Optoacoustic cooling of traveling hypersound waves

We experimentally demonstrate optoacoustic cooling via stimulated Brillouin-Mandelstam scattering in a 50 cm-long tapered photonic crystal fiber. For a 7.38 GHz acoustic mode, a cooling rate of 219 K from room temperature has been achieved. As anti-Stokes and Stokes Brillouin processes naturally break the symmetry of phonon cooling and heating, resolved sideband schemes are not necessary. The experiments pave the way to explore the classical to quantum transition for macroscopic objects and could enable new quantum technologies in terms of storage and repeater schemes.


I. INTRODUCTION
Cooling mechanical vibrations to the quantum ground state has recently been achieved in diverse optomechanical cavity configurations, such as micromechanical bulk resonators [1], drums embedded in superconducting microwave circuits [2], silicon optomechanical nanocavities [3], levitated nanoparticles [4] or bulk acoustic wave resonators [5].The experimental approaches used so far to achieve this state start with the system at cryogenic temperatures, in the mK regime inside a dilution refrigerator, decreasing the thermal population of phonons severely.Nonetheless, reaching the quantum ground state has been made possible only by the use of additional laser-based techniques, such as coupling to highly dampened solid state systems [6,7], feedback cooling [8][9][10][11] or resolved sideband cooling [12][13][14][15][16][17].The latter, also known as dynamical backaction, is based on the engineering of both cavity resonances and laser pump frequencies to favor the cooling over the heating process of a given phonon mode.
Reaching the quantum ground state is often a prerequisite for the study of quantum phenomena and enables applications in precision metrology [18,19], phonon thermometry [20,21], quantum state generation [22][23][24] or tests of fundamental physics [25,26].So far, however, the research focus was on optomechanical cavity structures, in which distinct mechanical resonances (standing density waves), interact with specific optical frequencies, thus providing efficient coupling.Despite this fact, a process bound to narrow mechanical resonances limits both the bandwidth of the optomechanical interaction and its application to parallel multi-frequency quantum operations.
An alternative approach is pursued by waveguide optomechanics, in which light can interact with traveling acoustic waves.The optical waves, which can be transmitted at any wavelength in the transparency window of the material, usually interact with a broad continuum of acoustic phonons.This strong interaction is enabled by electrostriction, radiation pressure and photoelasticity, the physical phenomena behind stimulated Brillouin-Mandelstam scattering (SBS) [27].SBS has received great interest for its broad range of applications in optical fibers or integrated photonic waveguides, such as sensing [28][29][30], signal processing [31][32][33], light storage [34][35][36], integrated microwave photonics [37][38][39] and lasing [40][41][42].The viable application of SBS for active cooling in waveguides has been theoretically proposed [43,44].First experimental results show cooling via SBS in optomechanical cavities [45] and integrated silicon waveguides [46], with cooling rates in the order of tens of Kelvin.Nonetheless, for continuous systems such as waveguides, high SBS cooling rates, particularly high enough to reach the quantum ground state, are still an open challenge.
Here, we experimentally demonstrate optoacoustic cooling of a band of continuous traveling acoustic waves in a waveguide system at room temperature.Without using a cryogenic environment, the acoustic phonons at 7.38 GHz are cooled by 219 K, reaching an effective mode temperature of 74 K.The cooled acoustic waves extend over a macroscopic length of 50 cm in a tapered chalcogenide glass photonic crystal fiber (PCF).The asymmetry of anti-Stokes and Stokes processes in the considered backward SBS interaction provides natural symmetry breaking in the heating-cooling of phonons and therefore no sideband cooling is necessary.We underpin our experimental results with a theoretical model that reproduces the replenishing of acoustic phonons by dissipation in a strong Brillouin-Mandelstam cooling regime.Given the long interaction length of 50 cm, the mechanical object addressed in the interaction is more massive than standard micro resonators.Achieving ground state cooling of such a macroscopic phonon would pave the way towards exploring the transition of classical to quantum physics.

