Large quantum dot energy level shifts in anomalous photon-assisted tunneling

This study reveals that singlet-triplet splittings in Ge/SiGe double quantum dots exhibit a surprising, strong dependence on top gate voltages, leading to anomalous photon-assisted tunneling measurements that are successfully modeled by a linear gate-voltage relationship.

The Gist

Imagine a quantum computer as a tiny, ultra-precise orchestra. In this orchestra, the musicians are "holes" (missing electrons) trapped inside tiny cages called quantum dots. To make music (or in this case, perform calculations), the conductor needs to know exactly how much energy each musician has.

In the world of quantum dots, there are two main types of energy levels:

1. Spin: Which way the musician is facing (up or down).
2. Orbital: How big the cage is and how the musician moves inside it.

The paper focuses on a specific "duet" between two musicians in two neighboring cages (a Double Quantum Dot). The researchers are studying the energy gap between two specific states of this duet, called the Singlet-Triplet (ST) splitting. Think of this gap as the "distance" between two notes the duet can play. If this distance is just right, the conductor can easily switch between notes to perform a calculation.

The Old Belief vs. The New Discovery

The Old Belief:
Scientists used to think that if you tweaked the "volume knobs" (called plunger gates) to control the musicians, the size of the cages and the energy gap between the notes would stay perfectly steady. They assumed the gap was like a fixed piano key: no matter how you adjusted the volume, the key's pitch wouldn't change. This made the math for controlling quantum computers very simple.

The New Discovery:
The researchers found out this assumption was wrong. They discovered that these energy gaps are actually very sensitive to the volume knobs.

• The Analogy: Imagine you are tuning a guitar. You expect that turning the tuning peg (the gate voltage) only changes the tension of the string. But in this quantum world, turning the peg actually changes the shape of the guitar body itself, which drastically shifts the pitch of the note in a way no one expected.
• The Result: Small, tiny adjustments to the gate voltages caused huge shifts in the energy gap.

How They Found It: The "Microwave Flashlight"

To see this hidden behavior, the team used a technique called Photon-Assisted Tunneling (PAT).

• The Metaphor: Imagine the two quantum dots are two rooms separated by a wall. The musicians (holes) can't jump the wall unless they have enough energy. The researchers shine a "microwave flashlight" (microwaves) at the wall.
• The Process: If the energy gap between the two rooms matches the energy of the flashlight photons, the musician can suddenly jump across the wall.
• The Surprise: Usually, if you draw a map of where these jumps happen, you get straight lines. But in this experiment, the lines were curved. This curvature was the "smoking gun" that proved the energy gap was changing as they moved the volume knobs. It was like seeing a straight road suddenly bend, telling them the ground beneath it was shifting.

They also used a second method called pulsed-gate spectroscopy (like taking a quick snapshot of the energy levels) to confirm that the gaps were indeed changing linearly with the voltage.

Why This Matters (According to the Paper)

The paper states that this discovery is crucial for building hole spin qubits (the musicians) in Germanium/Silicon-germanium materials.

• The Problem: If you are trying to control a quantum computer, you need to know exactly where your energy levels are. If you think they are fixed, but they are actually sliding around based on your control knobs, your calculations will be wrong.
• The Solution: The researchers built a new mathematical model that accounts for this "sliding." They showed that if you treat the energy gap as something that changes linearly with the voltage, their model perfectly matches the experimental data.

Summary

In short, this paper reveals that in these tiny quantum cages, the "notes" the musicians play are not fixed. They wiggle and shift significantly when you try to control them. The team proved this by watching how the musicians jumped between cages under microwave light, and they created a new rulebook (model) to predict exactly how those notes will shift. This is essential for anyone trying to tune these quantum instruments to play the right music.

Technical Summary

1. Problem Statement

In semiconductor spin qubits, particularly hole spin qubits in Ge/SiGe heterostructures, the orbital energy splittings are critical parameters. These splittings dictate the singlet-triplet (ST) energy splitting, which determines qubit properties such as readout windows and manipulation frequencies.

• The Assumption: Conventional effective theories and previous experimental observations (particularly near triple points in charge stability diagrams) assume that orbital energy levels and ST splittings remain constant when small adjustments are made to plunger gate voltages (the primary control parameter for qubits).
• The Gap: While it is known that orbital splittings depend on lateral confinement, the sensitivity of these splittings to small gate voltage changes in the operational regime of double quantum dots (DQDs) has not been fully characterized. If ST splittings are not tunable or behave unpredictably, it complicates the design and control of hybrid qubits and ST qubits.

