Coherent optical wavelength conversion via cavity optomechanics

Abstract

Both classical and quantum systems utilize the interaction of light and matter across a wide range of energies. These systems are often not naturally compatible with one another and require a means of converting photons of dissimilar wavelengths to combine and exploit their different strengths. Here we theoretically propose and experimentally demonstrate coherent wavelength conversion of optical photons using photon-phonon translation in a cavity-optomechanical system. For an engineered silicon optomechanical crystal nanocavity supporting a 4-GHz localized phonon mode, optical signals in a 1.5 MHz bandwidth are coherently converted over a 11.2 THz frequency span between one cavity mode at wavelength 1,460 nm and a second cavity mode at 1,545 nm with a 93% internal (2% external) peak efficiency. The thermal- and quantum-limiting noise involved in the conversion process is also analysed, and in terms of an equivalent photon number signal level are found to correspond to an internal noise level of only 6 and 4 × 10^-3 quanta, respectively.

ARTICLE

Received 6 Aug 2012 | Accepted 12 Oct 2012 | Published 13 Nov 2012

Both classical and quantum systems utilize the interaction of light and matter across a wide range of energies. These systems are often not naturally compatible with one another and require a means of converting photons of dissimilar wavelengths to combine and exploit their different strengths. Here we theoretically propose and experimentally demonstrate coherent wavelength conversion of optical photons using photonphonon translation in a cavity-optomechanical system. For an engineered silicon optomechanical crystal nanocavity supporting a 4-GHz localized phonon mode, optical signals in a 1.5 MHz bandwidth are coherently converted over a 11.2 THz frequency span between one cavity mode at wavelength 1,460 nm and a second cavity mode at 1,545 nm with a 93% internal (2% external) peak efciency. The thermal- and quantum-limiting noise involved in the conversion process is also analysed, and in terms of an equivalent photon number signal level are found to correspond to an internal noise level of only 6 and 4 10 3 quanta, respectively.

DOI: 10.1038/ncomms2201

Coherent optical wavelength conversion via cavity optomechanics

Jeff T. Hill1,*, Amir H. Safavi-Naeini1,*, Jasper Chan1 & Oskar Painter1

1 Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, 1200 E. California Blvd., MS 128-95, Pasadena, California 91125, USA. * These authors contributed equally to this work. Correspondence and requests for materials should be addressed to O.P. (email: mailto:[email protected]

Web End [email protected] ).

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The ability to coherently convert photons between disparate wavelengths has broad technological implications, not only for classical communication systems but also future

quantum networks13. For example, hybrid quantum networks require a low loss interface capable of maintaining quantum coherence while connecting spatially separate systems operating at incompatible frequencies4. For this reason, photons operating in the low loss telecommunications band are often proposed as a conduit for connecting different physical quantum systems5.

It has also been realized that a wide variety of quantum systems lend themselves to coupling with mechanical elements. A coherent interface between mechanics and optics, then, could provide the required quantum links of a hybrid quantum network6.

Until now, most experiments demonstrating both classical and quantum wavelength conversion have utilized intrinsic optical nonlinearities of materials712. The nonlinear interaction of light with acoustic or molecular mechanical vibrations of materials, for instance, enables a great many optical functions used in high-speed optical communication systems today8. With the technological advancements in the elds of nanomechanics and nanophotonics, it is now possible to engineer interactions of light and mechanics. Progress in this area has included enhanced nonlinear optical interactions in structured silica bres13, near

quantum-limited detection of nanomechanical motion14, and the

radiation pressure cooling of a mesoscopic mechanical resonator to its quantum ground state of motion14,15. Coupling of

electromagnetic and mechanical degrees of freedom, in which the coherent interaction rate is larger than the thermal decoherence rate of the system, as realized in the ground-state cooling experiments, opens up an array of new applications in classical and quantum optics. This realization, along with the inherently broadband nature of radiation pressure, has spawned a variety of proposals for converting between photons of disparate frequencies6,1618 through interaction with a mechanical degree of freedomproposals that are but one of many possible expressions of hybrid optomechanical systems4. Such cavity-optomechanical systems are not just limited to the optical frequency domain, but may also nd application to the

interconversion of microwave and optical photons17,19, enabling

a quantumoptical interface to superconducting quantum circuits20.

