Case Study 7.2: Rabi Oscillations — From NMR to Quantum Computing

One equation, three Nobel Prizes, and the technology that images your brain, keeps GPS accurate, and will (perhaps) break modern cryptography

The Universal Two-Level System

In 1937, Isidor Isaac Rabi was studying the magnetic properties of atomic nuclei using molecular beams — streams of molecules flying through vacuum past carefully shaped magnets. He and his colleagues discovered that when a beam of lithium chloride molecules was exposed to a weak oscillating magnetic field tuned to a specific frequency, the nuclei would absorb energy and flip their magnetic orientation.

The frequency was not arbitrary. It had to match the energy gap between the two spin orientations of the nucleus in the static magnetic field — a condition now called magnetic resonance.

What Rabi discovered was not merely a technique for measuring nuclear magnetic moments (though it earned him the 1944 Nobel Prize for that). He had found a universal phenomenon: any quantum two-level system driven by a resonant oscillating field will undergo coherent oscillations between its two states. The specific system — a nuclear spin, an atomic transition, a superconducting qubit — is almost irrelevant. The mathematics is always the same.

The Physics in Three Systems

System 1: Nuclear Magnetic Resonance (NMR)

A proton (spin-1/2) in a static magnetic field $\vec{B}_0 = B_0\hat{z}$ has two energy levels:

• $|{\uparrow}\rangle$: spin aligned with field, energy $E_\uparrow = -\gamma\hbar B_0/2$
• $|{\downarrow}\rangle$: spin anti-aligned, energy $E_\downarrow = +\gamma\hbar B_0/2$

where $\gamma$ is the gyromagnetic ratio. The transition frequency is:

$$\omega_0 = \gamma B_0$$

For protons in a 3 Tesla MRI magnet: $\omega_0/(2\pi) = 42.58 \times 3 = 127.74$ MHz (in the radio-frequency range).

An oscillating transverse field $\vec{B}_1(t) = B_1\cos(\omega t)\hat{x}$ drives the transition. The Rabi frequency is:

$$\Omega_R = \gamma B_1$$

With $B_1 = 10^{-4}$ T: $\Omega_R/(2\pi) \approx 4.26$ kHz. A $\pi/2$-pulse takes about 59 microseconds; a $\pi$-pulse takes about 117 microseconds.

In MRI, the $\pi/2$-pulse tips the net magnetization of billions of protons into the transverse plane, where they precess and emit a detectable radio-frequency signal. The frequency of this signal depends on the local magnetic field, which is made spatially dependent using gradient coils. By analyzing the frequencies, a computer reconstructs a 3D image of the body — all based on Rabi's equation.

Felix Bloch and Edward Purcell independently developed NMR spectroscopy in 1946, earning the 1952 Nobel Prize in Physics. Paul Lauterbur and Peter Mansfield developed MRI in the 1970s, earning the 2003 Nobel Prize in Physiology or Medicine.

System 2: Atomic Clocks

The most accurate timekeeping devices ever built are atomic clocks based on the Rabi oscillation between two hyperfine levels of cesium-133 atoms.

The cesium atom's ground state has two hyperfine levels separated by exactly:

$$\nu_{\text{Cs}} = 9\,192\,631\,770 \text{ Hz}$$

(This is not an experimental number — it is the definition of the SI second since 1967.)

In a cesium fountain clock: 1. Cold cesium atoms are prepared in state $|F=3\rangle$ 2. A $\pi/2$ microwave pulse at $\nu_{\text{Cs}}$ creates a superposition $\frac{1}{\sqrt{2}}(|F=3\rangle + |F=4\rangle)$ 3. The atoms are tossed upward (about 1 meter) and fall back under gravity — a free evolution time $T \approx 1$ s 4. A second $\pi/2$ pulse is applied 5. The population in $|F=4\rangle$ is measured

This is Ramsey interferometry — essentially the Mach-Zehnder interferometer from Section 7.1, but with atomic states instead of photon paths and microwaves instead of beam splitters. The transition probability depends on the detuning as:

$$P_{3\to 4} = \cos^2\left(\frac{\delta T}{2}\right)$$

producing a narrow resonance peak (the "Ramsey fringe") of width $\Delta\nu \sim 1/(2T) \approx 0.5$ Hz. This allows the microwave frequency to be locked to the atomic transition with extraordinary precision.

Modern cesium fountain clocks achieve fractional frequency uncertainties of $\sim 10^{-16}$, corresponding to an error of about 1 second in 300 million years. Optical lattice clocks using strontium atoms now reach $\sim 10^{-18}$ — accurate enough to detect the gravitational redshift from raising the clock by 2 centimeters.

System 3: Superconducting Qubits

In a superconducting transmon qubit (the type used by IBM, Google, and others), two energy levels are formed by the quantized charge oscillations in a circuit containing a Josephson junction — a tunnel junction between two superconductors.

The transition frequency is typically $\omega_0/(2\pi) \sim 5$ GHz (microwave range). The system is driven by microwave pulses delivered through a coaxial cable.

