Random Wave Functions Research Articles

Computation of the Spatial Distribution of Charge-Carrier Density in Disordered Media.

The space- and temperature-dependent electron distribution n(r,T) determines optoelectronic properties of disordered semiconductors. It is a challenging task to get access to n(r,T) in random potentials, while avoiding the time-consuming numerical solution of the Schrödinger equation. We present several numerical techniques targeted to fulfill this task. For a degenerate system with Fermi statistics, a numerical approach based on a matrix inversion and one based on a system of linear equations are developed. For a non-degenerate system with Boltzmann statistics, a numerical technique based on a universal low-pass filter and one based on random wave functions are introduced. The high accuracy of the approximate calculations are checked by comparison with the exact quantum-mechanical solutions.

Weak second-order quantum state diffusion unraveling of the Lindblad master equation

Simulating mixed-state evolution in open quantum systems is crucial for various chemical physics, quantum optics, and computer science applications. These simulations typically follow the Lindblad master equation dynamics. An alternative approach known as quantum state diffusion unraveling is based on the trajectories of pure states generated by random wave functions, which evolve according to a nonlinear Itô–Schrödinger equation (ISE). This study introduces weak first-order and second-order solvers for the ISE based on directly applying the Itô–Taylor expansion with exact derivatives in the interaction picture. We tested the method on free and driven Morse oscillators coupled to a thermal environment and found that both orders allowed practical estimation with a few dozen iterations. The variance was relatively small compared to the linear unraveling and did not grow with time. The second-order solver delivers a much higher accuracy and stability with bigger time steps than the first-order scheme, with a small additional workload. However, the second-order algorithm has quadratic complexity with the number of Lindblad operators as opposed to the linear complexity of the first-order algorithm.

Semi-Markov processes in open quantum systems. II. Counting statistics with resetting.

A semi-Markov process method for obtaining general counting statistics for open quantum systems is extended to the scenario of resetting. The simultaneous presence of random resets and wave function collapses means that the quantum jump trajectories are no longer semi-Markov. However, focusing on trajectories and using simple probability formulas, general counting statistics can still be constructed from reset-free statistics. An exact tilted matrix equationis also obtained. The inputs of these methods are the survival distributions and waiting-time density distributions instead of quantum operators. In addition, a continuous-time cloning algorithm is introduced to simulate the large-deviation properties of open quantum systems. Several quantum optics systems are used to demonstrate these results.

Quantum states in disordered media. II. Spatial charge carrier distribution

The space- and temperature-dependent electron distribution $n(\mathbf r,T)$ is essential for the theoretical description of the opto-electronic properties of disordered semiconductors. We present two powerful techniques to access $n(\mathbf r,T)$ without solving the Schr\"odinger equation. First, we derive the density for non-degenerate electrons by applying the Hamiltonian recursively to random wave functions (RWF). Second, we obtain a temperature-dependent effective potential from the application of a universal low-pass filter (ULF) to the random potential acting on the charge carriers in disordered media. Thereby, the full quantum-mechanical problem is reduced to the quasi-classical description of $n(\mathbf r,T)$ in an effective potential. We numerically verify both approaches by comparison with the exact quantum-mechanical solution. Both approaches prove superior to the widely used localization landscape theory (LLT) when we compare our approximate results for the charge carrier density and mobility at elevated temperatures obtained by RWF, ULF, and LLT with those from the exact solution of the Schr\"odinger equation.

Gradient-based reconstruction of molecular Hamiltonians and density matrices from time-dependent quantum observables

We consider a quantum system with a time-independent Hamiltonian parametrized by a set of unknown parameters $\alpha$. The system is prepared in a general quantum state by an evolution operator that depends on a set of unknown parameters $P$. After the preparation, the system evolves in time, and it is characterized by a time-dependent observable ${\cal O}(t)$. We show that it is possible to obtain closed-form expressions for the gradients of the distance between ${\cal O}(t)$ and a calculated observable with respect to $\alpha$, $P$ and all elements of the system density matrix, whether for pure or mixed states. These gradients can be used in projected gradient descent to infer $\alpha$, $P$ and the relevant density matrix from dynamical observables. We combine this approach with random phase wave function approximation to obtain closed-form expressions for gradients that can be used to infer population distributions from averaged time-dependent observables in problems with a large number of quantum states participating in dynamics. The approach is illustrated by determining the temperature of molecular gas (initially, in thermal equilibrium at room temperature) from the laser-induced time-dependent molecular alignment.

