Semiclassical Spectrum Research Articles

Regularising spectral curves for homogeneous Yang-Baxter strings

In this Letter, we study the semi-classical spectrum of integrable worldsheet σ-models using the Spectral Curve. We consider a Homogeneous Yang-Baxter deformation of the AdS5×S5 superstring, understood as the composition of a Jordanian with a “non-diagonal” TsT deformation. We derive its type IIB supergravity solution, whose isometry algebra features zero supercharges and a non-relativistic conformal algebra in 0+1 dimensions. While the Spectral Curves of non-diagonal TsT models are ill-defined, we demonstrate that the composition with a Jordanian model regularises this issue. From the regularised Curve, we derive the one-loop shift of the classical energy and the semi-classical spectrum of excitations of a point-like string. In the TsT limit, the one-loop shift vanishes despite the loss of supersymmetry. Our results suggest that it may be possible to use standard Bethe Ansatze on spin chain pictures of deformed N=4 Super-Yang-Mills theory dual to non-diagonal TsT models.

Semiclassical spectra of the single- and two-photon Tavis-Cummings models

We obtain explicit analytical expressions for the energy spectra of the resonant single- and two-photon Tavis-Cummings models in the asymptotic limit of a large number of atoms, A≫1, and a large number of total excitations, N≫1. We show that both the spectral singularities in the regime N/A∼1 and the equidistant spectra in the regime N/A≫1 can be accurately described in the semiclassical framework.

A time averaged semiclassical approach to IR spectroscopy.

We propose a new semiclassical approach to the calculation of molecular IR spectra. The method employs the time averaging technique of Kaledin and Miller upon symmetrization of the quantum dipole-dipole autocorrelation function. Spectra at high and low temperatures are investigated. In the first case, we are able to point out the possible presence of hot bands in the molecular absorption line shape. In the second case, we are able to reproduce accurate IR spectra as demonstrated by a calculation of the IR spectrum of the water molecule, which is within 4% of the exact intensity. Our time averaged IR spectra can be directly compared to time averaged semiclassical power spectra as shown in an application to the CO2molecule, which points out the differences between IR and power spectra and demonstrates that our new approach can identify active IR transitions correctly. Overall, the method features excellent accuracy in calculating absorption intensities and provides estimates for the frequencies of vibrations in agreement with the corresponding power spectra. In perspective, this work opens up the possibility to interface the new method with the semiclassical techniques developed for power spectra, such as the divide-and-conquer one, to get accurate IR spectra of complex and high-dimensional molecular systems.

Semiclassical theory and the Koopman-van Hove equation * *Contribution to the Special Collection ‘Koopman Methods in Classical and Quantum-Classical Mechanics’.

The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations. Semiclassical theory yields the Hamilton–Jacobi equation for the complex phase factor and the transport equation for the amplitude. These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space. There is a natural injection of configuration space solutions into phase space and a natural projection of phase space solutions onto configuration space. Hence, every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton–Jacobi constraint. For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space. For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum. If the classical spectrum is eliminated, then, with the correct choice of Jeffreys–Wentzel–Kramers–Brillouin (JWKB) matching conditions, the semiclassical spectrum satisfies the Einstein–Brillouin–Keller quantization conditions which include the correction due to the Maslov index. However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space. Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are approximated as local operators on phase space. Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.

Quantization of Nonsmooth Curves and the Semiclassical Spectrum of the One-Dimensional Schrödinger Operator with a Localized Perturbation of the Potential

Quantization of Nonsmooth Curves and the Semiclassical Spectrum of the One-Dimensional Schrödinger Operator with a Localized Perturbation of the Potential

Non-relativistic string monodromies

Spectral curve methods proved to be powerful techniques in the context of relativistic integrable string theories, since they allow us to derive the semiclassical spectrum from the minimal knowledge of a Lax pair and a classical string solution. In this paper we initiate the study of the spectral curve for non-relativistic strings in AdS5 × S5. First, we show that for string solutions whose Lax connection is independent of σ, the eigenvalues of the monodromy matrix do not have any spectral parameter dependence. We remark that this particular behaviour also appears for relativistic strings in flat space. Second, for some simple non-relativistic string solutions where the path ordered exponential of the Lax connection can be computed, we show that the monodromy matrix is either diagonalisable with quasi-momenta independent of the spectral parameter, or non-diagonalisable. For the latter case, we propose a notion of generalised quasi-momenta, based on maximal abelian subalgebras, which retain a dependence on the spectral parameter.

Signatures of Exciton Orbits in Quantum Mechanical Recurrence Spectra of Cu_{2}O.

The seminal work by Kazimierczuk etal. [Nature 514, 343 (2014)10.1038/nature13832] has shown the existence of highly excited exciton states in a regime, where the correspondence principle is applicable and quantum mechanics turns into classical mechanics; however, any interpretation of exciton spectra based on a classical approach to excitons is still missing. Here, we close this gap by computing and comparing quantum mechanical and semiclassical recurrence spectra of cuprous oxide. We show that the quantum mechanical recurrence spectra exhibit peaks, which, by application of semiclassical theories and a scaling transformation, can be directly related to classical periodic exciton orbits. The application of semiclassical theories to exciton physics requires the detailed analysis of the classical exciton dynamics, including three-dimensional orbits, which strongly deviate from hydrogenlike Keplerian orbits. Our findings illuminate important aspects of excitons in semiconductors by directly relating the quantum mechanical band structure splittings of excitons to the corresponding classical exciton dynamics.

