Open Quantum systems

Jérémie Leonard (IPCMS) and Anja Metelmann (KIT/CESQ)

This course introduces the formalism for characterizing the interaction between a quantum system and a thermal bath. Central to this study are foundational concepts such as the density operator and the Quantum Master Equation (QME). Notable within the QME framework are the Redfield and Lindblad formulations, which delve into essential aspects including populations, coherences, damping, and dephasing phenomena.

A significant component of this course entails modeling the bath as an ensemble of harmonic oscillators, a prevalent approach in quantum dynamics. Here, the examination of bath correlation functions emerges as pivotal in elucidating the system-bath interaction dynamics. Through theoretical exploration and practical applications, students will gain a comprehensive understanding of these fundamental concepts and their implications in quantum mechanics.

This course applies foundational quantum concepts to the study of light-matter interaction, focusing on two main areas:

  1. Condensed phase physics, emphasizing molecules and spectroscopy, including linear response theory and pure dephasing.
  2. Quantum technologies, covering circuit quantum electrodynamics, qubit state preparation, dephasing, and qubit lifetime analysis.

Through these studies, students gain insights into both fundamental physics and emerging technological applications of quantum mechanics.

Objectives: To grasp and apply fundamental tools and concepts for describing the interaction between a quantum system and its environment. These skills are essential for both condensed matter physics and broader quantum sciences and technologies.

Contents:

1 General concepts and theoretical background (10h)

Keywords: bath & system-bath coupling; master equation, quantum regression theorem, noise spectra

  • Using the Schrödinger equation: Free Evolution & interaction Picture; the Wigner-Weisskopf model (= coupling of a single state to a continuum of states): Markov approximation, Fermi Golden Rule, exponential population decay, Lorentzian spectral shape. The bath of independent harmonic oscillators; Application to spontaneous emission (exercise).
  • Using the density operator formalism: interaction picture and partial trace: the Quantum Master Equation (QME) in the Born and Markov approximations; bath correlation functions.
  • The Redfield and Lindblad formulations of the QME; Decoherence by population transfer and/or pure dephasing.
  • Illustration on simple models (exercise): bath correlation functions; the open 2-level system; the open harmonic oscillator, expectation values of operators
  • Quantum-classical correspondence (Wigner & P-distribution), quantum regression theorem, Wiener Khinchin theorem, noise spectral density, standard quantum limit on detection, shot noise

 2 Application to light-matter interaction. I. Condensed phase (4h)

Keywords: electron-phonon coupling; linear response theory, dipole-dipole correlation function, lineshape function, pure dephasing.

  • The « molecular » (or « condensed-phase ») Hamiltonian : Born-Oppenheimer separation between electronic and vibrational degrees of freedom; the two-state molecule / spin-boson model. Interaction with light and Condon approximation: ex: fluorescence emission from a cold/hot molecule.
  • Linear response & spectroscopy: dipole-dipole correlation function and lineshape function; vibrational motions, bath fluctuations & electronic decoherence: Pure dephasing.

3 Application to light-matter interaction. II. Quantum technologies (4h)

 Keywords: Jaynes Cummings Model, superconducting circuits, qubit measurement, dissipation engineering

 Circuit QED (Blais, Rev. Mod. Phys 2021): quantization of LC oscillator; need for a nonlinearity: Kerr Oscillator; Jaynes-Cummings model, dispersive regime, Schrieffer-Wolff transformation, coupling to environment (qubit decay/dephasing time);

  • Purcell effect/filter, control and read-out, signal-to-noise ratio, measurement induced dephasing, example: dissipation engineering for cooling and state preparation of a qubit