Optomechanically amplified wavelength conversion in diamond microcavities

TL;DR

This paper demonstrates efficient optomechanical wavelength conversion and amplification in diamond microcavities, enabling low-noise interfacing of quantum emitters at different wavelengths with high efficiency.

Contribution

It introduces the first demonstration of optomechanical frequency conversion with amplification in diamond microcavities, achieving high internal conversion efficiency.

Findings

Internal conversion efficiency of ~45%

First demonstration of optomechanical frequency conversion with amplification

Applicable to quantum technologies with diverse emitters

Abstract

Efficient, low noise conversion between different colors of light is a necessary tool for interfacing quantum optical technologies that have different operating wavelengths. Optomechanically mediated wavelength conversion and amplification is a potential method for realizing this technology, and is demonstrated here in microdisks fabricated from single crystal diamond--a material that can host a wide range of quantum emitters. Frequency up--conversion is demonstrated with internal conversion efficiency of $\sim$45% using both narrow and broadband probe fields, and optomechanical frequency conversion with amplification is demonstrated in the optical regime for the first time.

Figures

Equations

$$A_{\text{x}}=2\left(\frac{{\kappa }_{\text{e}}}{\kappa }\frac{4G_1{G}_2/\kappa }{{\Gamma }_{\text{m}}-4{\overline{G}}^2/\kappa }\right),$$

$$S_{\text{add,m}}=\frac{{\Gamma }_{\text{m}}{\kappa }_{\text{e}}{G}_1^2}{{\left|{\Gamma }_{\text{m}}\kappa /4-{\overline{G}}^2\right|}^2}\left(n_{\text{th}}+\frac{1}{2}\right),$$

$$S_{\text{add}}=\frac{{\Gamma }_{\text{m}}\kappa }{4G_2^2}\frac{\kappa }{{\kappa }_{\text{e}}}\left(n_{\text{th}}+\frac{1}{2}\right)+\frac{{\kappa }_{\text{i}}}{{\kappa }_{\text{e}}}+\frac{1}{2}.$$

$${\alpha }_{\text{in}}=\alpha \left(e^{i\varphi }+\frac{\beta }{2}{e}^{i\omega t}+\frac{\beta }{2}{e}^{-i\omega t}\right),$$

$$t_{\text{out}}=t_{\text{out}}^{+}+t_{\text{out}}^0+t_{\text{out}}^{-}$$

$$r_{\text{out}}=r_{\text{out}}^{+}+r_{\text{out}}^0+r_{\text{out}}^{-}$$

$$t_{\text{out}}^{+}$$

$$t_{\text{out}}^0$$

$$t_{\text{out}}^{-}$$

$$r_{\text{out}}^{+}$$

$$r_{\text{out}}^0$$

$$r_{\text{out}}^{-}$$

$${\left|t_{\text{out}}\right|}^2=|t^0{|}^2|{\alpha }_{\text{in}}^0{|}^2+t^{0\ast }{t}^{+}{\alpha }_{\text{in}}^{0\ast }{\alpha }_{\text{in}}^{+}+t^0{t}^{-\ast }{\alpha }_{\text{in}}^0{\alpha }_{\text{in}}^{-\ast }+c.c$$

$${\left|r_{\text{out}}\right|}^2=|r^0{|}^2|{\alpha }_{\text{in}}^0{|}^2+r^{0\ast }{r}^{+}{\alpha }_{\text{in}}^{0\ast }{\alpha }_{\text{in}}^{+}+r^0{r}^{-\ast }{\alpha }_{\text{in}}^0{\alpha }_{\text{in}}^{-\ast }+c.c,$$

$$\begin{gathered} {\left|t_{\text{out}}\right|}^2=|t^0{|}^2{\alpha }^2+{\alpha }^2\frac{\beta }{2}[t^{0\ast }{t}^{+}{e}^{-i\varphi }+t^0{t}^{-\ast }{e}^{i\varphi }]e^{-i\omega t} \\ +{\alpha }^2\frac{\beta }{2}[t^0{t}^{+\ast }{e}^{i\varphi }+t^{0\ast }{t}^{-}{e}^{-i\varphi }]e^{i\omega t} \end{gathered}$$

