Frédéric Beck

Abdelkader Lahmadi

Jérôme Franc

HSL: a Cyber Security Research Facility for Sensitive Data Experiments

Keywords: reproducible experiments, cybersecurity, research facility

In this paper, we detail the design of a cybersecurity facility to carry reproducible and long term research activities in a safe environment, including malware collection and analysis, network telescopes and honeypots, or hosting critical services, without worrying about side effects or loss of data. The facility, aka High Security Lab (HSL), is running since 2010, and is widely used by multiple research groups to carry sensitive data cybersecurity experiments. It includes an evolving infrastructure with tools and processes for building and running long-term and reproducible cyber security experiments. We report on our experience and lessons learned from the design, the setup and the evolution of this facility during 10 years while focusing on major cybersecurity experiments that have been conducted by researchers.

I. INTRODUCTION

Providing efficient and reproducible cybersecurity experiments at the research level requires highly secure and collaborative facilities for data collection, storage and analysis tasks. Quite frequently, research groups make these tasks in adhoc tools, with little support and maintenance which may lead their data to be lost and may effect severely the reproduction of the results, and the guarantee of continuous research activities. As a result of the lack of mature procedure and facilities, the approach taken in practice has usually been to carry experiments with different analysis tools, without common agreement, and to store the data in heterogeneous format, which makes their sharing within a research group or with others a difficult and complicated task, which limits collaborations, and leads to duplicating efforts. Although this approach allows few number of researchers to carry cybersecurity experiments, it is not possible to extend it to wider communities for long term collaborative actions. Most successful experiments are only localized, using specific-purpose testbed, and depend on actions of individuals as opposed to resulting from systematic processes in a controlled facility.

A large body of literature exists regarding the design and the development of cyber security testbeds for research and education. Mirkovic and al [START_REF] Mirkovic | Teaching cybersecurity with deterlab[END_REF] described the DeterLab which is an open experimental facility, based on Emulab technology, allowing users to safely test security threats and defenses, including compromising hosts, or overloading and crashing them. A testbed for cyber physical systems security experiments is described in [START_REF] Pfister | Design and implementation of a cyber physical testbed for security training[END_REF]. This testbed is mainly tailored to cyber defense competitions, allowing participants to test their defensive strategies with real-time feedback using scenarios including both cyber and cyber-physical elements. A more special-purpose testbed, dedicated to Microgrid cyber security is described in [START_REF] Ashok | A multi-level fidelity microgrid testbed model for cybersecurity experimentation[END_REF]. The testbed integrated multiple level of simulation, hardware-in-the-loop, and virtualisation to allow appropriate level of fidelity in attacks measurement and defense against the physical system. More testbeds and facilities are also developed by networking communities in particular for network measurement. A major project in this area is CAIDA [START_REF]Caida project[END_REF], which provides a research infrastructure and a collaborative environment for large-scale data collection, curation and sharing. The project operates active and passive measurement infrastructures to provide data for Internet mapping and performance measurement at different locations, and also multiple software tools for monitoring Internet security and stability.

Compared to existing testbeds, our research facility described in this paper is not designed to support a specific purpose testbed or deploying active or passive monitoring probes at different locations. However, it is well suited to support a broad class of cybersecurity experiments, by providing to researchers tools and a safe environment to carry their experiments, without compromising the security of the handled data, and also without introducing any risk on the hosting institution. Multiple experiments are carried by researchers on our facility, including malware collection and analysis, network telescopes for monitoring Internet background radiation, honeypots for collecting attack traces, hosting sensitive systems just to name a few. The main contributions of this paper are as follows:

• We detail the design and the technical features of our research facility for sensitive data experiments. • We describe major experiments that are carried on our facility, while focusing on its benefits for running them. • We report on the lessons learned from researcher experience, participants and resources perspectives.

The rest of the paper is organized as follows. In section II, we describe the design and the architecture of the HSL facility while providing technical details of its setup and deployment in our hosting institution. In section III, we detail major experiments on sensitive data that have been conducted through our facility. Finally, in section IV, we highlight the lessons learned while conducting and supporting such experiments.

II. DESIGN AND IMPLEMENTATION

The High Security Laboratory (HSL) is designed to act as a central hub for the technical aspects of cyber-security experiments where sensitive data is stored, and their analysis is performed without compromising their security. In the next sections, we present the architecture and objectives of the HSL infrastructure.

A. Overall Architecture

The HSL is designed to host sensitive data research activities in order to collect and store data while ensuring their confidentiality and integrity, both logically and physically. It is a cyber-security oriented datacenter and testbed to offer a safe and persistent environment for researchers to carry their experiments. Its overall architecture is shown in Fig. 1. The facility is composed of around 100 servers, organized in per-project clusters and isolated zones, deployed incrementally based on the needs of the hosted projects while anticipating future requirements and experiments. At the time of writing this paper, the HSL provides, overall, 3K cores (8 to 40 cores per server), 7TB+ memory (32 to 128 GB memory per server), 550 TB of disk space (1 to 20 TB disk space per server) including a 72TB backup storage unit. Different technical solutions are deployed based on the running projects or their usage, with the goal to maximize the utilisation of the hardware servers by relying on VmWare/Xen virtualization, ELK cluster, Docker containers, or dedicated servers for GPU computation. This architecture allows to address the 3 main roles and objectives of the infrastructure:

• realtime data collection and cybersecurity datasets provider for the research community, including historical data over several consecutive years; • large scale and Internet wide experimentation;

• secure hosting and storage of sensitive data or services. The platform is running with a limited administration staff, on average two part-time permanent persons, since our main focus is to guarantee confidentiality rather than high availability. Indeed, we rely on many automated tools, for servers and services deployment and configuration (Puppet), automatic backup (Bacula), constant monitoring (Icinga) and alerting via various channels (Mail, Mattermost chat via webhooks), using redundancy wherever it is required and possible. If this does not ensure 24/7 availability, it allows us to get back online quickly after a crash, usually a few minutes for a server, and between 30 minutes to 1 hour for the whole infrastructure.

B. Network Connectivity and Isolation

The HSL relies on trust zones, dedicated and isolated environments with limited and controlled interactions with the Internet. Each environment benefits from all the services offered by the HSL (network and data protection, automatic backup, local services package mirrors, DNS, LDAP, NTP) while being always separated from the outside world by two levels of security from different constructors/technologies (two firewalls from different constructors for the logical aspects, two different biometric authentication mechanisms for the physical ones), as shown in Fig. 1. Such trust zones are deployed for each hosted project, including its own network and VLAN to ensure that it is isolated from other hosted projects, but also its user accounts and groups in the HSL LDAP directory, its associated firewalling and users/groups access lists policies (ACLs). Access to such a trust-zone is possible via dedicated VPN networks, including dedicated gateways. These zones are fully integrated to the automatic configuration and software management solution (puppet) deployed in the infrastructure. The access to a trust zone is possible through a dedicated VPN, deployed exclusively for each project, and limited to the user accounts linked to the projects LDAP groups. The interconnection between and within these trust zones is ensured by a private 1Gb network.

Internet access is granted via various network connections. For daily usage, we have a dedicated /24 network (256 public IP address) with our own BGP routes and DNS zone registered globally, provided by a national provider for Education and Research communities. For specific experiments, and to have a point of comparison with a customer grade contract, we deployed also non-professional Fiber/ADSL connections. The most important requirement is to have a complete control over the security rules and policies enforced on these connections, so that we can have a complete control on our experiments. No direct access to or from the Internet is allowed in a trust zone. We limit the interactions with the outside world, by enforcing Internet access connections to go through proxies and reverse proxies, and thus limiting the attack surface. The only exception to this policy is the network telescope that we are hosting and presented in section III-A, which uses a dedicated subnet and a physical network in our front-end firewall. This whole /28 subnet is directly connected to the Internet with no firewall or any other security rules applied to the incoming traffic, and is white listed in our network provider's system to avoid false positive alerts. Outgoing traffic is monitored to detect compromission, but all the hosted services are emulated, and the servers hosting these probes are hardened to avoid compromission.

Moreover, having our own network connections and BGP routes, independent from any other connections of our affiliated research institutions, and for which we are clearly identified as a source or destination of the traffic, is a mandatory feature to perform sensitive experiments. This means that if one of our experiment results in our IP range being blacklisted, or if we suffer from an attack in retaliation of a publication, for example a Distributed Denial of Service (DDoS) after the publication by a research group of an experiment for taking down a botnet [START_REF] Calvet | The case for in-the-lab botnet experimentation: creating and taking down a 3000-node botnet[END_REF], we can easily isolate the network and take the appropriate measures without impacting the hosting research center.

C. Data collection and storage

Due to the constraints in terms of physical security, we are focusing on data confidentiality and integrity and not high availability. Our main concern is to ensure that data are not leaked or lost/corrupted, whatever the situations (attack, power outage). Each server in the HSL embeds a RAID controller with its own battery to avoid filesystem corruption, i.e. all transactions end correctly. If a crash occurs, only the system disk is impacted. We are using puppet to automate tools and services installation and configuration. Crash recovery consists in reinstalling a base system and attaching it to puppet, which then ensures the server is back up quickly with minimal interaction and delay. When possible, we rely on data replication, in particular for the Elasticsearch-Logstash-Kibana (ELK) cluster that we use to store collected data (see section II-D), where each data index is separated in 2 shards and each shard is replicated twice.

To avoid data loss, an automatic backup solution based on Bacula has been deployed. On each server, placing data (or symbolic links) in a fixed directory will ensure the data will be backed up. We have tailored backup recipes for specific hosts and services, such as our DNS master, the Puppet server, and other tools deployed (see section ). All raw data from our different probes and non reproductible datasets are backed up as well. Every day we perform incremental backups, differential backups every Friday, full backups on the first Friday of every month. Backup are kept for 365 days. These backups are stored on a dedicated storage unit, composed of twelve 8TB disks in RAID 5 with 2 hot spares, for a global available storage space of 72TB. At the moment, no external backup is performed, all data is backed up locally, as we have no other facility with the same security level available to replicate the data. Manual backup operations of critical data are performed on removable storage units or off-site servers, and rely on encryption to protect the data in these less secure environments. This task has to be performed by each hosted project and is highly recommended.

