Light Squeezed at Band-Gap Frequency in New States

Researchers are increasingly investigating high-harmonic generation (HHG) through the lens of strong-field quantum optics, demonstrating that generated radiation often exhibits nonclassical light characteristics. However, a comprehensive quantum optical understanding of HHG originating from topological insulators remains elusive. Christian Saugbjerg Lange and Lars Bojer Madsen, both from the Department of Physics and Astronomy at Aarhus University, have addressed this knowledge gap by examining HHG responses within the Su-Schrieffer-Heeger model, a finite atomic chain exhibiting both trivial and nontrivial insulating phases supporting edge states. Their findings reveal squeezed light generation at the band-gap frequency for both phases, with harmonic spectra differentiating the phases, although this distinction weakens with increasing chain length due to increased overlap between bulk and edge states. This work elucidates the role of dipole coupling strength in governing nonclassical HHG and opens new avenues for exploring the protected generation of quantum light in strong-field physics.

Imagine building a complex electrical circuit where the very edges conduct power differently to the interior. New work explores how light emission from materials with unusual electronic properties, specifically those supporting edge states, exhibits a unique quantum character. This investigation demonstrates squeezed light at the material’s band-gap frequency, offering a pathway to control non-classical light generation.

Scientists are increasingly applying quantum mechanical descriptions to the interaction between light and matter, a field known as strong-field quantum optics. Recent work has demonstrated that light generated through high-harmonic generation (HHG), a process where intense laser fields create new frequencies of light, often exhibits nonclassical properties.

A complete quantum optical understanding of HHG originating from materials with unusual electronic structures has remained elusive. Researchers have investigated HHG from a model system exhibiting distinct topological phases, revealing how these phases influence the quantum characteristics of the emitted light. At the heart of this investigation lies the Su-Schrieffer-Heeger (SSH) model, a simplified representation of a chain of atoms capable of existing in both a standard, or trivial, insulating state and a topologically nontrivial state.

Unlike conventional insulators, the nontrivial phase supports edge states, which are electronic states confined to the boundaries of the material. These edge states possess unique properties, offering potential for applications in quantum technologies. Findings indicate that HHG occurring in both the trivial and nontrivial phases results in squeezed light at the material’s band-gap frequency, a measure of quantum noise reduction.

A key distinction emerges when examining the harmonic spectrum, the distribution of generated frequencies. The spectrum effectively differentiates between the two topological phases. However, the degree of squeezing, a more direct indicator of nonclassical light, only distinguishes the phases for shorter chains of atoms. Once systems become larger, this ability diminishes.

By attributing this behaviour to increased overlap between the bulk electronic states and the edge states in the nontrivial phase for smaller systems, the research highlights the importance of dipole couplings in governing the nonclassical response of HHG. The study of these quantum optical properties could lead to new methods for generating and controlling quantum light sources.

Investigations are now focusing on how to protect the generation of quantum light within strong-field physics, potentially opening avenues for advanced quantum devices and technologies. Beyond fundamental physics, these results offer a pathway to engineer materials with tailored quantum optical properties for future applications.

Semiclassical dynamics and perturbative treatment quantify harmonic squeezing

A perturbative Heisenberg dynamics (PHD) formulation underpins the methodology employed to examine the quantum optical properties of high-harmonic generation (HHG) radiation. This approach connects optical observables to semiclassical dynamics induced within the system, proceeding in two stages. Initially, an intense driving field, characterised by a coherent-state parameter exceeding one, is addressed exactly through a transformation yielding semiclassical dynamics.

Then, coupling to the quantized field is treated perturbatively to second order, considering a weak light-matter coupling constant of 4×10−8 atomic units. Calculations extended beyond the harmonic spectrum to quantify the degree of squeezing, a key indicator of non-classical light, in decibels using ηk,σ = −10 log10 4min θk,σ∈(0,π]) ∆Xk,σ(θk,σ) 2.

This involved minimising the quadrature variance, calculated via a complex expression incorporating the semiclassical current operator and its fluctuations, ∆jσ,sc(t). The PHD predicts a linear scaling of quadrature variance with the number of emitter systems, N, so both the signal and degree of squeezing are expected to increase alongside N. To apply these equations to the Su-Schrieffer-Heeger (SSH) model, a classical laser field with polarization aligned along the chain was used to drive the system.

