Semiclassical Results Research Articles

Hawking radiation of a charged black hole in quantum gravity

We study black hole radiation of a Reissner-Nordstrom black hole with an electric charge in the framework of quantum gravity. Based on a canonical quantization for a spherically symmetric geometry, under physically plausible assumptions, we solve the Wheeler-De Witt equation in the regions not only between the outer apparent horizon and the spatial innity but also between the spacetime singularity and the inner apparent horizon, and then show that the mass loss rate of an evaporating black hole due to thermal radiation agrees with the semiclassical result when we choose an integration constant properly by physical reasoning. Furthermore, we also solve the Wheeler-De Witt equation in the region between the inner Cauchy horizon and the outer apparent horizon, and show that the mass loss rate of an evaporating black hole has the same expression. The present study is the natural generalization of the case of a Schwarzschild black hole to that of a charged Reissner-Nordstrom black hole.

Efficient semiclassical approach for time delays

The Wigner time delay, defined by the energy derivative of the total scattering phase shift, is an important spectral measure of an open quantum system characterizing the duration of the scattering event. It is proportional to the trace of the Wigner–Smith matrix Q that also encodes other time-delay characteristics. For chaotic cavities, these quantities exhibit universal fluctuations that are commonly described within random matrix theory. Here, we develop a new semiclassical approach to the time-delay matrix which is formulated in terms of the classical trajectories that connect the exterior and interior regions of the system. This approach is superior to previous treatments because it avoids the energy derivative. We demonstrate the methodʼs efficiency by going beyond previous work in establishing the universality of time-delay statistics for chaotic cavities with perfectly connected leads. In particular, the moment generating function of the proper time-delays (eigenvalues of Q) is found semiclassically for the first five orders in the inverse number of scattering channels for systems with and without time-reversal symmetry. We also show the equivalence of random matrix and semiclassical results for the second moments and for the variance of the Wigner time delay at any channel number.

Indefinite theta functions for counting attractor backgrounds

In this note, we employ indefinite theta functions to regularize canonical par- tition functions for single-center dyonic BPS black holes. These partition functions count dyonic degeneracies in the Hilbert space of four-dimensional toroidally compactified het- erotic string theory, graded by electric and magnetic charges. The regularization is achieved by viewing the weighted sums of degeneracies as sums over charge excitations in the near- horizon attractor geometry of an arbitrarily chosen black hole background, and eliminating the unstable modes. This enables us to rewrite these sums in terms of indefinite theta func- tions. Background independence is then implemented by using the transformation property of indefinite theta functions under elliptic transformations, while modular transformations are used to make contact with semi-classical results in supergravity.

First-principles binary diffusion coefficients for H, H₂, and four normal alkanes + N₂.

Collision integrals related to binary (dilute gas) diffusion are calculated classically for six species colliding with N2. The most detailed calculations make no assumptions regarding the complexity of the potential energy surface, and the resulting classical collision integrals are in excellent agreement with previous semiclassical results for H + N2 and H2 + N2 and with recent experimental results for CnH(2n+2) + N2, n = 2-4. The detailed classical results are used to test the accuracy of three simplifying assumptions typically made when calculating collision integrals: (1) approximating the intermolecular potential as isotropic, (2) neglecting the internal structure of the colliders (i.e., neglecting inelasticity), and (3) employing unphysical R(-12) repulsive interactions. The effect of anisotropy is found to be negligible for H + N2 and H2 + N2 (in agreement with previous quantum mechanical and semiclassical results for systems involving atomic and diatomic species) but is more significant for larger species at low temperatures. For example, the neglect of anisotropy decreases the diffusion coefficient for butane + N2 by 15% at 300 K. The neglect of inelasticity, in contrast, introduces only very small errors. Approximating the repulsive wall as an unphysical R(-12) interaction is a significant source of error at all temperatures for the weakly interacting systems H + N2 and H2 + N2, with errors as large as 40%. For the normal alkanes in N2, which feature stronger interactions, the 12/6 Lennard-Jones approximation is found to be accurate, particularly at temperatures above ∼700 K where it predicts the full-dimensional result to within 5% (although with somewhat different temperature dependence). Overall, the typical practical approach of assuming isotropic 12/6 Lennard-Jones interactions is confirmed to be suitable for combustion applications except for weakly interacting systems, such as H + N2. For these systems, anisotropy and inelasticity can safely be neglected but a more detailed description of the repulsive wall is required for quantitative predictions. A straightforward approach for calculating effective isotropic potentials with realistic repulsive walls is described. An analytic expression for the calculated diffusion coefficient for H + N2 is presented and is estimated to have a 2-sigma error bar of only 0.7%.

