We aim at identifying how multiple non-linear mechanisms impact the generation of non-classical state of light, such as squeezed states [1].

Non-linear interactions are necessary to generate quantum states of light. Through such interaction, mediated by non-linear optical matter, a set of photons are converted. The quantum nature of the resulting light derived directly from this conversion. For example, optical parametrical amplificator (a four-wave mixing process) produces a correlated pair of beams (signal and idler) from a strongly pumped one [2,3]. But non-linear optical devices encompass more than a single desired non-linear mechanisms. Many competing processes take place simultaneously in these devices coupling many modes together. If in principle, all these processes can lead to non-classical states of light their competition may in contrary leads to a reduction of the quantum nature of the light produced. In a recent work, it was shown that disorder in the non-linear optical devices can lead to a reduction of the weight of unwanted non-linear process and so protect the quantum nature of the state of light generated [4].

The project will start with a simplistic model where only a single non-linear process is present, for which we will quantify the non-classical nature: the squeezing. Then we will complexity the model introducing more modes and more non-linear mechanisms.

[1] D. Walls, ”Squeezed states of light,”

Interested candidates should contact Simon Pigeon (simon.pigeon@lkb.upmc.fr)

Theoretical internship.

Duration: 3 to 6 months.

When light propagates in linear medium (vacuum, air etc), photon behaves as a non-interacting gas, but when the medium is non-linear (atomic vapor or semiconductor microcavity for examples), they start flowing as a liquid called fluid of light. From the medium, photons acquire an effective interaction that may lead to macroscopic coherent behavior. Among the different system allowing to generate such particular fluids, semiconductor microcavities are surely the more advance allowing for the observation of superfluid of light [1,2].

As such vortex pair can appear spontaneously when the superfluid flow through a local potential barrier [3]. In the context of atomic superfluid the dynamics of these topological entities rather are well understood, but in the context of superfluid of light, the picture is blurred. This derived mainly from the open nature of fluids of light in cavities where photons lifetime is so short that the system needs to be continuously pumped with photons. This driving dissipative scheme and its interaction with vortex pair will be at the center of this internship. The student will investigate numerically with a code of its own, the dynamics of these pairs in present of a coherent driving in realistic experimental conditions.

[1] A. Amo, J. Lefrère, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdré, E. Giacobino, and A. Bramati, “Superfluidity of polaritons in semiconductor microcavities,”Interested candidates should contact Simon Pigeon (simon.pigeon@lkb.upmc.fr)

Numerical internship.

Duration: 3 to 6 months.

When light propagates in linear medium (vacuum, air etc), photon behaves as a non-interacting gas, but when the medium is non-linear (atomic vapor or semiconductor microcavity for examples), they start flowing as a liquid called fluid of light. From the medium, photons acquire an effective interaction that may lead to macroscopic coherent behavior such as superfluidity [1,2]. This surprising behavior connected to quantum states of matter raises the question about the quantumness in fluids of light. Among the features of quantum mechanics, quantum entanglement is surely one of the most infringing and most promising. It is the fundamental resource to quantum technologies mandatory to quantum computation or quantum cryptography.

Despite its importance, determining if fluids of light encompass some form of quantum entanglement remains elusive until now. Entanglement in its most intricate version was only recently evidenced in atomic superfluids through the violation of a Bell’s inequality [3.4]. Here we want to determinate if same results can be obtained in a superfluid of light?

To that purpose, the student will have to reformulate the Bell’s inequality used in atomic context to fluids of light context. After that tow option are available depending on the student wish : (i) performing a realistic quantum Monte Carlo simulation to evaluate Bell’s inequality or (ii) used a simplified model to calculate if Bell’s inequality is violated in fluids of light context.

[1] A. Amo, J. Lefrère, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdré, E. Giacobino, and A. Bramati, “Superfluidity of polaritons in semiconductor microcavities,”Interested candidates should contact Simon Pigeon (simon.pigeon@lkb.upmc.fr)

Numerical and/or Theoretical internship.

