Open quantum systems

Description

Short description:

1 General concepts and theoretical background (10h)

• 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)

• 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.

• Light-matter interaction & Linear response: 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)

• 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.

Syllabus

Key topics:

• Bath & system-bath coupling; master equation, quantum regression theorem, noise spectra

• Electron-phonon coupling; linear response theory, dipole-dipole correlation function, lineshape function, pure dephasing.

• Jaynes Cummings Model, superconducting circuits, qubit measurement, dissipation engineering

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