⚛️ quantum physics

Direct measurement of the energy spectrum of a quantum dot qubit

This paper introduces delta-axis spectroscopy (DAXS), a Hamiltonian-agnostic technique that enables the direct measurement of the full energy spectrum of a Si/SiGe double quantum dot across a wide detuning range, allowing for the precise extraction of diagonal and off-diagonal couplings in a 15-level Hubbard-like Hamiltonian.

Original authors: J. Reily, Daniel J. King, Jonathan C. Marcks, M. A. Wolfe, Piotr Marciniec, E. S. Joseph, Tyler J. Kovach, Brighton X. Coe, Mark Friesen, Benjamin D. Woods, M. A. Eriksson

Published 2026-04-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: J. Reily, Daniel J. King, Jonathan C. Marcks, M. A. Wolfe, Piotr Marciniec, E. S. Joseph, Tyler J. Kovach, Brighton X. Coe, Mark Friesen, Benjamin D. Woods, M. A. Eriksson

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Mapping the Quantum Playground

Imagine you have a tiny, invisible playground made of two adjacent rooms (these are the quantum dots). Inside these rooms, electrons (tiny charged particles) are playing. Sometimes, an electron stays in the left room; sometimes it hops to the right; sometimes they share a room.

To build a quantum computer, we need to control these electrons perfectly. But to do that, we need a map. We need to know exactly how much energy it takes for an electron to be in a specific room, how easily it can jump between rooms, and how the "rules" of the playground change when we tweak the walls.

In physics, this map is called a Hamiltonian. It's a giant mathematical spreadsheet that lists every possible state the electrons can be in and how they interact.

The Problem: The Old Maps Were Incomplete

For a long time, scientists had ways to peek at this map, but they were like looking at a city through a keyhole.

Old Method (PGS/DAPS): Imagine you are trying to map a mountain range, but you can only stand at one specific spot and look straight ahead. You can see the peak right in front of you, but you can't see the valleys to the side or the peaks behind the fog. You get a "slice" of the data, but you miss the big picture.
The Limitation: These old methods could tell you about the ground floor (the lowest energy states), but they struggled to see the "attic" (excited states) or how the rooms connect when the walls move.

The New Solution: "Delta-Axis Spectroscopy" (DAXS)

The authors of this paper invented a new way to look at the playground called DAXS.

The Analogy: The Flashlight in the Fog
Imagine the two quantum dots are two rooms connected by a door.

The Old Way: You stand in the hallway and shout "Hello!" (a voltage pulse). You listen for an echo to see if the door is open. You only hear what happens right when you shout.
The DAXS Way: Instead of just shouting, you start wiggling the door back and forth while simultaneously sweeping a flashlight across the whole room.
- By wiggling the door (pulsing the voltage) in a very specific direction (called the "delta axis"), you force the electrons to reveal all their possible positions, not just the ones they are currently sitting in.
- As you sweep the flashlight (changing the voltage), you see a complete "movie" of the energy levels. You see the ground floor, the attic, and exactly how the electrons jump between rooms as the door opens and closes.

How It Works (The "Magic" Trick)

The Pulse: They apply a square-wave voltage (like a rhythmic tapping) to the gates controlling the dots.
The Sweep: They slowly change the "detuning" (which basically means tilting the energy landscape so one room is lower than the other).
The Result: Instead of seeing just one line, they see a complex web of lines. Some lines are straight (easy paths), and some lines curve and avoid each other (these are called anticrossings, where two energy states get close but repel each other like magnets).

This web of lines is the energy spectrum. It's a complete map of the quantum playground.

What Did They Find?

Using this new map, the scientists were able to:

Measure the "Tunnel Coupling": This is how easily an electron can jump from one dot to the other. Think of it as the width of the door between the rooms. They found that the door gets wider or narrower depending on how they tune the system.
See the "Attic": They could see excited states (higher energy levels) that previous methods missed. This is crucial because if you accidentally hit these high-energy states while doing calculations, your quantum computer might make a mistake (leakage).
Filter Out the Noise: Sometimes, the "rooms" (dots) are connected to "hallways" (reservoirs) that have their own weird echoes. The scientists developed a clever trick to tell the difference between a signal coming from the quantum dot and a signal coming from the hallway, ensuring their map is accurate.

