3D phase-matching conditions for the generation of entangled triplets by χ<sup>(2)</sup> interlinked interactions

Maria Bondani; Alessia Allevi; Eleonora Gevinti; Andrea Agliati; Alessandra Andreoni

doi:10.1364/OE.14.009838

1. Introduction

The production of multipartite entangled states by means of multiple nonlinear interactions occurring in a single nonlinear crystal has been recently suggested [1, 2, 3] as an alternative to the use of single-mode squeezed states [4, 5, 6] or of two-mode entangled states and linear optical elements [7]. As in the case of bipartite states produced by spontaneous parametric downconversion, such multipartite states display entanglement also in the macroscopic regime in which bright outputs are generated: for this reason they are particularly interesting for all the applications of continuous-variable entanglement [8, 9].

The possibility of realizing the simultaneous phase-matching (PM) of two traveling-wave parametric processes in a single crystal in a seeded configuration has been already demonstrated [10, 11]. In this paper we present the experimental realization of the same interlinked interactions starting from vacuum fluctuations. The output of the nonlinear crystal displays the entire ensemble of inseparable tripartite entangled states satisfying the PM conditions. In order to identify a triplet, we develop a 3D calculation of the characteristics of interlinked PM that reproduce the experimental phenomenology. Finally we perform a preliminary verification of the number of photons conservation law by means of intensity correlations.

2. Theory

We consider the Hamiltonian describing the simultaneous PM of a downconversion and an upconversion processes

H_{int} = γ_{1} a_{1}^{†} a_{3}^{†} + γ_{2} a_{2}^{†} a_{3} + h . c .,

where γ ₁∝a ₄ and γ ₂∝a ₅ are coupling coefficients. The two interlinked interactions involve five fields a_j , two of which, say a ₄ and a ₅, enter the crystal and act as non-evolving pumps. When acting on the vacuum as the initial state, H _int admits the following conservation law

N_{1} (t) = N_{2} (t) + N_{3} (t),

being N_j (t)=〈 $a_{j}^{†}$ (t)a_j (t)〉 the mean number of photons in the j-th mode, and yields a fully inseparable tripartite state [1]. We consider the realization of Eq. (1) in a negative uniaxial crystal in type-I non-collinear PM interaction geometry. The processes must satisfy energy-matching (ω ₄=ω ₁+ω ₃, ω ₂=ω ₃+ω ₅) and PM conditions ( $k_{4}^{e}$ = $k_{1}^{o}$ + $k_{3}^{o}$ , $k_{2}^{e}$ = $k_{3}^{o}$ + $k_{5}^{o}$ ), where ω_j are the angular frequencies, k _j are the wavevectors and ^o,e indicate ordinary and extraordinary field polarizations. In order to investigate the geometrical constraints imposed by the PM conditions, we analytically calculate the output angles of each interacting field.

To simplify calculations, we assume, in the reference frame depicted in Fig. 1, that the two pumps lie in the (y, z)-plane containing the optical axis (OA) and the normal to the crystal entrance face and that the pump field a ₄ propagates along the normal, z. The solutions corresponding to the effective experimental orientation of the crystal can be obtained by simply calculating the refraction of the beams at the crystal entrance/output faces. Accordingly, the PM conditions for the two interactions simultaneously phase-matched can be written as

k_{1} \sin β_{1} + k_{3} \sin β_{3} = 0

k_{1} \cos β_{1} \sin ϑ_{1} + k_{3} \cos β_{3} \sin ϑ_{3} = 0

k_{1} \cos β_{1} \cos ϑ_{1} + k_{3} \cos β_{3} \cos ϑ_{3} = k_{4}

k_{2} \sin β_{2} = k_{3} \sin β_{3}

k_{2} \cos β_{2} \sin ϑ_{2} = k_{3} \cos β_{3} \sin ϑ_{3} + k_{5} \sin ϑ_{5}

k_{2} \cos β_{2} \cos ϑ_{2} = k_{3} \cos β_{3} \cos ϑ_{3} + k_{5} \cos ϑ_{5}

where the angles are defined as in Fig. 1. The wavevectors k_j are defined as k_j =n_j (ω_j,k̂_j)ω_j/c, being c the speed of light in the vacuum and n _j(ω_j,k̂_j ) the refraction indices of the medium

n_{j} (ω_{j}) = n_{o} (ω_{j}) j = 1, 3, 5

n_{2} (ω_{2}, φ) = {[\frac{\cos^{2} φ}{n_{o}^{2} (ω_{2})} + \frac{\sin^{2} φ}{n_{e}^{2} (ω_{2})}]}^{- 1 ⁄ 2}; n_{4} (ω_{4}, α) = {[\frac{\cos^{2} α}{n_{o}^{2} (ω_{4})} + \frac{\sin^{2} α}{n_{e}^{2} (ω_{4})}]}^{- 1 ⁄ 2},

