Drive a qubit with resonant light and watch it oscillate on the Bloch sphere—add decoherence to see damped oscillations
Rabi oscillations are one of the most fundamental phenomena in quantum mechanics. When a two-level quantum system (like an atom or qubit) is driven by a resonant electromagnetic field, the population oscillates between the ground state |0⟩ and excited state |1⟩. This sinusoidal exchange of probability is called Rabi flopping, and the oscillation frequency—the Rabi frequency—is proportional to the strength of the driving field.
The blue arrow (Bloch vector) represents the quantum state of the qubit. The north pole is |0⟩ (ground state), the south pole is |1⟩ (excited state). When driven on resonance (δ=0), the Bloch vector rotates around the equator, smoothly oscillating between |0⟩ and |1⟩. The trajectory traces a path on the sphere—a great circle when there's no damping, a spiral inward when decoherence is present.
The time-series plot shows P₀(t) and P₁(t)—the probabilities of measuring |0⟩ or |1⟩. For resonant driving (δ=0), these oscillate as P₁(t) = sin²(Ωt/2), where Ω is the Rabi frequency. Starting from |0⟩, the population smoothly transfers to |1⟩, then back to |0⟩, repeating indefinitely. With detuning (δ≠0), the oscillations become incomplete—the qubit doesn't fully reach |1⟩.
Detuning δ is the difference between the driving frequency and the qubit's natural frequency. When δ=0, the drive is on resonance and produces complete Rabi flopping. When δ≠0, the effective Rabi frequency becomes Ω' = √(Ω² + δ²), and the maximum population transfer decreases. The Bloch vector rotates around a tilted axis instead of the equator. This is why laser cooling and NMR require precise frequency control.
Real quantum systems lose coherence due to environmental interactions. T₁ (energy relaxation time) causes the qubit to decay from |1⟩ to |0⟩—the Bloch vector drifts toward the north pole. T₂ (dephasing time) causes loss of phase coherence—the Bloch vector shrinks toward the origin. For pure dephasing, T₂ ≤ 2T₁. When damping is enabled, you'll see the oscillations decay exponentially, and the Bloch vector spirals inward. This is why quantum computers need error correction.
The driven two-level system is described by the Hamiltonian (in the rotating frame):
H = ½ ℏδσ_z + ½ ℏΩσ_x
where σ_x and σ_z are Pauli matrices, δ is the detuning, and Ω is the Rabi frequency. This drives coherent rotation of the Bloch vector around the axis (Ω, 0, δ) in the rotating frame. The dynamics are described by the Schrödinger equation, which produces sinusoidal oscillations with frequency √(Ω² + δ²).
At resonance (δ=0), the drive purely induces σ_x rotations—the Bloch vector circles the equator. The population oscillates completely: P₁(t) = sin²(Ωt/2). Off resonance, the effective rotation axis tilts out of the equator. The population oscillates with amplitude Ω²/(Ω² + δ²), which decreases as |δ| increases. Far off resonance (|δ| ≫ Ω), the drive barely affects the qubit. This selectivity is exploited in quantum gates to address specific transitions without disturbing others.
Decoherence is described by the Lindblad master equation, which adds dissipative terms to the unitary evolution. Energy relaxation (T₁) is modeled by a Lindblad operator L = √(γ₁)σ₋, where γ₁ = 1/T₁. Dephasing (T₂) adds γ_φ = 1/T₂ - 1/(2T₁). These drive the density matrix toward a mixed state: the Bloch vector shrinks. In superconducting qubits, typical T₁ is tens of microseconds; in trapped ions, it can be minutes. The challenge of quantum computing is performing gates faster than T₂.
Perfect Rabi Flopping: Set δ=0 (on resonance), Ω=1, and disable damping. Watch the Bloch vector trace a perfect circle on the equator. The populations oscillate sinusoidally with period 2π/Ω. This is the textbook case—coherent, reversible, periodic.
