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arXiv:2606.10055

Identical Bosons, large occupation numbers and classical field description

Gaurav Goswami

quant-phastro-ph.COhep-phhep-thSubmitted Jun 8, 2026• Announced Jun 10, 2026• 0 citations
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Abstract

For a system with a large number of identical Bosons, it is common to claim, often without any additional justifications, that, when the mean occupation number in a single particle state is sufficiently large, classical field description will be applicable. This is why e.g. for ultra-light dark matter, the classical field equations are used to compute its dynamics. In this work, we test the validity and robustness of this assumption based on the criterion $2 σ_\varphi < |\langle \varphi \rangle| $ for classical field behaviour and applying it to aribtrary quantum states. We find that an arbitrary state with large occupation number doesn't behave classically while imposing some restrictions on the state vectors can improve the classical behavior. Since coherent states are known to have quasi-classical behaviour, we also ask how much deviation from coherent state can spoil the classical behaviour. Based on this analysis, we find that it is the proximity of the state to a large occupation coherent state rather than large occupation number itself which ensures validity of classical description. Implications of this for ultra light dark matter are discussed.

Comments
9 pages, 3 figures

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For a system with a large number of identical Bosons, it is common to claim, often without any additional justifications, that, when the mean occupation number in a single particle state is sufficiently large, classical field description will be applicable. This is why e.g. for ultra-light dark matter, the classical field equations are used to compute its dynamics. In this work, we test the validity and robustness of this assumption based on the criterion 2σφ<∣⟨φ⟩∣2 σ_\varphi < |\langle \varphi \rangle| 2σφ​<∣⟨φ⟩∣ for classical field behaviour and applying it to aribtrary quantum states. We find that an arbitrary state with large occupation number doesn't behave classically while imposing some restrictions on the state vectors can improve the classical behavior. Since coherent states are known to have quasi-classical behaviour, we also ask how much deviation from coherent state can spoil the classical behaviour. Based on this analysis, we find that it is the proximity of the state to a large occupation coherent state rather than large occupation number itself which ensures validity of classical description. Implications of this for ultra light dark matter are discussed.
It is understood that in free bosonic theories, the classical field theory accurately describes the full quantum theory when the occupancy numbers of systems are very large. However, the situation is less understood in interacting theories, especially on time scales longer than the dynamical relaxation time. Recently there have been claims that the quantum theory deviates spectacularly from the classical theory on this time scale, even if the occupancy numbers are extremely large. Furthermore, it is claimed that the quantum theory quickly thermalizes while the classical theory does not. The evidence for these claims comes from noticing a spectacular difference in the time evolution of expectation values of quantum operators compared to the classical micro-state evolution. If true, this would have dramatic consequences for many important phenomena, including laboratory studies of interacting BECs, dark matter axions, preheating after inflation, etc. In this work we critically examine these claims. We show that in fact the classical theory can describe the quantum behavior in the high occupancy regime, even when interactions are large. The connection is that the expectation values of quantum operators in a single quantum micro-state are approximated by a corresponding classical ensemble average over many classical micro-states. Furthermore, by the ergodic theorem, a classical ensemble average of local fields with statistical translation invariance is the spatial average of a single micro-state. So the correlation functions of the quantum and classical field theories of a single micro-state approximately agree at high occupancy, even in interacting systems. Furthermore, both quantum and classical field theories can thermalize, when appropriate coarse graining is introduced, with the classical case requiring a cutoff on low occupancy UV modes. We discuss applications of our results.
In any bosonic lattice system, which is not dominated by local interactions and thus ``frozen'' in a Mott-type state, numerical methods have to cope with the infinite size of the corresponding Hilbert space even for finite lattice sizes. While it is common practice to restrict the local occupation number basis to Nc{N}_{c}Nc​ lowest occupied states, the presence of a finite condensate fraction requires the complete number basis for an exact representation of the many-body ground state. In this work we present a truncation scheme to account for contributions from higher number states. By simply adding a single coherent-tail state to this common truncation, we demonstrate increased numerical accuracy and the possible increase in numerical efficiency of this method for the Gutzwiller variational wave function and within dynamical mean-field theory.
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The dynamics of a quantum system with a large number NNN of identical bosonic particles interacting by means of weak two-body potentials can be simplified by using mean-field equations in which all interactions to any one body have been replaced with an average or effective interaction in the mean-field limit N→∞N \rightarrow \inftyN→∞. In order to show these mean-field equations are accurate, one needs to show convergence of the quantum NNN-body dynamics to these equations in the mean-field limit. Previous results on convergence in the mean field limit have been derived for certain initial conditions in the case of one species of bosonic particles, but no results have yet been shown for multi-species. In this thesis, we look at a quantum bosonic system with two species of particles. For this system, we derive a formula for the rate of convergence in the mean-field limit in the case of an initial coherent state, and we also show convergence in the mean-field limit for the case of an initial factorized state. The analysis for two species can then be extended to multiple species.
A description of a large system of particles is often sought in a derivation from the detailed behaviour of just a few of the particles. The present thesis deals with the connection between such microscopic features and the nature of a collection of many particles. The study of two-body correlations yields insight into mechanisms that in the dilute limit can be accounted for by a mean field, and yields deviations especially in the presence of a two-body bound state or a resonance. Contents: Chapter 2 presents the hyperspherical frame for studying few-body correlations within a many-boson system. Chapter 3 contains a discussion of analytical and numerical solutions to the hyperangular part of the Hamiltonian. Chapter 4 contains a discussion of the hyperradial part of the description of a many-boson system, as well as a discussion of Bose-Einstein condensation in the hyperspherical model. In chapter 5 we compare results from the previous chapters to the results from the mean-field Gross-Pitaevskii equation and ranges of validity are estimated. In chapter 6 we discuss stability criteria, approaches to the dynamical evolution of the many-boson system, and relevant time scales. Chapter 7 deals with the effects of a deformed external trap.
Progress in the reliable preparation, coherent propagation and efficient detection of many-body states has recently brought collective quantum phenomena of many identical particles into the spotlight. This tutorial introduces the physics of many-boson and many-fermion interference required for the description of current experiments and for the understanding of novel approaches to quantum computing. The field is motivated via the two-particle case, for which the uncorrelated, classical dynamics of distinguishable particles is compared to the quantum behaviour of identical bosons and fermions. Bunching of bosons is opposed to anti-bunching of fermions, while both species constitute equivalent sources of bipartite two-level entanglement. The realms of indistinguishable and distinguishable particles are connected by a monotonic transition, on a scale defined by the coherence length of the interfering particles. As we move to larger systems, any attempt to understand many particles via the two-particle paradigm fails: In contrast to two-particle bunching and anti-bunching, the very same signatures can be exhibited by bosons and fermions, and coherent effects dominate over statistical behaviour. The simulation of many-boson interference, termed Boson-Sampling, entails a qualitatively superior computational complexity when compared to fermions. The hierarchy between bosons and fermions also characterises multipartite entanglement generation, for which bosons again clearly outmatch fermions. Finally, the quantum-to-classical transition between many indistinguishable and many distinguishable particles features non-monotonic structures. While the same physical principles govern small and large systems, the deployment of the intrinsic complexity of collective many-body interference makes more particles behave differently.