Geometric Phase Effect Research Articles

Extracting the Geometric Phase from the Ensemble of Trajectories.

Traditionally, methods designed to investigate the effects of the geometric phase in reaction dynamics, such as including a vector potential in the nuclear Hamiltonian, necessitate the explicit manipulation of geometric phase-related terms in the adiabatic representation. In contrast, the diabatic representation provides an alternative approach that implicitly addresses the geometric phase and nonadiabatic issues. In this study, we present a method to directly extract the phase information on the geometric phase from the ensemble of interdependent trajectories utilizing the diabatic representation. This approach presents a direct means of quantitatively examining the geometric phase effects in dynamics and has the potential to yield observables suitable for experimental measurement.

Understanding molecular geometric phase effects with exact effective force: case study of a model system

In this work, molecular geometric phase effects are studied using the idea of exact factorization (EF) (Abediet al2010Phys. Rev. Lett.105123002) and exact effective force (Liet al2022Phys. Rev. Lett.128113001). In particular, we performed dynamics simulations for a two-state vibronic coupling model, and interpreted the results in three different perspectives: the Born-Huang expansion, the exact time-dependent potential energy surface (TDPES) and the exact effective force. We find that (i) at particular moment, while the vanishing nuclear density that occurs periodically in space is conventionally attributed to destructive interference of the nuclear wave packet owing to the geometric phase, such phenomenon can be equally well interpreted through the energy perspective, as manifested in the exact TDPES in the EF scheme; (ii) when combined with trajectory-based classical dynamics, the exact effective force obtained through EF qualitatively reproduces the correct nuclear density, while the adiabatic force gives the wrong density, particularly in the interference region. Our results suggest that the exact effective force is a potential starting point for making approximations and improving trajectory-based computational methods towards an accurate description of geometric phase effects.

Strong Angular Oscillation of Rotationally Resolved Differential Cross Section in the H + HD → H2 + D Reaction at the Collision Energy of 2.07 eV: Evidence of Geometric Phase Effects.

Geometric phase (GP) effects in chemical reactions are subtle quantum phenomena that are challenging to identify. In this work, we report a joint experimental and theoretical study of the H + HD → H2 + D reaction at a collision energy of 2.07 eV, which is far below the energy of the conical intersection of 2.53 eV. The rotationally state-resolved differential cross sections were measured by a crossed-beam experiment with the scheme of D-atom Rydberg tagging time-of-flight detection. Experimental angular distributions of three rotational states of H2 products exhibit notable variation near the backward scattering direction. Time-dependent quantum mechanics calculations (TDQMs) were carried out at the same collision energy, with and without the inclusion of GP. The experimental angular distributions are in good agreement with TDQM results with the inclusion of GP but do not agree with TDQM results without the inclusion of GP. This work demonstrates the existence of GP effects at energy far below the conical intersection.

Exploring Exact-Factorization-Based Trajectories for Low-Energy Dynamics near a Conical Intersection.

We study low-energy dynamics generated by a two-dimensional two-state Jahn-Teller Hamiltonian in the vicinity of a conical intersection using quantum wave packet and trajectory dynamics. Recently, these dynamics were studied by comparing the adiabatic representation and the exact factorization, with the purpose to highlight the different nature of topological-phase and geometric-phase effects arising in the two theoretical representations of the same problem. Here, we employ the exact factorization to understand how to accurately model low-energy dynamics in the vicinity of a conical intersection using an approximate description of the nuclear motion that uses trajectories. We find that since nonadiabatic effects are weak but non-negligible, the trajectory-based description that invokes the classical approximation struggles to capture the correct behavior.

Treating geometric phase effects in nonadiabatic dynamics

Treating geometric phase effects in nonadiabatic dynamics

Observation of geometric phase effect through backward angular oscillations in the H + HD → H2 + D reaction

Quantum interference between reaction pathways around a conical intersection (CI) is an ultrasensitive probe of detailed chemical reaction dynamics. Yet, for the hydrogen exchange reaction, the difference between contributions of the two reaction pathways increases substantially as the energy decreases, making the experimental observation of interference features at low energy exceedingly challenging. We report in this paper a combined experimental and theoretical study on the H + HD → H2 + D reaction at the collision energy of 1.72 eV. Although the roaming insertion pathway constitutes only a small fraction (0.088%) of the overall contribution, angular oscillatory patterns arising from the interference of reaction pathways were clearly observed in the backward scattering direction, providing direct evidence of the geometric phase effect at an energy of 0.81 eV below the CI. Furthermore, theoretical analysis reveals that the backward interference patterns are mainly contributed by two distinct groups of partial waves (J ~ 10 and J ~ 19). The well-separated partial waves and the geometric phase collectively influence the quantum reaction dynamics.