Detuning Effect: Increase δ from 0 to 3 while keeping Ω=1. Notice how the maximum population transfer to |1⟩ decreases. The Bloch vector no longer circles the equator—it rotates around a tilted axis. The oscillation frequency increases to √(Ω² + δ²). At large δ, the drive barely affects the qubit.
Damping Destroys Oscillations: Enable damping with moderate T₁=10 and T₂=8. The oscillations decay exponentially. The Bloch vector spirals inward toward a steady state. For strong damping (small T₁, T₂), the oscillations may disappear entirely—the system reaches equilibrium faster than it can oscillate. This is the overdamped regime.
T₁ vs T₂: Set T₁=20 and T₂=5 (strong dephasing, weak relaxation). The Bloch vector shrinks rapidly toward the z-axis but stays near the equator. Now reverse: T₁=5, T₂=20. The vector drifts toward the north pole (ground state) while maintaining length. T₁ causes energy loss; T₂ causes phase loss. Both destroy quantum coherence, but in different ways.
π-Pulse: Set Ω=1, δ=0, and watch until t = π (one half-cycle). The qubit has flipped from |0⟩ to |1⟩. This is a π-pulse (or NOT gate in quantum computing). Now wait until t = 2π. You're back at |0⟩. A π-pulse inverts the state; a 2π-pulse does nothing. These are universal gates for qubits.
Decoherence-Limited Gates: Enable damping with T₂=10. Set Ω to achieve a π-pulse (Ωt=π) in much less than T₂. The gate succeeds. Now decrease Ω so the π-pulse takes longer than T₂. The gate fails—decoherence wins. This is why quantum computers need fast gates and long coherence times. The ratio T₂/t_gate is a key figure of merit.
Nutations (Fast Rabi): Increase Ω to 5. The Bloch vector spins rapidly. In experiments, you'd see the qubit oscillate many times within the measurement window. These fast oscillations (nutations) are visible in NMR and atomic physics. The Rabi frequency is directly measurable—it's proportional to the driving field amplitude.
Approach to Steady State: Enable strong damping (T₁=5, T₂=3) and wait. The system reaches a steady state where driving and damping balance. At resonance, the steady-state population depends on the ratio Ω/γ₁. For weak driving, the qubit stays mostly in |0⟩. For strong driving, it's pumped partially into |1⟩. This is the physics of optical pumping and laser cooling.
Rabi oscillations are not an esoteric detail—they're the beating heart of quantum control. Here's why:
Quantum gates are Rabi pulses: The fundamental operations in quantum computing (X, Y, Hadamard) are implemented by applying resonant drives for precisely calibrated durations. An X gate is a π-pulse; a Hadamard is a π/2-pulse followed by phase shifts. All single-qubit gates reduce to Rabi flopping.
Spectroscopy measures Rabi frequency: In atomic physics, NMR, and ESR, you measure the Rabi frequency to determine coupling strengths, transition dipole moments, and field intensities. It's a direct probe of the atom-field interaction.
Decoherence sets the quantum speed limit: The competition between coherent Rabi oscillations and incoherent damping determines what's possible. If T₂ is short, you can't perform many gates. If T₁ is short, your qubit resets to |0⟩. Understanding and extending T₁ and T₂ is the central challenge in building quantum computers.
Dressed states and AC Stark shifts: The eigenstates of the driven Hamiltonian (dressed states) are split by the Rabi frequency. This energy-level structure underlies phenomena like Autler-Townes splitting and light shifts, used in laser cooling and quantum control.
Adiabatic passage and coherent control: By slowly varying Ω and δ, you can adiabatically transfer population without oscillations (STIRAP). By shaping pulses, you can suppress errors and optimize gate fidelity. The Rabi model is the foundation for these advanced control techniques.
From NMR to superconducting qubits to trapped ions, every two-level quantum system exhibits Rabi oscillations when driven. Mastering this toy means mastering the language of quantum control—the physics that makes quantum computing, atomic clocks, and quantum sensors possible. Rabi's 1937 molecular beam experiments were the genesis; today's quantum computers are the culmination.