Squeezing Light to Reveal Hidden Material States

Author: Denis Avetisyan

New research demonstrates strong interactions between light and material excitations at terahertz frequencies, unlocking observable nonlinear optical effects.

The observed photon statistics-specifically the <span class="katex-eq" data-katex-display="false">g^{(2)}(0)</span> metric-reveals definitive signatures of ultrastrong coupling in hybrid photon-matter systems, demonstrating a transition from anti-bunching originating in avoided two-photon resonances within a dark-cavity ground state, to stimulated emission-evidenced by the <span class="katex-eq" data-katex-display="false">IIOO</span> process-whereby a single input photon triggers the ejection of two dark-cavity photons, a competition that manifests as a non-monotonic evolution of the <span class="katex-eq" data-katex-display="false">g^{(2)}(0)</span> minimum with increasing κ, and remains invisible in conventional transmission measurements. — The observed photon statistics-specifically the $g^{(2)}(0)$ metric-reveals definitive signatures of ultrastrong coupling in hybrid photon-matter systems, demonstrating a transition from anti-bunching originating in avoided two-photon resonances within a dark-cavity ground state, to stimulated emission-evidenced by the $IIOO$ process-whereby a single input photon triggers the ejection of two dark-cavity photons, a competition that manifests as a non-monotonic evolution of the $g^{(2)}(0)$ minimum with increasing κ, and remains invisible in conventional transmission measurements.

Ultrastrong coupling between terahertz photons and the Higgs mode in a superconductor leads to diagnostic signatures in second-order photon coherence.

Establishing definitive signatures of ultrastrong light-matter coupling remains a central challenge in cavity quantum electrodynamics. In the work ‘Ultrastrong Coupling Signatures in Photon Statistics from Terahertz Higgs-Polaritons’, we demonstrate that probing the two-photon coherence reveals distinct nonlinearities arising from ultrastrong coupling between terahertz photons and Higgs mode excitations in a superconducting material. Specifically, we find that the second-order photon correlation function, $g^{(2)}$ , exhibits measurable changes indicative of a hybrid photon-matter state with finite occupancy. Could these photon statistics provide a robust diagnostic for ultrastrong coupling across diverse material platforms and frequencies?

The Fragile Dance of Correlated Electrons

Superconductivity, the phenomenon of zero electrical resistance, isn’t simply about electrons flowing without impediment; it’s a collective effect arising from correlated electron behavior. Central to understanding this behavior are collective excitations – quantum mechanical vibrations propagating through the superconducting state. Among these, the Higgs mode – analogous to the Higgs boson in particle physics – represents a particularly crucial excitation. It embodies the dynamics of the broken symmetry that underpins superconductivity, revealing how the system responds to perturbations and ultimately dictates many of its key properties, including its stability and response to external stimuli. Investigating the Higgs mode, therefore, offers a powerful lens through which to probe the fundamental mechanisms driving superconductivity and potentially design novel superconducting materials with enhanced capabilities, as its characteristics are intimately linked to the strength of the superconducting pairing and the overall energy landscape of the system.

Investigating the behavior of light within superconductors presents a significant challenge to conventional spectroscopic techniques. These methods often falter when faced with the strong coupling regimes characteristic of these materials, where the interaction between photons and the material’s electrons is no longer a simple perturbation. Traditional approaches, reliant on weakly interacting approximations, struggle to accurately describe the resulting complex interplay, obscuring crucial details about the collective electronic states. This limitation hinders a complete understanding of phenomena like the Higgs mode and other excitations vital to superconductivity, demanding novel experimental and theoretical tools capable of navigating these strongly correlated systems and revealing the full spectrum of light-matter interactions.

