Régis Bismuth

The Ambiguous Relations between Corporate Compliance and Sovereignty

Introduction

Is compliance redefining the notion of sovereignty as we know it in public international law? At first, the question may seem incongruous, given that corporate compliance has to some extend stood away from the realm of international law. It is not so much that corporate compliance does not have a transnational dimension; it is rather that they both have their own instruments and methods, as well as ways of looking at the world. A closer examination reveals nonetheless that confronting corporate compliance and the international law understanding of sovereignty proves useful in shedding a light on the relatively ambiguous relations between them.

The starting point of the analysis reveals a certain dissonance. At first sight, the notion of sovereignty seems quite incompatible with that of corporate compliance. While sovereignty traditionally endorses a territorial approach to jurisdiction, corporate compliance has used methods and instruments which largely ignore borders and which do not take due account of their compatibility with international law. Indeed, if corporate compliance is perceived as a new form of regulation 1 , this new kind of imperium seems to unfold with international law rules on State jurisdiction remaining in a blindspot as well as with international law not necessarily able to detect how and to what extent corporate compliance may infringe international law. This largely silenced dissonance between corporate compliance and sovereignty must be assessed in light of the objectives and "monumental goals" that corporate compliance seeks to pursue 2 . Most of these "monumental goals" reflect imperatives that have been defined in a multilateral framework (fight against corruption, money laundering or terrorism financing, etc.). This has implied a convergence between international norms and the objectives of corporate compliance. This convergence does not however imply a perfect continuity and harmony between both frameworks. Indeed, if corporate compliance seems to unfold as a mere declination of what has been jointly defined by sovereign States, it operates nonetheless with its own mechanisms and set of rules3 . The situation is actually more complex given that some "monumental goals" collectively recognized as legitimate goals by States, do not necessarily translate into a similar set of rules in each national legal system (or regional for the European Union). This is the case, for example, of international human rights law and the due diligence obligations in global value chains, data protection or the exchange of information for tax purposes. In such situations, the risks of tension between corporate compliance and sovereignty are even more amplified.

The situation can be even tenser when the "monumental goals" of corporate compliance follow a strictly unilateral approach. Economic sanctions provide an interesting case in this regard. States can jointly adopt, under the auspices of the United Nations, multilateral economic sanctions that will eventually be complied with by economic actors. Some States have also developed a practice of unilateral economic sanctions, in view of pursuing global objectives (such as human rights protection, protection of democracy, etc.). Some of these sanctions, in particular US ones, may have an extraterritorial reach which possibly exposes economic actors to considerable conflicts of obligations. This has been particularly the case with the EU blocking regulation prohibiting EU businesses from complying with US extraterritorial sanctions. The US strategy could be more insidious when in some specific situations (for example in sanctioning Iran), it amalgamated unilateral sanctions with multilateral disciplines relating to the fight against terrorism financing and money laundering4 .

Corporate compliance seeks to ensure the effectivity of a nuanced range of objectives, from authentically multilateral "monumental goals" to strictly unilateral goals. But corporate compliance expands through methods and instruments (risk mapping, codes of conduct, internal investigations, negotiated justice settlements, etc.) that one may consider as "neutral" since they can be deployed interchangeably, for multilateral or unilateral objectives. Once corporate compliance internalizes those different objectives, this distinction between multilateral or unilateral dimensions quickly fades out, and the company has to choose between different options on the basis of various factors (sector of activity, exposure to the US market, etc.). This has subsequently amplified the existing tensions between corporate compliance and sovereignty. Indeed, corporate compliance tools tend to habituate the company to arbitrate between conflicting obligations in a transnational context (complying with economic sanctions or with a blocking regulation, transferring data to a foreign authority or complying with the requirements of data protection, etc.). This has fed a progressive shift from corporate compliance as a process of norm implementation to a process of norm generation.

In this sense, the ambiguous relations between corporate compliance and sovereignty seem to have three dimensions, which are non-mutually exclusive: extension, confrontation, and transformation. Compliance may first and foremost be conceived as a tool enabling States to indirectly extend the limits of their territorial sovereignty (I). But corporate compliance as a tool can nevertheless lead to several frictions, or even conflicts, with the notion of sovereignty. An illustration of this can be found when the "monumental goals" of corporate compliance are not defined in a multilateral fashion. This characterizes the confrontational dimension (II). Finally, by accustoming companies to methods and instruments reminiscent of sovereign functions, corporate compliance has the potential to facilitate the emergence of a corporate sovereignty, beyond State sovereignty. This is the transformational dimension, that we will discuss in our conclusion (III).

I. Extension: Pushing the Limits of Sovereignty through Corporate Compliance

State action is constrained by international law, and in particular, its principles on prescriptive, adjudicative and enforcement jurisdiction5 . Leaving aside cases of universal jurisdiction (for instance for international crimes) and protective jurisdiction when the fundamental interests of the State are at stake (for example for currency counterfeiting), the State cannot, in theory, extend the applicability of its regulation (prescriptive jurisdiction) or the jurisdiction of its courts (adjudicative jurisdiction) to situations taking place in a foreign country and not involving its nationals, whether natural or legal persons. Nor can the State, in principle, extend the enforcement of its decisions (enforcement jurisdiction) outside its territory, even where the situation concerns a national. This is the basic framework upon which the discourse on the extraterritoriality of law has been constructed over the past years6 . Of course, this framework does not allow us to grasp all social realities, in particular those where territoriality appears to be more elusive. This is the case for internet and data regulation for which location cannot fully be based on an objective territorial basis, and which may give rise to competing claims or conflicts of jurisdiction 7 .

The limits set by international law on prescriptive, adjudicative or enforcement jurisdiction can, almost silently, be circumvented by corporate compliance. This is not an issue of the extraterritoriality of US law (anti-corruption, economic sanctions, etc.) on the basis of which non-US corporations (and in particular European ones) have been sanctioned by US authorities [START_REF] Bismuth | Pour une appréhension nuancée de l'extraterritorialité du droit américain -Quelques réflexions autour des procédures et sanctions visant Alstom et BNP Paribas[END_REF] . It is rather a discussion about the ways through which instruments and methods of corporate compliance have in themselves generated some forms of extraterritoriality. Three instruments deserve to be analysed here: negotiated justice settlements, corporate monitorship, and risk mapping.