2. Methodology

The authors investigated a double quantum dot (DQD) device fabricated in a Ge/SiGe heterostructure using two complementary spectroscopic techniques to measure ST splittings under varying gate voltages:

• Device Architecture: A 2D quantum dot array device where a specific DQD is tuned using plunger gates (P3, P5) and barrier gates. Holes are manipulated via a charge sensor (W) and reservoirs (N).
• Photon-Assisted Tunneling (PAT):
  • Used to measure the energy dispersion near the $(2,1) \leftrightarrow (1,2)$ charge anticrossing.
  • Microwave photons drive tunneling when the energy difference ($\Delta E$) between charge states matches the photon energy ($hf$).
  • Anomaly Detection: The team looked for deviations from the standard linear PAT resonance lines, which would indicate a dependence of energy levels on gate voltages beyond simple detuning.
• Pulsed-Gate Spectroscopy:
  • Used to directly measure the ST splitting of individual dots away from the charge anticrossing (at larger detunings where PAT is ineffective).
  • A square-wave pulse is applied to a plunger gate to access excited triplet states. The energy gap between ground-state and excited-state loading lines provides the ST splitting value.
• Modeling:
  • A modified Hamiltonian model was developed where the ST splittings ($E_L$ and $E_R$) are not constants but linear functions of the plunger gate voltages ($V_{P3}, V_{P5}$).
  • The model includes crosstalk terms, acknowledging that a gate controlling one dot can influence the ST splitting of the adjacent dot.

3. Key Contributions

• Discovery of Anomalous PAT Resonances: The authors identified a previously unobserved phenomenon where PAT resonance lines in charge stability diagrams are curved rather than straight. This curvature indicates that the energy difference between charge states changes non-linearly with gate voltage due to shifting ST splittings.
• Quantification of Gate-Voltage Dependence: They demonstrated that small changes in plunger gate voltages (a few millivolts) result in large, linear shifts in ST splittings (tens of $\mu$eV).
• Unified Modeling Framework: They successfully combined PAT and pulsed-gate spectroscopy data into a single global fit model. This model treats ST splittings as linearly dependent on gate voltages, resolving the "anomalous" PAT data and providing a comprehensive description of the DQD energy landscape.
• Robustness Across Tunings: The phenomenon was observed across multiple device tunings with significantly different ratios of ST splittings between the two dots, confirming it is a general feature of the Ge/SiGe platform rather than an artifact of a specific tuning.

4. Key Results

• Magnitude of Shifts:
  • In one tuning, the left-dot ST splitting increased from 146 $\mu$eV to 206 $\mu$eV, and the right-dot from 81 $\mu$eV to 133 $\mu$eV as gate voltages were varied.
  • In another tuning, the left-dot ST splitting was tuned from 34 $\mu$eV to 68 $\mu$eV (very small values), while the right dot shifted from 316 $\mu$eV to 418 $\mu$eV.
• Anomalous PAT Features:
  • When excited state anticrossings are close to the ground state anticrossing, PAT resonance lines become curved.
  • The curvature direction and slope directly correlate with the gate-voltage dependence of the ST splitting. For example, if the right-dot ST splitting is small and voltage-dependent, the $(1,2)$ resonance line curves.
• Model Validation:
  • The linear dependence model showed excellent agreement with experimental data across all frequencies and device tunings.
  • The model successfully predicted the curvature of PAT lines and the linear trends observed in pulsed-gate spectroscopy.
• Crosstalk Effects: The model revealed that crosstalk (a gate affecting the other dot's ST splitting) is significant when ST splittings are small enough to be visible in PAT measurements.

5. Significance

• Qubit Control and Readout: The findings challenge the assumption that orbital parameters are static. For hole spin qubits (especially ST and hybrid qubits), this gate-voltage dependence means that qubit energy splittings can be actively tuned by adjusting plunger gates. This offers a new degree of freedom for optimizing qubit frequencies and readout windows.
• Platform Advantages: Unlike Si/SiGe electron qubits, which suffer from uncontrollable valley states that can vanish ST splittings, Ge/SiGe hole qubits naturally avoid valley states. This work confirms that Ge/SiGe offers large, tunable ST splittings, making it a superior platform for hybrid qubit implementations.
• Experimental Calibration: The study provides a necessary correction to effective theories used in qubit control. Future experiments must account for the linear shift of ST splittings with gate voltage to accurately predict qubit behavior, especially during fast gate operations or when sweeping parameters for initialization.
• Fundamental Physics: The results offer insight into how the local electrostatic environment in Ge/SiGe heterostructures influences orbital wavefunctions and hole confinement, suggesting strong sensitivity to gate-induced potential changes.

In conclusion, this paper establishes that ST splittings in Ge/SiGe double quantum dots are highly sensitive to plunger gate voltages, leading to anomalous PAT signatures. This tunability is a critical feature for the future manipulation and readout of hole spin qubits.