Single quantum emitter Dicke enhancement

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Single quantum emitter Dicke enhancement

Tommaso Tufarelli, Daniel Friedrich, Heiko Groß, Joachim Hamm, Ortwin Hess, and Bert Hecht

Phys. Rev. Research 3, 033103 – Published 29 July 2021

Abstract

Coupling $N$ identical emitters to the same field mode is a well-established method to enhance light-matter interaction. However, the resulting $\sqrt{N}$ boost of the coupling strength comes at the cost of a “linearized” (effectively semiclassical) dynamics. Here, we instead demonstrate a new approach for enhancing the coupling constant of a single quantum emitter, while retaining the nonlinear character of the light-matter interaction. We consider a single quantum emitter with $N$ nearly degenerate transitions that are collectively coupled to the same field mode. We show that in such conditions an effective Jaynes-Cummings model emerges with a boosted coupling constant of order $\sqrt{N}$ . The validity and consequences of our general conclusions are analytically demonstrated for the instructive case $N = 2$ . We further observe that our system can closely match the spectral line shapes and photon autocorrelation functions typical of Jaynes-Cummings physics, proving that quantum optical nonlinearities are retained. Our findings match up very well with recent broadband plasmonic nanoresonator strong-coupling experiments and will, therefore, facilitate the control and detection of single-photon nonlinearities at ambient conditions.

Received 29 October 2020
Revised 25 March 2021
Accepted 2 June 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.033103

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Collective effects in quantum optics Quantum optics with artificial atoms Superradiance & subradiance

Atomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Tommaso Tufarelli ^1,*, Daniel Friedrich², Heiko Groß², Joachim Hamm³, Ortwin Hess ^3,4,†, and Bert Hecht ^2,‡

¹School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
²Nano-Optics and Biophotonics Group, Experimentelle Physik 5, Physikalisches Institut, Universität Würzburg, D-97074 Würzburg, Germany
³The Blackett Laboratory, Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom
⁴School of Physics and CRANN Institute, Trinity College Dublin, Dublin 2, Ireland

^*tommaso.tufarelli@nottingham.ac.uk
^†ortwin.hess@tcd.ie
^‡hecht@physik.uni-wuerzburg.de

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Vol. 3, Iss. 3 — July - September 2021

