Semiclassical Distribution Research Articles

Semiclassical asymptotic distribution of resonances created by homoclinic and heteroclinic trajectories

Here we review a part of the joint work with Jean-François Bony, Thierry Ramond and Maher Zerzeri [Astérisque 405 (2018), pp. vii+314] on the semiclassical distribution of resonances for Schrödinger operators in a complex neighborhood of a fixed energy, assuming that the trapped set of the underlying classical dynamics on the energy surface consists of hyperbolic fixed points and associated homoclinic and heteroclinic trajectories. Based on a microlocal theory of solutions near a hyperbolic fixed point established by Bony, Fujiié, Ramond, and Zerzeri [J. Funct. Anal. 252 (2007), pp. 68–125], we compute the quantization condition of resonances and prove its accuracy. We will see in particular that the principal term of the imaginary part of resonances is − D 0 h -D_0 h , where D 0 D_0 is the minimum over all ‘cycles’ in the trapped set of what we call damping index defined in terms of the exponents at the hyperbolic fixed points therein. We will also see the subprincipal term of order h / | log ⁡ h | h/|\log h| . In this scale, resonances present various interesting distributions reflecting the geometry of the classical dynamics.

Questioning the adequacy of certain quantum arrival-time distributions

It is shown that a class of exponentially decaying time-of-arrival probability distributions, suggested by W\l{}odarz [Phys. Rev. A 65, 044103 (2002)], Marchewka and Schuss [Phys. Lett. A 240, 177 (1998), Phys. Rev. A 63, 032108 (2001); Phys. Rev. A 65, 042112 (2002)], and Jurman and Nikoli\ifmmode \acute{c}\else \'{c}\fi{} [Phys. Lett. A 396, 127247 (2021)], and a semiclassical distribution implicit in time-of-flight momentum measurements do not show the expected behavior for a Gaussian wave train. This casts doubts on the physical adequacy of these arrival-time proposals. In contrast, the quantum flux distribution (a special case of the Bohmian arrival-time distribution) displays the expected behavior.

Caustics of Weakly Lagrangian Distributions

We study semiclassical sequences of distributions \(u_h\) associated with a Lagrangian submanifold of phase space \(\mathcal {L}\subset T^*X\). If \(u_h\) is a semiclassical Lagrangian distribution, which concentrates at a maximal rate on \(\mathcal {L},\) then the asymptotics of \(u_h\) are well understood by work of Arnol’d, provided \(\mathcal {L}\) projects to X with a stable simple Lagrangian singularity. We establish sup-norm estimates on \(u_h\) under much more general hypotheses on the rate at which it is concentrating on \(\mathcal {L}\) (again assuming a stable simple projection). These estimates apply to sequences of eigenfunctions of integrable and KAM Hamiltonians.

Bound State Soliton Gas Dynamics Underlying the Spontaneous Modulational Instability.

We investigate the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The statistical properties of the noise-induced MI, observed previously in numerical simulations and in experiments, have not been explained theoretically. In this Letter, using the inverse scattering transform (IST) formalism, we propose a theoretical model of the asymptotic stage of the noise-induced MI based on N-soliton solutions of the focusing one-dimensional nonlinear Schrödinger equation. Specifically, we use ensembles of N-soliton bound states having a special semiclassical distribution of the IST eigenvalues, together with random phases for norming constants. To verify our model, we employ a recently developed numerical approach to construct an ensemble of N-soliton solutions with a large number of solitons, N∼100. Our investigation reveals a remarkable agreement between spectral (Fourier) and statistical properties of the long-term evolution of the MI and those of the constructed multisoliton, random-phase bound states. Our results can be generalized to a broad class of strongly nonlinear integrable turbulence problems.

Semiclassical asymptotic distribution of resonances created by homoclinic trajectories

Semiclassical asymptotic distribution of resonances created by homoclinic trajectories

Dissociation dynamics of the water dication following one-photon double ionization. I. Theory

The measurement of the triple differential cross section in the body frame for double photoionization of a molecule can be made in principle by detecting the ionic fragments and the two photoelectrons in coincidence---but only if the dynamics and geometry of dissociation of the doubly charged molecular ion are known. A classical trajectory study of the nine lowest states of the water dication is presented using high quality ab initio potential-energy surfaces. Sampling from a semiclassical initial distribution of positions and momenta is used to approximate ionization from the Frank-Condon region of the ground vibrational state of neutral ${\mathrm{H}}_{2}\mathrm{O}$. Excellent agreement in comparison with preliminary experimental momentum imaging measurements of double photoionization of water show that eight dication states can be unambiguously identified in the experiment with the aid of theory. The theoretical trajectory results allow body frame measurements of double photoionization to yield all eight states even though the usual assumption of direct dissociation, the ``axial recoil'' approximation, breaks down for three of the dication electronic states seen in the experiment.

