Exact open dynamics of a generalized class of dephasing-type spin-boson models

Read-out of Quasi-periodic Systems using Qubits

“…Finally, while we have focused on the exactly solvable pure dephasing coupling between the qubits and the quasi-periodic system, it would interesting to explore if we can obtain direct signatures of transport such as the current by considering couplings that allow energy exchange between the qubit and the system [72]. In this appendix, for the sake of completeness and clarity, we adapt to the setting in this paper and present the derivation of the spin-boson dynamics for dephasing baths presented in previous works [58][59][60][61][62][63][64]. We will work out the calculation for the situation with two qubits coupled to a Bosonic bath and obtain the one qubit situation as a simple limiting case.…”

“…with g i k = gS i,k , and σi τ with τ = z, ± denoting the usual Pauli and ladder operators for the qubit. From the above equation it is clear that the hamiltonian of interest here falls into the familiar family of spin-Boson type hamiltonians and more specifically the spin-Boson problem with a dephasing bath that has been extensively studied especially in the context of decoherence in quantum computation [58][59][60][61][62][63][64].The key advantage of this model is its exact solubility which arises from the fact that the qubit hamiltonian ω A 2 σi z commutes with the term representing the interaction with the bosonic modes. For the sake of completeness, we provide the details of this calculation (for both the one and two qubit situations) in the Appendix and present only the results here.…”

“…which is still in the class of spin-boson models with dephasing baths. As a result, the dynamics of this model can again be exactly solved [59,62,63] and we present the details in the Appendix . As before, working in the Heisenberg picture, we note that due to the nature of the coupling we have that σi z (t) = σi z (0), σj z (t) = σj z (0), and the only non-trivial dynamics is for the ladder operators of the individual qubits and is given by: σi…”

“…Hence, from the point of view of the qubits the system is essentially a bosonic bath with energy of the modes given by the single particle eigenenergies of the quasi-periodic system. In this light, we will use known results concerning spin-Boson type models especially the so called dephasing bath models as we will consider a form of coupling between the qubits and quasi-periodic system that commutes with the free hamiltonian of the qubits [58][59][60][61][62][63][64]. Thus, when a single qubit is coupled to the system it results in the pure dephasing or decoherence of the qubit state.…”