II. EXPERIMENTAL SETUP
The setup used for the experiment is shown in Fig. 1(a).The output of a continuous wave (CW) laser at wave- length λ p = 1550 nm is divided in two branches, pump and local oscillator (LO).The pump light is modulated into 100 ns long square pulses, with 25 % duty cycle, amplified via an erbium-doped fiber amplifier (EDFA) and filtered with a band-pass filter (BPF).The fixed output of the EDFA can be controlled via a variable attenuator, allowing to study the SBS resonances as a function of pump power.The coupling into the sample, a tapered chalcogenide glass solid core single mode PCF from Selenoptics, is done via free space.The glass composition is Ge 10 As 22 Se 68 , with a SBS resonance of Ω B /2π (1550 nm) = 7.38 GHz.The PCF air-filling ratio is 0.48 (Fig. 1(b)) and the insertion and transmission loss through the fiber is -3.64 dB in total.The initial core diameter is 12 µm, which is tapered down to 3 µm at the waist, with a length of 50 cm (Fig. 1(c)).An infrared (IR) camera allows to visualize the optical mode propagating through the core and optimize coupling.As the SBS signal is backscattered, a circulator stops it from going back into the laser, redirecting it into the detection part of the setup.Given the high Fresnel coefficients of the fiber facets, another BPF is used to filter out the strong elastic pump back-reflection.The filtered signal is mixed with a frequency-shifted LO, via a 200 MHz acousto-optic modulator (AOM), to perform heterodyne detection.The resulting optical interference is detected with a photodi- ode and the transduced electrical signal measured with an electrical spectrum analyzer (ESA).The experiment is performed at room temperature (293 K).

III. EXPERIMENTAL RESULTS
A diagram of the different mechanisms affecting the population of SBS-resonant anti-Stokes phonons Ω B is shown in Fig. 2(a).In this type of scattering, phonons are annihilated.Therefore, the interaction can be understood as a loss mechanism, present only while the system is optically pumped.The other two processes that affect the phonon occupation are thermal heating and acoustic dissipation.The population of a phonon mode (n th ) at a given frequency (Ω) is given by the Bose-Einstein statistics n th (Ω) = 1/(e ℏΩ/kBT − 1) and will depend on the temperature of the system (T).As the system thermalises, the phonon levels are filled with in-coherent phonons.Acoustic dissipation, on the other hand, describes the decay of the traveling density fluctuation.While a SBS interaction takes place, energy is being transferred from the acoustic into the optical field, with a coupling strength g om .This results in an effective temperature decrease of the resonant phonon mode.
The shape of the SBS resonances is described by ) where g B is the intrinsic nonlinear gain of the sample and the effective dissipation rate (Γ eff ) is given by the full width at half maximum (FWHM).The effective dissipation rate is defined as Γ eff = Γ m + Γ opt , where Γ m is the natural acoustic dissipation rate and Γ opt is the optically induced loss resulting from the SBS interaction.The cooling rate (R) is defined as the ratio between final and initial phonon occupation, nf and n0 respectively.In the weak coupling regime R is given by From the final phonon population, the effective temperature of the mode after the active cooling can be obtained using the Bose-Einstein equation.In Fig. 2(b) the experimental results for the tapered chalcogenide PCF are shown.From an initial phonon population of 830 at 293 K, a final population of 212 phonons is measured.This corresponds to an effective temperature of 74 K, resulting in a decrease of 219 K or 74.7 % from room temperature.Most of the SBS response from the sample comes from the waist of the taper, which is 50 cm long.The phonons addressed in the nonlinear interaction extend all over the active part, resulting thus in a phonon of macroscopic mass being cooled down to cryogenic temperatures.In an optomechanical resonator, both the Stokes and anti-Stokes processes address the same phonon field.For backward SBS in a waveguide system, such as an optical fiber, this symmetry is broken.The resonant longitudinal phonons involved in a Stokes process are co-propagating with the pump, while for the anti-Stokes process, they are counter-propagating.This inherent symmetry breaking allows to perform the cooling shown without working in the resolved sideband regime.Additionally, both resonances can be studied simultaneously, allowing to compare their different behaviours (Fig. 3(a) and 3(b)).The SBS peaks, with a Lorentzian shape, are defined by two parameters, height and linewidth (Γ eff ).The evolution of the peak height as a function of pump power is shown in Fig. 3(c).For low pump powers, both resonances increase in a parallel way, as the initial equilibrium population of the addressed phonon baths are almost equal.After a threshold of 15 mW, the Stokes peak increases exponentially.This behaviour is characteristic of stimulated scattering, in which the interference between pump and scattered light and the acoustic waves drive each other, creating a feedback loop.The linear fit of the exponential increase provides the SBS gain of the sample, g B = (1.32 ± 0.18) • 10 −9 m/W, comparable with literature values [47,48].The anti-Stokes peak, on the contrary, saturates in height after this threshold.
Regarding the linewidths, different behaviours are observed (Fig. 3(d)).The Stokes resonance narrows, as expected for SBS, while the anti-Stokes broadens.Both the broadening and saturation are footprints of the cooling of phonons via SBS.In the case of a Stokes interaction, the energy is transferred from the optical to the acoustic field, and the amount of phonons that can be created is limited by pump depletion or the damage threshold of the sample.Therefore, an increase in peak height is observed.This is not the case for the anti-Stokes resonance.The number of scattered photons depends on the available number of phonons, fixed by the initial temperature of the thermal bath.As the pump power is increased, more phonons are actively removed from the system, but a limit will be reached.In the ideal case, an observation of peak height saturation would mean that the system has entered a new equilibrium state, in which the thermal bath replenishes the mode at the same speed at which the phonons are actively annihilated.In our experiment this is not the case, as pump depletion arising from the strong Stokes interaction was observed to limit the cooling power [49].The linewidth broadening describes how broad the resonance condition is, i.e., how far from perfect phase-matching the process can still occur.
As the pump is increased, more perfectly phase-matched phonons are removed, yet more photons are present.The probability of scattering with an off-resonance phonon therefore broadens.

