Past exam of the laser science and quantum information processing course of the University of Oxford / 2021 / 6

Quantize a single electromagnetic normal mode of angular frequency ${\omega }_c$ and represent one isolated atomic transition as a two-level system with raising operator ${\sigma }_{+}=|e⟩⟨g|$. The bare Hamiltonian operator isThe first main approximation is the electric-dipole approximation: the field varies negligibly across an atom, $k{a}_{\mathrm{atom}}\ll 1$, and the leading coupling is $H_d=-\mathbf{d}\cdot \mathbf{E}({\mathbf{r}}_0)$. For states of definite opposite parity, the relevant dipole part is off-diagonal. After choosing operator phases, this gives $H_d=\hslash g(a+a^{†})({\sigma }_{+}+{\sigma }_{-})$, with $g$ fixed by the transition dipole and the one-photon field wave amplitude.

In the interaction picture, $a{\sigma }_{+}$ and $a^{†}{\sigma }_{-}$ oscillate at the difference ${\omega }_a-{\omega }_c$, whereas $a{\sigma }_{-}$ and $a^{†}{\sigma }_{+}$ oscillate at the sum ${\omega }_a+{\omega }_c$. The second main approximation is the rotating-wave approximation: near resonance and for $g\sqrt{n+1}\ll {\omega }_a+{\omega }_c$, the latter terms average away on the interaction time scale. The resulting Jaynes-Cummings model hasThe two-level system and single-normal mode truncations are additional assumptions: other atomic transitions and optical normal modes must be sufficiently far off resonance or weakly coupled. Dissipation is neglected over the interaction time. The retained terms exchange one atomic excitation for one photon, conserving $a^{†}a+{\sigma }_{+}{\sigma }_{-}$.

Use the resonant interaction picture and define $g>0$ by the one-atom Jaynes-Cummings model in part (a). Equal coupling to both atoms gives the Tavis-Cummings modelThe initial state has two excitations and is symmetric under atom exchange. Excitation conservation and this symmetry confine the evolution toThe antisymmetric atomic spin singlet state is uncoupled and unpopulated. Applying the annihilation operator and creation operator givesThe first off-diagonal entry combines two indistinguishable atomic emission paths; the second also includes the $\sqrt{2}$ enhancement for creating the second photon. Its eigenvalues are $0,\pm \sqrt{6}g$. Put $\Omega =\sqrt{6}g$ and $K=H_I/\hslash$, so $K^3={\Omega }^{2}K$. Its exponential isApplying this to $|A⟩$ yieldsAll three states have the same bare energy on resonance, so restoring the free Schrödinger-picture evolution adds only a common global phase. The expression starts at $|ee;0⟩$, has initial derivative $-i\sqrt{2}g|B⟩$, and its three squared probability amplitudes sum to one.

An ideal projective measurement of the field's photon number selects the coefficient of $|1⟩$. ThereforeWhenever this probability is nonzero, normalization of the projected joint quantum state gives the atomic Bell stateup to a common global phase. The maximum heralding probability is $1/3$, attained at $\sqrt{6}gt=\pi /2+n\pi$. At a time with $P_1=0$, that conditioning event is impossible, so a conditional state is not defined there. The interpretation is a photon number measurement at the selected time; continuous monitoring of escaping photons would instead change the dynamics and require a different model.

Suppose the conditional atomic pure state were a product stateComparing its four probability amplitudes with $(|eg⟩+|ge⟩)/\sqrt{2}$ givesThe last two equations make all four factors nonzero, contradicting either of the first two. Equivalently, the coefficient matrix has determinant $-1/2$, while a product state has a rank-one coefficient matrix and zero determinant. Thus the atoms are entangled; indeed each reduced density matrix is $I/2$, making the conditional state maximally entangled.

An ideal zero-photon number result projects onto the first term in part (b). ConsequentlyThis is a product state, not an entangled success. The remaining ideal outcome hasand leaves $|gg⟩$. In particular, simply discarding the field outcome does not reproduce the pure heralded Bell state.

The practical requirements are strong coherent coupling of both atoms to the same optical normal mode with stable relative coupling phases, preparation of $|ee;0⟩$, and interaction timing on the scale $g^{-1}$ before spontaneous emission or cavity loss destroys quantum coherence. The field must then be measured or extracted after halting the coherent exchange, with enough photon number resolution to distinguish one from two. Otherwise an inefficient detector can mistake a two-photon event for the desired herald. Even ideally the one-photon success probability is at most $1/3$, so trials must be repeated.

“No photons detected” implies $|ee⟩$ only for the ideal zero-number projection specified above. With loss or finite detector efficiency, no click also includes missed one- and two-photon events and generally leaves a conditional mixed state. Detecting a leaking photon during the interaction is also a different measurement protocol from the end-time projective measurement analyzed here.