Advances in Non-Hermitian Physics Enable Realistic Responses Beyond Closed-System Descriptions

The behaviour of quantum systems that lose energy to their surroundings, known as non-Hermitian systems, presents a fundamental challenge to predicting their properties, and understanding how these systems respond to external stimuli is crucial for developing new technologies. Milosz Matraszek, Wojciech J. Jankowski, and Jan Behrends, from the University of Cambridge and Quantinuum, now establish fundamental geometric limits on measurable quantities within these complex systems. Their work reveals previously unknown constraints on key properties, including how these systems conduct electricity and respond to external forces, even in materials exhibiting unusual topological characteristics. This achievement extends beyond theoretical descriptions, offering insights directly applicable to experiments involving open quantum systems and paving the way for more accurate modelling of realistic, real-world devices.

Findings in topological systems with non-Hermitian Chern numbers demonstrate that the non-Hermitian geometric constraints on response functions naturally arise in open quantum systems governed by out-of-equilibrium Lindbladian dynamics. These findings are relevant to experimental observables and responses under realistic setups that fall beyond idealized closed-system descriptions. Geometry plays a pivotal role in physics, extending from classical dynamics and general relativity to statistical thermodynamics and quantum mechanics.

Quantum Geometry and Open System Response

This work establishes a framework for understanding open quantum systems, which interact with an external environment. The research leverages quantum geometry and topology to characterize these systems, even when described by non-Hermitian Hamiltonians, because these properties reveal fundamental aspects of system behavior, such as robustness to perturbations and unusual transport phenomena. The ultimate aim is to calculate how the system responds to external stimuli, crucial for understanding its physical properties. The method involves a mathematical framework utilizing non-Hermitian Hamiltonians and the Keldysh formalism, a technique for dealing with time-dependent, non-equilibrium systems.

Key elements include Green’s functions that describe the propagation of quantum states and a self-energy term encapsulating the environment’s effects on the system. The research simplifies calculations by demonstrating that if the Hamiltonian commutes with the real part of the self-energy, the Green’s functions become simultaneously diagonalizable. Results show that response functions are expressed in terms of Lorentzian functions, reflecting the finite lifetime of quantum states due to dissipation. The team derived general bounds on these response functions based on the system’s geometry and the strength of coupling to the environment, and applied this framework to the Rice-Mele Hamiltonian, a model for topological insulators with disorder. This connection clarifies the relationship between the anti-Hermitian part of the Hamiltonian and jump operators describing transitions between states due to the environment.

Geometric Bounds on Non-Hermitian Response Functions

Scientists have identified geometric bounds for measurable quantities within non-Hermitian quantum systems, establishing constraints on properties like non-Hermitian geometric tensors, response correlations, conductivity, and weighting factors. The research demonstrates that these geometric constraints on response functions naturally emerge in open systems governed by out-of-equilibrium Lindbladian dynamics, offering insights into realistic physical scenarios beyond idealized closed-system models. These findings are highly relevant to experimental platforms such as quantum-optical systems and circuit-quantum electrodynamics, where dissipation can be precisely engineered. The team confirmed that the absorptive part of two-point correlation functions is positive semidefinite, leading to a general inequality relating the real and imaginary parts.

Measurements of frequency-dependent Lorentzian kernels confirmed their non-negative values, crucial for establishing the geometric bounds, particularly when energy differences are less than zero. The work also establishes a local bound on non-Hermitian Berry curvature, demonstrating a relationship between quantum geometric tensors and Berry curvature. Investigations into non-Hermitian Rice-Mele models, exhibiting non-trivial non-Hermitian Chern invariants, showcase these bounds, validating the local bound even with specific parameter settings. This breakthrough delivers a framework for understanding quantum geometric bounds in open quantum systems, specifically as constraints on externally coupled baths, utilizing the Keldysh formalism to reveal the emergence of these bounds in bubble diagrams intrinsic to response functions.

Geometric Limits in Non-Hermitian Quantum Systems

This research establishes geometric limits for measurable properties in non-Hermitian systems, extending beyond the constraints of traditional, closed-system quantum mechanics. Scientists identified specific boundaries applicable to non-Hermitian geometric tensors, response correlations, conductivity, and weighting factors, demonstrating these principles within topological systems characterized by non-Hermitian Chern numbers. The team reveals that these geometric constraints on response functions emerge naturally in open systems described by Lindbladian dynamics, offering a framework relevant to realistic experimental conditions. The investigation demonstrates that maintaining these quantum geometric bounds requires the positivity of functions related to the system’s Green’s functions, a condition satisfied by dynamics consistent with the non-Hermitian Rice-Mele model.

While broadly applicable to systems with multiple energy bands, the researchers note that the bounds can break down if complex eigenvalues appear with differing signs, potentially due to gain compensating for losses. They validated these general bounds using diagrams representing open quantum system responses, confirming their applicability to simulated non-Hermitian systems such as those found in photonics and optomechanics. Future work could explore the implications of inverted population distributions in the system’s environment, which may lead to a breakdown of these geometric bounds, and further investigate the behavior of these bounds in more complex multi-band systems.