Negotiated justice settlements have been a new instrument for sanctioning corporate misconduct [START_REF]L'entreprise et les conflits internationaux de loi[END_REF] . If they have led in certain cases to criticisms regarding their extraterritorial dimension 10 , it is precisely because such mechanisms may silence any debate relating to the issue on jurisdiction. The prosecuted company is faced with an alternative: either it chooses to benefit from all procedural guarantees in a criminal trial that can lead to a maximum financial sanction and market restrictions, or it waives these guarantees by agreeing to conclude a settlement with the prosecuting authorities, which will determine the facts and result in a negotiated sanction, usually a financial penalty. Choosing the negotiated solution makes the sanction more predictable, which is essential in terms of risk management. But this option nonetheless neutralizes any room for discussion regarding the question of legitimacy of the competence of the prosecuting State. States generally rely on their "doing-business jurisdiction" 11 which in most cases would have made it possible to assert their competence in the context of a criminal trial. The corporation's consent to the negotiated procedure allows the authorities to avoid complex discussions in situations where the jurisdictional nexus is more debatable. Such consent is also useful to the extent that it facilitates the implementation of the settlement. Negotiated justice has thus enabled the United States to use in economic sanction cases -and without explicitly admitting it -a US-dollar jurisdictional nexus when targeting transactions concluded outside the country between foreign persons, as long as they are concluded in dollars and cleared through the US financial system. This dollar-based competence lies on a very extensive interpretation of territorial jurisdiction, which has been adamantly criticized since there is no substantial connection between the targeted transaction and the US territory and given that the clearing transaction is merely incidental [START_REF] Bismuth | Pour une appréhension nuancée de l'extraterritorialité du droit américain -Quelques réflexions autour des procédures et sanctions visant Alstom et BNP Paribas[END_REF] . It is therefore not surprising to see that, to date and to our knowledge, only individuals -not facing the same constraints as corporations in negotiated justice settlements -have chosen to challenge the legality of this jurisdictional nexus [START_REF] Emmenegger | The Extraterritorial Application of U.S. Sanctions Law[END_REF] . In any case, by hindering any possible discussion on the issue of jurisdiction, negotiated justice settlements has enabled States -in particular, the USto adopt a very extensive conception of their jurisdiction.

What has just been pointed out concerning negotiated justice can be extended to what is commonly referred to as corporate monitorship. Negotiated settlements can give rise to sanctions with a retrospective dimension (for instance financial penalties) and others with a prospective dimension (behavioural remedies) [START_REF]L'entreprise et les conflits internationaux de loi[END_REF] . This type of sanctions can lead the prosecuting authorities to appoint a corporate monitor who is in charge of controlling the implementation of the company's commitments. This practice emerged in the United States. Corporate monitors are deemed to be independent from both the authorities and the sanctioned company. They are endowed with extensive prerogatives and conduct investigations and assessments of the company's compliance programs. They have to issue periodic reports to the company's management and to the prosecuting authorities. Corporate monitors are independent but with somehow a public function when it comes to control the internal management of the company 15 . It would of course be inconceivable for the States in which the sanctioned corporations operate that foreign public agents could freely exercise their imperium on their territory without their consent, for example, by conducting investigations to control the implementation of negotiated settlements. This precisely constitutes the exercise of enforcement jurisdiction on the territory of a foreign State, which is in principle prohibited under international law. We can apply a similar analysis to the context of corporate monitorship. Indeed, while being independent from the appointing State, corporate monitors remain nonetheless a State agent from an international law perspective, as they exercise by delegation "elements of governmental authority" under the international law rules on State responsibility [START_REF] Crawford | State Responsibility: The General Part[END_REF] . The monitorship programs implemented in highly sensitive cases offer compelling evidence of the problems that the monitor's functions raise. Some of these programs have led to intergovernmental agreements in view of facilitating the transfer of information to prosecuting authorities which would have been otherwise contrary to domestic blocking statutes. This happened for instance in the Alcatel-Lucent case [START_REF] Cohen-Tanugi | The Independent Corporate Monitor: Who, What, When and How?[END_REF] . However, to our knowledge, such a practice remains rare. It is generally recommended that corporate monitors first identify any thorny issues raised by the blocking statutes, and make sure that their activity does not blatantly breach them [START_REF] Soffer | US-Ordered Cross-Border Monitorships[END_REF] . This is another example of how, at the margins of international legality, corporate compliance tools allow prosecuting authorities to extend the spatial boundaries of their executive competence, when relying on mechanisms based on the consent of the targeted company and involving private persons in the monitoring process.

Risk mapping constitutes another instrument of corporate compliance which facilitates risk assessment for companies. Risk mapping must not be simply conceived as a way for a company to meet a legal obligation (in data protection, the fight against corruption, human rights protection, etc.). It also constitutes an instrument through which the company may ex ante adjust its internal processes 19 so as not to be held liable, in an ex-post approach, for failures to comply, for instance, with a due diligence obligation. In this respect, it is interesting to note that risk mapping and due diligence obligations leads the company through compliance processes to extend the application of the rules of the regulating State at the corporate group level, therefore with cross-border effects. The French statute on the devoir de vigilance 20 provides an interesting illustration. It applies exclusively to certain French companies and requires them to carry out risk mapping in order to take reasonable vigilance measures to prevent severe violations of human rights or environmental damage or health risks resulting directly or indirectly from the operations of the company and of the companies it controls as well as from the operations of the subcontractors or suppliers with whom they maintain an established business relationship. French law does not however require subsidiaries, suppliers and subcontractors located abroad to comply with certain standards. This would constitute an extraterritorial exercise of prescriptive jurisdiction. Rather, it indirectly requires French companies to use their economic leverage to ensure that these actors comply with certain standards stemming from the risk mapping process. In this sense, this duty of vigilance obligation involves a cross-border regulation of global value chains 21 . One may argue that the extraterritoriality of this mechanism is "potentially wider than what the letter of the law would have suggested" 22 . To some extent, the French statute on the devoir de vigilance builds on corporations' compliance tools to extend its territorial reach 23 .

In all of the above cases, such compliance tools largely fall outside the radar of international law because most of the key decisions have not formally been taken unilaterally by States but by companies. This also leads to some form of normative pluralism, as companies must not only implement the rules resulting from their compliance processes but also those required by domestic law. And these two categories of rules are not necessarily compatible. As previously shown, there is a risk that one of the corporate compliance tools (namely monitorship) may in practice violates the laws (namely the blocking statutes) of the States in which the investigations of the monitor are conducted. This allows us to delve deeper into the processes of corporate compliance and how they may conflict with domestic regulations. In this regard, the conflict arises not only because corporate compliance processes implement a foreign law (referred to as the law of the "home State") or an international standard, which may contradict the law of the State where it is implemented (referred to as the law of the "host State"), but also because they may go beyond the law of the home State or the international standard, and generate their own set of rules, which in themselves may clash with those of the host State.