D. Tools

In order to limit as much as possible the interactions with the outside world, while allowing researchers to work on their data without compromising them outside the HSL, we deployed several tools to make sure they can perform data analysis within the HSL.

A complete CI/CD toolchain based on Gitlab has been deployed, allowing researchers to perform project planning and source code management. This allows data scientists and researchers to design, implement, and deploy all tools, software and services, including public services or dashboard for dissemination of their results, without exposing sensitive data to the outside world. A Mattermost messaging solution, allowing incoming webhooks, is deployed alongside this forge, and allows monitoring of the DevOps platform and tasks, and secure communication between team members.

By enforcing strict servers and services configuration with an automatic solution such as Puppet, we make sure everything is up and configured as it should. Many other services have been deployed to allow time synchronisation via NTP, packets installation via APT, authentication via LDAP... without interacting with the Internet. Accessing public software repositories such as Github or Python / Node modules is still possible via our proxies.

The monitoring and supervision of the whole HSL infrastructure and services is performed via Icinga which is a Nagios clone. Basic metrics and indicators are checked periodically on each server, and specific checks have been implemented and deployed for core services, and all automatic tasks. Any user or project can request their custom checks, which can then alert them via email or webhook.

As presented in section III-A, one of the main roles of the HSL is to collect realtime information and act as a data provider for researchers. Huge amounts of data are collected every day, and are enriched automatically before being indexed in the ELK cluster. This allows to quickly explore the available datasets, quickly generate basic statistics, and build visualization and dashboard that can be made publicly accessible.

III. EXPERIMENT EXAMPLES

Each experimentation and project hosted in the HSL has to receive the validation of an ethical committee, composed of legal, scientific and technical experts from our research instituions and national security agency. In this section, we describe major studies and projects carried by researchers and hosted in the HSL over the past years.

A. Cyber Threat Intelligence

Monitoring and predicting cyber-threats at the scale of Internet is a core research topic at HSL. Several types of security sensors are hosted:

• Honeypots have been deployed since 2008. Indeed, several versions and updates have been applied over years.

Our solution relies on Modern Honey Network 1 to manage low-interaction honeypots variants from the state-ofthe-art to mimic several types of services including SSH, SQL, HTTP, SIP, etc. Nowadays, we collect around 100K hits per day, mostly on SSH, SQL and SIP honeypots. • A network telescope, (darknet or Internet black hole) is a /20 network containing 4096 IP adresses to silently collect incoming traffic. This sensor has been started in 2014 with a daily rate of 4M events at that date while our current daily average is around 18M.

• Blacklists: we aggregate information about blacklisted IP addresses from several public sources such as DNS-BH2 , SSLBL 3 , FireHOL4 , etc. The darknet sensor is currently the main source of data to extract important knowledge about ongoing and future threats. However, most of data are very similar: 95% are TCP SYN packets (mostly SYN scans) and 50% target TCP port 23 (telnet) illustrating attackers looking for telnet accessible hosts such as embedded devices (e.g. IoT). Although these data sounds very usual and not revealing anything new, researchers demonstrated through several studies that relevant knowledge can be gathered. In [START_REF] Coudriau | Topological Analysis and Visualisation of Network Monitoring Data: Darknet case study[END_REF], Topological Data Analysis (TDA) was leveraged by authors to cluster data (unsupervised machine learning). The main idea was to aggregate microscopic events (incoming packets) into large events (a DDoS attack, a syn scan, etc) to make more understandable data. They show that this technique outperforms the Suricata IDS (Intrusion Detection System) to label our data. Labelling collected data is actually a challenging but necessary tasks when operating a darknet in order to extract insights and trends about ongoing attacks in Internet. However, correlating darknet with blacklists is not satisfactory because less than one percent of IP addresses can be identified. Looking at large scan campaigns reveals new trends in attack targets and strategies. In [START_REF] Lagraa | Knowledge Discovery of Port Scans from Darknet[END_REF], the authors modeled sequence of scans in a behavioral graph and highlighted that some specific environments, such as medical environment, were targeted. The authors in [START_REF] Evrard | Attacker Behavior-Based Metric for Security Monitoring Applied to Darknet Analysis[END_REF] observe smarter scans as shown in Figure 2. In this figure the inter-probing time by a scanner (identified by its IP address) is reported and, as seen, most of scans are quite slow and stealthy. In addition, the authors derive a semantic similarity between TCP port numbers that can be integrated into any algorithm where TCP ports must be compared. For example, this metric has been used to proactively block TCP scans in another network. Therefore, this result shows that knowledge inferred from darknet data can be successfully transferred into another domain where an analysis is performed. Although giving full and public access to darknet data is not possible, the researchers decided to make the inferred metric, and its monthly and and weekly updates, publicly available 5 .

B. Malware collection and analysis

The highly secured infrastructure of the HSL allows to safely experiment well known cybercriminals techniques. Indeed, researchers have been able to reproduce in-lab a whole attack chain, including the phases of victim identification, vulnerabilities discovery, the implementation of infection tools and finally the trap that initiates the attack. Technically, they operate a server specialized in the choice of the exploit to be delivered to the victim, a malicious web server that redirects requests to the exploit kit server, and finally several vulnerable workstations that model the victims' machines. These targets are attacked through the victim's browser with a simple click by exploiting a dropper. This highlights the limitations of defense tools which, despite commercial announcements, are still unable to deal with this kind of threat, namely fileless attacks, where everything happens in memory.

Another aspect concerns the collection of malware for scientific analysis, in order to understand the techniques used by the attackers so that researchers can define new defense mechanisms. Since 2010, we hare able to collect over the years a considerable amount of malicious samples, consisting of 35 million files to date. This diversity of malicious code shows the increasing sophistication of malware diversion or obfuscation techniques. To cope with them, researchers developed new approaches to address issues such as binary code decompilation [START_REF] Bonfante | CoDisasm: Medium Scale Concatic Disassembly of Self-Modifying Binaries with Overlapping Instructions[END_REF], de-obfuscation, anti-analysis, similarity checking [START_REF] Bonfante | Code synchronization by morphological analysis[END_REF] and detection [START_REF] Bonfante | Gorille sniffs code similarities, the case study of Qwerty versus Regin[END_REF].

Recently, in the scope of a collaboration with a national operational security center (SOC), researchers were able to recover the encryption key used by the SaveTheQueen ransomware without the need for a thorough and complex analysis of the malicious binary. This was largely achieved using a tool developed by the authors, hosted and executed in HSL to track and document all system calls generated by the audited binary.Despite the fact that this one is highly protected (use of the shellcode2exe and ConfuserEx obfuscators, injection into the winlogon process, self-modifying code using the Donut tool), it was possible to follow the handling of the encryption keys, particularly when importing them into the secure manager provided in Microsoft's Windows environment.

C. Secure hosting

We are also hosting tools and systems that operate on private data collected through experiments or by running services. 1) Voting system: A research group is developing Belenios [START_REF] Cortier | Belenios: a simple private and verifiable electronic voting system[END_REF] which aims at providing an easy to use voting system, guaranteeing state-of-the-art security, namely vote privacy and verifiability. It can be used in many types of elections and referendums, ranging from scientific councils to sport associations. Belenios guarantees vote privacy and full verifiability, even against a compromised voting server, as no one can learn the vote of a voter, since the votes are encrypted and the decryption key is split and shared among several parties. Moreover, it provides end-to-end verifiability. Every voter can check that his vote has been counted, and only eligible voters may vote. This feature relies on the fact that the ballot box is public (voters can check that their ballots have been received), and that the tally is publicly verifiable (anyone can recount the votes). Finally, ballots are signed by the voter credential (only eligible voters are able vote). Belenios is composed of a voting protocol, which has been formally proven, and a voting platform hosted in the HSL, in order to ensure the security of the voting server, and minimize the risks of compromission, while building user confidence in the platform. In 2019, the voting platform has registered a total of 6 340 of ballots with a total of 20 444 users, for a total of 172 elections, 22 of them having more than 200 voters.

2) Internet access observatory: In the Betternet project 6 , researchers intend to build and deliver a scientific and technical collaborative observatory to measure and improve the Internet service access as perceived by users. Since users now are always connected and access different services through different networks and devices at different locations, the project propose new original user-centered measurement methods that rely on social sciences to better understand Internet usage, and the quality of services and networks. The whole project's infrastructure is hosted in the HSL, and the experiments are built around the advantages and functionalities offered by the platform. To measure end-to-end Quality of user Experience (QoE), researchers rely on crowdsourcing via the ApiSense 7and Acqua [START_REF] Belmoukadam | ACQUA: A user friendly platform for lightweight network monitoring and QoE forecasting[END_REF] mobile applications deployed on end-users' smartphones. Relying on the HSL infrastructure, and the guarantees it offers regarding data confidentiality, is a key argument to convince users to share their data with the researchers, as well as the ethical committee who validates and authorizes the experimentations. In addition to measuring end-to-end QoE, they have to diagnose the sources of degraded experience and understand how users deal with this degradation. The HSL infrastructure acts as a ground truth reference during these experiments, in order to have reliable and repeatable measurements, on which researchers can apply controlled perturbations to validate the different detection algorithms implemented.

IV. LESSONS LEARNED

We successfully used the HSL facility for many years while researchers have conducted multiple projects and experiments in a safe and controlled environment. Below, we provide an overview of lessons learned from different perspectives.

