Abstract
To date, microscale and nanoscale optomechanical systems have enabled many proof-of-principle quantum operations through access to high-frequency (gigahertz) phonon modes that are readily cooled to their thermal ground state. However, minuscule amounts of absorbed light produce excessive heating that can jeopardize robust ground-state operation within these microstructures. In contrast, we demonstrate an alternative strategy for accessing high-frequency (13 GHz) phonons within macroscopic systems (centimeter scale) using phase-matched Brillouin interactions between two distinct optical cavity modes. Counterintuitively, we show that these macroscopic systems, with motional masses that are 1 million to 100 million times larger than those of microscale counterparts, offer a complementary path toward robust ground-state operation. We perform both optomechanically induced amplification/transparency measurements and demonstrate parametric instability of bulk phonon modes. This is an important step toward using these beam splitter and two-mode squeezing interactions within bulk acoustic systems for applications ranging from quantum memories and microwave-to-optical conversion to high-power laser oscillators.
INTRODUCTION
The coherent control of mechanical objects (1–4) can enable applications ranging from sensitive metrology (5) to quantum information processing (6, 7). An array of devices (8–14) ranging from nano-optomechanical crystals to suspended micromirrors have been used to manipulate mechanical degrees of freedom using light. Central to the field of cavity optomechanics—and more generally to quantum information science—is the ability to harness long-lived mechanical excitations at high frequencies (15). Long-lived high-frequency phonons can be initialized deep in their quantum ground states and can preserve coherent information for extended periods of time in the presence of various decoherence channels. Optomechanical systems using high-frequency (gigahertz) phonons have enabled ground-state cooling (16), quantum control at the single-phonon level (17, 18), and remote entanglement between mechanical resonators (19–21). However, it may be difficult to implement more sophisticated quantum protocols using these systems because of spurious forms of laser heating that threaten robust ground-state operation within microscale and nanoscale systems (16, 18, 22). New device strategies may be necessary to mitigate these decoherence mechanisms for robust quantum optical control of high-frequency phonons.
Efficient optomechanical coupling to high-frequency bulk acoustic waves within macroscopic systems offers an alternative path to robust quantum control of phonons. Optical access to these phonons within pristine crystalline solids yield greatly reduced surface interactions and favorable thermal characteristics to mitigate spurious laser heating (23, 24). These resonators also grant access to world-record f ⋅ Q products (23), a key figure of merit that characterizes decoupling of a resonator from its thermal environment (15). While these bulk acoustic phonon modes have been accessed through Brillouin-like coupling using a free-space laser beam (24, 25), new strategies are needed to translate this physics into cavity optomechanical systems as the basis for phonon counting (17, 26), generation of nonclassical mechanical states (18), and efficient transduction of information between optical and phononic domains (27).
Here, we realize a cavity-optomechanical system that harnesses high-frequency (13 GHz) bulk acoustic phonon modes within a macroscopic crystal, offering an array of properties that open the door to robust quantum manipulation of phonons. These high-frequency phonons, with large (20 μg) motional masses, are used to mediate resonant coupling between two distinct modes of an optical cavity through Brillouin interactions. To enable beam splitter and entanglement interactions as the basis for quantum optical control of phonons, we engineer the multimode spectrum of the optical cavity to break the symmetry between the Stokes and anti-Stokes processes while allowing resonant driving of a chosen optical mode. Resonant driving allows us to attain intracavity photon numbers that are more than six orders of magnitude larger than those of high-frequency microscale counterparts (16). These resonantly enhanced photon numbers permit large (megahertz) optomechanical coupling rates and greater than unity cooperativities, necessary for efficient control of phonons using light. In addition, we demonstrate that phase-matched Brillouin interactions produce controllable coupling to one or more phonon modes, opening the door to new forms of multimode entanglement (28). Looking beyond the field of cavity quantum optomechanics, this device strategy presents new opportunities for sensitive materials spectroscopy and oscillator technologies.
