Landau-Zener-Stückelberg interferometry in dissipative circuit quantum electrodynamics

We study Landau-Zener-Stückelberg (LZS) interferometry in a cQED architecture under the effects of dissipation. To be specific, we consider a superconducting qubit driven by a $\mathrm{dc}+\mathrm{ac}$ signal and coupled to a transmission line resonator, but our results are valid for general qubit-resonators devices. To take the environment into account, we assume that the resonator is coupled to an Ohmic quantum bath. The Floquet-Born-Markov master equation is numerically solved to obtain the dynamics of the system for an arbitrary amplitude of the drive and different timescales. We unveil important differences in the resonant patterns between the strong coupling and ultrastrong coupling regimes in the qubit-resonator interaction, which are mainly due to the magnitude of photonic gaps in the energy spectrum of the system. We identify in the LZS patterns the contribution of the qubit gap and the photonic gaps, showing that for large driving amplitudes the patterns present a weaving structure due to the combined intercrossing of the different gaps contributions.

Figure 1

Lowest energy levels of the Rabi Hamiltonian Eq. (1) without driving ($A=0$), as a function of the dc bias $\varepsilon ={\varepsilon }_0$, for the parameters $\mathrm{\Delta }/{\omega }_r=0.0038$. (a) $g/{\omega }_r=0.0019$ (SC) and (b) $g/{\omega }_r=0.1125$ (USC) (solid lines). The qubit gap is at $\varepsilon =0$ while the photonic gaps are at $\varepsilon =\pm {\omega }_r$. The color (dashed) lines are the energies $\pm \varepsilon /2$ ($\pm \varepsilon /2+{\omega }_r$) of the product states $|\uparrow ,0\rangle$ and $|\downarrow ,0\rangle$ ($|\uparrow ,1\rangle$ and $|\downarrow ,1\rangle$) in the absence of qubit-resonator coupling ($g=0$) and for $\mathrm{\Delta }=0$.

Figure 2

Numerically obtained LZS interference patterns for the DJC Hamiltonian, Eq. (3). Plots of ${\overline{P}}_{|\downarrow ,n+1\rangle \to |\uparrow ,n\rangle }$ as a function of the driving amplitude $A$ and dc bias ${\delta }_0$ in units of $\omega$, for $g/{\omega }_r=0.0019$ and $n=3$ (a) $[n=10$ (b)] and $g/{\omega }_r=0.1125$ and $n=0$ (c) $[n=1$ (d)].

Figure 3

Instantaneous transition probability $P_{|\downarrow ,n+1\rangle \to |\uparrow ,n\rangle }\left(t\right)$ for driving amplitude $A/\omega =3$ and coupling $g/{\omega }_r=0.0019$. (Top) Photon number $n=3$ (a) and 10 (b) for resonance ${\delta }_0/\omega =0$. (Bottom) Photon number $n=3$ (c) and $n=10$ (d) for resonance ${\delta }_0/\omega =1$. The fast oscillations exhibited in the numerical results (solid line), are not captured by the RWA approximation, Eq. (5) (dashed line).

Figure 4

Intensity plots of the LZS interference patterns for the driven Rabi Hamiltonian (2). Plots of ${\overline{P}}_{|\uparrow \rangle }$ as a function of the driving amplitude $A$ and ${\varepsilon }_0$, for (a) $g/{\omega }_r=0.0019$ and (b) 0.1125. The calculations were performed for $\omega /{\omega }_r=0.0375$ and $\mathrm{\Delta }/{\omega }_r=0.0038$. The insets in both panels show the resonances patterns in more detail.

Figure 5

(Solid lines) Scheme of the boundaries of the regions where the LZS resonances mediated by the different avoided crossings occur in plots of ${\overline{P}}_{|\uparrow \rangle }$, as a function of the driving parameters ${\varepsilon }_0$ and $A$. States involved in the transitions are indicated. A rectangle in dashed lines shows the range of the parameters considered along this work.

Figure 6

LZS interference patterns for the Rabi Hamiltonian in the SC Regime ($g/{\omega }_r=0.0019$) considering effects of dissipation. Plots of ${\overline{P}}_{|\uparrow \rangle }\left(t\right)$ as a function of the driving parameters $A$ and ${\varepsilon }_0$, at finite time $t=1000\tau$ (a) and in the asymptotic regime $t=\infty$ (b). The calculations were performed for $\omega /{\omega }_r=0.0375$, $\mathrm{\Delta }/{\omega }_r=0.0038$, $T=0.0175{\omega }_r/k_B$, and $\kappa =0.001$ (see text for details).

Figure 7

(a) Avareged transition probability for the closed system ${\overline{P}}_{|\uparrow \rangle }$ (dashed line) and considering effects of dissipation at finite time ${\overline{P}}_{|\uparrow \rangle }(t\to 1000\tau )$ (solid line) as a function of ${\varepsilon }_0$ for $g/{\omega }_r=0.0019$ and $A=35\omega$. (b) Avareged transition probability in the asymptotic regime ${\overline{P}}_{|\uparrow \rangle }(t\to \infty )$ as a function of ${\varepsilon }_0$ for $g/{\omega }_r=0.0019$ and $A=35\omega$. Numerical results when the system is taken as qubit plus resonator (DR Hamiltonian) coupled to an Ohmic bath (solid line) and when we regard the driven qubit coupled to a structured bath (dashed lines).

Figure 8

LZS interference patterns for the Rabi Hamiltonian in the USC regime ($g/{\omega }_r=0.1125$) considering effects of dissipation. Plots of ${\overline{P}}_{|\uparrow \rangle }\left(t\right)$ as a function of the driving parameters $A$ and ${\varepsilon }_0$, at finite time $t=50\tau$ (a) and in the asymptotic regime $t=\infty$ (b). The calculations were performed for $\omega /{\omega }_r=0.0375$, $\mathrm{\Delta }/{\omega }_r=0.0038$, $T=0.0175{\omega }_r/k_B$, and $\kappa =0.001$ (see text for details).