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CRC Press eBooks

Nonclassical states of light propagating in Kerr media R. Tanas

V.V. Dodonov, V.I. Man'ko

Published Mar 13, 2003• 1 citation
Publisher

Abstract

In this chapter we review a number of quantum features of optical fields associated with propagation of intense light through nonlinear, isotropic media. We begin with the classical description of field propagation, introducing the nonlinear polarization of the medium which enters as the source term into the approximate field equations obtained from the Maxwell equations in the slowly varying amplitude approximation. This establishes the classical background for further quantum considerations. Next, we quantize the field and construct the effective Hamiltonian from which we get equations of motion for the quantum fields. We discuss a number of quantum effects such as photon antibunching, squeezing, the formation of Schrodinger cats and kittens, changes in field polarization due to the quantum nature of the field, as well as the quantum description of the field phase. The characteristic feature of quantum evolution — the periodicity — is strongly affected by dissipation. We give exact analytical formulae describing the quantum evolution of the field, including dissipation. We have collected results illustrating various aspects of quantum evolution and we believe that this review, although far from complete, will be a useful source of information on the subject.

DOI
10.1201/9781482288223-11

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In this chapter we review a number of quantum features of optical fields associated with propagation of intense light through nonlinear, isotropic media. We begin with the classical description of field propagation, introducing the nonlinear polarization of the medium which enters as the source term into the approximate field equations obtained from the Maxwell equations in the slowly varying amplitude approximation. This establishes the classical background for further quantum considerations. Next, we quantize the field and construct the effective Hamiltonian from which we get equations of motion for the quantum fields. We discuss a number of quantum effects such as photon antibunching, squeezing, the formation of Schrodinger cats and kittens, changes in field polarization due to the quantum nature of the field, as well as the quantum description of the field phase. The characteristic feature of quantum evolution — the periodicity — is strongly affected by dissipation. We give exact analytical formulae describing the quantum evolution of the field, including dissipation. We have collected results illustrating various aspects of quantum evolution and we believe that this review, although far from complete, will be a useful source of information on the subject.
We consider quantum-mechanically partially polarized light propagating through a Kerr-like medium. Using the usual form of the induced polarization P=A(E\ensuremath{\cdot}E∗{\mathrm{E}}^{\mathrm{*}}E∗)E+B(E\ensuremath{\cdot}E)E∗{\mathrm{E})\mathrm{E}}^{\mathrm{*}}E)E∗, the theory is formulated in terms of an effective Hamiltonian which is quartic in terms of the operators for two orthogonally polarized modes. Exact solutions in closed form for the Heisenberg equations of motion are obtained. These solutions are used to evaluate the physical behavior of various observables as the field propagates through a nonlinear medium. We also present explicit results for the time evolution of the input coherent and Fock states of the field. We show the generation of states that are macroscopic superposition of coherent states. We also find that if the input field is completely polarized, then due to quantum effects the output field becomes partially polarized. This is in contrast to the classical prediction and can have an important bearing on questions like topological phases of light propagating through a nonlinear medium. Numerical results for the energy in each mode, the correlation between two modes, and the higher-order correlations are presented. The input photon statistics is found to make a considerable difference in the dynamics.
Nonlinear phenomena in quantum optics are reviewed, with particular emphasis on scientific activities in quantum optics at the Natural Science Faculty of Palacký University ln Olomouc (Czech Republic). The first section is devoted to quantum and classical methods to obtain photon statistics in higher-harmonics generation, including a method of classical trajectories. It is followed by a discussion of photon bunching and antibunching of nonstationary fields. The chapter continues with a discussion of methods for reconstruction of quantum states. Other topics covered include the process of frequency downconversion with pulses, Zeno and anti-Zeno effects based on optical parametric processes, Raman nonlinear couplers as switching elements and sources of nonclassical light for optoelectronics and photonics, and quantum cryptography. Finally, quantum spatial propagation of a two-mode squeezed light in a cubically behaved quadratic medium is treated as another feature of a two-field squeezed steady-state light, and this approach is compared with the conventional one.
This chapter describes the formation, development, and major subjects of nonlinear optics and photonics. The semiclassical theory treats the light as a classical electromagnetic field governed by Maxwell equations, while the optical medium is a molecular system governed by quantum mechanics; the key parameters for this theory are various-order nonlinear susceptibilities. The quantum theory of radiation treats light as an ensemble of photons and the medium as a molecular system, both of which are described by quantum mechanics, and the key aspect is the concept of the so-called intermediate state represented by a virtual energy state level. The semiclassical theory gives a concise quantitative description of various nonlinear multi-wave mixing effects as well as those nonlinear processes related to the induced refractive-index changes. In contrast, the quantum theory of radiation can provide a conceptual interpretation for almost all known nonlinear optical effects and phenomena.
The aim of this work is to study how losses modify quantum effects in a system governed by the spatial nonlinear Schrodinger equation. Quantum effects in space, such as quantum noise, squeezing, and bunching, have been studied in a lossless case.1 We will consider the propagation of a field through a planar nonlinear waveguide,2,3 where confinement is provided in one transverse dimension (y direction) by a linear refractive index and in the orthogonal dimension (x direction) by the intensity dependence of the refractive index.1,4 Attenuation is given to the spatial distribution in the x-z plane. In this work we want to extend the dissipation approach described in Ref. 5 to a system governed by the spatiai-nonlinear Sclirodinger equation (SNLS). We use a quantum-field description1 in which the spatial variable of propagation z plays the role of the time in the standard quantum theory, and we apply it to the case of a strong plane wave that can be described as a classical field propagating through a nonlinear waveguide in the z direction.4 By using both paraxial and slowly-varying-envelope approximations, the propagation of the field in a planar waveguide is described by the NLS.1
The standard process for the production of strongly squeezed states of light is degenerate optical parametric amplification (OPA) below threshold in nonlinear dielectric media such as LiNbO3 or periodically poled potassium titanyl phosphate (KTP). Here, we present a graphical description of squeezed-light generation via OPA, visualizing the interaction between the nonlinear dielectric polarization of the medium and the electromagnetic quantum field. We explicitly focus on the transfer from the field's ground state to a squeezed vacuum state and from a coherent state to a bright squeezed state by the medium's second-order nonlinearity, respectively. Our pictures illustrate the phase-dependent amplification and deamplification of quantum uncertainties and give the phase relations between all propagating electromagnetic fields as well as the internally induced dielectric polarizations. The graphical description can also be used to describe the generation of nonclassical states of light via higher-order effects of the nonlinear dielectric polarization such as four-wave mixing and the optical Kerr effect.
This chapter begins by reviewing the semi-classical theory of weakly nonlinear optics, in which the atomic polarization is a power series in the field amplitude. After replacing the classical amplitudes by field operators, the expansion for the classical electromagnetic energy becomes the Hamiltonian. The third- and fourth-order classical terms (respectively representing three- and four-wave mixing) become three- and four-photon interaction terms. The three-photon terms describe sum-frequency generation and spontaneous down conversion. Down conversion is the principal modern source of entangled photons pairs. It has also been applied to yield absolute measurements of the quantum efficiency of detectors and as a source of hyper-entangled photons. The four-photon interaction terms describe frequency tripling, three-photon down conversion, and photon-photon scattering. The description of an experimental demonstration of photon-photon scattering in a rubidium vapor is followed by a discussion of Kerr media.