Collective atom-cavity coupling and nonlinear dynamics with atoms with multilevel ground states

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Collective atom-cavity coupling and nonlinear dynamics with atoms with multilevel ground states

Elmer Suarez, Federico Carollo, Igor Lesanovsky, Beatriz Olmos, Philippe W. Courteille, and Sebastian Slama

Phys. Rev. A 107, 023714 – Published 15 February 2023

Abstract

We investigate experimentally and theoretically the collective coupling between atoms with multilevel ground-state manifolds and an optical cavity mode. In our setup the cavity field optically pumps populations among the ground states. The ensuing dynamics can be conveniently described by means of an effective dynamical atom-cavity coupling strength that depends on the occupation of the individual states and their coupling strengths with the cavity mode. This leads to a dynamical backaction of the atomic populations on the atom-cavity coupling strength which results in a nonexponential relaxation dynamics. We experimentally observe this effect with laser-cooled ${}^{87}{Rb}$ atoms, for which we monitor the collective normal-mode splitting in real time. Our results show that the multilevel structure of electronic ground states can significantly alter the relaxation behavior in atom-cavity settings as compared to ensembles of two-level atoms.

Received 12 October 2022
Accepted 19 January 2023

DOI:https://doi.org/10.1103/PhysRevA.107.023714

Physics Subject Headings (PhySH)

Atoms, ions, & molecules in cavities Cavity quantum electrodynamics Collective effects in atomic physics Collective effects in quantum optics Optical pumping

Atomic ensemble Atoms Collective dynamics

Cavity resonators

Atomic, Molecular & OpticalNonlinear Dynamics

Authors & Affiliations

Elmer Suarez¹, Federico Carollo ², Igor Lesanovsky ^2,3, Beatriz Olmos ^2,3, Philippe W. Courteille⁴, and Sebastian Slama ¹

¹Center for Quantum Science and Physikalisches Institut, Eberhard-Karls Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
²Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
³School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
⁴Centro de Pesquisa em Óptica é Fotônica, Instituto de Física de São Carlos, Universidade de São Paulo, São Paulo, São Paulo 05508-220, Brazil

sebastian.slama@uni-tuebingen.de

Ground-state bistability of cold atoms in a cavity

B. Gábor, D. Nagy, A. Dombi, T. W. Clark, F. I. B. Williams, K. V. Adwaith, A. Vukics, and P. Domokos

Phys. Rev. A 107, 023713 (2023)

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Vol. 107, Iss. 2 — February 2023

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Images

Figure 1
Multilevel ground-state systems coupled to a single-mode cavity. (a) In our experiment, the $5 S_{1 / 2}, F = 2$ ground states are coupled to the $5 P_{3 / 2}, F^{'} = 3$ states by a linearly polarized cavity field driving $π$ transitions with Clebsch-Gordan coefficients $c_{m}$ . The measured atom-cavity spectrum indicates the detunings $δ_{p}$ used in the experiment (black open circles). (b) The model system describes a simpler situation with two ground states used for analytical analysis in Sec. 4. Here the cavity field drives $σ^{+}$ transitions. The closed red dots in (a) and (b) indicate the steady-state occupations and the red circle in (b) shows the initial occupation. (c) The sketch of the experiment shows a free-falling cloud of cold ${}^{87}{Rb}$ atoms inside the cavity with round-trip length $l$ , pumped with probe laser frequency $ω_{p}$ . The transmission is detected on an avalanche photo diode (APD).
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Figure 2
Effective coupling strength $g_{eff}^{2}$ in units of the coupling constant for various ground-state populations with cavity-driven $π$ transitions. The effective coupling strength of the steady state is larger than that of the state with equal populations.
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Figure 3
Measured dynamics. (a) Measured and simulated cavity transmission for probe detuning $δ_{p} = + 24 MHz$ , scaled to the empty cavity maximum of $P_{cav} = 2.4$ nW and an atom number of $N = 11 200$ . Two time intervals can be identified, where the signal is mainly dominated by (i) redistribution of atoms among the sublevels by pumping and (ii) loss of atoms from the cavity. (b) Dynamics of the collectively enhanced effective coupling strength corresponding to the simulation in (a). (c) Time evolution of the populations $P_{m}$ of the ground-state levels, corresponding to the simulation in (a). The inset shows the normal-mode spectrum at different times of the dynamics. The dashed vertical line indicates the detuning used in the measurement shown in (a).
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Figure 4
Detuning-dependent transmitted power. Measured and simulated cavity transmission as explained in Fig. 3 for different values of $δ_{p}$ . Measurements are averages over ten single tracks. Fitted atom numbers are $N_{- 24 MHz} = 10 600$ , $N_{- 15 MHz} = 9200$ , $N_{+ 15 MHz} = 10 500$ , and $N_{+ 24 MHz} = 11 200$ for the corresponding detunings $δ_{p}$ , consistent with the typical atoms number determined from the normal-mode splitting. The fitted intracavity power $P_{cav} = 2.4$ nW is equal for all curves.
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Figure 5
Nonlinear dynamics. (a) Parameters $α$ and $β$ appearing in the differential equation (18), calculated for strong coupling with $g_{0} \sqrt{N} / Γ = 10$ (red and blue solid lines) and weak coupling with $g_{0} \sqrt{N} / Γ = 0.01$ (black solid lines, on top of each other). In the simulation we set $Γ = κ$ and $Δ_{c} = Δ_{a}$ . The black dashed line is the sum $α + β$ and determines whether the initial dynamics for $P_{-} = 1$ is accelerated (for positive values) or slowed down (for negative values). (b) Simulated time dynamics of $P_{-} (t)$ for weak and strong coupling. In the weakly coupled case (black curve) the decay follows an exponential function. Strong coupling can lead to acceleration (cyan) or deceleration (pink) of this dynamics. The corresponding detunings of $Δ_{a} = 0$ , $Δ_{a} = g_{0} \sqrt{N}$ , and $Δ_{a} = 1.5 g_{0} \sqrt{N}$ have been chosen as indicated by the symbols in (a). The laser power in the different cases has been adjusted such that all curves start with the same slope.
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