Phase space is a useful tool to represent physical systems, such as the position and momentum of a particle, or the electric field of propagating light. It is common to describe quantum states of such systems by their Wigner function: the quantum generalization of the joint probability distribution. Of particular interest are the states for which the Wigner function reaches negative values. This “Wigner negativity” is a hallmark of quantum phenomena, and is known to be necessary for quantum computing. In our recent publications [1,2], we theoretically study a method to remotely generate Wigner negativity. First, we divide our system into two subsystems. Then, we show that upon appropriate measurements on one subsystem one can induce Wigner negativity in the other subsystem, provided there are strong quantum correlations between the two. We precisely quantify which level of correlations and measurements are necessary and sufficient for success.

More specifically, we start from a Gaussian state and provide a general expression for the Wigner function of one part of this system (kept with Alice), when it is conditioned upon a measurement performed on another part of the system (kept with Bob). We prove [1] that Alice can only acquire Wigner negativity when they initially share a state with the following property: upon Alice’s measurements of position and momentum, Bob obtains conditional measurement statistics that violate Heisenberg’s inequality. Alice can then perform Einstein-Podolsky-Rosen steering on Bob’s subsystem.

Our work [1] thus fundamentally connects Wigner negativity and Einstein-Podolsky-Rosen steering, having crucial consequences for quantum state engineering. In previous work [2], we had already shown a complementary result: when there is Einstein-Podolsky-Rosen steering in the initial Gaussian state, Bob can always induce Wigner negativity in Alice’s subsystems by performing a clever operation known as “photon subtraction”. **This means that we have found a new definition for Einstein-Podolsky-Rosen steering. It is the quantum correlation in the Gaussian state, shared by Alice and Bob, that allows Bob to induce Wigner negativity in Alice’s subsystem.**

- M. Walschaers, V. Parigi, and N. Treps,
*Practical Framework for Conditional Non-Gaussian Quantum State Preparation*, PRX Quantum**1**, 020305 (2020) - M. Walschaers and N. Treps,
*Remote generation of Wigner-negativity through Einstein-Podolsky-Rosen steering,*Phys. Rev. Lett.**124**, 150501 (2020).

Our group, the multimode quantum optics group of Laboratoire Kastler Brossel, pioneered many aspects of continuous variable (CV) approach to quantum optics. Our main objects of interest are therefore the quadratures of the electric field, which are typically measured through homodyne detection. Our activities generally span both spatial and spectral modes, which we manipulate to develop tools for quantum computation, communication, and metrology.

Our general objective is the creation of multimode squeezed states of optical pulses, either by a synchronously pumped optical parametric oscillator (SPOPO) [1]., or through nonlinear waveguides. As such, our sources can create big entangled Gaussian states, which can be probed in arbitrary modes by shaping the local oscillator of the homodyne detector. In recent years, we have gradually explored mode-selective photon subtraction and addition, which allows us to generate multimode non-Gaussian states of light in a highly versatile way [2].

The group has a strong experimental focus, but also has purely theoretical activities where the framework of CV quantum optics is further developed. In particular for non-Gaussian quantum states, there are still many fundamental questions that remain unanswered [3]. The interplay between theoretical work and experiments is a key element of our group.

In the proposed post-doctoral project, you will work primarily on the SPOPO experiment where you will manipulate the spectral modes of a frequency comb to engineer entangled Gaussian states, and use photon-subtraction to induce non-Gaussian features in these states. Your primary goal is to develop new measurement techniques to extract more information from these intricate quantum states.

On the one hand, you will optimize a multi-pixel setup, which will allow for frequency-resolved multimode homodyne detection. On the other hand, you will implement a double homodyne detection scheme (also referred to as heterodyne detection in some literature), to implement a projective measurement on coherent states.

You will then use these new detection schemes to perform multi-parameter estimation, an important subject in quantum metrology. Furthermore, you will get the chance to work in a collaboration with the neighbouring computer science laboratory (LIP6) to experimentally implement new verification protocols for quantum computation [4].

