Odinger Equation Research Articles

Dynamic interference in strong-field multiphoton excitation and ionization

By numerically solving the time-dependent Schr\"odinger equation, we theoretically investigated the dynamic interference in multiphoton excitation and ionization of hydrogen irradiated by 400-nm linearly polarized laser pulses. Interference of the multiphoton excitation paths at the rising and falling edges of the laser pulses gives rise to the fast oscillations of the intensity-dependent yield of the excited states. Our results show that the period of the oscillation increases abruptly when the laser intensity increases above the eight-photon ionization threshold. By tracing the population evolution of the ground state, we find that the depletion of the ground state on the rising edge of the laser pulse closes the multiphoton excitation path at the falling edge of the pulse. The slower oscillation comes from the interference of the excitation paths at different instants during the rising edge of the laser pulses. The change in the interference paths in the high-laser-intensity region also occurs in multiphoton ionization. The photoelectrons released at different instants of the rising edge of the laser pulses possess different parities. This difference is imprinted in the up-down asymmetry along the laser polarization direction in the photoelectron momentum distribution. The dynamic interference of the Rydberg states provides rich information about the dynamics in strong-field ionization.

Hidden-nucleons neural-network quantum states for the nuclear many-body problem

Understanding how protons and neutrons self-organize to form atomic nuclei requires solving the nuclear many-body Schr\"odinger equation. This work addresses this nontrivial task through a neural network quantum states ansatz that involves additional ``hidden'' degrees of freedom to improve the accuracy of the solution systemically. Light and medium-mass nuclei's energy and spatial density distributions are in excellent agreement with those obtained utilizing exact-diagonalization and diffusion Monte Carlo approaches.

Depletion in fermionic chains with inhomogeneous hoppings

The ground state of a free-fermionic chain with inhomogeneous hoppings at half-filling can be mapped into the Dirac vacuum on a static curved space-time, which presents exactly homogeneous occupations due to particle-hole symmetry. Yet, far from half-filling we observe density modulations and depletion effects. The system can be described by a 1D Schr\"odinger equation on a different static space-time, with an effective potential which accounts for the depleted regions. We provide a semiclassical expression for the single-particle modes and the density profiles associated to different hopping patterns and filling fractions. Moreover, we show that the depletion effects can be compensated for all filling fractions by adding a chemical potential proportional to the hoppings. Interestingly, we can obtain exactly the same density profiles on a homogeneous chain if we introduce a chemical potential which is inverse to the hopping intensities, even though the ground state is different from the original one.

Imprinting Chirality on Atoms Using Synthetic Chiral Light Fields.

Atoms are usually thought of as achiral objects. However, one can construct superpositions of atomic states that are chiral [1]. Here, we show how to excite such superpositions with tailored light fields both in the weak-field and strong-field regimes, using realistic laser parameters. First, we use time-dependent Schrödinger equation simulations to demonstrate the creation of a time-dependent bound chiral wave packet in sodium atoms. Second, we show how the time-dependent handedness of this wave packet can be probed by photoelectron circular dichroism, in spite of the central symmetry of the core potential. Third, we use time-dependent Schrödinger equation simulations to show how chirality can be directly imprinted on a photoelectron wave packet created by strong-field ionization and introduce an unambiguous chiral measure that allows us to characterize its handedness.

Wave localization in number-theoretic landscapes

We investigate the localization of waves in aperiodic structures that manifest the characteristic multiscale complexity of certain arithmetic functions with a central role in number theory. In particular, we study the eigenspectra and wave localization properties of tight-binding Schr\"{o}dinger equation models with on-site potentials distributed according to the Liouville function $\lambda(n)$, the M\"{o}bius function $\mu(n)$, and the Legendre sequence of quadratic residues modulo a prime (QRs). We employ Multifractal Detrended Fluctuation Analysis (MDFA) and establish the multifractal scaling properties of the energy spectra in these systems. Moreover, by systematically analyzing the spatial eigenmodes and their level spacing distributions, we show the absence of level repulsion with broadband localization across the entire energy spectra. Our study introduces deterministic aperiodic systems whose eigenmodes are all strongly localized in realistic finite one-dimensional systems and provides opportunities for novel quantum and classical devices of particular importance to cold-atom experiments in engineered speckle potentials and enhanced light-matter interactions.