The volume of the black hole interior at late times

Understanding the fate of semi-classical black hole solutions at very late times is one of the most important open questions in quantum gravity. In this paper, we provide a path integral definition of the volume of the black hole interior and study it at arbitrarily late times for black holes in various models of two-dimensional gravity. Because of a novel universal cancellation between the contributions of the semi-classical black hole spectrum and some of its non-perturbative corrections, we find that, after a linear growth at early times, the length of the interior saturates at a time, and towards a value, that is exponentially large in the entropy of the black hole. This provides a non-perturbative test of the complexity equals volume proposal since complexity is also expected to plateau at the same value and at the same time.

Semiclassical spectrum of the Dirichlet–Pauli operator on an annulus

This paper is devoted to the semiclassical analysis of the spectrum of the Dirichlet–Pauli operator on an annulus. We assume that the magnetic field is strictly positive and radial. We give an explicit asymptotic expansion at the first order of the lowest eigenvalues of this operator in the semiclassical limit. In particular, we exhibit the Aharonov–Bohm effect that has been revealed, for a constant magnetic field, in a recent paper by Helffer and Sundqvist.

Semiclassical and quantum nonlinear spectra of a strongly coupled single Λ-type three-level atom-cavity QED system

We present detailed numerical simulations of semiclassical and quantum spectra of a cavity QED system consisting of a single three-level atom in Λ-configuration interacting with a quantized cavity mode through one of its transitions while its other transition is driven by a coherent classical field. We compute the semiclassical and quantum spectra of the system under various levels of external driving field strengths. In the semiclassical approach, we neglect the correlations between the cavity and atomic operators, whereas in the quantum case no such assumption is made. Our results show that the two approaches yield identical results under sufficiently weak driving field conditions. However, the two approaches yield starkly different results at stronger driving field intensities: the fully quantum approach yields a well-defined multiphoton spectrum whereas the semiclassical approach results in a bistable spectrum. Furthermore, our calculations reveal a complex Raman structure at the position of the dark-state transition.

Unsupervised Machine Learning Neural Gas Algorithm for Accurate Evaluations of the Hessian Matrix in Molecular Dynamics.

The Hessian matrix of the potential energy of molecular systems is employed not only in geometry optimizations or high-order molecular dynamics integrators but also in many other molecular procedures, such as instantaneous normal mode analysis, force field construction, instanton calculations, and semiclassical initial value representation molecular dynamics, to name a few. Here, we present an algorithm for the calculation of the approximated Hessian in molecular dynamics. The algorithm belongs to the family of unsupervised machine learning methods, and it is based on the neural gas idea, where neurons are molecular configurations whose Hessians are adopted for groups of molecular dynamics configurations with similar geometries. The method is tested on several molecular systems of different dimensionalities both in terms of accuracy and computational time versus calculating the Hessian matrix at each time-step, that is, without any approximation, and other Hessian approximation schemes. Finally, the method is applied to the on-the-fly, full-dimensional simulation of a small synthetic peptide (the 46 atom N-acetyl-l-phenylalaninyl-l-methionine amide) at the level of DFT-B3LYP-D/6-31G* theory, from which the semiclassical vibrational power spectrum is calculated.

On the semiclassical spectrum of the Dirichlet–Pauli operator

This paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set with Dirichlet conditions on the boundary. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Segal–Bargmann and Hardy spaces associated with the magnetic field.

Semiclassical asymptotic spectrum of the two-dimensional Hartree operator near a local maximum of the eigenvalues in a spectral cluster

We consider the eigenvalue problem for the two-dimensional Hartree operator with a small nonlinearity coefficient. We find the asymptotic eigenvalues and asymptotic eigenfunctions near a local maximum of the eigenvalues in spectral clusters formed near the eigenvalues of the unperturbed operator.

Machine learning for vibrational spectroscopy via divide-and-conquer semiclassical initial value representation molecular dynamics with application to N-methylacetamide.

A machine learning algorithm for partitioning the nuclear vibrational space into subspaces is introduced. The subdivision criterion is based on Liouville's theorem, i.e., the best preservation of the unitary of the reduced dimensionality Jacobian determinant within each subspace along a probe full-dimensional classical trajectory. The algorithm is based on the idea of evolutionary selection, and it is implemented through a probability graph representation of the vibrational space partitioning. We interface this customized version of genetic algorithms with our divide-and-conquer semiclassical initial value representation method for the calculation of molecular power spectra. First, we benchmark the algorithm by calculating the vibrational power spectra of two model systems, for which the exact subspace division is known. Then, we apply it to the calculation of the power spectrum of methane. Exact calculations and full-dimensional semiclassical spectra of this small molecule are available and provide an additional test of the accuracy of the new approach. Finally, the algorithm is applied to the divide-and-conquer semiclassical calculation of the power spectrum of 12-atom trans-N-methylacetamide.

