Lecture 2.6: Time Evolution and Rabi Oscillations

Review: The Bloch Sphere and Rotations¶

Qubit States on the Bloch Sphere¶

Any qubit state can be written as:

where $\theta$ and $\phi$ are the polar and azimuthal angles on the Bloch sphere.

The six cardinal states are:

• z-axis: $|0\rangle$ (north pole), $|1\rangle$ (south pole)

• x-axis: $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, $|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$

• y-axis: $|+i\rangle = \frac{1}{\sqrt{2}}(|0\rangle + i|1\rangle)$, $|-i\rangle = \frac{1}{\sqrt{2}}(|0\rangle - i|1\rangle)$

Rotations: SU(2)¶

Last lecture we learned that SU(2) is the group of rotations on the Bloch sphere.

Rotation about the z-axis:

where $\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.

Rotation about an arbitrary axis $\hat{n}$:

where $\hat{n}\cdot\vec{\sigma} = n_x\sigma_x + n_y\sigma_y + n_z\sigma_z$.

iClicker Question 1¶

Consider the rotation $R_z(\theta) = e^{-i\sigma_z\theta/2}$.

Which pair of states are unchanged by this rotation (up to a global phase)?

• (A) $|+\rangle$ and $|-\rangle$ (the ±x states)

• (B) $|+i\rangle$ and $|-i\rangle$ (the ±y states)

• (C) $|0\rangle$ and $|1\rangle$ (the ±z states) ✓

• (D) All six cardinal states

Answer: (C). The states $|0\rangle$ and $|1\rangle$ are eigenstates of $\sigma_z$, so they only pick up a phase under $R_z(\theta)$:

States on the equator ($|+\rangle$, $|-\rangle$, $|+i\rangle$, $|-i\rangle$) are not eigenstates of $\sigma_z$ — they precess around the z-axis.

General principle: Eigenstates of a generator are invariant under the transformation it generates.

The Stern-Gerlach Experiment: Discovering Spin¶

The 1922 Experiment¶

Before we write down the spin Hamiltonian, let’s see where it comes from experimentally.

In 1922, Otto Stern and Walther Gerlach performed one of the most important experiments in quantum mechanics. They sent a beam of silver atoms through an inhomogeneous magnetic field and observed where the atoms landed on a detector screen.

Classical expectation: A magnetic dipole in a non-uniform field feels a force proportional to its component along the field gradient. If the atomic magnetic moments were oriented randomly (like tiny compass needles pointing in all directions), the beam should spread into a continuous smear.

What they observed: The beam split into exactly two discrete spots.

Why Deflection Measures Spin¶

The key physics is straightforward:

1. Energy of a dipole in a field: A magnetic moment $\vec{\mu}$ in a magnetic field has energy

   $$U = -\vec{\mu}\cdot\vec{B}$$

   (5)

2. Force from a gradient: Force is the negative gradient of energy. For a field pointing in the z-direction with a gradient:

   $$F_z = -\frac{\partial U}{\partial z} = \mu_z\frac{\partial B_z}{\partial z}$$

   (6)

3. Magnetic moment comes from spin: For an electron, $\mu_z = \gamma S_z$ where $\gamma$ is the gyromagnetic ratio.

4. Putting it together:

   $$F_z = \gamma S_z \frac{\partial B_z}{\partial z}$$

   (7)

The force is directly proportional to $S_z$. Classically, $S_z$ could be anything, giving a continuous range of forces and a smeared spot. But quantum mechanics says $S_z$ can only be $+\hbar/2$ or $-\hbar/2$ — nothing in between.

Two allowed values of spin → two allowed forces → two spots.

This was shocking. The atoms behaved as if their magnetic moment could only point in two directions — “up” or “down” relative to the field — with nothing in between.

The Interpretation: Spin-1/2¶

We now understand that:

1. Silver has one unpaired electron in its outer shell (configuration: [Kr] 4d¹⁰ 5s¹)

2. The electron has an intrinsic angular momentum called spin, with quantum number $s = 1/2$

3. When measured along any axis, spin-1/2 can only yield two outcomes: $+\hbar/2$ (“spin up”) or $-\hbar/2$ (“spin down”)

4. The two spots correspond to these two eigenstates

The Stern-Gerlach apparatus is a qubit measurement device. It projects the electron spin onto the $|0\rangle$ or $|1\rangle$ state (relative to the field direction) and physically separates them.

Why This Matters for Us¶

The Stern-Gerlach experiment shows that:

• Quantum measurements give discrete outcomes, not continuous values

• The spin state collapses upon measurement

• A spin-1/2 particle is a natural physical qubit

When we write $|\psi\rangle = c_0|0\rangle + c_1|1\rangle$ and talk about the Bloch sphere, we’re describing something real: the quantum state of an electron spin.

