Many-body Evolution Research Articles

Dynamics of mean-field bosons at positive temperature

We study the time evolution of an initially trapped weakly interacting Bose gas at positive temperature, after the trapping potential has been switched off. It has been recently shown in Deuchert–Seiringer (2021) that the one-particle density matrix of Gibbs states of the interacting trapped gas is given, to leading order in N , as N \rightarrow \infty , by that of the ideal gas, with the condensate wave function replaced by the minimizer of the Hartree energy functional. We show that this structure is stable with respect to the many-body evolution in the following sense: the dynamics can be approximated in terms of the time-dependent Hartree equation for the condensate wave function and in terms of the free evolution for the thermally excited particles. The main technical novelty of our work is the use of the Hartree–Fock–Bogoliubov equations to define a fluctuation dynamics.

Effective Dynamics of Extended Fermi Gases in the High-Density Regime

We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus on the high-density regime, in the semiclassical scaling, and we consider a class of initial data describing zero-temperature states. In the non-relativistic case we prove that, as the density goes to infinity, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation, for short macroscopic times. In the case of relativistic dispersion, we show convergence of the many-body evolution to the relativistic Hartree equation for all macroscopic times. With respect to previous work, the rate of convergence does not depend on the total number of particles, but only on the density: in particular, our result allows us to study the quantum dynamics of extensive many-body Fermi gases.

A superconducting quantum simulator based on a photonic-bandgap metamaterial.

Synthesizing many-body quantum systems with various ranges of interactions facilitates the study of quantum chaotic dynamics. Such extended interaction range can be enabled by using nonlocal degrees of freedom such as photonic modes in an otherwise locally connected structure. Here, we present a superconducting quantum simulator in which qubits are connected through an extensible photonic-bandgap metamaterial, thus realizing a one-dimensional Bose-Hubbard model with tunable hopping range and on-site interaction. Using individual site control and readout, we characterize the statistics of measurement outcomes from many-body quench dynamics, which enables in situ Hamiltonian learning. Further, the outcome statistics reveal the effect of increased hopping range, showing the predicted crossover from integrability to ergodicity. Our work enables the study of emergent randomness from chaotic many-body evolution and, more broadly, expands the accessible Hamiltonians for quantum simulation using superconducting circuits.

Metastable discrete time-crystal resonances in a dissipative central spin system

We consider the non-equilibrium behavior of a central spin system where the central spin is periodically reset to its ground state. The quantum mechanical evolution under this effectively dissipative dynamics is described by a discrete-time quantum map. Despite its simplicity this problem shows surprisingly complex dynamical features. In particular, we identify several metastable time-crystal resonances. Here the system does not relax rapidly to a stationary state but undergoes long-lived oscillations with a period that is an integer multiple of the reset period. At these resonances the evolution becomes restricted to a low-dimensional state space within which the system undergoes a periodic motion. Generalizing the theory of metastability in open quantum systems, we develop an effective description for the evolution within this long-lived metastable subspace and show that in the long-time limit a non-equilibrium stationary state is approached. Our study links to timely questions concerning emergent collective behavior in the 'prethermal' stage of a dissipative quantum many-body evolution and may establish an intriguing link to the phenomenon of quantum synchronization.

