arXiv:1910.11395v2 [physics.optics] 7 Aug 2020

Molecular platform for frequency upconversion at the single-photon level

Philippe Roelli,1 Diego Martin-Cano,2 Tobias J. Kippenberg,1, ∗ and Christophe Galland1, †

1EPFL, Swiss Federal Institute of Technology, Institute of Physics, Lausanne, Switzerland2Max Planck Institute for the Science of Light, Erlangen, Germany

(Dated: August 10, 2020)

Direct detection of single photons at wavelengths beyond 2 microns under ambient conditionsremains an outstanding technological challenge. One promising approach is frequency upconversioninto the visible (VIS) or near-infrared (NIR) domain, where single photon detectors are readilyavailable. Here, we propose a nanoscale solution based on a molecular optomechanical platformto up-convert photons from the far and mid-infrared (covering part of the THz gap) into the VIS-NIR domain. We perform a detailed analysis of its outgoing noise spectral density and conversionefficiency with a full quantum model. Our platform consists in doubly resonant nanoantennasfocusing both the incoming long-wavelength radiation and the short-wavelength pump laser fieldinto the same active region. There, infrared active vibrational modes are resonantly excited andcouple through their Raman polarizability to the pump field. This optomechanical interaction isenhanced by the antenna and leads to the coherent transfer of the incoming low-frequency signalonto the anti-Stokes sideband of the pump laser. Our calculations demonstrate that our scheme isrealizable with current technology and that optimized platforms can reach single photon sensitivityin a spectral region where this capability remains unavailable to date.

I. INTRODUCTION

Many applications in security, material science andhealthcare would benefit from the development of newtechnologies for far and mid-infrared (FIR and MIR, re-spectively) detection and thermal imaging [1]. Driven byapplications in astronomy, novel cryogenic detectors inthe FIR range appeared in the past few years [2, 3]. How-ever the ability to efficiently manipulate such electromag-netic signals at room temperature is still lacking [4, 5]. Inparticular, single photon detection, which is now routinein the VIS-NIR region (wavelength in vacuum from 400to 2000 nm), remains impossible or unpractical at longerwavelengths. The development of new detection devicesoperating without complex cryogenic apparatus, and fea-turing improved sensitivity, lower noise and reduced foot-print, would significantly impact sensing, imaging, spec-troscopy and communication technologies.

In this work we propose a new route to achieve low-noise detection of non-coherent radiation between 5-50THz by leveraging optomechanical transduction withmolecules [6], whose natural oscillation frequencies areresonant with the incoming field. Our strategy consistsin converting the incoming low-frequency signal onto theanti-Stokes sideband of a pump laser in the VIS-NIR do-main, where detectors with single photon sensitivity arereadily available [7, 8]. This approach is inspired by therecent realization of coherent frequency conversion usingdifferent types of optomechanical cavities [9–15] and isconceptually distinct from a recently demonstrated de-tection scheme assisted by a microfabricated resonator[16]. As an outlook, we propose to leverage construc-tive interference between signals coming from an array

∗ corresponding authors: [email protected]† [email protected]

of coherently pumped up-converters in order to increasefurther the strength of the converted signal over the in-coherent thermal noise.

While coherent conversion from the MIR to the VIS-NIR domain has so far been achieved by sum-frequencygeneration in bulk nonlinear crystals [17–20], theseschemes operate under several watts of pump power andrequire phase-matching between the different fields prop-agating in the crystal. Our scheme, on the contrary,relies solely on the spatial overlap of the two incomingfields. Indeed, we use a nanometer-scale dual antennathat confines both electromagnetic fields into similar sub-wavelength mode volumes. The optomechanical interac-tion with the vibrational system takes place in the nearfield, without need to fulfill a phase matching condition.Moreover, thanks to the giant field enhancement pro-vided by plasmonic nanogaps, the required pump powerto achieve efficient conversion is dramatically reduced.

The protocol that we introduce leverages the intrinsicability of specific molecular vibrations to interact bothresonantly with MIR-FIR fields and parametrically withVIS-NIR fields, as routinely observed in infrared absorp-tion and Raman spectroscopy, respectively. The wealthof accessible vibrational modes and frequencies [21, 22]offers a convenient toolbox to realize efficient frequencyupconversion in the technologically appealing region ofthermal imaging.

We first introduce the framework describing the inter-action between a molecular vibration and two electro-magnetic fields, one that is resonant with the vibrationalfrequency, the other one that is parametrically coupledto it through the molecular polarization. We computethe conversion efficiency and the noise figures of meritof our novel device as a function of the optical pumpdetuning and power. We illustrate the achievable perfor-mance with a device operating at 30 THz (10 µm) andfind internal conversion efficiencies on the order of a few

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percent and noise-equivalent power below 10−12 W/√

Hz.Finally, we demonstrate how our approach may be usedto reach single-photon detection at frequencies down to∼ 5 THz.

II. OPTICAL CONVERSION SCHEME

We start with the description of the two types of inter-actions leveraged in the conversion process and describethe relevant parameters. For simplicity, we now use theabbreviation IR to denote MIR or FIR fields, dependingon the vibrational frequency considered. First we modelthe resonant absorption process. We assume that thevibrational system is weakly driven, meaning that theaverage number of excited collective vibrational quantais much smaller that the total number of molecular oscil-lators coupled to the incoming field, Nir. At the single-molecule level, this easily satisfied condition correspondsto neglecting transitions beyond the ground and first ex-cited vibrational states. The collective excitation of anensemble of vibrational modes can thus be treated as anensemble of two-level systems [23].

The interaction part of the Hamiltonian is correspond-ingly approximated by :

Hint = −i~g(Nir)ir,0

√nir(a†irσ

−ν + airσ

+ν

), (1)

with a†ir, air the IR field bosonic ladder operators andσ+ν , σ

−ν the raising and lowering operators of the collec-

tive two level system described by a transition frequency

ων . g(Nir)ir,0 =

√Nirgir,0 is the collective resonant vacuum

coupling rate of the vibrational mode ν for Nir identi-cal molecules, and nir the mean occupation of the IRantenna mode.

The incoming IR field at frequency ωir is enhancedby a frequency-matched antenna and performs work on

the collective transition dipole ~dν of the molecular vibra-tion [24]. On resonance (ωir = ων) the average number ofcreated phonons is (see Appendix C for a detailed deriva-tion)

nirb =

(2g

(Nir)ir,0

Γtot

)2

ηirκir|〈ain

ir〉|2 (2)

with |〈ainir〉|2 the incoming IR photon flux. In this ex-

pression κir = κirex + κir0 is the loss rate of the antennaat the incoming frequency, which is the sum of the exter-nal decay rate κirex (by radiative coupling to the far-fieldmodes) and the internal decay rate κir0 (by absorptionin the metal). ηir = κirex/κ

ir is the coupling ratio of theantenna and Γtot the total vibrational decay rate, wherethe intrinsic vibrational linewidth Γν is modified by itscoupling to the IR antenna [25] and the optomechanicalinteraction with the pump laser, as explained below.

As pictured in Fig. 1, we employ a second antennaresonant at ωc (a frequency in the VIS-NIR domain,

𝜔𝜔IR

𝜔𝜔p

𝜔𝜔p + 𝜔𝜔𝜈𝜈

𝜔𝜔p − 𝜔𝜔𝜈𝜈

𝜔𝜔p

(a)

𝜔𝜔IR

𝜔𝜔p

𝜔𝜔p + 𝜔𝜔𝜈𝜈

𝜔𝜔p − 𝜔𝜔𝜈𝜈

𝜔𝜔p

(b)

Figure 1. (a) Illustration of the envisioned up-conversiondevice. Both electromagnetic modes are collected with thehelp of the dual antenna and confined within a volume wheremolecules are located. (b) Frequency picture of the optome-chanical conversion mechanism involving both IR absorptionand Raman scattering by specific vibrational modes. Herethe pump tone (ωp) is red-detuned from the optical resonance(ωc ' ωp +ωir) while the incoming IR signal is resonant witha specific vibrational mode (ωir = ων).

which we call “optical” domain from here on for brevity),

whose decay rates κoptex , κ

opt0 are defined in the same

way as the IR antenna parameters. The optical antennaenhances the parametric optomechanical interaction ofthe molecular vibration with a pump laser in the op-tical domain, as described in Ref. [6]. Concisely theinteraction between an optical field and Nopt molecu-lar oscillators leads to a dispersive interaction described

by the Hamiltonian Hint = −~g(Nopt)opt,0 a†optaopt

(bν + b†ν

)with g

(Nopt)opt,0 =

√Noptgopt,0 the collective optomechan-

ical vacuum coupling rate and a†opt, aopt

(b†ν , bν

)the

optical pump field bosonic ladder operators (the vibra-tional phononic operators at frequency ν). The numberof molecules Nopt participating in the optomechanical in-teraction with the pump laser may be different from thenumberNIR participating to IR absorption, as elaboratedin Sec. III below (cf. also Appendix D Sec. 5,6).

