Quantum Three-body Problem Research Articles

Soft IsoGeometric analysis of the bound states of a quantum three-body problem in 1D

The study of quantum three-body problems has been centred on low-energy states that rely on accurate numerical approximation. Recently, isogeometric analysis (IGA) has been adopted to solve the problem as an alternative but more robust (with respect to atom mass ratios) method that outperforms the classical Born–Oppenheimer (BO) approximation, especially for the cases with small mass ratios. In this paper, we focus on the performance of IGA and apply the recently-developed softIGA to further reduce the spectral errors of the low-energy bound states. This is an extension to the recent work that was published as an ICCS conference paper in Deng (2022). The main idea of softIGA is to add high-order derivative-jump terms with a penalty parameter to the IGA bilinear forms. With an optimal choice of the penalty parameter, we observe eigenvalue error superconvergence. Herein, the optimal parameter coincides with the ones for the Laplace operator (zero potential) and can be heuristically computed for a general elliptic operator. We focus on linear and quadratic elements and demonstrate the outperformance of softIGA over IGA through a variety of examples including both two- and three-body problems in 1D.

Impurity-induced pairing in two-dimensional Fermi gases

We study induced pairing between two identical fermions mediated by an attractively interacting quantum impurity in two-dimensional systems. Based on a stochastic variational method (SVM), we investigate the influence of confinement and finite interaction range effects on the mass ratio beyond which the ground state of the quantum three-body problem undergoes a transition from a composite bosonic trimer to an unbound dimer-fermion state. We find that confinement as well as a finite interaction range can greatly enhance trimer stability, bringing it within reach of experimental implementations such as found in ultracold atom systems. In the context of solid-state physics, our solution of the confined three-body problem shows that exciton-mediated interactions can become so dominant that they can even overcome detrimental Coulomb repulsion between electrons in atomically-thin semiconductors. Our paper thus paves the way towards a universal understanding of boson-induced pairing across various fermionic systems at finite density, and opens perspectives towards realizing unexplored forms of electron pairing beyond the conventional paradigm of Cooper pair formation.

Solution of the Quantum Three-Body Problem in a Neighborhood of the Three-Particle Forward Scattering Direction

Solution of the Quantum Three-Body Problem in a Neighborhood of the Three-Particle Forward Scattering Direction

Efimov-DNA phase diagram: Three stranded DNA on a cubic lattice.

We define a generalized model for three-stranded DNA consisting of two chains of one type and a third chain of a different type. The DNA strands are modeled by random walks on the three-dimensional cubic lattice with different interactions between two chains of the same type and two chains of different types. This model may be thought of as a classical analog of the quantum three-body problem. In the quantum situation, it is known that three identical quantum particles will form a triplet with an infinite tower of bound states at the point where any pair of particles would have zero binding energy. The phase diagram is mapped out, and the different phase transitions are examined using finite-size scaling. We look particularly at the scaling of the DNA model at the equivalent Efimov point for chains up to 10000 steps in length. We find clear evidence of several bound states in the finite-size scaling. We compare these states with the expected Efimov behavior.

New Concept for Studying the Classical and Quantum Three-Body Problem: Fundamental Irreversibility and Time’s Arrow of Dynamical Systems

The article formulates the classical three-body problem in conformal-Euclidean space (Riemannian manifold), and its equivalence to the Newton three-body problem is mathematically rigorously proved. It is shown that a curved space with a local coordinate system allows us to detect new hidden symmetries of the internal motion of a dynamical system, which allows us to reduce the three-body problem to the 6th order system. A new approach makes the system of geodesic equations with respect to the evolution parameter of a dynamical system (internal time) fundamentally irreversible. To describe the motion of three-body system in different random environments, the corresponding stochastic differential equations (SDEs) are obtained. Using these SDEs, Fokker-Planck-type equations are obtained that describe the joint probability distributions of geodesic flows in phase and configuration spaces. The paper also formulates the quantum three-body problem in conformal-Euclidean space. In particular, the corresponding wave equations have been obtained for studying the three-body bound states, as well as for investigating multichannel quantum scattering in the framework of the concept of internal time. This allows us to solve the extremely important quantum-classical correspondence problem for dynamical Poincaré systems.

Active learning algorithm for computational physics

In large-scale computation of physics problems, one often encounters the problem of determining a multi-dimensional function, which can be time-consuming when computing each point in this multi-dimensional space is already time-demanding. In the work, we propose that the active learning algorithm can speed up such calculations. The basic idea is to fit a multi-dimensional function by neural networks, and the key point is to make the query of labeled data economically by using a stratagem called "query by committee". We present the general protocol of this fitting scheme, as well as the procedure of how to further compute physical observables with the fitted functions. We show that this method can work well with two examples, which are quantum three-body problem in atomic physics and the anomalous Hall conductivity in condensed matter physics, respectively. In these examples, we show that one reaches an accuracy of few percent error for computing physical observables with less than $10\%$ of total data points compared with uniform sampling. With these two examples, we also visualize that by using the active learning algorithm, the required data are added mostly in the regime where the function varies most rapidly, which explains the mechanism for the efficiency of the algorithm. We expect broad applications of our method on various kind of computational physics problems.

