Long-lived guided phonons in fiber by manipulating two-level systems

R. O. Behunin,1 P. Kharel,1 W. H. Renninger,1 H. Shin,1 F. W. Carter,1 E. Kittlaus,1 and P. T. Rakich1

1Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA(Dated: February 29, 2016)

The synthesis of ultra-long lived acoustic phonons in a variety of materials and device geometriescould enable a range of new coherent information processing and sensing technologies; many formsof phonon dissipation pose a barrier to this goal. We explore linear and nonlinear contributionsto phonon dissipation in silica at cryogenic temperatures using fiber-optic structures that tightlyconfine both photons and phonons to the fiber-optic core. When immersed in helium, this fibersystem supports nearly perfect guidance of 9 GHz acoustic phonons; strong electrostrictively me-diated photon-phonon coupling (or guided-wave stimulated Brillouin scattering) permits a flexibleform of laser-based phonon spectroscopy. Through linear and nonlinear phonon spectroscopy, weisolate the effects of disorder-induced two-level tunneling states as a source of phononic dissipationin this system. We show that an ensemble of such two-level tunneling states can be driven intotransparency–virtually eliminating this source of phonon dissipation over a broad range of frequen-cies. Experimental studies of phononic self-frequency saturation show excellent agreement with atheoretical model accounting for the phonon coupling to an ensemble of two-level tunneling states.Extending these results, we demonstrate a general approach to suppress dissipation produced bytwo-level tunneling states via cross-saturation, where the lifetime of a phonons at one frequencycan be extended by the presence of a high intensity acoustic beam at another frequency. Our mod-eling and measurements suggest that Rayleigh scattering dominates phonon losses for the longestlifetimes achieved in our system. Although these studies were carried out in silica, our findings arequite general, and can be applied to a range of materials systems and device geometries.

I. INTRODUCTION

Access to new regimes of classical and quantum dy-namics hinge upon our ability to create and manipu-late ultra-long lived coherent excitations in electromag-netic, optical, and phononic domains [1–10]. In partic-ular, ultra long-lived phonon modes have been identi-fied as a crucial new resource by quantum information,optomechanics, and precision metrology communities[3, 8, 9, 11–19]. To this end, a variety of systems, rangingfrom nano- and micro-scale phononic devices to resonatortechnologies of centimeter-scale, have harnessed remark-able phonon coherence times [2, 3, 8, 12, 20–23]; however,radical improvements in performance are possible if tech-nical and fundamental sources of dissipation are mastered[3, 8, 10, 23–25]. This realization has spawned a resur-gence of interest in the fundamental origins of phonondissipation at cryogenic temperatures [22, 24–29].

A ubiquitous source of dissipation arises from disorder-induced defects. Some of such defects possess quantizedenergy spectra, and can exchange energy with electro-magnetic, optical, and phononic fields [22, 23, 30–38].When phonon-active, such defect states can absorb andemit phonons just as atoms absorb and emit light. Dis-sipation by such defects pose a fundamental limit tophonon lifetimes in acoustic media at cryogenic temper-atures. Phonon-active defect states have been exten-sively studied in (highly disordered) amorphous media[26, 27, 29, 36–42]; however, their deleterious effects alsoappear within highly ordered crystalline systems and insystems with material interfaces [8, 10, 22]. This formof dissipation bars access to new regimes of classical andquantum dynamics, central to a range of emerging tech-

nologies [8, 10].

In this work, we examine the dynamics of an ensem-ble of phonon-active defects using a guided wave geom-etry that produces tight confinement of both light andacoustic waves (Fig 1). Strong photon-phonon couplingwithin this system permits noninvasive optical excita-tion and interrogation of high frequency (∼ 9.2 GHz)phonons. Tight confinement of acoustic modes pro-duce high phonon intensities (600 W/m2) in the core ofthis waveguide, permitting frequency selective nonlinearphonon spectroscopy with modest (∼mW) optical pow-ers over a range of temperatures (1.1-300 K). Building onestablished models, we elucidate the nature of phonon-defect interactions in our system, allowing us to extractdefect density, coupling strength, and a range of other pa-rameters that capture the dynamics of the defect ensem-ble. We show that this ensemble of defects can be driveninto transparency in the strong-field limit yielding an es-timated factor 45 suppression of defect-induced dissipa-tion at 1.1 Kelvin. In this limit, defect-induced dissipa-tion is a negligible source of loss within our guided-wavesystem; remarkably high phononic Q-factors (>12,000)and decay lengths (>1mm) are achieved. Building onthese findings, material engineering and more sophisti-cated schemes could provide a path toward radically en-hanced coherence times [43] in a range of systems, as thebasis for emerging quantum information technologies.

Phonon dissipation can be suppressed by lowering thesystem temperature (below 150K in our system), andmay in itself be a viable strategy to achieve low phononlosses in crystalline material which have low defect den-sities. However, for amorphous (and even crystalline)materials such as silica or silicon nitride, a tempera-

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ture will be reached below which the acoustic damp-ing will cease to decrease as the temperature is loweredfurther [8–10, 45–47]. This effect is believed to arisefrom resonant absorption by hypothetical two-level tun-neling states (TLSs) [39–42]; the dominant contributionto phonon dissipation in many materials at low temper-atures.

Tunneling states are hypothesized to arise from a sub-set of atoms that inhabit asymmetric double-well poten-tials [39–42] (see Fig.1). A perturbation of this potentialby strain enables phonon absorption, and at low tem-peratures when these TLSs condense into their groundstates (see Figs. 1 and 2) they are capable of resonantabsorption of phonons with energy E. The two-level na-ture of this decay channel leads to non-linearity in thephonon dynamics [36, 39–42] and provides a way to breakthrough the dissipation floor established by resonant ab-sorption.

We demonstrate that by working with high phononintensities resonant absorption can be saturated, signif-icantly extending phonon lifetimes. This self-frequencysaturation is accomplished by increasing the amplitudeof an acoustic mode (at a single frequency) to drive theTLSs into transparency.

Two-level tunneling states have generated much con-temporary interest in their own right for the possibil-ity of beneficial implications. In particular, it has beenshown that they can engender the dynamics of nano-electromechanical systems with nonlinearity, even in thesingle phonon regime [44], and their relatively long coher-ence times have facilitated their use as a quantum mem-ory [34]. In light of this recent interest, our explorationof TLSs and their manipulation may provide valuableinformation for a variety fields.

II. OVERVIEW

A. Stimulated Brillouin scattering

Our system is a 2.2 cm segment of ge-doped NufernUHNA-3 optical fiber. This fiber’s high germanium con-centration (∼ 44 wt%) guides light exceptionally andproduces a large longitudinal sound speed contrast be-tween core vL,core = 4, 740 ± 68 m/s [48] and claddingvL,clad = 5, 944 m/s [49] that promotes acoustic guid-ance. The strong confinement and guidance of opticaland acoustic modes (see Figs. 1 & 3) allows for the effi-cient excitation of phonons via stimulated Brillouin scat-tering (SBS) and which provides a means to characterizeour system [55–57].

Stimulated Brillouin scattering is a resonant processinvolving the interaction of light and sound. It occursin media exhibiting photoelasticity and its complementelectrostriction which are respectively characterized by achange in refractive index upon strain and by an opticalintensity-induced elastic deformation. Hence, the beattone formed by two counter propagating optical fields can

drive acoustic waves through electrostriction, and oncecreated, acoustic waves act as moving Bragg gratingswhich reflect and Doppler shift incident optical beamsas a consequence of photoelasticity.

