Limit Of Large Particle Numbers Research Articles

Optimal rate of condensation for trapped bosons in the Gross–Pitaevskii regime

We study the Bose-Einstein condensates of trapped Bose gases in the Gross-Pitaevskii regime. We show that the ground state energy and ground states of the many-body quantum system are correctly described by the Gross-Pitaevskii equation in the large particle number limit, and provide the optimal convergence rate. Our work extends the previous results of Lieb, Seiringer and Yngvason on the leading order convergence, and of Boccato, Brennecke, Cenatiempo and Schlein on the homogeneous gas. Our method relies on the idea of 'completing the square', inspired by recent works of Brietzke, Fournais and Solovej on the Lee-Huang-Yang formula, and a general estimate for Bogoliubov quadratic Hamiltonians on Fock space.

A McKean--Vlasov SDE and Particle System with Interaction from Reflecting Boundaries

We consider a one-dimensional McKean--Vlasov SDE on a domain and the associated mean-field interacting particle system. The peculiarity of this system is the combination of the interaction, which keeps the average position prescribed, and the reflection at the boundaries; these two factors make the effect of reflection nonlocal. We show pathwise well-posedness for the McKean--Vlasov SDE and convergence for the particle system in the limit of large particle number.

Can a bright soliton model reveal a genuine time crystal for a finite number of bosons?

We analyze time crystal effects in a finite system of bosons which form a bright soliton clump on the Aharonov-Bohm ring. In the large particle number limit, N → ∞, this setup corresponds to the Wilczek model, where it is known that the time crystal behavior cannot be observed in the ground state of the system because a spontaneously formed soliton does not move. Here, we show that while the spontaneous formation of a moving soliton in the ground state can occur for , the soliton decays before it makes a single revolution along the ring and the time crystal dynamics is impossible.

Scaling limits of bosonic ground states, from many-body to non-linear Schrödinger

How and why could an interacting system of many particles be described as if all particles were independent and identically distributed? This question is at least as old as statistical mechanics itself. Its quantum version has been rejuvenated by the birth of cold atoms physics. In particular, the experimental creation of Bose–Einstein condensates leads to the following variant: why and how can a large assembly of very cold interacting bosons (quantum particles deprived of the Pauli exclusion principle) all populate the same quantum state? In this text I review the various mathematical techniques allowing to prove that the lowest energy state of a bosonic system forms, in a reasonable macroscopic limit of large particle number, a Bose–Einstein condensate. This means that indeed in the relevant limit all particles approximately behave as if independent and identically distributed, according to a law determined by minimizing a non-linear Schrödinger energy functional. This is a particular instance of the justification of the mean-field approximation in statistical mechanics, starting from the basic many-body Schrödinger Hamiltonian.

Asymptotic population imbalance of an ultracold bosonic ensemble in a driven double well

We demonstrate that an ultracold many-body bosonic ensemble confined in an one-dimensional (1D) double well potential exhibits a population imbalance between the two wells at large timescales, when the depth of the wells are modulated by a time-dependent driving force. The specific form of the driving force is shown to break spatial parity and time-reversal symmetries, which leads to such an asymptotic population imbalance (API). The value of the API can be flexibly controlled by changing the phase of the driving force and the total number of particles. While the API is highly sensitive to the initial state in the few-particle regime, this dependence on the initial state is lost as we approach the classical limit of large particle numbers. We perform a Floquet analysis in the few-particle regime and an analysis based on a driven classical non-rigid pendulum in the many-particle regime. Although the obtained API values in the many-particle regime agree very well with that obtained in the classical limit, we show that there exists a significant disagreement in the corresponding real-time population imbalance due to quantum correlations.

