Dynamics of a dispersively coupled transmon qubit in the presence of a\n noise source embedded in the control line

Abstract

We describe transmon qubit dynamics in the presence of noise introduced by an\nimpedance-matched resistor ($50\\,\\mathrm{\\Omega}$) that is embedded in the\nqubit control line. To obtain the time evolution, we rigorously derive the\ncircuit Hamiltonian of the qubit, readout resonator and resistor by describing\nthe latter as an infinite collection of bosonic modes through the\nCaldeira-Leggett model. Starting from this Jaynes-Cummings Hamiltonian with\ninductive coupling to the remote bath comprised of the resistor, we\nconsistently obtain the Lindblad master equation for the qubit and resonator in\nthe dispersive regime. We exploit the underlying symmetries of the master\nequation to transform the Liouvillian superoperator into a block diagonal\nmatrix. The block diagonalization method reveals that the rate of exponential\ndecoherence of the qubit is well-captured by the slowest decaying eigenmode of\na single block of the Liouvillian superoperator, which can be easily computed.\nThe model captures the often used dispersive strong limit approximation of the\nqubit decoherence rate being linearly proportional to the number of thermal\nphotons in the readout resonator but predicts remarkably better decoherence\nrates when the dissipation rate of the resonator is increased beyond the\ndispersive strong regime. Our work provides a full quantitative description of\nthe contribution to the qubit decoherence rate coming from the control line in\nchips that are currently employed in circuit QED laboratories, and suggests\ndifferent possible ways to reduce this source of noise.\n