In this Article, we demonstrate optical wavelength conversion utilizing a simple hybrid optomechanical system consisting of an acoustic and optical resonator formed from the top silicon device layer of a silicon-on-insulator wafer typically used in the microelectronics industry. Through nanoscale lithographic patterning, an optomechanical crystal (OMC) resonator21 is formed, which supports a 4-GHz mechanical resonance colocalized with two optical resonances in the S and C telecommunications bands. The extreme localization of both acoustic and optical energy in this OMC resonator results in a strong radiation pressure interaction between both optical modes and the mechanical motion of the resonator, enabling conversion of light between the two optical cavity modes, which span a frequency of 11.2 THz. Optical wavelength conversion is demonstrated over the 1.5 MHz bandwidth of the mechanical resonator at a peak internal efciency exceeding 90% (the end-to-end measured efciency is limited to 2% by the optical bre coupling efciency to the optical cavities), and with a thermal-limited noise of only 6 quanta, well above the quantum-limited noise of 4 10 3 quanta.

ResultsTheoretical description. As illustrated in Fig. 1a, conceptually the proposed system for wavelength conversion may be thought of as consisting of two optical cavities coupled to the same mechanical resonator.

Motion of the mechanical resonator induces a shift proportional to the amplitude of motion in the resonance frequency of each of the optical cavities, corresponding to the usual radiation pressure interaction in a cavity-optomechanical system. In this case, the two optical cavities are assumed to have different resonant frequencies, between which the desired optical wavelength conversion occurs. For each of the optical cavities, a pump laser beam red-detuned by the mechanical resonance frequency is used to couple the optical cavity modes to the mechanical resonator via the radiation pressure interaction, resulting in a

1

a

ain aout

[afii9837]i,1 [afii9837]i,2

c

a1 a2

b

[afii9837]e,1

[afii9837]e,2

G1 G2

0

1

[afii9828]i

b

d

11.2 THz

[afii9825]2

[afii9825]1

aout

ain

e

f

0

[afii9853]l,2

[afii9853]m [afii9853]m (= 4 GHz)

[afii9853]l,1+1

[afii9853]l,2+1

[afii9853]l,1

Figure 1 | System model and physical realization. (a) Diagram of the wavelength conversion process as realized via two separate FabryPerot cavities. The two optical cavity modes, a1 and a2, are coupled to the same mechanical mode, b, with coupling strengths G1 and G2, respectively. The optical cavity modes are each coupled to an external waveguide (with coupling strengths ke,1 and ke,2), through which optical input and output signals are sent. The optical cavities also have parasitic (intrinsic) loss channels, labelled ki,1 and ki,2, whereas the mechanical mode is coupled to its thermal bath at rate gi.

(b) Schematic indicating the relevant optical frequencies involved in the wavelength conversion process. The cavity control laser beams, labelled a1 and a2, are tuned to a mechanical frequency red of the corresponding optical cavity resonances. An input signal (ain) is sent into the input cavity at frequency ol,1

D1. The input signal is converted into an output signal (aout) at frequency ol,2 D1 via the optomechanical interaction. (c) Scanning electron

micrograph of the fabricated silicon nanobeam optomechanical cavity. Scale bar, 1 mm. (d,e) Finite element method (FEM) simulation of the electromagnetic energy density, normalized to the maximum modal energy, of the rst- (d) and (e) second-order optical cavity modes of the silicon nanobeam. (f) FEM simulation of the displacement eld of the colocalized mechanical mode, normalized to the maximum modal displacement.

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coherent mixing of optical and mechanical degrees of freedom. Under these conditions, the shared mechanical resonator acts as a bridge between the two optical cavities, whereby incident optical radiation resonant with one optical mode is converted into mechanical motion, which is then converted back into optical radiation at the frequency of the other optical mode. This process, akin to the four-wave mixing found in coherent anti-Stokes Raman spectroscopy22, can result in near-unity conversion efciency by both matching the pump-induced coupling rates of the mechanical mode to each of the optical cavity modes and increasing the optomechanical coupling rate above that of the intrinsic loss rate of the mechanical resonator.