$$\begin{gathered} {\left|r_{\text{out}}\right|}^2=|r^0{|}^2{\alpha }^2+{\alpha }^2\frac{\beta }{2}[r^{0\ast }{r}^{+}{e}^{-i\varphi }+r^0{r}^{-\ast }{e}^{i\varphi }]e^{-i\omega t} \\ +{\alpha }^2\frac{\beta }{2}[r^0{r}^{+\ast }{e}^{i\varphi }+r^{0\ast }{r}^{-}{e}^{-i\varphi }]e^{i\omega t}. \end{gathered}$$

$${\left|t_{\text{out}}\right|}^2=|t^0{|}^2{\alpha }^2+{\alpha }^2\beta |A|\cos (\omega t-\arg \{A\})$$

$${\left|r_{\text{out}}\right|}^2=|r^0{|}^2{\alpha }^2+{\alpha }^2\beta |B|\cos (\omega t-\arg \{B\}),$$

$$A=(t^{0\ast }{t}^{+}{e}^{-i\varphi }+t^0{t}^{-\ast }{e}^{i\varphi })$$

$$B=(r^{0\ast }{r}^{+}{e}^{-i\varphi }+r^0{r}^{-\ast }{e}^{i\varphi }).$$

$$t^0({\Delta }_{\text{oc}})=1-\frac{{\kappa }_{\text{e}}/2}{i({\Delta }_{\text{oc}})+\kappa /2}$$

$$t^{\pm }(\pm {\Delta }_{\text{pc}})=1-\frac{{\kappa }_{\text{e}}/2}{i({\Delta }_{\text{oc}}-(\pm {\Delta }_{\text{pc}}))+\kappa /2+\frac{N{g}_0^2}{i({\omega }_{\text{m}}-(\pm {\Delta }_{\text{pc}}))+{\Gamma }_{\text{m}}/2}}.$$

$$\tan \theta \approx {\alpha }_{\text{chirp}}\cot \left(\frac{\phi }{2}\right)$$

$$t^{-}(-{\Delta }_{\text{pc}})=1-\sqrt{{\kappa }_{\text{e}}/2}(a_{\text{s}}(-{\Delta }_{\text{pc}})+a_{\text{a}}(-{\Delta }_{\text{pc}})),$$

$$a_{\text{s,a}}(-{\Delta }_{\text{pc}})=\frac{-\sqrt{{\kappa }_{\text{e}}/2}}{-{\kappa }_{\text{s,a}}/2+i(({\Delta }_{\text{oc}}-(-{\Delta }_{\text{pc}}))\pm {\kappa }_{\text{bs}}/2)}$$

$$g_0=\frac{\beta {\omega }_{\text{cal}}}{2}\sqrt{\frac{1}{n_{\text{th}}}\frac{S_{\text{mech}}({\omega }_{\text{m}})}{S_{\text{cal}}({\omega }_{\text{cal}})}}.$$

$$S_2={\alpha }_{{\lambda }_2}^2\beta |B|,$$

$$S_2={\alpha }_{{\lambda }_2}^2\beta |r_{{\lambda }_2}^{+}||r_{{\lambda }_2}^0|,$$

$$r_{\text{out}}^{+}$$

$$r_{\text{out}}^0$$

$$r_{\text{out}}^{-}$$

$$S_1=2|r_{{\lambda }_1}^0|{\alpha }_{{\lambda }_1}{\alpha }_{{\lambda }_{1,\text{conv}}},$$

$$A_{\text{x,exp}}=\frac{P_{\text{out}}}{P_{\text{in}}}=\frac{{\alpha }_{{\lambda }_2}^2|r_{{\lambda }_2}^0{|}^2}{{\alpha }_{{\lambda }_1}^2|r_{{\lambda }_1}^0{|}^2}{\left(\frac{S_1}{S_2}\right)}^2|r_{{\lambda }_2}^{+}{|}^2.$$