Bulk crystalline optomechanical system
In what follows, we explore optomechanical interactions mediated by macroscopic phonon modes within the bulk crystalline cavity optomechanical system of Fig. 1A. A 5.2-mm-thick flat-flat quartz crystal is placed within a nearly hemispherical Fabry-Pérot cavity having high-reflectivity (98%) mirrors; this optomechanical assembly is cooled to ~8 K temperatures to greatly extend the phonon lifetimes within crystalline quartz. At room temperature, high-frequency phonons (>10 GHz) have a mean free path (~100 μm) that is much smaller than the crystal dimension. Just as the optical Fabry-Pérot resonator supports a series of standing electromagnetic waves (red and blue) seen in Fig. 1A, the planar surfaces of the quartz crystal produce acoustic reflections to form an acoustic Fabry-Pérot resonator that supports a series of standing-wave elastic modes (green) at these low temperatures. To avoid anchoring losses, an annular spacer prevents the crystal surface from contacting the mirror; this same spacer sets the position of the crystal (d) within the cavity.
(A) Schematic of an optomechanical system that consists of a bulk acoustic wave resonator that is placed inside an optical cavity. Two distinct standing-wave longitudinal optical modes interact through electrostrictive coupling with a standing-wave longitudinal phonon mode formed within the planar crystalline quartz crystal at cryogenic temperatures (~8 K). Large mode volumes for both light and sound result from macroscopic geometrical parameters: Lac ≃ 5.2 mm, Lopt ≃ 9.1 mm, and d ≃ 0.15 mm. (B) Assuming no optical reflections in the quartz-vacuum interface, one obtains equally spaced longitudinal optical modes. Therefore, a phonon mode mediating intermodal optomechanical interactions can scatter incident laser light at ωj to adjacent optical modes ωj+1 and ωj−1 with nearly equal scattering rates. (C) However, modest optical reflections (~4%) in the quartz-vacuum interface, which lead to notable nonuniformity in optical mode spacing, can be exploited to bias our system to strongly favor one scattering process (ωj→ωj−1) over the other (ωj→ωj+1). The variation in cavity resonances arises because of dispersive shifts of cavity resonances arising from multipath interference. (D) The reflection spectrum of the optical cavity obtained by frequency sweeping an incident laser light reveals this variation in optical mode spacing (or FSR) between adjacent fundamental Gaussian modes. a.u., arbitrary units. (E) Measurement of the FSR over a wider frequency range by sweeping the laser from 1548- to 1552-nm wavelength shows large undulation (2.6 GHz) in the FSR as a function of the optical cavity frequency. Consequently, we find multiple pairs of resonances with frequency differences that are equal to the Brillouin frequency, which is a necessary requirement for intermodal optomechanical coupling in our system.
These high-frequency bulk acoustic phonon modes are used to mediate efficient coupling between distinct longitudinal optical modes of the Fabry-Pérot cavity through a multiresonant (or multimode) optomechanical process. Coupling occurs through a Brillouin-like optomechanical process when the time-modulated electrostrictive optical force distribution, produced by the interference between distinct modes of the Fabry-Pérot cavity, matches the elastic profile (and frequency) of a bulk acoustic phonon mode. The same intrinsic photoelastic response that generates the optical forces within the crystal also modulates the refractive index of the crystal via elastic-wave motion. Through the formation of a time-modulated photoelastic grating, these “Brillouin-active” elastic waves mediate dynamical Bragg scattering (or energy transfer) between distinct longitudinal modes of the optical Fabry-Pérot cavity.
