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Physica Scripta

Quantum Fluctuations and Squeezing in the Interaction of an Atom with a Single Field Mode

P. L. Knight

Published Jan 1, 1986• 186 citations
Publisher

Abstract

The interaction of a single atom with a single quantized field mode is one of the simplest quantum optical models of resonance, yet exhibits a wide variety of non-trivial fundamental features which depend for their existence on field quantization. We discuss such problems with special emphasis on reversible spontaneous emission and one-photon Rabi oscillations, photon statistical collapses and revivals of coherent evolution, and the existence of phase-dependent squeezed quantum noise.

DOI
10.1088/0031-8949/1986/t12/007

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•May 25, 2004•0 citations
The interaction of a single atom with a single quantized field mode is one of the simplest quantum optical models of resonance, yet exhibits a wide variety of non-trivial fundamental features which depend for their existence on field quantization. We discuss such problems with special emphasis on reversible spontaneous emission and one-photon Rabi oscillations, photon statistical collapses and revivals of coherent evolution, and the existence of phase-dependent squeezed quantum noise.
Light and atoms couple together via the electromagnetic interaction. The simplest form of this interaction is the electric dipole interaction, and in its quantised form it provides a useful starting point for the investigation of quantum effects in optics. Two examples of quantum noise manipulation of the light field due to interaction with atoms will be presented, as well as an analysis of a simple model for laser cooling of trapped atoms. The first example of quantum noise manipulation is the investigation of a quantum non-demolition measurement scheme based on a three-level atomic system in the ladder configuration. An effective two-level model of the atomic system is used, which enables the inclusion of spontaneous emission noise from the upper atomic level. The system is found to perform well, when detuned far from resonance. The second example is the treatment of squeezing in the intensity difference between two modes coupled by a three-level atomic system in the ladder configuration. The noise correlations are similar to those occurring in the optical parametric oscillator, and give rise to good squeezing when the system is well detuned from the intermediate level. The simple model of laser cooling consists of a single two-level atom with quantised centre-of-mass motion constrained to move in a one-dimensional harmonic potential while interacting with a single-mode classical travelling light field. It is shown that there is an analogy between this model and the Jaynes-Cummings model. This gives rise to interesting coherent effects including quantum collapses and revivals in the atomic inversion. Sideband cooling occurs for this model when the light field is tuned to the atom's first lower vibrational sideband. The strong sideband and Lamb-Dicke perturbation regimes are defined. Analytic results have previously been obtained for the latter regime, but we carry out a numerical investigation of the steady state and time evolution behaviour in the former regime. Differences in the behaviour in the two regimes are discussed. Finally the possibility of observing quantum jumps between trap levels is discussed.
In the preceding chapters concerning the interaction of a radiation field with matter, we assumed the field to be classical. In many situations this assumption is valid. There are, however, many instances where a classical field fails to explain experimentally observed results and a quantized description of the field is required. This is, for example, true of spontaneous emission in an atomic system which was described phenomenologically in Chapter 5. For a rigorous treatment of the atomic level decay in free space, we need to consider the interaction of the atom with the vacuum modes of the universe. Even in the simplest system involving the interaction of a single-mode radiation field with a single two-level atom, the predictions for the dynamics of the atom are quite different in the semiclassical theory and the fully quantum theory. In the absence of the decay process, the semiclassical theory predicts Rabi oscillations for the atomic inversion whereas the quantum theory predicts certain collapse and revival phenomena due to the quantum aspects of the field. These interesting quantum field theoretical predictions have been experimentally verified. In this chapter we discuss the interaction of the quantized radiation field with the two-level atomic system described by a Hamiltonian in the dipole and the rotating-wave approximations. For a single-mode field it reduces to a particularly simple form. This is a very interesting Hamiltonian in quantum optics for several reasons. First, it can be solved exactly for arbitrary coupling constants and exhibits some true quantum dynamical effects such as collapse followed by periodic revivals of the atomic inversion.
We investigate the phenomena of single-mode and two-mode squeezing for a three-level atom and two modes. The field modes are initially taken in coherent states. The effects of detuning on these phenomena are studied for one-photon and two-photon interactions. The relation between the collapse and revival phenomena and single-mode and two-mode squeezing is shown. It is found that the two-mode squeezing is much more effective than the single-mode squeezing for the one-photon interaction. While for the two-photon interaction the reverse is true.
Abstract This chapter presents some of the models describing the time-dependent interaction between a simple atom and a field, which may be described either classically or fully quantum mechanically. A number of equivalent interaction pictures and the important rotating-wave approximation are introduced. The two-state dynamics of an atom coupled to a classical field is solved, starting with perturbation theory. Non-perturbative behaviour is analysed, leading to Rabi oscillations of the atomic probabilities. The Jaynes–Cummings model of the dynamics of a single two-state atom coupled to a single quantized mode of the electromagnetic field is solved leading to collapses and revivals of the excited state probability. An exactly soluble model of the coupling of a single state to a Bixon–Jortner quasi-continuum of discrete levels is analysed. Finally, the coherent interactions between two field modes, of the type common in nonlinear optics, are discussed.
Quantum Rabi oscillation in cavity quantum electrodynamics is at the very heart of the interaction of an atomic system with a single electromagnetic field in a resonator. It has proved a powerful tool to entangle the atom with the field in the case of an empty cavity (vacuum Rabi oscillation). On the other side, as soon as the field is macroscopic, its state seems unaffected and one observes the well-known phenomenon of classical Rabi oscillation which only affects the atomic populations. We propose to experimentally investigate the frontier between those two behaviours, when one has to take into account the fluctations of the photon number. In order to do so, we let interact circular Rydberg atoms with a microwave resonator in which a mesocopic field is injected (a few tens of photons). We observe a strong correlation between specific atomic states and the phase of the injected field, which is measured by a homodyne technique. Those correlations explain in an easy way all the standard features of quantum Rabi oscillations (especially the collapses and revivals). Moreover they allow for the preparation of atom-field entangled state as well as non classical Schroedinger cat like states for the field.