Full text

The interaction of light and vibration of matter is a rich area of physics dating to Brillouin and Raman’s studies of scattering phenomena Brillouin (1922); Raman (1928). With the advent of quantum mechanics, the concept of vibrational energy quanta, phonons Tamm (1932), spurred the study of photon–phonon scattering processes, leading to many technological breakthroughs in spectroscopy, medicine, and communications. Photon-phonon interactions are becoming increasing relevant for solid state quantum technologies, which utilize phonons for transducing Gustafsson et al. (2014); Chu et al. (2017), storing Lee et al. (2011); England et al. (2015), and transmitting information Patel et al. (2018). Cavity optomechanics aims to enhance coherent phonon-photon interactions through co–localization of mechanical and optical resonances coupled via radiation pressure or other optical forces Aspelmeyer, Kippenberg, and Marquardt (2014). These systems provide a platform that could have application in a quantum network or quantum internet Kimble (2008), namely as a phonon mediated way of transducing quantum information from visible or microwave photons to telecommunication wavelength photons Andrews et al. (2014); Habraken et al. (2012); Simon (2017). Phonon mediated wavelength conversion has been demonstrated in both optical Hill et al. (2012); Dong et al. (2012); Liu et al. (2013) and microwave Palomaki et al. (2013); Andrews et al. (2014) regimes, and many current efforts are focused on optical to microwave transduction Bochmann et al. (2013); Rueda et al. (2016); Vainsencher et al. (2016); Reed et al. (2017); Higginbotham et al. (2018); Fan et al. (2018).

Recently, nanofabrication advances have enabled initial investigations of wide electronic bandgap materials such as single–crystal diamond (SCD) for cavity optomechanics Mitchell et al. (2016); Burek et al. (2016); Lake et al. (2018). Diamond does not suffer from nonlinear absorption at telecommunications wavelengths, which can limit achievable local field strength and corresponding optomechanical coupling rates in smaller bandgap materials such as silicon Barclay, Srinivasan, and Painter (2005); Chan et al. (2011). A bonus of working with diamond is that it also hosts highly coherent artificial atoms such as the nitrogen vacancy (NV) and silicon vacancy (SiV) color centers Aharonovich, Englund, and Toth (2016), which have been used in demonstrations of quantum memory Pfender et al. (2017); Sukachev et al. (2017), quantum entanglement Togan et al. (2010); Bernien et al. (2013), quantum teleportation Pfaff et al. (2014), loophole-free Bell’s inequality violation Hensen et al. (2015), and the transfer of phase information from microwave to optical fields Lekavicius et al. (2017). Diamond cavity optomechanics can in principle interface these color centers with light and other quantum systems in new ways Habraken et al. (2012); Simon (2017). One such application is conversion of color center emission to telecommunication wavelengths. A purely optomechanical approach allows high efficiency wavelength conversion between any two wavelengths of a cavity’s mode spectrum Safavi-Naeini and Painter (2011); Hill et al. (2012); Dong et al. (2012); Liu et al. (2013) for relatively low input power, negating the need for material dependent nonlinear optical processes Dréau et al. (2018); Farrera et al. (2016); Maring et al. (2017); Rakher et al. (2010); Pelc et al. (2012); De Greve et al. (2012); Radnaev et al. (2010); Li, Davanço, and Srinivasan (2016). Here we demonstrate optomechanical wavelength conversion in a diamond cavity for the first time, and show that this scheme can be operated in a regime where the converted signal is optomechanically amplified. We also probe the bandwidth and gain properties of this process using a novel broadband coherent spectroscopy technique.

Cavity optomechanical wavelength conversion coherently couples two optical cavity modes at frequencies $\omega_{\text{o,1}}$ and $\omega_{\text{o,2}}$ via their independent optomechanical coupling to a common cavity mechanical resonance at frequency $\omega_{\text{m}}$. In microdisk cavities, such as the diamond cavity studied here and illustrated in Fig. 1(a), the optomechanical coupling is created by the radiation pressure force exerted by optical whispering gallery modes on the microdisk. This interaction can coherently exchange energy between optical and mechanical domains when a strong control laser is input to the cavity at a frequency $\omega_{\text{c}}$ red-detuned $\Delta_{\text{oc}}=\omega_{\text{o}}-\omega_{\text{c}}=\omega_{\text{m}}$ from the cavity mode by the mechanical resonance frequency. The resulting system can be represented by a photon-phonon beamsplitter Hamiltonian Aspelmeyer, Kippenberg, and Marquardt (2014), and has been used to demonstrate optomechanically induced transparency (OMIT) and slow light Weis et al. (2010); Safavi-Naeini et al. (2011) in a variety of devices, including in diamond microdisks Lake et al. (2018).

