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Annual Meeting Optical Society of America

Three-photon optical excitation of a slightly relativistic single electron

A. E. Kaplan

Published Jan 1, 1985• 0 citations
Publisher

Abstract

We showed earlier1 that a single electron in a magnetic field can exhibit hysteretic cyclotron resonance when the frequency of driving EM field is close to the cyclotron frequency (usually in the microwave range). This effect is based on a slightly relativistic mass effect of the electron; recently it has been observed experimentally.2 Here we show that based on the same small relativistic dependence, yet another nonlinear optical effect can be predicted: large excitation of microwave (or rf) cyclotron motion of a single electron in the Penning trap by two optical waves, when their respective frequencies ω1 and ω2 differ approximately by the double cyclotron frequency ω0, i.e., |ω1 − ω2| ≃ 2ω0. The critical amplitudes of the optical fields required to attain such an excitation are very low, e.g., E1,2 ≃ 10−2V/cm if λ1,2 ≃ 10 μm and λ0 ≃ 2 mm. From the quantum electronics standpoint this effect may be regarded either as a three-photon process (ω0 = |ω1 − ω2| −ω0) of as a higher-order stimulated Raman scattering. Since the conduction electrons in some semiconductors may also exhibit pseudo-relativistic nonlinear cyclotron resonance,3 the analogous three-photon optical excitation may also be expected in semiconductors.

