Bulk crystalline optomechanics

TL;DR

This paper demonstrates that low-temperature bulk crystalline solids can host macroscopic phonon modes with high Q-factors and strong optomechanical coupling, enabling cavity-less phonon control and new applications in quantum and classical technologies.

Contribution

It introduces a method to create macroscopic, high-Q phonon modes in bulk crystals at low temperatures, significantly enhancing optomechanical interactions without nanostructuring.

Findings

Achieved large motional masses (~60 μg) for phonon modes.

Enhanced optical nonlinearities by over four orders of magnitude at low temperatures.

Demonstrated cavity-less phonon cooling and heating through dispersive symmetry breaking.

Abstract

Brillouin processes couple light and sound through optomechanical three-wave interactions. Within bulk solids, this coupling is mediated by the intrinsic photo-elastic material response yielding coherent emission of high frequency (GHz) acoustic phonons. This same interaction produces strong optical nonlinearities that overtake both Raman or Kerr nonlinearities in practically all solids. In this paper, we show that the strength and character of Brillouin interactions are radically altered at low temperatures when the phonon coherence length surpasses the system size. In this limit, the solid becomes a coherent optomechanical system with macroscopic (cm-scale) phonon modes possessing large ($60\ \mu \rm{g}$) motional masses. These phonon modes, which are formed by shaping the surfaces of the crystal into a confocal phononic resonator, yield appreciable optomechanical coupling rates (${\sim}100$ Hz), providing access to ultra-high $Q$-factor ($4.2{\times}10^7$) phonon modes at high ($12$ GHz) carrier frequencies. The single-pass nonlinear optical susceptibility is enhanced from its room temperature value by more than four orders of magnitude. Through use of bulk properties, rather than nano-structural control, this comparatively simple approach is enticing for the ability to engineer optomechanical coupling at high frequencies and with high power handling. In contrast to cavity optomechanics, we show that this system yields a unique form of dispersive symmetry breaking that enables selective phonon heating or cooling without an optical cavity (i.e., cavity-less optomechanics). Extending these results, practically any transparent crystalline material can be shaped into an optomechanical system as the basis for materials spectroscopy, new regimes of laser physics, precision metrology, quantum information processing, and for studies of macroscopic quantum coherence.

Equations

$$H_m^{\text{int}}=\frac{\hslash }{2\pi }{\int }_k{\int }_{k^{\prime }}\left[g_m(k,k^{\prime })a_{p,k}^{†}{a}_{s,k^{\prime }}{b}_m+\text{H.c.}\right]dkd{k}^{\prime },$$

Full text

I Introduction

In recent years, high-coherence phonons have become an attractive resource for applications like precision measurements and high-fidelity information processing. Such applications, in conjunction with new strategies for the control of phonons using hybrid opto-mechanical Aspelmeyer2014 ; Meystre2013 ; Marquardt2009 ; Favero2009a ; DeLorenzo2014 ; Lee2012a ; Safavi-Naeini2012 ; Safavi-Naeini2013 ; Andrews2014 ; Wilson2014 ; Chan2011 ; Poggio2007 ; Jayich2008a ; Bahl2011 ; Cohen2014 , electro-mechanical Ekinci2005 ; Regal2008 ; Bochmann2013 ; Mahboob2013 ; Macquarrie2013 ; MacQuarrie2014 ; Weinstein2014 ; Woolley2016 ; Bagci2014 , and superconducting circuit QED systems Chu2017a ; Schuetz2015 ; Cleland2004 ; Teufel2011a ; Fink2015 have sparked a surge of interest in phononic device physics and technologies. In the pursuit of new mesoscopic quantum phenomena, it has become commonplace to operate these systems at cryogenic temperatures to avoid thermal noise. Low temperatures have another advantage; internal sources of phonon dissipation plummet within crystalline solids permitting acoustic phonons to live for an astounding number ($10^{7}{-}10^{10}$) of cycles Thaxter1966 ; Blair1966 . When harnessed to create high quality-factor ($Q$) modes, such long-lived phonons become a tremendous resource for new hybrid technologies Chu2017a ; Aspelmeyer2014 ; Bochmann2013 . Within a variety of micro- and nanoscale systems, high $Q$-factor phonon modes have enabled new platforms for technologies like squeezing Safavi-Naeini2013 , single phonon detection Cohen2014 , and microwave to optical conversion Bochmann2013 ; Pitanti2015 ; Balram2015 .

Recently, the highest $Q$-factor (${\sim}10^{9}$) microwave frequency (${\sim}200\,$MHz) phonon modes have been created using elegant high quality bulk-crystalline resonator structures with minimal material interface loss Goryachev2013b ; Goryachev2012 ; Galliou2013 . These ultrahigh $Q$-factor bulk-acoustic modes have enabled new precision metrology studies that examine anomalously strong nonlinearities Goryachev2014b , gravitational wave detection Goryachev2014 , and Lorentz symmetry Lo2016 . To date, these high-$Q$ bulk acoustic modes have only been accessible through electro-mechanical (piezoelectric) interactions within crystalline quartz structures. Alternatively, a range of new optomechanical interactions may be possible in an expanded array of materials if these phonon modes were accessible with light.

In this paper, we demonstrate a new form of optomechanical coupling that grants access to ultra long-lived bulk-acoustic modes at cryogenic temperatures. Our system consists of a pristine crystalline solid whose surfaces are shaped to create a stable phonon resonator geometry that traps ultra-high $Q$-factor ($4.2{\times}10^{7}$) phonon modes. By engineering the optomechanical coupling between these high $Q$-factor phonon modes and an incident laser-field, we create Brillouin-like optomechanical interactions which are greatly enhanced compared to traditional Brillouin interactions. Using only the intrinsic photoelastic material response, we engineer efficient optomechanical coupling to massive ($60\ \upmu$g) high frequency (${>}12$ GHz) bulk acoustic phonon modes. This bulk crystalline system supports high optical powers (${>}$kW) and can be designed with practically any transparent crystalline medium. These unique properties open the door to new forms of cryogenic phonon spectroscopy, optomechanical parametric oscillators, and precision metrology techniques.

