Large-amplitude harmonic driving of highly coherent flux qubits

The device for the Josephson flux qubit can be considered as a solid-state artificial atom with multiple energy levels. When a large-amplitude harmonic excitation is applied to the system, transitions at the energy levels avoided crossings produce visible changes in the qubit population over many driven periods that are accompanied by a rich pattern of interference phenomena. We present a Floquet treatment of the periodically time-dependent Schrödinger equation of the strongly driven qubit beyond the standard two-level approach. For low amplitudes, the average probability of a given sign of the persistent current qubit exhibits, as a function of the static flux detuning and the driving amplitude, Landau-Zener-Stückelberg (LZS) interference patterns that evolve into complex diamondlike patterns for large amplitudes. In the case of highly coherent flux qubits we show that the higher-order diamonds can not be simply described relying on LZS transitions in each avoided crossing considered separately. In addition we propose a spectroscopic method based on starting the system in the first excited state instead of in the ground state, which can give further information on the energy-level spectrum and dynamics in the case of highly coherent flux qubits. We compare our numerical results with recent experiments that perform amplitude spectroscopy to probe the energy spectrum of the artificial atom.

Figure 1

Circuit for the DJFQ as described in the text. Josepshon junctions 1 and 2 have Josepshon energy $E_J$ and capacitance $C$, and junction 3 has Josepshon energy and capacitance $\alpha$ times smaller. The arrows indicate the sign convention for defining the gauge-invariant phase differences. The circuit encloses a magnetic flux $\Phi =f{\Phi }_0$.

Figure 2

Lowest seven energy levels of the DJFQ as a function of flux $f_0$ for $\eta =0.25$ and $\alpha =0.80$. Arrows indicate the position of the avoided level crossings ${\Delta }_{ij}$ measured from $f_0=0.497$ (indicated by the vertical dot-dashed line). Calculations were done using $M=256$ (see the text for details). Energy is measured in units of $E_J$ and flux in units of ${\Phi }_0$.

Figure 3

(Color online) Large-amplitude spectroscopic diamonds obtained for the Josephson flux qubit with $\eta =0.25$ and $\alpha =0.8$. The intensity of ${\overline{P}}_{-}$ is plotted as a function of flux detuning $\delta f$ and rf amplitude $f_p$. The system is driven at a frequency $\omega =0.001$ (in units of $E_J/\hslash$). Numerical calculations were done by direct numerical integration using $M=128$ and $\Delta t=0.1t_J$. Data points correspond to a grid of $\Delta f_0=1\times {10}^{-3}$ and $\Delta f_p=1\times {10}^{-4}$.

Figure 4

(Color online) $P_{-}$ as a function of time $\tau =\omega t/2\pi$ for $\eta =0.25$ and $\alpha =0.8$. (a) ${ϵ}_0=\omega$, for $\omega =0.001$, $\delta f=-0.11\times {10}^{-3}$, and $f_p=0.24\times {10}^{-3}$ (black solid line). The fast driving TLS Eq. (6) for $n=1$ with ${\Omega }_1={\Delta }_{00}{J}_1\left(4\pi S_{01}{f}_p/\omega \right)=1.87\times {10}^{-4}$ is plotted for comparison (red dashed line). (b) ${ϵ}_0=3\omega$, for $\omega =0.001$, $\delta f=-0.33\times {10}^{-3}$, and $f_p=0.35\times {10}^{-3}$ (black solid line). The fast driving TLS Eq. (6) for $n=3$ with ${\Omega }_3={\Delta }_{00}{J}_3\left(4\pi S_{01}{f}_p/\omega \right)=1.14\times {10}^{-4}$ is plotted for comparison (red dashed line). (c) ${ϵ}_0=3\omega$, for a higher frequency $\omega =0.003$, $\delta f=-0.99\times {10}^{-3}$, and $f_p=1.05\times {10}^{-3}$ (black solid line). Notice that the oscillations on time scales $\sim \pi /\omega$ are smaller for $\omega =0.003$, as described in the text. The fast driving TLS Eq. (6) for $n=3$ is plotted for comparison (red dashed line). (d) Out of $\left(n=3\right)$ resonance, for $\omega =0.001$, $\delta f=-0.32\times {10}^{-3}$, and $f_p=0.35\times {10}^{-3}$ (black solid line). Notice that the oscillations on time scales $\sim \pi /\omega$ persist. Numerical calculations were done with $N_l=6$ levels, $K=180$ and $M=1024$. Calculations with $N_l=2$ levels overlap almost exactly with the $N_l=6$ calculations in this case.

