🔬 mesoscale physics

Avoided crossings in spin-boson systems: consequences for adiabatic state preparation

This paper investigates how avoided crossings in the many-body spectrum of finite spin-boson systems constrain adiabatic state preparation times and induce non-monotonic behavior in local observables, providing essential benchmarks for quantum simulation and control.

Original authors: Joona Marjamäki, Robert van Leeuwen, Riku Tuovinen

Published 2026-09-25

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CC BY 4.0

Original authors: Joona Marjamäki, Robert van Leeuwen, Riku Tuovinen

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

In the microscopic world of quantum physics, the boundary between a system and its surroundings is often blurred. Imagine a tiny particle, like a qubit used in quantum computers, existing in a state of delicate balance. This particle does not exist in isolation; it is constantly interacting with a sea of vibrations and electromagnetic fluctuations known as a "bath." Understanding how this particle settles into a stable, calm state while surrounded by this noisy environment is a central challenge. If the particle is disturbed too quickly, it gets excited and behaves unpredictably. However, if the interaction with its environment is turned on very slowly, the particle can adjust smoothly, settling into a correlated state where it and the environment are inextricably linked. This process, known as adiabatic state preparation, is the theoretical foundation for preparing the initial states needed for quantum simulations and computing. The difficulty lies in knowing exactly how slow "slow" needs to be, as the rules change depending on the strength of the connection between the particle and its surroundings.

A team of researchers at the University of Jyväskylä in Finland has investigated these rules by simulating a specific, simplified version of this problem: a two-level system, or qubit, coupled to a small number of vibrating modes. By running precise computer simulations, they mapped out the energy landscape of this system as the connection between the qubit and the vibrations was gradually strengthened. They discovered that the path to a stable state is not a smooth, straight line. Instead, the energy levels of the system twist and turn, occasionally coming very close to one another without quite touching. These moments, called avoided crossings, act like narrow bridges in a mountain pass. When the system passes through these narrow gaps, the energy difference between the stable state and the next possible state becomes tiny. The researchers found that these narrow gaps are the primary bottleneck for preparation. To successfully guide the system into its final state without causing errors, the process must be slowed down significantly whenever these avoided crossings are encountered. The stronger the interaction between the qubit and the environment, the more frequent and complex these narrow gaps become, demanding even longer preparation times.

The study revealed a surprising complexity when looking at different ways to measure the system's success. If one only tracks the total energy of the system, the time required to reach stability increases steadily and predictably as the interaction gets stronger. This aligns with the expectation that stronger connections require more careful handling. However, when the researchers looked at specific local properties, such as the orientation of the qubit itself, the story changed. In certain ranges of interaction strength, the time needed to stabilize these local properties did not simply keep increasing. Instead, it fluctuated, sometimes requiring less time and sometimes much more, depending on the precise details of the energy landscape. This non-monotonic behavior arises because the qubit and the environment engage in a coherent, rhythmic exchange of energy that persists in these finite systems. Even if the total energy looks stable, these local oscillations can linger, requiring the preparation process to run longer to ensure the qubit has truly settled.

The researchers tested these findings across different scenarios, varying the number of vibrational modes and the strength of the tunneling between the qubit's two states. They found that systems with more vibrational modes created a denser landscape of these narrow gaps, making the preparation process more demanding. In cases where the tunneling between states was very weak, the system required significantly longer times to prepare, roughly three times longer than in other regimes, because the energy gaps became exceptionally narrow. The simulations showed that to achieve a high degree of accuracy, the switching time needed to be increased to suppress unwanted jumps between states. For instance, in one specific configuration, the required time to stabilize the system's local orientation exceeded the time needed to stabilize its total energy, highlighting that different aspects of the system settle at different rates.

These findings offer a concrete guide for anyone attempting to control quantum systems. The work demonstrates that simply turning up the interaction strength is not enough; one must account for the specific structure of the energy levels. The presence of avoided crossings dictates the speed limit for preparation. If the process is rushed, the system will fail to reach the desired state, leading to errors in quantum simulations or computations. By identifying exactly where these bottlenecks occur, the study provides a benchmark for designing better control protocols. It suggests that for practical applications in quantum technology, such as manipulating qubits in a processor, the preparation phase must be tailored to the specific spectral features of the device. The ability to predict these requirements based on the underlying energy structure means that future quantum experiments can be designed with greater precision, ensuring that the delicate quantum states are prepared correctly before any further operations are attempted.

Technical Summary: Avoided Crossings in Spin-Boson Systems

Problem Statement
The preparation of correlated equilibrium states in interacting quantum systems is a fundamental challenge, particularly for the spin-boson model, which describes a two-level system (qubit) coupled to a bosonic environment. In practical applications, such as quantum computing, systems are often initialized in an uncoupled state and then subjected to a "quench" or sudden interaction switch-on. This approach generates difficult-to-disentangle excitations and non-physical transients. While adiabatic switching protocols—gradually increasing the coupling strength—are a standard method to prepare the true correlated ground state, their efficiency is limited by the instantaneous many-body spectrum. Specifically, regions of "avoided crossings" in the energy spectrum create small excitation gaps, imposing stringent conditions on the switching speed required to maintain adiabaticity. The precise relationship between these spectral features, the coupling strength, and the necessary switching times in finite systems remains a critical area of investigation.