Dispersion Interactions in Exciton-Localized States. Theory and Applications to π-π* and n-π* Excited States.

We address the problem of intermolecular interaction energy calculations in molecular complexes with localized excitons. Our focus is on the correct representation of the dispersion energy. We derive an extended Casimir-Polder formula for direct computation of this contribution through second order in the intermolecular interaction operator V̂. An alternative formula, accurate to infinite order in V̂, is derived within the framework of the adiabatic connection (AC) theory. We also propose a new parametrization of the VV10 nonlocal correlation density functional, so that it corrects the CASSCF energy for the dispersion contribution and can be applied to excited-state complexes. A numerical investigation is carried out for benzene, pyridine, and peptide complexes with the local exciton corresponding to the lowest π–π* or n– π* states. The extended Casimir-Polder formula is implemented in the framework of multiconfigurational symmetry-adapted perturbation theory, SAPT(MC). A SAPT(MC) analysis shows that the creation of a localized exciton affects mostly the electrostatic component of the interaction energy of investigated complexes. Nevertheless, the changes in Pauli repulsion and dispersion energies cannot be neglected. We verify the performance of several perturbation- and AC-based methods. Best results are obtained with a range-separated variant of an approximate AC approach employing extended random phase approximation and CASSCF wave functions.

Topological Distillation by Principal Component Analysis in Disordered Fractional Quantum Hall States* *Supported by the National Natural Science Foundation of China (Grant No. 11674282), and the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000).

We study the behavior of two-dimensional electron gas in the fractional quantum Hall (FQH) regime in the presence of disorder potential. The principal component analysis is applied to a set of disordered Laughlin ground state model wave function to enable us to distill the model wave function of the pure Laughlin state. With increasing the disorder strength, the ground state wave function is expected to deviate from the Laughlin state and eventually leave the FQH phase. We investigate the phase transition from the Laughlin state to a topologically trivial state by analyzing the overlap between the random sample wave functions and the distilled ground state wave function. It is proposed that the cross point of the principal component amplitude and its counterpart is the critical disorder strength, which marks the collapse of the FQH regime.

Off-diagonal observable elements from random matrix theory: distributions, fluctuations, and eigenstate thermalization

We derive the eigenstate thermalization hypothesis (ETH) from a random matrix Hamiltonian by extending the model introduced by Deutsch (1991 Phys. Rev. A 43 2046). We approximate the coupling between a subsystem and a many-body environment by means of a random Gaussian matrix. We show that a common assumption in the analysis of quantum chaotic systems, namely the treatment of eigenstates as independent random vectors, leads to inconsistent results. However, a consistent approach to the ETH can be developed by introducing an interaction between random wave-functions that arises as a result of the orthonormality condition. This approach leads to a consistent form for off-diagonal matrix elements of observables. From there we obtain the scaling of time-averaged fluctuations of generic observables with system size for which we calculate an analytic form in terms of the inverse participation ratio. The analytic results are compared to exact diagonalizations of a quantum spin chain for different physical observables in multiple parameter regimes.

Quantum Dynamics as Landau–Lifshitz-Type Dynamics and Random Wave Function Collapse

It is first observed that quantum dynamics can be viewed as Landau-Lifshitz-type dynamics. Then, by using this fact, instability-induced random branching of deterministic dynamics is discussed as a possible mechanism of random wave function collapse. In the case of two-level systems important in quantum computing and spintronics, the Born probability rule emerges as the simplest solution to the equation for measurement probabilities.

Nonergodic Phases in Strongly Disordered Random Regular Graphs.

We combine numerical diagonalization with semianalytical calculations to prove the existence of the intermediate nonergodic but delocalized phase in the Anderson model on disordered hierarchical lattices. We suggest a new generalized population dynamics that is able to detect the violation of ergodicity of the delocalized states within the Abou-Chakra, Anderson, and Thouless recursive scheme. This result is supplemented by statistics of random wave functions extracted from exact diagonalization of the Anderson model on ensemble of disordered random regular graphs (RRG) of N sites with the connectivity K=2. By extrapolation of the results of both approaches to N→∞ we obtain the fractal dimensions D_{1}(W) and D_{2}(W) as well as the population dynamics exponent D(W) with the accuracy sufficient to claim that they are nontrivial in the broad interval of disorder strength W_{E}<W<W_{c}. The thorough analysis of the exact diagonalization results for RRG with N>10^{5} reveals a singularity in D_{1,2}(W) dependencies which provides clear evidence for the first order transition between the two delocalized phases on RRG at W_{E}≈10.0. We discuss the implications of these results for quantum and classical nonintegrable and many-body systems.