In prior work, Ohno et al. demonstrated that extended phonon coherence alters the character of the Brillouin interactions within crystalline substrates at cryogenic temperatures Ohno2006 ; Ohno2015 . In this paper, we bridge the gap between cryogenic Brillouin physics and mesoscopic optomechanical interactions by engineering bulk phonons and their interaction with light. Using new optomechanical design principles, simulation techniques, and precision spectroscopy methods, we create ultra high $Q$-factor macroscopic (cm-scale) phonon modes that can be addressed individually using a laser field, expanding the range of possible optomechanical interactions in bulk crystals. We also show that, in contrast to cavity optomechanics, this system exhibits a new form of symmetry breaking that enables phonon heating or cooling without an optical cavity (or cavity-less optomechanics).

II Optomechanical coupling in crystals

II.1 Physical System and Approach

In what follows, we examine the physics of light-sound coupling in a bulk crystalline system (Fig. 1a). The surfaces of this crystalline solid are fashioned into a plano-convex geometry that produces stable Hermite-Gaussian-like acoustic phonon modes (Fig. 1b-c) with ultra-high $Q$-factors at cryogenic temperatures. These phononic cavity-modes are designed to permit strong optomechanical coupling with an incident laser beam.

At room temperatures, intrinsic forms of phonon dissipation limit the phonon mean-free path $(l_{ph})$ to ${\sim}100$ microns Braginsky1985 ; Cleland2003 . Since this mean free path ($l_{ph}$) is much smaller than the crystal length ($L=5\,$mm), the phonons decay before a round trip oscillation can be completed. However, at cryogenic temperatures, intrinsic phonon dissipation is dramatically reduced Braginsky1985 ; Cleland2003 ; Thaxter1966 ; Blair1966 , permitting the phonons to traverse multiple cavity round-trips, forming ultra-high $Q$-factor phonon modes Goryachev2012 ; Galliou2013 . In the limit as the phonon coherence is much greater than the system length ($l_{ph}\gg L$), the dynamics of photon-phonon coupling becomes quite different from what has historically been termed Brillouin interactions. To differentiate these two regimes of interaction we term them the Brillouin-limit ($l_{ph}\ll L$) and the coherent-phonon limit ($l_{ph}\gg L$) respectively. In the coherent-phonon limit we will show that the nature of this coupling is altered, giving rise to greatly enhanced nonlinear susceptibilities as well as a form of dispersive symmetry breaking that enable new approaches for phonon self-oscillation or phonon cooling within this bulk crystalline system.

We experimentally explore the strength and character of the optomechanical coupling using continuous-wave pump-probe spectroscopy. We quantify the phonon-mediated energy transfer between counter-propagating pump-wave, with frequency $\omega_{p}$, and probe-wave, with frequency $\omega_{s}$, using the basic experimental configuration illustrated in Fig 2a. Both the pump- and probe-beams pass once through the phononic cavity; the light-field is not resonantly enhanced within the crystal. In this experiment, stimulated emission of a phonon is observed as energy transfer between the two incident optical beams via the three-wave interaction of Fig. 2b-c. Optomechanical coupling is mediated by photoelastic interactions within the bulk of the crystal; however, energy transfer only occurs when the phase matching and energy conservation conditions are met, as described in the following section.

II.2 Optomechanical Coupling

In general, acoustic phonons can scatter an incident pump photon ($\omega_{p}$) through either a Stokes process (creating a phonon) or an anti-Stokes process (annihilating a phonon). Of these interactions, only the Stokes process produces stimulated energy transfer between counter-propagating waves as the basis for the driven pump-probe studies employed here. Through a Stokes process, a right-moving pump photon, with frequency and wavevector ($\omega_{p}$, $k_{p}$), scatters to a red-shifted left-moving Stokes photon ($\omega_{s}$, $-k_{s}$) and a right-moving Stokes phonon ($\Omega_{s}$, $q_{s}$). Here, the optical wavevector ($k_{j}$) and frequency ($\omega_{j}$) are related by the optical dispersion relation of the solid, $k(\omega)$, as $k_{j}=k(\omega_{j})$; the phonon wavevector ($q_{s}$) and frequency ($\Omega_{s}$) are related by the acoustic dispersion relation, $q(\Omega_{s})$, as $q_{s}=q(\Omega_{s})$. For this scattering process to occur, both energy conservation ($\Omega_{s}=\omega_{p}-\omega_{s}$) and phase-matching ($q_{s}=k_{p}+k_{s}$) must be satisfied (Fig. 2b-c). Combined, these conditions require that the Stokes phonon have a frequency and wavevector component ($\Omega_{s}$, $q_{s}$) that satisfy the equation, $q(\Omega_{s})=\Delta k(\Omega_{s})$, where $\Delta k(\Omega)=k(\omega_{p})+k(\omega_{p}-\Omega)$ is the optical wavevector mismatch. We use this requirement as the basis for a succinct diagrammatic representation that includes both phase-matching and energy conservation.

For example, within an infinite (lossless) system, the optical waves and phonon-modes can be viewed as plane-waves with perfectly well-defined frequencies and wavevectors; Fig. 2d shows the phonon dispersion relation, $q(\Omega)$ (blue), plotted atop the optical wavevector mismatch, $\Delta k(\Omega)$ (orange). Energy conservation and phase-matching are satisfied by the set of states where these curves intersect. In this idealized case, $q(\Omega_{s})=\Delta k(\Omega_{s})$ is satisfied at a single point, $(\Omega_{s},q_{s})$, meaning that Brillouin coupling occurs at only a single frequency. However, in practice, we will see that the system geometry and phonon dissipation shape the coupling bandwidth.

