Günther Schuh

Andreas Gützlaff

Frederick Sauermann

Judith Maibaum
email: j.maibaum@wzl.rwth-aachen.de

Digital Shadows as an Enabler for the Internet of Production

Keywords: Digital shadow, Internet of production, Production planning and control

Due to increasing atomization, manufacturing companies generate increasing amounts of production data. Most of this data is domain-specific, heterogeneous and unstructured. This complicates the access, interpretation, analysis and usage for efficiency improvement, faster reaction to change and weaknesses identification. To overcome this challenge, the idea of an "internet of production" is to link all kind of production relevant data by a data lake. Based on this data lake, digital shadows aggregate data for a specific purpose. For example, digital shadows in production planning and control help to manage the dynamic changes like delays in production or machine break-downs. This paper examines the existing research in the field of digital twins and digital shadows in manufacturing and gives a brief overview of the historical development. In particular, the potential and possible applications of digital shadows in production planning and control are analyzed. A top-down-bottom-up approach is developed to support the design of digital shadows in production planning and control.

Introduction

The purpose of digital shadows (DS) is to provide all information needed to successfully perform a task and make a decision. The objective is to improve decision-quality and the performance of manufacturing. In practice the use of DS gains relevance as the quantity of information as well as the complexity of information systems increase and DS reduce the complexity for users by providing the relevant information. [START_REF] Bauernhansl | The Digital Shadow of production -A concept for the effective and efficient information supply in dynamic industrial environments[END_REF][START_REF] Cheng | Data and knowledge mining with big data towards smart production[END_REF] Advantages in information and communication technologies (ICT) enable companies to collect, use and store data to integrate the physical and the virtual world. [START_REF] Belli | Toward Industry 4.0 With IoT: Optimizing Business Processes in an Evolving Manufacturing Factory[END_REF][START_REF] Wagner | Challenges and Potentials of Digital Twins and Industry 4.0 in Product Design and Production for High Performance Products[END_REF][START_REF] Brecher | Control from the Cloud: Edge Computing, Services and Digital Shadow for Automation Technologies[END_REF] Using ICT enables e.g. virtual product and process planning or predictive maintenance. [START_REF] Kritzinger | Digital Twin in manufacturing: A categorical literature review and classification[END_REF][START_REF] Qi | Digital Twin and Big Data Towards Smart Manufacturing and Industry 4.0: 360 Degree Comparison[END_REF] In many different industries the use of data analysis to support decision making is increasing. [START_REF] Qi | Digital Twin and Big Data Towards Smart Manufacturing and Industry 4.0: 360 Degree Comparison[END_REF] Hence, a goal for producing companies is to identify an efficient way to use data and information. [START_REF] Bauernhansl | The Digital Shadow of production -A concept for the effective and efficient information supply in dynamic industrial environments[END_REF] In manufacturing, main benefits are a faster reaction to unexpected events and the identification of weaknesses in order to increase the efficiency of manufacturing. [START_REF] Rosen | About The Importance of Autonomy and Digital Twins for the Future of Manufacturing[END_REF] This is of high relevance as manufacturing companies face challenges like increasing requirements of the customers on the products, rapidly changing markets, shorter product life cycles and sustainability. [START_REF] Bauernhansl | The Digital Shadow of production -A concept for the effective and efficient information supply in dynamic industrial environments[END_REF][START_REF] Wagner | Challenges and Potentials of Digital Twins and Industry 4.0 in Product Design and Production for High Performance Products[END_REF] Manufacturing companies collect tremendous volumes of structured, unstructured and semi-structured data from different sources along the product life cycle. [START_REF] Qi | Digital Twin and Big Data Towards Smart Manufacturing and Industry 4.0: 360 Degree Comparison[END_REF][START_REF] Riesener | The Digital Shadow as Enabler for Data Analytics in Product Life Cycle Managemen[END_REF] The increasing speed and volume of data collection simultaneously increase the challenges of data quality. [START_REF] Riesener | The Digital Shadow as Enabler for Data Analytics in Product Life Cycle Managemen[END_REF] Additionally, in current manufacturing environment there are numerous domains that use different IT-systems, containing specific data and models [START_REF] Schuh | Internet of Production[END_REF]. This data is stored in different and redundant sources, is heterogeneous and hardly integrated. Therefore, access, interpretation and analysis of production data is difficult [START_REF] Riesener | The Digital Shadow as Enabler for Data Analytics in Product Life Cycle Managemen[END_REF][START_REF] Schuh | The Digital Shadow of Services: A Reference Model for Comprehensive Data Collection in MRO Services of Machine Manufacturers[END_REF].

Production planning and control (PPC) allocates activities of employees, production capacities and materials to fulfil customer orders. Therefore, PPC significantly influences the performance of a manufacturing system. [START_REF] Frazzon | Data-driven production control for complex and dynamic manufacturing systems[END_REF] One challenge for PPC is the reaction to dynamic changes, e.g. operation times deviating from the plan or machines break-downs. [13,[START_REF] Yang | A Flexible Simulation Support for Production Planning and Control in Small and Medium Enterprises[END_REF] Additionally, the necessary information is collected manually from different IT-systems causing delays, additional search efforts and missing information. [START_REF] Belli | Toward Industry 4.0 With IoT: Optimizing Business Processes in an Evolving Manufacturing Factory[END_REF][START_REF] Uhlemann | The Digital Twin: Realizing the Cyber-Physical Production System for Industry 4.0[END_REF] Often PPC is dependent on the expertise of the planner. [START_REF] Yang | A Flexible Simulation Support for Production Planning and Control in Small and Medium Enterprises[END_REF] However, even experienced decision makers cannot estimate the full impact of their decisions on the overall manufacturing system, which only achieves local optimization [START_REF] Kunath | Integrating the Digital Twin of the manufacturing system into a decision support system for improving the order management process[END_REF]. Therefore, the use of DS in PPS is promising to overcome these challenges and realize dynamic real-time PPC.

In section 2, a literature review on digital shadows is conducted, including a distinction to the related digital twins (DT). Next, the application of DS for PPC and challenges and benefits are described. Section 4 summarizes and discusses the paper.

Literature review on digital shadows and digital twins in manufacturing

This section provides a general overview of the historical development of publications and definitions, applications and benefits of DS and DT in manufacturing.

Historical development of publications

In the field of engineering and computer science Google Scholar is the most extensive search engine. [START_REF] Martín-Martín | Google Scholar, Web of Science, and Scopus: A systematic comparison of citations in 252 subject categories[END_REF] Therefore, an analysis of metadata of publications extracted from Google Scholar gives an overview about the historical development of research. DS and DT are a relatively new research field. Prior to 2010, total 208 contributions to the topic DT and 291 contributions to the topic DS were published. Less than 100 each in the context of manufacturing. The development of new publications since 2014 shows that research on the topic of DT has grown strongly. The number of publications per year has grown from under 100 in 2014 to more than 4,500 in 2019. One main research focus is DT in production and manufacturing as Fig. 1 shows. In comparison, the number of publications on the topic of DS has only grown slightly, and has grown from about 100 in 2014 to more than 250 in 2019. The main research focus is also on DT in manufacturing, see Fig. 1. A related field of research is data storage e.g. in data lakes or data warehouses.