Convergence over fractals for the periodic Schrödinger equation

We consider a fractal refinement of Carleson's problem for pointwise convergence of solutions to the periodic Schr\"odinger equation to their initial datum. For $\alpha \in (0,d]$ and \[ s < \frac{d}{2(d+1)} (d + 1 - \alpha), \] we find a function in $H^s(\mathbb{T}^d)$ whose corresponding solution diverges in the limit $t \to 0$ on a set with strictly positive $\alpha$-Hausdorff measure. We conjecture this regularity threshold to be optimal. We also prove that \[ s > \frac{d}{2(d+2)}\left( d+2-\alpha \right) \] is sufficient for the solution corresponding to every datum in $H^s(\mathbb T^d)$ to converge to such datum $\alpha$-almost everywhere.

Ionization dynamics and gauge invariance

Photoionization is one of the most fundamental processes in laser-matter interaction. It plays a crucial role also from the practical point of view since the electron-density dynamics in a gaseous medium is a process that affects the field propagation. Photoionization has been addressed in different regimes: linear, multiphoton, and tunneling. The ideal tool that allows for the description of ionization, in all aforementioned regimes, is the numerical evaluation of the time-dependent Schr\"odinger equation. The determination of the electron density needs the computation of the time-dependent ionization probability which unfortunately is an ambiguous quantity due to the gauge dependence of the latter. In this paper, we show how to overcome this difficulty by properly defining the time-dependent ionization probability in the context of the resolvent operator method. We show in particular that the velocity gauge allows for a definition of adiabatic states that is suitable to define an ionization threshold at all times during the interaction to compute ionization probability. Applications to linear, multiphoton, and tunneling regimes are presented for the one-dimensional problem. The extension to the nondipole case is discussed and we show that time-dependent ionization probability cannot be defined unambiguously due to the introduction of the magnetic-field component. We also discuss the case of gauge invariance in a subspace of the eigenbasis defined by the Hamiltonian.

Quantum and classical branching flow in space and time

Branching flow, a phenomenon known for steady wave propagation in two-dimensional weakly correlated random potential, is also present in the time-dependent Schr\"odinger equation for a single particle in one dimension, moving in a fluctuating random potential. We explore the two-dimensional parameter space of this model using numerical simulations and identify its classical regions, where just one classical parameter is sufficient for its specification, and its quantum region, where such a simplification is not possible. We also identify the region of the parameter space where known analytical results of a classical white noise model are relevant. Qualitative behavior of quantum and classical particle dynamics is discussed in terms of branching time scale and a new time scale related to a particle's kinetic energy.

Calculation of Coulomb breakup cross sections using a new Coulomb dynamical polarization potential

A new method for calculating the Coulomb breakup of unstable neutron-rich isotopes at high energies is presented. The calculations employ the eikonal approximation and use a new Coulomb dynamical polarization potential (CDPP), calculated by solving the Schr\"odinger equation for the entire motion of the exotic projectile as a two-body cluster structure using the adiabatic approximation and incorporating excitations to the continuum. Calculations for some exotic isotopes are compared with Coulomb dissociation cross section data and found to be in good agreement.

Photoelectron momentum distributions in the strong-field ionization of atomic hydrogen by few-cycle elliptically polarized optical pulses

We investigate the strong-field ionization of atomic hydrogen in a few-cycle elliptically polarized infrared pulse by solving the time-dependent Schr\"odinger equation. The dependence of the photoelectron momentum distribution on the pulse intensity, ellipticity, length, envelope, and carrier envelope phase is analyzed. In particular, we explain the variation of the electron offset angle with asymptotic electron energy through the combined action of the field and the Coulomb potential, and demonstrate that low-ellipticity pulses make it possible to access the electron release time.

Crossover between quantum and classical waves and high-frequency localization landscapes

Anderson localization is a universal interference phenomenon occurring when a wave evolves through a random medium and it has been observed in a great variety of physical systems, either quantum or classical. The recently developed localization landscape theory offers a computationally affordable way to obtain useful information on localized modes, such as their location or size. Here we examine this theory in the context of classical waves exhibiting high-frequency localization and for which the original localization landscape approach is no longer informative. Using the so-called Webster's transformation, to convert a classical wave equation into a Schr\"odinger equation with the same localization properties, and combining a set of frequency-shifted operators, we introduce an optimized localization landscape. This optimized localization landscape offers an affordable way to reveal key information on mode localization across the frequency spectrum.

