The challenge of controlling transitions in quantum systems undergoing avoided crossings, as described by the Landau‑Zener model, has long occupied physicists. The authors address this problem by exploring counterdiabatic driving, a technique to engineer control fields that effectively restore adiabaticity. Their research demonstrates a method for managing transitions across a range of energy gaps—a scenario common in many physical systems. This work is significant because it moves beyond controlling single transitions to statistically minimising the probability of unwanted transitions across an ensemble of systems, offering a more robust approach to quantum control. The team’s findings reveal a fundamental trade‑off between maintaining instantaneous adiabaticity and achieving a low final transition probability, with analytical results supported by comprehensive numerical simulations.

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Statistical Control of Landau‑Zener Transition Probabilities

Scientists demonstrate a novel approach to counterdiabatic driving, successfully constructing a single control field capable of managing an ensemble of Landau‑Zener (LZ) type Hamiltonians, each possessing a varying energy gap. This research addresses a critical limitation of conventional counterdiabatic protocols, which typically require precise knowledge of a fixed energy gap, and instead focuses on statistical optimisation to minimise the average transition probability across a distribution of gaps.

The team achieved this by focusing on a specific class of control fields inspired by established counterdiabatic driving techniques, revealing a systematic trade‑off between instantaneous adiabaticity and the ultimate transition probability. The study unveils a generalized LZ system defined by a single gap and control parameter, allowing for both analytical and numerical investigation of the control field’s efficacy.

Researchers found that certain limiting cases, including a scenario employing a Dirac‑delta function, are analytically tractable, providing valuable insight into the optimisation process. Comprehensive numerical simulations systematically support and extend these analytical findings, confirming the effectiveness of the proposed control strategy across a range of gap distributions.

The work establishes a clear connection between control‑field design and the resulting adiabaticity, demonstrating that deviation from strictly adiabatic paths can, in fact, lower the average transition probability. Experiments show a significant performance difference between control fields based on Pauli operators, with σ₁ controls consistently outperforming the σ₂ fields commonly used in standard counterdiabatic driving.

This is particularly advantageous as the σ₁ operator is already inherent within the LZ Hamiltonian itself, simplifying experimental implementation by eliminating the need for additional engineered control Hamiltonians. The research further reveals that the optimal temporal function for the control field deviates from the traditional Lorentzian shape, suggesting a more nuanced approach to achieving robust adiabatic control.

The breakthrough reveals a pathway towards more robust quantum control in systems with inherent or induced variations in energy gaps, such as many‑body spin chains, thermal ensembles, or systems with spatially varying parameters. The work opens possibilities for improved adiabatic quantum computing and the development of more resilient quantum technologies, where maintaining coherence despite parameter fluctuations is paramount. By prioritising statistical minimisation of transition probability, the team has provided a valuable tool for manipulating quantum states in complex and realistic scenarios.

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LZ Transitions and Adiabatic Control Optimisation

The study investigated adiabatic transfer processes within the Landau‑Zener (LZ) model, focusing on systems experiencing avoided crossings. Researchers engineered a control field designed to drive ensembles of LZ‑type Hamiltonians, each possessing a distribution of energy gaps, with the aim of statistically minimising the average transition probability. This work restricted controls to a specific class, building upon prior investigations, and revealed a systematic trade‑off between instantaneous adiabaticity and the ultimate transition probability achieved.

Analytical treatment of limiting cases, including a linear sweep and the LZ system with a Dirac‑function, provided a foundation for comprehensive numerical validation and expansion of these results.

Scientists harnessed adiabatic transfers to create robustness against variations in the gap parameter, potentially enabling control of spin‑collision dynamics in optically trapped thermal atoms. This approach mirrors earlier experiments utilising nuclear magnetic resonance to control qubits with counterdiabatic control, suggesting practical applications given the likelihood of random parameter fluctuations in real‑world implementations.

The research employed a static limit, assuming parameter fluctuations occur much slower than the system’s typical timescale, and lays the groundwork for future investigations into the interplay between noise with finite‑time correlations and control via counterdiabatic fields. Detailed analysis involved calculating time‑evolution matrices, denoted as U₀ for the standard LZ system, and extending this notation to generalized Hamiltonians.

