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Optomechanically Induced Transparency

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Science  10 Dec 2010:
Vol. 330, Issue 6010, pp. 1520-1523
DOI: 10.1126/science.1195596

Abstract

Electromagnetically induced transparency is a quantum interference effect observed in atoms and molecules, in which the optical response of an atomic medium is controlled by an electromagnetic field. We demonstrated a form of induced transparency enabled by radiation-pressure coupling of an optical and a mechanical mode. A control optical beam tuned to a sideband transition of a micro-optomechanical system leads to destructive interference for the excitation of an intracavity probe field, inducing a tunable transparency window for the probe beam. Optomechanically induced transparency may be used for slowing and on-chip storage of light pulses via microfabricated optomechanical arrays.

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 (36) 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 (1013) 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 ωc(x) = ωc + Gx, where ωc is the unperturbed resonance frequency. A control laser (frequency ωl) maintains a control field a¯eiω1t, 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 Ωm is the mechanical (angular) resonance frequency. A second, weak laser oscillating at ωp = ωl + Ω, is subsequently used to probe the (modified) cavity resonance by driving an intracavity probe field contained in a perturbation term δa(t).

Fig. 1

Optomechanically induced transparency. (A) A generic optomechanical system consists of an optical cavity with a movable boundary, illustrated here as a Fabry-Perot–type resonator in which one mirror acts like a mass-on-a-spring movable along x. The cavity has an intrinsic photon loss rate κ0 and is coupled to an external propagating mode at the rate κex. Through the external mode, the resonator is populated with a control field (only intracavity field is shown). The response of this driven optomechanical system is probed by a weak probe field sent toward the cavity, the transmission of which (i.e., the returned field “Probe out”) is analyzed here. (B) The frequency of the control field is detuned by Embedded Image from the cavity resonance frequency, where a detuning close to the lower mechanical sideband, Embedded Image, is chosen. The probe laser’s frequency is offset by the tunable radio frequency Ω from the control laser. The dynamics of interest occur when the probe laser is tuned over the optical resonance of the cavity, which has a linewidth of κ = κ0 + κex. (C) Level scheme of the optomechanical system. The control field is tuned close to red-sideband transitions, in which a mechanical excitation quantum is annihilated (mechanical occupation nmnm − 1) when a photon is added to the cavity (optical occupation npnp + 1), therefore coupling the corresponding energy eigenstates. The probe field probes transitions in which the mechanical oscillator occupation is unchanged. (D) Transmission of the probe laser power through the optomechanical system in the case of a critically coupled cavity κ0 = κex as a function of normalized probe laser frequency offset, when the control field is off (blue lines) and on (green lines). Dashed and full lines correspond to the models based on the full (Eq. 1) and approximative (Eq. 3) calculations, respectively.

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)]eiω1taround 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=|11+if(Ω)i(Δ¯+Ω)+κ/2+2Δ¯f(Ω)ηcκ|2(1)withf(Ω)=G2a¯2χ(Ω)i(Δ¯Ω)+κ/2(2)can be derived [see (1618) and supporting online material (SOM) Eq. S25]. Here, χ(Ω)=[meff(Ωm2Ω2iΓmΩ)]1 is the susceptibility of the mechanical oscillator of effective mass meff and damping rate Γm. The optical mode is characterized by a total loss rate κ = κ0 + κex and the cavity coupling parameter ηc = κex/(κ0 + κex). 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 ≈ Ωm 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 ηc = 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 Ωm, the mechanical mode starts to oscillate coherently, δx(t) = 2Re[X eiΩt]. 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 << Ωm, Stokes scattering (to the optical frequency ωl − 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 eiΩt. However, this field of frequency ωp = ωl + 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+ηcκδsin(3)2meffΩm(iΔ+Γm/2)X=iGa¯A(4)where δsin is the amplitude of the probe field drive, and we abbreviate Δ′ ≡ Ω − Ωm. We have assumed a high-quality factor of the mechanical oscillator (Γm << Ωm) and the control beam detuning Δ¯=Ωm. The solution for the intracavity probe field amplitude readsA=ηcκ(iΔ+κ/2)+Ωc2/4iΔ+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=/2meffΩm designates the spread of the ground-state wave function of the mechanical oscillator. For Ωc > Γm, 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 Ωc in their energy.

