Quantum theoretical aspects of triple-photon statesproduced by a 𝜒(3) process

Edgar A. Rojas-Gonzales∗, Audrey Dot†, Adrien Borne†, Richard Simon‡, Patricia Segonds†,Ariel Levenson∗, Benoıt Boulanger†, Kamel Bencheikh ∗,

∗Laboratoire de Photonique et de Nanostructures, CNRS–UPR20, Route de Nozay, 91460 Marcoussis, France†Institut Neel/CNRS – Universite Joseph Fourier, Boıte Postale 166, F-38042 Grenoble, Cedex 9, France

‡Innovation Department, Horiba Jobin Yvon, 91380 Chilly Mazarin, FranceEmail: kamel.bencheikh@lpn.cnrs.fr

Abstract—We address here the quantum properties of triplephotons generated by a third order nonlinear interaction. Thoughthey have not yet been observed, triple-photon states constitutenew quantum states of light showing three-body quantum prop-erties and non-Gaussian statistics.

I. INTRODUCTION

Twin photons have played an important role over the last40 years in the field of quantum mechanics, deeply influencingnonlinear and quantum optics by their multitude applicationsbased mainly on the two-body quantum entanglement that twinphotons exhibit when created from the annihilation of a pumpphoton. Triple photons, obtained from a same parent pumpphoton, can play a role similar to that of twin photons, openingthe way to new possibilities and offering new resources forquantum optics taking benefit from the three-body quantumproperties they carry. The early attempt to generate triplephotons, or more precisely to obtain three photons exhibitingthree-body quantum entanglement, is done through two effi-cient twin-photon generation processes [1]. When two pairsof photons, created simultaenously, are detected with single-photon detectors in a clever experimental arrangement, it isfound that three of them are in the three-body Greenberger-Horne-Zeilinger state [2]. Combining three squeezed electro-magnetic fields generated by optical parametric oscillators,tripartite entanglement has been observed by T. Aoki et al.in 2003 [3]. More recently, three-photon generation (TPG)[4] and entanglement [5] have been reported using cascadedtwin-photon generation interactions. However, the most directway to generate triple photons is based on a third ordernonlinear interaction where a pump photon ℎ𝜔𝑝 splits intothree photons with respective energies ℎ𝜔1, ℎ𝜔2, and ℎ𝜔3 in amaterial exhibiting a third order electrical susceptibility. But,the generation of triple photons based on a 𝜒(3) interactionis still a real challenge as the magnitude of the susceptibilityis very weak in most of the known nonlinear materials. In2004, this limitation has been overcome by stimulating the𝜒(3)-based TPG by the injection of two modes among thetriple in a phase-matched KTP crystal along with the pumpfield [6]. The observation of about 1013 photons per pulsein the third mode was a clear signature that triple photonshave been generated. This encouraging result pushes us to payattention to the quantum properties of the triple photons andto investigate them.

In this paper, we will discuss some of the quantum aspectsof the triple photons. We will first consider the emblematic

fully degenerate TPG case where the three photons are indistin-guishable. Then we will address the non-degenerate situationthat is more compatible with the pioneering work of Douadyand Boulanger [6].

II. QUANTUM PROPERTIES OF TPG

A. Fully degenerate triple-photon generation

In this situation, each photon of the triple is indistinguish-able from the two others. They have all the same energy ℎ𝜔,same polarisation and the same wavevector k. The TPG isdescribed by the hamiltonian given by

�� = ℎ𝑏(��†𝑝��3 + ��𝑝��

†3), (1)

where ��𝑝 and �� are the annihilation operator associated withthe pump field and the triple-photon field. The quantumproperties of the field associated with the triple photons areobtained by considering the time evolution of the quantumstate under the Hamiltonian given by eq. (1). Initially, the stateis ∣𝜓(𝑡 = 0)⟩ = ∣𝛼⟩ ⊗ ∣0⟩ = ∣𝛼, 0⟩, with the pump field inthe coherent state ∣𝛼⟩ and having an average photon number∣𝛼∣2, and the triple-photon state is the vacuum state ∣0⟩. Theevolution of the quantum state is described by the Schodingerequation, which is solved by expending the quantum state intothe photon number ”Fock” states [7]. We obtain

∣𝜓(𝑡)⟩ = 𝑒−∣𝛼∣2/2∞∑𝑛

𝑛+1∑𝑙=1

𝑛∑𝑘=0

𝑐𝑛,𝑙,𝑘∣𝑛− 𝑘, 3𝑘⟩ (2)

where the coefficients 𝑐𝑛,𝑙,𝑘 are complex numbers. Expression(2) shows that when 𝑘 pump photons are annihilated in thenonlinear medium, 3𝑘 triples are generated. The full quantumdescription of the three-photon state is given by the Wignerfunction defined as

𝑊 (𝑞, 𝑝) =1

2𝜋

∫𝑒𝚤𝑝𝑥⟨𝑞 − 𝑥/2∣𝜌∣𝑞 + 𝑥/2⟩𝑑𝑥. (3)

The Wigner function is very helpful to visualize in the phasespace (amplitude 𝑞 and phase 𝑝 quadratures) a quantum me-chanical system defined by its matrix density 𝜌 = ∣𝜓(𝑡)⟩⟨𝜓(𝑡)∣.For a classical state, the Wigner function is positive. Figure1 shows the Wigner function a triple-photon state calculatedfor ∣𝜓(𝑡 = 0)⟩ = ∣3, 0⟩. It exhibits a star shape withinterferences and negative values which are the signature ofthe nonclassical properties of the three-photon quantum states.