The transition probability P₀ was determined from the matrix elements A₀ and B₀, utilizing Parabolic Cylinder Functions D_ν(z) and their asymptotic expansions for large time intervals. Specifically, the team derived expressions for A₀ and B₀, and subsequently the Landau‑Zener formula

[

P₀(a)=e^{-\pi a^{2}},

]

to quantify transition probabilities as a function of the gap parameter.

Furthermore, the study established key symmetries within the Hamiltonian, demonstrating that time‑evolution matrices U(t,0) and U(0,-t) are related, and that the LZ formula is even in the gap parameter a. These symmetries, expressed through Pauli‑matrix relations, simplified calculations and provided a deeper understanding of the system’s behaviour. This methodological precision enabled the researchers to accurately model and control quantum transitions, paving the way for advancements in quantum technologies and robust quantum‑control schemes.

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Ensemble Control Minimises Landau‑Zener Transition Probability

Scientists have achieved a significant breakthrough in controlling quantum systems undergoing Landau‑Zener (LZ) transitions, demonstrating a novel approach to counterdiabatic driving for ensembles of systems with varying energy gaps. The research focuses on minimising the average transition probability across a distribution of LZ‑type Hamiltonians, a challenge previously unmet by standard adiabatic‑control methods. Experiments revealed a systematic trade‑off between instantaneous adiabaticity and the final transition probability, highlighting the complex interplay between maintaining coherence and achieving rapid evolution.

The team constructed a single control field, designated H₁, designed to statistically minimise the average transition probability across the ensemble, rather than driving each system transitionlessly. This work restricts attention to a specific class of H₁ controls, inspired by established counterdiabatic driving techniques, leading to the definition of a Generalised LZ System characterised by a single gap and control parameter.

Analytical solutions were obtained for limiting cases, including the LZ system driven by a Dirac‑delta function, providing valuable insight into the optimisation process and revealing a control field that differs significantly from standard counterdiabatic theory. Measurements confirm that employing σ₁ control fields, utilizing the Pauli operator σ₁, generally outperforms those based on σ₂, a key finding with practical implications for experimental implementation. This is because σ₁ is already inherent within the LZ Hamiltonian, eliminating the need for additional engineered control terms.

Furthermore, tests prove the ideal temporal function for the control field is no longer a standard Lorentzian shape, indicating a departure from conventional adiabatic‑control protocols. Results demonstrate that allowing controlled deviations from the adiabatic path can, on average, deliver a lower final transition probability.

The research establishes a foundation for robust quantum control in systems with inherent parameter variations, such as many‑body spin chains or thermal ensembles, where random energy gaps are commonplace. This breakthrough delivers a pathway towards faster and more reliable quantum‑state manipulation, with potential applications in superconducting qubits and NV centres.

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Optimising Landau‑Zener Transitions with Control Fields

This work presents a novel approach to controlling transitions in Landau‑Zener (LZ) systems, specifically addressing ensembles of Hamiltonians with varying energy gaps. Researchers successfully designed a single control field capable of driving these systems, statistically minimising the average transition probability. This was achieved by systematically investigating the trade‑off between instantaneous adiabaticity and the final transition probability, with analytical solutions found for certain limiting cases, including a connection to the standard LZ system with a Dirac‑delta function.

The central finding demonstrates that a control field with φ = 0 minimises the average transition probability across all gap parameters, and, importantly, remains effective even as the gap distribution broadens infinitely. This is supported by both analytical derivations and comprehensive numerical simulations.

While acknowledging limitations inherent in restricting attention to a specific class of control fields, the authors highlight the potential for extending these results to more general scenarios. Future research could explore the application of these control strategies to more complex systems and investigate the robustness of the findings under different noise conditions.

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More information

Counterdiabatic driving for random‑gap Landau‑Zener transitions – arXiv pre‑print: https://arxiv.org/abs/2601.10659