Spectral dimension of bosonic string theory

Given that the scale of quantum gravity is not experimentally accessible, one naturally resorts to mathematical consistency as a measure for a good candidate theory to replace General Relativity at high energies. Reproducing the semi-classical results of black hole entropy has become a standard test for any prospective theory of quantum gravity. It is often argued that another such commonality, albeit less known, is the similar fractal behaviour. It is shown that many, if not all, approaches to quantum gravity predict a spectral dimension of 2 in ultraviolet regime. In this paper, by computing the heat kernel, we show that the spectral dimension of closed bosonic string theory is 26. We discuss the implications of this disparity.

Time-dependent semiclassics for ultracold bosons

We study the out-of-equilibrium dynamics of ultracold bosons in a double- and triple-well potential within the Bose-Hubbard model by means of the semiclassical Herman-Kluk propagator and compare the results to the frequently applied "classical dynamics" calculation in terms of the truncated Wigner approximation (TWA). For the double-well system we find the semiclassical results in excellent agreement with the numerically exact ones, while the TWA is not able to reproduce any revivals of the wave function. The triple-well system turns out to be more difficult to handle due to the irregularity of the corresponding classical phase space. Here, deviations of the TWA from the exact dynamics appear even for short times, while better agreement is obtained using the semiclassical approach presented in this article.

Interference nature of quantum breather oscillation

In a system of coupled nonlinear oscillators, the breather (or local mode) solution is studied fully quantum mechanically, as well as by a semiclassical initial value representation of the propagator and classical Wigner dynamics. We show that the initial breather state is a superposition of almost degenerate eigenstates. From this simple observation it follows that the breather must decay and revive (i.e., oscillate with energy localization for extended times). Numerical results are shown for a two degree of freedom system. The fact that the semiclassical real-time result reproduces the full quantum one to a large degree, whereas the classical Wigner dynamics based on a similar set of trajectories does not, indicates that the breather oscillation can be viewed as an interference phenomenon.

Comparative quantum and semiclassical analysis of atom-field systems. I. Density of states and excited-state quantum phase transitions

We study the non-integrable Dicke model, and its integrable approximation, the Tavis-Cummings model, as functions of both the coupling constant and the excitation energy. Excited-state quantum phase transitions (ESQPT) are found analyzing the density of states in the semi-classical limit and comparing it with numerical results for the quantum case in large Hilbert spaces, taking advantage of efficient methods recently developed. Two different ESQPTs are identified in both models, which are signaled as singularities in the semi-classical density of states, one {\em static} ESQPT occurs for any coupling, whereas a dynamic ESQPT is observed only in the superradiant phase. The role of the unstable fixed points of the Hamiltonian semi-classical flux in the occurrence of the ESQPTs is discussed and determined. Numerical evidence is provided that shows that the semi-classical result describes very well the tendency of the quantum energy spectrum for any coupling in both models. Therefore the semi-classical density of states can be used to study the statistical properties of the fluctuation in the spectra, a study that is presented in a companion paper.

Spontaneous Photon Emission in Cavities

We investigate spontaneous photon emission processes of two-level atoms in parabolic and ellipsoidal cavities thereby taking into account the full multimode scenario. In particular, we calculate the excitation probabilities of the atoms and the energy density of the resulting few-photon electromagnetic radiation field by using semiclassical methods for the description of the multimode scenario. Based on this approach photon path representations are developed for relevant transition probability amplitudes which are valid in the optical frequency regime where the dipole and the rotating-wave approximations apply. Comparisons with numerical results demonstrate the quality of these semiclassical results even in cases in which the wave length of a spontaneously emitted photon becomes comparable or even larger than characteristic length scales of the cavity. This is the dynamical regime in which diffraction effects become important so that geometric optical considerations are typically not applicable.

