Decoherence and the Bloch equations

where the relaxation rateand excitation rateΓ ↓ = S B (ω 0 ) ≈ 2πg 2 ω 0(1 + n ω0 )Γ ↑ = S B (−ω 0 ) ≈ 2πg 2 ω 0n ω0are given by the bath noise spectral density at the frequency of the atom transition.We see that the ration between the relaxation and excitation is given bythe temperature of the bath onlyΓ ↑Γ ↓= n ω 01 + n ω0= e −¯hω0/k BT .Both off-diagonal elements also decay exponentially due to transverse noise withthe rate (Γ ↑ + Γ ↓ )/2.3 Bloch equationsThe Bloch equations were introduced in 1946 in the context of nuclear magneticresonance (NMR). They describe the motion of a spin-1/2 on the Bloch sphere(r(t)) under the influence of control fields B(t) and dissipation described byphenomenological timescales for relaxation/mixing T 1 and dephasing T 2 :ṙ = −B × r − 1 (rz − r 0 ) 1z ẑ − (r xˆx + r y ŷ) , (22)T 1 T 2where r 0 z is the steady state projection on the z-axis. In the previous section wederived:Γ ↑Γ ↑ = S B (−ω 0 ), Γ ↓ = S B (ω 0 ), = e −¯hω0/k BTΓ ↓1= Γ ↓ + Γ ↑ , rz 0 = Γ ↓ − Γ ↑= tanh ¯hω 0T 1 Γ ↓ + Γ ↑ 2k B T ,11T2∗ = 2S B (0), = 1T 2 T2∗ + 1 , (23)2T 1where we assumed the noise arising from a source at temperature T .4 Driven Bloch equations - describing a driventransmon in a transmission lineWe start from the hamiltonian of a transmon with frequency ω 0 , driven by acoherent flux field of frequency ω dH = −¯hω 02 σ z − A cos (ω d t)σ x ,6