Coherent response of inhomogeneously broadened and spatially localized emitter ensembles in waveguide QED

L. Ruks, X. Xu, R. Ohta, W. J. Munro, and V. M. Bastidas

Phys. Rev. A 109, 023706 – Published 9 February 2024

Abstract

Spectrally and spatially varying ensembles of emitters embedded into waveguides are ever-present in both well-established and emerging technologies. If control of collective excitations can be attained, a plethora of coherent quantum dynamics and applications may be realized on-chip in the scalable paradigm of waveguide quantum electrodynamics (WQED). Here, we investigate inhomogeneously broadened ensembles embedded with subwavelength spatial extent into waveguides employed as single effective and coherent emitters. We develop a method permitting the approximate analysis and simulation of such mesoscopic systems featuring many emitters, and show how collective resonances are observable within the waveguide transmission spectrum once their linewidth exceeds the inhomogeneous line. In particular, this allows for near-unity and tailorable non-Lorentzian extinction of waveguide photons overcoming large inhomogeneous broadening present in current state-of-the-art implementations. As a particular illustration possible in such existing experiments, we consider the classic emulation of the cavity QED (CQED) paradigm here using ensembles of rare-earth ions as coherent mirrors and qubits and demonstrate the possibility of strong coupling given existing restrictions on inhomogeneous broadening and ensemble spatial extent. This work introduces coherent ensemble dynamics in the solid state to WQED and extends the realm to spectrally tailorable emitters.

Received 13 September 2023
Revised 16 January 2024
Accepted 22 January 2024

DOI:https://doi.org/10.1103/PhysRevA.109.023706

Physics Subject Headings (PhySH)

Collective effects in atomic physics Effects of atomic coherence on light propagation Light-matter interaction

Atomic ensemble Rare-earth doped crystals Waveguides

Dipole approximation Two-level models

Atomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

L. Ruks^1,2,*, X. Xu ¹, R. Ohta¹, W. J. Munro², and V. M. Bastidas^1,2

¹NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan
²NTT Research Center for Theoretical Quantum Physics, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan

^*Lewis.ruks@ntt.com

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Vol. 109, Iss. 2 — February 2024

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Images

Figure 1
Binning and formation of a single effective emitter (large blue circle, bottom) for a localized ensemble of emitters (small red circles, top).
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Figure 2
Single-ensemble transmission. (a) Transmission through an ensemble featuring no positional inhomogeneity as the collective waveguide coupling efficiency $N Γ_{1D} / γ_{inh}$ is increased for each of the three prototypical spectral distributions. We take $Γ^{'} = 10^{- 6} γ_{inh}$ . (b) Transmission profile for an ensemble of $N = 10^{9}$ emitters with $γ_{inh} / (2 π)$ = 50 GHz, $Γ_{1D} / (2 π) = Γ^{'} / (2 π) = 100 Hz$ . We consider $m = 10^{3}$ positional bins in Eq. (4) for one realization of $z_{p} \sim U (0, δ z)$ .
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Figure 3
Strong coupling in the emitter-based cavity system. (a) Development of the peak splitting for side-illumination with increasing emitter number $N_{C} = 2 N_{Q}$ in the cavity ensembles. We take $γ_{inh} / (2 π) = 10 GHz, Γ_{1D} / (2 π) = Γ^{'} / (2 π) = 100 Hz$ , and $δ z = 0$ within each ensemble. (b) Transmission spectrum for the side-illumination scheme [27]. We consider $m = 10^{3}$ positional bins for each spatially localized ensemble and show for one realization of positions uniformly distributed over a width $δ z = 0.1 λ$ within each ensemble. Parameters are $γ_{inh} / (2 π) = 10 GHz, 2 N_{Q} = N_{C} = 4 \times 10^{8} .$ The shaded regions give the bounds of transmission obtained over 100 realizations of positions. The qubit ensemble is additionally detuned (in practice using, e.g., surface acoustic waves [3, 49]) to counter the mirror-qubit detuning that arises according to according to Eq. (8). (c) Rabi oscillations of the qubit population $P = | B_{Q}^{-} |^{2}$ with $m = 10^{3}$ positional bins sampled from uniform distributions of width $δ z$ . A single realization of positions is chosen in each case. Parameters are identical to (b), except $2 N_{Q} = N_{C} = 1 \times 10^{9} .$
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Figure 4
Transmission through an ensemble of size $N = 1000$ using the exact expression [40] (blue line), and using the binning procedure with $m = 50$ (orange dots) for $Γ^{'} = Γ_{1D}$ and varying $γ_{inh}, δ z$ . For the exact calculation we sample each emitter position $z_{j} \sim U (0, δ z)$ and each detuning $Δ_{j} \sim N (0, γ_{inh} / \sqrt{ln 2})$ from the Gaussian distribution N giving a FWHM $γ_{inh} .$ For the approximation, the binning procedure is applied according to the main text wiith $z_{p} \sim U (0, δ z) .$
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Figure 5
(a) First eight wave vectors $k_{μ}$ of the spin-wave eigenmodes calculated using Eq. (D5) for spatially extended ensembles. The colorbar denotes decay rate of the collective mode [calculated using Eq. (D4)] normalized to the collective decay rate of a point-like ensemble. The dashed line denotes $k_{μ} = ν$ , and highlights that the eigenmode with maximal decay rate has a wave vector approximately lying on this line (i.e., the eigenmode wavelength is approximately the length of the ensemble sample). For large $ν ≫ 1$ the eigenmode with maximal decay observes a local maximum of its decay rate at $k_{μ} \sim ν \sim (n + 1 / 2) π .$ However, this maximum is decreasing with increasing $ν .$ (b) Normalized eigenvalues in the complex plane as a function of $ν$ . Away from $ν \approx 0$ , there is, in general, more than one eigenmode with appreciable (normalized) decay rate.
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