Photon-assisted tunnelling with nonclassical microwaves in hybrid circuit QED systems

Motivated by recent experiments where superconducting microwave circuits have been coupled to electrons in semiconductor nanostructures [1-3], we study theoretically the interplay of non-classical light produced in a cavity with electron transport through a tunnel junction [4]. We demonstrate that this basic light-matter interaction is naturally characterized by non-positive definite quasi-probability distributions which are intimately connected to the Glauber-Sudarshan P-function. We further demonstrate that this negative quasiprobability has unequivocal signatures on the differential conductance that should be easily detectable in state of art experiments. This thus turns the tunnel junction into a non-trivial probe of the microwave state. We also discuss the non-trivial backaction of the junction current on the cavity.\\[4pt] [1] T. Frey, P. Leek, M. Beck, A. Blais, T. Ihn, K. Ensslin, and A. Wallraff, Phys Rev Lett 108 046807(2012)\\[0pt] [2] M. Delbecq, V. Schmitt, F. Parmentier, N. Roch, J. Viennot, G. Feve, B. Huard, C. Mora, A. Cottet, and T. Kontos, Phys. Rev. Lett. 107, 256804 (2011)\\[0pt] [3] O. Parlavecchio, C. Altimiras, J.R. Souquet, P. Simon, I. Safi, P. Joyez, D. Vion, P. Roche, D. Est\`eve, F. Portier arXiv:1409.6696\\[0pt] [4] Nature Communications 5, 5562 (2014)