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Optomechanical Dark Mode

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Science  21 Dec 2012:
Vol. 338, Issue 6114, pp. 1609-1613
DOI: 10.1126/science.1228370

Abstract

Thermal mechanical motion hinders the use of a mechanical system in applications such as quantum information processing. Whereas the thermal motion can be overcome by cooling a mechanical oscillator to its motional ground state, an alternative approach is to exploit the use of a mechanically dark mode that can protect the system from mechanical dissipation. We have realized such a dark mode by coupling two optical modes in a silica resonator to one of its mechanical breathing modes in the regime of weak optomechanical coupling. The dark mode, which is a superposition of the two optical modes and is decoupled from the mechanical oscillator, can still mediate an effective optomechanical coupling between the two optical modes. We show that the formation of the dark mode enables the transfer of optical fields between the two optical modes. Optomechanical dark mode opens the possibility of using mechanically mediated coupling in quantum applications without cooling the mechanical oscillator to its motional ground state.

In an optomechanical resonator, circulating optical fields can couple to the motion of a mechanical oscillator via radiation pressure (1, 2). Studies of optomechanical interactions have led to the experimental realization of a number of remarkable phenomena, including strong coupling between an optical and a mechanical mode (35), optomechanically induced transparency (OMIT) (57), and coherent interconversion between optical and mechanical excitations (4, 8). These advances have opened up avenues in coupling hybrid quantum systems, by taking advantage of unique properties of an optomechanical system, and especially in developing a new type of light-matter quantum interface (9, 10). A major obstacle, however, is the inherent thermal motion of a mechanical oscillator. A straightforward, but technically challenging, approach to overcome the thermal motion is to cool the mechanical oscillator to its motional ground state (4, 1113). An alternative approach, as proposed recently, is to exploit the use of a mechanically dark mode, which is decoupled from the mechanical oscillator and thus is robust against mechanical dissipation (14, 15). A similar process that exploits mechanical coupling, while circumventing effects of thermal mechanical motion, has also been explored for trapped ions (16, 17).

A dark mode is analogous to the well-known coherent-population-trapped state or the dark state in atomic physics (18, 19). Figure 1A illustrates a Λ-type three-level atomic system, in which the two lower states, |1> and |2>, couple to an upper state via two dipole optical transitions. The formation of a dark state, which is a special coherent superposition of |1> and |2>, prevents the optical excitation to the upper state through destructive interference, thus protecting the system from dissipation or decoherence associated with the upper state. In addition to atomic vapors, dark states have also been demonstrated in solid-state systems of single electron spins, such as quantum dots and diamond nitrogen vacancy (NV) centers (2022).

Fig. 1

Concept of the experiment. (A) A Λ-type three-level system that can lead to the formation of a dark state. (B) An optomechanical system in which two optical modes couple to a mechanical oscillator via radiation pressure, with respective optomechanical coupling rates G1 and G2. (C) Two optical fields, E1 and E2, at the red side band of the respective optical resonance drive the respective optomechanical coupling. (D) A simplified schematic of the experimental setup, with Ein exciting mode 1 in a silica microsphere.

Here, we report an experimental demonstration of a mechanically dark mode by coupling two optical whispering gallery modes (WGMs) to a mechanical breathing mode in a silica resonator in the regime of weak optomechanical coupling. This dark mode is a special coherent superposition of the two optical modes. The cancellation in the mechanical coupling induced by the superposition decouples the dark mode from the mechanical oscillator. The formation of the dark mode, however, also induces a conversion of optical fields from one optical mode to the other, effectively mediating an optomechanical coupling between the two optical modes. This type of mechanically mediated coupling can be immune to thermal mechanical motion, providing a promising mechanism for interfacing hybrid quantum systems (9, 14, 15).

To introduce the optomechanical dark mode, we consider an optomechanical system, in which two optical modes couple to a mechanical oscillator with optomechanical coupling rates G1 and G2, respectively (Fig. 1B). As illustrated in Fig. 1C, the optomechanical coupling is driven by two strong laser fields, E1 and E2, with frequencies, ωl1 and ωl2, that are each one mechanical frequency, ωm, below the respective cavity resonance, ω1 and ω2. In analogy to atomic dark and bright states, we define a mechanically dark mode, a^D=(G2a^1G1a^2)/G˜, and a mechanically bright mode, a^B=(G1a^1+G2a^2)/G˜, with G˜=G12+G22, where a^1 and a^2 are the annihilation operators for the signal field in the two optical modes in the respective rotating frame of the external driving field. The optomechanical interaction Hamiltonian in terms of these superposition optical modes is then given byH=ωm(b^+b^+a^B+a^B+a^D+a^D)+G˜(a^B+b+a^Bb+) (1)where b^ is the annihilation operator for the mechanical oscillator. As shown in Eq. 1, the dark mode is decoupled from the mechanical oscillator (14, 15). The dark mode is spectrally separated from the bright mode in the limit of ultrastrong optomechanical coupling, for which G1 and G2 far exceed κ1 and κ2, the decay rates of mode 1 and mode 2, as well as γm, the mechanical damping rate. In this limit, the coupling between the bright mode and the mechanical oscillator leads to the formation of two normal modes with frequencies given by ωm±G˜.