The concert of these two effects transfer energy be-tween the two participating pump and Stokes opticalfields, and the acoustic field with respective angular fre-quencies ωp, ωS, Ω and wavevector magnitudes kp, kS, q.The transfer of energy is only significant when the threefields are phase matched (see Fig. 1 g-h); for backwardSBS in 1D [56]

ωp = ωS + Ω (1)

kp = q − kS, (2)

alternatively the phase-matching conditions can beviewed as energy and ‘momentum’ conservation. Theminus sign in front of kS in the second equation aboveindicates that the Stokes photon and the pump counter-propagate (the phonon and the pump copropagate). Fora given pump frequency we can approximate the phononfrequency, excited via SBS, by assuming linear dispersionfor light (ω = (c/n)k) and sound (Ω = vq)

Ω ≈ 2nv

cωp (3)

where n is the effective index of refraction, v is the modalsound speed, and c is the speed of light.

In our system SBS results in the transfer of energyfrom the pump to Stokes beam. Net Stokes amplifica-tion can be measured by balanced detection. For suchmeasurements it is prudent to work in the weak signalregime, given by gBPpL 1, where gB is the Brillouingain (gB ≈ 0.6(Wm)−1 at room temperature in UHNA-3fiber), Pp is the pump power, and L is the length of thefiber under test (FUT). In this case the power transferredto the Stokes beam ∆PS, which is directly acquired bybalanced detection, is small and is given by

∆PS ≈(Γ/2)2gBPpPSL

(Ω− ωIM)2 + (Γ/2)2(4)

where we have dropped a negligible correction arisingfrom the optical loss, PS is the input power of the Stokesbeam, the angular frequency ωIM is the angular frequencydetuning between the pump and Stokes beams which isexperimentally set by an intensity modulator (IM) seeFig. 4, and 1/Γ is the phonon lifetime. By sweepingωIM balanced detection supplies the peak Brillouin gain,and the phonon frequency and lifetime which characterizeSBS’s Lorentzian frequency response.

B. Phonon dissipation in glasses

Dissipation of phonons in glasses has several origins:from multi-phonon interactions to scattering by defects.Among the latter of these are hypothesized TLSs that at-tenuate acoustic waves through the processes of resonant

3

CladdingCore

Acoustic wave

Pump light

Probe lightb)

h)g)

f)

d)

x

Pote

ntia

l ene

rgy

OSia)

(1) (2)

(3)

c)

e)LHe

LHe

x

Rcore

Rclad

MFD

MFD

a-SiO2

a-SiO2

ge-doped

FIG. 1. a) Illustration of the microscopic structure of silica glass and some candidate defects proposed as the origin of two-level tunneling states (1)-(3). Individual TLSs are characterized by an asymmetric double-well potential for the generalizedcoordinate x with well-separation d, asymmetry ∆, barrier height V , and the average oscillation frequency of the two individualwells ωc. Tunneling between the two-wells is characterized by the tunneling strength ∆0 (defined below). Excited |e〉 and

ground |g〉 energy eigenstates of an uncoupled TLS are split by energy E =√

∆2 + ∆20. b) Guided optical and acoustic beams

used for the stimulated Brillouin spectroscopy of our system. c) & e) define the material properties and geometry of the fibersystem under study. LHe stands for liquid helium. Optical d) and acoustic f) intensity profile for the modes participating inBrillouin scattering. MFD is mode field diameter. g) spatial and h) temporal phase matching conditions between the opticaland acoustic waves in SBS (see text).

scattering and relaxation absorption. To understand theorigins of these two mechanisms we lay out the tunnelingstate model and discuss its consequences for the dynam-ics of phonons. For those seeking more details, completederivations of all results in this section can be found inthe Appendix.

A tunneling state is characterized by a double-well po-tential of asymmetry ∆ and the overlap energy ∆0 =~ωce−λ where ~ωc is roughly the average zero point en-

ergy of the two wells and λ =√

2mV d~ characterizes the

extent of wave function overlap between the two wellswith m the mass of the atom(s) comprising the TLS, Vthe barrier height, d the ‘distance’ between the double-well’s minima (see Fig. 1), and where ~ is Planck’s con-stant divided by 2π. Given a finite barrier height theatom(s) may tunnel between the two minima of the po-tential. The parameters ∆ and λ are assumed to be uni-formly distributed over the ensemble of TLSs resulting ina constant density of states G(∆, λ) = P , or in terms ofthe overlap energy f(∆,∆0) = P/∆0. This assumptionof uniformity correctly predicts the linear in temperaturebehavior of the specific heat and the anomalous thermalconductivity of glasses at low temperatures [39–42].

At low temperatures the TLS+phonon system, includ-ing the strain-induced perturbation of the asymmetry,can be modeled with the following Hamiltonian

H = Hph +∑j

[1

2(Ej +Dj · ξ(rj))σz,j +Mj · ξ(rj)σx,j

](5)

[42] where the physics of the jth TLS is approximatedas an effective two-level system using the Pauli matri-ces σi,j with i = x, y, z. The elastic strain field ofpolarization η is labeled as ξη, the sum over j counts allTLSs in the glass at various positions rj with energy split-ting Ej and coupling parameters Dη,j ≡ 2(∆j/Ej)γη andMη,j ≡ (∆0,j/Ej)γη, Hph is the free Hamiltonian for thephonons (see Appendix), and the ‘dot product’ Dj · ξ ab-stractly represents a sum over polarizations

∑ηDη,jξη.

Generally, the deformation potential tensor, quantifyingthe TLS-strain coupling, is unique to each TLS and ori-entation dependent; we have ignored this complicationabove where the TLS-phonon interaction is character-ized by a polarization dependent deformation potentialconstant γη.

C. Resonant phonon absorption by TLSs

Tunneling states can resonantly interact with phononsthough three processes: stimulated absorption, and spon-taneous and stimulated emission. The relative magnitudeof each of these processes determines the phonon dissipa-tion rate and depends on temperature and the intensityof the acoustic field.

4

a) High temperature transparency

b) Low temperature absorption

c) Saturation-induced transparency

acoustic wave amplitude

acoustic damping due to resonant absorption

resonant absorptionsaturated at high intensity

- TLSs resonant with acoustic wave

FIG. 2. Resonant absorption of phonons of angular frequencyΩ by TLSs (illustrated by green spheres). a) At high tem-peratures (kBT ~Ω) the TLSs addressed by the acousticwave have almost equal probabilities of being in the excitedor ground states, and hence resonant absorption is compen-sated by stimulated phonon emission. b) At low tempera-tures (kBT ~Ω) the TLSs addressed by the acoustic wavecondense to their ground states. At sufficiently low acousticintensities TLSs, having resonantly absorbed a phonon, de-cay via spontaneous emission thus attenuating the incidentacoustic wave in the process. c) At low temperatures andhigh phonon intensities a given TLS is interrogated by mul-tiple phonons during its excited state lifetime. Hence, an ex-cited TLS may decay via stimulated phonon emission whichpreserves the coherence of the incident phonon. At high in-tensities resonant absorption is compensated by stimulatedemission.

0 1 2 3

0

1 Optical SimulationAcoustic AnalyticsAcoustic Simulationn

Optical

1.48271

Radius (µm)

Nor

mal

ized

inte

nsity

1.4447

Acoustic

Ref

ract

ive

inde

x(n

)

0.9 µm core

FIG. 3. Normalized fundamental mode acoustic and opticalintensity, and index profile in UHNA3 fiber. Solid lines areobtained from finite element simulations, and red dots fromanalytical calculations of the acoustic modes. Inset: Crosssectional view of the acoustic and optical intensity. The smallcore size and good opto-acoustic mode overlap ensures highoptical and acoustic intensities as well strong optomechanicalcoupling.