Why Does This Matter?

Think of building a quantum computer like building a high-speed train.

Without DAXS: You are trying to lay tracks in the dark, guessing where the curves are. You might hit a bump or derail.
With DAXS: You have a perfect 3D blueprint of the terrain. You know exactly where the curves are, how steep the hills are, and where the tunnels are.

This technique allows scientists to tune their quantum dots with much higher precision. It helps them avoid "hotspots" (places where the system gets confused) and ensures that the quantum bits (qubits) stay stable and reliable.

The Bottom Line

This paper introduces a new "flashlight" technique (DAXS) that lets scientists see the entire energy landscape of a quantum dot system, not just a tiny slice of it. By mapping out every possible state and how they connect, they can build better, more reliable quantum computers. It's a move from guessing the shape of the mountain to having a full, detailed topographical map.

1. Problem Statement

Quantum dot (QD) spin qubits, particularly in Si/SiGe systems, are a leading platform for scalable quantum computing. To coherently manipulate these qubits, researchers must precisely map the relationship between applied gate voltages and the system's Hamiltonian parameters (e.g., detuning, tunnel couplings, and energy splittings).

Existing techniques for characterizing these parameters have significant limitations:

Partial Information: Methods like Pulsed-Gate Spectroscopy (PGS) and Detuning Axis Pulsed Spectroscopy (DAPS) often reveal only specific energy splittings or ground-state couplings, failing to capture the full Hamiltonian matrix.
Complexity: Techniques like Landau-Zener-Stückelberg interferometry require complex pulsing schemes, especially for excited states.
Limited Scope: Many methods rely on resonators, elevated temperatures, or only provide data at fixed detuning points, missing the full energy dispersion landscape.
Ambiguity: Distinguishing between true quantum dot eigenstates and artifacts caused by reservoir density-of-states (DOS) modulations (quasi-1D resonances) is difficult.

The authors aim to develop a technique that provides a Hamiltonian-agnostic, comprehensive map of the double quantum dot (DQD) energy spectrum across a wide energy range and detuning values, allowing for the extraction of both diagonal and off-diagonal Hamiltonian elements.

2. Methodology: Delta-Axis Spectroscopy (DAXS)

The authors introduce Delta-Axis Spectroscopy (DAXS), a technique that utilizes simple square-wave voltage pulses to map the energy spectrum.

Device Setup: The experiment uses a Si/SiGe double quantum dot device (Intel Tunnel Falls design) with plunger gates ( $P_L, P_R$ ), barrier gates ( $B_L, B_C, B_R$ ), and accumulation gates ( $A_L, A_R$ ). The system is tuned to the $(1,3)-(0,4)$ electron occupation regime.
Measurement Axes:
- Detuning ( $\varepsilon$ ): Defined as $\mu_2 - \mu_1$ (difference in chemical potentials).
- Delta ( $\delta$ ): Defined as $(\mu_1 + \mu_2)/2$ (average chemical potential).
Pulsing Scheme: Unlike PGS, which sweeps pulse amplitude along a single gate, DAXS applies simultaneous square-wave pulses to both plunger gates ( $P_L$ and $P_R$ ) along the $\delta$ axis (parallel to the polarization line).
Data Acquisition:
- The charge sensor current is measured using lock-in amplifiers while the system is pulsed.
- To distinguish DQD states from reservoir resonances, the authors perform measurements at multiple reservoir gate voltages ( $A_R$ ) and average the results. Reservoir resonances shift with $A_R$ voltage, while DQD eigenstates remain fixed, allowing them to be averaged out.
Fitting Procedure:
1. Peak Extraction: Visible energy states in the DAXS plot are identified. A Savitzky-Golay filter reduces noise, and Lorentzian functions are fitted to the peaks in each column of data to determine the center energy ( $\delta$ ) for a given detuning ( $\varepsilon$ ).
2. Hamiltonian Fitting: The extracted $(\varepsilon, \delta)$ points are fitted to the eigenvalues of a 15-level Hubbard-like Hamiltonian. This matrix includes singlet and triplet sub-matrices for various orbital states ( $L_n, R_n$ ).
3. Parameter Extraction: The fitting process extracts diagonal terms (energy levels) and off-diagonal terms (tunnel couplings $t_{ij}$ ) simultaneously.

