www.sciencemag.org/cgi/content/full/science.1195596/DC1

Supporting Online Material for

Optomechanically Induced Transparency

Stefan Weis, Rémi Rivière, Samuel Deléglise, Emanuel Gavartin, Olivier Arcizet, Albert Schliesser, Tobias J. Kippenberg*

*To whom correspondence should be addressed. E-mail: tobias.kippenberg@epfl.ch

Published 11 November 2010 on Science Express DOI: 10.1126/science.1195596

This PDF file includes:

SOM Text Figs. S1 to S3 References

Supporting Online Material for Optomechanicallyinduced transparency

Stefan Weis1,2,†, Remi Riviere2,†, Samuel Deleglise1,2,†, Emanuel Gavartin1,Olivier Arcizet3, Albert Schliesser1,2, Tobias J. Kippenberg1∗

1Ecole Polytechnique Federale de Lausanne, EPFL, 1015 Lausanne, Switzerland2Max-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany

3Institut Neel, 25 rue des Martyrs, 38042 Grenoble, France.

∗To whom correspondence should be addressed; E-mail: tobias.kippenberg@epfl.ch.†These authors contributed equally to this work.

Contents1 Derivation of OMIT 2

2 Comparison with EIT in Atomic Physics 6

3 Simplifications in the weak coupling case 7

4 Experimental setup 9

5 Measurement using the phase modulation scheme 9

6 Group delay 13

7 Theoretical treatment of a splitted resonance 13

8 Relation of OMIT to coherent mixing of mechanical excitations 16

9 Table of symbols 17

1

1 Derivation of OMITIn the following, we derive the expressions describing the optomechanical equivalent of Elec-tromagnetically Induced Transparency (EIT), as discussed in (1,2), as well as more recent anal-ysis (3). The starting point of the following analysis is the Hamiltonian formulation of a genericoptomechanical system put forward by Law (4).

1.1 HamiltonianIf the free spectral range of the cavity is much larger than the mechanical oscillation frequency,such that only one optical mode is coupled to the mechanical mode, the optomechanical Hamil-tonian can be written as:

H = Hmech + Hopt + Hint + Hdrive (S1)

Hmech =p2

2meff+

1

2meffΩ

2mx

2 (S2)

Hopt = ~ωc

(a†a+

1

2

)(S3)

Hint = ~Gx a†a (S4)

Hdrive = i~√ηcκ(sin(t)a† − s∗in(t)a

), (S5)

where x and p are the position and momentum operators of the mechanical degree of freedomhaving effective mass meff and angular frequency Ωm, and sin(t) is the drive amplitude normal-ized to a photon flux at the input of the cavity. a and a† are the annihilation and creation opera-tors of the cavity mode. We have furthermore used the coupling parameter ηc ≡ κex/κ0 + κex,where κ0 denotes the intrinsec loss rate and κex the external loss rate (i.e. wave guide coupling).Experimentally, the parameter ηc can be continuously adjusted by tuning the taper-resonatorgap (5, 6).

We will solve this problem for a driving field sin(t) = (sin + δsin(t)) e−iωlt, where ωl isthe driving laser frequency, and we deliberately identify sin = sl. We will then first derive thelinearized Langevin equations (7) for a generic perturbation term δsin(t) before identifying itwith the probe field δsin(t) = spe

−i(ωp−ωl)t.

2

1.2 Langevin equationsIn a frame rotating at ωl with ∆ = ωl − ωc, we obtain:

d

dta(t) =

(+i∆− κ

2

)a(t)− iGx(t)a(t) +

√ηcκ sin(t) +

√(1− ηc)κ δsvac(t) (S6)

d

dtx(t) =

p(t)

meff(S7)

d

dtp(t) = −meffΩ

2mx(t)− ~Ga†(t)a(t)− Γmp(t) + δFth(t), (S8)

where the decay rates for the optical (κ) and mechanical oscillators (Γm) have been introducedclassically, and δsvac(t) and δFth(t) are the quantum and thermal noise terms (8). We firstdenote a and x the intracavity field and mechanical displacement for the static solution, inwhich all time derivatives vanish and sp → 0. From (S6)(S8), it follows immediately that a andx must fulfill the self consistent equations:

a =

√ηcκ

−i(∆−Gx) + κ/2sin (S9)

x =a2

meffΩ2m

, (S10)

where we have assumed a to be real and positive. This system can give rise to bistability forsufficiently strong control fields (9) (7). However, for weak and detuned control fields, onlyone solution exists and |a|2 ∝ ηcsin

