Multiphoton quantum optics and quantum state engineering
Introduction
In this report we review and discuss recent developments in the physics of multiphoton processes in nonlinear optical media and optical cavities, and their manipulation in the presence of passive and active optical elements. We review as well effective Hamiltonian models and Hamiltonian dynamics of nonquadratic (anharmonic) multiphoton interactions, and the associated engineering of nonclassical states of light beyond the standard coherent and two-photon squeezed states of linear quantum optics.
We present a detailed analysis of the methods and techniques for the production of genuinely quantum multiphoton processes in nonlinear media, and the corresponding models of multiphoton effective nonlinear interactions. Our main goal is to introduce the reader to the fascinating field of quantum nonlinear optical effects (such as, e.g., quantized Kerr interactions, quantized four-wave mixing, multiphoton down conversion, and electromagnetically induced transparency (EIT) and their application to the engineering of (generally nonGaussian), nonclassical states of the quantized electromagnetic field, optical Fock states, macroscopic superposition states such as, e.g., optical Schrödinger cat states, multiphoton squeezed states and generalized coherent states, and multiphoton entangled states.
This review is mainly devoted to the theoretical aspects of multiphoton quantum optics in nonlinear media and cavities, and theoretical models of quantum state engineering. However, whenever possible, we tried to keep contact with experimental achievements and the more promising experimental setups proposals. We tried to provide a self-contained introduction to some of the most relevant and appropriate theoretical tools in the physics of multiphoton quantum optics. In particular, we have devoted a somewhat detailed discussion to the recently introduced formalism of canonical multiphoton quantum optics, a systematic and consistent multiphoton generalization of standard one- and two-photon quantum optics. We have included as well an introduction to group-theoretical techniques and nonlinear operatorial generalizations for the definition of some types of nonclassical multiphoton states. Our review is completed by a self-contained discussion of very recent advances that by combining linear and nonlinear optical devices have lead to the realization of some multiphoton entangled states of the electromagnetic field. This multiphoton entanglement, that has been realized either on discrete or on continuous variables systems, is relevant for applications in efficient quantum computation, quantum teleportation, and related problems in quantum communication and information.
Multiphoton processes occur in a large variety of phenomena in the physics of matter–radiation interactions. Clearly, it is a task beyond our abilities and incompatible with the requirements that a review article should be of a reasonable length extension, and sufficiently self-contained. We thus had to make a selection of topics, that was dictated partly by our personal competences and tastes, and partly because of the rapidly growing importance of research fields including engineering and control of nonclassical states of light, quantum entanglement, and quantum information. Therefore, our review is concerned with that part of multiphoton processes that leans towards the “deep quantum” side of quantum optics, and it does not cover such topics as Rydberg states and atoms, intense fields, multiple ionization, and molecular processes, that are all, in some sense, on the “semiclassical” side of the discipline. Moreover, we have not included sections or discussions specifically devoted to quantum noise, quantum dissipative effects, and decoherence. A very brief “framing” discussion with some essential bibliography on these topics is included in the conclusions.
The plan of the paper is the following. In Section 2 we give a short review of linear quantum optics, introduce the formalism of quasi-probabilities in phase space, and discuss the basics of homodyne and heterodyne detections and of quantum state tomography. In Section 3 we introduce the theory of quantized macroscopic fields in nonlinear media, and we discuss the basic properties of multiphoton parametric processes, including the requirements of energy conservation and phase matching, and the different experimental techniques for the realization of these requirements and for the enhancement of the parametric processes corresponding to higher-order nonlinear susceptibilities. In Section 4 we discuss in detail some of the most important and used parametric processes associated to second- and third-order optical nonlinearities, the realization of concurring interactions, including three- and four-wave mixing, Kerr and Kerr-like interactions, three-photon down conversion, and a first introduction to the engineering of mesoscopic quantum superpositions, and multiphoton entangled states. In Section 5 we describe group theoretical methods for the definition of generalized (multiphoton) coherent states, Hamiltonian models of higher-order nonlinear processes, including degenerate k-photon down conversions with classical and quantized pumps, Fock state generation in multiphoton parametric processes, displaced–squeezed number states and Kerr states, intermediate (binomial) states of the radiation field, photon-added and photon-subtracted states, higher-power coherent and squeezed states, and general n-photon schemes for the engineering of arbitrary nonclassical states. As already mentioned, in Section 6 we report on a recently established general canonical formulation of multiphoton quantum optics, that allows to introduce multiphoton squeezed states associated to exact canonical structures and diagonalizable Hamiltonians (multiphoton normal modes), we study their two-mode extensions defining nonGaussian entangled states, and we discuss some proposed setups for their experimental realization. In Section 7 we give a bird-eye view on the most relevant theoretical and experimental applications of multiphoton quantum processes and multiphoton nonclassical states in fields of quantum communication and information. Finally, in Section 8 we present our conclusions and discuss future perspectives.
