Past exam of the laser science and quantum information processing course of the University of Oxford / 2017 / 6

Use $a|n⟩=\sqrt{n}|n-1⟩$, $a^{†}|n⟩=\sqrt{n+1}|n+1⟩$, ${\sigma }_z|e⟩=|e⟩$ and ${\sigma }_z|g⟩=-|g⟩$. Only one interaction term acts on each of the two quantum states. HenceThe interaction exchanges one atomic excitation with one field quantum. The square-root factor is the photon-number enhancement of the transition matrix element.

The total excitation operator $\mathcal{N}=a^{†}a+({\sigma }_z+1)/2$ commutes with the Hamiltonian. Each sector with $\mathcal{N}=n+1\ge 1$ is therefore spanned by $|e,n⟩,|g,n+1⟩$, and no term connects it to a sector of different excitation number.

In that ordered basis, with the source's Rabi detuning $\Delta =\omega -{\omega }_0$,The first term is the common sector energy; the remaining diagonal entries record the bare Rabi detuning and the off-diagonal terms exchange the excitation.

The isolated ground sector of the Jaynes-Cummings model is an important exception to a statement that the entire Hilbert space consists of such pairs: $|g,0⟩$ has no partner and is annihilated by both interaction terms. It is a one-dimensional invariant sector with energy $-\hslash {\omega }_0/2$. The full decomposition is this one-dimensional sector plus all the two-dimensional sectors above.

Subtract the common sector energy and call the remaining eigenvalue $\hslash \lambda$. Its characteristic equation isThe dressed-state energies are consequentlyOn resonance their splitting is $2\hslash |g|\sqrt{n+1}$; far off resonance the eigenstates approach the uncoupled quantum states with small dispersive energy shifts. The eigenvalue splitting is an angular-frequency difference ${\Omega }_n$, so an ordinary frequency would be ${\Omega }_n/(2\pi )$.

The requested bare labels require $n\ge 1$. Their four energies areAmong these four, the pair that is fully present in one interacting sector is $|e,n-1⟩,|g,n⟩$. Its center is $C=\hslash \omega (n-1/2)$ and its dressed branches areTheir endpoint values at ${\omega }_0=0$ and $2\omega$ are the same, $C\pm \hslash \sqrt{{\omega }^2+4g^{2}n}/2$. At ${\omega }_0=\omega$ they are $C\pm \hslash |g|\sqrt{n}$. The avoided crossing therefore has resonant gapAt the three horizontal-axis landmarks, the bare energies divided by $\hslash \omega$ are respectivelywhich specifies the vertical-axis labeling without assigning an unstated numerical $g$. The original plot below uses illustrative $n=3$ and $g/\omega =0.08$ and shows this complete central doublet together with all four bare lines.

Sector qualification: the outer displayed state $|e,n⟩$ couples to $|g,n+1⟩$, outside the requested bare set. Likewise $|g,n-1⟩$ couples to $|e,n-2⟩$ if $n\ge 2$; for $n=1$ it is the isolated $|g,0⟩$. Thus the four-state set cannot be diagonalized as a closed four-dimensional block. If branches associated with these outer states are also desired, complete those neighboring sectors and use the formula in part c with indices $n$ and $n-2$.

Finally, “adjacent dressed states” within the shown resonant pair have the gap just boxed. Adjacent rungs of the dressed ladder are not equally spaced: at resonance,When neighboring doublets remain ordered, the gap between their nearest branches is $\hslash \omega -\hslash |g|(\sqrt{n+1}+\sqrt{n})$. This distinguishes the within-doublet Rabi splitting from differences between adjacent excitation sectors.