Decoherence and the Bloch equations

The field emitted by the transmon into the transmission line is proportional toΦ out ∝ ( ρ 01 e iω dt + ρ 10 e −iω dt )remembering that the density matrix was obtained in the frame rotating withthe drive.In the experiment, it is more appropriate to talk about voltage amplitudethan flux fields. In summary, we send down a voltageV in = V d sin (ω d t),and the average reflected voltage signal is the field emitted by the transmon, i.e()Γ sin (ω d t) + ∆ ↓Γ 2cos (ω d t)V r = −V d .2Γ 2 1 + ∆2 + A2Γ 2 Γ 2 ↓ Γ 2For resonant drive ∆ = 0 the reflection isΓ ↓ sin (ω d t)V r = −V d2Γ 2 1 + A2Γ ↓ Γ 2,with maximum reflectionV rV d= − Γ ↓2Γ 2=Γ ↓Γ ↓ + 2Γ ∗ 21=1 + 2Γ ∗ 2 /Γ ,↓for vanishing drive strength. The transmitted field V t is the sum of the drivefield and the field emitted by the transmon V t = V d + V r .8