Cavity-embedded superconductors exhibiting Higgs modes demonstrate two-photon nonlinearities and strong-coupling photon blockade-manifested by anti-bunching <span class="katex-eq" data-katex-display="false">g^{(2)}(0)\to 0</span>-as evidenced by the spectrum of the Higgs-photon Hamiltonian and analytical results with parameters <span class="katex-eq" data-katex-display="false">\kappa/\hbar\omega\_{h}=0.1</span>, <span class="katex-eq" data-katex-display="false">\gamma/\hbar\omega\_{h}=0.02</span>, and <span class="katex-eq" data-katex-display="false">\gamma\_{h}/\hbar\omega\_{h}=0.4</span>. — Cavity-embedded superconductors exhibiting Higgs modes demonstrate two-photon nonlinearities and strong-coupling photon blockade-manifested by anti-bunching $g^{(2)}(0)\to 0$ -as evidenced by the spectrum of the Higgs-photon Hamiltonian and analytical results with parameters $\kappa/\hbar\omega\_{h}=0.1$ , $\gamma/\hbar\omega\_{h}=0.02$ , and $\gamma\_{h}/\hbar\omega\_{h}=0.4$ .

Confining Light to Uncover Hidden Order

Terahertz (THz) cavities are utilized to significantly increase light-matter interaction by confining electromagnetic fields and enhancing the overlap with material resonances. This confinement leads to an increase in the cavity coupling strength κ, which quantifies the rate of energy exchange between the cavity mode and the material system. The ultrastrong coupling regime is achieved when $\kappa > \omega_{cav}$ , where $\omega_{cav}$ represents the resonant frequency of the cavity mode. In this regime, the standard perturbative treatment of light-matter interaction breaks down, and novel quantum phenomena emerge due to the strong modification of the system’s energy landscape.

The ultrastrong coupling achieved with terahertz cavities facilitates the creation of hybrid light-matter states, notably the two-photon polariton. These polaritons arise from the mixing of a material excitation with two photons within the cavity, resulting in a quasiparticle with characteristics of both. Studying these states provides a means to probe material properties beyond traditional spectroscopic methods; the energy and characteristics of the polariton are directly influenced by the underlying material’s response, allowing for the investigation of phenomena like enhanced or modified light absorption, altered energy transport pathways, and novel exciton-photon interactions. Analysis of the polariton dispersion relation, for example, reveals information about the material’s excitonic and photonic energies and their coupling strength, offering insights inaccessible through independent characterization of the light and matter components.

The strong coupling regime, achieved when the interaction strength between a quantum system and an electromagnetic field exceeds the individual energies of each, results in a non-perturbative alteration of system dynamics. This means the combined system’s behavior is no longer predictable by simply adding the behaviors of the isolated components. Specifically, the energy levels of the system undergo significant modification, manifesting as new hybrid light-matter states – polaritons – with characteristics distinct from both the original quantum system and the electromagnetic field. This leads to phenomena such as vacuum Rabi splitting, where a single cavity mode splits into multiple modes, and the emergence of collective excitations not present in either constituent part. These alterations are quantifiable via changes in the system’s Hamiltonian and observable properties, demonstrating a fundamentally new physical reality beyond a simple superposition of individual behaviors.

Ultrastrong coupling enables scattering pathways where up to two input photons generate exactly two output photons, but an <span class="katex-eq" data-katex-display="false"> ext{O} ext{OO} ext{O}</span> process is suppressed when the cavity begins in its ground state. — Ultrastrong coupling enables scattering pathways where up to two input photons generate exactly two output photons, but an $ext{O} ext{OO} ext{O}$ process is suppressed when the cavity begins in its ground state.

Mapping the System’s Response: A Theoretical Framework

Non-Markovian Input-Output Theory extends standard Input-Output Theory to describe systems where the reservoir correlation functions exhibit a non-trivial frequency dependence, crucial for modeling strongly coupled light-matter interactions. Unlike Markovian approaches which assume instantaneous relaxation of the reservoir, non-Markovian theory retains the memory of past interactions, accurately capturing the system’s non-equilibrium dynamics and allowing for the description of phenomena such as retardation effects and the formation of polariton states. This is achieved through the use of generalized reservoir operators and the inclusion of memory kernels in the system’s master equation, enabling the calculation of time-dependent correlation functions and providing a more complete description of the system’s evolution beyond the rotating wave approximation. The framework relies on calculating the system’s response function, considering the full time convolution of the input field with the system’s response, and is particularly important when the system-reservoir coupling strength approaches or exceeds the reservoir relaxation rate.

