The Exercise

A spin coupled with equal strength to three equivalent spins has its resonance shifted in proportion to the total magnetization of the three: the sum of their spin values along the field, $+1$ for a spin up and $-1$ for a spin down. Each of the eight basis states of the three spins gives one shift. The distinct shifts are the lines of the spectrum, and the number of basis states giving a shift is the relative magnitude of its line. The first part of the exercise asks how many lines there are and in what proportions.

The second part concerns two coupled spins in a static field, in thermal equilibrium at high temperature. A $\pi/2$ pulse on the first spin turns its magnetization into the transverse plane, the pair evolves under the coupling, and the transverse magnetization of the first spin is read out as the signal, its free induction decay. The exercise asks for that signal as a function of time.

Three Spins

The magnetization $m(A)$ of the basis state of $n$ spins in which the spins of a set $A \subseteq \{1, \dots, n\}$ point down and the others point up is the integer $$m(A) = n - 2|A|.$$(1)^✦

A Coupled Pair

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$, and an observable $A$ on $S$ is a self adjoint operator on that space, $A^\dagger = A$. A unitary $U$ of $S$ is a unitary operator on the state space, $U^\dagger U = U U^\dagger = 1$; it takes a state $\rho$ to the state $U \rho U^\dagger$, and its adjoint $U^\dagger$ is again a unitary of $S$. The Hamiltonian propagator of an observable $H$ from time $t_1$ to time $t_2$ is the unitary, with $\hbar$ the reduced Planck constant, $$U_H(t_1, t_2) = \exp\bigl[-iH(t_2 - t_1)/\hbar\bigr].$$(4)^✦

Each spin is a qubit, the system whose state space is $\mathbb{C}^2$, and the pair is the composite system whose state space is the tensor product $\mathbb{C}^2 \otimes \mathbb{C}^2$. The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(5) and for a real vector $\vec{n}$ in three dimensions the Pauli observable $\vec{n} \cdot \vec{\sigma} = n_x X + n_y Y + n_z Z$ is the observable of the qubit with that matrix as its operator. The observables $X$, $Y$ and $Z$ are the Pauli observables along the three coordinate axes. On the pair, the tensor product $A \otimes B$ of two observables measures $A$ on the first spin and $B$ on the second, and with the identity observable $I$ we write $X_1 = X \otimes I$, $Y_1 = Y \otimes I$, $Z_1 = Z \otimes I$ and $Z_2 = I \otimes Z$.

The Zeeman Hamiltonian of the pair is $\hbar\omega(Z_1 + Z_2)$ and the coupling Hamiltonian is $J Z_1 Z_2 = J\, Z \otimes Z$, for real $\hbar$, $\omega$ and $J$. The high temperature thermal state of an observable $H$ of a system with a $d$ dimensional state space, at inverse temperature $\beta$, is the state $(I - \beta H)/d$, defined when this operator is positive of unit trace.^✦ For the Zeeman Hamiltonian of the pair and $(2\beta\hbar\omega)^2 \le 1$ it is $$\rho_0 = \frac{1}{4}\bigl[I - \beta\hbar\omega(Z_1 + Z_2)\bigr].$$(6)^✦

The read pulse is the adjoint of the propagator (4) of the pulse generator $Y_1$ from time $0$ to time $\pi/4$, with $\hbar = 1$, $$P = U_{Y_1}(0, \pi/4)^\dagger = e^{i\pi Y_1/4},$$(7)^✦ a $\pi/2$ pulse on the first spin about the $y$ axis. The pulsed state $\rho_1$ of the pair is the state $P \rho_0 P^\dagger$ into which the pulse $P$ takes the thermal state $\rho_0$. The pair then evolves by the propagator of the coupling from time $0$ to time $t$, again with $\hbar = 1$, which is $U(t) = e^{-iJtZ_1Z_2}$, and reaches the state $\rho(t) = U(t) \rho_1 U(t)^\dagger$. The signal is read through the operator $iX_1 + Y_1$ on the first spin: the free induction decay of the pair, of amplitude $V_0$, is $$V(t) = V_0 \, \mathrm{tr}\bigl[\rho(t)\,(iX_1 + Y_1)\bigr].$$(8)^✦

The Signal

What Is Checked

The number of lines (2) and their magnitudes (3) are stated over the magnetization (1) of the basis states of three spins. The signal (9) is stated over the free induction decay (8) of two qubits, built from the thermal state (6) of the Zeeman Hamiltonian, the read pulse (7), the propagator of the coupling $J Z_1 Z_2$ and the readout $iX_1 + Y_1$.

The library proves that for $(2\beta\hbar\omega)^2 \le 1$ the operator (6) is positive with unit trace, so that it is a state of the pair.