Observation of Floquet States in a Strongly Driven Artificial Atom

We present experiments on the driven dynamics of a two-level superconducting artificial atom. The driving strength reaches 4.78 GHz, significantly exceeding the transition frequency of 2.288 GHz. The observed dynamics is described in terms of quasienergies and quasienergy states, in agreement with Floquet theory. In addition, we observe the role of pulse shaping in the dynamics, as determined by nonadiabatic transitions between Floquet states, and we implement subnanosecond single-qubit operations. These results pave the way to quantum control using strong driving with applications in quantum technologies.

Figure 1

(a) Schematic representation of the experimental setup. The qubit, formed by a superconducting loop interrupted by Josephson junctions (cross symbols), is coupled to a superconducting coplanar waveguide resonator. Readout is based on the transmission of a microwave pulse from the resonator input (left) to its output port (right). A waveguide (bottom) is used to couple microwave control pulses to the qubit. (b) Representation of qubit control pulses, with rise and fall times $t_r$ and $t_f$, respectively, and maximum-amplitude duration $t_p$. The thick line indicates the pulse envelope, which reaches a maximum $A_m$. During rise and fall, the envelope is shaped as $(A_m/2)[1-\cos (\pi t/t_r)]$ and $(A_m/2)\{1+\cos [\pi (t-t_p-t_r)/t_f]\}$. (c) Qubit transition frequency ${\omega }_{01}$ from spectroscopy measurements versus the static magnetic flux ${\mathrm{\Phi }}_s$. The continuous line is a fit of the transition frequency, yielding the parameters $\mathrm{\Delta }=2\pi \times 2.288\mathrm{GHz}$ and $I_p=690\mathrm{nA}$.

Figure 2

Coherent oscillations versus driving amplitude for (a)–(d) resonant driving ($\omega =\mathrm{\Delta }$) and (e)–(h) off-resonance driving ($\omega =0.6\mathrm{\Delta }$). (a),(e) Qubit excited state probability $P_1$ versus control pulse duration $t_p$ for (a) $A_m=0.10$ and 1.00 GHz and (e) $A_m=0.30$ and 1.44 GHz. (b),(f) Discrete Fourier transforms of the oscillations in (a) and (e), respectively. (c),(g) Color plot of the Fourier transform of population oscillations versus frequency and driving pulse amplitude. (d),(h) Positions of peaks in the Fourier transform of coherent oscillations versus driving amplitude extracted from the data in (c) and (g) (dots). The lines are plots of $n\omega$, $n\omega -\mathrm{\Delta }\epsilon$, and $n\omega +\mathrm{\Delta }\epsilon$, respectively, with the quasienergy difference $\mathrm{\Delta }\epsilon$ determined numerically and $n$ an even integer. For the resonant driving case (d), we show in addition corresponding curves (dashed) based on the analytical approximation for $\mathrm{\Delta }\epsilon$ discussed in the text (see also Ref. [31]).

Figure 3

Measurements and simulations of the evolution of the Bloch vector components, given by the average values of the Pauli ${\sigma }_{\alpha }$ ($\alpha =x,y,x$) operators, after a pulse with zero rise and fall time, versus the length of the pulse for (a) $A_m=2\pi \times 0.10\mathrm{GHz}$ and (b) $A_m=2\pi \times 0.46\mathrm{GHz}$. The experimental results are in excellent agreement with the results of numerical simulations.

Figure 4

(a),(b) Qubit excited state probability $P_1$ versus pulse duration $t_p$ for various rise (fall) times $t_r$ ($t_f$) and equal maximum amplitude $A_m=2\pi \times 1.33\mathrm{GHz}$. Panels (a) and (b) show data with symmetric or asymmetric rise and fall. (c) The measured (dots and squares) and simulated (continuous and dashed lines) fast oscillation amplitudes, at frequencies $2\omega -\mathrm{\Delta }\epsilon$ and $2\omega +\mathrm{\Delta }\epsilon$, respectively. (d) Adiabatic and (e) nonadiabatic evolution in the Floquet picture. The state is represented on the Bloch sphere, with a basis chosen such that the instantaneous quasienergy states $|u_{0,1}⟩$ are in the equatorial plane. The state vector evolution (thin arrows) is a rotation around a fictitious field (thick arrows). The initial qubit state is $(|u_0⟩+|u_1⟩)/\sqrt{2}$. In the adiabatic case (d), the state evolution is described by a phase $-{\int }_0^t\mathrm{\Delta }\epsilon (t)dt$ applied to $|u_1⟩$, which is equivalent to rotation around the fictitious field $\overrightarrow{\mathrm{\Delta }}\epsilon (t)$ aligned with $|u_0⟩$. In this picture, the evolution of the qubit is a simple rotation, although in the energy eigenbasis the qubit undergoes complex dynamics. In the nonadiabatic case (e), transitions between Floquet states arise during pulse turn-on and turn-off, characterized by the unitary transformations $U_{F,\mathrm{rise}}$ and $U_{F,\mathrm{fall}}$, respectively, which correspond, in general, to rotations around axes that are not parallel to $\overrightarrow{\mathrm{\Delta }}\epsilon (t)$. The asterisk (${}^{*}$) indicates that the actual rise or fall time is around 20 ps, determined by the analog bandwidth of the AWG.