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arXiv:1907.08092 · Journal of Physics Condensed Matter

Theory of valley-resolved spectroscopy of a Si triple quantum dot coupled to a microwave resonator

Maximilian Rimbach-Russ, Csaba G. Péterfalvi, Guido Burkard

Published Dec 12, 2019• 11 citations
arXivHTMLPDF

Abstract

We theoretically study a silicon triple quantum dot (TQD) system coupled to a superconducting microwave resonator. The response signal of an injected probe signal can be used to extract information about the level structure by measuring the transmission and phase shift of the output field. This information can further be used to gain knowledge about the valley splittings and valley phases in the individual dots. Since relevant valley states are typically split by several [Formula: see text], a finite temperature or an applied external bias voltage is required to populate energetically excited states. The theoretical methods in this paper include a capacitor model to fit experimental charging energies, an extended Hubbard model to describe the tunneling dynamics, a rate equation model to find the occupation probabilities, and an input-output model to determine the response signal of the resonator.

DOI
10.1088/1361-648x/ab613f

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cond-mat.mes-hallCross-listedarXiv:2604.00232•Apr 2, 2026
We theoretically study a silicon triple quantum dot (TQD) system coupled to a superconducting microwave resonator. The response signal of an injected probe signal can be used to extract information about the level structure by measuring the transmission and phase shift of the output field. This information can further be used to gain knowledge about the valley splittings and valley phases in the individual dots. Since relevant valley states are typically split by several [Formula: see text], a finite temperature or an applied external bias voltage is required to populate energetically excited states. The theoretical methods in this paper include a capacitor model to fit experimental charging energies, an extended Hubbard model to describe the tunneling dynamics, a rate equation model to find the occupation probabilities, and an input-output model to determine the response signal of the resonator.
The band structure of bulk silicon has a sixfold valley degeneracy. Strain in the Si/SiGe quantum well system partially lifts the valley degeneracy, but the materials factors that set the splitting of the two lowest lying valleys are still under intense investigation. Using cavity input-output theory, we propose a method for accurately determining the valley splitting in Si/SiGe double quantum dots embedded in a superconducting microwave resonator. We show that low lying valley states in the double quantum dot energy level spectrum lead to readily observable features in the cavity transmission. These features generate a ``fingerprint'' of the microscopic energy level structure of a semiconductor double quantum dot, providing useful information on valley splittings and intervalley coupling rates.
We study an accumulation mode Si/SiGe double quantum dot (DQD) containing a single electron that is dipole coupled to microwave photons in a superconducting cavity. Measurements of the cavity transmission reveal dispersive features due to the DQD valley states in Si. The occupation of the valley states can be increased by raising the temperature or applying a finite source-drain bias across the DQD, resulting in an increased signal. Using the cavity input-output theory and a four-level model of the DQD, it is possible to efficiently extract valley splittings and the inter- and intravalley tunnel couplings.
Both quantum transport measurements in the Pauli blockade regime and microwave cavity transmission measurements are important tools for spin-qubit readout and characterization. Based on a generalized input-output theory we derive a theoretical framework to investigate how a double quantum dot (DQD) in a transport setup interacts with a coupled microwave resonator while the current through the DQD is rectified by Pauli blockade. We show that the output field of the resonator can be used to infer the leakage current and thus obtain insight into the blockade mechanisms. In the case of a silicon DQD, we show how the valley quasidegeneracy can impose limitations on this scheme. We also demonstrate that a large number of unknown DQD parameters including (but not limited to) the valley splitting can be estimated from the resonator response simultaneous to a transport experiment, providing more detailed knowledge about the microscopic environment of the DQD. Furthermore, we describe and quantify a back action of the resonator photons on the steady-state leakage current.
As an application in circuit quantum electrodynamics coupled systems, superconducting resonators play an important role in high-sensitivity measurements in a superconducting-semiconductor hybrid architecture. Taking advantage of a high-impedance $\mathrm{Nb}\text{\ensuremath{-}}\mathrm{Ti}\text{\ensuremath{-}}\mathrm{N}$ resonator, we perform excited-state spectroscopy on a GaAs\mathrm{Ga}\mathrm{As}GaAs double quantum dot (DQD) by applying voltage pulses to one gate electrode. The pulse train modulates the DQD energy detuning and gives rise to charge state transitions at zero detuning. Benefiting from the outstanding sensitivity of the resonator, we distinguish different spin-state transitions in the energy spectrum according to the Pauli exclusion principle. Furthermore, we experimentally study how the interdot tunneling rate modifies the resonator response. The experimental results are consistent with the simulated spectra based on our model.
Abstract Scaling up spin qubits in silicon-based quantum dots is one of the pivotal challenges in achieving large-scale semiconductor quantum computation. To satisfy the connectivity requirements and reduce the lithographic complexity, utilizing the qubit array structure and the circuit quantum electrodynamics (cQED) architecture together is expected to be a feasible scaling scheme. A triple-quantum dot (TQD) coupled with a superconducting resonator is regarded as a basic cell to demonstrate this extension scheme. In this article, we investigate a system consisting of a silicon TQD and a high-impedance TiN coplanar waveguide (CPW) resonator. The TQD can couple to the resonator via the right double-quantum dot (RDQD), which reaches the strong coupling regime with a charge–photon coupling strength of g 0/(2π) = 175 MHz. Moreover, we illustrate the high tunability of the TQD through the characterization of stability diagrams, quadruple points (QPs), and the quantum cellular automata (QCA) process. Our results contribute to fostering the exploration of silicon-based qubit integration.
The orbital, spin and valley degrees of freedom in silicon quantum dots support many modes of spin qubit operation. However, it is generally challenging to obtain information about the energy level spectrum over large ranges of parameter space. We demonstrate a form of spectroscopy that is capable of mapping the energy level structure of a double quantum dot as a function of level detuning, interdot tunnel coupling, and magnetic field. In the one electron regime, we directly observe the transition from the atom like energy levels of isolated quantum dots to molecular like bonding and anti bonding states with increasing interdot tunnel coupling. We also resolve the Zeeman splitting of ground and excited valley states in a magnetic field. In the two electron regime, we gain access to the detuning dependent singlet triplet splitting. Our work may be extended to a broader class of systems, such as strong spin-orbit materials or proximitized quantum dots, allowing the direct extraction of various energy gaps.