5.2 Time-Dependent Perturbation Theory

5.2 Time-Dependent Perturbation Theory#

Overview#

When a quantum system is driven by a time-varying external field, transitions between energy levels occur. Time-dependent perturbation theory provides the framework for computing transition amplitudes, rates, and response functions. The interaction picture separates fast free evolution from slow perturbative dynamics, leading to the Dyson series and Green’s functions. Fermi’s golden rule converts these amplitudes into measurable transition rates, while the adiabatic theorem describes the opposite limit of slow changes. Applications range from atomic spectroscopy and spontaneous emission to the Kubo formula for quantum Hall conductance.

Topics#

Lesson

Title

Core Question

5.2.1

Interaction Picture

How do we separate fast free evolution from slow perturbative dynamics?

5.2.2

Fermi’s Golden Rule

When does a transition probability become a constant rate?

5.2.3

Applications

How does perturbation theory predict spectra, lifetimes, and conductance?

Key Concepts#

  • Interaction picture: Rotating frame that factors out \(H_0\) evolution, exposing slow \(V(t)\) dynamics

  • Dyson series: Systematic expansion of the time-evolution operator in powers of the perturbation

  • Green’s function: Resolvent operator encoding bound states, resonances, and scattering continuum

  • Fermi’s golden rule: Transition rate \(\Gamma = (2\pi/\hbar)\vert\langle f\vert V\vert i\rangle\vert^2 \rho(E_i)\) for transitions into a continuum

  • Selection rules: Symmetry constraints that forbid certain matrix elements

  • Adiabatic theorem: Slow parameter changes keep the system in its instantaneous eigenstate, acquiring a Berry phase

  • Kubo formula: Linear response expression for conductivity from current-current correlations

Learning Objectives#

  • Transform between Schr”{o}dinger, Heisenberg, and interaction pictures, and construct the Dyson series for transition amplitudes.

  • Derive Fermi’s golden rule and apply it to compute transition rates, lifetimes, and selection rules.

  • Explain the adiabatic theorem and its connection to the Berry phase.

  • Apply harmonic perturbation theory to atomic transitions and spectroscopy.

  • Use the Kubo formula to compute the Hall conductance for filled Landau levels.

Project#

Project: Cavity QED and the Jaynes-Cummings Model: Rabi Splitting and Collapse-Revival#

The Jaynes-Cummings model describes a two-level atom coupled to a single cavity mode. It exhibits rich dynamics: Rabi oscillations, collapse of oscillations due to photon decay, and surprising revival behavior. This project uses time-dependent perturbation theory and exact solvers to explore the frontier of quantum optics.

Objective: Study the dynamics of atom-cavity coupling, compute Rabi splitting, and discover collapse-revival phenomena—physics at the heart of cavity QED and superconducting qubit systems.

Suggested Approach:

  1. Setup: Write the Jaynes-Cummings Hamiltonian $\(H = \hbar \omega_0 \hat{\sigma}^+\hat{\sigma}^- + \hbar \omega_c \hat{a}^\dagger\hat{a} + \hbar g (\hat{\sigma}^+ \hat{a} + \hat{\sigma}^- \hat{a}^\dagger)\)$

    where \(g\) is the coupling strength, and \(\omega_0, \omega_c\) are atomic and cavity frequencies.

  2. Weak coupling limit (\(g \ll |\omega_0 - \omega_c|\)): Apply time-dependent perturbation theory to compute transition rates and predict Rabi frequency.

  3. Strong coupling regime: Solve the Hamiltonian exactly (analytically or numerically). Observe:

    • Rabi splitting: vacuum Rabi oscillations at frequency \(\Omega_R \approx 2g\)

    • Collapse-revival: oscillations decay (collapse) due to photon leakage, then partially revive

  4. Dissipation model: Include cavity decay \(\kappa\) (photon escape) and atomic spontaneous emission \(\gamma\). Use the Lindblad master equation.

  5. Experimental connections: Compare to superconducting qubit data (circuit QED, transmons) or optical cavity QED (Rydberg atoms, trapped ions).

Expected Deliverable:

  • Code implementing: Rabi model, Jaynes-Cummings Hamiltonian, Lindblad dissipation

  • Plots: (1) time evolution of atom and cavity populations; (2) collapse and revival of Rabi oscillations; (3) dissipation effects on dynamics

  • Analysis: extract Rabi frequency from oscillations; measure collapse and revival timescales; compare weak-coupling perturbative prediction to exact result

  • Brief report (2–3 pages) explaining the physics, experimental applications, and limitations of perturbation theory in the strong-coupling regime

Frontier relevance: Cavity QED is a testbed for quantum technologies. The Jaynes-Cummings model and its extensions underpin superconducting qubit systems (IBM, Google, Rigetti), trapped-ion quantum computers, and quantum simulators. Understanding how perturbation theory breaks down and exact solutions emerge is essential for engineering quantum devices.