By Peter D. Drummond, Mark Hillery

Taking part in a widespread position in communications, quantum technological know-how and laser physics, quantum nonlinear optics is an more and more very important box. This e-book offers a self-contained therapy of box quantization and covers subject matters akin to the canonical formalism for fields, phase-space representations and the surrounding challenge of quantization of electrodynamics in linear and nonlinear media. beginning with a precis of classical nonlinear optics, it then explains intimately the calculation options for quantum nonlinear optical structures and their functions, quantum and classical noise assets in optical fibers and functions of nonlinear optics to quantum details technological know-how. Supplemented through end-of-chapter workouts and specified examples of calculation strategies in numerous structures, this booklet is a useful source for graduate scholars and researchers in nonlinear optics, condensed subject physics, quantum info and atomic physics. a fantastic beginning in quantum mechanics and classical electrodynamics is thought, yet no past wisdom of nonlinear optics is needed.

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**Extra info for Quantum theory of nonlinear optics**

**Sample text**

The energy eigenstates of this quantum nonlinear Schroedinger equation are known from early theoretical work on interacting Bose gases[16, 69, 47] . The simplest of these eigenstates or quantum solitons is a cluster of photons with well-defined energy and photon-number – a photonic atom. 1. 4) 2 24n2 Since this is an energy eigenstate, such a state is predicted to propagate stably, without distortion or breakup into its component photons, just as an atom does in free space. Of course, by Heisenberg’s uncertainty principal, a wavepacket of these quantum solitons - combining different linear momenta (p) together - is required to obtain a state localized in any given spatial region.

The time-dependent classical solution is given by: EN ψ0 (τ, ζ) = sech(ζ)exp[i(θ0 + τ /2)]. 5) Here θ0 is the initial phase. Thus, the only change predicted classically is that the total phase θ(τ ) = θ0 + τ /2 increases linearly with distance down the fiber. We also note that the parameter n corresponds to half the mean photon number N = N with this average pulse amplitude. The classical result of a soliton as a phase-shifted, undistorted pulse, never occurs in quantum theory. Even for a pure quantum soliton the center-ofmass position obeys the Heisenberg uncertainty principle.

These can be easily treated on extending the stochastic form of the equations to include Raman effects. By taking the equation for the density operator ρˆ describing the above Hamiltonian, and introducing a coherent state expansion in the form of the positive-P representation[60, 61] , a Fokker- Planck equation can be readily derived. Thus, equivalent stochastic differential equations for the associated c-number fields are obtainable. Substituting the integrated phonon variables into the equations for the photon field, gives the equation for φ.

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