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Implementation of quantum and classical discrete fractional Fourier transforms

View Article: PubMed Central - PubMed

ABSTRACT

Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.

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Experimental visualization of the discrete Hermite–Gauss polynomials.(a–d) Evolution of single-site inputs into the magnitudes of the respective eigensolutions, as predicted theoretically (methods). The experimental data (blue crosses) is compared with the analytic DFrFT (red curves).
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f3: Experimental visualization of the discrete Hermite–Gauss polynomials.(a–d) Evolution of single-site inputs into the magnitudes of the respective eigensolutions, as predicted theoretically (methods). The experimental data (blue crosses) is compared with the analytic DFrFT (red curves).

Mentions: An unequivocal criterion, for the functionality of devices that perform the DFrFT, equation (2), can be formulated by evaluating . At this particular distance, point-like excitations will give rise to signal magnitudes that perfectly resemble the magnitudes of one of the eigenstates of the transform. More specifically, for transforms such as equation (2), one finds that an excitation of the qth site excites the qth system eigenstate up to local phases (see the Methods section for explanations). The experimental demonstration of this intriguing effect is shown in the subpanels of Fig. 3a–d along with the theoretical predictions. It can be argued that for any point-like excitation, the continuous limit cannot be met experimentally (see the Methods for discussion). Instead, equation (2) creates a non-uniform amplitude distribution with a phase difference of π/2 between adjacent sites. Nevertheless, in the continuous limit, tends to the usual FT kernel12. At this point, it is worth emphasizing the formal equivalence to the quantum Heisenberg XY model in condensed matter physics2223. In this respect, our observations demonstrate the capability of the here-presented systems to store quantum information in XY Hamiltonians by converting specific inputs into eigenstates of the system16. To our knowledge, this rather rare property has never been thoroughly investigated before.


Implementation of quantum and classical discrete fractional Fourier transforms
Experimental visualization of the discrete Hermite–Gauss polynomials.(a–d) Evolution of single-site inputs into the magnitudes of the respective eigensolutions, as predicted theoretically (methods). The experimental data (blue crosses) is compared with the analytic DFrFT (red curves).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4814576&req=5

f3: Experimental visualization of the discrete Hermite–Gauss polynomials.(a–d) Evolution of single-site inputs into the magnitudes of the respective eigensolutions, as predicted theoretically (methods). The experimental data (blue crosses) is compared with the analytic DFrFT (red curves).
Mentions: An unequivocal criterion, for the functionality of devices that perform the DFrFT, equation (2), can be formulated by evaluating . At this particular distance, point-like excitations will give rise to signal magnitudes that perfectly resemble the magnitudes of one of the eigenstates of the transform. More specifically, for transforms such as equation (2), one finds that an excitation of the qth site excites the qth system eigenstate up to local phases (see the Methods section for explanations). The experimental demonstration of this intriguing effect is shown in the subpanels of Fig. 3a–d along with the theoretical predictions. It can be argued that for any point-like excitation, the continuous limit cannot be met experimentally (see the Methods for discussion). Instead, equation (2) creates a non-uniform amplitude distribution with a phase difference of π/2 between adjacent sites. Nevertheless, in the continuous limit, tends to the usual FT kernel12. At this point, it is worth emphasizing the formal equivalence to the quantum Heisenberg XY model in condensed matter physics2223. In this respect, our observations demonstrate the capability of the here-presented systems to store quantum information in XY Hamiltonians by converting specific inputs into eigenstates of the system16. To our knowledge, this rather rare property has never been thoroughly investigated before.

View Article: PubMed Central - PubMed

ABSTRACT

Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.

No MeSH data available.


Related in: MedlinePlus