Two-colour high-purity Einstein-Podolsky-Rosen photonic state

We report a high-purity Einstein-Podolsky-Rosen (EPR) state between light modes with the wavelengths separated by more than 200 nm. We demonstrate highly efficient EPR-steering between the modes with the product of conditional variances \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{{\mathcal{E}}}}}}}}}^{2}=0.11\pm 0.01\ll 1$$\end{document}E2=0.11±0.01≪1. The modes display − 7.7 ± 0.5 dB of two-mode squeezing and an overall state purity of 0.63 ± 0.16. EPR-steering is observed over five octaves of sideband frequencies from RF down to audio-band. The demonstrated combination of high state purity, strong quantum correlations, and extended frequency range enables new matter-light quantum protocols.

Entanglement is the backbone of quantum information science and its applications [1].Entangled states of light are necessary for distributed quantum protocols, quantum sensing [2] and quantum internet [3].A distributed quantum network requires entanglement between light modes of different colour optimized for interaction with the nodes as well as for communication between them.Here we demonstrate high-purity Einstein-Podolsky-Rosen (EPR) entangled state between light modes with the wavelengths separated by more than 200 nm.The modes display −7.7 ± 0.5 dB of two-mode entanglement and an overall state purity of 0.63 ± 0.16.Entanglement is observed over five octaves of sideband frequencies from rf down to audio-band.In the context of two-colour entanglement, the demonstrated combination of high state purity, strong entanglement, and extended frequency range paves the way to new matter-light quantum protocols, such as teleportation between disparate quantum systems [4], quantum sensing [2,5,6] and quantumenhanced gravitational wave interferometry [7,8].The scheme demonstrated here can be readily applied towards entanglement between telecom wavelengths and atomic quantum memories [9,10].
The nonlinear optics toolbox is the primary resource for generating photonic entangled states in continuous variables (CV).The χ (2) nonlinearity is usually explored to produce single-mode squeezed states, where the noise in one quadrature is suppressed beyond the vacuum field fluctuations.The interference of single-mode squeezed states have been used to demonstrate strong two-mode entanglement [11,12], but this method limits the EPR states to be monochromatic.
As alternatives, second-harmonic generation (SHG) [13] and the process of non-degenerate parametric downconversion [14,15] allow for the generation of correlations between modes at different frequencies.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 case where Previous approaches to generation of multi-colour quantum correlations with frequency non-degenerate optical parametric oscillators (NOPO) differ significantly from the degenerate case due to the lack of frequencymatched local oscillators (LOs).Operation above the oscillation threshold has been used to overcome this problem, but it increases the complexity of the system making it more susceptible to noise contamination and resulting in modest levels of entanglement [16][17][18].Noteworthy, to the best of our knowledge, low-frequency (< 500 kHz) two-colour entanglement relevant for sensing applications has never been demonstrated.
Here we demonstrate a high-quality, tunable and versatile two-colour entanglement source enabled by a novel experimental scheme.Two coherent laser sources are upconverted via the sum-frequency generation (SFG) ω 1 + ω 2 = ω SFG , and the output is used as a pump beam for the NOPO below threshold, see Figure 1.The entangled output modes of the NOPO, Ω 1,i and Ω 2,i centered around two different colours of the lasers, ω 1 , ω 2 , respectively, are separated and superimposed with the coherent states at ω 1 and ω 2 enabling independent detection and entanglement verification.Optoelectronic control of the double-resonance NOPO and wide tunability of the relative phases of the four quadrature operators allow for generation of robust and strong two-colour EPR entanglement.The particular choice of the wavelengths of the entangled modes at 852 nm and 1064 nm has been motivated by the envisioned application for entanglementenhanced gravitational wave interferometry [7,8].
To demonstrate the quantum correlations, we apply the EPR-paradox framework [19,20] of Reid's EPR criterion [21].In this context, we reproduce the original EPR-paradox situation if by measurements on one 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 [21,22] as the measure of the EPR entanglement as follows where the conditional uncertainty is defined as , with the parameter w O ∈ R.This was later recognized as an EPRsteering criterion and is sufficient and necessary for Gaussian states [22], ruling out the local state description of the system (1) or (2).Moreover, E 2 can be used as a quantifier for the degree of EPR entanglement in a system [20].Another necessary and sufficient entanglement criterion for Gaussian states is based on the inequality [23,24] where the sum of variances V EPR allows finding the entanglement of formation (EoF) [25] which defines a number of e-bits that can be distilled from the state.The theory comprising the dynamics of the EPR variables from a NOPO can be found in [15,26].In a nutshell, the generalized quadrature operator x 1 (θ) ≡ e iθ a † 1 + e −iθ 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 ) in case of symmetric losses are given by the well-known expressions [26] where σ = P/P th is the pump power (P ) normalized by the threshold power (P th ), Ω = Ω/δν is the measured noise sideband frequency (Ω) normalized by the cavity bandwidth (δν), and η tot is the total efficiency [15,26].
