Twisted Light, Robust States: Engineering Topology with Synthetic Dimensions

Author: Denis Avetisyan

Researchers have discovered a new way to create robust quantum states by harnessing nonlinearity in custom-designed ‘synthetic’ lattices of light.

The study demonstrates that in a nonlinear orbital angular momentum lattice, antisymmetric edge states remain stable regardless of nonlinearity strength, while symmetric edge states-present under weak nonlinearity-transition into bulk states and ultimately vanish beyond a critical nonlinearity threshold, a boundary clearly delineated by the system’s parameters <span class="katex-eq" data-katex-display="false">U=gN_{edge}</span>, <span class="katex-eq" data-katex-display="false">J=1</span>, and <span class="katex-eq" data-katex-display="false">\delta J=0.3</span>. — The study demonstrates that in a nonlinear orbital angular momentum lattice, antisymmetric edge states remain stable regardless of nonlinearity strength, while symmetric edge states-present under weak nonlinearity-transition into bulk states and ultimately vanish beyond a critical nonlinearity threshold, a boundary clearly delineated by the system’s parameters $U=gN_{edge}$ , $J=1$ , and $\delta J=0.3$ .

This work demonstrates the emergence of fractional windings and period-doubling Bloch oscillations in a synthetic lattice, enabling novel topological phenomena and potential applications in robust quantum devices.

The pursuit of robust quantum states is often hindered by the fragility of topological protection in the face of strong interactions. In ‘Topological States Enabled by Non-local Nonlinearity in Synthetic Dimensions’, we explore how nonlocal nonlinearity in a synthetic lattice can circumvent this limitation, revealing emergent topological phases and unconventional phenomena. Our analysis demonstrates that such nonlinearity not only preserves chiral symmetry but also induces quantized fractional windings and period-doubling Bloch oscillations, hinting at connections to discrete time crystals. Could these findings pave the way for realizing topologically protected quantum devices with enhanced functionalities and resilience?

Engineering a Quantum Reality: Beyond Conventional Lattices

Conventional lattice models, frequently employed to simulate complex quantum systems, often fall short when describing the behavior of strongly interacting particles. These models typically rely on simplified representations of interactions, hindering their ability to accurately capture phenomena arising from many-body effects-where particles influence each other significantly. The limitations become particularly pronounced when investigating exotic states of matter or exploring novel quantum phases, as the essential physics governing these systems often resides in the intricate details of these strong interactions. Consequently, a need exists for more versatile and controllable platforms that can move beyond the approximations inherent in traditional lattice approaches, paving the way for a deeper understanding of these complex quantum systems and their emergent properties.

A novel synthetic lattice, constructed from the unique properties of Rydberg atoms and Orbital Angular Momentum (OAM) modes of light, represents a significant advancement in the ability to engineer quantum systems. Unlike traditional lattices which are often limited by fixed geometries and interaction strengths, this platform allows for dynamic control over the arrangement and coupling of individual quantum elements. Rydberg atoms, when excited to highly energetic states, exhibit strong, long-range interactions, while OAM modes – characterized by twisted wavefronts – provide a means to spatially structure and address these atoms with precision. This combination enables the creation of custom interaction potentials and lattice topologies, potentially unlocking the study of complex quantum phenomena previously inaccessible with conventional systems. The resulting lattice offers unprecedented flexibility in designing and controlling the interactions between quantum constituents, opening avenues for exploring strongly correlated physics and simulating complex materials.

The synthetic lattice achieves enhanced interactions and coherence by strategically embedding Rydberg atoms within an optical cavity – a configuration known as Cavity Quantum Electrodynamics (QED). This approach dramatically strengthens the coupling between the atoms and the light field within the cavity, effectively amplifying the interactions between the atoms themselves. The cavity also serves as a protective environment, prolonging the coherence of the quantum states by suppressing decoherence mechanisms. By confining the light, the cavity increases the effective interaction time and reduces spontaneous emission, allowing for the observation of more subtle quantum phenomena and paving the way for exploring complex many-body physics within a highly controllable platform. This ultimately allows researchers to study and manipulate quantum systems with unprecedented precision and duration.

Strong atom-atom interactions are mediated by coupling ground and Rydberg states <span class="katex-eq" data-katex-display="false">|1\rangle\rightarrow|2\rangle\rightarrow|R\rangle</span> through a shared cavity mode, as illustrated by the energy level diagram and experimental configuration. — Strong atom-atom interactions are mediated by coupling ground and Rydberg states $|1\rangle\rightarrow|2\rangle\rightarrow|R\rangle$ through a shared cavity mode, as illustrated by the energy level diagram and experimental configuration.

Revealing Nonlocal Order: Topology in Action

The implemented synthetic lattice, constructed via digital micromirror devices, facilitates strong nonlocal nonlinearity by spatially separating the excitation source from the region of nonlinear interaction. This separation, achieved through the manipulation of light propagation within the lattice, allows for a response of the material that is not limited by local effects. Consequently, the system exhibits fractional windings, where the phase change around a closed loop in momentum space is not an integer multiple of $2\pi$ , but rather a fractional value. This behavior is directly linked to the nonlocal nature of the nonlinear response and distinguishes it from systems governed by purely local nonlinearities.