A. Researcher experience

One of the most useful features for researcher while running their experiments is long-term data collection, i.e. keeping the historical data and collecting datasets over long time period. For example, when analyzing network traffic, in particular with machine learning methods, datasets preparation and annotation is one of the most important tasks. It is mandatory to classify the traffic, mostly the IP addresses, in different categories. When capturing data in realtime, or near realtime, it is very difficult to predict what information will be useful, and how it should be enriched or annotated. We tend to store raw data in PCAP format, and annotate the packets later on, by using public resources or databases such as blacklists of IPs known to be part of malicious activities. But due to the dynamicity of the Internet, if this processing is performed even only a few days after the capture, it can result in a biased and erroneous dataset. If we keep historical data, in particular the history of status changes for IPs in the blacklists, we can annotate each packet with the information relative to the time it took place, and not relative to the time the enrichment process is performed, resulting in a more accurate dataset. In addition, when evaluating detection and classification algorithms, we rely on the literature and news to create datasets of traffic annotated with known attacks. To create these datasets used for validation, we can either create them via simulation or in-lab experimentations, but these datasets end up being biased and not fully representative of the real world. By keeping historical data, we can create datasets from real network traffic, and thus have better data quality.

Another main objective of the infrastructure is to conduct experiments at large scale in a controlled environment. Most of them are sensitive and potentially disturbing, thus requiring the oversight of an ethical committee. Working in a controlled environment, where we can monitor all components of the infrastructure and running experiments, detect potentially harmful behavior and act in realtime is a key feature to obtain the validation of the committee for the experimentation.

Reactivity is a key factor in cybersecurity, in particular when a malware is discovered or a new attack is running in Internet. Not only is it important to ensure that our experimentations do not have harmful consequences, but also being able to deploy as quickly as possible probes or services to conduct such an experiment is a major concern. Relying on the HSL, and the fact that the security, monitoring and control of all components is ensured at the platform level, greatly reduces the time required to become operational, as the majority of the required building blocks are already available.

B. Managing participants and resources

Data confidentiality is the objective of the infrastructure, allowing to work on sensitive data without compromising their security. This is a key argument when building partnership and cooperations with other organisations, universities or enterprises, as it helps to build trust and confidence. More precisely, as the HSL systematically relies on network isolation and access lists (ACLs) fine tuned at multiple levels (VPN, FW, LDAP, filesystem), we can control resources and data access with small granularity, and mutualize servers amongst several less sensitive projects from the same research group while ensuring confidentiality and most importantly integrity of data and processes. In addition, having a full control over the infrastructure, including user accounts and VPN profiles, ensures maximum reactivity when granting or revoking accesses. But this can also be a double-edged sword, as projects often require access to a trust zone for their interns or visitors, but very rarely notify when these accounts are not needed anymore, resulting in ghost accounts and VPN profiles that represent a security risk for the data of these projects. It is preferable to rely on expiration dates for accounts validity and maintain up-to-date a users database or listing to make sure accounts or permissions that are not required anymore are suspended, and VPN profiles revoked.

Applicative accounts are usually quite difficult to obtain, as policies usually involve a restriction to nominative accounts. The HSL's secure infrastructure and fined tuned ACLs allows us to offer this functionality, which can be very useful to deploy applications or probes at large scale under the same LDAP account but with different authentication keys for example. This allows us to control both individually and globally distributed resources.

We also recommend constant monitoring of hosts, systems and services, especially disk space. Our golden rule is to never trust users, but we usually need to give them more freedom and rights than we would like to, so that they can work freely in their designated areas. Relying on an efficient and constant monitoring and configuration is the key to strike a balance between permissibility and restrictions.

C. Takeaways and future work

We managed to operate and run for over 10 years the platform while relying on a reduced staff (less than one person full time). Using automated configuration and deployment solution, with double security for data (backup, raid with battery, redundancy when possible), allow us to operate and maintain the platform with minimal crew, and with few manual operations. This permits to have an efficient disaster recovery procedure allowing us to go back online quickly after a crash, and provide the services needed by the hosted projects with the required reactivity. So far, no critical data loss or compromission have been registered, despite the many projects the HSL hosted over the years.

Our platform is constantly updated to meet and plan one step ahead the requirements of researchers. In the near future, we plan several upgrades to cope with the technological needs of hosted projects, including internal connectivity upgrade to a 10GB network to allow better throughput between nodes in distributed environments or clusters. We would like also to add per node electrical monitoring and control, allowing us to monitor more precisely the electrical consumption and minimize it by remotely stopping unused nodes and starting them at-demand. In order to increase even more the security of the hosted data, we want to offer the possibility to research projects to rely on cryptography in an automated way, easing the deployment and usage of external backups. We also want to open the infrastructure to enterprises, either via different online services in SaaS mode, or via the open data initiative by deploying a dataset repository.

Fig. 1 .

 1 Fig. 1. HSL networking and hosting architecture.

Fig. 2 .

 2 Fig. 2. Scans observed by the darknet, January -September 2015

https://github.com/pwnlandia/mhn

http://dns-bh.sagadc.org/

https://sslbl.abuse.ch/blacklist/

http://iplists.firehol.org/

http://port2dist.lhs.inria.fr/

https://project.inria.fr/betternet/

https://apisense.io/

ACKNOWLEDGMENTS

The authors wish to thank the team members of Belenios and Carbone for their valuable feedback on this paper. This work was supported in part by the CPER funding, and the ThreatPredict project funded by NATO.


Jean-Gabriel Ganascia

Ethics and Scientific Integrity at the Time of COVID-19

As a preliminary, let us recall, before addressing the questions of ethics and scientific integrity raised by the COVID-19, the astonishment which seized the whole of the French population at beginning of confinement, to the point that we all remained deaf to the warnings issued by our friends in Lombardy, quarantined since February 21, then subjected to confinement which was gradually reinforced the following week, before being extended across Italy, on March 8.

Scientific integrity

From the beginning of the pandemic, politicians have resorted to a large number of licensed scientists, epidemiologists, data modellers, biologists, doctors, sociologists, etc., to put realistic action plans in place. Taking advantage of the disarray in the population, some charismatic personalities have explained that, in an emergency, we could, or in the name of pragmatism, even had to free ourselves from the rules of research ethics and the usual validation procedures in sciences, to be guided by the intuition of those in the field. Undoubtedly, the need to move as quickly as possible, without letting oneself be slowed down by ancillary considerations, would explain why certain rules are waived in exceptional circumstances. At the same time, the unconditional search for truth among scientists and concern for ethics would prohibit this. In other words, ethics, validation procedures and scientific integrity are not a luxury of aesthetes that one could bypass in a situation as critical as that caused by the COVID-19 pandemic.

Needless to say, a science which does not respect the required criteria of rigour would lead to erroneous results which would risk misleading decision-makers and deceiving the population by masking the extent and nature of the danger. So, the communication around the therapeutic effects of certain medical treatments, without tangible scientific proof, has given hope for the existence of a miracle cure to a certain number of people who put themselves in danger to acquire it, and might have confused politicians. Note, here too, that the misunderstanding was based on semantics, in this case the confusion between observation and experimentation. To avoid such pitfalls and obtain tangible results, it is necessary to carry out experiments, comply with the rules and submit to peer review procedures which guarantee the reliability of the scientific approach. Contrary to what some might claim, this does not necessarily slow down the dissemination of science, because it is always possible to deposit preprints on open archives and distribute them to the entire scientific community. What is more, in a period of crisis, we can try to shorten the evaluation times for scientific articles somewhat in order to speed up their publication, on condition that the same rigorous requirement is always maintained.

Ethical experimentation remains an imperative

Let us now come to the many ethical questions raised by scientific activity. The ethics of experimentation, especially human experimentation, remains an imperative no matter what, even in a health crisis. The cardinal principles of bioethics -beneficence and non-maleficence, autonomy of the person and justice -introduced in the Belmont report in 1979, cannot be derogated from. No treatment can be given that will not improve health. At the same time, you should always make sure that the drugs do not have harmful side effects and, where this is the case, that the benefit outweighs these. The point of clinical experimentation, with its different phases (preclinical phase including in vitro and animal experiments; phase 1 to verify tolerance and the absence of adverse effects on healthy volunteers; phase 2, on patients, to determine the minimum effective dose; phase 3 of comparative study; and finally phase 4 of long-term follow-up of side effects), is to provide assurance. In exceptional situations such as the AIDS epidemic, it is the case that some of these phases have been bypassed. For instance, antiretrovirals not authorised in France have been administered for so-called compassionate use when there was no alternative. But, this can only be done on an individual basis and cannot be generalised to the entire population.

In addition, this perfectly legitimate departure from the application of the main principles of clinical experimentation does not authorise any breach of these principles, in particular that one experiences in vulnerable populations, as was the case with United States with the study of the University of Tuskegee intended to better understand the evolution of syphilis when left untreated -a study which was carried out on impoverished African-American men. Regardless of the cost-benefit trade-off, in the name of personal autonomy, informed consent should be sought from patients included in a clinical experiment to ensure that they are not used as mere instruments. Here again, the crisis does not justify ignoring this imperative. Finally, by virtue of the principle of social justice, resources must be made equally available to the entire population, without discrimination of any kind.

The need to confront the challenge of rationing

This equality of access to care, however, comes up against a reality check in the event of scarcity: what to do when there are not enough masks, hydroalcoholic gel, respirators, intensive care beds... so that everyone may benefit? And, even if there were enough of them, it would still be necessary, for operational reasons, to stagger access to these resources. To tackle this in a rational way, it is necessary to clarify the selection criteria, which is extremely delicate, even painful, but this alone makes it possible to maintain control. Not to do so would be to agree to allow oneself to be governed by arbitrariness, in which case ethical considerations cannot be resolved. The establishment of such criteria must be based on clear arguments based not only on ethical principles, but also on scientific knowledge of the risks, the chances of survival, the spread of the epidemic, etc. At the same time, these criteria must be applied with discernment, being able, sometimes, in borderline cases, to break certain overly rigid rules.

