Cavity quantum electrodynamics in superconducting circuits: Susceptibility at elevated temperatures

Abstract

We study the properties of superconducting electrical circuits, realizing cavity QED. In particular we explore the limit of strong coupling, low dissipation, and elevated temperatures relevant for current and future experiments. We concentrate on the cavity susceptibility as it can be directly experimentally addressed, i.e., as the impedance or the reflection coefficient of the cavity. To this end we investigate the dissipative Jaynes-Cummings model in the strong coupling regime at high temperatures. The dynamics is investigated within the Bloch-Redfield formalism. At low temperatures, when only the few lowest levels are occupied the susceptibility can be presented as a sum of contributions from independent level-to-level transitions. This corresponds to the secular (random phase) approximation in the Bloch-Redfield formalism. At temperatures comparable to and higher than the oscillator frequency, many transitions become important and a multiple-peak structure appears. We show that in this regime the secular approximation breaks down, as soon as the peaks start to overlap. In other words, the susceptibility is no longer a sum of contributions from independent transitions. We treat the dynamics of the system numerically by exact diagonalization of the Hamiltonian of the qubit plus up to 200 states of the oscillator. We compare the results obtained with and without the secular approximation and find a qualitative discrepancy already at moderate temperatures.

Applications of the Fokker-Planck equation in circuit quantum electrodynamics

“…We show that that despite the apparent restrictiveness of this process, we retain much of the important behaviour of the system in our effective single oscillator system, even when the cavity and qubit are resonant and this approach would seem most likely to break down. The Jaynes-Cummings model, which approximates the transmon as a two-level system, has been studied extensively using numerical solutions at low occupation and semiclassical models in the limit of strong driving [31] and in the presence of non-zero temperature [32,33]. In the case of strong driving, however, the higher levels of the transmon become relevant to the dynamics and no analytical solution exists in this regime.…”

State-dependent impedance of a strongly coupledoscillator−qubitsystem

“…In a nonsteady state the emission and absorption between the system energy levels does not obey the detailed balance condition and χ(ω, t) may differ qualitatively from the steady-state susceptibility χ(ω) studied in Ref. 13. However, the oscillator-qubit system approaches the thermal equilibrium irrespective of the initial state and, therefore,…”

“…The propagator contains 16N 4 terms. Fortunately, the interaction with the bath couples only states of the reduced system differing roughly byhω 0 in energy 13 . For this reason, the effective number of propagator elements is proportional to N 2 .…”

Circuit QED with a flux qubit strongly coupled to a coplanar transmission line resonator

“…This justifies ignoring thermal effects in our simulations for now. That said, even a moderate increase in temperature can significantly change the outcome of an experiment 27 . One further simplifying assumption in the model is that T 1 and T 2 do not change as the qubit is detuned from the optimal bias point Φ 0 /2.…”