2. We then linearize the problem for δa |a|, plugging theansatz a(t) = a + δa(t) and x(t) = x + δx(t) into equations (S6)(S7)(S8) and retain only firstorder terms in the small quantities δa, δa† and δx. We then obtain

d

dtδa(t) =

(+i∆− κ

2

)δa(t)− iGaδx(t) +

√ηcκδsin(t) +

√(1− ηc)κ δsvac(t) (S11)

d2

dt2δx(t) + Γm

d

dtδx(t) + Ω2

mδx(t) = − ~Gmeff

a(δa(t) + δa†(t)

)+ δFth(t), (S12)

where we used the Hermitian property δx(t) = δx†(t) in equation (S11), and introduced thecorrected detuning ∆ = ∆ − Gx. Since the drives are weak, but classical coherent fields, wewill identify all operators with their expectation values y(t) ≡ 〈y(t)〉, and drop the quantumand thermal noise terms: which average to 0.

3

1.3 Solution1.3.1 General solution

We now have to solve the equations of the expectation values for the drive (in the rotating frame)δsin(t) = spe

−i(ωp−ωl)t. For a given Ω = ωp − ωl we use the ansatz

δa(t) = A− e−iΩt + A+ e+iΩt (S13)

δa∗(t) = (A+)∗ e−iΩt + (A−)∗ e+iΩt (S14)

δx(t) = X e−iΩt +X∗ e+iΩt. (S15)

If sorted by rotation terms, this yields six equations. However, the probe field’s transmission atfrequency ωl + Ω only depends on A−. In this sense, the three equations of interest are:(

−i(∆ + Ω) + κ/2)A− = −iGaX +

√ηcκsp (S16)(

+i(∆− Ω) + κ/2)

(A+)∗ = +iGaX (S17)

meff(Ω2

m − Ω2 − iΓmΩ)X = −~Ga

(A− + (A+)∗

). (S18)

The solution of interest is

A− =1 + if(Ω)

−i(∆ + Ω) + κ/2 + 2∆f(Ω)

√ηcκsp, (S19)

with

f(Ω) = ~G2a2 χ(Ω)

i(∆− Ω) + κ/2(S20)

and the mechanical susceptibility

χ(Ω) =1

meff

1

Ω2m − Ω2 − iΩΓm

. (S21)

1.3.2 Spectrum of the transmitted light

Using the input-output relation (10), one obtains:

sout(t) = sin(t)−√ηcκ a(t) (S22)

= (sc −√ηcκa)e−iωct + (sp −

√ηcκA

−)e−i(ωc+Ω)t −√ηcκA+e−i(ωc−Ω)t. (S23)

The transmission of the probe field is then given by:

tp =sp −

√ηcκA

−

sp

(S24)

= 1− 1 + if(Ω)

−i(∆ + Ω) + κ/2 + 2∆f(Ω)ηcκ. (S25)

4

Here, we refer to the probe transmission in the general sense of the ratio of the probe fieldreturned from the system divided by the sent probe field. Depending on the cavity geometry,the returned probe field corresponds to the probe field reflected from the input coupler (Fabry-Perot-type resonator) or the probe field transmitted through the fiber taper used for evanescentcoupling of an ideal WGM resonator. In general, the ”returned probe field” refers to the probefield in the spatial mode which is fed by the probe if it is tuned off-resonance from the cavitymode.

1.3.3 Resolved-sideband limit

In the resolved sideband regime (11) (κ Ωm), the lower sideband, far off-resonance, can beneglected:

A+ ≈ 0.

In addition, we can linearize the mechanical susceptibility for small values of the parameter∆′ = Ω− Ωm:

meff(Ω2m − Ω2 − iΓmΩ) ≈ Ωm(2∆′ − iΓm).