Figure 5

(Color online) $P_{-}$ as a function of time $\tau =\omega t/2\pi$ for $\eta =0.25$ and $\alpha =0.8$. The flux qubit is driven at a strong amplitude with $f_p=16\times {10}^{-3}$. (a) Resonance condition for the TLS in the fast driving regime: ${ϵ}_0=3\omega$, for $\omega =0.001$ and $\delta f=-0.33\times {10}^{-3}$. (b) ${ϵ}_0\ne n\omega$, for $\omega =0.001$ and $\delta f=-0.32\times {10}^{-3}$. Numerical calculations were done with $N_l=6$ levels, $K=300$ and $M=1024$. Calculations with $N_l=2$ levels are plotted for comparison (lower curve, red line).

Figure 6

(Color online) Large-amplitude spectroscopic (half) diamonds obtained for the Josephson flux qubit with $\eta =0.25$ and $\alpha =0.8$. The intensity of ${\overline{P}}_{-}$ is plotted as a function of flux detuning $\delta f$ and rf amplitude $f_p$. (a) The system is driven at a frequency $\omega =0.001$. Calculations were done using $N_l=6$ (six levels), $150\le K\le 300$ and $M=1024$. The black solid lines indicate the edges of the first diamond D1 and the beginning of the second one D2. The black dot indicates a particular value of flux detuning $\delta f\simeq -0.003$ for $f_p=0$. See text for a detailed analysis. (b) The flux qubit is driven at a frequency $\omega =0.002$. Calculations were done using $N_l=6$, $150\le K\le 250$ and $M=1024$. Data points correspond to a fine grid of $\Delta f_0=2\times {10}^{-5}$ and $\Delta f_p=3\times {10}^{-5}$.

Figure 7

(Color online) ${\overline{P}}_{-}$ (black solid line) calculated employing the lowest six levels and ${\overline{P}}_{-}^{\left(2\right)}$ (red dashed line) calculated employing the lowest two levels as a function of the driving amplitude. Calculations were done using $K=150–300$ and $M=1024$. (a) $\delta f=-0.00309$ and (b) $\delta f=-0.00308$. The circle, square, and triangle denote the beginning of the first diamond D1, the end of D1, and the beginning of the second diamond D2, respectively, for the present value of $\delta f$. See text for details.

Figure 8

(Color online) ${\overline{P}}_{-}$ (black solid line) as a function of the static flux $f_0$ for two different values of the driving amplitude $f_p=0.001$ (upper panel) and $f_p=0.018$ (lower panel) calculated using $N_l=6$. In both panels, a dashed line denotes the value of $f_0$ that for $f_p=0.001$ $\left(f_p=0.018\right)$ gives the beginning of the first (second) diamond D1 (D2). As a comparison we plotted in red dashed line the ${\overline{P}}_{-}^{\left(2\right)}$ computed employing the lowest two levels. Numerical calculations were done with $K=150–300$ and $M=1024$. See Fig. 6 and text for further details.

Figure 9

(Color online) ${\overline{P}}_{-}$ for $\delta f=-0.002$ calculated employing the lowest eight levels as a function of the driving amplitude. Calculations were done using $K=150–400$ and $M=1024$. The red symbols in the axis show the values of $f_p$ for which the different avoided crossings are reached (see Fig. 2). The vertical dotted lines indicate the borders of the different spectroscopic diamonds D1, D2, and D3.

Figure 10

(Color online) ${\overline{P}}_{-}$ (red line) and $1-{\overline{P}}_{-}^e$ (black line) as a function of the driving amplitude for $\delta f=-0.00066$. Calculations were done employing the lowest six levels using $K=150–300$ and $M=1024$.

Figure 11

(Color online) Large-amplitude excited-state spectroscopic (half) diamonds obtained for the DJFQ with $\eta =0.25$ and $\alpha =0.8$ at a driven frequency $\omega =0.001$. Calculations were done employing $N_l=6$ (six levels), $150\le K\le 300$ and $M=1024$. (a) Intensity plot of $1-{\overline{P}}_{-}^e$. (b) Intensity plot of ${\overline{P}}_{-}-\left(1-{\overline{P}}_{-}^e\right)$.

Figure 12

(Color online) ${\overline{P}}_{-}^e$ calculated employing the lowest eight levels as a function of the driving amplitude for $\delta f=-0.014$. The system is driven at a frequency $\omega =0.001$, $K=150–300$ and $M=1024$. The red symbols show the position of the different avoided crossings.