Appendix G: interaction between a monochromatic field and two-level atom. The problem is treated first in the case of a classical field and a quantum two-level system (semiclassical approach): It is characterised by a rotation of the Bloch vector (Rabi ocillation) and allows us to generate any qubit state by applying a field of well-controlled duration and amplitude. One then includes spontaneous emission to the model, and finally obtains the set of Bloch equations that are used in many different problems of light–matter interaction. One then considers the full quantum case of cavity quantum electrodynamics (CQED), where the field is single mode and fully quantum: this is the Jaynes–Cummings Hamiltonian approach, which is fully solvable when one negelcts spontaneous emission: quantum oscillations and revivals are predicted. Damping is then introduced in the model, and two regimes of strong and weak couplings are predicted in this case.

Typically, we are interested in a small system (a few particles, or some region in space), entangled with its surroundings. Exact equations (Dyson or Nakajima–Zwanzig) for the reduced system dynamics are readily derived using suitable projection operators. They are rarely solvable, however, since full account is taken for mutual influences between the system and its environment. Nevertheless, for weak mutual coupling, environment-induced dynamics in the (small) system can be much slower than system-induced dynamics in the (large) environment. This justifies the Born–Markov approximation, leading to closed equations for the system. In Hilbert space, we demonstrate the emergence of irreversible dynamics and exponential decay for pure sates. In Liouville space, we derive the Redfield equation. Invoking the secular (or, rotating-wave) approximation, we derive Pauli’s master equations, which properly account for relaxation to equilibrium. Rates of spontaneous emission, coherence transfer (Bloch equations), and pure dephasing are derived and analyzed for a dissipative qubit.

We learn time-dependent perturbation theory, where we focus on finding the probability that an applied perturbation causes a transition between energy levels of the unperturbed Hamiltonian. We calculate the probability amplitude for a transition from an initial state to a final state subject to a time-dependent perturbation. We learn that an excited state in an atom has a finite lifetime due to spontaneous emission. We learn that electric dipole transitions obey selection rules.

This chapter discusses the spin degrees of freedom in an atom. We first review how atomic energy levels can be classified in terms of the electron spin and orbital angular momenta and how this couples to the nuclear spin. We then describe how atoms interact with each other, and how the spins affect this interaction. The effect of electromagnetic radiation on the energy levels of an atom is described, and the Hamiltonian for energy levels transitions . After briefly describing how the important phenomena of the ac Stark shift and Feshbach resonances occur, we then turn to describing how dissipative dynamics affect atomic systems. Specifically, we examine master equations for spontaneous emission and atom loss, and look at these can be solved. Finally, we consider an alternative framework for solving such open systems using the quantum jump method, which allows for a stochastic approach to solving the dynamics

Applications of near-IR fibre amplifiers and fibre lasers in gas spectroscopy are reviewed. Examples are given where fibre amplifiers may be employed to boost the optical power, for example, in photoacoustic spectroscopy or when splitting a single laser output over multiple fibre optic paths in tomographic imaging. The use of mode-locked fibre lasers for the generation of high-performance frequency combs is discussed and examples given of the state-of-the-art in compact, field-deployable erbium fibre laser combs. The method of dual comb spectroscopy is explained and illustrated with applications in the monitoring of atmospheric trace gases, pollution and exhaust emissions. Several techniques are considered for enhancing sensitivity by means of a high-finesse fibre laser cavity, such as by fibre ring-down spectroscopy or through use of the amplified spontaneous emission present within the laser cavity. Intra-cavity laser absorption spectroscopy, where the fibre laser’s spectral distribution is monitored during the transient period, is discussed in detail with examples given of its potential application for the simultaneous measurement of several gas species in various environments.