Optomechanically Induced Transparency

Coherent interaction of laser radiation with multilevel atoms and molecules can lead to quantum interference in the electronic excitation pathways (1). A prominent example observed in atomic three-level systems is the phenomenon of electromagnetically induced transparency (EIT), in which a control laser induces a narrow spectral transparency window for a weak probe laser beam. When this generic EIT effect had first been observed in an atomic gas (2), its relevance in nonlinear optics and optical (quantum) information processing was quickly recognized. In particular, the rapid variation of the refractive index concomitant with the opening of the transparency window gives rise to a dramatic reduction of the group velocity of a propagating optical pulse (3, 4). Dynamic control of EIT with the control laser enables even a complete stop, that is, storage, of the pulse in an atomic medium (5, 6). The experimental demonstration of slowing and stopping light (3–6) has attracted strong attention, because it provides a route to implement a photonic quantum memory (7) or a classical optical buffer. EIT has subsequently been studied in a wide variety of atomic media, but also in several solid-state systems (8, 9) with a well-suited level structure.

Recent experiments with optomechanical systems have demonstrated that the mechanical response to thermal forces can be controlled by an optical field. This effect has been exploited, for example, to implement optomechanical laser cooling and amplification (10–13) as well as normal mode splitting (14). In other work, the mechanical response was optically tailored to exhibit destructive interference between different mechanical excitation pathways (15). Whereas in these studies, the mechanical response to thermal Langevin force was modified, we demonstrate here, as recently suggested (16, 17), that the system’s optical response to a weak “probe” laser can be controlled by a second “control” laser driving the lower motional sideband. A window of transparency arises from the destructive interference of excitation pathways for the intracavity probe field when a two-photon resonance condition is met. As pointed out independently, this effect can be considered a strict optomechanical analog of EIT (18), originating from a similar effective interaction Hamiltonian (19). Advantageously, this form of induced transparency does not rely on naturally occurring resonances and could therefore also be applied to previously inaccessible wavelength regions such as the technologically important near-infrared. Furthermore, a single optomechanical element can already achieve unity contrast, which in the atomic case is only possible within the setting of cavity quantum electrodynamics (20).

Our experiment (Fig. 1) consists of an optomechanical system featuring linear optomechanical coupling G in the sense that the cavity resonance frequency is given by ω′c(x) = ω_c + Gx, where ω_c is the unperturbed resonance frequency. A control laser (frequency ω_l) maintains a control field a¯e−iω1t, containing |a¯|2photons, in the cavity. The static radiation pressure originating from this field displaces the mechanical mode by x¯, leading to an effective detuning from the cavity resonance Δ¯=ω1−(ωc+Gx¯). We consider the situation where the control laser is tuned close to the lower motional sideband, i.e., Δ¯≈−Ωm where Ω_m is the mechanical (angular) resonance frequency. A second, weak laser oscillating at ω_p = ω_l + Ω, is subsequently used to probe the (modified) cavity resonance by driving an intracavity probe field contained in a perturbation term δa(t).

• Download high-res image
• Open in new tab
• Download Powerpoint

In the case of a weak probe field (compared to the control field), one can linearize the optomechanical dynamics (21) for the mechanical displacement x(t)=x¯+δx(t) and the intracavity field a(t)=[a¯+δa(t)]e−iω1taround the steady-state values (x¯,a¯). For the probe power transmission—that is, the ratio of the probe power returned from the system divided by the input probe power—the general expression|tp|2=|1−1+if(Ω)−i(Δ¯+Ω)+κ/2+2Δ¯f(Ω)ηcκ|2(1)withf(Ω)=ℏG2a¯2 χ(Ω)i(Δ¯−Ω)+κ/2(2)can be derived [see (16–18) and supporting online material (SOM) Eq. S25]. Here, χ(Ω)=[meff(Ωm2−Ω2−iΓmΩ)]−1 is the susceptibility of the mechanical oscillator of effective mass m_eff and damping rate Γ_m. The optical mode is characterized by a total loss rate κ = κ₀ + κ_ex and the cavity coupling parameter η_c = κ_ex/(κ₀ + κ_ex). The presence of a control field a¯ (tuned to the lower sideband) induces a transmission window for the probe beam when the resonance condition Ω ≈ Ω_m is met (Fig. 1). The depth and the width of this transmission window are tunable by the power of the control beam, as in the case of atomic EIT, with the best contrast achieved in the case of critical coupling η_c = 1/2.