1. Weak fields

First, we consider weak fields which is roughly charac-terized by a mean free time between TLS-phonon inter-actions that is long compared to the excited state lifetimeof the TLS, and which will be defined quantitatively be-low. For such low intensity acoustic waves of angularfrequency Ω the golden rule can be used to find the TLS-induced decay rate (see Appendix)

Γres =πPγ2Ω

ρv2tanh

(~Ω

2kBT

)(6)

=πPγ2Ω

ρv2( Pg︸︷︷︸

(i)

− Pe︸︷︷︸(ii)

)

[39–42] where the polarization dependence of the defor-mation potential γ and the sound speed v have beensuppressed, kB is the Boltzmann constant, and ρ is thematerial density. Γres is characterized by two regimesoccurring at high and low temperatures T . This is elu-cidated by the temperature dependence of the phonondecay rate on the thermal equilibrium population inver-sion Pe−Pg of the TLSs at energy ~Ω and temperature T ,where Pe and Pg are the probabilities to find a TLS in theexcited and ground state, respectively. At low tempera-tures ~Ω kBT the decay rate is maximized since theTLSs are found entirely in their ground state (Pg → 1),and thus (i) stimulated absorption dominates. In the lowintensity regime TLSs that have absorbed phonons decaythrough spontaneous emission which attenuates coherentphonon beams. As the temperature is raised the prob-

5

ability to find the TLS in its excited state grows whichopens the possibility for (ii) stimulated phonon emissionwhich coherently amplifies the sound amplitude. At hightemperatures these two processes compensate each otherexactly and resonant absorption is suppressed.

2. Strong fields

At high acoustic intensities perturbation theory is nolonger adequate to calculate the phonon decay rate. Inthis regime one must solve the coupled Heisenberg equa-tions of motion for the TLSs and the acoustic field (seethe Appendix for a complete derivation). The backreac-tion of the TLSs on the phonons is characterized by acomplex susceptibility that modifies the phonon dynam-ics in two ways: by resonant absorption induced dissipa-tion and a frequency shift. At high acoustic intensitiesthe dissipation rate due to resonant absorption is givenby

Γres ≈ πPγ2Ω

ρv2

tanh(~Ω/2kBT )√1 + J/Jc

. (7)

[42] where J is the acoustic intensity, and Jc is the criticalintensity that demarcates the boundary between weakand strong fields. The critical intensity

Jc ≈~2ρv3

2γ2T1T2(8)

is characterized by the time scales T1 and T2 whichare phenomenological upper state lifetime and dephasingtimes for the TLSs with energy E ≈ ~Ω (see Appendix).

As with the case of weak fields, this result for thephonon dissipation can be interpreted as a competitionbetween stimulated emission and absorption. In thestrong field case however the effective population inver-sion for the TLSs which interact with a phonon of fre-quency Ω is given by the nonequilibrium steady-statevalue

Pe − Pg = − tanh(~Ω/2kBT )√1 + J/Jc

. (9)

At high intensities J/Jc 1 a TLS, excited through res-onant absorption, will encounter several phonons withinits upper state lifetime. Thus, stimulated emission canbecome a dominant decay channel and resonant absorp-tion will be suppressed.

Resonant absorption also leads to a (acoustic intensity-insensitive) frequency shift given by

∆ωres(T )−∆ωres(T0) ≈ −Pγ2Ω

ρv2

[ln

(~Ω

kBT

)− ReΨ

(1

2+

~Ω

2πikBT

)]− (T → T0), (10)

where Ψ is the digamma function [42].

D. Relaxation absorption

In addition to resonant absorption TLSs can atten-uate coherent acoustic waves via relaxation absorption.This process results from a modulation of the TLS’s en-ergy splitting by the presence of a time-dependent strainfield, i.e. Ej(t)→ Ej +Dη,jξη(t). As Ej(t) changes, theTLS can equilibrate with the surroundings by absorbingor releasing energy. Thus, the instantaneous ‘equilib-rium’ population inversion is modulated in time. For oursystem (see Appendix) relaxation-absorption results indissipation of phonons characterized by the decay rate

Γrel ≈ π3

24

Pγ2η

ρ2v2η~4

∑η′

γ2η′

v5η′

(kBT )3, (11)

and a temperature-independent frequency shift.

III. EXPERIMENTAL SETUP

The system is characterized by pump-probe measure-ments performed using balanced detection and lock-inamplification. Our apparatus consists of a 1548.963 nmsource that is split into two optical lines (see Fig. 4);one reserved to act as a pump, and the second to act asa probe. The pump line is subsequently amplified andpump and probe polarizations are aligned. The pumpbeam is then power modulated at the fixed frequencyΩmod for lock-in detection before being sent through theFUT. The probe beam is sent through an intensity mod-ulator which upon exit is filtered to isolate a single side-band. To achieve common mode noise rejection using bal-anced detection, the probe beam is split into two arms.One arm passes probe light through the FUT and is am-plified via SBS whereas the other arm acts as a reference.A variable attenuator on the reference arm is adjusted sothat both probe arms have the same optical power in theabsence of gain via SBS. The difference in power of thebalanced probe arms yields the net gain experienced bythe probe beam passing through the sample.

Modulation of the pump power generates a fixed-frequency side-band of the amplified probe which is de-tected with a lock-in amplifier. To measure Brillouin gainspectra (BGS) the modulation frequency ωIM is sweptacross the Brillouin resonance generating a Lorentzianresponse.

The temperature of the fiber is controlled using adouble walled cryostat, and a large copper heat sink.The samples are mounted in a shallow mail slot passingthrough the block. The block size was chosen to ensuretemperature stability, and to allow a minimum of 10 Bril-louin lineshape measurements per 100 mK as the blockwarms slowly to room temperature. The system is cooledusing liquid Helium and evaporative cooling resulting inlowest achievable temperatures in the neighborhood of 1K.

6

EDFA EDFA VOA

EDFAIM

Lock-in

PC

Circulator

FBG

VOA

Cryostat

FUT

PC

Circulator

Circulator

FBG

BD

Pump

Probe

LASER

FIG. 4. Experimental setup to measure SBS with balanceddetection. The acronyms above are defined as: EDFA Erbiumdoped fiber amplifier, VOA variable optical attenuator, RFradio frequency signal generator, BD balance detector, andFBG fiber Bragg grating. We use a DFB (distributed feed-back) laser with a wavelength of 1548.963 nm and a linewidthof 5 kHz. Our probe line is synthesized by sending pumplaser light through an IM modulated in the neighborhood of9.2 GHz, the carrier and the high frequency side-band areremoved by operating at the null point bias point of the IMand band-pass filtering. Probe light is split in two arms, oneacting as a reference and the other traversing the FUT. Thepower modulated pump results in a time harmonic amplifi-cation of the Stokes beam at Ωmod which is acquired withlock-in detection.

To study the saturation of losses arising from the TLSsthe intensity of the sound field inside the FUT was sweptat fixed temperature. In the weak signal regime thesteady-state phonon intensity follows the powers of theoptical fields

J(ωIM) ≈ v

Γ

ωIM

ωS

1

AeffGB(ωIM)PpPS, (12)

where Aeff ≈ 1.6µm2 (estimated from simulations, seeFig. 3) is the phonon mode area. Hence, the phononintensity can be controlled by changing the power of theoptical driving fields. By sequential adjustment of thevariable optical attenuators on pump and probe lines thephonon intensity generated through SBS can be sweptthrough more than 4 decades of dynamics range. Themaximum pump and probe powers were used to deter-mine the 2.2 cm length of the FUT; this length en-sures that all measurements were performed in the weak-signal regime. This fiber length allowed access to highphonon intensities while simultaneously preventing in-homogeneous broadening from non-linearity induced bystrong backward scattering via SBS.

