Optical response of atom chains beyond the limit of low light intensity: The validity of the linear classical oscillator model

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Optical response of atom chains beyond the limit of low light intensity: The validity of the linear classical oscillator model

L. A. Williamson and J. Ruostekoski

Phys. Rev. Research 2, 023273 – Published 3 June 2020

Abstract

Atoms subject to weak coherent incident light can be treated as coupled classical linear oscillators, supporting subradiant and superradiant collective excitation eigenmodes. We identify the limits of validity of this linear classical oscillator model at increasing intensities of the drive by solving the quantum many-body master equation for coherent and incoherent scattering from a chain of trapped atoms. We show that deviations from the linear classical oscillator model depend sensitively on the resonance linewidths $υ_{α}$ of the collective eigenmodes excited by light, with the intensity at which substantial deviation occurs scaling as a power law of $υ_{α}$ . The linear classical oscillator model then becomes inaccurate at much lower intensities for subradiant collective excitations than superradiant ones, with an example system of seven atoms resulting in critical incident light intensities differing by a factor of 30 between the two cases. By individually exciting eigenmodes, we find that this critical intensity has a $υ_{α}^{2.5}$ scaling for narrower resonances and more strongly interacting systems, while it approaches a $υ_{α}^{3}$ scaling for broader resonances and when the dipole-dipole interactions are reduced. The $υ_{α}^{3}$ scaling also corresponds to the semiclassical result whereby quantum fluctuations between the atoms have been neglected. We study both the case of perfectly mode-matched drives and the case of standing-wave drives, with significant differences between the two cases appearing only at very subradiant modes and positions of Fano resonances.

Received 4 February 2020
Accepted 30 April 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.023273

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)

Collective effects in quantum optics Long-range interactions Quantum description of light-matter interaction

Atomic, Molecular & Optical

Authors & Affiliations

L. A. Williamson and J. Ruostekoski

Physics Department, Lancaster University, Lancaster LA1 4YB, United Kingdom

Article Text

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Vol. 2, Iss. 2 — June - August 2020

