Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 60 / 4

The terminal term $\mathcal{A}$ is the objective reward. With the Hilbert-Schmidt inner product $⟨⟨A|\rho ⟩⟩=\mathrm{Tr}(A^{†}\rho )$, a Hermitian target observable gives its final expected value. The vector ${\rho }_v$ is the density operator represented as a vector in operator space; it retains the same physical content as the density matrix, not a wavefunction of the system.

The term $\mathcal{D}$ imposes the dynamical equation by a time-dependent costate $A_v$. It is a Lagrange-multiplier operator in the same operator space, propagated backward from a terminal condition. It is not required to be a positive, trace-one density matrix. DefineThen the enforced state equation is ${\dot{\rho }}_v=K[f]{\rho }_v$. The uncontrolled coherent part is ${\mathcal{L}}_0$, the dissipative contribution is ${\mathcal{L}}_D$, and each $f_m{\mathcal{L}}_m$ describes a control coupling. For Quantum Hamiltonian control, ${\mathcal{L}}_m\rho =[H_m,\rho ]$.

The term $\mathcal{C}$ is a quadratic field-energy or fluence penalty. For the positive scale $p_0$ and positive weights ${\lambda }_m$, larger ${\lambda }_m$ make a given control amplitude more costly. Thus maximizing $J$ trades terminal performance against control fluence while enforcing the dynamics. The multiplier term vanishes on any dynamically feasible trajectory. The weights do not represent decoherence rates; they specify control-resource preferences.

At an unconstrained differentiable extremum the first variation vanishes independently with respect to the state, costate and real controls, with initial state fixed. For the real functional, take the real Hilbert-Schmidt pairing when necessary. The Euler-Lagrange equations are most clearly written using the forward generator $K$:Here the dagger is the adjoint for the Hilbert-Schmidt inner product, and ${\hat{A}}_v$ is the vectorized terminal observable. Equivalently,The adjoint is essential if ${\mathcal{L}}_D$ is not self-adjoint: dissipative costate evolution must not be replaced by the forward dissipator.

For each unconstrained real field, stationarity givesThus $f_m=\mathrm{Im}⟨⟨A_v|{\mathcal{L}}_m|{\rho }_v⟩⟩/(p_0{\lambda }_m)$. For Hermitian state and costate with Quantum Hamiltonian couplings, the contraction before taking its imaginary part is purely imaginary, and this is a real field. These are the Liouville-space costate equations for quantum optimal control.

Forward state propagation, backward costate propagation, and the control stationarity equations must be solved together. They give necessary conditions, not a certificate of global optimality. For a local maximum the second variation on admissible perturbations is nonpositive. If amplitude or bandwidth constraints are imposed, the unrestricted pointwise field equation must be replaced by the corresponding constrained stationarity condition.

For the first two targets assume $N\ge 5$, so the indexed levels exist. A possible level population reward isIt selects $\hat{A}=|5⟩⟨5|$ and $t_F=10$.

For the coherent target, use its rank-one orthogonal projector, not the sum of its two level populations:The imaginary quantum coherence term fixes the desired relative phase $-i$. Maximizing level populations alone would not select this superposition.

For energy, choose a specified final energy observable $H_F$:For stored energy after the control is switched off, $H_F=H_0$ is the natural choice. If a fixed external field remains, use that known final Quantum Hamiltonian instead. If the terminal Quantum Hamiltonian itself is varied through the controls, the objective has explicit control dependence and additional terminal derivatives must be included. Each reward can be combined with the given dynamics and fluence penalty as $J_j={\mathcal{A}}_j-\mathcal{D}-\mathcal{C}$.

Use spectral pulse shaping of a broadband coherent laser pulse. A 4f pulse shaper disperses the frequency components with a diffraction grating, maps them to different transverse positions with a lens, applies a programmable complex mask at the Fourier plane, and recombines them with a second lens and grating. A spatial light modulator or suitable amplitude-and-phase modulation stages impose the mask.

Compute the desired complex spectral field by Fourier transform of the required temporal envelope, including its phase. In the available spectral band, chooseThe mask amplitude changes the spectral weights, and its phase changes temporal interference and chirp. Recombining the masked spectrum produces the shaped temporal pulse. A phase-only device needs additional optics to implement independent amplitude control. Calibrate the mask-to-frequency mapping and verify the output pulse experimentally, correcting it iteratively if needed.

The source bandwidth must contain the requested spectral components; a passive mask attenuates rather than supplies missing energy, so an overall rescaling may be needed. Finite spectral resolution determines the accessible time window, and finite bandwidth limits rapid temporal features.