The optical antenna field can be split into an aver-age coherent amplitude α and a fluctuating term so thataopt = α + δaopt. Expanding to first order in α theoptomechanical interaction we obtain the linearized in-teraction

Hlin = −~g(Nopt)opt,0

√nopt

(δa†opt + δaopt

)(bν + b†ν

), (3)

with nopt = |α|2 the mean occupation of the optical an-tenna mode (see Appendix A).

3

The spectral density of the output field on the opti-cal port in [photons/(Hz·s)] can be evaluated throughthe calculation of the two-time correlations of the opticaloutput field operators [26, 27]:

Sout(ω) ∝ κoptex

2π

∫ ∞−∞

dτeiωτ 〈δa†opt(τ)δaopt(0)〉. (4)

Following previous works in optomechanics [28] andtheir extension to molecular optomechanics [29, 30]we can write an analytical expression of the out-going spectral density at the anti-Stokes sidebandSout(ωaS) ∝ A−nf/ (Γ∗ν + Γopt) (at the Stokes sidebandSout(ωS) ∝ A+ (nf + 1) / (Γ∗ν + Γopt)) with nf the meanfinal phonon number of the vibrational mode. Due tothe IR and optomechanical interactions the intrinsic vi-brational damping rate is modified Γtot = Γ∗ν +Γopt withΓ∗ν the IR antenna-assisted damping rate and Γopt =A−−A+ the additional damping rate of electromagneticorigin characterized by the imbalance between the opti-cal antenna-assisted transition rates to the ground A−

and excited vibrational states A+ (see Appendix A).The final phonon number in the vibrational mode, nf ,

is given by the expression [26]

nf =Γ∗ν

Γ∗ν + Γoptnb +

A+

Γ∗ν + Γopt, (5)

where nb = nirb + nth is the total phonon number in theabsence of optical drive. It is the incoherent sum of theIR-induced vibrational excitation (Eq. 2) and the ther-mal noise, nth = 1/ (exp [~ων/kBTbath]− 1) for a bathtemperature Tbath. We assume here that the pump laserdoes not lead to significant Ohmic heating of the sys-tem. It is however straightforward to model laser-inducedheating by introducing a pump-power-dependent bathtemperature Tbath.

The resulting spectral density Soutopt in the absence of

incoming IR radiation (nirb = 0) should be integrated overthe device’s operational bandwidth (BW ≡ Γtot) to ob-

tain its dark-count rate Soutopt =

∫BW

Soutopt dω. The dark-

count rate arising from the thermal contribution to thefirst term in Eq. (5) can be reduced by cooling the bath,whereas the second term describes a minimal noise levelresulting from phonon creation by spontaneous Stokesscattering of the pump laser, a process equivalent toquantum backaction in cavity optomechanics. Thereforean optimal power that maximizes the signal-to-noise ratio(SNR) exists, akin to the standard quantum limit (SQL)in position measurements.

From these expressions we are also able to describe theconversion efficiency from an incoming rate of IR photonscoupled to the antenna into an outgoing rate of opti-cal photons emitted by the antenna into free space, asSoutir→opt = ηext|〈ain

ir〉|2 where ηext = ηopt · ηint · ηir is an

external conversion efficiency 1 with ηopt = κoptex /κ

opt.

1 In this expression, we did not include the frequency dependence

The internal conversion efficiency ηint can in turn bedivided into a power-dependent part ηOM(nopt) and apart describing the spatial overlap between the IR nearfield, the optical near field and the molecular ensemble,which we write ηoverlap. To approximate this last termwe factorize it into two contributions: the spatial overlapbetween the IR and optical near fields (ηmode) and thevectorial overlap between the near-field polarization (typ-ically normal to the antenna surface) and the molecularorientation, which we name ηpol; so that we can write

ηint ' ηpol · ηmode · ηOM(nopt). (6)

The evaluation of ηoverlap = ηpolηmode is detailed in Ap-pendix D (Sec. 5,6). The power and detuning dependenceof the optomechanical efficiency term ηOM are depicted inFig. 3 (b) and its exact calculation is given in AppendixA.

III. MOLECULAR TRANSDUCER

The electric dipole moment ~µν and polarizability ανof a vibrational mode can be extracted from experimen-tal data of resonant light absorption and inelastic lightscattering, respectively [21, 22]. For vibrational modes

Detection

𝜕𝜕𝜕𝜕𝜕𝜕𝑄𝑄𝜈𝜈

𝜕𝜕𝜕𝜕𝜕𝜕𝑄𝑄𝜈𝜈

(a)

Detection

optical domainIR domain

(b)

Figure 2. (a) Polar plots of both the electric moment deriva-tive (left) and the projection of the Raman tensor on themain plane of the molecule (right) for the vibrational modeν = 1002 cm−1 of the thiophenol molecule (background im-age). (b) Local density of states distribution inside the dualantenna for an IR mode at 32 THz (left) and a NIR mode at374 THz (right).

of the photonic density of state in free space, nor the factorsrelated to the specific optical design used to couple light in andout of the structure.

4

lacking centrosymmetry the derivatives of both quanti-ties with respect to the displacement coordinate can benonvanishing [31]. We show such a situation in Fig. 2where we plot the projections of the derivatives of theelectric moment and of the polarizability with respect tothe molecular coordinate Qν of the 1002 cm−1 mode ofthiophenol, which we choose as an example in our calcu-

lations. We note that the projection of the tensor(∂αν

∂Qν

)onto an axis perpendicular to the principal axis of the

electronic moment derivative(∂~µν

∂Qν

)can be nonvanish-

ing. Several polarization directions for in- and outcou-pling of resonant and up-converted fields are thus con-ceivable.

The calculations leading to the parametric optome-chanical vacuum coupling rate gopt,0 between the an-tenna field and the vibrational mode has been previ-ously described [6] and its value is given by gopt,0 =

ωc

(~eopt · ∂αν

∂Qν· ~eopt

)(1

Voptε0

)√~

2ωνwith αν the polar-

izability tensor, Qν the reduced displacement coordinateof the vibrational mode labeled by ν, Vopt the opticalmode volume and ~eopt the unit polarization vector of theoptical antenna mode.

The coupling rate gir,0 associated with a vibrational

mode ν is linked to an effective transition dipole ~dνthat can be numerically computed using e.g. densityfunctional theory (DFT; cf. Appendix D). Its value is

gir,0 = 1~~dν · ~E0 where the electric field per photon is

given by ~E0 =√

~ων

2ε0Vir~eir with Vir the mode volume and

~eir the unit polarization vector of the IR mode [27, 32].

Since g(Nir)ir,0 scales with

√Nir/Vir, it can be indepen-

dent of the mode volume as long as this volume is filledwith molecules. On the contrary the interaction of the

vibration with the VIS-NIR optical field g(Nopt)opt,0 scales as√

Nopt/Vopt advocating for a device confining stronglythis field, thereby reducing the optical power required toreach an efficient conversion process.

IV. DUAL PLASMONIC ANTENNA

Nanoantennas have proven to be instrumental in en-hancing the interaction of molecules with off-resonantVIS-NIR optical fields (e.g. for surface-enhanced Ramanscattering, SERS) [33, 34] and resonant IR fields (e.g. forsurface-enhanced infrared absorption, SEIRA) [35, 36].We now present the design of a new dual-resonant an-tenna (see Figs. 1 and 2) and compute the interaction ofthe local fields at the VIS-NIR and IR resonances with aspecific vibrational mode of an ensemble of molecules cov-ering the nanostructure. We assume that molecules areattached with their main axis perpendicular to the metal-lic surfaces, and extract from DFT calculations the rele-vant components of the derivatives of the electronic mo-ment and polarizability with respect to the normal modecoordinate. We note that calculations for specific self-

assembled monolayer orientations [37, 38] or randomlyoriented molecules can be achieved from the full knowl-

edge of(∂αν

∂Qν

)and

(∂~µν

∂Qν

).