Quantum simulation of conical intersections.

We explore the simulation of conical intersections (CIs) on quantum devices, setting the groundwork for potential applications in nonadiabatic quantum dynamics within molecular systems. The intersecting potential energy surfaces of H3+ are computed from a variance-based contracted quantum eigensolver. We show how the CIs can be correctly described on quantum devices using wavefunctions generated by the anti-Hermitian contracted Schrödinger equation ansatz, which is a unitary transformation of wavefunctions that preserves the topography of CIs. A hybrid quantum-classical procedure is used to locate the seam of CIs. Additionally, we discuss the quantum implementation of the adiabatic to diabatic transformation and its relation to the geometric phase effect. Results on noisy intermediate-scale quantum devices showcase the potential of quantum computers in dealing with problems in nonadiabatic chemistry.

New diabatic potential energy surfaces of KH2 system constructed using neural network method

New diabatic potential energy surfaces (PESs) for the KH2 system, related to both ground and first excited states, were constructed using only adiabatic electronic energies. The elements of the diabatic potential energy matrix were determined using neural network (NN) method. Prior to calculation, the symmetry constraints of the coupling term were incorporated into the NN code. A comprehensive comparison was undertaken between the topographic characteristics of the newly constructed diabatic PESs and those previously developed using the molecular properties method. To validate the accuracy of the new diabatic PESs, the dynamics calculations of K(4p2P) + H2 (v 0 = 1, j 0 = 0) reaction were performed based on both new and old diabatic PESs with same parameters. The results indicate that the values based on the new diabatic PESs generally agree well with those from the old ones. In addition, compared with the previous diabatic PESs, the new diabatic PESs can accurately describe the geometric phase effect.

Geometric Phase Effects in the H + H2+ Reaction on the Lowest Triplet Ground State.

To our knowledge, this is the first time geometric phase (GP) effects in the H + H2+ reaction on its lowest triplet ground state with collision energies lower than 1.85 eV have been studied using the quantum wave packet method and vector potential approach. We obtained the total reaction probabilities including and not including GP (NGP) effects for J ≤ 4. Visible GP effects could be seen at the lower energy regime but are tiny at the higher one. Moreover, they are more obvious in the product rovibrational state-resolved reaction probabilities, and the relative resonance magnitudes between GP and NGP results change with product rotational state values alternatively. The main reasons are the interferences between the one- and two-transition-state (1-TS and 2-TS) reaction paths, in that at the lower energy regime, the reaction probabilities from the 2-TS pathway show peaks of comparable probabilities compared with that of the 1-TS pathway. In addition, the "out-of-phase" trend observed in the H + H2 reaction does not exist rigorously in this system. Importantly, the visible GP effects exist in this H3+ system, which makes it a very useful candidate reaction for nonadiabatic investigations in both theory and experiment.

A metasurface-based direct-reading linear polarization analyzer

Polarization state is one of the fundamental properties of electromagnetic (EM) wave, which has been widely investigated in fields of information encryption, remote sensing and multifunctional devices etc The existing methods for polarization detection are generally based on the measurement of Stokes parameters or the amplitude and phase difference between orthogonal components or polarization-dependent imaging. However, they generally requires post-possessing process to identify the polarization state or suffer from the shortcomings of limited detection states, which results in less intuitiveness and limited practical values of those methods. Here, we proposed a metasurface-based and direct-reading linear polarization analyzer, which could reveal the polarization angle of linearly incident plane wave in imaging way by utilizing geometric phase effect. Different linear polarization angles lead to the different positions of the brightest spot, and the polarization state of incident wave could be directly read out by comparing the position of the brightest spot and the index of reference spots above. Besides, this method could be generalized to simplify the process of detecting arbitrary polarization states, with which only the intensity ratio between orthogonal circularly polarized components needs to be measured. Moreover, a digital linear polarization analyzer is designed based on the similar method, which helps to reveal the linear polarization angle digitally and is much more straightforward for linear polarization detection. The proposed direct-reading linear polarization analyzer has the potential of being applied in fields of smart sensing and the development of human-computer interaction system etc.