Semiclassical inverse spectral problem for seismic surface waves in isotropic media: part I. Love waves

We analyze the inverse spectral problem on the half line associated with elastic surface waves. Here, we focus on Love waves. Under certain generic conditions, we establish uniqueness and present a reconstruction scheme for the S-wavespeed with multiple wells from the semiclassical spectrum of these waves.

Semiclassical inverse spectral problem for seismic surface waves in isotropic media: part II. Rayleigh waves

We analyze the inverse spectral problem on the half line associated with elastic surface waves. Here, we extend the treatment of Love waves [] to Rayleigh waves. Under certain conditions, and assuming that the Poisson ratio is constant, we establish uniqueness and present a reconstruction scheme for the S-wave speed with multiple wells from the semiclassical spectrum of these waves.

Semiclassical vibrational spectroscopy with Hessian databases.

We report on a new approach to ease the computational overhead of ab initio "on-the-fly" semiclassical dynamics simulations for vibrational spectroscopy. The well known bottleneck of such computations lies in the necessity to estimate the Hessian matrix for propagating the semiclassical pre-exponential factor at each step along the dynamics. The procedure proposed here is based on the creation of a dynamical database of Hessians and associated molecular geometries able to speed up calculations while preserving the accuracy of results at a satisfactory level. This new approach can be interfaced to both analytical potential energy surfaces and on-the-fly dynamics, allowing one to study even large systems previously not achievable. We present results obtained for semiclassical vibrational power spectra of methane, glycine, and N-acetyl-L-phenylalaninyl-L-methionine-amide, a molecule of biological interest made of 46 atoms.

The one dimensional semi-classical Bogoliubov-de Gennes Hamiltonian with PT symmetry: generalized Bohr-Sommerfeld quantization rules

We present a method for computing first order asymptotics of semiclassical spectra for 1-D Bogoliubov-de Gennes (BdG) Hamiltonian from Supraconductivity, which models the electron/hole scattering through two SNS junctions. This involves: 1) reducing the system to Weber equation near the branching point at the junctions; 2) constructing local sections of the fibre bundle of microlocal solutions; 3) normalizing these solutions for the “flux norm” associated to the microlocal Wronskians; 4) finding the relative monodromy matrices in the gauge group that leaves invariant the flux norm; 5) from this we deduce Bohr-Sommerfeld (BS) quantization rules that hold precisely when the fibre bundle of microlocal solutions (depending on the energy parameter E) has trivial holonomy. Such a semi-classical treatement reveals interesting continuous symetries related to monodromy.

On-the-fly ab initio semiclassical evaluation of time-resolved electronic spectra.

We present a methodology for computing vibrationally and time-resolved pump-probe spectra, which takes into account all vibrational degrees of freedom and is based on the combination of the thawed Gaussian approximation with on-the-fly ab initio evaluation of the electronic structure. The method is applied to the phenyl radical and compared with two more approximate approaches based on the global harmonic approximation-the global harmonic method expands both the ground- and excited-state potential energy surfaces to the second order about the corresponding minima, while the combined global harmonic/on-the-fly method retains the on-the-fly scheme for the excited-state wavepacket propagation. We also compare the spectra by considering their means and widths, and show analytically how these measures are related to the properties of the semiclassical wavepacket. We find that the combined approach is better than the global harmonic one in describing the vibrational structure, while the global harmonic approximation estimates better the overall means and widths of the spectra due to a partial cancellation of errors. Although the full-dimensional on-the-fly ab initio result seems to reflect the dynamics of only one mode, we show, by performing exact quantum calculations, that this simple structure cannot be recovered using a one-dimensional model. Yet, the agreement between the quantum and semiclassical spectra in this simple, but anharmonic model lends additional support for the full-dimensional ab initio thawed Gaussian calculation of the phenyl radical spectra. We conclude that the thawed Gaussian approximation provides a viable alternative to the expensive or unfeasible exact quantum calculations in cases, where low-dimensional models are not sufficiently accurate to represent the full system.

Anharmonic vibrational eigenfunctions and infrared spectra from semiclassical molecular dynamics.

We describe a new approach based on semiclassical molecular dynamics that allows simulating infrared absorption or emission spectra of molecular systems with inclusion of anharmonic intensities. This is achieved from semiclassical power spectra by computing first the vibrational eigenfunctions as a linear combination of harmonic states, and then the oscillator strengths associated with the vibrational transitions. We test the approach against a 1D Morse potential and apply it to the water molecule with results in excellent agreement with discrete variable representation quantum benchmarks. The method does not require any grid calculations, and it is directly extendable to high dimensional systems. The usual exponential scaling of the basis set size with the dimensionality of the system can be avoided by means of an appropriate truncation scheme. Furthermore, the approach has the advantage to provide IR spectra beyond the harmonic approximation without losing the possibility of an intuitive assignment of absorption peaks in terms of normal modes of vibration.