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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum carpets and fractals for superpositions of states of photons

Alexander V. Gorokhov, Alexander V. Shaikin

Published Sep 26, 2003• 0 citations
Publisher

Abstract

Examples of Berrys and WBBZ - (D.Wojcik, I.Bialynicki-Birula and K.Zyczkowski) quantum constructions are considered. The behavior of dimensions of these objects is investigated in going from coordinate representation to momentum and coherent state representations. The mean value of quanta and dispersion for isolated quantum harmonic oscillator prepared in fractal state and interacting with another oscillator are calculated.© (2003) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

DOI
10.1117/12.518613

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Examples of Berrys and WBBZ - (D.Wojcik, I.Bialynicki-Birula and K.Zyczkowski) quantum constructions are considered. The behavior of dimensions of these objects is investigated in going from coordinate representation to momentum and coherent state representations. The mean value of quanta and dispersion for isolated quantum harmonic oscillator prepared in fractal state and interacting with another oscillator are calculated.© (2003) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
Examples of Berry’s and WBBZ - (D.Wojcik, I.Bialynicki-Birula and K.Zyczkowski) quantum fractal constructions are considered. The behavior of fractal dimensions of these very special superpositions of basic states of a particle in a potential box and harmonic oscillator is investigated in going from coordinate representation to momentum and coherent state representations. The mean value of quanta and dispersion for isolated quantum harmonic oscillator prepared in “fractal” state and interacting with another oscillator are calculated.
A review is given of the properties of two types of quantum states, which have great uncertainty in the coordinate and momentum.Both are obtained from the states of a harmonic oscillator by means of certain transformations.The first type is correlated states, they are obtained from coherent states with a help of the Bogolyubov transform.The variances of the coordinate and momentum of such a state depend on the Bogolyubov transform parameters and can, in general, take arbitrarily large values.Their specific values are determined by the physical processes with which the Bogolyubov transform is realized.A concrete example of such physical process is considered.Another type are stretched states.Formally, they arise when the n-partial state of a harmonic oscillator undergoes a transform associated with a scale transformation of the phase space.The dispersion of the coordinate and momentum of these states depends on the scale transformation parameter and can also take arbitrarily large values.Physically stretched states can be obtained by passing n-photon states through a quantum amplifier.The role of the scale transformation of the phase space is played by the gain of the quantum amplifier.
We propose a general construction of wave functions of arbitrary prescribed fractal dimension, for a wide class of quantum problems, including the infinite potential well, harmonic oscillator, linear potential, and free particle. The box-counting dimension of the probability density ${P}_{t}(x)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}|\ensuremath{\Psi}(x,t){|}^{2}$ is shown not to change during the time evolution. We prove a universal relation Dt=1+Dx/2{D}_{t}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1+D}_{x}/2Dt​=1+Dx​/2 linking the dimensions of space cross sections Dx{D}_{x}Dx​ and time cross sections Dt{D}_{t}Dt​ of the fractal quantum carpets.
The notion of f-oscillators generalizing q-oscillators is introduced. For classical and quantum cases, an interpretation of the f-oscillator is provided as corresponding to a special nonlinearity of vibration for which the frequency of oscillation depends on the energy. The f-coherent states (nonlinear coherent states) generalizing q-coherent states are constructed. Applied to quantum optics, photon distribution function, photon number means, and dispersions are calculated for the f-coherent states as well as the Wigner function and Q-function. As an example, it is shown how this nonlinearity may affect the Planck distribution formula.
The notion of f-oscillators generalizing q-oscillators is introduced. For classical and quantum cases, an interpretation of the f-oscillator is provided as corresponding to a special nonlinearity of vibration for which the frequency of oscillation depends on the energy. The f-coherent states (nonlinear coherent states) generalizing q-coherent states are constructed. Applied to quantum optics, photon distribution function, photon number means, and dispersions are calculated for the f-coherent states as well as the Wigner function and Q -function. As an example, it is shown how this nonlinearity may affect the Planck distribution formula.