IV. RESULTS

A. Temperature and Intensity Dependence ofPhonon Losses

Brillouin gain spectra corresponding to the fundamen-tal acoustic mode in a 2.2 cm segment of UHNA-3 fiberwere continuously acquired as the system slowly warmedto room temperature from 4.17 K. Pump and probe pow-ers were set to 35 mW and 0.550 mW, respectively to en-sure SBS measurements in the weak signal regime. TheBGS were binned in 100 mK steps, averaged, and fit tothe Lorentzian model given by Eq. 4. This analysis pro-vides the phonon frequency Ω and the dissipation rateΓ as a function of temperature. The red data points inFigure 5a show the dissipation rate of the fundamentalacoustic mode as a function of temperature. Three rep-resentative BGS are shown in Fig. 5d. To judge therelative importance of resonant absorption by TLSs wehave plotted the ratio Pe/Pg as a red curve (top).

For low phonon intensities (J Jc) the linewidth be-gins to level off below 4K (red points in Figs. 5a & 5b)and then begins to increase as the temperature is lowered.This gray region in Figs. 5a & 5b, where Pe/Pg < 0.9, in-dicates the temperature range where resonant absorptionby TLSs begins to dominate the acoustic damping.

After cooling the system to 1.1 K, we let the fiber sys-tem slowly warm up to 4.17 K. For each 100 mK risein temperature we acquire BGS as a function of phononintensity. The data is analyzed by binning and averagingas described above, and results in dissipation rate as afunction of both temperature and phonon intensity. Thisdata is presented in 5b & 5c: 5b shows phonon dissipa-tion rate as a function of temperature. The three sets ofdata correspond with three distinct settings of the opti-cal powers that generate phonons of low, moderate, andhigh intensities (as compared to Jc). Normalized BGSfor the three settings is shown in Fig. 5e for the lowesttemperature of 1.1 K. Figure 5c shows phonon dissipationrate as a function of phonon intensity for three differenttemperatures (black, orange and tan points). It is qual-itatively clear from Figs. 5b & 5c that at temperatureslower than 4 K phonon dissipation is suppressed at highphonon intensity; this indicates saturation of the reso-nant absorption due to TLSs and is made evident by thetheoretical plot of Pe/Pg (at top), computed using Eq.9.

In Figs. 5c, f, & g we compare our measurements withthe tunneling state model. In Fig. 5c a model of thephonon dissipation rate given by

Γ = Γres + Γ0 (13)

is fit to the data where Γ0 is an offset parameter thatrepresents all intensity independent background acousticlosses including relaxation absorption, Rayleigh scatter-ing, phonon-phonon interactions, etc. Three parametersare employed to fit the data: 1) Pγ2

L (γL being the de-formation potential for longitudinal waves), 2) Jc, and

7

1 10 100

10

100

1

10

100

Temperature (K)

Dis

sipa

tion

rate

(MH

z)

Linewidth (M

Hz)

150 K

25 K

4.4 K

1 2 3 4 5

10

1

2

3

45

Temperature (K)

Dis

sipa

tion

rate

(MH

z) Linewidth (M

Hz)

J3

J2

J1

5

.

.

.

.

.

J (W/m2)

Dis

sipa

tion

rate

(MH

z)

9050 9100 9150 9200

25 K150 K

4.4 K

1.1 1.2 1.3 1.4

0.0

0.2

0.4

0.6

1

d) f )

ii

iii

sweep i

i

sweep iiisweep ii

J3>J2>J1

Linewidth (M

Hz)

2 3 4 50.70.80.91.0

10 1000.70.80.91.0

T = 1.1 K

9184 9186 9188 9190

J1J2J3

e)

a)

.

.

.

.c)

fully saturated

b)

6

12

18

Temperature (K)

Temperature (K)Frequency (MHz)

Frequency (MHz)

BG

S (a

.u.)

Freq

. Shi

ft (M

Hz)

Nm

. BG

S (a

.u.)

(W/m

2)

(MH

z)5

10

15

0.5

1.0

1.5

2.0

2.5

/

1.1 K2.0 K3.3 K

0.7 MHz 4.5 MHz

dissipation floor at 1.1 K

sweep i

sweep ii

sweep iii

0.70.80.91 0

at 1.1 K

1 2 3

g)

Temperature (K)0.1 1 10 100

0.1 1 10010

1251020

FIG. 5. a) Dissipation rate of the fundamental acoustic mode at low intensity as a function of temperature. b) Phonondissipation rate as a function of temperatures below 6 K. The three data sets below 4 K correspond with different pumpand probe power settings corresponding with low, moderate, and high phonon intensities, as compared to Jc. c) Phonondissipation rate as a function of acoustic intensity. Each curve has a fixed temperature. d) Brillouin gain spectra as a functionof temperature. e) Normalized Brillouin gain spectra as a function of intensity at 1.1 K. f) Comparison of the measuredBrillouin frequency vs. temperature with the theoretical prediction given by Eq. 10. g) Critical intensity Jc and backgroundlosses Γ0 as a function of temperature. The parameters for the theory can be found in Tab.1

3) the offset Γ0. The parameters P and γL are assumedto be constant. In distinction, Jc and Γ0 are expectedto depend upon temperature. In particular, Jc scales in-versely with the effective decay rates T1T2. The upperstate lifetime T1 can be approximated with the Fermi’sgolden rule (see Eq. C1 of the Appendix), but an esti-mation of the temperature dependence of T2 is beyondthe scope of this work. Instead we fit Jc (gray dots inFig 5g) to a power law Jc = aT b that is shown as a solidgray line in 5g. In Fig. 5f the measured shift in Bril-louin frequency (gray points) referenced to T0 = 1.09 Kis compared to Eq. 10. The fitted values obtained fromFig. 5c were used as inputs. The offset Γ0 is plotted as

blue points in Figs. 5g and contains all background losseswhich we model as

Γ0 = Γrel + ΓBG. (14)

where ΓBG is a constant representing the background dis-sipation floor for our system. The model is fit to the databy adjusting the offset and the value of the longitudinaldeformation potential γL (we assume that the transversedeformation potential is given by γ2

T ≈ γ2L/2 [38]). Pγ2

Lis held fixed to the value obtained from the data analysisin Fig. 5c.