Because of the extended nature of the optomechanical interaction within the crystal, phase matching and energy conservation determine the set of phonon modes (Ωm, qm) that can mediate resonant coupling between adjacent optical modes (ωj, kj) and (ωj−1, kj−1) of the Fabry-Pérot cavity (see Fig. 1A). One finds that the Brillouin-active phonons must satisfy the conditions qm = kj−1 + kj and Ωm = ωj − ωj−1 to mediate dynamical Bragg scattering within the crystal. Neglecting some details pertaining to modal overlaps, one expects that phonons in a narrow band of frequencies near the Brillouin frequency (ΩB ≃ 2ωjυa/υo) can meet this condition; here, υa(υo) is the speed of sound (light) in the quartz crystal. Within the z-cut crystalline quartz substrate, this expression leads us to expect intermodal coupling at ΩB ≃ 2π × 12.7 GHz when driving the optical cavity with 1.55-μm wavelength light. On the basis of the resonance condition, we seek a Fabry-Pérot design whose free spectral range (FSR), defined as
Most applications of optomechanical interactions require a means of selecting between the Stokes and anti-Stokes interactions. Conventional single-mode cavity optomechanical systems use the detuning of an external drive field from a single-cavity resonance to produce this asymmetry. By comparison, this multimode system offers the possibility for resonant pumping, which carries many advantages (see Discussion). However, in the case when the optical Fabry-Pérot resonator has regular mode spacing (i.e.,
The introduction of the quartz crystal into the optical Fabry-Pérot resonator provides an elegant means of solving this problem; modest optical reflections (~4%) produced by the surfaces of the crystal shift the modes of the Fabry-Pérot cavity. As a result, the typically uniform density of modes (Fig. 1B) is transformed into a highly nonuniform density of modes (Fig. 1C) such that
This modification to the density of modes can be seen from reflection measurements of this bulk crystalline optomechanical system (Fig. 1D), which reveal a large (~21%) undulation in the FSR of the optical Fabry-Pérot resonator; Fig. 1E shows the measured frequency spacing between adjacent optical resonances (
Even with resonant driving of the optical mode, this form of symmetry breaking results in a large (>1000-fold) difference between the Stokes and anti-Stokes scattering rates. In conventional single-mode cavity optomechanical systems, this large Stokes/anti-Stokes asymmetry is produced by far-detuning the drive from an optical cavity at the expense of the intracavity photon number. In contrast, this multimode optomechanical system permits greatly enhanced intracavity photon numbers for the same input power through resonant driving (29). The relative strength of the Stokes/anti-Stokes scattering rates in the case of resonant driving is determined by the ratio (2Δω/κ)2, where
The multimode coupling described above can be represented by the interaction Hamiltonian
To calculate the single-photon coupling rate,
Using the parameters of our system, Eq. 2 predicts a maximum single-photon coupling rate of
The wave vector–selective nature of this coupling enables new approaches to precisely tailor interaction with one or more phonon modes. This system differs from conventional Brillouin scattering because even phonon modes that satisfy both energy conservation and phase-matching requirements can have vanishing optomechanical coupling rates. This intriguing new feature arises because the coupling to a particular phonon mode also depends on the location of the crystal (d) inside the optical cavity. For instance, the overlap integral in Eq. 2 yields a coupling rate of zero when the position of the crystal is such that nodes of a phonon mode coincide with the anti-nodes of the optical forcing function.
Despite the fact that this current apparatus does not permit us to change the crystal position, we can still control the number of phonon modes that participate in the optomechanical interaction. Independent of crystal location, a pair of adjacent optical modes is guaranteed to couple to at least one phonon mode near the Brillouin frequency because the phase-matching condition is relaxed by the crystal’s finite length. However, by using optical resonances at different wavelengths, one can change the position of the nodes inside the crystal, thereby selecting a different group of phonons to mediate optomechanical coupling (see section S2D).
RESULTS
Probing coherent optomechanical response
We explore the optomechanical coupling in our system by probing its coherent response to an optical drive. Specifically, we use the well-known techniques called optomechanically induced amplification (OMIA) and optomechanically induced transparency (OMIT) (32).
