Quantum Mechanics · Chapter 20

Driven two-level systems

The simplest quantum system pushed from outside, which is how atoms are driven by lasers and qubits by microwaves. Two-level Hamiltonians, avoided crossings, Rabi oscillations, the rotating frame, detuning, and the Landau–Zener formula for a level crossing swept at a finite speed.

H^= \frac{1}{2} (ϵ σ_{z} + Δ σ_{x})

E_{\pm} = \pm \frac{1}{2} ϵ^{2} + Δ^{2}

P_{e} (t) = sin^{2} \frac{Ω t}{2}

Ω_{R} = Ω^{2} + δ^{2}

Lessons

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05DetuningDrive the atom a little off its frequency and the Rabi oscillation changes in two ways. It no longer reaches the upper level: the largest probability is $\Omega^{2}/(\Omega^{2}+\delta^{2})$. And it swings faster, at the generalised Rabi frequency $\sqrt{\Omega^{2}+\delta^{2}}$. On the Bloch sphere the state turns about a tilted axis, so it never reaches the south pole. Swept through the resonance, the largest probability traces a peak of half-width $\Omega$: a strong drive broadens the line it drives.Read lesson →

06Landau–Zener transitionsNow sweep the bias through an avoided crossing at a steady rate. A slow sweep follows the adiabatic level and ends in the other diabatic state; a fast one does not notice the coupling and stays in the diabatic state it started in, jumping the gap. Landau and Zener found the exact probability of the jump: $e^{-\pi\Delta^{2}/2\hbar v}$, where $\Delta$ is the smallest gap and $v$ the rate at which the bias changes. It depends on one number, the square of the gap over the sweep rate. It is exact, and it comes from summing the whole Dyson series of the problem.Read lesson →

07Adiabatic and diabatic evolutionA slow sweep through an avoided crossing moves a system from one diabatic state to the other with near certainty, and it does not care about the details: the exact rate, the exact coupling, the shape of the pulse, as long as the adiabatic theorem holds. That robustness is why adiabatic passage is used where a $\pi$ pulse, which needs its length and strength just right, would be fragile. A fast sweep, by contrast, leaves the state behind — the diabatic limit, the sudden approximation of the chapter on time-dependent perturbations. Between them the Landau–Zener formula interpolates, and a sweep can be tuned to split a state into any chosen superposition.Read lesson →

08Chapter practiceBring the chapter together: two-level Hamiltonians, avoided crossings, Rabi oscillations, the rotating frame, detuning, the Landau–Zener formula and adiabatic passage. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Driven two-level systems

Build the foundation