DOI
10.1364/oam.1985.fg3

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We showed earlier1 that a single electron in a magnetic field can exhibit hysteretic cyclotron resonance when the frequency of driving EM field is close to the cyclotron frequency (usually in the microwave range). This effect is based on a slightly relativistic mass effect of the electron; recently it has been observed experimentally.2 Here we show that based on the same small relativistic dependence, yet another nonlinear optical effect can be predicted: large excitation of microwave (or rf) cyclotron motion of a single electron in the Penning trap by two optical waves, when their respective frequencies ω1 and ω2 differ approximately by the double cyclotron frequency ω0, i.e., |ω1 − ω2| ≃ 2ω0. The critical amplitudes of the optical fields required to attain such an excitation are very low, e.g., E1,2 ≃ 10−2V/cm if λ1,2 ≃ 10 μm and λ0 ≃ 2 mm. From the quantum electronics standpoint this effect may be regarded either as a three-photon process (ω0 = |ω1 − ω2| −ω0) of as a higher-order stimulated Raman scattering. Since the conduction electrons in some semiconductors may also exhibit pseudo-relativistic nonlinear cyclotron resonance,3 the analogous three-photon optical excitation may also be expected in semiconductors.
It was recently predicted by us that cyclotron resonance of free electrons in vacuum and conduction electrons in semiconductors may exhibit bistable and hysteretic behavior which is due to the relativistic mass-effect (or pseudo relativistic-in semiconductors). Consistent with this prediction, the hysteretic cyclotron resonance of a trapped single electron in vacuum has recently been experimentally observed by Gabrielse et al. A preliminary estimate shows that their experimental results are consistent with the relativistic nature of the observed hysteresis. In this paper we consider this phenomenon as ultimate bistability since it is based on the most fundamental mechanism of nonlinearity (the relativistic mass-effect), involves the interaction of an EM wave with the simplest single elementary particle, and exhibits the first known intrinsic bistability with no macroscopic optical feedback. We also show that a hysteretic resonance of a Single electron based on relativistic effects is feasible also in a parabolic potential (with no magnetic field required to attain a resonance).
The feasibility of an optical cyclotron bistability of free electrons in vacuum as well as of conductive electrons in semiconductors is discussed. It was recently shown by us1 that the cyclotron bistability of the free electrons can be based on the dependence of a cyclotron frequency of forced oscillations on the relativistic mass of the electron and, hence, on its momentum (or kinetic energy). It was also suggested that1 the analogous effect could be feasible in semiconductors whose conductive electrons have a particularly strong dependence of their effective masses on energy (or momentum) of their excitation. The main advantageous features of the cyclotron bistability in semiconductors is that (i) the effective mass of electrons in some semiconductors (e.g. InSb) is very small which results in a considerable decrease of a required magnet field for a given resonant frequency, and (ii) the nonlinearity associated with conductive electrons in semiconductors is by many orders of magnitude larger than the relativistic nonlinearity of free electrons, which allows for a fairly low critical pumping intensity, even if taking into consideration a faster relaxation time in semiconductors.
``Cyclo-Raman resonance'' of a single electron (i.e., cyclotron motion of an electron driven by optical fields) in the nth order at the cyclotron frequency ${\ensuremath{\Omega}}_{c}$ can be excited by biharmonic laser beams with their frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$ such that ${\ensuremath{\omega}}_{1}$-${\ensuremath{\omega}}_{2}$\ensuremath{\approxeq}n${\ensuremath{\Omega}}_{c}$. This effect exhibits hysteresis, ``prohibited'' and ``allowed'' orbits of excitation, phase multistability, and the optical Stark effect.
A single magnetized electron driven by the EM wave in the vicinity of the cyclotron frequency can experience hysteretic resonance1,2 caused by a tiny relativistic change of its mass. Here we develop a quantum theory of this effect considering a quantum anharmonic oscillator driven by a periodic force. Assuming the radiation relaxation and that the quantum transitions occur only between neighboring slightly nonequidistant eigenstates of the oscillator, we describe the dynamics of the system by the infinite number of coupled kinetic equations for the density matrix elements and derive a procedure for solving them.
We investigate theoretically the use of an electron in a Penning trap as a detector of single microwave photons. At the University of Sussex we are developing a chip Penning trap technology, designed to be integrated within quantum circuits. Microwave photons are guided into the trap and interact with the electron’s quantum cyclotron motion. This is an electric dipole transition, where the near field of the microwave radiation induces quantum jumps of the cyclotron harmonic oscillator. The quantum jumps can be monitored using the continuous Stern-Gerlach effect, providing the quantum non demolition signal of the microwave quanta. We calculate the quantum efficiency of photon detection and discuss the main features and technical challenges for the trapped electron as a quantum microwave sensor.
An atomic electron, in the vacuum of photons, can emit and reabsorb photons of any frequency. Such processes give rise to well known radiative corrections (Lamb—shift, g — 2, natural width of excited states...). We review in this paper similar effects which can be induced by the interaction of an atomic electron with an incident electromagnetic wave, and we try to get some physical insight in radiative processes by comparing these two types of « spontaneous » and « induced » effects. Optical pumping methods have been very useful for studying how atoms are perturbed by optical or RF photons. The Zeeman sublevels of an atomic ground state are broadened and shifted by a resonant or quasiresonant optical irradiation. The corresponding light-shifts (or « Lamp-shifts » as A. Kastler called them) are due to virtual absorptions and reemissions of incident photons. Another example of an induced effect is the modification of the magnetic properties of an atomic level due to a high frequency non resonant RF irradiation. The Larmor precession is slowed down and becomes anisotropic. The two previous examples correspond to two situations leading to simple calculations. For a resonant or quasiresonant excitation, a single atomic transition needs to be considered. Non perturbative treatments can be given which include the effect of the atom field coupling to all orders. For example, the « dressed atom » approach allows a unified treatment of various effects such as light shifts, Autler-Townes doublets resonance fluorescence triplets... On the other hand, for a high frequency irradiation, all atomic transitions must be included. It is possible in this case to derive an effective hamiltonian describing, to lowest order in the fine structure constant α, how electronic energy levels are perturbed in the presence of N photons ω. The remaining part of the paper is devoted to the analysis of such a situation. The terms of the effective hamiltonian proportional to the number N of incident photons correspond to the induced effects. They can be fully interpreted in semiclassical terms and they describe how the slow motion of the electron in the external static fields is modified by the high frequency vibration in the incident monochromatic wave. For example, the vibrating electron averages the static potential over a finite length, the angular oscillation of the spin reduces the effective magnetic moment. The remaining terms, which are independent of N, exist even in the vacuum of photons (N = 0) and describe the contribution of the mode ω to spontaneous radiative corrections. It is possible to split them into two parts. The first one has the same structure as the N-dependent terms, with N replaced by 1/2. It describes the corrections due to the vibration of the electron in vacuum fluctuations which have a spectral power density ħω/2 per mode. The second part represents the effect of radiation reaction, i.e. the interaction of the electron with its self field. Such an interpretation is confirmed by a Heisenberg equations approach where the total force acting upon the electron is split into two physical (i.e. hermitian) parts representing respectively the effect of the vacuum « free » field and the effect of the « source » field produced by the electron itself. These considerations are finally applied to the problem of the positive sign of g - 2. The g-factor can be written as 2 ωL/ωc, where ω L is the Larmor frequency of the spin and ωc the cyclotron frequency of the charge for an electron in a weak static field B 0. In the absence of radiative corrections, g = 2, and ω L = ωc. To lowest order in 1/c (order zero), radiation reaction slows down ωc but not ωL (a charge is more coupled to its self field than a magnetic moment). The effect of vacuum fluctuations appears only at order 2 in 1/c and corresponds to relativistic and magnetic effects which slow down both ωL and ωc. Therefore, it appears that, in the non relativistic domain, the main effect is a reduction of ωc by radiation reaction and this explains why 2 ωL/ωc becomes larger than 2. A complete relativistic calculation (to all orders in 1/ c, but to lowest order in α) does not change this conclusion since the contribution of non relativistic modes (ħω < mc2) is predominant in the integral giving g — 2.