The normalized phase-amplitude representation and the Darboux figure are first defined. Then as an example, we establish the quantum optics description for the field in photonic crystal fibers in the defined representation. The field is evolved from one Darboux state to another state in the Darboux figure. By the continuous inputting (pump) of light, the waves induced by nonlinear effects become stronger (the wave frequency number remains constant at the same distance, and their intensities accumulate). The blueshifted waves induced by the third-order resonant susceptibility are not strong enough to cover other nonlinear effect and turn weak along the length (pump turns weak).
The normalized phase-amplitude representation and the Darboux figure are first defined. Then as an example, we establish the quantum optics description for the field in photonic crystal fibers in the defined representation. The field is evolved from one Darboux state to another state in the Darboux figure. By the continuous inputting (pump) of light, the waves induced by nonlinear effects become stronger (the wave frequency number remains constant at the same distance, and their intensities accumulate). The blueshifted waves induced by the third-order resonant susceptibility are not strong enough to cover other nonlinear effect and turn weak along the length (pump turns weak).
The formalism of Maker, Terhune, and Savage [Phys. Rev. Lett. 12, 507 (1964)] for third-order nonlinear susceptibilities is extended to account for nonlocal response. In the case of counterpropagating beams, we show that five constants instead of two are required in general to describe the nonlinearity. In the particular case of optical pumping nonlinearities, which we study extensively, we show that four constants, which describe the magnetization and electric-quadrupole moment of the medium induced by the forward and backward beams, are sufficient. We evaluate these constants for various values of the angular momenta of the atomic levels connected by the incident field. We finally show how this formalism can be applied to several problems such as induced focusing, four-wave-mixing generation, optical instability, and polarization properties of phase-conjugate and phase-contrast mirrors.
Nonlinear optics is the study of the response of dielectric media to strong optical fields. The fields are sufficiently strong that the response of the medium is, as its name implies, nonlinear. That is, the polarization, which is the dipole moment per unit volume in the medium, is not a linear function of the applied electric field. In the equation for the polarization, there is a linear term, but, in addition, there are terms containing higher powers of the electric field. This leads to significant new types of behavior, one of the most notable being that frequencies different from that of the incident electromagnetic wave, such as harmonics or subharmonics, can be generated. Linear media do not change the frequency of light incident upon them. The first observation of a nonlinear optical effect was, in fact, second-harmonic generation – a laser beam entering a nonlinear medium produced a second beam at twice the frequency of the original. Another type of behavior that becomes possible in nonlinear media is that the index of refraction, rather than being a constant, is a function of the intensity of the light. For a light beam with a nonuniform intensity profile, this can lead to self-focusing of the beam. Most nonlinear optical effects can be described using classical electromagnetic fields, and, in fact, the initial theory of nonlinear optics was formulated assuming the fields were classical. When the fields are quantized, however, a number of new effects emerge. Quantized fields are necessary if we want to describe fields that originate from spontaneous emission. For example, in a process known as spontaneous parametric down-conversion, a beam of light at one frequency, the pump, produces a beam at half the original frequency, the signal. This second beam is a result of spontaneous emission.
We explore the frequency translation of single-photon states of light and the creation of photon pairs by four-wave mixing in optical fiber. Frequency translation refers to changing the central frequency of a field, while photon pair creation refers to the creation of two individual photons at the same time. We demonstrate these effects in third-order nonlinear optical fiber. While both phenomena have previously been shown by three-wave mixing in second-order nonlinear media, there are compelling reasons to develop these tasks in third-order media. Most importantly, frequency translation in third-order material allows for the practical implementation of both small and large frequency shifts, while second-order material only practically allows for large shifts. Photon creation in third-order media often permits more flexible phase-matching conditions, allowing for the creation of a wider variety of quantum states than is often possible in second-order media. In our theoretical study of photon pair creation, we focus on the spectral correlations of the photon pairs. We pay particular attention to the creation of quantum states of high purity, where the photons are not spectrally correlated with one another. High purity photons are a requisite resource for several different quantum information processing applications, such as linear-optical quantum computing. We find that states with high purity can be realized with a minimal amount of spectral filtering. Experimentally, we study photon frequency translation in photonic crystal fiber. The central wavelength of the input photons was translated from 683 nm to 659 nm. We perform second-order intensity correlation measurements on both channels to demonstrate their quantum nature. This resulted in values of 0.21 ± 0.02 and 0.19 ± 0.05 for the 683-nm and 659-nm channels, respectively, demonstrating that those fields were dominated by their single-photon component. The efficiency at which the process occurred was 29 percent. Theoretically, we develop a Green function formalism to describe the translation process and develop a computational model to calculate the solution to the governing equations. Also, in a related experiment, we demonstrate classical frequency translation from 851 nm to 641 nm, a record translation in both wavelength and frequency, at an efficiency of 0.2 percent in a birefringent fiber.
Squeezing is a fundamental quantum optical phenomenon - induced by various nonlinear optical processes - in which a special state of the electromagnetic field is generated, for which quantum noise fluctuations are reduced below the shot noise level for certain frequency ranges. Two qualitatively different kinds of squeezing are currently known to take place in optical fibers. The first is the four-wave mixing (FWM) induced quadrature squeezing, in which an intense continuous wave (CW) pump initiates a spontaneous emission of signal and idler waves from vacuum fluctuations at symmetric detunings from the pump. The noise components at the signal and idler frequencies become correlated via the fiber nonlinearity, leading to a reduction of the noise power spectrum below the shot noise level. All the waves participating to the FWM-induced squeezing are monochromatic waves. The second kind of squeezing is the so called self-phase modulation (SPM) induced squeezing, in which the spectral components of an intense short pulse in a coherent state become correlated via the SPM during propagation. We demonstrate theoretically that a third kind of squeezing is possible in optical fibers, in which the noise spectrum of the quasi-monochromatic resonant radiation (RR) emitted by optical solitons when subject to perturbations shows evidence of photon correlation and a reduction of noise below the shot-noise level.
A nonlinear response theory of a quantum-mechanical system undergoing arbitrary relaxation and interacting with fields, some of which may be strong enough to saturate optical transitions, is developed. Explicit expressions for second-order and third-order susceptibilities are obtained. If the fields are weak, then these expressions show the existence of additional resonant contributions to ${\ensuremath{\chi}}^{(n)}$ which arise due to inelastic collisions. Various applications of these ${\ensuremath{\chi}}^{(n)}$'s to modulation spectroscopy, four-wave mixing, and pump-probe experiments are discussed. The general structure suggests how the additional resonances can be used to determine inelastic rates in a Doppler-broadened medium. For saturating fields, ${\ensuremath{\chi}}^{(n)}$'s become dependent on the intensity of such fields and can be formally obtained from weak-field ${\ensuremath{\chi}}^{(n)}$'s, provided proper identification of unperturbed eigenfunctions, eigenvalues, and relaxation times is made. Such intensity-dependent ${\ensuremath{\chi}}^{(n)}$'s have resonant denominators which lead to resonances at Rabi frequencies and submultiples of these frequencies, the widths of which are also dependent on the strength of the field. In the special cases intensity-dependent ${\ensuremath{\chi}}^{(1)}$ agrees with the work of Cohen-Tannoudji and co-workers.