Classical and Quantum Parametric Phenomena provides an overview of the phenomena arising when parametric pumping is applied to oscillators. These phenomena include parametric amplification, noise squeezing, spontaneous symmetry breaking, activated switching, cat states, and synthetic Ising spin lattices. To understand these effects, topics such as nonlinear and stochastic dynamics, coupled systems, and quantum mechanics are introduced. Throughout the book, introductions are kept as succinct as possible and attention is focused on understanding parametric oscillators. As a result, the text helps readers to familiarize themselves with many aspects of parametric systems and understand the common theoretical origin of nanomechanical sensors, optical amplifiers, and superconducting qubits. Parametric phenomena have enabled important scientific breakthroughs over the last decades and are still at the focus of intense research efforts. This book provides a resource for experimental and theoretical physicists entering the field or wishing to gain a deeper understanding of the underlying connections. This includes combining formal and intuitive explanations, accompanied by exercises based on numerical Python codes. This combination allows readers to experience parametric phenomena from various directions and apply their understanding directly to their own projects.

Table of Contents:
Introduction 0.1: Historical Review 0.2: Present and Future 1 The Harmonic Resonator 1.1: Newton's Equation of Motion 1.2: Response of the Driven Resonator 1.3: Matrix Formulation 1.4: Parametric Modulation 1.5: Floquet Theory 1.6: Summary of Chapter 1 1.7: Exercises for Chapter 1 2 The Duffing Resonator 2.1: The Quartic Potential 2.2: The Cubic Potential 2.3: Summary of Chapter 2 2.4: Exercises for Chapter 2 3 Degenerate Parametric Pumping 3.1: The Nonlinear Parametric Resonator 3.2: Parametric Pumping via Three-Wave Mixing 3.3: Summary of Chapter 3 3.4: Exercises for Chapter 3 4 Dissipation and Force Fluctuations 4.1: The Role of Force Noise 4.2: The Fluctuation-Dissipation Theorem 4.3: The Probability Distribution Approach 4.4: Summary of Chapter 4 4.5: Exercises for Chapter 4 5 Parametric Resonators with Force Noise 5.1: Multistability and Quasi-Stable Solutions 5.2: Parametric Amplification Below Threshold 5.3: Parametric Pumping Above Threshold 5.4: Hierarchy of Relevant Timescales 5.5: Summary of Chapter 5 5.6: Exercises for Chapter 5 6 Coupled Harmonic Resonators 6.1: Static Coupling 6.2: Nondegenerate Three-Wave Mixing 6.3: Alternative Types of Coupling 6.4: Summary of Chapter 6 6.5: Exercises for Chapter 6 7 Coupled Parametric Oscillators 7.1: Equations for N Coupled Parametric Oscillators 7.2: Examples for N = 2 7.3: Networks with N > 2 7.4: Summary of Chapter 7 7.5: Exercises for Chapter 7 8 The Quantum Harmonic Oscillator 8.1: From Classical to Quantum Fluctuations 8.2: From First to Second Quantization 8.3: Quantum State Representations 8.4: Summary of Chapter 8 8.5: Exercises for Chapter 8 9 From Closed to Open Quantum Systems 9.1: Coupling to a Thermal Environment 9.2: The Driven Quantum Resonator 9.3: Summary of Chapter 9 9.4: Exercises for Chapter 9 10 The Quantum Parametric Oscillator 10.1: General Hamiltonian 10.2: Quantum Parametric Phenomena 10.3: Coupled Quantum Parametric Oscillators 10.4: Summary of Chapter 10 10.5: Exercises for Chapter 10 11 Experimental Systems 11.1: Mechanical Resonator Example 11.2: Electrical Resonator Example 11.3: Optical Resonator Example 11.4: Rescaling of the Numerical Values 11.5: Summary of Chapter 11 11.6: Exercises for Chapter 11 List of Important Symbols