squeezed subspaces.
Several factors may affect the observation of optimum correlations. Asymmetric losses may require optimization of the quadrature combination to achieve the best value of crosscorrelations^{19}. Another limitation is due to the angular jitter of the noise ellipse, leading to a projection of antisqueezing onto the squeezed quadrature^{37}. The effect of the phase noise of an arbitrary quadrature operator Q(θ) can be modeled by \({V}_{Q}(\delta {\theta }_{n})={\cos }^{2}(\delta {\theta }_{n}){V}_{Q(\theta )}+{\sin }^{2}(\delta {\theta }_{n}){V}_{Q(\theta+\pi /2)}\), where δθ_{n} is the RMS phase noise. Due to a complex architecture of phases required for observation of twocolour EPR correlations combined with low losses and high parametric gain, δθ_{n} is a dominant factor limiting the degree of quantum steering. Therefore, below we use V_{Q} and V_{Q}(δθ_{n}) interchangeably.
Results
Experimental scheme
The layout of the experimental setup is presented in Fig. 1a. We measure the field quadratures from the NOPO output to observe correlations and to determine the witness of EPRsteering between the signal and idler beams. Each of the two modes is superimposed with the corresponding local oscillator and directed to a balanced homodyne detector. Control of the relative phases between the local oscillators and the corresponding quantum modes selects which quadrature is projected into the photocurrents i_{1} ∝ x_{1}(θ_{1}) and i_{2} ∝ x_{2}(θ_{2}). Fig. 1b, c show the experimental realizations of the photocurrents presenting real time strong nonclassical correlations between the quadrature measurements of the fields at two colours separated by 200 nm.
The relative phase \({\theta }_{j}={\phi }_{j}^{{{{{{{{\rm{OPO}}}}}}}}}{\phi }_{j}^{{{{{{{{\rm{LO}}}}}}}}}\) is monitored through the interference between the local oscillators and weak back reflection of the locking beams (Fig. 1a) from the nonlinear crystal. We use those interference signals to control the phases in each LO path with a PZT, thereby selecting the quadratures to be measured (see Supplementary Note 1 and 5). The observables X^{±} (Y^{±}) are recorded by initially setting of θ_{1} = θ_{2} = 0(π/2) and by the subsequent fine adjustment of one of the phases to maximize the measured correlations.
Quantum correlations and EPRsteering
The pumping power corresponding to the maximal EPR correlations corresponded to σ ≈ 0.25 with respect to the oscillation threshold. For these pumping conditions, we observed \({V}_{{X}^{}}=7.1\pm 0.5\) dB; \({V}_{{Y}^{+}}=6.2\pm 0.5\) dB for the frequency range 50–300 kHz (see Fig. 2). Operation closer to the threshold gain does not improve the level of quantum correlations due to the enhanced influence of the phase noise. Further down in the audio frequency band, the correlations are even more vulnerable to the phase noise. Still, a combination of passive stability and active optoelectronic control allows us to achieve the EPR correlations of \({V}_{{X}^{}}=5.7\pm 0.6\) dB; \({V}_{{Y}^{+}}=5.2\pm 0.6\) dB down to 10 kHz (Fig. 2).
We obtain the spectra of the quadratures of the optical fields \({V}_{Q}^{{{{{{{{\rm{o}}}}}}}}}\) required for characterizing the EPRsteering by correcting the measured variances V_{Q} by the nonunity quantum efficiency of the detectors (See Methods) \({\eta }^{\det }\) as \({V}_{Q}={\eta }^{\det }({V}_{Q}^{{{{{{{{\rm{o}}}}}}}}}1)+1\), Using the average detector efficiency^{19} (see Methods) \({\eta }^{\det }=\sqrt{{\eta }_{1}^{\det }{\eta }_{2}^{\det }}=0.945\), we obtain the following variances of the light modes for the data presented in Fig. 2: \({V}_{{X}^{}}^{{{{{{{{\rm{o}}}}}}}}}=8.3\pm 0.6\) dB; \({V}_{{Y}^{+}}^{{{{{{{{\rm{o}}}}}}}}}=7.1\pm 0.5\) dB and \({V}_{{X}^{+}}^{{{{{{{{\rm{o}}}}}}}}}=10.0\pm 0.5\) dB; \({V}_{{Y}^{}}^{{{{{{{{\rm{o}}}}}}}}}=9.3\pm 0.6\) dB. From these data, using w_{x} = − w_{y} = 1, we obtain the witness of the efficient EPRsteering for the two optical fields: \({{{{{{{{\mathcal{E}}}}}}}}}_{12}^{2}={{{{{{{{\mathcal{E}}}}}}}}}_{21}^{2}={{{{{{{{\mathcal{E}}}}}}}}}^{2}=0.11\pm 0.01\ll 1\). To the best of our knowledge, this is by far the highest level of EPRsteering achieved between continuous variable modes at disparate wavelengths.
