Landau-Zener-Stückelberg Interference in a Multimode Electromechanical System in the Quantum Regime

The studies of mechanical resonators in the quantum regime not only provide insight into the fundamental nature of quantum mechanics of massive objects, but also introduce promising platforms for novel hybrid quantum technologies. Here we demonstrate a configurable interaction between a superconducting qubit and many acoustic modes in the quantum regime. Specifically, we show how consecutive Landau-Zener-Stückelberg (LZS) tunneling type of transitions, which take place when a system is tuned through an avoided crossing of the coupled energy levels, interfere in a multimode system. The work progresses experimental LZS interference to cover a new class of systems where the coupled levels are those of a quantum two-level system interacting with a multitude of mechanical oscillators. The work opens up applications in controlling multiple acoustic modes via parametric modulation.

Figure 1

Photon-assisted Landau-Zener-Stückelberg interference in a multimode qubit-oscillator system. The solid lines represent eigenvalues from Eq. (3). We restrict for simplicity to the lowest excitation manifold; for example, $|g,1^{(3)}⟩$ means that the qubit is in the ground state, and the harmonic mode number 3 has one photon, and the rest of the oscillators are in the ground state. The arrow sketches a slow modulation of the bare qubit energy splitting represented by the blue and red lines. The dashed lines are the energies of three harmonic modes.

Figure 2

(a) Schematic cross section of the high-overtone bulk acoustic modes that are located inside the massive substrate. (b) Photograph of the device shows a transmon qubit that has an irregular pentagon shape. (c) Simulation of the mode displacement profile of one overtone acoustic mode. (d) Circuit diagram of the hybrid system. The piezoelectric GaN is effectively sandwiched between the capacitor plates of the transmon. (e) Two-tone spectroscopy showing the vacuum Rabi splitting in the qubit on resonance with a mechanical mode number $i=319$ at ${\omega }_0/2\pi \simeq 5.554\mathrm{GHz}$ and flux $V_{\mathrm{\Phi }}=0.37\mathrm{V}$. (f) Spectroscopy as a function of flux bias, and a sketch of the slow bias modulation.

Figure 3

LZS dynamics in the multimode electromechanical system. (a) Experimental data depicting the qubit population when the slow modulation amplitude is varied. (b) Simulation of the qubit population without presence of the acoustic modes in an otherwise similar situation. In both (a) and (b), the dashed black (black-white) lines display the respective LZS resonance conditions [Eq. (1a), Eq. (1b)]. The modulation frequency is ${\omega }_{\mathrm{rf}}/2\pi =60\mathrm{MHz}$.

Figure 4

Resonances in the rotating frame. (a) Excited state probability of the qubit under low-frequency modulation with ${\omega }_{\mathrm{rf}}/2\pi =139$; (b) with ${\omega }_{\mathrm{rf}}/2\pi =145\mathrm{MHz}$. In both (a),(b), the qubit frequency ${\omega }_0/2\pi \simeq 5.484\mathrm{GHz}$, flux $V_{\mathrm{\Phi }}=0.343\mathrm{V}$, $A/2\pi =210\mathrm{MHz}$, and $\mathrm{\Omega }/2\pi =3\mathrm{MHz}$. The black solid lines are numerical simulation with $\gamma /2\pi =8$, $g_m/2\pi =5.5\mathrm{MHz}$. The vertical axis scaling from the measured phase into $p_e$ is used as an adjustable parameter. We can infer that in (b), the sideband acoustic modes exhibit entanglement characterized by logarithmic negativity on the order 0.07.

Figure 5

Multiphoton transitions in the multimode system. (a) The measured phase shift shows how the qubit and acoustic modes experience resolved sidebands that are detuned from the original resonance by ${\omega }_{\mathrm{rf}}$. (b) Numerical simulation showing the qubit population. The parameters in both (a),(b) are as in Fig. 4 except $A/2\pi =110\mathrm{MHz}$. The qubit sidebands are marked with $n$, and the mechanical sidebands as an example for ${\omega }_m^{(307)}/2\pi \simeq 5.345\mathrm{GHz}$ by $k$.