Time optimal observability for Grushin Schrödinger equation

We consider two dimensional Grushin Schr\"odinger equation posed on a finite cylinder $\Omega=(-1,1)_x\times \T_y$ with Dirichlet boundary condition. We obtain the sharp observability by any horizontal strip, with the optimal time $T_*>0$ depending on the size of the strip. Consequently, we prove the exact controllability for Grushin Schr\"odinger equation. By exploiting the concentration of eigenfunctions of harmonic oscillator at $x=0$, we also show that the observability fails for any $T\leq T_*$.

Unitary evolution for a two-level quantum system in fractional-time scenario.

The time-evolution operator obtained from the fractional-time Schrödinger equation(FTSE) is said to be nonunitary since it does not preserve the norm of the vector state in time. As done in the time-dependent non-Hermitian quantum formalism, for a traceless non-Hermitian two-level quantum system, we demonstrate that it is possible to map the nonunitary time-evolution operator in a unitary one. It is done by considering a dynamical Hilbert space with a time-dependent metric operator, constructed from a Hermitian time-dependent Dyson map, in respect to which the system evolves in a unitary way, and the standard quantum mechanics interpretation can be made properly. To elucidate our approach, we consider three examples of Hamiltonian operators and their corresponding unitary dynamics obtained from the solutions of FTSE, and the respective Dyson maps.

Transparent boundary condition for simulating rogue wave solutions in the nonlinear Schrödinger equation.

This paper addresses the construction of numerical boundary conditions for simulating rogue wave solutions in the nonlinear Schrödinger equation. While three kinds of commonly used boundary conditions require a big enough computational domain to reproduce solutions faithfully in the central domain, we propose transparent boundary conditions for the Peregrine soliton and Kuznetsov-Ma breather solutions, respectively. For both solutions, these boundary conditions require a smaller computational domain than other boundary conditions to attain the best accuracy of the Crank-Nicolson scheme and selected mesh size, which will be referred to as the "acceptable accuracy" below. In particular, the computational domain with these boundary conditions is only 1/16 as small as others in the simulations of the Peregrine soliton solution. As a result, they reduce both the memory requirement and the computing time for the Peregrine soliton solution.

Radial-momentum-resolved measurement of the tunneling ionization time in attoclock experiments

We studied the tunneling ionization time in the elliptically polarized laser field with the recently proposed phase-of-the-phase spectroscopy, wherein a weak linearly polarized second-harmonic field was introduced to the attoclock frame. By monitoring the oscillation of the photoelectron yield with the relative phase between the two fields, the ionization time of the photoelectrons can be determined without any effort on handling the Coulomb interaction between the escaping electron and the parent ion. This advantage enables us to resolve the ionization time in the full photoelectron momentum distribution. By solving the time-dependent Schr\"odinger equation, we showed that the ionization time depends on the radial momentum of photoelectron. The photoelectron with larger radial momentum ionizes earlier at the emission angle before the most probable emission angle, while this radial-momentum dependence is reversed at the emission angle after the most probable emission angle. These results are confirmed by our experiments.