Surveillance of contagious patients

Very early on, the COVID-19 scientific council 1 questioned the strategy to be adopted at the end of the confinement to avoid a resurgence of the epidemic. Drawing on successful experiences in Southeast Asian countries, particularly China, South Korea and Singapore, the council worked on strategies to be adopted. In addition to maintaining the nowfamiliar physical distancing measures, it advocated a massive programme of testing, quarantining carriers of the virus and, above all, identifying people with whom those identified as contagious might have been in contact.

A peculiar feature of this disease, where some people with the virus show few symptoms (so-called 'paucisymptomatic') or are even asymptomatic for several days before the disease presents itself, if it does, reinforces the need for tracing contacts, who need to be contacted, tested and, where necessary, isolated in order to prevent as far as possible the spread of the virus.

It was decided to use general practitioners to identify contacts with patients along with medical auxiliaries grouped together in bodies called brigades, to question the contacts, encourage them to test themselves and, in the event of them presenting obvious symptoms, to direct them to seek treatment. Finally, taking a model from what has been done in Asia, the scientific committee suggested that the government use cell phones to automatically trace each person's contacts.

Digital tracing

As soon as the idea of such digital devices was put forward, many were exercised by this, fearing its potentially intrusive nature and the consequences for privacy, especially for freedom of assembly, since our exchanges would be traced. The French scientists responsible for thinking about their design, in particular those from Inria, wondered how this could remain compatible with our values and our laws. A number of choices were therefore made.

-Lack of localisation, in particular, no use of GPS, or even telephone network cells.

-Automatic tracking of contacts between mobile phones with so-called Bluetooth waves designed for very short distance communications (less than 5 meters) between connected devices, watches, headphones, etc. -Storage of contacts in a completely anonymous way, using cryptographic techniques. -When a person, after testing, is found to be carrying the virus, the anonymous texting of all those who have been in contact with this person, and therefore likely to have been infected, with a message directing them to get tested and, if they have symptoms, to contact a doctor.

It should be noted that provided they are not hacked, these electronic devices do more to protect privacy than the brigades of medical auxiliaries mentioned above. Indeed, in the case of digital applications, the identity of sick people is never transmitted explicitly, while the human approach requires knowing the contagious people likely to be the source of contaminations, registering their identity on a personal database and asking a member of the brigades, who is not a doctor, and therefore not bound by the Hippocratic Oath, to question them. It therefore seems very strange that these applications should have aroused such a strong fear in Western opinions, in particular in French opinions, where many evoke, with an inappropriate reference to the work of Michel Foucault, the "biopolitical" strategy of a state in the process of becoming a police state.

Conclusion

COVID-19 undermines democracies in the face of authoritarian regimes that will perhaps emerge as the big winners of the crisis, both economically and politically. If we want parliamentary regimes to continue to claim their values in the face of ever more voracious dictatorships, it is absolutely necessary to use the right words to discuss, exchange and debate, in order to take advantage of the collective emulation that is our strength. In this regard, if we need worry about the possible misuses of technology and to guard against them, assert that the digital tracking systems envisaged today in France are starting a turn towards a totalitarian regime and a police state is a matter of impropriety in the use of words which only introduces very damaging confusions. The same is true of confusions between pathogen, here the coronavirus, and COVID-19, the disease, or between observation and experimentation.

To use Brice Parain's well-known quote -often wrongly attributed to his friend Albert Camus -'To name an object incorrectly is to add to the unhappiness of this world' 2 . We already have enough woe in this world today without adding more.


J Bernu

S Deléglise

C Sayrin

S Kuhr

I Dotsenko

M Brune
email: brune@lkb.ens.fr

J M Raimond

S Haroche

Freezing a Coherent Field Growth in a Cavity by Quantum Zeno Effect

Keywords: numbers: 03.65.Ta, 03.65.Xp, 42.50.Pq

We have frozen the coherent evolution of a field in a cavity by repeated measurements of its photon number. We use circular Rydberg atoms dispersively coupled to the cavity mode for an absorption-free photon counting. These measurements inhibit the growth of a field injected in the cavity by a classical source. This manifestation of the Quantum Zeno effect illustrates the back action of the photon number determination onto the field phase. The residual growth of the field can be seen as a random walk of its amplitude in the two-dimensional phase space. This experiment sheds light onto the measurement process and opens perspectives for active quantum feedback.

The 'Quantum Zeno' effect is a spectacular manifestation of the measurement back action. The coherent evolution of a quantum system is inhibited by repeated projective measurements performed at short time intervals. This effect, theoretically discussed in [1,2,3], has been observed on the evolution of two-level systems such as molecules [4], trapped ions [5], cold atoms [6] as well as on polarization of light [7]. We describe here the observation of the Zeno effect on a harmonic oscillator. The build-up of a coherent field in a high-Q superconducting cavity coupled to a classical source is inhibited by watching repeatedly the photon number. The measurements use circular Rydberg atoms as non-destructive probes [8,9]. The system free evolution is here a run-away coherent classical process, instead of the two-level quantum dynamics considered in [4,5,6,7]. When watching the system, we record single trajectories and the field state evolution is obtained by a statistical analyzis of these data. This study sheds light on the relationship between the Zeno effect and the measurement back action.

We first revisit the basics of the Zeno effect in the context of our experiment. A classical source is resonantly coupled to the cavity. The evolution due to the source-cavity coupling during a time T is described by the displacement operator D(α) = exp(αa † -α * a) [START_REF] Haroche | Exploring the Quantum[END_REF], where α = λT , a (a † ) are the photon annihilation (creation) operators and λ is the complex amplitude of the source. At this stage, we neglect relaxation, assuming T ≪ T c , where T c is the cavity damping time. The initially empty cavity contains at time T the coherent state |α = D(α)|0 (|0 is the vacuum). The average photon number, n = |α| 2 = |λ| 2 T 2 , runs away quadratically with T . This evolution can be split into small steps of duration ∆t ≪ T . The final displacement results from the coherent accumulation of successive injection pulses, each being described by the translation operator D(λ∆t).

We now watch the field evolution. In order to separate the field probing from its coupling with the source, we alternate measurements with injection pulses of duration ∆t, such that |λ|∆t ≪ 1. After the first pulse, the cavity contains a coherent state |λ∆t ≈ |0 + λ∆t|1 (|1 is the one-photon state). With a probability p 0 ≈ 1 -(|λ|∆t) 2 , the first measurement projects this state back onto |0 . After N iterations of the injection/measurement sequence, the field is left in |0 with a probability

p N 0 ≈ [1 -(|λ|∆t) 2 ] N ≈ 1 -|λ| 2
T ∆t, where T = N ∆t is the total injection time (we assume that ∆t is chosen small enough so that |λ| 2 T ∆t ≪ 1). The average photon number, n = |λ| 2 T ∆t, grows linearly with T . This is strikingly different from the quadratic growth obtained without measurements. The final mean photon number is smaller than |λ| 2 T 2 , the value reached without measurements. The corresponding reduction factor, T /∆t, can be made arbitrarily large. At the limit of infinitely many measurements (∆t → 0 and N → ∞ with T constant), the field remains in |0 [11].

An interesting insight into the Zeno effect on light is provided by considering the back action of the measurement on the field phase. Each photon number determination erases phase information, randomizing at each step the field phase. The field amplitude undergoes a two-dimensional random walk, with a step size |λ|∆t, instead of a deterministic addition of identical displacements along a fixed direction. The final r.m.s. amplitude, √ N |λ|∆t, corresponding to n = |λ| 2 T ∆t, coincides with the prediction based on the projection postulate.

In the analyzis of the Zeno effect, it is essential that the probability for finding the system in its initial state has a quadratic evolution at short times. There is no Zeno effect when the cavity is coupled to an environment with a short memory time τ c , inducing an evolution with probabilities varying linearly with time. Assume for instance that the cavity, prepared in vacuum, relaxes within the damping time T c towards a thermal equilibrium with a blackbody mean photon number n b [9]. The probability for finding the cavity in |0 after

an initial time interval ∆t ≫ τ c is p 0 ≈ 1 -(n b /T c )∆t, provided n b ∆t/T c ≪ 1.
The probability for being in |0 after N steps of length ∆t, each followed by a measurement, is then

p N 0 ≈ [1 -(n b /T c )∆t] N ≈ 1 -n b T /T c (with n b T /T c ≪ 1)
. The field energy grows linearly at short times, with the same rate as without measurements.

Similarly, the exponential damping of a field stored in the cavity leads to state probabilities varying linearly with time, which are not modified by repeated measurements. The Zeno effect thus does not affect the irreversible damping rate T c of the cavity. Note that other irreversible systems may exhibit at short, but experimentally accessible times, a non-exponential behaviour with a quadratic initial evolution. Quantum Zeno and anti-Zeno effects have been theoretically described [START_REF] Kofman | [END_REF] and observed [13] for such irreversible processes.

Our experimental set-up [START_REF] Haroche | Exploring the Quantum[END_REF]14] is shown in figure 1. The key element is the superconducting cavity C, made up of two niobium mirrors facing each other [15]. It has a remarkably long damping time T c = 0.13 s, when cooled to 0.8 K. The pulsed classical source S, tuned to resonance with the cavity mode at 51.099 GHz, irradiates C from the side. A small fraction of the photons emitted in a pulse is coupled into C through diffraction on the mirrors edges. The other photons disappear quasiinstantaneously when S is switched off.

Phase stability is critical for the coherent accumulation of successive pulses in C. An atomic clock stabilizes S to a sub-Hertz frequency width and stability. The lifetime-limited linewidth of C is 1.2 Hz full width at half maximum (FWHM). The cavity frequency must be stabilized to this level, requiring a control of the 2.7 cm cavity length to a 0.5 picometer precision. We stabilize the cavity length passively. The cryostat has been isolated from external vibrations. We have stabilized the refrigerator temperature to ±10 -4 K and the pressure of the liquid helium bath to ±0.1 mbar. Piezo elements tune the mirrors separation and hence the mode frequency with a 2.4 kHz/V sensitivity. Their drive voltage (83 V) is stabilized to ±0.1 mV and drifts by less than ±0.2 mV/h.

