Floquet Spectroscopy of a Strongly Driven Quantum Dot Charge Qubit with a Microwave Resonator

J. V. Koski, A. J. Landig, A. Pályi, P. Scarlino, C. Reichl, W. Wegscheider, G. Burkard, A. Wallraff, K. Ensslin, and T. Ihn

Phys. Rev. Lett. 121, 043603 – Published 23 July 2018

Abstract

We experimentally investigate a strongly driven GaAs double quantum dot charge qubit weakly coupled to a superconducting microwave resonator. The Floquet states emerging from strong driving are probed by tracing the qubit-resonator resonance condition. In this way, we probe the resonance of a qubit that is driven in an adiabatic, a nonadiabatic, or an intermediate rate, showing distinct quantum features of multiphoton processes and a fringe pattern similar to Landau-Zener-Stückelberg interference. Our resonant detection scheme enables the investigation of novel features when the drive frequency is comparable to the resonator frequency. Models based on the adiabatic approximation, rotating wave approximation, and Floquet theory explain our experimental observations.

Received 8 February 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.043603

Physics Subject Headings (PhySH)

Floquet systems Semiconductor quantum optics

Double quantum dots

Microwave techniques

Condensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

J. V. Koski¹, A. J. Landig¹, A. Pályi^2,3, P. Scarlino¹, C. Reichl¹, W. Wegscheider¹, G. Burkard⁴, A. Wallraff¹, K. Ensslin¹, and T. Ihn¹

¹Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland
²Department of Physics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary
³MTA-BME Exotic Quantum Phases “Momentum” Research Group, Budapest University of Technology and Economics, H-1111 Budapest, Hungary
⁴Department of Physics, University of Konstanz, D-78457 Konstanz, Germany

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Issue

Vol. 121, Iss. 4 — 27 July 2018

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Images

Figure 1
(a) A scanning electron micrograph of the device. Two quantum dots are defined in the positions indicated by the orange dots. A superconducting resonator is connected to the leftmost gate, and the qubit is driven by applying a continuous tone to the rightmost gate. Unused gate electrodes are greyed out. (b) Stability diagram of the double quantum dot. The black rectangle marks the operation regime. (c) Charge qubit (solid black line) and resonator photon energy (solid blue line) as a function of $δ_{0}$ . (d) Resonator transmission $T$ as a function of $V_{L}$ and $V_{R}$ . The $δ_{0}$ axis is depicted with an arrow.
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Figure 2
(a) Effective energy of an adiabatically driven qubit as a function of $δ_{0}$ with drive amplitudes $A_{d} / h = 0 GHz$ , 5 GHz, and $\sim 7 GHz$ in order of increasing energy (from black to red). (b) Illustration of the expected resonances as a function of $δ_{0}$ and $A_{d}$ for $ν_{d} = 1 GHz$ and $Δ / h = 6.8 GHz$ . The resonance positions satisfying $ϵ_{q, ad} = h ν_{r} + N h ν_{d}$ are shown as blue lines. The stars in panels (a) and (b) mark the corresponding resonance positions. The lines with a unit slope emerging from the zero- $A_{d}$ resonances approximate the regions (marked in grey) in which resonances are not visible since $A_{d} < | N | h ν_{d}$ . (c)–(f) Transmission as a function of $δ_{0}$ and $A_{d}$ for low $ν_{d}$ . The $N = - 1$ resonance is indicated with a dashed ellipse in panel (f). Yellow dots show the theoretically predicted resonance positions for the $δ_{0} > 0$ half, denoting where one of the resonance conditions in Eq. (2) is fulfilled. The radius of each yellow dot is proportional to the corresponding transition strength $M$ in Eq. (3). The predicted resonances are symmetric in $δ_{0}$ .
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Figure 3
(a)–(d) Transmission for increasing drive frequency. Yellow dots show the resonance conditions in Eq. (2), and their radius proportional to the corresponding transition strength $M$ in Eq. (3). The black lines show the RWA resonance conditions by Eq. (5), and orange lines [in panels (a) and (b)] show the adiabatic resonance conditions $ϵ_{q, ad} = h ν_{r}$ with Eq. (1). (e) The energy of the states $| g, 0 ⟩$ , $| g, 1 ⟩$ , and $| e, 0 ⟩$ by RWA as a function of drive amplitude for parameters in (c). Black solid lines correspond to $δ_{0} / h = 0 GHz$ with $ϵ_{q, 0} / h = Δ / h = 5.5 GHz$ , and black dashed lines correspond to $δ_{0} / h = 4 GHz$ with $ϵ_{q, 0} / h = (\sqrt{Δ^{2} + δ_{0}^{2}}) / h \approx 6.8 GHz$ . The splitting $Δ ϵ = \sqrt{{(A_{d} \sin (θ) / 2)}^{2} + {(ϵ_{q, 0} - h ν_{d})}^{2}}$ between $| g, 1 ⟩$ and $| e, 0 ⟩$ is determined by Eq. (4). The resonance condition for $| g, 0 ⟩$ and $| e, 0 ⟩$ with $h ν_{r}$ (dashed blue line) is marked by stars. (f) Corresponding plot for parameters in (d) and $δ_{0} / h = 6 GHz$ , $ϵ_{q, 0} / h \approx 10.7 GHz$ .
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Figure 4
Transmission with (a) near-resonant driving $ν_{d} ≃ ν_{r}$ and (b) near-half-resonant driving $ν_{d} ≃ ν_{r} / 2$ as a function of detuning and drive amplitude. Yellow dots show the resonance conditions in Eq. (2), and their radius is proportional to the transition strength in Eq. (3). The black lines show the RWA resonance conditions by Eq. (5). (c) Energy levels for parameters of panel (a) with $δ_{0} / h = 4 GHz$ and $ϵ_{q, 0} / h \approx 7.5 GHz$ . (d) Energy levels for parameters of panel (b) with $δ_{0} / h = 4 GHz$ and $ϵ_{q, 0} / h \approx 7.5 GHz$ . The RWA results are shown as dashed lines. Second order effects [32] form an avoided crossing $Δ ε_{2} \propto A_{d}^{2} \cos (θ) \sin (θ)$ between the RWA states.
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