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Precise measurements of the frequencies of light waves have become common with mode-locked laser frequency combs1. Despite their huge success, optical frequency combs currently remain bulky and expensive laboratory devices. Integrated photonic microresonators are promising candidates for comb generators in out-of-the-lab applications, with the potential for reductions in cost, power consumption and size. Such advances will significantly impact fields ranging from spectroscopy and trace gas sensing to astronomy, communications and atomic time-keeping. Yet, in spite of the remarkable progress shown over recent years, microresonator frequency combs (‘microcombs’) have been without the key function of direct f–2f self-referencing, which enables precise determination of the absolute frequency of each comb line. Here, we realize this missing element using a 16.4 GHz microcomb that is coherently broadened to an octave-spanning spectrum and subsequently fully phase-stabilized to an atomic clock. We show phase-coherent control of the comb and demonstrate its low-noise operation.

Subharmonic resonance behaviors are investigated for the classical hydrogen atom, with classical radiation damping and circularly polarized light acting on the classical electron. This study is intended for both potential experimental applications as well as for deeper theoretical purposes. Long resonant states are predicted for realistic Rydberg atoms and highly excited hydrogen states. Several previously undiscovered physical effects are predicted. First, the semimajor axis remains relatively constant when in subharmonic resonance; second, the eccentricity steadily increases until a maximum, critical value is reached, at which point orbital decay sets in. If the initial orbit is circular, this critical eccentricity value is shown to always be the same for each subharmonic condition, regardless of the initial orbital radius. An analytic derivation for this result is presented. The illustrated dynamics are of interest for the classical theory of stochastic electrodynamics (SED) regarding whether SED can fundamentally describe more of quantum phenomena, particularly atomic excited state behavior and related emission and absorption spectra. Also of interest are how classical resonances can be imposed on a near continuum of quantum states. Finally, there may be future technological applications, such as “reading” and “writing” information into Rydberg atoms in the form of subharmonic resonances.

Previously unexplored resonance conditions are shown to exist for the classical hydrogen atomic system, where the electron is treated as a classical charged point particle following the nonrelativistic Lorentz-Dirac equation of motion about a stationary nucleus of opposite charge. For circularly polarized (CP) light directed normal to the orbit, very pronounced subharmonic resonance behavior is shown to occur with a variety of interesting properties. In particular, only if the amplitude of the CP light exceeds a critical value, will the resonance continue without radius and energy decay. A perturbation analysis is carried out to illustrate the main features of the behavior. The present phenomena adds to a growing list of other nonlinear dynamical behaviors of this simple system, that may well be important for more deeply understanding classical and quantum connections.

The classical hydrogen atom is examined for the situation where a circularly polarized electromagnetic plane wave acts on a classical charged point particle in a near-circular orbit about an infinitely massive nucleus, with the plane wave normally incident to the plane of the orbit. The effect of the phase α of the polarized wave in relation to the velocity vector of the classical electron is examined in detail by carrying out a perturbation analysis and then comparing results using simulation methods. By expanding the variational parts of the radius and angular velocity about their average values, simpler nonlinear differential equations of motion are obtained that still retain the key features of the oscillating amplitude, namely, the gradual increase of the envelope of the oscillating amplitude and the point of rapid orbital decay. Also, as shown here, these key features carry over nicely to conventional quantities of interest such as energy and angular momentum. The phase α is shown here to have both subtle yet very significant effects on the quasistability of the orbital motion. A far wider range of phase conditions are found to provide stability than might intuitively be expected, with the time to orbital decay, td, varying by orders of magnitude for any plane wave with an amplitude A above a critical value, Ac.

The present study examines the behavior of a classical charged point particle in near-elliptic orbits about an infinitely massive and oppositely charged nucleus, while acted upon by applied electromagnetic radiation. As recently shown for near-circular orbits, and now extended here to the elliptical case, rather surprising nonlinear dynamical effects are readily produced for this simple system. A broad range of stability-like conditions can be achieved by applying radiation to this classical atom. A perfect balance condition is examined, which requires an infinite number of plane waves representing harmonics of the orbital motion. By applying a scale factor to this radiation, stability-like conditions are produced where periodic variations in semimajor and semiminor axes occur for extended periods of time, before orbital decay eventually takes over due to the effects of radiation reaction. This work is expected to lead to both practical suggestions on experimental ideas involving controlling ionization and stabilization conditions, as well as hopefully aiding in theoretical explorations of stochastic electrodynamics.

