Momentum-space Density Research Articles

On the relation between nodal structures in quantum wave functions and particle correlation

We study the influence of nodal structures in two-dimensional quantum mechanical densities on wave packet entanglement. This is motivated by our recent study [Entropy, 25, 970 (2023)], which showed that the mutual information derived from the momentum-space probability density of a coupled two-particle system exhibits an unusual time dependence, which is not encountered if the position-space density is employed in the calculation. In studying a model density, here, we identify cases where the mutual information increases with the number of nodes in the wave function and approaches a finite value, whereas in this limit, the linear correlation vanishes. The results of the analytical model are then applied to interpret the correlation measures for coupled electron-nuclear dynamics, which are treated by numerically solving the time-dependent Schrödinger equation.

Differential Shannon Entropies Characterizing Electron-Nuclear Dynamics and Correlation: Momentum-Space Versus Coordinate-Space Wave Packet Motion.

We calculate differential Shannon entropies derived from time-dependent coordinate-space and momentum-space probability densities. This is performed for a prototype system of a coupled electron-nuclear motion. Two situations are considered, where one is a Born-Oppenheimer adiabatic dynamics, and the other is a diabatic motion involving strong non-adiabatic transitions. The information about coordinate- and momentum-space dynamics derived from the total and single-particle entropies is discussed and interpreted with the help of analytical models. From the entropies, we derive mutual information, which is a measure for the electron-nuclear correlation. In the adiabatic case, it is found that such correlations are manifested differently in coordinate- and momentum space. For the diabatic dynamics, we show that it is possible to decompose the entropies into state-specific contributions.

Wilsonian renormalization as a quantum channel and the separability of fixed points

We show that the Wilsonian formulation of the renormalization group (RG) defines a quantum channel acting on the momentum-space density matrices of a quantum field theory. This information theoretical property of the RG allows us to derive a remarkable consequence for the vacuum of theories at a fixed point: they have no entanglement between momentum scales. Our result can be understood as deriving from the scale symmetry of such theories and leads to constraints on the form of the ground state and on expectation values of momentum-space operators.

Itinerant ferromagnetism entrenched by the anisotropy of spin–orbit coupling in a dipolar Fermi gas

We investigate the itinerant ferromagnetism in a dipolar Fermi atomic system with the anisotropic spin–orbit coupling (SOC), which is traditionally explored with isotropic contact interaction. We first study the ferromagnetism transition boundaries and the properties of the ground states through the density and spin-flip distribution in momentum space, and we find that both the anisotropy and the magnitude of the SOC play an important role in this process. We propose a helpful scheme and a quantum control method which can be applied to conquering the difficulties of previous experimental observation of itinerant ferromagnetism. Our further study reveals that exotic Fermi surfaces and an abnormal phase region can exist in this system by controlling the anisotropy of SOC, which can provide constructive suggestions for the research and the application of a dipolar Fermi gas. Furthermore, we also calculate the ferromagnetism transition temperature and novel distributions in momentum space at finite temperature beyond the ground states from the perspective of experiment.

Densities and momentum distributions in A≤12 nuclei from chiral effective field theory interactions

This study presents calculations of one- and two-body densities in both coordinate and momentum space for various nuclei up to ${}^{12}$C, using the phenomenological AV18+UX and various Norfolk $N\phantom{\rule{0}{0ex}}N$ + $N\phantom{\rule{0}{0ex}}N\phantom{\rule{0}{0ex}}N$ $\ensuremath{\chi}$EFT interactions. It features new calculations of the pair density as a function of both pair separation and center-of-mass, as well as the two-body momentum distribution for short- and long-range pairs differentiated by a pair separation boundary. The full set of results are available online for general use by the nuclear physics community and are intended to provide useful insights into the short-range structure of nuclei and various reaction processes.

Shannon Information Entropy Sum of a Free Particle in Three Dimensions Using Cubical and Spherical Symmetry

In this paper, the plane wave solutions of a free particle in three dimensions for Cubical and Spherical Symmetry have been considered. The coordinate space wave functions for the Cubical and Spherical Symmetry are obtained by solving the Schrdinger differential equation. The momentum space wave function is obtained by using the operator form of an observable in the case of Cubical Symmetry. For Spherical Symmetry, the same is obtained by taking the Fourier transform of the respective coordinate space wave function. The wave functions have been used to constitute probability densities in coordinate and momentum space for both the symmetries. Further, the Shannon information entropy has been computed both in coordinate and momentum space respectively for (L is the length of the side of the cubical box) values for Cubical Symmetry and for values in Spherical Symmetry keeping (k is the wave vector and p is the momentum of the free particle) constant. The values obtained for the Shannon information entropies are found to satisfy the Bialynicki-Birula and Myceilski (BBM) inequality at larger values () in case of Cubical Symmetry and for values of and in Spherical Symmetry.