In multimode cavity optomechanical systems, coherent photon-phonon coupling is harnessed to convert probe photons input on resonance with a mode at $\omega_{\text{o,1}}$ to a mechanical excitation that is in turn converted to signal photons resonant with the second cavity mode $\omega_{\text{o,2}}$. This process requires inputting two control lasers, referred to as write and read fields, to the cavity. Typically both control lasers are red–detuned by $\Delta_{\text{oc}}=\omega_{\text{m}}$ from their respective cavity modes as illustrated in Figs. 1(b,c). This setup, which has been used in previous optical domain wavelength conversion demonstrations Hill et al. (2012); Dong et al. (2012); Liu et al. (2013), can in principle be amplified by blue-detuning the read laser from the cavity ($\Delta_{\text{oc}}=-\omega_{\text{m}}$), as shown in Fig. 1(d). This enables low–noise amplification of the converted photons, as recently demonstrated by Ockeloen-Korppi et al. in the microwave regime Ockeloen-Korppi et al. (2016) and reported below for the first time in the optical domain. Such two–port amplification avoids optomechanical self oscillation by balancing optomechanical amplification with damping for the blue– and red–detuned optical modes, respectively, and unlike traditional optomechanical amplification Massel et al. (2011); McRae and Bowen (2012); Li et al. (2012) has a fundamentally unlimited gain–bandwidth product.

The diamond microdisks used here for wavelength conversion, an example of which is shown in the scanning electron micrograph image in Fig. 2(a), support whispering gallery modes with optical quality factor $Q_{\text{o}}$ sufficiently high for operation in the resolved sideband regime of cavity optomechanics. These devices were fabricated using an optimized reactive ion undercutting process described in Ref. Mitchell, Lake, and Barclay (2019) that results in a smooth diamond microdisk supported by a thin diamond pillar, and have higher $Q_{\text{o}}$ than those previously used for single-mode diamond optomechanics studies Lake et al. (2018).

All of the measurements presented below involve two sets of microdisk optical modes widely separated in wavelength and mutually coupled to the same mechanical radial breathing mode (RBM) of the microdisk, whose mechanical displacement profile is illustrated in Fig. 1(a). Their optomechanical properties were probed using coherent fiber taper optical mode spectroscopy as in previous work Mitchell et al. (2016); Mitchell, Lake, and Barclay (2019); Lake et al. (2018) using the apparatus described in more detail below. Fiber taper transmission spectra for the two modes are shown in Figs. 2(b,c), revealing resonances with frequencies $\omega_{\text{o,1}}/2\pi=195$ THz ($\lambda_{1}=1535$ nm) and $\omega_{\text{o,2}}/2\pi=191$ THz ($\lambda_{2}=1566$ nm), and unloaded $Q_{\text{o}}^{\text{i}}\sim 1.4\times 10^{5}$ and $2.1\times 10^{5}$, respectively. The doublet nature of the resonances indicates that they are standing wave modes Borselli, Johnson, and Painter (2005); the red mode of each doublet was used for all of the measurements described below. The $\lambda_{1}$ mode was found to be TM–like, while the $\lambda_{2}$ mode was TE–like, which is reflected in the different splitting of each doublet, but is not an essential ingredient for the results presented here. These modes are used to optomechanically transduce the motion of the microdisk mechanical resonances, with a typical photodetected power spectral density of the thermomechanical motion of the microdisk’s RBM shown in Fig. 2(d), revealing a resonance frequency $\omega_{\text{m}}/2\pi=2.27$ GHz and mechanical quality factor $Q_{\text{m}}=\omega_{\text{m}}/\Gamma_{\text{m}}\sim 5,800$ in the ambient conditions used for all of the reported measurements. The device’s combination of high mechanical frequency and high $Q_{\text{o}}$ places it in the resolved sideband regime $\kappa/\omega_{\text{m}}\leq$ 1, where $\kappa=\omega_{\text{o}}/Q_{\text{o}}$ is the total cavity photon decay rate.

The experimental setup used to coherently couple light at multiple wavelengths to the microdisk’s mechanical motion is shown in Fig. 3(a). Two independent tunable diode lasers (Newport TLB-6700) generate the control fields. An amplitude or phase electro–optic modulator (EOSpace) driven by a vector network analyzer (Keysight E5063A) at a swept frequency $\delta\omega_{\text{p}}$ generates the probe field by adding sidebands to the $\omega_{\text{c,1}}$ or $\omega_{\text{c,2}}$ control field, respectively. An RF switch selects which control laser will be modulated, determining whether to carry out frequency down- or up–conversion, respectively. The control field intensities are amplified via an erbium–doped fiber amplifier (EDFA, Pritel) before being coupled to the microdisk via the optical fiber taper. The transmitted signal is used for the optical mode spectroscopy described above, while the reflected signal is routed via an optical circulator through a tunable band pass filter (TBF, Optoplex, 100 GHz) that separates the converted signal and the read control field from the write control and the input probe fields. The beat note from the converted signal field interfering with the read control field is then demodulated from the photodetector output by the vector network analyzer.