Duration: 3 to 6 months

Turbulence remains up today among the most puzzling of phenomena in physics. One of the key discoveries made since then was the energy cascade mechanism and its scaling by A. Kolmogorov, which explains how large-scale turbulence decays into small-scale turbulence. Small size turbulences are subsequently dissipated through viscosity. But superfluids have no viscosity! So how do they dissipate energy coming from turbulence? Recent advances in helium superfluids have provided some insight in terms of Kelvin wave perturbations of a vortex filament. These results apply to 3D equilibrium superfluids, whereas in 2D flows, energy cascades present a distinct scaling but also inverse energy cascade (toward large scale) [1]. Can this rich physic be explored with light?

Indeed, when light propagates in linear medium (vacuum, air etc), photon behaves as a non-interacting gas, but when the medium is non-linear (atomic vapor or semiconductor microcavity for examples), they start flowing as a liquid called fluid of light. From the medium, photons acquire an effective interaction that may lead to macroscopic coherent behavior such as superfluidity [2,3].

We want to numerically observe energy cascade in these 2D superfluids of light. To that purpose, the student will develop a code to simulate a 2D turbulent flow of light based on generalized Gross Pitaevskii equation.

[1] M. T. Reeves, T. P. Billam, B. P. Anderson, and A. S. Bradley, “Inverse Energy Cascade in Forced Two-Dimensional Quantum Turbulence,“

Interested candidates should contact Simon Pigeon (simon.pigeon@lkb.upmc.fr)

Numerical internship.

Duration: 3 to 6 months.

Classic thermodynamics is the phenomenological theory of the average behavior of heat and work for systems in thermal equilibrium. At the nanoscale, fluctuations dominate and systems generically operate far from thermal equilibrium [1]. The extension of thermodynamics to such situations was achieved only rather recently with the discovery of so-called fluctuations theorems. These theorems are statements of the second law of thermodynamics, and they constitute one of the most important breakthroughs in modern theoretical physics. At their core, they quantify that entropy only increases on average, and that processes with negative entropy production do occur, but they are exponentially unlikely.

The inception of these fluctuation theorems effectively opened a new field of research, which quickly adapted the name ‘stochastic thermodynamics’. In contrast to conventional thermodynamics, in stochastic thermodynamics work, heat, and also entropy production are defined along single trajectories of a classical system [2]. Naturally, the generalization of this new version of thermodynamics to quantum systems has posed a formidable task [3], which had lead to the development of rapidly growing field of research known as quantum thermodynamics. Intensive works were done along this line considering quantum close system exploring the role of quantum information, thermalisation or even quantum thermal machine [4].

However much more remain to do regarding quantum open systems even so they are the most commonly used in experiment to explore the quantum features. Among the large variety of those systems quantum optical devices are definitively those that allows the more accurate and profound exploration of quantum realm. This theoretical internship will be centred along this line and more precisely along the use of thermodynamics of trajectories [5] in quantum optical context.

[1] C. Jarzynski, Diverse phenomena, common themes

Interested candidates should contact Simon Pigeon (simon.pigeon@lkb.upmc.fr).

Theoretical internship.

Duration: 3 to 6 months.

Most work in many body-physics (theoretical and experimental) are dealing with massive material particules (atoms, electrons…). However, we know since the early days of quantum mechanics that photons in a box can be interpreted as a massless Bose gas of non-interacting particules and this interpretation leads to a correct calculation of the black-body radiation. Recently, it has been realized that, under suitable circumstances, photons can acquire an effective mass and will behave as a quantum fluid of light with photon-photon interactions. Striking experimental demonstrations of superfluidity and other quantum hydrodynamics effects such as quantized vortices and solitons have been performed using semi-conductor planar micro-cavities. Building on these experiments done by the LKB group, we propose to use a different geometry (propagating light instead of confined) to study quantum fluids of light.