In bulk media, the Brillouin frequency and coupling strength are determined by the material properties. Making the approximation $\Delta k(\Omega_{s})\cong 2k(\omega_{p})$, the Brillouin frequency is estimated from our phase-matching conditions as $\Omega_{s}\cong 2\,\omega_{p}(v_{a}/v_{o})$, where $v_{o}$ ($v_{a}$) is the phase velocity of light (sound) within the crystal. In typical materials (and with near-infrared light), the Brillouin frequency ($\Omega_{s}/2\pi$) is between 10 and 50 GHz, and the photoelastic tensor predominantly couples to longitudinal acoustic phonons. Through stimulated Stokes scattering, phonon emission can be viewed as the result of time modulated optical forces that are produced by electrostriction (or photoelastic response) within the crystal. Through phase-matched coupling, pump- and probe optical fields interfere to a produce a time-modulated force distribution (of beat-frequency $\Omega$) with a wavevector $\Delta k(\Omega)$. When phase-matching is satisfied, these forces resonantly drive phonons with wavevector $q(\Omega_{s})$.

Next, we consider optomechanical coupling within the finite crystalline system of Fig. 2e. At room temperatures high-frequency ($10$ GHz) Brillouin-active elastic waves decay rapidly (${\sim}1000$ cycles), meaning that $l_{ph}\ll L$ and the system is in the Brillouin regime of interaction (Fig. 2e). Rapid phonon decay ($l_{ph}\sim 100$ microns) permits us to neglect propagation of the envelope of the phonon field in the theoretical description of the gain dynamics; the dynamics of photon-phonon coupling is faithfully represented using the local nonlinear optical susceptibility following conventional Brillouin treatments boyd . In other words, each volume element within the crystal can be viewed as an independent system, which supplies distributed Brillouin gain as the pump and probe waves interact within the crystal.

In the Brillouin-limit, because elastic waves are heavily damped in comparison to the system size ($l_{ph}\ll L$), the density of phonon modes (or density of states) can be viewed as continuous. Damping also effectively broadens the phonon dispersion curve. This is because the damped system has complex eigen-values that correspond to exponential temporal (spatial) decay. A Fourier transform reveals that these exponentially damped elastic-wave solutions take on a Lorentzian frequency bandwidth given by the phonon decay rate, $\Gamma$. In other words, the linear phononic dispersion curve of Fig. 2d effectively broadens as seen in Fig. 2f. This 2D intensity map now represents the magnitude of the phonon response as a function of the wavevector and frequency of an applied forcing function (i.e. dissipation permits coupling to phonons with a finite phase-mismatch). In this picture, a band of frequencies over which coupling occurs is identified by the intersection of the loss-broadened dispersion curve (blue intensity map) with the optical wavevector mismatch (orange), as seen in Fig. 2f. Projecting these points of intersection to frequency axis of Fig. 2f, we find a Lorentzian line-shape $\propto[(\Omega-\Omega_{s})^{2}+(\Gamma/2)^{2}]^{-1}$; since phonons and Stokes photons are generated synchronously, the phase-matched optical response (Fig. 2f-g) is identical in form. This well-known Lorentzian response is characteristic of Brillouin interactions boyd . See Supplement Section 1.1 for more information.

At cryogenic temperatures, new behaviors emerge as we enter the coherent-phonon limit ($l_{ph}\gg L$). In the absence of phonon dissipation, standing-wave phonon modes are formed between the front and back faces of the crystal (Fig. 2h) creating a discrete set of cavity modes $\{\Omega_{m}\}$ with characteristic wavevectors $\{q_{m}\}$. In this case, the continuous dispersion curve (Fig. 2f) becomes discretized by the formation of cavity modes, with mode spacing $\Delta\Omega/2\pi\cong v_{a}(2L)^{-1}$, where $v_{a}$ is the acoustic velocity (see Fig. 2i). As before, each cavity mode response will also have Lorentzian frequency broadening due to exponential temporal decay. The $m^{th}$ cavity mode, with decay rate $\Gamma_{m}$, will have have a Lorentzian width ($\Gamma_{m}$) in the frequency domain. However, in contrast with the Brillouin limit, because the spatial extent of the standing-wave is finite (i.e., abruptly terminates at the crystal faces), the phonon wavevector occupies a band of spatial frequencies and takes on a $\textrm{sinc}^{2}$ form. This can be seen from the spatial Fourier transform of a truncated sinusoid. Therefore, each acoustic cavity mode has a frequency/wavevector response depicted by the lower right inset of Fig. 2i. Following the same procedure as described above, the optical response of this acousto-optical interaction is given by the intersection of the discretized phonon response curve (blue) with the optical wavevector mismatch (orange). The formation of cavity modes within this finite system then yields a response of the form $\sim\sum_{m}[(\Omega-\Omega_{m})^{2}+(\Gamma_{m}/2)^{2}]^{-1}\textrm{sinc}^{2}[(q(\Omega_{m})-\Delta k_{s}(\Omega))L/2]$. Hence, the finite extent of the crystal shapes the response into a series discrete modes which are modulated by a $\textrm{sinc}^{2}$ distribution (Fig. 2j).

The coherent-phonon limit produces a new characteristic nonlinear susceptibility (e.g. compare Figs. 2g with 2j). When phonons are trapped to produce high $Q$-factor phonon modes, we will show that in comparison to the Brillouin limit, the nonlinear optical coupling strength can be enhanced by several orders of magnitude, and the dynamics of photon-phonon coupling become nontrivial and depends on phonon propagation. Moreover, this interaction opens the door to new regimes of dynamics including optomechanical self-oscillation and mode-cooling processes, akin to cavity optomechanical systems. A key requirement for reaching these new regimes of dynamics is low-loss phonon modes that exhibit appreciable coupling to the light-field. Therefore, as detailed in the following section, the geometry of the crystal and the overlap between the phonon mode and the light-field become crucial considerations.