IV. THEORY OF BRILLOUIN COOLING IN WAVEGUIDES
An analysis using the theory of waveguide Brillouin optomechanics [44,50,51] of the experimental results is presented in this section.In a typical SBS-active waveguide, backward SBS describes an optoacoustic interaction where two light fields are coherently coupled to an acoustic field.By absorbing a pump photon, the frequency of the backward-scattered photons can be upshifted or downshifted, which corresponds to the Stokes and anti-Stokes processes, respectively.The Stokes process is a parametric down-conversion interaction.It causes heating for acoustic phonons and enables the generation of entangled photon-phonon pairs.The anti-Stokes process is a beam-splitter interaction between scattered photons and acoustic phonons.It can produce phonon cooling, i.e., broadening of the acoustic linewidth, as shown in Figs.3(b) and 3(d).In addition, the natural dispersive symmetry breaking between the Stokes and anti-Stokes processes in the backward SBS scattering in waveguides allows to study the anti-Stokes process individually.By giving the system Hamiltonian derived previously in [52,53] and considering the undepleted pump approximation [54], the dynamics of the linearized optoacoustic anti-Stokes interaction in the momentum space can be given by [49] where a as (b ac ) denote the photon (phonon) annihilation operator for the k th anti-Stokes mode (acoustic mode) with wavenumber k. ∆ L is the frequency detuning between pump and anti-Stokes fields.γ o and g om correspond to the optical loss rate and the pump-enhanced optoacoustic coupling strength between anti-Stokes photons and acoustic phonons.∆ 1 = kυ as and ∆ 2 = kυ ac represent wavenumber-induced frequency shifts for the anti-Stokes photons and acoustic phonons, where υ as (υ ac ) is the group velocity of the anti-Stokes (acoustic) wave.ξ as denotes the quantum zero-mean Gaussian noise of the anti-Stokes mode and ξ ac corresponds to the acoustic thermal noise which obeys relations ⟨ξ ac (t)⟩ = 0 and ⟨ξ † ac (t 1 )ξ ac (t 2 )⟩ = n th δ(t 1 − t 2 ), where n th is the thermal phonon occupation under the environment temperature.
For simplicity, only the case where the anti-Stokes mode and acoustic mode are phase-matched with pump mode, i.e., ∆ 1 = ∆ 2 = 0 and ∆ L = −Ω B is discussed.By switching to a frame rotating with frequency Ω B and considering relations of Langevin noises ξ as,ac , the dynamics of the mean phonon number and photon number can be given by [43] where N a = ⟨a † as a as ⟩ and N b = ⟨a † ac a ac ⟩ correspond to the mean photon and phonon numbers, respectively.Eq. ( 3) can be solved to obtain the phonon occupation at the steady state [49] From Eq. ( 4), the cooling rate can be enhanced by increasing the coupling strength g om , i.e., increasing the pump power, as shown in Fig. 2 (b).However, this cooling rate will be limited by the ratio Γ m /(γ o +Γ m ), similar to the case of sideband cooling in cavity optomechanics [55][56][57].The optically-enhanced acoustic damping rate is defined thus as which can be seen in Figs.3(b) and 3(d), as the anti-Stokes resonance broades with increasing input power.Eq. ( 5) shows a saturation in linewidth for high pump powers, indicating a physical limit for the phonon cooling achievable through this process.Given the system parameters in this experiment, the minimum phonon population achievable from room temperature is around 100 phonons, corresponding to an effective temperature of 36 K (R = 0.1).It should be noted that this system is a continuous optomechanical system, which provides cooling for groups of phonons [43,46], instead of single-mode or multi-mode mechanical cooling in cavity optomechanical systems.In Eq. ( 3), only the phase-matching case with zero wavenumber is considered.For acoustic modes with non-zero wavenumber, the cooling rate at steadystate can be calculated by including the effects of the wavenumber-induced frequency shifts ∆ 1,2 [49].

