Semiclassical Approximation Research Articles

On the global approximate controllability in small time of semiclassical 1-D Schrödinger equations between two states with positive quantum densities

In this paper, we study, in the semiclassical sense, the global approximate controllability in small time of the quantum density and quantum momentum of the 1-D semiclassical cubic Schrödinger equation with two controls between two states with positive quantum densities. We first control the asymptotic expansions of the zeroth and first order of the physical observables via the Agrachev–Sarychev method. Then we conclude the proof through techniques of semiclassical approximation of the nonlinear Schrödinger equation.

Particle Creation and the Schwinger Model

We study the particle creation process in the Schwinger model coupled with an external classical source. One can approach the problem by taking advantage of the fact that the full quantized model is solvable and equivalent to a (massive) gauge field with a non-local effective action. Alternatively, one can also face the problem by following the standard semiclassical route. This means quantizing the massless Dirac field and considering the electromagnetic field as a classical background. We evaluate the energy created by a generic, homogeneous, and time-dependent source. The results match exactly in both approaches. This proves in a very direct and economical way the validity of the semiclassical approach for the (massless) Schwinger model, in agreement with a previous analysis based on the linear response equation. Our discussion suggests that a similar analysis for the massive Schwinger model could be used as a non-trivial laboratory to confront a fully quantized solvable model with its semiclassical approximation, therefore mimicking the long-standing confrontation of quantum gravity with quantum field theory in curved spacetime.

Overlap of parafermionic zero modes at a finite distance

Parafermion bound states (PBSs) are generalizations of Majorana bound states (MBSs) and have been predicted to exist as zero-energy eigenstates in proximitized fractional quantum Hall edge states. Similarly to MBSs, a finite distance between the PBS can split the ground state degeneracy. However, parafermionic modes have a richer exchange statistics than MBSs, so several interaction terms are allowed by the underlying $\mathbb{Z}_{2n}$ symmetry, rendering the effective Hamiltonian governing a pair of PBSs at a finite distance nontrivial. Here, we use a combination of analytical techniques (semiclassical instanton approximation) and numerical techniques (quantum Monte Carlo simulations) to determine the effective coupling Hamiltonian. For this purpose, we go beyond the dilute one-instanton gas approximation and show how finite-size effects can give rise to higher-order parafermion interactions. We find excellent agreement between the analytical results and Monte Carlo simulations. We estimate that these finite-size corrections should be observable in some of the recently proposed experiments to observe PBSs in strongly correlated systems.

Semiclassical strong-field theory of phase delays in ω−2ω above-threshold ionization

Phase and time delays of atomic above-threshold ionization were recently experimentally explored in an $\omega -2\omega$ setting [Zipp et al, Optica 1, 361 (2014)]. The phases of wavepackets ejected from argon by a strong $2\omega$ pulse were probed as a function of the relative phase of a weaker $\omega$ probe pulse. Numerical simulations solving the time-dependent Schr\"{o}dinger equation (TDSE) displayed a sensitive dependence of the doubly differential momentum distribution on the relative phase between the $\omega$ and $2\omega$ fields. Moreover, a surprisingly strong variation of the extracted phase delays on the intensity of the probe pulse was found. We present a semiclassical strong-field description of the phase delays in the emission of electrons in an $\omega -2\omega$ setting and apply it to atomic hydrogen. Non-perturbative effects in both the $2\omega$ pump and the $\omega$ probe field are included. The semiclassical description allows tracing phase delays to path interferences between emission during different points in time of emission within the temporal unit cell of the two-color laser field. We find good agreement between the semiclassical saddle-point approximation, the full strong field approximation (SFA), and previous results applicable in the perturbative limit of probe fields. We show that the RABBIT-like perturbative description of phase delays breaks down for stronger fields and higher-energy electron emission. In this regime, characterization of the ionization signal requires an entire ensemble of phase delays {$\delta_i(E)$} with $i=1,2,\ldots$ the difference in photon numbers of the strong $2\omega$ field involved in the interfering paths. Comparison between SFA and TDSE calculations reveals the influence of the Coulomb field even in this strong-field scenario.

Transfer Learning for Affordable and High-Quality Tunneling Splittings from Instanton Calculations.