Rotational Control of Asymmetric Molecules: Dipole- versus Polarizability-Driven Rotational Dynamics.

We experimentally study the optical- and terahertz-induced rotational dynamics of asymmetric molecules in the gas phase. Terahertz and optical fields are identified as two distinct control handles over asymmetric molecules, as they couple to the rotational degrees of freedom via the molecular dipole and polarizability selectively. The distinction between those two rotational handles is highlighted by different types of quantum revivals observed in long-duration (>100 ps) field-free rotational evolution. The experimental results are in excellent agreement with random phase wave function (RPWF) simulations [Phys. Rev. A 91, 063420 (2015)] and provide verification of the RPWF as an efficient method for calculating asymmetric molecular dynamics at ambient temperatures, where exact calculation methods are practically not feasible. Our observations and analysis pave the way for orchestrated excitations by both optical and terahertz fields as complementary rotational handles that enable a plethora of new possibilities in three-dimensional rotational control of asymmetric molecules.

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment

A quantum system (with Hilbert space $\mathscr{H}_1$) entangled with its environment (with Hilbert space $\mathscr{H}_2$) is usually not attributed a wave function but only a reduced density matrix $\rho_1$. Nevertheless, there is a precise way of attributing to it a random wave function $\psi_1$, called its conditional wave function, whose probability distribution $\mu_1$ depends on the entangled wave function $\psi\in\mathscr{H}_1\otimes\mathscr{H}_2$ in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of $\mathscr{H}_2$ but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about $\mu_1$, e.g., that if the environment is sufficiently large then for every orthonormal basis of $\mathscr{H}_2$, most entangled states $\psi$ with given reduced density matrix $\rho_1$ are such that $\mu_1$ is close to one of the so-called GAP (Gaussian adjusted projected) measures, $GAP(\rho_1)$. We also show that, for most entangled states $\psi$ from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval $[E,E+\delta E]$) and most orthonormal bases of $\mathscr{H}_2$, $\mu_1$ is close to $GAP(\mathrm{tr}_2 \rho_{mc})$ with $\rho_{mc}$ the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then $\mu_1$ is close to $GAP(\rho_\beta)$ with $\rho_\beta$ the canonical density matrix on $\mathscr{H}_1$ at inverse temperature $\beta=\beta(E)$. This provides the mathematical justification of our claim in [J. Statist. Phys. 125:1193 (2006), http://arxiv.org/abs/quant-ph/0309021] that $GAP$ measures describe the thermal equilibrium distribution of the wave function.

Erratum: Orientation dynamics of asymmetric rotors using random phase wave functions [Phys. Rev. A 91, 063420 (2015)

Intense terahertz-frequency pulses induce coherent rotational dynamics and orientation of polar molecular ensembles. Exact numerical methods for rotational dynamics are computationally not feasible for the vast majority of molecular rotors - the asymmetric top molecules at ambient temperatures. We exemplify the use of Random Phase Wave Functions (RPWF) by calculating the terahertz-induced rotational dynamics of sulfur dioxide (SO2) at ambient temperatures and high field strengths and show that the RPWF method gains efficiency with the increase in temperature and in the THz-field strengths. The presented method provides wide-ranging computational access to rotational dynamical responses of molecules at experimental conditions which are far beyond the reach of exact numerical methods.

