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Strong Coupling Between Phonons and Optical Beating in Backward Brillouin Scattering Kien Phan Huy, Jean-Charles Beugnot, Joël-Cabrel Tchahame, Thibaut Sylvestre

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Strong Coupling Between Phonons and Optical Beating in Backward Brillouin Scattering Kien Phan Huy, Jean-Charles Beugnot, Joël-Cabrel Tchahame, Thibaut Sylvestre Strong coupling between phonons and optical beating in backward Brillouin scattering Kien Phan Huy, Jean-Charles Beugnot, Joël-Cabrel Tchahame, Thibaut Sylvestre To cite this version: Kien Phan Huy, Jean-Charles Beugnot, Joël-Cabrel Tchahame, Thibaut Sylvestre. Strong coupling between phonons and optical beating in backward Brillouin scattering. Physical Review A, American Physical Society, 2016, 94, pp.43847 - 43847. 10.1103/PhysRevA.94.043847. hal-01469831 HAL Id: hal-01469831 https://hal.archives-ouvertes.fr/hal-01469831 Submitted on 16 Feb 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PHYSICAL REVIEW A 94, 043847 (2016) Strong coupling between phonons and optical beating in backward Brillouin scattering Kien Phan Huy,* Jean-Charles Beugnot, Joel-Cabrel¨ Tchahame, and Thibaut Sylvestre Institut FEMTO-ST UMR 6174, Universite´ Bourgogne Franche-Comte,´ CNRS, Besanc¸on, France (Received 12 July 2016; published 25 October 2016) Brillouin scattering is a fundamental nonlinear interaction between two optical waves and an acoustic wave mediated by electrostriction and photoelasticity. In this paper, we revisit the usual theory of this inelastic scattering to get a joint system in which the acoustic wave is strongly coupled to the interference pattern between the optical waves. We show in particular that when the coupling rate exceeds the phonon damping rate, the system enters the strong-coupling regime, giving rise to anticrossing in the dispersion relation and Rabi-like splitting. We further find numerically that strong coupling can, in principle, be observed using backward Brillouin scattering in subwavelength-diameter optical waveguides. DOI: 10.1103/PhysRevA.94.043847 I. INTRODUCTION differs from other regimes because it is cavityless and does not rely on the optical and acoustic field variations. In our study, Cavity optomechanics, a branch of physics that studies the this is the phase difference between the pump and Stokes interaction of light with tiny mechanical objects, has recently waves that govern the strong coupling. When the proper phase drawn widespread interest because of key fundamental obser- conditions are fulfilled, strong coupling and Rabi splitting vations such as resolved-sideband cooling, optomechanically can be numerically observed in subwavelength waveguides induced transparency, quantum coherent coupling, and Rabi providing that the coupling rate exceeds the phonon damping oscillations [1–4]. Rabi oscillations were initially described rate. as damped periodic oscillations of an excited atom coupled The paper is organized as follows: First, we describe the to an electromagnetic cavity in which the atom alternately principle and methodology of the photon-phonon interaction emits and reabsorbs photons [5]. From a theoretical point under consideration. Then we derive from the standard theory of view, this remarkable and ubiquitous phenomenon can be of BS a joint two-level system, as commonly used in the Rabi readily described as a joint system of two coupled oscillators. If problem. Finally, we investigate from this system the specific both oscillators are uncoupled, they share the same degenerate conditions that allow the strong optoacoustic coupling regime eigenfrequency. When they are strongly coupled, however, to be achieved in small optical waveguides. the degeneracy is removed and the frequency is split into two distinguishable eigenfrequencies corresponding to the odd and even supermodes of the joint system. The frequency difference II. PRINCIPLE AND METHODOLOGY between the two eigenfrequencies is called the Rabi frequency Let us first describe the photon-phonon interaction under [5,6]. When the same effect occurs at the quantum level, consideration. When a coherent laser light is coupled and e.g., an atom and a photon embedded in a cavity, the new guided into a thin optical fiber, as shown schematically in eigensolutions are coherent superpositions of the two particles. Fig. 1, light generates and interacts with several types of When the two particles are not interacting with each other, they acoustic waves [13–15]. Here we assume only one acoustic have their own dispersion relations ki (ω) that may cross each longitudinal wave interacting with the guided light (this other. However, when the strong-coupling regime is reached, assumption will be discussed later on). As sketched in Fig. 1, the joint system dispersion relation displays an avoided BS is an inelastic scattering whereby two frequency-detuned crossing and Rabi splitting. Since its first