The system (S16)(S17)(S18) then simplifies to:(−i(∆ + Ωm + ∆′) + κ/2

)A− = −iGaX +

√ηcκsp (S26)

Ωm(2∆′ − iΓm)X = −~GaA−. (S27)

The solution for the intracavity field is:

A− =

√ηcκsp

−i(∆ + Ωm + ∆′) + κ/2 + Ω2c/4

(−i∆′+Γm/2)

, (S28)

where we have introduced the coupling between the mechanical and optical resonators:

Ωc = 2Gaxzpf ,

with

xzpf =

√~

2meffΩm

,

the zero point fluctuations amplitude of the mechanical oscillator. This formula becomes

A− =

√ηcκsp

−i∆′ + κ/2 + Ω2c/4

(−i∆′+Γm/2)

(S29)

for a control laser tuned to the lower motional sideband (∆ = −Ωm).The achievable coupling rate is only limited by the appearence of radiation pressure bista-

bility for high control power. In the resolved sideband regime, the upper limit for the couplingrate is found to be Ωc < Ωm (12, 13).

We will now briefly review the formalism used to describe EIT in the context of atomicphysics to emphasize the analogy between the two phenomena.

5

2 Comparison with EIT in Atomic PhysicsFor atomic EIT, we essentially revisit the well-known derivation developed in (14), which willassist in identifying the close resemblance between OMIT and atomic EIT. We consider a Λ-system, consisting of a common upper state |3〉 and two (long-lived) ground states |1〉 and |2〉.In a semiclassical treatment (15), the relevant Hamiltonian is given by:

H =∑j

~ωjσjj −~2µ23(σ23 + σ32)E(t)− ~

2µ13(σ13 + σ31)E(t), (S30)

where σij = |i〉〈j| are the atomic projection operators and i, j ∈ 1, 2, 3 label the three in-volved levels. ωij and µij are the frequency and dipole moment along the electric field’s di-rection for the i → j transition. The (classical) field contains the two (coupling and probe)components

E(t) =1

2Ec

(e−iωct + e+iωct

)+

1

2Ep

(e−iωpt + e+iωpt

), (S31)

where ωc is tuned close to ω32 and ωp close to ω31. The usual Heisenberg equations of motionfor the operators σij can then be derived using i~dσij

dt= [σij, H]. Retaining only near-resonant

terms, the equations of motion can be written as

˙σ12 = −iω21σ12 +i

2~µ23Ecσ13e

+iωct − i

2~µ13Epσ32e

−iωpt (S32)

˙σ23 = −iω32σ23 +i

2~µ23Ec(σ22 − σ33)e−iωct +

i

2~µ13Epσ21e

−iωpt (S33)

˙σ13 = −iω31σ13 +i

2~µ23Ecσ12e

−iωct +i

2~µ13Ep(σ11 − σ33)e−iωpt. (S34)

We emphasize that the rotating wave approximation (neglecting all non-resonant contributions)is analogous to the resolved sideband approximation presented in the context of OMIT. For asufficiently weak probe field, the expectation values σij = 〈σij〉 can further be approximated toobey σ11 ≈ 1 and σ22 ≈ σ33 ≈ σ23 ≈ σ32 ≈ 0 at all times, while the remaining expectationvalues obey

σ12 = −i(ω21 − iγ12/2)σ12 +i

2~µ23Ecσ13e

+iωct (S35)

σ13 = −i(ω31 − iγ13/2)σ13 +i

2~µ23Ecσ12e

−iωct +i

2~µ13Epe

−iωpt, (S36)

where damping rates γ12 and γ13 were introduced classically. Changing to a rotating frameσ12 = S12e

−iΩt, σ13 = S13e−i(ωc+Ω)t and Epe

−iωpt = Epe−i(ωc+Ω)t with ωp = ωc + Ω, we obtain

in the steady state

(−i(Ω− ω21) + γ12/2)S12 = +i

2~µ23EcS13 (S37)

(−i(Ω + ωc − ω31) + γ13/2)S13 = +i

2~µ23EcS12 +

i

2~µ13Ep, (S38)

6

which is solved by

S13 =iµ13Ep/2~

−i(∆′ + ωc − ω32) + γ13/2 + Ω2c/4

−i∆′+γ12/2

, (S39)

where we now abbreviate ∆′ = Ω − ω21 = ωp − ω31. We note as an aside that an equivalentcalculation can be made for the atomic coherences ρ12 and ρ13, yielding essentially the sameresult (15). This result simplifies for a control field on resonance (ωc = ω32):

S13 =iµ13Ep/2~

−i∆′ + γ13/2 + Ω2c/4

−i∆′+γ12/2

. (S40)

The induced dipole moment along the electric field’s direction is given by p = µ13(σ13 + σ31)so that the polarizibility α of the medium at the probe frequency in the presence of the couplingbeam can be directly given by

α =µ13S13

Ep/2=

iµ213/~

(−i∆′ + γ13/2) + Ω2c/4

(−i∆′+γ12/2)

. (S41)

Evidently one can identify a formal correspondence between the physical entities involved inEIT in atomic physics and OMIT in optomechanical systems. Equations (S37)(S38)(S39)(S40)are perfectly equivalent to (S26)(S27)(S28)(S29) by applying the identifications listed in thefollowing table.