In our design the incoming field (to be up-converted)and the pump laser field are each resonant with a dif-ferent component of the antennas arranged in a crossedconfiguration (cf. Appendix E for additional informationon the design and values of its parameters). At their in-tersection, the near-field polarizations of the two fieldsare collinear (~eir ' ~eopt), and we obtain ηpol = 33 %for the vibrational mode illustrated in Fig. 2, when ac-counting for the specificities of the corresponding Ramantensor and IR transition dipole. Electromagnetic simu-lations demonstrate that the two fields, despite differ-ing in frequency by more than one order of magnitudein that particular example, are confined within a verysimilar volume inside the nanogaps separating the twostructures. This results in a spatial overlap of the twomain electromagnetic field components within the dualantenna reaching ηmode = 44 %.

From our numerical calculations we find that theantenna-assisted IR coupling rate for the vibrationalmode at wavenumber ν = 1002 cm−1 reachesg

(Nir)ir,0 /(2π) ∼ 186 GHz as Vir is decreased by several

orders of magnitude below its diffraction limit (the cal-

culation of Vir and ~dν are detailed in Appendix C). Asthe cavity damping rate remains larger than the collec-

tive vacuum IR coupling rate (2g(Nir)ir,0 < κir/2 — Purcell

regime) the antenna-enhanced damping rate for this vi-brational mode can be approximated by the expression

[32] : Γ∗ν ' Γν + κir/2

(1−

√1− (2g

(Nir)ir,0 )2/(κir/2)2

).

We compute in Appendix D the coupling rate of an-other vibrational mode that has a larger IR dipole mo-ment: Under optimal molecular orientation and fillingconditions, that mode is at the onset of the collectivestrong coupling regime with the IR antenna mode [23].While future work is needed to properly describe thisregime in the context of wavelength conversion, we notethat our design offers new perspectives to realize a sourceof IR photons. Indeed, in the strong coupling regimeand under optomechanical parametric amplification [6],the optically pumped population is shared between thecollective vibrational mode and the corresponding IR an-tenna mode, resulting in the fast emission of IR radiation.

V. OPTICAL NOISE CONTRIBUTIONS

A useful figure of merit to compare the performanceof detectors independently of their respective opera-tional bandwidth is the noise-equivalent power, NEP =

Pmin,inir /

√BW [W ·Hz−1/2] where Pmin,in

ir is the incom-ing power at which the detector reaches a unity signal-to-noise ratio (SNR). This definition can be translatedto a our converter by defining the SNR as SNR(ω) =

Soutir→opt(ω)/Sout

opt(ω). If we assume that the dark noise of

5

(a)

(b)

(c)

Figure 3. (a-b) Left axis: Noise-equivalent power (NEP)(black solid line); right axis: Power-dependent part of theinternal conversion efficiency ηOM(nopt) (red solid line) andnoise equivalent photon number per gate nG (blue solid line)plotted on a logarithmic vertical scale (a) as a function of thedetuning of the optical pump laser with respect to the plas-monic resonance for a fixed intracavity optical photon numbernopt = 1000; and (b) plotted as a function of intracavity op-tical photon number nopt for a fixed optimal red detuning∆ = −Ων . We use the parameters for the dual antenna andmolecular system described in the text. All their numericalvalues can be found in the Appendixes. (c) nG as a functionof the thermal occupancy of the vibrational mode for severalintracavity optical photon numbers. The dashed vertical linedenotes the thermal occupancy of the mode considered in thetext ων/(2π) = 30 THz at room temperature.

a detector in the VIS-NIR range is negligible compared tothe up-conversion noise, the NEP of the converter is byextension that of a detector built upon it. We show thecomputed NEP in Fig. 3 as a function of the optical pumppower and laser detuning from the cavity resonance. Re-markably, the NEP reaches values that improve on thestate of the art for uncooled commercial devices (Ta-ble I) and compares favorably with more recently demon-strated room-temperature platforms [39, 40] operating at

the higher end of the frequency range achievable with ourmolecular up-converter.

Detector type NEP O(...) [

W ·Hz−1/2]

Golay cell O(

10−9)

Pyroelectric O(

10−10)

MCT photodiode O(

10−10)

Microbolometer O(

10−11)

Molecular device O(

10−(11..12))

Table I. Noise-equivalent power of commercially available un-cooled devices in the 5–50 THz region [4, 5] and comparisonwith the device presented in this manuscript (molecular de-vice).

When operating with a pump laser red-detuned fromthe optical resonance (∆ = ωp − ωc = −ων) in the re-solved sideband regime κopt/2 < ων we can simplify theinteraction of Eq. 3 and obtain

Heff = −~g(Nopt)opt,0

√nopt

(δa†optbν + h. c.

), (7)

This regime provides maximal efficiency and optimalNEP, as seen in Fig. 3(a). We note that for low-frequencyvibrational modes the condition κopt/2 < ων could beachieved with the help of hybrid cavities that feature nar-rower linewidths [41, 42].

Keeping this optimal detuning, we investigate inFig. 3(b) how the NEP depends on optical pump power.As the intracavity pump photon number is increased,the efficiency initially grows linearly, while the noise re-mains constant, limited by the thermally generated anti-Stokes signal. This yields a square-root decrease of NEPwith pump power. Interestingly, at high intracavity pho-ton number the contribution of optomechanical quantumbackaction to the dark-count rate surpasses the ther-mal contribution, and the NEP degrades with increas-ing power. This behavior is reminiscent of the standardquantum limit for displacement detection in optomechan-ical cavities.

Another unique feature of the up-conversion schemeis its compatibility with single-photon detectors alreadyavailable in the VIS-NIR range. To assess more preciselythe feasibility of operating our device in single-photoncounting mode, we introduce the noise-equivalent pho-ton rate, i.e. Sout

opt/ηext ≡ |〈ainir〉|2/SNR. This quantity

corresponds to the incoming IR photon rate at the in-put of the device that would generate an output rate ofup-converted photons equal to the dark-count rate.

In practice, noisy single-photon detectors are best op-erated in gated mode. In our approach, this mode iseasily realized using a pulsed optical pump laser, witha pulse duration ∆t of a few picoseconds that ide-ally matches the molecular vibrational linewidth; ∆t '

6

(Γ∗ν)−1

. This mode of operation provides not only bet-ter noise rejection and higher intracavity photon num-bers (therefore better efficiency), but also ultrafast tim-ing resolution that is not otherwise achievable due to theintrinsic timing jitter of VIS-NIR single-photon counters(typically several tens of picoseconds) [8]. We thereforedefine the noise-equivalent photon number per gate nG =Sout

opt/ (ηextΓ∗ν) which translates the noise-equivalent pho-

ton rate into an average incoming IR noise photons pertime gate.

VI. CONVERTER ARRAYS

To gain more insight on the limiting factors constrain-ing single-photon operation, we plot nG as a function ofthe thermal occupancy of the vibration and of the in-tracavity pump photon number in Fig. 3(c). This graphshows that moderate cooling of the device to 100C belowambient (achieved with thermoelectric cooling systems)would bring the thermal occupancy of this vibrationalmode down to nth = 5.6 · 10−4 allowing to reach a noiseas low as nG ' 2 · 10−2, making single-photon countingwith picosecond time resolution in the MIR and FIR do-mains a realistic prospect. For the lower-frequency range(5–20 THz) reducing the temperature of the moleculesin a cryogenic environment would be required to allowsingle-photon operation.

A promising way to further reduce the gated dark-count level consists in designing an array of molecularconverters, sufficiently distant from each other so as notto interact by near-field coupling. We assume that thearray is illuminated by spatially coherent IR signal andoptical pump beam, which is achievable when using ahigh f -number lens due to the sub-wavelength dimen-sions of the antennas. The key advantage of this schemeis that the anti-Stokes fields of thermal origin from dif-ferent antennas would not exhibit any mutual phase co-herence; they will add up incoherently in the far field.On the contrary, the up-converted (sum-frequency) anti-Stokes fields would be phase coherent and interfere con-structively in specific directions, in analogy with a phasedemitter array [43, 44]. Considering a simple linear array,as demonstrated in Appendix F, this effect would jointlydecrease the thermal contribution to the dark-count rateand dilute the intracavity photon number per device, en-abling single-photon operation with improved sensitivity.