We will address a new way of testing light superfluidity by studying the drag force that a photon fluid exerts on a mobile obstacle. This experiment is based on a movable defect, a nanofiber, immersed in an atomic vapor so that the refractive-index mismatch provides a constant potential.

A light beam hitting the nanofiber tip at a small angle, will create a flow around the impurity, or in optics language, a radiation pressure force resulting in the opto-mechanical deformation of the obstacle. We expect to observe a cancellation of this “optical drag force” at high intensity indicating a superfluid flow of photons around the obstacle.

In terms of optics, this leads to a non-intuitive cancellation of the radiation pressure thanks to non-linear interactions.

We are currently building a setup to detect with very high precision (about 20 nm) the position of the nanofiber. By monitoring the position of the nanofiber, we will measure the small radiation pressure force exerted by a laser beam on the nano-object. First, we are going to perform preliminary measurements with light propagating in air (linear medium). In this condition, the radiation pressure force should be an increasing function of the incident laser power. The second step of the project will be to immerse the nanofiber in an atomic vapor. When light propagates through this nonlinear medium, it becomes superfluid above a critical laser power (i.e. above a critical photon density). In that case, we will measure the force acting on the the nanofiber as a function of the incident laser power. Contrary to air, we expect that, above a critical value of the intensity, the force is going to vanish, revealing the superfluid nature of light.

Interested candidates should contact Quentin Glorieux (quentin.glorieux@lkb.upmc.fr) and Alberto Bra- mati (bramati@lkb.upmc.fr).

PDF Version : Mechanical_Signature_Superfluidity_2017

Quantum fluids Physics is the study of hydrodynamic systems which demonstrate a quantum behavior. A large range of many-particle systems are currently under intense investigation, from liquid Helium, to electrons in solids, to quark-gluon plasma and trapped gases of ultra-cold atoms. Surprisingly, these very different systems show similar behavior when the thermal de Broglie wavelength becomes comparable or larger than the average inter-particle spacing. In this regime, the Bose versus Fermi statistics of the particules starts playing a critical role in determining the properties of the fluid. For exemple, in a non-interacting Bose gas a macroscopic fraction of the particules will accumulates in the lowest-energy state leading to a Bose-Einstein condensate. When interactions between particules are not negligible, the Physics become even more fascinating with the appearance of purely quantum effects such as supra-conductivity and superfluidity.

Historically, most work in many body-physics (theoretical and experimental) are dealing with massive material particules (atoms, electrons…). However, we know since the early days of quantum mechanics, that photons in a box can be interpreted as a massless Bose gas of non-interacting particules and this interpretation leads to a correct calculation of the black-body radiation. Recently, it has been realized that under suitable circumstances photons can acquire an effective mass and will behave as a quantum fluid of light with photon-photon interactions. Striking experimental demonstrations of superfluidity and other quantum hydrodynamics effects such as quantized vortices and solitons have been performed using semi-conductor planar micro-cavities. Building on these experiments done by the LKB group, we propose to use a different geometry (propagating light instead of confined) to study quantum fluids of light.

Quantum wells embedded in microcavities are systems where the interaction between matter (excitons) and the light (photons) exhibits unusual characteristics. In the regime of strong coupling between excitons and the photons, the fields can be described as a pseudo-particle, which is a mixed state of light and matter. These pseudo-particles are called polaritons.

These polaritons are bosons that are confined in a 2D structure (transverse plane of the microcavity). Due to Coulombian interactions, fascinating phenomena such as superfluidity and Bose-Einstein condensation can be observed.

The LKB Quantum Optics group has a recognize expertise in this domain and is looking forward on developing a new experimental setup to study the propagation of polariton superfluids in an optically controlable potential landscape. As we already demonstrated in the group, a defect in the microcavity acts like a potential well and induces the formation of vortices, anti-vortices and solitons. With a non-resonant light source, one can induce artificial defects in the cavity by injecting excitons. We aim at building an experiment in which the transverse beam profile can be set arbitrarily and reconfigure easily. This can be achieved using a Spatial Light Modulator (SLM) in addition with an algorithm that generate a numerical hologram in the Fourier plan.