Itinerant ferromagnetism of a dipolar Fermi gas with Raman-induced spin-orbit coupling

We elucidate the itinerant ferromagnetism of a dipolar Fermi gas with a Raman-induced spin-orbit coupling by investigating the exotic phase diagrams. It is revealed that the dipolar interaction along with spin-orbit coupling can corroborate the formation of ferromagnetism and that the Raman coupling adversely eliminates the tendency to this ferromagnetism transition, which greatly transcends the general understanding of this subject with contact interaction only. We explore the ground states through the density and spin-flip distribution in momentum space. We calculate the transition temperatures well within the reach of an experimental system when altering the dipolar and spin-orbit-coupling strength, which paves a way to the further experimental realization.

Bogoliubov theory of a Bose–Einstein condensate of rigid rotor molecules

We consider a BEC of rigid rotor molecules confined to quasi-2D through harmonic trapping. The molecules are subjected to an external electric field which polarizes the gas, and the molecules interact via dipole–dipole interactions. We present a description of the ground state and low-energy excitations of the system including an analysis of the mean-field energy, polarization, and stability. Under large electric fields the gas becomes fully polarized and we reproduce a well known density-wave instability which arises in polar BECs. Under smaller applied electric fields the gas develops an in-plane polarization leading to the emergence of a new global instability as the molecules ‘tilt’. The character of these instabilities is clarified by means of momentum-space density–density structure factors. A peak at zero momentum in the spin–spin structure factor for the in-plane component of the polarization indicates that the tilt instability is a global phonon-like instability.

Studying the low-entropy Mott transition of bosons in a three-dimensional optical lattice by measuring the full momentum-space density

We report on a combined experimental and theoretical study of the low-entropy Mott transition for interacting bosons trapped in a three-dimensional (3D) cubic lattice -- namely, the interaction-induced superfluid-to-normal phase transition in the vicinity of the zero-temperature Mott transition. Our analysis relies on the measurement of the 3D momentum distribution, which allows us to extract the momentum-space density $\rho({\bf k}={\bf 0})$ at the center of the Brillouin zone. Upon varying the ratio between the interaction $U$ and the tunnelling energy $J$ across the superfluid transition, we observe that $\rho({\bf k}={\bf 0})$ exhibits a sharp transition at a value of $U/J$ consistent with the bulk prediction from quantum Monte Carlo. In addition, the variation of $\rho({\bf k}={\bf 0})$ with $U/J$ exhibits a critical behavior consistent with the expected 3D XY universality class. Our results show that the tomographic reconstruction of the momentum distribution of ultracold bosons can reveal traits of the critical behavior of the superfluid transition even in an inhomogeneous trapped system.

Certifying the Adiabatic Preparation of Ultracold Lattice Bosons in the Vicinity of the Mott Transition.

We present a joint experimental and theoretical analysis to assess the adiabatic experimental preparation of ultracold bosons in optical lattices aimed at simulating the three-dimensional Bose-Hubbard model. Thermometry of lattice gases is realized from the superfluid to the Mott regime by combining the measurement of three-dimensional momentum-space densities with abinitio quantum MonteCarlo (QMC) calculations of the same quantity. The measured temperatures are in agreement with isentropic lines reconstructed via QMC for the experimental parameters of interest, with a conserved entropy per particle of S/N=0.8(1)k_{B}. In addition, the Fisher information associated with this thermometry method shows that the latter is most accurate in the critical regime close to the Mott transition, as confirmed in the experiment. These results prove that equilibrium states of the Bose-Hubbard model-including those in the quantum-critical regime above the Mott transition-can be adiabatically prepared in cold-atom apparatus.