The combination of transfer learning (TL) a low-level potential energy surface (PES) to a higher level of electronic structure theory together with ring-polymer instanton (RPI) theory is explored and applied to malonaldehyde. The RPI approach provides a semiclassical approximation of the tunneling splitting and depends sensitively on the accuracy of the PES. With second-order Møller-Plesset perturbation theory (MP2) as the low-level model and energies and forces from coupled cluster singles, doubles, and perturbative triples [CCSD(T)] as the high-level (HL) model, it is demonstrated that CCSD(T) information from only 25-50 judiciously selected structures along and around the instanton path suffice to reach HL accuracy for the tunneling splitting. In addition, the global quality of the HL-PES is demonstrated through a mean average error of 0.3 kcal/mol for energies up to 40 kcal/mol above the minimum energy structure (a factor of 2 higher than the energies employed during TL) and <2 cm-1 for harmonic frequencies compared with computationally challenging normal mode calculations at the CCSD(T) level.

Condensation temperature of strongly interacting 39K condensates in the mean-field and semi-classical approximations

We consider the effect of inter-atom interactions on the condensation temperature Tc of an atomic Bose-Einstein condensate. We find an analytic expression of the shift in Tc induced by interactions with respect the ideal non-interacting case, in the mean-field and semi-classical approximations. Such a shift is expressed in terms of the ratio a/λTc between the s-wave scattering length a and the thermal wavelength λTc. This result is used to discuss the tension between mean-field predictions and observations in strongly interacting K39 condensates. It is shown that such a tension is solved taking into account the details of the Feshbach resonance used to tune a in the experiments.

Laser-induced dynamic alignment of the HD molecule without the Born-Oppenheimer approximation.

Laser-induced molecular alignment is well understood within the framework of the Born-Oppenheimer (BO) approximation. Without the BO approximation, however, the concept of molecular structure is lost, making it hard to precisely define alignment. In this work, we demonstrate the emergence of alignment from the first-ever non-BO quantum dynamics simulations, using the HD molecule exposed to ultrashort laser pulses as a few-body test case. We extract the degree of alignment from the non-BO wave function by means of an operator expressed in terms of pseudo-proton coordinates that mimics the BO-based definition of alignment. The only essential approximation, in addition to the semiclassical electric-dipole approximation for the matter-field interaction, is the choice of time-independent explicitly correlated Gaussian basis functions. We use a variational, electric-field-dependent basis-set construction procedure, which allows us to keep the basis-set dimension low while capturing the main effects of electric polarization on the nuclear and electronic degrees of freedom. The basis-set construction procedure is validated by comparing with virtually exact grid-based simulations for two one-dimensional model systems: laser-driven electron dynamics in a soft attractive Coulomb potential and nuclear rovibrational dynamics in a Morse potential.

Exponentially small quantum correction to conductance

When time-reversal symmetry is broken, the average conductance through a chaotic cavity, from an entrance lead with N 1 open channels to an exit lead with N 2 open channels, is given by N 1 N 2/M, where M = N 1 + N 2. We show that, when tunnel barriers of reflectivity γ are placed on the leads, two correction terms appear in the average conductance, and that one of them is proportional to γ M . Since M ∼ ℏ −1, this correction is exponentially small in the semiclassical limit. We derive this term from a semiclassical approximation, generally expected to give only leading orders in powers of ℏ. Even though the theory is built perturbatively both in γ and in 1/M, the final result is exact.

Satellite and hypersatellite structure of Mα1,2 x‐ray transitions in thorium excited by fast oxygen ions

AbstractThe high‐resolution spectra of thorium Mα1,2 (M5 → N6,7) and Mβ1 (M4 → N6) x‐ray lines induced by an impact of O ions with energy 376.0 MeV were measured using a von Hamos crystal spectrometer in order to study the x‐ray emission from multiple ionized atoms. The observed x‐rays have been interpreted in terms of the relativistic multiconfiguration Dirac‐Fock (MCDF) calculations performed for dominating vacancy configurations expected to contribute to the measured spectra. Due to the complexity of such calculations for selected multi‐vacancy configurations the simplified average &lt;MCDF&gt; model, combining the MCDF calculations and binomial distribution of vacancies has been used. The measured spectra clearly exhibit a rich structure of Mα1,2 x‐ray satellites of thorium, including the transitions from the multivacancy initial configurations M−1N−n as well as the hypersatellites corresponding to the transitions in complex multivacancy initial configurations M−mN−n (m,n ≤ 4). The present interpretation of the measured satellite/hypersatellite structure of Mα1,2 (M5 → N6,7) x‐ray transitions allows us to determine the ionization probabilities for the M‐ and N‐shell in central collisions. The measured ionization probabilities are compared with the theoretical predictions of the geometrical model (GM), and the semiclassical approximation (SCA) using both the Dirac hydrogenic and Dirac‐Hartree‐Fock selfconsistent wave functions.