Fig. 1. Wigner function of a three-photon state obtained with a pump incoherent state ∣3⟩.

We can experimentally bring to light such properties by mea-suring the quantum fluctuations of the three-photon mode fieldalong the amplitude 𝑞 quadrature or the phase 𝑝 quadratureusing balanced homodyne detections [8]. Indeed, due to thestar-shape of the Wigner function with negative values, theprobability distributions will be non-Gaussian unlike for thecoherent state describing the electromagnetic field emitted bya single mode laser source.

An experimental approach to generate a three-photon star-shape state is to use optical fibers. Though they have relativelylow susceptibility values (𝜒(3) ∼ 10−22 m2/V2), we can benefitof their long interaction lengths and the high optical densitieswe can couple into the few 𝜇m radius fiber cores. We haverecently estimated that about 0.2 triples/s can be generatedat 1596 nm in a 1-meter-long fiber when pumped with 1 Wcw laser at 532 nm [9]. The key issue then is to fulfill thephase-matching condition. This is achieved for a smart designof a Germanium-doped optical fiber [10], where the phase-matching condition is fulfilled between the LP03 fiber modeat 532 nm and the mode LP01 at 1596 nm. TPG requires thenpumping the germanium-doped fiber at 532 nm in the particularLP03 mode. In order to evaluate if the fiber is well designedand if the phase-matching condition is fulfilled, we havedemonstrated third harmonic generation (THG) the reverseprocess of TPG by injecting into the fiber a TEM00 (close tothe LP01 mode) laser beam at the 1596 nm. As shown in fig. 2,the observation of a green light around 532 nm in the LP03 thatgrows following a cubic law with the injected power for lowvalues is a clear signature that the phase matching conditionis fulfilled. TPG in the fiber is however more complicate toobtain since it requires first to shape the pump field at 532 nminto the LP03 mode.

B. Non-degenerate triple-photon generation

In the non-degenerate TPG case, the triple photons haveeither different energies, polarisation or propagation directions.In the following, we suppose they have different energies. It isthen experimentally possible to separate them. This situationis more appropriate to describe the seeded TPG obtained byDouady and Boulanger in 2004 [6]. As a first approach, weconsider protocols based on the recombination of the generatedtriple photons in a nonlinear crystal using sum-frequencygeneration interactions in order to give prominence to the

Fig. 2. Third harmonic power as a function of the fundamental injectedpower. Inset: mode profile (LP03) of the generated third harmonic field.

nonclassical behavior of the three photons. Such approach wasused by Abram et al. [11] in order to study the coherence oftwin-photons. We have conducted analog theoretical studiesfor triple photons [12], considering third-order sum-frequencygeneration of the three photons, Ω = 𝜔1 + 𝜔2 + 𝜔3, andsecond-order sum-frequency generation with three possibili-ties: Ω = 𝜔1 + 𝜔2, Ω = 𝜔1 + 𝜔3, and Ω = 𝜔2 + 𝜔3. Howeverthe quantum properties of the triple photons in the non-degenerate case can be addressed by simply measuring the fieldassociated with each mode. Unfortunately, such experimentalstudy is not yet feasible in the case of seeded-TPG case asthe optical powers in play, even the generated one during thenonlinear interaction, are strong and above the saturation levelof any existing high-sensitivity infrared detector. Despite thissevere experimental limitation, we have theoretical analyzedthe quantum properties of the triple photons in order topredict if they exhibit three-body quantum entanglement. Toquantify this entanglement or inseparability, we have adoptedthe quantum tools developed by Van Loock and Furusawa [13]and already used experimentally in different situations [3], [5],[14].

The protocole to evaluate the three-body inseparability isbased on the evaluation of the criterium 𝑆 = ⟨Δ��2⟩+ ⟨Δ𝑣2⟩,where ⟨Δ��2⟩, respectively ⟨Δ𝑣2⟩, is the variance of thequadrature ��, respectively 𝑣, a linear combination of thequadratures 𝑞 = ��+ ��†, respectively 𝑝 = 𝚤(��− ��†), describingthe modes 1, 2 and 3 of the triple photons. These linear combi-nations can be written as �� = ℎ𝑖𝑞𝑖 and 𝑣 = 𝑔𝑖𝑝𝑖, 𝑖 = {1, 2, 3},where ℎ𝑖 and 𝑔𝑖 are arbitrary real numbers that can be fixedexperimentally by adjusting for example the electronic gainsor attenuations. When 𝑆 ≤ 2(∣ℎ𝑛𝑔𝑛∣ + ∣ℎ𝑚𝑔𝑚 + ℎ𝑘𝑔𝑘∣), thequantum system is said inseparable or equivalently, the modes1, 2 and 3 are entangled.