The model parameter values obtained from this analy-sis are tabulated in Tab. I and their values are compared

8

44% wt. ge-doped silica vitreous silicaρ 2, 666 kg m−3 2, 202 kg m−3 [49]vL 4, 760± 68 m s−1 [48] 5, 944 m s−1 [49]

a 0.9 W m−2 K−b

b 2.6Jc 1.2 W m−2

Pγ2L 1.6× 107 J m−3 1.3× 107 J m−3 [58]

P 23 6.85 [58]γL 0.5 eV 0.86 eV [58]√T1T2 10 ns (est. with Eq.8) ∗53 µs [37]

T1 ≈ 79 ns (est. with Eq.C1) ∗200 µs [37]T2 ≈ 1.3 ns ∗14 µs [37]

TABLE I. Comparison of the properties of ge-doped silica andsilica. All temperature and frequency dependent quantitiesare listed for T = 1.1 K and Ω = (2π)9.188 GHz. The productT1T2 is extracted from Jc defined by Eq. 8. The parameterT1 above is calculated from Eq. C1, and then subsequentlyemployed to extract T2 from Jc. P is in units of 1044 J−1 m−3.∗The values of T1 and T2 for vitreous silica were measured atT = 20 mK and for frequency 0.68 GHz. Hence, these valuesshould not be directly compared the values listed for ge-dopedsilica. Using Eq. C1 and the fitted tunneling state parametersT1 at 20 mK and for 0.68 GHz phonons is estimated at 665µs in UHNA-3, this compares well with the listed value forvitreous silica.

with those of vitreous silica. There are several remarksin order. The data of Fig. 5c is well-described by thetunneling state model, and the computed Brillouin fre-quency shift using the fitted value of Pγ2

L from Fig. 5ccompares well with our measurements, however the the-ory and experiment begin to diverge at higher tempera-tures. Relaxation absorption accounts well for the inten-sity independent background, and the fitted value for γLis comparable to silica. The fitted parameters and theirtheoretical relationships can also be used to estimate thedensity of TLSs P and the dephasing rate T2.

As a final note, the values of T1 and T2 should not bedirectly compared with those of vitreous silica listed inTab. I. The phonon frequencies and the temperatureswere distinct from our measurements; for an appropriatecomparison T1 can be estimated from the minimum TLSupper state lifetime computed using perturbation theoryEq. C1 with ∆0 → E, taking T = 20 mK and Ω/2π tobe 0.68 GHz, and using the fitted TLS parameters for ge-doped silica. We find T1 ≈ 665 µs which is comparableto the reported values for vitreous silica in Tab. I

V. DISCUSSION

Acoustic dissipation imposed by defects presents a bar-rier to attaining low loss acoustic modes that are criti-cal to a range of technologies. To overcome/understandthese challenges we explored defect-induced dissipationin a guided wave system using nonlinear phonon spec-troscopy. Theory-experiment comparison using the tun-

neling state model quantifies the influence of defects inour system. We found that the large phonon intensities,made possible by the tight acoustic confinement, permitaccess to the TLS-induced regime of nonlinear phonondynamics. We demonstrated self-frequency saturation,where TLS resonant absorption is driven to transparencyas the intensity of the acoustic beam becomes large.

At the highest intensities we encounter a dissipationfloor Γ0 due to relaxation absorption Γrel and a roughlytemperature independent offset ΓBG. At the lowest tem-peratures and highest intensities we estimate that relax-ation absorption makes up only 6% of the dissipation ratefloor, and thus we believe we have reached the neighbor-hood of the smallest phonon dissipation rates possiblein UHNA3 fiber estimated to be (2π)650 kHz.There areseveral possible explanations for ΓBG such as; phonon-phonon scattering, anchor losses, inhomogeneous broad-ening due to irregularities in the fiber geometry along itslength, acoustic leakage due to imperfect guiding, andRayleigh scattering due to disorder and scattering bydopants. Phonon-phonon scattering can be estimatedfrom the Landau-Rumer theory using Fig. 2 of [9] andgives negligible dissipation, and anchor losses are likelyunimportant as the mode is highly confined to the core.Large amounts of inhomogeneous broadening are unlikelygiven the relatively short length of the fiber and the ob-served Lorentzian shape of the BGS, and acoustic leakagewas estimated in simulations to be too small.

We attribute the dissipation floor to Rayleigh scatter-ing. There are several pieces of corroborating evidencefor this hypothesis; the fiber is doped with a large con-centration of germanium that will produce large stochas-tic variations of density and sound speed in the fibercore, ΓBG is insensitive to temperature and intensity, andan estimation of Rayleigh scattering in Top High Qual-ity alpha synthetic quartz for 9.2 GHz phonons givesa lower bound for the Rayleigh scattering dissipationrate (2π)200 kHz. By working at lower frequencies (asRayleigh scattering scales with the fourth power of thefrequency) or with fibers with lower defect densities muchlonger phonon lifetimes may be possible.

Acknowledgements: Primary support for this workprovided by NSF grant DMR 1119826.

Appendix A: Acoustic guidance in fiber

The high germanium concentration (∼ 44 wt%) inUHNA-3 fiber produces a large longitudinal sound speedcontrast between core vL,core = 4, 740± 68 m/s [48] andcladding vL,clad = 5, 944 m/s [49] that promotes acousticguidance. However, ideal guidance is not expected sincethe shear wave velocity in the cladding vS,clad = 3, 764m/s [49] is smaller than vL,core, and hence leads to modeconversion at the core-clad interface [50] that leads toacoustic leakage.

To understand the limits of acoustic guidance we stud-ied the axial-radial acoustic modes of our system with full

9

vectorial simulations and semi-analytical calculations.We solved for acoustic eigenfrequencies of a core region(diameter 1.8 µm) embedded in a finite cladding (diam-eter 125 µm) surrounded by liquid helium. A perfectlyabsorbing boundary condition was implemented adiabat-ically in the liquid region at large separation from thecore to prevent spurious reflections from the simulationboundaries. We employed a shear velocity in the coreregion vS,core ∼ 3092 m/s which was estimated by inter-polating between the values of pure silica and germania,and the liquid region was ascribed the acoustic proper-ties of liquid helium. The core, cladding and liquid wereassumed lossless other than the absorbing layer used tomodel mode leakage.

Acoustic energy that enters the liquid region leavesthe system irreversibly and hence reduces the acousticguidance. The signature of this leakage is an imaginarypart to the eigenfrequencies that quantifies the dissipa-tion rate. Our simulations correctly predict the acousticfrequency and yield negligible leakage (much smaller thanthe dissipation rate observed in experiment). Hence, weconclude that the dissipation of phonons in our system isdue almost entirely to internal friction of the fiber as op-posed to leakage. In this sense we describe our phononsas well-guided.

It should be emphasized that acoustic guiding in fiberis not guaranteed. Germanium doping has the fortunateconsequence that increased doping leads to higher indexof refraction while simultaneously decreasing sound speed[49, 51], thus enhancing the confinement of light andsound simultaneously. Titanium dioxide and phospho-rous pentoxide are other dopants which simultaneouslyenhance guiding of light and sound, while boron triox-ide and fluorine doping enhance the confinement of lightwhile raising the sound speed [52]. Fibers doped with theformer will possess leaky acoustic modes. At room tem-perature, where phonons are strongly damped in silica,one may posit that the guiding of sound in fiber playsa minor role as the mean free path is on the order ofthe acoustic mode field diameter. However, the existenceof higher order guided phonon modes can be observedat room temperature [53, 54] which shows that acous-tic modes in fiber cannot be completely understood interms of bulk properties without considering boundaries.We have also simulated axial-radial modes in a systemcomposed of a core embedded in an infinite cladding re-gion. In this case the computed acoustic leakage rateexceeds the observed phonon dissipation rate suggestingthat the cladding liquid interface plays an important rolein the observed acoustic guidance. In light of this fact itis important note that the leakage rate we derived fromsimulation for the finite fiber case is only a lower bound aswe have not accounted for nonuniformity in the geometryof the fiber over its length. Irregularities in cladding andcore diameters will contribute to inhomogenous broad-ening that will manifest as an effective increase in theleakage rate.

Appendix B: Tunneling state theory of phonondissipation in glasses

In this section we summarize the underpinnings of thetunneling state theory.

