Dynamically induced robust phonon transport and chiral cooling in an optomechanical system

The transport of sound and heat, in the form of phonons, can be limited by disorder-induced scattering. In electronic and optical settings the introduction of chiral transport, in which carrier propagation exhibits parity asymmetry, can remove elastic backscattering and provides robustness against disorder. However, suppression of disorder-induced scattering has never been demonstrated in non-topological phononic systems. Here we experimentally demonstrate a path for achieving robust phonon transport in the presence of material disorder, by explicitly inducing chirality through parity-selective optomechanical coupling. We show that asymmetric optical pumping of a symmetric resonator enables a dramatic chiral cooling of clockwise and counterclockwise phonons, while simultaneously suppressing the hidden action of disorder. Surprisingly, this passive mechanism is also accompanied by a chiral reduction in heat load leading to optical cooling of the mechanics without added damping, an effect that has no optical analog. This technique can potentially improve upon the fundamental thermal limits of resonant mechanical sensors, which cannot be attained through sideband cooling.

symbol meaning c p,σ , c σ Two frequency-adjacent optical modes; the pump mode (c p,σ ) and the anti-Stokes mode (c σ ) c +(−) Annihilation operator of the optical mode in the cw(ccw) direction a +(−) Annihilation operator of the high-Q phonon mode in the cw(ccw) direction b k Annihilation operator of the phonon modes in the system λ k Optomechanical single-photon coupling strengths to the two optical modes; the c σ and c p,σ modes Λ k Pump-enhanced optomechanical coupling constant, Λ k |αλ k |. Optomechanical single-photon coupling strengths, h 2 0 + g 2 0 = 1 V 0 Coupling strength of disorder-induced scattering between the a ± modes to the b ∓ mode ω 1 Cavity resonance frequency of the pump optical mode ω 2 Cavity resonance frequency of the anti-Stokes optical mode ω L Pump laser frequency ω m Mechanical resonance frequency δ Detuning of the pump mode from the pump laser, δ = ω 1 − ω L ∆ Detuning of the anti-Stokes mode from the pump laser, ∆ = ω 2 − ω L ∆ 2 Detuning of the anti-Stokes mode from the scattered light, Intrinsic loss rate of the anti-Stokes optical mode κ ex Loss rate associated with the external coupling κ Measurable optical linewidth of the anti-Stokes mode, κ = κ 0 + κ ex κ p Measurable optical linewidth of the pump mode γ Mechanical damping rate of the high-Q phonon modes Γ Mechanical damping rate of the phonon quasi-modes n + = |α| 2 Intracavity photon number in the cw optical mode c + n − = |β| 2 Intracavity photon number in the ccw optical mode c − C α Cooperativity of the c + mode, C α = 4α 2 g 2 0 /Γ κ C β Cooperativity of the c − mode, C β = 4β 2 g 2 0 /Γ κ χ −1 a± (ω) Mechanical susceptibilities of the a ± modes T a± Bath temperatures of the a ± modes T eff a± Effective temperatures of the a ± modes n L Intracavity photon number in the pump optical mode driven by the pump laser n Effective phonon occupation number of the a − phonon mode S a− (ω) Quantum noise spectrum of a − mode, S a− (ω) =