An analysis is carried out involving reversible thermodynamic operations on arbitrarily shaped small cavities in perfectly conducting material. These operations consist of quasistatic deformations and displacements of cavity walls and objects within the cavity. This analysis necessarily involves the consideration of Casimir-like forces. Typically, even for the simplest of geometrical structures, such calculations become quite complex, as they need to take into account changes in singular quantities. Much of this complexity is reduced significantly here by working directly with the change in electromagnetic fields as a result of the deformation and displacement changes. A key result of this work is the derivation that for such cavity structures, classical electromagnetic zero-point radiation is the appropriate spectrum at a temperature of absolute zero to ensure that the reversible deformation operations obey both isothermal and adiabatic conditions. In addition, a generalized Wien displacement law is obtained from the demand that the change in entropy of the radiation in these arbitrarily shaped structures must be a state function of temperature and frequency.

The thermodynamic behavior is analyzed of a single classical charged particle in thermal equilibrium with classical electromagnetic thermal radiation, while electrostatically bound by a fixed charge distribution of opposite sign. A quasistatic displacement of this system in an applied electrostatic potential is investigated. Treating the system nonrelativistically, the change in internal energy, the work done, and the change in caloric entropy are all shown to be expressible in terms of averages involving the distribution of the position coordinates alone. A convenient representation for the probability distribution is shown to be the ensemble average of the absolute square value of an expansion over the eigenstates of a Schrodinger-like equation, since the heat flow is shown to vanish for each hypothetical “state.” Subject to key assumptions highlighted here, the demand that the entropy be a function of state results in statistical averages in agreement with the form in quantum statistical mechanics. Examining the very low and very high temperature situations yields Planck's and Boltzmann's constants. The blackbody radiation spectrum is then deduced. From the viewpoint of the theory explored here, the method in quantum statistical mechanics of statistically counting the “states” at thermal equilibrium by using the energy eigenvalue structure, is simply a convenient counting scheme, rather than actually representing averages involving physically discrete energy states.

Cross-term conservation relationships for electromagnetic energy, linear momentum, and angular momentum are derived and discussed here. When two or more sources of electromagnetic fields are present, these relationships connect the cross terms that appear in the traditional expressions for the electromagnetic (1) energy, (2) linear momentum, and (3) angular momentum, over to, respectively, (1) the sum of the rates of work, (2) the sum of the forces, and (3) the sum of the torques, that are due to the fields of each charge or current source acting upon the other charge and current sources. These relationships, although not new, appear to be rarely recognized and used in the physics literature. As shown here, they can be extremely helpful for solving and gaining a deeper physical understanding into a rather diverse range of interesting problems in electrodynamics, including (1) aspects of Poynting's theorem when applied to charged point particles, (2) the detailed physical basis of electrostatic analysis, (3) understanding the connection between different techniques used in the past for solving Casimir force problems, and (4) reconciling the invalidity of Newton's third law in electrodynamics.

In the past, a few researchers have presented arguments indicating that a statistical equilibrium state of classical charged particles necessarily demands the existence of a temperature-independent, incident classical electromagnetic random radiation. Indeed, when classical electromagnetic zero-point radiation is included in the analysis of problems with macroscopic boundaries, or in the analysis of charged particles in linear force fields, then good agreement with nature is obtained. In general, however, this agreement has not been found to hold for charged particles bound in nonlinear force fields. The point is raised here that this disagreement arising for nonlinear force fields may be a premature conclusion on this classical theory for describing atomic systems, because past calculations have not directed strict attention to electromagnetic interactions between charges. This point is illustrated here by examining the classical hydrogen atom and showing that this problem has still not been adequately solved.