The internship is proposed to Master 2 students and can be followed by a PhD in the group.

Interested candidates should contact Quentin Glorieux (quentin.glorieux@lkb.upmc.fr) and Alberto Bramati (bramati@lkb.upmc.fr) for more details. For more details, please check our website: www.quantumoptics.fr.

The goal of this project is to increase the theoretical understanding of multimode photon-added and -subtracted states in the continuous variable regime. The basis for the work is this a theoretical formalism [1] that was recently developed in the group. This work is carried out in connection to experimental work on optical frequency combs.

Optical frequency combs have an enormous potential in a wide range of fields, among which we find metrology and data processing. In recent years, our group has experimentally developed setups where this optical frequency combs have been used to execute quantum protocols. As a concrete goal, we strive to use such light source to execute quantum computations. Important steps in this direction were the experimental realisation of continuous variable cluster states [2], and the verification of large scale entanglement [3].

The potential of these light sources stems from their highly multimode character. Indeed, the different frequencies that comprise the light are more generally associated with time-frequency modes which can be squeezed and entangled. These procedures are useful for rendering the light non-classical and for quantum data encoding, but they are not sufficient to perform computations that cannot be efficiently simulated by classical computers. The reason is that the statistics measurement outcomes for the field quadratures follows a Gaussian distribution which is easily simulated by a classical computer.

An experimentally feasible [4] way to fundamentally alter the statistics of measurement outcomes is to subtract or add a photon to the beam of light. This procedure can be implemented in a mode-tunable way, such that we can select in which superposition of modes the photon is added or subtracted [5]. The process of photon addition and subtraction can, under certain conditions, make the Wigner function of the light negative for some point in optical phase space. This means that the statistics of the measurement outcome can no longer be described by a normal probability distribution, and thus it can no longer be straightforwardly simulated by classical computers.

However, such photon-added or subtracted multimode light is not not very well understood on a theoretical level. The goal of this internship is to help fill this void. Our group recently developed a theoretical framework [1,6] to study multimode photon-added or -subtracted states of light. In the internship, the intern will use these theoretical tools to investigate properties of these states, and if necessary (s)he will help extend the theoretical formalism. Potential topics than can be investigated include (but are not limited to):

**Characterising and measuring entanglement properties (in particular for mixed states)**- Measurement-based quantum computation with photon-added and subtracted states
- Techniques for measuring negativity of the Wigner function
- Quantifying quantum supremacy: how hard is it to simulate these states.
- Quantum transport with photon-added and subtracted states: transferring properties from one mode to the other
- Manipulating photon-added and -subtracted states with Gaussian operations
- Quantum metrology with photon-added and -subtracted states

Alternatively, interns may also choose to work on more general problems related to quantum optics on phase space, in particular in the Wigner function representation. One important question which is still widely open is that of the physicality of a non-Gaussian Wigner function: Under which conditions does a normalised function on phase space correspond to a well-defined quantum state?

*References*

[1] M. Walschaers, C. Fabre, V. Parigi, and N. Treps “*Entanglement and Wigner function negativity of multimode non-Gaussian states*” arXiv:1707.02285 (2017). – Accepted for publication in Phys. Rev. Lett.

[2] Y. Cai, J. Roslund, G. Ferrini, F. Arzani, X. Xu, C. Fabre, and N. Treps “*Multimode entanglement in reconfigurable graph states using optical frequency combs*” Nat Commun. **8**, 15645 (2017).

[3] S. Gerke, J. Sperling, W. Vogel, Y. Cai, J. Roslund, N. Treps, and C. Fabre “*Full Multipartite Entanglement of Frequency-Comb Gaussian States*” Phys. Rev. Lett. **114**, 050501 (2015).

