A unified approach, for solving a wide class of single and many-body quantum problems, commonly encountered in literature is developed based on a recently proposed method for finding solutions of linear differential equations. Apart from dealing with exactly and quasi-exactly solvable problems, the present approach makes transparent various properties of the familiar orthogonal polynomials and also the construction of their respective ladder operators. We illustrate the procedure for finding the approximate eigenvalues and eigenfunctions of non-exactly solvable problems.
The three-body scattering problem in Coulombic systems is widespread, however yet unresolved problem by the mathematically rigorous methods. In this work this long term challenge has been undertaken by combining distorted waves and Faddeev-Merkuriev equation formalisms in conjunction with the complex scaling technique to overcome difficulties related with the boundary conditions. Unlike the common belief, it is demonstrated that the smooth complex scaling method can be applied to solve three-body Coulomb scattering problem in a wide energy region, including fully elastic domain and extending t…
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.
The challenge posed by the many-body problem in quantum physics originates from the difficulty of describing the non-trivial correlations encoded in the exponential complexity of the many-body wave function. Here we demonstrate that systematic machine learning of the wave function can reduce this complexity to a tractable computational form, for some notable cases of physical interest. We introduce a variational representation of quantum states based on artificial neural networks with variable number of hidden neurons. A reinforcement-learning scheme is then demonstrated, capable of either fin…
Knowledge of all correlation functions of a system is equivalent to solving the corresponding many-body problem. Already a finite set of correlation functions can be sufficient to describe a quantum many-body system if correlations factorise, at least approximately. While being a powerful theoretical concept, an implementation based on experimental data has so far remained elusive. Here, this is achieved by applying it to a non-trivial quantum many-body problem: A pair of tunnel-coupled one-dimensional atomic superfluids. From measured interference patterns we extract phase correlation functio…
Bell non-locality represents the ultimate consequence of quantum entanglement, fundamentally undermining the classical tenet that spatially-separated degrees of freedom possess objective attributes independently of the act of their measurement. Despite its importance, probing Bell non-locality in many-body systems is considered to be a formidable challenge, with a computational cost scaling exponentially with system size. Here we propose and validate an efficient variational scheme, based on the solution of inverse classical Ising problems, which in polynomial time can probe whether an arbitra…
The exact solution of the interacting many-body system is important and is difficult to solve. In this paper, we introduce a group method to solve the interacting many-body problem using the relation between the permutation group and the unitary group. We prove a group theorem first, then using the theorem, we represent the Hamiltonian of the interacting many-body system by the Casimir operators of unitary group. The eigenvalues of Casimir operators could give the exact values of energy and thus solve those problems exactly. This method maps the interacting many-body system onto an intermediat…
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+…
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…
We construct a family of quasi-solvable quantum many-body systems by an algebraic method. The models contain up to two-body interactions and have permutation symmetry. We classify these models under the consideration of invariance property. It turns out that this family includes the rational, hyperbolic (trigonometric) and elliptic Inozemtsev models as the particular cases.
Solving The Quantum Many-Body Hamiltonian Learning Problem with Neural Differential Equations
ICFO-Institut de Ciencies Fotoniques（ICFO光子科学研究所）；The Barcelona Institute of Science and Technology（巴塞罗那科学技术学院）；Quside Technologies SL（Quside技术有限公司）；ICREA-Institucio Catalana de Recerca i Estudis Avançats（加泰罗尼亚研究与高级研究学院）
Spectrum of the Pauli projector of a quantum many-body system is studied. It is proven that the kern of the complete many-body projector is identical to the kern of the sum of two-body projectors. Since the kern of the many-body Pauli projector defines an allowed subspace of the complete Hilbert space, it is argued that a truncation of the many-body model space following the two-body Pauli projectors is a natural way when solving the Schrödinger equation for the many-body system. These relations clarify a role of the many-body Pauli forces in a multicluster system.
Two quantum quartic anharmonic many-body oscillators are introduced. One of them is the celebrated Calogero model (rational $A_n$ model) modified by quartic anharmonic two-body interactions which support the same symmetry as the Calogero model. Another model is the three-body Wolfes model (rational $G_2$ model) with quartic anharmonic interaction added which has the same symmetry as the Wolfes model. Both models are studied in the framework of algebraic perturbation theory and by the variational method.
