**Overview**

A fundamental task in quantum metrology is to identify the ultimate sensitivity limit in the estimation of a parameter encoded into a quantum state. Even under ideal conditions, when all technical noise sources are removed, quantum noise poses unavoidable limitations to such estimation. In spite of that, quantum parameter estimation theory provides the tools to reduce noise by optimizing the output measurements. This optimization leads to the quantum Cramér-Rao lower bound, which gives the minimal uncertainty of the estimator of a parameter, and that can be further optimized by finding quantum states that, for a given parameter, maximize the value of the quantum Fisher information.

In the optical scenario, would it be in imaging, remote sensing or interferometric measurement, the parameter of interest does not only modify the quantum state of the probe light, but also its spatio-temporal distribution. This distribution is conveniently described in terms of *modes* i.e. normalized solutions of Maxwell’s equations in vacuum [1]. Optimal quantum parameter estimation is thus at the crossroads between quantum information theory and optical mode manipulation, where only by taking into account both classical and quantum optimization one can derive efficient and practical estimators.

The multimode quantum optics group of Laboratoire Kastler Brossel, pioneered many aspects of optical quantum parameter estimation, in particular in the continuous variable (CV) approach [2,3]. Our activities generally span both spatial and spectral modes, which we manipulate to develop tools for quantum computation, communication, and metrology. The group has a strong experimental focus, but also has purely theoretical activities where the framework of CV quantum optics is further developed. The interplay between theoretical work and experiments is a key element of our group.

**Project**

When dealing with extracting a few parameters from an image, using prior information allows for going far beyond the Rayleigh limit. In practice, this can be achieved using the correlation of electromagnetic field amplitudes at different transverse positions of the image plane, and not only the intensity distribution. Technically, this involves decomposing the incoming field into an orthonormal basis of spatial modes (typically, Hermite-Gaussian) and measuring the amplitude (or intensity) of each basis component. This method of spatial demultiplexing, or SpaDe [4], enables us to not only achieve sub-Rayleigh precision, but also, in some cases, reach the ultimate resolution limits allowed by quantum mechanics [5,6]. The objectives of the group are to:

- Determine the physical limits of multi-parameter estimation to quantum estimation theory.
- Demonstrate practical parameter estimation with mode demultiplexing
- Implement a Bayesian framework for static and dynamic superresolution imaging.

Furthermore, the group has an ongoing collaboration with Thales Research and Technology to study distant imaging with frequency conversion, whose aim is to bring the system to new applications.

The postdoctoral fellow will have to coordinate the group activities in optical parameter estimation to reach the above objectives. She/He will have a background either in theoretical or experimental physics, but an interest to combine both in order to bring modal approach to parameter estimation to a practical device and to apprehend the fundamental limits imposed by the quantum nature of light in multi-parameter estimation in the presence of experimental imperfection. The work can have extension either in the more fundamental studies of the group, in particular regarding the link between Quantum Fisher information and quantum non-gaussian states of light [7], or to more practical considerations in particular within the framework of the collaboration with Thales.

**Environment**

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 activity, and you are responsible for the everyday organization of the work. You will be involved in several European programs, which will enlarge your scientific network and provide opportunities for international collaborations.

*Application process:*Send CV and motivation letter to nicolas.treps@sorbonne-universite.fr*Application deadline:*Preferentially apply before 31^{st}of March 2023 (late application will be considered as long as the position has not been filled).

**References**

[1] C. Fabre and N. Treps, Modes and states in quantum optics. Rev. Mod. Phys. 92, 035005 (2020). https://doi.org/10.1103/RevModPhys.92.035005

[2] N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, A Quantum Laser Pointer, Science 301, 940 (2003).

