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Annual Meeting Optical Society of America

Quantum theory of relativistic hysteresis of a single driven electron

A. E. Kaplan

Published Jan 1, 1987• 0 citations
Publisher

Abstract

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.

DOI
10.1364/oam.1987.wu7

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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.

Annual Meeting Optical Society of America•Jan 1, 1985•0 citations
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 consider a driven damped anharmonic oscillator that classically leads to a bistable steady state and to hysteresis. The quantum counterpart for this system has an exact analytical solution in the steady state that does not display any bistability or hysteresis. We use quantum-state diffusion theory to describe this system and to provide a new perspective on the lack of hysteresis in the quantum regime so as to study in detail the quantum to classical transition. The analysis is also relevant to measurements of a single periodically driven electron in a Penning trap where hysteresis has been observed.
We demonstrate that the stochastic resonance phenomenon can occur in a pure quantum regime for a wider class of microscopic systems described by a dissipative anharmonic oscillator driven by two periodic forces. We apply a quantum-state diffusion method, and display the synchronization of quantum trajectories by the stochastic resonance process. The model proposed is accessible for experiments, and results in the formation of quantum states of the anharmonic oscillator, namely sub- and super-Poissonian statistics for time intervals exceeding the relaxation rate. We show that stochastic resonance phenomenon can be described in terms of the minimization of the quantum von Neuman entropy, and demonstrate the possibility of controlling both the dissipative dynamics and quantum statistics of anharmonic oscillator by two external periodic forces.
A quantum cyclotron is one trapped electron or positron that occupies only its lowest cyclotron and spin states. A master equation is solved for a driven quantum cyclotron with a QND (quantum nondemolition) coupling to a detection oscillator in thermal equilibrium, the latter making this an open quantum system. The predicted rate of a cyclotron and spin quantum jumps as a function of drive frequency, for a small coupling between the detection motion and its thermal reservoir, differs sharply from what has been predicted and used for past measurements. The calculation suggests a ten times more precise electron magnetic moment measurement is possible, as needed to investigate current differences between the most precise prediction of the standard model of particle physics, and the most accurate measurement of a property of an elementary particle.
The hysteresis curve of the bistable cyclotron motion of a single electron in a magnetic field driven by external microwave radiation is obtained by only varying the phase of a phase-sensitive back-action-evading experimental setup. Thus the steady-state solution of the cyclotron motion includes two branches which could be switched by only changing the phase of the device. This effect is of particular interest as an example of the influence of the noise properties of the measurement apparatus on a microscopic system subjected to measurement.
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).
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.