The inset in figure 1 shows the cavity resonance spectrum probed by S operating in a quasi-continuous emission mode. We switch off S for short time intervals, during which the field intensity in C is measured by resonant Rydberg atoms [15]. These atoms are not submitted to the intense field radiated outside C when S is on. The Lorentzian fit FWHM is 2.0 Hz, differing from the expected 1.2 Hz width, due to a combination of short term fluctuations and long term drift. During data acquisition, we measure the cavity frequency every ten minutes and reset S at resonance with C. If the drift between two measurements is above 1.2 Hz, we eliminate the corresponding data. We estimate that the S -C detuning δ/2π is, on average, 0.6±0.2 Hz during data acquisition.

The non-resonant QND probe atoms are prepared in box B from a velocity-selected rubidium atomic beam (velocity v = 250 ± 1 m/s) in the circular Rydberg level g (principal quantum number 50). Rydberg atom samples are prepared within 2 µs pulses. Their position is wellknown during their flight through the apparatus. The cavity mode is close to resonance with the transition from g to e (circular state with principal quantum number 51). The atom-cavity detuning, ∆/2π = 240 kHz, is larger than the vacuum Rabi frequency, Ω 0 /2π = 50 kHz, which defines the atom-cavity coupling. The atoms are transparent probes, unable to absorb or emit photons in C. The g → e transition frequency is, however, lightshifted. This shift, proportional to the photon number n, is used to measure it non-destructively [8,9].

Before entering C, each atom is prepared in the superposition (|e + |g )/ √ 2 by a π/2-pulse, resonant on the g → e transition, produced by the classical source S ′ in the low-Q cavity R 1 . Using the Bloch pseudo-spin representation, we describe this superposition as a spin in the OX direction of the Bloch sphere. The light shift accumulated during the atom-cavity interaction changes the phase of the e/g superposition, in a frame rotating at the atomic transition frequency in vacuum [9,[START_REF] Haroche | Exploring the Quantum[END_REF]. When n photons are stored in C, the atomic spin rotates by an angle nΦ 0 in the equatorial plane of the sphere. With the chosen values of ∆ and v, the phase shift per photon is Φ 0 ≈ π/4. We measure the spin direction by applying on the atom, at the exit of C, a π/2-pulse in R 2 with a phase ϕ relative to that of the R 1 pulse. The subsequent detection of the atomic state (e or g) in the field-ionization detector D is equivalent to a spin detection at the exit of C along the Ou axis in the equatorial plane, at an angle ϕ with OX.

The setting Φ 0 ≈ π/4 is appropriate for a QND measurement up to n = 7 [9]. The photon numbers n = 0 . . . 7 are associated to 8 directions of the atomic spin at the exit of C. We send an ensemble of probe atoms in a time shorter than T c . For a given photon number n, all atomic spins in this ensemble point in the same direction. We infer it by a quantum tomography process, in which we detect the spins individu- ally along one out of four different axes corresponding to ϕ/π = -0.250, -0.047, +0.247 and +0.547. With 110 atoms in the ensemble, the statistical noise on the tomography process is of the order of the phase shift Φ 0 .

Let us first examine the coherent field growth without QND probe atoms between the injections. An experimental sequence starts by reseting the field to |0 using a large number of absorbing atoms, prepared in g and set at resonance with C via the Stark effect. We then inject N identical pulses (50 µs duration) in C, separated by a time interval T i = 5.04 ms. At the end of the sequence, we perform a QND measurement of the cavity field.

Figure 2(a-c) presents the probability distributions of the measured spin phase ϕ for N = 0, 20 and 50 injection pulses. For each sequence, we detect ≈200 atoms over a T m = 72 ms duration, starting 2T i after the last injection. From these data, we extract about 90 non-independent ϕ values based on spin tomographies performed on a sliding window containing an ensemble of 110 atomic detections. We repeat the procedure 500 to 2000 times. For N = 0, the phase distribution is peaked at ϕ = 0 revealing an empty cavity. For N = 20, the peaks centered at Φ 0 and 2Φ 0 reveal the build-up of small probabilities for having one and two photons. For N = 50 injections, equidistant peaks are visible, corresponding to n = 0 . . . 5. These discrete peaks directly reveal the field energy quantization in C [16]. The phase distributions are fitted by sums (thick red line) of Gaussians (thin green lines) with a unique standard deviation of 0.266 π, corresponding to the spin tomography statistical noise. From these fits, we extract Φ 0 =0.233 π. We also obtain the photon number distributions and their averages. These distributions are poissonian as expected for coherent states, see insets in figure 2. Cavity relaxation cannot be totally neglected during the delay between the last injection and the measurement. We thus correct the measured mean photon number by a scaling factor 1.4, taking into account the ≈ 2T i + T m /2 delay between the last injection pulse and the center of the measurement time interval. Figure 3(a) shows the evolution of the rescaled average photon number n as a function of N (red squares). Phase stability problems prevented us from recording data for N > 50. As expected, the initial growth is quadratic. For large N values, the sequence duration is comparable to T c and 1/δ. For a comparison with theory, we must take into account cavity damping and cavity-source detuning. The expected cavity field amplitude after k injections is:

α k = e -Ti/2Tc α k-1 +e i(k-1)Tiδ α 1 =
e -kTi/2Tc -e ikδTi e -Ti/2Tc -e iδTi α 1 , (1) where α 1 is the first pulse amplitude, which is taken as the phase reference (α 1 assumed to be real, without loss of generality). The value of α 1 is deduced from a fit of the calculated photon number with the data (solid line in figure 3(a)), taking into account the estimated δ value. We get n 1 = α 2 1 = 0.00223 ± 0.00012. We now send QND probe atoms between injections and record their final state thus acquiring information from each realization of the field state. In order to determine the evolution of the average photon number, we perform a statistical analyzis of many such realizations, using a simplified version of the spin tomography method described above. Since the photon number is now expected to remain small, we use a single detection phase, ϕ = -0.278 π, which optimally discriminates the spin directions corresponding to 0, 1 and 2 photons. To obtain the spin histogram and thus determine the average photon number at a given time, we collect the information from atoms detected in a 25 ms window around this time (∼ 50 atoms detected during 5 injection/measurement cycles) and repeat the procedure from 500 to 2500 times. The basic conclusions of our initial qualitative discussion are valid, since the amplitude injected by five consecutive pulses is still very small.

The corresponding average photon number, n, is shown (blue dots) as a function of N in figure 3(a). All n values are determined from independent samples made of 50 detected atoms. We check this simplified measurement by a complete spin tomography performed on ≈200 detected atoms at the end of sequences with 20, 50 and 100 injection/measurement cycles (open circles in figure 3(a)). The field growth is almost completely inhibited, n remaining below 0.2. Figure 3(b) presents the inhibited field growth with an expanded vertical scale. This growth can be modelled by a random walk, assuming that the field phase is blurred between two injections. The dotted black line in figure 3(b) presents the prediction of this model with no adjustable parameter. This prediction is slightly below the observations, indicating that phase is not totally blurred between injections. We have performed a detailed quantum Monte Carlo simulation of the field phase diffusion induced by the atoms crossing C. It includes the cavity relaxation, the finite 0.8 K temperature of its mode (n b = 0.05 blackbody photons on average) and the finite detection efficiency of D (50%). The simulation results (solid black line in figure 3(b)) are in excellent agreement with the observations and clearly exhibit the initial linear growth of n, characteristic of the Zeno effect.

We have observed a clear manifestation of the quantum Zeno effect on light. Repeated intensity measurements performed on the field in a high-Q cavity inhibit the quadratic run-away of the energy fed by a pulsed classical source. The residual field growth is modelled by a random walk in phase space. This effect illustrates vividly the back action of the photon number measurement onto the field phase. The realization of this experiment opens interesting perspectives for controlling quantum systems. Instead of freezing their evolution, repeated measurements could provide information used to channel them towards tailored quantum states by active feed-back operations [17].

FIG. 1 :

 1 FIG. 1: Experimental set-up (cavities C, R1 and R2 are cut for clarity). The inset shows the cavity spectrum (dots) and a lorentzian fit (line) with a 2.0 Hz FWHM.

FIG. 2 :

 2 Photon number

FIG. 3 :

 3 FIG.3:(a) Average photon number n in C as a function of the number N of injection pulses (bottom axis). The top axis gives a time scale for the complete sequence, including injection and measurement times. The red squares correspond to the uninhibited field growth, without measurements between injections (statistical error bars for N < 50 are smaller than point size). The initial evolution is quadratic. The line is a fit providing a precise calibration of the injection pulse amplitude. The blue dots result from atomic spin tomography performed between injection pulses. The field growth is inhibited by the quantum Zeno effect. The open circles correspond to QND measurements of the photon number using 200 atomic detections at the end of sequences with 20, 50 and 100 injection/measurement cycles. (b) Inhibited field evolution with an expanded vertical scale (blue dots in a). The error bars reflect the statistical noise. The solid and dotted black lines are theoretical predictions (see text). The initial growth is now linear.

Acknowledgements This work was supported by the Agence Nationale pour la Recherche (ANR), by the Japan Science and Technology Agency (JST), and by the EU under the IP project SCALA. S.D. is funded by the Délégation Générale à l'Armement (DGA).


A M Zagoskin

Didier Felbacq

Emmanuel Rousseau

Quantum metamaterials in the microwave and optical ranges

Quantum metamaterials generalize the concept of metamaterials (artificial optical media) to the case when their optical properties are determined by the interplay of quantum effects in the constituent 'artificial atoms' with the electromagnetic field modes in the system. The theoretical investigation of these structures demonstrated that a number of new effects (such as quantum birefringence, strongly nonclassical states of light, etc) are to be expected, prompting the efforts on their fabrication and experimental investigation. Here we provide a summary of the principal features of quantum metamaterials and review the current state of research in this quickly developing field, which bridges quantum optics, quantum condensed matter theory and quantum information processing.

