🔬 mesoscale physics

Valley transitions during spin shuttling in Si/SiGe heterostructures

This paper theoretically demonstrates that coherent spin shuttling in Si/SiGe heterostructures can be achieved by operating at specific velocities (tens of m/s) that preserve quantum information despite valley excitations, and proposes a Partial Landau-Zener strategy to significantly enhance shuttling fidelity compared to standard models.

Original authors: Jonas R F Lima, Guido Burkard

Published 2026-10-08

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Jonas R F Lima, Guido Burkard

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

In the quest to build a quantum computer, scientists are turning to silicon, the same material that powers the smartphones in our pockets. Within tiny, engineered traps called quantum dots, individual electrons can be coaxed into acting as bits of information, or qubits. Because silicon is so pure and its natural magnetic interactions are so weak, these electron spins can hold their quantum state for a remarkably long time, making them excellent candidates for storing data. However, for a quantum computer to be useful, these qubits must be able to talk to one another over long distances. The challenge is that the forces that connect them are very short-range, so researchers have developed a method to physically move, or "shuttle," these electrons across the chip while keeping their delicate information intact.

The obstacle to this movement is a hidden feature of silicon known as the "valley." In the atomic structure of silicon, electrons can settle into one of several nearly identical energy states, much like a ball resting in one of several shallow dips on a bumpy surface. As an electron is moved across the chip, it encounters a landscape where the depth of these dips changes randomly due to imperfections in the material. If the electron accidentally jumps from one dip to another during the journey, it loses its connection to the information it was carrying, causing the quantum data to degrade. For years, the prevailing wisdom suggested that to prevent these jumps, one must move the electron very slowly, carefully navigating the bumps, or try to smooth out the terrain itself.

A team of researchers at the University of Konstanz has now challenged this long-held assumption. By simulating the movement of an electron through a silicon device, they discovered that the relationship between speed and stability is far more complex than a simple trade-off. Their work reveals that the electron's journey is not just about avoiding the bumps, but about how the electron's internal state rotates as it moves. They found that there are three distinct ways the electron behaves depending on how fast it travels, and surprisingly, a specific moderate speed offers the best protection for the quantum information.

The researchers modeled the journey of an electron moving through a silicon-germanium structure, a common material used in these devices. They tracked how the electron's state evolved as it traveled a distance of 300 nanometers, a journey that involves passing through thousands of tiny atomic irregularities. When they simulated a slow journey, moving at speeds less than one meter per second, they observed a steady loss of information. In this slow regime, the electron gradually leaked out of its intended state, and the longer the distance, the more information was lost. This happens because the random variations in the material's landscape cause the electron to drift away from its starting point over time.

They also looked at what happens at extremely high speeds, hundreds of meters per second. Here, the electron becomes a blur, existing in a chaotic mix of both possible states simultaneously. This creates a different kind of noise that also destroys the information, and the longer the electron travels, the worse this confusion becomes. However, between these two extremes lies a "sweet spot" where the physics works in the electron's favor. When the electron is shuttled at speeds of tens of meters per second, it undergoes a series of complete, predictable flips between the two states.

In this middle regime, the electron does not get stuck in a confused mix, nor does it slowly leak away. Instead, it flips back and forth so cleanly that if it starts in the correct state, it will end in the correct state, regardless of how far it travels. The researchers found that the only thing that could ruin this perfect journey is if the electron gets stuck in the wrong state for too long and relaxes, but the simulations show that the time spent in the wrong state is far too short for this to happen. This means that at these specific speeds, the quantum information is preserved perfectly, and the distance traveled becomes irrelevant.

This finding aligns with recent experimental results where scientists achieved high-quality data transfer after moving electrons quickly over long distances, a result that was previously difficult to explain. The study suggests that the old strategy of moving slowly to avoid errors is actually counterproductive. Instead, the key to success is moving at a moderate pace that enters this specific regime where the electron's internal rhythm stays synchronized with the journey, avoiding both the slow leakage and the chaotic mixing found at extreme speeds.

The team also explored ways to improve the process further, testing strategies that had been proposed based on older theories. They found that trying to slow down the electron only when it passed over a specific "bump" in the landscape was not very effective, because the problem is not just the height of the bump, but the direction the electron's state is pointing as it moves. They proposed a new method of adjusting the speed based on this direction, which proved to be much more efficient in their simulations. They also tested a strategy where the electron takes a detour around the bumps, moving sideways to avoid the worst areas. While this helped, it was not as effective as simply controlling the speed correctly.

Ultimately, the research points to a clear path forward for building quantum computers. The most reliable way to move quantum information is not to tiptoe through the atomic landscape, but to stride through it at a steady, moderate pace. By moving at speeds of tens of meters per second, the electron's internal state remains robust against the random imperfections of the silicon, preserving the data over any distance. This insight transforms a major hurdle in quantum computing from a problem of avoidance into a problem of timing, offering a practical and powerful solution for the future of the technology.

