ℏRabi Oscillations & The Jc Model

LIGHT-MATTER INTERACTION & QUANTUM CONTROL RABI OSCILLATIONS & THE JC MODEL Coherent Coaxial Dynamics of Atom-Cavity Coupled Systems An axiomatic guide detailing semi-classical drives, fully quantized fields, and the Rotating Wave Approximation.

1. SEMI-CLASSICAL CONTROL: RABI OSCILLATIONS When a two-level quantum system (qubit) interacts with a classical monochromatic electromagnetic drive, the state vector undergoes coherent periodic population transfers known as Rabi Oscillations. The Driven Qubit Hamiltonian (Rotating Frame): By applying the Rotating Wave Approximation (RWA) to drop counter-rotating terms oscillating rapidly at 2ω, the system energy matrix under a drive frequency ωd simplifies to: ℏ ℏ ℏ ∆ Ω Hrot = 2∆Z + 2ΩX = 2 Ω −∆ Where: ∆= ωq −ωd is the drive detuning, Ωis the Rabi frequency (drive amplitude), and X, Z are Pauli matrices. Transition Probability Evolution: Preparing the register in |0⟩at t = 0 and applying the drive yields a time-dependent probability of flipping the state to |1⟩: Ω2 Ωefft P0→1(t) = sin2 where the effective frequency is Ωeff = p Ω2 + ∆2 Ω2eff 2 ✦On-Resonance (∆= 0): Ensures a complete population swap (Pmax = 1) at a pulse time of t = π/Ω(a π-pulse). ✦Off-Resonance (∆̸= 0): Destructive interference restricts full population transfer (Pmax < 1), speeding up the oscillation velocity but minimizing maximum state-flip yield.

2. FULLY QUANTIZED INTERACTION: THE JAYNES-CUMMINGS HAMILTO- NIAN The Jaynes-Cummings (JC) Model upgrades the classical drive to a fully quantized electromagnetic mode inside an optical or superconducting cavity, forming the mathematical backbone of Cavity Quantum Electrodynamics (CQED). The Total Hamiltonian: Combining the bare qubit energy, the quantized photon field, and the dipole interaction yields: 1 1 HJC = ℏωr a†a + + + ℏg a†σ−+ aσ+ 2 2ℏωqZ Where a†/a are the photon creation/annihilation operators, σ+/σ−are the qubit raising/lowering operators, and g is the vacuum coupling strength. The interaction terms conserve total excitation number N = a†a + 12Z.

3. ENERGY SPLITTING: DRESSED STATE SUBSPACES Because total excitation is conserved, the JC matrix diagonalizes into uncoupled two-dimensional manifolds spanned by the unperturbed basis blocks {|e, n⟩, |g, n + 1⟩}. THE DRESSED EIGENSTATES & VACUUM SPLITTING Diagonalizing the n-photon interaction manifold yields the strongly coupled “dressed” eigenstates: |+, n⟩= cos θn |e, n⟩+ sin θn |g, n + 1⟩ |−, n⟩= −sin θn |e, n⟩+ cos θn |g, n + 1⟩ 2g√n + 1 Where the mixing angle matches: tan(2θn) = ∆ The Vacuum Rabi Splitting: On resonance (∆= 0), the energy separation between these states expands to ∆E = 2ℏg√n + 1. Crucially, even with an empty cavity (n = 0), a splitting of 2ℏg persists due to interaction with vacuum fluctuations.

4. THE DISPERSIVE REGIME & QUBIT READOUT When the qubit and cavity are far detuned relative to their coupling matrix (|∆| = |ωq −ωr| ≫g), actual real photon exchanges are suppressed. Applying a second-order Schrieffer-Wolff transformation reveals the dispersive Hamiltonian: 1 g2 Hdisp ≈ℏ(ωr + χZ) a†a + 2ℏω′qZ where the dispersive shift is χ = ∆ The Computational Realization: The cavity resonance frequency shifts by +χ or −χ conditional on the internal state of the qubit (Z = ±1). Measuring this microwave transmission shift enables non-destructive dispersive qubit state readout.