Quantum lattice-gas models for the many-body Schrodinger equation
Bruce M. Boghosian, Washington Taylor
备注：
13 pages LaTeX, with 2 postscript figures; talk given by WT at the Sixth International Conference on Discrete Fluid Mechanics, BU, Boston MA, August 1996. Several minor errors corrected
A general class of discrete unitary models are described whose behavior in the continuum limit corresponds to a many-body Schrodinger equation. On a quantum computer, these models could be used to simulate quantum many-body systems with an exponential speedup over analogous simulations on classical computers. On a classical computer, these models give an explicitly unitary and local prescription for discretizing the Schrodinger equation. It is shown that models of this type can be constructed for an arbitrary number of particles moving in an arbitrary number of dimensions with an arbitrary interparticle interaction.
The three-body Schrödinger operator in the space of square integrable functions is found to be a certain extension of operators which generate the exponential unitary group containing a subgroup with nilpotent Lie algebra of length $κ+1$, $κ=0,1,...$ As a result, the solutions to the three-body Schrödinger equation with decaying potentials are shown to exist in the commutator subalgebras. For the Coulomb three-body system, it turns out that the task is to solve - in these subalgebras - the radial Schrödinger equation in three dimensions with the inverse power potential of the form $r^{-κ-1}$. …
We argue that the "reduced wave function", proposed recently [Phys.Rev.Lett. {\bf 75}, 2255 (1995)], contains conditional and restricted information on the reduced system. The concept of "reduced wave function" can thus not represent a relevant alternative to the common reduced dynamics methods.
There are efficient many-body methods, such as the (symmetry-restored) generator coordinate method in nuclear physics, that formulate the A-body Schrödinger equation within a set of nonorthogonal many-body states. Solving the corresponding secular equation requires the evaluation of the norm matrix and thus the capacity to compute its entries consistently and without any phase ambiguity. This is not always a trivial task, e.g. it remained a long-standing problem for methods based on general Bogoliubov product states. While a solution to this problem was found recently in Ref. [L. M. Robledo, P…
A simple argument is presented which, based on the minimal coupling Lagrangian for a many-body system, keeps the gauge invariance of the many-body Schrödinger equation with explicit Coulomb potential. The elimination of longitudinal electric field does not necessarily lead to the breakdown of gauge invariance. The total time derivative term in the matter-EM field interaction in the Lagrangian is canceled out by the choice of Coulomb gauge. The remaining interaction is described by transverse vector potential and the longitudinal electric field which is the homogeneous solution of Gauss law. Th…
We first derive the Rayleigh-Schrödinger many-body perturbation theory up to third order (RSPT3) for Hamiltonians with three-body interaction. The structure of closed-shell nuclei in a wide mass range from 4He to 48Ca has been investigated by the RSPT3 with explicit NN+3N Hamiltonian. The RSPT3 calculations are performed within Hartree-Fock bases. The perturbative contribution of antisymmetrized Goldstone diagrams (diagrammatic expansion for RSPT) with normal-ordered interaction has been analyzed. We demonstrate that the normal-ordered two-body level (NO2B) approximation which neglects the res…
We discuss a recent test of the performance of the kinetic energy partition method (KEP) through its application to two separable quantum-mechanical models. We argue that one of the benchmark models is exceedingly simple for testing any realistic approximate method and that almost any reasonable approach yields better results. In the second example our exact benchmark eigenvalues disagree considerably with those chosen by the authors for comparison, which casts doubts on the accuracy of their KEP approach.
Schrödinger equation belongs to the most fundamental differential equations in quantum physics. However, the exact solutions are extremely rare, and many analytical methods are applicable only to the cases with small perturbations or weak correlations. Solving the many-body Schrödinger equation in the continuous spaces with the presence of strong correlations is an extremely important and challenging issue. In this work, we propose the functional tensor network (FTN) approach to solve the many-body Schrödinger equation. Provided the orthonormal functional bases, we represent the coefficients o…
A method to solve the Schrödinger equation based on the use of constant particle-particle interaction potential surfaces is proposed. The many-body wave function is presented in configuration interaction form with coefficients - configuration weight functions - dependent on the total interaction potential. A set of linear ordinary differential equations for the configuration weight functions was developed and solved for particles in a infinite well and He-like ions. The results demonstrate that the method is variational and provides upper bound for energy of the ground state; even in its lowes…
In the reversible Schrodinger-Newton equation a complex Newton coupling G*exp(-i*alpha) is proposed in place of G. The equation becomes irreversible and all initial one-body states are expected to converge to solitonic stationary states. This feature is verified numerically. For two-body solutions we point out that an effective Newtonian interaction is induced by the imaginary mean-fields as if they were real. The effective strength of such induced gravity depends on the local wave functions of the participating distant bodies.