Characterization of digital shadows in manufacturing

The following paragraphs provide an overview of DS by describing different definitions, the main elements and benefits of DS. DS contain data traces and models. [START_REF] Brecher | Control from the Cloud: Edge Computing, Services and Digital Shadow for Automation Technologies[END_REF] Data traces consists of time-variant data and metadata like information about the source or recording time, which are needed for further data processing. [START_REF] Bauernhansl | The Digital Shadow of production -A concept for the effective and efficient information supply in dynamic industrial environments[END_REF] The data is generated by e.g. sensors or simulation and provided in domain-specific real-time. [START_REF] Landherr | The Application Center Industrie 4.0 -Industrydriven Manufacturing, Research and Development[END_REF] DS contain both information about past and current conditions and provide information about future conditions. [START_REF] Stecken | Digital Shadow Platform as an Innovative Business Model[END_REF] DS provide near real-time data and information to enable an organization to control a permanently changing production [START_REF] Landherr | The Application Center Industrie 4.0 -Industrydriven Manufacturing, Research and Development[END_REF]. When the system changes, DS are updated. [START_REF] Brecher | Control from the Cloud: Edge Computing, Services and Digital Shadow for Automation Technologies[END_REF] DS serve as a platform to integrate information from different sources to overcome semantic heterogeneity. [START_REF] Riesener | The Digital Shadow as Enabler for Data Analytics in Product Life Cycle Managemen[END_REF] For this purpose, DS link data and models [START_REF] Stark | Innovations in digital modelling for next generation manufacturing system design[END_REF]. Data exchange between different domains improve the manufacturing, e.g. the manufacturer creates plant models which the operator needs when operating the plant. Typical tasks supported by DS are order processing, service or production planning. Since DS are task-specific, many different DS exist. [13,[START_REF] Kubenke | Assessing the Efficiency of Information Retrieval from the Digital Shadow at the Shop Floor using IT Assistive Systems[END_REF] Additionally, DS can be used through the whole product life cycle. [START_REF] Stark | Innovations in digital modelling for next generation manufacturing system design[END_REF] DS support decision makers and operators by providing the right information at the right place in the right quality in a sufficient way. [START_REF] Bauernhansl | The Digital Shadow of production -A concept for the effective and efficient information supply in dynamic industrial environments[END_REF]13] As DS provide only the taskspecific relevant data, the identification of necessary data, selection, aggregating and determination of granularity level of data are core part of designing DS. [START_REF] Riesener | Framework for defining information quality based on data attributes within the digital shadow using LDA[END_REF][START_REF] Schuh | Methodology for the derivation of a digital shadow for engineering management[END_REF] Data of past and current conditions is analyzed, e.g. for forecasts or simulations, to support decision making [START_REF] Stecken | Digital Shadow Platform as an Innovative Business Model[END_REF].

In conclusion a DS is a set of aggregated data traces and models providing the necessary information for a task. As the required information is provided in near real-time, DS enable a fast reaction to the dynamic and complex manufacturing environment.

Characterization of digital twins in manufacturing

As in the previous paragraph, the following paragraphs provide an overview of DT by describing different definitions, the main elements and benefits of DT. Due to the increasing research in the field of DT, different definitions of a DT exist [START_REF] Jones | Characterising the Digital Twin: A systematic literature review[END_REF]. NASA provided the initial definition of a DT as "an integrated multi-physics, multi-scale, probabilistic simulation of a vehicle or system that uses the best available physical models, sensor updates, fleet history, etc., to mirror the life of its flying twin". [START_REF] Shafto | Medeling, Simulation, Information Technology & Processing Roadmap[END_REF] Likewise, many authors describe the DT in manufacturing as a virtual, computer-based complete representation or counterpart of a physical system which allows various analyses of the system based on real-time production data, e.g. [START_REF] Stecken | Digital Shadow Platform as an Innovative Business Model[END_REF][START_REF] Vachalek | The digital twin of an industrial production line within the industry 4.0 concept[END_REF][START_REF] Negri | A Review of the Roles of Digital Twin in CPS-based Production Systems[END_REF][START_REF] Weyer | Future Modeling and Simulation of CPS-based Factories: an Example from the Automotive Industry[END_REF][START_REF] Brenner | Digital Twin as Enabler for an Innovative Digital Shopfloor Management System in the ESB Logistics Learning Factory at Reutlingen -University[END_REF]. DT support the analysis of static and dynamic systems allowing real-time optimizations, decision support and predictions. [START_REF] Negri | A Review of the Roles of Digital Twin in CPS-based Production Systems[END_REF] DT should improve and shorten production cycles and reveal inefficiencies. [START_REF] Vachalek | The digital twin of an industrial production line within the industry 4.0 concept[END_REF] The comprehensive literature review on DT by [START_REF] Kritzinger | Digital Twin in manufacturing: A categorical literature review and classification[END_REF] emphasizes that in existing research the terms digital model, DS and DT are often used synonymously, as all are a digital counterpart of a physical system. Therefore, they are distinguished based on the data flow. Both DT and DS have an automatic data flow from the physical and the digital system. The data flow from the digital to the physical system is automatic for DT and manual for DS. [START_REF] Kritzinger | Digital Twin in manufacturing: A categorical literature review and classification[END_REF] In the newest literature review of DT, [START_REF] Jones | Characterising the Digital Twin: A systematic literature review[END_REF] differ between a virtual-tophysical and a physical-to-virtual twinning process. Changes in the physical or virtual entity are registered by metrology methods and lead to a change of the corresponding virtual or physical twin. Jones et al. identify reduction of costs, risk, complexity and reconfiguration time and the improvement of efficiency, decision making, manufacturing processes and competitiveness as perceived benefits described in literature and industry. However, these assumptions are often not based on quantitative evaluations. [START_REF] Jones | Characterising the Digital Twin: A systematic literature review[END_REF] Summarized, a DT of a manufacturing system consists of a physical and virtual part. Both parts are a complete representation of the counterpart allowing analyses of the original system. The aim is to support decision making and improve the production.

The objective of both DS and DT is to process data in order to support decision making. The comparison of DS and DT shows that they differ in the granularity of the data base. DT are a complete representation of a physical system, whereas DS only represent those parts of a system that are relevant for a specific task. In addition, a core element of DS is the linkage of data from different domains. In the following sections, the focus is on DS as they have similar functionalities as DT but require less data capacity and provide better decision support due to the task-specific information.

Application of digital shadows in production planning and control

Based on the current research in the following section an approach for designing digital shadows is developed and specified using the example of PPC. Additional potentials of DS for PPC are further analyzed. PPC is chosen as an example because it is a core element in manufacturing companies. [START_REF] Frazzon | Data-driven production control for complex and dynamic manufacturing systems[END_REF] Recent trends like volatile markets, increasing complexity and shortened product lifecycles also affect PPC [13]. Therefore, PPC must cope with increasing dynamic and complexity. Existing PPC systems, like Enterprise Resource Planning (ERP) system or Manufacturing Execution System (MES), support different tasks of PPC but the support in everyday, short-term decisionmaking situations could be further increased [START_REF] Kunath | Integrating the Digital Twin of the manufacturing system into a decision support system for improving the order management process[END_REF].

Approach for designing digital shadows

To extend the existing research on DS a top-down-bottom-up approach to design DS in PPC was developed. The aim is to fulfill the information requirements of production planner and controller by providing the necessary information. As a result the PPC should improve through better decision support and reduced information search efforts. Therefore the information needed (information requirements) of the users are identified top-down. Additionally, the data base is described bottom-up to describe the information provided (information offers). In the following, this approach is presented in detail, Fig. 2 gives an overview of the approach. In the first step of the top-down analysis, the tasks in the PPC are analyzed to identify the user requirements. According to Schuh et al. the core tasks of PPC are production program planning, production requirements planning and production control. [START_REF] Schuh | Produktionsplanung und -steuerung: Grundlagen, Gestaltung Und Konzepte[END_REF] At each of these tasks, production planners and controllers perform different sub-tasks with different requirements. The derived user requirements can be divided in functional and data quality requirements. All user requirements are documented in a specification sheet. In the subsequent user requirements analysis, the customer requirements are evaluated, classified and structured. This serves as the basis for the information requirement derivation. The user requirements are formulated as information requirements for the DS and entered in a requirements specification. This serves as the basis for modelling the information requirements. The result of the top-down analysis is a detailed description of the information requirements from the production planner and controller perspective.

At the same time, the data base is analyzed bottom-up to identify the information offers. For this purpose, the data base and different data analysis methods are investigated. In the first step data is described and characterized (in the context of PPC). The result is a morphology of PPC data describing the data type and context information like unit or change-frequency. In the next step, relations within the data are analyzed and described. The relations simplify the aggregation, linkage, selection and evaluation of the data in a DS. In the next step, suitable data analysis models are selected. First the properties of data analysis methods are described. Based on this, it is investigated which of the described data can be evaluated and how. The aim is to describe the possible analyses and to use them to derive the information offers. Based on the prior steps the information offers are modeled. The result of the bottom-up analysis is a detailed description of the information offers in the context of PPC. All in all, this procedure enables the description of DS for all tasks of the PPC. These DS provide production planners and controllers with all information they need for their tasks, thus enabling them to make well-founded decisions.