A configuration with multiple converters within theIR spot could alternatively be leveraged for on-chip IRmultiplexing [45–48] with distinct converters respond-ing to distinct IR frequencies by the proper choiceof molecular vibrations and antenna design, therebybypassing the limited detection bandwidth of a singleconverter. This sub-wavelength platform benefits fromthe coherent nature of the conversion process and opensthe route to IR spectroscopy, IR hyperspectral imagingand recognition technologies.

VII. CONCLUSION

In summary, we presented a new concept for frequencyupconversion from the mid-infrared to the visible domainbased on the interaction of both fields with molecular vi-brations coupled to a dual-resonant nanoantenna. Weconsidered an incoming long-wavelength infrared radia-tion that resonantly excites a vibrational mode, which issimultaneously coupled through its Raman polarizabil-ity to a coherent pump field at shorter wavelength (vis-ible or near-infrared), resulting in up-conversion of theIR signal onto the anti-Stokes sideband of the pump.Thanks to the recently developed framework of molec-ular cavity optomechanics, we were able to treat theproblem in a full quantum model, and thereby predictthe internal quantum efficiency of our device, as well asits outgoing noise spectral density. We showed that thenoise added in the conversion process has two main ori-gins: thermal noise and backaction noise (including quan-tum and dynamical backaction), the latter increasing su-perlinearly with pump power and eventually becomingdominant. We analyzed the dependence of the noise-equivalent power (NEP)on the intracavity pump photonnumber and pump-cavity detuning, and predicted thatunder the optimal condition of red-sideband excitation,

the NEP can be as low as few pW · Hz−1/2, improvingon the state of the art for devices operating at ambientconditions.

We stress that our numerical estimates are based ona realistic nanoantenna design and a common simplemolecule (thiophenol). Although the intracavity photonnumbers required to reach optimal performance appearto be large, they can be achieved under pulsed excitation[49]. Moreover requirements on the intracavity powerwould be lowered by further reducing the gap size (downto 1–2 nm) and by chemical engineering of the molecu-lar converter toward higher Raman activity. Our studyalso shows that by moderately increasing the resonantcoupling rate between molecular vibration and IR an-tenna, the system would enter the IR strong couplingregime, with the formation of vibrational polaritons [23].We leave the study of the conversion process in this newregime for future investigation.

ACKNOWLEDGEMENTS

The authors thank Wen Chen and the reviewers ofthe article for valuable comments. C. G. acknowledgessupport from the Swiss National Science Foundationthrough Grant No. PP00P2-170684. This work hasreceived funding from the European Union’s Horizon2020 research and innovation programme with grantagreement No. 732894 (HOT) and No. 829067 (THOR).P. R. acknowledges support from the Max Planck-EPFLCenter for Molecular Nanoscience and Technology. D.M.-C. thanks Vahid Sandoghdar and acknowledgesfinancial support from the Max Planck Society.

7

Appendix A: Optomechanical framework

A detailed description of the optomechanical frame-work can be found elsewhere [6, 28–30]. Here we just

remind the readers of the few definitions and relation-ships used in the paper.

The average number of intracavity excitations in the VIS-NIR (label ‘opt’) or IR antenna mode is related to the in-coming photon flux |〈ain

opt/ir〉| and incoming power P inopt/ir

by

nopt/ir = |〈ainopt/ir〉|

2 κopt/irex

∆2 +(κopt/ir/2

)2 =P in

opt/ir

~ωp/irκ

opt/irex

∆2 +(κopt/ir/2

)2 . (A1)

When considering the molecular vibrational levelsand their parametric coupling to the optical field, theantenna-assisted transition rate to a lower excited level(anti-Stokes transition) is given by

A− =(g

(Nopt)opt,0

)2

noptκopt

(ων −∆)2

+ (κopt/2)2 . (A2)

The antenna-assisted transition rate to a higher excitedvibrational level (Stokes transition) is given by

A+ =(g

(Nopt)opt,0

)2

noptκopt

(ων + ∆)2

+ (κopt/2)2 . (A3)

The interested reader can find the complete derivation

of the outgoing spectral density in Ref. [26]. In thismanuscript we are interested in the signal arising on theanti-Stokes sideband. Starting from Eq. (4) in the maintext and following the same calculation steps we arriveat the final expression

Souttot (ωaS) =

2

π

ηoptA−

Γ∗ν + Γoptnf (A4)

For convenience we label the different components of theoutgoing noise spectral density according to the originof the vibrational population from which they result (cf.Eq. (5) in the main text):

Souttot (ωaS) ∝ A−

Γ∗ν + Γoptnth︸ ︷︷ ︸

Soutth

+A−Γ∗ν

(Γ∗ν + Γopt)2 ηoverlapn

irb︸ ︷︷ ︸

Soutir→opt

+A−

(Γ∗ν + Γopt)2

[A+ − Γoptnth

]︸ ︷︷ ︸

Soutba

. (A5)

With this notation the total noise quanta in the outgoingoptical field is Sout

opt = Soutth + Sout

ba .

We can then derive the expression for the conver-sion efficiency defined as Sout

ir→opt = ηext|〈ainir〉|2 and

obtain

ηext = ηopt · ηoverlap ·A−Γ∗ν

Γ∗ν + Γopt

1

κir

(2g

(Nir)ir,0

Γ∗ν + Γopt

)2

︸ ︷︷ ︸ηint

·ηir.

(A6)This expression comprises the different factors constitut-ing Eq. (6) of the main text.

For a pump field red-detuned from the optical antennaresonance (∆ = ωp − ωc = −ων) this expression canbe further developed to evidence the dependence of theinternal conversion efficiency on IR and optical collective

vacuum coupling rates:

ηint = ηoverlap·

(2g

(Nopt)opt,0

)2

nopt

κopt (Γ∗ν + Γopt)

Γ∗ν(Γ∗ν + Γopt)

(2g

(Nir)ir,0

)2

κir (Γ∗ν + Γopt).

(A7)Using this conversion efficiency, we have an alternate wayto calculate the NEP directly from the dark-count rate

and efficiency of the device as NEP = ~ων

ηext

√Sout

opt [8].

This method gives identical results as the one presentedin the main text.

In Fig. 4 we show the computed conversion efficiencyand NEP for the case ∆ = −ων . In this red-detunedconfiguration our model assumptions remain valid for alarge range of optical intracavity photon numbers. Athigh optical power we observe that the efficiency and theNEP reach an extremal value when the backaction con-tribution to the outgoing noise equals that of thermalvibrations.

8

In Fig. 4b,c we also show how these extremal pointsmove when changing the IR absorption cross-section (inb) or Raman activity (in c) of the molecules. As can beexpected, improving the Raman activity of the moleculesonly displaces the curves to the left without modifyingthe extremal values – meaning that lower optical pumppower is required to reach the same NEP and conversionefficiency. In contrast, improving the IR absorption canlead to a lower achievable NEP, in a certain range. Fortoo large IR cross-sections, the vibration starts to enterthe strong coupling regime with the IR antenna, and adifferent treatment would be needed to provide accuratepredictions.

Appendix B: Zero-temperature limit

The limit of vanishing thermal occupancy of the vi-brational mode is relevant for specific applications, andit demonstrates how the backaction noise acting onto thevibration sets a fundamental lower bound on the achiev-able NEP of the converter. When (nth ∼ 0) the outgoingnoise spectral density of eq. (A5) can be simplified to

Sout0K (ωaS) =

2

πηopt

A−A+

(Γ∗ν + Γopt)2 . (B1)

Taking into account eq. (A6) for the expression of theexternal conversion efficiency, the NEP and the nG at0 K can be calculated.