The interaction Hamiltonian describing the radiation pressure interaction between the two cavity modes and the mechanical resonator is given by H

Pk hgk^awk^ak^b ^bw, where ^ak ( ^b) are the annihilation operators for the optical cavity modes (common mechanical mode), and gk is the optomechanical coupling rate between the mechanical mode and the kth cavity mode (k 1,2). Physically, gk represents the frequency shift of

cavity mode k owing to the zero-point motion of the mechanical resonator. Wavelength conversion is driven by two control laser beams (ak in Fig. 1b), of frequency ol,k and nominal detuning dk ok ol,k om to the red of cavity resonance at

frequency ok. In the resolved sideband regime, where omckk (kk the bandwidth of the kth cavity mode), the spectral ltering of each cavity preferentially enhances photonphonon exchange. The resulting beam-splitter-like Hamiltonian is

H Pk hGk^awk^b ^ak^bw (refs 17, 23), where Gk gk

p

gOM;2gOM;1

piom o g/2

; 1

where g gi gOM,2 gOM,1 is the total mechanical damping rate

and equal to the bandwidth of the conversion process. From this expression, the spectral density of a converted signal Sout,2(o),

given the input signal spectral density Sin,1(o), may be found and

is given by

Sout;2o Z2Z1

gOM;2gOM;1

o om2 g/22

nadded Sin;1o: 2

These spectral densities have units of photons per hertz per second and are proportional to optical power. The added noise, nadded, arises from thermal uctuations of the mechanical

system and the quantum back-action noise of light present in each optical mode. From here, we see that in a system with ideal cavitywaveguide coupling, (Z1,Z2 1), the peak internal photon

conversion efciency is given by

Zmax;int

4C1C2

1 C1 C22

: 3

This efciency only depends on the internal coupling of the optomechanical system, and for both C1 C2 and C1,C2c1,

approaches unity. The latter condition can be understood from requiring the coupling between the optical and mechanical modes to overtake the intrinsic mechanical loss rate, while the rst requirement is owing to impedance-like matching17. The total

system efciency is Zmax Z1Z2Z

max,int.

nc;k

p is

the parametrically enhanced optomechanical coupling rate owing to the ak control beam (nc,k is the control beam-induced intracavity photon number). In the weak-coupling limit, Gk{kk, this interaction effectively leads to an additional mechanical damping rate, gOM;k 4G2k/kk. The degree to which

this optomechanical loss rate dominates the intrinsic mechanical loss is called the cooperativity, Ck gOM,k/gi. At large

cooperativities, the optomechanical damping has been used as a nearly noiseless loss channel to cool the mechanical mode to its ground state14,15. In the case of a single optical cavity system, it

has also been used as a coherent channel allowing inter-conversion of photons and phonons leading to the observation of electromagnetically induced transparency (EIT)24,25. In the

double optical cavity system presented here, coherent wavelength conversion of photons results.

As shown in Fig. 1a, each optical cavity is coupled not just to a common mechanical mode, but also to an optical bath at rate ki,k

and to an external photonic waveguide at rate ke,k (the total cavity linewidth is kk ki,k ke,k). The external waveguide coupling

provides an optical interface to the wavelength converter, and in this work consists of a single-transverse-mode waveguide, bidirectionally coupled to each cavity mode. The efciency of the input/output coupling is dened as Zk ke,k/2kk, half that of

the total bidirectional rate. Although the wavelength converter operates symmetrically, here we will designate the higher frequency cavity mode (k 1) as the input cavity and the lower

frequency cavity (k 2) as the output cavity. As shown in Fig. 1b,

photons sent into the wavelength converter with detuning D1Bom from the control laser a1, are converted to photons D1 detuned from the control laser a2, an 11.2 THz frequency span for the device studied here.

The details of the conversion process can be understood by solving the HeisenbergLangevin equations (see Supplementary Methods and refs 17,18,26). We linearize the system and work in the frequency domain, obtaining through some algebra the scattering matrix element s21(o), which is the complex,

frequency-dependent conversion coefcient between the input eld at cavity ^a1 and the output eld at cavity ^a2. This coefcient is given by the expression

s21o

Z2Z1

Device description and characterization. The optomechanical system used in this work, shown in Fig. 1cf, consists not of a separate set of optical cavities, but rather of a single OMC nanobeam cavity. The OMC nanobeam is fabricated from a silicon-on-insulator microchip21, in which the top, 220 nm thick, silicon device layer is patterned with a quasi-periodic linear array of etched air holes. The resulting silicon beam is 10 mm long and 600 nm wide. The hole radii and positions are designed by numerical optimization and repeated simulations, as discussed in detail in ref. 27. The larger air holes on either end of the beam induce Bragg-like reection of both guided optical and acoustic waves, resulting in strongly conned optical and mechanical resonances at the beams centre. The device used in this work is designed to have both a rst-order (^a1, lE1,460 nm) and a second-order (^a2, lE1,545 nm) optical cavity resonance of high quality factor. Both of these optical cavity modes are dispersively coupled to the same gigahertz-frequency mechanical resonance, depicted in Fig. 1f (^b, om/2p 3.993 GHz, Qm 87 103 at

TE14 K). A tapered optical bre waveguide, placed in close proximity (B100 nm) to the OMC nanobeam cavity28 using precision stages, is used to evanescently couple light into and out of the cavity modes.