Table S1: Comparison of physical entities relevant for EIT and OMIT.

EIT OMITprojection operator σ13 (coherence ρ13) intracavity field amplitude A−

projection operator σ12 (coherence ρ12) mechanical displacement amplitude Xenergy difference between ground states~ω21

phonon energy ~Ωm

Rabi frequency µ23Ec/~ optomechanical coupling rate 2Gaxzpf

3 Simplifications in the weak coupling caseIn addition to the resolved sideband approximation, we will consider the case where the op-tomechanical coupling is weak compared to the optical losses (Ωc,Γm κ). We also assumethat the control laser is tuned on the lower sideband (∆ = −Ωm). Then, the EIT feature is very

7

well described by a Lorentzian transmission window in the optical transmission spectrum. Thiscan be seen by applying the simplification −i∆′ + κ/2 ≈ κ/2 in equation (S29):

A− ≈4√ηcκ(−i∆′ + Γm/2)

2κ(Γm/2− i∆′) + Ω2c

sp. (S42)

Plugging the corresponding value of A− in (S24), one obtains:

tp = 1− 2ηc +2ηcΩ

2c

Ω2c + Γmκ− 2i∆′κ

. (S43)

In order to isolate the interesting physics of OMIT from the well-understood waveguide-cavitycoupling effects, we introduce the normalized transmission:

t′p =tp − tr1− tr

, (S44)

where tr is the residual on resonance transmission in the absence of a coupling laser:

tr = tp(∆′ = 0,Ωc = 0) (S45)

= 1− 2ηc. (S46)

The normalized transmission is then independant of ηc:

t′p =Ω2c

Ω2c + Γmκ− 2i∆′κ

. (S47)

This corresponds to the transmission in the case of critical coupling ηc = 1/2. The optome-chanically induced transparency window is hence given by:

|t′p|2 =Ω4c/κ

2

(Ω2c/κ+ Γm)2 + (2∆′)2 . (S48)

A Lorentzian of width

ΓOMIT = Γm + Ω2c/κ (S49)

(S50)

and peak value

|t′p(∆′ = 0)|2 =

(Ω2c/κ

Γm + Ω2c/κ

)2

. (S51)

These two quantities can be expressed very simply by introducing the cooperativity of the cou-pled systems C = Ω2

c/(Γmκ) (16):

ΓOMIT = Γm(1 + C) (S52)

|t′p(∆′ = 0)|2 =

(C

1 + C

)2

. (S53)

8

4 Experimental setupAs shown in Figure S1, the experiment was carried out at cryogenic temperatures using aHelium-3 buffer gas cryostat. In addition to reducing the thermal Brownian motion of themechanical oscillator this allows eliminating thermo-optical nonlinearities (17) which can im-pede driving the lower motional sideband with a strong control laser. The sample is mountedon a cryogenic head, which allows approaching a tapered fiber for near field evanescent cou-pling using piezoelectric positioners. While an external-cavity diode laser was used for initialcharacterization, a low-noise, continuous-wave Titanium Sapphire laser operating at a wave-length of λ ≈ 775 nm is employed for the actual OMIT experiments. The Ti:sapphire laser’slinewidth is reduced below 30 kHz by stabilization to a temperature-controlled reference cavityusing the Pound-Drever-Hall technique. This approach furthermore proved to provide sufficientmutual frequency stability of the cryogenic microresonator and the cavity-stabilized laser on therelevant scale of the cavity linewidth κ.

In order to detect the optical response of the optomechanical system with a high sensitivity,we use a homodyne detection scheme: The RF beat of the beam transmitted through the op-tomechanical system with a strong local oscillator is detected using fast balanced photodiodes.