Supplementary Note 1. Defining the modes and their coupling
Our system is composed of cw and ccw optical modes, high-Q phonon modes, as well as vibrational excitations inside the material, i.e. the phonon bath. As described in the main text, we focus on a scenario in which high-Q modes of clockwise and counterclockwise circulation with annihilation operators a σ are coupled via disorder to a quasi-mode (broad mode representing many actual mechanical modes) circulating in the opposite direction with annihilation operators bσ. In the experiment, for each circulation we have a pair of optical modes c p,σ and c σ . The optomechanical coupling allows for transfer of light from c p,σ to c σ with a corresponding annihilation of a phonon that is phase matched. This process overlaps with both the high-Q modes and the quasi-modes.
With the above basic picture, we examine this model using the rotating wave approximation (RWA) as the experimental configuration is all narrowband. We can then use the input-operator language to describe the open system dynamics. After displacing the optical cavity fields by the pump amplitudes in the cavity, c σ → √ n σ +c σ with √ n + = α, √ n − = β, the c p,σ fluctuations decouple from the rest of the system. Working in the frame rotating with the pump laser frequency, we write the linearized Heisenberg-Langevin equations for the mechanical and optical modes in the Fourier domain with Fourier frequency ω: where V 0 is the coupling strength of disorder-induced scattering between the cw(ccw) high-Q phonon modes a σ to the ccw(cw) phonon quasi-modes bσ. In contrast to the usual quantum optics literature, we here define detuning to be the mode frequency minus the signal frequency. Thus a positive detuning is red detuned. This makes comparison to mechanical motion as transparent as possible. Meanwhile, the coupling of the quasi-modes b σ to the c σ modes is given by g 0 √ n σ , while h 0 √ n σ captures the coupling between a σ and c σ . For any given ω, there is a self-consistent ω b ≈ ω that describes the relevant portion of the bath modes. Thus we take |ω b − ω| Γ in what follows. Specifically, we have the following main assumptions for this simple model: 1. Phonon backscattering occurs between high-Q phonon modes and the phonon quasimodes, i.e. between a + ←→ b − and a − ←→ b + , with strength V 0 .
2. The cw(ccw) optical mode c +(−) couples to the high-Q phonon mode a +(−) and the cw(ccw) phonon quasi-mode b +(−) with different strengths. The cw optical mode c + couples to the cw high-Q mode a + via direct optomechanical interaction with strength αh 0 and couples to the quasi-mode with strength αg 0 . Likewise, the ccw optical mode c − couples to the ccw high-Q mode a − with strength βh 0 and couples to the ccw quasi-mode b − with strength βg 0 . Here are n + and n − are the number of intracavity photon in the cw optical mode c + and the ccw mode c − , respectively.
3. The high-Q phonon modes a +(−) and the phonon quasi-modes b +(−) have the intrinsic damping rates γ and Γ , respectively (γ Γ ). The cw modes and ccw modes have symmetry with respect to the origin. We also assume that the damping rate Γ is in the same order as the optical loss rate κ.
We exclude a simpler model, of two degenerate mechanical modes and no additional quasimodes, as it fails to produce two key features of the data. First, at low pump power, we would experimentally observe some mode splitting, representing a breaking of circular symmetry from disorder-induced scattering. Second, at high pump power, the lowest linewidth the backward mode could achieve would be equivalent to its initial linewidth, and its temperature would be equal to the bath temperature. Optical coupling to multiple mechanical modes is the next best alternative, and as we show here, describes these phenomena.
Based on these assumptions, we can obtain the simplified continuum model as shown in Supplementary Fig. 1.a. In principle, the dynamics of the system can be solved numerically. However the loop structure in this coupled six-mode system will complicate the result, rendering interpretation difficult. To better capture the main physics, we can make the following approximation: we assume the g 0 parameter is larger than h 0 so that the optical field couples more strongly to the bulk modes b ± . We note that this assumption is not actually that important for our main result, as the crucial point is that the dominant mechanical damping mechanism is coupling of the high-Q modes to the quasi-modes -this is independent of g 0 and h 0 . We can then break the loop into two pieces (see Supplementary  Fig. 1.a-c).  Figure 1: (a) Full description of the optomechanical coupling of the c ± modes to a ± modes and b ± modes, and the disorder-induced scattering between the a ± modes and b ∓ modes. Separated optomechanical coupling descriptions by coupling directions (b)-(c). (b) The ccw-side coupling that the optical mode c − is mainly coupled to the a − mode with the disorder-induced scattering. (c) Likewise, the cw-side coupling that the cw mode c − is coupled to the a + mode.

SN2.2 Linewidths of the a ± modes
We can define the cooperativities as C α = 4α 2 g 2 0 /Γ κ and C β = 4β 2 g 2 0 /Γ κ, which are both dimensionless parameters describing the strength of optomechanical coupling relative to cavity decay rate and mechanical damping rate. Under the phase matching condition that the pump laser and its scattered light are near the two frequency-adjacent optical modes, we can expect ∆ ≈ ω m (see Supplementary Fig. 2). To evaluate the linewidth of the high-Q phonon modes, we set ω ≈ ω m . Then we have: where γ a + and γ a − are the linewidths of the high-Q phonon modes a ± . Note that the linewidths γ a ± are larger than their minimum measurable linewidth γ m = γ + 4V 2 0 Γ (obtained when optical power is zero, i.e. α = 0, β = 0) due to the disorder induced backscattering to the counter-propagating quasimode (see Eq. 2)

