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Submitted - 06 November 2022
Reviewed - 02 February 2023
Accepted - 20 March 2023

Sunday, March 26, 2023 at 6:30:00 AM UTC

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Quantum Dot Source-Drain Transport Response at Microwave Frequencies

Harald Havir,1 Subhomoy Haldar,1 Waqar Khan,1 Sebastian Lehmann,1 Kimberly A. Dick, 1, 2 Claes Thelander,1 Peter Samuelsson,3 Ville F. Maisi1
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quantum, quantum technologies, centre for quantum technologies, quantum computer (literature subject)

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Abstract
Quantum dots are frequently used as charge sensitive devices in low temperature experiments to probe electric charge in mesoscopic conductors where the current running through the quantum dot is modulated by the nearby charge environment. Recent experiments have been operating these detectors using reflectometry measurements up to GHz frequencies rather than probing the low frequency current through the dot. In this work, we use an on-chip coplanar waveguide resonator to measure the source-drain transport response of two quantum dots at a frequency of 6 GHz, further increasing the bandwidth limit for charge detection. Similar to the low frequency domain, the response is here predominantly dissipative. For large tunnel coupling, the response is still governed by the low frequency conductance, in line with Landauer-Büttiker theory. For smaller couplings, our devices showcase two regimes where the high frequency response deviates from the low frequency limit and Landauer-Büttiker theory: When the photon energy exceeds the quantum dot resonance linewidth, degeneracy dependent plateaus emerge. These are reproduced by sequential tunneling calculations. In the other case with large asymmetry in the tunnel couplings, the high frequency response is two orders of magnitude larger than the low frequency conductance G, favoring the high frequency readout.

Inroduction
The ability to detect single electrons in the solid state is useful for a variety of applications, including spin qubit readout [1–4], electrical current and capacitance standards [5, 6], studying cooper pair breaking [7–9], single-shot photodetection [10–13], and nanothermodynamics and fluctuations [14– 19]. While many methods exist to detect charge, one of the main ways are by utilizing quantum dots (QD). These systems make excellent charge detectors due to their high sensitivity and well-established transport theory [20, 21], allowing detectors to be made predictable and with a well-understood operation principle. Originally, measurements were performed at DC, relying on a difference in current for the readout resulting in a bandwidth up to some kHz [6, 22]. In the last two decades, the readout methods have moved towards measuring the reflected power in a high-frequency tank circuit with resonant frequency in the 100 MHz - 1 GHz range. This results in bandwidths in the MHz range allowing for µs time resolution [23–25]. The response of the system in these studies is still governed by the low frequency response of the system, i.e. the admittance Y(ω) is equal to the DC conductance G of the system. In this article, we increase the QD sensor frequency to the 4 - 8 GHz frequency range where the cavity photon energy h¯ω is greater than the thermal energy kT [26]. This opens up the avenue to increase the bandwidth correspondingly by an order of magnitude, yielding possibly a time resolution sufficient to probe the electron position in DQD systems within the recently achieved coherence times [27, 28]. The pioneering works have considered the dispersive response of the QD at these frequencies motivated mostly by quantum capacitance effects [29]. In this article, we focus on the dissipative part that yields a stronger response, making it useful for charge readout [26]. We present experimental results for two devices and show that for both of them at sufficiently large tunnel couplings that we are lifetime broadened, Γ > kT, the low frequency result of Y(ω) = G still applies. However, when the device is tuned to the thermally broadened limit where the tunnel couplings Γ < kT, the measured admittance is qualitatively different from the DC conductance, displaying a linewidth of 2h¯ω in the QD level tuning and a factor two difference in admittance depending on the direction of the level shift of the quantum dot relative to the leads ε, attributed to spin degeneracy. These results are well captured by sequential tunneling theory, directly evaluating the admittance for a QD subjected to a time-periodic drive [30], or using P(E) theory in which the admittance is inferred from the absorption in the cavity [31, 32]. Lastly, we show in the other device which exhibits asymmetric tunnel couplings where the DC transport is suppressed while remaining lifetime broadened, the AC response in this device remains large, in line with Ref. 26, indicating a potentially useful consequence of probing QD devices at high frequencies. This response falls in a regime where neither non-interacting scattering theory nor sequential tunneling models are applicable.

Conclusion
In summary, we studied the high frequency source-drain response of a quantum dot. We showed experimentally that the low frequency result of Y(ω) = G holds for quantum dots tuned to sufficiently large tunnel couplings in line with the slow-drive limit. However, when the tunnel couplings are tuned to be smaller than the photon energy, the measured linewidth of the admittance Y(ω) is set by the photon energy. This response is well-described by sequential tunneling theory. Additionally, the low-frequency limit does not hold when the drive amplitude is made sufficiently large or with large asymmetry in tunnel couplings of the junctions. For the highly asymmetric case, it is also shown that the admittance Y(ω) can be orders of magnitude larger than the conductance G, indicating a potential benefit of measuring at high frequencies, as the readout strength remains large even for weakly conducting dots.