III Experimental Results

Above, we have considered the simplified case of 1D wave propagation in a finite crystal with varying degrees of acoustic loss. In practice, spatial dispersion and the geometry of our 3D crystal strongly impact the lifetime and character of the resonantly excited phonon modes. For instance, enhancement of the susceptibility is severely limited within a flat-flat crystal because this planar geometry forms an unstable resonator. In this case, phonon modes excited by a finite laser beam suffer appreciable diffractive losses, limiting the phonon $Q$-factor and nonlinear enhancement at cryogenic temperatures (Supplement Section 5).

To greatly enhance this coupling, we shape the surfaces of the crystal to form ultra-high $Q$-factor phonon modes that are tightly confined, permitting efficient coupling to a laser beam with a small (${\sim}30\ \upmu$m) spot size at ${\sim}10$ GHz frequencies. A plano-convex resonator geometry is known to support stable Hermite-Gaussian modes for optical waves propagating in an isotropic medium siegmanbook . However, for the complex elastic wave dynamics in anisotropic crystals, it is not clear that a comparable stable cavity exists. Therefore, we develop an acoustic beam propagation algorithm which accounts for the anisotropic elastic properties of the crystal and the curvature of the cavity end-faces. We demonstrate stable acoustic resonators in a plano-convex geometry, and by adapting optical design principles to the case of anisotropic elastic wave propagation, we create stable resonator geometries (Fig. 1) that support Hermite-Gaussian-like modes within z-cut quartz and TeO2 crystals. Simulated spatial profiles for the fundamental (L0), and higher order transverse (L1, L2) modes are shown for quartz in Fig. 3c.

These phonon modes have many unusual properties. Absent device imperfections and material dissipation, acoustic mode simulations reveal that the $Q$-factor of the fundamental (L0) mode exceeds $10^{9}$ for each crystal type. These tightly confined phonon modes have negligible anchoring losses because the transverse mode size is more than a hundred times smaller than the crystal diameter. The modal mass Pinard1999 , given by the mode size and the mass density is 60 $\upmu$g (Supplement Section 2.4). This large modal mass represents a unique and valuable aspect of this system compared to less massive cavity optimechanical systems for its value for studies of quantum decoherence (see further information in the discussion). For device parameters and simulation methods, see Supplement Section 5.

To explore optomechanical coupling in both the Brillouin and coherent-phonon limits of operation, we perform phonon-mediated energy transfer measurements in both crystal species. Stimulated energy transfer between counter-propagating pump and probe beams is measured with the basic experimental configuration seen in Fig. 2a both at room temperature and at cryogenic temperatures (see Supplement Sections 3-4, 6). As discussed previously, stimulated Stokes energy transfer between pump ($\omega_{p})$ and probe ($\omega_{s}$) beams provides a direct measurement of the effective nonlinear susceptibility of the system.

The stimulated Stokes energy transfer spectrum is obtained by fixing the pump frequency ($\omega_{p}$) while the pump-probe frequency detuning ($\Omega=\omega_{p}-\omega_{s}$) is swept through the Brillouin resonance ($\Omega_{s}$). Stimulated Stokes scattering is observed as a fractional increase in the probe power ($\Delta P_{s}/P_{s}$) while the probe frequency is tuned. Since each Stokes-scattering event corresponds to emission of a phonon, the Stokes energy transfer ($\Delta P_{s}$) also provides a direct measure of the phonon emission rate. Estimation of the Brillouin frequency from phase-matching considerations ($\Omega_{s}\cong 2\,\omega_{p}(v_{a}/v_{o})$) leads us to expect resonant coupling to longitudinal phonons at $\sim$12 GHz frequencies within both quartz and TeO2 when using a pump wavelength of $1549$ nm.

We begin by examining the stimulated Stokes spectrum at room temperature (i.e., in the Brillouin-limit). The Stokes energy transfer measurements seen in Fig. 3g(d) reveal broad Lorentzian line-shapes at 12.43 GHz (11.82 GHz), corresponding to a peak fractional change in the probe intensity of 13 ppm (80 ppm) for z-cut quartz (TeO2); these results agree well with the predicted resonance frequencies. Note that the fractional change in probe power (or gain) produced by TeO2 is larger than that of quartz because TeO2 possesses a larger photo-elastic response. The measured $7$-MHz ($10$-MHz) line-widths for quartz (TeO2) agree with known dissipation rates and correspond to a spatial decay length of ${\sim}140$ (${\sim}70$) microns Sonehara2007 .

At cryogenic temperatures (5-15 Kelvin) the system enters the coherent-phonon limit ($l_{ph}\gg L$). In this regime, photon-phonon coupling is mediated by a series of standing-wave cavity modes. Fig. 3b shows the stimulated Stokes spectrum generated by quartz at 9 Kelvin temperatures. The wide spectral scan (inset) reveals a series of regularly spaced resonances modulated by a sinc2 profile. The calculated phase-matching bandwidth (dashed line) is shown atop the data, demonstrating good agreement of the predicted geometric bandwidth with the observed spectral trend. Note that the central frequency of the quartz (TeO2) spectrum shifts to 12.63 GHz (12.21 GHz) due to changes in elastic and optical properties at cryogenic temperatures (e.g. see Ref. Ohmachi1970a ; McSkimin1965 ).