Back to Physics: Spin in a Magnetic Field¶

The Physical Setup¶

An electron (or any spin-1/2 particle) in a magnetic field $\vec{B}$ has Hamiltonian:

where $\gamma$ is the gyromagnetic ratio.

The spin precesses around the magnetic field direction — this is the quantum analog of a gyroscope.

The Question¶

Given an initial state $|\psi(0)\rangle$, how do we find $|\psi(t)\rangle$?

The Schrödinger Equation¶

The Fundamental Law¶

Quantum states evolve according to:

Solving with the Time Evolution Operator¶

Define the time evolution operator $U(t)$ by:

Substituting into the Schrödinger equation:

Since this must hold for any initial state:

The solution is:

You can verify this solves the equation: $\frac{\partial}{\partial t}e^{-iHt/\hbar} = \frac{-iH}{\hbar}e^{-iHt/\hbar}$.

The Hamiltonian as a Generator¶

In the infinitesimal limit ($\Delta t$ small):

Compare to our rotation formula: $R(\delta\theta) \approx I + \delta\theta\, G$.

The Hamiltonian is the generator of time evolution.

Just as the Pauli matrices generate rotations in space, the Hamiltonian generates translations in time.

$U(t)$ is Unitary¶

Why Unitarity Matters¶

We start with a normalized state: $\langle\psi(0)|\psi(0)\rangle = 1$.

We need it to stay normalized:

For this to equal 1 for any initial state, we need:

This is the definition of a unitary operator.

Verification¶

For $U(t) = e^{-iHt/\hbar}$ with Hermitian $H$ (i.e., $H^\dagger = H$):

Unitarity guarantees probability conservation.

iClicker Questions¶

iClicker A: The Generator Pattern¶

The Hamiltonian generates time evolution: $e^{-iHt/\hbar}|\psi(0)\rangle = |\psi(t)\rangle$.

What does the operator $e^{-i\hat{p}x_0/\hbar}$ do to a wavefunction $\psi(x)$?

• (A) Translates it forward in time

• (B) Translates it in position by $x_0$ ✓

• (C) Rotates it on the Bloch sphere

• (D) Decreases its amplitude

• (E) Inverts it

Answer: (B). Just as the Hamiltonian $H$ generates time translations, the momentum operator $\hat{p}$ generates spatial translations:

This is the same pattern: Hermitian operator in the exponent -> unitary transformation that “moves” the state.

The Generator Pattern¶

We’ve now seen the same mathematical structure three times:

The pattern: Hermitian operators generate unitary transformations through exponentiation.

This is one of the deepest ideas in physics: symmetries (unitary transformations) are generated by observables (Hermitian operators).

Example 1: Magnetic Field Along z — The Phase Gate¶

Setup¶

where $\omega_0 = -\gamma B_z$ is the Larmor frequency.

Time Evolution¶

This is rotation around the z-axis at angular frequency $\omega_0$.

What Happens to Different States?¶

Starting in $|0\rangle$:

Just a global phase — the state doesn’t move on the Bloch sphere.

Starting in $|+\rangle$:

The relative phase changes — the state precesses around the equator.

This is precession: the quantum version of a spinning top in a gravitational field.

Example 2: Magnetic Field Along x — Rabi Oscillations¶

Setup¶

where $\Omega$ is the Rabi frequency (proportional to $B_x$).

Time Evolution¶

Using the rotation formula:

Evolution of $|0\rangle$¶

Starting in the ground state $|0\rangle$:

The Probabilities Oscillate!¶

The population oscillates between $|0\rangle$ and $|1\rangle$. This is fundamentally different from precession (Example 1), where the populations stayed constant.

Special Pulses¶

π-pulse ($\Omega t = \pi$):

Complete population inversion — the qubit flips from ground to excited state.

π/2-pulse ($\Omega t = \pi/2$):

Creates an equal superposition — the state moves from the pole to the equator.

Eigenstates: States That Don’t Change¶

Definition¶

An eigenstate of the Hamiltonian is a state that evolves only by a global phase:

Under time evolution:

The state just picks up a phase factor — it doesn’t move on the Bloch sphere.

Examples¶

For $H = \frac{\hbar\omega_0}{2}\sigma_z$:

• Eigenstates are $|0\rangle$ and $|1\rangle$ (the poles)

• These don’t precess — they’re aligned with the field

For $H = \frac{\hbar\Omega}{2}\sigma_x$:

• Eigenstates are $|+\rangle$ and $|-\rangle$ (the x-axis states)

• Under a $B_x$ field, $|+\rangle$ doesn’t undergo Rabi oscillations — it’s aligned with the drive

General rule: Eigenstates of the Hamiltonian are stationary. Superpositions of different energy eigenstates oscillate.