To demonstrate wavelength conversion the lasers were first setup in the standard up-conversion configuration illustrated in Fig. 1(c): both read and write control fields red–detuned from their respective cavity modes by $\Delta_{\text{oc,1}}=\Delta_{\text{oc,2}}=\omega_{\text{m}}$, and the write control field weakly modulated at $\delta\omega_{\text{p}}$ to generate the probe sideband (wavelength $\lambda_{2}$). The resulting measured beatnote, $S_{\text{C}}(\delta\omega_{\text{p}})$, between the up-converted field at $\lambda_{1}$ and the read control field, is shown in Fig. 3(b). Efficient conversion is observed when $\delta\omega_{\text{p}}=\omega_{\text{m}}$, i.e. when the probe field is on-resonance with the cavity mode, and is observed over a bandwidth defined by the optomechanically broadened mechanical linewidth.

Calibrating $S_{\text{C}}$ and writing it as a wavelength conversion efficiency requires careful characterization of the detectors and electronics used in the experiment. Instead, following previous reports Hill et al. (2012); Liu et al. (2013), we infer an external conversion efficiency $\eta_{\text{ext}}=\eta_{1}\eta_{2}\eta_{\text{int}}$ between $\lambda_{2}$ and $\lambda_{1}$ from the optomechanical properties of our device. Here $\eta_{1}=13\%$ and $\eta_{2}=21\%$ are the experimentally measured waveguide-cavity coupling efficiencies at wavelengths $\lambda_{1}$ and $\lambda_{2}$ respectively, extracted by determining the coupling rate $\kappa_{\text{e}}$ between the microdisk standing wave modes and the forward propagating fiber mode from fits to the doublet resonances, taking into account the standing wave modes’ equal coupling to the backwards propagating fiber mode. The internal conversion efficiency, $\eta_{\text{int}}=4C_{1}C_{2}/(1+C_{1}+C_{2})^{2}$, is determined solely by the optomechanical cooperativity, $C_{j}$, of each mode $\lambda_{j}$ Hill et al. (2012). Here $C_{j}=4N_{j}g_{0,j}^{2}/\kappa_{j}\Gamma_{\text{m}}$ was measured from the modes’ OMIT dips shown in the insets to Fig. 3(b), where $N_{j}$ and $g_{0,j}$ are the respective control field intracavity photon number and the vacuum optomechanical coupling rate of mode $j$. These single color OMIT measurements record the beatnote between control field $j$ in the red detuned configuration ($\Delta\omega_{\text{oc},j}=\omega_{\text{m}}$) and its modulator generated sideband at $\omega_{\text{c},j}+\delta\omega_{\text{p}}$. Note that while both control lasers were kept on during this measurement, only one OMIT process was carried out at a time, by placing the other laser off resonance. By fitting the OMIT spectra in Fig. 3(b) to the model described in Supplement 1 we extract $C_{1}\sim 3.94$ and $C_{2}\sim 1.02$, for $N_{1}\sim 3.1\times 10^{6}$ and $N_{2}\sim 2.2\times 10^{5}$, respectively, corresponding to $\eta_{\text{int}}\sim 45\%$. Here $N_{j}$ and $\Delta_{\text{oc},j}$ were used as the only fitting parameters where all other required parameters were measured independently. Measurements of $g_{0}$ were performed using the phase tone calibration method Gorodetksy et al. (2010), giving $g_{0}/2\pi\sim$ 15 kHz and 27 kHz for the $\lambda_{1}$ and $\lambda_{2}$ modes, respectively, as described in Supplement 1. Note that the Fano–shape of the $\lambda_{1}$ OMIT dip is related to a non-zero phase imparted by the amplitude modulator, and the demodulation process of the VNA, as described in detail in Supplement 1. These measured values for $C_{j}$ are consistent with the estimated fiber taper input power for each control field of $P_{\text{in}}\sim$ 16 mW and $\sim$ 3.8 mW.