Potentials of digital shadows for production planning and control

The following paragraph analyzes the potential benefits DS offers for PPC. As DS provide real-time data, one potential is the ability to react to dynamic changes. [13,[START_REF] Yang | A Flexible Simulation Support for Production Planning and Control in Small and Medium Enterprises[END_REF] Especially for scheduling of orders and production control DS enables task-specific decision support by data analyses. [START_REF] Fang | Digital-Twin-Based Job Shop Scheduling Toward Smart Manufacturing[END_REF] One example for improving existing PPC systems is the lead time prediction as in [START_REF] Schuh | Databased prediction of order-specific transition times[END_REF]. The integration of historical and real-time data, static and dynamic data into PPC and the analysis by e.g. data mining methods increase the quality and reliability of planning results [START_REF] Rosen | About The Importance of Autonomy and Digital Twins for the Future of Manufacturing[END_REF]. Additionally, the DS as single source of truth reduces efforts for manual data acquisition from different systems. [START_REF] Uhlemann | The Digital Twin: Realizing the Cyber-Physical Production System for Industry 4.0[END_REF] The targeted use of information from other areas in real time can improve the PPC [START_REF] Shao | DIGITAL MANUFACTURING: REQUIREMENTS AND CHALLENGES FOR IMPLEMENTING DIGITAL SURROGATES[END_REF]. The decision support for production planners and controllers can be increased by further analyses and simulation and considering a larger data basis. [START_REF] Rosen | About The Importance of Autonomy and Digital Twins for the Future of Manufacturing[END_REF][START_REF] Yang | A Flexible Simulation Support for Production Planning and Control in Small and Medium Enterprises[END_REF] Another potential of DS for PPC is the development of an autonomous, self-controlling PPC. [START_REF] Rosen | About The Importance of Autonomy and Digital Twins for the Future of Manufacturing[END_REF] In summary, the main advantages of DS are an improved decision support for production planners and controllers in short-term decision situations and the reduction of information search efforts and the provision of real-time information. An approach to design DS for PPC must describe how the information can be provided and derived from the data and models in such a way that it increases the decision quality.

Conclusion and further research

A DS is a set of aggregated data traces and models that provide necessary information for a task. This enables decision support in manufacturing with low latency times. The use of DS in the dynamic and complex PPC therefore offers several potentials for improvement, like enhanced decision support or reduced information search efforts. Additionally, a concept for designing DS in PPC was described. The concept describes a top-down bottom-up approach, with which the information needs and the information offers are linked.

In further research, the procedure for describing specific DS, e.g. in PPC, should be further detailed. Future research should also focus on the further development and application of DS for PPC in industry. This includes the modelling of data structures and information requirements.

Fig. 1 .

 1 Fig. 1. Evolution of research on digital shadows and digital twins

Fig. 2 .

 2 Fig. 2. Top-down-bottom-up approach for designing DS

4 .

 4 Modelling of information offers 2. Description of relations within the data 1. Characterization of the data 1. Identification of user requirements 2. Analysis of user requirements 4. Modelling of information requirements Bottom-up Information offers Top-down Information requirements 3. Derivation of information requirements 3. Selection of suitable data analysis models

Acknowledgement

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany ś Excellence Strategy -EXC-2023 Internet of Production -390621612


K Kolasiński

Hermann Sellier

Bartlomiej Szafran

Conductance measurement of spin-orbit coupling in the two-dimensional electron systems with in-plane magnetic field

published or not. The documents may come

Conductance measurement of spin-orbit coupling in two-dimensional electron systems with an in-plane magnetic field

Introduction. Charge carriers in semiconductor devices are subject to spin-orbit (SO) interactions [1] stemming from the anisotropy of the crystal lattice and/or the device structure. The SO interactions translate the carrier motion into an effective magnetic field leading to carrier spin relaxation and dephasing [2][3][4], spin Hall effects [5][6][7], formation of topological insulators [8], persistent spin helix states [9][10][11], Majorana fermions [12]. Moreover, the SO coupling paves the way to spin-active devices, including spin-filters based on quantum point contacts (QPCs) [13] or spin transistors [14][15][16][17][18], which exploit the precession of the electron spin in the effective magnetic field [19]. The most popular playground for studies of spin effects and construction of spin-active devices is the two-dimensional electron gas (2DEG) confined at an interface of an asymmetrically doped III-V heterostructure, with a strong built-in electric fields in the confinement layer giving rise to the Rashba SO coupling [20] and with the Dresselhaus coupling due to the anisotropy of the lattice which is enhanced by a strong localization of the electron gas in the growth direction [START_REF] Winkler | Spin-orbit Coupling Effects in Two-Dimensional Electron and Hole Systems[END_REF].

The SO interaction is sample-dependent and its characterization is of a basic importance for description of spin-related phenomena and devices. The SO coupling constant are derived from the Shubnikov-de Haas [START_REF] Nitta | [END_REF][23][24][25][26][27][28][29] oscillations, antilocalization in the magnetotransport [30], photocurrents [31], or precession of optically polarized electron spins as a function of their drift momentum [32]. Usually both the Rashba and Dresselhaus interactions contribute to the overall SO coupling. Separation of contributions of both types of SO coupling is challenging and requires procedures based on optical polarization of the electron spins [31][32][33]. In this Letter we investigate the possibility for extraction of the Rashba and Dresselhaus constants from a purely electric measurement of the two-terminal conductance. The proposed method does not involve application of optical excitation [31,32] or a particularly complex gating [32]. The procedure given below requires rotation of the sample in an external in-plane magnetic field [34], which is straightforward as compared to application of the rotated electric field to 2DEG [32]. Also, the present approach is suitable for high mobility samples and goes without analysis of the localization effects in the magnetotransport [30].

The procedure which is proposed below bases on an idea that the effects of the SO coupling related to the wave vector component in the direction of the current flow can be excluded by a properly oriented external inplane magnetic field. The procedure exploits spin effects of backscattering -due to intentionally introduced potentials -or simply to intrinsic imperfections within the sample. In particular we show that the linear conductance of a disordered sample reveals an oscillatory behavior as a function of the magnetic field direction and amplitude. The dependence allows one to determine the strength of the SO interaction as compared to the spin Zeeman effect as well as the relative strength of both Rashba and Dresselhaus contributions.

Spin-dependent scattering model. Let us start from a simple model of electron scattering (see Fig. 1). The electron is injected to the system from e.g. a QPC and comes to the potential defect from the left. The defect is taken as an infinite potential step, so that the scattering probability is 1. The incident and backscattered waves are denoted by |k + σ where k ± σ stands for the absolute value of the wave vector for the spin state σ and the superscript sign indicates the electron incoming from the left (+) or backscattered (-). Only the backscattering which returns the carriers to the QPC can alter the conductance, so we consider the scattering wave function along the line between the QPC and the defect

|Ψ σ = e ik + σ r k + σ + Σ σ a σσ e -ik - σ r k - σ , (1)
where a σσ stand for the scattering amplitudes. Scattering at other angles does not decrease the conductance and is neglected for a moment. Within the 2DEG, outside the scattering center and the QPC channel the 2D electron Hamiltonian for in-plane field B = r=(0,0) Figure 1. Sketch of considered scattering process. The electron wave is incoming from the left from a source (a QPC for instance) in spin state σ, propagates to the right and is backscattered at position r = (0, 0) by the potential barrier induced by the impurity.

(B x , B y , 0) reads

H = E kin I+σ x (αk y -βk x +b x )+σ y (βk y -αk x +b y ), (2)
where

E kin = 2 k 2 2m eff , b x/y = 1 2 gµ B B x/y
, m eff is the electron effective mass, α and β are the Rashba and Dresselhaus constants. The spin Zeeman effect is introduced via Pauli matrices σ x/y and the Zeeman energy will be denoted below by

E B = 1 2 gµ B |B| = b 2 x + b 2 y .
Note, that we use the symmetric gauge A = (B y z, -B x z, 0) then by choosing the plane of the 2DEG confinement to be located at z = 0, we get A = 0, and the magnetic field enters the Hamiltonian only via the spin Zeeman term i.e. the orbital effects do not affect the electron transport.