[3] O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, C. Fabre, Ultimate sensitivity of precision measurements with in- tense Gaussian quantum light: a multi-modal approach, Phys. Rev. A 85, 010101(R) (2012). https://journals.aps.org/pra/abstract/10.1103/PhysRevA.85.010101

[4] Labroille, G. et al. Efficient and mode selective spatial mode multiplexer based on multi-plane light conversion. Opt. Express 22, 15599–15607 (2014). https://doi.org/10.1364/OE.22.015599

[5] Tsang, M., Nair, R. & Lu, X.-M. Quantum theory of superresolution for two incoherent optical point sources. Phys. Rev. X 6, 031033 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031033 .

[6] P. Boucher, C. Fabre, G. Labroille, and N. Treps, Spatial Optical Mode Demultiplexing as a Practical Tool for Optimal Transverse Distance Estimation, Optica, OPTICA **7**, 1621 (2020). https://doi.org/10.1364/OPTICA.404746

[7] C. E. Lopetegui, M. Gessner, M. Fadel, N. Treps, and M. Walschaers, Homodyne Detection of Non-Gaussian Quantum Steering, PRX Quantum 3, 030347 (2022). https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.3.030347

- Research Field
- Physics » Quantum mechanics
- Education Level
- PhD or equivalent

- Languages
- ENGLISH
- Level
- Good

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 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].

Apart from our experimental focus, the group 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.

**Project**

These theoretical activities are at the core the proposed post-doctoral project, which aim to unravel properties of non-Gaussian states. In particular, we focus on the study of non-Gaussian entanglement, i.e., quantum states where the quantum correlations themselves are of a non-Gaussian nature. This entanglement may be understood as a type of quantum correlations that cannot be witness based on the covariance matrix of the state. We previously showed the existence of such non-Gaussian quantum correlations in pure photon-subtracted states [4], but are currently lacking a more general theoretical framework to study its properties.

Your role as a post-doc is to help develop witnesses and measures for non-Gaussian entanglement, tailored towards multipartite systems ideally compatible with the group’s experimental capabilities. You will then apply these techniques to explore applications of non-Gaussian entanglement. Depending on your background, the applications could be aimed at various quantum technologies, ranging from quantum computing to quantum metrology.

Possible starting points for the work include the use of metrology-based entanglement witnesses [5], phase space methods [3], techniques from network theory [6,7], and more brute-force machine-learning techniques [8]. However, we encourage candidates to use their past experience the pursue alternative routes.

** **

**Environment**

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 participate in the supervision of PhD students who work on theoretical topics and interact with the experimentalists in the group. Your work fits in the ANR project “NoRdiC”, which provides support to enlarge your scientific network and establish new international collaborations.

**Practical information**

*Application process:*CV and motivation letter to be uploaded to CNRS portal*Application deadline:*16/02/2022.*Starting date:*we aim for 01/06/2022 but are flexible*Duration of contract:*Two years*Salary:*Monthly net salary between 2100€ and 2800€, depending on experience*For more information:*walschaers@lkb.upmc.fr

**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 **16**, 144–147 (2020).

[3] M. Walschaers, PRX Quantum 2, 030204 (2021).

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

[5] M. Gessner, L. Pezzè, and A. Smerzi, Quantum 1, 17 (2017).

[6] G. García-Pérez, M. A. C. Rossi, B. Sokolov, E.-M. Borrelli, and S. Maniscalco, Phys. Rev. Research 2, 023393 (2020).

[7] M. Walschaers, N. Treps, B. Sundar, L. D. Carr, V. Parigi arXiv:2012.15608

[8] V. Cimini, M. Barbieri, N. Treps, M. Walschaers, and V. Parigi, Phys. Rev. Lett. 125, 160504 (2020).

]]>Quantum communication experiments developed for communication networks require single photon sources with a tunable frequency and a narrow linewidth. Semiconductor quantum dots are good candidates for realizing such sources, but they need to be coupled to high finesse cavities to improve their coherence properties. Enhancement of light-matter interaction at the nanometer scale is a key issue of modern photonics. It is possible to tailor the properties of individual quantum emitters by modifying their electromagnetic environment. However, in order to achieve a strong influence of the environment on fluorescent emission, it is critical to control the position of the emitter with nanometer accuracy in the near field of a nanostructure. Our group is working to control the nano-positioning of single photon emitters on photonic nanostructures with high precision.

