The work done during abrupt changes to quantum systems can be predicted using the mathematics of random matrices, according to new research. Scientists have found that the distribution of energy changes during these changes mirrors the statistics of traces of powers of Haar unitaries, which are random unitary matrices.
For translation-invariant quadratic fermionic chains, the work variable W is calculated as a sum of coefficients multiplied by the real and imaginary parts of the trace of a unitary matrix raised to the power r. This work demonstrates that non-Gaussian fluctuations in the work distribution, or unexpected energy spikes, arise specifically from systems with many interaction terms or interactions that decay slowly, and are characterized by the appearance of Fisher-Hartwig singularities.
Work Distribution via Random Matrix Theory & Haar Unitaries
This unexpected connection, detailed in a recent publication in Quantum, suggests that the seemingly chaotic energy fluctuations during these changes exhibit a mathematical order akin to pure chance within a defined framework. Researchers demonstrated this correspondence by modeling the work done during these quenches using the established tools of random matrix theory, a field originally developed to study the spectra of heavy nuclei.
This model provides a quantifiable link between microscopic interactions within the quantum chain and the work performed during the quench, allowing for a detailed analysis of energy transfer at the quantum level. The model also accounts for systems exhibiting superconducting pairing, adding further terms to the calculation.
These non-Gaussian fluctuations are further characterized by the presence of Fisher-Hartwig singularities, pinpointing the specific conditions under which these quantum systems deviate from predictable behavior. The authors note that “a growing width does not by itself spoil the Gaussian shape, which can survive once the width is scaled out,” indicating that the structure of the underlying model dictates the fluctuations in work. This work establishes a direct link between the shape and size of these fluctuations and the model’s architecture, offering a new avenue for understanding and controlling quantum systems undergoing rapid change.
Gaussianity & Deviations in Quantum Quench Work Statistics
The shape of work fluctuations in abruptly changed quantum systems directly correlates to the structure of the model governing them, according to recent findings. The study demonstrates that a Gaussian distribution, describing the statistical spread of work done, can persist even as the width of that distribution increases, provided that width is appropriately scaled.
This resilience of the Gaussian shape is illustrated through examples in the XY chain and verified with numerical tests. The research further clarifies conditions under which deviations from this Gaussian behavior occur; a model’s structure, specifically the number of interaction terms or the rate at which those interactions decay, dictates when these deviations, manifesting as unexpected energy spikes, emerge.
This work builds on earlier theoretical foundations, referencing the work of Talkner, Lutz and Hänggi on fluctuation theorems and Heyl’s review of dynamical quantum phase transitions. The findings, published in Quantum, provide a framework for understanding and predicting the behavior of quantum systems undergoing rapid change.
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