Increasing $\eta_{\text{int}}$ could be achieved by operating in the $C_{j}\gg 1$ regime and by matching $C_{1}=C_{2}$. For the diamond cavities studied here, in practice $N_{j}$ can be increased above $10^{6}$ until becoming limited by thermal instabilities within the cavity. For example, C ¿ 3.9 is demonstrated for measurements presented in Supplement 1. In this experiment we were unable to balance $C_{1}$ and $C_{2}$ while maintaining $C_{j}>1$ as the input laser used to drive $\lambda_{2}$ mode was less efficiently amplified in the EDFA. Combined with greater loss in the system at $\lambda_{2}$, this resulted in $N_{2}<N_{1}$. The use of a second EDFA with a gain maximum near $\lambda_{2}$ would allow independent control of $N$ for each mode. Assuming each mode could reach $C\sim 2$, $\eta_{\text{int}}\sim 64\%$ would be achievable. Finally, the relatively small $\eta_{\text{ext}}=1.23\%$ demonstrated here could be improved by operating in the over coupled regime, which requires either higher $Q_{\text{o}}$ or improved coupling via, for example, an external on-chip waveguide. Unfortunately the $\omega_{\text{o,2}}$ mode was outside the range of the TBF, prohibiting measurement of down–conversion for this device. However, down–conversion was measured using a different device, with lower conversion efficiency, as described in Supplement 1.

These devices are also promising for quantum limited amplification of the wavelength converted signal field, as previously demonstrated in the microwave domain Ockeloen-Korppi et al. (2016). As described above and shown in Fig. 1(d), signal amplification is achieved by placing the $\lambda_{1}$ read control laser blue–detuned from the cavity, $\Delta_{\text{oc,1}}=-\omega_{\text{m}}$, while keeping the $\lambda_{2}$ write control laser red–detuned as before, $\Delta_{\text{oc,2}}=\omega_{\text{m}}$. This results in what can be thought of as a two step process involving a beam–splitter interaction between the $\lambda_{2}$ mode and the mechanical mode, and parametric amplification between the mechanical mode and the $\lambda_{1}$ mode Ockeloen-Korppi et al. (2016). The converted–amplified signal is detected in the same fashion as above, where the beat note $S_{\text{C}}^{\text{A}}$ produced by the amplified converted light and the read control field is shown in Fig. 3(b). The frequency and linewidth of the amplified, $S_{\text{C}}^{\text{A}}$, and unamplified, $S_{\text{C}}$, spectra is governed by the optomechanical damping ($\Gamma_{\text{opt}}$) and optical spring effect ($\delta\omega_{\text{m}}$) induced on the mechanical resonance dynamics by each control laser. As both control lasers are red–detuned for $S_{\text{C}}$, its effective mechanical linewidth is broader and the center frequency is red–shifted compared to its intrinsic value shown in Fig. 2(d). In contrast, as described above, for $S_{\text{C}}^{\text{A}}$ the $\lambda_{1}$ laser is blue–detuned while $\lambda_{2}$ is red–detuned. As the photon assisted optomechanical coupling rates, $G_{j}=g_{0,j}\sqrt{N_{j}}$, are such that $G_{1}>G_{2}$ ($G_{1}=2\pi\times 26.4\ \text{MHz},G_{2}=2\pi\times 12.5\ \text{MHz}$), this initially resulted in the microdisk being in the self-oscillation regime ($\Gamma_{\text{m}}+\Gamma_{\text{opt}}<0$). To obtain the amplified spectra in Fig. 3(b) $\Delta_{\text{oc,1}}$ was adjusted such that the device was no longer in the self–oscillation regime, although a narrowed linewidth and blue–shift of the center frequency is still observed due to the imbalance of $G_{1}$ and $G_{2}$. Balancing $G_{1}$ and $G_{2}$ was a technical limitation in our experiment, which could be remedied by using a second EDFA to independently tune $G_{1}$ and $G_{2}$, enabling operation in the large gain–bandwidth product regime. Ideally, in the low–noise amplification scheme described in Ref. Ockeloen-Korppi et al. (2016) the system should be operated in the $G_{2}\gtrsim G_{1}$ regime to avoid instabilities associated with optomechanical self–oscillation.