A schematic of the experimental set-up used to characterize the OMC wavelength converter is shown in Fig. 2a. Measurements were performed in vacuum and at low temperature inside a continuous ow cryostat with a cold nger temperature of 9 K (the corresponding sample temperature, as inferred from the thermal bath temperature of the localized mechanical mode of the OMC cavity, is 14 K (ref. 15)). Initial characterization of the optical cavity modes is performed by scanning the tunable control lasers across a wide bandwidth and recording the transmitted optical intensity through the optical bre taper waveguide. Such a wavelength scan is shown in Fig. 2b, from which the resonance

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a

b

Optical wavelength (nm)

Output laser (2)

1,450 1,460 1,470 1,480 1,490 1,500 1,510 1,520 1,530 1,540 1,550 1,560

Normalized transmission Reflection (%)

1.00.80.60.40.2

~1,545 nm

a-m

-mux

Q1= 392,000

[afii9834]1 = 0.1

Q2= 112,000

[afii9834]2 = 0.21

Cryostat

Spectroscopy,

thermometry

[afii9838]1= 1,461.12 nm

[afii9838]2= 1,545.76 nm

OMC Taper

PD2

c

2.0

d

0 3

3 (2-[afii9853]m)/2[afii9843] (MHz)

12

~1,460 nm

a-m

1.5

PD1

0 3

3 (1-[afii9853]m)/2[afii9843] (MHz)

8

1.0

Input laser (1)

aom

0.5

4

Input signal

calibration

EIT reflection

spectroscopy

0

3.2 3.6 4 4.4 4.8

0 3.2 3.6 4 4.4 4.8

1/2[afii9843] (GHz) 2/2[afii9843] (GHz)

Figure 2 | Experimental set-up and optical spectroscopy. (a) Two tunable external cavity diode lasers are used as control beams driving the wavelength conversion process. The input (output) control laser is locked as a mechanical frequency red-detuned from the rst-order (second-order) cavity mode at lE1,460 nm (lE1,545 nm). Both control beams can be amplitude modulated (a-m) to perform EIT-like spectroscopy of the cavity modes, or in the case of the input laser, to generate the input sideband signal for the wavelength conversion process. As described in the Methods, an AOM is used to calibrate the input sideband signal. The light from both lasers is combined using a wavelength multiplexer (l-mux), and then sent into a dimpled optical bre taper that is coupled to the OMC cavity. The cavity sample is placed in a cryostat to precool it down to TE14 K. The transmitted light from the cavity is sent to a high-speed photodetector (PD2), which is connected to a spectrum analyser to measure the converted signal on the output laser. The reected optical signal from the cavity is directed via an optical circulator to a second high-speed photodetector (PD1) to probe the EIT-like spectrum of each cavity mode. (b) Broad wavelength transmission scan showing bre taper coupling to both the rst-order (lE1,460 nm) and second-order (lE1,545 nm) cavity modes.

(c) EIT scan of cavity mode ^

a1, with inset showing a zoomed-in scan of the transparency window. The solid black lines correspond to ts to a theoretical model allowing extraction of system parameters (k1, g, d1 and om). (d) Corresponding EIT scan of cavity mode ^a2. For measurements in both (c,d) the control beam intensities were set such that gOM,1EgOM,244gi, with detunings d1\om and d2Eom.

frequency (o1/2p 205.3 THz, o2/2p 194.1 THz), cavity

linewidth (k1/2p 520 MHz, k2 1.73 GHz) and waveguide

coupling efciency (Z1 0.10, Z2 0.21) of the two nanobeam

cavity modes are determined.

Further characterization of the optomechanical cavity is performed by using the control laser beams in conjunction with a weak sideband probe. With control beams a1 and a2 detuned a mechanical frequency to the red of their respective cavity modes (d1,2 om), a weak sideband signal generated from a1 is swept

across the rst-order cavity mode. The resulting reected sideband signal versus sideband frequency shift D1 is plotted in Fig. 2c, showing the broad cavity resonance along with a narrow central reection dip (see Fig. 2d inset). The narrow reection dip, akin to the EIT transparency window in atomic systems, is due to the interference between light coupled directly into the cavity mode and light coupled indirectly through the mechanical mode24,25.