Exploiting Peierls’ phase substitution, time-dependent hopping parameters were obtained, transforming the SSH Hamiltonian into a semiclassical time-dependent form. The semiclassical current operator was then defined, utilising single-particle eigenstates expressed as a superposition of amplitudes on each site within the unit cell. The many-body state was constructed as a Slater determinant, representing a specific combination of single-particle states.

Considering the system at half-filling, all states below the energy gap were initially occupied in the trivial phase, while one edge state was occupied in the nontrivial phase. To determine the quadrature variance, a complete set of many-body eigenstates was inserted, simplifying the expectation value of current correlations to a summation over transition currents between many-body states. The SSH model comprises independent particles and both the Hamiltonian and current operator are single-particle operators, meaning only single-particle excitations need consideration.

Topological phase transitions evidenced by enhanced harmonic generation and spectral signatures

Harmonic spectra obtained from the Su-Schrieffer-Heeger (SSH) model reveal distinctions between trivial and nontrivial topological phases. The research demonstrates that interband mechanisms efficiently generate harmonics above the band-gap energy, resulting in an increased signal around ωk/ωL ≈22 and 23 for long and short chains, respectively. The presence of edge states within the band gap allows harmonic radiation generation down to the 11’th and 12’th harmonic in the topological phase for both chain lengths.

In particular, harmonic peaks appear below the band-gap energies for both topological phases, originating from propagation within the band between spectrally close-lying states. The topological nontrivial phase consistently exhibits a stronger signal across all frequencies, attributed to transitions via zero-energy states and a resultant higher population of states above the gap.

Calculations of the degree of squeezing, a measure of the nonclassicality of the generated light, show little difference between phases for a long chain of 50 cells. However, for a shorter chain of 12 cells, the topological phase displays a larger degree of squeezing at frequencies below the band-gap energy, specifically around ω/ωL ∼10−20. Detailed analysis reveals that excluding edge-state transition currents leaves the degree of squeezing unchanged for the long chain.

By contrast, omitting these currents eliminates the discrimination between phases in the short chain, confirming the edge states’ responsibility for the observed signal at lower frequencies. At a chain length of 12 cells, the overlap between bulk and edge states becomes significant, allowing the edge states to contribute to the signal. Once the bulk-edge contribution is negligible, the degree of squeezing does not differ between the two topological phases of the SSH model. These findings demonstrate that strong-field quantum optics provides access to nonclassical electronic responses and that the strength of dipole couplings governs the nonclassical high-harmonic generation response.

Squeezed light generation via topological states in solid-state harmonic materials

Once considered a purely gaseous phenomenon, high-harmonic generation is now being actively explored in solid-state materials, opening avenues for compact and efficient sources of extreme ultraviolet and x-ray radiation. Realising the full potential of solid-state harmonics demands precise control over the quantum properties of the emitted light, a challenge complicated by the inherent disorder and many-body effects within materials.

This research offers a step forward by examining harmonic generation not simply as a response to a strong field, but as a distinctly quantum optical process influenced by the material’s underlying electronic structure. The investigation of topological materials, those with unique edge states, introduces a new layer of complexity. By modelling harmonic generation within a Su-Schrieffer-Heeger chain, researchers demonstrate that the generated light exhibits squeezed quantum states, a non-classical property.

Discerning the material’s topological phase based solely on the degree of squeezing proves difficult for smaller systems, as edge and bulk states increasingly overlap. For years, a key obstacle has been identifying strong signatures of topology in harmonic spectra. This study highlights the importance of quantifying the quantum state of the emitted light itself, unlike previous work focusing on spectral features.

Maintaining these squeezed states outside the confines of the simulation remains a significant hurdle. Future work might concentrate on designing materials where edge states are more spatially separated, or on exploring alternative topological phases that produce a more pronounced quantum signature. Bridging the gap between these theoretical models and experimental observations is a critical direction, demanding careful consideration of material imperfections and decoherence effects.