We propose a general framework of quantum kinetic Monte Carlo algorithm, based on a stochastic representation of a series expansion of the quantum evolution. Two approaches have been developed in the context of quantum many-body spin dynamics, using different decomposition of the Hamiltonian. The effectiveness of the methods is tested for many-body spin systems up to 40 spins.
We present a variational Monte Carlo method that solves the nuclear many-body problem in the occupation number formalism exploiting an artificial neural network representation of the ground-state wave function. A memory-efficient version of the stochastic reconfiguration algorithm is developed to train the network by minimizing the expectation value of the Hamiltonian. We benchmark this approach against widely used nuclear many-body methods by solving a model used to describe pairing in nuclei for different types of interaction and different values of the interaction strength. Despite its poly…
Thermodynamics of quantum systems and quantum thermal machines are rapidly developing fields, which have already delivered several promising results, as well as raised many intriguing questions. Many-body quantum machines present new opportunities stemming from many-body effects. At the same time, they pose new challenges related to many-body physics. In this short review we discuss some of the recent developments on technologies based on many-body quantum systems. We mainly focus on many-body effects in quantum thermal machines. We also briefly address the role played by many-body systems in …
The spectral form factor (SFF), characterizing statistics of energy eigenvalues, is a key diagnostic of many-body quantum chaos. In addition, partial spectral form factors (PSFFs) can be defined which refer to subsystems of the many-body system. They provide unique insights into energy eigenstate statistics of many-body systems, as we show in an analysis on the basis of random matrix theory and of the eigenstate thermalization hypothesis. We propose a protocol that allows the measurement of the SFF and PSFFs in quantum many-body spin models, within the framework of randomized measurements. Aim…
We investigate many-body localization of interacting spinless fermions in a one-dimensional disordered and tilted lattice. The fermions undergo energy-dependent transitions from ergodic to Stark many-body localization driven by the tilted potential, which are manifested by the appearance of mobility edges between delocalized states and Stark many-body localized states even when the disorder is weak. We can concretely diagnose these transitions rather than crossovers by finite-size scaling of energy-level statistics. Moreover, in the Stark many-body localization, the entanglement entropy obeys …
We study a spin-gas model, where N_S system qubits are interacting with N_B bath qubits via many-body interactions. We consider multipartite Ising interactions and show how the effect of decoherence depends on the specific coupling between the system and its environment. For instance, we analyze the influence of decohenerce induced by k-body interactions for different values of k. Moreover, we study how the effect of decoherence depends on the correlation between baths that are coupled to different individual system qubit and compare Markovian with non-Markovian interactions. As examples we co…
We investigate the collective aspects of Rydberg excitation in ultracold mesoscopic systems. Strong interactions between Rydberg atoms influence the excitation process and impose correlations between excited atoms. The manifestations of the collective behavior of Rydberg excitation are the many-body Rabi oscillations, spatial correlations between atoms as well as the fluctuations of the number of excited atoms. We study these phenomena in detail by numerically solving the many-body Schrëdinger equation.
Ultra-cold atoms in optical lattices provide an ideal platform for exploring many-body physics of a large system arising from the coupling among a series of small identical systems whose few-body dynamics is exactly solvable. Using Landau-Zener (LZ) transition of bosonic atoms in double well optical lattices as an experimentally realizable model, we investigate such few to many body route by exploring the relation and difference between the small few-body (in one double well) and the large many-body (in double well lattice) non-equilibrium dynamics of cold atoms in optical lattices. We find th…
We show how coherences between identical constituents of a many-body quantum state can be interrogated by suitable correlation functions, and identify sufficient conditions under which low-order correlators fully characterize many-body coherences, as controlled by the constituents' mutual distinguishability. Comparison of correlators of different order detects many-body entanglement.
Many-body forces are sometimes a relevant ingredient in various fields, such as atomic, nuclear or hadronic physics. Their precise structure is generally difficult to uncover. So, phenomenological effective forces are often used in practice. Nevertheless, they are always very heavy to treat numerically. The envelope theory, also known as the auxiliary field method, is a very efficient technique to obtain approximate, but reliable, solutions of many-body systems interacting via one- or two-body forces. It is adapted here to allow the treatment of a special form of many-body forces. In the most …
We discuss the possibility to modify many-body Hilbert quantum formalism that is necessary for the representation of quantum systems dynamics. The notion of effective classical algorithm and visualization of quantum dynamics play the key role.