A similar EIT response is shown in Fig. 2d for the second-order cavity mode. Each of the transparency windows occur at Dk om,

with a bandwidth equal to the optically damped mechanical linewidth (g gi gOM,1 gOM,2). By tting the optical resonance

lineshape and transparency windows to theory, one can also extract the control beam detunings dk. In what follows, we use this sort of

EIT reection spectroscopy, time-multiplexed in between wavelength conversion measurements, to set and stabilize the frequency of the control beams to dk om. We note the EIT

reection spectroscopy, though convenient for full characterization and elucidation of the conversion process, is not necessarily performed in a real system application given that both the laser and the cavity resonance have frequency drift, over the measured period of days, that is much smaller than the optical cavity linewidth k of the devices studied here.

Efciency. Coherent wavelength conversion can be thought of as occurring between the input transparency window of ^a1 and the

output transparency window of ^a2, with the phonon transition mediating the conversion. As shown above, conversion efciency is theoretically optimized for matched cooperativities of control beam a1 and a2 (or equivalently gOM,1 gOM,2). Calibration of the

optically induced mechanical damping by the control beams can be performed as in the EIT spectroscopy described above, or by measuring the spectral content of the photodetected transmission intensity of ^a2 in the absence of any optical input. Such a measurement (see set-up in Fig. 2a and Supplementary Fig. S1) measures the mechanical resonator linewidth through the noise spectrum generated by the beating of noise sideband photons generated by thermal motion of the mechanical resonator with the control beam a2. Fig. 3a plots the inferred optomechanically induced damping of the mechanical resonator as a function of the power (intracavity photon number, nc) of control beams a1 and a2, the slope of which gives the zero-point optomechanical coupling for both cavity modes (g1/2p 960 kHz and g2/2p 430 kHz).

To quantify the efciency of the wavelength conversion process, the input cavity control beam a1 is held xed at a detuning d1

om and a power producing an intracavity photon population

of nc,1 100, corresponding to large cooperativity C1E16. The

input signal, ain, is generated as an upper sideband of a1 using an electro-optic intensity modulator (the lower sideband is rejected by the wavelength converter as it is detuned to D1E om).

Conversion of the ain sideband to an output signal emanating from cavity mode ^a2 is completed by applying control beam a2 with detuning d2 om. The converted output signal sideband

emitted from cavity ^a2 is beat against the transmitted control beam a2 on a high-speed photodiode. The amplitude of the input signal tone near resonance with cavity mode ^a1 is calibrated using a second reference signal generated through acousto-optic modulation of a1, whereas the amplitude of the output signal sideband is inferred from calibration of the control beam intracavity number nc,2 and the optical transmission and detection chain (see

Fig. 2a and Supplementary Fig. S1). Figure 3b plots the resulting

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a

106

g 1=960 kHz

105

[afii9828] OM

104

g 2=430 kHz

103

100 101 102 103

Intracavity photon number

b

0.020

0.015

0.010

0

0.005

2

0.020

2

|S 21|

0.010

0 8 (1[afii9853]m)/2[afii9843] (MHz)

4 4 8

0

|S 21|

0

0.5

1.0

1.5

2.0

C2 /C1

c

123.0

95

PSD (dBm/Hz)

123.6

105

124.2

!

1

4 0 4

1

2

k1

4om

2

1

2

k2

4om

2

115

; 4

where the rst term arises owing to thermal noise of the cooled and damped mechanical resonator, and the last two terms are quantum noise resulting from the spontaneous scattering of the control beams (quantum back-action noise29). These sources of noise are independent and uncorrelated and as such, add together incoherently. Figure 4a pictorially indicates the various cooling, heating and spontaneous scattering mechanisms that lead to the output noise of ^a2. A plot showing the measured output noise spectral density, calibrated in units of photon number, is given in Fig. 4b for the optimal conversion efciency of Fig. 3b. From the peak of the output noise spectral density ( Z1Z2n

added) in this plot, the added noise referred to the input of ^a1 is estimated to be naddedE60 quanta. The corresponding internal added noise (for Z 1) is only n

125

10

8

6

4 8 10

2 0 2 4

6

(2[afii9853]m)/2[afii9843] (MHz)