During the internship, the main goal will be to study experimentally a hybrid apparatus that couples the light from a single photon emitter to an optical nanofiber. In the long term, we will nano-position with an atomic-force microscope tip a single quantum emitter inside a cavity directly embedded in the nanofiber itself.

Optical cavities are often employed to increase the interaction between light and matter. A natural step forward for nanofiber-based hybrid systems is therefore the use of a cavity that can substantially enhance the coupling of the nano-emitter emission into the nanofiber. Indeed the interaction is enhanced by the transverse confinement of the field in the fiber core as well as the longitudinal confinement of the field between the mirrors. It has been theoretically predicted that the collection efficiency into the nanofiber guided modes can be enhanced up to 94% by incorporating a moderate finesse cavity structure to the nanofiber.

The nanofiber is home-made at LKB by heating up a commercial optical fiber that can be stretched in order to reduce its diameter down to 300 nm. This diameter is less than the wavelength of the light and will therefore induce an strong evanescent electromagnetic field around the nanofiber. An emitter placed on the nanofiber will then interact with this evanescent field and be coupled to the guided mode of the fiber.

Recent results [1] have been obtained on this topic and we are looking for a motivated, smart and curious candidate who wants to join the Quantum Optics group and to conduct high impact research in the field of NanoOptics.

1. S. Pierini, M. D’Amato, M. Goyal, Q. Glorieux, E. Giacobino, E. Lhuillier, C. Couteau, A. Bramati, ACS Photonics 7, 2265-2272 (2020),

Highly photo-stable Perovskite nanocubes: towards integrated single photon sources based on tapered nanofibers

Analogue gravity is a type of analogue quantum simulation that enables the study of gravitational effects in the laboratory [1]: it is possible to create conditions in which waves in media propagate as though they were in the vicinity of a black hole [2] or on an expanding universe [3], for example.

In the Quantum Optics Group at Laboratoire Kastler Brossel, we study excitons-polaritons in semiconductor microcavities and make them behave as “fluids of light”. At present, we are interested in engineering the flow profile of the fluid of light to create event horizon (the point of no-return that bounds the interior of the black hole) for excitations of the fluid. We aim to observe the Hawking effect at the horizon, that is the spontaneous emission of entangled pairs of excitations of the fluid at the horizon [4]. Specifically, the internship is concerned with setting up and using homodyne detection to this end.

We have recently obtained promising theoretical results and are currently assembling a new experiment to create the horizon and observe the Hawking effect. The M2 internship would consist in using this new experimental platform to collect data and in analysing this data by comparing it with theoretical predictions.

The student would work with the Polariton team (2 PhD students and a postdoc) under the supervision of Prof Alberto Bramati, who has strong expertise in quantum optics techniques like homodyne detection. This M2 internship could lead to a PhD project depending on funding availability. The research community at LKB is composed of people from all around the world and we strive to promote an inclusive environment. We encourage applications from female candidates and candidates from under-represented groups.

- [1] W. G. Unruh, Physical Review Letters 46, 1351 (1981).
- [2] L.-P. Euve ́, F. Michel, R. Parentani, T. Philbin, and G. Rousseaux, Physical Review Letters 117, 1079 (2016).
- [3] S. Eckel, A. Kumar, T. Jacobson, I. B. Spielman, and G. K. Campbell, Physical Review X 8, 021021 (2018).
- [4] M.J.Jacquet, T.Boulier,F.Claude, A.Maïtre, E.Cancellieri, C.Adrados, A.Amo, S.Pigeon, Q.Glorieux,

More details here

The project aims at the implementation of networks based on continuous-variable Gaussian correlations and non-Gaussian operations, and their exploitation for quantum simulations and optimized quantum information tasks. The post-doc will work on/supervise one of two running experimental setups at near infrared and telecom wavelengths based on multimode parametric processes in waveguides pumped via femtosecond laser. Protocols will concern: quantum simulation, quantum reservoir computing, multiparty quantum communication in complex networks.

