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Quantum Optics Journal of the European Optical Society Part B

Quantum state evolution of the fundamental mode in the process of second-harmonic generation

Sergei P. Nikitin, Anatoly Viktorovich Masalov

Published Apr 1, 1991• 42 citations
Publisher

Abstract

By the computer calculations of the quasiprobability distribution the authors have shown that in a process of second-harmonic generation at phase matching conditions the quantum state of the fundamental mode evolves into a superposition of two macroscopically distinguishable states. Creation of the superposition destroys the squeezing formed in the initial stage of the evolution of the fundamental mode.

DOI
10.1088/0954-8998/3/2/003

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By the computer calculations of the quasiprobability distribution the authors have shown that in a process of second-harmonic generation at phase matching conditions the quantum state of the fundamental mode evolves into a superposition of two macroscopically distinguishable states. Creation of the superposition destroys the squeezing formed in the initial stage of the evolution of the fundamental mode.
For the process of intracavity second-harmonic generation in the region of unstable dynamics, the joint fluctuations of numbers of photons and phases of the fundamental mode and second harmonic are studied in positive P representation. It is shown that above the bifurcation point of the optical system the distribution functions of the phases of the fundamental mode and second harmonic have a two-component structure. The quantum trajectories of the phases of these modes are not localized in the components of their states, which demonstrates the presence of quantum-mechanical interference between these components. In both components of each state the same values of numbers of photons are realized. It is shown that in the case of coherent initial states of both modes the joint distribution functions of the numbers of photons and phases are in the oscillatory regime. With time, the modes pass from a two-component delocalized state to a one-component one and then back to the two-component state. The time of oscillation of the distribution function is of the order of the photon's lifetime in the cavity volume. Self-matching of phases of interacting modes is investigated above the bifurcation point.
The production of squeezed states in second-harmonic generation is discussed theoretically from the viewpoint of the structure of the nonlinear medium, phase-matching conditions, and the polarization state of the fields. The nonlinear coupling constants are derived in explicit form for all 102 magnetic symmetry groups for crystalline as well as electrically polarized isotropic media. The phase-matching conditions are discussed in detail. Phase mismatch is shown to accumulate along the optical path, and is calculated in full detail for various media and geometries. We refer to this as the mismatch accumulation effect. General formulae for normally ordered variances of the second-harmonic and fundamental fields are derived by perturbative procedure, and are specified for some special cases of practical interest.
Photon number and phase fluctuations and correlations in the second harmonic generation are discussed. The new Pegg-Barnett Hermitian phase formalism is used to deal with the phase properties of the field. The method of numerical diagonalization of the interaction Hamiltonian is applied to find the state evolution and, consequently, the number and phase quantum fluctuations. It is shown that the joint phase probability distribution evolves into a multi-peak structure indicating clearly the transition from the second harmonic to the down-conversion regime, and back. The evolution of all relevant quantities is illustrated graphically, and their dependence on the mean number of photons of the initial field is shown explicitly. The number-phase uncertainty product for the fundamental mode is plotted making evident the abrupt transition from the low to the high level of quantum fluctuations.
We analyze squeezing of both the fundamental and harmonic fields undergoing traveling-wave, second-harmonic generation (SHG) in second-order (${\mathrm{\ensuremath{\chi}}}^{(2)}$) nonlinear media. We take into account depletion of the fundamental field as well as the phase mismatch between the fundamental and harmonic fields. The behavior of the quantum noises on the propagating fields is studied by linearizing the nonlinear operator equations around the mean-field values. We first consider the degenerate case that is applicable to type-I phase-matching geometries, obtaining expressions for squeezing in both the fundamental and harmonic fields under the conditions of perfect phase matching and large phase mismatch. We show that in the case of a large phase mismatch, the intensity-dependent self-phase shift of the fundamental field, arising due to cascading of the ${\mathrm{\ensuremath{\chi}}}^{(2)}$ nonlinearity, is responsible for the squeezing generation. We also numerically solve the linearized quadrature-operator equations together with the nonlinear mean-field equations. We find that in the case of a finite phase mismatch the harmonic field can be highly squeezed. This is in contrast to the perfectly phase-matched case where the maximum squeezing is limited to 50%. Finally, we analyze the nondegenerate case that applies to type-II phase-matching geometries. Here we show that the commonly used type-II phase-matched SHG process with the input harmonic field in the vacuum state is equivalent to a type-I SHG process in parallel with a degenerate optical-parametric process. The latter causes squeezing in the mode that is polarized orthogonal to the fundamental beam. In the perfect phase-matching case, the squeezing in the orthogonally polarized mode follows the simple expression S=1-\ensuremath{\gamma}, where \ensuremath{\gamma} is the harmonic conversion frequency.