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IEEE Journal of Quantum Electronics

Ultimate bistability: Hysteretic resonance of a slightly-relativistic electron

A. E. Kaplan

Published Sep 1, 1985• 10 citations
Publisher

Abstract

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).

DOI
10.1109/jqe.1985.1072813

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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.
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.
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.
In nonlinear optical interactions, one expects the transverse spatial intensity dependence to be modified. Nonlinear interactions within an optical resonator, in the absence of a mode-selective element such as an aperture, will thus in general couple the energy into various transverse modes of the resonator. One attractive analytical model incorporating transverse effects into optical bistability assumes, in spite of the observations above, that the transverse intensity profile remains a single-Gaussian (TEM00) mode. This model seems to be supported by experimental results for bistability with two-level sodium atoms in a confocal ring resonator. To test the generality of the single-Gaussian mode model, we have performed bistability experiments with two-level atoms in standing-wave resonators. Two resonators, differing only in length, were used. The confocal resonator is transverse-mode-degenerate and will support a combination of modes, as would the ring; the slightly longer resonator will allow only a single-transverse mode to be excited. Preliminary results indicate that there may be a difference in the two cases; although the behavior near the critical onset of bistability seems to be the same in the two cases, there seems to be a difference in the evolution of hysteresis with increasing cooperativity parameter. Further results, including a quantitative comparison with the single-Gaussian mode theory for standing-wave resonators, are presented.
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.
The response of a nonlinear oscillator to an external electromagnetic field is studied in the quantum regime. It is demonstrated that bistability occurs when the shift of the resonance frequency, induced by the external field and modified by fluctuations (quantum and thermal), surpasses the natural linewidth. This theory can be applied to the study of the nonlinear response, near resonance, of a single electron, trapped in a magnetic field, to an external electromagnetic field. It is shown that, even at low quantum levels of the oscillator, bistability can be observed due to the interplay between the small relativistic nonlinearity, the external electromagnetic field, and the heat bath.