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Fibonacci Lattice Condensates as FCCL Simulators: Born Rule Measurement and Golden-Ratio Revivals in Bose-Einstein Condensates

Rémi Leroy

Published Sep 7, 2026• 0 citations
Publisher

Abstract

We show that a Bose-Einstein condensate (BEC) loaded into a Fibonacci-modulated optical lattice constitutes a direct experimental implementation of the Fibonacci Cascade Conformal Layer (FCCL). The FCCL Born rule 𝑃(𝑛) = 𝜑^{−2𝑛}/√5 manifests as the timeof-flight momentum distribution after ballistic expansion: peaks at momenta 𝑘_𝑛 = 𝑛𝑘₀ carry amplitudes decaying as 𝜑−𝑛/5^{1/4}, with consecutive peak ratios exactly equal to 𝜑. Three additional predictions follow from the FCCL cascade structure: (i) quantum revival times spaced by the golden ratio, 𝑡_𝑛 = 𝑡₀𝜑^𝑛; (ii) critical temperatures for successive subbands scaling as 𝑇_𝑐(𝑛)/𝑇_𝑐(0) = 𝜑^{−𝑛/3}, with 𝑇_𝑐(3)/𝑇_𝑐(0) = 1/𝜑 exactly; (iii) a spectral bandwidth exactly equal to 2𝐽₀ (independent of the ratio 𝐽₁/𝐽₂), from the Fibonacci identity 𝜑 −1 + 𝜑−2 = 1. These predictions are accessible with current cold-atom technology and constitute the first direct laboratory test of the FCCL Born rule that does not require a particle collider.

