# Theory of photon-subtracted quantum states of multimode light

*Theory of photon-subtracted quantum states of multimode light.*

*Theory of photon-subtracted quantum states of multimode light.*

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.