Inspection of Fig. 3b reveals three families of resonances (color-coded) that repeat at the acoustic free spectral range ($v_{a}/(2L)\cong 630$ kHz). Elastic mode simulations of Fig. 3a,c show that this mode structure is consistent with coupling to the fundamental (L0) and higher order (L1, L2) modes supported by the plano-convex geometry. The experimental results of Fig. 3b show the Stokes spectrum obtained when the laser-field is intentionally misaligned from the cavity-axis; this enhances coupling with higher-order spatial modes, increasing their visibility. The simulated phonon spectrum (Fig. 3a) reproduces the experimental spectrum by seeding the cavity with an acoustic beam with a lateral center offset of $10\ \upmu$m horizontally and $15\ \upmu$m vertically. As anticipated, improved axial alignment of the laser fields with the plano-convex geometry enhances coupling to the fundamental (L0) mode and suppresses higher order modes (L1,L2), as seen in Fig. 3h. A similar comparison of mode structure and spectrum of TeO2 shows good agreement with elastic wave simulations. For further details see Supplement Section 5.

High resolution spectral analysis of individual L0 cavity modes (Fig. 3f,i) reveals a spectral width, $\Gamma_{m}/2\pi$, of 300 Hz (23 kHz) in the quartz (TeO2) system, corresponding to phonon $Q$-factors of 42 million (0.5 million). Since the peak susceptibility scales inversely with dissipation rate (or proportional to $Q$-factor), these high-$Q$ cavity modes enhance the peak stimulated Stokes scattering rates by orders of magnitude. Within quartz (TeO2), Stokes scattering rates are increased by factors of more than $10^{4}$ ($10^{2}$) over their room temperature values. In quartz, these ultra-long lived phonon modes are consistent with a coherence lengths ($l_{coh}$) of ${\sim}3$ meters. Moreover, the demonstrated quality-factors in quartz are among the highest measured for phonon frequencies above $10$ GHz Blair1966 , corresponding to a $(\textrm{frequency})\times(Q\textrm{-factor})$ product of $4.2{\times}10^{17}$ Hz, which is comparable with world-class electromechanical devices demonstrated in Refs. Goryachev2012 ; Galliou2013 .

This drastically modified susceptibility (of Fig. 3) highlights the fact that the nature and dynamics of coupling in this bulk optomechanical system are radically altered from that of conventional Brillouin scattering. We see that the laser-field can now be used to selectively couple to an array of macroscopic phonon modes. As we explore the physics of this system further, we will show that this bulk crystalline system admits new regimes of dynamics–including phonon self-oscillation and mode cooling–which ordinarily require an optical cavity in the context of cavity optomechanics.

IV Bulk Crystal As Optomechanical System

To analyze this new system and to leverage the extensive body of work in the context of cavity optomechanics, it is useful to develop a quantum description of optomechanical coupling. Since the laser field couples to a macroscopic phonon mode, we can define a coupling rate between the optical fields and the quantized phonon-modes, as is typical in cavity optomechanics Aspelmeyer2014 ; Meystre2013 ; Marquardt2009 ; Favero2009a .

It is instructive to draw an analogy between our bulk crystalline interaction and a multimode cavity optomechanical interaction, wherein a single phonon mode ($b$) mediates coupling between two distinct optical modes ($a_{p}$) and ($a_{s}$). This process is captured by an interaction Hamiltonian of the form $\hbar(g_{12}a_{p}a_{s}^{\dagger}b^{\dagger}+g^{*}_{12}a_{p}^{\dagger}a_{s}b)$ Aspelmeyer2014 ; Grudinin2010 ; Braginsky2001 ; Law1995 , where $b$, $a_{p}$, and $a_{s}$ are phonon, pump-photon, and Stokes-photon annihilation-operators, respectively. This interaction has been used to describe optomechanical coupling in the case when a phonon mode is frequency-matched to the free spectral range (FSR) of an optical Fabry-P$\acute{\text{e}}$rot resonator; in this case, the phonon mediates coupling between two different longitudinal optical modes.

Coupling between counter-propagating laser beams in our bulk crystalline system (Fig. 1a) is reminiscent of this multimode cavity optomechanical interaction because a single phonon couples two distinct optical modes–each with different wavevectors. However, since light travels directly (or ballistically) through our bulk crystalline device, no optical cavity modes are formed. As a consequence, the $m^{th}$ phonon mode admits coupling to a continuum of optical waves.

The Hamiltonian of the phonon field, $H^{\text{ph}}=\sum_{m}\hbar\Omega_{m}b_{m}^{\dagger}b_{m}$, is a sum over the set of phonon-modes, whereas the Hamiltonian in the optical field is given by an integral over all possible wavevectors for both the Stokes- and the pump-waves. Assuming that the traveling optical waves are sharply peaked about their carrier wavevectors, the Hamiltonian for the optical fields is $H^{\text{opt}}\!\!=\!\!\int dk\hbar\omega_{p}(k)a_{p,k}^{\dagger}a_{p,k}+\int dk^{\prime}\hbar\omega_{s}(k^{\prime})a_{s,k^{\prime}}^{\dagger}a_{s,k^{\prime}}$. In this framework, the optomechanical coupling to the $m^{th}$ phonon-mode is given by the interaction Hamiltonian

[TABLE]

where, ${g}_{m}(k,k^{\prime})=g^{m}_{0}L\textrm{sinc}[(k^{\prime}-k+q_{m})L/2]e^{i\phi}$ is termed the geometric coupling rate, $\phi=((k^{\prime}-k+q_{m})L/2)+\pi/2$, and $g^{m}_{o}$ is a bare coupling rate produced by the zero-point motion of the $m^{th}$ phonon mode. The total interaction, $H^{int}=\sum_{m}H^{int}_{m}$, includes contributions from all of the optomechanically-coupled phonon modes supported by the crystalline resonator. Note that this formulation (Supplement Section 2-2.4) assumes that the optical beam profiles are uniform, and the propagating optical fields do not grow rapidly in space (For a more general treatment, not subject to these constraints, see Supplement Section 2.6). This newly defined coupling rate, $g^{m}_{0}$, can be related directly to the multimode cavity optomechanical coupling rate, $g_{12}$, under certain limits in which a the cavity mode is equivalent to the traveling wave as detailed in the Section 2.5 of the Supplement.