with n_o,e (ω) given by the dispersion relations of the medium. Note that cosϕ=cosβ₂ cos(ϑ₂-α) and β₅=β₄=ϑ₄=0. For fixed frequencies and propagation directions of the pump fields, we have 11 variables (ω₁, ω₂, ω₃, ϑ₁, ϑ₂, ϑ₃, ϑ₅, β₁, β₂, β₃, α) and 8 equations (energy conservation and Eqs. (3) to (8)). The problem can thus be analytically solved by choosing three of the variables (say ω₁, ϑ₅ and α) as parameters (see the Appendix for an outline of the procedure). In order to efficiently handle the dependence of the solutions on the free parameters, we implemented the calculation by using the software Mathematica (Wolfram Research).

Fig. 1. Scheme of the phase-matched interlinked interactions: (x,y)-plane coincides with the crystal entrance face; α, tuning angle; β_j’s, angles to (y, z)-plane; ϑ_j’s, angles on the (y,z)-plane; φ, angle to the optical axis (OA).

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In Fig. 2 we show one picture of the movie containing the comparison between measured and calculated results for the output of the crystal as a function of the tuning angle α for a fixed value of the external angle between the two pumps. Note that not all the calculated output frequencies appear in the experimental part due to the sensitivity of the photocamera sensor. As the pictures of the experimental outputs were taken on a screen located beyond the crystal normally to the direction of pump a4, the calculations take into account both the rotation of the crystal and the refraction of the beams at the entrance and exit faces of the crystal. In particular, refraction at the exit face gives: sinβ_j,out=n _j sinβ_j and sinϑ_j,out=n _j/(1- $n_{j}^{2}$ sin²β_j)^1/2cosβ_j sinϑ_j.

3. Experiment

For the realization of the interaction described by Eq. (1), the pump fields were provided by the outputs of a continuous-wave mode-locked Nd:YLF laser regeneratively amplified at the repetition rate of 500 Hz (High Q Laser Production): the third-harmonic pulse (λ ₄=349 nm, ~4.45 ps) was used as the a₄ field producing the downconversion cones and the fundamental pulse (λ₅=1047 nm, ~7.7 ps) as the a₅ field pumping the upconversion process. As depicted in Fig. 3 a), both pumps were focused and injected, at the angle ϑ₅ _,ext=-34.8 deg with respect to each other, into a β-BaB₂O₄ crystal (BBO, Fujian Castech Crystals, 10 mm×10 mm cross section, 4 mm thickness) cut for type-I interaction (ϑ_cut=34 deg). The required superposition of the two pumps in time was obtained with a variable delay line. For alignment purposes, we used the light of a He-Ne laser (Melles-Griot, 5 mW max output power) to seed the process at λ₁=632.8 nm. The beam was collimated and sent to the BBO in the plane containing the pumps at the external angle ϑ_1,ext=-2.54 deg with respect to a₄. The seeded interactions produced two new fields: a₃ (λ₃=778.2 nm, ϑ_3,ext=3.35 deg) generated as the difference-frequency of a₄ and a₁, and a₂ (λ₂=446.4 nm, ϑ_2,ext=-12.78 deg) generated as the sum-frequency of a₃ and a₅. The intensity values of the pump pulses on BBO were ~16 GW/cm² for field a₄ and ~0.24 GW/cm² for field a₅. As shown by the picture in Fig. 3 b), on a white screen located beyond the crystal it was possible to see both the tunable bright downconversion cones and the two polychromatic half-moon-shaped states generated by the upconversion process together with the spots of the fields generated by the seeded interactions.

Fig. 2. (1.271 MB) Movie showing the variation of the measured and calculated outputs of the crystal as a function of the tuning angle α for fixed external angle between the pumps. Left panel: picture of the output of the crystal taken with a commercial digital photocamera (no color correction applied). LHS: a portion of the downconversion cones. RHS: output of the upconversion process. Right panel: calculated output of the upconversion process.

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As a first characterization of the process, we realized intensity-correlation measurements by filtering each portion of light in frequency and aligning three pin-holes of suitable diameters [12] on the spots of the seeded process (the seeding He-Ne light was switched-off once completed the alignment). The light selected by the pin-holes was focused on three p-i-n photodiodes (two, D_1,2 in Fig. 3 a), S5973-02 and one, D₃, S3883, Hamamatsu). Each current output was integrated by a synchronous gated-integrator in external trigger modality, digitized by a 13-bit converter (SR250, Stanford Research Systems) and recorded in a PC-based multichannel analyzer. Data acquisition and analysis were performed with software LabView (National Instruments). The mean numbers of the detected photons in the three parties of the triplet were M ₁=1.082×10⁸, M ₂=2.3×10⁶ and M ₃=1.115×10⁸. In order to obtain a comparison between Eq. (2) and the mean numbers of the detected photons, we must take into account the overall detection efficiencies in the three detection arms (η₁~0.44, η₂~0.72 and η₃~0.43). As we could measure them with a low accuracy we can only assess that Eq. (2) is qualitatively satisfied. To be independent of the detection efficiencies, we evaluate the correlation function of the detected photons on a set of K subsequent laser shots