Methodology
The authors investigate finite spin-boson systems containing either one or three bosonic modes. The study employs a dual approach combining static spectral analysis with dynamic time-evolution simulations:

Exact Diagonalization: The authors numerically diagonalize the many-body Hamiltonian to obtain the exact ground state and the full energy spectrum. To ensure numerical exactness, the Hilbert space is truncated based on boson occupation numbers, with convergence verified against a tolerance of $10^{-6}$ . The truncation size is increased monotonically with the effective coupling strength ( $\gamma$ ) to account for stronger qubit-bath dressing.
Adiabatic Switching Protocol: The system is initialized in the ground state of the non-interacting Hamiltonian ( $g_k=0$ ). The interaction is turned on slowly using a sinusoidal ramp function $f(t) = \sin^2(\pi t / 2t_0)$ over a switch-on time $t_0$ .
Adiabatic Condition Analysis: The theoretical requirement for adiabaticity is quantified using the condition $\max_s R_{m0}(s) \ll t_0$ , where $R_{m0}(s)$ depends on the matrix elements of the interaction Hamiltonian and the inverse square of the energy gaps between the ground state and excited states.
Benchmarking: The time-evolved state is compared against the exact ground state obtained via diagonalization. The "switch-on time" ( $t_0$ ) is determined as the minimum time required for specific observables (ground-state energy and Pauli matrix expectation values) to converge to the exact results within a specified tolerance ( $10^{-6}$ for energy, $10^{-4}$ for oscillation amplitude).

Key Contributions and Results

Spectral Structure and Avoided Crossings: The study confirms that the low-energy many-body spectrum exhibits a complex structure of avoided crossings. The positions and gap sizes of these crossings depend on the tunneling amplitude ( $\epsilon$ ) and the coupling strength ( $\gamma$ ).
Constraints on Switching Times: The authors demonstrate that regions with small many-body level spacings (associated with avoided crossings) impose the strongest constraints on adiabatic preparation.
- For $\epsilon = 1$ (tunneling equal to qubit energy), the required switch-on time increases monotonically with $\gamma$ , consistent with the monotonic growth of the adiabatic ratio $R_{m0}(s)$ .
- For smaller tunneling amplitudes ( $\epsilon = 0.5, 0.1$ ), the switch-on time exhibits non-monotonic behavior. Local reductions in the required time coincide with specific avoided crossings and level bunching in the spectrum, where the dominant contribution to the adiabatic condition shifts between different excited states.
- The case of $\epsilon = 0.1$ requires significantly longer switch-on times (approximately three times longer than other regimes) due to a slow approach to saturation of observables, rather than strong non-adiabatic oscillations.
Observable-Dependent Convergence: A critical finding is the discrepancy between global and local observables. While the ground-state energy follows the overall spectral trends and converges monotonically with increasing $t_0$ $t_{0}$ , local observables (specifically Pauli matrix expectation values $\langle \sigma_z \rangle$ $⟨ σ_{z} ⟩$ and $\langle \sigma_x \rangle$ $⟨ σ_{x} ⟩$ ) exhibit non-monotonic switching-time behavior.
- In the single-mode case with $\epsilon = 1$ , the switch-on time for Pauli matrices peaks at a critical coupling ( $\gamma \approx 3.3$ ) and then decreases. This peak corresponds to a near-resonant regime where coherent oscillations between near-degenerate excited states persist.
- These local observables sustain coherent oscillations near narrow gaps, requiring longer switch-on times to average out the oscillation amplitude below the convergence tolerance, even when the ground-state energy has already converged.

Significance and Claims
The paper claims to provide concrete benchmarks for the theoretical treatment of open quantum systems and practical implementations in quantum simulation and qubit control. By isolating intrinsic quantum dynamics in finite environments, the study clarifies how correlated equilibrium states emerge without the approximations often used for continuum baths.

The authors emphasize that the efficiency of adiabatic state preparation is not solely determined by the global energy gap but is highly sensitive to the specific observable being measured. Local quantities can equilibrate on substantially different timescales than global energetic measures due to coherent oscillations sustained by finite-size effects and avoided crossings. Consequently, protocols for qubit control and quantum information processing must account for these local dynamical features to avoid transient effects and ensure the system reaches the intended correlated equilibrium state. The findings suggest that starting from a correlated equilibrium state, rather than an uncoupled configuration, offers a controlled route for investigating system responses to external driving.

Technical Summary: Avoided Crossings in Spin-Boson Systems

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