X(3872) ’s excitation and its connection with production at hadron colliders

The LHCb collaboration has found that the production rate of $X(3872)$ in proton-proton collisions decreases as final state particle multiplicity increases. Moreover, the ALICE experiment at CERN has observed that the number of deuterons produced increases with multiplicity, a behavior that is qualitatively different from that of the $X(3872)$. These experimental findings may point to a compact structure for the $X(3872)$ or, at least, that its hadronization could proceed through a charm-anticharm core. We have recently used a diffusion Monte Carlo method to solve the many-body Schr\"odinger equation that describes the $X(3872)$ as a $c \bar c q \bar q$ tetraquark system with quantum numbers $I^G(J^{PC})=0^+(1^{++})$ and $1^-(1^{++})$. According to our structural analysis, the quark--(anti-)quark correlations resemble light-meson--heavy-meson molecules of type $\omega J/\psi$ and $\rho J/\psi$, rather than the most extended $D\bar D^{\ast}$ interpretation. It was argued that this fact may be the key to make compatible the molecular features of the $X(3872)$ with its production observables. The same formalism allows us to compute the first color excited $c \bar c q \bar q$ tetraquark state with either $I^G(J^{PC})=0^+(1^{++})$ or $1^-(1^{++})$. A bound-state is found in each channel, their masses are around 4.0 GeV which is an energy region where many new exotic candidates have been collected by the Particle Data Group. Concerning their structural properties, these states cluster in a compact diquark-antidiquark arrangement which matches perfectly with a so-called Born-Oppenheimer tetraquark configuration. The promptly production rates of these states in proton-proton, proton-nucleus and nucleus-nucleus collisions should fall off equal to or even faster than those of the $X(3872)$.

Dipole-forbidden transitions induced by the gradient of optical near fields

The electric-field spatial gradient is a very important property of an optical near-field (ONF). In this study, its effect on the single-photon ionization is theoretically investigated via the three-dimensional time-dependent Schr\"odinger equation and perturbation theory. The results show that the field gradient can lead to dipole-forbidden transitions, which competes with the usual dipole transition and results in changes of the photoelectron momentum distribution. Additionally, we establish a relationship between the field gradient with the angular distribution of the photoelectrons. Through this mapping, the field gradient can be accurately retrieved. Furthermore, a way for extracting the phase difference between the partial waves with odd and even parity from the photoelectron momentum distribution is proved to be feasible. We find that the Cooper minimum can enhance the effect of the field gradient and thus improves the accuracy of our proposed retrieval procedure. Our results pave the way to exploit atomic structural properties as a probe of the gradient of the ONF.

Generalized Ellis-Bronnikov bilayer graphene wormholelike surface

In this paper, we investigate the spinless stationary Schr\"odinger equation for the electron when it is permanently bound to a generalized Ellis-Bronnikov bilayer graphene wormholelike surface. The curvature gives rise to a geometric potential affecting thus the electronic dynamics. The geometry of the wormhole's shape is controlled by the parameter $n$ which assumes even values. We discuss the role played by the parameter $n$ and the orbital angular momentum on bound states and probability density for the electron.

Ultrafast valley polarization of graphene nanorings

We study theoretically electron dynamics of a graphene nanoring placed in the field of an ultrashort optical pulse. We describe the graphene nanoring within an effective model with infinite mass boundary conditions. For an optical pulse with a duration of just a few femtoseconds, the electron dynamics is coherent and is described by a time-dependent Schr\"odinger equation. If the optical pulse is circularly polarized, then two valleys of graphene are populated differently, resulting in a finite valley polarization of the system after the pulse. Such a valley polarization is a unique property of graphene nanoscale systems, while for a graphene monolayer, a circularly polarized pulse does not produce any valley polarization. The valley polarization of the graphene nanoring depends on parameters of the system, such as inner and outer radii. With the system's size increasing, the valley polarization monotonically decreases, converging to its zero value for the infinite graphene monolayer.

Detection-time distribution for several quantum particles

We address the question of how to compute the probability distribution of the time at which a detector clicks in the situation of $n$ nonrelativistic quantum particles in a volume $\mathrm{\ensuremath{\Omega}}\ensuremath{\subset}{\mathbb{R}}^{3}$ in physical space and detectors placed along the boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Omega}}$ of $\mathrm{\ensuremath{\Omega}}$. We previously [Tumulka, Ann. Phys. (NY) 442, 168910 (2022)] argued in favor of a rule for the one-particle case that involves a Schr\"odinger equation with an absorbing boundary condition on $\ensuremath{\partial}\mathrm{\ensuremath{\Omega}}$ introduced by Werner; we call this rule the ``absorbing boundary rule.'' Here, we describe the natural extension of the absorbing boundary rule to the $n$-particle case. A key element of this extension is that, upon a detection event, the wave function gets collapsed by inserting the detected position, at the time of detection, into the wave function, thus yielding a wave function of $n\ensuremath{-}1$ particles. We also describe an extension of the absorbing boundary rule to the case of moving detectors.