Thus the sum and the difference of the quadratures behave as two independent single-mode squeezed subspaces.Several factors may affect the observation of optimum entanglement.Squeezing in X − and Y + is the signature of EPR correlations, but asymmetric losses may require optimization of the quadrature combination to achieve the best value of cross-correlations [15].Another limitation is due to the angular jitter of the noise ellipse, leading to a projection of anti-squeezing onto the squeezed quadrature [27].The effect of the phase noise of an arbitrary quadrature operator Q(θ) can be modeled by where δθ n is the RMS phase noise.Therefore, for a finite δθ n the highest level of noise reduction is achieved at a specific pump power.Because phase noise is always present, below we use The layout of the experimental setup is presented in Fig. 1a.We measure the field quadratures from the NOPO output to observe correlations and determine the EPR entanglement between the signal and idler beams.Each of the two entangled modes is directed to a balanced homodyne detector to be mixed with the corresponding local oscillator -control of the relative phase between the local oscillator and the entangled mode selects which quadrature is projected into the photocurrents i 1 ∝ x 1 (θ 1 ) and i 2 ∝ x 2 (θ 2 ).Figs. 1 (b-c) show the experimental realizations of the photocurrents presenting strong non-classical correlations between the quadrature measurements by the two detectors.
The relative phase θ j = φ OPO j − φ LO j is recovered from the interference between the local oscillator and a weak back reflection of the cavity lock beam (Fig. 1a) from the intracavity nonlinear crystal.We use this interference information to control the phase in each LO path with a PZT and to select which quadrature to measure (see Supplementary Information).The observables X ± (Y ± )  are recorded by initially setting of θ 1 = θ 2 = 0 (π/2) and the subsequent fine adjustment of one of the phases to maximize the measured correlations.We found the optimal pumping power to be σ ≈ 0.25, corresponding to operation well below the threshold.For these pumping conditions, we measured V X − = −7.1±0.5 dB; V Y + = −6.2± 0.5 dB for the frequency range 50 to 300 kHz (see Fig. 2).Increasing the pump power does not improve the entanglement level due to the enhanced phase noise influence.Further down in the audio frequency band, the entanglement is extremely vulnerable to phase noise.The combination of passive stability and active optoelectronic control allows us to achieve entanglement down to below 10 kHz (Fig. 2).However, even the optimized low-frequency noise limited the EPR correlations to V X − = −5.7 ± 0.6 dB; V Y + = −5.2± 0.6 dB.
We obtain the quadratures of the optical field V o Q characterizing its entanglement by correcting the measured variances V Q by the non-unity quantum efficiency of the detectors η det as Using the average detector efficiency [15] (see Methods) .945 , we obtain the following variances of the light modes for the data presented in Fig. 2: V From that we obtain the entanglement measures for the optical fields: E 2 c = 0.029 ± 0.007 1 and V EPR,c = 0.34 ± 0.04 2 (−7.7 ± 0.5 dB below the entanglement boundary).To the best of our knowledge, this is the highest level of EPR entanglement achieved between beams at disparate wavelengths.
To fully characterize our EPR state, we focus on its pu-rity, which is directly related to the twin-photon nature of the parametric down-conversion process.For a Gaussian state with the covariance matrix V the state purity is given by µ ] which yields µ = 0.63 ± 0.16.Our results compare favorably with the highest to date two-colour entanglement with V EPR,c = 0.38 and with purity 0.11 (corrected by the reported η det ) observed in the MHz range [18].High purity is especially relevant for quantum enhancement of interferometry where both quadratures can contain useful information [29,30].
We have presented experimental realization of the EPR entangled state of light between modes of different colours with unprecedented degree of entanglement and purity and have expanded the entanglement into the acoustic frequency range.
The present limitations to the observed entanglement are primarily due to the imperfect phase-locking affected by spurious non-linear mixing.Improvement of phase control would make possible to observe even higher levels of entanglement, preserving the state purity.Changing the phase control scheme to a coherent phase-lock [31] and locking the cavity by frequency-shifted modes would eliminate classical noise injection, making it possible to observe two-colour entanglement down to the Hz domain.
The present choice of wavelengths of 1064 nm and 852 nm is motivated by the current laser choice of LIGO and is geared towards the broadband quantum noise reduction in gravitational wave detectors using an ensemble of Caesium atoms [7,8].Our approach is readily applicable to entanglement generation between other colours, for example, between 852 nm compatible with the atomic quantum memory and a telecom wavelength.