Notice that phase-matching is implicit in our geometric coupling rate, ${g}_{m}(k,k^{\prime})$. In the case when the incident fields are monochromatic, and the coupling is weak, ${a}_{p,k}$ and ${a}_{s,k^{\prime}}$ can be replaced with sharply peaked distributions consistent with the frequency and wavevector of each laser-field; in this limit one can readily show that the nonlinear optical susceptibility takes on the form $\sum_{m}|g^{m}_{0}|^{2}\,[(\Omega-\Omega_{m})^{2}+(\Gamma_{m}/2)^{2}]^{-1}\textrm{sinc}^{2}[(q(\Omega_{m})-\Delta k_{s}(\Omega))L/2]$ (see Supplement Section 2.2). This form of the susceptibility is identical to that obtained through phase-matching considerations (Section II.2), demonstrating the equivalence between the two pictures. See Supplement Section 2 for more information.

Using this form of the susceptibility to analyze our stimulated Stokes spectra for both quartz and TeO2 systems, we find zero-point coupling rates ($g^{m}_{0}/2\pi$) of $31$ Hz and $82$ Hz, respectively. These values agree with theoretical values of $36$ Hz and $66$ Hz obtained using known material properties and device geometry (within experimental error). See Supplement Section 6.

The Hamiltonian framework can now be used to derive and interpret the optomechanical cooperativity, an important metric for optomechanical interactions (Supplement Section 2.3). The cooperativity, which compares the photon-phonon coupling rate to the photon and phonon dissipation rates, is defined as $\mathcal{C}^{\text{om}}=4n_{p}|g_{12}|^{2}/(\Gamma_{m}\kappa)$ in the context of cavity-optomechanics Aspelmeyer2014 ; Meystre2013 ; Marquardt2009 ; Safavi-Naeini2013a ; here $n_{p}$ is the number of pump photons in the optical cavity and $\kappa$ is decay rate for the photons. However, since our system does not have cavity modes, it is more natural to define a free-space cooperativity ($\mathcal{C}^{\text{fs}}$) in terms of the incident pump-power ($P_{p}$) as $\mathcal{C}^{\text{fs}}=P_{p}|g^{m}_{0}|^{2}L^{2}/(\Gamma_{m}\hbar\omega_{p}v_{gs}v_{gp})$. Here, $v_{gs}$ ($v_{gp}$) is the group velocity of the Stokes (pump) wave. The connection between $\mathcal{C}^{\text{om}}$ and $\mathcal{C}^{\text{fs}}$ are discussed in Supplement Section 2.5. When $\mathcal{C}^{\text{fs}}$ reaches unity, phonon self-oscillation and appreciable mode-cooling are possible. For phonon self-oscillation, within our quartz crystal, a dissipation rate of 300 Hz translates to a threshold power of 7.2 Watts. Interestingly, the combination of high power handling and ultra-high $Q$-factor phonon modes within this bulk crystalline system could lead to extraordinary spectral-narrowing as the basis for new oscillator technologies Vahala2008 .

V symmetry breaking

Through optomechanical interactions, a phonon mode can typically mediate Stokes and anti-Stokes processes simultaneously; to engineer useful optomechanical operations, we typically seek to control which process will occur. Hence, as we contemplate the range of possible interactions, it is crucial to identify the relative probability of Stokes and anti-Stokes scattering by these ultra high $Q$-factor phonon modes. For instance, in order to selectively heat or cool a phonon mode it is necessary to bias the system, giving preference to either Stokes and anti-Stokes processes. In the context of cavity optomechanics, Stokes and anti-Stokes symmetry breaking is achieved by detuning the frequency of incident pump-wave from the optical cavity resonance. However, our bulk crystalline system does not possess an optical cavity, and we will see that a cavity is unnecessary to bias the system.

In stark contrast to cavity optomechanical systems, this bulk crystalline optomechanical system produces a form of dispersive symmetry breaking, arising from distinct phase-matching requirements imposed by Stokes and anti-Stokes scattering. In other words, this system exhibits a new form of symmetry breaking that enables phonon heating or cooling without an optical cavity (which we refer to as cavity-less optomechanics).

Energy conservation ($\omega_{as}=\omega_{p}+\Omega_{as}$) and phase-matching ($k_{as}=k_{p}+q_{as}$) conditions for the anti-Stokes process combine to require $q(\Omega_{as})=k(\omega_{p})+k(\omega_{p}+\Omega_{as})$, whereas Stokes scattering requires $q(\Omega_{s})=k(\omega_{p})+k(\omega_{p}-\Omega_{s})$. The sign-difference in these two relations results in a difference between the peak-frequency of the Stokes ($\Omega_{s}$) and anti-Stokes ($\Omega_{as}$) processes. The Stokes scattering is a maximum for phonon-modes of frequency $\Omega_{s}=\Omega_{0}(1+\frac{v_{a}}{v_{o}})^{-1}$ whereas anti-Stokes scattering is maximum for $\Omega_{as}=\Omega_{0}(1-\frac{v_{a}}{v_{o}})^{-1}$, where $\Omega_{0}\equiv 2\,\omega_{p}(v_{a}/v_{o})$. For typical parameters (e.g. 5-mm long z-cut quartz), this gives a dispersive frequency difference, $(\Omega_{as}-\Omega_{s})/2\pi$ of ${\sim}1$ MHz, which becomes significant in the coherent-phonon limit.

