Catastrophic transition between dynamical patterns in a phonon laser

Letter
Open Access

Catastrophic transition between dynamical patterns in a phonon laser

Qing Lin, Bing He, and Min Xiao

Phys. Rev. Research 3, L032018 – Published 26 July 2021

Abstract

The bifurcations or transitions of the classical nonlinear systems with finite degrees of freedom exhibit some dynamical behaviors that are similar to those in the phase transitions of many-body systems. A dynamical type of them in the form of a sudden transition of the amplitudes between two harmonic oscillations, which are stable under periodically driving force, is not so common to see. We show that a sudden transition of this type can be realized in a system of coupling a microresonator that supports a mechanical vibration to another microresonator, and the system was experimentally demonstrated as a phonon laser. If the pump laser power or the intercavity coupling strength is adjusted to a critical value, the cavity field patterns will suddenly change, together with a jump of the stabilized mechanical amplitude. Such transition can be applied to a precise measurement of the optomechanical coupling constant, which is hard to measure due to its proportion to a tiny zero-point mechanical fluctuation amplitude.

Received 5 January 2021
Accepted 12 July 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.L032018

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)

Bifurcations Coupled oscillators Nonlinear optics Optomechanics

Nonlinear DynamicsAtomic, Molecular & Optical

Authors & Affiliations

Qing Lin ¹, Bing He ^2,*, and Min Xiao ³

¹Fujian Key Laboratory of Light Propagation and Transformation, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
²Center for Quantum Optics and Quantum Information, Universidad Mayor, Camino La Pirámide 5750, Huechuraba, Chile
³Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA

^*bing.he@umayor.cl

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Issue

Vol. 3, Iss. 3 — July - September 2021

Subject Areas

Optics

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Images

Figure 1
(a) A system of coupled microresonators similar to some experimental phonon laser setups [49, 56, 57]. The mechanical motion $x_{m} (t)$ on $μ R_{1}$ stabilizes to an oscillation (an exaggerated view is given here). The coupling strength $J$ is adjustable by the gap distance between $μ R_{1}$ and $μ R_{2}$ . (b) The change of the stable cavity field patterns due to a small increase from $E = 1 837 165.781 145 κ$ to $E = 1 837 165.781 146 κ$ . The single-frequency pattern before the transition is due to the optical gain, which makes $I_{2}$ higher than $I_{1}$ on both sides of the transition. The simulations are performed with an initially static mechanical mode of $E_{m} (0) = 0$ . $κ t$ is the dimensionless time scale. (c) The corresponding effect on the evolution paths for the mechanical energy $E_{m} = \frac{1}{2} (X_{m}^{2} + P_{m}^{2})$ , a dimensionless quantity equivalent to a phonon number. The used system parameters: $g_{m} = 2 \times 10^{- 6} κ, ω_{m} = 10 κ, γ = 0.01 κ, g_{0} = 2 κ, I_{S} = 10^{10}, γ_{m} = 0.01 κ$ , and $J = 0.3 κ$ .
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Figure 2
(a) The evolved time-average mechanical energy under a fixed cavity coupling $J = 0.3 κ$ , for two different values of $E_{m} (0)$ . (b) The evolved time-average mechanical energy under the condition $g_{m} E = 10 κ^{2}$ , for the same values of $E_{m} (0)$ in (a). (c1) The distribution of the evolved time-average mechanical energy from $E_{m} (0) = 0$ . (c2) The changed distribution of the sudden transition boundary due to $E_{m} (0) = 10^{14}$ . The fixed parameters are the same as those in Fig. 1.
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Figure 3
(a1),(a2) The comparison of the critical slowing-down tendencies between those of a single-cavity setup with $J = 0$ and a two-coupled-cavity setup with $J = κ$ . The multimode pattern for $E_{m} (t)$ at $J = κ$ is due to the large optomechanical force that renders the small displacement $d$ in Eq. (4) time dependent. The evolutions are from a static initial condition and with the system constants in Fig. 1 except that $g_{0} = 0$ . (b1),(b2) The numerically obtained relations for determining the critical exponents.
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Figure 4
(a) The transition locations in terms of $g_{m}$ and $E_{c}$ . The equations of the three curves (from the left to the right): $g_{m} E_{c} = 1.5 κ^{2}, 2.0 κ^{2}$ , and $2.5 κ^{2}$ . The straight lines in the inset: $g_{m} E_{c} / κ^{2} = 12.1 J_{c} / κ + 0.094$ (pink), $g_{m} E_{c} / κ^{2} = 6.415 J_{c} / κ + 0.4156$ (green), and $g_{m} E_{c} / κ^{2} = 26.49 J_{c} / κ + 1.773$ (black). At low $J / κ$ , there are small deviations of such straight lines from the values of the product $g_{m} E_{c} / κ^{2}$ . These relations are obtained with $g_{0} = 0$ . (b1),(b2) An example of discriminating two close values of the constant $g_{m}$ . In a single cavity almost no difference exists in the field intensity $I_{c}$ , but their field patterns will separate simply by coupling to another microresonator as in (b2). To show the clearer oscillation patterns, we here include the optical gain among the used system parameters from Fig. 1.
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