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Atoms and Photons Atoms and photons J.M. Raimond Universit´ePierre et Marie Curie Laboratoire Kastler Brossel D´epartement de physique de l'Ecole´ normale sup´erieure updated by H´el`enePerrin Laboratoire de physique des lasers, CNRS Universit´eSorbonne Paris Nord helene.perrin @ sorbonne-paris-nord.fr October 13, 2020 2 Contents 1 Classical and phenomenological approach 9 1.1 A classical model: the harmonically bound electron . 9 1.1.1 Equations of motion . 9 1.1.2 Polarizability . 11 1.1.3 Scattering by an atom . 11 1.1.4 Propagation in matter . 13 1.2 Einstein's coefficients . 14 1.2.1 The three coefficients . 14 1.2.2 Relations between the coefficients . 15 1.2.3 Stimulated emission: the laser . 16 1.2.4 A simple model of laser radiation . 17 1.2.5 Properties of laser radiation . 18 2 Semi-classical approach 21 2.1 Interaction Hamiltonian . 21 2.1.1 Gauge choice . 22 2.1.2 A · P interaction . 23 2.1.3 D · E interaction . 24 2.2 Non-resonant interaction: Perturbative approach . 25 2.3 Free atom and resonant field . 28 2.3.1 Atomic system and interaction . 28 2.3.2 Rabi precession . 31 2.3.3 Ramsey separated oscillatory fields method . 34 2.4 Relaxing atom and resonant field . 36 2.4.1 Description of the atomic relaxation . 36 2.4.2 Quantum jumps . 39 2.4.3 Quantum Monte Carlo . 41 2.5 Optical Bloch equations . 44 2.5.1 The equations . 44 2.5.2 Two limit cases: back to Einstein's coefficients . 47 2.6 Applications of the optical Bloch equations . 53 2.6.1 Steady-state and saturation . 53 2.6.2 Optical pumping . 57 2.6.3 Dark resonances and EIT . 59 2.6.4 Maxwell-Bloch equations . 62 3 Field quantization 65 3.1 Introduction . 65 3.1.1 Planck 1900 . 65 3.1.2 Einstein 1905 . 68 3 4 CONTENTS 3.2 Field eigenmodes . 69 3.2.1 Mode decomposition of the electromagnetic field . 69 3.2.2 Normal variables . 72 3.2.3 Field energy: canonical variables . 73 3.2.4 Field momentum and angular momentum . 76 3.3 Field quantization . 78 3.3.1 Creation and annihilation operators, Hamiltonian . 78 3.3.2 Field operators . 80 3.4 Field quantum states . 84 3.4.1 Fock states . 84 3.4.2 Coherent states . 85 3.4.3 Phase space representations . 89 3.5 Coupling field modes: the beamsplitter . 99 3.5.1 Hamiltonian approach . 99 3.5.2 State transformations . 102 3.6 Field relaxation . 105 3.6.1 Lindblad equations . 105 3.6.2 Evolution of some states . 106 4 Coupling quantum field to matter 111 4.1 Interaction Hamiltonians . 111 4.1.1 Quantum field and classical charges . 111 4.1.2 Quantum field and quantized atom . 114 4.2 Spontaneous emission in free space . 114 4.2.1 Fermi Golden Rule . 114 4.2.2 Wigner-Weisskopf . 116 4.3 Photodetection . 118 4.3.1 Photodetector model . 118 4.3.2 Intensity correlations . 120 4.4 The dressed atom model . 124 4.4.1 The atom-field eigenstates . 125 4.4.2 Resonant case: Rabi oscillation induced by n photons . 127 4.4.3 Non-resonant coupling and single-atom index effects . 129 4.4.4 Two applications . 131 4.5 Cavity Quantum electrodynamics . 132 Introduction The atom-field interaction is certainly one of the most important problems in quantum physics at low energies. Most of the information we get on the visible universe, most of the actions we may have on it are mediated by the interactions of the atoms with the electromagnetic field. The quantum theory of these interactions, the \quantum electrodynamics" is one of the key successes of last century physics. It provides an extremely detailed insight, with an impressive compatibility with the available experimental data. For instance, the spectrum of the simplest atom, hydrogen, can be computed and compared with experiments with a precision at the 10−12 level, only limited by our imperfect knowledge of the structure of the proton. Quantum optics, the manipulation of fields with a few light quanta only, has also led to very impressive vindications of the most intimate aspects of quantum physics, for instance with the direct evidence of quantum non-locality in the famous experiments led by A. Aspect in the 80s. The matter-field interaction at the quantum level has also led to a flurry of important practical applications. The laser, at the heart of the modern information society (any message on the internet at one time transits as a collection of light pulses in an optical fibre), is a direct application of our understanding of the atom-field coupling. The atomic clocks are the heart of the GPS system. Their precision is considerably improvedp by the cold atom and trapped ions techniques, leading to an impressive stability (in the 10−17= Hz range) and accuracy (below 10−18 as reported in 2018 for the NIST aluminium ion clock interrogated using quantum logic spectroscopy). The cold atoms science reflects our ability