Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 60 / 3

Take the drift propagator with the correct initial normalization, $U_0(t,t_0)=e^{-i(t-t_0)H_0}$, and define the interaction picture by $U=U_0{U}_I$. Differentiate this product and substitute the Schrödinger equation:Since $i{\dot{U}}_0=H_0{U}_0$, the drift terms cancel. Left multiplication by $U_0^{†}$ givesThe printed $e^{-it{H}_0}$ is this expression with time origin $t_0=0$. Its differential identity remains valid at arbitrary $t_0$, but its interaction-picture initial operator would then be $e^{i{t}_0{H}_0}$ rather than the identity. Using $t-t_0$ resolves that normalization without changing the derivation.

Use $t_0=0$ as in the displayed carrier-phase convention. The drift is diagonal, and conjugation givesMultiplying by $A(t)\cos (\omega t+\varphi )$ and expanding the cosine into exponentials yieldsThe rotating-wave approximation averages away the components oscillating at $2\omega$, retainingThe envelope is slowly varying and the effective coupling is weak compared with the carrier frequency. The pulse phase selects the transverse rotation axis on the Bloch sphere; its pulse area determines the angle. Counter-rotating corrections are neglected under this approximation, not identically zero.

The original Figure 1 has the allowed chain $1\leftrightarrow 2\leftrightarrow 3\leftrightarrow 4$. Its positive resonance frequencies are ${\omega }_1=E_2-E_1$, ${\omega }_2=E_2-E_3$ and ${\omega }_3=E_4-E_3$ in units $\hslash =1$; level 2 lies above level 3. Use a pulse at ${\omega }_1$ to transfer level population from 1 to 2, then one at ${\omega }_2$ to transfer it from 2 to 3, and finally one at ${\omega }_3$ to transfer it from 3 to 4.

For a selected pair, choose the optical phase so the effective Quantum Hamiltonian has a fixed off-diagonal rotation axis. If its coupling coefficient is $g_{jk}(t)=A_{jk}(t)|d_{jk}|/2$ in this normalization, the active two-dimensional propagator iswhere $K_{jk}^2=I_{jk}$. A full level population transfer has $\beta =\pi /2$. Equivalently, in the Rabi-frequency convention ${\Omega }_R=2g_{jk}$, the required area is $\int {\Omega }_{R}dt=\pi$. On spectator levels the embedded two-level quantum rotation is the identity.

Thus three selective $\pi$ pulses at ${\omega }_1,{\omega }_2,{\omega }_3$, in that order, transfer $|1⟩$ to $|4⟩$ up to global phase. A strongly regular quantum Hamiltonian separates the resonances; envelopes long enough that their bandwidth and coupling strength are small relative to unwanted frequency gaps make off-resonant excitation arbitrarily small within the selective-pulse limit. The phases accumulated along this path do not affect the final level population.

Use geometric quantum state synthesis by adjacent rotations, rather than three full level population swaps. Each resonant pulse implements a complex Givens rotation on one allowed two-state plane. Start with the target amplitude vector, use such rotations to eliminate entries back to the first basis vector, and reverse that sequence. Its angles are fixed by the ratios of target amplitudes, and optical phases supply their relative complex phases.

Here is an explicit real-coordinate construction in the interaction picture. Define a phase-calibrated rotation byA resonant Quantum Hamiltonian proportional to $-i|j⟩⟨k|+i|k⟩⟨j|$ realizes this rotation; the optical phase chooses that quadrature. Let $\beta =\arccos (1/\sqrt{3})$. ThenThe pulse areas equal these rotation parameters in the direct Hamiltonian-coefficient convention of quantum pulse area. A given sign or phase choice for a physical transition depends on its dipole and basis conventions; calibration realizes the specified rotation without assuming every adjacent transition has the same energy ordering.

If the requested amplitudes refer to the laboratory frame at $t_F$, the target for this construction is $U_0(t_F,0{)}^{†}|\Psi ⟩$, not simply $|\Psi ⟩$. Set the rotation phases accordingly, or add the equivalent accessible diagonal phase corrections. This compensates the known drift phases and gives the stated laboratory superposition, not merely the correct level populations. Strong regularity and connected allowed transitions guarantee the required controllability; bandwidth restrictions set how slowly the selective approximation must be implemented.