Γ (k) = \frac{1}{K} \sum_{i = 1}^{K} \frac{[m_{1} (i) - 〈 m_{1} 〉] [m_{2} (i + k) + m_{3} (i + k) - 〈 m_{2} + m_{3} 〉]}{σ (m_{1}) σ (m_{2} + m_{3})},

in which m _j is the number of detected photons and σ(x)=(〈x ²〉-〈x〉²)^1/2 is the standard deviation and the averages, 〈...〉, are evaluated on the same set of laser shots. We measured a value Γ(0)=0.916 for the correlation coefficient. The existence of strong intensity correlations among the generated fields, implied by the conservation law of Eq. (2), is necessary but not sufficient to demonstrate the entangled nature of the triplet. In fact, by extending to tripartite states the calculations made on bipartite states [12], it can be demonstrated [13] that for bright output beams the correlation coefficient must approach unity regardless the quantum or classical nature of the correlations. A discrimination between classical correlations and entanglement could, for instance, be done in terms of the difference photocurrent detected on the three output beams [12]. Moreover, a sufficient criterion for the full inseparability of the state described by Eq. (1) is given by the possibility of realize a true tripartite quantum protocol, such as the telecloning scheme described in [1].

Fig. 3. Upper panel: scheme of the experimental setup. BBO, nonlinear crystal; F_1–3, filters; P_1–3, pin-holes; f_1–5,s, lenses; D_1–3, p-i-n photodiodes; SGI, synchronous gated-integrator; MCA+PC, computer integrated multichannel analyzer. Lower panel: picture of the visible portion of the output states on a screen located beyond the nonlinear crystal.

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4. Discussion and conclusions

The perfect agreement between calculations and phenomenology of the generated states demonstrates the correctness of the model and enables us to forecast different interaction schemes that in the future will allow to overcome the experimental difficulties experienced in the present setup. In fact, our measurements are promising but not yet fully satisfactory for the identification of the triplets: we have to maximize the experimental value of Γ(0). Its non unitary value was probably due to the imperfect spatial selection of the parties by means of the pin-holes. In addition, in the interaction scheme we realized, portions of the signal and idler cones were superimposed so that some spurious light was present in the measurements thus determining a decrease in the correlation coefficient. This problem can be overcome by changing the parameters of interaction scheme (tuning angle and internal angles between the fields). Finally, to achieve a demonstration of the quantum nature of the correlations among the parties of the triplet, it would be necessary to decrease the number of photons in the generated fields in order to minimize the presence of noise coming from the laser source [12].

Appendix

Outline of the procedure to solve system (3)–(8). From the first of Eqs. (10) we write

k_{2} = {[G_{2} \cos^{2} (ϑ_{2} - α) \cos^{2} β_{2} + L_{2}]}^{1 ⁄ 2},

with G ₂=c ²/ $ω_{2}^{2}$ [1/ $n_{o}^{2}$ (ω₂)-1/ $n_{e}^{2}$ (ω₂)]=1/ $k_{2, o}^{2}$ -1/ $k_{2, e}^{2}$ and L ₂=c ²/ $ω_{2}^{2}$ 1/ $n_{e}^{2}$ (ω₂)=1/ $k_{2, e}^{2}$ . Squaring and summing Eqs. (3), (4) and (5) and defining A=( $k_{4}^{2}$ + $k_{3}^{2}$ - $k_{1}^{2}$ )/(2k ₃ k ₄) we get

\cos β_{3} = \frac{A}{\cos ϑ_{3}},

whereas squaring and summing Eqs. (6), (7) and (8) yield

k_{2}^{2} = k_{3}^{2} + k_{5}^{2} + 2 k_{3} k_{5} \cos β_{3} (\sin β_{3} \sin ϑ_{5} + \cos β_{3} \cos ϑ_{5})

k_{2}^{2} \cos^{2} β_{2} = k_{3}^{2} \cos^{2} β_{3} + k_{5}^{2} + 2 k_{3} k_{5} \cos β_{3} (\sin β_{3} \sin ϑ_{5} + \cos β_{3} \cos ϑ_{5}) .

By eliminating k ₂ from Eqs. (14) and (15) and using Eq. (13) we find

\cos^{2} β_{2} = \frac{k_{3}^{2} A^{2} \tan^{2} ϑ_{3} + k_{5}^{2} + 2 k_{3} k_{5} A (\tan ϑ_{3} \sin ϑ_{5} + \cos ϑ_{5})}{k_{3}^{2} + k_{5}^{2} + 2 k_{3} k_{5} A (\tan ϑ_{3} \sin ϑ_{5} + \cos ϑ_{5})} .