An optical realization of the phenomenon of many-body coherent destruction of tunneling, recently predicted for interacting many-boson systems by Gong, Molina and Haenggi [Phys. Rev. Lett. 103, 133002 (2009)], is proposed for light transport in engineered waveguide arrays. The optical system enables a direct visualization in Fock space of the many-body tunneling control process
The zero-range potential is customarily employed in various mean-field calculations of many-body systems in atomic and nuclear physics within, correspondingly, Gross-Pitaevskii and Skyrme-Hartree-Fock approach. We argue, however, that a many-body system with zero-range potentials is unstable against clusterization into collapsed three-body subsystems. We show that neither the density dependence of the potential nor an additional repulsive three-body potential can prevent this unexpected correlational collapse if the potentials are of zero range. Therefore the zero-range potential can only be u…
Adiabatic quantum computing is an analog quantum computing scheme with various applications in solving optimization problems. In the parity picture of quantum optimization, the problem is encoded in local fields that act on qubits which are connected via local 4-body terms. We present an implementation of a parity annealer with Transmon qubits with a specifically tailored Ising interaction from Josephson ring modulators.
The geometric phase induced in an auxiliary qubit by a many-body system is calculated and discussed. Two kinds of coupling between the auxiliary qubit and the many-body system are considered, which lead to dephasing and dissipation in the qubit, respectively. As an example, we consider the XY spin-chain dephasingly couple to a qubit, the geometric phase induced in the qubit is presented and discussed. The results show that the geometric phase might be used to signal the critical points of the many-body system, and it tends to zero with the parameters of the many-body system going away from the…
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.
Many-body localization is shown to suppress imaginary parts of complex eigenenergies for general non-Hermitian Hamiltonians having time-reversal symmetry. We demonstrate that a real-complex transition, which we conjecture occurs upon many-body localization, profoundly affects the dynamical stability of non-Hermitian interacting systems with asymmetric hopping that respect time-reversal symmetry. Moreover, the real-complex transition is shown to be absent in non-Hermitian many-body systems with gain and/or loss that breaks time-reversal symmetry, even though the many-body localization transitio…
Using the trajectory conception of state we give a simple demonstration that the quantum state of a many-body system may be expressed as a set of states in three-dimensional space, one associated with each particle. It follows that the many-body wavefunction may be derived from a set of waves in 3-space. Entanglement is represented in the trajectory picture by the mutual dependence of the 3-states on the trajectory labels.
Ergodicity in quantum many-body systems is - despite its fundamental importance - still an open problem. Many-body localization provides a general framework for quantum ergodicity, and may therefore offer important insights. However, the characterization of many-body localization through simple observables is a difficult task. In this article, we introduce a measure for distances in Hilbert space for spin-1/2 systems that can be interpreted as a generalization of the Anderson localization length to the many-body Hilbert space. We show that this many-body localization length is equivalent to a …
Coupling a quantum many-body system to an external environment dramatically changes its dynamics and offers novel possibilities not found in closed systems. Of special interest are the properties of the steady state of such open quantum many-body systems, as well as the relaxation dynamics towards the steady state. However, new computational tools are required to simulate open quantum many-body systems, as methods developed for closed systems cannot be readily applied. We review several approaches to simulate open many-body systems and point out the advances made in recent years towards the si…
Using a separable many-body variational wavefunction, we formulate a self-consistent effective Hamiltonian theory for fermionic many-body system. The theory is applied to the two-dimensional Hubbard model as an example to demonstrate its capability and computational effectiveness. Most remarkably for the Hubbard model in 2-d, a highly unconventional quadruple-fermion non-Cooper-pair order parameter is discovered.
Quantum many-body scarring is believed to be the mechanism behind long-lived coherent oscillations in interacting Rydberg atom chains. These persistent oscillations are due to the large overlap of the many-body scars with certain initial states. We show that the "effective dimension" is a useful measure for identifying non-thermalising initial states in many-body scarred systems. By minimising the effective dimension we find physically reasonable initial states of the Rydberg chain that lead to more pronounced and longer lived oscillations, accentuating the effect of the many-body scars on the…
Short review on entanglement, as seen from a quantum information perspective, and some simple applications to many-body quantum systems. Special emphasis in area laws, cold atoms, and efficient descriptions using tensor network states.
