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IEEE Journal of Quantum Electronics

Quantum theory of an internally phase-modulated laser - I

P. Titterton

Published Mar 1, 1968• 8 citations
Publisher

Abstract

A rigorous quantum-mechanical treatment of the internally phase-modulated laser is performed in terms of traveling-wave modes of the quantized electromagnetic field. A detailed treatment of the modulation process is consistent not only with the phase-locked and FM behavior of the laser, but also describes the external (to the laser cavity) frequency shifting of phase-locked light and single-frequency ("super-mode") operation of the FM laser.

DOI
10.1109/jqe.1968.1075031

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A rigorous quantum-mechanical treatment of the internally phase-modulated laser is performed in terms of traveling-wave modes of the quantized electromagnetic field. A detailed treatment of the modulation process is consistent not only with the phase-locked and FM behavior of the laser, but also describes the external (to the laser cavity) frequency shifting of phase-locked light and single-frequency ("super-mode") operation of the FM laser.
The paper presents a detailed analysis of FM laser oscillation which includes the effect of arbitrary atomic lineshape, saturation, and mode pulling. Such oscillation is achieved by means of an intracavity phase perturbation, and is a parametric oscillation wherein the laser modes oscillate with FM phases and nearly Bessel function amplitudes. One principal idea is that of the competition between different FM oscillations. The effect of the intracavity phase perturbation is to associate a set of sidebands with each of the previously free-running laser modes. While the free-running laser modes experienced their gain from essentially independent atomic populations, the competing FM oscillations to a large extent see the same atomic population; and in cases of interest the strongest of these oscillations is able to quench the competing weaker oscillations and establish the desired steady state condition. Results of the analysis include the following: threshold and power output, amplitudes and phases of all sidebands, frequency pulling of the entire oscillation, time domain behavior, distortion, super-mode conversion efficiency, and effect of mirror motion. Results of numerical application of the theory to a number of specific cases are given.
We present a steady-state analysis of a quantum-mechanical model of an atom laser. A single-mode atomic trap coupled to a continuum of external modes is driven by a saturable pumping mechanism. In the dilute flux regime, where atom-atom interactions are negligible in the output, we find an analytic form for the linewidth and frequency shift of the laser. This result does not make the Born-Markov approximation, but is based on the far less restrictive ``self-consistent Markov approximation.'' The more exact treatment has a different effective damping rate and occupation of the lasing mode, as well as a shifted frequency and linewidth of the output. We examine gravitational damping numerically, finding linewidths and frequency shifts for a range of pumping rates. We treat mean-field damping analytically, finding a memory function for the Thomas-Fermi regime. The occupation and linewidth are found to have a nonlinear scaling behavior which has implications for the stability of atom lasers.
A novel phase-modulation technique which permits subkilohertz-laser stability and new levels of precision in laser spectroscopy was reported recently. For spectroscopy, the basic arrangement consists of a combination of an optical pump and a probe field which is phase modulated. The pump prepares the atomic sample by burning a narrow hole within the atom's inhomogeneous line shape, and the probe beam samples the prepared hole when its modulation sidebands are swept into resonance. Off resonance, the probe is balanced as pairs of sidebands produce heterodyne beat signals of opposite phase which just cancel. On resonance, the balance is upset and yields a nonvanishing beat signal with a Lorentzian absorption or dispersion line shape and with residual noise approaching the shot noise limit. Here we investigate the theory of phase-modulation spectroscopy. We treat the nonlinear response of an atomic two-level quantum system subject to an intense pump and a weak copropagating or counterpropagating phase-modulated probe beam. The density-matrix equations of motion are solved by a Laplace-transform method and by the novel use of a translation operator which allows the infinite hierarchy of coupled equations to close. A solution equivalent to the rate-equation result is developed and coherence corrections are found which predict new resonances that have just been detected in this laboratory. The delayed pump-probe technique encountered in solid-state laser spectroscopy is analyzed in this context for two- and three-level quantum systems. The response of a Fabry-Perot cavity to a phase-modulated light wave is examined also and reveals an unexpected absorption feature.
The propagation of light in a one-dimensional cavity bounded by moving mirrors is studied in the context of the theory of gas lasers. A canonical quantization of the free field is carried out by utilizing the symplectic structure of the space of solutions of the wave equation. It is shown that no Hamiltonian and no Schrodinger picture exists and that the photon number operator is in general not conserved. Changes in the number operator are, however, ordinarily too small to be observed experimentally. Automorphic coordinate transformations are used to develop adiabatic expansions for the mode functions of cavities with one fixed and one moving mirror. The coupling of the field to two-level atoms is done both quantum-mechanically and semi-classically. For the interacting quantum field an interaction Hamiltonian does exist, even though there is no total Hamiltonian. The semi-classical equations treat the atomic density matrix as a distribution in phase space. They possess traveling wave solutions which are expected to be important in understanding the behavior of mode-locked lasers. (Author)
We study the nonlinear interaction of a cold sample of 88Sr^{88}\mathrm{Sr}88Sr atoms coupled to a single mode of a low finesse optical cavity in the so-called bad cavity limit, and we investigate the implications for applications to laser stabilization. The atoms are probed on the weak intercombination line $|5{s}^{2}{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}\ensuremath{\rangle}\phantom{\rule{0.16em}{0ex}}\ensuremath{-}\phantom{\rule{0.16em}{0ex}}|5s5p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{1}\ensuremath{\rangle}$ at 689 nm in a strongly saturated regime. Our measured observables include the atomic induced phase shift and absorption of the light field transmitted through the cavity represented by the complex cavity transmission coefficient. We demonstrate high signal-to-noise-ratio measurements of both quadratures---the cavity transmitted phase and absorption---by employing frequency modulation (FM) spectroscopy (noise-immune cavity-enhanced optical-heterodyne molecular spectroscopy). We also show that when FM spectroscopy is employed in connection with a cavity locked to the probe light, observables are substantially modified compared to the free-space situation in which no cavity is present. Furthermore, the nonlinear dynamics of the phase dispersion slope is experimentally investigated, and the optimal conditions for laser stabilization are established. Our experimental results are compared to state-of-the-art cavity QED theoretical calculations.