Typical parameters: - $\pi$-pulse duration: 10-20 ns - Rabi frequency: $\Omega_R/(2\pi) \sim 25$-$50$ MHz - Coherence time $T_2$: 50-300 $\mu$s (current state-of-the-art) - Number of Rabi oscillations before decoherence: $T_2 \cdot \Omega_R \sim 10^3$-$10^4$

A single qubit gate is literally a calibrated Rabi pulse: - X gate (bit flip) = $\pi$-pulse: $|0\rangle \to |1\rangle$, $|1\rangle \to |0\rangle$ - Hadamard gate $\approx$ $\pi/2$-pulse (with appropriate phase): $|0\rangle \to (|0\rangle + |1\rangle)/\sqrt{2}$ - Phase gate = wait for free precession: $|1\rangle$ picks up phase relative to $|0\rangle$

Every quantum computing algorithm — from Shor's factoring algorithm to quantum error correction — is built from sequences of these pulses. The theory of Rabi oscillations from Section 7.9 is not an academic exercise: it is the literal operating principle of quantum computers.

The Bloch Sphere: Geometry of Two-Level Dynamics

The state of any two-level system can be written as:

$$|\psi\rangle = \cos(\theta/2)|1\rangle + e^{i\phi}\sin(\theta/2)|2\rangle$$

This maps to a point on a sphere of unit radius — the Bloch sphere (formally introduced in Chapter 13): - North pole ($\theta = 0$): $|1\rangle$ (ground state) - South pole ($\theta = \pi$): $|2\rangle$ (excited state) - Equator ($\theta = \pi/2$): equal superpositions with different relative phases

On this sphere: - Free precession ($\hat{V} = 0$) rotates the state around the $z$-axis at frequency $\omega_0$ - A resonant Rabi pulse rotates the state around an axis in the equatorial plane at frequency $\Omega_R$ - A $\pi$-pulse is a 180-degree rotation (north pole to south pole) - A $\pi/2$-pulse is a 90-degree rotation (north pole to equator)

This geometric picture makes it obvious why the Mach-Zehnder interferometer, NMR, and quantum computing all use the same mathematics: they are all rotations on the Bloch sphere, just performed by different physical mechanisms.

Decoherence: The Enemy of Rabi Oscillations

In a perfect isolated system, Rabi oscillations continue indefinitely. In reality, the oscillations decay due to interactions with the environment — a process called decoherence.

Two characteristic times govern the decay: - $T_1$ (longitudinal relaxation): the timescale for the excited state to decay to the ground state (energy loss) - $T_2$ (transverse relaxation): the timescale for the loss of phase coherence between $|1\rangle$ and $|2\rangle$

$T_2 \leq 2T_1$ always (you cannot maintain phase coherence longer than you maintain population).

The ratio of coherence time to gate time is the fundamental figure of merit for quantum computing. If this ratio is large enough (roughly $> 10^4$ with quantum error correction), fault-tolerant quantum computation becomes possible.

The Road from Rabi to Quantum Advantage

The sequence of ideas is remarkably direct:

1. Rabi (1937): Discovers resonant transitions in two-level systems
2. Bloch, Purcell (1946): Develop NMR spectroscopy
3. Ramsey (1950): Invents separated oscillatory fields (Ramsey interferometry)
4. Feynman (1982): Proposes quantum simulation
5. Deutsch (1985): Proposes quantum computation
6. Shor (1994): Discovers quantum factoring algorithm
7. Cirac, Zoller (1995): Propose trapped-ion quantum computer
8. Google (2019): Claims quantum supremacy with superconducting qubits

Every step in this chain depends on understanding and controlling Rabi oscillations. The ability to perform precise rotations on a two-level system — to deliver exactly the right pulse for exactly the right duration — is the atomic skill from which all of quantum technology is built.

Discussion Questions

1. In NMR, $\Omega_R/(2\pi) \sim 10$ kHz while $\omega_0/(2\pi) \sim 100$ MHz. The ratio $\omega_0/\Omega_R \sim 10^4$. Explain why this large ratio is essential for the validity of the rotating wave approximation.

2. The cesium clock definition $\nu_{\text{Cs}} = 9\,192\,631\,770$ Hz means the second is defined in terms of a quantum mechanical transition. Discuss the philosophical implications: our unit of time is fundamentally quantum.

3. In MRI, different tissues have different $T_1$ and $T_2$ values (e.g., gray matter: $T_1 \approx 1.0$ s, $T_2 \approx 100$ ms; fat: $T_1 \approx 0.25$ s, $T_2 \approx 80$ ms). Explain qualitatively how these differences produce image contrast.

4. Current quantum computers have $\sim 10^3$-$10^4$ operations per coherence time. Shor's algorithm to factor a 2048-bit number requires $\sim 10^{10}$ operations. How does quantum error correction bridge this gap? (Preview of Chapter 35.)

5. Both the Mach-Zehnder interferometer (photon paths) and Ramsey interferometry (atomic states) have the same mathematical structure. Identify the correspondence: what plays the role of the beam splitter? The mirrors? The path length difference?