Resonance fluorescence of an artificial atom with a time-delayed coherent feedback

Abstract

The model of light-matter interaction in quantum electrodynamics typically relies on the Markovian approximation, which assumes that the system's future evolution depends solely on its current state, effectively treating it as a ``memoryless" process. However, this approximation is not valid in scenarios when retardation effects are significant. These memory and retardation effects have the potential to improve existing quantum technologies (e.g., large-scale quantum networks, quantum information processing) and unlock new phenomena for future applications. In this work, we show theory and experiments of a time-delayed coherent feedback system using a transmon artificial atom (treated as a qubit) embedded in a superconducting circuit waveguide, in both linear and nonlinear excitation regimes. By using a feedback loop with a delay time comparable to the qubit relaxation time, pronounced non-Markovian effects appear in the dynamics of the qubit evolution. We also show how the resonance fluorescence spectrum, including elastic and inelastic scattering (such as the well-known Mollow triplet), can be significantly modified through the interaction between the qubit and feedback loop to show genuine non-Markovian and quantum nonlinear phenomena that cannot be explained with instantaneous coupling parameters. This work presents the first experimental report of Mollow triplets in the non-Markovian regime.

Figures (3)

• Figure 1: (a) Schematic of the qubit-mirror in a quasi-1D system. The round trip length is $L_{0}$ and the qubit couples to the waveguide symmetrically with total relaxation rate $\Gamma$. The qubit is pumped with frequency $\omega_{\rm{p}}$ and Rabi frequency $\Omega_{\rm{L}}(\Omega_{\rm{NL}})$ directly through the waveguide. (b) Optical microscope images of the transmon qubit koch2007charge and the experimental setup. The blue dashed rectangle shows the qubit capacitively coupled to a 1D transmission line and the red dashed square shows the superconducting quantum interference device (SQUID) with a tunable frequency through the applied external magnetic flux with the superconducting magnet. The structure above the red dashed square is the flux line but unused in this experiment. One end of the transmission line couples to the feedback loop terminated by a connection to the grounding plane that acts as the mirror, and the material of the superconducting coaxial cable is NbTi. The other end connects to a three port circulator through which the input signal enters the waveguide and the output signal is detected. The details of the waveguide-mirror system and the setup can be found in Fig. S3 and Fig. S2 of Sec. II of the SM SuppMaterial, respectively.
• Figure 2: Single-tone spectroscopic measurement of the reflection coefficient $r$ as a function of $\phi$ and $\omega_{\rm{p}}$ using a weak pump (${\color{black} \Omega_{\rm{L}}} \ll \gamma_{\rm{d}}$). (a) and (b) show the magnitude response, $|r|$, and (c) and (d) show the phase response, $\theta_r$, of the experimental and simulated $r$, respectively (the full spectroscopy is shown in Fig. S4 of the SM SuppMaterial. The node and antinode frequencies of the pump field are indicated by the green and red arrows respectively. The solid black line in these graphs indicate the expected qubit bare resonance frequency $\omega_0$ without coupling to the continuous photonic mode, calculated by the extracted anharmonicity and maximum qubit frequency. The yellow dashed curve shows the frequency shift in experiment. (e) and (f) respectively show experimental and simulated line cuts from (a)-(d) of $r$ in the complex plane for fixed $\omega_0$ indicated by the red ($\omega_0/2\pi = 4.9 \,\, \rm{GHz}$, $\phi = 0$) and orange ($\omega_0/2\pi = 4.822 \,\, \rm{GHz}$, $\phi = -0.632\pi$) arrows. The orange curved arrows show the tilted angle of the complex plane. (g) shows $(\widetilde{\omega}_0 - \omega_0)\tau$ of the experimental (blue data points) and simulated (blue solid curve) results as well as the tilted angle $\phi_{\widetilde{\omega}_{\rm{0}}}-\phi$ directly extracted from the complex plane (red crosses data points) versus $\omega_0$. The orange arrow points out the corresponding $\phi_{\widetilde{\omega}_{\rm{0}}}-\phi$ of the orange complex plane in (e) and (f). The disagreement between data and theory around $4.9\,\, \rm{GHz}$ is likely due to the qubit coupling to a stray resonance en1.
• Figure 3: Resonant fluorescence emission spectra of the qubit as a function of the pump power, $P_{\rm{p}}$ and the detected radiation $\omega$, with $\omega_{\rm{p}}=\widetilde{\omega}_{0}$. For (a)-(c) $\phi = 0$ and (d)-(f) $\phi = -0.632 \pi$. In (a) and (d) the experimental results are shown while in (b) and (e) we show the corresponding simulated results. The bright emission at center peak is due to a multi-level effect (see Sec. V of the SM SuppMaterial). The central peak in (a), (b) and (d), (e) correlate with sideband brightness. For comparison, the simulated spectral response with a Markovian feedback loop is shown in (c) and (f). Under the strong pump, the qubit energy level split is $\Omega_{\rm{eff}}$, creating three transition frequencies, $\widetilde{\omega}_{\rm{0}} \pm \Omega_{\rm{eff}}$ and $\widetilde{\omega}_{\rm{0}}$. In (g)-(j) we show individual spectra with the experimental data as colored dots and the simulated response as a solid curve. For $\phi = 0$, in (g) $P_{\rm{p}} = -105.4 \, \, \rm{dBm}$ and in (h) $P_{\rm{p}} = -108.4 \, \, \rm{dBm}$. For $\phi = -0.632\pi$, in (i) $P_{\rm{p}} = -102.3 \, \, \rm{dBm}$ and in (j) $P_{\rm{p}} = -112.3 \, \, \rm{dBm}$. The QTDW simulation results use 1000 trajectories per spectrum in (b), (c), (e), and (f) and 10,000 trajectories per simulated spectra in (g)-(j).