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Physical Review A

Photonic bands and normal mode splitting in optical lattices interacting with cavities

Ph. W. Courteille, Yajaira Dalila Rivero Jerez, Gustavo Henrique de França, C. A. Pessoa, A. Cipris, Mayerlin Núñez Portela, R Teixeira, Sebastian Slama

Published Jan 17, 2025• 2 citations
Publisher

Abstract

The strong collective interaction of atoms with an optical cavity causes normal mode splitting of the cavity's resonances, whose width is given by the collective coupling strength. At low optical density of the atomic cloud, the intensity distribution of light in the cavity is ruled by the cavity's mode function, which is determined solely by its geometry. In this regime the dynamics of the coupled atom-cavity system is conveniently described by the open Dicke model, which we apply to calculating normal mode splitting generated by periodically ordered clouds in linear and ring cavities. We also show how to use normal mode splitting as a witness for Wannier-Bloch oscillations in the tight-binding limit. At high optical density the atomic distribution contributes to shaping the mode function. This regime escapes the open Dicke model, but can be treated by a transfer matrix model provided the saturation parameter is low. Applying this latter model to an atomic cloud periodically ordered into a one-dimensional lattice, we observe the formation of photonic bands gaps competing with the normal mode splitting. We discuss the limitations of both models and point out possible pathways to generalized theories.