5 Measurement using the phase modulation schemeFor technical reasons, the optical response was probed using a frequency modulation technique:the coupling laser is phase modulated using an EOM (Fig. S1) at frequency Ω, hence creatingtwo sidebands at ωl + Ω and ωl −Ω. In the resolved sideband regime, only the upper sideband,close to resonance interacts with the cavity, acting as a weak probe beam. The lower one andcarrier are transmitted unchanged through the tapered fiber. However, one has to take them intoaccount in order to understand quantitatively the obtained results. We will show here that themeasured signal is linked to the transmission at the probe frequency through a direct relation.

The incident fields at the homodyne beamsplitter are a carrier and two sidebands of thelocal oscillator, and the carrier and two sidebands of the beam entering the cavity (Fig. S1).We note tc, tus and tls the complex transmission coefficient across the cavity for the carrier,upper sideband and lower sideband respectively. The phase of the local oscillator Φ is activelyadjusted so that it matches the phase of the control beam emerging from the cavity.

At one exit of the beamsplitter the optical power is proportional to∣∣∣∣Ecave−iωlt

(tc + i

β

2e−iΩttus + i

β

2e+Ωttls

)+ iELOe

−iωlteiΦ(

1 + iβ

2e−iΩt + i

β

2eiΩt)∣∣∣∣2 ,

(S54)

where β is the depth of the modulation induced by the EOM, ELO and Ecav are the field ampli-tudes in the local oscillator and signal arms of the homodyne setup. The interesting terms are

9

Mod

ulat

ion W

BalancedhomodynedetectorPhase

control

3Hecryostat

Referenceresonator

Ti:Sapphire laser

Networkanalyzer

out in

Frequencystabilization

Figure S1: Experimental setup. An optomechanical system consisting of a toroid microres-onator held at cryogenic temperatures in a Helium-3 buffer gas cryostat. The control and probefields are derived from a single Ti:sapphire laser, which is stabilized to an external referenceresonator using the Pound-Drever-Hall technique. While the laser carrier is used as controlbeam, the probe beam is created by a phase modulator driven at the radio frequency Ω. Thisoptical input is split into two arms, one of which is sent to a tapered fiber in the cryostat, whichallows optical coupling to the whispering gallery mode of the toroidal resonator as shown inmicrograph with a ∼ 60µm-diameter toroid. The other arm serves as the local oscillator in abalanced homodyne receiver used to analyze the light returned from the optomechanical sys-tem. While the receiver’s DC component is used to lock the phase of the local oscillator, theAC component is analyzed using a network analyzer.

10

FTi:Sap

phire

Laser

ELO

Ecav

wl

wl

wlwl

Figure S2: The optical setup as described in Figure S1. The laser is phase modulated, creatingtwo sidebands at frequency ωl ±Ω. The local oscillator field is transmitted unchanged whereasthe field in the signal arm is affected by the cavity transmission. In the RSB regime, lowersideband and carrier, off resonant by approximately 2Ωm and Ωm are not affected.

11

the modulated cross-terms, they are given by

2 Re

[(Ecave

−iωlttc)·(iELOe

−iωlteiΦ(iβ

2e−iΩt + i

β

2e+iΩt

))∗+

(Ecave

−iωlt

(iβ

2e−iΩttus + i

β

2e+iΩttls

))·(iELOe

−iωlteiΦ)∗]

= βEcavELO Re[tc ·(−e−iΦ

(e+iΩt + e−iΩt

))+(e−iΩttus + e+iΩttls

)·(e−iΦ

)]= βEcavELO Re

[(e−iΦ

) (−tc2 cos(Ωt) +

(e−iΩttus + e+iΩttls

))](S55)

Now writing real and imaginary parts of the used functions as

e−iΦ ≡ Φ′ + iΦ′′ (S56)tc ≡ t′c + it′′c (S57)tus ≡ t′us + it′′us (S58)tls ≡ t′ls + it′′ls (S59)

we get (omitting the prefactor βEcavELO)

cos(Ωt) (−2Φ′t′c + 2Φ′′t′′c + (t′us + t′ls)Φ′ − (t′′us + t′′ls)Φ

′′)︸ ︷︷ ︸A

+ sin(Ωt) (−(t′′us − t′′ls)Φ′ − (t′us − t′ls)Φ′′)︸ ︷︷ ︸B

(S60)