Probabilistic hysteresis in an isolated quantum system: The microscopic onset of irreversibility from a quantum perspective

Recently probabilistic hysteresis in isolated Hamiltonian systems of ultracold atoms has been studied in the limit of large particle numbers, where a semiclassical treatment is adequate. The origin of irreversibility in these sweep experiments, where a control parameter is slowly (adiabatically) tuned back and forth, turned out to be a passage blue back and forth across a separatrix (integrable case) or a passage in and out of a chaotic sea in phase space (chaotic case). Here we focus on the full quantum mechanical description of the integrable system and show how the semiclassical results emerge in the limit of large particle numbers. Instead of the crossing of a separatrix in phase space, where classical adiabaticity fails, the origin of irreversibility in the quantum system is a series of avoided crossings of the adiabatic energy levels: they become so close that already for modest particle numbers the change of the external parameter has to be unrealistically slow to reach the quantum adiabatic limit of perfectly reversible evolution. For a slow but finite sweep rate we find a broad regime where the quantum results agree with the semiclassical results, but only if besides the limit $N\to \infty$ an initial ensemble of states is considered, with sufficient initial energy width. For a single initial energy eigenstate we find in contrast that the backward sweep reveals strong quantum effects even for very large particle numbers.

Dynamical quantum phase transitions in interacting atomic interferometers

Particle-wave duality has allowed physicists to establish atomic interferometers as celebrated complements to their optical counterparts in a broad range of quantum devices. However, interactions naturally lead to decoherence and have been considered as a longstanding obstacle in implementing atomic interferometers in precision measurements. Here, we show that interactions lead to dynamical quantum phase transitions between Schr\"{o}dinger's cats in an atomic interferometer. These transition points result from zeros of Loschmidt echo, which approach the real axis of the complex time plane in the large particle number limit, and signify pair condensates, another type of exotic quantum states featured with prevailing two-body correlations. Our work suggests interacting atomic interferometers as a new tool for exploring dynamical quantum phase transitions and creating highly entangled states to beat the standard quantum limit.

Two-state model for vortex nucleation in a rotating Bose-Einstein condensate

It is well-known that a rotating Bose-Einstein condensate forms vortices to carry the angular momentum. For a first vortex to nucleate at the trap center, the rotational frequency must become larger than a certain critical value. The vortex nucleation process, however, is sensitive to the trap shape. It was shown earlier that for a symmetry-breaking potential that preserves parity, at criticality the leading natural orbitals may become degenerate, giving rise to a maximally entangled quantum state, found from exact solutions for just a few bosons. Developing an effective two-state model, we here show that in the limit of large particle numbers, the many-body ground state becomes either a so-called twin-like or a Schr\"odinger cat-like state. We corroborate this finding by a direct comparison to the exact numerical solution of the problem, feasible for moderate particle numbers within the lowest Landau level approximation. We show that the nature of the quantum state at criticality can be controlled by both the quadrupolar deformation and the flatness of the confining potential.

Finite-size and finite-time effects in large deviation functions near dynamical symmetry breaking transitions

We introduce and study a class of particle hopping models consisting of a single box coupled to a pair of reservoirs. Despite being zero-dimensional, in the limit of large particle number and long observation time, the current and activity large deviation functions of the models can exhibit symmetry-breaking dynamical phase transitions. We characterize exactly the critical properties of these transitions, showing them to be direct analogues of previously studied phase transitions in extended systems. The simplicity of the model allows us to study features of dynamical phase transitions which are not readily accessible for extended systems. In particular, we quantify finite-size and finite-time scaling exponents using both numerical and theoretical arguments. Importantly, we identify an analogue of critical slowing near symmetry breaking transitions and suggest how this can be used in the numerical studies of large deviations. All of our results are also expected to hold for extended systems.

Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation

Data assimilation refers to the methodology of combining dynamical models and observed data with the objective of improving state estimation. Most data assimilation algorithms are viewed as approximations of the Bayesian posterior (filtering distribution) on the signal given the observations. Some of these approximations are controlled, such as particle filters which may be refined to produce the true filtering distribution in the large particle number limit, and some are uncontrolled, such as ensemble Kalman filter methods which do not recover the true filtering distribution in the large ensemble limit. Other data assimilation algorithms, such as cycled 3DVAR methods, may be thought of as controlled estimators of the state, in the small observational noise scenario, but are also uncontrolled in general in relation to the true filtering distribution. For particle filters and ensemble Kalman filters it is of practical importance to understand how and why data assimilation methods can be effective when used with a fixed small number of particles, since for many large-scale applications it is not practical to deploy algorithms close to the large particle limit asymptotic. In this paper, the authors address this question for particle filters and, in particular, study their accuracy (in the small noise limit) and ergodicity (for noisy signal and observation) without appealing to the large particle number limit. The authors first overview the accuracy and minorization properties for the true filtering distribution, working in the setting of conditional Gaussianity for the dynamics-observation model. They then show that these properties are inherited by optimal particle filters for any fixed number of particles, and use the minorization to establish ergodicity of the filters. For completeness we also prove large particle number consistency results for the optimal particle filters, by writing the update equations for the underlying distributions as recursions. In addition to looking at the optimal particle filter with standard resampling, they derive all the above results for (what they term) the Gaussianized optimal particle filter and show that the theoretical properties are favorable for this method, when compared to the standard optimal particle filter.