MCTDH-X: The multiconfigurational time-dependent Hartree method for indistinguishable particles software

We introduce and describe the multiconfigurational time-depenent Hartree for indistinguishable particles (MCTDH-X) software, which is hosted, documented, and distributed at http://ultracold.org. This powerful tool allows the investigation of ground state properties and dynamics of interacting quantum many-body systems in different spatial dimensions. The MCTDH-X software is a set of programs and scripts to compute, analyze, and visualize solutions for the time-dependent and time-independent many-body Schrödinger equation for indistinguishable quantum particles. As the MCTDH-X software represents a general solver for the Schrödinger equation, it is applicable to a wide range of problems in the fields of atomic, optical, molecular physics, and condensed matter systems. In particular, it can be used to study light–matter interactions, correlated dynamics of electrons in the solid state as well as some aspects related to quantum information and computing. The MCTDH-X software solves a set of nonlinear coupled working equations based on the application of the time-dependent variational principle to the Schrödinger equation. These equations are obtained by using an ansatz for the many-body wavefunction that is a expansion in a set of time-dependent, fully symmetrized bosonic (X = B) or fully anti-symmetrized fermionic (X = F) many-body basis states. It is the time-dependence of the basis set that enables MCTDH-X to deal with quantum dynamics at a superior accuracy as compared to, for instance, exact diagonalization approaches with a static basis, where the number of basis states necessary to capture the dynamics of the wavefunction typically grows rapidly with time. Herein, we give an introduction to the MCTDH-X software via an easy-to-follow tutorial with a focus on accessibility. The illustrated exemplary problems are hosted at http://ultracold.org/tutorial and consider the physics of a few interacting bosons or fermions in a double-well potential. We explore computationally the position-space and momentum-space density, the one-body reduced density matrix, Glauber correlation functions, phases, (dynamical) phase transitions, and the imaging of the quantum systems in single-shot images. Although a few particles in a double well potential represent a minimal model system, we are able to demonstrate a rich variety of phenomena with it. We use the double well to illustrate the fermionization of bosonic particles, the crystallization of fermionic particles, characteristics of the superfluid and Mott-insulator quantum phases in Hubbard models, and even dynamical phase transitions. We provide a complete set of input files and scripts to redo all computations in this paper at http://ultracold.org/data/tutorial_input_files.zip, accompanied by tutorial videos at https://tinyurl.com/tjx35sq. Our tutorial should guide the potential users to apply the MCTDH-X software also to more complex systems.

Momentum-Space Probability Density of $${}^6$$He in Halo Effective Field Theory

We compute the momentum-space probability density of ${}^6$He at leading order in Halo EFT. In this framework, the ${}^6$He nucleus is treated as a three-body problem with a ${}^4$He core ($c$) and two valence neutrons ($n$). This requires the $nn$ and $nc$ t-matrices as well as a $cnn$ force as input in the Faddeev equations. Since the $nc$ t-matrix corresponds to an energy-dependent potential, we consider the consequent modifications to the standard normalization and orthogonality conditions. We find that these are small for momenta within the domain of validity of Halo EFT. In this regime, the ${}^6$He probability density is regulator independent, provided the cutoff is significantly above the EFT breakdown scale.

Identifying quantum phase transitions using artificial neural networks on experimental data

Machine learning techniques such as artificial neural networks are currently revolutionizing many technological areas and have also proven successful in quantum physics applications. Here we employ an artificial neural network and deep learning techniques to identify quantum phase transitions from single-shot experimental momentum-space density images of ultracold quantum gases and obtain results, which were not feasible with conventional methods. We map out the complete two-dimensional topological phase diagram of the Haldane model and provide an accurate characterization of the superfluid-to-Mott-insulator transition in an inhomogeneous Bose-Hubbard system. Our work points the way to unravel complex phase diagrams of general experimental systems, where the Hamiltonian and the order parameters might not be known.

Highlights from NA61/SHINE: Proton Intermittency Analysis

The NA61/SHINE experiment at CERN SPS searches for the critical point of strongly interacting matter via scanning the phase diagram by changing beam momenta (13A–150A GeV/c) and system size (p + p, p + Pb, Be + Be, Ar + Sc, Xe + La). An observation of local proton-density fluctuations that scale as a power law of the appropriate universality class as a function of phase space bin size would signal the approach of the system to the vicinity of the possible critical point. An investigation of this phenomenon was performed in terms of the second-scaled factorial moments (SSFMs) of proton density in transverse momentum space with subtraction of a noncritical background. New NA61/SHINE preliminary analysis of Ar + Sc data at 150A GeV/c revealed a nontrivial intermittent behavior of proton moments. A similar effect was observed by NA49 in “Si” + Si data at 158A GeV/c. At the same time, no intermittency signal was detected in “C” + C and Pb + Pb events by NA49, as well as in Be + Be collisions by NA61/SHINE. EPOS1.99 also fails to describe the power-law scaling of SSFMs in Ar + Sc. Qualitatively, the effect is more pronounced with the increase of collision-peripherality and proton-purity thresholds, but a quantitative estimate is to be properly done via power-law exponent fit using the bootstrap method and compared to intermittency critical index ϕ 2 , derived from 3D-Ising effective action.