[4] Y.-S. Ra, C. Jacquard, A. Dufour, C. Fabre, and N. Treps “*Tomography of a Mode-Tunable Coherent Single-Photon Subtractor*” Phys. Rev. X **7**, 031012 (2017).

[5] V. Averchenko, C. Jacquard, V. Thiel, C. Fabre, and N. Treps “*Multimode theory of single-photon subtraction*” New J. Phys. **18** 083042 (2016).

[6] M. Walschaers, C. Fabre, V. Parigi, and N. Treps “*Statistical signatures of multimode single-photon added and subtracted states of light*” arXiv:1708.08412 (2017). To be published in Phys. Rev. A

Interested candidates should contact Nicolas Treps (nicolas.treps@lkb.upmc.fr), Mattia Walschaers (mattia.walschaers@lkb.upmc.fr) or Valentina Parigi (valentina.parigi@lkb.upmc.fr)

For more details, please check our website: www.quantumoptics.fr.

Theoretical internship.

Duration: 3 to 6 months.

** **The last 20 years have seen a rapid development of ideas and proof-of-principle experiments regarding quantum technologies: quantum communication, quantum computation, quantum metrology and quantum simulation. If it is already clear that in principle the quantum world can provide useful advantages in terms of security, efficiency, speed, and sensitivity, we still need to solve the scalability problem. Indeed, generating simple systems for coding and manipulating quantum information is relatively easy. However, it is still not evident how to handle large quantum systems involving many degrees of freedom, in order, for instance, to implement a complete architecture where the quantum approach provides a real advantage compared to conventional systems.

The Quantum Optics group at Laboratoire Kastler Brossel developed in the last years an all-optical solution using the different wavelengths of light as quantum channels and, within each quantum channel, the optical field quadratures to encode quantum information. In this regime, we demonstrated the scalability of multimode entanglement[1,2]. Furthermore, this resource can be tuned at the measurement stage to allow for the on-demand construction of quantum networks suitable for measurement based quantum computing[3,4].

This highly scalable entanglement is of Gaussian nature, in the sense that all measured quantities display Gaussian statistics. While this is sufficient for quantum metrology, it can be proved that any calculation performed using these states can be efficiently simulated using a classical computer. Thus, moving away from Gaussian statistics is a requirement to demonstrate quantum supremacy. We have implemented this *de-gaussification* operation using mode dependent photon subtraction[5], while demonstrated theoretically that it can lead to a type of entanglement fundamentally different from the Gaussian one[6].

*Internship project*

The internship will aim at characterizing spectrally broaden femtosecond pulses and use them to perform quantum measurements demonstrating large multimode entanglement compatible with measurement based quantum computing protocols.

In the current experimental setup, the intrinsic number of modes are involved in the generated network could be of the order of 50. All measurements we have implemented so far are able to address a maximum of 16 modes. These are performed using multimode homodyne detection, which is based on the interferences between the signal beam and an intense local oscillator. The oscillator originates from the ultrafast laser source, whose spectral extension is not large enough to efficiently interact with all the generated quantum modes.

We have recently implemented spectral broadening of the local oscillator using self-phase modulation in a short photonic crystal (PC) fiber, expanding the spectrum of the original laser by a factor of two. This source will be characterized, improved, and used for multimode measurement to demonstrated unprecedented number of quantum channels simultaneously available.

*PhD project*

The internship can be naturally followed by a PhD project that will be based on this experimental setup. The main objectives are to, on the one hand, explore the possibilities of highly multimode non-Gaussian light, a largely unknown regime so far both theoretically and experimentally, and on the other hand to implement non-Gaussian quantum protocols, as for instance quantum verification. This last part could be performed within the collaboration of the group with ATOS Company, in the quest of the future quantum computing architecture.