SN2.3 Effective temperature of the phonon modes
Another important feature that comes from the continuum model is the reduction in the effective temperature of the a − mode, because of coherent damping of the σ = + mechanical modes. When the right-hand side of the equation (2) is considered with an assumption that the optical noise c in σ is negligible compared to the thermal noise source, we have the effective noise on a − as: The effective temperature of mode a − is then given by: When near resonance, ω ≈ ∆, we have: It reveals that the second term in equation (8) decreases with increasing C α . This fact indicates the effective temperature of the ccw a − mode reduces with increase of the cw pump laser (∝ |α| 2 ). We can derive the effective temperature of the a + mode in the same manner.
In the experiment shown in the main paper, the ccw probe β is much smaller compared to the cw pump α, thus this effective temperature T eff a + change is not significant for the a + mode.

SN2.4 Analysis of the direct coupling model
For the direct coupling model, i.e. disorder only coupling a − and a + via V 1 and no additional quasi-modes, we can find striking differences from the observations (See Supplementary Fig. 3). After adiabatic elimination of c ± , we have Heisenberg-Langevin equations for V 0 = 0 oḟ with Γ α ≡ 4h 2 0 |α| 2 /κ the optically-induced damping. We see that the normal modes of these equations have resonance conditions corresponding to two poles: where is the average damping and δΓ = |Γ α − Γ β | is the difference in damping. Thus at zero power the two poles are split on the real axis by ±V 1 , leading to mode splitting which is not observed in the experiment. Furthermore, as δΓ increases to be larger than V 1 , the damping rates start to differ, whereas in the experiment the damping is different for all optical powers. Finally, at high δΓ , the imaginary (damping) part of the pole is still always ≥ γ, the value of the damping at zero optical power in this model, counter to the observed behavior in the experiment. Regarding the temperature of the a − mode, working in the large δΓ limit, we do see some cooling of a − at intermediate powers, as predicted in Supplementary Ref. [1].  Figure 3: The 'direct coupling model' consists of the optomechanical coupling of the c + optical mode to the a + high-Q mode, and direct disorder-induced coupling between the a ± modes. αh 0 is the light-enhanced optomechanical coupling strength and V 1 is the direct coupling rate.

Supplementary Note 3. Experimental details
In Supplementary Fig. 4, we illustrate the detailed experimental setup used to measure the spectra of the cw (ccw) high-Q phonon modes a ± . The experiment is performed using a silica microsphere resonator optical Q > 10 8 that is evanescently coupled to a tapered fiber waveguide. The scattered light that results from the opto-acoustic coupling is sent to the photodetector through the same waveguide that also carries the pump laser. A tunable External Cavity Diode Laser (ECDL) spanning 1520 -1570 nm drives light into the waveguide. A 90/10 fiber optic splitter separates this source into the cw pump and the ccw probe laser. In the cw direction, the pump laser is amplified by an Erbium-Doped Fiber Amplifier (EDFA). Thus, the EDFA affects the cw pump laser only, not the ccw probe laser.
In order to measure anti-Stokes light detuning from the optical mode, ∆ 2 = ω 2 − ω AS , we employ a Brillouin Scattering Induced Transparency measurement [2]. Here ω 2 is the resonant frequency of the anti-Stokes optical mode, and ω AS is the frequency of anti-Stokes scattered field via Brillouin scattering as illustrated in Supplementary Fig. 2. An electrooptic modulator (EOM) is used to generate the required probe sidebands relative to the cw pump laser. The upper sideband is used to probe the acousto-optic interference within anti-Stokes mode.
1 % of the signal after the EOM output is used as a reference to a network analyzer (NA) to measure the transfer function for this optical probe. The remaining light passes through a fiber polarization controller (FPC) to maximize coupling between the taper and the resonator. Circulators are employed for performing analysis of the cw and ccw scattered light. An oscilloscope (OSC) and a real-time spectrum analyzer (RSA) are used to measure these optical signals on a photodetector.  Figure 4: Detailed experimental setup. The blue lines indicate the optical paths (fiber), while the black narrow lines indicate the electrical signal paths. A fiber-coupled tunable external cavity diode laser (ECDL) provides light through a 90/10 optical coupler that splits the light into cw and ccw directions. The cw pump power is controlled by an Erbium-doped fiber amplifier (EDFA). An electro optic modulator (EOM) is employed for probing detuning of anti-Stokes light from its optical mode. Scattered light is captured at a photodetector and is sent to an oscilloscope (OSC) and a real-time spectrum analyzer (RSA) for analysis.