Revealing the system-bath coupling via Landau-Zener-Stückelberg interferometry in superconducting qubits

In this work we propose a way to unveil the type of environmental noise in strongly driven superconducting flux qubits through the analysis of the Landau-Zener-Stückelberg (LZS) interferometry. We study both the two-level and multilevel dynamics of the flux qubit driven by a $\mathrm{dc}+\mathrm{ac}$ magnetic field. We find that the LZS interference patterns exhibit well-defined multiphoton resonances whose shape strongly depends on the timescale and the type of coupling to a quantum bath. For the case of transverse system-bath coupling, the $n$-photon resonances are narrow and nearly symmetric with respect to the dc magnetic field for almost all timescales, while in the case of longitudinal coupling they exhibit a change from a wide symmetric to an antisymmetric shape for times of the order of the relaxation time. We find this dynamic behavior relevant for the interpretation of several LZS interferometry experiments in which the stationary regime is not completely reached.

Figure 1

Population $P_{+}$ as a function of the dc flux detuning ${\tilde{f}}_{\mathrm{dc}}$ (normalized by $f_{\omega }={\omega }_0/4\pi I_p$) for the FQ restricted to TLS, driven with $\mathrm{amplitude}{f}_{\mathrm{ac}}=0.003$ and ${\omega }_0=2\pi /\tau =0.003E_J/\hslash$. The Ohmic bath is at $T=0.0014$ (20 mK for $E_J/h\approx 300\mathrm{GHz}$). Dotted line: ${\overline{P}}_{+}$ for the isolated system; dashed line: $P_{+}(t=1000\tau )$; solid line: asymptotic ($t\to \infty$) ${\overline{P}}_{+}$. The horizontal solid line indicates the $P_{+}=0.5$ value. (a) Longitudinal coupling for an Ohmic bath with ${\gamma }^{\left(f\right)}=0.001$ and (b) enlarged view of (a) around $n=4$ photon resonance. (c) Transverse coupling for an Ohmic bath with ${\gamma }^{\left(ch\right)}=0.001$ and (d) enlarged view of (c) around $n=4$ photon resonance.

Figure 2

(a) Relaxation time $t_r$ for the longitudinal (dash-double-dotted line) and transverse (dashed line) couplings. (Half) decoherence time $t_d/2$ for the longitudinal (dash-dotted line) and transverse (solid line) couplings. The flux detuning is normalized by $f_{\omega }={\omega }_0/4\pi I_p$, such that the $n$-photon resonances are at $\tilde{f}=n{f}_{\omega }$. (b) Enlarged view of (a) around the $n=4$ resonance. The experimental time $t_{\exp }/\tau =$ 1000 is plotted by the dotted line.

Figure 3

Intensity plots of the population $P_{+}$ as a function of ${\tilde{f}}_{\mathrm{dc}}$ and driving time $t$. (a) Longitudinal coupling for ${\gamma }^{\left(f\right)}=0.001$. (b) Transverse coupling for ${\gamma }^{\left(ch\right)}=0.001$. See text for details.

Figure 4

Intensity plots of the population $P_{+}$ as a function of ${\tilde{f}}_{\mathrm{dc}}$ and the mixing parameter ${cos}^2\theta$. ${\gamma }^{\left(f\right)}=\gamma {cos}^2\theta$ and ${\gamma }^{\left(ch\right)}=\gamma {sin}^2\theta$, with $\gamma =0.001$. (a) $t=\infty$, (b) $t=1000\tau$. (c) Intensity plot of the ratio of ${log}_{10}\left(2t_r/t_d\right)$.

Figure 5

LZS interference patterns. Plots of $P_{+}$ as a function of the driving amplitude $f_{\mathrm{ac}}$ and dc detuning ${\tilde{f}}_{\mathrm{dc}}$ for $t=1000\tau$. (a) Flux noise. (b) Charge noise. The calculations were performed for ${\omega }_0=2\pi /\tau =0.003E_J/\hslash$, Ohmic bath at $T=0.0014E_J/k_B\sim 20\mathrm{mK}$, and $\gamma =0.001$ (see text for details).

Figure 6

LZS interference patterns. Plots of $P_{+}$ as a function of the driving amplitude $f_{\mathrm{ac}}$ and dc detuning ${\tilde{f}}_{\mathrm{dc}}$ for the asymptotic regime, $t\to \infty$. (a) Flux noise. (b) Charge noise. Parameters are the same as in Fig. 5.

Figure 7

$P_{+}$ vs $f_{\mathrm{ac}}$ for ${\tilde{f}}_{\mathrm{dc}}\equiv 2.7f_{\omega }$ for $t=1000\tau$ (red dashed line), for the asymptotic state $t\to \infty$, (blue solid line), and for the isolated case (dots). (a) Flux noise. (b) Charge noise. (c) $t_r/\tau$ for flux noise (dash-double-dotted line) and charge noise (dashed line), $t_d/2\tau$ for flux noise (dot-dashed line) and charge noise (solid line), and $t_{\exp }=1000\tau$ (dotted line). The vertical dashed lines are guides for the eyes to show the boundaries of diamonds D1 and D2 and the region in between, named D12, for the value of ${\tilde{f}}_{\mathrm{dc}}\equiv 2.7f_{\omega }$.