Past exam of the atomic and Optical Physics course of the University of Cambridge / 2013 / Paper 1 (170: Atomic and Optical Physics)

For a free-space electric dipole transition, the spontaneous emission rate in SI units isHere $|\mathbf{d}{|}^2$ includes the relevant dipole components; a specified single polarization requires its corresponding matrix element.

A complete scattering cycle returns the internal state to the ground state and changes the atomic momentum to $\mathbf{P}+\mathbf{p}-{\mathbf{p}}^{\prime }$. ThusFor photon-scattering recoil heating with isotropic emission of fixed momentum magnitude, $⟨{\mathbf{p}}^{\prime }⟩=0$, givingThe first term is the velocity-dependent work of absorption and can cool an atom if the photon opposes its motion. The last two are positive absorption and emission recoil energies. If the actual absorbed directions are also isotropic and independent of $\mathbf{P}$, the first term averages to zero. For approximately equal incoming and outgoing photon momenta $\hslash k$,Isotropic incident illumination alone would not guarantee isotropic absorption if Doppler-dependent excitation selected directions.

Source qualification for the trapped calculation. Set $m=M$ for the same atom. The printed Gaussian $\exp [-x^2/(2l_0{)}^2]$ has the wrong width for the specified harmonic oscillator: the printed ground and first-excited functions have squared norms $\sqrt{2}$ and $2\sqrt{2}$, respectively. The normalized oscillator states fixed by the given $l_0=\sqrt{\hslash /(m{\omega }_{\mathrm{HO}})}$ instead useAlso, one common complex phase multiplying both off-diagonal terms does not make the Hamiltonian a Hermitian operator. Use an absorption matrix element $(\hslash \Omega /2)e^{-ikx}$ and its Hermitian conjugate for emission; choosing the opposite propagation convention reverses the imaginary sideband phase but not a transition probability. The physical results below use these explicitly stated corrections. Parts (i) and (ii) also give the literal integrals of the printed functions.

Conditional motional heating and recoil averaging. With $\eta =k{l}_0/\sqrt{2}$, an oscillator eigenstate has $⟨K{⟩}_n=⟨V{⟩}_n=(n+1/2)\hslash {\omega }_{\mathrm{HO}}/2$. Therefore the absorption outcomes haveThe same statements hold for the specified emission outcomes $|e,0⟩\to |g,0⟩$ and $|e,0⟩\to |g,1⟩$. Internal excitation energy goes into, or comes from, the photon and is distinct from this external energy change. Emission into the first motional state makes the emitted photon correspondingly less energetic.

A sudden recoil from $|0⟩$ gives probabilities $P_n=e^{-{\eta }^2}{\eta }^{2n}/n!$, so $P_1\simeq {\eta }^2\ll 1$ while $P_0\simeq 1$. The rare first sideband transfers a whole trap quantum, much larger than $E_R$; the mean is neverthelessThe emission average is the same for a one-dimensional recoil of the same magnitude. Projecting isotropic three-dimensional emission onto one axis instead gives a factor $⟨{\cos }^2\theta ⟩=1/3$ for that axis.

A sudden unobserved kick initially raises kinetic energy by $E_R$ and leaves the position distribution, hence potential energy, unchanged. It creates a coherent superposition of motional states. The equal kinetic/potential increments above concern energy-eigenstate-conditioned outcomes, or time-averaged/dephased motion; these are different measurement conditions. There is no contradiction.

With $\Gamma >{\omega }_{\mathrm{HO}}$, motional sidebands are not cleanly resolved, and a short-lived excited atom has little time to move before decay. In the stronger sudden limit $\Gamma \gg {\omega }_{\mathrm{HO}}$ the recoil-kick picture is direct. For only the stated inequality, finite linewidth weights can change numerical factors; it does not fix an exact continuous-wave scattering rate. The average heating remains of recoil order per absorption or emission, and of order $2E_R$ when both absorption and emission kicks are counted at full magnitude. For the axial component of isotropic three-dimensional emission, the emission contribution instead carries the factor $1/3$ noted above. Strong confinement suppresses the chance of changing $n$, but does not make its averaged recoil energy disappear. This agrees with free-space recoil heating while replacing small continuous energy changes by rare quantized motional changes.

Write $g=\hslash \Omega /2$, with Rabi frequency $\Omega$, for the internal coupling and use normalized oscillator states. The motional carrier transition matrix element is the Gaussian characteristic function,In the Lamb-Dicke regime, $M_{00}=g[1-{\eta }^2/2+O({\eta }^4)]$. The diagonal internal-energy terms cannot connect $g$ to $e$.