Gradually, more and more non-Gaussian elements will be added to the experimental setup. On the level of state engineering, you will have the possibility to work on the subtraction or addition of multiple photons. On the detection stage of the experiment, you can incorporate photon-number detection schemes, and test a new type of mesoscopic detector. All of these elements can be used to experimentally explore non-Gaussian quantum steering and to perform ultra-sensitive parameter estimation. Our ultimate goal is then to experimentally establish a relation between parameter estimation and quantum steering [5].

As a whole, the group has a tradition of working together with a diverse range of people from very varied backgrounds. This diversity often leads to fruitful scientific input from different points of view, and it allows the group to explore new avenues. This has, for example, led to a growing activity in theoretical work over the past few years. The strength of our group is the constructive interplay between all these different points of view. Furthermore, the moderate size of our group gives PhD students and postdocs the opportunity to discuss with PIs on a daily basis. This gives rise to a dynamical atmosphere with a lot of space for discussion.

In your day to day activities, you will supervise PhD students who work on the same experimental setup, and you are responsible for the everyday organisation of the experimental work. You will be involved in the European FET Open project “STORMYTUNE”, which will enlarge your scientific network and provide opportunities for international collaborations.

Send CV and motivation letter to nicolas.treps@upmc.fr*Application process:*Preferentially apply before November 1st (late application will be considered as long as the position has not been filled).*Application deadline:*Monthly net salary between 2100€ and 2800€, depending on experience*Salary:*flexible*Starting date:*

References

[1] J. Roslund, R. M. De Araujo, S. Jiang, and C. Fabre, Nature Photonics **8**, 109 (2014)

[2] Y.-S. Ra, A. Dufour, M. Walschaers, C. Jacquard, T. Michel, C. Fabre, and N. Treps, Nature Physics **11**, 1 (2019).

[3] M. Walschaers, C. Fabre, V. Parigi, and N. Treps, Phys Rev Lett **119**, 183601 (2017).

[4] U. Chabaud, D. Markham, and F. Grosshans. Phys Rev Lett **124** 063605 (2020).

[5] B. Yadin, M. Fadel and M. Gessner, arXiv:2099.08440 (2020).

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Complex network theory has provided a deep insight of complex systems, assembling theoretical tools able to the describe dynamical behavior of biological, social and technological structures. During the recent years a new area applying network theory and complex networks to quantum physical systems has emerged [1,2,4]. May complex networks structures help us to have a better understanding of the quantum world? Which kind of complex networks will be used in future quantum information technologies? The ERC project COQCOoN is going to tackle the subject via theory and experiments based on multimode optical quantum networks.

The Multimode quantum optics group at Laboratoire Kastler Brossel (C. Fabre, N. Treps, V. Parigi and M. Walschaers) is one of the main actors in devising experimental setups for producing cluster states, i.e. large entangled networks useful in quantum information protocol on a large scale. We recently demonstrated that these networks can be reshaped at will, and they can even take the complex structure of the real-world information networks, like internet [3,4,5,6].

The PhD project concerns the experimental implementation of quantum complex networks via femtosecond laser sources at telecom wavelengths, which are the most suitable for long-range fiber-based quantum communications and allow for the exploitation of the already existent integrated components developed in classical communications. The goal is the implementation of advanced quantum information protocols; theoretical activity can also be included in the project.

Practical information: applicants should have a Master diploma in Physics. Familiarity with quantum information and/or experimental optics will be valuable.

Starting date: Fall 2020. Location: Laboratoire Kastler Brossel (Paris).

For inquires, expression of interest and applications write to valentina.parigi@lkb.upmc.fr. Application should include a CV, a motivation letter and reference names and should be sent not later than 30^{th} of June 2020.

[1] G. Bianconi “Interdisciplinary and physics challenges of network theory” Europhys. Lett. 11156001 (2015)

[2]J. Biamonte, M. Faccin, and M. De Domenico, Complex networks from classical to quantum, Communications Physics 2, 53 (2019).

