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

Electron interference in the presence of nonclassical microwaves with frequency ω1 and classical RF radiation with frequency ω2, 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 ω1/ω2, 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 ω1, to M photons with frequency ω2. 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. copyright 1996 The American Physical Society