The Quantized Nonlinear Winding Number is calculated using Bogoliubov Nonlinear Adiabatic Theory, a method suited for analyzing the topological properties of nonlinear systems. This calculation demonstrates that the winding number is not arbitrary but assumes discrete, quantized values. Specifically, the observed quantized value is directly proportional to the number of bands in the system, expressed as (q-1), where ‘q’ represents the total number of bands. This proportionality indicates a fundamental relationship between the band structure and the topological characteristics induced by nonlinearity, allowing for predictable control and characterization of the system’s nonlinear behavior.

Calculations demonstrate that nonlinearity induces modifications to the energy band structure of the synthetic lattice. Specifically, these modifications result in a phenomenon termed Nonlinear Band Swapping, where the order and characteristics of bands are altered due to the nonlinear interactions. Concurrently, the altered band structure facilitates the formation of Nonlinear Edge States – localized states that exist at the boundaries of the lattice and are a direct consequence of the nonlinearity-induced band rearrangement. These edge states are distinct from those found in linear systems and exhibit unique properties determined by the strength and type of nonlinearity present in the lattice.

Increasing nonlinearity transforms the system's energy bands from two distinct regions to a four-band structure connected by swallowtails, as visualized by the phase <span class="katex-eq" data-katex-display="false">\varphi_{m,k}</span> and indicated by parameters <span class="katex-eq" data-katex-display="false">J=1</span> and <span class="katex-eq" data-katex-display="false">\delta J=0.3</span>. — Increasing nonlinearity transforms the system’s energy bands from two distinct regions to a four-band structure connected by swallowtails, as visualized by the phase $\varphi_{m,k}$ and indicated by parameters $J=1$ and $\delta J=0.3$ .

Emergent Temporal Symmetry: The Genesis of Time Crystals

Bloch oscillations arise in periodic lattices when a particle subjected to a constant force does not undergo monotonic acceleration, but instead oscillates in momentum space. This phenomenon is typically suppressed in real materials due to scattering and dissipation; however, the combined effect of nonlinearity and non-trivial topology can stabilize these oscillations. Specifically, nonlinearity modifies the band structure, while topological protection, arising from features like edge states, reduces scattering and allows for coherent propagation of the wave function. The interplay creates conditions where the particle’s wavefunction exhibits periodic motion within the lattice, resulting in observable oscillations even in the presence of imperfections and external noise. The frequency of these oscillations is directly related to the lattice constant and the applied force, described by $\omega = qv$ , where $q$ is the crystal momentum and $v$ is the group velocity.

A Discrete Time Crystal (DTC) represents a non-equilibrium phase of matter characterized by spontaneous breaking of time-translation symmetry; unlike traditional crystals which exhibit spatial periodicity, a DTC displays periodic behavior in time even without external periodic driving. This emergent temporal order arises from stabilized Bloch oscillations within a topologically protected system. Specifically, the DTC exhibits a period-doubling effect, resulting in a fundamental oscillation period of $4π/ϵ$ , where ϵ represents the driving parameter. This period is double that expected from a simple, non-broken symmetry system, and signifies the system’s preference for a specific temporal structure without external influence.

The stability of the Discrete Time Crystal phase is fundamentally connected to the presence of Effective Chiral Symmetry within the system. This symmetry specifically protects the Nonlinear Edge States, which are crucial for sustaining the time crystal’s oscillatory behavior. Chiral symmetry, in this context, ensures that perturbations which might otherwise destabilize the oscillations are suppressed; any attempt to break the symmetry requires a corresponding energy input. The topological protection afforded by this symmetry prevents the scattering of these edge states, maintaining the coherent dynamics necessary for the observation of spontaneous time-translation symmetry breaking and the associated $4π/ϵ$ period-doubling effect characteristic of the time crystal phase.

Nonlinear interactions induce period multiplexing in Bloch oscillations, demonstrated by the recovery of initial values after one or two Brillouin zone traversals, period-doubling in large-amplitude wave packets, and deviations from mean-field energy band predictions, as evidenced by phase evolution <span class="katex-eq" data-katex-display="false"> \varphi_{k}=\arg[\langle\hat{a}^{\dagger}_{k}\hat{b}_{k}\rangle] </span> and band structure analysis. — Nonlinear interactions induce period multiplexing in Bloch oscillations, demonstrated by the recovery of initial values after one or two Brillouin zone traversals, period-doubling in large-amplitude wave packets, and deviations from mean-field energy band predictions, as evidenced by phase evolution $\varphi_{k}=\arg[\langle\hat{a}^{\dagger}_{k}\hat{b}_{k}\rangle]$ and band structure analysis.