Optical bistability in strong-coupling cavity QED with a few atoms

We present exact numerical solutions of the damped-driven Jaynes–Cummings model adapted to describe absorptive optical bistability in the limit of a few atoms strongly coupled to a high-finesse resonator. We show that the simplifying semiclassical result for many physical quantities of interest is well reproduced by the quantum model including even with only a few atoms in the strongly coupled system. Non-trivial atom–field quantum cross-correlations show up in the strong-driving limit.

Stark broadening measurement of Al II lines in a laser-induced plasma

Laser induced plasma was a light source for the study of Stark broadening parameters of singly charged aluminum ion lines. Plasma electron number density in the range (0.3–2.3)×1023m−3 was measured from the Stark width of the hydrogen Hα impurity line, while the electron temperature between 6500 and 17,500K was determined from relative intensities of Fe II, Mg I and Al II spectral lines using the Boltzmann plot technique. The experimental Stark widths were compared with other experiments and theories, which include semiclassical results and data evaluated from the modified semiempirical formula.

Quantum Stark broadening of Ar XV lines. Strong collision and quadrupolar potential contributions

We present in this paper electron impact broadening for six Ar XV lines using our quantum mechanical formalism and the semiclassical perturbation one. Additionally, our calculations of the corresponding atomic structure data (energy levels and oscillator strengths) and collision strengths are given as well. The lines considered here are divided into two sets: a first set of four lines involving the ground level: 1s22s21S0– 1s22snp 1P1o where 2⩽n⩽5 and a second set of two lines involving excited levels: 1s22s2p 1P1o–1s22s3s 1S0 and 1s22s2p 3P0o–1s22s3s 3S1. An extensive comparison between the quantum and the semiclassical results was performed in order to analyze the reason for differences between quantum and semiclassical results up to the factor of two. It has been shown that the difference between the two results may be due to the evaluation of strong collision contributions by the semiclassical formalism. Except few semiclassical results, the present results are the first to be published. After the recent discovery of the far UV lines of Ar VII in the spectra of very hot central stars of planetary nebulae and white dwarfs, the present -and may be further- results can be used also for the corresponding future spectral analysis.

The effect of the Coriolis coupling on H+ND reaction: A time dependent wave packet study

We present a quantum scattering calculation of initial-state-resolved reaction probabilities, integral cross section, initial state selected reaction rate constants and thermal rate constant for exchange and depletion channels of the H+ND reaction. The calculations are carried out using Coriolis coupling (CC) method on the modified NH2Ã2A1 potential energy surface(PES). The initial state selected reaction probabilities are calculated for 0.0–0.5eV of collision energy range. The reaction probabilities for depletion channel shows some sharp and large resonances associated with long-lived collision complexes, as compared to exchange channel. Integral cross sections for both reactions depend strongly on initial rotational states. The calculated thermal reaction rate constant for the depletion channel is in a good agreement with the values previously obtained by the centrifugal sudden approximation, experimental and semiclassical results.

Tunneling conductance due to a discrete spectrum of Andreev states

We study tunneling spectroscopy of subgap Andreev states in a superconducting system and discuss the general situation when the discrete nature of these levels is relevant and thus the standard semiclassical result for tunneling conductance being proportional to the density of states is not applicable. If the tunneling coupling is weak, individual levels are resolved and the conductance G(V ) at low temperatures is composed of a set of resonant Lorentz peaks which cannot be described within perturbation theory over tunneling strength. We establish a general formula for the peak widths and heights and show that the width of any peak scales as normal-state tunnel conductance, while its height is ≲2e2 h−1 and depends only on contact geometry and the spatial profile of the resonant Andreev level. We also establish an exact formula for the single-channel conductance that takes the whole Andreev spectrum into account, and use it to study the interference of Andreev reflection processes through different levels. We study tunneling conductance at finite bias G(eV >0) for a system with a pair of almost decoupled Majorana fermions and derive the conditions for the ‘universal’ zero-bias peak with the height 2e2/h to be observed in a realistic system which always hosts an even number of Majorana fermions.

Semi-classical theory for quantum quenches in the O(3) non-linear sigma model

We use the semi-classical approach to study the non-equilibrium dynamics of the O(3) non-linear sigma model. For a class of quenches defined in the text, we obtain the order-parameter dynamical correlator in the thermodynamic limit. In particular we predict quench-dependent relaxation times and correlation lengths. The approach developed for the O(3) non-linear sigma model can also be applied to the transverse field Ising chain, where the semi-classical results can be directly compared to both the exact and the numerical ones, revealing the limits of the method.