to use light to change the motion of atoms. It culminates in the Bose Einstein condensation, observed for the first time in 1995. This new state of matter can be used as a fully controllable prototype of a condensed matter system. The addition of a periodic potential created by a light standing wave realizes a `quantum simulator' of various properties (quantum phase transitions, topological phases...) too complex to be computed or modelled accurately. These lectures will not of course cover all aspects of an extremely rich problem. We will focus on the basic aspects of quantum light-matter interaction. How does a single or a few atoms interact with near-resonant light? We will pass progressively from the simplest classical or qualitative approaches to the final explicit treatment of the interaction of a quantum atom with a quantum field. The first Chapter will be devoted to the interaction of atoms, classically modelled as a charged harmonic oscillator, with a classical field. This model is of course extraordinarily naive, but we will see in the following Chapters that its conclusions are surprisingly close to those of fuller quantum models. We will also introduce in this Chapter the Einstein coefficients describing phenomenologically the rate of energy exchange between the atoms and the field. The consistency of this model requires the introduction of a stimulated emission process, which is at the heart of the laser (or maser) principle. We will later recover these coefficients in a more rigorous way and get insight into their validity domain. The second Chapter will be devoted to the interaction of a quantized atom with a classical field. When this field is far off-resonance from any atomic transition, we will recover, mutatis mutandis, the basic conclusions of the charged harmonic oscillator model. At resonance, we will describe the new and important phenomenon of Rabi rotation, and discuss a few of its applications such as Ramsey interferometry, a key component of high-resolution spectral measurements. This first discussion will assume that the atom is, besides its interaction with the field, isolated from the outside world. This 5 6 INTRODUCTION is never the case in practice, be it only due to the spontaneous emission phenomenon, which makes all atomic excited states unstable (the rate of this emission will be precisely computed in Chapter 4 by a fully quantum approach). We will thus introduce in Chapter 2 the appropriate tools for the description of quantum relaxation, or decoherence, i.e. the coupling of a quantum system to a complex reservoir at thermal equilibrium. We will do so in a framework (Kraus operators) inspired by the recent developments of quantum information theory and providing a very clear and intuitive picture of the relaxation phenomenon. We will later show how this approach leads to the standard master equation, describing the time evolution of the reduced density matrix of the atom. Equipped with these equations, we will be ready to treat the interaction of a relaxing atom with a classical field, and to establish the so-called optical Bloch equations. We will discuss a variety of applications of these equations, from high- resolution spectroscopic methods based on the saturated absorption, to electromagnetically induced transparency (EIT) and slow light. This semi-classical approach, if extremely useful, does not tell the whole story. We now need to properly quantize the electromagnetic field. We will do so in Chapter 3. The first step is to find a convenient mode basis for the classical field, solution of the Maxwell equations with appropriate limiting conditions. We will then write the Hamiltonian of these classical modes and identify them with those of 1-dimensional harmonic oscillators. The canonical quantization of the field will then be a simple problem. We will introduce the field operators, the field energy eigenstates and the coherent states, which are the closest possible analogue of a classical field. Finally we will introduce a few pictorial representations of these states in the phase plane (the so-called Fresnel plane). We will also treat the coupling of two field modes by a semi-transparent mirror (a beam-splitter) and recognize that this simple device produces extraordinarily complex entangled quantum states. The final part of this Chapter will be devoted to the exploration of field decoherence. We will in particular show that the coherent states are particularly robust with respect to decoherence. The final Chapter will be devoted to the interaction of a quantum field with a quantum atom, completing our initial goals. We will begin with the spontaneous emission problem and show how its rate can be precisely computed.
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