These measurements are performed using a tunable laser at frequency ωl, whose output is split into two arms (see Fig. 2A). Laser light in one arm is intensity-modulated at a variable frequency (Ω) using a microwave generator. Light in the other arm is frequency-shifted to ωl + 2π × 44.0 MHz using an acousto-optic modulator (AOM). This AOM-shifted light acts as a local oscillator (LO) such that the Stokes and anti-Stokes signals appear as distinct tones in the radio-frequency (rf) spectrum analyzer during heterodyne detection. The tones at frequencies ωl and ωl − Ω serve as control and probe lasers, respectively. The strong control laser (ωl) drives the higher-frequency optical mode at ωj+1, whereas the weak probe laser (ωp = ωl − Ω) is swept near the lower-frequency optical mode at ωj (see inset i of Fig. 2A). The third tone at ωl + Ω is irrelevant, as it is not resonant with the optical cavity modes. Light is delivered to and collected from the optical cavity through a combination of fiber collimators and free-space optics. Light transmitted through the optical cavity is combined with the LO and detected using a photoreceiver, which is connected to an rf spectrum analyzer. This spectrum analyzer monitors the beat note between the transmitted probe laser and the LO by tracking the frequency (Ω) of the microwave generator. Heterodyne detection of the probe light transmitted through the optical cavity provides a direct measurement of the intracavity probe power.
(A) Schematic of the measurement setup used to perform OMIA. A strong control and a weak probe laser are synthesized from the same tunable laser source using an intensity modulator (IM). This light is coupled into and out of the optical cavity using fiber-optic collimators and free-space lenses. The frequency of the tunable laser is set so that the control laser is directly on resonance with the high-frequency optical mode at ωj+1 (see inset i). The probe laser is swept near the lower cavity mode at frequency ωj by sweeping the rf drive frequency (Ω) of the intensity modulator. To examine the coherent response of the intracavity probe field due to optomechanical coupling, a heterodyne measurement is performed between the transmitted probe light and the AOM-shifted laser light. (B) We observe coherent buildup of the intracavity photon number as the probe laser is scanned near the optical mode ωj. The optical cavity decay rate of κj ≃ 2π × 73 MHz obtained for this mode is consistent with the losses at the two mirrors. In addition, we see sharp resonances on the optical mode spectrum corresponding to the phonon-mediated transfer of energy from the high-frequency optical mode at frequency ωj+1 to the lower-frequency optical mode at ωj. A close zoom in of the optical mode spectrum near the center of the optical resonance reveals three high-frequency acoustic modes around 12.661 GHz with a frequency spacing of ~612 kHz (see inset ii). This frequency spacing is consistent with the acoustic FSR of υa/2Lac for the standing-wave longitudinal modes formed in crystalline quartz along the z axis. These acoustic modes have very high longitudinal mode numbers (or overtone number) of m ≈ 2.08 × 104.
This OMIA measurement (Fig. 2B) reveals a broad optical cavity resonance of linewidth κj ≃ 2π × 73 MHz, consistent with the mirrors’ reflectivities. Near the center of the optical resonance, we find three narrow resonances (see inset ii of Fig. 2B) corresponding to phonon modes near the Brillouin frequency of 12.661 GHz. These resonances are equally spaced by ~612 kHz, the expected acoustic FSR υa/2Lac.This result demonstrates optomechanical coupling to multiple high-frequency longitudinal phonon modes, in a manner consistent with the phase matching described by Eq. 2.