From Eq. (3) we get sinβ₁=-k ₃/k ₁ sinβ₃ that, substituted into Eq. (4) together with Eq. (13), gives: $ϑ_{1} = - k_{3} A ⁄ \sqrt{k_{1}^{2} - k_{3}^{2} \sin^{2} β_{3}}$ . Dividing Eq. (7) by Eq. (8) and using Eq. (13) we get

\tan ϑ_{2} = \frac{k_{3} A \tan ϑ_{3} + k_{5} \sin ϑ_{5}}{k_{3} A + k_{5} \cos ϑ_{5}} .

By squaring Eq. (12) and inserting Eq. (14) and Eq. (16) into it we obtain

\cos^{2} (ϑ_{2} - α) = \frac{1 - L_{2} [k_{3}^{2} + k_{5}^{2} + 2 k_{3} k_{5} \cos β_{3} (\sin β_{3} \sin ϑ_{5} + \cos β_{3} \cos ϑ_{5})]}{G_{2} [k_{3}^{2} \cos^{2} β_{3} + k_{5}^{2} + 2 k_{3} k_{5} \cos β_{3} (\sin β_{3} \sin ϑ_{5} + \cos β_{3} \cos ϑ_{5})]} .

By substituting Eq. (13) into Eq. (18), expanding cos² (ϑ₂-α) in Eq. (18) and using Eq. (17) to eliminate ϑ₂, we get a quadratic algebraic equation for the variable tanϑ₃

a \tan^{2} ϑ_{3} + b \tan ϑ_{3} + c = 0,

whose coefficients are

a=G ₂ A ² $k_{3}^{2}$ sin²α

b=2G ₂ Ak ₃(Ak ₃+k ₅cosϑ₅) sinαcosα+2G ₂ Ak ₃ k ₅sinϑ₅ sin²α+2L ₂ Ak ₃ k ₅sinϑ₅

c=G ₂(Ak ₃+k ₅cosϑ₅)²(cos²α-sin²α)+2G ₂ k ₅ sinϑ₅(Ak ₃+k ₅cosϑ₅) sinαcosα+G ₂(A ² $k_{3}^{2}$ + $k_{5}^{2}$ +2A k ₃ k ₅cosϑ₅) sin²α+L ₂( $k_{3}^{2}$ + $k_{5}^{2}$ +2Ak ₃ k ₅ cosϑ₅)-1.

Solving Eq. (19) gives ϑ₃ and then all the other variables as a function of the free parameters ω₁, α and ϑ₅.

Acknowledgements

This work was supported by the Italian Ministry for University Research through the FIRB Project n. RBAU014CLC-002. The Authors thank M.G.A. Paris (Universitè degli Studi, Milano) for theoretical support on correlations.

Present addresses: E. Gevinti, STMicroelectronics, via Olivetti, 2 - 20041 Agrate Brianza (MI), Italy; A. Agliati, Quanta System, via IV Novembre, 116 - 21058 Solbiate Olona (VA), Italy.

References and links

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10. Chao-Kuei Lee, Jing-Yuan Zhang, J. Huang, and Ci-Ling Pan, “Generation of femtosecond laser pulses tunable from 380 nm to 465 nm via cascaded nonlinear optical mixing in a noncollinear optical parametric amplifier with a type-I phase matched BBO crystal,” Opt. Express 11, 1702–1708 (2003). http://www.opticsexpress.org/abstract.cfm?id=73362 [CrossRef] [PubMed]

11. M. Bondani, A. Allevi, E. Puddu, A. Andreoni, A. Ferraro, and M. G. A. Paris, “Properties of two interlinked χ⁽²⁾ interactions in noncollinear phase matching,” Opt. Letters29, 180–182 (2004), and erratum Opt. Letters29, 1417–1417 (2004). [CrossRef]

12. A. Agliati, M. Bondani, A. Andreoni, G. De Cillis, and M. G. A. Paris, “Quantum and classical correlations of intense beams of light via joint photodetection,” J. Opt. B - Quantum Semiclass. Opt. 7, S652–S663 (2005). [CrossRef]

13. M. G. A. Paris, A. Allevi, M. Bondani, and A. Andreoni are preparing a manuscript to be called “Quantum and classical correlations in tripartite states of light”.

3D phase-matching conditions for the generation of entangled triplets by χ⁽²⁾ interlinked interactions

Abstract

1. Introduction

2. Theory

3. Experiment

4. Discussion and conclusions

Appendix

Acknowledgements

References and links

Supplementary Material (1)

Cited By

Figures (3)

Equations (19)

Optics Express