Semiclassical approach to the work distribution

Work in closed quantum systems is usually defined by a two-point measurement. This definition of work is compatible with quantum fluctuation theorems but it fundamentally differs from its classical counterpart. In this paper, we study the correspondence principle in quantum chaotic systems. We derive a semiclassical expression of the work distribution for chaotic systems undergoing a general, finite time, process. This semiclassical distribution converges to the classical distribution in the usual classical limit. We show numerically that, for a particle inside a chaotic cavity, the semiclassical distribution provides a good approximation to quantum distribution.

Numerical solutions of ideal quantum gas dynamical flows governed by semiclassical ellipsoidal-statistical distribution

The ideal quantum gas dynamics as manifested by the semiclassical ellipsoidal-statistical (ES) equilibrium distribution derived in Wu et al. (Wu et al. 2012 Proc. R. Soc. A468, 1799-1823 (doi:10.1098/rspa.2011.0673)) is numerically studied for particles of three statistics. This anisotropic ES equilibrium distribution was derived using the maximum entropy principle and conserves the mass, momentum and energy, but differs from the standard Fermi-Dirac or Bose-Einstein distribution. The present numerical method combines the discrete velocity (or momentum) ordinate method in momentum space and the high-resolution shock-capturing method in physical space. A decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. Computations of two-dimensional Riemann problems are presented, and various contours of the quantities unique to this ES model are illustrated. The main flow features, such as shock waves, expansion waves and slip lines and their complex nonlinear interactions, are depicted and found to be consistent with existing calculations for a classical gas.

Complex Zeros of Eigenfunctions of 1D Schrodinger Operators

In this article we study the semiclassical distribution of the complex zeros of the eigenfunctions of the 1D Schrodinger operators for the class of real polynomial potentials of even degree, with fixed energy level, E. We show that as h n → 0 the zeros tend to concentrate on the union of some level curves ℜ (S (z m , z)) = c m where S(zm,z)=∫zmzV(t)−Edt is the complex action, and z m is a turning point. We also calculate these curves for some symmetric and nonsymmetric one-well and double-well potentials. The example of the nonsymmetric double-well potential shows that we can obtain different pictures of complex zeros for different subsequences of h n .

Semiclassical Wigner distribution for a two-mode entangled state generated by an optical parametric oscillator

We derive the steady state solution of the Fokker-Planck equation that describes the dynamics of the nondegenerate optical parametric oscillator in the truncated Wigner representation of the density operator. We assume that the pump mode is strongly damped, which permits its adiabatic elimination. When the elimination is correctly executed, the resulting stochastic equations contain multiplicative noise terms, and do not admit a potential solution. However, we develop an heuristic scheme leading to a satisfactory steady-state solution. This provides a clear view of the intracavity two-mode entangled state valid in all operating regimes of the OPO. A nongaussian distribution is obtained for the above threshold solution.

Spin Accumulation with Spin-Orbit Interaction

Spin accumulation is a crucial but imprecise concept in spintronics. In metal-based spintronics it is characterized in terms of semiclassical distribution functions. In semiconductors with a strong spin-orbit coupling the spin accumulation is interpreted as a superposition of coherent eigenstates. Both views can be reconciled by taking into account the electron-electron interaction: a sufficiently strong self-consistent exchange field reduces a spin accumulation to a chemical potential difference between the two spin bands even in the presence of spin-orbit coupling. We demonstrate the idea on a clean two-dimensional electron gas by showing how the exchange field protects a spin accumulation from dephasing and introduces an easy-plane anisotropy.

Quantum corrections for Boltzmann equation

We present the lowest order quantum correction to the semiclassical Boltzmann distribution function, and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation, and the quantum Wigner distribution function is expanded in powers of Planck constant, too. The negative quantum correlation in the Wigner distribution function which is just the quantum correction terms is naturally singled out, thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework.

Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory

We derive an asymptotic formula for the amplitude distribution in a fully nonlinear shallow-water solitary wave train which is formed as the long-time outcome of the initial-value problem for the Su–Gardner (or one-dimensional Green–Naghdi) system. Our analysis is based on the properties of the characteristics of the associated Whitham modulation system which describes an intermediate “undular bore” stage of the evolution. The resulting formula represents a “non-integrable” analogue of the well-known semi-classical distribution for the Korteweg–de Vries equation, which is usually obtained through the inverse scattering transform. Our analytical results are shown to agree with the results of direct numerical simulations of the Su–Gardner system. Our analysis can be generalised to other weakly dispersive, fully nonlinear systems which are not necessarily completely integrable.