In what follows, we tune the optical wavelength to excite a different pair of optical resonances, ensuring that only a single-phonon mode mediates the optomechanical interaction. A typical OMIA spectrum in which the optomechanical coupling is mediated by a single-phonon mode is shown in Fig. 3A. We perform both OMIA and OMIT measurements to determine the associated single-photon coupling rate and the intrinsic mechanical damping rate. When the system is driven by a strong control laser and a weak probe, we can describe the OMIA (OMIT) phenomena with a linearized interaction Hamiltonian
(A) We perform OMIA measurements on a single-phonon mode using a blue-detuned control laser that is resonant with the high-frequency optical mode (see inset). (B) As we increase the power in the control laser, the height of the peak corresponding to the OMIA signal increases nonlinearly with the control laser power, as displayed in (C). The linewidth of the OMIA peak, however, decreases with the control power, as seen in (G). (D) OMIT measurements are performed with the same phonon mode by simply red-detuning the control laser. Note that, because the control laser has a slight offset relative to the low-frequency optical mode at ωj, the center of the cavity resonance shifts, but the OMIT dip is observed exactly at a phonon frequency of 12.645 GHz. (E) The dip of the OMIT signal decreases nonlinearly as a function of the control power, as displayed in (F). The linewidth of the OMIT signal, however, increases linearly with the control power, as seen in (G). From the dependence of linewidth as a function of the control laser power, we obtain a cold cavity linewidth Γm/2π ≃ 86 kHz for the phonon mode at 12.645 GHz.
To measure Γm and
The cooperativities (C) and intrinsic mechanical damping rate (Γm) obtained from experiments along with nc calculated from measuring input control laser power Pin give
Thermal fluctuations and phonon self-oscillation
We measure thermal fluctuations of the mechanical mode through spontaneous light-scattering measurements. We use a control laser that is resonant with the higher-frequency optical mode ωj+1 (see Fig. 4A); no probe field is supplied for these measurements. The thermally populated phonon mode mediates scattering of incident control photons from frequency ωj+1 to ωj through the Stokes process. Heterodyne detection is used to monitor the power spectrum of this spontaneously scattered Stokes light, as shown in the inset i of Fig. 4B. As the control laser power is increased, we observe a sharp increase in the magnitude of the scattered Stokes light accompanied by spectral narrowing of the heterodyne beat tone (inset ii of Fig. 4B). Optomechanical systems in the blue-detuned regime display these lasing transitions at C = 1, when Γeff = Γm(1 − C) = 0. When Γeff < 0 (i.e., C > 1), parametric instability initially causes thermal fluctuations to grow exponentially in time and eventually saturate to reach coherent self-sustained oscillations. This phenomenon is well studied in many optomechanical systems (29, 33) and is also commonly known as “phonon lasing” (15). The measurement of total backscattered light from the cavity using a power meter as a function of the input control laser power (Fig. 4B) reveals a self-oscillation threshold of 137 mW, consistent with the threshold (~140 mW) predicted from the measured values of nc,
(A) To observe thermal fluctuations of the phonon mode, we turn off the probe laser and tune a strong control laser directly on resonance with the high-frequency optical mode. The thermally populated phonon mode spontaneously scatters light from the higher-energy optical mode to the lower-energy optical mode (Stokes field); this light exits our optical system through both mirrors. (B) From the measurement of back-reflected optical power as a function of the input control laser power, we observe a clear threshold behavior at 137 mW corresponding to regenerative self-oscillation of the phonon mode (i.e., C = 1). Before threshold, a small (~2.6%) back-reflection results from imperfect coupling of the control laser to the optical mode at ω2. However, once self-oscillating, phonons scatter a large fraction of the input control laser into the Stokes light, which exits the optical cavity through both forward and backward directions, a substantial increase in the back-reflected optical power occurs. Furthermore, from the heterodyne beat tone of the scattered Stokes light with the frequency-shifted version of the input control laser, we observe a dramatic line narrowing of phonons as we cross the lasing threshold, as displayed in insets i and ii.
DISCUSSION
These results lay the foundation for a promising new class of macroscopic cavity optomechanical systems that rely on bulk acoustic modes of a crystalline solid—rather than subwavelength structural control—to achieve high-frequency multimode interactions. Because phase matching determines the phonon frequency, this approach permits coupling to massive (20 μg), high-frequency phonon modes without size reduction. Resonant optical driving of this multimode system produces appreciable coupling rates (1.5 MHz) to high-frequency (13 GHz) phonons and C > 1. These results are obtained using the simplest of flat-flat crystal geometries, meaning that these same principles can be readily used to transform practically any transparent crystal into a high-frequency cavity optomechanical system. Because the Brillouin frequency depends on the optical wavelength and material parameters, these same strategies can be used to harness phonons over a tremendous range of frequencies (e.g., 5 to 100 GHz) by designing the system around different wavelengths and materials. The versatility and robustness of this strategy should lend itself to new types of hybrid quantum systems, new forms of materials spectroscopy, and studies of laser-oscillator physics.