If the two unnormalized printed functions are used verbatim, the literal ground-state integral is instead $g\sqrt{2}\exp (-k^2{l}_0^2/2)$. This is not a matrix element between normalized eigenstates of the stated trap; even its zero-recoil value is incorrectly $\sqrt{2}g$.

For the harmonic-oscillator recoil matrix element, using ${\psi }_1=\sqrt{2}(x/l_0){\psi }_0$ and differentiating the Gaussian transform with respect to $k$,Consequently $|M_{01}{|}^2/|M_{00}{|}^2={\eta }^2$. Reversing the travelling-wave phase changes the sign of the imaginary amplitude. For small $\eta$, the sideband is weaker by $\eta$ in amplitude and ${\eta }^2$ in probability.

With the printed unnormalized Gaussians, the literal integral would be $-2igk{l}_0\exp (-k^2{l}_0^2/2)$. It is distinguished from the physical normalized-oscillator result rather than silently treating inconsistent wavefunctions as normalized.

For the Bogoliubov spectrum, define ${ϵ}_k={\hslash }^2{k}^2/(2m)$ and $E_k=\sqrt{{ϵ}_k({ϵ}_k+2{\mu }_0)}$. For repulsive interactions and ${\mu }_0>0$,Thus the low-momentum excitations are gapless phonons and the high-momentum branch approaches free-particle curvature. The crossover occurs when kinetic and interaction energy are comparable. The Landau criterion follows because an obstacle moving at velocity $v$ can supply energy $\mathbf{v}\cdot \mathbf{p}$ to an excitation. HereThis is the homogeneous phonon kinematic threshold, not a guarantee that every perturbation dissipates immediately above it.

In two dimensions, $n=|\psi {|}^2$ has units of inverse area and ${\hslash }^2/m$ has units of energy times area. Therefore $\tilde{g}$ is dimensionless. Neglecting the kinetic term in the Gross–Pitaevskii equation, with positive coupling, giveswith $n=0$ outside the radius in the Thomas–Fermi approximation for a condensate. Multiplication givesThe sharp edge is an approximation; the neglected gradient term rounds it over a healing length scale.

Using the local sound-speed criterion for circular condensate stirring, the moving beam has speed $v=\Omega D$. The local interaction chemical potential is ${\mu }_0-V(D)=m{\omega }_r^2(R^2-D^2)/2$, soThe onset equality is $\Omega D=c(D)$ and the permitted excitation condition isThis uses the beam speed as the local relative-flow scale. A finite hole can distort the surrounding flow or nucleate vortices, so the result is the requested local phonon estimate, not a universal necessary condition for all experimental stirring.

For the loss-driven onset of condensate stirring, first integrate the density:At fixed coupling and trap frequency, ${\mu }_0\propto N^{1/2}$ and $R\propto N^{1/4}$. Hence $R(t)=R(0)e^{-t/(4\tau )}$ while $D$ stays fixed. For $\Omega ={\omega }_r/\sqrt{2}$, threshold requires $D/R=1/\sqrt{2}$. Starting at $D/R(0)=1/2$,At onset, $N=N_0/4$, $R=R(0)/\sqrt{2}$ and the beam is still inside the cloud. Excitations become kinematically allowed just beyond this equality, provided the stipulated adiabatic regime of the Thomas–Fermi approximation for a condensate remains valid.

The Lamb shift is a radiative displacement of atomic energy levels beyond the values from the Dirac equation in a Coulomb potential. In hydrogen, the $2S_{1/2}$ and $2P_{1/2}$ levels are degenerate for a point Coulomb potential in the Dirac equation, yet the $2S_{1/2}$ level lies higher by roughly a gigahertz. This is distinct from the fine structure splitting between different $J$ levels and from nuclear-spin hyperfine structure.

Electron self-energy in the quantized electromagnetic field is the dominant contribution to the familiar hydrogen splitting; vacuum polarization contributes too. Renormalization separates the physical electron mass from its divergent free-electron self-energy. A useful leading-log scale for a hydrogenic $nS$ level isnot an exact numerical formula. The enhanced sensitivity of S states follows from their nonzero wavefunction at the nucleus; leading short-distance corrections act like contact perturbations. A precision prediction also includes finite nuclear size and recoil, and must consistently separate those effects from the radiative contribution.^[ref]