II. Confrontation: Compliance versus

Corporate compliance operates as a "delegation of normativity" [START_REF] Racine | Propos introductifs[END_REF] . The company implements the "public" standard through its "private" compliance processes, which may then apply beyond the home State's territory. The potential conflict between the public standard (whether foreign or international) implemented by the company and the host State's domestic law has already been contemplated, for instance in the field of corporate social responsibility. The key instrument in this regard, the United Nations Guiding Principles on Business and Human Rights (UNGPs) -adopted in 2011 by the UN Human Rights Council 25 and which has inspired the French statute on the devoir de vigilance -provides that if businesses must comply with domestic law, they must "seek ways to honour the principles of internationally recognized human rights when faced with conflicting requirements" 26 . The Commentary to the UNGPs adds that "where the domestic context renders it impossible to meet this responsibility fully, business enterprises are expected to respect the principles of internationally recognized human rights to the greatest extent possible in the circumstances, and to be able to demonstrate their efforts in this regard" [START_REF][END_REF] . This situation is also contemplated by the OCDE Guidelines on Multinational Enterprises, with a similar recommendation 28 . Such conflicts are not hypothetical and are likely to deal with highly sensitive topics. This is the case for example of a domestic law that would prohibit women from accessing senior positions [START_REF]BIICL, When National Law Conflicts with International Human Rights Standards: Recommendations for Business[END_REF] . We can feel some form of discomfort when reading the UNGPs or OECD Guidelines since they oscillate between on the one hand an imperative to respect and protect human rights and on the other hand the required compliance with domestic rules (which is commonly referred to as the "When in Rome Approach"). This could lead to embarrassing accommodations within compliance procedures (e.g., allowing remote work for an employee facing discrimination in the host State) 30 .

We have just envisaged how a conflict between the law of the host State and corporate compliance processes materializes and where the former overrides the latter. In some cases, a company may choose to implement, at its own risk, its own standards without even taking into consideration the possibility of a violation of the law of the host State. It should be noted in this regard that the compliance standard may be specific to the company and not simply dictated by the law of the home State. Indeed, compliance processes are not a mere mechanical transposition of legal obligations. The company can devise its own compliance process on the basis the different rules or requirements it must take into account and according to its own risk assessment 31 . If the rules of the home State offer a wide margin of appreciation to the prosecuting authorities and if the risk of financial sanctions is significant, the company will be tempted to embark on a maximalist compliance policy aimed at eliminating any possible risk of non-compliance. This compliance policy is known as "derisking" or "overcompliance".

The implementation of US economic sanctions regularly gives rise to this type of behaviour. This was the case some time ago when Wal Mart's Canadian subsidiary, fearing US sanctions, removed Cuban-made pyjamas from sale 32 and when the Hilton Group's subsidiary in Norway rejected hotel reservations made by Cuban officials for the same reason -the latter situation gave rise to protests given that it was likely to constitute discrimination based on nationality 33 . These practices of overcompliance have taken an excessive dimension within financial institutions in the face of significant financial penalties that some of them have faced. As a result, several banks have adopted a derisking policy, choosing to reject any transaction involving Iran or Cuba regardless of whether they were specifically falling within the scope of US sanctions.

This overcompliance policy can also lead to particularly grotesque situations. This has been the case for instance in the field of international tax cooperation. In the wake of tax scandals involving Swiss banks, the United States adopted in 2010 the FATCA statute (Foreign Account Tax Compliance Act), which requires all foreign financial institutions to transfer to the US tax authorities information regarding financial assets held by US persons abroad. Given that transferring this information would have been for banks in blatant breach of domestic blocking statutes, more than a hundred States have agreed to enter into bilateral agreements with the US to allow such a transfer. Under this framework, financial institutions are obliged to identify US taxpayers by using indicia of US citizenship, including a place of birth in the US. This has led several foreign banks not willing to deal with the implications and bear the cost of these requirements to simply refuse US-born clients -including those who have no ties to the US other than their place of birth and who have as second nationality that of their State of residence [START_REF] Bismuth | L'extraterritorialité du FATCA et le problème des "américains accidentels[END_REF] . This discrimination on grounds of nationality is another evidence that compliance processes can lead to corporate conducts conflicting with the domestic law of the host State. Besides, such overcompliance practices sometimes spreads into market practices without raising any concerns about their legality. What is even more disconcerting is the institutionalisation of such discriminatory practices against US-born persons in boilerplate clauses found inEU Commission's bond issue contracts [START_REF] Bismuth | Emissions obligataires européennes : La Commission participe-t-elle à une entreprise de discrimination fondée sur la nationalité ?[END_REF] -the same Commission that intends to neutralise the extraterritorial reach of US laws.

Both examples, of FATCA and economic sanctions eventually show that overcompliance does not necessarily aim at preventing risks of non-compliance with legal obligations. Companies seem inclined to no longer pursue a wide range of activities considered too risky and costly from a corporate compliance perspective [START_REF] Artingstalli | Drivers & Impacts of Derisking, Financial Conduct Authority[END_REF] . Although some institutions have already identified and started addressing the overcompliance issue, including the Financial Action Task Force (TATF) in the fight against money laundering and terrorism financing [START_REF] Fatf | Drivers for "De-Risking[END_REF] , others are fully aware of it and have proven to anticipate an overcompliance based on companies' risk aversion to extending the reach of their regulations. In that regard, the US Treasury official in charge of designing the US strategy for on economic sanctions, fight against money laundering and terrorism financing, noted that "the ultimate application of American financial power extraterritorially […] relied on the decisions and actions of the private sector" [START_REF] Zarate | Treasury's War -The Unleashing of a New Era of Financial Warfare[END_REF] and predicted that financial institutions would way beyond what was prescribed under US law.

III. Transformation: Compliance and the Legitimisation of "Corporate Sovereignty"?

Corporations, more particularly those of a certain size or those operating in certain sectors, are the addressees of an increasing number of obligations to reach specific "monumental goals" that States are no longer able to assume on their own in an increasingly complex transnational context. Vesting companies with missions of general interest -for instance ensuring the proper functioning of the market -is hardly new. We have indeed witnessed this trend with the so-called "gatekeepers" in the area of financial regulation and corporate governance (rating agencies, auditors, etc.)39 .

Corporate compliance, whether due to its instruments or methods or the objectives it seeks to pursue, has moved corporations into a whole new dimension that entirely redefines how they deal with rules. Companies are no longer the mere addressees of regulations, they have become an essential part of their interpretation and implementation: "the legal subject has become the agent of legality […] because of its position and its power". Besides, as it has been highlighted, corporate compliance processes may lead companies to devise their specific standards in a normative environment where they are subject to several (and sometimes conflicting) obligations.