To characterize the noise properties of the $S_{\text{C}}^{\text{A}}$ spectrum we follow the analysis in Ref. Ockeloen-Korppi et al. (2016), starting with expected frequency–converted gain, given by

[TABLE]

where the optical cavity dissipation constants have been taken to be equal and $\overline{G}^{2}=G_{1}^{2}-G_{2}^{2}$. The adjusted $G_{1}$ value was determined from the linewidth of $S_{\text{C}}^{\text{A}}$ where $G_{2}$ was held constant, giving $G_{1}\sim 2\pi\times 14.1$ MHz. For the operating conditions here, and taking $\kappa=\kappa_{2}$, Eqn. 1 gives $A_{\text{x}}\sim 1.2$. For these operating conditions the amplifier is not in a low–noise regime as the added noise to $S_{\text{C}}^{\text{A}}$ is dominated by the contribution from the mechanical bath, whose equivalent added noise on resonance is given by $S_{\text{add,m}}/|A_{\text{x}}|^{2}$:

[TABLE]

where $n_{\text{th}}\sim k_{\text{B}}T/\hbar\omega_{\text{m}}$ is the thermal occupation of the RBM. For this experiment this results in $\sim 6.52\times 10^{3}$ added quanta to the $S_{\text{C}}^{\text{A}}$ spectra. To predict future performance of this system within the context of low–noise single photon wavelength conversion, we calculate the added noise to the amplified optical signal in the ideal high gain ($G_{2}\gtrsim G_{1}$) and large $\Gamma_{\text{opt}}$ regime, given by

[TABLE]

Here $\kappa_{\text{i}}=\kappa-2\kappa_{\text{e}}$ is the intrinsic cavity photon decay rate, where the optical decay rates have been taken to be similar for each cavity. The predicted added noise as a function of $N_{2}$ is shown in Fig. 3(c) for the parameters of the device under study and for various temperatures. We predict that the minimum added noise of $\sim$ 2 quanta, limited by the coupling efficiency achieved here, can be approached for $N_{2}\sim 1\times 10^{6}$ at 10 mK and $N_{2}\sim 1\times 10^{8}$ at 4 K for this device, provided that operation in the ($G_{2}\gtrsim G_{1}$) regime is enabled by overcoming technical limitations in our experiment. Furthermore, amplification at the SQL of 1/2 quanta could be possible by improving the coupling to the device such that $\kappa_{\text{e}}\gg\kappa_{\text{i}}$.

To further characterize the converted amplified signal we used a broadband (compared to $\kappa$) RF noise source to drive the phase modulator generating the OMIT probe field, similar to the approach described in Ref. Zou et al. (2017). Here an arbitrary waveform generator (AWG:Tektronix 70002A) outputting a pseudo Gaussian white noise signal with a sampling rate and length of 25 GS/s and 2 GS, respectively, drove the RF input of the phase EOM (20 GHz bandwidth) to create a broadband optical probe field from the write field control laser. When the control laser is tuned to $\Delta\omega_{\text{oc,2}}=\omega_{\text{m}}$, observation of both OMIT and wavelength conversion on a real time spectrum analyzer (Tektronix RSA5106B) is possible. Figure 4(a) shows the measured probe field’s power spectral density, revealing the cavity response for frequency components of the probe that do not coherently drive the mechanics, which in these measurements is the broad peak in the demodulator signal with a line width of $\kappa/2\pi=1.4$ GHz, and an OMIT dip at $\delta\omega_{\text{p}}=\omega_{\text{m}}$, i.e. where the OMIT condition is satisfied. This demonstrates a straightforward method for measuring OMIT and the cavity response simultaneously in frequency space.

Wavelength conversion of the $\lambda_{2}$ broadband probe generated by the RF noise source, in both the unamplified and amplified configurations, measured in reflection, is shown in Figs. 4(b) and 4(c) respectively. The plots in Figs. 4(b) and 4(c) show the detected power spectral density of the converted signal at $\lambda_{1}$ (blue trace) compared for reference with the corresponding measurement when the $\lambda_{2}$ probe is off (orange trace) so that the detected spectrum is derived entirely from the optomechanically modified motion of the microdisk due to the $\lambda_{1}$ and $\lambda_{2}$ control fields. As above, when both control lasers are red–detuned in Fig. 4(b) , the effective mechanical linewidth is broader and the center frequency is red–shifted compared to its intrinsic value (purple trace), which was measured at low power with a single optical mode to avoid optomechanical back action. In the amplified conversion case shown in Fig. 4(c) the linewidth is slightly narrowed and shifted to higher frequency due to the fact that $G_{1}>G_{2}$. In principle this technique could be utilized in balancing $G_{j}$ by adjusting $N_{j}$ such that the linewidth of the observed spectra with the broadband probe off is equal to the intrinsic linewidth observed in the absence of the strong control fields. Here the ratio of the amplified to unamplified peak height is $\sim 2.4\times$ larger than what was observed for the coherently driven case in Fig. 3(b). While the control field input power of each mode was held constant between these measurements, $\Delta_{\text{oc,1}}$ and $\Delta_{\text{oc,2}}$ were manually set for each case, which could account for this discrepancy.