The determination of the inseparability criteria 𝑆 requiresthe computation of the variance of the quantum fluctuationsof the field associated with each mode of the triples andconsequently the computation of the quadrature �� and itsconjugate 𝑣. For this study, we adopte the Heiseinberg pointof view, by calculating first the evolution of the annihilationoperator of the triple photons, namely ��1, ��2, and ��3, underthe hamiltonian

�� = ℎ𝑏(��†𝑝��1��2��3 + ��𝑝��†1��

†2��

†3). (4)

We end up with a set of differential equations

𝑑��𝑖𝑑𝑡

=𝚤

ℎ[��, ��𝑖] = −𝚤𝑏��†𝑗 ��

†𝑘, (5)

Fig. 3. Parametric plot of the gain versus the relative phase en degrees in thecase of fully seeded TPG. The arrows indicate the amplification (gain > 1)for Δ𝜙 = 𝜋/2 and deamplification (gain < 1) for Δ𝜙 = 5𝜋/6.

where {𝑖, 𝑗, 𝑘} = {1, 2, 3}. The set of equations (5) hasno trivial analytical solution. Therefore, the Baker-Haussdorfexpansion is used to obtain a limited development of theoperators at time 𝑡. The order of the development and itsvalidity depend on both the average photon number seeding theTPG process and on the strength of the nonlinear interactioncoefficient 𝑏 which is related to 𝜒(3) and the interaction time𝑡. Once we have obtained a solution of the equations (5),any quantum property of the triple photons can be analyzed.We have thus considered different situations, starting withfluorescente TPG (no seeding) ending with the triple-seedingcase where fields oscillating at 𝜔1, 𝜔2 and 𝜔3 are also presentwith the pump field at the input of the nonlinear medium.This last situation is particularly interesting as the outputproperties depend on the relative phase between the pump andthe injected modes of triple photons. Figure 3 shows the opticalgain which is the ratio of the output average photon numberover the seeding average photon number at the input for agiven triple-photon mode. We can see that the gain dependson the relative phase Δ𝜙 = 𝜙𝑝 − 𝜙1 − 𝜙2 − 𝜙3 between thepump and the three modes. An amplification is obtained forΔ𝜙 = 𝜋/2 + 𝑘(2𝜋/3), where 𝑘 = {0, 1, 2}, correspondingto efficient TPG where pump photons are annihilated givingbirth to triple photons. For Δ𝜙 = 𝜋/6 + 𝑘(2𝜋/3), the gain isbelow unity, corresponding to a deamplification. In this case,the output average photon number is smaller than the seeding.We have in fact favored the reverse process of TPG, wheretriple photons are annihilated creating a pump photon throughthird-order sum-frequency generation.

Figure 4 shows the the inseparability criterium 𝑆 calculatedfor the fully seeded TPG, where we have considered theparticular generalized quadratures �� = 𝑞1 + (𝑞2 + 𝑞3)/

√2 and

𝑣 = 𝑝1 − (𝑝2 + 𝑝3)/√2. The green line indicates the classical

limit. The blue line represents the 𝑆 parameter calculatedwhen the relative phase is Δ𝜙 = 𝜋/2, corresponding to theamplification case. We see that as the seeding average photonnumber ��𝑖𝑛 is increased, the 𝑆 parameter goes well below theclassical limit with 𝑆 → 0, reviling the inseparability of thethree modes of the triple photons. The red line is obtainedfor Δ𝜙 = 5𝜋/6 when the gain is below unity however staysalways above the classical limit.

Fig. 4. The inseparability criterium 𝑆 as a function the seeding averagephoton number ��𝑖𝑛. The green line is the classical below which the quantumstate is said inseparable. The red line is obtained for Δ𝜙 = 5𝜋/6 and theblue line for Δ𝜙 = 𝜋/2.

III. CONCLUSION

Two-body entanglement is obtained from the generationof twin-photon, a nonlinear process based on a second or-der electrical susceptibility. As a result several proposalsand experimental demonstrations, related to the possibility ofhaving two distant bodies, photons here, quantum entangled,appeared in fields like quantum cryptography, teleportation,or dense coding. More generally, a considerable amount ofexciting and non-intuitive results were found studying bipartiteentanglement of twin photons.

A logical but not trivial step foward seems to be the tri-partite entanglement. In principle, third order nonlinear opticalprocesses are the most natural way of generating three-bodyentanglement. Indeed, a pump photon incident on a nonlinearcrystal with a third order electrical susceptibility is annihilated,giving birth to three photons with non classical properties. Wepresented here our study on the quantum properties of the triplephotons, reviling their quantum behavior. We show that triple-photon quantum states exhibit interferences in phase spacewhere the Wigner function can even take negative values. Thisleeds to non-gaussian probability distributions of the quantumfluctuations of the electromagnetic field associated with thetriple-photon modes. When looking to the triple-photon modesseparately we find that they exhibit three-body entanglement,which will constitute the fundamental ingredient for newunveiled quantum phenomena waiting to be discovered.

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