A and B represent the in-phase and quadrature response of the system to the input modulation.In the resolved sideband regime, only the upper sideband is affected by the cavity. In this caseΦ′ = t′c = t′ls = 1 and Φ′′ = t′′c = t′′ls = 0. Moreover, the upper sideband, close to resonance, isprobing the OMIT signal tus = tp. The quadratures measured by the network analyzer are then:

A ≈ 1− t′us = 1− Re(tp) (S61)B ≈ −t′′us = − Im(tp). (S62)

The complex amplitude response thom = A+ iB as measured on the network analyzer is hencegiven in good approximation by:

thom ≈ 1− tp. (S63)

The normalized response t′hom is directly related to the normalized transmission tp′:

t′hom =thom

1− tr(S64)

= 1− tp′. (S65)

In particular, if we consider the form (S47) for the probe beam transmission, the measuredsignal is then given by:

t′hom =Γmκ− 2i∆′κ

Ω2c + Γmκ− 2i∆′κ

. (S66)

12

We can easily calculate the normalized transmitted power:

|t′hom|2 = 1− Ω2c/κ (Ω2

c/κ+ 2Γm)

(Γm + Ω2c/κ)2 + (2∆′)2

(S67)

The measured signal is hence an inverted lorentzian peak with width ΓOMIT (same width as|tp|2). The minimum value of the dip |t′hom(∆′ = 0)|2 can be linked very easily to the maximumvalue of the transmission window |t′p(∆′ = 0)|2 by remarking that for ∆′ = 0 the transmissioncoefficients tp and thom are real. The relation (S63) gives then:

|t′hom(∆′ = 0)|2 =(

1−√|t′p(∆′ = 0)|2

)2

(S68)

6 Group delayEIT is the underlying phenomenon allowing for slowing down of light pulses. Indeed, the sharptransparency window in the medium is accompanied by a very abrupt change of its refractiveindex leading to slow group velocities (see (14) for a detailed analysis of the phenomenon).In the case of a single optically active element like an optomechanical device, the rapid phasedispersion φ(ω) = arg(tp(ω)) leads to a ‘group delay’ τg given by:

τg = −dφdω. (S69)

A full calculation based on the expression (S29) shows that the group delay diverges for smallvalues of the transparency. However, in the regime C & 1, where the medium is not completelyopaque, a simple calculation based on expression (S47) is perfectly valid:

φ(∆′) = arctan

(2∆′κ

Ωc + Γmκ

). (S70)

This gives for the middle of the transparency window (∆′ = 0):

τg(∆′ = 0) =

2κ

Ω2c + Γmκ

(S71)

=1

Γm

(2

C + 1

)(S72)

=2

ΓOMIT

. (S73)

7 Theoretical treatment of a splitted resonanceOur ring cavity can support two counterpropagating modes which are frequency degeneratefor symmetry reasons. The propagation direction of the light in the coupling region therefore

13

determines which mode is excited. However, as noted in early work on microspheres (18), andin theoretical as well as experimental work (19–21), due to residual scattering of light at thesurface or in the bulk glass, the counterpropagating mode can also be significantly populated.

accw

probe in probe out

probe frequency

1

|tr|2

acw

Figure S3: In a ring cavity, two frequency-degenerate counterpropagating modes acw and accw

coexist. Only accw is coupled to the waveguide, however, because of scattering into the counter-propagating mode, acw and accw are coupled at a rate γ. For γ κ, the frequency degeneracyof the new eigenmodes a+ = (accw + acw)/

√2 and a− = (accw − acw)/

√2 is lifted and the

optical resonance splits up.

The essence of the phenomenon can be described by a coupled mode theory: if the twomodes acw and accw (see figure S3) are coupled by a coupling rate γ, the equations of motionbecome:

accw(t) = (i(∆−Gx)− κ/2)accw(t) + iγ

2acw(t) +

√ηcκsin(t) (S74)

acw(t) = (i(∆−Gx)− κ/2)acw(t) + iγ

2accw(t), (S75)

and the radiation pressure force is now described by the equation:

d2

dt2x(t) + Γm

d

dtx(t) + Ω2

mx(t)+ = − ~Gmeff

(|accw|2 + |acw|2). (S76)

Indeed, because of the symmetry of the radial breathing mode, the oscillator is not driven bythe cross term 2Re(a∗ccwacw). We can then easily rewrite these equations in terms of the two

14

stationary modes a+ = (accw + acw)/√

2 and a− = (accw − acw)/√

2:

a+(t) =[i(∆−Gx+

γ

2)− κ/2

]a+(t) +

√ηcκ

2sin(t) (S77)

a−(t) =[i(∆−Gx− γ

2)− κ/2

]a−(t) +

√ηcκ

2sin(t) (S78)

d2

dt2x(t) + Γm

d

dtx(t) + Ω2

mx(t) = − ~Gmeff

(|a+|2 + |a−|2). (S79)

The two stationary modes are the eigenmodes of the evolution and the degeneracy is lifted bythe coupling rate γ.

