$N$-Phonon Bundle Emission via the Stokes Process

$N$ -Phonon Bundle Emission via the Stokes Process

Qian Bin, Xin-You Lü, Fabrice P. Laussy, Franco Nori, and Ying Wu

Phys. Rev. Lett. 124, 053601 – Published 3 February 2020

Abstract

We demonstrate theoretically the bundle emission of $n$ strongly correlated phonons in an acoustic cavity QED system. The mechanism relies on Stokes resonances that generate super-Rabi oscillations between states with a large difference in their number of excitations, which, combined with dissipation, transfer coherently pure $n$ -phonon states outside of the cavity. This process works with close to perfect purity over a wide range of parameters and is tunable optically with well-resolved operation conditions. This broadens the realm of quantum phononics, with potential applications for on-chip quantum information processing, quantum metrology, and engineering of new types of quantum devices, such as optically heralded $n$ -phonon guns.

Received 2 August 2019
Accepted 8 January 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.053601

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Mechanical effects of light on material media Phonons Stimulated Raman scattering

Perturbative methods

Atomic, Molecular & OpticalParticles & Fields

Authors & Affiliations

Qian Bin¹, Xin-You Lü^1,*, Fabrice P. Laussy ^2,3, Franco Nori^4,5, and Ying Wu^1,†

¹School of Physics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China
²Faculty of Science and Engineering, University of Wolverhampton, Wulfruna Street, Wolverhampton WV1 1LY, United Kingdom
³Russian Quantum Center, Novaya 100, 143025 Skolkovo, Moscow Region, Russia
⁴Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
⁵Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

^*xinyoulu@hust.edu.cn
^†yingwu2@126.com

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Vol. 124, Iss. 5 — 7 February 2020

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Images

Figure 1
(a) Scheme of the model and of the Stokes resonances in the frequency domain. The Stokes resonance is realized when a laser pumps the $n$ th-order phonon sideband, which corresponds to (b) an ideal Raman process. For large electron-phonon coupling and pumping-laser intensity, the energy structure changes to that of the strong coupling regime $λ \sim ω_{b}$ (c) and of the strong driving (Mollow) regime $Ω \sim ω_{b}$ (d). Panel (d) shows the case of two-phonon resonance, where the states $| n, + ⟩$ and $| n + 2, - ⟩$ are degenerate. The dashed arrows in (d) represent the energy gap between two manifolds. Here, $x = (b + b^{†}) / 2$ is the position quadrature of the cavity mode, and $x_{ZPF}$ is the corresponding zero-point fluctuation.
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Figure 2
Super-Rabi oscillations as seen through the population dynamics $P_{j k} (t) = | ⟨ j, k | ψ (t) ⟩ |^{2}$ ( $j = n, \tilde{n}$ and $k = c, v, +, -$ ) with $n = 2$ , 3 corresponding to the main and inset parts, respectively. The solid lines and dots correspond to the exact numerical results and the analytical approximate solutions based on $Ω_{eff}^{(n)}$ [42], respectively. System parameters are (a) $λ / ω_{b} = 0.03$ , $Ω / ω_{b} = 0.003$ , (b) $λ / ω_{b} = 0.1$ , $Ω / ω_{b} = 0.003$ , (c) $λ / ω_{b} = 0.03$ , $Ω / ω_{b} = 0.8$ , corresponding to the regimes of Figs. 1, 1, and 1, respectively.
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Figure 3
(a) Equal-time $n$ th-order correlation functions $g^{(n)}$ as a function of $Δ / ω_{b}$ . (b)–(c) Correlation function $g^{(2)}$ for different $λ / ω_{b}$ (b) and $Ω / ω_{b}$ (c). The dashed lines indicate the $n$ -phonon resonances $Δ = - n ω_{b}$ , $Δ = Δ_{n} (λ)$ , and $Δ = Δ_{n} (Ω)$ . System parameters are (a) $λ / ω_{b} = 0.03$ , $Ω / ω_{b} = 0.003$ , (b) $Ω / ω_{b} = 0.003$ , (c) $λ / ω_{b} = 0.03$ , and $κ / ω_{b} = 0.002$ , $γ / ω_{b} = 0.0002$ , and $γ_{ϕ} / ω_{b} = 0.0004$ for panels (a)–(c).
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Figure 4
(a)–(c) Small fraction of a quantum trajectory in the regime of two-phonon emission. The inset bar graphs show the total number of single-, two- and $2 n$ -phonon ( $n = 2$ ) clicks over 25 quantum trajectories. (d)–(f) Full density matrices of the system at the times indicated with purple arrows in the trajectory, showing the cascade-emission process. System parameters are $λ / ω_{b} = 0.1$ , $Ω / ω_{b} = 0.2$ , $κ / ω_{b} = 0.002$ , $γ / ω_{b} = 0.0002$ , and $γ_{ϕ} / ω_{b} = 0.0004$ , satisfying the regime around the sweet point $κ \approx 10 Ω_{eff}^{(2)}$ [42].
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Figure 5
Purities of (a) two-phonon and (b) three-phonon emissions vs $λ / ω_{b}$ and $κ / ω_{b}$ when $Ω / ω_{b} = 0.2$ . (c) Vertical cuts of the purity along the arrows in (a) and (b). The solid and dashed lines correspond to $λ / ω_{b} = 0.06$ and $λ / ω_{b} = 0.1$ , respectively. (d) Horizontal cuts of the purity along the arrows in (a) and (b). Inset: Purities vs $Ω / ω_{b}$ when $λ / ω_{b} = 0.1$ and $κ / ω_{b} = 0.002$ . System parameters are $γ / ω_{b} = 0.0002$ and $γ_{ϕ} / ω_{b} = 0.0004$ .
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