In the regime of weak optical pumping, Γopt < Γ∗ν , weobtain a linear scaling of nG as a function of the intracav-ity photon number (appearing in the transition rates A−

and A+) while the NEP remains constant (cf. Fig. 5).The value of the NEP at this plateau, which correspondsto an intrinsic quantum limit due to measurement back-action, is given by the following expression

NEP0K =~ωνκir

ηoverlap√ηoptηir

√A+

A−(Γ∗ν + Γopt)

(5/2)

Γ∗ν

(2g

(Nir)ir,0

)2 . (B2)

Appendix C: Absorption of incoming IR radiationby molecular vibration

We describe in the following the coupling between aresonant field and a single molecular oscillator inside acavity (i.e. antenna). We also derive the expression givenin the main text for the number of excited phonons in thesteady state. Our treatment is inspired by that of Ref.[24].

The interaction between an external monochromaticfield of frequency ωir and the molecular vibration insidea cavity is given in the dipole approximation by

Hint = −~d · ~Eir, (C1)

(a)

(b)

(c)

Figure 4. (a) Relative contributions to outgoing opticalnoise (Sout

opt) on the anti-Stokes sideband as a function of in-tracavity photon number for a pump tone red-detuned fromthe resonance of the optical antenna (∆ = −ων). The contri-bution from the thermal population of the vibrational modeto the dark-count rate is depicted in green and the backac-tion noise in blue. (b-c) Noise-equivalent power (NEP, leftaxis, black solid line) and power-dependent part of the inter-nal conversion efficiency (ηOM, right axis, red solid line) as afunction of intracavity optical photon number for ∆ = −ων .The parameters used to plot the lines are described in theAppendixes. The dots indicate the extremal values in NEPand ηOM when varying the absorption intensity [0.1:10]I irν in(b) or the Raman activity [0.1:10]RLLν in (c). The coloredareas denote the NEP and ηOM achievable when sweeping ei-ther parameter and the dashed arrows indicate the directionof evolution of the extremum when increasing the parametervalue.

where ~Eir = i√nir

[e−iωirte−i(φ+φ0)t~E0 − eiωirtei(φ+φ0)t~E∗0

]with φ the phase offset between the field and the dipole,φ0 an adjustable phase parameter of the driving field.

~E0 =√

~ωir

2ε0Vir~ek is the vacuum field, with Vir the mode

volume and ~ek the unit polarization vector of the IRmode.

This interaction can be written in terms of the bosonicladder operators describing the IR mode inside the cavity

a†ir, air and the vibrational phononic operators b†ν , bν atfrequency ν. For a weak IR drive the vibrational Hilbert

9

Figure 5. Zero-temperature quantum limit on the NEP.Noise-equivalent power (NEP, black solid line, left axis),power-dependent part of the internal conversion efficiency(ηOM, red solid line, right axis) and noise-equivalent photonnumber per gate (nG, blue solid line, right axis) as a functionof the intracavity optical photon number for ∆ = −ων . Theparameters used here are described in the Appendixes.

space of each molecule can be reduced to ground andfirst excited state |0〉, |1〉 and described like a two levelsystem (TLS) with creation and annihilation operatorsσ+ν , σ

−ν [27]. We note that the validity of the TLS de-

scription for a collective vibrational mode of N oscillatorswould break down only for a number of excitations of or-der N [32].

The dipolar transition is purely off-diagonal in this

basis and described as ~d = dν (σ−ν ~eν + σ+ν ~e∗ν ). The

field inside the cavity is in turn described by ~Eir =

−i√nirE0

[e−iφ0 air~ek − eiφ0 a†ir~e

∗k

][32]. In a frame ro-

tating at the frequency of the IR driving field and keep-

ing only the resonant processes (a†irσ−ν , airσ

+ν ) we obtain

the interaction Hamiltonian :

Hint = −i~gir(e−iφ0 a†irσ

−ν + eiφ0 airσ

+ν

), (C2)

with gir =dν · E0

~√nir~e

∗ν ~eke

iφ0 = gir,0√nir.

Here we choose the additional phase term of the drivingfield in order for the coupling to be real positive, withoutloss of generality.

We follow the dynamics of the TLS in this rotatingframe. Introducing the rate Γtot which describes the totaldamping of the vibrational mode as described in the main

text, we obtain

ρ11 = igir2

(ρ10 − ρ01)− Γtotρ11 (C3a)

ρ00 =− igir2

(ρ10 − ρ01) + Γtotρ11 (C3b)

ρ01 =− iδρ01 − igir2

(ρ11 − ρ00)− Γtot

2ρ01 (C3c)

ρ10 = iδρ10 + igir2

(ρ11 − ρ00)− Γtot

2ρ10, (C3d)

with δ = ωir − ων the detuning between the IR driveand the vibrational resonance.

These equations are often described with the help of theBloch vector components:

u =1

2(ρ01 + ρ10) (C4a)

v =1

2i(ρ01 − ρ10) (C4b)

w =1

2(ρ11 − ρ00) . (C4c)

The components u, v of the Bloch vectors are re-

lated to the average dipole value [24]: 〈~d 〉 =

2~dν (u cosωirt− v sinωirt). We derive the master equa-tions as a function of these components:

u = δ v − Γtot

2u (C5a)

v =− δ u− girw −Γtot

2v (C5b)

w = girv − Γtotw −Γtot

2. (C5c)

The steady-state solutions of these equations are

u =gir2

δ

δ2 + (Γ2tot/4) + (g2

ir/2)(C6a)

v =gir2

Γ/2

δ2 + (Γ2tot/4) + (g2

ir/2)(C6b)

w +1

2=g2ir

4

1

δ2 + (Γ2tot/4) + (g2

ir/2). (C6c)

The average number of photons absorbed per unit timeby the vibrational dipole is given by

dnirbdt

=dW ir

dt

1

~ωir=~En cosωirt · 〈 ~d 〉

~ωir. (C7)

If the detuning and coupling are much smaller than thevibrational damping rate (δ, gir < Γtot), the averagenumber of absorptions over an IR period can be writ-ten as

dnirbdt

= girv =g2ir

2

Γtot/2

δ2 + (Γ2tot/4) + (g2

ir/2)' g2

ir

Γtot. (C8)

In the steady state the rate of photons absorbed by thevibrational mode equals the phonon damping rate so thatthe average number of excited phonons is

10

nirb =dnirbdt

1

Γtot=

g2ir

Γ2tot

=g2ir,0

Γ2tot

nir =g2ir,0

Γ2tot

κirex

δ2 + (κir/2)2 S

inirδκir

=4g2

ir,0

Γ2tot

ηirκir|〈ain

ir〉|2. (C9)

The condition δ κir is satisfied in realistic scenariossince the IR antenna decay rate κir is typically fasterthan the vibrational damping rate Γtot. We note thatthe average number of excited phonons nirb can also besimply derived from the steady-state population of theupper TLS state nirb = w + 1

2 .

Appendix D: Simulation of molecular parameters

The infrared absorption intensity of fundamental vi-brational transitions I irν can be obtained by DFT calcu-lations [31, 50] of the derivatives of the electric momentcomponents µiν with respect to the normal coordinatesrepresenting the vibrational mode of interest. They areusually expressed in [km·mol−1]. For a non-degenerateand harmonic vibrational mode the absorption intensityaveraged over all orientations is given by

〈I irν 〉 =NA

12ε0c2

3∑i=1

(∂µiν∂Qν

)2

, (D1)

with NA the Avogadro number.

1. Gaussian calculations

The procedure is well described in the context of Ra-man calculations in the book of Le Ru & Etchegoin [51].The Gaussian software gives access to the derivativesof the electric dipole with respect to the ith componentof the displacement in Cartesian coordinates of the nthatom. These derivatives can then be converted to deriva-tives with respect to the normal coordinates of a vibra-tional mode ν and are given in atomic units, i.e. the elec-tric moment is given in bohr-electron (2.54 D / 8.48·10−30

C·m) and the displacement in bohr (0.529 A). The quan-

tities ∂~µν

∂Qνcan be converted to other systems of units:

(∂~µν∂Qν

)2

[D2 · A−2 · amu−1] =

[2.54

0.53

]2

·(∂~µν∂Qν

)2

[atomic units], (D2)

and finally linked to the absorption intensity I irν of anincoming field of polarization ~ei

I irν [km ·mol−1] = 126.8 ·(~ei ·

∂~µν∂Qν

)2

[D2 · A−2 · amu−1].