Vibration-vibration and vibration-translation energy transfer in H2-H2 collisions: A critical test of experiment with full-dimensional quantum dynamics

Quantum scattering calculations of vibration-vibration (VV) and vibration-translation (VT) energy transfer for non-reactive H2-H2 collisions on a full-dimensional potential energy surface are reported for energies ranging from the ultracold to the thermal regime. The efficiency of VV and VT transfer is known to strongly correlate with the energy gap between the initial and final states. In H2(v = 1, j = 0) + H2(v = 0, j = 1) collisions, the inelastic cross section at low energies is dominated by a VV process leading to H2(v = 0, j = 0) + H2(v = 1, j = 1) products. At energies above the opening of the v = 1, j = 2 rotational channel, pure rotational excitation of the para-H2 molecule leading to the formation of H2(v = 1, j = 2) + H2(v = 0, j = 1) dominates the inelastic cross section. For vibrationally excited H2 in the v = 2 vibrational level colliding with H2(v = 0), the efficiency of both VV and VT process is examined. It is found that the VV process leading to the formation of 2H2(v = 1) molecules dominates over the VT process leading to H2(v = 1) + H2(v = 0) products, consistent with available experimental data, but in contrast to earlier semiclassical results. Overall, VV processes are found to be more efficient than VT processes, for both distinguishable and indistinguishable H2-H2 collisions confirming room temperature measurements for v = 1 and v = 2.

Systematics of fine structure in the α decay of deformed even-even nuclei

We present detailed systematics of the results for α transitions to excited 2+, 4+, and 6+ states in the α decay of deformed even-even nuclei. Both the semiclassical and coupled-channel results are analyzed in the valence nucleon scheme. This provides a good test of the reliability of the various theoretical models for the α-decay fine structure. It is found that the coupled-channel results agree well with the experimental data in both systematic behavior and magnitude, while the semiclassical results have relatively large deviations from the experimental data. This confirms the good reliability of the coupled-channel studies.

One-dimensional hydrogen atom with minimal length uncertainty and maximal momentum

We present exact energy eigenvalues and eigenfunctions of the one-dimensional hydrogen atom in the framework of the Generalized (Gravitational) Uncertainty Principle (GUP). This form of GUP is consistent with various theories of quantum gravity such as string theory, loop quantum gravity, black-hole physics, and doubly special relativity and implies a minimal length uncertainty and a maximal momentum. We show that the quantized energy spectrum exactly agree with the semiclassical results.

Transport moments and Andreev billiards with tunnel barriers

Open chaotic systems are expected to possess universal transport statistics and recently there have been many advances in understanding and obtaining expressions for their transport moments. However, when tunnel barriers are added, which represents the situation in more general experimental physical systems, much less is known about the behaviour of the moments. By incorporating tunnel barriers in the recursive semiclassical diagrammatic approach, we obtain the moment generating function of the transmission eigenvalues at leading and subleading orders. For reflection quantities, quantum mechanical tunnelling phases play an essential role and we introduce new structures to deal with them. This allows us to obtain the moment generating function of the reflection eigenvalues and the Wigner delay times at a leading order. Our semiclassical results are in complementary regimes to the leading order results derived from the random matrix theory expanding the range of theoretically known moments. As a further application, we derive to the leading order the density of states of Andreev billiards coupled to a superconductor through tunnel barriers.

Semiclassical approach to the quantum Loschmidt echo in deep quantum regions: From validity to breakdown

Semiclassical results are usually expected to be valid in the semiclassical regime. An interesting question is, in models in which appropriate effective Planck constants can be introduced, to what extent will a semiclassical prediction stay valid when the effective Planck constant is increased? In this paper, we numerically study this problem, focusing on semiclassical predictions for the decay of the quantum Loschmidt echo in deep quantum regions. Our numerical simulations, carried out in the chaotic regime in the sawtooth model and in the kicked rotator model and also in the critical region of a one-dimensional Ising chain in transverse field, show that the semiclassical predictions may work even in deep quantum regions, particularly for perturbation strength in the so-called Fermi-Golden-Rule regime.