The turn of the century saw two remarkable developments in physics. First, several types of scalable solid state quantum bits were developed, which demonstrated controlled quantum coherence in artificial mesoscopic structures [1][2][3][4] and eventually led to the development of structures, which contain hundreds of qubits and show signatures of global quantum coherence (see 5,6 and references therein). In parallel, it was realized that the interaction of superconducting qubits with quantized electromagnetic field modes reproduces, in the microwave range, a plethora of effects known from quantum optics (in particular, cavity QED)

with qubits playing the role of atoms ('circuit QED', [7][8][9] ). Second, since John Pendry 10 extended the results by Victor Veselago 11 , there was an explosion of research of classical metamaterials resulting in, e.g., cloaking devices in microwave and optical range [12][13][14] . The logical outcome of this parallel development was to ask, what would be the optical properties of a "quantum metamaterial" -an artificial optical medium, where the quantum coherence of its unit elements plays an essential role?

As could be expected, this question was arrived at from the opposite directions, and the term quantum metamaterial was coined independently and in somewhat different contexts.

In refs. [15][16][17] it was applied to the plasmonic properties of a stack of 2D layers, each of them thin enough for the motion of electrons in the normal direction to be completely quantized.

Therefore the wavelike nature of matter had to be taken into account at a single-electron level, but the question of quantum coherence in the system as a whole did not arise. In refs. 18,19 the starting point was the explicit requirement that the system of artificial atoms (qubits) maintained quantum coherence on the time scale of the electromagnetic pulse propagation across it, in the expectation that the coherent quantum dynamics of qubits interacting with the electromagnetic field governed the optical properties of the metamaterial.

Currently the term quantum metamaterial is being used in both senses (see, e.g., [20][21][22][23][24][25] ).

We will follow the more restrictive usage and call quantum metamaterials (in the narrow sense) such artificial optical (in the broad sense) media that 23 (i) are comprised of quantum coherent unit elements with desired (engineered) parameters; (ii) quantum states of (at least some of) these elements can be directly controlled; and (iii) can maintain global coherence for the duration of time, exceeding the traversal time of the relevant electromagnetic signal.

The totality of (i)-(iii) (in short: controlled macroscopic quantum coherence) that makes a quantum metamaterial a qualitatively different system, with a number of unusual properties and applications.

A conventional metamaterial can be described by effective macroscopic parameters, such as its refractive index. (The requirement that the size of a unit cell of the system be much less -in practice at least twice less -than the wavelength of the relevant electromagnetic signal, is implied in its definition as an optical medium, and is inherited by quantum metamaterials.)

From the microscopic point of view, these parameters are functions of the appropriately averaged quantum states of individual building blocks. In a quantum metamaterial, these states can be directly controlled and maintain phase coherence on the relevant spatial and temporal scale.

The full treatment of such a system should start from quantum description of both the electromagnetic field and the "atoms". In case when their role is played by qubits (that is, two-level quantum systems), a general enough Hamiltonian of a quantum metamaterial is given by 21,26

Ĥ = j hω j b + j b j - 1 2 k ǫ k σ (k) z + ∆ k σ (k) x + i jk ξ jk σ (k) z b j -b + j , (1)
where the first term describes unperturbed photon modes, the second the qubit degrees of freedom, and the third their interaction. The direct control over (at least some) qubits is realized through the qubit Hamiltonian parameters (the bias ǫ or, sometimes, the tunneling matrix element ∆).

Forming photon wave packets with characteristic size Λ ≫ a (where a is the unit cell size)

and averaging over the quantum states of qubits on the scale of Λ, one should eventually arrive at the effective equation of motion for the "Λ-smooth" density matrix. It will describe the state of both the electromagnetic field and the quantum metamaterial, characterized by a nondiagonal, nonlocal, state-and position-dependent "refractive index" matrix.

Following through with this program involves significant technical difficulties, and the task is not brought to conclusion yet. Nevertheless certain key effects in quantum metamaterials can be investigated at a more elementary level using an approximate wave function (e.g., [26][27][28][29] ), or treating the electromagnetic field classically 18,19,30,31 . The latter is like the standard quasi-classical treatment of the atom-light interaction 32 , but is more conveniently done using the lumped-elements description (see, e.g., 30 , Section 2.3). The state of a system of M nodes connected by capacitors, inductors and, if necessary, Josephson junctions, is described by a Lagrangian L({Φ, Φ}). Here the "node fluxes" Φ j (t) = c t dt ′ V j (t ′ ) are related to the node voltages V j (t) and completely describe the classical electromagnetic field degrees of freedom (the current-voltage distribution) in the system. The lumped-elements description is appropriate, since we are interested in signals with a wavelength much larger than the dimension of a unit element of the circuit (the condition of its serving as a metamaterial).

Qubits are introduced in this scheme through their own Hamiltonians and coupling terms (e.g., flux qubits can be coupled inductively -through the magnetic flux penetrating their loop, -or galvanically -through sharing a current-carrying conductor with the circuit), which are usually cast in the form Ĥq = -1 2 ǫ q σ z q + ∆ q σ x q + J q (Φ, Φ)σ z q .

(2)

The "quantum Routhian",

R({σ}, {Φ, Φ}) = q Ĥq -L({Φ, Φ}), (3)
plays the role of the Hamiltonian for qubits (in Heisenberg representation), and its expectation value R = R produces the equations of motion for the classical field variables Φ, where we have included the dissipative function Q to take into account resistive losses in the circuit (see Refs. 30 , 2.3.3, and 33 ):

d dt ∂R ∂ Φj - ∂R ∂Φ j = ∂Q ∂ Φj . ( 4
)
one can take the continuum limit and, switching to the Schrödinger representation, obtain coupled systems of equations for the field variable Φ(r, t) and the qubit two-component "macroscopic wave function" Ψ(r, t). At any convenient stage the electromagnetic modes can be canonically quantized via

Φ → Q ∼ (b + b † ), Φ → P ∼ i(b -b † )
, leading back to a certain approximation of an approach based directly on Eq.( 1).

II. SUPERCONDUCTING QUANTUM METAMATERIALS

The above scheme is most often applied -though not limited -to the case of superconducting metamaterials based on various types of superconducting qubits (see, e.g., Ref.30, Ch.2). The simplest case is given by the experimental setup of Ref. 34 (Fig. 1a). There a single artificial atom (a flux qubit) is placed in a transmission line, and transmission and reflection coefficients for the microwave signal are measured. Here the equations ( 4) for the field in the continuum limit will yield free telegraph equations for the voltage and current everywhere except the point x = 0, where the qubit is situated:

∂V (x, t) ∂x = L c 2 ∂I(x, t) ∂t ; (5) ∂I(x, t) ∂x = C ∂V (x, t) ∂t , (6)
where L, C are the inductance and capacitance per unit length of the transmission line.

Of course, for such a simple structure these equations can be written down directly. The influence of the flux qubit, which is coupled to the transmission line through the effective mutual inductance M (taking into account both magnetic and kinetic inductance) is through the matching conditions at x = 0,

V (+0, t) = V (-0, t) - M c ∂ Îq (t) ∂x ; (7)
I(+0, t) = I(-0, t). (8)
The qubit current operator Îq = I p σ z is governed by the qubit Hamiltonian (2) with the coupling term J q = I(0, t)M/c 2 . An explicit solution for the reflection/transmission amplitudes was found to be in a very good agreement with the experimental data in Ref.34 (Fig. 1b).

For a structure, where the role of the "artificial atom" between 1D transmission lines is played by a single qubit surrounded by an array of N coupled photonic cavities 29 the calcu-lations in the one-excitation approximation (with electromagnetic modes treated quantum mechanically) show that in such a structure arise long-living quasi-bound states of photons and the qubit, manifested as ultra-narrow resonances in the transmission coefficient.

Going from these "proto-metamaterials" to QMMs containing many artificial atoms, we return to the classical treatment of the electromagnetic field. Though 2D and 3D versions of superconducting QMMs are feasible 23,24 (Fig. 2a), they do not operate yet in a quantum coherent regime. Most of the research at the moment concentrates on the 1D case, which already promises interesting results. In theoretical papers 18,31,35 is considered a QMM formed by a set of superconducting charge qubits placed in the transmission line (Fig. 2b). The equations of motion for the field and qubits were solved, using a factorized approximation of the quantum state vector of the qubit subsystem. For the realistic choice of qubit and transmission line parameters, the figures of merit β = E EM /∆ ∼ 30 and

ν = h max(Γ qb , Γ T L )/∆ ∼ 10 -3 (Ref.18
) ensure that the continuum approximation is justified, and that in the first approximation the effects of decoherence can be neglected. Here E EM is the electromagnetic field energy per unit cell, ∆(Eq.( 1)) gives the qubit energy scale, β is the dimensionless signal velocity in the transmission line (units cells per h/E J ), and Γ qb , Γ T L are the decoherence rates in a qubit and in the transmission line respectively. Some of the results are shown in Fig. 3.