(D3)

2. Cross-section

This expression can be linked to the absorptioncross-section I irν = NA

∫σν′,abs dν′. If we assume a

Lorentzian profile for the transition considered, the on-resonance value of the cross-section is

∫σν′,abs dν′ =

π2σabs (ν′ = ν) δν = π

2σabs (ν′ = ν) Γtot

2πc .

Thus, the absorption cross-section can be inferred fromDFT calculations:

σν,abs [cm2] =4c

NAΓtotI irν [km ·mol−1] · 107. (D4)

3. Effective dipole moment

Accordingly we can also describe an effective dipolemoment dν to characterize the vibrational transition andlink it explicitly to the electronic moment derivativesfound in DFT calculations:

dν =

√3~ε0cΓtotσν,abs

2ων=

√6~ε0c2103

NA

√I irν [km ·mol−1]

ων(D5)

4. Raman activity of an ensemble of molecules

We refer the interested reader to Refs. [6, 51] for de-tailed descriptions of the Raman activity, its connectionwith the optomechanical coupling rate and its calculationthrough DFT. For completeness we reproduce here a fewexpressions of the tensorial quantity ∂αν

∂Qνaveraged over

randomly oriented molecules. To simplify the notationwe introduce the Raman tensor Rν = ∂αν

∂Qνand we refer

to the scalar (~ei ·Rν · ~ej) as Rijν . Taking ~ei ⊥ ~ej andaveraging over random orientations of the molecules one

11

Mode [cm−1] I irν (〈I irν 〉) [km ·mol−1] RLLν(〈RLLν 〉

)[A

4 · amu−1] ηpol (〈ηpol〉) Coverage g(Nir)ir,0 /κir

mode 1002 0.52 (0.51) 2.40 (0.96) 0.33 (0.14)monolayer 0.01

volume 0.06

mode 1093 86.95 (28.52) 0.85 (0.31) 0.97 (0.18)monolayer 0.17

volume 0.75

Table II. Molecular parameters of interest for our conversion scheme for two vibrational modes of the thiophenol molecule.Calculations are obtained for a molecule oriented vertically with respect to both IR and VIS/NIR local fields (values averagedover all molecular orientations are given in parentheses for completeness). The resulting resonant coupling terms are calculatedfor two different coverages of the nanostructure by the molecules and given in units of κir.

can obtain

〈∣∣Riiν ∣∣2〉 =

√45αν2 + 4γν2

45(D6)

〈∣∣Rjiν ∣∣2〉 =

√3γν2

45, (D7)

with α2ν = 1

9 [Rxxν +Ryyν +Rzzν ]2

and γ2ν =

12

[(Rxxν −Ryyν )

2+ (Ryyν −Rzzν )

2+ (Rzzν −Rxxν )

2]

+

3[(Rxyν )

2+ (Rxzν )

2+ (Ryzν )

2]. These quantities do not

depend on the two orthogonal orientations of the fieldchosen as polarization basis but only on the intrinsicproperties of the molecule. In that situation Raman scat-tering can be described by a scalar named the magnitude

of the Raman tensor R2 = 〈∣∣Riiν ∣∣2〉+〈∣∣Rjiν ∣∣2〉 =

45α2ν+7γ2

ν

45and can be derived directly from DFT calculations.We also introduce the depolarization ratio ρ =

〈∣∣Rjiν ∣∣2〉/〈∣∣Riiν ∣∣2〉 that evaluates the importance of the

cross-polarized component of the Raman-scattered field(with respect to the incoming field) and that is boundedby 0 ≤ ρ ≤ 3/4. In the SERS scenario the outgoing fieldis solely polarized along the direction of the local cavityfield ~eL. For randomly oriented molecules the magnitudeof the Raman tensor is thus rescaled by a factor depen-dent on the depolarization ratio:

〈∣∣RLLν ∣∣2〉 = R2 1

1 + ρ, (D8)

5. Local overlap - ηpol

The factor ηpol describes the local overlap between thetwo fields involved in our conversion scheme, on the onehand, and the IR dipole and Raman tensor of the molec-ular vibration, on the other hand. It is defined in thefollowing way:

ηpol =~eL · ∂~µν

∂Qν

‖ ∂~µν

∂Qν‖RLLν‖Rν‖

(D9)

with the label L designating the direction of the near fieldat the location of the molecule. To compute 〈ηpol〉) (see

Table II) we numerically average ηpol over all possibleorientations of the molecule, while keeping the IR andoptical local field collinear.

6. Orientation and number of moleculescontributing to the IR/optical process

From the DFT calculations we compute the molecularparameters for several cases of interest and report theirvalues in Table II. Two orientations (main axis of themolecule parallel to both local fields or fully randommolecular orientation) are considered. Two options arealso considered for the coverage: one monolayer coveringthe planar parts of the metallic nanostructure or a su-perposition of layers filling the entire volume where thefields are localized. We use the IR/optical mode volumesVIR/opt (given below), the molar mass (M = 0.1102

kg/mol), volume density (ρ = 1077 kg/m3) or surfacedensity (ρS = 6.8 · 1018 m−2) of thiophenol to estimatethe number Nir (Nopt) of molecules participating in theIR (optical) process.

Appendix E: Numerical simulation of the antenna’soptical response

Our dual antenna consists of two gold bowtie struc-tures. We set the gap between the tips of both antennas(S = 25 nm) so that the design can be fabricated usingcurrent nanofabrication techniques such as focusedion beam milling or advanced e-beam lithography.We select the other structural parameters (Fig. 6) inorder to obtain appropriate resonances both in themid-IR (length L and width W ) and in the opticaldomain (short length l). In our design the 24 nm highnanostructure lies on top of a gold substrate. Theyare separated by an inactive dielectric layer (n = 1.47)of 8 µm thickness. This substrate reflects the incom-ing IR field and creates an interference pattern thatimproves IR absorption as shown in a previous study [35].

12

LSl

W

XY-view

Zoom in(a)

(b) (c)

Figure 6. (a) Geometry of the dual antenna considered in our manuscript. The parameters W = 70 nm, L = 2.25 µm, l = 90nm, S = 25 nm are used to run the numerical simulations and evaluate the performance of our conversion scheme. The antennaedges are rounded with 4 and 1.3 nm radii of curvature for the optical and IR antenna, respectively. (b-c) Normalized decayrate of an emitter placed at the center of the dual antenna in the mid-IR (b) and in the visible range (c).

1. Numerical calculations

We use a 3D FEM software (Comsol Multiphysics)to evaluate our dual-antenna design. A Drude-Lorentzmodel describes the electromagnetic response of gold fit-ted from experimental data [52]. For the calculation inthe optical range a dielectric layer (ITO) with refractiveindex n = 1.94 and thickness 52 nm is added below theantenna [53].

A dipolar emitter is placed in the center of our struc-ture to evaluate the local density of electromagneticstates inside the antenna. Figure 6 shows the modifica-tion of the radiated power as a function of the oscillationfrequency of the dipole. Based on these plots, we modelthe response of the structure to an incoming optical fieldand to an incoming mid-IR field (around 32 THz) as be-ing each dominated by a single resonance with Lorentzianprofile. We thus use a multi-Lorentzian fit to extractthe relevant linewidths and total decay rates. Throughthe Purcell formula [30], we could estimate the corre-sponding effective mode volumes. Additional integralsare computed to determine the losses originating fromabsorption in the metal and determine the ratio betweenintrinsic and radiative losses at the resonance frequencies.All parameters are shown in Table III.

Parameter Mid-IR VIS/NIR

κir/opt/2π [THz] 3.2 26.7

ηir/opt 0.52 0.73

Vir/opt [m3] 2.6 · 10−21 1.0 · 10−21

Table III. Antenna parameters as obtained from our FEMsimulations. κir/opt are the total decay rates of IR, respec-tively optical, energy stored in the antenna, ηir/opt are the

respective radiative efficiencies defined as κir/optex /κir/opt, and

Vir/opt are the mode volumes.

2. Spatial overlap - ηmode

The spatial overlap between the two electromagneticmodes is computed numerically from the field distribu-tions of both modes according to

ηmode =

(∫ ∣∣∣ ~Eir · ~Eopt

∣∣∣ dA)2

(∫ ∣∣∣ ~Eir · ~Eir

∣∣∣dA)(∫ ∣∣∣ ~Eopt · ~Eopt

∣∣∣dA)(E1)

Imperfect overlap can result from polarization and/orconfinement mismatch. In our case the dominant mis-match is that between the spatial extents of the twomodes. In the regions where both fields are confinedtheir polarization mismatch is on the contrary negligible.

