Past exam of the laser science and quantum information processing course of the University of Oxford / 2017 / 3

Let $p=|\alpha {|}^2$, so $|\beta {|}^2=1-p$. The projective measurement has outcome 0 with probability $p$ and leaves the state $|0⟩$; outcome 1 has probability $1-p$ and leaves $|1⟩$. Alice knows her actual result, so her conditional description is a pure state. Her fidelity of quantum states with the initial state is $p$ after outcome 0 or $1-p$ after outcome 1. Its outcome average is $p^2+(1-p{)}^2$.

Bob knows the preparation but not the outcome, so his density matrix is the nonselective postmeasurement stateIt is mixed for $0<p<1$ and pure only at the endpoints. The relative phase has disappeared because the classical outcome record is unavailable to him.

The undisclosed qubit measurement outcome with an unspecified preparation prior requires care: Charlene has no specified prior over preparations. Consequently the information given does not determine a unique posterior probability for the two outcomes. Under a symmetric ignorance or maximum-entropy prior, her best assignment isMore generally, if her prior mean of $|\alpha {|}^2$ is $q$, her assignment is ${\rho }_C=\mathrm{diag}(q,1-q)$ and its fidelity of quantum states with the actual initial state is $pq+(1-p)(1-q)$. Knowing that a measurement occurred guarantees diagonality in its basis, but does not by itself prove $q=1/2$. Different descriptions here reflect different available records, rather than different physical outcomes. ^[ref]

Use the common $D_{3/2}$ level to subtract the two upper-state energies. The indicated transition satisfiesThe value inherits the rounding of the supplied wavelengths. It is shorter than 397 nm because $P_{3/2}$ lies above $P_{1/2}$.

A static electric potential alone cannot confine a charge stably in all three dimensions in a charge-free region. An RF quadrupole ion trap uses a rapidly alternating electric field to provide effective radial confinement; DC endcap fields can supply axial confinement. Near a stable equilibrium, the secular motion sees an approximately harmonic pseudopotential. It has quantized vibrational levels $E={\sum }_i\hslash {\omega }_i(n_i+1/2)$, with separate normal modes along the principal directions. RF Paul trap micromotion is superposed on this slower secular oscillation. For several trapped ions, Coulomb interactions instead give collective normal modes. ^[ref]

The two $D$ states are metastable, so a quantum superposition can retain its relative phase much longer than one involving a rapidly decaying $P$ state. In the stipulated transition diagram they have no allowed downward electric-dipole transition; real weaker decay channels limit, rather than eliminate, this lifetime advantage. Their distinct energies permit resolved state-selective optical control.

Both $D$ levels couple to $P_{3/2}$. Apply coherent beams near 850 nm and 854 nm, detuned from that common intermediate level by a large Rabi detuning ${\Delta }_R$. Absorption from one beam and stimulated emission into the other transfer population between the two $D$ states without appreciably populating the short-lived intermediate level. Tune the frequency difference to their splitting, including differential AC Stark shifts. In the far-detuned limit,for an initially pure $D_{3/2}$ state on two-photon resonance. The controllable relative optical phase sets the rotation axis; the large Rabi detuning suppresses spontaneous-scattering errors. Suitable polarizations and a chosen pair of magnetic sublevels are required in an actual experiment.

Use state-dependent fluorescence. With 397 nm light on $S_{1/2}\leftrightarrow P_{1/2}$ and an 866 nm repumper on $D_{3/2}\leftrightarrow P_{1/2}$, an ion initially in $D_{3/2}$ is brought into a bright cycling manifold. The repumper prevents population trapping in $D_{3/2}$. An ion in $D_{5/2}$ remains dark while the 854 nm light is off, since it has no allowed link to this cycle. A detected bright or dark record therefore measures the qubit in its logical basis by electron shelving.

To reset, apply 854 nm light to empty the shelved state, and use 397 nm plus repumping for Doppler cooling and internal-state preparation. Doppler cooling alone generally does not reach the ground state. In the resolved-sideband and Lamb-Dicke regime, drive the red motional Raman sidebandand then optically repump to $D_{3/2}$ with little change in motional occupation. Each cycle removes approximately one vibrational quantum. The $n=0$ state is dark to this red sideband because its matrix element contains $\sqrt{n}$. Repeat with appropriate wavevector projections and frequency differences for all modes to be cooled, then optically pump into the selected $D_{3/2}$ sublevel. Small recoil during repumping is necessary for a high ground-state probability. ^[ref]