Let us first neglect the Dresselhaus coupling (β = 0). Plane wave solution for the eigenvalues of the Schrödinger equation gives

E σ = 2 k 2 2m eff + σ |p| , (3)
with σ = {+, -} denoting projections of the spin on the direction of polarization p = (αk y + b x , -αk x + b y ), and eigenvectors

k ± σ = 1 √ 2 1 σ p ± x +ip ± y p ± ≡ 1 √ 2 1 σe iφ(k ± σ ,B) , (4)
for the incident (+) and backscattered (-) directions of the electron motion with p ± = |p ± |. Due to the assumed infinite scattering potential, the wave function in Eq. (1) has to vanish at r = 0 (see Fig. 1),

Ψ σ (r = 0) = |k + σ + Σ σ a σσ k - σ = 0, hence a σσ = -σ σe iφ(k + σ ,B) + σ e iφ k - -σ ,B e iφ(k - + ,B) + e iφ(k - -,B) . (5)
In the following we use In 0.5 Ga 0.5 As material parameters with m = 0.0465m 0 , Landé factor g = 9, and the Fermi energy E F = 20meV. For the bulk Rashba [35] constant α 3D = 57.2 Å 2 , the 2D value is α = α 3D F z , where F z is the electric field in the growth-direction. The Rashba constant can be controlled by the external voltages [START_REF] Nitta | [END_REF] and for In 0.5 Ga 0.5 As SO coupling constants of the order of 5 to 10 meV nm [START_REF] Nitta | [END_REF] were recorded. In Fig. 2(a) we present the scattering amplitudes a σσ obtained from Eq. ( 5) as a function of the direction of the magnetic field B = (B cos (θ) , B sin (θ)), with B = 5T for scattering along the x direction, k = (k x , 0). Note that for the magnetic field oriented in the y direction θ = π/2, i.e. for B = (0, B y ) the diagonal elements of the scattering amplitudes are zero. This is a special case for which the spinor in Eq. ( 4) can be written in

form |k ± σ = 1 iσd ±
, where d ± = sign (-αk ± x + b y ). Let us now include the Dresselhaus SO coupling. The 2D Dresselhaus constant is given by β

= β 3D k 2 z = β 3D π 2 d 2
, where β 3D is the bulk constant and d is the width of the 2DEG confinement in the growth direction. We consider β values from 0 to α [9,10]. The cubic Dresselhaus interaction is neglected as a small effect [10]. In the absence of the B field, for the electron incident along the x direction i.e. k y = 0, the polarization di-rection is p = (-βk x , -αk x ) with the energy eigenvalues

E σ = 2 k 2 x 2m eff + σk x α 2 + β 2 .
As a result the Dresselhaus interaction sets the direction of the electron spin polarization to θ = arctan α β and increases the effective SO coupling constant to

γ eff = α 2 + β 2 .
The above conclusions can also be reached by a direct inspection of the off-diagonal part of Hamiltonian (2) for the electron transport along the x direction (k y = 0, k x = k F ). The effective magnetic field in Eq. ( 2) is (-βk F +b x , -αk F +b y ). Both components of the effective magnetic field vanish for

tan θ = b y b x = α β (6)
and

E B = 1 2 gµ B B = b 2 x + b 2 y = k F γ eff ≡ E SO . (7)
For illustration we calculated the electron density at the source position -including the incident and backscattered waves using Eqs. (1,5) as ρ = σ Ψ σ |Ψ σ . The backscattering probability is roughly proportional to the electron density at the QPC [36]. The electron density -is depicted in Fig. 3(b-d) for α = 12meVnm, β = 0; and Fig. 3(c-d) α = 9meVnm, β = 8meVnm. These values produce the same effective coupling constant γ eff ≈ 12meVnm. h -G from its quantized value for M subbands [EF =20 meV] passing across the QPC for a potential defect at a distance of 1500 nm from the QPC as a function of the inplane magnetic field value and orientation. The results were calculated numerically within the Landauer approach. The QPC gate potential modeled with analytical formulas for a rectangle gate adapted from Ref. [START_REF] Davies | we apply the formula for the finite rectangle gates[END_REF]. (b) Charge density at the entrance to the QPC calculated with a simple model of Eq.( 5) for the QPC at 1500 nm from the scatterer. The results of Fig. 3(b-d) contain a distinct circular pattern in the θ, B plane. The position of the center is given by Eqs. (6,7). The angular coordinate of the center allows one to determine the ratio of the Rashba and Dresselhaus constants and the SO coupling constant γ eff can be read-out from the position of the center of the pattern on the B scale, provided that the Fermi wave vector is known. In presence of SO coupling and / or the Zeeman effect k F is spin dependent [START_REF] Winkler | Spin-orbit Coupling Effects in Two-Dimensional Electron and Hole Systems[END_REF]. However, for the E B = E SO the off-diagonal terms of the Hamiltonian (2) vanish and the Fermi wavevector is directly related to the Fermi energy E F = 2 k 2 F 2m eff , which for the adopted parameters gives k F = 0.156/nm. For γ eff = 12 meV nm one obtains E SO = 1.875 meV, which coincides with E B for B = 7.2 T (see Fig. 3(b-d)).

In Fig. 3(c) one notices a reduction of the period with respect to 3(b) with the source-impurity distance increased to 2000 nm from 1500 nm. The period of the oscillations ∆ B is ∆ Coherent quantum transport calculations. With the intuitions gained by the simple analytical model we can pass to the calculations of the coherent transport using a standard numerical method [START_REF] Kolasiński | [END_REF], based on the quantum transmitting boundary solution of the quantum scattering equation at the Fermi level implemented in the finite difference approach, which produces the electron transfer probability used in the Landauer formula for conductance summed over the subbands of the channels far from the scattering area. Zero temperature is assumed. For the numerical calculations we consider a channel extended along the x direction, hence k x in Eq. ( 2) remains a quantum number characterizing the asymptotic states of the channel. Within the computational box the wave vector is replaced by an operator k = (k x , k y ) = -i∇.

We consider a QPC/defect system of Fig. 1. The results presented in Fig. 3(a) indicate a reduction of the conductance below the maximal value M e 2 h for M subbands passing across the QPC. The central position of the pattern nearly coincides with the one of Fig. 3(c). The local extremum of conductance in the center of the pattern indicates the value and orientation of the external magnetic field which lifts the SO interaction effects. The angular position of Fig. 3(c) is exactly reproduced, and the amplitude of the field is B = 6.5 T instead of B = 7.2 T. The deviation in the location of the central point in the B axis in Fig. 3(a) results from the confinement in the QPC channel which is not included in our free particle model (see below).

The effects described so far dealt with interference of the electron waves between the source (QPC) and the defect. In fact, the role of the source can be played by any scattering center, and the extraction of the SO coupling constant requires a presence of two or more scatterers to allow for formation of standing waves described in the previous section. For the rest of the paper we consider a channel of homogenous width W , which carries M transport modes at the Fermi level. In Fig. 4(d) we presented the conductance results for a clean channel of width W = 180nm and the computational box of length L = 1.6µm. A smooth potential barrier is introduced across the channel with height 10 meV and width 200nm. Depending on the orientation of the magnetic field the number of transport modes varies between M = 17 and M = 18. The simulation was performed for α = 9meVnm and β = 8meVnm as in Fig. 3(b,c). The conductance plot possesses an extremum precisely at the angle of θ = arctan 9 8 . The magnetic field of the extremum is slightly shifted to lower values than 7.2 T -which is a result of the reduction of k x within the potential barrier. The lack of conductance oscillations that were observed above in Fig. 3 results from a small barrier length (d s-i =200nm).