Our quartz-based optomechanical system has many promising features in the context of cavity quantum optomechanics. Large optomechanical coupling rates (>60 MHz) to these high-frequency phonons could be achieved even without miniaturization because this macroscopic system can store a large number of intracavity photons (>1013) through resonant driving (35). The demonstrated coupling rate of 1.5 MHz is already large enough to enter the strong-coupling regime (gm > κ/2, Γm/2) if the optical finesse is boosted from its current value (
Hybrid quantum systems using long-lived phonons within bulk acoustic resonators could be a valuable resource for quantum information processing. It is possible to increase the Q factor of these high-frequency phonons within quartz from their present values (1.5 × 105) to 4 × 107 by shaping the acoustic resonator into a plano-convex geometry (24). These highly coherent bulk acoustic modes could then be useful as quantum memories. Moreover, these high-Q phonon modes could readily permit strong coupling to individual defect centers for quantum information processing (see section S5), a result that is perhaps surprising given the large acoustic mode volumes relative to previous proposals (37, 38). Simultaneous optomechanical and electromechanical control of bulk acoustic wave phonons within piezoelectric crystals also offers a path toward high-fidelity microwave-to-optical conversion (39, 40).
Beyond the conventional goals of quantum optomechanics, this optomechanical system presents new opportunities for materials spectroscopy and precision measurement. Sensitive metrology of cryogenic phonon physics and defect centers can be performed in a wide array of materials to understand various decoherence channels for phonons. Moreover, bulk acoustic resonators show great potential for quantum noise–limited optomechanical oscillators with ultranarrow fundamental linewidth (<1 nHz) as they can support large coherent phonon populations (>1012) (see section S6). These highly coherent oscillators could be used for precision sensing (41), time keeping (42), and the exploration of new physics (43).
MATERIALS AND METHODS
The planar crystalline z-cut α-quartz crystal (Aa grade) used for this experiment was obtained commercially from Rocky Mountain Instrument Co. For the theoretical estimate of the coupling rate, we used the following parameters: n = 1.55, p13 = 0.27, ρ = 2648 kg/m3, Lopt = 9.13 mm, Lac = 5.19 mm, A ≃ π × (61 μm)2, ωj = 2π × 193.4 THz, and Ωm = 2π × 12.65 GHz (see section S2D).
The Gaussian optical modes, with beam waist wopt ≃ 61 μm, drive a longitudinal standing-wave phonon mode with Gaussian transverse profile with beam waist wac ≃ 43 μm. Therefore, the effective motional mass of the phonon mode in our system
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/4/eaav0582/DC1
Section S1. Asymmetric cavity mode spacing
Section S2. Hamiltonian treatment
Section S3. Optomechanically induced amplification
Section S4. Optomechanically induced transparency
Section S5. Acoustic diffraction loss
Section S6. Thermal fluctuations and phonon lasing
Section S7. Relative scattering rate
Section S8. Phonon counting sensitivity
Fig. S1. Scattering matrix treatment to determine optical mode spectrum.
Fig. S2. Variation of optical FSR with crystal displacement.
Fig. S3. Optical cavity reflection spectrum at cryogenic temperature.
Fig. S4. Normalized coupling rate.
Fig. S5. Cartoons depicting OMIA and OMIT measurements.
Fig. S6. OMIA measurements.
Fig. S7. Acoustic diffraction loss.
Fig. S8. Slope efficiency of the phonon laser.
Fig. S9. Phonon counting in a multiresonant optomechanical system.
Table S1. Optical and acoustic parameters.
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REFERENCES AND NOTES
- Copyright © 2019 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).
