Tunable fractional Fourier transform implementation of electronic wave functions in atomically thin materials

Area of Science:

• Quantum mechanics
• Condensed matter physics
• Materials science

Background:

• The fractional Fourier transform (FrFT) is a generalization of the Fourier transform with applications in signal processing and quantum mechanics.
• Implementing quantum wave function manipulations in solid-state systems is crucial for developing quantum technologies.

Purpose of the Study:

• To propose and theoretically investigate a method for implementing a tunable fractional Fourier transform of electron wave functions in atomically thin materials.
• To explore the connection between the FrFT implementation and the Berry phase for electrons governed by Schrödinger and Dirac equations.
• To assess the potential of this system as a coprocessor for integrated logic circuits.

Main Methods:

• Theoretical modeling of electron wave functions in atomically thin materials subjected to a transverse parabolic potential.
• Analysis of electron propagation dynamics under the influence of the applied potential.
• Investigation of the conditions for achieving fractional Fourier transforms of various orders.
• Examination of the Berry phase difference arising from Schrödinger and Dirac equations.

Main Results:

• A tunable fractional Fourier transform of electron wave functions can be achieved by controlling the parabolic potential and propagation length.
• The difference in propagation lengths for a given FrFT order between Schrödinger and Dirac electrons serves as a manifestation of the Berry phase.
• The system can perform discrete fractional Fourier transforms, including the discrete Fourier transform, in a single step.

Conclusions:

• Atomically thin materials with parabolic potentials offer a viable platform for implementing tunable fractional Fourier transforms of electron wave functions.
• The observed Berry phase difference highlights fundamental distinctions between relativistic and non-relativistic electron dynamics.
• This approach paves the way for novel integrated quantum electronic devices and coprocessors.