Figure 3 | Wavelength conversion efciency and bandwidth. (a) Plot of the optically induced mechanical damping versus control beam intensity (intracavity photon number) for each cavity mode, as measured through the mechanical thermal noise spectrum imprinted on the optical output intensity of ^

a2. The calibration curve for cavity mode ^a2 is performed with a1 turned off, whereas the curve for cavity mode ^

a1 is generated with a weak a2 such that gOM,2oogOM,1 for all measured points. (b) End-to-end power conversion efciency of input signal to output signal for D1 0 as a function

of the ratio of the cooperativities of the control beams. In this plot, the control beam intensity for the rst-order cavity mode is held xed (with C1E16), while the intensity of the control beam of the second-order cavity mode is swept from C2ooC1 to C24C1. The blue circles correspond to measured data points, whereas the solid red line is a theoretical curve using independently measured system parameters. Inset shows the conversion efciency versus input signal detuning for matched control beamsC1 C2E16, indicating a conversion bandwidth of B1.55 MHz.

(c) Measured power spectrum of the output optical channel forC1 C2E16, showing a series (D1 sweep) of converted input tones (green)

sitting on top of a much smaller thermal noise pedestal (blue). Inset shows a zoom-in of the thermal noise pedestal.

power conversion efciency, given by the magnitude squared of equation (2), versus the ratio of C2/C1 as the power of control beam a2 is varied and for an input signal exactly resonant with cavity mode ^a1 (D1 om). The power conversion is referred

directly to the input and output of the optomechanical cavity (that is, not including additional losses in the optical link). The solid red line shows the theoretical model for the conversion efciency using the independently measured system parameters, and taking into account power-dependent effects on the optical losses in the silicon cavity modes (see Methods). Good correspondence is seen between the measured data and theory, with a maximum of the conversion occurring at C1EC2 as expected from the matching condition.

Bandwidth. The bandwidth of the wavelength conversion process is also probed, using matched conditions (C1 C2E16), and as

above, with control beam detunings dk om. The modulation

frequency generating ain is now varied from (D1 om)/2p 10 MHz to 10 MHz, sweeping the narrow-band input

sideband signal across the transparency window of cavity mode ^a1. The corresponding measured output signal frequency shift follows that of the input signal (D2 D1), with the resulting

power conversion efciency plotted in the inset of Fig. 3b. A peak efciency, measured from the input to the output port at the optomechanical cavity system, is Z 2.2%, corresponding to an

internal conversion efciency of Zmax,int 93%. Fitting the results

to a Lorentzian as in equation (1), we nd the conversion bandwidth to be 1.55 MHz, equal to the optically damped mechanical linewidth of the mechanical resonance, g/2p.

Noise. The spectrum of the converted output signal for a series of signal frequencies D1 are shown in Fig. 3c. As can be seen from this plot and the zoomed-in inset, the narrow-band-converted photons sit atop a noise pedestal of bandwidth corresponding to the damped mechanical resonator. This noise is quantied by the added noise quanta number component of equation (2). As shown in the Supplementary Methods, under matched conditions the theoretical added noise is,

nadded 2Z 11

ginb

g

addedE6, predominantly owing to the cooled phonon occupation of the mechanical resonator (/nSE3). Owing to the strong ltering provided by the high-Q cavities (kk/omoo1), the quantum back-action noise contributes an insignicant 4 10 2 (4 10 3) added input (internal) noise

photons. The effectiveness of noise reduction in the optomechanical cavity system is highlighted in Fig. 4b by showing the estimated thermal output noise in the absence of sideband cooling by the control beams (dashed red curve) and the quantum back-action noise in the absence of cavity ltering (green dashed curve).

DiscussionAlthough the demonstrated wavelength conversion in this work represents an important proof of principle, there are still considerable practical challenges to utilizing such cavity-optomechanical devices for classical or quantum wavelength translation2,9. With a thermal phonon occupancy /nSrZ1/2 required for single-photon conversion at unity signal-to-noise, a primary concern for quantum networking1 is the thermal energy stored in the mechanical resonator. Although resolved sideband cooling of nanomechanical resonators to occupancies below one has recently been demonstrated14,15, this relation emphasizes the additional

need for efcient cavity coupling. For chip-scale photonic devices similar to the one presented here, the technology already exists for efcient optical connectivity, with optical bre-to-chip coupling of

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where b1 is the modulation index of the input signal and f accounts for any phase difference between the sidebands (f 0 if there is pure amplitude modulation).