In the limit γ κ, the two modes are well resolved and only one of them (a−) has tobe considered since a+ is non resonant and hence not populated (also, the experimental condi-tions have to be chosen in order to avoid a mechanical sideband coinciding with the resonancefrequency of a+). In this limit, the optomechanical system reads:

a−(t) =[i(∆−Gx− γ

2)− κ/2

]a−(t) +

√ηcκ

2sin(t) (S80)

d2

dt2x(t) + Γm

d

dtx(t) + Ω2

mx(t) = − ~Gmeff

|a−|2. (S81)

This system is equivalent to (S11) and (S12) after formally replacing the coupling parameter ηc

by ηc ≡ ηc/2, and ∆ by ∆ ≡ ∆− γ/2. This equivalence also holds for the transmitted fields

sout(t) = sin(t)−√ηcκ accw(t) = sin(t)−√ηcκ/2 (a−(t) + a+(t))

≈ sin(t)−√ηcκ/2 a−(t) (S82)

as illustrated in figure S3. The reduced effective coupling parameter arises from the scattering ofhalf of the intracavity power to the uncoupled mode. Note that in this case part of the intracavitylight can be dissipated to a reflected field

sref(t) = −√ηcκ acw(t) = −√ηcκ/2 (−a−(t) + a+(t))

≈√ηcκ/2 a−(t), (S83)

which can consitute another channel of optical dissipation. In the regime we consider, an opticalbeam resonant with the a− cavity eigenmode dissipates a fraction of 2ηc(1−ηc) of the launchedpower due to the intrinsic losses of the cavity, a fraction of (1−ηc)

2 is transmitted and a fractionof η2

c is reflected.With our present settings, we measured a residual transmission of |tr|2 ≈ 0.5, we can hence

infer the effective coupling parameter ηc by solving

|tr|2 = 1/2 = (1− 2ηc)2, (S84)

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leading to

ηc =2−√

2

4≈ 0.15. (S85)

The intracavity power is hence smaller than the one calculated in the “standard” situation ηc =1/2 and γ = 0. In the calculation of the coupling rate Ωc, we took this factor into account;an additional reduction factor of 1.9 had to be introduced to account for taper losses in thisexperiment.

8 Relation of OMIT to coherent mixing of mechanical exci-tations

Due to a significant overlap in the employed terminology, we would like to briefly explainthe main differences between OMIT and the recently reported coherent mixing of mechanicalexcitations (22).

The systems used to demonstrate coherent mixing of mechanical excitations feature a pair ofmechanical oscillators, coupled by a coupling rate γ (denoted as κ in the original manuscript).The displacement of one oscillator can be optically read out using a cavity, it is therefore referredto as “bright” mode (displacement xb). The other mechanical mode does not interact with lightand is therefore designated as “dark” mode (displacement xd). The (mechanical) dynamics ofthe two coupled mechanical modes can be described by the Hamiltonian

Hm =p2b

2mb

+1

2mbΩ

2mbx

2b +

p2d

2md

+1

2mdΩ

2mdx

2d + γxbxd, (S86)

where pb (pd), Ωb (Ωd) and mb (md) are the momentum, mechanical frequency, and effectivemass of the bright (dark) mode.

Such coupled mechanical systems have been interpreted as a mechanical analog to EIT(23), as for sufficiently close mechanical resonance frequencies Ωmb and Ωmd, a cancellationof the mechanical response to an external force at intermediate Fourier frequencies can occur.This cancellation can be interpreted as a mechanical ‘transparency’, arising from destructiveinterference of mechanical excitation pathways. Importantly, however, here the interferenceoccurs exclusively in the mechanical domain.