Optimal control of mode-selective femtochemistry in multidimensional systems

We present a strategy for optimal control of the ground-state dynamics in multidimensional systems based on a combination of the semiclassical Wigner distribution approach with direct quantum chemical molecular dynamics (MD) ``on the fly.'' This allows one to treat all degrees of freedom without the need for precalculation of global potential energy surfaces. We demonstrate the scope of our theoretical procedure on two prototype systems representing rigid symmetrical molecules $({\mathrm{Na}}_{3}\mathrm{F})$ and flexible biomolecules with low-frequency modes (glycine). We show that the ground-state isomerization process can be selectively driven by ultrashort laser pulses with different shapes which are characteristic of the prototype systems. Thus, our method opens perspectives for control of the functionality of biomolecules. Moreover, assignment of the underlying processes to pulse shapes based on MD allows one to use optimal control as a tool for analysis.

Switching via quantum activation: A parametrically modulated oscillator

We study switching between period-two states of an underdamped quantum oscillator modulated at nearly twice its natural frequency. For all temperatures and parameter values switching occurs via quantum activation: it is determined by diffusion over oscillator quasienergy, provided the relaxation rate exceeds the rate of interstate tunneling. The diffusion has quantum origin and accompanies relaxation to the stable state. We find the semiclassical distribution over quasienergy. For T=0, where the system has detailed balance, this distribution differs from the distribution for $T\to 0$; the T=0 distribution is also destroyed by small dephasing of the oscillator. The characteristic quantum activation energy of switching displays a typical dependence on temperature and scaling behavior near the bifurcation point where period doubling occurs.

Mapping the Wigner distribution function of the Morse oscillator onto a semiclassical distribution function

The mapping of the Wigner distribution function (WDF) for a given bound-state onto a semiclassical distribution function (SDF) satisfying the Liouville equation introduced previously by us is applied to the ground state of the Morse oscillator. Here we give results showing that the SDF gets closer to the corresponding WDF as the number of levels of the Morse oscillator increases. We find that for a Morse oscillator with one level only, the agreement between the WDF and the mapped SDF is very poor but for a Morse oscillator of ten levels it becomes satisfactory.

Semiclassical distribution of eigenvalues for elliptic operators with Hölder continuous coefficients, part i: non-critical case

We consider a version of the Weyl formula describing the asymptotic behaviour of the counting function of eigenvalues in the semiclassical approximation for self-adjoint elliptic differential operators under weak regularity hypotheses. Our aim is to treat

Quantum corrections to the semiclassical temperature scale in the structured emission of tetrahedral complexes

Tetrahedral complexes with an excited state consisting of two closely lying triplet states are considered together with the full interaction Hamiltonian, including ligand field, Jahn-Teller, pseudo-Jahn-Teller, and spin-orbit interactions. The emission band shape is numerically calculated using Monte Carlo integration. The numerical calculations are applied to the ${\mathrm{PbWO}}_{4}$ system. Previous results using a semiclassical distribution function gave reasonable line shapes, but a mismatch with physical temperature. Using the correct quantum distribution function full correspondence with physical temperature is achieved.

Semiclassical approximations to real-time quantum-mechanical effects in correlation functions of complex molecular systems

Semiclassical approximation of real-time quantum effects is analyzed with the aid of the semiclassical initial value representation (SC-IVR) and Wigner distribution functions. We utilize these two ingredients to propose a new version of the semiclassical correlation function that contains, in principle, all quantum-mechanical effects. The advantage of this formulation is that it allows for a stepwise approximation specifically for real-time quantum effects based on a gradual inclusion of more degrees of freedom into the integral responsible for interference. From numerical calculations, this procedure does not seem to depend significantly on the choice coordinates if all degrees of freedom are coupled. This freedom from the coordinate choice removes possible ambiguities in applying this method. Several example cases are presented to demonstrate the usefulness of this approach.

Mapping Wigner distribution functions into semiclassical distribution functions

A mapping that relates the Wigner phase-space distribution function of a given stationary quantum mechanical wave function, a solution of the Schr\"odinger equation, to a specific solution of the Liouville equation, both subject to the same potential, is studied. By making this mapping, bound states are described by semiclassical distribution functions still depending on Planck's constant, whereas for elastic scattering of a particle by a potential they do not depend on it, the classical limit being reached in this case. Following this method, the mapped distributions of a particle bound in the P\"oschl-Teller potential and also in a modified oscillator potential are obtained.