Relaxation and dephasing

In this chapter I derive the coupling of the system to an external bath following Ithieret al.[44] to take into account relaxation and dephasing processes. Recall

that the Hamiltonian contains ωa†aand µb†a+h.c[Eq. (3.35) and (3.47)]. For

simplicity, let me consider its analogy to a two-level system. Consider the follow- ing Hamiltonian, which describes a two-level system not coupled to any external field.

H=~n(0)·~σ =nzσz+n⊥σ⊥, (3.53)

whereσ⊥ contains σx andσy orσ− and σ+ and~n is a field that contains infor-

mation about how the system interacts with its environment. At~n(0)it does not interact at all.

Let~n be a function that depends on two parameters,~n(λ,ρ), where λ is a

variable related to the coupling to the transmission lines andρis a variable related

to the coupling to an external source of decoherence. In the Hamiltonian, if the coupling of the transmon with the external bath is neglected, then~n(0,0) =n_zand n_⊥=0. A perturbation of the Hamiltonian byδ λ –but keepingρ =0– gives

~n(δ λ,0) =~n(0,0) + d~n dλδ λ+ O δ λ2 =n_z(0,0) +dnz dλδ λ·ˆ n_z+dn⊥ dλ δ λ·ˆ n_⊥+O δ λ2. (3.54)

During the derivation of the Hamiltonian the following quantities are obtained

δ λ =b+b† (3.55) dn_⊥ dλ = 1 √ π τ (3.56) dn_z dλ =0 (3.57) n_z(0,0) =ω. (3.58)

Now let’s do the same withρ andλ simultaneously. By perturbing our Hamil-

tonian, we obtain ~n(δ λ,δ ρ) =n_z(0,0) +dn⊥ dλ δ λ·ˆ n_⊥+dnz dρδ ρ·ˆ n_z+dn⊥ dρ δ ρ·ˆ n_⊥+O δ2. (3.59) The first two terms of this expansion give the Hamiltonian previously found. The other two terms describe the interaction with the external bath. Let _dρd~n be a bath coupling rate~K(p) and δ ρ the operators that describe this decoherence in

the formd+d†. The Hamiltonian describing the decoherence is given by H_d =

Z

d p K_z(p)(d+d†)σz+K⊥(p)(d+d†)σ⊥

Let me analyze how this Hamiltonian acts on any state in the z-basis. The first term of the Hamiltonian acting on a state|↑i (|↓i) gives a constant times d†|↑i

(d†|↓i). Thus the initial state is only modified by a phase change plus the creation or annihilation of an external bath mode. This term describes a dephasing pro- cess. The second term flips the initial state from|↑i(or|↓i) to|↓i(or|↑i) and also gives an external mode excitation. Thus, it describes the relaxation/excitation of the two-level system.

Now this has to be generalized to a multilevel system. In our proposed de- vice, whose energy levels are schematically represented in FIG.3.1, there are six possible processes involving the relaxation of the transmon. These are

|ωT3i −→ |ωT2i |ωT2i −→ |ωT1i

|ωT3i −→ |ωT1i |ωT2i −→ |GSi

|ωT3i −→ |GSi |ωT1i −→ |GSi,

where |ωT3i, |ωT2i and |ωT1i are kets representing the third, second and third

excited states of the transmon, respectively. |GSiis the ground state.

For simplicity, I only consider the process on the last row. The reason for in- troducing this extra Hamiltonian is to quantify the effect of an external bath on the successful operation of the device. By considering only two out of the six possible processes, nothing relevant is being omitted, because the relaxation ratio can be increased so that the relaxation of the excited states described by these two pro- cesses actually contains the sum of all the possible relaxation processes of these states. With this simplification, the information on what state the transmon decays into is lost, but I am not interested in the final state. I only want to know whether the device works (no relaxation) or it does not (relaxation).

The relaxation terms involve the creation of an external bath mode and an operator or product of operators that describe the annihilation of one excitation (and its hermitian conjugate). Other possible combinations are suppressed by the rotating wave approximation. The dephasing terms contain a creation of an external bath mode and a projector operator of an excited state. Furthermore, commutation of the dephasing and relaxation operators has to be imposed, since these elements belong to different Hilbert spaces. The resulting Hamiltonian is

H_d = Z d p

_∑

wherer(p)are the relaxation modes andd(p)are the dephasing modes. Here the indexiruns over{T1,T2,2a,2b,3a,3b}. It is important to notice that the vacuum

|0iis still an eigenstate of the Hamiltonian with zero energy. The coupling func- tionsK(p)can be approximated as constants [45] (first Markov approximation).

K_r(p)≈ r γr 2π (3.62) K_d(p)≈ r γd 4π. (3.63)

Up to this point I have derived all the formulas with all the factors of ¯halthough in some equations I have set implicitly ¯h=1 –by giving to the frequenciesωiunits

of energy. From now on I will consider ¯h=1 to simplify the notation.