[1] J. Roslund, R. M. De Araujo, S. Jiang, and C. Fabre, Nature PhotonicsInterested candidates should contact Nicolas Treps (nicolas.treps@lkb.upmc.fr), Claude Fabre (claude.fabre@lkb.upmc.fr) or Valentina Parigi (valentina.parigi@lkb.upmc.fr)

For more details, please check our website: www.quantumoptics.fr.

Experimental internship.

Duration: 3 to 6 months. Possible PhD.

Optical frequency combs have revolutionized frequency metrology and found application from fundamental constants measurements and optical clocks, to spectroscopy and telecommunications. They are becoming key elements of more and more advanced experiments as they offer unprecedented properties in terms of spectral purity, pulse repetition rate stability and spectrum spanning. It is thus crucial to get a better understanding of the noise sources and coupling mechanisms that directly impact the performances.

On the other hand, quantum optics has been, from its beginning, the driving force both for the exploration of fundamental limits of the quantum world and for conceiving seminal ideas and applications of the so-called quantum technologies. The last 20 years have seen a rapid development of ideas and proof-of-principle experiments involving the fields of quantum communication, quantum computation, quantum metrology and quantum simulation.

In the photonic scenario single photons are the traditional carriers of quantum information, but the scalability and adaptability of quantum resources is still an open problem. The system developed in the Quantum Optics group of the Laboratoire Kastler Brossel is based on a complementary approach where the information is not encoded on discrete variables, like the number of photons, but in continuous variable of the electromagnetic field, i.e. its amplitude and phase. In our approach, in the case of Gaussian quantum states, the resource to be scaled-up is the number of modes in which these states of light are simultaneously and deterministically generated[1].

This project is based on ultra stable femtosecond mode locked laser (∼ 30 fs pulses, repetition rate of 150 MHz). We first apply the quantum optics tools to give a modal description of classical femtosecond lasers, shining new light on laser dynamics[2]. This study will be pursued and applied to other type of lasers, using further developments achieved in a quantum context. This part of the project is supported by an ANR-DGA project in collaboration with Thales and Laboratoire Aimé Coton.

In order to develop quantum complex network, we use this laser to pump a non-linear crystal. Each pump pulse generates a couple of entangled pulses in two separate spatial channels. While the two will maintain the spectral multi-mode decomposition that can already be arranged as a quantum network, one of the two channels will be delayed in the temporal domain by the time between two consecutive pulses, before being mixed with the other channel on a 50:50 beam-splitter. This will create a dual-rail temporal structure characterized by non-trivial entanglement, which can involve up to 10^{6} temporal-modes. The interplay between the spectral and the temporal structure of entanglement connections will be used for generating quantum networks[3] with more elaborated geometry involving community structures. Optimization of quantum information protocols and quantum transport phenomena will be studied. Beyond the already identified quantum strategies involving multi-mode quantum states, the variety of experimental implementations opens new fundamental questions on how to describe the quantum nature of a large multiple system and which kind of advantages can be obtained from these resources.

We specifically propose an internship on the detection, at the quantum level, of single pulses with a 150MHz repetition rate. The general aim is being able, in real time, to extract the multimode quantum structure in time and frequency of both the classical and quantum sources available.

- [1] J. Roslund, R. M. De Araujo, S. Jiang, and C. Fabre, Nature Photonics
**8**, 109 (2014). - [2] R. Schmeissner, J. Roslund, C. Fabre, and N. Treps, Phys Rev Lett
**113**, 263906 (2014). - [3] J. Nokkala, F. Galve, R. Zambrini, S. Maniscalco, and J. Piilo, Sci. Rep.
**6**, 26861 (2016).

Interested candidates should contact Nicolas Treps (nicolas.treps@lkb.upmc.fr), Claude Fabre (claude.fabre@lkb.upmc.fr) or Valentina Parigi (valentina.parigi@lkb.upmc.fr)

For more details, please check our website: www.quantumoptics.fr.

Experimental internship.

Duration: 3 to 6 months. Possible PhD.