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

[4] J. Nokkala, F. Arzani, F. Galve, R. Zambrini, S. Maniscalco, J. Piilo, N. Treps, V. Parigi, Reconfigurable optical implementation of quantum complex networks, New J. Phys. **20, **053024 (2018)

[5] F. Sansavini and V. Parigi “Continuous variables graph states shaped as complex networks: optimization and manipulation” Entropy 22, 26 (2020)

[6] M. Walschaers, S. Sarkar, V. Parigi, and N. Treps “Tailoring Non-Gaussian Continuous-Variable Graph States”, Phys. Rev. Lett. 121, 220501 (2018)

]]>In our group, we use light to develop basic building blocks for such a quantum computer. In particular, we use our *quantum frequency comb* as a platform for quantum information processing. The different frequencies can be entangled in a controllable way, which makes the platform scalable and programable. However, in this platform it is hard to access the class of quantum states that are known as *non-Gaussian states *(see textbox for further details), which means that the quantum computer is not universal.

In our recent Nature Physics paper, the research team developed a technique known as “mode-selective photon subtraction”, where one photon is literally taken out of the light beam to create such non-Gaussian states. A crucial element of the experiment is the control of the frequency of the subtracted photon, as we even managed to perform photon subtraction in a superposition of frequencies. Due to this degree of control, we could explore the interplay between the non-Gaussian effects that are induced by removing a photon, and the quantum entanglement that is present in the quantum frequency comb. This allowed us to verify a previous theoretical prediction that non-Gaussian features spread out because of the entanglement.

We work in the so-called continuous-variable approach, which means that the quantities which we measure can take any possible real value (even in the quantum regime). In practice, what we measure are the amplitude and the phase of the electric fields that comprise our quantum frequency comb.

When we do not subtract a photon, the statistics of these measurements will always lead to a normal (Gaussian) probability distribution. Non-Gaussian states, on the other hand, are much wilder and can have more exotic measurement statistics form the phase and the amplitude of the field. The measurement can be so exotic, that we can no longer represent the probability of measuring a certain amplitude and a certain phase by one joint probability distribution. We can follow a mathematical procedure to construct such a joint probability distribution, but the results will be quite strange, and we will find what seem to be negative probabilities. This resulting function that describes the joint measurement statistics of the phase and amplitude of the light field is known as the Wigner function.

The fact that it reaches negative values (and thus is not a probability distribution) reflects the fact that the amplitude and the phase of the field are complementary observables. Their measurements are constrained by Heisenberg’s uncertainty relation; hence they cannot be measured precisely at the same time. Therefore, it is logical that weird things can happen when we try to deduce the joint measurement statistics! These negative values of the Wigner function are a real hallmark of quantum physics. We need them to violate Bell inequalities and to achieve them construct universal quantum computers. We showed that by photon subtraction we can induce these properties in a large system.

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Light offers a vast potential in the development of modern quantum technologies due to its intrinsic resilience to decoherence effects that tend to scramble quantum information in matter-based setups. One avenue for employing light to process quantum information focuses on the continuous variable regime, where the observables of interest are the quadratures of the electric field. These continuous variables have proven their worth as a platform for creating huge entangled states (entangling up to one million optical modes). Additionally, this entanglement can be created in a deterministic fashion, and the resulting states are easily manipulated with standard techniques in optics [1,2]

To reach a quantum advantage, and perform a task that cannot be efficiently simulated with a classical device, we require more than just entanglement. The additional ingredient is non-Gaussian statistics in the outcomes of the quadrature measurements. More specifically, we must create quantum states with a negative Wigner functions. At LKB, we have recently developed a mode-tunable photon subtractor as a device for creating such states [3,4]. As such, we now have the possibility to produce large entangled states and to render them non-Gaussian. This opens up a whole new realm of research, where a vast amount of questions on the interplay between entanglement and non-Gaussian effects remain unanswered [4,5]. Within this internship, we will explore some of these questions.

In particular, we will investigate the interplay between non-Gaussian aspects of quantum states and quantum steering (also known as Einstein-Podolsky-Rosen steering). The latter is a special feature of certain quantum correlations, and is in some sense “stronger” than quantum entanglement. In general, when two subsystems, **A **(Alice) and **B **(Bob) are correlated, a measurement on **A **improves the precision of predictions for a measurement on **B**. In classical statistical theories, there are limits on the amount of information that can be extracted in this way. However, in quantum physics, these limits can be overcome, and in some cases we can find quantum correlations that allow us to make predictions about system **B **that are more precise than possible with any classical correlation. Such quantum correlations are said to be steerable, and **A **is said to be able to steer **B**. In continuous variable quantum optics, a profound example of a steerable quantum correlation can be found in Einstein-Podolsky-Rosen states.