Qiskit Demonstration: Rabi Oscillations¶

import numpy as np
import matplotlib.pyplot as plt
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

# Rabi oscillations: Rx(θ) applied to |0⟩
thetas = np.linspace(0, 4*np.pi, 100)
P0 = []
P1 = []

for theta in thetas:
    qc = QuantumCircuit(1)
    qc.rx(theta, 0)  # Rotation around x-axis by angle theta
    state = Statevector(qc)
    probs = state.probabilities()
    P0.append(probs[0])
    P1.append(probs[1])

# Plot
plt.figure(figsize=(10, 5))
plt.plot(thetas/np.pi, P0, 'b-', linewidth=2, label='$P_0(t) = \\cos^2(\\Omega t/2)$')
plt.plot(thetas/np.pi, P1, 'r-', linewidth=2, label='$P_1(t) = \\sin^2(\\Omega t/2)$')
plt.xlabel('$\\Omega t / \\pi$', fontsize=14)
plt.ylabel('Probability', fontsize=14)
plt.title('Rabi Oscillations: Population Transfer Between $|0\\rangle$ and $|1\\rangle$', fontsize=14)
plt.legend(fontsize=12)
plt.grid(True, alpha=0.3)
plt.axhline(y=0.5, color='gray', linestyle='--', alpha=0.5)
plt.axvline(x=1, color='green', linestyle=':', alpha=0.7, label='π-pulse')
plt.axvline(x=0.5, color='orange', linestyle=':', alpha=0.7, label='π/2-pulse')
plt.show()

Summary¶

1. Time evolution is governed by the Schrödinger equation: $|\psi(t)\rangle = e^{-iHt/\hbar}|\psi(0)\rangle$

2. The Hamiltonian generates time evolution, just as Pauli matrices generate spatial rotations.

3. Unitarity ($U^\dagger U = I$) guarantees probability conservation.

4. Precession ($H \propto \sigma_z$): States rotate around the z-axis. Poles are stationary; equator precesses.

5. Rabi oscillations ($H \propto \sigma_x$): Population oscillates between $|0\rangle$ and $|1\rangle$. π-pulse inverts; π/2-pulse creates superposition.

6. Eigenstates of the Hamiltonian are stationary — they evolve only by a global phase.

iClicker E: Energy Eigenstates¶

A qubit has Hamiltonian $H = \frac{\hbar\omega}{2}\sigma_z$, so the energy eigenstates are $|0\rangle$ (energy $+\hbar\omega/2$) and $|1\rangle$ (energy $-\hbar\omega/2$).

The state $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ is prepared at $t=0$. At time $t$, the state is:

• (A) Still $|+\rangle$ (stationary)

• (B) $\frac{1}{\sqrt{2}}(e^{-i\omega t/2}|0\rangle + e^{+i\omega t/2}|1\rangle)$ ✓

• (C) $|0\rangle$

• (D) $|1\rangle$

• (E) A mixed state

Answer: (B). Energy eigenstates each pick up their own phase factor $e^{-iE_n t/\hbar}$. Since $|+\rangle$ is a superposition of two energy eigenstates with different energies, the relative phase changes with time. The state precesses around the z-axis.

iClicker F: What Doesn’t Change?¶

Under the Hamiltonian $H = \frac{\hbar\Omega}{2}\sigma_x$, which quantity is CONSTANT in time?

• (A) $\langle\sigma_z\rangle$ (z-component of spin)

• (B) $\langle\sigma_x\rangle$ (x-component of spin) ✓

• (C) $\langle\sigma_y\rangle$ (y-component of spin)

• (D) The state $|\psi\rangle$

• (E) None of these

Answer: (B). The Hamiltonian $H \propto \sigma_x$ generates rotations around the x-axis. Under such rotations, the x-component of any vector is unchanged — only the y and z components rotate into each other.

More formally: $[H, \sigma_x] = [\sigma_x, \sigma_x] = 0$, so $\langle\sigma_x\rangle$ is a conserved quantity.

Homework 2.6¶

Problem: Time Evolution Practice¶

A qubit has Hamiltonian $H = \frac{\hbar\omega}{2}\sigma_z$.

(a) Write the time evolution operator $U(t) = e^{-iHt/\hbar}$ as an explicit $2\times 2$ matrix.