State purity
To characterize our EPR state, we focus on its purity, which is directly related to the twinphoton nature of the parametric downconversion process. For a Gaussian state with the covariance matrix \({\mathbb{V}}\), assuming no phaseamplitude correlations between different modes^{38}, the state purity is given by \(\mu=1/\sqrt{{{{{{{{\rm{Det}}}}}}}}{\mathbb{V}}}=1/\sqrt{{V}_{{X}^{}}^{{{{{{{{\rm{o}}}}}}}}}{V}_{{Y}^{}}^{{{{{{{{\rm{o}}}}}}}}}{V}_{{X}^{+}}^{{{{{{{{\rm{o}}}}}}}}}{V}_{{Y}^{+}}^{{{{{{{{\rm{o}}}}}}}}}}\) which yields μ = 0.63 ± 0.16. Our results compare favorably with the highest to date twocolour entanglement with purity 0.11 (corrected by the reported \({\eta }^{\det }\)) observed in the MHz range^{22}. High purity is especially relevant for quantum enhancement of interferometry where both quadratures can contain useful information^{39,40}.
Further improvement of the phase control would make it possible to observe an even higher degree of EPRsteering, while preserving the state purity. Coherent phaselock^{41} would eliminate classical noise injection and enable observation of twocolour entanglement down to the Hz domain.
Discussion
To summarize, we have presented the experimental realization of the EPR state of light between modes of different colours with the unprecedented degree of continuous variable EPRsteering and purity. Those properties extend over a wide signal frequency range into the acoustic frequency band.
Our approach is readily applicable to entanglement generation between modes with vastly different and tunable wavelengths, thus making it a valuable tool for quantum networks combining long distance propagation with quantum memories.
Methods
NOPO design
The NOPO cavity is designed and tuned to be resonant for both signal (852 nm) and idler (1064 nm) beams while the pump beam (473 nm) is used in a singlepass regime. The cavity has a bowtie configuration to reduce the negative influence of backscattered light and to improve the escape efficiency^{18}. Quantum light emerges through the output coupling mirror with the transmission coefficient T = 12% for both 852 nm and 1064 nm modes. Thus, the cavity bandwidth, free spectral range, and finesse are very similar for both wavelengths. The main NOPO parameters are given in Table 1. To minimize astigmatism and contamination from highorder transverse modes, we finetune the cavity size and angles of incidence on the mirrors. The NOPO is built in a monolithic aluminum box for better mechanical stability. We use a type0 periodically poled KTP (PPKTP) crystal (Raicol Crystals Ltd) as the nonlinear medium with an antireflection (AR) coating for 473 nm, 852 nm and, 1064 nm. The desired phase matching is achieved by setting the crystal temperature to ≈ 63 ^{∘}C and stabilizing it to ± 1 mK. The passive intracavity losses \({{{{{{{{\mathcal{L}}}}}}}}}_{j}\) are dominated by the PPKTP bulk losses, Table 1.
Estimated efficiencies
The measured efficiencies in our system are shown in Table 2. The single beam escape efficiency is given by \({\eta }_{j}^{{{{{{{{\rm{esc}}}}}}}}}=T/(T+{{{{{{{{\mathcal{L}}}}}}}}}_{j})\), and is the most significant parameter to guarantee highpurity state generation. We achieve the overall escape efficiency \({\eta }^{{{{{{{{\rm{esc}}}}}}}}}=\sqrt{{\eta }_{1}^{{{{{{{{\rm{esc}}}}}}}}}{\eta }_{2}^{{{{{{{{\rm{esc}}}}}}}}}}=98.5\, \pm \,0.2\%\)^{19}. We have also explored the effect of the bluelightinduced infrared absorption (BLIIRA)^{42} on the overall escape efficiency and found it to be negligible under our pumping conditions. The combination of ultralow intracavity losses and BLIIRAfree operation allows us to achieve the escape efficiency for a twocolour system comparable to the stateofart degenerate OPO^{11}.
Table 2 presents the propagation efficiency \({\eta }_{j}^{{{{{{{{\rm{pro}}}}}}}}}\) from the NOPO output to the detectors, the homodyne efficiency η^{mm} of the signalLO modematching, and the photodiodes’ quantum efficiency \({\eta }_{j}^{\det }\) (see Supplementary Note 4).
Data availability
The data presented in the figures have been deposited in the University of Copenhagen depository under the link: erda.ku.dk/archives/035f072dbb7c48095ed9d78fecd92d81/publishedarchive.html.
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Acknowledgements
We gratefully acknowledge discussions with J. H. Müller and J. Appel. This project has been funded by the InnoFond Denmark through the Eureka Turbo program, by the European Research Council (ERC) under the EU’s Horizon 2020 research and innovation programme (grant agreement No 787520), and by the VILLUM FONDEN under the Villum Investigator Grant no. 25880. T. B. B. was partially supported by CAPES and CNPQ.
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E.S.P. conceived and led the project. T.B.B. and V.N. designed and built the experiment with the help of H.K. and M.L. T.B.B. and V.N. obtained the main experimental results. The paper was written by E.S.P., T.B.B. and V.N., with contributions from H.K. and M.L. E.S.P. and M.L. supervised the research.
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Brasil, T.B., Novikov, V., Kerdoncuff, H. et al. Twocolour highpurity EinsteinPodolskyRosen photonic state. Nat Commun 13, 4815 (2022). https://doi.org/10.1038/s41467022324957
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DOI: https://doi.org/10.1038/s41467022324957
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