The Jaynes-Cummings block of excitation number

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Quantum Information and Technology, Ch Superconducting qubits and circuit QED

Problem

n

, in the basis

\{|n+1,0\rangle,|n,1\rangle\}

(photon number, qubit state,

|0\rangle

the qubit ground state), has exact eigenvalues

(n+\tfrac12)\hbar\omega_r \pm \hbar\sqrt{(n+1)g^2+\Delta^2/4},

with

\Delta = \omega_q-\omega_r > 0

the qubit-resonator detuning and

g

the coupling strength. ((a)) For

g\sqrt{n+1} \ll \Delta

, expand the square root to first order in

(g/\Delta)^2

and find the energy shift, relative to its uncoupled value, of the dressed level that reduces to

|n+1,0\rangle

g\to0

, and of the dressed level that reduces to

|n,1\rangle

. ((b)) Using the

n=0

block, find the shift of the level connected to

|1,0\rangle

and hence the resonator frequency when the qubit is in

|0\rangle

. ((c)) Using the

n=0

block (for the shift of

|0,1\rangle

) and the

n=1

block (for the shift of

|1,1\rangle

), find the resonator frequency when the qubit is in

|1\rangle

, and hence show the dispersive shift is

\chi = g^2/\Delta

. ((d)) Evaluate

\chi/2\pi

for

g/2\pi = 45.0\,\mathrm{MHz}

and

\Delta/2\pi = 900\,\mathrm{MHz}

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