The oscillations reappear when one replaces the barrier by a random disorder due to the random nonmagnetic and spin-diagonal potential fluctuations. The fluctuations simulate inhomogeneity of the doping of the potential barrier which provides the charge to the 2DEG. In 2DEG in III-V's due to the spatial separation of the impurities of the 2DEG, the defects do not introduce any significant contribution to the spin-orbit interaction (see Ref. [39] and the Supplement [40]). Figure 4(c) displays the conductance for the channel of the same width and length. The potential -displayed in Fig. 4(a) is locally varied within the range of (-0.5E F ,0.5E F ). The perturbation induces a multitude of scattering evens -the local density of states at the Fermi level for B = 0 is displayed in Fig. 4(b). In spite of the complexity of the density of states the angular shift is still arctan (9/8) ≈ π/4. The shift of the G extremum along the B scale with respect to 7.2 T is detectable -but small and of an opposite sign than in Fig. 4(d). This shift is related to the fact that for a finite width channel k y is an operator that mixes the subbands. The wave vector k y is a well-defined quantum number for electrons moving in an unconfined space. The small -but detectable -effects of a finite W disappear completely for a wider channel -which is illustrated in Fig. 4(e) for W = 0.8 µm. Here, the number of conducting bands varies between 80 and 81. The local extremum of conductance appears exactly at the positions indicated in the previous section. Note, that although the number of subbands change by 1 in Fig. 4(c,e), the variation of conductance is as large as ∼ 3e 2 /h in Fig. 4(c) and ∼ 6e 2 /h in Fig. 4(e). The conductance variation in Fig. 3(a) was very small -since the defect was far away from the QPC, for the disordered channel it is no longer the case. For completeness in Fig. 4(f) we presented calculations for a twice smaller SO coupling constants α = 4.5 meVnm, β = 4 meVnm, and γ eff = 6meVnm. The position of the maximal G along the B scale is consistently reduced from 7.2 T to 3.6 T, and the orientation of the magnetic field vector corresponding to the extremum is unchanged. Summary. We have shown that the in-plane magnetic field can lift the off-diagonal terms of the transport Hamiltonian for the two-dimensional electron gas that result from the Zeeman effect and the SO interaction. The effect appears only for a value and orientation of the external magnetic field which excludes the spin mixing effects that accompany the backscattering in presence of the SO coupling. In consequence the conductance maps for a system containing two or more scatterers -intentionally introduced -or inherently present in a disordered sample exhibit a pronounced extremum as a function of the magnetic field modulus B and orientation θ. An experimental value of B -for which the Zeeman energy is equal to the SO coupling energy -should allow one to extract the effective SO coupling constant including both the Rashba and Dresselhaus terms, and the orientation field indicates the relative contributions of both. The results indicate the ratio of the Dresselhaus and Rashba constants is exactly resolved by the procedure, and the amplitude of the magnetic field -hence the effective SO constant varies only within a 10% from the exact value depending on the channel width and disorder profile.

Supplemental Material for Conductance measurement of spin-orbit coupling in the two-dimensional electron systems with in-plane magnetic field The Letter indicated a well resolved extremum of conductance for the conditions given by Eq. ( 6) and ( 7), for which the external magnetic field cancels the SO effective magnetic field that depends on the electron wave vector. The presented analysis is based on the electron transport along the x direction. The conductance extremum was found for a quantum point contact with a single defect [Fig. 3(a)] and for a potential barrier [Fig. 3(d)]. These two systems corresponded to a nearly 1D transport, since the potential barrier was introduced in a separable manner, and in the quantum point contact an appearance of the non-zero k y implies removal of the electron from the beam of backscattered electrons. However, for the perturbed sample [Fig. 4(c,e,f)] the electrons undergo multiple scattering and acquire non-zero k y values between the scattering events. Results of Fig. 4(c,e,f) obtained for a disorder covering basically the entire width of the sample indicate that the effects of temporarily non-zero k y for the spin precession averages out to zero and the conductance extremum stays in the position defined by Eqs. (6,7). In order to support the discussion further we present in this document the case with a smaller number of defects, where the averaging may not be complete. We take a channel of width W =400nm and length L =1200nm. The simulation was performed for α = 9meVnm and β = 8meVnm as in Fig. 3(b,c). In the first column of Fig. S1 we show the obtained conductance images for different configurations of disorder potential V dis in the channel (second column). The third column present an image of LDOS computed for B = 0.

K. Kolasiński,
One notices the angle shift θ = arctan α β , equal to the ratio between Rashba and Dresselhaus strengths, remains constant for different densities and configurations of disorder. Hence still for a limited number of defects the measurement of θ should provide good information about the α β . However, the position of the vertical line for which E B = E SO can be estimated roughly from the obtained images to be between 7 and 8 T which agrees with the assumed value of the SO interaction in simulated device. The exact position of the center on the B scale for the adopted parameters is 7.2 T.

To conclude, in general we have found that detecting the angle shift θ is more robust against the sample variations than finding the magnetic field strength symmetry point.

II. THE EFFECT OF THE IMPURITIES ON SO COUPLING CONSTANT

The disorder considered in the Letter is a potential fluctuation, which results from e.g. non-ideally homogenous doping of the barrier layer under which the twodimensional electron gas is confined. In the 2DEG heterostructures the electron gas is separated from the impurities by an undoped spacer of a few to about 20 nm. The potential fluctuation affects the electric field in the growth direction, so an estimation of the resulting local change of the Rashba SO coupling constant is needed.

The Rashba SO interaction strength α = α 3D F z results from the structural inversion asymmetry in the 2DEG confinement plane and is proportional to the electric field in the growth direction F z , where for the bulk In 0.5 Ga 0.5 As alloy Rashba [1] constant is α 3D = 57.2 Å 2 . The strength of the electric gradient F z can be tuned by external voltages [2] and may be varied from 50 to 100 kV/cm in In 0.5 Ga 0.5 As as reported in Ref. [2]. The value of the electric field can be also changed by random impurities in the structure or result from the inhomogeneity of the distribution of the impurities, which may locally change the value of F z → F z + ∆F z . The vertical position of the 2DEG sheet is set at z = 0. In order to estimate the strength of the electric gradient induced by arXiv:1604.01270v2 [cond-mat.mes-hall] 30 Sep 2016 single impurity we computed the electric field

∆F z (x, y) = - ∂ ∂z e 4π * |r imp -r | r =(x,y,0) (1)
induced by negatively charged impurity located 10nm from the 2DEG surface (see Fig. S2(c)). The correction to the electric field due to a single impurity located is negligibly weak as compared to the nominal value of F z , hence it can be neglected in our simulations.

III. FRIEDEL OSCILLATIONS AND THE EFFECTIVE POTENTIAL

In strongly perturbed systems the charge density forms ripples with the periodicity of half of the Fermi wavelength. We considered the effect for the present perturbation -an impurity at a distance from the 2DEG. We simulated the effect of the single negatively charged impurity on the electron density using the DFT method described in Ref. [3]. In simulations we put α = 10meVnm and β = 0 i.e. only the Rashba term is active. The impurity is introduced 10nm from the the 2DEG computational sheet in the middle of the computational box. The average electron density in the simulation is 1×10 12 [ 1 cm 2 ]. The self-consistent DFT solution is reached for temperature T = 1K. The rest of the parameters is the same as in the transport simulations.

In Fig. S2(a) we show the self consistent electron density obtained from our DFT solver. One may see that the effect of introduced impurity is relatively weak for a given configuration. The electron density for E F = 20 meV is large, hence the screening is quite efficient, resulting in small depletion in the electron density. The change of the effective potential energy due to the presence of the defect is well visible (see Fig. S2(b)) and vary from about 42meV to 46meV.

In Fig. S3(a) we plot the cross section of the electron density and effective potential from Fig. S2(a-b) with peak in the V eff and deep in electron density n at x = 200nm. Due to the relatively weak perturbation of the impurity (∼ 25% of depletion) the amplitude of the Friedel oscillations are very small but still visible in the electron density plot. However, due to the smearing property of the Coulomb kernel V (r) = ´dr n(r )/ |rr | the weak Friedel oscillations in n become even weaker in the resulting potential energy, hence the lack of the Friedel oscillation in the obtained V eff plot. In order to enhance the oscillation of the charge density in Fig. S3(b) we put the impurity at 2.5nm only from the electron gas. Now, the ripples on the density are more pronounced, but their are not resolved in the self-consistent potential anyway. To conclude, the effect of Friedel oscillation on the DFT potential for the considered perturbation is weak, and the modification of the charge density by the rota- tion of the magnetic field in presence of the SO coupling should not alter the results for the electron transport.