V. CONCLUSIONS AND OUTLOOK
This experiment has demonstrated that SBS is a promising tool in the challenge of bringing waveguide modes to their quantum ground state of mechanical motion.A massive 50 cm-long phonon with frequency 7.38 GHz is brought to cryogenic temperatures from room temperature, reducing the effective mode temperature by 219 K, one order of magnitude higher than previously reported [46].With our novel theoretical description of the cooling process, the physical cooling limit was calculated to be 90 % of population decrease.This opens the path to the realistic achievement of reaching the quantum ground state in waveguides, given the high frequency of the resonant phonons addressed in this experiment.Performing the experiment in a cryogenic environment, such as a liquid helium cryostat at 4 K, paired with the efficient cooling present in the fiber, would produce occupations of few or even less than one phonon.These results therefore pave the way towards accessing the quantum nature of massive objects.A similar work was published on the arXiv [58] showing cooling by 21 K in a liquid-core fiber.
This Supplementary Material provides additional analyses and derivations on the following topics: (A) motion equation of linearized Brillouin interaction, (B) analyzing Brillouin cooling via the covariance approach and (C) pump depletion limiting cooling.where we assume that the initial state of the acoustic mode is the thermal state.Thus the effective acoustic damping rate can be given by , we show the linewidth of the acoustic mode with respect to pump power in Fig. 4, where the blue-solid curve denotes the effective acoustic damping rate calculated in Eq. (B15).Here the optomechanical coupling strength can be evaluated as g om = G B Γ m P Lc/(4n), where c is the speed of light and n is the refractive index.
Appendix C: Pump depletion limiting cooling Experimentally, it is observed that the lowest phonon mode effective temperature achieved is 74 K, higher than the minimum temperature expected from the theory (around 36 K).This is caused by a depletion of the pump via the strong Stokes wave.All the theoretical derivations in this paper are made under the assumption of undepleted pump, i.e. the amount of energy scattered in the Brillouin-Mandelstam process being negligible compared to the incident pump energy.Given the extremely high gain of the sample, this assumption was observed to not apply to the whole measurement range.
As shown in Eq. (B15), higher pump power results in a higher degree of cooling.On the other hand, higher pump power will also result in a stronger Stokes response from the system [8].As the Stokes response increases exponentially with input power, after a certain threshold, the amount of Stokes light will start to deplete the pump wave [9].This is shown in Fig. 5.For pump powers higher than 240 mW, the height of the Stokes resonance stops increasing exponentially, indicating the start of the pump depletion regime.This depletion of the pump can also be observed in the pump signal transmitted through the sample (Fig. 6).Below the depletion threshold, the pump pulses maintain their initial square shape.As the power is increased, a dip in the amplitude can be observed after the sample.This depletion of the pump via the Stokes process limits the maximum cooling power achievable from the system.As more pump power is scattered into the Stokes wave, less energy is available to further cool the anti-Stokes resonant phonons.

FIG. 2 .
FIG. 2. (Color online) (a) Diagram of the different processes affecting the population of resonant anti-Stokes phonons at ΩB.(b) Light blue crosses show the experimentally measured decrease of resonant anti-Stokes phonon population (ΩB/2π = 7.38 GHz) and its respective effective temperature as a function of pump power.From an initial population at room temperature (293 K) of 830 phonons, a final population of 212 is measured, corresponding to 74 K.This results in a temperature decrease of 219 K, or 74.7 %.The solid blue line shows the theoretical decrease of phonon population according to Eq. (4).The horizontal black dashed lines are added as visual aids and indicate the initial and final temperature of the phonon mode.

FIG. 3 .
FIG. 3. (Color online) (a) In red, behaviour of the Stokes resonance for three different pump powers.(b) In blue, anti-Stokes resonance for the same powers as (a).Note the different y-axis scales.(c) Peak height of the Stokes (red triangles) and anti-Stokes (blue crosses) resonances in logarithmic scale as a function of pump power.(d) Peak linewidth (Γ eff ) of the Stokes (red triangles) and anti-Stokes (blue crosses) resonances as a function of pump power.The Stokes resonance narrows, but the anti-Stokes peaks broadens.This indicates an increase of the effective dissipation rate (Γ eff = Γm + Γopt) of the addressed phonons, caused by the SBS interaction and a relaxation of the phase-matching condition.

FIG. 5 .FIG. 6 .
FIG. 5. (Color online).Peak height of the Stokes (red triangles) and anti-Stokes (blue crosses) resonances in logarithmic scale as a function of pump power.The vertical dashed line indicates the threshold for pump depletion, in which the Stokes peak no longer increases logarithmically.