(b) If the initial state is $|0\rangle$, find $|\psi(t)\rangle$. Does the state change physically, or just by a global phase?

(c) If the initial state is $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, find $|\psi(t)\rangle$.

(d) For part (c), compute $P_0(t) = |\langle 0|\psi(t)\rangle|^2$ and $P_1(t) = |\langle 1|\psi(t)\rangle|^2$. Do the populations change with time?

(e) For part (c), compute $P_+(t) = |\langle +|\psi(t)\rangle|^2$. At what time $t$ does the state first become orthogonal to $|+\rangle$? What state is it at that moment?

Problem: Rabi Oscillations¶

A qubit has Hamiltonian $H = \frac{\hbar\Omega}{2}\sigma_x$ and starts in state $|0\rangle$.

(a) Using $R_x(\theta) = \cos\frac{\theta}{2}I - i\sin\frac{\theta}{2}\sigma_x$, show that:

(b) Compute $P_1(t) = |\langle 1|\psi(t)\rangle|^2$. Sketch this function from $t=0$ to $t = 4\pi/\Omega$.

(c) At what time does the qubit first have a 50% probability of being in $|1\rangle$? What is this pulse called?

(d) At what time does the qubit first reach $|1\rangle$ with certainty? What is this pulse called?

(e) If $\Omega = 2\pi \times 10$ MHz, calculate the duration of a π-pulse in nanoseconds.

Problem: The Hadamard Gate¶

The Hadamard gate is one of the most important single-qubit gates:

(a) Verify that $H$ is unitary: compute $H^\dagger H$ and show it equals $I$.

(b) Verify that $H^2 = I$. What does this tell you geometrically about the rotation?

(c) The general rotation formula is:

For $H^2 = I$, we need $\theta = \pi$ (a 180° rotation). So $H$ should equal $-i(\hat{n}\cdot\vec{\sigma})$ for some axis $\hat{n}$, up to a global phase.

Show that:

(d) From part (c), what is the rotation axis $\hat{n}$? Express it as a unit vector.

(e) Describe in words: where is this axis on the Bloch sphere? (Hint: it’s in the xz-plane, halfway between...)

(f) The Hadamard swaps the z-basis and x-basis:

• $H|0\rangle = |+\rangle$

• $H|1\rangle = |-\rangle$

• $H|+\rangle = |0\rangle$

• $H|-\rangle = |1\rangle$

Verify one of these by explicit matrix multiplication.

(g) Why do you think the Hadamard gate is preferred over, say, $R_y(\pi/2)$ for creating superpositions? (Hint: look at the matrix elements of each.)

Problem: Unitarity Implies Hermitian Generator¶

This problem proves the deep connection between unitary operators and Hermitian generators.

(a) Let $U = e^{iG}$ for some operator $G$. Write an expression for $U^\dagger$ in terms of $G^\dagger$.

(b) The condition for $U$ to be unitary is $U^\dagger U = I$. Substitute your expression from (a) and show that this requires:

(c) For the equation in (b) to hold, we need $G^\dagger = G$. Explain why.

Hint: If $A$ and $B$ are matrices, $e^A e^B = e^{A+B}$ only when $[A,B] = 0$. What does $e^{-iG^\dagger}e^{iG} = I = e^0$ tell you?

(d) Conclude: If $U = e^{iG}$ is unitary, then $G$ must be Hermitian.

(e) We’ve been writing time evolution as $U(t) = e^{-iHt/\hbar}$. Using your result, explain why the Hamiltonian $H$ must be Hermitian for time evolution to be unitary.

(f) The Pauli matrices $\sigma_x$, $\sigma_y$, $\sigma_z$ are Hermitian. Verify that $\sigma_x^\dagger = \sigma_x$ by writing out the matrix and its conjugate transpose.

(g) Using your result from (d), explain why $R_z(\theta) = e^{-i\sigma_z\theta/2}$ is guaranteed to be unitary without doing any explicit calculation.

Problem: Space Translation Operator¶

The momentum operator in one dimension is $\hat{p} = -i\hbar\frac{d}{dx}$.

(a) Consider the translation operator $T(a) = e^{-i\hat{p}a/\hbar}$. Using $\hat{p} = -i\hbar\frac{d}{dx}$, show that:

(b) Taylor expand $e^{-a\frac{d}{dx}}$ acting on a function $\psi(x)$:

(c) Recognize this Taylor series as the expansion of $\psi(x-a)$ around $x$. Conclude:

(Note: The sign convention varies. With $T(a) = e^{+i\hat{p}a/\hbar}$, you get $\psi(x+a)$.)

(d) Verify that $T(a)$ is unitary using the result from the previous problem.