Baryon masses in the three-state Potts field theory in a weak magnetic field

The 3-state Potts field theory describes the scaling limit of the 3-state Potts model on the two-dimensional lattice near its continuous phase transition point. In the presence of thermal and magnetic field perturbations, the 3-state Potts field theory in the ordered phase exhibits confinement of kinks, which allows both mesons and baryons. We calculate the masses of light baryons in this model in the weak confinement regime in leading order of the small magnetic field. In leading order of perturbation theory, the light baryons can be viewed as bound states of three quantum particles—the kinks, which move on a line and interact via a linear potential. We determine the masses of the lightest baryons by numerical solution of the associated non-relativistic one-dimensional quantum three-body problem.

Reconstruction and control of a time-dependent two-electron wave packet.

The concerted motion of two or more bound electrons governs atomic and molecular non-equilibrium processes including chemical reactions, and hence there is much interest in developing a detailed understanding of such electron dynamics in the quantum regime. However, there is no exact solution for the quantum three-body problem, and as a result even the minimal system of two active electrons and a nucleus is analytically intractable. This makes experimental measurements of the dynamics of two bound and correlated electrons, as found in the helium atom, an attractive prospect. However, although the motion of single active electrons and holes has been observed with attosecond time resolution, comparable experiments on two-electron motion have so far remained out of reach. Here we show that a correlated two-electron wave packet can be reconstructed from a 1.2-femtosecond quantum beat among low-lying doubly excited states in helium. The beat appears in attosecond transient-absorption spectra measured with unprecedentedly high spectral resolution and in the presence of an intensity-tunable visible laser field. We tune the coupling between the two low-lying quantum states by adjusting the visible laser intensity, and use the Fano resonance as a phase-sensitive quantum interferometer to achieve coherent control of the two correlated electrons. Given the excellent agreement with large-scale quantum-mechanical calculations for the helium atom, we anticipate that multidimensional spectroscopy experiments of the type we report here will provide benchmark data for testing fundamental few-body quantum dynamics theory in more complex systems. They might also provide a route to the site-specific measurement and control of metastable electronic transition states that are at the heart of fundamental chemical reactions.

Eckart frame Hamiltonians in the three-body problem

The Eckart frame is used to separate out the collective rotations in the quantum three-body problem. Explicit expressions for the corresponding rotational and vibro-rotational (i.e. Coriolis) Hamiltonians are derived. Special attention is paid to the situation when two principal moments of inertia are equal in the equilibrium configuration.

Collective Multipole Expansions and the Perturbation Theory in the Quantum Three-Body Problem

The perturbation theory with respect to the potential energy of three particles is considered. The first-order correction to the continuum wave function of three free particles is derived. It is shown that the use of the collective multipole expansion of the free three-body Green function over the set of Wigner $D$-functions can reduce the dimensionality of perturbative matrix elements from twelve to six. The explicit expressions for the coefficients of the collective multipole expansion of the free Green function are derived. It is found that the $S$-wave multipole coefficient depends only upon three variables instead of six as higher multipoles do. The possible applications of the developed theory to the three-body molecular break-up processes are discussed.

Recovery of classical chaoticlike behavior in a conservative quantum three-body problem

Recovering trajectories of quantum systems whose classical counterparts display chaotic behavior has been a subject that has received a lot of interest over the last decade. However, most of these studies have focused on driven and dissipative systems. The relevance and impact of chaotic-like phenomena to quantum systems has been highlighted in recent studies which have shown that quantum chaos is significant in some aspects of quantum computation and information processing. In this paper we study a three-body system comprising of identical particles arranged so that the system's classical trajectories exhibit Hamiltonian chaos. Here we show that it is possible to recover very nearly classical-like, conservative, chaotic trajectories from such a system through an unravelling of the master equation. First, this is done through continuous measurement of the position of each system. Second, and perhaps somewhat surprisingly, we demonstrate that we still obtain a very good match between the classical and quantum dynamics by weakly measuring the position of only one of the oscillators.

Kinetic energy operator approach to the quantum three-body problem with Coulomb interactions

We present a non-variational approach to the solution of the quantum three-body problem, based on the decomposition of the three-body Laplacian operator through the use of its intrinsic symmetries. With the judicious choice of angular momentum eigenfunctions, a clean separation of spatial rotation from kinematic rotation is achieved, leading to a finite set of coupled PDEs in terms of the canonical variables. Numerical implementation of this approach to the three-body Coulomb problem is shown to yield accurate ground state eigenvalues and wavefunctions, together with those of low-lying excited states. We present results on some typical three-body systems. In particular, the eigenvalues and wavefunctions of the even-parity P e 3 state of the negative hydrogen ion are detailed for the first time. The issue of computational efficiency is also briefly discussed.