Technical Summary: Valley Transitions during Spin Shuttling in Si/SiGe Heterostructures

Problem Statement
Coherent spin shuttling in Si/SiGe heterostructures is a critical requirement for scalable semiconductor quantum computing architectures, enabling the connection of distant spin qubits. While isotopically purified silicon offers long relaxation and dephasing times, a primary challenge for long-range transport remains the management of valley excitations. The valley degree of freedom in silicon arises from six degenerate conduction band minima, partially lifted by strain and interface effects in Si/SiGe to form a two-fold degeneracy. The energy splitting between these low-lying valley states ( $E_{VS}$ ) is determined by the random distribution of Germanium atoms at the interface, causing $E_{VS}$ to fluctuate randomly as a quantum dot moves through the device. These fluctuations can induce uncontrolled transitions between quasi-degenerate valley states. Since valley states possess different g-factors, such transitions lead to spin-valley entanglement and dephasing in the presence of a magnetic field, potentially destroying quantum information. Previous strategies to mitigate this, such as increasing valley splitting or slowing shuttling velocities, have largely relied on the standard Landau-Zener (LZ) model, which assumes a linear Hamiltonian trajectory and ignores the complex phase evolution of the intervalley coupling.

Methodology
The authors theoretically investigate valley dynamics by modeling the system as a two-level system comprising the two low-lying valley states of a mobile quantum dot.

Hamiltonian Construction: Using effective mass theory, the authors calculate the valley splitting ( $E_{VS}$ ) and the complex intervalley coupling ( $\Delta$ ) as functions of the quantum dot's position. The potential landscape includes a disordered SiGe barrier modeled by a sum of delta functions representing Ge atoms.
Phase Dynamics: Crucially, the study tracks not only the magnitude of the valley splitting but also the phase $\theta = \arg(\Delta)$ of the complex intervalley coupling. The authors demonstrate that the Hamiltonian curve in the parameter space $(x, y, z)$ is not a straight line, rendering the standard LZ formula invalid. Instead, they utilize a generalization known as the Partial Landau-Zener (PLZ) model, where the transition probability depends on the angular velocity $\dot{\theta}$ of the valley phase.
Simulation: The time-dependent Schrödinger equation is solved numerically for a constant shuttling velocity $v$ . The valley shuttling fidelity ( $F_V$ ) is defined as the probability of the electron remaining in the valley ground state after transport. Simulations are averaged over 1000 realizations of the disordered interface for various quantum dot sizes and interface widths.

Key Contributions and Results
The study identifies three distinct dynamical regimes for valley evolution based on the shuttling velocity, challenging the conventional binary view of adiabatic vs. non-adiabatic transport:

Distance-Dependent Valley Leakage (DVL) Regime ( $v < v_c$ , where $v_c \approx 10^{-2} - 10^{-1}$ m/s):
Contrary to standard LZ predictions which suggest slow transport is adiabatic, the authors find that below a critical velocity $v_c$ , valley leakage increases with shuttling distance. This is attributed to the accumulation of valley phase changes. In this regime, reducing the quantum dot size (to increase average valley splitting) paradoxically increases leakage by enhancing the variability of the splitting.
Full Valley Transitions (FVT) Regime ( $10^{-1}$ m/s $\lesssim v \lesssim 10$ m/s):
In this intermediate velocity range, the system undergoes a sequence of full transitions between the valley ground and excited states. Crucially, the authors find that the quantum information is preserved in this regime. Because the electron spends negligible time in the excited state (shorter than the valley relaxation time), no spin-valley entanglement accumulates. The transitions result only in a deterministic spin rotation, which can be corrected post-shuttling. This regime allows for coherent spin shuttling independent of distance, aligning with recent experimental observations of high fidelity at fast velocities.
Combined Valley States (CVS) Regime ( $v \gtrsim 10$ m/s):
At very high velocities, the electron exists in a linear combination of both valley states throughout the transport. This leads to spin-valley entanglement and distance-dependent dephasing, making this regime unsuitable for coherent shuttling.

Strategies for Fidelity Enhancement
The paper proposes and evaluates three strategies to enhance valley shuttling fidelity, comparing those based on the standard LZ model against those based on the PLZ model:

Controlled Slow-down: Reducing velocity in a short region at the end of the path to ensure the system ends in the ground state. This is effective for the FVT regime but requires knowledge of transition locations.
Adapted Velocity:
- Standard LZ approach: Adjusts velocity based on $E_{VS}$ magnitude. This yields high fidelity only at very low average velocities ( $\sim$ mm/s), which is impractical for fault-tolerant computing.
- Partial LZ approach: Adjusts velocity to keep the angular velocity of the valley phase ( $\dot{\theta}$ ) constant. This strategy achieves high fidelity ( $>98\%$ ) at average velocities of several m/s to tens of m/s, significantly outperforming the standard LZ approach.
Lateral Avoidance: Deviating the shuttling path in the lateral ( $y$ ) direction to avoid valley splitting minima. While this suppresses leakage, the PLZ-based optimization (avoiding phase changes) is less efficient than the adapted velocity strategy because it cannot fully prevent the intervalley coupling from traversing the complex plane quadrants.

Significance
The paper claims that the Partial Landau-Zener model provides a more accurate description of valley transitions than the standard model, primarily due to the critical role of the valley phase. The most significant finding is the identification of the Full Valley Transitions (FVT) regime as the ideal operating window for spin shuttling. In this regime, fast shuttling (tens of m/s) preserves quantum information without requiring complex error correction for valley-induced dephasing, provided the system avoids the CVS regime. This theoretical result offers a consistent explanation for recent experimental successes in achieving high-fidelity spin shuttling over long distances at high speeds. Furthermore, the proposed adapted velocity strategy based on the PLZ model offers a practical pathway to maintain high fidelity at experimentally desirable velocities, surpassing previous strategies that relied on the standard LZ framework.

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