Hilbert Space Fragmentation and Controlled Dephasing in Quantum Many-Body Systems.

The behaviour of complex quantum systems, particularly those with strong interactions, often deviates from the expected ergodic behaviour, becoming constrained to limited regions within their vast quantum state space, known as Hilbert space. This fragmentation of Hilbert space restricts the system’s evolution, hindering experimental observation of complete sector properties. Researchers at Boston University and Stanford University, alongside colleagues at the Max-Planck-Institut für Physik komplexer Systeme, now demonstrate a method to reliably mix quantum states within these fragmented sectors using controlled dephasing, enabling clearer measurement of sector characteristics. Dominik Vuina, Robin Schäfer, David M. Long, and Anushya Chandran detail their findings in a paper entitled ‘Probing Hilbert space fragmentation using controlled dephasing’, where they show that simple observables, such as spin imbalance in the strongly interacting XXZ chain, can differentiate these sectors and reveal information about their average properties.

Researchers are investigating the influence of controlled dephasing on quantum many-body systems, with a specific focus on the XXZ chain and the behaviour observed within what are termed Krylov sectors. Krylov sectors represent fragmented regions of the system’s Hilbert space, which is the complete set of all possible quantum states. This fragmentation restricts the system’s evolution to smaller subspaces, often resulting in slow or non-ergodic dynamics, meaning the system does not explore all accessible states over time. The study demonstrates that introducing controlled dephasing effectively mixes the system within a single Krylov sector, enabling differentiation between these sectors through simple observable measurements and providing a pathway to characterise these fragmented Hilbert spaces.

The authors establish that the spin imbalance between even and odd sublattices serves as a key observable for distinguishing Krylov sectors in the strongly interacting XXZ chain. The XXZ chain is a model in quantum physics used to describe interacting spins, and ‘strongly interacting’ refers to a regime where the interactions between these spins are significant. Initialising the system in a specific state results in a characteristic imbalance evolution, beginning positive, decaying to a negative minimum, and ultimately returning to zero. This minimum value directly reflects the average imbalance inherent within the Krylov sector associated with the initial state, offering a quantitative link between control parameters and system behaviour.

This research highlights how dephasing acts as a mechanism to overcome the slow dynamics often observed within individual Krylov sectors, promoting mixing within a sector and facilitating the measurement of sector properties. By actively controlling dephasing, scientists gain access to sector-averaged properties, overcoming limitations imposed by slow intra-sector dynamics and offering a more complete understanding of many-body quantum systems exhibiting Hilbert space fragmentation.

Researchers analytically determine the magnitude of this minimum in the strong interaction limit, validating these findings through numerical simulations performed at interaction strengths relevant to current experimental capabilities. These simulations confirm the theoretical predictions, strengthening the validity of the model and establishing a clear connection between the dephasing rate and the observed dynamics, allowing for precise characterisation of the fragmented Hilbert space and providing insights into the underlying dynamics of strongly interacting quantum systems.

The research highlights the utility of dephasing not as a source of decoherence to be avoided, but as a controllable tool for probing and characterising the complex dynamics within fragmented quantum systems. Decoherence refers to the loss of quantum information due to interaction with the environment. By actively mixing states within a Krylov sector, scientists facilitate the measurement of sector-averaged properties, overcoming the limitations imposed by slow intra-sector dynamics. This approach offers a pathway towards a more complete understanding of many-body quantum systems exhibiting Hilbert space fragmentation and provides valuable insights into the underlying physics.

Future research will focus on extending these techniques to more complex systems and exploring the potential for using controlled dephasing to manipulate and control quantum states. Scientists plan to investigate the effects of different dephasing protocols on the system’s dynamics and to develop new methods for characterising the fragmented Hilbert space. They also aim to explore the potential applications of these techniques in quantum information processing and quantum simulation.