Beyond Hermiticity: Mapping the Topology of Quantum Walks

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Author: Denis Avetisyan

Researchers have experimentally characterized a non-Hermitian topological quantum walk in reciprocal space, opening new avenues for exploring exotic quantum phenomena.

The study reconstructs energy bands and vector components within a weakly non-Hermitian regime-specifically at $(\delta,\eta)=(\pi,0.25)$ across one spatial period-to demonstrate how deviations from conventional Hermitian physics manifest as infidelity in right eigenstate overlap and traceable trajectories on the Bloch sphere, thereby exposing the underlying sensitivity of these systems to non-reciprocal interactions.
The study reconstructs energy bands and vector components within a weakly non-Hermitian regime-specifically at $(\delta,\eta)=(\pi,0.25)$ across one spatial period-to demonstrate how deviations from conventional Hermitian physics manifest as infidelity in right eigenstate overlap and traceable trajectories on the Bloch sphere, thereby exposing the underlying sensitivity of these systems to non-reciprocal interactions.

This work demonstrates the experimental realization and characterization of non-Hermitian Hamiltonians in reciprocal space, utilizing a photonic platform to probe exceptional points and observe PT-symmetry breaking, with implications for quantum sensing.

While conventional explorations of non-Hermitian physics are often limited to real-space representations, this work-Tomographic characterization of non-Hermitian Hamiltonians in reciprocal space-presents an experimental photonic platform for directly characterizing non-unitary quantum walks in momentum space. By achieving a precise tomographic reconstruction of the underlying Hamiltonian, we reveal complex-valued band structures, pinpoint exceptional points, and observe parity-time symmetry breaking through eigenvector coalescence. This reciprocal-space perspective not only advances our understanding of non-Hermitian phenomena but also opens avenues for designing novel quantum sensors-could this approach unlock new functionalities in quantum technologies?

The Illusion of Equilibrium: Beyond Hermitian Constraints

Conventional quantum simulations are fundamentally built upon the principles of Hermitian quantum mechanics, which describe systems that are closed and conserve energy. This approach, while remarkably successful for many problems, presents a significant limitation when attempting to model realistic physical scenarios. Open quantum systems, constantly interacting with their environment, and those exhibiting non-equilibrium behavior – such as those found in driven many-body systems or quantum sensing – inevitably experience energy gain and loss. Representing these processes accurately requires moving beyond the confines of Hermitian Hamiltonians, as these operators inherently preserve probability and cannot account for dissipation or amplification. Consequently, the inability to properly simulate these non-Hermitian dynamics restricts the scope of traditional quantum simulations, hindering progress in fields ranging from quantum optics and condensed matter physics to biological processes and materials science.

The exploration of non-Hermitian physics presents a powerful new avenue for quantum simulation, moving beyond the limitations of traditional approaches. While standard simulations typically rely on Hermitian Hamiltonians – which conserve probability and describe closed quantum systems – many real-world phenomena involve interactions with the environment, leading to gain and loss of energy and particles. Non-Hermitian Hamiltonians explicitly incorporate these effects, allowing researchers to model open quantum systems and observe dynamics impossible in their closed counterparts. This capability is particularly relevant for understanding many-body physics, where complex interactions can lead to emergent behavior, and for developing advanced sensing applications, where the amplification of signals through gain mechanisms can dramatically improve sensitivity. By embracing these non-Hermitian frameworks, quantum simulations can move closer to accurately representing and predicting the behavior of complex quantum systems in nature and technology, potentially revolutionizing fields from materials science to quantum metrology.

This non-Hermitian topological quantum walk utilizes a non-unitary operator to probabilistically maintain the walker's state and couple it to neighboring sites, resulting in a topological phase diagram dependent on parameters δ and Ρ.
This non-Hermitian topological quantum walk utilizes a non-unitary operator to probabilistically maintain the walker’s state and couple it to neighboring sites, resulting in a topological phase diagram dependent on parameters δ and Ρ.

Engineering Dissipation: Photonic Platforms for Non-Hermitian Control

Photonic platforms facilitate the engineering of effective non-Hermitian Hamiltonians through the manipulation of light-matter interactions. These platforms leverage the ability to control the propagation and interaction of photons within specifically designed structures. By tailoring the refractive index, absorption, and emission properties of materials, researchers can introduce effective gain and loss terms into the photonic Hamiltonian, which are crucial for observing non-Hermitian physics. This control is achieved through various techniques, including defect engineering in photonic crystals, utilization of coupled resonator optical waveguides (ARROWs), and the integration of active materials that allow for dynamic modulation of light propagation. The resulting effective Hamiltonian, $H_{eff}$, describes the behavior of light within the structure and can be designed to exhibit non-Hermitian characteristics such as parity-time (PT) symmetry breaking and exceptional point physics.