Since phase matching is implicit in our definition of the geometric coupling rate, ${g}_{m}(k,k^{\prime})$, the effect of the relative Stokes/anti-Stokes frequency shift can be seen by comparing the Stokes (${g}_{m}(k_{p},k_{s})$) and anti-Stokes (${g}_{m}(k_{p},k_{as})$) coupling rates. Fig. 4a(i-iii) compare the Stokes/anti-Stokes coupling rates as a function of phonon frequency (mode number) for crystal lengths ($L$) of 1, 5, 20 mm. Since the coupling rate for these processes are peaked at distinct frequencies, entire families of modes can be addressed, each with differing degrees of heating and cooling. In the limiting case of a large $L$, we see that the Stokes and anti-Stokes process are mediated by distinct sets of phonon-modes (Fig. 4a(iii)).

Since dispersive symmetry breaking results from the frequency dependence of our geometric coupling, we can quantify the frequency dependence of ${g}_{m}(k_{p},k_{s})$ by varying the pump frequency as we perform stimulated Stokes scattering measurements. By tuning the pump frequency from $\omega_{p}$ to $\omega^{\prime}_{p}\rightarrow\omega_{p}+\Omega_{as}$, we can quantify the effective geometric couping rate associated with the anti-Stokes processes. The measured geometric couplings associated with the Stokes and anti-Stokes processes are shown in Fig. 4c for z-cut quartz. Comparison of these two measurements reveals a dispersive frequency shift of 830 kHz, which agrees well with calculations.

In conjunction with theory, these measurements reveal that dispersive symmetry breaking produces a very strong differential gain, as the basis for selective mode heating/cooling in this system. For instance, the data of Fig. 4c reveal that phonon mode $m_{-1}$ is $70$ times more likely to experience Stokes scattering, whereas mode $m_{2}$ is $20$ times more likely to experience anti-Stokes scattering. Hence, these measurements illustrate how dispersive symmetry breaking permits mode selective heating or cooling within this bulk crystalline system. See Supplement Section 7 for more information.

VI Discussion and Conclusions

In the preceding section, we have shown that mode-cooling and mode-heating can be readily performed using this cavity-less optomechanical system. As we examine the range of possible uses for this system, it is also intriguing to note that this crystalline quartz system can support immense optical powers (kW to MW) without adverse effects Chiao1964a . Since the dispersive symmetry breaking (Section V) is compatible with the use of relatively short pulses ($<1$ns), we can use pulses to reach very large cooperativities ($10^{3}-10^{6}$) within this cavity-less system. The idea of using pulses to perform quantum control has been recognized as a fertile direction in numerous prior works Vanner2011 ; Vanner2013 ; Meenehan2015 ; Bennett2016 , and phase-matched optomechanical coupling within this bulk crystalline system may offer a different way of approaching these ideas. Building on these concepts, one can now consider using this system to perform quantum control or entanglement operations using these ultra-high $Q$-factor phonon modes.

Pulses are not necessary to reach new regimes of nonlinear dynamics, however. Even under continuous-wave (CW) operation, mode cooling and phonon self-oscillation are within reach if cooperativities are increased from their current values ($\mathcal{C}^{\text{fs}}\approx 0.03$) to $\mathcal{C}^{\text{fs}}>1$. The cooperativity, $\mathcal{C}^{\text{fs}}=P_{p}|g^{m}_{0}|^{2}L^{2}/(\Gamma_{m}\hbar\omega_{p}v_{gs}v_{gp})$, is increased by reducing the phonon dissipation rate ($\Gamma_{m}$) while increasing the coupling rate ($g^{m}_{0}$) and the incident pump-power ($P_{p}$). With fixed material properties (e.g. with quartz) cooperativities of unity can be attained by improving the acousto-optic overlap and boosting the pump power to 2 Watts. It is also interesting to note that greatly reduced phonon dissipation rates ($\Gamma_{m}$) may be possible with further device refinement. For example, with improved surface quality and crystal purity, significant reductions in phonon dissipation rates are possible. Since the roughness-induced diffraction losses scale with the root-mean-squared roughness to the 2${}^{\text{nd}}$ power, a modest (2-fold) reduction in the surface roughness could translate to a 4-fold reduction in dissipation rate (assuming that surface roughness plays a dominate role in dissipation), potentially reducing the phonon self-oscillation optical threshold ($\mathcal{C}^{\text{fs}}=1$) to $\sim 0.5$ Watts.

Cooperativities can also be drastically enhanced by constructing our optomechanical system using media with a larger photo-elastic response. For a fixed optical wavelength and mode-matched optical and acoustic waves, the cooperativity scales as $\mathcal{C}^{\text{fs}}\propto P_{p}n^{7}p_{13}^{2}/(\Gamma_{m}\rho v_{a})$. Hence, the coupling rate is enhanced by reducing the acoustic dissipation ($\Gamma_{m}$) or improving the material photo-elastic response. For fixed crystal geometry ($L=5$ mm), enhanced photo-elastic response produced by PbMoO4, Ge, and GaAs produce bare coupling rates ($g^{m}_{0}$) of $\sim$144, $\sim$266, and $\sim$292 Hz at center frequencies of 11, 26, and 25 GHz, respectively. Hence, comparable phonon dissipation rates ($\Gamma_{m}=300\times 2\pi$ Hz) within these media translate to self-oscillation threshold ($\mathcal{C}^{\text{fs}}=1)$ powers of $\sim$175, $\sim$40 and $\sim$33 mW, respectively.