Energy transitions driven by phase space reflection operators

Phase space reflection operators lie at the core of the Wigner–Weyl representation of density operators and observables. The role of the corresponding classical reflections is known in the construction of semiclassical approximations to Wigner functions of pure eigenstates and their coarsegrained microcanonical superpositions, which are not restricted to classically integrable systems. In their active role as unitary operators, they generate transitions between pairs of eigenstates specified by transition Wigner functions (or cross-Wigner functions): the square modulus of the transition Wigner function at each point in phase space is the transition probability for the reflection through that point.Coarsegraining the initial and final energies provides a transition probability density as a phase space path integral. It is here investigated in the simplest classical approximation involving microcanonical Wigner functions. A reflection operator generates a transition between a pair of energy shells with a probability density given by the integral of the inverse modulus of a Poisson bracket over the intersection of a shell with the reflection of its pair. The singularity of the pair of Wigner functions at their dominant caustics is nicely integrable over their intersection, except for a single degree of freedom. Even though this case is not directly relevant for future investigations of chaotic systems, it is shown here how the improved approximation of the spectral Wigner functions in terms of Airy functions resolves the singularity.

Semiclassical theory for plasmons in spatially inhomogeneous media

Recent progress in experimental techniques has made the quantum regime in plasmonics accessible. Since plasmons correspond to collective electron excitations, the electron–electron interaction plays an essential role in their theoretical description. Within the Random Phase Approximation, this interaction is incorporated through a system of equations of motion, which has to be solved self-consistently. For homogeneous media, an analytical solution can be found using the Fourier transform, giving rise to Lindhard theory. When the medium is spatially inhomogeneous, this is no longer possible and one often uses numerical approaches, which are however limited to smaller systems. In this paper, we present a novel semi-analytical approach for bulk plasmons in inhomogeneous media based on the semiclassical (or WKB) approximation, which is applicable when the charge density varies smoothly. By solving the equations of motion self-consistently, we obtain the expressions of Lindhard theory with a spatially varying Fermi wavevector. The derivation involves passing from the operators to their symbols, which can be thought of as classical observables on phase space. In this way we obtain effective (Hamiltonian) equations of motion for plasmons. We then find the quantized energy levels and the plasmon spectrum using Einstein–Brilllouin–Keller quantization. Our results provide a theoretical basis to describe different setups in quantum plasmonics, such as nanoparticles, quantum dots and waveguides.

Possible quantum effects at the transition from cosmological deceleration to acceleration

The recent transition from decelerated to accelerated expansion can be seen as a reflection (or ``bounce'') in the connection variable, defined by the inverse comoving Hubble length ($b=\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{a}$, on shell). We study the quantum cosmology of this process. We use a formalism for obtaining relational time variables either through the demotion of the constants of nature to integration constants, or by identifying fluid constants of motion. We extend its previous application to a toy model (radiation and $\mathrm{\ensuremath{\Lambda}}$) to the realistic setting of a transition from dust matter to $\mathrm{\ensuremath{\Lambda}}$ domination. In the dust and $\mathrm{\ensuremath{\Lambda}}$ model two time variables may be defined, conjugate to $\mathrm{\ensuremath{\Lambda}}$ and to the dust constant of motion, and we work out the monochromatic solutions to the Schr\"odinger equation representing the Hamiltonian constraint. As for their radiation and $\mathrm{\ensuremath{\Lambda}}$ counterparts, these solutions exhibit ``ringing,'' whereby the incident and reflected waves interfere, leading to oscillations in the amplitude. In the semiclassical approximation we find that, close to the bounce, the probability distribution becomes double peaked, one peak following a trajectory close to the classical limit but with a Hubble parameter slightly shifted downwards, the other with a value of $b$ stuck at its minimum $b={b}_{\ensuremath{\star}}$. Still closer to the transition, the distribution is better approximated by an exponential distribution, with a single peak at $b={b}_{\ensuremath{\star}}$, and a (more representative) average $b$ biased towards a value higher than the classical trajectory. Thus, we obtain a distinctive prediction for the average Hubble parameter with redshift: slightly lower than its classical value when $z\ensuremath{\approx}0$, but potentially much higher than the classical prediction around $z\ensuremath{\sim}0.64$, where the bounce most likely occurred. The implications for the ``Hubble tension'' have not escaped us.