DOI
10.1103/physreva.111.013310

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The strong collective interaction of atoms with an optical cavity causes normal mode splitting of the cavity's resonances, whose width is given by the collective coupling strength. At low optical density of the atomic cloud, the intensity distribution of light in the cavity is ruled by the cavity's mode function, which is determined solely by its geometry. In this regime the dynamics of the coupled atom-cavity system is conveniently described by the open Dicke model, which we apply to calculating normal mode splitting generated by periodically ordered clouds in linear and ring cavities. We also show how to use normal mode splitting as a witness for Wannier-Bloch oscillations in the tight-binding limit. At high optical density the atomic distribution contributes to shaping the mode function. This regime escapes the open Dicke model, but can be treated by a transfer matrix model provided the saturation parameter is low. Applying this latter model to an atomic cloud periodically ordered into a one-dimensional lattice, we observe the formation of photonic bands gaps competing with the normal mode splitting. We discuss the limitations of both models and point out possible pathways to generalized theories.
Strong collective interaction of atoms with an optical cavity causes normal mode splitting of the cavity's resonances, whose width is given by the collective coupling strength. At low optical density of the atomic cloud the intensity distribution of light in the cavity is ruled by the cavity's mode function, which is solely determined by its geometry. In this regime the dynamics of the coupled atom-cavity system is conveniently described by the open Dicke model, which we apply to calculating normal mode splitting generated by periodically ordered clouds in linear and ring cavities. We also show how to use normal mode splitting as witness for Wannier-Bloch oscillations in the tight-binding limit. At high optical density the atomic distribution contributes to shaping the mode function. This regime escapes the open Dicke model, but can be treated by a transfer matrix model provided the saturation parameter is low. Applying this latter model to an atomic cloud periodically ordered into a one-dimensional lattice, we observe the formation of photonic bands gaps competing with the normal mode splitting. We discuss the limitations of both models and point out possible pathways to generalized theories.
A novel regime of atom-cavity physics is explored, arising when large atom samples dispersively interact with high-finesse optical cavities. A stable far-detuned optical lattice of several million rubidium atoms is formed inside an optical ring resonator by coupling equal amounts of laser light to each propagation direction of a longitudinal cavity mode. An adjacent longitudinal mode, detuned by about 3 GHz, is used to perform probe transmission spectroscopy of the system. The atom-cavity coupling for the lattice beams and the probe is dispersive and dissipation results only from the finite photon-storage time. The observation of two well-resolved normal modes demonstrates the regime of strong cooperative coupling. The details of the normal mode spectrum reveal mechanical effects associated with the retroaction of the probe upon the optical lattice.
This paper deals with atomic systems coupled to a structured reservoir of quantum EM field modes, with particular relevance to atoms interacting with the field in photonic band gap materials. The case of high Q cavities has been treated elsewhere using Fano diagonalization based on a quasimode approach, showing that the cavity quasimodes are responsible for pseudomodes introduced to treat non-Markovian behaviour. The paper considers a simple model of a photonic band gap case, where the spatially dependent permittivity consists of a constant term plus a small spatially periodic term that leads to a narrow band gap in the spectrum of mode frequencies. Most treatments of photonic band gap materials are based on the true modes, obtained numerically by solving the Helmholtz equation for the actual spatially periodic permittivity. Here the field modes are first treated in terms of a simpler quasimode approach, in which the quasimodes are plane waves associated with the constant permittivity term. Couplings between the quasimodes occur owing to the small periodic term in the permittivity, with selection rules for the coupled modes being related to the reciprocal lattice vectors. This produces a field Hamiltonian in quasimode form. A matrix diagonalization method may be applied to relate true mode annihilation operators to those for quasimodes. The atomic transitions are coupled to all the quasimodes, and the true mode atom-EM field coupling constants (one-photon Rabi frequencies) are related to those for the quasimodes and also expressions are obtained for the true mode density. The results for the one-photon Rabi frequencies differ from those assumed in other work. Expressions for atomic decay rates are obtained using the Fermi Golden rule, although these are valid only well away from the band gaps.
The electromagnetic mode density of the vacuum can be dramatically \nmodified by the presence of an optical resonator. In the strong coupling \nregime, spontaneous emission in a cavity becomes a reversible process and \nthe intracavity photon number undergoes Rabi oscillations. We load up \nto 200x10^3 ^133Cs atoms into a nearly confocal lossy cavity and reach the \ncollective strong coupling regime. Normal mode splitting, the hallmark of this regime, is observed and cooperativities up to C_coll = (186±5) are measured. In a second experiment we investigate for the first time \nthe multi-mode character of the coupled cavity-atom system. In a confocal \ncavity the higher-order transverse cavity modes are degenerate in frequency and accessible to the spontaneous emission of the atomic ensemble. \nWe observe an increase of the coupling constant measured via modal decomposed transmission analysis, which could be attributed to \nthe presence of the higher-order modes. Normal mode splitting proportional to the square root of the atom number was visible for all of the \ndifferent mode components. Furthermore, we observe a redistribution of the relative weights in the modal transmission composition, which scales with the atom number in the cavity mode. \nIn a second set of experiments, ^87Rb atoms were loaded into a dissipative \nlin ⊥ lin lattice. By driving the lattice with a biharmonic force, transport can be observed when the systems symmetries are broken: the so called ratchet effect. Research in this area is concerned with the appearance \nof current reversals. We were able to identify dissipation related symmetry breaking as the underlying cause of an observed current reversal, which occurs as a function of the driving frequency. Furthermore, in a \nsecond experiment, we use the ratchet effect as a probe of the optical potential depths. We show that an oscillating force with a frequency far above any other system-inherent timescale, can be used to renormalize \nthe optical potential. The ^87Rb atoms experience an average position dependent force, which becomes controllable over the amplitude of the \napplied driving.
We investigate the control of interacting matter through strong coupling to a single electromagnetic mode, such as the photon mode in a Fabry-Perot or split-ring cavity. For this purpose, we analyze the exact effective theory for the collective light-matter hybrid modes of a generic system of NNN transition dipoles within an interacting solid. The approach allows to predict properties of the coupled light-matter system from the nonlinear response functions of the uncoupled matter ``outside the cavity''. The limit of large NNN corresponds to a conventional macroscopic description based on the polarizability of matter. In this limit, the cavity does not affect the static ferroelectric response. Corrections, which are needed to understand finite size systems and to obtain the nonlinear light-matter response, can be obtained from the non-linear susceptibilities of the matter outside the cavity. The theory is benchmarked for the Dicke model, and for a quantum Ising model which serves as a minimal mean-field model for a quantum paraelectric material like SrTiO3.
Multi-normal-mode splitting peaks are experimentally observed in a system with Doppler-broadened two-level atoms inside a relatively long optical cavity. In this system, the atom-cavity interaction can reach the ``superstrong-coupling'' condition with atom-cavity coupling strength gNg\sqrt{N}gN​ to be near or larger than the cavity free-spectral range ${\ensuremath{\Delta}}_{\text{FSR}}$. In such case, normal-mode splitting can occur in many cavity longitudinal modes to generate the multi-normal-mode splitting peaks, which can be well explained by the linear-dispersion enhancement due to the largely increased atomic density in the cavity. Many new interesting phenomena might come out of this superstrong atom-cavity coupling regime.