Quantum optomechanics of a two-dimensional atomic array

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Quantum optomechanics of a two-dimensional atomic array

Ephraim Shahmoon, Mikhail D. Lukin, and Susanne F. Yelin

Phys. Rev. A 101, 063833 – Published 26 June 2020

Abstract

We demonstrate that a two-dimensional atomic array can be used as a platform for quantum optomechanics. Such arrays feature both nearly perfect reflectivity and ultralight mass, leading to significantly enhanced optomechanical phenomena. Considering the collective atom-array motion under continuous laser illumination, we study the nonlinear optical response of the array. We find that the spectrum of light scattered by the array develops multiple sidebands, corresponding to collective mechanical resonances, and exhibits nearly perfect quantum-noise squeezing. Possible extensions and applications for quantum nonlinear optomechanics are discussed.

Received 24 March 2020
Accepted 20 May 2020

DOI:https://doi.org/10.1103/PhysRevA.101.063833

Physics Subject Headings (PhySH)

Collective effects in quantum optics Light-matter interaction Mechanical effects of light on material media Optomechanics Quantum description of light-matter interaction Quantum optics Superradiance & subradiance

Atomic, Molecular & Optical

Authors & Affiliations

Ephraim Shahmoon^1,*, Mikhail D. Lukin¹, and Susanne F. Yelin^1,2

¹Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
²Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA

^*Present address: Department of Chemical & Biological Physics, Weizmann Institute of Science, Rehovot 761001, Israel.

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Vol. 101, Iss. 6 — June 2020

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Figure 1
Light scattering and optomechanics in an ordered 2D atomic array. (a) The atoms are spanning the $x y$ plane at equilibrium position $z = 0$ for all atoms, with interatomic spacing $a ≲ λ$ , $λ$ being the wavelength of the incident light. For nonsaturated atoms (linear response), and ignoring their motion, full reflection is observed ( $r = - 1$ , $t = 0$ ) when the frequency of the incident light matches the cooperative resonance of the array [28]. (b) With longitudinal, light-induced atomic motion ( $z_{n}$ for an atom $n$ ), a nonlinear component ( $E_{NL}$ ) is added to the reflected field, due to the optomechanical coupling.
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Figure 2
Collective mechanical response of the array. (a) Coupled mechanical oscillators model corresponding to Eq. (1): each atom is found in a potential with trap frequency $ν$ (thin black “springs”) and coupled via laser-induced dipolar interactions $K_{n m}$ to the rest of the atoms (thick blue “springs” connecting different atoms). (b) Eigenfrequencies $ν_{j}$ of the resulting mechanical collective normal modes $j = 1, ..., 196$ , for $N = 14^{2}$ atoms, lattice constant $a / λ = 0.2$ and detuning $δ_{L} - Δ = - (γ + Γ) / 4$ . (c, d) Spatial profile of the mechanical collective modes $j = 2$ (c) and $j = 196$ (d) from the example in panel (b). The incident beam, with waist $w_{0} / λ = 1.5$ , is smaller than the array; therefore the profile of the highest frequency mode $j = 196$ is highly oscillatory around the center of the array, where the beam intensity and the interactions it induces are strongest. Other physical parameters: incident beam strength at the center $Ω = 1.1 γ$ ( $Ω ≪ γ + Γ$ ), recoil energy $E_{R} / (ℏ γ) = 1 / 1620$ (corresponding to ${}^{87}{Rb}$ ), and Lamb-Dicke parameter $η = q x_{0} = 0.12$ (corresponding to, e.g., a potential depth $V = 1500 E_{R}$ , trap length $l = 400 nm$ , and wavelength $λ = 780 nm$ ).
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Figure 3
Comparison with cavity optomechanics. (a) Standard cavity optomechanics model, Eq. (4): a cavity mode $\hat{c}$ (resonant frequency $ω_{c}$ and width $κ$ ) is formed by two mirrors, one of which is movable (coordinate $\hat{z}$ ). (b) 2D atom array: dipole-dipole interactions between the atoms form a collective dipole of the array atoms ( $\hat{σ}$ ) with a resonant frequency shifted by $Δ$ and widened by $Γ$ from the single-atom resonance $ω_{a}$ and $γ$ . Longitudinal motion of an atom $n$ inside the trap is described by the coordinate ${\hat{z}}_{n}$ . The analogy between the systems is captured by the mapping from Table 1.
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Figure 4
Collective mechanical sidebands in reflected light. (a) Intensity spectrum of the normal spatial component of the reflected light [Eq. (9) for $k_{⊥} = 0$ , divided by the experiment time $L / c$ ]. The central sharp peak at the laser frequency ( $ω = 0$ ) is due to the strong linear reflection ${| r |}^{2}$ [with the $δ (ω)$ peak approximated here by a Gaussian of width $c / L \to α_{0} / 3$ ]. The incident light excites the collective mechanical modes from Fig. 2, which are revealed by the appearance of sidebands centered around their eigenfrequencies $ν_{j}$ (same parameters as in Fig. 2, yielding ${| r |}^{2} = 0.8$ ). (b) The $| k_{⊥} | > 0$ nonlinear components of the spectrum can be detected at the corresponding far-field angles. (c) Zoom-in of the sidebands for different detection angles: $k_{⊥} = 0$ [blue, same as Fig. 4], $k_{⊥} = (0.2 q, 0.2 q)$ (green). Sidebands at higher frequencies $ν_{j}$ are associated with mechanical mode profiles of higher spatial frequencies and therefore become more pronounced for larger detection angles $| k_{⊥} |$ .
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Figure 5
Quantum squeezing of the reflected field. (a) The correlation between spatiotemporal modes $(\pm k_{⊥}, \pm ω)$ can be observed by homodyne detection at both angles $\pm k_{⊥}$ . (b) Squeezing spectrum, Eq. (13), as a function of $ω$ for fixed $\pm k_{⊥} = 0$ ( $S < 1$ signifies quantum noise squeezing). The two dips at the corresponding mechanical resonance $\pm ν_{k_{⊥} = 0} = \pm ν$ exhibit very strong squeezing, as expected from Eq. (14) [e.g., reaching $S \sim 10^{- 3}$ (see panel d) for the specific parameters considered here: $a / λ = 0.5$ , $V / E_{R} = 1500$ , trap length $400 nm$ (yielding $η \approx 0.12$ ), $δ_{L} - Δ = - (γ + Γ) / 4$ , and uniform illumination with $Ω / γ = 0.2$ ]. (c) The bandwidth of the squeezing-spectrum dip is determined by the parameter $W_{k_{⊥}}$ and can be increased by increasing either $a$ or $Ω$ [Eqs. (14) and (15)]. This is demonstrated by the red (dashed) curve ( $a / λ$ increased to 0.55) and the green (dash-dot) curve ( $Ω / γ$ increased to 0.25), as compared to the blue curve [same as panel (b)]. (d) The dependence of the squeezing on $k_{⊥}$ (spatial squeezing) signifies the quantum correlations generated between the spatial modes $\pm k_{⊥}$ , reflecting the unique multimode and nonlocal optomechanical response of the array. Here this dependence is plotted as a function of $k_{x}$ for fixed $ω = ν - 12 α$ and $k_{y} = 0$ .
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Figure 6
Balanced scheme for the generation and detection of quantum optical squeezing beyond the nearly perfectly reflecting case. Laser drive is incident from both sides with equal magnitude $Ω$ and a phase difference $ϕ$ . The detected output field, ${\tilde{a}}_{k_{⊥} k}$ , is a superposition of the outputs fields from both sides, with an adjustable interference phase $φ$ .
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