He2+ impact single ionization cross sections of Cu atom

Semi-classical binary encounter approximation has been used for theoretical calculations of single ionization cross sections of Cu atom at ground state by alpha particle impact in energy range varying from threshold 35 keV/amu to 425 keV/amu. An accurate expression of cross section for energy transfer ΔΕ (σΔΕ) as given by Vriens and quantum mechanical Hartree-Fock velocity distributions for target electrons have been used in the calculation. Major contribution to the total single ionization cross sections of Cu are from 4s and 3d subshells. The ionization cross sections decrease with the increase of impact energy same as experimental data reported. The theoretical and experimental results of single ionization cross sections have same trends against the increase of impact energy. The ratio factor falls within 2, varying from 1.26 to 1.86, for given energy range. Theoretical results are under valid range. About 50% of total theoretical results of single ionization cross sections have ratio factor (R) . Major contribution to total ionization cross section is from 3d and 4s subshells-electrons whose contributions varies from 62 to 71% and 41.7% to 23.8% respectively. The higher value of linear correlation coefficient (=0.9647) and lower value of standard deviation (=0.822) shows that results calculated are close to the experimental data in the intermediate and high energy range. BIBECHANA 19 (2022) 111-118

Propagating-wave approximation in two-dimensional potential scattering

We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of the evanescent waves to the scattering amplitude. This corresponds to replacing the interaction potential $v$ with an associated energy-dependent nonlocal potential ${\mathscr{V}}_k$ that does not couple to the evanescent waves. The scattering solutions $\psi(\mathbf{r})$ of the Schr\"odinger equation, $(-\nabla^2+{\mathscr{V}}_k)\psi(\mathbf{r})=k^2\psi(\mathbf{r})$, has the remarkable property that their Fourier transform $\tilde\psi(\mathbf{p})$ vanishes unless $\mathbf{p}$ corresponds to the momentum of a classical particle whose magnitude equals $k$. We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective non-unitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials, and similarly to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.

A Semiclassical Framework for Mixed Quantum Classical Dynamics.

Semiclassical (SC) approximations for quantum dynamic simulations in complex chemical systems range from rigorously accurate methods that are computationally expensive to methods that exhibit near-classical scaling with system size but are limited in their ability to describe quantum effects. In practical studies of high-dimensional reactions, neither extreme is the best choice: frequently a high-level quantum mechanical description is only required for a handful of modes, while the majority of environment modes that do not play a key role in the reactive event of interest are well served with a lower level of theory. In this feature, we introduce modified Filinov filtration as a powerful tool to construct mixed quantum-classical SC theories where different subsystems can be quantized to different extents without introducing ad hoc intersubsystem interaction terms. We demonstrate that these Filinov-based SC methods can systematically tune between quantum and classical limit SC behavior, offering a practical way forward to accurate and computationally efficient simulations of high-dimensional quantum processes.