The nonextensive statistical ensembles with dual thermodynamic interpretations

The nonextensive statistical ensembles are revisited for the complex systems with long-range interactions and long-range correlations. An approximation, the value of nonextensive parameter (1-q) is assumed to be very tiny, is adopted for the limit of large particle number for most normal systems. In this case, Tsallis entropy can be expanded as a function of energy and particle number fluctuation, and thus the power-law forms of the generalized Gibbs distribution and grand canonical distribution can be derived. These new distribution functions can be applied to derive the free energy and grand thermodynamic potential in nonextensive thermodynamics. In order to establish appropriate nonextensive thermodynamic formalism, the dual thermodynamic interpretations are necessary for thermodynamic relations and thermodynamic quantities. By using a new technique of parameter transformation, the single-particle distribution can be deduced from the power-law Gibbs distribution. This technique produces a link between the statistical ensemble and the quasi-independent system with two kinds of nonextensive parameter having quite different physical explanations. Furthermore, the technique is used to construct nonextensive quantum statistics and effectively to avoid the factorization difficulty in the power-law grand canonical distribution.

Exact dynamics of a one dimensional Bose gas in a periodic time-dependent harmonic trap

We study the unitary dynamics of a 1D gas of hard-core bosons trapped into a harmonic potential which varies periodically in time with frequency . Such periodic systems can be classified into orbits of different monodromies corresponding to two different physical situations, namely the case in which the bosonic cloud remains stable during the time-evolution and the case where it turns out to be unstable. In the present work we derive in the large particle number limit exact results for the stroboscopic evolution of the energy and particle densities in both physical situations.

Hyperspherical lowest-order constrained-variational approximation to resonant Bose-Einstein condensates

We study the ground state properties of a system of $N$ harmonically trapped bosons of mass $m$ interacting with two-body contact interactions, from small to large scattering lengths. This is accomplished in a hyperspherical coordinate system that is flexible enough to describe both the overall scale of the gas and two-body correlations. By adapting the lowest-order constrained variational (LOCV) method, we are able to semi-quantitatively attain Bose-Einstein condensate ground state energies even for gases with infinite scattering length. In the large particle number limit, our method provides analytical estimates for the energy per particle $E_0/N \approx 2.5 N^{1/3} \hbar \omega$ and two-body contact $C_2/N \approx 16 N^{1/6}\sqrt{m\omega/\hbar}$ for a Bose gas on resonance, where $\omega$ is the trap frequency.

Understanding quantum work in a quantum many-body system.

Based on previous studies in a single-particle system in both the integrable [Jarzynski, Quan, and Rahav, Phys. Rev. X 5, 031038 (2015)2160-330810.1103/PhysRevX.5.031038] and the chaotic systems [Zhu, Gong, Wu, and Quan, Phys. Rev. E 93, 062108 (2016)1539-375510.1103/PhysRevE.93.062108], we study the the correspondence principle between quantum and classical work distributions in a quantum many-body system. Even though the interaction and the indistinguishability of identical particles increase the complexity of the system, we find that for a quantum many-body system the quantum work distribution still converges to its classical counterpart in the semiclassical limit. Our results imply that there exists a correspondence principle between quantum and classical work distributions in an interacting quantum many-body system, especially in the large particle number limit, and further justify the definition of quantum work via two-point energy measurements in quantum many-body systems.

