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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.

No MeSH data available.


Set-up used to carry out spatial correlation measurements of photonic quantum states.The set-up consists of three parts—the state preparation, the execution of the DFrFT and the correlation measurement.
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f5: Set-up used to carry out spatial correlation measurements of photonic quantum states.The set-up consists of three parts—the state preparation, the execution of the DFrFT and the correlation measurement.

Mentions: As a second case, we consider a fully symmetric path-entangled two-photon state of the form . Physically, both photons are entering together into the array at either site j or −j with equal probability282930. The correlations are determined by , from which we infer that the probability of measuring photon coincidences at coordinates (k, l) vanishes at sites where the sum (k+l) is odd. In contrast, at coordinates where (k+l) is even, the correlation function collapses to the expression . This indicates that in this path-entangled case the correlation map appears rotated by 90° with respect to the matrix obtained with separable two-photon states. We performed an experiment to demonstrate these predictions using states of the type , which were prepared using a 50:50 directional coupler acting as a beam splitter29. The whole experiment is achieved using a single chip containing both the state preparation stage followed by a Jx-system, yielding high interferometric control over the field dynamics (Fig. 5). The experimental measurements are presented in Fig. 4b. Similarly, suppression of states occurs as a result of destructive quantum interference. As predicted, a closer look into the correlation pattern reveals that indeed the correlation map appears rotated by 90° with respect to the matrix obtained with separable two-photon states.


Implementation of quantum and classical discrete fractional Fourier transforms
Set-up used to carry out spatial correlation measurements of photonic quantum states.The set-up consists of three parts—the state preparation, the execution of the DFrFT and the correlation measurement.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f5: Set-up used to carry out spatial correlation measurements of photonic quantum states.The set-up consists of three parts—the state preparation, the execution of the DFrFT and the correlation measurement.
Mentions: As a second case, we consider a fully symmetric path-entangled two-photon state of the form . Physically, both photons are entering together into the array at either site j or −j with equal probability282930. The correlations are determined by , from which we infer that the probability of measuring photon coincidences at coordinates (k, l) vanishes at sites where the sum (k+l) is odd. In contrast, at coordinates where (k+l) is even, the correlation function collapses to the expression . This indicates that in this path-entangled case the correlation map appears rotated by 90° with respect to the matrix obtained with separable two-photon states. We performed an experiment to demonstrate these predictions using states of the type , which were prepared using a 50:50 directional coupler acting as a beam splitter29. The whole experiment is achieved using a single chip containing both the state preparation stage followed by a Jx-system, yielding high interferometric control over the field dynamics (Fig. 5). The experimental measurements are presented in Fig. 4b. Similarly, suppression of states occurs as a result of destructive quantum interference. As predicted, a closer look into the correlation pattern reveals that indeed the correlation map appears rotated by 90° with respect to the matrix obtained with separable two-photon states.

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.