Classical and quantum evolution in a simple coherent neutrino problem

The extraordinary neutrino flux produced in extreme astrophysical environments like the early Universe, core-collapse supernovae and neutron star mergers may produce coherent quantum neutrino oscillations on macroscopic length scales. The Hamiltonian describing this evolution can be mapped into quantum spin models with all-to-all couplings arising from neutrino-neutrino forward scattering. To date many studies of these oscillations have been performed in a mean-field limit where the neutrinos time evolve in a product state. In this paper we examine a simple two-beam model evolving from an initial product state and compare the mean-field and many-body evolution. The symmetries in this model allow us to solve the real-time evolution for the quantum many-body system for hundreds or thousands of spins, far beyond what would be possible in a more general case with an exponential number (${2}^{N}$) of quantum states. We compare mean-field and many-body solutions for different initial product states and ratios of one- and two-body couplings, and find that in all cases in the limit of infinite spins the mean-field (product state) and many-body solutions coincide for simple observables. This agreement can be understood as a consequence of the fact that the typical initial condition represents a very local but dense distribution about a mean energy in the spectrum of the Hamiltonian. We explore quantum information measures like entanglement entropy and purity of the many-body solutions, finding intriguing relationships between the quantum information measures and the dynamical behavior of simple physical observables.

Bosonization of Fermionic Many-Body Dynamics

We consider the quantum many-body evolution of a homogeneous Fermi gas in three dimensions in the coupled semiclassical and mean-field scaling regime. We study a class of initial data describing collective particle-hole pair excitations on the Fermi ball. Using a rigorous version of approximate bosonization, we prove that the many-body evolution can be approximated in Fock space norm by a quasifree bosonic evolution of the collective particle-hole excitations.

Many-body dark solitons in one-dimensional hard-core Bose gases

The existence and stability of solitonic states in one-dimensional repulsive Bose-Einstein condensates is investigated within a fully many-body framework by considering the limit of infinite repulsion (Tonks-Girardeau gas). A class of stationary, shape-invariant states propagating at constant velocity are explicitly found and compared to the known solution of the Gross-Pitaevskii equation. The typical features attributed to nonlinearity are thus recovered in a purely linear theory, provided the full many-particle physics is correctly accounted for. However, the formation dynamics predicted by the Gross-Pitaevskii approximation considerably differs from the exact many-body evolution.

Measurement of a one-dimensional matter-wave quantum breather

Employing the Bethe ansatz approach and numerical simulations of measurements of particles' positions we investigate a post-quench many-body dynamics of attractively interacting bosons on a ring, which in the mean-field approach corresponds to the so-called breather solution. Despite the fact that the initial many-body ground state is translationally invariant, the measurements reveal breather dynamics if quantum fluctuations of the center of mass of the system are extracted. Moreover, the analysis of the many-body evolution shows signatures of dissociation of the solitons that form the breather.

Gross–Pitaevskii dynamics for Bose–Einstein condensates

We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. Under a physically motivated assumption on the energy of the initial data, we show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross-Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e. on the number of excitations of the Bose-Einstein condensate). To reach this goal, we combine the method of \cite{LNS}, where fluctuations around the Hartree dynamics for $N$-particle initial data in the mean-field regime have been analyzed, with ideas from \cite{BDS}, where the evolution of Fock-space initial data in the Gross-Pitaevskii regime has been considered.

Fluctuations of N-particle quantum dynamics around the nonlinear Schrödinger equation

We consider a system of N bosons interacting through a singular two-body potential scaling with N and having the form N3β−1V(Nβx), for an arbitrary parameter β∈(0,1). We provide a norm-approximation for the many-body evolution of initial data exhibiting Bose–Einstein condensation in terms of a cubic nonlinear Schrödinger equation for the condensate wave function and of a unitary Fock space evolution with a generator quadratic in creation and annihilation operators for the fluctuations.

Transient and persistent particle subdiffusion in a disordered chain coupled to bosons

We consider the propagation of a single particle in a random chain, assisted by the coupling to dispersive bosons. Time evolution treated with rate equations for hopping between localized states reveals a qualitative difference between dynamics due to noninteracting bosons and hard-core bosons. In the first case the transient dynamics is subdiffusive, but multi-boson processes allow for long-time normal diffusion, while hard-core effects suppress multi-boson processes leading to persistent subdiffusive transport, consistent with numerical results for a full many-body evolution. In contrast, analogous study for a quasiperiodic potential reveals a stable long-time diffusion.