The input sideband at frequency ol,1 D1 is nearly resonant with the rst cavity

mode at o1, whereas the lower frequency sideband at ol,1 D1 is detuned by

B 2om. As such, only the upper frequency sideband is resonant with the

transparency window of the rst cavity mode, and only this sideband is converted by the OMC cavity into a sideband at the second, output cavity mode ^a2. The converted sideband is generated from the control laser beam a2 of the second cavity mode with frequency ol,2 D1. Thus, the output eld amplitude near the fre

quency of the second control laser beam (amplitude A2) is, ignoring f,

A2 ! A2 s21D1

b1

a

b

a2

[afii9828]OM,2n

Cooling

Cavity filtering

[afii9828]OM,k ([afii9837]k /4[afii9853]m)2

Output noise (photons/sHz)

102

101

100

a1

[afii9828]OM,1n

101

102

103

104

105

4 eiD t: 6 The power measured by a photodetector from the optical signal eminating from the output cavity mode is proportional to

P2 jA2 j 2 1 js21D1j

b1A1

b

[afii9828]n

4A2 cos D1t

: 7

We thus dene b2 |s21(D1)|b1A1/4A2 as the modulation index of the output

signal,

P2 jA2 j 2 1 b2 cos D1t

: 8

By careful calibration of b1 and b2, and measurement of the control beam power (Pa ), the conversion scattering matrix element can be determined:

s21D1 2

b2A2

b1A1 2

[afii9828]inb

4

2

0

2

Detuning (MHz)

4

Figure 4 | Output noise. (a) Schematic showing the relevant input noise terms contributing to the total noise at the output. The rates of noisy phonon generation are ginb owing to the thermal bath and gOM,(1,2) (k(1,2)/

4om)2 owing to spontaneous Stokes scattering from the two control beams. Cooling of the mechanical mode is also performed by the control beams, resulting in a theoretically cooled phonon occupancy of /nS (1/g)

(ginb SjgOM,j(kj/4om)2). The output noise power is proportional to the

output cavity cooling rate, gOM,2/nS. (b) Output noise power spectral

density. The measured noise data (red circles) is shown along with several theoretical noise curves. The dashed red curve is the corresponding output noise power in the absence of control beam cooling. The solid (dashed) green curve is the output noise power owing to spontaneous Stokes scattering in the presence (absence) of cavity ltering.

b2 b1

s

Pa Pa

: 9

Calibration of the input signal. The appearance of f in equation (5), along with its sensitivity to effects such as chromatic dispersion in the optical bre and components, make calibration of b1 more challenging owing to interference between the upper and lower frequency sidebands. Direct photodetection of the optical input signal results in a f-dependent measured beat signal between the carrier and the two sidebands. To bypass this problem, we generate an additional single sideband using an acousto-optic modulator (AOM), and beat it against each of the input optical sidebands. This is accomplished by splitting off a portion of the optical input signal before creating the EOM sidebands, and frequency shifting it by DAOM (see Supplementary Fig. S1). This AOM-shifted signal is then recombined with the main input signal, giving an overall signal equal to

Aout;1 A1

b1

4 A1e iD t

b1

B95% (ref. 30) and waveguide-to-cavity coupling efciency of Z1\0.999 having been realized31. Further reduction in the output noise may also be realized by simply reducing the temperature of operation; for the 4 GHz phonon frequency of the device in this study, nbt10 3 for a bath temperature of TB10 mK realizable in a helium dilution refrigerator. Integration of cavity-optomechanical devices into milliKelvin experiments is particularly relevant for superconducting quantum circuits, which themselves have already been strongly coupled to mechanical resonators32. Addition of an optical cavity coupled directly, or via an additional acoustic waveguide, to the same mechanical resonator would provide a microwave-to-optical quantum interface17,19 similar to the optical-to-optical interface shown here. In the case of solid-state qubits with optical transitions far from the telecommunication bands, such as the nitrogen-vacancy electron spin in diamond33, the radiation pressure nonlinearity of cavity optomechanics could be substituted for intrinsic materials nonlinearities in proposed cavity-based schemes for single-photon wavelength conversion and pulse shaping34. Owing to the excellent mechanical properties of diamond and recent advancements in thin-lm diamond photonics35, a cavity-optomechanical wavelength converter could be fabricated around an nitrogen-vacancy qubit from the same diamond host.