OMIT is realized using toroidal whispering-gallery-mode microresonators (Fig. 2A) (10, 17). The cavity is operated in the undercoupled regime (ηc < 1/2), which together with modal coupling between counterpropagating modes (SOM Sec. 7) leads to a nonzero probe (amplitude) transmission tr = tp(Δ′ = 0, Ωc = 0) at resonance (Fig. 2B), even in the absence of the control beam. In the case of the present device, |tr|2 ≈ 0.5. [Note, however, that |tr|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 tp=(tptr)/(1tr).

Fig. 2

Optomechanical system. (Top) A toroidal microcavity is used to demonstrate OMIT: The resonator is coupled to the control and probe fields using a tapered fiber. The optical mode couples through radiation pressure force to the mechanical radial breathing mode of the structure. In this ring geometry, the cavity transmission, defined by the ratio of the returned probe-field amplitude divided by the incoming probe field is simply given by the transmission through the tapered fiber. (Bottom) Under the chosen waveguide-toroid coupling conditions, there is a nonzero probe power transmission |tr|2 at resonance. The control field induces an additional transparency window with a contrast up to 1 − |tr|2.

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 (meff, G/2π, Γm/2 π, Ωm/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 ωl + Ω = ωp (probe field) and ωl − 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 thom, which, in the RSB regime, is related to the probe transmission by the simple relation thom ≈ 1 − tp.

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 ΩΔ¯, because the probe laser then matches the cavity resonance [ωpωc(x¯)]. Notably, however, the sharp OMIT window occurs only when the two-photon resonance condition Ω = Ωm (with Ω = ωp − ωl) is met, independent of the detuning Δ¯ of the control beam, giving clear evidence to the theoretically suggested underlying mechanism.

Fig. 3

Observation of OMIT. (A) Theoretically expected intracavity probe power, oscillation amplitude X, normalized probe power transmission Embedded Image, and the normalized homodyne signal Embedded Image as a function of the modulation frequency Ω/2π (top to bottom panels). The first two panels have additionally been normalized to unity. When the two-photon resonance condition Δ′ = 0 is met, the mechanical oscillator is excited, giving rise to destructive interference of excitation pathways for an intracavity probe field. The probe transmission therefore exhibits an inverted dip, which can be easily identified in the homodyne signal. (B) Experimentally observed normalized homodyne traces when the probe frequency is scanned by sweeping the phase modulator frequency Ω for different values of control beam detuning Embedded Image. Whereas the center of the response of the bare optical cavity shifts correspondingly, the sharp dip characteristic of OMIT occurs always for Δ′ = 0. The power of the control beam sent to the cavity is 0.5 mW in these measurements. The middle panel shows the operating conditions where the control beam is tuned to the lower motional sideband Embedded Image ≈ −Ωm = −2π · 51.8 MHz. The region around the central dip (orange background) is studied in more detail in a dedicated experimental series (see Fig. 4).

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 |tp|2=(1|thom|)2, where thom=thom/(1tr) 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 ΓOMIT ≈ Γm (1 + C), where CΩc2mκ is an equivalent optomechanical cooperativity parameter (14). From the model (Eq. 5), the expected probe transmission on resonance is simply given by tp(Δ′ = 0) = C/(C + 1). Our data match these expectations well if we allow for a linear correction factor in the optomechanical coupling frequency Ωc due to modal coupling (SOM Sec. 7) and losses in the fiber taper. We have reached probe power transmission |tp|2 up to 81%, indicating the high contrast achievable in OMIT.

Fig. 4

Controlling OMIT. (A) Experimental normalized homodyne traces in the presence of a control beam (circles) for four different powers in the control beam from 0.125 mW up to 6.5 mW, and Lorentzian models. The minimum homodyne signal (measured at Δ′ = 0) directly indicates the maximum probe power transmission achieved in this case. These values are given in the right panel for a larger set of probe scans, together with the theoretical model developed in this work. (B) Width of the transparency window extracted from the same set of probe scans. Good agreement with the theoretical prediction is found over the entire power range.

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 τg ≈ 2/ΓOMIT 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 τg ΓOMIT ≈ 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).

Supporting Online Material

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

SOM Text

Figs. S1 to S3

References

References and Notes

  1. T.J.K. acknowledges financial support by the Max Planck Society, European Research Council Starting Grant (SiMP), the European Union’s MINOS, a Marie Curie Excellence Grant, the Swiss National Science Foundation, National Centre of Competence and Research of Quantum Photonics, and the Defense Advanced Research Projects Agency. S.D. is funded by a Marie Curie Individual Fellowship.
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