Mean Field Evolution of Fermions with Coulomb Interaction

We study the many body Schr\"odinger evolution of weakly coupled fermions interacting through a Coulomb potential. We are interested in a joint mean field and semiclassical scaling, that emerges naturally for initially confined particles. For initial data describing approximate Slater determinants, we prove convergence of the many-body evolution towards Hartree-Fock dynamics. Our result holds under a condition on the solution of the Hartree-Fock equation, that we can only show in a very special situation (translation invariant data, whose Hartree-Fock evolution is trivial), but that we expect to hold more generally.

Bogoliubov correction to the mean-field dynamics of interacting bosons

We consider the dynamics of a large quantum system of $N$ identical bosons in 3D interacting via a two-body potential of the form $N^{3\beta-1} w(N^\beta(x-y))$. For fixed $0\leq \beta <1/3$ and large $N$, we obtain a norm approximation to the many-body evolution in the $N$-particle Hilbert space. The leading order behaviour of the dynamics is determined by Hartree theory while the second order is given by Bogoliubov theory.

The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-Body Evolution

We consider a 2D time-dependent quantum system of $N$-bosons with harmonic external confining and \emph{attractive} interparticle interaction in the Gross-Pitaevskii scaling. We derive stability of matter type estimates showing that the $k$-th power of the energy controls the $H^{1}$ Sobolev norm of the solution over $k$-particles. This estimate is new and more difficult for attractive interactions than repulsive interactions. For the proof, we use a version of the finite-dimensional quantum di Finetti theorem from [49]. A high particle-number averaging effect is at play in the proof, which is not needed for the corresponding estimate in the repulsive case. This a priori bound allows us to prove that the corresponding BBGKY hierarchy converges to the GP limit as was done in many previous works treating the case of repulsive interactions. As a result, we obtain that the \emph{focusing} nonlinear Schr\"{o}dinger equation is the mean-field limit of the 2D time-dependent quantum many-body system with attractive interatomic interaction and asymptotically factorized initial data. An assumption on the size of the $L^{1}$-norm of the interatomic interaction potential is needed that corresponds to the sharp constant in the 2D Gagliardo-Nirenberg inequality though the inequality is not directly relevant because we are dealing with a trace instead of a power.

Correlation Structures, Many-Body Scattering Processes, and the Derivation of the Gross–Pitaevskii Hierarchy

We consider the dynamics of $N$ bosons in three dimensions. We assume the pair interaction is given by $N^{3\beta -1}V(N^{\beta }\cdot )$ . By studying an associated many-body wave operator, we introduce a BBGKY hierarchy which takes into account all of the interparticle singular correlation structures developed by the many-body evolution from the beginning. Assuming energy conditions on the $N$-body wave function, for $\beta \in \left( 0,1\right] $, we derive the Gross-Pitaevskii hierarchy with $2$-body interaction. In particular, we establish that, in the $N\rightarrow \infty $ limit, all $k$-body scattering processes vanishes if $k\geqslant 3$ and thus provide a direct answer to a question raised by Erd\"{o}s, Schlein, and Yau in [31]. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for $\beta \in \left( 0,1\right) $ and weakly for $\beta =1$. Since this new BBGKY hierarchy converts the problem from a two-body estimate to a weaker three body estimate for which we have the estimates to achieve $\beta <1$, it then allows us to prove that all limit points of the ordinary BBGKY hierarchy satisfy the space-time bound conjectured by Klainerman and Machedon in [47] for $\beta \in \left( 0,1\right) $.