At low temperatures kBT ~ωc a TLS can be effec-tively represented by the Hamiltonian

HTLS =1

2

(∆ ∆0

∆0 −∆

)(B1)

where the states 〈L| = (0, 1) and 〈R| = (1, 0) correspondwith position states with the particle localized in the left(‘L’) or right (‘R’) well and which we approximate asbeing orthogonal. Diagonalizing the Hamiltonian abovegives the corresponding energy eigenstates

|e〉 =∆0√

2E(E + ∆)

[|L〉+

∆ + E

∆0|R〉

]|g〉 =

∆0√2E(E −∆)

[|L〉+

∆− E∆0

|R〉]

(B2)

which reveal that the stationary states of a free TLS arespatial superpositions between the wells and have ener-gies ± 1

2E ≡ ±12

√∆2 + ∆2

0, see Fig. 1b. Such a TLS isperturbed by a local strain; the dominant effect of this isa shift in the asymmetry ∆ that perturbs the diagonal el-ements of the Hamiltonian in Eq. B1 as ∆→ ∆+ ∂∆

∂ξabξab

where ξab is the elastic strain, 12∂∆∂ξab

≡ γab is the defor-

mation potential characterizing the linear response of theTLS potential to strain, ‘a’ and ‘b’ denote spatial com-ponents of the strain tensor, and the Einstein summationconvention for repeated indices is used. Upon applyingthe unitary transformation which diagonalizes the Eq.B1 to the full Hamiltonian including the strain-inducedperturbation of the asymmetry and averaging over TLSorientations, we arrive at Eq. 5. The free Hamiltonianfor the phonons is given by∑

q,η

~Ωqη(b†qηbqη + 1/2) (B3)

where the sum counts modes with wavevector and po-larization q and η, and bqη and Ωqη are the qη-modeannihilation operator and frequency, respectively. Thedecomposition of the strain field into normal modes, herein plane waves, reveal the connection with the annihila-tion operator bqη

ξab(x) =i

2

∑q,η

(qarηb(q) + qbrηa(q))

√~

2ΩqηρVeiq·xbqη

+H.c. (B4)

where V is volume of the system, ρ is the material density,rηb(q) is a unit vector for η-polarized phonons, and H.c.stands for Hermitian conjugate.

10

Appendix C: TLS excited state lifetime

Before we begin our investigation of TLS-induced ef-fects upon the phonons we outline some of the phonon-induced effects upon the TLSs which play a role in ouranalysis. The most important of these effects is that in-teraction with the phonons leads to a finite TLS upperstate lifetime.

For a TLS of asymmetry ∆ and overlap energy ∆0

Fermi’s golden rule gives the excited state decay rate

1

τ=∑η

γ2η

v5η

E∆20

2πρ~4coth

(E

2kBT

)(C1)

[39–42] where the sum over η counts decay channels cor-responding with the various phonon polarizations, and Tis the temperature of the phonon bath. The polarizationdependence of the deformation potential and the soundspeed are accounted for in the η suffices. For fixed energyE the lifetime τ has the minimum value τmin obtainedfrom Eq. C1 by taking ∆0 → E.

The lifetime of the jth TLS is derived using Fermi’sgolden rule. We begin with matrix elements for upwardtransition of the TLS from the ground |gj〉 to the excitedstate |ej〉

〈ej , nqη − 1|Mj : ξ(rj)σx,j |gj , nqη〉 =

iqγη

√~

2ΩqηρVeiq·rj

∆0

E

√nqη (C2)

where nqη is the number of phonon quanta in the qη-mode. The matrix element for downward transitions isgiven by

〈gj , nqη + 1|Mj : ξ(rj)σx,j |ej , nqη〉 =

− iqγη

√~

2ΩqηρVe−iq·rj

∆0

E

√nqη + 1.

(C3)

After averaging over the initial (thermal) phonon stateand summing over all final states that contribute to thetwo processes the golden rule gives the upper transitionrate as

Cg→e =∑η

γ2η

v5η

Ej∆20,j

2πρ~4

1

eEjkBT − 1

(C4)

where the Debye phonon density of states has been usedgη(E) = E2/(2π2~3v3

η)V . The rate Ce→g can simi-larly be computed and is identical to Cg→e if the factor(exp(Ej/kBT )−1)−1 is taken to (exp(Ej/kBT )−1)−1+1.

These two rates can be combined to give the time rateof change for probability of the TLS to be in its excitedstate

Pe =Cg→ePg − Ce→gPe=− (Cg→e + Ce→g)Pe + Cg→e (C5)

since Pe + Pg = 1 which gives the decay rate for theexcited state as Cg→e + Ce→g written explicitly in Eq.C1.

Appendix D: Resonant phonon absorption by TLSs

1. Derivation of weak field phonon decay rate

Eq. 6 is derived using the golden rule. The transitionrates for an increase in the number of quanta in the qη-phonon mode from nqη to nqη + 1 as well as the decayrate from nqη to nqη − 1 are given by

Cnqη→nqη+1 =1

V

∑j

Pe∆2

0,jγ2η

E2j

πq2

Ωqηρ(nqη + 1)δ(Ej − ~Ωqη)

Cnqη→nqη−1 =1

V

∑j

Pg∆2

0,jγ2η

E2j

πq2

Ωqηρnqηδ(Ej − ~Ωqη).

(D1)

The sum over the various TLSs can be performedby assuming the validity of the ergodic theorem whichstates that the volume average is equal to the ensem-ble average of the TLSs in the thermodynamic limit i.e.1V

∑j F(∆j ,∆0,j , Ej) →

∫d∆∫d∆0P/∆0 F(∆,∆0, E)

where F represents the summand and recall that P/∆0

is the TLS density of states.

Converting the sum over j to an integral over the TLSdistribution and completing the integrals over ∆ and ∆0

gives time rate of change of nqη

nqη =λ[Pe(nqη︸︷︷︸(i)

+ 1︸︷︷︸(ii)

)− Pg nqη︸︷︷︸(iii)

] (D2)

where λ(Pe − Pg) is given by Eq. 6. The under-braces denote the terms contributing to; (i) stimulatedphonon emission, (ii) spontaneous phonon emission, and(iii) stimulated phonon absorption. From the expressionabove it is clear that when Pg → 1 that the TLSs pre-dominantly attenuate the acoustic wave. Resonant ab-sorption is suppressed at either high temperature wherePe = Pg, or high intensities where nqη 1 and Pe → Pg.However, the physics of the latter case can only be elu-cidated by working with the full dynamics governed bythe Bloch equations.

2. Strong fields

At high acoustic intensities perturbation theory is nolonger adequate to calculate the phonon decay rate. Inthis regime one must solve the coupled Heisenberg equa-tions of motion for the TLSs and the acoustic field given

11

by

σz,j =2

~Mj · ξ(rj)σy,j (D3)

σy,j =1

~(Ej +Dj · ξ(rj))σx,j −

2

~Mj · ξ(rj)σz,j (D4)

σx,j = −1

~(Ej +Dj · ξ(rj))σy,j (D5)

bqη = −iΩqηbqη −1

2

∑j

gqη,j(∆jσz,j + 2∆0,jσx,j)

(D6)

where gqη,j ≡ 1~γηEjq√

~2ΩqηρV

e−iq·rj .

When the acoustic field is driven strongly at angularfrequency ω ≈ Ωqη the physics can be greatly simpli-fied. Under these conditions we focus our attention solelyon the classical steady-state dynamics of the qη-mode.We account for the remaining, thermally populated,phonon modes with phenomenological damping termsand ‘Langevin’ forces that influence the TLS dynamics.Strong driving also allows the use of the rotating waveapproximation (RWA) that we implement by decompos-ing the strain field and the x and y Pauli operators intopositive and negative frequencies i.e. ξ = ξ(+) + ξ(−)

where ξ(±) ∝ e∓iωt, and similarly σx,j = σ(+)x,j + σ

(−)x,j

where σ(±)x,j ∝ e∓iωt. The operator σy,j is similarly de-

composed and σz,j is assumed to be time-independent inthe steady-state limit.

