Circuit Quantum Electrodynamics - Yale School of Engineering ...

CHAPTER 2. CAVITY QUANTUM ELECTRODYNAMICS 39
tions, it is relatively straightforward to go beyond perturbation theory diagonalizing the Hamiltonian
exactly, to calculate the energy levels and eigenstates (see appendix B). A plot of the transition
frequency between the ground state and one excitation manifold based on these calculations is shown
in figure 2.3. The energies are:
E±,n = �nωr ± ��
4ng2 + ∆2 (2.4)
2
Eg,0 = − �∆
2
Where the n used in Eq. 2.4 is the total number of excitations 1 in the system not the number of
photons and ± refer to the higher energy or lower energy state in the n excitation manifold not the
atom state. Well into the dispersive limit, one can Taylor expand the square root in eq. 2.4 in the
small dispersive energy shift, n (g/∆) 2 , yielding
�
E±,n ≈ �nωr ± ∆ + �
2 ng2 �
/∆
The coupling effectively scales with the number of excitations, so for a given detuning there is a
critical number of excitations, ncrit, which will make the dispersive limit break down (and the Taylor
expansion diverge).
ncrit = ∆2
4g 2
While ncrit sets an upper limit on the number of photons in the dispersive limit, a more quantitative
measure of dispersiveness is the orthogonality of the wavefunctions. The eigenstates can be expressed
as:
(2.5)
(2.6)
|−, n〉 = cosθn |g, n〉 − sin θn |e, n − 1〉 (2.7)
|+, n〉 = sinθn |g, n〉 + cosθn |e, n − 1〉
θn = 1
2 arctan
� √ �
2g n
∆
If the system is well into the dispersive limit, then these can be approximated by (or computed
directly using perturbation theory)
1 This also differs slightly from the convention in [Blais2004]. My n is his n + 1 so that n = 1 is one excitation.
(2.8)