The multimode quantum optics group at LKB carried out leading research in experimental generation of cluster states, i.e. large entangled networks useful in quantum information protocol on a large scale. The group has a strong experimental focus, but is also engaged in purely theoretical activities aiming at developing quantum optics in the CV framework.

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 and Continuous Variable systems would be an advantage.

Expected starting date: fall/winter 2021-22 (with some flexibility)

__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.

References

- Nokkala, R. Martínez-Peña, G. L. Giorgi, V. Parigi, M. C Soriano, R. Zambrini,
*Gaussian states of continuous-variable quantum systems provide universal and versatile reservoir computing*, Communications Physics volume 4, Article number: 53 (2021) - Roman-Rodriguez, B. Brecht, S. Kaali, C. Silberhorn, N. Treps, E. Diamanti, V. Parigi,
*Continuous variable multimode quantum states via symmetric group velocity matching*, New J. Phys. 23 043012 (2021) - Sansavini and V. Parigi,
*Continuous Variables Graph States Shaped as Complex Networks: Optimization and Manipulation*, Entropy 22, 26 (2020) - Mattia Walschaers, Nicolas Treps, Bhuvanesh Sundar, Lincoln D Carr, Valentina Parigi
*Emergent complex quantum networks in continuous-variables non-Gaussian states*arXiv preprint arXiv:2012.15608 - Cimini, M. Barbieri, N. Treps, M. Walschaers, and V. Parigi,
*Neural Networks for Detecting Multimode Wigner Negativity*, Phys. Rev. Lett. 125, 160504 (2020)

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Analogue gravity enables the study of fields on curved spacetimes in the laboratory [1]: it is possible to create conditions in which waves in media propagate as though they were in the vicinity of a black hole [2] or on an expanding universe [3], for example.

In the Quantum Optics Group at Laboratoire Kastler Brossel, we study exciton- polaritons in semiconductor microcavities and make them behave as “fluids of light”. At present, we are interested in engineering the flow profile of the fluid of light to create the analogue of a rotating black hole — an effective spacetime characterised by two intangible surfaces: the event horizon (the point of no-return that bounds the interior of the black hole) and, further out, the ergosurface (a point beyond which waves and particles cannot remain at rest with respect to an outside observer).

This can be done by pumping the microcavity with a Laguerre-Gauss beam, thus inducing a vortex flow in the fluid of light [4]. We want to observe the propagation of small amplitude waves (e.g. density perturbations) as well as phase singularities (vortices and dark solitons) on this rotating spacetime. This could lead to the observation of effects such as the Hawking effect, rotational superradiance or the black hole bomb.

We have recently gathered promising preliminary experimental results with rotating spacetimes and are currently assembling a new experiment to push these investigations further.

We are looking for talented and motivated post-doc researchers. The selected candidate will work with the Polariton Team under the supervision of Prof Alberto Bramati.

Contact: Prof. Alberto Bramati, alberto.bramati@lkb.upmc.fr

[1] W. G. Unruh, Physical Review Letters 46, 1351 (1981).

[2] L.P. Euve, F. Michel, R. Parentani, T. Philbin, and G. Rousseaux, Physical Review Letters 117, 1079 (2016).

[3] S. Eckel, A. Kumar, T. Jacobson, I. B. Spielman, and G. K. Campbell, Physical Review X 8, 021021 (2018).

[4] M. J. Jacquet, T. Boulier, F. Claude, A. Maıtre, E. Cancellieri, C. Adrados, A. Amo, S. Pigeon, Q. Glorieux, A. Bramati, et al., Philosophical Transactions of the Royal Society A 378, 20190225 (2020), arXiv: 2002.00043.

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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