DOI
10.5281/zenodo.22608381

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We show that a Bose-Einstein condensate (BEC) loaded into a Fibonacci-modulated optical lattice constitutes a direct experimental implementation of the Fibonacci Cascade Conformal Layer (FCCL). The FCCL Born rule 𝑃(𝑛) = 𝜑^{−2𝑛}/√5 manifests as the timeof-flight momentum distribution after ballistic expansion: peaks at momenta 𝑘_𝑛 = 𝑛𝑘₀ carry amplitudes decaying as 𝜑−𝑛/5^{1/4}, with consecutive peak ratios exactly equal to 𝜑. Three additional predictions follow from the FCCL cascade structure: (i) quantum revival times spaced by the golden ratio, 𝑡_𝑛 = 𝑡₀𝜑^𝑛; (ii) critical temperatures for successive subbands scaling as 𝑇_𝑐(𝑛)/𝑇_𝑐(0) = 𝜑^{−𝑛/3}, with 𝑇_𝑐(3)/𝑇_𝑐(0) = 1/𝜑 exactly; (iii) a spectral bandwidth exactly equal to 2𝐽₀ (independent of the ratio 𝐽₁/𝐽₂), from the Fibonacci identity 𝜑 −1 + 𝜑−2 = 1. These predictions are accessible with current cold-atom technology and constitute the first direct laboratory test of the FCCL Born rule that does not require a particle collider.
The first successful experimental realization of a Bose-Einstein condensate (BEC) in 1995 sparked a still growing research activity that in the meantime has led to beautiful and surprising results, such as the interference of matter waves [4], the Mott-superfluid phase transition [49] or the direct observation of Anderson localization of matter waves [10]. The strong interest in BECs arises, among others, from the possibility to investigate known but hard to study quantum systems on a rather macroscopic scale: for example one can load ultra-cold bosons into an optical lattice created by a standing light wave, and thereby simulate a solid state device. When it comes to a measurement on the BEC, however, essentially all measurement techniques result in the destruction of the condensate: for example in the so called “time of flight method” the optical lattice is switched off to observe the expansion of the trapped atoms and thereby directly visualize their momentum distribution, which highlights the condensed component [11]. In this thesis we elaborate on a means to probe a Bose-Einstein condensate from outside, with a scattering particle. The interaction between the probe particle and the bosons allows energy exchange between them, i.e., this setup includes inelastic scattering. After the scattering process, the condensate is still confined in the optical lattice and can therefore be used for further investigations. Of course, this method of probing a BEC is only then a serious alternative if the properties of the BEC can be inferred from the scattering signal. Besides outlining the scattering formalism, the central aim of this work is to show how the properties of the BEC can be deduced from those quantities accessible in a scattering experiment, like for example the transmission or the inelastic cross section. The other, equally strong motivation of this work concerns the signature of the interaction between the bosons in the scattering quantities. The dynamics of an analog classical system depends strongly on the interaction strength, which can induce a complex structure of the underlying phase space like, e.g., the appearance of a separatrix or even the emergence of chaotic dynamics in a globally mixed regular-chaotic phase space. We are interested to which extent such phase-space metamorphosis is reflected in the properties of the scattering particle. Currently light scattering (Bragg spectroscopy) on an optical lattice [78, 77] or rather on an optical lattice embedded into a cavity [62,68,19] is under intense investigation. These efforts predominantly aim at a characterization of the Mott-superfluid transition by specific scattering properties. In the present thesis we focus on condensates loaded in two and three lattice sites, with a large number of bosons. These systems exhibit a diversity of important physical effects (as we will see below) and thus are of paradigmatic relevance for a detailed understanding of multiparticle quantum dynamics in periodic potentials.
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La thèse traite de la dynamique d'un Condensat de Bose Einstein (CBE) dans un réseau optique modulé en phase et en amplitude. Dans un premier temps, je présente le dispositif expérimental permettant d'obtenir le CBE ainsi que le réseau optique, et décris les différents contrôles de la phase et de l'amplitude que nous avons mis en place. La première expérience que nous avons effectuée repose sur le changement brusque de la phase du réseau permettant de déplacer celui-ci de quelques dizaines de nm. Cette expérience nous a permis de mettre au point une méthode de calibration de la profondeur du potentiel périodique se basant sur la mesure de la période de la micro-oscillation de la chaîne de condensats induite par le déplacement soudain du potentiel. Il existe ainsi une bijection entre profondeur et période de l'oscillation faisant de cette dernière une candidate idéale comme méthode de calibration. Cette oscillation présente également de l'effet tunnel entre puits du réseau. Nous avons pu mesurer le temps tunnel i.e. le retard entre un paquet d'atomes ayant traversé une barrière de potentiel par effet tunnel et un paquet d'atomes ayant continué son oscillation. La dernière partie de ce manuscrit présente une étude de la dynamique du condensat dans un réseau dont la phase ou l'amplitude sont modulées sinusoïdalement. Nous avons étudié trois régimes de fréquences de modulation présentant des comportements très différents. Les régimes de fréquences sont définies par rapport à la fréquence résonante entre la bande fondamentale du réseau et la première bande excitée. Pour la modulation à basse fréquence, le taux tunnel intersite est renormalisé et peut atteindre des valeurs effectives négatives. Ce phénomène engendre une instabilité dynamique qui est à l'origine d'une transition de phase quantique. Nous avons effectué une étude théorique, numérique et expérimentale qui nous a permis de déterminer le rôle des fluctuations quantiques et thermiques dans la cinétique de cette transition. Pour le régime de modulation haute fréquence, la profondeur du potentiel du réseau est renormalisée par une fonction de Bessel. Nous nous sommes servis de cet effet pour placer les atomes dans une situation très loin de l'équilibre avec un grand contrôle. Enfin, en nous plaçant dans le régime où la fréquence de modulation est de l'ordre de la fréquence résonante entre la bande fondamentale du réseau et la première bande excitée, nous induisons des transitions interbandes. Nos données révèlent les règles de sélection qui sont différentes pour la modulation de phase et d'amplitude, et le rôle des interactions dans les excitations résonantes.
We derive the Bell correlations of the Fibonacci Cascade Conformal Layer (FCCL) bilateral state — the quantum state whose Born rule probability distribution follows inverse powers of the golden ratio, normalized by the square root of five. The central result is the Fibonacci Correlation Theorem: the FCCL correlation function, evaluated at angles that are simple fractions of a full rotation (one-sixth, one-quarter, one-half), takes exact reciprocal Fibonacci values (one-half, one-third, one-fifth). These identities follow algebraically from the Fibonacci recursion and hold without approximation. The CHSH Bell inequality analysis reveals a sharp classical-quantum boundary determined by the measurement type. With diagonal measurements — functions of the cascade level, such as energy or position — the CHSH value is exactly 2 (the classical bound). The bilateral cascade state has perfect classical correlations in the cascade position basis: the cascade level acts as a local hidden variable for diagonal observations. With off-diagonal measurements — hopping operators that create coherence between adjacent cascade levels — the CHSH value reaches the quantum bound of 2 times the square root of 2 (Tsirelson). This boundary is not an approximation: classical physics corresponds to cascade-diagonal measurements; quantum mechanics to cascade-off-diagonal ones. The quantum-classical transition is a geometric property of the Fibonacci cascade, not a scale-dependent phenomenon. In Fibonacci-lattice Bose-Einstein condensates, the two measurement types correspond to time-of-flight imaging (diagonal, classical correlations) and Ramsey spectroscopy (off-diagonal, quantum correlations). The ratio of consecutive Fibonacci correlations is itself a Fibonacci ratio, providing a precision spectroscopic test accessible with current cold-atom technology.
We derive quantum measurement theory from the Fibonacci Cascade Conformal Layer (FCCL). Measurement is identified as a Cardy projection: the probability of obtaining outcome 𝑛 when measuring a cascade state |𝜓⟩ equals |⟨𝑛|𝐵Planck⟩|² = 𝜑^{−2𝑛}/√5 — the Born rule, derived from the Cardy formula without postulation. The preferred “pointer basis” of decoherence consists of the FCCL Cardy boundary states {|𝐵𝑛⟩} — energy eigenstates of the cascade Hamiltonian, selected by the environment because they minimize the cascade free energy. The coherence between cascade levels 𝑛1 and 𝑛2 decays at rate Γ₁₂ = (𝑘_𝐵𝑇 /ℏ) × 𝑁_modes × (1 − 𝜑^{−2|𝑛1−𝑛2|} / √5), recovering standard decoherence scaling for macroscopic objects while protecting quantum coherence for microscopic systems. Quantum Darwinism emerges naturally: the probability that 𝑘 independent environmental modes all record the same measurement outcome is (𝜑^{−2𝑛}/√5)^𝑘 — the Born rule raised to the 𝑘-th power.