Neutrino magnetic moment,CPviolation, and flavor oscillations in matter

We consider collective oscillations of neutrinos, which are emergent nonlinear flavor evolution phenomena instigated by neutrino-neutrino interactions in astrophysical environments with sufficiently high neutrino densities. We investigate the symmetries of the problem in the full three flavor mixing scheme and in the exact many-body formulation by including the effects of CP violation and neutrino magnetic moment. We show that, similar to the two flavor scheme, several dynamical symmetries exist for three flavors in the single-angle approximation if the net electron background in the environment and the effects of the neutrino magnetic moment are negligible. Moreover, we show that these dynamical symmetries are present even when the CP symmetry is violated in neutrino oscillations. We explicitly write down the constants of motion through which these dynamical symmetries manifest themselves in terms of the generators of the SU(3) flavor transformations. We also show that the effects due to the CP-violating Dirac phase factor out of the many-body evolution operator and evolve independently of nonlinear flavor transformations if neutrino electromagnetic interactions are ignored. In the presence of a strong magnetic field, CP-violating effects can still be considered independently provided that an effective definition for neutrino magnetic moment is used.

Entropy and correlation functions of a driven quantum spin chain

We present an exact solution for a quantum spin chain driven through its critical points. Our approach is based on a many-body generalization of the Landau-Zener transition theory, applied to a fermionized spin Hamiltonian. The resulting nonequilibrium state of the system, while being a pure quantum state, has local properties of a mixed state characterized by finite entropy density associated with Kibble-Zurek defects. The entropy and the finite spin correlation length are functions of the rate of sweep through the critical point. We analyze the anisotropic $XY$ spin-$1∕2$ model evolved with a full many-body evolution operator. With the help of Toeplitz determinant calculus, we obtain an exact form of correlation functions. The properties of the evolved system undergo an abrupt change at a certain critical sweep rate, signaling the formation of ordered domains. We link this phenomenon to the behavior of complex singularities of the Toeplitz generating function.

On a New Formulation of the Real-Time Propagator

The development of theoretical tools for the study of dynamical phenomena of many-particle systems on the quantum level is a fundamental challenge since many decades. A lot of efforts have been invested on Feynman's path integral approach, however, no computationally tractable method for investigating realistic systems could be developed up to now. In this paper we propose an alternative representation of the real-time many-body evolution operator formulated within the framework of the auxiliary field formalism. Our goal is to derive a new auxiliary field functional integral representation, in which the large oscillations of the functional integrand are reduced, in order to render the auxiliary field approach more attractive for real-time computation. This objective is attained using a generalized version of the method of Gaussian equivalent representation of Efimov and Ganbold [Phys. Stat. Sol. 168 (1991) 165], which eliminates the low-order fluctuations of the auxiliary field from the interaction functional.

Auxiliary Field Functional Integral Representation of the Many-Body Evolution Operator

Finding an appropriate functional integral representation of the many-body evolution operator is a crucial task for performing efficient calculations of fermionic systems within the auxiliary field approach. In this paper we derive a new field representation of the imaginary-time evolution operator using the method of Gaussian equivalent representation of Efimov and Ganbold (1991, Physica Status Solidi168, 165). The goal is to obtain a functional integral representation, in which the main divergences caused by the tadpole Feynman diagrams are efficiently eliminated. These diagrams provide the main contributions to the ground state of the system under consideration, and therefore it is important to take them into account adequately, especially at lower temperatures. In addition, we show that the well-known mean field representation of the imaginary-time evolution operator is only the limiting case of the Gaussian equivalent representation in the small time-step regime.

RESULTS FROM SHELL-MODEL MONTE CARLO STUDIES

▪ Abstract We review results obtained using Shell-Model Monte Carlo (SMMC) techniques. These methods reduce the imaginary-time many-body evolution operator to a coherent superposition of one-body evolutions in fluctuating one-body fields; the resultant path integral is evaluated stochastically. After a brief review of the methods, we discuss a variety of nuclear-physics applications. These include studies of the ground-state properties of pf-shell nuclei, Gamow-Teller strength distributions, thermal and rotational pairing properties of nuclei near N = Z, γ-soft nuclei, and ββ-decay in76Ge. Several other illustrative calculations are also reviewed. Finally, we discuss prospects for further progress in SMMC and related calculations.