Klein–Gordon oscillator under gravitational effects in a topologically charged Ellis–Bronnikov wormhole

In this paper, we investigate the phenomenon of a relativistic quantum oscillator in non-massive wormhole spacetime, known in the literature as Ellis–Bronnikov-type wormhole, for a spin-zero particle subjected to the effects of a scalar field. We have obtained analytically the bound state solutions and your respective energy spectrum. The energy profile of this scalar field is modified by the wormhole throat as well as the scalar coupling constant of curvature. In the second part of the work, we investigate how the energy spectrum for a spin-zero particle is modified by an Aharonov–Bohm geometric phase effect. We also built a graphical representation to try to visualize this modification generated by the magnetic flux constant.

Geometric phase and photonic spin Hall effect in thin films with architected columnar morphology

We have detected spin-dependent splitting of light, the signature of the photonic spin Hall effect (PSHE), via quantum weak measurements on two types of thin films with architected columnar morphology. Specifically, we fabricated columnar thin films comprising parallel tilted nanocolumns and chiral sculptured thin films comprising parallel upright nanohelices by resistively heating zinc selenide (ZnSe) in a low-pressure chamber and collecting the collimated vapor flux of ZnSe on planar substrates with dynamically varying orientation. The architected columnar morphology creates a spin-dependent geometric Pancharatnam–Berry (PB) phase corresponding to the evolution of polarization states on the Poincaré sphere. Morphology-controlled anisotropy and spatial inhomogeneity result in the depolarization and optical rotation of incident plane-polarized light, and intrinsic spin-precession coupling with the propagation vector, contributing to the efficient generation and two-dimensional manipulation of both in-plane and out-of-plane spin splitting and steering the PB phase in the propagation process. The first experimental observations of widely tailorable PSHE and PB phase in thin films with architected columnar morphology may lead to new applications ranging from spin-controlled nanophotonics to optoelectronic devices for quantum information processing and optical communication.

Effects of quantum geometric phase of a particle in an oscillating hard-wall spherical trap

Effects of quantum geometric phase of a particle in an oscillating hard-wall spherical trap

Geometric phase in coupled cluster theory.

It has been well-established that the topography around conical intersections between excited electronic states is incorrectly described by coupled cluster and many other single reference theories (the intersections are "defective"). Despite this, we show both analytically and numerically that the geometric phase effect (GPE) is correctly reproduced upon traversing a path around a defective excited-state conical intersection (CI) in coupled cluster theory. The theoretical analysis is carried out by using a non-Hermitian generalization of the linear vibronic coupling approach. Interestingly, the approach qualitatively explains the characteristic (incorrect) shape of the defective CIs and CI seams. Moreover, the validity of the approach and the presence of the GPE indicate that defective CIs are local (and not global) artifacts. This implies that a sufficiently accurate coupled cluster method could predict nuclear dynamics, including geometric phase effects, as long as the nuclear wavepacket never gets too close to the conical intersections.

Universal anomaly of dynamics at phase transition points induced by Pancharatnam-Berry phase

Dynamical anomalies are often observed near both the continuous and first-order phase transition points. We propose that the universal anomalies could originate from the geometric phase effects. A Pancharatnam-Berry phase is accumulated continuously in quantum states with the variation of tuning parameters. Phase transitions are supposed to induce an abrupt shift of the geometric phase. In our multi-level quantum model, the quantum interference induced by the geometric phase could prolong or shorten the relaxation times of excited states at phase transition points, which agrees with the experiments, models under sudden quenches and our semi-classical model. Furthermore, we find that by setting a phase shift of π, the excited state could be decoupled from the ground state by quantum cancellation so that the relaxation time even could diverge to infinity. Our work introduces the geometric phase to the study of conventional phase transitions as well as quantum phase transition, and could substantially extend the dephasing time of qubits for quantum computing.

Geometric Phase Effect in Attosecond Stimulated X-ray Raman Spectroscopy

Conical intersections (CIs) are diabolical points in the potential energy surfaces generally caused by point-wise degeneracy of different electronic states, and give rise to the geometric phases (GPs) of molecular wave functions. Here we theoretically propose and demonstrate that the transient redistribution of ultrafast electronic coherence in attosecond Raman signal (TRUECARS) spectroscopy is capable of detecting the GP effect in excited state molecules by applying two probe pulses including an attosecond and a femtosecond X-ray pulse. The mechanism is based on a set of symmetry selection rules in the presence of nontrivial GPs. The model of this work can be realized for probing the geometric phase effect in the excited state dynamics of complex molecules with appropriate symmetries, using attosecond light sources such as free-electron X-ray lasers.