Thermal and quantum fluctuations of confined Bose–Einstein condensate beyond the Bogoliubov approximation

The formulation for zero mode of a Bose–Einstein condensate beyond the Bogoliubov approximation at zero temperature [Y. Nakamura et al., Phys. Rev. A 89 (2014) 013613] is extended to finite temperature. Both thermal and quantum fluctuations are considered in a manner consistent with a concept of spontaneous symmetry breakdown for a finite-size system. Therefore, we need a proper treatment of the zero mode operators, which invoke non-trivial enhancements in depletion condensate and thermodynamical quantities such as the specific heat. The enhancements are visible in the weak interaction case. Our approach reproduces the results of a homogeneous system in the Bogoliubov approximation in a large particle number limit.

Classical-quantum correspondence in bosonic two-mode conversion systems: Polynomial algebras and Kummer shapes

Bosonic quantum conversion systems can be modeled by many-particle single-mode Hamiltonians describing a conversion of $n$ molecules of type A into $m$ molecules of type B and vice versa. These Hamiltonians are analyzed in terms of generators of a polynomially deformed $su(2)$ algebra. In the mean-field limit of large particle numbers, these systems become classical and their Hamiltonian dynamics can again be described by polynomial deformations of a Lie algebra, where quantum commutators are replaced by Poisson brackets. The Casimir operator restricts the motion to Kummer shapes, deformed Bloch spheres with cusp singularities depending on $m$ and $n$. It is demonstrated that the many-particle eigenvalues can be recovered from the mean-field dynamics using a WKB type quantization condition. The many-particle state densities can be semiclassically approximated by the time-periods of periodic orbits, which show characteristic steps and singularities related to the fixed points, whose bifurcation properties are analyzed.

Large-Nground state of the Lieb-Liniger model and Yang-Mills theory on a two-sphere

We derive the large particle number limit of the Bethe equations for the ground state of the attractive one-dimensional Bose gas (Lieb-Liniger model) on a ring and solve it for arbitrary coupling. We show that the ground state of this system can be mapped to the large-N saddle point of Euclidean Yang-Mills theory on a two-sphere with a U(N) gauge group, and the phase transition that interpolates between the homogeneous and solitonic regime is dual to the Douglas-Kazakov confimenent-deconfinement phase transition.

Periodic mean-field solutions and the spectra of discrete bosonic fields: Trace formula for Bose-Hubbard models

We consider the many-body spectra of interacting bosonic quantum fields on a lattice in the semiclassical limit of large particle number N. We show that the many-body density of states can be expressed as a coherent sum over oscillating long-wavelength contributions given by periodic, nonperturbative solutions of the, typically nonlinear, wave equation of the classical (mean-field) limit. To this end, we construct the semiclassical approximation for both the smooth and oscillatory parts of the many-body density of states in terms of a trace formula starting from the exact path integral form of the propagator between many-body quadrature states. We therefore avoid the use of a complexified classical limit characteristic of the coherent state representation. While quantum effects such as vacuum fluctuations and gauge invariance are exactly accounted for, our semiclassical approach captures quantum interference and therefore is valid well beyond the Ehrenfest time where naive quantum-classical correspondence breaks down. Remarkably, due to a special feature of harmonic systems with incommensurable frequencies, our formulas are generically valid also in the free-field case of noninteracting bosons.

The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases

We study the ground state of a trapped Bose gas, starting from the full many-body Schrodinger Hamiltonian, and derive the nonlinear Schrodinger energy functional in the limit of large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive nonlinear Schrodinger ground state.

Semiclassical quantization for a bosonic atom-molecule conversion system

We consider a simple quantum model of atom-molecule conversion where bosonic atoms can combine into diatomic molecules and vice versa. The many-particle system can be expressed in terms of the generators a deformed $SU(2)$ algebra, and the mean-field dynamics takes place on a deformed version of the Bloch sphere, a teardrop shaped surface with a cusp singularity. We analyse the mean-field and many-particle correspondence, which shows typical features of quantum-classical correspondence. We demonstrate that semiclassical methods can be employed to recover full many-particle features from the mean-field description in cold atom systems with atom-molecule conversion, and derive an analytic expression for the many-particle density of states in the limit of large particle numbers.