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

## TL;DR

This paper presents a theoretical framework for valley-resolved spectroscopy of a silicon triple quantum dot coupled to a microwave resonator, enabling extraction of valley splittings and phases through transmission and phase shift measurements.

## Contribution

It introduces a comprehensive theoretical approach combining capacitor, Hubbard, rate equation, and input-output models to analyze valley states in a silicon TQD system.

## Key findings

- Response signal reveals valley splittings and phases.
- Finite temperature or bias voltage populates excited valley states.
- The models accurately fit experimental charging energies.

## 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 $\mu\text{eV}$, 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.