We consider a many-body localized system coupled globally to a central $d$-level system. Under an appropriate scaling of $d$ and $L$, we find evidence that the localized phase survives. We argue for two possible thermalizing phases, depending on whether the qudit becomes fully ergodic. This system provides one of the first examples of many-body localization in the presence of long-range (non-confining) interactions.
Motivated by recent experimental observations of coherent many-body revivals in a constrained Rydberg atom chain, we construct a weak quasi-local deformation of the Rydberg blockade Hamiltonian, which makes the revivals virtually perfect. Our analysis suggests the existence of an underlying non-integrable Hamiltonian which supports an emergent SU(2)-spin dynamics within a small subspace of the many-body Hilbert space. We show that such perfect dynamics necessitates the existence of atypical, nonergodic energy eigenstates - quantum many-body scars. Furthermore, using these insights, we construc…
We demonstrate that a many-body nonlocality is a resource for ultra-precise metrology. This result is achieved by linking the sensitivity of a quantum sensor with a combination of many-body correlation functions that witness the nonlocality. We illustrate our findings with some prominent examples---a collection of spins forming an Ising chain and a gas of ultra-cold atoms in any two-mode configuration.
Manipulating many body quantum systems is a challenge. A useful way to achieve it would be to entangle the system to a diluted system, with a small particle number. Preparation of such entangled states can be facilitated as ground state of a many body Hamiltonian or the steady state of a many body open quantum system. Here we study two-site lattice models with a conserved boson number, biased to display a large occupancy in one of the sites. The Von Neumann entanglement entropy as well as the Logarithmic negativity show a typical power law decay in $R$, the occupancy ratio between the two site…
Here, I focus on the use of microscopic, few-body techniques that are relevant in the many-body problem. These methods can be divided into indirect and direct. In particular, indirect methods are concerned with the simplification of the many-body problem by substituting the full, microscopic interactions by pseudopotentials which are designed to reproduce collisional information at specified energies, or binding energies in the few-body sector. These simplified interactions yield more tractable theories of the many-body problem, and are equivalent to effective field theory of interactions. Dir…
A new convenient method to diagonalize the non-relativistic many-body Schroedinger equation with two-body central potentials is derived. It combines kinematic rotations (democracy transformations) and exact calculation of overlap integrals between bases with different sets of mass-scaled Jacobi coordinates, thereby allowing for a great simplification of this formidable problem. We validate our method by obtaining a perfect correspondence with the exactly solvable three-body ($N=3$) Calogero model in 1D.
Hamiltonians which are inaccessible in static systems can be engineered in periodically driven many-body systems, i.e., Floquet many-body systems. We propose to use interacting particles in a one-dimensional (1D) harmonic potential with periodic kicking to investigate two-dimensional (2D) topological and many-body physics. Depending on the driving parameters, the Floquet Hamiltonian of single kicked harmonic oscillator has various lattice structures in phase space. The noncommutative geometry of phase space gives rise to the topology of the system. We investigate the effective interactions of …
We investigate dynamical quantum phase transitions in disordered quantum many-body models that can support many-body localized phases. Employing $l$-bits formalism, we lay out the conditions for which singularities indicative of the transitions appear in the context of many-body localization. Using the combination of the mapping onto $l$-bits and exact diagonalization results, we explicitly demonstrate the presence of these singularities for a candidate model that features many-body localization. Our work paves the way for understanding dynamical quantum phase transitions in the context of man…
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…
We study an experimentally realizable paradigm of complex many-body quantum systems, a two-band Wannier-Stark model, for which diffusion in Hilbert space as well as many-body Landau-Zener processes can be engineered. A cross-over between regular to quantum chaotic spectra is found within the many-body avoided crossings at resonant tunneling conditions. The spectral properties are shown to determine the evolution of states across a cascade of Landau-Zener events. We apply the obtained spectral information to study the non-equilibrium dynamics of our many-body system in different parameter regim…