To gain more physical insight into this phenomenon, it is instructive to consider this effect in a sideband picture. The simultaneous presence of control and probe fields generates a radiation-pressure force oscillating at the frequency difference Ω. If this driving force oscillates close to the mechanical resonance frequency Ω_m, the mechanical mode starts to oscillate coherently, δx(t) = 2Re[X e^−iΩt]. This in turn gives rise to Stokes- and anti-Stokes scattering of light from the strong intracavity control field. If the system resides deep enough in the resolved-sideband (RSB) regime with κ << Ω_m, Stokes scattering (to the optical frequency ω_l − Ω) is strongly suppressed because it is highly off-resonant with the optical cavity. We can therefore assume that only an anti-Stokes field builds up inside the cavity, δa(t) ≈ A⁻ e^−iΩt. However, this field of frequency ω_p = ω_l + Ω is degenerate with the near-resonant probe field sent to the cavity. Destructive interference of these two driving waves can suppress the build-up of an intracavity probe field. These processes are captured by the Langevin equations of motion for the complex amplitudes A⁻ and X, which require in the steady state (SOM Eqs. S26 and S27)(−iΔ′+κ/2) A−=−iGa¯X+ηcκ δsin(3)2meffΩm(−iΔ′+Γm/2) X=−iℏGa¯A−(4)where δs_in is the amplitude of the probe field drive, and we abbreviate Δ′ ≡ Ω − Ω_m. We have assumed a high-quality factor of the mechanical oscillator (Γ_m << Ω_m) and the control beam detuning Δ¯=−Ωm. The solution for the intracavity probe field amplitude readsA−=ηcκ(−iΔ′+κ/2)+Ωc2/4−iΔ′+Γm/2δsin(5)This solution has a form well known from the response of an EIT medium to a probe field (1). The coherence between the two ground states of an atomic Λ system, and the coherence between the levels probed by the probe laser undergo the same evolution as do the mechanical oscillation amplitude and the intracavity probe field in the case of optomechanically induced transparency (OMIT). The role of the control laser’s Rabi frequency in an atomic system is taken by the optomechanical coupling rate Ωc=2a¯Gxzpf, where xzpf=ℏ/2meffΩm designates the spread of the ground-state wave function of the mechanical oscillator. For Ω_c > Γ_m, κ the system enters the strong coupling regime (22, 23) investigated recently in the mechanical domain (14), in which the optical and mechanical systems are hybridized to dressed states that differ by ℏΩc in their energy.

OMIT is realized using toroidal whispering-gallery-mode microresonators (Fig. 2A) (10, 17). The cavity is operated in the undercoupled regime (η_c < 1/2), which together with modal coupling between counterpropagating modes (SOM Sec. 7) leads to a nonzero probe (amplitude) transmission t_r = t_p(Δ′ = 0, Ω_c = 0) at resonance (Fig. 2B), even in the absence of the control beam. In the case of the present device, |t_r|² ≈ 0.5. [Note, however, that |t_r|² < 0.01 can be achieved with silica toroids (24).] To separate the effects of this residual transmission from OMIT, we introduce the normalized transmission of the probe t′p=(tp−tr)/(1−tr).

• Download high-res image
• Open in new tab
• Download Powerpoint

The mechanical motion was detected using a balanced homodyne detection scheme (fig. S1) measuring the phase quadrature of the field emerging from the cavity (25). This allows extracting the parameters of the device used in these experiments, which are given by (m_eff, G/2π, Γ_m/2 π, Ω_m/2 π, κ/2 π) ≈ (20 ng, −12 GHz/nm, 41 kHz, 51.8 MHz, 15 MHz), placing it well into the resolved sideband regime (25). To probe the cavity transmission spectrum in the presence of a control beam, the Ti:sapphire control laser is frequency modulated at frequency Ω using a broadband phase modulator, creating two sidebands at ω_l + Ω = ω_p (probe field) and ω_l − Ω. In the resolved sideband regime, only the upper sideband, nearly resonant, interacts with the optomechanical system. Keeping the laser detuned to the lower motional sideband of the cavity (Δ¯≈−Ωm), a sweep of the modulation frequency Ω scans the probe field through the cavity resonance. As shown in detail in the SOM (Sec. 5), demodulation of the total homodyne signal at the modulation frequency Ω using a network analyzer (NA) allows extracting a “transmission” homodyne signal t_hom, which, in the RSB regime, is related to the probe transmission by the simple relation t_hom ≈ 1 − t_p.