Analysis of the phase images obtained during the collection of a holographic registration system based on the geometric phase effect and a polarization camera

The results of measuring the surface depth of the test object using digital holography are presented. The resulting image was compared to a model based on the calibration slide documentation. In the presented holographic microscope, instead of an eyepiece, a lens with a geometric phase effect is used, which converts a beam with linear polarization into a pair of beams with circular polarizations (diverging and converging). The parallel phase shift method was used to obtain phase distribution. Using a polarization camera, four interferograms corresponding to four different linear projections of interfering waves with right and left circular polarizations were recorded in one exposure. Holograms of a phase object-micrometer were obtained, according to which, by the method of parallel phase shift, the distribution of phase lag introduced by the object was restored. To correct the aberration, subtraction of the recorded phase raid of the illuminating wave — the experimentally obtained phase of the wavefront without an object is used. The developed digital holographic phase microscope based on a geometric phase lens and a polarization camera makes it possible to correctly visualize the surface relief profile. The microscope can be used as a tool for monitoring the state of biological objects exposed to external effects.

A Holographic Principle for Non-Relativistic Quantum Mechanics

The quantum-classical isomorphism for self-consistent field theory, which allows quantum particles in space-time to be represented as classical one-dimensional threads embedded in a five dimensional thermal-space-time, is summarized and used to explain a selection of quantum phenomena. Introduced by Feynman, and used for modern quantum simulations, the isomorphism, when phrased in a field-theoretic way, has been shown to be the same as quantum density functional theory, the theorems of which guarantee equivalent predictions with non-relativistic quantum mechanics. If the Feynman dimension is considered to be real, there is a duality between classical threads in five dimensions and quantum particles in four dimensions. Using the 5D picture, intuitive explanations are given for quantum phenomena including the uncertainty principle, tunnelling, geometric phase, and interference effects. Advantages of the 5D picture are presented, which include fewer postulates, no measurement problem, and the need for only classical concepts in the higher dimensional space. Limitations of the approach such as the interpretation of entanglement and spin are discussed.

Analytical derivative couplings within the framework of time-dependent density functional theory coupled with conductor-like polarizable continuum model: Formalism, implementation, and applications.

The nonadiabatic phenomena, which are characterized by a strong coupling between electronic and nuclear motions, are ubiquitous. The nonadiabatic effect of the studied system can be significantly affected by the surrounding environment, such as solvents, in which such nonadiabatic process takes place. It is essential to develop the theoretical models to simulate these processes while accurately modeling the solvent environment. The time-dependent density functional theory (TDDFT) is currently the most efficient approach to describe the electronic structures and dynamics of complex systems, while the polarizable continuum model (PCM) represents one of the most successful examples among continuum solvation models. Here, we formulate the first-order derivative couplings (DCs) between the ground and excited states as well as between two excited states by utilizing time-independent equation of motion formalism within the framework of both linear response and spin flip formulations of TDDFT/CPCM (the conductor-like PCM), and implement the analytical DCs into the Q-CHEM electronic structure software package. The analytic implementation is validated by the comparison of the analytical and finite-difference results, and reproducing geometric phase effect in the protonated formaldimine test case. Taking 4-(N,N-dimethylamino)benzonitrile and uracil in the gas phase and solution as an example, we demonstrate that the solvent effect is essential not only for the excitation energies of the low-lying excited-states but also for the DCs between these states. Finally, we calculate the internal conversion rate of benzophenone in a solvent with DC being used. The current implementation of analytical DCs together with the existing analytical gradient and Hessian of TDDFT/PCM excited states allows one to study the nonadiabatic effects of relatively large systems in solutions with low computational cost.

Machine learning of double-valued nonadiabatic coupling vectors around conical intersections.

In recent years, machine learning has had an enormous success in fitting ab initio potential-energy surfaces to enable efficient simulations of molecules in their ground electronic state. In order to extend this approach to excited-state dynamics, one must not only learn the potentials but also nonadiabatic coupling vectors (NACs). There is a particular difficulty in learning NACs in systems that exhibit conical intersections, as due to the geometric-phase effect, the NACs may be double-valued and are, thus, not suitable as training data for standard machine-learning techniques. In this work, we introduce a set of auxiliary single-valued functions from which the NACs can be reconstructed, thus enabling a reliable machine-learning approach.