The converted signal power in both amplified and unamplified cases can be computed directly from the area under the spectrum. When the probe is off we see the added noise from the thermally driven signal level that would have to be overcome by increasing $C_{j}$ to allow operation at the single–photon level. We estimate the gain utilizing measurement of the cavity transmission profile of the $\lambda_{2}$ mode on the RSA utilizing the broadband probe and the data in Figs. 4(c) as outlined in the Supplementary Material. Using this method we estimate a gain of $A_{\text{x, exp}}\leq 3.2$, which combined with the total linewidth, $\Gamma_{\text{tot}}=\Gamma_{\text{m}}+\Gamma_{\text{opt}}=2\pi\ \times 5.8$ kHz give a gain–bandwidth product, GBW $\leq$ 19 kHz. Possible explanations for the mismatch of $A_{\text{x}}$ and $A_{\text{x, exp}}$ include the approximation of similar $\kappa$ and $\kappa_{\text{e}}$ for each optical mode in calculating $A_{\text{x}}$, uncertainty in the measured optical output power from the cavity, and variation in $\Delta_{\text{oc,1}}$ and $\Delta_{\text{oc,2}}$, as mentioned above. With the data available a measurement of the added noise to the converted signal was not possible, however the thermal component is expected to dominate, as discussed above. To compare the amplified vs. unamplified wavelength conversion, where we have defined the signal ratio, as the difference in integrated power with the broadband probe field on and off, to isolate the converted signal from the thermal component, which is then divided by the off–resonant noise floor. We find that the amplified conversion signal is 1.62 $\times$ larger than without amplification, for the same input broadband probe. With calibration of the probe spectrum, this broadband probe technique could also be used to determine the conversion efficiency, similar to what has been suggested by Liu et al. in Ref. Liu et al. (2013) where a coherent drive and RSA were used to characterize the converted signal.

In summary, we have demonstrated a diamond cavity optomechanical system capable of phonon–mediated wavelength conversion with an internal conversion efficiency of $\sim 45\%$. While we were limited by an imbalance of $G_{1}$ and $G_{2}$, and $\kappa_{\text{e}}<\kappa$ in this experiment, operation in the stable, high-gain regime $G_{2}\gtrsim G_{1}$ should be possible by utilizing a second EDFA to allow separate control of $G_{1}$ and $G_{2}$, and improving waveguide–cavity coupling would enable $A_{\text{x}}\gg 1$. To the best of our knowledge, this is also the first demonstration of optomechanical amplification, albeit not operating in the low–noise regime, of the converted light in the optical domain. In this configuration this system has promise for reaching a minimum added noise of $\sim 2$ quanta during the amplification process by operating at larger photon number and with $G_{2}\gtrsim G_{1}$. This device can in principle operate at the standard quantum limit (SQL) where a minimum of half an energy quanta of noise is added to the signal by improving the waveguide–cavity coupling efficiency, and has immediate practical use in increasing signal to noise when measuring quantum optomechanical effects Caves (1982). In addition, use of a broadband noisy optical probe field was introduced for both OMIT and wavelength conversion. Finally, as previous studies in these structures Mitchell et al. (2016) have demonstrated the existence of high-$Q_{\text{o}}$ optical modes at $\sim 738$ nm and $\sim 637$ nm, near the zero phonon lines of SiV and NV color centers respectively, these devices have potential as a converter of diamond color center emission to telecommunications wavelengths for application in quantum networks.

1 Funding

This work was supported by National Research Council Canada (NRC), Alberta Innovates, National Sciences and Engineering Research Council of Canada (NSERC), and Canada Foundation for Innovation (CFI).

See Supplement 1 for supporting content.