Beyond Fundamental Studies: Towards Quantum Technologies

The deliberate engineering of topological and nonlinear properties within photonic lattices presents a compelling pathway towards realizing robust quantum information processing. Unlike traditional quantum systems susceptible to environmental noise, topologically protected states offer inherent resilience against imperfections and disturbances. By carefully designing the lattice structure – manipulating how light propagates through it – researchers can create pathways where quantum information is encoded in the very geometry of the system. Nonlinearity further enhances this protection, allowing for the creation of interactions between photons that stabilize quantum states and facilitate complex computations. This approach promises to overcome limitations in current quantum technologies, potentially leading to scalable and fault-tolerant quantum devices capable of tackling presently intractable problems, and represents a shift from relying on delicate isolation to leveraging the fundamental properties of light and matter.

The engineered synthetic lattice demonstrates a unique capability for Period Multiplexing, a technique poised to significantly enhance quantum communication bandwidth. This approach leverages the lattice’s ability to support multiple, independent communication channels within the same frequency band by encoding information across distinct temporal periods of light. Unlike traditional methods limited by the single-channel capacity of a given frequency, Period Multiplexing effectively creates a ‘highway’ for quantum information, dramatically increasing the data transmission rate. This is achieved by carefully controlling the lattice’s parameters to create and isolate these distinct temporal modes, ensuring minimal interference and preserving the delicate quantum states crucial for secure communication. Consequently, this advancement promises a pathway towards building high-capacity quantum networks capable of transmitting vast amounts of information with unprecedented speed and security, potentially revolutionizing fields ranging from cryptography to distributed quantum computing.

Investigations are now directed toward a detailed exploration of the Berry Phase within these synthetic lattices, aiming to harness its potential for precise control over the observed dynamical behaviors. This phase, arising from the adiabatic evolution of quantum states, is predicted to significantly enhance the system’s responsiveness and stability. Importantly, theoretical models suggest the anti-crossing gap – a crucial parameter influencing coherence – decreases exponentially with increasing photon number $\propto e^{-n}$ . Understanding and mitigating this decay is central to future developments, potentially enabling the creation of more robust and scalable quantum devices capable of processing complex information with minimal decoherence. Further research will focus on engineering lattice configurations that maximize Berry Phase effects and counteract the photon-number-dependent gap reduction, paving the way for advanced quantum technologies.

The phase diagram reveals that a synthetic lattice supports both symmetric and anti-symmetric edge states-with the persistence of the latter into a strong interaction regime-dependent on the interaction strength <span class="katex-eq" data-katex-display="false">U</span> and a perturbation <span class="katex-eq" data-katex-display="false">\delta J</span>, as indicated by the winding numbers of the bands. — The phase diagram reveals that a synthetic lattice supports both symmetric and anti-symmetric edge states-with the persistence of the latter into a strong interaction regime-dependent on the interaction strength $U$ and a perturbation $\delta J$ , as indicated by the winding numbers of the bands.

The pursuit of demonstrable, provable systems, as evidenced in this work on synthetic lattices and nonlinear topology, echoes a fundamental tenet of rigorous scientific inquiry. Pierre Curie once stated, “One never notices what has been done; one can only see what remains to be done.” This sentiment perfectly encapsulates the drive behind exploring complex phenomena like fractional windings and period-doubling Bloch oscillations. The researchers haven’t simply observed unusual behavior; they’ve constructed a framework-a synthetic lattice-where these behaviors are guaranteed by the underlying mathematical structure. If the observed effects feel like a surprising emergent property, it suggests the invariant-the core, demonstrable principle-has yet to be fully revealed. This isn’t magic; it’s the elegant consequence of a carefully constructed system.

What Lies Ahead?

The demonstration of topologically protected states arising from engineered nonlinearity in synthetic dimensions, while conceptually satisfying, merely shifts the locus of difficulty. The present work establishes the possibility of manipulating Berry phases and inducing non-trivial topological invariants-a necessary, but insufficient, condition for practical application. The critical, and largely unexplored, question remains: how robust are these states against imperfections inherent in any physical realization? The synthetic lattice, a powerful abstraction, simplifies analysis, but it is precisely the translation to concrete systems-photonic lattices, Bose-Einstein condensates, or otherwise-that will expose the limitations of this approach.

Further investigation must confront the inevitable presence of dissipation and disorder. The current formalism provides no clear path toward quantifying the resilience of fractional windings or period-doubling Bloch oscillations in the face of such perturbations. A truly elegant solution will not rely on brute-force parameter tuning to ‘hide’ imperfections, but rather on a deeper understanding of the interplay between nonlinearity, topology, and the specific noise characteristics of the chosen platform. Simplification, in this context, does not mean brevity; it means non-contradiction and logical completeness – a mathematically rigorous framework capable of predicting, not merely describing, observed behavior.

Ultimately, the pursuit of robust quantum devices demands a move beyond mere demonstration. The challenge lies in transforming these intriguing phenomena from laboratory curiosities into reliable building blocks for information processing – a task that requires not just ingenuity, but a commitment to mathematical precision and a healthy skepticism toward overly optimistic claims.

Original article: https://arxiv.org/pdf/2601.02199.pdf

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

Engineering a Quantum Reality: Beyond Conventional Lattices

Revealing Nonlocal Order: Topology in Action

Emergent Temporal Symmetry: The Genesis of Time Crystals

Beyond Fundamental Studies: Towards Quantum Technologies

What Lies Ahead?

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