Section snippets
A short review of linear quantum optics
In 1927, Dirac [1] was the first to carry out successfully the (nonrelativistic) quantization of the free electromagnetic field, by associating each mode of the radiation field with a quantized harmonic oscillator. Progress then followed with the inclusion of matter–radiation interaction [2], the definition of the general theory of the interacting matter and radiation fields [3], [4], and, after two decades of strenuous efforts, the final construction of divergence-free quantum electrodynamics
Parametric processes in nonlinear media
The advent of laser technology allowed to begin the study of nonlinear optical phenomena related to the interaction of matter with intense coherent light, and extended the field of conventional linear optics (classical and quantum) to nonlinear optics (classical and quantum). Historically, the fundamental events, which marked such a passage, were the realization of the first laser device (a pulsed ruby laser) in 1960 [12] and the production of the second harmonic, through a pulsed laser
Second and third order optical parametric processes
Moving on from the basic aspects introduced in Section 3, we now begin to discuss in some detail the most important multiphoton processes occurring in nonlinear media. In this section we restrict the analysis to processes generated by the strongest optical nonlinearities, i.e. those associated to the second- and third-order susceptibilities as described by the trilinear Hamiltonian (86) and the quadrilinear Hamiltonian (89). Historically, after the experimental generation of the second harmonic
Models of multiphoton interactions and engineering of multiphoton nonclassical states
The many fascinating properties of nonclassical states of light allow, at least in principle, innovative and far-reaching applications in quantum control, metrology, interferometry, quantum information and communication. In a way, although the very concept of nonclassicality presents many subtleties and its quantification is somehow still controversial, one might go so far to say that it should be considered a physical resource to be exploited, much in the same sense as energy and entropy. In
Canonical multiphoton quantum optics
In Section 5 we have reviewed several interesting approaches sharing a common goal: the generalization of two-photon interactions to multiphoton processes associated to higher order nonlinearities, for the engineering of multiphoton nonclassical states of the electromagnetic field. However relevant in various aspects, all these methods fail in extending the elegant canonical formalism of two-photon processes, that, via the linear Bogoliubov transformation (or the equivalent unitary squeezing
Introduction and general overview
In the last two decades, systems of quantum optics are progressively emerging as an important test ground for the study and the experimental realization of quantum computation protocols and quantum information processes. In this section, after a very brief resume of some essential concepts of quantum information theory, we discuss the role and possible applications of nonlinear interactions and multiphoton states in this rapidly growing discipline. Clearly, in this short summary, we will only
Conclusions and outlook
Quantum optics plays a key role in several branches of modern physics, involving conceptual foundations and applicative fallouts as well, ranging from fundamental quantum mechanics to lasers and astronomical observation. To understand the vast and complex structure of quantum optics, it is sufficient to list the phenomena that heavily involve multiphoton processes: among them, ionization processes, spontaneous emission, photo-association/dissociation of molecules, quantum interference, quantum
Acknowledgements
We acknowledge financial support from MIUR under project ex 60%, INFN, and Coherentia CNR-INFM.
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