Circuit Quantum Electrodynamics - Yale School of Engineering ...

CHAPTER 2. CAVITY QUANTUM ELECTRODYNAMICS 41
(i.e. dislike the BH Lemma) you may expand the exponential to the appropriate order and compute
the Hamiltonian directly to the requested order in g/∆.
Exercise 2.1.2. Applying the unitary transformation found in exercise 2.1.1 instead to the wave-
functions, find the eigenstates of the dispersive Hamiltonian to first order in g/∆. Show that these
agree with the exact eigenstates first order.
Exercise 2.1.3. Apply (to first order) the unitary transformation found in exercise 2.1.1 to the
photon annihilation operator UaU † . What does this say about the radiative decay of the qubit?
Exercise 2.1.4. Using Fermi’s Golden Rule find the decay rate of a qubit-like excitation in the
dispersive Jaynes-Cummings Hamiltonian with a decay Hamiltonian Hκ = √ κ � b † a + a † b � . The
solution can be found in appendix B. b. Along similar lines derive the rate of photon loss through
the atom decay channel.
Exercise 2.1.5. a. Derive the energy levels of Eq. 2.4 by diagonalizing the n-excitation manifold
exactly. See appendix B for solution. b. By taking appropriate energy differences to second order
find the cavity, Stark, and Lamb shifts.
Exercise 2.1.6. The 1/2 in the harmonic oscillator Hamiltonian H = �ωr(a † a + 1/2) is often
neglected as it appears to be a constant offset which can be eliminated by appropriate gauge choice.
The Lamb shift arises from this 1/2 term. Using CQED, design an experiment to measure the Lamb
shift.
2.2 Strong Dispersive Interactions
In the previous section, nothing was mentioned about the strength of the coupling relative to decay,
the only constraint was that ng/∆ ≪ 1 (outside of blue region in Fig. 2.4). In fact in the dispersive
limit the old definition of strong coupling, that the rate of atom-photon oscillations were faster than
decay, contradicts the notion of dispersiveness, that no (very little) mixing occurs between atom and
photon. Instead one can compare the manifestation of the dipole coupling, the dispersive frequency
shifts, to the decay rates. To access the strong dispersive regime one must have χ = g 2 /∆ > γ, κ
(white region in Fig. 2.4). The hallmark of this type of strong coupling is that the qubit spectrum
resolves into individual photon number peaks. Also above this threshold, the cavity shift is larger
than the cavity linewidth, “splitting” the cavity, based on the state of the atom (see Fig. 2.5). When