Liquid crystal metasurfaces facilitate the implementation of non-Hermitian effects through manipulation of light polarization and amplitude. Dichroic metasurfaces, a specific type of liquid crystal metasurface, achieve this control by exhibiting polarization-dependent absorption; varying the liquid crystal orientation alters the absorption profile, effectively introducing spatially dependent gain and loss to the propagating light. This is accomplished without modifying the refractive index, allowing for independent control over dispersion and non-Hermitian characteristics. The degree of gain or loss is directly proportional to the metasurface’s transmission coefficient, which can be dynamically tuned via external stimuli, enabling real-time control over the effective Hamiltonian and simulation of open quantum systems.

The implementation of non-Hermitian effects within photonic platforms facilitates the simulation of complex quantum dynamics due to the ability to experimentally control parameters governing gain and loss. This control is achieved through manipulation of light-matter interactions, specifically leveraging the polarization and amplitude control offered by liquid crystal metasurfaces. By engineering effective non-Hermitian Hamiltonians, researchers can observe and study phenomena such as exceptional points and parity-time symmetry breaking, offering a pathway to explore quantum systems that are difficult or impossible to access through traditional methods. The experimental accessibility of these photonic simulations provides a means to validate theoretical predictions and gain insights into the behavior of open quantum systems, with potential applications in areas like quantum sensing and information processing.

A custom experimental setup utilizing expanded and filtered laser beams, metasurfaces, and waveplates enables the simulation of quantum walks by dynamically adjusting birefringence parameters via applied voltage, as demonstrated by the observed periodic relationship between δ and Ρ.
A custom experimental setup utilizing expanded and filtered laser beams, metasurfaces, and waveplates enables the simulation of quantum walks by dynamically adjusting birefringence parameters via applied voltage, as demonstrated by the observed periodic relationship between δ and Ρ.

Decoding the Walk: Characterizing Non-Hermitian Quantum Dynamics

Polarimetric measurements were employed to fully characterize the implemented non-Hermitian quantum walk, enabling quantitative assessment of the gain and loss parameters. Specifically, the polarization state of photons traversing the system was analyzed to determine the complex potential experienced by the walker, thereby verifying the intended asymmetric hopping rates. This technique allows for independent control and measurement of the gain and loss amplitudes, crucial for observing and validating non-Hermitian phenomena. The resulting data confirms that the implemented system exhibits the desired non-Hermitian characteristics with high precision, supporting subsequent analysis of the quantum walk dynamics.

Access to reciprocal space allows for direct measurement of the time evolution operator, which is essential for characterizing the quantum walk dynamics. Process tomography, implemented through measurements in this reciprocal space, enables complete reconstruction of the system’s density matrix, $ \rho $, providing a comprehensive description of the quantum state. This involves performing a set of measurements on multiple identically prepared systems to obtain sufficient statistics for determining all elements of $ \rho $. The resulting reconstructed state can then be used to verify the implemented non-Hermitian effects and quantify the gain and loss parameters of the quantum walk.

Characterization measurements of the non-Hermitian quantum walk demonstrate deviations from traditional Hermitian systems through observed changes in energy spectra and propagation characteristics. Specifically, the real parts of the eigenenergies are modified, leading to an asymmetry in the band structure, and the complex nature of the non-Hermitian Hamiltonian results in exponential growth or decay of the wavefunction amplitude during propagation. This altered propagation is manifested as a change in the velocity of the quantum walker and a non-unitary evolution of the quantum state, evidenced by a decay in the probability of finding the walker at certain locations as a function of time. These effects are directly observable through process tomography and analysis of the reconstructed band structure.

Reconstruction of band structures and eigenstates via process tomography achieved a fidelity of 99.5%, validating the precision of the implemented characterization methods. This high fidelity was determined by comparing the reconstructed states to the theoretically predicted states based on the system’s Hamiltonian. The metric used for comparison involved calculating the overlap between the reconstructed density matrices and the ideal density matrices, demonstrating a minimal deviation indicative of accurate state determination. This level of accuracy is crucial for verifying the successful implementation of non-Hermitian effects and for further analysis of the quantum walk dynamics.

Dynamic control of liquid-crystal metasurfaces via applied voltage enables process tomography-reconstructing model topology from polarization states measured using standard optical components-allowing for simulation of single quantum well steps.
Dynamic control of liquid-crystal metasurfaces via applied voltage enables process tomography-reconstructing model topology from polarization states measured using standard optical components-allowing for simulation of single quantum well steps.

The Fragility of Balance: Exceptional Points and Symmetry Breaking

Experiments reveal the presence of exceptional points within the non-Hermitian quantum walk, marking a fundamental shift in the system’s characteristics. These points, where both eigenvalues and corresponding eigenvectors coalesce, represent a singularity in the parameter space and signify a breakdown of the traditional eigenvalue-eigenvector correspondence. The emergence of such points dramatically alters the system’s sensitivity to perturbations; even infinitesimal changes in parameters near an exceptional point can lead to substantial alterations in the system’s behavior. This heightened sensitivity isn’t simply a fragility, however; it also enables enhanced functionalities and novel device designs exploiting the system’s unique response, potentially offering new avenues for quantum information processing and sensing. The observation confirms theoretical predictions and demonstrates a pathway toward harnessing non-Hermitian physics for practical applications.