= 1 2 ( 1 +Figure 2 .

 212 Figure 2. (a) Scattering amplitudes calculated from Eq. (5) for α = 12meVnm and β = 0 as a function of angle formed by the magnetic field vector and the x axis θ. (b) Same as in (a), but for a fixed angle θ = π as a function of Rashba constant α. In (a) and (b) the solid lines show the result for B = 5T and the black dashed lines for B = 7.2T. In Fig. 2(b) we show the evolution of the scattering amplitudes for the orientation of magnetic field fixed at θ = π, as a function of the Rashba constant α. Then, in Eq. (2) k y = 0 and b y = 0. The scattering amplitudes cross near α ≈ 8meVnm [Fig. 2(b)]. In this point the Zeeman energy E B is equal to the SO coupling energy E SO = αk x . For αk x = E B the off-diagonal terms are: -b x σ x +(E B +b y )σ y , for which the scattering amplitudes (5) for eigenvectors (4) simplify to |a σσ | 2 = 1 2 for any σσ and for any in-plane orientation of B vector. The E B ≈ E SO case is presented in Fig. 2(a) where the black dashed lines show the scattering amplitudes for B = 7.2T, which shows that almost complete spin mixing |a σσ | 2 ≈ 1 2 is present for any angle.Let us now include the Dresselhaus SO coupling. The 2D Dresselhaus constant is given by β = β 3D k 2

  Figure 3. (a) Reduction of QPC [Fig. 1] conductance G = M e 2h -G from its quantized value for M subbands [EF =20 meV] passing across the QPC for a potential defect at a distance of 1500 nm from the QPC as a function of the inplane magnetic field value and orientation. The results were calculated numerically within the Landauer approach. The QPC gate potential modeled with analytical formulas for a rectangle gate adapted from Ref.[START_REF] Davies | we apply the formula for the finite rectangle gates[END_REF]. (b) Charge density at the entrance to the QPC calculated with a simple model of Eq.(5) for the QPC at 1500 nm from the scatterer. (c) Same as (b) but with Rashba and Dresselhaus SO interactions present. (d) Same as (c) but with the source at 2000nm from the scatterer. The values of the coupling constants α and β are given in meV×nm units. The vertical dashed line in(b-d) indicates B = 7.2 T for which the Zeeman energy is equal to the SO coupling energy, EB = ESO, see text. The horizontal dashed line shows the angle arctan α

  Figure 3. (a) Reduction of QPC [Fig. 1] conductance G = M e 2h -G from its quantized value for M subbands [EF =20 meV] passing across the QPC for a potential defect at a distance of 1500 nm from the QPC as a function of the inplane magnetic field value and orientation. The results were calculated numerically within the Landauer approach. The QPC gate potential modeled with analytical formulas for a rectangle gate adapted from Ref.[START_REF] Davies | we apply the formula for the finite rectangle gates[END_REF]. (b) Charge density at the entrance to the QPC calculated with a simple model of Eq.(5) for the QPC at 1500 nm from the scatterer. (c) Same as (b) but with Rashba and Dresselhaus SO interactions present. (d) Same as (c) but with the source at 2000nm from the scatterer. The values of the coupling constants α and β are given in meV×nm units. The vertical dashed line in(b-d) indicates B = 7.2 T for which the Zeeman energy is equal to the SO coupling energy, EB = ESO, see text. The horizontal dashed line shows the angle arctan α β

B ≈ 1 .

 1 5T in (c), and ∆ (b) B ≈ 2.0T in (b). The ratio ∆ (c) B /∆ (b) B ≈ 3/4, is exactly an inverse of the source-impurity d s-i distance ratio.

Figure 4

 4 Figure 4. a) Potential disorder in simulated quantum wire. b) Local density of states obtained for channel from (a) at B=0. c) Conductance through the wire in (a) as a function of magnetic field amplitude B and direction angle. d) Same as c) but with one potential barrier in the middle of channel instead of random disorder. e) Same as (c) but for the wire with length L = 4000nm and W = 800nm. f) Same as (e) but for SO couplings twice smaller γ eff = 6meVnm. For comparison of (c,e) see Fig. 3(b). The values M show the number of non-degenerated modes in the channel.

Figure S1 .

 S1 Figure S1. First column -conductance as a function of magnetic field amplitude and angle. Second column -the random disorder potential energy in the channel in units of meV. Third column -resulting LDOS for B = 0 T in arbitrary units.

  Figure S2. (a) The self consistent electron density in the channel of width 200nm and length 400nm. (b) DFT self-consistent potential energy. (c) The bare electric field ∆Fz(x, y) at z = 0 induced by the impurity (see Eq. (1)).

  Figure S3. (a) The cross section of normalized electron density for y = 100 nm from Fig. S2(a) (black line) and the effective potential -potential of the impurity screened by the deformation of the electron gas (red line). (b) Same as (a) only for the impurity at 2.5 nm from the electron gas.

  1, 2 H. Sellier, 2 and B. Szafran 1 1 AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, al. Mickiewicza 30, 30-059 Kraków, Poland 2 Institut Néel, CNRS and Université Joseph Fourier, BP 166, 38042 Grenoble, France

	I. STABILITY OF THE CONDUCTANCE PATTERN

Acknowledgments

This work was supported by National Science Centre (NCN) grant DEC-2015/17/N/ST3/02266, and by PL-Grid Infrastructure. The first author is supported by the Smoluchowski scholarship from the KNOW funding and by the NCN Etiuda stipend DEC-2015/16/T/ST3/00310.


E A Rojas González

A Borne

B Boulanger

J A Levenson

K Bencheikh

Continuous variables triple-photon states quantum entanglement

Keywords: numbers: 42.50.-p, 42.65.-k, 42.50.Dv, 42.50.Lc

We investigate the quantum entanglement of the three modes associated with the three-photon states obtained by triple-photon generation in a phase-matched third-order nonlinear optical interaction. Although the second order processes have been extensively dealt with, there is no direct analogy between the second and third-order mechanisms. We show for example the absence of quantum entanglement between the quadratures of the three modes in the case of spontaneous parametric triple-photon generation. However, we show that genuine triple-photon entanglement is obtained in the fully seeded case, and its efficiency increases with the seeding level.

Twin-photon states produced by second-order nonlinear optical parametric interactions have deeply influenced the history of quantum optics and are the main ingredient in several quantum protocols. Their ubiquity contrasts with the case of triple-photon states (TPS). The unique quantum properties of TPS were anticipated in the 1989 seminal paper of Greenberger, Horne and Zeilinger, from which a particular category of TPS is called GHZ [1]. Their tripartite entanglement is identified as a direct way to contradict local realism theory for non-statistical predictions of quantum mechanics. Later on, W states have been predicted to retain maximally bipartite entanglement when any one of the three qubits is traced out [2]. Very recently, anisotropy was identified as a new invariant for pure three-qubit states and was found identical for each of their pairs, opening several promising applications [3]. More generally, one can expect that TPS plays in quantum optics a similar role as twin-photon states did over the last 40 years and allow the development of novel and efficient quantum protocols based on triple-photon entanglement or announced entangled twins. This is provided that clever optimization of both material and nonlinear interaction is achieved to overcome the well-known weakness of third-order nonlinear processes. Until now, only configurations based on second-order nonlinear interactions have succeed in producing TPS [4][5][6][7][8] at the expense of increased complexity, low efficiency or detection conditioned protocols preventing their use. However, the most direct way to produce TPS is indeed in materials exhibiting a pure third-order nonlinear susceptibility, χ (3) , in which a pump photon with energy ω p is down-converted simultaneously into a triplet ω 1 , ω 2 and ω 3 with the subsequent energy conservation, ω p = ω 1 + ω 2 + ω 3 , and momentum conservation k p = k 1 + k 2 + k 3 . Only one experimental demonstration of Triple-Photon Generation (TPG) was reported in the regime of bi-state seeding, i.e., two over the three fields associated to the out-coming component of the triple-photons are injected together with the pump field [9]. This pioneering work at the classical level clearly demonstrates a phase-matched third-order nonlinear process. Its transposition to spontaneous down-conversion is however precluded by the small χ (3) value. For instance, in the fluorescence regime, less than a triplet per day is expected [10]. There is an active research for novel materials and geometrical configurations pushing nowadays the efficiency of the reverse nonlinear process, namely third-harmonic generation, from bulk to waveguide operation [11][12][13][14]. One can reasonably conjecture this will contribute to increase TPG efficiency as predicted by semiclassical theories [15][16][17]. However, one can wonder if our comprehension at the quantum level of the process is sufficient to optimize the interaction. Until now, this optimization is strongly driven by the analogy with the well-known and largely investigated χ (2) process.