Appendix F: Linear array of converters

We discuss the different contributions to the opticalnoise starting from the expression for nf , Eq. (5) inthe main text. When Γopt Γ∗ν the equation for thevibrational population splits into three different factorsidentified as thermal nth, dynamical backaction ndba andquantum backaction noises nqba, respectively:

nf ' nth −Γopt

Γ∗νnth +

A+

Γ∗ν

(1− Γopt

Γ∗ν

). (F1)

For sensing applications it is enlightening to study howthe contributions from the different noise terms are af-fected when considering an array of converters coherentlyilluminated by the IR field and the pump laser. We de-scribe a linear array of Nconv optomechanical convert-ers. For simplicity we consider identical converters sep-arated uniformly with a spacing d < λopt < λir in or-der to avoid multiple maxima in the radiation patternof the array. If all converters are excited in phase, the

13

described configuration is known as the broadside config-uration and the maximum radiation is directed normal tothe array axis. We assume that the optical pump poweris split among the antennas2, so that the pump powerper antenna is diluted according to |α(i)|2 = 1

Nconv|α(0)|2

and the optomechanical coupling rate scales as 1Nconv

, i.e.

A± (i) = 1Nconv

A± (0). In this section, we use the super-

scripts (0) vs. (i) to designate quantities computed un-der single-converter operation vs. array operation (bothof them refer to single-converter quantities).

a. Thermal regime : If the backaction effects are

weak at a single-converter level Γ(0)opt < Γ

∗,(0)ν , the power

dilution leads to Γ(i)opt Γ

∗,(i)ν . The expression of the

final population nf (Eq. F1) shows that in this casethermal noise is the main contribution to the total noise.

In the far field, constructive interference among thefields emitted from individual antennas sharpens the pat-tern of coherent radiation [43] so that the total IR con-

verted signal in this direction scales as [54] Sout,(Nconv)ir→opt =

(array factor)2 · Sout,(i)

ir→opt which results in Sout,(Nconv)ir→opt '

N2convS

out,(i)ir→opt along the direction of maximum radiation

for a broadside array. On the contrary if the converters

are sufficiently spaced to avoid any near-field coupling thethermal emission remains incoherent and quasi-isotropic.

We combine the factors related to the power dilutionand to the directivity of the linear array to describe theSNR of the array in the regime dominated by thermalnoise:

SNR(Nconv) ' N2conv ·

Sout,(i)ir→opt

Sout,(i)th

= N2conv ·

1

Nconv·S

out,(0)ir→opt

Sout,(0)th

.

(F2)

b. Zero-temperature limit : In the case where thethermal population of the vibrational mode is negligible(nth ∼ 0), corresponding to backaction noise dominatingover thermal noise, the incoming power dilution lowersequivalently the converted signal and the output noiseper antenna, so that the SNR of a sufficiently large arrayin this regime is given by :

SNR(Nconv) ' N2conv ·

Sout,(i)ir→opt

Sout,(i)ba

= N2conv ·1 ·

Sout,(0)ir→opt

Sout,(0)ba

. (F3)

[1] Masayoshi Tonouchi, “Cutting-edge terahertz technol-ogy,” Nature Photonics 1, 97–105 (2007).

[2] S. Ariyoshi, K. Nakajima, A. Saito, T. Taino, C. Otani,H. Yamada, S. Ohshima, J. Bae, and S. Tanaka, “Ter-ahertz response of NbN-based microwave kinetic induc-tance detectors with rewound spiral resonator,” Super-conductor Science and Technology 29, 035012 (2016).

[3] J. Bueno, O. Yurduseven, S. J. C. Yates, N. Llombart,V. Murugesan, D. J. Thoen, A. M. Baryshev, A. Neto,and J. J. A. Baselmans, “Full characterisation of a back-ground limited antenna coupled KID over an octave ofbandwidth for THz radiation,” Applied Physics Letters110, 233503 (2017).

[4] F. Sizov, “Terahertz radiation detectors: the state-of-the-art,” Semiconductor Science and Technology 33, 123001(2018).

[5] A. Rogalski, M. Kopytko, and P. Martyniuk, “Two-dimensional infrared and terahertz detectors: Outlookand status,” Applied Physics Reviews 6, 021316 (2019).

[6] Philippe Roelli, Christophe Galland, Nicolas Piro, andTobias J. Kippenberg, “Molecular cavity optomechanicsas a theory of plasmon-enhanced Raman scattering,” Na-ture Nanotechnology 11, 164–169 (2016).

[7] G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov,A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov,C. Williams, and Roman Sobolewski, “Picosecond su-perconducting single-photon optical detector,” AppliedPhysics Letters 79, 705–707 (2001).

2 The diffraction limit for both beams being largely different, wenote that multiple converters fit under a focused IR spot. In thatcase the IR power per converter would not scale down.

[8] Robert H. Hadfield, “Single-photon detectors for opticalquantum information applications,” Nature Photonics 3,696–705 (2009).

[9] L. Tian and Hailin Wang, “Optical wavelength conver-sion of quantum states with optomechanics,” PhysicalReview A 82, 053806 (2010).

[10] Chunhua Dong, Victor Fiore, Mark C. Kuzyk, andHailin Wang, “Optomechanical Dark Mode,” Science338, 1609–1613 (2012).

[11] Jeff T. Hill, Amir H. Safavi-Naeini, Jasper Chan, andOskar Painter, “Coherent optical wavelength conversionvia cavity optomechanics,” Nature Communications 3,1196 (2012).

[12] T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Sim-monds, and K. W. Lehnert, “Coherent state transferbetween itinerant microwave fields and a mechanical os-cillator,” Nature 495, 210–214 (2013).

[13] Joerg Bochmann, Amit Vainsencher, David D.Awschalom, and Andrew N. Cleland, “Nanome-chanical coupling between microwave and opticalphotons,” Nature Physics 9, 712–716 (2013).

[14] R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Ci-cak, R. W. Simmonds, C. A. Regal, and K. W. Lehn-ert, “Bidirectional and efficient conversion between mi-crowave and optical light,” Nature Physics 10, 321–326(2014).

[15] Moritz Forsch, Robert Stockill, Andreas Wallucks, IgorMarinkovic, Claus Gartner, Richard A. Norte, Frank vanOtten, Andrea Fiore, Kartik Srinivasan, and SimonGroblacher, “Microwave-to-optics conversion using a me-chanical oscillator in its quantum ground state,” NaturePhysics 16, 69–74 (2020).

[16] Cherif Belacel, Yanko Todorov, Stefano Barbieri, DjamalGacemi, Ivan Favero, and Carlo Sirtori, “Optomechani-

14

cal terahertz detection with single meta-atom resonator,”Nature Communications 8, 1578 (2017).

[17] R.W. Boyd and D. Prato, Nonlinear Optics (Elsevier Sci-ence, 2008).

[18] Ketil Karstad, Andre Stefanov, Mark Wegmuller, HugoZbinden, Nicolas Gisin, Thierry Aellen, Mattias Beck,and Jerome Faist, “Detection of mid-IR radiation bysum frequency generation for free space optical commu-nication,” Optics and Lasers in Engineering Optics inSwitzerland, 43, 537–544 (2005).

[19] P. Tidemand-Lichtenberg, J. S. Dam, H. V. Andersen,L. Høgstedt, and C. Pedersen, “Mid-infrared upconver-sion spectroscopy,” JOSA B 33, D28–D35 (2016).

[20] Yu-Pei Tseng, Christian Pedersen, and Peter Tidemand-Lichtenberg, “Upconversion detection of long-wave in-frared radiation from a quantum cascade laser,” OpticalMaterials Express 8, 1313–1321 (2018).

[21] Gerhard Herzberg, Molecular Spectra and MolecularStructure II. Infrared and Raman Spectra of PolyatomicMolecules (Van Nostrand, 1960).

[22] Norman B. Colthup, Lawrence H. Daly, and Stephen E.Wiberley, Introduction to Infrared and Raman Spec-troscopy: 3rd edition (Elsevier Science, 1990).