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Figure 1
Visual summary of the light-matter interaction models treated here. For multilevel, TC, and Dicke models, only energy levels that have a direct correspondence in the Jaynes-Cummings model (JCM) are shown. For the TC model, the limit $N ≫ 1$ is considered. Left: Standard Jaynes-Cummings model in which a single two-level emitter couples resonantly to a single-mode field with strength $g$ . The resulting anharmonic spectrum is a cornerstone of modern quantum optics but challenging to probe experimentally: The required regime $g ≫ κ$ is difficult to achieve, where $κ$ is the cavity decay rate. Center: Multilevel model, the focus of this paper. A quasidegenerate multilevel emitter with $N$ closely spaced excited states interacts with a single-mode field. When all the transitions are approximately resonant with the field, an effective coupling constant $g_{eff}$ , enhanced by a factor of order $\sqrt{N}$ , is established. In principle, the resulting low-energy spectrum can closely approximate the JC spectrum for any value of $N$ . Right: Tavis-Cummings and Dicke models in which $N ≫ 1$ nearly identical two-level systems interact with the same field. A boosted effective coupling constant $g_{eff} \propto \sqrt{N}$ is established also in this case, but the spectrum becomes approximately harmonic as $N$ is increased.
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Figure 2
Illustration of the transformation between the bare emitter basis and the radiation basis, for $N = 2$ sublevels, and the resulting appearance of a boosted effective coupling. The same principles hold for larger values of $N$ . (a) In the bare emitter basis the two levels $| e_{1, 2} 〉$ are not directly coupled to each other, whereas $ɛ$ plays the role of a relative detuning between them. Both $| G 〉 \leftrightarrow | e_{1, 2} 〉$ transitions couple to the cavity field, and we are assuming for simplicity that the associated coupling constants are the same: $g_{1} = g_{2} = g$ . (b) In the radiation basis, the bright- and dark-states $| B 〉, | D 〉$ have the same frequency but are coupled to each other with strength $ɛ / 2$ . The transition $| G 〉 \leftrightarrow | B 〉$ is coupled to the field with enhanced strength $g_{eff} = \sqrt{2} g$ , whereas the dark-state $| D 〉$ does not directly interact with the field.
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Figure 3
Schematic of the investigated setup. The system may be excited through an incoherent pump of intensity $Λ$ applied to the emitter or via coherent driving of the (plasmonic) cavity with field amplitude $Ω$ and frequency $ω_{L}$ . The bright state of the emitter is coupled to the cavity with the effective strength $g_{eff}$ , and the emitter sublevels are spread in frequency over a range $ɛ$ . $κ$ is the loss rate of the cavity, and $γ$ is the decay rate of all the emitter sublevels, which are also subject to dephasing at rate $γ_{d}$ (not shown). All the observables discussed in this paper can be measured by collecting the cavity output field.
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Figure 4
Eigenvalues of $H_{1}$ as a function of the average detuning $Δ$ for $ɛ = 0.25 g_{eff}$ (blue continuous lines). The anticrossing behavior of eigenvalues $E_{1, \pm}$ around $Δ = 0$ approximates well (the maximum relative error in the plotted range being $\sim 1.3 %$ ) the reference JCM with coupling $g_{eff}$ (red circles). The energy $E_{1, D}$ of the quasidark state is instead approximately linear in the detuning and sits in between the JCM-like energies. An elementary perturbation theory calculation indeed yields $E_{n, D} ≃ Δ (1 - ɛ^{2} / 4 g_{eff}^{2} n)$ , which agrees well with the exact value in the considered regime (relative error below 0.2%). Qualitatively similar results are obtained for higher excitation numbers $n \geq 2$ , and, in fact, it is easy to see that the relative weight of $ɛ$ and $Δ$ decreases with $n$ .
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Figure 5
Comparison of cavity emission spectra under incoherent pumping ( $Λ \neq 0, Ω = 0$ ). Spectra $S_{a} (ω)$ are plotted for the multilevel model with $N = 3, ɛ = 0.05 g_{eff}$ (solid blue), as well as reference JC (dashed red) and TC models (dot-dashed green). Plots (a)–(c) are in the single-excitation regime with $κ = g_{eff}$ and $Λ = 0.01 κ$ . Plots (d)–(f) are in the biexcitation regime with $κ = 0.05 g_{eff}$ and $Λ = 0.02 κ$ . Dephasing increases from left to right: (a) $γ_{d} = 0,$ (b) $γ_{d} = 0.1 κ,$ (c) $γ_{d} = κ$ , (d) $γ_{d} = 0,$ (e) $γ_{d} = 0.2 κ$ , and (f) $γ_{d} = κ$ . Spectra for the TC and JC models are rescaled by a factor $(1 - p_{dark})$ . Gray vertical lines indicate the resonances corresponding to the first (dot dashed) and second rung (dotted) of the JC ladder. See the Appendix for further details on the TC and JC models employed here.
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Figure 6
Steady-state photon autocorrelation, $g^{(2)} (0)$ , versus driving field detuning. Results are plotted for the multilevel model with $N = 3, ɛ = 0.05 g_{eff}$ (solid blue), reference JC (dashed red), TC (dot-dashed green), and coupled-oscillator models (gray dotted). Driving strengths are set to $Λ = 0, Ω = 0.01 g_{eff}$ in all cases. Plots (a)–(c) are in the single-excitation regime with $κ = g_{eff}$ . Plots (d)–(f) are in the biexcitation regime with $κ = 0.05 g_{eff}$ . Dephasing increases from left to right as in Fig. 5: Dephasing values are (a) $γ_{d} = 0,$ (b) $γ_{d} = 0.1 κ,$ (c) $γ_{d} = κ$ , (d) $γ_{d} = 0,$ (e) $γ_{d} = 0.2 κ$ , and (f) $γ_{d} = κ$ . See the Appendix for further details on JC, TC, and coupled-oscillator models.
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