In the limit of weak optomechanical coupling, the dark mode can no longer be spectrally separated from the bright mode. The system, however, can still be driven optically into the dark mode via suppression of the bright-mode excitation. In contrast to the dark mode, an optical excitation of the bright mode induces a mechanical excitation. Anti-Stokes scattering of the strong driving field off this mechanical excitation in turn generates an optical field that interferes destructively with the optical excitation field in the bright mode. This OMIT process can effectively prevent the excitation of the bright mode. Specifically, when the optomechanical system shown in Fig. 1B is excited by a signal field resonant with mode 1, the OMIT suppresses the bright-mode amplitude by a factor of (1+C˜), where C˜=C1+C2, with Ci=4Gi2/γmκi (i = 1, 2) being the optomechanical cooperativity (23, 24). For simplicity, κ1=κ2 is also assumed. The ratio of dark- to bright-mode population in the steady state is then given by (G2/G1)2(1+C˜)2 (24). Hence, a large cooperativity is sufficient in preventing the excitation of the bright mode via OMIT, effectively driving the system into the dark mode. Similar results can also be obtained when κ1κ2, with the dark-to-bright-population ratio modified as (G2/G1)2[1+C2+C1(κ1/κ2)]2 (24). In a typical optomechanical system, the optical linewidth is orders of magnitude greater than the mechanical linewidth. It is thus more practical to realize large cooperativity than ultrastrong coupling.

The dark mode can be probed through the excitation of the two individual optical modes. In the above case, the intracavity field amplitudes of mode 1 and mode 2 are, respectively,α1=α0[C1/(1+C˜)+C2]/C˜ (2A)α2=α0C1C2[1/(1+C˜)1]/C˜ (2B)where α0 is the field amplitude in mode 1 in the absence of optomechanical coupling. In both equations, the first term in the bracket is due to the bright mode, and the second term is due to the dark mode (24). As expected from the suppression of the bright-mode amplitude by OMIT, the bright-mode term scales with 1/(1+C˜). Equation 2B also shows that the bright- and dark-mode contributions interfere destructively in mode 2. In this context, the excitation of mode 2 results directly from the suppression of the bright-mode amplitude.

We used silica microspheres with a diameter near 30 μm as a model optomechanical resonator (25). Two WGMs, with mode 1 near 637 nm and mode 2 near 800 nm, coupled to the (1, 0) mechanical breathing mode of a silica microsphere. Two samples were used, with (κ1,,κ2,,ωm,,γm)/2π19,16,150,0.055 MHz and (κ1,,κ2,,ωm,,γm)/2π15,15,154,0.06 MHz for sample A (used for Fig. 2) and B (used for Fig. 3), respectively. All experiments were carried out at room temperature.

Fig. 2

Excitation of the dark mode. (A and B) Optical emission from mode 1 as a function of detuning, Δ = ωin − ωl1, with C1 = 1.4 (P1 = 2.5 mW) and Pin = 10 μW. The emission power is normalized to that obtained at the cavity resonance with C1 = C2 = 0. (C) Optical emission from mode 2 as a function of Δ with C1 = 1.4 and Pin = 20 μW. Care was taken in normalizing the emission power to the input signal power (24). (D) Emission powers from mode 1 (squares) and mode 2 (circles) at Δ = ωm as a function of C2, derived from (B) and (C). Solid lines in (A) to (D) are the theoretical calculations discussed in the text. (E) Calculated dark-mode fraction. The diamonds correspond to the experimental results shown in (D). (Inset) The timing of the detection gate used for the experiment. Pin, P1, and P2 are incident optical powers for Ein, E1, and E2, respectively.

Fig. 3

Heterodyne-detected optical emission from mode 2 obtained with Pin = 0.1 mW, C1 = 0.25, and C2 = 0.4. A driving field at the red side band of mode 2 served as the local oscillator. The dashed line plots the calculated envelope for the heterodyne signal, with an adjustable offset. (Inset) The beat signal (squares) with an expanded time scale. Solid red line shows for reference a periodic oscillation with ωm/2π = 154 MHz.

