Modes and states in quantum optics

A few decades ago, quantum optics stood out as a new domain of physics by exhibiting states of light with no classical equivalent. The first investigations concerned single photons, squeezed states, twin beams, and Einstein-Podolsky-Rosen states, which involve only one or two modes of the electromagnetic field. The study of the properties of quantum light then evolved in the direction of more and more complex and rich situations, involving many modes of the spatial, temporal, frequency, or polarization type. Actually, each mode of the electromagnetic field can be considered as an individual quantum degree of freedom. It is then possible, using the techniques of nonlinear optics, to couple different modes and thus build in a controlled way a quantum network [H. Jeff Kimble, Nature (London) 453, 1023 (2008)] in which the nodes are optical modes, and that is endowed with a strong multipartite entanglement. In addition, such networks can be easily reconfigurable and are subject only to weak decoherence. They indeed open many promising perspectives for optical communications and computation. Because of the linearity of Maxwell equations a linear superposition of two modes is another mode. This means that a "modal superposition principle" exists hand in hand with the regular quantum state superposition principle. The purpose of this review is to show the interest of considering these two aspects of multimode quantum light in a global way. Indeed, using different sets of modes allows one to consider the same quantum state under different perspectives: a given state can be entangled in one basis and factorized in another. It is shown that there exist some properties that are invariant over a change in the choice of the basis of modes. The method of finding the minimal set of modes that are needed to describe a given multimode quantum state is also presented. It is then shown how to produce, characterize, tailor, and use multimode quantum light while also considering the effect of loss and amplification on such light and the modal aspects of the two-photon coincidences. Switching to applications to quantum technologies, this review shows that it is possible to find not only quantum states that are likely to improve parameter estimation but also the optimal modes in which these states "live." Finally, details on how to use such quantum modal networks for measurement-based quantum computation are presented.