Fractional Shapiro steps in electron interference in the presence of nonclassical microwaves

Abstract

Electron interference in the presence of nonclassical microwaves with frequency ${\mathrm{\ensuremath{\omega}}}_{1}$ and classical RF radiation with frequency ${\mathrm{\ensuremath{\omega}}}_{2}$, is studied. The relative phase factor between the two electron beams is a quantum-mechanical operator, whose expectation value with regard to the density matrix describing the nonclassical microwaves, determines the interference. It is shown that the visibility of the time-averaged intensity is a constant for all irrational values of ${\mathrm{\ensuremath{\omega}}}_{1}$/${\mathrm{\ensuremath{\omega}}}_{2}$, and shows peaks (fractional Shapiro steps) at all rational values. These peaks can provide direct experimental evidence of the highly nonlinear processes of frequency conversion from N photons with frequency ${\mathrm{\ensuremath{\omega}}}_{1}$, to M photons with frequency ${\mathrm{\ensuremath{\omega}}}_{2}$. Results for various types of nonclassical microwaves (e.g., coherent states, squeezed states, number eigenstates, etc.) are derived and a comparison with the corresponding classical case is made. The difference is due to partial interference destruction from the quantum fluctuations in the nonclassical microwaves, and demonstrates the sensitivity of the interference to these quantum fluctuations. \textcopyright{} 1996 The American Physical Society.