These processes of norm creation, interpretation and implementation actually echo the classification of State competences under international law principles of State jurisdiction (prescriptive, adjudicative and enforcement jurisdiction). It is thus not a surprise to witness an emerging discourse on "corporate sovereignty" which would compete or coexist with State sovereignty. Since the beginning of this century, some literature on the issue has been published [START_REF]Leviathans -Multinational Corporations and the New Global History[END_REF] , underlining in some cases the necessity to constitutionalise this form of private power [START_REF] Backer | The Concept of Constitutionalization and the Multi-Corporate Enterprise: From Body Corporate to Sovereign Enterprise[END_REF] . This literature on corporate sovereignty has also more recently relied on the development of digital giants. Not only are these digital actors establishing their own transnational supreme courts, like the Facebook Oversight Board, suggesting that even managerial decisions are also judicially reviewable by internal corporate courts [START_REF] Klonick | The Facebook Oversight Board: Creating an Independent Institution to Adjudicate Online Free Expression[END_REF] . But States have also recognized their quasi-sovereign functions by appointing so-called "digital ambassadors" so as to initiate a dialogue on an equal basis with the giants of the Silicon Valley [START_REF] Satariano | The World's First Ambassador to the Tech Industry[END_REF] . We may not be yet at the stage to recognise immunities and privileges for these corporations, but using the vocabulary of diplomatic relations nonetheless has an undoubted symbolic dimension.

Corporate compliance may not be at the source of all these evolutions, but it has nonetheless significantly facilitated and legitimised these changes.

  1 M.-A. Frison-Roche, « Du droit de la régulation au droit de la compliance », in M.-A. Frison-Roche (dir.), Régulation, supervision, compliance, Paris, Journal of Regulation & Compliance & Dalloz, 2017, p. 12. 2 M.-A. Frison-Roche, « Legal Approach to Compliance Tools », in M.-A. Frison-Roche (ed.), Compliance Tools, Paris/Bruxelles, Journal of Regulation & Compliance & Bruylant, 2021, p. 35.

  M. Rathbone, P. Jeydel et A. Lentz, « Sanctions, Sanctions Everywhere: Forging a Path through Complex Transnational Sanctions Laws », Georgetown Journal of International Law, 2013, vol. 44, p. 1120.

	30 Ibid., p. 26.
	31 J.-B. Auby, « Le dialogue de la norme étatique et de la compliance », in M.-A. Frison-Roche (dir.), Régulation,
	supervision, compliance, Paris, Dalloz, 2017, p. 103.

32

33 Ibid., p. 1121.

Regarding anti-corruption law, some illustrations may be found in : R. Bismuth, J. Dunin-Wasowicz et P.N. Nichols (eds.), The Transnationalization of Anti-Corruption Law, Transnational Law and Governance, Abingdon/New York, Routledge, 2021.

On the US Treasury's strategy, see J.C. Zarate, Treasury's War -The Unleashing of a New Era of Financial Warfare, New York, PublicAffairs, 2013.

C. Staker, « Jurisdiction », in M.D. Evans (ed.), International Law, 5th éd., Oxford, Oxford University Press, 2018, p. 289.

R. Bismuth, « Pour une appréhension nuancée de l'extraterritorialité du droit américain -Quelques réflexions autour des procédures et sanctions visant Alstom et BNP Paribas », Annuaire Français de Droit International (2015), 2016, p.

7 J. Daskal, « Borders and Bits », Vanderbilt Law Review, 2018, vol. 71, p. 179 ; R. Bismuth, « Le Cloud Act face au projet européen e-evidence : Confrontation ou coopération ? », Revue Critique de Droit International Privé, 2019, n° 3, p. 681.

R. Bismuth, « Au-delà de l'extraterritorialité, une compétence économique », in SFDI, Extraterritorialités et droit international, Paris, Pedone, 2020, p. 118.

J.C. Coffee Jr, Gatekeepers: The Professions and Corporate Governance, New York, Oxford University Press, 2006.


H Bartolomei

R Bisognin

H Kamata

J.-M Berroir

E Bocquillon

G Ménard

B Plac ¸ais

A Cavanna

U Gennser

Y Jin

P Degiovanni

C Mora

G Fève

Observation of edge magnetoplasmon squeezing in a quantum Hall conductor

Squeezing of the quadratures of the electromagnetic field has been extensively studied in optics and microwaves. However, previous works focused on the generation of squeezed states in a low impedance (Z0 ≈ 50Ω) environment. We report here on the demonstration of the squeezing of bosonic edge magnetoplasmon modes in a quantum Hall conductor whose characteristic impedance is set by the quantum of resistance (RK ≈ 25kΩ), offering the possibility of an enhanced coupling to low-dimensional quantum conductors. By applying a combination of dc and ac drives to a quantum point contact, we demonstrate squeezing and observe a noise reduction 18% below the vacuum fluctuations. This level of squeezing can be improved by using more complex conductors, such as ac driven quantum dots or mesoscopic capacitors.

In quantum Hall conductors, charge excitations propagate ballistically along one dimensional chiral channels. This ballistic propagation has been exploited in electron quantum optics experiments 1,2 focusing on the generation and manipulation of elementary electron and hole excitations of the Fermi sea. These are particle-like fermionic excitations, but the dynamics of charge propagation along one-dimensional edge channels can be equivalently described in terms of collective bosonic excitations called edge magnetoplasmons (EMP), which consist of coherent superpositions of electron-hole pairs on top of the Fermi sea.