Entanglement and correlations in fast collective neutrino flavor oscillations

Collective neutrino oscillations play a crucial role in transporting lepton flavor in astrophysical settings like supernovae and neutron star binary merger remnants, which are characterized by large neutrino densities. In these settings, simulations in the mean-field approximation show that neutrino-neutrino interactions can overtake vacuum oscillations and give rise to fast collective flavor evolution on timescales $t\ensuremath{\propto}{\ensuremath{\mu}}^{\ensuremath{-}1}$, with $\ensuremath{\mu}$ proportional to the local neutrino density. In this work, we study the full out-of-equilibrium flavor dynamics in simple multiangle geometries displaying fast oscillations in the mean field linear stability analysis. Focusing on simple initial conditions, we analyze the production of pair correlations and entanglement in the complete many-body-dynamics as a function of the number $N$ of neutrinos in the system, for up to thousands of neutrinos. Similarly to simpler geometries with only two neutrino beams, we identify three regimes: stable configurations with vanishing flavor oscillations, marginally unstable configurations with evolution occurring on long timescales $\ensuremath{\tau}\ensuremath{\approx}{\ensuremath{\mu}}^{\ensuremath{-}1}\sqrt{N}$, and unstable configurations showing flavor evolution on short timescales $\ensuremath{\tau}\ensuremath{\approx}{\ensuremath{\mu}}^{\ensuremath{-}1}\mathrm{log}(N)$. We present evidence that these fast collective modes are generated by the same dynamical phase transition which leads to the slow bipolar oscillations, establishing a connection between these two phenomena and explaining the difference in their timescales. We conclude by discussing a semiclassical approximation which reproduces the entanglement entropy at short to medium timescales and can be potentially useful in situations with more complicated geometries where classical simulation methods starts to become inefficient.

Modification method of NUT-Kerr-Newman-de Sitter black hole entropy by Lorentz symmetry breaking and beyond the semi-classical approximation

According to Lorentz symmetry breaking and black hole quantum tunneling radiation theory, aether-like field is correctly introduced into the general stationary black hole space-time, and the modified boson dynamic equation is obtained. On this basis, the Hawking temperature and black hole entropy of NUT-Kerr-Newman-de Sitter black hole with general stationary characteristics are modified, and the method used in this paper and the significance of the conclusions are deeply discussed.

The cosmological constant as a boundary term

We compare the path integral for transition functions in unimodular gravity and in general relativity. In unimodular gravity the cosmological constant is a property of states that are specified at the boundaries whereas in general relativity the cosmological constant is a parameter of the action. Unimodular gravity with a nondynamical background spacetime volume element has a time variable that is canonically conjugate to the cosmological constant. Wave functions depend on time and satisfy a Schrödinger equation. On the contrary, in the covariant version of unimodular gravity with a 3-form gauge field, proposed by Henneaux and Teitelboim, wave functions are time independent and satisfy a Wheeler-DeWitt equation, as in general relativity. The 3-form gauge field integrated over spacelike hypersurfaces becomes a “cosmic time” only in the semiclassical approximation. In unimodular gravity the smallness of the observed cosmological constant has to be explained as a property of the initial state.

Electronic transport in three-terminal chaotic systems with a tunnel barrier

We consider the problem of electronic quantum transport through ballistic mesoscopic systems with chaotic dynamics, connected to a three-terminal architecture in which one of the terminals has a tunnel barrier. Using a semiclassical approximation based on matrix integrals, we calculate several transport statistics, such as average and variance of conductance, average shot-noise power, among others, that give access to the extreme quantum regime (small channel numbers in the terminal) for broken and intact time-reversal symmetry, which the traditional random matrix approach does not access. As an application, we treat the dephasing regime.

Functional Properties of Tetrameric Molecular Cells for Quantum Cellular Automata: A Quantum-Mechanical Treatment Extended to the Range of Arbitrary Coulomb Repulsion

We discuss the problem of electron transfer (ET) in mixed valence (MV) molecules that is at the core of molecular Quantum Cellular Automata (QCA) functioning. Theoretical modelling of tetrameric bi-electronic MV molecular square (prototype of basic QCA cell) is reported. The model involves interelectronic Coulomb repulsion, vibronic coupling and ET between the neighboring redox sites. Unlike the majority of previous studies in which molecular QCA have been analyzed only for particular case when the Coulomb repulsion energy significantly exceeds the ET energy, here we do not imply assumptions on the relative strength of these two interactions. Moreover, in the present work we go beyond the adiabatic semiclassical approximation often used in theoretical analysis of such systems in spite of the fact that this approximation ignores such an important phenomenon as quantum tunneling. By analyzing the electronic density distributions in the cells and the ell-cell response functions obtained from a quantum-mechanical solution of a complex multimode vibronic problem we have concluded that such key features of QCA cell as bistability and switchability can be achieved even under failure of the condition of strong Coulomb repulsion provided that the vibronic coupling is strong enough. We also show that the semiclassical description of the cell-cell response functions loses its accuracy in the region of strong non-linearity, while the quantum-mechanical approach provides correct results for this critically important region.