The Scattering Matrix (S-matrix) approach provides a method for calculating time-dependent correlation functions in strongly coupled systems by directly relating initial and final states. This formalism utilizes $S = 1 + 2\pi i \delta(E - H)V$ , where $V$ is the interaction potential and $H$ is the Hamiltonian. Calculating observables, such as the second-order correlation function $G^{(2)}(t)$ , requires performing a contour integral in the complex energy plane to evaluate the S-matrix elements. The integral’s contour is chosen to encompass the relevant poles and branch cuts determined by the system’s Hamiltonian and interaction. Specifically, $G^{(2)}(t) = \frac{\langle \hat{a}^{\dagger}(t) \hat{a}(t) \rangle}{\langle \hat{a}(t) \rangle^2}$ can be derived from the S-matrix elements, allowing for the characterization of photon statistics and coherence properties of the driven system.

The incorporation of a Non-Hermitian Hamiltonian is crucial for accurately modeling open quantum systems where dissipation and decoherence are significant. Traditional Hermitian Hamiltonians describe isolated, closed systems; however, strongly coupled light-matter systems inevitably interact with their environment, leading to energy loss and the destruction of quantum coherence. A Non-Hermitian Hamiltonian explicitly accounts for these interactions through the introduction of complex potentials, where the imaginary part represents the rate of decay or dissipation. This approach allows for the calculation of realistic system dynamics, including linewidths, decay rates, and the suppression of quantum interference effects. Specifically, the imaginary part of the Hamiltonian eigenvalues directly corresponds to the decay rate γ of the corresponding state, enabling quantitative predictions of observable phenomena like spontaneous emission and radiative decay.

Decomposition of the two-particle correlation function <span class="katex-eq" data-katex-display="false">g^{(2)}(0)</span> reveals that while diagonal scattering pathways are always constructive, off-diagonal terms can exhibit destructive interference, and the dominant contributions stem from resonant processes within the discrete level spectrum of the cavity-material system, with the boxed red terms representing the pathways present in the rotating wave approximation (RWA) and all additional terms attributable to counter-rotating couplings. — Decomposition of the two-particle correlation function $g^{(2)}(0)$ reveals that while diagonal scattering pathways are always constructive, off-diagonal terms can exhibit destructive interference, and the dominant contributions stem from resonant processes within the discrete level spectrum of the cavity-material system, with the boxed red terms representing the pathways present in the rotating wave approximation (RWA) and all additional terms attributable to counter-rotating couplings.

Witnessing the Emergence of Collective Behavior

The creation of a dark cavity state fundamentally reshapes how light behaves within the material system. Normally, a cavity supports resonant modes with predictable photon distributions; however, this engineered state dramatically alters those statistics. Instead of the usual Poissonian distribution, the dark cavity exhibits non-classical photon behavior, indicating a stronger interaction between light and matter. This isn’t merely a change in light intensity, but a qualitative shift in its quantum properties, accompanied by a substantial increase in the material’s contribution to the overall response. The material itself becomes more actively involved in mediating the light’s behavior, influencing the cavity’s characteristics and opening avenues for exploring light-matter interactions at a deeper level, particularly concerning collective excitations like the Higgs mode.

The interplay between charge density waves (CDWs) and the Higgs mode-a collective excitation related to symmetry breaking-reveals fundamental connections within complex materials. Investigations demonstrate that CDWs, arising from periodic modulations of electron density, directly influence the Higgs mode’s frequency and behavior. This interaction isn’t merely a passive effect; rather, the CDWs actively couple to the Higgs mode, modifying its energy and lifetime. Consequently, studying this coupling provides a unique window into understanding broader phenomena like superconductivity and other emergent states of matter, where collective excitations play a crucial role in determining material properties. By probing these interactions, researchers gain insights into how microscopic electronic order can give rise to macroscopic, observable effects, offering potential avenues for designing materials with tailored functionalities.