In this Letter, we develop a full quantum description of TPS in the framework of Continuous Variables (CV) corresponding to the manipulation of the electromagnetic fields. To the best of our knowledge, the CV quantum properties [START_REF] Braunstein | Quantum information with continuous variables[END_REF] remain unexplored for TPS generated by a pure third-order nonlinear interaction. We discuss spontaneous, single, double, and triple-seeded configurations and demonstrate that using twin-photon knowledge is often a wrong strategy for TPS optimization. As a result, no entanglement is found in the spontaneous emission regime. More importantly, tripartite entanglement is found in the triple seeding configuration and is found strongly increasing with the seeding, becoming a novel avenue for TPS optimization.

The starting point of our theoretical analysis is the interaction Hamiltonian describing TPG in a nonlinear material exhibiting a third-order nonlinear susceptibility [START_REF] Dot | [END_REF]. It reduces to

Ĥ = κ(â † 1 â † 2 â † 3 + â1 â2 â3 ), (1)
for a monochromatic strong non-depleted classical pump.

arXiv:1709.03317v1 [quant-ph] 11 Sep 2017

κ is proportional to the pump amplitude and to the nonlinear susceptibility χ (3) . The annihilation operators âk (k = 1, 2, 3) describe the triple photons modes. The Hamiltonian describes the annihilation of a pump photon and the simultaneous creation of triple photons, and the inverse process. We now consider the evolution of the system in the Heisenberg picture. The operators time evolution is given by:

dâ k (t) dt = ı [ Ĥ, âk (t)]. (2)
Let Âk (  † k ) be the annihilation (creation) operators of mode k after the nonlinear interaction. We define the amplitude and phase quadratures pk = Âk +  † k and qk = ı( Âk - † k ), satisfying the canonical commutation relations [q k , pk ] = 2ı. Their quantum fluctuations are measured with balanced homodyne detections. By sweeping the phase θ k of the local oscillator, used to measure mode k, one has access to the generalized quadratures Pk (θ k ) = e -ıθ k  †

k + e ıθ k Âk and Qk (θ k ) = Pk (θ k + π/2). Let us finally define the following linear combinations:

û = h 1 P1 + h 2 P2 + h 3 P3 v = g 1 Q1 + g 2 Q2 + g 3 Q3 , (3)
where h k and g k (k = 1, 2, 3) are arbitrary real parameters introduced experimentally as electrical attenuations or amplifications of the photocurrents generated by the homodyne detection photodetectors. Phase dependence is intentionally omitted in Eq.(3) for simplicity. We quantify multi-body quantum entanglement of TPS using the non-separability criterion S introduced by P. van Loock and A. Furusawa in Ref. [20], defined as:

S = ∆û 2 + ∆v 2 , (4)
which can be measured experimentally using homodyne detection technics. Expanding Eq. ( 4) and using Eq. ( 3) we find:

S = 3 k=1 h 2 k ∆ Q2 k + 3 k=1 g 2 k ∆ P 2 k + 3 k=1 3 m=1 m =k h k h m ( Qk Qm -Qk Qm ) + 3 k=1 3 m=1 m =k g k g m ( Pk Pm -Pk Pm ).
(

) 5
The two first terms are a linear summation of the generalized quadrature variances. The two last terms describe cross-correlations between the output modes at the origin of the entanglement. Indeed, whereas the two first terms are always positive, the two last can be negative and can interfere destructively with the first ones. The criterion S is a function of the real parameters h k and g k . If

S < f p = 2(|h k g k | + |h l g l + h m g m |) (6)
for a given permutation {k, l, m} of {1, 2, 3}, then the quantum system is fully inseparable and exhibits genuine tripartite quantum entanglement [20]. However, if

S < f s = 2(|h k g k | + |h l g l | + |h m g m |), (7)
which is less restrictive than Eq. ( 6), the system is at least partially separable. If f p ≤ S < f s , two among the three modes could be entangled, without being entangled to the third part. According to [20], the parameters h k and g k are chosen such that 0 < f p ≤ f s and [û, v] = 0, to allow S → 0 and the existence of simultaneous eigenstates of û and v.

The determination of S in the case of TPS relies on the resolution of Eq. ( 2). Unfortunately, they have no known analytical solutions when solved in the frame of quantum mechanics, though it has been shown that the exact classical solutions are Jacobi elliptic functions [21].

Here, to obtain an approximate solution of the operators at a time t, we use the Baker-Hausdorff expansion

Âk = âk + ∞ n=1 (ıξ) n n! Ωn;klm , (8)
to a finite order, where ξ = κt is the interaction strength and

Ωn;klm = 1 ( κ) n [ Ĥ, [ Ĥ, [• • • Ĥ, [ Ĥ n-times , âk ]]]], (9)
with {k, l, m} being permutations of {1, 2, 3}. Expanding the commutators in Eq. ( 9), we find that Ωn;klm is a function of the creation and annihilation operators of the three modes raised at different powers. We also find a symmetry with respect to the indexes l and m, i.e. Ωn;klm = Ωn;kml , which is helpful in the subsequent calculations. The expansion order of the operators in Eq. ( 8) is crucial for the validity of our analysis. It depends on the interaction strength ξ and on the average number of seeding photons. In the following, we will analyze different situations of TPG expanding Eq. ( 8) to a finite order and calculating the criterion S using a combination of both symbolic and numerical computational methods. For all numerical analysis we take |ξ| = 1.75 × 10 -6 , deduced from [21], which is very representative of the third-order nonlinearity of nowadays materials.

Triple-photon parametric fluorescence. In the fluorescence case, there is no seeding and since |ξ| 1, the expansion is valid at any order. It is well known that Spontaneous Parametric Down-Conversion (SPDC) induces strong quantum entanglement between the twin modes. Surprisingly, our analysis shows that a spontaneous triple-photon generation does not exhibit such quantum entanglement in the CV regime for any chosen linear combination of û and v. We find that the crosscorrelation terms in Eq. ( 5) between the output modes vanish and that S reduces to

S = (1 + 2 Â † Â ) 3 k=1 (h 2 k + g 2 k ), (10)
when assuming identical average photon number  †  in each mode. The full derivation of Eq. ( 10) is given in the supplemental material [22]. According to Eq. ( 10), the criterion S only depends on the strength of the interaction ξ, which is contained in  †  and on the parameters h k and g k . We compare the result given by Eq. ( 10) to the classical limit f s given in Eq. ( 7) by analyzing the difference Sf s . After some mathematical manipulations, we end up with S -

f s = 2Γ 2 Â † Â + 3 k=1 (|h k | - |g k |) 2 , where Γ 2 = (h 2 1 + h 2 2 + h 2 3 + g 2 1 + g 2 2 + g 2 3 )
. Each term being positive, S f s for any choice of h k and g k . This result clearly demonstrates that the three modes are fully independent, meaning that their quantum fluctuations are totally uncorrelated. Though we were expecting a similarity with respect to the twin-photon fluorescence where an entanglement is found, this result could be easily understood as follows. The triplets are generated from the vacuum quantum fluctuations as the twins. However, for a given eigenvalue outcome of the observable, Q1 for example, the two others, Q2 and Q3 , can still take any random pair of eigenvalues. Each realization is thus independent from the previous one.

Partially seeded triple-photon generation. The validity of our analysis in the seeded cases is discussed in details in the supplemental material [22], treating particularly the fully seeded TPG which is the most restrictive case. We show that a sufficient condition for the expansion to be valid is |ξα| 1, where N in = |α| 2 is the incident average photon number per mode, chosen accordingly to always fulfill the condition. All subsequent numerical analysis are done using N in 10 11 and a fifthorder expansion of the operators Eq. ( 8). The associated errors on the operators is about 6.4 % and even smaller on the estimation of the gains, variances and S.