In ref. (22) such a mechanical transparency is reported for two different mechanical oscilla-tor systems. In both cases, the mechanical response of the bright mode to the thermal Langevinforce is read out using the optical cavity transducer. At the same time, it is demonstrated that thecancellation of a mechanical response can be induced also in systems in which Ωmb and Ωmd areoriginally too different for the interference to be observed, by tuning the resonance frequencyΩmb of the bright mode using the optical spring effect in the cavity.

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However, this mechanical interference and the one responsible for OMIT are very differentin their nature. Evidently, OMIT is based on an optical interference effect. A second mechan-ical mode is therefore not necessary to implement the scheme. In OMIT, the control beamcouples the optical probe field to a mechanical mode, and thus generates multiple excitationpathways for an intracavity probe field, as described in (1–3). As an important consequenceOMIT provides a direct possibility to control the transmission of an optical probe beam.

In this context it is important to note that the term “cavity transmission” is used in a differentway in ref. (22) and this work: In ref. (22), the “cavity transmission spectrum” refers to theRF/microwave-spectrum of the power fluctuations of the light emerging from the cavity, as itis induced by the thermally driven motion of the mechanical bright mode. In our manuscript,we refer to the transmission of the probe field as the ratio of the probe field amplitude returnedfrom the cavity divided by the sent probe field amplitude, directly providing the required transferfunction for light slowing and storage protocols (24, 25).

9 Table of symbols

17

symbol meaning definitionωl laser frequencyωc cavity resonance frequencyωp probe frequency∆ detuning of the control field ∆ = ∆−Gxκ optical linewidth (FWHM)κ0 intrinsic loss rateκex coupling rate to the waveguideηc coupling parameter ηc = κex/(κ0 + κex)a mean intracavity mode amplitudesin mean drive amplitudeG optomechanical coupling dω′c/dx

Ωm mechanical resonance frequencyΓm mechanical damping ratemeff effective massx equilibrium displacement

∆′ detuning of the probe from the center of theOMIT feature

∆′ = ωp − ωl − Ωm

xzpf zero-point fluctuations xzpf =√~/ (2meffωm)

Ωc optomechanical coupling rate Ωc = 2Gaxzpf

C cooperativity C = Ω2c/(Γmκ)

χ(Ω) mechanical susceptibility χ(Ω) = (meff(Ω2m − Ω2 − iΓmΩ))−1

tp complex amplitude transmission at probefrequency

thom complex transmission signal measured in thehomodyne receiver

β Modulation depthtus complex transmission of the upper sidebandtc complex transmission of the carriertls complex transmission of the lower sidebandacw amplitude of the clockwise propagating

modeaccw amplitude of the counterclockwise propagat-

ing modeγ coupling between the two counter propagat-

ing modesa+ symmetric stationary mode a+ = (acw + accw) /

√2

a− antisymmetric stationary mode a− = (acw − accw) /√

2

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References and NotesS1. A. Schliesser, Cavity optomechanics and optical frequency comb generation with silica

whispering-gallery-mode microresonators, Ph.D. thesis, Ludwig-Maximilians-UniversitatMunchen (2009). Http://edoc.ub.uni-muenchen.de/10940/, page 121ff.

S2. A. Schliesser, T. J. Kippenberg, Advances in atomic, molecular and optical physics,E. Arimondo, P. Berman, C. C. Lin, eds. (Elsevier Academic Press, 2010), vol. 58, chap. 5,pp. 207–323.

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S12. F. Marquardt, J. P. Chen, A. A. Clerk, S. M. Girvin, Physical Review Letters 99, 093902(2007).

S13. J. M. Dobrindt, I. Wilson-Rae, T. J. Kippenberg, Physical Review Letters 101, 263602(2008).

S14. P. W. Milonni, Fast light, slow light and left-handed light (Taylor and Francis, 2005).

S15. M. O. Scully, M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997).

S16. H. J. Kimble, Cavity Quantum Electrodynamics, P. R. Berman, ed., Advances in Atomic,Molecular and Optical Physics (Academic Press, 1994).

S17. O. Arcizet, R. Riviere, A. Schliesser, G. Anetsberger, T. J. Kippenberg, Physical ReviewA 80, 021803(R) (2009).

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S18. D. S. Weiss, et al., Optics Letters 20, 1835 (1995).

S19. M. L. Gorodetsky, A. D. Pryamikov, V. S. Ilchenko, Journal of the optical society ofAmerica B - Optical physics 17, 1051 (2000).

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