After these simplifications the coupled equations ofmotion for the driven phonon amplitude βqη, and themean values for the TLS operators, 〈σi,j〉 ≡ Si,j , reduceto

Sz,j = − 1

T1(Sz,j − w0,j) +

2

~Mη,j

[ξ(−)qη (rj)S

(+)y,j +H.c.

](D7)

S(±)y,j = − 1

T2S

(±)y,j +

1

~EjS

(±)x,j −

2

~Mη,jξ

(±)qη (rj)Sz,j

S(±)x,j = − 1

T2S

(±)x,j −

1

~EjS

(±)y,j

βqη = −iΩqηβqη + Fqη −∑j

∆0,jgqη,jS(+)x,j

where the TLS phenomenological decay rates 1/T1, quan-tifying the upper state lifetime, and 1/T2, characterizingthe dephasing rate, arise from the interaction with thethermal phonon field, and additionally in the latter fromspectral diffusion of a given TLS’s oscillation frequency1~ (Ej − Dj · ξ(rj)) as the static background strain fieldis modified by spin flips of neighboring TLSs [58]. The

strain amplitude of the qη-mode ξ(+)qη is given by the co-

efficient of bqη in Eq. B4 with bqη replaced by βqη, Mη,j

is γη(∆j/Ej), and Fqη is the magnitude of an externaldrive at angular frequency ω. The thermal equilibriumvalue of Sz,j given by w0,j ≡ − tanh(Ej/2kBT ) plays therole of a ‘Langevin’ force in this system of equations by

ensuring the return to thermal equilibrium in the absenceof driving.

In steady-state and noting that ξ(+) = ξ(−)† the equa-

tion for Sz,j can be solved in terms of ξ(+)qη giving

Sz,j =w0,j

1 +4M2

η,jT1T2

~2 |ξ(+)qη |2

(1

1+δ2jT22

+ 11+Σ2

jT22

) (D8)

where δj ≡ Ej/~ − ω and Σj ≡ Ej/~ + ω. Eq. D8 isthe nonequilibrium steady-state value for the populationinversion at energy Ej under driving by an acoustic beamat angular frequency ω. Eq. D8 can be used to find thesteady-state solution for Sx,j which can then be pluggedinto the equation of motion for the phonon annihilationoperator to give the TLS-influenced phonon dynamics

i(Ωqη − ω)βqη ≈ Fqη

+∑j

M2η,jq

2T1

2~ΩqηρV

[1

1 + iT2δj− 1

1− iT2Σj

]Sz,j︸ ︷︷ ︸

−i∆ωqη− 12 Γres

qη

βqη.

(D9)

The backreaction of the TLSs on the phonons is charac-terized by a complex susceptibility −i∆ωqη − 1

2Γresqη that

modifies the phonon dynamics in two ways: The real partresults in dissipation of the phonon beam by resonantabsorption, and the imaginary part induces a frequencyshift (to be discussed below).

Assuming the validity of the ergodic hypothesis thesum over j can be converted to an integral. By assumingthat T2 is insensitive to ∆0, and by working in ‘polar’coordinates (E, φ), i.e ∆ = E cosφ and ∆0 = E sinφ,the dissipation rate can be expressed as

Γresqη ≈

Pγ2ηΩqη

~ρv2η

∫ ∞−∞

dET2 tanh(E/2kBT )

1 + (E~ − ω)2T 22 +

4γ2ητminT2

~2 |ξ(+)qη |2

,

(D10)

where ωT2 1 has been used, and T1 is taken to beapproximately given by Eq. C1. For ωT2 1, the in-tegrand is sharply peaked for E ≈ ω. Assuming thatT2, τmin, and tanh(E/2kBT ) vary little over the rangeE ∈ ~(ω − 1/T2, ω + 1/T2) the integral is given approx-imately by Eq. 7 where Ωqη = Ω, ω = Ω, the expres-

sion for the acoustic intensity J = 2ρv3|ξ(+)qη |2 has been

used, and the critical intensity is actually inversely pro-portional to τmin as opposed to T1. With f(∆,∆0) andEq. C1 the density of TLSs as a function of E and τ canbe derived. This distribution is highly peaked near τmin

(see Eq. 3.8 of [42]), and thus we approximate T1 & τmin

in Eq. 8.

Appendix E: Relaxation absorption

Relaxation absorption results from the modulation ofthe TLS’s energy splitting by a time-dependent strain

12

field. Such an effect was neglected in taking the RWAto arrive at Eqs. D7 where we have dropped a Sz,j-dependent drive term in the equation of motion for thephonon amplitude. The time dependence of Sz,j can beproperly accounted for, and its effect on the phonon dy-namics can be taken into account, leading to dissipationand a frequency shift of the phonons.

The steady-state amplitude of the time-dependentcomponent of Sz,j that dominantly couples to βqη is givenby

δS(+)z,j =

1

1− iωT1

∂w0,j

∂EjDη,jξ

(+)qη (rj) (E1)

where δSz,j ≡ Sz,j−w0,j . Accounting for this additionaldrive term in the equation of motion for the phononsleads to

(i(Ωqη − ω + ∆ωqη)+

1

2Γresqη

)βqη = Fqη

− i

V

∑j

D2η,j

4ρv2η

Ωqη

1− iωT1

∂w0,j

∂Ej︸ ︷︷ ︸−i∆ωrel

qη− 12 Γrel

qη

βqη.

(E2)

Just as in the case of resonant absorption, the effect of re-laxation absorption is quantified by the complex suscep-tibility −i∆ωrel

qη− 12Γrel

qη. Here we focus on the relaxation-induced damping coefficient

Γrelqη = − 2

V

∑j

D2η,j

4ρv2η

ΩqηωT1

1 + ω2T 21

∂w0,j

∂Ej. (E3)

To evaluate Γrelqη we employ the ergodic hypothesis and

note an important observation about the relative magni-tudes of the probing frequency and the inversion decayin our experiment. Assuming that T1 is given by the up-per state lifetime of the TLSs then it takes a minimumvalue calculated by taking ∆0 → E in Eq. C1. Fur-thermore, the factor ∂w0/∂E exponentially suppressescontributions to the integrals from E bigger than kBT .Hence, for the frequencies of interest in our experimentit can be shown that ωT1 1 over the integration rangecontributing to Γrel

qη. Thus, a Taylor expansion of the

integrand of Γrelqη is justified for small 1/ωT1 and results

in

Γrelqη =

Pγ2η

ρv2ηkBT

∫d∆

∫d∆0

∆2

∆0E2

1

τ

1

cosh2 E2kBT

(E4)

where we’ve taken Ωqη/ω = 1 and where we’ve replacedT1 with τ . One arrives at Eq. 11 after performing theintegrals.

[1] J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina,Phys. Rev. Lett. 89, 117901 (2002).

[2] Cavity Optomechanics: Nano-and Micromechanical Res-onators Interacting with Light., M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt (eds), (Springer,Berlin/Heidelber, 2014).

[3] M. Aspelmeyer, T. J. Kippenberg and F. Marquardt,arXiv preprint arXiv:1303.0733

[4] R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak,R. W. Simmonds, C. a. Regal, and K. W. Lehnert, Nat.Phys. 10, 321 (2014).

