Spontaneous emission of a three-level artificial atom in a one-dimensional open waveguide

TL;DR

This work tackles spontaneous emission from a ladder-type three-level emitter in a one-dimensional open waveguide, focusing on the two-excitation subspace relevant for two-photon processes. An analytic real-space method yields closed-form amplitudes for the fully excited initial state, enabling exact expressions for the single- and two-photon components and their long-time spectra. The main findings show that the emission dynamics crucially depend on the decay-rate ratio $\Gamma_2/\Gamma_1$; under strong coupling, emitted photons exhibit frequency correlations and can become identical in frequency, with tunability via the anharmonicity parameter $\alpha_r$ and the decay-rate ratio. These results provide design principles for photon-pair generation and frequency-structured quanta in superconducting waveguide QED, with broad relevance to transmon-based platforms and other 1D emitter systems.

Abstract

We study the dynamical and spectral characteristics of a quantum three-level ladder system, interacting with a continuous electromagnetic field in one-dimensional open waveguide. Common realization of such systems is a waveguide QED setup ~ -- a superconducting artificial atom (transmon), coupled to an open microwave transmission line. We derive an analytical solution for spontaneous emission of initially excited atom, and use it to study the probability of state detection and spectral density of output photon states. We find that for strong coupling of transmon to a waveguide emitted photons show correlation in frequency and can have the same energies, even if the three-level system is anharmonic.

Figures (4)

• Figure 1: (a) System under study - three-level emitter (3LE), coupled to the center of one-dimensional waveguide at point $x=0$. Coupling of first and second excited states are $V_1$ and $V_2$, respectfully. Excited atom can emit photons through spontaneous decay, which are propagate only forward due to usage of chiral approximation (see text for details). (b) Energy levels diagram of the bare transmon and corresponding resonant frequencies.
• Figure 2: Probabilities to detect a certain state in entire waveguide, plotted with expressions (\ref{['prob']}): red solid line - transmon fully excited in level $\left| f \right\rangle$ and no photons presented; blue dashed line - transmon in first-excited state $\left| e \right\rangle$ and single photon in a waveguide; green dash-dotted line - transmon in a ground state $\left| g \right\rangle$, and two photons in a waveguide. Sum of all probabilities is equals to one and shown by thin dotted line. Different ratios of $\Gamma_2/\Gamma_1$ are shown above plots. Time is measured in $\Gamma_2$ units. Decay of upper level is constant, $\Gamma_2/\Omega_1 = 0.02$.
• Figure 3: Two-photon spectral density of spontaneous emission from three-level transmon $S_{sp}(\omega_1,\omega_2)\Gamma_2^2$, plotted using expression (\ref{['spectra2']}). (a) high anharmonicity $\alpha_r = -5 \%$ and typical decay rates relation $\Gamma_2/ \Gamma_1 = 3/2$; (b) smaller anharmonicity $\alpha_r = -3 \%$, and $\Gamma_2/ \Gamma_1 = 3/2$; (c) faster decay of upper level $\Gamma_2/ \Gamma_1 = 3$ and $\alpha_r = -3 \%$; (d) same decay $\Gamma_2/ \Gamma_1 = 3$ and smaller anharmonicity $\alpha_r = -2 \%$. Decay of upper level taken constant $\Gamma_2 / \Omega_1 = 0.02$, except for (e), where weak-coupling is analyzed for $\Gamma_2 / \Omega_1 = 0.001$, $\Gamma_2/\Gamma_1 = 1.5$ and $\alpha_r=-3 \%$.
• Figure 4: Spectral density of two identical photons (\ref{['spectra1']}) vs frequency detuning - (a) and (b), and vs detuning and anharmonicity - (c) and (d). (a) and (c) are plotted for $\Gamma_2/\Gamma_1 = 3/2$ ratio, and (b) and (d) for $\Gamma_2/\Gamma_1 = 3$. In (a), (b) solid magenta line corresponds to $\alpha_r=-3 \%$, green dashed line to $\alpha_r = -4\%$, and black dotted line to $\alpha_r=-6\%$. Vertical grey lines show points for $\delta_\omega/\Omega_1 = \alpha_r/2$. In (c), (d) thin white lines indicate resonant frequency $\omega=\Omega_1$. Decay of upper level taken constant $\Gamma_2 / \Omega_1 = 0.02$.