Unitary Quantum Three-Body Problem in a Harmonic Trap

We consider either 3 spinless bosons or 3 equal mass spin-1/2 fermions, interacting via a short-range potential of infinite scattering length and trapped in an isotropic harmonic potential. For a zero-range model, we obtain analytically the exact spectrum and eigenfunctions: for fermions all the states are universal; for bosons there is a coexistence of decoupled universal and efimovian states. All the universal states, even the bosonic ones, have a tiny 3-body loss rate. For a finite range model, we numerically find for bosons a coupling between zero angular momentum universal and efimovian states; the coupling is so weak that, for realistic values of the interaction range, these bosonic universal states remain long-lived and observable.

Precise numerical results for limit cycles in the quantum three-body problem

The study of the three-body problem with short-range attractive two-body forces has a rich history going back to the 1930s. Recent applications of effective field theory methods to atomic and nuclear physics have produced a much improved understanding of this problem, and we elucidate some of the issues using renormalization group ideas applied to precise nonperturbative calculations. These calculations provide 11–12 digits of precision for the binding energies in the infinite cutoff limit. The method starts with this limit as an approximation to an effective theory and allows cutoff dependence to be systematically computed as an expansion in powers of inverse cutoffs and logarithms of the cutoff. Renormalization of three-body bound states requires a short range three-body interaction, with a coupling that is governed by a precisely mapped limit cycle of the renormalization group. Additional three-body irrelevant interactions must be determined to control subleading dependence on the cutoff and this control is essential for an effective field theory since the continuum limit is not likely to match physical systems (e.g., few-nucleon bound and scattering states at low energy). Leading order calculations precise to 11–12 digits allow clear identification of subleading corrections, but these corrections have not been computed.

SEPARATION OF ROTATIONAL DEGREES OF FREEDOM IN A QUANTUM THREE-BODY PROBLEM

We demonstrate explicitly how the rotational degrees of freedom in a quantum three-body problem is separated completely from the internal ones. It is shown that only finite partial angular momentum states are involved in constructing the base-functions with the given angular momentum and parity, and the contributions by the remaining partial angular momentum states are incorporated into the contributions by the generalized radial functions.

Analytical Structure Matching and Very Precise Approach to the Coulombic Quantum Three-Body Problem

A powerful approach to solve the Coulombic quantum three-body problem is proposed. The approach is exponentially convergent and more efficient than the hyperspherical coordinate method and the correlation-function hyperspherical harmonic method. This approach is numerically competitive with the variational methods, such as that using the Hylleraas-type basis functions. Numerical comparisons are made to demonstrate the efficiency of this approach, by calculating the nonrelativistic and infinite-nuclear-mass limit of the ground state energy of the helium atom. The exponential convergency of this approach is due to the full matching between the analytical structure of the basis functions that are used in this paper and the true wavefunction. This full matching was not reached by most other methods. For example, the variational method using the Hylleraas-type basis does not reflects the logarithmic singularity of the true wavefunction at the origin as predicted by Bartlett and Fock. Two important approaches are proposed in this work to reach this full matching: the coordinate transformation method and the asymptotic series method. Besides these, this work makes use of the least square method to substitute complicated numerical integrations in solving the Schrödinger equation without much loss of accuracy, which is routinely used by people to fit a theoretical curve with discrete experimental data, but here is used to simplify the computation.

Quantum three-body problems

A scheme for dealing with the quantum three-body problem is presented to separate the rotational degrees of freedom completely from the internal ones. In this method, the three-body Schrodinger equation is reduced to a system of coupled partial differential equations, depending only upon three internal variables. For arbitrary total orbital angular momentum / and the parity (− 1)l+λ (λ = 0 or 1), the number of the equations in this system isl = 1 −λ. By expanding the wavefunction with respect to a complete set of orthonormal basis functions, the system of equations is further reduced to a system of linear algebraic equations.

EXACT SOLUTION TO THE SCHRODINGER EQUATION FOR THE QUANTUM RIGID BODY

The exact solution to the Schrodinger equation for the rigid body with the given angular momentum and parity is obtained. Since the quantum rigid body can be thought of as the simplest quantum three-body problem where the internal motion is frozen, this calculation method is a good starting point for solving the quantum three-body problems.

Extended class of Dubrovin's equations related to the one-dimensional quantum three-body problem

The relation of the quantum 1D three-body problems with zero-range interaction to the matrix Riemann-Hilbert problem with meromorphic coefficient is shown. The solution of this problem is discussed using the exact analytic diagonalization of the coefficient. The problem is reduced to the boundary value problem on the Riemann surface. The solution of this problem is expressed in terms of the Riemann theta-functions. An extended class of integrable Dubrovin's type ordinary differential equations related to the one-dimensional quantum three-body problem is derived.

Conformally euclidean internal coordinate space in the quantum three-body problem

The internal coordinate space for three particles of arbitrary mass is shown to be a three-dimensional conformally Euclidean space. This gives the internal kinetic energy operator a simple form and provides a unified view of the different hyperspherical coordinates used in quantum three-body problems.