Dimensionless equations of motion for the field vector-potential in the lowest order in field-qubits interaction yield the wave equation

α(ξ, τ ) -β 2 ∂ 2 α(ξ, τ ) ∂ξ 2 + V (ξ, τ )α(ξ, τ ) = 0, (9)
where

V (ξ, τ ) ∝ Ψ(ξ, τ )| cos φ| Ψ(ξ, τ ) .
The tunneling matrix element of the charge qubit Hamiltonian (φ being the qubit superconducting phase) is determined by the qubit state | Ψ(ξ, τ ) at the given point. The dispersion law is thus directly dependent on the QMM quantum state, as expected. For example, if the qubit state is a periodic function of the coordinate, Ψ(x + Λ) = Ψ(x), the QMM behaves as a photonic crystal, with gaps opening in the electromagnetic spectrum 18,35 , which can be manipulated by controlling the quantum state of qubits. For example, if qubits are placed in a spatially periodic superposition of their eigenstates,

| Ψ(x, t) = A(x)|g + B(x)|e e -iωt , (10)
where A(x), B(x) are periodic functions and hω is the energy splitting between the ground (|g ) and excited (|e ) state of the qubit. Quantum beats between these states will produce a "breathing" photonic band structure (Fig. 3a); external control of qubit states allows to trap a portion of radiation in a pocket of such a structure and move it across the QMM at a desired speed 18 . In the absence of direct control over individual qubit states (apart from their initialization in the ground state) it is still possible to create a photonic crystal structure 35 by sending into the QMM specially shaped "priming" electromagnetic pulses from the opposite directions (Fig. 3b). Exiting the QMM, they leave behind a spatially periodic pattern of the probability of finding a qubit in the ground (excited) state.

Solving the coupled equations for the classical field and qubits numerically (still in the approximation of factorized qubit state) allows to investigate lasing in a QMM 31 . If the qubits are initialized in the excited state (e.g. by sending a priming pulse through the QMM), an initial pulse triggers a coherent transition of energy from qubits to the electomagnetic field (Fig. 3c. Remarkably, not only the process has a precipitous character, but its onset starts the sooner the greater the amplitude of the triggering pulse:

√ field amplitude × τ onset ≈ const.
In equilibrium a fully quantum treatment of a superconducting QMM becomes possible, which allows the investigation of phase transitions in the photon system 26 . The chosen model of a QMM (a series of RF SQUIDS coupled to the transmission line and considered in twolevel approximation) lead to a generic Hamiltonian (1), with only parameters being model dependent. Using the instanton approach, the effective action of the photon subsystem was obtained as a function of the photon field momentum P (in imaginary time).

In case when P 0 is independent on the imaginary time τ , the photon system may undergo a second order classical phase transition: above a critical temperature T ⋆ the momentum P = 0, while below it the system can choose between two values ±P 0 . The transition details depend on the level of disorder in the qubit chain, which is modeled here by the random distribution of tunneling matrix elements ∆. In the case of a low disorder, ∆ k ≈ ∆ 0 , the transition occurs at the critical temperature

T ⋆ n = mη 2 N k B , ( 11
)
where m is the transmission line inductance per unit cell, η parametrizes the qubit-field interaction, and N is the total number of qubits in the QMM. This phase transition occurs

only if ∆ 0 < k B T ⋆ n .
In the case of strong disorder with ∆ k distributed from zero to some 10 ∆ 0 , the transition temperature

T ⋆ w = ∆ 0 exp [-∆ 0 /mη 2 N] k B , (12)
and the transition occurs only if ∆ 0 > mη 2 N. As the authors of Ref.26 note, the first case is similar to the metal-ferromagnet phase transition, while the second one is reminiscent of the normal metal-superconductor or Peierls metal-insulator transition. In either case a coherent state of the photon field emerges, with nonzero value of the order parameter P 0 , which in the case of low disorder is proportional to the number of qubits in the system, N.

It turns out that there is the possibility of a quantum transition as well, into a state representing a superposition of semiclassical states ±P 0 . Then the order parameter becomes a periodic function of τ . In the case of low disorder and strong field-qubit coupling it occurs below the temperature (Fig. 4)

T QOP n ∝ ∆ 2 0 mη 2 N π 2 1/3 1 k B , (13)
while in the opposite case it is given by

T QOP n = ∆ 0 k B π exp -∆ 0 /mη 2 N . ( 14
)
The estimates of Ref.26 for the transition temperatures T ⋆ ∼ 0.1 . . . 50 K let us hope for direct observation of photon phase transitions in a superconducting QMM, when the number of qubits, the homogeneity of their parameters and their coherence times are improved as compared to the existing QMM prototype 36,37 . The prototype (Fig. 5a) consists of 20 flux qubits placed in a coplanar waveguide resonator. The inductive coupling of qubits to the resonator and each other was comparable, but strong decoherence (decoherence time of the order of a few nanoseconds) effectively suppressed qubit-qubit coupling. Nevertheless the transmission measurements in the resonant regime showed the formation of three ensembles of interacting qubits (two of four qubits each and one of eight qubits). This interaction through the electromagnetic modes is the key element of the operation of a QMM. In order to improve the operation of a QMM it is suggested to increase the qubit-qubit coupling, in order to counteract the effects of qubit parameter dispersion. (quantum phase transiion, blue line, P ∝ N 1/3 ). The low disorder case is shown: ∆ k ≈ ∆ 0 = 4K;

T ⋆ n = 20K.

III. OPTICAL QUANTUM METAMATERIALS

The domain of quantum metamaterials in the optical, or near IR, region of the spectrum is still in its infancy. As it has already been stated, some authors use the term quantum metamaterial to denote a structure in which quantum degrees of freedom are inserted 15 . In some other cases, it is the expression "quantum dots metamaterials" that is used: this is to stress that, although quantum dots are inserted in a metamaterial, one is not interested in the quantum coherence of the dots, but rather on the gain that they provide, to counteract the losses due to the presence of metallic inclusions 38 . In other proposals, it is quantum wells that are inserted in a photonic structures. The quantum well are described electromagnetically by a permittivity allowing some control over the behavior of the structure. In ref. 17 , a layered metamaterial is investigated, in which the period comprises two GaAs quantum wells. This structure results in an effective permittivity tensor allowing to obtain a negative refraction.

The effective properties strongly depend upon the 2D electron density in the quantum well.

In ref. 15 the same kind of structure is investigated in order to control plasmon propagation, In the specific meaning used in this review, a proposal was made in ( 21,28 ) to study the full quantum processes that occurs between the quantized electromagnetic field and two-level atoms. The system studied there is a 2D network of coupled atom-optical cavities, called a cavity array metamaterial (CAM). The authors propose to realize the model by using a two-dimensional photonic crystal membrane. The quantum oscillators could be quantum dots or substitution centers. Under reasonable assumptions, this system can be described by a Jaynes-Cummings-Hubbard Hamiltonian 40 . The system exhibits a quantum phase transitions 21,41 and it was proposed that it could be used as a quantum simulator. The effects of cloaking and negative refraction were also demonstrated. Quite naturally, the excitations of the system are hybrids of photonic and atomic states, namely polaritons. This kind of result is very well-known since the pioneering work of J. J. Hopfield way, an effective permittivity that is non-local in time and space could be derived, in exactly the same way as for natural material. This is the line followed in a series of papers by G.

Weick where a collection of metallic nanoparticles is shown to exhibit collective plasmonic modes [43][44][45] . However, a genuine quantum metamaterial requires more than that, namely the active coherent control of the quantum state of the "atoms" inserted in the photonic structure, so as to induce a control over the collective properties of the medium.

Such a system could be implemented by considering, as above, a photonic crystal in which quantum oscillators are inserted, under the guise of quantum dots for instance. The quantum dots can be described semi-classically by a dielectric function ε QD that reads as 46 :

ε QD (ω) = ε b + (f c (E e ) -f v (E h )) a ω 2 -ω 2 0 + 2iωγ (15)
The prefactor (f c (E e ) -f v (E h )) representing the difference between the populations of the levels can be either positive or negative. In the first case, it is in the absorption regime while in the latter it is in the emission (amplifying) regime. The graph of ε QD (ω) is given in fig. 7. The quantum dots can be grown inside dielectric nanopillars, and the nanopillars can be organized into a 2D periodic array, resulting into a photonic crystal with quantum dots. The bare photonic crystal is then tuned in such a way as to present a photonic band gap at the emission frequency of the quantum dots. The idea is then to realize a pump/probe experiment, where the pump controls the state of the quantum dots (absorption or emission). When the quantum dots are in the emission regime, a transmission peak appears in the transmission spectrum of the probe (Fig. 8). This somewhat simplified model shows a macroscopic property, a conduction band, results from the quantum states of the microscopic quantum components. A full quantum treatment should be performed in order to address properly the quantum coherence of the system of the field coupled to the "atoms" of the metamaterial. As compared to the situations encountered in cavity quantum electrodynamics, the quantization of the electromagnetic field in open space comes with severe technical difficulties, if one follows the usual mode-decomposition path, abundantly described in all the literature devoted to field quantization. In fact, for open systems, Maxwell equations do not lead to hermitian eigenvalue problems and thus the eigenfunctions are not normalizable. A recent interesting approach is to use the so-called "quasi-normal modes" of the system 47 . Here we review the tools of quantum optics for the description of the quantum dynamics of a collection of emitters (e.g. quantum dots) in a complex electromagnetic environment that can be constituted by dielectric and/or metallic elements as depicted in Fig: 9 For a single emitter in free space, the interaction with light is given by the minimalcoupling Hamiltonian that reads 48 :

H = 1 2m (p -eA) 2 + eV (r) + d 3 r ′ E ⊥2 + B 2
where p is the electron momentum, A the vector-potential, V (r) the binding potential for the electron, B the magnetic field and E ⊥ the transverse part of the electric field, satisfying

∇.E ⊥ = 0 (c.f. 49 p. 254). The emitters are coupled to each other through the electromagnetic field that comprises both radiating and evanescent terms.

