Circuit Quantum Electrodynamics - Yale School of Engineering ...

CHAPTER 2. CAVITY QUANTUM ELECTRODYNAMICS 37
universe forward! A canonical example of an irreversible process, cooling of trapped atoms relies on
spontaneous decay to break the symmetry between heating and cooling. Long decay times associated
with atomic microwave transitions, while good for coherence, make it difficult to cool and trap such
atoms based on their microwave transitions. Because excitations are equally shared between atoms
and the cavity, they decay at the average of the two rates, Γeff = (γ⊥ + κ)/2. When κ ≫ γ the
cavity speeds up the atom decay, allowing faster extraction of energy. Future experiments might
couple circuit based cavities to increase the cooling efficiency of atom trapping/cooling. In fact, this
particular trick known as the Purcell effect [Purcell1946] works even if g < κ, where the effective
decay rate becomes Γeff = g 2 /κ. In addition, the Purcell effect allows one to redirect decay from
lossy internal modes (γ⊥) to controlled radiative modes (κ), a property sometimes used to enhance
the emission of LED’s [Boroditsky1999].
Exercise 2.0.1. Using degenerate perturbation theory find the eigenstates and energies for the res-
onant (ωa = ωr) Jaynes-Cummings Hamiltonian in Eq. 2.1.
Exercise 2.0.2. Derive the Purcell effect, by calculating the overlap of the cavity and qubit wave-
functions. Does the decay depend on the photon number? Why or why not?
2.1 Dispersive Limit
In the resonant limit (ωa −ωr ≪ g), the atom and photon can exchange energy causing them to lose
their individual character, and forming a strongly coupled system. In many cases it is preferable for
the interaction to be dispersive, where no actual photons are absorbed by the atom. This can be
achieved by detuning the atom and cavity by ∆ = ωa − ωr where g ≪ ∆. In this dispersive limit,
the effect of the dipole interaction can be modeled using (non-degenerate) perturbation theory (see
Ex. 2.1.1)). Expanding in powers of g/∆ to second order yields the approximate Hamiltonian:
�
H ≈ � ωr + g2
∆ σz
�
�a � †
a + 1/2 + �ωa σz/2 (2.2)
The atom and cavity still interact through the dispersive shift term proportional to g 2 /∆. Because
it commutes with the rest of the Hamiltonian it conserves both the photon number and the atom
state, but shifts the energies of both the atom and the photons (see Fig. 2.2b). The first term
remains the Hamiltonian of a harmonic oscillator, but the effective frequency ω ′ r = ωr ± g 2 /∆,
depends on the state of atom (see Fig 2.2b). This atom state dependent cavity shift can be used to