During the preparation of this manuscript, a similar work appeared on the arXiv36.

Methods

Generation and conversion of the input signal. We begin by considering the input signal to the rst optical cavity mode. This signal is generated by weakly modulating the a1 control beam (represented by amplitude A1) at a frequency D1 using an electro-optic modulator (EOM) producing

A1 ! A1

b1

4 A1e iD t

b1

4 A1eiD teif; 5

4 A1eiD t f A3eiD

t; 10 where A3 is the eld amplitude of the signal split off from the input signal before the EOM. The total photodetected signal is then given by

jAout;1 j 2 jA1 j2 jA3 j 2

j A1 j2 b1 cosD1t f/2 cosf/2

2 jA1A3 j cosDAOMt

j A1A3 j

b1

2 cosD1 DAOMt

jA1A3 j

b1

2 cosD1 DAOMt f : 11 From this equation, we see that the optical power at the detector consists of components at zero frequency (DC) as well as four modulated tones (see Supplementary Fig. S2). By taking the ratio of the tone at DAOM and D1 DAOM,

the value of b1 can be accurately determined, independent of f. Note that in order to determine b2 no such additional calibration signal is required. The ltering properties of the input and output cavities results in a single output sideband, with no phase dependence of the photodetected intensity spectrum. In this case, careful optical and electronic calibration of the measured output cavity (^a2) optical transmission intensity, similar to that described in the Supplementary Information ofref. 15, provides an accurate estimate of b2 and the wavelength-converted signal strength.

Power-dependent optical cavity loss. To model the wavelength conversion process, all of the optical cavity mode and mechanical resonator parameters entering into equation 1 must be independently determined. Most of these parameters are measured as described in the main text; however, additional measurements were performed to determine the power-dependent optical cavity loss in the silicon OMC device. Owing to two-photon absorption in silicon, the parasitic optical cavity loss of both cavity modes is not static, but rather depends upon both control beam powers. This effect is particularly acute for the higher Q-factor rst-order cavity mode (^a1), where in Supplementary Fig. S3a we plot the normalized on-resonant optical transmission (T0,1) of the rst-order cavity mode versus the

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power (as represented by the intracavity photon population, nc,2) of control beam

a2 feeding the second-order cavity mode. The steady rise in the on-resonance transmission versus nc,2 is attributable to the increased optical absorption loss stemming from free carriers generated by two-photon absorption of the control beam a2. The corresponding change in coupling efciency, Z1 ke,1/2k1, is shown

on the right-side axis of Supplementary Fig. S3a. Neglecting the effects of nonlinear optical absorption and free carriers on the intrinsic optical quality factor of ^a1 leads to a theoretical model of conversion efciency denoted by the dashed black line in Supplementary Fig. S3b. Taking into account the nonlinear optical loss measured in Supplementary Fig. S3a leads to the theoretical conversion efciency denoted by the solid red line, showing much better correspondence with the data (the solid red line is the model t shown in the main text). The effect of two-photon absorption and free-carrier absorption is much smaller on the second-order cavity mode (^a2)

owing to its already lower optical Q-factor, and we neglect it here. The control-beam-generated free carriers also affect the intrinsic mechanical quality factor (gi)

as was shown inref. 15; however, this effect is small for the control beam powers studied here (the efciency curve of Supplementary Fig. S3b is also not modied as we plot it versus the ratio of cooperativities for which gi cancels out).

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Acknowledgements

This work was supported by the DARPA/MTO MESO programme, the Institute for Quantum Information and Matter, an NSF Physics Frontiers Centre with support of the Gordon and Betty Moore Foundation, and the Kavli Nanoscience Institute at Caltech. J.C. and A.H.S.-N. gratefully acknowledge support from NSERC.

Author contributions

A.H.S.-N., J.C. and J.T.H. designed and fabricated the devices. J.T.H., along with support from A.H.S.-N. and J.C. performed the measurements. A.H.S.-N. and J.T.H. performed the data analysis. A.H.S.-N. and O.P. developed the experimental concept, and O.P. supervised the measurements and analysis. All authors worked together on writing the manuscript.

Additional information

Supplementary Information accompanies this paper on http://www.nature.com/naturecommunications

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How to cite this article: Hill, J.T. et al. Coherent optical wavelength conversion via cavity optomechanics. Nat. Commun. 3:1196 doi: 10.1038/ncomms2201 (2012).

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