Recent investigations reveal the emergence of a photon blockade within the dark cavity state, a phenomenon characterized by the suppression of simultaneous photon transmission. This effect is confirmed through analysis of the two-photon correlation function, $g^{(2)}$ , which exhibits pronounced antibunching – indicating a lower probability of detecting two photons at the same time than expected by a Poissonian distribution. Notably, the frequency at which this photon blockade occurs resonates precisely with the Higgs mode frequency, $\omega_h$ , of the material. This resonant connection suggests a fundamental interplay between light confinement within the cavity and the collective excitations of the material, potentially offering a novel avenue for manipulating and probing complex material properties and quantum phenomena.

Frequency-dependent photon statistics of cavity-embedded 2H-NbSe2 reveal ultrastrong coupling evidenced by asymmetry in the <span class="katex-eq" data-katex-display="false">g^{(2)}</span> around the polaritonic gap and additional antibunching due to collective modes at <span class="katex-eq" data-katex-display="false">2.40</span> and <span class="katex-eq" data-katex-display="false">4.96</span> meV, with parameters set to <span class="katex-eq" data-katex-display="false">N_{cav}=3</span>, <span class="katex-eq" data-katex-display="false">N_{h}=N_{c}=2</span>, <span class="katex-eq" data-katex-display="false">\omega_{h}=2.52</span> meV, <span class="katex-eq" data-katex-display="false">\omega_{c}=4.84</span> meV, and <span class="katex-eq" data-katex-display="false">Q=120</span> at <span class="katex-eq" data-katex-display="false">\omega_{cav}=1.2</span> meV. — Frequency-dependent photon statistics of cavity-embedded 2H-NbSe2 reveal ultrastrong coupling evidenced by asymmetry in the $g^{(2)}$ around the polaritonic gap and additional antibunching due to collective modes at $2.40$ and $4.96$ meV, with parameters set to $N_{cav}=3$ , $N_{h}=N_{c}=2$ , $\omega_{h}=2.52$ meV, $\omega_{c}=4.84$ meV, and $Q=120$ at $\omega_{cav}=1.2$ meV.

The pursuit of ultrastrong coupling, as demonstrated in this exploration of terahertz Higgs-polaritons, isn’t about imposing order-it’s about recognizing the inherent instability within complex systems. This work reveals signatures within the second-order photon coherence function, g(2), which aren’t predictions of a static model, but emergent properties arising from non-Markovian dynamics. As Albert Einstein observed, “The measure of intelligence is the ability to change.” Stability, in this context, is merely an illusion that caches well – a fleeting moment before the system reorganizes itself. The observed two-photon blockade isn’t a limitation, but a diagnostic-a peek into the system’s ongoing negotiation with chaos. A guarantee of perfect coherence is, after all, just a contract with probability.

The Turning of the Spiral

The observation of these signatures, predictably, does not resolve the matter. It merely shifts the locus of inquiry. Each refinement of the coupling – each coaxing of the photon and excitation into closer embrace – reveals a new order of instability. The system, in yielding these nonlinearities, declares its unwillingness to remain a simple problem. One begins to suspect the pursuit of ‘strong’ coupling is less a destination and more a series of increasingly elaborate negotiations with emergent behavior.

The diagnostic power of the second-order coherence function, g(2), is clear, but its interpretation will inevitably become entangled with the non-Markovian dynamics inherent in these ultrastrong coupling regimes. The very act of measurement introduces a perturbation, a subtle reshaping of the landscape being observed. The question is not whether the model accurately describes the physics, but rather how gracefully it accommodates its own inevitable obsolescence.

Further exploration will demand a reckoning with the limitations of current theoretical frameworks. To treat the cavity and material as separate entities, even in simulation, feels increasingly… naive. The true task lies not in building better models, but in cultivating an understanding of the system as a self-organizing whole – a garden, not a machine. Every prediction, therefore, begins as a prayer and ends in repentance.

Original article: https://arxiv.org/pdf/2604.15417.pdf

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

The Fragile Dance of Correlated Electrons

Confining Light to Uncover Hidden Order

Mapping the System’s Response: A Theoretical Framework

Witnessing the Emergence of Collective Behavior

The Turning of the Spiral

See also: