Two-level approximation of transmons in quantum quench experiments
Quantum quench is a typical protocol in the study of nonequilibrium dynamics of quantum many-body systems. Recently, a number of experiments with superconducting transmon qubits are reported, in which the spin and hard-core boson models with two energy levels on individual sites are used. The transmons are a multilevel system and the coupled qubits are governed by the Bose-Hubbard model. How well they can be approximated by a two-level system has been discussed and analysed in different ways for specific experiments in the literature. Here, we numerically investigate the accuracy and validity of the two-level approximation for the multilevel transmons based on the concept of Loschmidt echo. Using this method, we are able to calculate the fidelity decay (i.e., the time-dependent overlap of evolving wave functions) due to the state leakage to transmon high energy levels. We present the results for different system Hamiltonians with various initial states, qubit coupling strength, and external driving, and for two kinds of quantum quench experiments with time reversal and time evolution in one direction. We show quantitatively the extent to which the fidelity decays with time for changing coupling strength (or on-site interaction over coupling strength) and filled particle number or locations in the initial states under specific system Hamiltonians, which may serve as a way for assessing the two-level approximation of transmons. Finally, we compare our results with the reported experiments using transmon qubits.