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Decay of transmon qubit strongly coupled with a continuum

Ya. S. Greenberg, A. A. Shtygashev, O. V. Kibis

TL;DR

This paper develops a resolvent-operator framework to study the decay of a three-level transmon qubit strongly coupled to a continuum of waveguide modes. By formulating a multimode Jaynes–Cummings model and transforming to the continuum, it derives analytical expressions for the resonance frequency shifts $\Delta_2(E)$ and widths $\Gamma_2(E)$ that govern the decay of the top level $|f\rangle$, with the second-level coupling to the ground state $|g\rangle$ playing a crucial role. In the weak-coupling regime the decay reduces to a Lorentzian two-level-like form, while in the strong regime the spectrum splits into multiple resonances, including long-lived quasi-stable states; introducing $V_1$ between $|e\rangle$ and $|g\rangle$ can suppress these coherent features. The analysis, aided by Gaussian densities of states for the continuum, yields quantitative predictions for resonance positions, heights, and lifetimes, providing a framework applicable to open quantum-system dynamics of artificial atoms in one-dimensional waveguides.

Abstract

We investigate the decay of three-level artificial atom, a superconducting transmon qubit which interacts with a continuum of modes in an open one-dimensional waveguide. For strong interaction of transmon with a continuum we obtain analytical expressions for the frequency shifts and widths of the resonances the values of which are calculated numerically for the Gaussian density of states. We show that the coupling between the second level and ground state of a transmon significantly influences the decay of the third transmon level.

Decay of transmon qubit strongly coupled with a continuum

TL;DR

This paper develops a resolvent-operator framework to study the decay of a three-level transmon qubit strongly coupled to a continuum of waveguide modes. By formulating a multimode Jaynes–Cummings model and transforming to the continuum, it derives analytical expressions for the resonance frequency shifts and widths that govern the decay of the top level , with the second-level coupling to the ground state playing a crucial role. In the weak-coupling regime the decay reduces to a Lorentzian two-level-like form, while in the strong regime the spectrum splits into multiple resonances, including long-lived quasi-stable states; introducing between and can suppress these coherent features. The analysis, aided by Gaussian densities of states for the continuum, yields quantitative predictions for resonance positions, heights, and lifetimes, providing a framework applicable to open quantum-system dynamics of artificial atoms in one-dimensional waveguides.

Abstract

We investigate the decay of three-level artificial atom, a superconducting transmon qubit which interacts with a continuum of modes in an open one-dimensional waveguide. For strong interaction of transmon with a continuum we obtain analytical expressions for the frequency shifts and widths of the resonances the values of which are calculated numerically for the Gaussian density of states. We show that the coupling between the second level and ground state of a transmon significantly influences the decay of the third transmon level.

Paper Structure

This paper contains 14 sections, 81 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Projection operators
  4. Green function
  5. Calculation of the matrix elements of the shift operator
  6. Calculation of $R_{ff} (z) = \left\langle {f0} \right|R(z)\left| {f0} \right\rangle$
  7. Calculation of $R_{ff}(z\pm i\eta)$ (\ref{['18']})
  8. Spectral function $U_{ff}(E)$
  9. Transformation to continuum modes
  10. The coupling between transmon and continuum modes
  11. Weak coupling
  12. Strong coupling
  13. Numerical results and discussions
  14. Conclusion

Figures (7)

  • Figure 1: Transmon levels
  • Figure 2: Transmon decay parameters for weak coupling, $L_2=0.1$. The left column describes the decay of the third level when the second level is stable ($V_1=0$). The influence of the interaction between $|e\rangle$ and $|g\rangle$ is shown in the right column where $L_1=\frac{2}{3}L_2$. The plots in left column are calculated from (\ref{['72']}), (\ref{['73']}), (\ref{['45']}, while those in right column are calculated from (\ref{['76']}), (\ref{['77']}), (\ref{['48']}).
  • Figure 3: Transmon decay parameters for relatively weak coupling, $L_2=0.3$. The left column describes the decay of the third level when the second level is stable ($V_1=0$). The influence of the interaction between $|e\rangle$ and $|g\rangle$ is shown in the right column where $L_1=\frac{2}{3}L_2$. The plots in left column are calculated from (\ref{['72']}), (\ref{['73']}), (\ref{['45']}, while those in right column are calculated from (\ref{['76']}), (\ref{['77']}), (\ref{['48']}).
  • Figure 4: Transmon decay parameters for strong coupling, $L_2=1$. The left column describes the decay of the third level when the second level is stable ($V_1=0$). The influence of the interaction between $|e\rangle$ and $|g\rangle$ is shown in the right column where $L_1=\frac{2}{3}L_2$. The plots in left column are calculated from (\ref{['72']}), (\ref{['73']}), (\ref{['45']}, while those in right column are calculated from (\ref{['76']}), (\ref{['77']}), (\ref{['48']}).
  • Figure 5: Transmon decay parameters for ultra coupling, $L_2=6$. The left column describes the decay of the third level when the second level is stable ($V_1=0$). The influence of the interaction between $|e\rangle$ and $|g\rangle$ is shown in the right column where $L_1=\frac{2}{3}L_2$. The plots in left column are calculated from (\ref{['72']}), (\ref{['73']}), (\ref{['45']}, while those in right column are calculated from (\ref{['76']}), (\ref{['77']}), (\ref{['48']}).
  • ...and 2 more figures