For the demonstration of the dark mode, Ein with frequency ωin excited mode 1 resonantly or near-resonantly. Optical emissions from mode 1 and mode 2, which are directly proportional to the respective intracavity intensity, were measured as a function of detuning, Δ = ωinωl1, with a special heterodyne-detection technique (24). For simplicity, we refer to these spectra as emission spectra. To avoid heating induced by the strong driving fields and to enable measurements on the behavior of the mechanical mode, we used 8-μs-long optical pulses for E1, E2, and Ein, each with the same timing and with a duty cycle below 5%. Figure 1D shows a simplified schematic of the experimental setup. In order to probe the steady-state behavior, emission spectra were obtained with time-gated detection, with a 1-μs detection gate positioned between 6 and 7 μs of the incident optical pulses (Fig. 2E inset) (24). At relatively high optical powers, spectral shifts of WGM resonances resulting from Kerr effects become substantial. For experiments in Fig. 2, care was taken to keep the frequencies of the two driving fields at ωm below the respective WGM resonances.

Figure 2A shows emission spectra from mode 1, obtained with C1 = 1.4 and C2 = 0. In this case, the mechanical oscillator couples only to mode 1. The resulting OMIT process prevents the excitation of mode 1, inducing a sharp dip at the anti-Stokes resonance, Δ = ωm, with a width determined by γm(1+C1) (6, 7). For our studies, C1 was determined from theoretical fitting of OMIT dips obtained with C2 = 0, whereas C2 was similarly determined from theoretical fitting of OMIT dips obtained with C1 = 0 and with mode 2 excited resonantly by an input signal field.

By turning on both E1 and E2, we coupled both optical modes to the mechanical oscillator. With increasing C2, the excitation of the dark mode should lead to an increasing excitation of mode 1 and thus the vanishing of the OMIT dip for mode 1 (see also Eq. 2A). Figure 2B shows emission spectra from mode 1 obtained with C1 = 1.4 and with increasing C2. The depth of the dip at Δ = ωm decreases with increasing C2, accompanied by a spectral broadening of the dip. Figure 2B also shows a slight spectral shift of the emission dip at relatively high C2. The shift is due to the optical spring effect, for which radiation pressure induces a shift in ωm.

The dark-mode formation necessitates the conversion of optical fields from mode 1 to mode 2, because Ein couples directly only to mode 1. Figure 2C shows the emission spectra from mode 2 obtained under nearly the same condition as that for Fig. 2B. At Δ = ωm, the emission from mode 2 increases simultaneously with the emission from mode 1 with increasing, but still relatively small C2 (Fig. 2D), which is a signature that the system is driven toward a dark mode.

For energy conservation, the optical mode conversion should induce a dip in the emission spectrum of mode 1. A pronounced dip in the emission spectra of mode 1 persists even at the highest C2 used (Fig. 2B). Under these conditions the system is nearly completely in the dark mode. With increasing C2, the dip in the emission spectra of mode 1 evolves from an OMIT dip (at C2 = 0) into a dip that reflects the process of optical mode conversion.

For a quantitative analysis, we used the coupled oscillator model to describe the coupling between the mechanical oscillator and the two optical modes (24). The solid curves in Fig. 2, A to C, show the calculated emission spectra from mode 1 and mode 2, with all parameters determined directly (κ1,,κ2,,ωm,,γm) or indirectly (C1, C2, η1η2 = 0.16) from experiments, with η1 and η2 being the output coupling ratio for the two optical modes. Figure 2D plots the calculated emission power at Δ = ωm for the two optical modes. Additional theoretical calculations also confirm that the experimental results shown in Fig. 2 reflect the steady-state behavior of the optomechanical system (24).

The agreement between experiment and theory shown in Fig. 2, A to D, enables us to determine the dark-mode fraction (the ratio of the dark-mode population over the total bright- and dark-mode population) by using the coupled oscillator model. The steady-state dark-mode fraction corresponding to the experimental results in Fig. 2D is calculated and plotted (Fig. 2E). With C1 = 1.4 and C2 = 3.5, the dark-mode fraction reaches 99%.

The excitation of the dark mode not only leads to the simultaneous rise of optical emissions from mode 1 and mode 2 with increasing (but relatively small) C2, as discussed earlier, but also accounts for the saturation of the optical mode conversion observed at relatively large C2. As shown in Fig. 2D, after the system is driven into a predominantly dark mode, a further increase in C2 leads to a saturation and then decrease in the emission from mode 2, whereas the emission from mode 1 continues to rise.