A many-body wave function is approximated by a product of two functions: the wave function $ϕ$ depending on the particle coordinates and the function $χ$ depending only on the value of interparticle interaction potential. For the given $ϕ$ an ordinary linear differential equation for $χ$ is derived by averaging the Hamiltonian over the constant interparticle interaction potential surface. Generalized Hartree-Fock equations containing correlation effects are obtained. To test the proposed technique the ground $1^{1}S_0$ and excited $2^{3}S_1$ states of two-electron ions from H$^-$ up to Ne$^{8+…
The design of efficient and robust pulse sequences is a fundamental requirement in quantum control. Numerical methods can be used for this purpose, but with relatively little insight into the control mechanism. Here, we show that the free rotation of a classical rigid body plays a fundamental role in the control of two-level quantum systems by means of external electromagnetic pulses. For a state to state transfer, we derive a family of control fields depending upon two free parameters, which allow us to adjust the efficiency, the time and the robustness of the control process. As an illustrat…
The many-body Schrödinger dynamics of a one-dimensional bosonic Josephson junction is investigated for up to ten thousand bosons and long times. The initial states are fully condensed and the interaction strength is weak. We report on a universal fragmentation dynamics on the many-body level: systems consisting of different numbers of particles fragment to the same value at constant mean-field interaction strength. The phenomenon manifests itself in observables such as the correlation functions of the system. We explain this universal fragmentation dynamics analytically based on the Bose-Hubba…
Analytical solution of the Schrödinger equation with $1/r^3$ and attractive $1/r^2$ potentials: Universal three-body parameter of mixed-dimensional Efimov states
University of Electro-Communications（电气通信大学）；Tohoku University（东北大学）
In this paper, we propose a new approach to the relativistic quantum mechanics for many-body, which is a self-consistent system constructed by juxtaposed but mutually coupled nonlinear Dirac's equations. The classical approximation of this approach provides the exact Newtonian dynamics for many-body, and the nonrelativistic approximation gives the complete Schrödinger equation for many-body.
Schrödinger Generator for High-Dimensional Integration and Sampling on Quantum Many-Body States
Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University（复旦大学现代物理研究所核物理与离子束应用教育部重点实验室）；Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University（国家自然科学基金委员会理论核物理研究中心（复旦大学））
Comment on a letter to Nature Physics, where Sakmann and Kasevich claim to solve the many-body time dependent Schrödinger equation to simulate single experimental runs of interacting quantum systems.
In their recent paper [Nature Physics 15, 451 (2006)], Sakmann and Kasevich study the formation of fringe patterns in ultra-cold Bose gases and claim: `Here, we show how single shots can be simulated from numerical solutions of the time-dependent many-body Schrödinger equation.' It would be remarkable if they had solved this exponentially complex equation. Instead they solve nonlinear equations with the aim to approximate the solution of the Schrödinger equation. The authors proceed to criticize phase-space approaches to simulating quantum dynamics and claim the impossibility of interpreting s…
Quantum systems composed of $N$ distinct particles in $\R^2$ with two-body contact interactions of TMS type are shown to arise as limits - in the norm resolvent sense - of Schrödinger operators with suitably rescaled pair potentials.
We derive the classical equations of hydrodynamic type (Euler equation and the continuity equation) from which the Schrodinger equation follows as a limit case. It is shown that the statistical ensemble corresponding to quantum system and described by the Schrodinger equation, can be considered as an inviscid gas obeying the ideal gas law with quickly oscillating sign-alternating temperature. This statistical ensemble performs the complex movements consisting of smooth average movement and fast oscillations. It is shown that average movements of statistical ensemble are described by Schrodinge…
Three-body recombination in quantum gases is traditionally associated with heating, but it was recently found that it can also cool the gas. We show that in a partially condensed three-dimensional homogeneous Bose gas three-body loss could even purify the sample, that is, reduce the entropy per particle and increase the condensed fraction $η$. We predict that the evolution of $η$ under continuous three-body loss can, depending on small changes in the initial conditions, exhibit two qualitatively different behaviours - if it is initially above a certain critical value, $η$ increases further, wh…
The four-body problem for an interacting two-species Fermi gas is solved analytically in a confined quasi-one-dimensional geometry, where the two-body atom-atom scattering length $a_{aa}$ displays a confinement-induced resonance. We compute the dimer-dimer scattering length $a_{dd}$, and show that this quantity completely determines the many-body solution of the associated BEC-BCS crossover phenomenon in terms of bosonic dimers.
We report on measurements of three-body recombination rates in an ultracold gas of $^7$Li atoms in the extremely nonuniversal regime where the two-body scattering length vanishes. We show that the rate is well defined and can be described by two-body parameters only: the scattering length $a$ and the effective range $R_e$. We find the rate to be energy independent, and, by connecting our results with previously reported measurements in the universal limit, we cover the behavior of the three-body recombination in the whole range from weak to strong two-body interactions. We identify a nontrivia…
We analyse the breathing-mode oscillations of a harmonically quenched Tonks-Giradeau (TG) gas using an exact finite-temperature dynamical theory. We predict a striking collective manifestation of impenetrability---a collective many-body bounce effect. The effect, while being invisible in the evolution of the in-situ density profile of the gas, can be revealed through a nontrivial periodic narrowing of its momentum distribution, taking place at twice the rate of the fundamental breathing-mode frequency. We identify physical regimes for observing the many-body bounce and construct the respective…
We experimentally demonstrate an original method to measure very accurately the density of a frozen Rydberg gas. It is based on the use of adiabatic transitions induced by the long-range dipole-dipole interaction in pairs of nearest neighbor Rydberg atoms by sweeping an electric field with time. The efficiency of this two-body process is experimentally tunable, depends strongly on the density of the gas and can be accurately calculated. The analysis of this efficiency leads to an accurate determination of the Rydberg gas density, and to a calibration of the Rydberg detection. Our method does n…