Electric quadrupole and magnetic dipole moments of odd nuclei near the magic ones in a self-consistent approach

We present a model which describes the properties of odd-even nuclei with one nucleon more, or less, with respect to the magic number. In addition to the effects related to the unpaired nucleon, we consider those produced by the excitation of the closed shell core. By using a single particle basis generated with Hartree-Fock calculations, we describe the polarization of the doubly magic-core with Random Phase Approximation collective wave functions. In every step of the calculation, and for all the nuclei considered, we use the same finite-range nucleon-nucleon interaction. We apply our model to the evaluation of electric quadrupole and magnetic dipole moments of odd-even nuclei around oxygen, calcium, zirconium, tin and lead isotopes. Our Random Phase Approximation description of the polarization of the core improves the agreement with experimental data with respect to the predictions of the independent particle model. We compare our results with those obtained in first-order perturbation theory, with those produced by Hartree-Fock-Bogolioubov calculations and with those generated within the Landau-Migdal theory of finite Fermi systems. The results of our universal, self-consistent, and parameter free approach have the same quality of those obtained with phenomenological approaches where the various terms of the nucleon-nucleon interaction are adapted to reproduce some specific experimental data. A critical discussion on the validity of the model is presented.

Orientation dynamics of asymmetric rotors using random phase wave functions

Intense terahertz-frequency pulses induce coherent rotational dynamics and orientation of polar molecular ensembles. Exact numerical methods for rotational dynamics at room temperature are computationally not feasible for the vast majority of molecular rotors: the asymmetric top molecules at ambient temperatures. We exemplify the use of random phase wave functions (RPWFs) by calculating the terahertz-induced rotational dynamics of sulfur dioxide at ambient temperatures and high-field strengths and show that the RPWF method gains efficiency with the increase in temperature and in the terahertz-field strengths. The present method provides wide-ranging computational access to rotational dynamical responses of molecules at experimental conditions that are far beyond the reach of exact numerical methods.

Spin and the Thermal Equilibrium Distribution of Wave Functions

Consider a quantum system S weakly interacting with a very large but finite system B called the heat bath, and suppose that the composite S∪B is in a pure state Ψ with participating energies between E and E+δ with small δ. Then, it is known that for most Ψ the reduced density matrix of S is (approximately) equal to the canonical density matrix. That is, the reduced density matrix is universal in the sense that it depends only on S’s Hamiltonian and the temperature but not on B’s Hamiltonian, on the interaction Hamiltonian, or on the details of Ψ. It has also been pointed out that S can also be attributed a random wave function ψ whose probability distribution is universal in the same sense. This distribution is known as the “Scrooge measure” or “Gaussian adjusted projected (GAP) measure”; we regard it as the thermal equilibrium distribution of wave functions. The relevant concept of the wave function of a subsystem is known as the “conditional wave function.” In this paper, we develop analogous considerations for particles with spin. One can either use some kind of conditional wave function or, more naturally, the “conditional density matrix,” which is in general different from the reduced density matrix. We ask what the thermal equilibrium distribution of the conditional density matrix is, and find the answer that for most Ψ the conditional density matrix is (approximately) deterministic, in fact (approximately) equal to the canonical density matrix.

Random wavefunctions and percolation

Recently it was conjectured that nodal domains of random wavefunctions are adequately described by critical percolation theory. In this paper we strengthen this conjecture in two respects. First, we show that, though wavefunction correlations decay slowly, a careful use of Harris' criterion confirms that these correlations are unessential and nodal domains of random wavefunctions belong to the same universality class as non-correlated critical percolation. Second, we argue that level domains of random wavefunctions are described by the non-critical percolation model.

Statistical properties of many particle eigenfunctions

Wavefunction correlations and density matrices for few or many particles are derived from the properties of semiclassical energy Green functions. Universal features of fixed energy (microcanonical) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as N → ∞. This arises through a little known asymptotic limit of Bessel functions. Constraints due to symmetries, boundaries and collisions between particles can be included.

Random wave functions with boundary and normalization constraints

We present an improved version of Berry's ansatz able to incorporate exactly the existence of boundaries and the correct normalization of the eigenfunction into an ensemble of random waves. We then reformulate the Random Wave conjecture showing that in its new version it is a statement about the universal nature of eigenfunction fluctuations in systems with chaotic classical dynamics. The emergence of the universal results requires the use of both semiclassical methods and a new expansion for a very old problem in quantum statistical physics.

Statistical properties of chaotic wavefunctions in two and more dimensions

Starting with Berry's hypothesis for fixed energy waves in a classically chaotic system, and casting it in a Green function form, we derive wavefunction correlations and density matrices for few or many particles. Universal features of fixed energy (microcanonical) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as N↦∞. This arises through a little known asymptotic limit of Bessel functions. The Berry random wave hypothesis in many dimensions may be viewed as an alternative approach to quantum statistical mechanics, when extended to include constraints and potentials.