Dark-mode formation can enable efficient transfer of optical fields between the two optical modes. The overall photon-conversion efficiency, defined as the ratio of the output-signal photon flux for mode 2 over the input-signal photon flux for mode 1, is given by χ=4η1η2C1C2/(1+C1+C2)2 (14, 15, 24). Near-unity photon conversion can thus be achieved in the limit that C1=C2>>1 and η1=η2=1. With C1=C2>>1, the dark mode features nearly equal photon populations in the two optical modes. Unity photon conversion can occur because a destructive interference prevents the escape of photons from mode 1 (14). The small output-coupling ratio (η1η2 = 0.16), along with the modest cooperativity used in our experiment, leads to the relatively small mode-conversion efficiency observed in Fig. 2.

The optical-mode conversion can also be described theoretically and completely with a scattering matrix approach and without resorting to the dark-mode concept (26, 27). In this approach, the condition of C1=C2>>1 can be understood simply in terms of impedance matching (26). By establishing a close connection between the weak and strong coupling regime, the dark-mode description provides important insights on why the mode-conversion process can be robust against thermal mechanical noise even in a weak coupling regime.

We further characterized the emission from mode 2 by measuring directly in the time domain the heterodyne signal that mixes the emission from mode 2 with a driving field E2. Figure 3 shows the transient heterodyne signal obtained with C1 = 0.25 and C2 = 0.4. The rise of the heterodyne signal with a rise time of order 1/[(1+C1+C2)γm] is in good agreement with the theoretical calculation based on the coupled oscillator model and on the use of the experimentally determined C1, C2, and γm. The heterodyne signal features a periodic oscillation with a frequency given by ωm (Fig. 3 inset), demonstrating the coherent nature of the optical mode conversion. Specifically, there is a well-defined relative phase between E2 and the converted optical field in mode 2.

We now turn to the behavior of the mechanical oscillator, which can serve as a probe for the OMIT process for the bright mode when the optical excitation is dominated by the dark mode. As discussed earlier, the OMIT arises from anti-Stokes scattering of the driving fields off the mechanical excitation induced by the bright mode excitation. To probe the mechanical excitation, we added a weak 3-μs probe pulse, which arrives 1 μs after E1 and is also at the same frequency as E1 (Fig. 4 inset). We used the probe pulse and time-gated detection, with the 1-μs gate positioned at the center of the probe pulse, to measure the displacement power density spectrum of the mechanical mode (24). The spectrally integrated area of the power density spectrum determines the average phonon number, <N>, of the mechanical mode. For the experiment, a relatively strong input signal was used such that <N0>, the average phonon number obtained with C2 = 0, is two orders of magnitude greater than the average thermal phonon number.

Fig. 4

Induced mechanical excitation underlying the OMIT for the bright mode, obtained as a function of C2 and with C1 = 0.7 (P1 = 1.25 mW) and Pin = 10 μW. At C2 << 1, (ω1 − ωl1)/2π and (ω2 − ωl2)/2π are estimated to be 150 and 145 MHz, respectively. The solid line shows the result of the theoretical calculation, as discussed in the text. (Inset) The pulse sequence used, with the shaded area indicating the timing of the detection gate.

<N>/<N0> obtained with C1 = 0.7 were plotted (Fig. 4) as a function of C2, for which sample A was used, and ωl1 and ωl2 were fixed and were near the respective red sideband. Other experimental conditions are the same as those for Fig. 2D. The experimental results are in good agreement with the theoretical calculation based on the coupled oscillator model. The calculation also includes corrections due to the Kerr effect with εi=ξiP2 (i = 1, 2), where ε1 and ε2 are the Kerr shift for mode 1 and mode 2 induced by E2, respectively, and (ξ1, ξ2) = (–0.1, –0.46) MHz/mW. The observation of the induced mechanical excitation when the system is predominantly in the dark mode confirms the underlying OMIT process for the bright mode. Figure 4 also shows that the anti-Stokes scattering of E2 damps the mechanical oscillation when the system is driven to the dark mode with increasing C2.

Although silica WGM resonators feature modest optomechanical cooperativity, much greater cooperativity (103 or greater) can be attained with membrane- or nanobeam-based optomechanical systems that feature ultrahigh mechanical Q factors (28, 29). With these systems, mechanically mediated processes, such as the optical mode conversion, can be pursued in a quantum regime at an elevated temperature. The concept of the dark mode can also be extended to other hybrid mechanical systems (30, 31), including the recently developed system of a mechanical resonator coupling to a single-electron spin in a diamond NV center (32).

Note added in proof: After the acceptance for publication of this work, optical mode conversion in an optomechanical crystal cavity was reported by Hill et al. (33).

Supplementary Materials

www.sciencemag.org/cgi/content/full/science.1228370/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S9

References

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. Acknowledgment: This work is supported by the Defense Advanced Research Projects Agency ORCHID (Optical Radiation Cooling and Heating in Integrated Devices) program through a grant from Air Force Office for Scientific Research and by NSF.

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