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Figure 1
System setup. A coherent field is incident on a chain of two-level atoms oriented along $\hat{z}$ with spacing $a < λ$ . A detector completely enclosing the atoms collects all of the scattered light $E_{s}^{+}$ , to give a photon scattering rate $n$ . Analyzing the coherent $n_{C}$ and incoherent $n_{I}$ photon scattering rates allows a systematic study of the limits of validity of the linear classical oscillator model, as a function of incident intensity $I_{in}$ .
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Figure 2
Deviation between the full quantum scattering and scattering predicted from the linear classical oscillator model. Parameters are $N = 7$ , $a = 0.4 λ$ , and $\hat{e} = (\hat{x} - i \hat{y}) / \sqrt{2}$ . The separate curves correspond to drive fields that are mode matched and resonant with different low light intensity collective modes $u_{α}$ (dots are numerical simulations, red curves are spline fits). In (a) and (b), the corresponding linewidths are, from shallowest curve to steepest curve, $υ_{α} = (1.41, 1.29, 1.15, 0.97, 1.03, 0.96) γ$ (main figures) and $υ_{α} = 0.18 γ$ (insets). (a) The relative deviation between the coherent scattering rate obtained from the linear classical oscillator model ( $n_{C}^{lin}$ ) and the full quantum solution ( $n_{C}$ ) grows proportional to $I_{in}$ . In all but one case, the slope of the curves depends inversely on the collective mode linewidth. The semiclassical model (dotted black curve) for a drive mode-matched and resonant with the most superradiant mode (main figure) and most subradiant mode (inset) captures the qualitative behavior of the scattering but is not quantitatively accurate. (b) The ratio of incoherent ( $n_{I}$ ) to coherent scattering displays the same behavior as $(n_{C}^{lin} - n_{C}) / n_{C}$ . The results for a single isolated atom, obtained from Eqs. (17), are shown by the gray dashed lines in (a) and (b) for comparison.
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Figure 3
Driving the atom chain with a field that is mode-matched and resonant with a low light intensity collective eigenmode $u_{α}$ results in a critical intensity $I_{C}$ that depends strongly on the collective mode linewidth $υ_{α}$ . The atom chains used are (a) $a = 0.4 λ$ , $\hat{e} = (\hat{x} - i \hat{y}) / \sqrt{2}$ , (b) $a = 0.4 λ$ , $\hat{e} = (\hat{z} - i \hat{x}) / \sqrt{2}$ , and (c) $a = 0.6 λ$ , $\hat{e} = (\hat{x} - i \hat{y}) / \sqrt{2}$ . In each case, we plot points for all collective modes of all atom numbers from $N = 2$ to $N = 10$ . A power law fit $I_{C} \propto υ_{α}^{2.5}$ (solid curves) describes well the data in (a), as well as modes $υ_{α} ≲ 0.5 γ$ in (b) (insets, with log-log scale). A power-law fit to $I_{C}$ calculated from the semiclassical model Eqs. (22) (dotted curves) gives good agreement for drives mode-matched to collective modes with $υ_{α} ≳ 0.5 γ$ in (b) and (c). In each figure, the grey dashed horizontal lines give the value of $I_{C}$ for a single isolated atom.
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Figure 4
(a), (b) The collective radiative linewidths $υ_{α}$ of an infinite chain of atoms for $\hat{e} = (\hat{x} - i \hat{y}) / \sqrt{2}$ for (a) $a = 0.4 λ$ and (b) $a = 0.6 λ$ . The spectrum drops abruptly to zero outside the light line, $K_{μ} > 2 π / λ$ . The linewidths of each mode of chains with ten atoms are shown for comparison (blue circles), at a wave vector (chosen to be positive) of the standing wave with maximum overlap with the mode. (c), (d) The critical intensity $I_{C}$ as a function of atom number, for drives mode-matched and resonant with each of the $N$ low light intensity collective modes $u_{α}$ (black dots). Results for lattice spacings (c) $a = 0.4 λ$ and (d) $a = 0.6 λ$ , for $\hat{e} = (\hat{x} - i \hat{y}) / \sqrt{2}$ . The solid curves join the collective modes of different atom numbers that have maximum overlap with the same $K_{μ}$ , for $K_{μ} = π / a$ (blue curve), $K_{μ} = 0$ (red curve), and $K_{μ} = 2 π / (3 a)$ for a cosine mode (green curve) and a sine mode (purple curve). The $K_{N / 2}$ mode in (c) lies outside the light line and the corresponding $I_{C}$ decreases exponentially with increasing $N$ (inset, with log vertical scale). The single atom value of $I_{C} / I_{s}$ is indicated by a thick horizontal marker on the vertical axis.
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Figure 5
Occupation measure $L_{β}$ of each of the collective modes $β$ with the steady-state polarization obtained in the limit of low light intensity, for a standing-wave drive targeting mode $α$ . The linewidth of the targeted modes are indicated in the figures, with occupations given by the unfilled red bar. The filled blue bars give the occupations of the remaining modes. In cases (a)–(c), the overlap is dominated by the targeted mode, even in (a) where the targeted mode lies outside the light line. An anomaly occurs in (d) where the most subradiant mode dominates the occupation despite the drive targeting a superradiant mode. This is due to a Fano resonance between the two modes, as discussed further in the main text.
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Figure 6
(a), (b) The critical intensity $I_{C}$ for both the optimized standing-wave drives (blue circles) and the perfectly mode-matched drives (red diamonds), as a function of the collective mode linewidth of the targeted mode. The optimized standing-wave drive agrees well for the perfectly mode-matched drive for most cases. Solid curves show the $υ_{α}^{2.5}$ scaling from Fig. 3. (a) $a = 0.4 λ$ . The most subradiant collective mode lies outside the light line, for which the standing-wave drive gives a substantially higher $I_{C}$ than the perfectly mode-matched drive. At $υ_{α} = 1.27 γ$ the standing-wave drive gives an anomalously low $I_{C}$ (point indicated by an arrow). (b) $a = 0.6 λ$ . All the collective modes lie inside the light line, and the standing-wave drive and perfectly mode-matched drive give comparable $I_{C}$ for all $υ_{α}$ . (c) Deviation between the full quantum scattering and scattering predicted by the linear classical oscillator model for the $a = 0.4 λ$ atom chain driven by a standing-wave drive overlapping with the $υ_{α} = 1.24 γ$ mode (monotonic curve) and the $υ_{α} = 1.27 γ$ mode (nonmonotonic curve) (dots are numerical simulations, red curves are spline fits). A Fano resonance at $υ_{α} = 1.27 γ$ results in an anomalously high $I_{C}$ .
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