The internship offers two possible directions of research to probe how quantum steering and non-Guassian effects are intertwined:

- On the one hand, we explore how Gaussian quantum states with steerable quantum correlations react to non-Gaussian operations such as photon subtraction. This part will be an extension of previous work [5] done by the group.
- On the other hand, we explore how quantum states with manifestly non-Gaussian entanglement can be steered in a systematic way (see Figure). Brute force numerical methods are available to check whether a state is steerable or not, but our goal is rather to acquire an analytical understanding that allows us to learn something about the properties of these non-Gaussian states.

Finally, we will explore the possible connection between non-Gaussian entanglement and remote preparation of negativity of the Wigner function. The above figure shows the scenario where all the entanglement between Alice and Bob is non-Gaussian. However, we might consider modifying the beam splitter after the single-photon detector, such that Alice and Bob are working in a different basis. In this sense, it is an important question what are the signatures of non-Gaussian entanglement in different mode bases. This may potentially make it possible to experimentally observe non-Gaussian entanglement and steering.

**Practical aspects: **The starting and ending dates of the internship are flexible, but the project is intended to last between three and six months (depending on the candidates prior knowledge in quantum optics). Ideally the internship will either end before or start after august 2020, in accordance with the organisation of the academic year in France.

**Subsequent PhD possibilities: **The subject of non-Gaussian states in quantum optics is vast, and interested student who choose this project have the possibility to apply for a PhD position in the group. Even though the internship is mainly theoretical, a potential PhD can also involve experimental work in the multimode quantum optics group. ** **

**Contact: Mattia Walschaers**

[1] J. Roslund, R. M. de Araujo, S. Jiang, C. Fabre, N. Treps, “Wavelength-multiplexed quantum networks with ultrafast frequency combs”, Nature Photonics 8, 109 (2014).

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

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

[4] Y.-S. Ra, A Dufour, M. Walschaers, C. Jacquard, T. Michel, C. Fabre, N. Treps, Non-Gaussian quantum states of a multimode light field, arXiv preprint arXiv:1901.10939, accepted Nat Physics

[5] M. Walschaers and N. Treps, Remote generation of Wigner-negativity through Einstein-Podolsky-Rosen steering, arXiv preprint arXiv:1912.02778.

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A first result discusses the remote preparation of Wigner-negativity by using Einstein-Podolsky-Rosen steering. The second result was obtained by Francesca Sansavini during her master thesis, where she showed how to shape continuous variable graph states as complex networks and how to optimise them.

]]>Mattia will continue working in the multimode quantum optics group, where he was previously active as a post-doc. His main research interest is theoretical quantum optics, with a particular interest for non-Gaussian quantum states and quantum entanglement. In the group he works together with the experimentalists to develop methods for detecting non-Gaussian quantum correlations and Wigner-negativity.

On the long term, he hopes to contribute to the development of continuous-variable quantum technologies in the lab.

]]>We will deal with Continuous Variables Quantum Complex Networks.

Follow future updates on our section Quantum Complex Networks. ]]>

This project concerns the implementation of complex quantum networks, i.e. quantum systems organized in a complex structure depicted as networks, and their exploitation for quantum simulations and optimized quantum information technologies.

The experimental implementation is based on multimode quantum process based on parametric interaction and time/frequency modes of femtosecond lasers (near-infrared and telecom wavelengths). Networks structures are encoded in the quantum correlations of continuous variables of the involved fields.

Candidates must hold an internationally recognized PhD in a field related to experimental quantum physics. A good background and past research/publication track record in experimental optics, and quantum physics is required. Knowledge of quantum information would be an advantage.

Application procedure: Inquiries and applications should be sent by email to Valentina Parigi (valentina.parigi@lkb.upmc.fr). Applications should include a detailed CV, a brief statement of research interests and two names of potential referees.

The position is founded by the ERC grant COQCOoN (COntinuous variables Quantum COmplex Networks) of Valentina Parigi, expected starting date: June 2019.

The team is part of the Multimode Quantum Optic team at Laboratoire Kastler Brossel (PIs: Nicolas Treps, Valentina Parigi and Claude Fabre)

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