Figure 3A shows the theoretically expected response of the optomechanical system and the detected signals. Clearly, the OMIT window is apparent in the intracavity probe power as described by Eq. 5, occurring simultaneously with the onset of radiation-pressure–driven mechanical oscillations. The excitation of the intracavity probe field therefore is suppressed, and the transmitted field nearly equals the probe field sent to the cavity. The lowest panel shows the corresponding homodyne signal, and the five panels in Fig. 3B show experimentally measured homodyne traces for detunings Δ¯/2π varying from −69.1 MHz to −35.4 MHz and a control laser power of 0.5 mW. The center of the probe extinction (maximum of the homodyne signal) always occurs for Ω≈−Δ¯, because the probe laser then matches the cavity resonance [ωp≈ω′c(x¯)]. Notably, however, the sharp OMIT window occurs only when the two-photon resonance condition Ω = Ω_m (with Ω = ω_p − ω_l) is met, independent of the detuning Δ¯ of the control beam, giving clear evidence to the theoretically suggested underlying mechanism.

• Download high-res image
• Open in new tab
• Download Powerpoint

To analyze the effect of the control beam more systematically, its detuning was fixed to the lower motional sideband. Varying its power from 0.125 to 6.5 mW, traces of the homodyne signal are taken in the vicinity of the two-photon resonance (Fig. 4). Dips of increasing depth and width are observed, which can be modeled by a simple Lorentzian function (SOM Sec. 3). The minimum homodyne signal is obtained under the condition of the two-photon resonance Δ′ = 0. In this case, the homodyne signal power and the probe power transmission are simply interrelated by |t′p|2=(1−|t′hom|)2, where t′hom=thom/(1−tr) is the normalized homodyne signal. The width of the measured dip in the normalized homodyne signal is equal to that of the coupling-induced transmission window Γ_OMIT ≈ Γ_m (1 + C), where C ≡ Ωc2/Γ_mκ is an equivalent optomechanical cooperativity parameter (14). From the model (Eq. 5), the expected probe transmission on resonance is simply given by t′p(Δ′ = 0) = C/(C + 1). Our data match these expectations well if we allow for a linear correction factor in the optomechanical coupling frequency Ω_c due to modal coupling (SOM Sec. 7) and losses in the fiber taper. We have reached probe power transmission |t′p|2 up to 81%, indicating the high contrast achievable in OMIT.

• Download high-res image
• Open in new tab
• Download Powerpoint

In fact, any optomechanical system reaching C ≳ 1 can realize an appreciable control-induced probe transmission, as desired, for example, in all-optical switches. Interestingly, the systems currently available reach C ≈ 1 with only thousands (26) or even hundreds (27) of control photons in the cavity, and recently emerging integrated nano-optomechanical structures (28) may be able to further reduce this number. The resulting extreme optical nonlinearities could be of interest for both fundamental and applied studies.

The tunable probe transmission window also modifies the propagation of a probe pulse due to the variation of the complex phase picked by its different frequency components. Indeed, a resonant probe pulse experiences a group delay of τ_g ≈ 2/Γ_OMIT in the regime C ≳ 1 of interest (SOM Sec. 6), a value exceeding several seconds in some available optomechanical systems (29). However, undistorted pulse propagation only occurs if the full probe-pulse spectrum is contained within the transparency window of the system. This restricts the effectiveness of such a delay due to the fixed delay-bandwidth product of τ_g Γ_OMIT ≈ 2. A cascade of systems may alleviate this shortcoming—the most interesting scenario being a large array of concatenated optomechanical systems, as suggested in the context of OMIT (16, 17, 30), and radio frequency/microwave photonics (15). The group delay could then be dynamically tuned while the probe pulse is propagating through the array. Such systems could be practically implemented in lithographically designed optomechanical systems both in the microwave (27) and optical (31) domain.

Note added in proof: After online publication of this work, OMIT has also been reported in microwave optomechanical systems by Teufel et al. (32).