The original Lamb-Retherford experiment used a beam of hydrogen atoms, electron-impact excitation into metastable $2S$, and microwave-induced transfer into short-lived $2P$ states. Those atoms decayed before reaching the detector. Metastable atoms surviving to a metal target ejected electrons, so a drop in the measured current signalled a resonance. Varying the microwave excitation and magnetic field established the displacement from the Dirac equation expectation. Microwave quenching converted a small energy difference into a measurable loss of a long-lived population, avoiding optical Doppler widths that obscure the splitting.^[ref]

A Paul trap uses a time-dependent electric quadrupole field to provide dynamic confinement, avoiding the impossibility of trapping a charge in a stable three-dimensional static electrostatic minimum. Its dynamics follows a Mathieu equation: stable drive parameters produce slow Paul trap secular motion with superposed fast Paul trap micromotion. If ${\mathbf{E}}_{\mathrm{rf}}$ denotes the peak oscillating field, the effective energy for small Paul trap micromotion isThe radiofrequency pseudopotential of a Paul trap is an averaged mechanical energy, not a static electrostatic potential. Doppler cooling and, where needed, resolved sideband cooling reduce Paul trap secular motion; the Lamb-Dicke regime suppresses higher motional sidebands.^[ref]

Two long-lived internal states encode a qubit, for example hyperfine states, while an optical metastable transition provides another option. Optical pumping initializes it. Resonant or Raman transition pulses make coherent single-qubit rotations, and state-dependent fluorescence distinguishes a bright state from a dark shelved state. Shared quantized normal modes let one ion's internal state influence another, providing entangling gates in the proposal of J. I. Cirac and P. Zoller; the motional bus must be returned to its intended state rather than left carrying unwanted information.^[ref]

A concrete early experiment demonstrated a controlled-NOT gate between the internal hyperfine and motional degrees of freedom of a single trapped ion. This was an internal–motion two-qubit gate, not yet a gate between two separate ions.^[ref]

Challenges include electric-field-noise heating of the motional bus, excess Paul trap micromotion from stray fields, magnetic and laser phase noise, off-resonant spontaneous scattering, imperfect preparation and readout, and unwanted excitation of spectator modes. Scaling to many ions adds mode crowding, individual-addressing crosstalk and transport requirements. Cooling and quantum error correction must remove entropy without erasing stored qubits. Long storage coherence is necessary but not sufficient: gates and measurements must also be accurate and reproducible. The original gate experiment explicitly identified laser, field, trap-drive and motional decoherence limitations.^[ref]

For a nucleus of spin $I>0$ and electronic $J=1/2$, the magnetic-dipole hyperfine interaction is $A\mathbf{I}\cdot \mathbf{J}$, using dimensionless angular-momentum operators. The electric-quadrupole hyperfine contribution vanishes for $J=1/2$. In zero field, $F=I\pm 1/2$ labels the levels:Their signed separation is $\Delta =A(I+1/2)$. For $A>0$, the larger-$F$ multiplet lies higher; negative $A$ reverses this order. If $I=0$ there is no such hyperfine splitting.

Let the field define $z$ and writeHere $g_I$ is the signed nuclear Zeeman coefficient expressed in Bohr-magneton units, including its nuclear-to-Bohr-magneton factor; it is not an unconverted nuclear $g$ factor. Total projection $m=m_I+m_J$ is conserved, while $F$ need not be. In the basis $|m-1/2,+1/2⟩$, $|m+1/2,-1/2⟩$, the two-state block isFor $A>0$, define $x=(g_J-g_I){\mu }_{B}B/\Delta$. Diagonalization gives the Breit-Rabi formula,For either sign of $A$, the block itself remains valid; its ordered eigenvalues are $-A/4+g_I{\mu }_{B}Bm\pm \frac{1}{2}\sqrt{{\Delta }^2+2Am(g_J-g_I){\mu }_{B}B+(g_J-g_I{)}^2{\mu }_B^2{B}^2}$.

For a stretched hyperfine Zeeman state at projection $m=\pm (I+1/2)$ there is only one state each. Their energies are exactlyAssigning two branches to each stretched projection would double-count states.

In weak fields the shift is $g_F{\mu }_{B}mB$, with $g_F=g_I\pm (g_J-g_I)/(2I+1)$ for $F=I\pm 1/2>0$; an $F=0$ state has no first-order Zeeman shift. At strong hyperfine-decoupling fields the eigenstates approach product states withThis is the hyperfine regime of the Paschen-Back effect, still assuming the field is weak enough not to invalidate the electronic $J=1/2$ manifold. States of the same $m$ mix and avoid crossing when their off-diagonal coupling is nonzero; different-$m$ levels can cross.