When only one of the three modes is excited by a bright coherent field |α containing N in photons in average, no three-body quantum entanglement was found for any of the analyzed cases. If for example mode 3 is seeded, we can replace the operator â3 by its classical amplitude field counterpart ıE 3 , chosen to be complex for convenience, assuming that E 3 is a real number. The Hamiltonian becomes Ĥ = ı κ(â

1 â2 -â † 1 â † 2 )
, where κ = gE 3 . This Hamiltonian describes SPDC, where a pump photon is converted into twin-photons. In the third-order configuration, however, the down-conversion process is proportional to an effective second-order susceptibility proportional to the product χ (3) E 3 , meaning that the efficiency depends on the seeding level. Figure 1 allows to analyze the bipartite nonseparability criterion S for these twins. The magenta continuous line with the (+) markers shows S for modes 1 and 2, using the quadratures û = Q1 + Q2 and v = P1 -P2 , as a function of the seeding average photon number. The quadratures are determined from the exact solutions of Eq. ( 2). The criterion S starts at 4, which is the classical limit, i.e. the sum of the four quadrature variances, then decreases as the seeding gets stronger. The gray area beneath S = 2 indicates the region in which quantum entanglement is present, meaning that the measurement of the quadrature Q2 (respectively P2 ) allows to know Q1 (respectively P1 ) better than the standard quantum limit associated with the shot noise of a coherent state [23]. Figure 1, shows that S gets into the quantum entanglement region. The correlations get stronger as the seeding increases. In a sense, the single mode seeded TPG is a re-configurable two-body entanglement source where the seeded field acts as a control parameter: it changes the strength of the interaction and selects which quadratures are entangled depending on the seeding phase. When E 3 → 0, it reaches the fluorescence case, which can be seen as an effective second-order process driven by the quantum fluctuations of mode 3. Actually this reasoning helps us to understand the origin of the missing three-body quantum entanglement. In- deed, each eigenvalue resulting from the measurement of the seeding mode, even at shot noise, will reset the twinphoton generation process. The case of two modes coherently seeded TPG is analyzed hereafter. This situation corresponds to the experimental arrangement described in [21]. The input state of the system is |ψ = |0 1 , α 2 , α 3 . The most trivial linear combinations in Eq. ( 3) to be considered is for

h 1 = g 1 = 1, h 2 = h 3 = 1/ √ 2 and g 2 = g 3 = -1/ √ 2.
All the local oscillator phases are θ k = 0. Any other phase does not influence the final result except by shifting the minimum value of S in the phase space. The chosen parameters suggest that the quantum properties of the non-injected mode 1 are investigated through the measurements of both seeded modes 2 and 3. The green line with (×) markers in Fig. 1 represents the corresponding S. Starting at 4, it decreases as the seeding grows. At N in 2 × 10 10 , S goes below f p = 2 indicating genuine three-body quantum entanglement. In this regime, measuring both modes 2 and 3 is necessary to know mode 1 better than the standard quantum limit.

Fully seeded triple-photon generation. This section focuses on the case where the three modes are initially excited. The input state of the system is |ψ = |α 1 , α 2 , α 3 . As for the fully seeded second-order parametric interaction, it is necessary to consider the phase of the different modes at the input of the nonlinear material relative to the pump phase. We also consider identical coherent states seeding the three modes, so that α i = |α|e ıφ for each mode.

We start by looking at the evolution of the gain G = N out /N in , where N out = Â † k Âk and N in = â † k âk are the average photon number in each mode at the output and the input of the nonlinear medium. We also calculate the variance of the quadratures Pk and Qk of each mode. Figure 2.(a) represents the gain G for N in = 4 × 10 10 in polar coordinates as a function of the phase φ of the seeding. It shows amplification (G > 1) and deamplification (G < 1) regimes and features three lobs subsequent to the e ı3φ dependance of N out . The maxima are located at φ = π/2, 7π/6 and 11π/6 corresponding to the phases for which the down-conversion TPG process ω p → ω 1 + ω 2 + ω 3 is enhanced and achieving thus gains greater than unity. The minima are obtained for φ = π/6, 5π/6 and 3π/2. They correspond to gains smaller than unity. Here, it is the sum frequency process ω 1 + ω 2 + ω 3 → ω p which predominates. Figure 2.(b) shows the variances of the quadratures Pk (red) and Qk (blue) in the same conditions of Fig. 2.(a). The most interesting feature, besides the presence of the three lobes, is that the variances are always above the shot noise (gray circle) when the modes are measured separately. This behavior is equivalent to non-degenerate twin-photon generation where signal and idler modes exhibit superpoissonian fluctuations.

Looking for tripartite quantum entanglement, we consider the following particular linear combinations of the operators of Eq. ( 3).

û = Q1 + 1 β √ 2 ( Q2 + Q3 ) v = P1 -β 1 √ 2 ( P2 + P3 ), (11)
where we have introduced an extra free parameter β for helpful optimization. Our analysis shows that for each phase θ 1 of the local oscillator used for detecting mode 1, it exists a couple of phases θ 2 and θ 3 for detecting modes 2 and 3 that minimizes S, reaching values below the non-separability limit f p = 2. We also show that further optimisation can be obtained through the choice of the adjustable parameter β. The red line with (•) markers in Fig. 1 shows the evolution of S as a function of the seeding average photon number per mode. It is obtained in the amplification regime. The local oscillator phases are θ 1 = 0 and θ 2 = θ 3 = π. As the seeding increases, the red curve goes into the grey region below the limit f p = 2, demonstrating genuine three-body quantum entanglement. This means that a measurement of both modes 2 and 3 is necessary to gain information on mode 1 better than the shot noise limit. Indeed, if one tries to determine mode 1 using only mode 2 for example by measuring the variances of the quadratures û = Q1 + Q2 and v = P1 -P2 , then S > 2, as shown by the blue line with ( ) markers in Fig. 1.

Conclusion.

In our analysis of parametric TPG, some counterintuitive results are found when comparing with the well-known twin-photon second-order equivalent. The first counterintuitive result is obtained for the pure spontaneous case for which we demonstrate in all cases the absence of three-body quantum entanglement. The more important finding is for the fully seeded TPG. It exhibits a phase-dependent gain, behaving similarly to the well-known second-order phase-sensitive parametric amplifier. But it also appears as an efficient way to generate genuine TPS quantum entanglement. Finally, an additional property corresponds to the increase in efficiency with the amplitude of the seeding classical field. All the parameters considered in these calculations are compatible with existing experimental results at the classical level for bulk materials. It is obvious that the overall efficiency could be greatly improved in the guided and integrated configurations.

This three-body genuine quantum entanglement constitutes a real asset for a new generation quantum cryptography and other quantum information protocols. We also stress that in the quest for the realization of continuous variable GHZ states, genuine entanglement of pure third-order TPS can offer the missing ingredient.

FIG. 1 . 2 .

 12 FIG. 1. Evolution of the non-separability criterion S as a function of the seeding average photon number for |ξ| = 1.75 × 10 -6 and for different injection cases. The gray area highlights the quantum entanglement region. The Triple-Photon Generation (TPG) boxes show the different schemes with the seeded (full arrows) and non-seeded (dashed arrows) modes. The S-boxes indicate the modes concerned by the S measurement. Magenta line with (+) markers: single seeding case. Green line with (×) markers: double seeding case. Red line with (•) markers for the fully seeded at θ1 = 0, θ2 = θ3 = π, and for β = √ 2. The phase of the seeding coherent state is φ = π/2. Blue line with ( ) markers: same as the red curve but for quadratures û = Q1 + Q2 and v = P1 -P2.

FIG. 2 .

 2 FIG. 2. Gain (a) and Pk (red) and Qk (blue) variances (b) calculated using the fifth-order expansion of the operators as a function of the phase of the seeding fields, for |ξ| = 1.75×10 -6 and N in = 4 × 10 10 . The gray circles indicate the unity gain in (a) and the standard quantum limit in (b).