Quantum Theory Elucidates Nonlinear Phononics with Analytical Solutions Considering Third and Fourth order Fluctuations

The ability to control crystal vibrations with brief bursts of energy presents exciting possibilities for materials science, but a comprehensive understanding of how these vibrations behave under strong excitation has remained elusive. Francesco Libbi and Boris Kozinsky now present a new theoretical framework that explains the dynamics of these vibrations, known as phonons, in a way that allows for precise mathematical solutions. Their work establishes that a powerful pulse can actively suppress, or squeeze, these vibrations, a phenomenon previously observed but not fully explained, and provides a systematic way to predict and control this effect. This breakthrough introduces a new approach to nonlinear phononics, potentially enabling the design of materials with tailored properties through the precise manipulation of lattice vibrations and offering a pathway to induce symmetry breaking in paraelectric materials.

Terahertz Phonon Control in Strontium Titanate

Research into nonlinear phononics explores how intense light, particularly terahertz radiation, manipulates atomic vibrations, or phonons, within materials like strontium titanate. This field aims to control material properties by driving these vibrations in a nonlinear manner, overcoming limitations of traditional approaches. Scientists combine theoretical modelling, using quantum mechanical simulations, with experimental observations to understand and harness these effects. A central idea is that manipulating phonons nonlinearly can unlock new functionalities in materials. Nonlinear phononics studies how strong excitation of lattice vibrations leads to deviations from simple harmonic behaviour, known as anharmonicities.

These anharmonicities can create new phonon modes, amplify existing ones, and ultimately drive structural changes within a material. Strontium titanate is a key focus due to its quantum paraelectric nature, meaning it is close to a ferroelectric phase transition and therefore sensitive to external stimuli. Researchers aim to use nonlinear phononics to induce or enhance ferroelectricity, or other desired properties, within this material. Terahertz pulses serve as the driving force for exciting phonons due to their frequency range matching many phonon modes in solids. Scientists rely heavily on ab initio calculations, based on density functional theory and beyond, to calculate phonon frequencies and anharmonicities, model material responses to terahertz pulses, and predict structural changes.

Machine learning techniques are also being integrated to improve the efficiency and accuracy of these complex calculations. Research suggests that strong optical excitation can quench, or suppress, lattice fluctuations in strontium titanate, potentially leading to enhanced functionality. Scientists are also investigating the possibility of using light to control the magnetic order in antiferromagnetic materials and amplify specific phonon modes through parametric amplification. This research extends to hydrogen-rich materials under extreme pressure, where quantum effects play a crucial role in determining crystal structure.

Gaussian Dynamics Predicts Nonlinear Phononic Behaviour

Scientists developed a novel theoretical framework to analyze nonlinear phononics, building upon the time-dependent self-consistent harmonic approximation. This method models the quantum density of a crystal as a Gaussian distribution within Wigner phase-space, parameterized by centroid positions, momenta, and covariance matrices representing fluctuations in position, momentum, and their coupling. The team focused on lattice fluctuations, represented by the covariance matrix, and how these fluctuations evolve under external stimuli. To advance the time-dependent self-consistent harmonic approximation, researchers analytically evaluated ensemble averages of forces and the curvature of the potential energy surface, enabling a fully atomistic and quantum generalization of existing nonlinear phononics models.

This analytical evaluation circumvents computationally demanding Monte Carlo integration typically required for large systems, providing a significant methodological innovation. The resulting theory accurately describes the evolution of key parameters, including atomic positions, momenta, and fluctuation matrices. The study demonstrates that the developed framework recovers previously established models when considering coupling with low-frequency acoustic modes and resonantly driven infrared modes. Furthermore, scientists derived a closed-form equation describing the dynamics of the diagonal components of lattice fluctuations, providing a simplified yet accurate representation of the system’s behaviour. This equation reveals a crucial mechanism: the sudden excitation of a phonon mode induces the quenching, or cooling, of its lattice fluctuations, reshaping the potential energy landscape.

Phonon Quenching Drives Symmetry Breaking in Crystals

This work presents a groundbreaking analytical theory of nonlinear phononics, establishing a framework for understanding and manipulating the dynamics of nuclear density in crystals. Scientists developed a method to explicitly analyze the influence of fluctuations on these dynamics, achieving exact solutions for nuclear time evolution while fully considering fluctuations arising from third- and fourth-order phonon couplings. The research demonstrates that a strong pulse displacing a phonon mode from equilibrium induces the quenching, or squeezing, of its lattice fluctuations, systematizing a previously observed mechanism. The team formulated a time-dependent self-consistent harmonic approximation framework, modelling the quantum density of a crystal as a Gaussian distribution in phase-space.

This approach allows for the calculation of the evolution of centroid positions, momenta, and covariance matrices representing position, momentum, and position-momentum fluctuations, with the latter specifically representing lattice fluctuations. Scientists derived equations governing the time evolution of these parameters, enabling the calculation of ensemble averages of forces and the curvature of the potential energy landscape. Through exact analytical calculations, the research demonstrates that when a fourth-order polynomial accurately models the interaction potential, ensemble averages of the potential energy and its derivatives can be computed analytically. This achievement significantly reduces computational cost and facilitates the use of high-order integration schemes. These calculations confirm that the quenching of lattice fluctuations of a specific phonon mode can be leveraged to induce symmetry breaking, opening new avenues for controlling material properties with light.