EMP have been largely investigated in the past by studying the propagation of the time dependent electrical current in the time [3][START_REF] Zhitenev | Time resolved measurements of transport in edge channels[END_REF][START_REF] Ernst | Acoustic Edge Modes of the Degenerate Two-Dimensional Electron Gas Studied by Time-Resolved Magnetotransport Measurements[END_REF][START_REF] Sukhodub | Observation of an Interedge magnetoplasmon mode in a degenerate two-dimensional electron gas[END_REF][START_REF] Kamata | Voltagecontrolled Group Velocity of Edge Magnetoplasmon in the Quantum Hall Regime[END_REF][START_REF] Kumada | Edge magnetoplasmon transport in gated and ungated quantum Hall systems[END_REF] or frequency domain [START_REF] Talyanskii | Spectroscopy of a twodimensional electron gas in the quantum-Hall-effect regime by use of low-frequency edge magnetoplasmons[END_REF][START_REF] Gabelli | Relaxation time of a chiral quantum RL circuit[END_REF][START_REF] Hashisaka | Distributed electrochemical capacitance evidenced in highfrequency admittance measurements on a quantum Hall device[END_REF][START_REF] Delgard | Coulomb interactions and effective quantum inertia of charge carriers in a macroscopic conductor[END_REF] . Experiments have highlighted the dependence of EMP propagation speed on the magnetic field and on the screening by nearby electrostatic gates. All these studies are based on a classical description of charge propagation along the edge channels which can be modeled as transmission lines [START_REF] Hashisaka | Distributed electrochemical capacitance evidenced in highfrequency admittance measurements on a quantum Hall device[END_REF] . However, chiral edge channels have three important differences with respect to standard 50 Ohms coaxial cables. Firstly, the chirality results in the separation between forward and backward propagating waves. Secondly, the speed of the EMP [START_REF] Kumada | Edge magnetoplasmon transport in gated and ungated quantum Hall systems[END_REF] is of the order of 10 5 m.s -1 , three orders of magnitude smaller than the speed of light, resulting in wavelengths in the µm range at GHz frequencies compared to the cm range in standard coaxial cables. EMPs would thus allow for more compact circuits. Finally, their characteristic impedance is of the order of the resistance quantum, R K ≈ 25 kΩ, much larger than the 50 Ohms standard, offering the possibility of a strong coupling to low dimensional quantum conductors of high impedance [START_REF] Parmentier | Strong back-action of a linear circuit on a single electronic quantum channel[END_REF] . These specificities motivated recent theoretical and experimental 1415 studies of EMP transmission lines for efficient coupling to on-chip high impedance quantum devices, such as charge or spin qubits, for the study of Coulomb interaction effects in one-dimensional edge channels [START_REF] Gourmelon | Characterization of helical Luttinger liquids in microwave stepped-impedance edge resonators[END_REF] , or for the realization of on-chip microwave circulators 17 . So far, these studies have focused on the classical regime, where EMP states can be described as coherent states. However, as for other bosonic modes, quantum EMP states can also be generated. In the last years there has been a strong interest for the generation of quantum radiation by quantum conductors [START_REF] Beenakker | Antibunched photons emitted by a quantum point contact out of equilibrium[END_REF][START_REF] Lebedev | Statistics of radiation emitted from a quantum point contact[END_REF][START_REF] Grimsmo | Quantum optics theory of electronic noise in coherent conductors[END_REF] and in particular of squeezed states [START_REF] Mendes | Cavity squeezing by a quantum conductor[END_REF][START_REF] Ferraro | Enhancing photon squeezing one leviton at a time[END_REF][START_REF] Rebora | Suppression of the radiation squeezing in interacting quantum Hall edge channels[END_REF] . So far, it has been limited to the study of low impedance (50 Ohms) transmission lines coupled to superconducting circuits [START_REF] Armour | Universal quantum fluctuations of a cavity mode driven by a josephson junction[END_REF][START_REF] Mendes | Parametric amplification and squeezing with an ac-and dc-voltage biased superconducting junction[END_REF][START_REF] Rolland | Antibunched photons emitted by a dc-biased Josephson junction[END_REF] or tunnel junctions [START_REF] Gasse | Observation of squeezing in the electron quantum shot noise of a tunnel junction[END_REF] . We report here on the generation of squeezed EMP states at the output of a quantum point contact used as an electronic beam-splitter in a GaAs quantum Hall conductor, as discussed in Ref. [23]. Using two-particle interference processes 2 occurring between electron and hole excitations colliding on the splitter, we generate a squeezed EMP vacuum state at frequency f = Ω 2π = 7.75 GHz at the splitter output with a noise minimum 18% below the vacuum fluctuations. The non-linear EMP scattering at the splitter breaks a 2f pump signal into coherent photon pairs, thereby achieving squeezing [START_REF] Yuen | Two-photon coherent states of the radiation field[END_REF] .

Squeezed EMP states could be used for quantum enhanced measurements in EMP interferometers [START_REF] Hiyama | An edgemagnetoplasmon Mach Zehnder interferometer[END_REF] , or to extend the study of low dimensional quantum conductors in the regime where they are driven by quantum voltage sources [START_REF] Souquet | Photon-assisted tunnelling with nonclassical light[END_REF] , exploiting the strong coupling of high impedance transmission lines to high impedance low-dimensional quantum circuits.

In the bosonic description of charge propagation, the charge density ρ(x, t) carried by a single edge channel can be expressed as a function of a chiral bosonic field Φ(x, t) with ρ(x, t) = -e √ π ∂ x Φ(x, t). The relation between the electrical current and the field can then be deduced directly from charge conservation: i(x, t) = e √ π ∂ t Φ(x, t). At low frequency (typically a few GHz), dispersion effects can be neglected, such that Φ(x, t) can be decomposed in terms of elementary plasmon excitations at pulsation ω propagating with constant speed velocity v:

Φ(x, t) = -i √ 4π ω 2π ωT meas [b ω e iω(x/v-t) -h.c.] (1) with [b ω , b † ω ] = δ ω,ω . (2)
b † ω is the operator which creates a single plasmon of energy ω and obeys the usual bosonic commutation relations. The long measurement time T meas sets the discretization of the plasmon modes by steps of 2π/T meas . In order to address the squeezing of EMP modes, it is useful to introduce the quadratures of the bosonic field at a given pulsation Ω defined for a phase 0 ≤ ϕ ≤ π:

X Ω,ϕ = b Ω e iϕ + b † Ω e -iϕ √ 2 (3)
Their fluctuations ∆X 2 Ω,ϕ can be decomposed into an isotropic ∆X 2 Ω,iso and an anisotropic, ϕ dependent, part ∆X 2 Ω,an given by:

∆X 2 Ω,iso = b † Ω b Ω -b † Ω b Ω + 1 2 (4)
∆X 2 Ω,an = ( b 2 Ω -b Ω 2 )e 2iϕ (5)
For classical coherent states,

∆X 2 Ω,ϕ
is isotropic ( ∆X 2 Ω,an = 0) and given by ∆X 2 Ω,ϕ = 1/2 which are called vacuum fluctuations. For squeezed states, the minimum value of the noise, obtained for a certain value ϕ = ϕ 0 of the angle, goes below the vacuum fluctuations, ∆X 2 Ω,ϕ0 < 1/2. As imposed by Heisenberg's uncertainty principle, the orthogonal quadrature then exhibits larger fluctuations: ∆X 2 Ω,ϕ0+π/2 > 1/2. The quadratures of the field and their fluctuations can be experimentally accessed from the measurements of the electrical current i(t) and its fluctuations at high frequency. More precisely, defining i Ω,ϕ (t) = cos(Ωt + ϕ) i(t), one has:

i Ω,ϕ (t) Tmeas = -2e Ω πT meas X Ω,ϕ (6)
S Ω,ϕ = 2 dτ δi Ω,ϕ (t + τ /2)δi Ω,ϕ (t -τ /2) Tmeas (7) = e 2 Ω 2π ∆X 2 Ω,ϕ (8)
where i Ω,ϕ (t) Tmeas denotes the average of i Ω,ϕ (t) over the measurement time T meas . Classical states are thus defined by current fluctuations S Ω,ϕ = e 2 Ω 4π = e 2 f 2 and squeezed states by S Ω,ϕ0 < e 2 f 2 . Experimentally, one measures the noise in excess of the equilibrium fluctuations ∆S Ω,ϕ = S Ω,ϕ -e 2 f 2 and squeezing occurs when ∆S Ω,ϕ0 < 0 31 .

The principle of the experiment is represented in Figure 1.A. A quantum point contact (QPC) is used as a beam-splitter for electronic excitations of transmission probability T (and reflection probability R = 1 -T ). By plugging two electronic sources at inputs 1 and 2 of the QPC, collisions between FIG. 1. A. Principle of the experiment: a QPC is used as an electronic beam-splitter of transmission T for the collision of electron and hole excitations generated at inputs 1 and 2. Input 1 is connected to a dc source, which shifts the chemical potential of the edge channel by -eVdc. Input 2 is connected to an ac sinusoidal source of amplitude Vac and frequency 2f . EMP squeezing is characterized at output 4 by measuring the correlations SΩ,ϕ of the current iΩ,ϕ. For a squeezed vacuum, SΩ,ϕ goes below the vacuum fluctuations for ϕ = 0 and above them for ϕ = π/2. B. Experimental setup: the low frequency noise ∆S0 is measured at output 3 and the high frequency noise ∆SΩ,ϕ at output 4. The EMP current i(t) is weakly transmitted to a Z0 = 50 Ω coaxial cable, amplified using two cryogenic amplifiers in a double balanced configuration [START_REF] Bisognin | Microwave photons emitted by fractionally charged quasiparticles[END_REF][START_REF] Parmentier | A high sensitivity ultralow temperature RF conductance and noise measurement setup[END_REF] and multiplied to the local oscillator V l (t) = V l cos (2πf t + ϕ) using high frequency mixers. The noise power is then integrated using a diode in a 800 MHz bandwidth set by a low pass filter. The noise is measured via a lock-in detection by applying a square modulation at a frequency fm = 234 Hz to the dc voltage Vdc. The resulting output excess noise, ∆S out Ω,ϕ is proportional to ∆SΩ,ϕ with a proportionality factor G(Ω) that needs to be calibrated. electron and hole excitations [START_REF] Liu | Quantum interference in electron collision[END_REF][START_REF] Bocquillon | Coherence and indistinguishability of single electron wavepackets emitted by inde-pendent sources[END_REF] emitted by each source occur at the beam-splitter. Previous implementations of high frequency noise measurements [START_REF] Bisognin | Microwave photons emitted by fractionally charged quasiparticles[END_REF][START_REF] Zakka-Bajjani | Experimental test of the high-frequency quantum shotnoise theory in a quantum point contact[END_REF] have focused on the single source configuration, where the quantum point contact is biased by a dc voltage V dc . These measurements have shown that there is a voltage threshold for the generation of high frequency noise V dc = hf /q which can be used to measure the charge q of the quasiparticles scattered by the splitter [START_REF] Bisognin | Microwave photons emitted by fractionally charged quasiparticles[END_REF] . However, in this configuration, the noise is isotropic and no squeezing is obtained. By adding an ac sinusoidal source at frequency 2f at input 2, EMP squeezing at frequency f can be obtained when both the dc voltage V dc and the ac voltage amplitude V ac are set close to the threshold hf /q. Squeezing is characterized by multiplying the electrical current i(t) at the output of the splitter with the local oscillator cos(Ωt + ϕ). As represented in Figure 1.A., the resulting low-frequency current correlations S Ω,ϕ are expected to go beyond (respectively above) the vacuum fluctuations when the phase of the local oscillator is set to ϕ = 0 (respectively ϕ = π/2). This approach has strong similarities with the one developed by Gasse et al. in Ref. [27] where squeezing is generated in a 50 Ohms coaxial line using a low impedance tunnel junction of resistance R = 70 Ω. As mentioned above, our work is different as it demonstrates squeezing in a high impedance Z ≈ R K EMP transmission line allowing for a strong in situ coupling to mesoscopic circuits.

The experimental setup is represented in Figure 1.B. The conductor is a two-dimensional electron gas in a GaAs/AlGaAs heterostructure, with charge density 1.9 × 10 15 m -2 and mobility µ = 2.4 × 10 6 cm -2 V -1 s -1 at 4 K. A magnetic field B = 2.6 T is applied perpendicularly to the sample in order to reach the quantum Hall effect at filling factor ν = 3 where three edge channels propagate along the edges of the sample. A quantum point contact is used to partition selectively the outer edge channel on which we focus in this study. The transmission is set to T ≈ 0.5 to generate the maximum partition noise. A dc current is generated at input 1 of the quantum point contact and the ac voltage V ac (t) = V ac cos (4πf t) is applied at input 2 with f = 7.75 GHz. The measurement frequency f = 7.75 GHz is chosen such that k B T el hf , where T el is the electronic temperature. The high frequency noise ∆S Ω,ϕ (V dc , V ac ) is measured by weakly transmitting the EMPs propagating at output 4 to a 50 Ohm coaxial line, where the weak coupling is ensured by the strong impedance mismatch between the 50 Ohms of the coaxial cable and the impedance R K /ν of the quantum Hall conductor. The signal is then amplified by a set of two double balanced cryogenic amplifiers [START_REF] Bisognin | Microwave photons emitted by fractionally charged quasiparticles[END_REF][START_REF] Parmentier | A high sensitivity ultralow temperature RF conductance and noise measurement setup[END_REF] . S Ω,ϕ is measured by multiplying the output signal with a local oscillator V l (t) = V l cos (Ωt + ϕ) using high frequency mixers. The local oscillator is locked in phase with the pump V ac (t) and the phase ϕ of the measured quadrature can be continuously varied. S Ω,ϕ is finally measured using a diode which integrates the power at the output of the mixer in a 800 MHz bandwidth set by a low pass filter. The typical resolution required on S Ω,ϕ is smaller than 10 -29 A 2 Hz -1 , which corresponds to the thermal noise generated by a variation of a few tens of microKelvins of a 50 Ohm resistor. This needs to be compared to the base temperature of the fridge (≈ 30 mK) and to the noise temperature of the cryogenic amplifiers (≈ 4 K). In order to mitigate longtime variations of these two noise temperatures which could easily overcome the signal, we use a lock-in detection of the noise by modulating in time the applied dc voltage on input 1 (between V dc and 0) at a frequency f m = 234 Hz. The output excess high frequency noise ∆S out Ω,ϕ (V dc , V ac ) is proportional to ∆S Ω,ϕ (V dc , V ac ) with a proportionality factor G(Ω) that takes into account both the weak coupling to the transmission line and the amplification chain. We calibrate G(Ω) by measuring simultaneously the easily calibrated 41 excess zero frequency noise ∆S 0 (V dc , V ac = 0) (at output 3 of the splitter) and the high frequency noise ∆S Ω,ϕ (V dc , V ac = 0) (at output 4) when the pump is off, V ac = 0. In the absence of the pump, ∆S Ω,ϕ (V dc , 0) is independent of the phase ϕ and is directly related to the excess zero frequency noise shifted by the voltage [START_REF] Roussel | Perturbative fluctuation dissipation relation for nonequilibrium finite-frequency noise in quantum circuits[END_REF][START_REF] Safi | Fluctuation-dissipation relations for strongly correlated out-of-equilibrium circuits[END_REF] ±hf /e:

∆S Ω,ϕ (V dc , 0) = ∆S 0 (V dc + hf /e) + ∆S 0 (V dc -hf /e) 4 , (9)
with ∆S 0 (V dc ± hf /e) = S 0 (V dc ± hf /e) -S 0 (±hf /e).

As can be seen from Eq. ( 9), for high voltages V dc > hf /e, ∆S Ω,ϕ (V dc , 0) varies linearly with the applied voltage with a slope e 3 h RT which can be used for the determination of G(Ω). The excess low frequency noise ∆S 0 (V dc , 0) and the high frequency noise 2∆S Ω,ϕ=0 (V dc , 0) for f = Ω 2π = 7.75 GHz are plotted in Figure 2, where the constant G(Ω) has been adjusted so that ∆S Ω,ϕ (V dc , 0) verifies Eq. ( 9). The black dotted line represents the standard low-frequency shot noise formula ∆S 0 (V dc , 0) = 2 e 3 h RT V dc [coth eVdc 2k B Tel -2k B Tel eVdc ] with T el = 40 mK and the transmission of the beam-splitter is set to T = 0.42. The blue dashed line represents the prediction from Eq.( 9) which agrees well with the data. As expected, the low frequency noise varies linearly for eV dc > k B T el which contrasts with the high frequency noise which is suppressed when V dc is smaller than the voltage threshold hf /e = 32 µV for f = 7.75 GHz. We now turn to the noise measurements performed when the pump is on, with V ac = 33 µV. The output signal is then directly proportional to S Ω,ϕ (V dc , V ac ) -S Ω,ϕ (V dc = 0, V ac ). In order to reconstruct the excess output noise, ∆S Ω,ϕ (V dc , V ac ) it is necessary to independently measure the excess high frequency noise generated by the pump: ∆S Ω,ϕ (V dc = 0, V ac ). This is performed by measuring the excess noise when the pump amplitude is modulated at f m = 234 Hz at zero dc voltage. By summing these two contributions, one obtains ∆S Ω,ϕ (V dc , V ac ) = S Ω,ϕ (V dc , V ac ) -S Ω,ϕ (V dc = 0, V ac ) + ∆S Ω,ϕ (V dc = 0, V ac ). We discuss first the low-frequency measurements, ∆S 0 (V dc , V ac ), represented by the red dots in Figure 2. For V dc = 0, the excess low-frequency noise is set by the partitioning of electron-hole pairs generated by the pump. For V dc > hf /e, we observe ∆S 0 (V dc , V ac ) = ∆S 0 (V dc , 0). This is due to two particle interferences occurring between the electrons emitted by the dc source and the electron-hole pairs generated by the ac pump which fully suppress the partition noise of the pump. Our data agree well with the red dashed line, which is the theoretical prediction for a pump amplitude V ac = 33 µV and T el = 40 mK 41 . The high frequency noise measurements plotted in Figure 3 show a completely different behavior. ∆S Ω,ϕ (V dc , V ac ) depends strongly on the phase ϕ as shown by the strong differences between ϕ = 0 (red points) and ϕ = π/4 (yellow points). ∆S Ω,ϕ=π/4 (V dc , V ac ) resembles the measurement in the absence of the pump with a slight shift upwards corresponding to the partition noise of the pump. In particular, it is symmetric for positive and negative biases V dc . ∆S Ω,ϕ=0 (V dc , V ac ) looks completely different. It is asymmetric as a function of V dc . It is larger for V dc > 0 compared to V dc < 0 and even goes below zero for V dc ≈ -V ac = -33µV. It shows that for this combination of dc and ac voltages, squeezing of the EMP mode at frequency f is obtained. The observed asymmetry can be easily explained. Due to the electronhole symmetry of the pump (the sine excitation is symmetric with respect to positive and negative energies), one has ∆S Ω,ϕ=0 (V dc , V ac ) = ∆S Ω,ϕ=π/2 (-V dc , V ac ). Both quadratures for a given sign of the dc bias can thus be accessed from the measurement of a single quadrature for positive and negative bias. For V dc ≈ -V ac , ∆S Ω,ϕ=0 (V dc , V ac ) is negative; the other quadrature ∆S Ω,ϕ=π/2 (V dc , V ac ) then shows an excess noise compared to the equilibrium situation as expected from Heisenberg uncertainty principle.

Finally, we can convert our noise measurements into the fluctuations ∆X 2 Ω,ϕ using Eq.( 8). We have plotted in Figure 4.A ∆X 2 Ω,ϕ as a function of ϕ for V dc = -33 µV. In order to emphasize the anisotropy of the noise, we have plotted the same data in polar coordinates in Figure 4.B. The classical isotropic fluctuations are represented by the black dashed line and the theoretical predictions by the blue dashed line which agrees very well with our data. As discussed before, a clear squeezing can be observed for ϕ = 0 with an 18% reduction compared to vacuum fluctuations.

To conclude, we have demonstrated squeezing of EMP modes at frequency f = 7.75 GHz by using two-particle interferences in an electronic beam-splitter between a dc and an ac sinusoidal electronic sources. Squeezed EMP states could be used in EMP interferometers for quantum enhanced sensors or in EMP cavities used as quantum buses to transmit quantum states between distant mesoscopic samples. For practical applications, it will be necessary to increase the degree of squeezing which could be achieved following two different ways. Firstly, one can structure the ac drive, replacing its sinusoidal temporal dependence by Lorentzian shaped current pulses [START_REF] Mendes | Cavity squeezing by a quantum conductor[END_REF][START_REF] Ferraro | Enhancing photon squeezing one leviton at a time[END_REF][START_REF] Rebora | Suppression of the radiation squeezing in interacting quantum Hall edge channels[END_REF] . Secondly, the squeezing demonstrated here is based on the non-linear evolution of the high frequency noise with the applied dc bias voltage. Much larger nonlinearities can be obtained using mesoscopic conductors such as ac driven mesoscopic capacitors [START_REF] Fève | An On-Demand Coherent Single Electron Source[END_REF] for much larger squeezing efficiency [START_REF] Mendes | Cavity squeezing by a quantum conductor[END_REF] . The versatility of quantum Hall conductors can also be exploited by coupling non-linear mesoscopic conductors to quantum Hall resonators for parametric amplifica-tion and squeezing. Many building blocks of edge magnetoplasmonics are already available, the present work demonstrates that they can be combined in order to generate quantum EMP states.

Observation of edge magnetoplasmon squeezing in a quantum Hall conductor: supplementary material H. Bartolomei Our experimental measurements of squeezing can be compared with theoretical predictions using scattering theory of photo-assisted electron transport 1 . As shown in Refs. [2] and [3], the current correlations in the time domain at the output of the QPC can be obtained from the input current correlations and from the incoming single electron coherence [START_REF] Zhitenev | Time resolved measurements of transport in edge channels[END_REF] . Going to frequency space, one can then extract the non stationary part, at pulsation 2Ω, of the time dependent current noise spectrum at Ω, thus leading to the anisotropic contribution ∆S Ω,an , whereas the time averaged current noise spectrum at Ω leads to the isotropic contribution ∆S Ω,iso . In the experimental configuration considered here, input 1 is d.c. biased, such that the incoming single electron coherence in input 1 can be expressed as a function of the excess electronic distribution function ∆f µ (Ω) = f µ (ω)f 0 (ω), where f µ (ω) is the Fermi-Dirac distribution for electrons at chemical potential µ = -eV and f 0 (ω) is the Fermi-Dirac distribution at the reference chemical potential µ = 0. Input 2 being a.c. biased, the excess electronic coherence at input 2 can be expressed as a function of the excess electronic Wigner distribution function ∆W (t, Ω) introduced in Ref. [5]. As a result, ∆S Ω,ϕ = ∆S Ω,iso + ∆S Ω,an is given by: ∆S Ω,iso RT e 2 = dΩ 2π ∆W (t, Ω ) t g µ (Ω, Ω ) + ∆f µ (Ω )g 0 (Ω, Ω )

∆S Ω,an RT e 2 = dΩ 2π e 2iΩt ∆W (t, Ω )

t (1 -2f µ (Ω ))e 2iϕ (2)
in which g µ (Ω, Ω ) = (1f µ (Ω -Ω)f µ (Ω + Ω)). In the case where source 2 is a periodic a.c. drive at frequency f , V ac (t), the excess Wigner distribution ∆W (t, ω) can be computed from the photo-assisted transition amplitudes p n [V ac ] defined as:

e -i e t -∞ Vac(t )dt = n p n [V ac ]e -2iπnf t (3)
In the case of a sinusoidal a.c. drive, V ac (t) = V ac cos (2πf t), one has p n [V ac ] = J n (eV ac /(hf )), where J n denotes the Bessel function of order n. The red and yellow dashed lines on Fig. S1 (Fig. 3 of the main manuscript) result from numerical evaluations of Eqs.( 1), ( 2) and (4).

28 FIG. 2 .

 282 FIG. 2. Measurements of ∆S0(Vdc, Vac = 0) (black circles) and ∆SΩ ϕ (Vdc, Vac = 0) (blue points) when the pump is switched off, and measurements of ∆S0(Vdc, Vac) when the pump is switched on with Vac = 33 µV (red points). The black and red dashed lines represent the theoretical predictions for ∆S0 with Vac = 0 (black dashed line) and Vac = 33 µV (red dashed line) with Tel = 40 mK. The blue dashed line is the theoretical prediction for ∆SΩ,ϕ with Vac = 0 and Tel = 30 mK.

6 AFIG. 3 .

 63 FIG. 3. A. Measurements of ∆SΩ,ϕ for ϕ = 0 (red points) and ϕ = π/4 (yellow points) as a function of the dc bias voltage for Vac = 33 µV. The red and yellow dashed lines represent the theoretical predictions for Vac = 33 µV. Best agreement is obtained using Tel = 30 mK slightly lower than Tel = 40 mK which gave the best agreement for low frequency noise measurements. B. Measurements of ∆X 2 Ω,ϕ as a function of ϕ for Vdc = -33 µV. The black dashed line represents the classical fluctuations and the blue dashed line the theory. C. Same data in polar coordinates.

  ∆W (t, ω) = n+,n-p n+ [V ac ]p n-[V ac ] * e 2iπ(n+-n-)f t × f 0 (ωπ(n + + n -)f )f 0 (ω) (4) ∆W (t, ω) t = n |p n [V ac ]| 2 f 0 (ω -2πnf )f 0 (ω)

  1 , R. Bisognin 1 , H. Kamata 1 , J.-M. Berroir 1 , E. Bocquillon 1,2 , G. Ménard 1 , B. Plac ¸ais 1 , A. Cavanna 3 , U. Gennser 3 , Y. Jin 3 , P. Degiovanni 4 , C. Mora 5 , and G. Fève 1 * 1 Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité, Paris, France 2 II. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937 Köln 3 Centre de Nanosciences et de Nanotechnologies (C2N), CNRS, Université Paris-Saclay, 91120 Palaiseau, France. 4 Univ Lyon, Ens de Lyon, Université Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France 5 Université de Paris, Laboratoire Matériaux et Phénomènes Quantiques, CNRS, F-75013 Paris, France.

C. Grenier, R. Hervé, E. Bocquillon, F.D. Parmentier, B. Plac ¸ais, J.M. Berroir, G. Fève, and P. Degiovanni, Single-electron quantum tomography in quantum Hall edge channels, New Journal of Physics 13, 093007 (2011).

5 D. Ferraro, A. Feller, A. Ghibaudo, E. Thibierge, E. Bocquillon, G. Fève, Ch. Grenier, P. Degiovanni, Wigner function approach to single electron coherence in quantum Hall edge channels, Physical Review B 88, 205303 (2013). arXiv:2210.04279v1 [cond-mat.mes-hall] 9 Oct 2022

We thank D. Ferraro and G. Rebora for useful discussions. This work is supported by the ANR grant "Qusig4Qusense", ANR-21-CE47-0012, the project EMPIR 17FUN04 SE-QUOIA, and the French RENATECH network.