Unconventional cavity optomechanics: Nonlinear control of phonons in the acoustic quantum vacuum

Xin Wang, Wei Qin, Adam Miranowicz, Salvatore Savasta, and Franco Nori

Phys. Rev. A 100, 063827 – Published 16 December 2019

Abstract

We study unconventional cavity optomechanics and the acoustic analog of radiation pressure to show the possibility of nonlinear coherent control of phonons in the acoustic quantum vacuum. Specifically, we study systems where a quantized optical field effectively modifies the frequency of an acoustic resonator. We present a general method to enhance such a nonlinear interaction by employing an intermediate qubit. Compared to conventional optomechanical systems, the roles of mechanical and optical resonators are interchanged, and the boundary condition of the phonon resonator can be modulated with an ultrahigh optical frequency. These differences allow to test some quantum effects with parameters which are far beyond the reach of conventional cavity optomechanics. Based on this interaction form, we show that various nonclassical quantum effects can be realized. Examples include an effective method for modulating the resonance frequency of a phonon resonator (e.g., a surface-acoustic-wave resonator), demonstrating mechanical parametric amplification, and the dynamical Casimir effect of phonons originating from the acoustic quantum vacuum. Our results demonstrate that unconventional optomechanics offers a versatile hybrid platform for quantum engineering of nonclassical phonon states in quantum acoustodynamics.

Received 26 February 2019

DOI:https://doi.org/10.1103/PhysRevA.100.063827

Physics Subject Headings (PhySH)

Casimir effect Optomechanics

Quantum Information, Science & TechnologyGeneral Physics

Authors & Affiliations

Xin Wang ^1,2, Wei Qin ², Adam Miranowicz ^2,3, Salvatore Savasta ^2,4, and Franco Nori ^2,5

¹Institute of Quantum Optics and Quantum Information, School of Science, Xi'an Jiaotong University, Xi'an 710049, China
²Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
³Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland
⁴Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina, I-98166 Messina, Italy
⁵Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

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Vol. 100, Iss. 6 — December 2019

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Images

Figure 1
(a) Diagrammatic sketch of an unconventional optomechanical (UOM) system: A localized mechanical oscillator (with a massive particle) is placed inside an optical cavity, and its spring constant is linearly modulated by the cavity field $E (x, t)$ . (b) Schematic diagram and (c) lumped-circuit layout describing an unconventional optomechanical (UOM) system based on a surface-acoustic-wave (SAW) resonator: A charge qubit, with two Josephson junctions (red bars) is placed into the SAW resonator (confined by two Bragg mirrors). Their interaction is mediated via two identical inter-digitated-transducers (IDTs) of capacitance $C_{q}$ via the piezoelectric effects. A transmission-line resonator (TLR) longitudinally couples to the charge qubit via the mutual inductance $M$ . The UOM Hamiltonian in Eq. (5) can be mapped as the field current operator $\hat{I} (t)$ effectively changes the SAW resonator boundary condition. One Bragg mirror acts as a fast “oscillating” mirror.
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Figure 2
(a) Mechanical frequency shift $({\bar{ω}}_{m} - ω_{m})$ versus the field operator average $〈 ξ 〉 = 〈 a^{†} + a 〉$ . The solid (asterisk) curve corresponds to the analytical (numerical) results. In numerical calculations, the shifted mechanical frequency is defined as the difference between the first and second eigenvalues in the subspace of the qubit ground state $| g 〉$ . The mechanical frequency can be suddenly (periodically) modulated via a step longitudinal bias (an oscillating electromagnetic field) along the green vertical arrow (the blue loop). (b) The Rabi oscillations between the states $| g, 0, 2 〉$ and $| g, 1, 0 〉$ . The solid curves (symbols) represent the evolution described by the original Hamiltonian $H_{T}$ (the reduced Hamiltonian $H_{Q}$ ). $P_{e}$ is the probability of finding the qubit in its excited state.
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Figure 3
The mechanical gain rate $G$ of the input signals changing with the relative phase $ϕ$ for MPA. The analytical results correspond to Eq. (30). We set $α / (2 π) = 0.045 MHz$ . The qubit and phonon decay rates are $Γ / (2 π) = 0.05 MHz$ and $κ / (2 π) = 0.2 MHz$ , respectively.
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Figure 4
(a) Phonon-output rate $P_{out}$ , and correlation function $g_{2} (0)$ , as functions of the TLR drive strength $ε$ . The dashed line is plotted at $ε / (2 π) = 0.05 MHz$ , which is set in plots (b) and (c). (b) Normalized correlation function $g_{2} (τ) / g_{2} (0)$ changing with the delay time $κ τ$ . (c) Phonon output spectrum density $S (ω)$ , for different detuning cases. The dotted line position corresponds to $ω = ω_{c} / 2$ . Here the decay rates are $Γ / (2 π) = 0.05 MHz$ , $κ / (2 π) = 0.2 MHz$ , and $γ / (2 π) = 0.1 MHz$ .
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Figure 5
(a) Signal-to-noise-ratio (SNR) and second-order correlation function $g_{2} (0)$ versus the thermal phonon number $n_{th}$ . The gray area corresponds to $SNR < 1$ . (b) The emitted phonon spectrum $log [S (ω)]$ changes with the modulating frequency $Ω$ . A bimodal spectrum structure distributes along the dashed line (which corresponds to $ω = Ω / 2$ ). Here we set the coherent drive amplitude on the qubit as $Ω_{d} / (2 π) = 100 MHz$ . Other parameters are the same as those in Fig. 4.
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