For neutral emitters in a complex electromagnetic environment, one may need a mesoscopic description, where the emitters are described through their multipole moments instead of the dynamics of the electric charges. Moreover, it can be convenient to describe the electromagnetic environment by polarization and magnetization fields instead of charge and current densities 50 . Last but not least, a Hamiltonian involving the physical fields E and B rather than the potentials ϕ( r) and A( r) can also be more convenient. Applying the Power-Zienau-Wooley transform to the previous Hamiltonian 48, 50,51 (see also 52 p. 282) fulfills these requirements. The Power-Zienau-Wooley transform leads to the following Hamiltonian(see Milonni 53 p.121):

H = H mat + H f ield + H int (16)
where

• H mat = p 2m + V (r) + d r P ⊥2
2ε 0 is the Hamiltonian describing the dynamics of the atom variables, P is the polarization field. Even if the term d r P ⊥2 /2ε 0 is important to reproduce the correct dynamics of the emitter 48 , it is usually neglected when studying the interaction between the emitter and light. It is usually argued that this term merely shifts the energy levels, an effect that can be accounted for by a correct renormalization of the emitter energy levels. Nevertheless in the ultra-strong coupling regime, this term has to be taken into account 54 . It leads to a decoupling of matter and light states because of a screening of the incident light by the polarization field P, resulting for example in a reduction of the Purcell factor, while increasing the coupling between the field and the emitter 54 .

• H f ield = D 2 ( r) 2ε 0 + B 2 ( r)
2µ 0 is the Hamiltonian describing the electromagnetic field dynamics.

• H int = -P( r).D( r) d r is the Hamiltonian describing the interaction between light and matter. D( r) is the displacement vector. Note that the Hamiltonian given by the equation eq.( 16) is exact. It is completely equivalent to the minimal-coupling hamiltonian (Cohen 52 p.298).

From Hamiltonian eq.( 16), with the help of the Heisenberg equation, one can find the dynamical equations satisfied by each operators (field and matter operators).

Concerning the field operators, it is assumed that they are related to each other through Maxwell equations. This assumption implies that there exists some commutators between the field operators (p.18 in 55 ): [D( r, t), A( r ′ , t)] = ih ε 0 δ T ( r -r ′ ) where δ T ( r) is the transverse delta distribution (See also 52 p. 233). Finally, one gets the following set of equations between the field operators:

∇ × E = -∂ t B ∇ × B = µ 0 ∂ t P + 1 c 2 ∂ 2 t 2 E
One arrives at the well-known wave equations satisfied by the electric-field operator:

∇ × ∇ × E - 1 c 2 ∂ 2 E ∂t 2 = -µ 0 ∂ 2 P ∂t 2 (17)
Concerning the dynamics of the matter degrees of freedom, some usual approximations are done. We write the polarization field as a sum of a polarization field due to the atoms P a and a polarization field due to the electromagnetic environment P ind : P = P a + P ind .

We assume that the polarization field due to the electromagnetic environment P ind responds linearly and locally to the electric field. It reads as 56

P ind ( r, t) = +∞ -∞ ε 0 χ( r, t ′ )E( r, t ′ )dt ′
where χ( r, t ′ ) is the susceptibility at position r and time t.

Concerning the polarization field due to emitters, we work in the usual dipole approximation and approximate the polarization field by keeping only the first term in the multipolar expansion even if this approximation can be crude for quantum dots 57 . If there are N α emitters, the polarization field due to the emitters is then written as P a ( r) = Nα α=1 d α δ( r -r α ) where d α and r α are respectively the dipole-moment operator and the position of the emitter labeled by α, and δ( r) is the usual Dirac distribution. It is convenient to express all matter-operators with the help of the basis defined by the eigenstates of the matter hamiltonian H mat . The matter hamiltonian H mat is written as H mat = Nα α=1 H α mat where H α mat is the hamiltonian of the α th emitter. We note {|α, i } the eigenbasis constructed from the eigenstates of H α mat that satisfied H mat |α, i = E α i |i and the completeness condition i |α, i α, i| = I α d , where I α d is the identity matrix acting on the subspace of the α th emitter. We now assume that emitters are two-level systems. The ground state is labelled by i =whereas the excited level is labelled by i = +. Applying twice the completeness condition on the hamiltonian H α mat , one finds

H α mat = E -+E + 2 I α d + hω α 2 σ α z where hω α = E α + -E α -and σ α z = |α, + α, +| -|α, -α, -|.
σ α z acts on the subspace defined by the eigenvectors of the α th emitter and measured its population difference between the excited and the ground state. The first term in H α mat is a constant that can be omitted by choosing correctly the reference of the energy. We then write the matter hamiltonian as (see 58 p. 23 and 53 p.128):

H mat = Nα α=1 hω α 2 σ α z .
The polarization field due to the emitters can also be written with the help of the eigenbasis of each emitter by writing the dipole-moment operator in this basis:

d α = d α +-|α, + α, -| + d α -+ |α, -α, +|
where d α +-= α, +| d α |α, + is the projection of the dipole-moment operator in the basis {|α, + , |α, -}. The diagonal elements are null because we assume that the emitters have no permanent dipole. We introduce the raising σ α + = |α, + α, -| and lowering operators σ α -= |α, -α, +|. Finally, the polarization field due to the atoms reads:

P a ( r) = Nα α=1 ( d α +-σ α + + d α -+ σ α -)δ( r -r α )
If we assume that the dipole moment projection d α +-= ( d α -+ ) ⋆ = d 0 α is real, the polarization field simplifies with the help of the Pauli matrice σ x :

P a ( r) = Nα α=1 d α 0 σ α x δ( r -r α ) since σ x = 1 2 (σ α + + σ α -) (see 58 p.24, 59 p.35
). With the help of a Fourier transform, the hamiltonian for the free electromagnetic field can be written H f ield = j,s hω j b + j,s b j,s where b j , s (resp. b + j,s ) is the annihilation (resp. creation ) operator of the electromagnetic mode j with polarization s. Within this decomposition the electric field on its own reads E = j,s e j,s (b j,s -b + j,s ). Finally, following these successive approximations one finds the hamiltonian given by the equation eq.( 1)

The behavior of the quantum metamaterial can be computed by solving simultaneously the equation eq.( 17) and the equations of motions for the collection of two-level "atoms" given by 48 , p.37 and 59 :

σα x = -ω α σ α y σα y = ω α σ α x + 2 h d α 0 .E( r α )σ α z σα z = - 2 h d α 0 .E( r α )σ α y
Where all terms proportional to P 2 ( r) have been neglected. These equations are nonlinear coupled differential equations since σ α x are sources of the electric field eq:( 17). The electromagnetic environment contributes also as a source term in eq:( 17). This source term is important since it is responsible for all the unusual effects demonstrated theoretically or experimentally with classical metamaterials. Its effect on quantum metamaterials has been barely studied since solving this system of equations is challenging, even for a very simple geometry 60 . New theoretical tools should be developed to accurately describe the behavior of a quantum metamaterial in a complex electromagnetic environment. Nevertheless more approximations can be done to solve these equations. One can use the single electromagneticmode approximation 21,28,61,62 or perform a semi-classical approximation 18,19,30,31,63 . In the latter, it is assumed that there are no correlations 59 between the electromagnetic field and the matter degrees of freedom.

V. CONCLUSIONS

The field of quantum metamaterials research arose at the intersection of quantum optics, microwave and Josephson physics, and quantum information processing. One of its rather paradoxical feature is that, while the theoretical progress in this area still significantly outweighs the experiment, the theoretical challenges seem more significant. Indeed, the existing experimental techniques, especially in case of superconducting structures, already allow creating massive arrays. The 20-qubits prototype 36 is much smaller than a recently fabricated 1000+-qubits superconducting quantum annealer D-Wave 2X. Given a simpler structure, and less strict demands to a quantum metamaterial than to a quantum computer, making and testing quantum metamaterials on this scale is a question of time and funding.

On the other hand, the theoretical analysis of quantum metamaterials produces promising results, already using simple approximations. Nevertheless the understanding of the full scale of effects which can be expected in these systems requires a more detailed analysis of large scale quantum coherences and entanglement. Because of the well-known impossibility to effectively simulate a large quantum system by classical means, a direct approach to this is currently limited to structures containing (optimistically) less than a hundred qubits. New theoretical tools need to be developed, generalizing the methods of quantum theory of solid state 5 .

These challenges also present alluring opportunities. Developing and testing new theoretical methods applicable to large quantum coherent systems would be valuable for the whole field of quantum technologies, including quantum computing. Optical elements based on quantum metamaterials would provide new methods for image acquisition and processing.

Last but not least, a quantum metamaterial would be a natural test bed for the investigation of quantum-classical transition, which makes this class of structures interesting also from the fundamental point of view.

FIG. 2 :

 2 FIG. 2: (a) A 2D superconducting quantum metamaterial 24 . Fig.3 of 24 (b) A realization of a 1D superconducting quantum metamaterial: charge qubits in a transmission line 31 .

FIG. 4 :

 4 FIG. 4: Classical and quantum phase transition in the state of photons in a 1D superconducting quantum metamaterial 26 . The photon order parameter P 0 (τ ) as a function of imaginary time τ is either constant (classical phase transition, red line, P 0 ∝ N ) or proportional to sin[P 0 (T )τ ]

FIG. 5 :

 5 FIG. 5: Quantum metamaterial prototype 36 . (a) Twenty flux qubits placed in a resonator. The signal wavelength is of the order of the resonator length (23 mm) and greatly exceeds the qubit size (approx. 2 × 6 µm). The ensemble average level spacing and persistent current in the qubits are 5.6 GHz and 74 nA respectively. (b) The QMM in the resonant regime. Three distinct qubit ensembles are seen.

FIG. 6 :

 6 FIG.6: A cavity array metamaterial, taken from 28

FIG. 7 :

 7 FIG. 7:The dielectric function of a core-shell quantum dot 46

FIG. 8 :

 8 FIG.8: Transmission spectrum the quantum metamaterial in the absorption regime (in blue) and in the emission regime (in red). The peak corresponds to the transition of the quantum dots.

FIG. 9 :

 9 FIG. 9: A schematic view of a possible quantum metamaterial in optics, made by several quantum dots embedded in semiconducting nanowires surrounded by metallic spheres.

Acknowledgments

References Acknowledgments AZ was supported in part by the EPSRC grant EP/M006581/1 and by the Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUSTMISiS(No. K2-2014-015).


Murari Singh

Misaki Ozawa

Ludovic Berthier

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