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 single-pass regime.The cavity has a bow-tie configuration to reduce the negative influence of back-scattered light and to improve the escape efficiency [14].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 I.To minimize astigmatism and contamination from high-order transverse modes, we fine-tune 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 type-0 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 L j are dominated by the PPKTP bulk losses, Table I.

Estimated efficiencies
The measured efficiencies in our system are shown in Table II.The single beam escape efficiency is given by η esc j = T /(T + L j ), and is the most significant parameter to guarantee high-purity state generation.This leads to an overall escape efficiency η esc = η esc 1 η esc 2 = 98.5 ± 0.2% [15].We also studied the effect of the bluelight-induced infrared absorption (BLIIRA) [32] on the overall escape efficiency and found it negligible under the normal pumping conditions.The combination of ultralow intracavity losses and BLIIRA-free operation are unprecedented for a two-colour system giving an escape ef-ficiency comparable to the state-of-art degenerate OPO [11].
Table II presents the propagation efficiency η pro j from the NOPO output to the detectors, the homodyne efficiency η mm of the signal-LO mode-matching, and the photodiodes' quantum efficiency η det j (see Supplementary Information).

Figure 1 .
Figure 1.(A) Scheme of the experimental setup.The 852 nm and 1064 nm lasers produce the local oscillators and the blue light used to pump the NOPO through the sum-frequency generation.The entangled modes at the two colours emerging from the NOPO are separated with a dichroic mirror, mixed with the LOs, and measured by the homodyne detectors.The photocurrents are recorded by the analog-to-digital converter (ADC) to obtain information on the joint system operators.(B-C) The experimental realizations of the photocurrents i1 and i2 showing strong non-classical correlations for {x1, x2} and {y1, −y2}.Here the signals were demodulated at 200 kHz and integrated by a 10 kHz low-pass filter.The quadrature values are in vacuum state units.

Figure 2 .
Figure2.Spectra of the EPR quadratures normalized to shot-noise level (SN) for the frequency range 10 to 300 kHz.The left traces show the quantum noise suppression optimized for low spectral frequencies (10-50 kHz), while the right part corresponds to the best entanglement level achieved in 50-300 kHz spectral range (see text below).The narrow peaks come from the phase noise of the lasers.The data are corrected for electronic noise which is 18.5 dB below the shot-noise level.

Table I .
NOPO main parameters.