Modes and states in Quantum Optics

Abstract

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 EPR states, that 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, either spatial, temporal, frequency, or polarization modes. 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 to build in a controlled way a quantum network (Kimble, 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 subject only to weak decoherence. They open indeed 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 the present 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 to consider the same quantum state under different perspectives: a given state can be entangled in one basis, factorized in another. We will show that there exist some properties that are invariant over a change in the choice of the basis of modes. We will also present the way to find the minimal set of modes that are needed to describe a given multimode quantum state. We will then show how to produce, characterize, tailor and use multimode quantum light, consider the effect of loss and of amplification on such light and the modal aspects of the two-photon coincidences. Switching to applications to quantum technologies, we will show in this review 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". We will finally present how to use such quantum modal networks for measurement-based quantum computation.

Quantum state engineering in arrays of nonlinear waveguides

“…Remarkably, the kind of equations that we find here for spatial modes are formally similar to those that appear in the context of SPDC in frequency combs [24,32,33]. We draw parallels between the spectral approach that leads to multimode entanglement and our spatial approach in ANW in Table I.…”

“…In the next section II B, we introduce both linear and nonlinear supermode bases, work out the corresponding propagation equations, and give the general solution to the propagation problem. Furthermore, we use the relationship between the two bases to draw mathematical parallels with SPDC frequency modes [24,32,33].…”

“…The measurement of quantum noise variances and correlations is carried out by multimode balanced homodyne detection (BHD) [33]. In a fully fibered approach the multimode squeezed state generated in the array can be collected in optical fibers through a V-groove array.…”

Detecting a Target With Quantum Entanglement

“…Let us start by defining a set of modes of the electromagnetic field [21] {f i (r, t)} as solutions of the wave equation…”

“…In the classical illumination scenario, we send the coherent state | √ N s to the target region and we measure the returned light, which is either in a strong thermal state (H 0 is true) or in a superposition of a very weak coherent state and a much stronger thermal state (H 1 is true). Therefore, under both hypothesis, the returned state is a Gaussian state of the form (21) with mean vectors x0 = (0, 0) (under H 0 ) and x1 = (2 √ κN s , 0) (under H 1 ). The covariance matrix of the return mode is the same under both hypothesis and is given by…”