Monitoring Spontaneous Charge-Density Fluctuations by Single-Molecule Diffraction of Quantum Light.

Homodyne X-ray diffraction signals produced by classical light and classical detectors are given by the modulus square of the charge density in momentum space |σ(q)|2, missing its phase, which is required in order to invert the signal to real space. We show that quantum detection of the radiation field yields a linear diffraction pattern that reveals σ(q) itself, including the phase. We further show that repeated diffraction measurements with variable delays constitute a novel multidimensional measure of spontaneous charge-density fluctuations. Classical diffraction, in contrast, only reveals a subclass of even-order correlation functions. Simulations of two-dimensional signals obtained by two diffraction events are presented for the amino acid cysteine.

Imaging electron-density fluctuations by multidimensional X-ray photon-coincidence diffraction

The ultrafast spontaneous electron-density fluctuation dynamics in molecules is studied theoretically by off-resonant multiple X-ray diffraction events. The time- and wavevector-resolved photon-coincidence signals give an image of electron-density fluctuations expressed through the four-point correlation function of the charge density in momentum space. A Fourier transform of the signal provides a real-space image of the multipoint charge-density correlation functions, which reveal snapshots of the evolving electron density in between the diffraction events. The proposed technique is illustrated by ab initio simulations of the momentum- and real-space inelastic scattering signals from a linear cyanotetracetylene molecule.

Wigner distributions for gluons

We investigate the gluon Wigner distributions for unpolarized, longitudinally polarized and transversely polarized target state. Instead of a nucleon, we take the target state to be a quark dressed with a gluon at one loop and investigate the gluon Wigner distributions at leading twist. Better numerical convergence is obtained compared to an earlier study, that removes the regulator dependence of the results. We present a first calculation of the Wigner distribution for the transversely polarized target and linearly polarized gluon. We study the spin densities in momentum and impact parameter space. We also investigate the quark and gluon helicity and orbital angular momentum distributions at small-x.

Shannon entropies and Fisher information of K-shell electrons of neutral atoms

We represent the two K-shell electrons of neutral atoms by Hylleraas-type wave function which fulfils the exact behavior at the electron–electron and electron-nucleus coalescence points and, derive a simple method to construct expressions for single-particle position- and momentum-space charge densities, ρ(r) and γ(p) respectively. We make use of the results for ρ(r) and γ(p) to critically examine the effect of correlation on bare (uncorrelated) values of Shannon information entropies (S) and of Fisher information (F) for the K-shell electrons of atoms from helium to neon. Due to inter-electronic repulsion the values of the uncorrelated Shannon position-space entropies are augmented while those of the momentum-space entropies are reduced. The corresponding Fisher information are found to exhibit opposite behavior in respect of this. Attempts are made to provide some plausible explanation for the observed response of S and F to electronic correlation.

Topological Fulde-Ferrell states in alkaline-earth-metal-like atoms near an orbital Feshbach resonance

We study the effects of synthetic spin-orbit coupling on the pairing physics in quasi-one-dimensional ultracold Fermi gases of alkaline-earth-metal-like atoms near an orbital Feshbach resonance (OFR). The interplay between spin-orbit coupling and pairing interactions near the OFR leads to an interesting topological Fulde-Ferrell state, where the nontrivial topology of the state is solely encoded in the closed channel with a topologically trivial Fulde-Ferrell pairing in the open channel. We confirm the topological property of the system by characterizing the Zak phase and the edge states. The topological Fulde-Ferrell state can be identified by the momentum-space density distribution obtained from time-of-flight images.

Momentum-Space Correlations of a One-Dimensional Bose Gas

Analyzing the noise in the momentum profiles of single realizations of one-dimensional Bose gases, we present the experimental measurement of the full momentum-space density correlations ⟨δn_{p}δn_{p^{'}}⟩, which are related to the two-body momentum correlation function. Our data span the weakly interacting region of the phase diagram, going from the ideal Bose gas regime to the quasicondensate regime. We show experimentally that the bunching phenomenon, which manifests itself as super-Poissonian local fluctuations in momentum space, is present in all regimes. The quasicondensate regime is, however, characterized by the presence of negative correlations between different momenta, in contrast to the Bogolyubov theory for Bose condensates, predicting positive correlations between opposite momenta. Our data are in good agreement with abinitio calculations.