observation, this optical pump (red right-pointing arrow) and Stokes (blue concept has recently been extended to cavity optomechanics left-pointing arrow) waves coherently interact in a dielectric where mechanical and optical modes of high-quality-factor waveguide, giving rise to an optical interference pattern resonators are strongly coupled [7,8], and to many other joint (purple) that generates an acoustic wave (green phonon arrow) systems including photons, phonons, excitons, and plasmons through the effect of electrostriction. Simultaneously, the [2,5,6,9–11]. photoelastic effect creates a moving index grating that travels In this work, we theoretically investigate strong coupling at the speed of hypersound (i.e., a few thousand of m s−1 in in backward Brillouin scattering (BS), which is a well-known silica and chalcogenide glasses). The index grating acts as a photon-phonon interaction involving two frequency-detuned moving Bragg mirror and reflects off the pump light with a counterpropagating optical waves and an acoustic wave. Doppler downshift matching the frequency detuning between Although similar ideas have recently been introduced by Van the two optical waves. When the frequency detuning is equal Laer et al. [12], our work specifically demonstrates that strong to the acoustic phonon frequency, phase matching is achieved coupling can be achieved between the acoustic phonon and and therefore one gets amplification of both the Stokes wave the optical beat note formed by the pump and Stokes waves. and the optical beat note. In the weak-coupling regime, both the The strong coupling investigated in this paper fundamentally optical beat note and acoustic wave grow along the propagation distance, while in the strong-coupling regime, they couple and alternately exchange energy over short periods of few tens of *[email protected] microns. 2469-9926/2016/94(4)/043847(11) 043847-1 ©2016 American Physical Society HUY, BEUGNOT, TCHAHAME, AND SYLVESTRE PHYSICAL REVIEW A 94, 043847 (2016) waveguide [16]. To this end, we solved the dispersion equation of longitudinal acoustic modes βa() in a silica fiber taper in order to select only one acoustic mode interacting with guided light. This dispersion relation reads [15] 2p q2 + β2 J (pa)J (qa) − q2 − β2 2J (pa)J (qa) a a 1 1 a 0 1 = 2 4βa pqJ1(pa)J0(qa), (1) = 2 − 2 = 2 − 2 with p 2 βa and q 2 βa , where VL and VT VL VT are the longitudinal and shear acoustic velocities, respectively, related to the longitudinal and shear components of the displacement. Ji are the Bessel functions. a is the fiber taper FIG. 1. Principle of strong coupling using backward Brillouin radius. We then do the same work for the optical modes, as scattering in an optical fiber. A strong pump wave (red arrow, pointing in Ref. [17]. The optical dispersion equation for a step index right) coherently interacts with a counterpropagating probe wave, fiber of core refractive index n1 and cladding index n2 is giving rise to an optical beat note which generates an acoustic J (γ a) n2 K (γ a) wave through electrostriction. Under strong coupling, the interference m 1 + 1 m 2 2 pattern and the phonon are coherently coupled and alternately γ1aJm(γ1a) n2 γ2aKm(γ2a) exchange energy over short periods of few tens of microns. Bottom m 1 1 1 n2 1 left: scheme of a two-level system with an exciton interacting with = + + 2 , (2) 2 2 2 2 2 2 a photon. Bottom right: scheme of the strong-coupling regime in a γ1 γ2 γ1 n1 γ2 Brillouin scattering. = 2 2 − 2 = 2 − 2 2 where γ1 k0n1 β , γ2 β k0n2, and Km denotes the modified Bessel function of the second kind, with the From a quantum point of view, a pump photon is annihilated prime denoting differentiation with respect to the argument. to create both a Stokes photon and an acoustic phonon, as k0 is the propagating wave vector in vacuum. For each depicted in the bottom left of Fig. 1. Similarly to the Rabi integer value m, the eigenvalue β is the effective propagation problem, the two pump and Stokes photons can be seen as constant along the fiber axis of the given mode. This leads the two levels of an artificial atom. In our case, however, the to the propagating constants of the pump (P ) and Stokes (S) two coupled objects are actually the acoustic phonon and the waves, βP (ω) and βS (ω). Then the phase-matching condition optical beat between the pump and Stokes waves. As a βP (ω + ) − βS (ω) = βa() sets the detuning frequency result, our joint system is fundamentally different from the at which Brillouin scattering occurs. conventional exciton-polariton picture. In the latter case, the Figure 2 typically shows a numerical simulation of the two-level system includes only one particle. It is described as acoustic wave spectrum generated in a silica fiber taper (see an exciton with a given resonant frequency, as sketched by Appendix A for chalcogenide), as a function of frequency the spring in the bottom left of Fig.
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