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Europhysics Letters (EPL)

Electron Interference in the Presence of AC Non-Classical Electromagnetic Fields

Apostol Vourdas

Published Nov 1, 1995• 19 citations
Publisher

Abstract

Electron interference in the presence of non-classical time-dependent magnetic flux is studied. The quantum uncertainty Δϕ in the value of the non-classical magnetic flux plays a destructive role in the interference. 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 non-classical electromagnetic field determines the interference. A method similar to the Shapiro experiment in the context of Josephson junctions is used for the study of the details of this interference. Coherent states and squeezed states are considered and various interesting characteristics of the interference, which are associated to the quantum nature of these fields, are studied. The results are compared and contrasted with those in the classical case.

DOI
10.1209/0295-5075/32/4/001

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Electron interference in the presence of non-classical time-dependent magnetic flux is studied. The quantum uncertainty Δϕ in the value of the non-classical magnetic flux plays a destructive role in the interference. 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 non-classical electromagnetic field determines the interference. A method similar to the Shapiro experiment in the context of Josephson junctions is used for the study of the details of this interference. Coherent states and squeezed states are considered and various interesting characteristics of the interference, which are associated to the quantum nature of these fields, are studied. The results are compared and contrasted with those in the classical case.
The interaction of mesoscopic interference devices with nonclassical electromagnetic fields is studied. The external quantum fields induce a phase factor in the electric charges. This phase factor, which is a generalization of the standard Aharonov–Bohm phase factor, is in the case of nonclassical electromagnetic fields a quantum mechanical operator. Its expectation value depends on the density matrix describing the nonclassical photons and determines the interference. Several examples are discussed, which show that the quantum noise of the nonclassical photons partly destroys the electron interference fringes. An interesting application arises in the context of distant electron interference devices, irradiated with entangled photons. In this case the interfering electrons in the two devices become entangled. The same ideas are applied in the context of SQUID rings irradiated with nonclassical electromagnetic fields. It is shown that the statistics of the Cooper pairs tunnelling through the Josephson junction depends on the statistics of the photons.
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.
Two electron interference experiments which are far from each other are considered. They are irradiated with correlated nonclassical electromagnetic fields, produced by the same source. The phase factors are in this case operators, and their expectation values with respect to the density matrix of the electromagnetic field quantify the observed electron fringes. The correlated photons create correlations between the observed electron intensities. Both cases of classically correlated (separable) and quantum mechanically correlated (entangled) electromagnetic fields are considered. It is shown that the induced correlation between the distant electron interferences is sensitive to the nature of the correlation between the irradiating photons.
Electron interference in mesoscopic devices irradiated by external monochromatic nonclassical microwaves is considered. The relative phase factor between two electron paths is a quantum mechanical operator whose expectation value with respect to the density matrix of the nonclassical microwaves is studied. It is shown that, due to both classical and quantum noise in the microwaves, the absolute value of the phase factor is less than 1,1,1, causing partial destruction of the interference. Separable and entangled two-mode nonclassical microwaves are also considered and their effect on the expectation value of the phase factor is studied.
Electron interference related to Aharonov–Bohm effect in the presence of the two-mode quantized radiation field is investigated. Characteristics of Shapiro steps, the visibility of the interference fringe and the temporal behavior of electron interference in correlated and uncorrelated two-mode radiation field are discussed. It is shown that for the correlated two-mode squeezed vacuum the integer Shapiro steps are obtained and the time evolution of the intensity of electron interference can exhibit very interesting periodic spontaneous collapse and revival (CR), while for the uncorrelated two-mode coherent state the fractional Shapiro steps are obtained and the CR phenomenon does not occur. In addition, the relation between CR and the quantum fluctuation in electron interference is also discussed.
We develop a quantum field-theoretic method to describe the effect of frequency mixing using Josephson junctions or using an electron beam traversing a succession of microwave cavities. One of our purposes is to discuss the basic mechanism of current modulation and the subsequent radiation in the complete quantum mechanical context. We use the Heisenberg picture, so that the field operators depend on time and state vectors are fixed. It is essential to take quantum mechanical expectation values only for the quantities that are actually observed. Thus, in the case of frequency mixing, we do not evaluate the current but rather the radiated power, which is observable. An evaluation of the current would destroy correlations between Fourier components of the current produced in the first cavity or in the first junction. Leaving the current as an operator permits the correlations produced in the first cavity (junction) to affect the radiation in the second cavity (junction). Another purpose of the approach is to find the possible dependence of the output radiation on the quantum state of the incident radiation preexisting in cavities or in Josephson junctions. In the case of frequency mixing using two microwave cavities where the fields are treated as independent, we show that it is possible to distinguish the case where both cavities are in definite phase-certain coherent states from the phase-uncertain number states by the value of the wave function renormalization constant. However, if the relative phase of the two cavities cannot be defined, this distinction disappears. This indicates that the radiation occurs at the beat frequency even when the cavity fields are taken to be in energy eigenstates, so that no phase whatever can be associated with the fields in those cavities and thus a classical velocity modulation picture does not apply.