[5] T. Bagci, a Simonsen, S. Schmid, L. G. Villanueva, E.Zeuthen, J. Appel, J. M. Taylor, a Srensen, K. Usami, aSchliesser, and E. S. Polzik, Nature, 507, 81 (2014).

[6] J. Bochmann, A. Vainsencher, D. D. Awschalom, and A.N. Cleland, Nat. Phys., 9, 712 (2013).

[7] A. Pitanti, J. M. Fink, A. H. Safavi-Naeini, C. U. Lei, J.T. Hill, A. Tredicucci, and O. Painter, arXiv:1407.2982.

[8] S. Galliou, M. Goryachev, R. Bourquin, P. Abb, J. P.Aubry, and M. E. Tobar, Sci. Reps. 3, 2132 (2013).

[9] M. Goryachev, D. L. Creedon, S. Galliou, and M. E.Tobat, Phys. Rev. Lett. 111, 085502 (2013).

[10] M. Goryachev, M. E. Tobar, and S. Galliou, Jt. Eur.Freq. Time Forum Int. Freq. Control Symp. 937 (2013).

[11] A. D. OConnell, M. Ansmann, R. C. Bialczak, M.Hofheinz, N. Katz, E. Lucero, C. McKenney, M. Neeley,H. Wang, E. M. Weig, a. N. Cleland, and J. M. Martinis,Appl. Phys. Lett. 92, 112903 (2008).

[12] A. O Connell, M. Hofheinz, M. Ansmann, R. Bialczak, M.Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M.Weides, J. Wenner, J. Martinis, and A. Cleland, Nature(London) 464, 697 (2010).

[13] C. A. Regal, J. D. Teufel, and K. W. Lehnert, Nat. Phys.4, 555 (2008).

[14] J. Teufel, T. Donner, M. Castellanos-Beltran, J. Harlow,and K. Lehnert, Nat. Nanotechnol. 4, 820 (2009).

[15] G. Anetsberger, O. Arcizet, Q. Unterreithmeier, R. Riv-iere, A. Schliesser, E. Weig, M. Gorodetsky, J. Kotthaus,and T. J. Kippenberg, Nat. Phys. 5, 909 (2009).

[16] T. Westphal, D. Friedrich, H. Kaufer, K. Yamamoto, S.Gossler, H. Mu ller-Eberhardt, S. Danilishin, F. Khalili,K. Danzmann, and R. Schnabel, Phys. Rev. A 85, 063806(2012).

[17] T. J. Kippenberg and K. J. Vahala, Opt. Exp. 15, 17172(2007).

[18] T. J. Kippenberg and K. J. Vahala, Science 321, 1172(2008).

[19] F. Marquardt and S. M. Girvin, Physics 2, 40 (2009)[20] H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A.

Starbuck, Z. Wang, and P. T. Rakich, Nat. Commun., 4,1944 (2013).

[21] J. C. Sankey, C. Yang, B. M. Zwicki, a. M. Jayich, andJ. G. E. Harris, Nat. Phys. 6, 707 (2010).

[22] S. M. Meenehan, J. D. Cohen, S. Grblacher, J. T. Hill, A.H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, Phys.Rev. A, 90, 011803 (2014).

13

[23] C. Seoanez, F. Guinea and A. H. Castro Neto, EPL 78,60002 (2007).

[24] M. Goryachev, D. L. Creedon, E. N. Ivanov, S. Galliou,R. Bourquin, and M. E. Tobar, Appl. Phys. Lett., 100,243504 (2012).

[25] M. Goryachev, W. G. Farr, E. N. Ivanov, and M. E.Tobar, J. Appl. Phys., 114, 094506 (2013).

[26] O. Arcizet, R. Riviere, A. Schliesser, G. Anetsberger, andT. J. Kippenberg, Phys. Rev. A 80, 021803(R) (2009).

[27] R. Riviere et. al., Phys. Rev. A 83, 063835 (2011).[28] S. Galliou, J. Imbaud, M. Goryachev, R. Bourquin, and

P. Abb, Appl. Phys. Lett., 98, 091911 (2011).[29] T. Faust, J. Rieger, M. J. Seitner, J. P. Kotthaus, and

E. M. Weig, Phys. Rev. B 89, 100102(R) (2014).[30] J. M. Martinis, K. B. Cooper, R. McDermott, M. Stef-

fan, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P.Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett95, 210503 (2005).

[31] M. Constantin, C. C. Yu, J. M. Martinis, Phys. Rev. B79, 094520 (2009).

[32] E. R. MacQuarrie, T. a. Gosavi, N. R. Jungwirth, S. a.Bhave, and G. D. Fuchs, Phys. Rev. Lett., 111, 227602,(2013).

[33] E. R. MacQuarrie, T. a. Gosavi, a. M. Moehle,N. R. Jungwirth, S. a. Bhave, and G. D. Fuchs,arXiv:1411.5325

[34] M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N.Katz, E. Lucero, a. OConnell, H. Wang, a. N. Cleland,and J. M. Martinis, Nat. Phys., 4, 523 (2008).

[35] G. Grabovskij, T. Peichl, J. Lisenfeld, G. Weiss, and A.V. Ustinov, Science 338, 232 (2012).

[36] B. Golding, J. E. Graebner, B. I. Halperin, R. J. Schutz,Phys. Rev. Lett. 30, 223 (1973).

[37] B. Golding and J. E. Graebner, Phys. Rev. Lett 37, 852(1976).

[38] B. Golding, J. E. Graebner, and R. J. Schutz, Phys. Rev.B 14, 1660 (1976).

[39] P. W. Anderson, B. I. Halperin, and C. M. Varma, Phil.Mag. 25, 1 (1972).

[40] J. Jackle, Z. Phys, 257, 212 (1972).[41] S. Hunklinger, Le Journal de Physique Colloques 43.C9,

C9-461 (1982).[42] W. A. Phillips, Rep. Prog. Phys 50, 1657 (1987).[43] D. O. Krimer, B. Hartl, S. Rotter, arXiv:1501.03487[44] T. Ramos, V. Sudhir, K. Stannigel, P. Zoller and T. J.

Kippenberg, Phys. Rev. Lett. 110, 193602 (2013).[45] R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys.

74, 991 (2002).[46] S. Le Floch and P. Cambon, Opt. Comm. 219, 395

(2003).[47] T. Sonehara, Y. Konno, H. Kaminaga and S. Saikan, J.

Korean Phys. Soc. 51, 836 (2007).[48] P. Dragic, J. Non. Crys. Sol. 355, 403 (2009).[49] C.-K. Jen, A. Safaai-Jazi and G. W. Farnell, IEEE

Trans. Ultrason. Ferroelectr. Freq. Control UFFC-33,634 (1986).

[50] J. Rose, Ultrasonic Waves in Solid Media, (CambridgeUniversity Press, Cambridge, 1999).

[51] F. Kong and L. Dong, Opt. Exp. 20, 27810 (2012).[52] R. N. Thurston, J. Sound and Vibration, 159, 441 (1992).[53] R. Shelby, M. Levenson, and P. Bayer, Phys. Rev. B 31,

5244 (1985).[54] A. Kobyakov, M. Sauer, and D. Chowdhury, Adv. Opt.

Photon. 2, 1 (2009).[55] E. Ippen and R. Stolen, Appl. Phys. Lett. 21, 539 (1972).[56] R. Boyd, Nonlinear Optics (Academic Press, Boston,

2009), 3rd ed.[57] G. Agrawal, Nonlinear Fiber Optics (Academic Press,

San Diego, 1995).[58] J. E. Graebner, B. Golding, and L. C. Allen, Phys. Rev.

B 34, 5696 (1986).