3. Key Contributions

Novel Spectroscopy Technique: DAXS is the first method demonstrated to map the full energy dispersion of a DQD as a function of detuning using simple low-frequency pulsing, revealing hybridized states and anticrossings that are invisible to standard PGS.
Full Hamiltonian Reconstruction: The technique successfully extracts both diagonal and off-diagonal elements of a 15-level Hamiltonian, providing a complete picture of the system's energy landscape.
Artifact Suppression: The paper provides a robust method for distinguishing true DQD eigenstates from quasi-1D reservoir resonances by analyzing the dependence of spectral lines on reservoir gate voltage and magnetic field.
Validation: The method is validated by comparing DAXS results with PGS data (showing good agreement where they overlap) and by demonstrating expected physical behaviors (e.g., Zeeman splitting in magnetic fields).

4. Key Results

Energy Spectrum Mapping: The DAXS plot clearly shows the theoretical energy dispersion, including the characteristic "V-shape" of ground states and the curvature of anticrossings for excited states.
Tunnel Coupling Extraction:
- The authors extracted tunnel couplings between ground and excited states (e.g., $t_{11}, t_{21}, t_{41}$ ).
- General Trend: Higher-energy states generally exhibit larger tunnel couplings due to reduced localization, though exceptions exist (e.g., $t_{41}$ was found to be small).
- Gate Dependence: Tunnel couplings generally increase monotonically with the center barrier gate voltage ( $B_C$ ), consistent with physical expectations.
State Identification:
- Magnetospectroscopy: By applying an in-plane magnetic field, the authors confirmed the spin nature of the states. The ground state showed no splitting (singlet), while the first excited state split into three lines (triplet), and higher states behaved consistently with singlet/triplet assignments.
- Reservoir vs. Dot: Lines that shifted with $A_R$ voltage were identified as reservoir resonances, while vertical lines were confirmed as DQD eigenstates.
Uncertainty Analysis:
- Repeated scans showed variability in extracted couplings ranging from 1.6 to 3.9 GHz (1-2 standard deviations).
- Some couplings (e.g., $t_{31}, t_{41}$ ) showed high variability due to obscured anticrossings or low signal-to-noise, highlighting the limits of the technique for very weak couplings.
- The sign of tunnel couplings was tested; while absolute signs are ambiguous without phase information, the magnitude differences between sign combinations were found to be small (<5% for most, <20% for others), validating the use of positive constraints in the fit.

5. Significance and Outlook

Scalability: DAXS is a "simple yet powerful" technique that does not require complex pulsing sequences or high-frequency resonators, making it highly suitable for scaling to large quantum dot arrays.
Qubit Control: By providing a complete map of the energy spectrum and tunnel couplings, DAXS enables more precise calibration of qubit operations, helping to avoid leakage to excited states and identifying "hotspots" (anticrossings) that could cause errors.
Future Applications: The authors suggest that DAXS could be adapted to use neighboring quantum dots as charge reservoirs instead of macroscopic leads, further integrating the technique into large-scale quantum processor architectures.
Physics Insight: The ability to resolve the full 15-level spectrum allows for deeper investigation into many-body physics, orbital state structures, and the interplay between valley and spin degrees of freedom in Si/SiGe systems.

In conclusion, this work establishes DAXS as a superior diagnostic tool for semiconductor spin qubits, offering a comprehensive, experimentally accessible route to characterizing the full Hamiltonian of double quantum dot systems.