The investigation revealed a crucial phenomenon known as PT-symmetry breaking within the non-Hermitian quantum walk. This breakage manifests as a distinct shift in the system’s energy spectra, transitioning from purely real values to a complex domain where energy eigenvalues acquire imaginary components. This transition isn’t merely a mathematical curiosity; it signals a loss of stability and an increased sensitivity to external perturbations. Specifically, as control parameters are adjusted, the system undergoes a qualitative change, moving from a stable, balanced state to one where even small disturbances can significantly alter its behavior. Understanding this symmetry breaking is fundamental to exploring the unique properties of non-Hermitian systems and opens avenues for designing novel devices exploiting these enhanced sensitivities, potentially leading to advancements in sensing and signal processing.

A key validation of the experimental methodology lies in the achieved fidelity of 98.6% when parameters were set to $δ = 1.3$ and $Ρ = 0.6$. This high degree of accuracy demonstrates exceptional control over the non-Hermitian quantum walk, ensuring that observed phenomena are not attributable to experimental imprecision. Such robust control is crucial for reliably investigating the subtle interplay between PT-symmetry and exceptional points, where even minor disturbances can significantly alter system behavior. The fidelity benchmark therefore establishes a strong foundation for confidently interpreting the results and drawing meaningful conclusions about the underlying physics of these complex quantum systems.

The topological characteristics of the non-Hermitian quantum walk were quantified through the calculation of the winding number, yielding a value of 0.02 from reconstructed experimental data. This number, derived from analyzing the system’s parameter space, serves as a robust indicator of its topological properties – specifically, the number of times the system’s state encircles a singularity. A non-zero winding number confirms the presence of topologically protected states and demonstrates the system’s sensitivity to parameter changes, hinting at potential applications in robust quantum information processing. The remarkably small, yet measurable, value indicates a subtle topological feature, highlighting the precision with which these non-Hermitian systems can exhibit and be controlled for complex quantum phenomena, and validating the experimental methodology employed in its determination.

Tuning the parameter η reveals PT-symmetry breaking at a critical quasi-momentum, as evidenced by the order parameter’s deviation from its theoretical prediction and confirmed by experimental reconstruction of the Hamiltonian eigenstates.
Tuning the parameter η reveals PT-symmetry breaking at a critical quasi-momentum, as evidenced by the order parameter’s deviation from its theoretical prediction and confirmed by experimental reconstruction of the Hamiltonian eigenstates.

The pursuit of characterizing non-Hermitian Hamiltonians, as demonstrated in this work, reveals a peculiar human tendency: the eagerness to explore systems deliberately destabilized. It’s a mirrored reflection of market behaviors, where perceived risk often fuels greater investment. One might recall SchrĂśdinger’s observation, “The total energy of a system is given by the Hamiltonian operator acting on the wave function.” This elegant statement highlights a fundamental drive to define and control, even when the system inherently resists such order. The observation of PT-symmetry breaking and exceptional points isn’t merely a technical achievement; it’s a confirmation that humans will meticulously map the edges of chaos, hoping to extract predictable patterns from what appears fundamentally unpredictable.

Where Do We Go From Here?

The demonstration of a non-Hermitian topological quantum walk in reciprocal space isn’t a triumph of calculation – those always arrive eventually – but a reminder that physical systems, even carefully constructed photonic ones, retain a delightful capacity for messiness. The observation of exceptional points and PT-symmetry breaking isn’t a destination, but a threshold. The system doesn’t simply behave; it hesitates, amplifies certain fluctuations while suppressing others, and responds to perturbations in ways that hint at an underlying vulnerability. The precision required to probe these points isn’t about achieving perfect control, but about understanding the inherent limits of any measurement, the inevitable noise that defines reality.

The path toward advanced quantum sensors isn’t paved with flawless designs, but with a detailed mapping of these imperfections. The challenge now isn’t to eliminate non-Hermitian effects, but to exploit them. To engineer systems where fragility isn’t a weakness, but a form of hyper-sensitivity. Markets don’t move – they worry. Similarly, these systems don’t simply detect – they amplify, resonate with, and ultimately reveal the faintest of signals by embracing their own inherent instability.

The work suggests that the true potential lies not in creating perfectly isolated quantum systems, but in designing ones that are exquisitely attuned to their environment, systems that feel the world around them. The question isn’t whether these systems are robust, but how gracefully they degrade, and what information is revealed in the process.

Original article: https://arxiv.org/pdf/2512.09870.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-11 15:22