[23] A. Shalabney, J. George, J. Hutchison, G. Pupillo,C. Genet, and T. W. Ebbesen, “Coherent coupling ofmolecular resonators with a microcavity mode,” NatureCommunications 6, 5981 (2015).

[24] Claude Cohen-Tannoudji, Jacques Dupont-Roc, andGilbert Grynberg, Atom-photon interactions: basic pro-cesses and applications (Wiley, 1992).

[25] Bernd Metzger, Eric Muller, Jun Nishida, Benjamin Pol-lard, Mario Hentschel, and Markus B. Raschke, “Purcell-enhanced spontaneous emission of molecular vibrations,”Physical Review Letters 123 (2019).

[26] I. Wilson-Rae, N. Nooshi, J. Dobrindt, T. J. Kippen-berg, and W. Zwerger, “Cavity-assisted backaction cool-ing of mechanical resonators,” New Journal of Physics10, 095007 (2008).

[27] Marlan O. Scully and Muhammad S. Zubairy, QuantumOptics (Cambridge University Press, 1997).

[28] Albert Schliesser and Tobias J Kippenberg, “Cavity op-tomechanics with whispering-gallery mode optical micro-resonators,” in Advances In Atomic, Molecular, and Op-tical Physics, Vol. 58 (Elsevier, 2010) pp. 207–323.

[29] Mikolaj K. Schmidt, Ruben Esteban, AlejandroGonzalez-Tudela, Geza Giedke, and Javier Aizpurua,“Quantum Mechanical Description of Raman Scatteringfrom Molecules in Plasmonic Cavities,” ACS Nano 10,6291–6298 (2016).

[30] Mikolaj K. Schmidt, Ruben Esteban, Felix Benz,Jeremy J. Baumberg, and Javier Aizpurua, “Linkingclassical and molecular optomechanics descriptions ofSERS,” Faraday Discussions 205, 31–65 (2017).

[31] Edgar Bright Wilson, J. C. Decius, and Paul C. Cross,Molecular Vibrations: The Theory of Infrared and Ra-man Vibrational Spectra (Courier Corporation, 1980).

[32] Serge Haroche and Jean-Michel Raimond, Exploring theQuantum: Atoms, Cavities, and Photons (Oxford Uni-versity Press, 1997).

[33] Wenqi Zhu and Kenneth B. Crozier, “Quantum me-chanical limit to plasmonic enhancement as observed bysurface-enhanced raman scattering,” Nature Communi-cations 5, 5228 (2014).

[34] Felix Benz, Mikolaj K. Schmidt, Alexander Dreismann,Rohit Chikkaraddy, Yao Zhang, Angela Demetriadou,Cloudy Carnegie, Hamid Ohadi, Bart de Nijs, RubenEsteban, Javier Aizpurua, and Jeremy J. Baumberg,“Single-molecule optomechanics in “picocavities”,” Sci-ence 354, 726–729 (2016).

[35] Lisa V. Brown, Xiao Yang, Ke Zhao, Bob Y. Zheng, Pe-ter Nordlander, and Naomi J. Halas, “Fan-Shaped GoldNanoantennas above Reflective Substrates for Surface-Enhanced Infrared Absorption (SEIRA),” Nano Letters15, 1272–1280 (2015).

[36] Frank Neubrech, Christian Huck, Ksenia Weber, An-nemarie Pucci, and Harald Giessen, “Surface-EnhancedInfrared Spectroscopy Using Resonant Nanoantennas,”Chemical Reviews 117, 5110–5145 (2017).

[37] Frank Schreiber, “Structure and growth of self-assembling monolayers,” Progress in Surface Science 65,151–257 (2000).

[38] J. Christopher Love, Lara A. Estroff, Jennah K. Kriebel,Ralph G. Nuzzo, and George M. Whitesides, “Self-Assembled Monolayers of Thiolates on Metals as a Formof Nanotechnology,” Chemical Reviews 105, 1103–1170(2005).

[39] Mingsheng Long, Anyuan Gao, Peng Wang, Hui Xia,Claudia Ott, Chen Pan, Yajun Fu, Erfu Liu, XiaoshuangChen, Wei Lu, Tom Nilges, Jianbin Xu, Xiaomu Wang,Weida Hu, and Feng Miao, “Room temperature high-detectivity mid-infrared photodetectors based on blackarsenic phosphorus,” Science Advances 3, e1700589(2017).

[40] Xiaolong Chen, Xiaobo Lu, Bingchen Deng, Ofer Sinai,Yuchuan Shao, Cheng Li, Shaofan Yuan, Vy Tran, KenjiWatanabe, Takashi Taniguchi, Doron Naveh, Li Yang,and Fengnian Xia, “Widely tunable black phosphorusmid-infrared photodetector,” Nature Communications 8,1–7 (2017).

[41] Hugo M. Doeleman, Ewold Verhagen, and A. FemiusKoenderink, “Antenna–cavity hybrids: Matching polaropposites for purcell enhancements at any linewidth,”ACS Photonics 3, 1943–1951 (2016).

[42] Burak Gurlek, Vahid Sandoghdar, and Diego Martin-Cano, “Manipulation of quenching in nanoantenna–emitter systems enabled by external detuned cavities: Apath to enhance strong-coupling,” ACS Photonics 5, 456–461 (2018).

[43] Daniel Dregely, Richard Taubert, Jens Dorfmuller, RalfVogelgesang, Klaus Kern, and Harald Giessen, “3d opti-cal Yagi–Uda nanoantenna array,” Nature Communica-tions 2, 267 (2011).

[44] Sebastian Busschaert, Nikolaus Flory, Sotirios Pa-padopoulos, Markus Parzefall, Sebastian Heeg, andLukas Novotny, “Beam Steering with a Nonlinear Opti-cal Phased Array Antenna,” Nano Letters 19, 6097–6103(2019).

[45] Benedikt Schwarz, Peter Reininger, Daniela Ristanic,Hermann Detz, Aaron Maxwell Andrews, WernerSchrenk, and Gottfried Strasser, “Monolithically inte-grated mid-infrared lab-on-a-chip using plasmonics andquantum cascade structures,” Nature Communications5, 4085 (2014).

[46] Hongtao Lin, Zhengqian Luo, Tian Gu, Lionel C. Kimer-ling, Kazumi Wada, Anu Agarwal, and Juejun Hu, “Mid-infrared integrated photonics on silicon: a perspective,”Nanophotonics 7, 393–420 (2018).

15

[47] Andreas Tittl, Aleksandrs Leitis, Mingkai Liu, FilizYesilkoy, Duk-Yong Choi, Dragomir N. Neshev, Yuri S.Kivshar, and Hatice Altug, “Imaging-based molecularbarcoding with pixelated dielectric metasurfaces,” Sci-ence 360, 1105–1109 (2018).

[48] Filiz Yesilkoy, Eduardo R. Arvelo, Yasaman Jahani,Mingkai Liu, Andreas Tittl, Volkan Cevher, YuriKivshar, and Hatice Altug, “Ultrasensitive hyperspec-tral imaging and biodetection enabled by dielectric meta-surfaces,” Nature Photonics , 390–396 (2019).

[49] Gelon Albrecht, Stefan Kaiser, Harald Giessen, andMario Hentschel, “Refractory plasmonics without refrac-tory materials,” Nano Letters 17, 6402–6408 (2017).

[50] Conrard Giresse Tetsassi Feugmo and Vincent Liegeois,“Analyzing the Vibrational Signatures of Thiophenol Ad-

sorbed on Small Gold Clusters by DFT Calculations,”ChemPhysChem 14, 1633–1645 (2013).

[51] Eric Le Ru and Pablo Etchegoin, Principles of Surface-Enhanced Raman Spectroscopy: and Related PlasmonicEffects (Elsevier, 2008).

[52] D. R. Lide, ed., CRC Handbook of Chemistry andPhysics, 87th ed. (CRC Press, Boca Raton, FL, 2006).

[53] Arvind Sundaramurthy, K. B. Crozier, G. S. Kino, D. P.Fromm, P. J. Schuck, and W. E. Moerner, “Field en-hancement and gap-dependent resonance in a system oftwo opposing tip-to-tip Au nanotriangles,” Physical Re-view B 72, 165409 (2005).

[54] Constantine A. Balanis, Antenna Theory: Analysis andDesign (Wiley-Interscience, 2005).