Research-level question in a solvable setting
Why does resonant excitation oscillate, and when is the rotating-wave approximation reliable?
A coherent near-resonant drive induces transitions between two energy levels. Control drive amplitude, detuning and duration, compare the exact rotating-wave approximation against a direct integration of the laboratory-frame Bloch dynamics, and identify when the approximation loses precision.
Prerequisites: quantum-state normalization, Schrödinger time evolution, matrix spectral problems, and the corresponding section of the English lecture notes. The mathematical explanation below is integral to the interactive exercise.
Quantitative predictions
Computed observables
Excited-state probability during a pulse
Compare the analytic RWA oscillation against integration of the explicitly oscillating laboratory Hamiltonian.
RWA and laboratory-frame excited-state probability versus pulse time
The exact rotating-wave expression is Pₑ=Ω²/(Ω²+Δ²)sin²[√(Ω²+Δ²)t/2]; the second curve integrates the full cosine drive with carrier ω₀=12.
Frequency selectivity
For a fixed pulse duration, the transition probability varies with detuning and can show sidelobes and shifted zeros.
Excited-state transition probability versus applied detuning
The RWA detuning scan is evaluated analytically at the selected pulse duration; it is not a Lorentzian equilibrium absorption spectrum.
Upper bound on off-resonant transfer
At a fixed detuning, the oscillatory probability never exceeds the envelope Ω²/(Ω²+Δ²).
Maximum possible RWA excitation amplitude versus detuning
The envelope describes peak population transfer achievable by choosing a suitable pulse duration within the RWA model.
Derive and compare
Mathematical model
Consider the driven spin Hamiltonian Hlab(t)=(ω₀/2)σz+Ω cos(ωdrive t)σx, with ℏ=1 and fixed carrier ω₀=12 for the lab-frame numerical calculation. The system begins in the lower σz eigenstate. In the frame rotating at ωdrive=ω₀−Δ, the rotating-wave approximation (RWA) discards the fast counter-rotating term and leaves the constant effective Hamiltonian Heff=(Δσz+Ωσx)/2.
At resonance Δ=0, the excited probability follows sin²(Ωt/2), reaching one at a π pulse. For nonzero detuning the maximum is Ω²/(Ω²+Δ²), so a finite off-resonant drive cannot fully transfer the population under the RWA. The effective oscillation angular frequency ΩR increases with |Δ| even as the excitation envelope decreases.
To assess the approximation, we independently integrate the laboratory-frame Bloch equation ṙ=B(t)×r with B(t)=[2Ωcos(ωdrive t),0,ω₀], initial r=(0,0,−1), and Pₑ=(1+rz)/2. The fourth-order Runge–Kutta integration uses small steps relative to the carrier oscillation, with no RWA invoked in the right-hand side. Differences between curves diagnose counter-rotating corrections, which grow as Ω or |Δ| becomes comparable to the carrier frequency.
A frequency scan at fixed pulse time should not be confused with a steady-state spectroscopic line shape. Both the coherent nutation angle and the detuning-dependent Rabi frequency influence that scan. Dissipation would create a different driven steady-state problem.
Test an exactly soluble physical limit
At zero drive Ω=0 the prepared lower state does not excite. At resonance and within the RWA, t=π/Ω yields unit excitation. At large |Δ|≫Ω the maximum excitation probability decreases quadratically as Ω²/Δ². For Ω≪ω₀, numerical laboratory-frame evolution approaches the RWA over moderate pulses.
Assumptions, uncertainties, and reliability
Isolated driven qubit with classical monochromatic field, no spontaneous emission, no decoherence, positive drive frequency, ℏ=1 and fixed ω₀=12. The analytical curve is RWA; laboratory integration gives a direct model comparison over finite time and is limited by the numerical time step.
Explore independently
Three worked investigations
- Find a π pulse. Set Δ=0 and adjust t to π/Ω. Record the RWA excitation and check the full laboratory-frame result.
- Study detuning suppression. Keep Ω and t fixed, move Δ and distinguish the oscillation frequency from the maximum transition envelope.
- Stress-test an approximation. Increase Ω/ω₀. Compare the two independently calculated curves and identify the parameter regime in which counter-rotating corrections become noticeable.
Self-check: On exact resonance, which RWA pulse duration first gives unit excitation?
Extended exercise: derive one prediction independently, identify the model approximation or physical symmetry that supports it, and test its numerical convergence. AI guidance is useful for generating further test cases after the derivation has been checked.
Interpretation
What survives beyond the model
Driven resonance connects state preparation to time-dependent perturbations, rotating frames and experimentally measurable pulse protocols. Comparing the analytically soluble RWA against full numerical motion makes the approximation itself an investigable physical object.
Lecture notes: 06_time_dependent_perturbation.tex ↗.