This bulk crystalline optomechanical system also provides optical access to a large range of phonon-mode sizes and masses. Phonon resonators can be formed in cavities as short as tens of microns to as long as tens of centimeters with corresponding phonon motional masses spanning from nanograms to grams. Counterintuitively, one finds that cooperativity ($\mathcal{C}^{\text{fs}}\propto P_{p}n^{7}p_{13}^{2}/(\Gamma_{m}\rho v_{a})$) is invariant with volume of the phonon mode (or modal mass). This is of considerable interest because the combination of coherent quantum states and large masses are required to test fundamental theories of quantum decoherence and quantum gravity (see discussion in Ref. Aspelmeyer2014 for more information).

It is instructive to contemplate the construction of a cavity-optomechanical system which harnesses the presented bulk crystalline phonon modes by, for example, placing the bulk crystalline system of Fig. 1a within an optical cavity. A cavity-optomechanical system of this type could offer intriguing opportunities. For instance, the extraordinarily high power handling of this crystalline system permits intracavity power enhancement to reach high cooperativies ($\mathcal{C}^{\text{om}}\gg 1$) with CW operation. Assuming that the crystal is placed within an optical cavity whose length matches the crystal length ($L$) and maintains the modal overlap, the coupling rate of this cavity-system will be equivalent to the coupling rates ($g^{m}_{o}$) measured in this paper (See Supplement Section 2.5). In this configuration, the intracavity power will be resonantly enhanced by a factor of the optical finesse ($\mathcal{F}$), permitting extraordinarily high cooperativities ($\mathcal{C}^{\text{om}}=1000$) within high finesse ($\mathcal{F}=10^{4}$) cavities using moderate incident powers (100 mW).

Brillouin-based optomechanical coupling in an optical cavity have been demonstrated in microcavities Bahl2011 ; Rokhsari2005 ; Bahl2012 ; Tomes2011 at room temperatures as well as in an ultra-cold (15 mK) superfluid Kashkanova2016 . The bulk crystalline system, owing to its simple configuration with minimal contribution from surfaces, allows for material-limited phononic dissipation at desirable high frequencies (${>}10$ GHz). Moreover, since practically any transparent crystalline solid can be shaped into a low loss phononic resonator, a wide range of materials provides a flexible design space for specific applications. Each material offers a different phonon frequency, wavelength transparency, and may permit coupling to a range of additional excitations. For example, by utilizing a piezoelectric material, the cavity optomechanical system could combine with the electromechanical coupling techniques of Ref. Goryachev2012 ; Galliou2013 for a high frequency RF-to-optical conversion platform Bochmann2013 ; Pitanti2015 ; Balram2015 . Phonon modes of the type from this bulk crystalline system can also be readily coupled with superconducting qubits Chu2017a .

The task of adapting this bulk crystalline system to a cavity-optomechanical system presents some challenges; one must exercise great care in the design of phonon modes and optical modes depending on the optical and elastic properties of the crystal. For instance, the cavities required to produce optimal phonon confinement and optical confinement are likely to be very different; this is because nontrivial dispersion surfaces of the elastic medium drastically alter the conditions for stable phonon cavity-mode formation (for further discussion see Supplement Section 5). Moreover, while it is tempting to consider creating an optical cavity by depositing multi-layer stacks directly on the crystal, one must also be cognizant of the fact that (perhaps large) excess phonon dissipation will be accompanied by deposition of layered media on the crystal surface Galliou2016b ; Galliou2016c . Whether the bulk crystalline system has an optical cavity or not, the techniques developed in the present work are essential for optimizing phonon dissipation, material type, material quality, and optomechanical coupling. In other words, here we have taken the crucial steps required before constructing more sophisticated cavity optomechanical systems.

Looking ahead, it is clear that the field of quantum information and quantum measurement have growing need for ultra-low dissipation modes within optical, acoustic, and electromagnetic systems. As we evaluate the best approach for obtaining low-loss phonon modes in bulk or micro-scale systems, it remains unclear which materials are most ideal as the basis for high performance optomechanical Aspelmeyer2014 , electro-optomechanical Bochmann2013 , or quantum-phononic systems Chu2017a ; improved understanding of cryogenic phonon dissipation could provide crucial upper bound for device performance. To date, we have a fragmented understanding of the limits of phonon dissipation; aside from crystalline quartz, very few crystalline media have been exhaustively studied at cryogenic temperatures. With our laser-based approach to analyzing phonon dissipation, we have demonstrated $(\textrm{frequency})\times(Q\textrm{-factor})$ products of $4.2{\times}10^{17}$ Hz, which are comparable to the world-class quartz electromechanical systems of Ref. Goryachev2012 ; Galliou2013 . In addition, since all media posses photoelastic response, we can use these non-invasive laser-based optomechanical techniques to study practically any bulk crystalline medium. Hence, beyond new device concepts, bulk crystalline optomechanics has the potential to greatly expand our knowledge of cryogenic phonon physics.

In summary, we have demonstrated bulk crystalline optomechanics with macroscopic pristine single-crystal shaped acoustic resonators. Through a Brillouin-like coupling, high frequency (${>}10$ GHz) ultra-high $Q$-factor phonon-modes are stimulated and detected with light. Photon-phonon coupling in this bulk crystalline system presents an array of opportunities for both spectroscopy and optomechanical device design. In the context of nonlinear optics, this process is an extreme limit of Brillouin interactions wherein excited phonons become highly non-local and coherent. In the context of optomechanical phenomena, coupling to such mesoscopic phonon modes can be used as the basis for new quantum-optomechanical interactions. With high frequency (${>}10$ GHz), ultra-high $Q$-factor ($4.2{\times}10^{7}$) phonon modes, with variable mass ($10^{-9}{-}1$g), and high power handling ($>$kW), it presents an intriguing avenue for optomechanical device design as the basis for quantum information processing, tests of quantum coherence and sensitive metrology. From either perspective, dramatic ($10^{4}$-fold) enhancement of the nonlinear coupling is achieved by trapping ultra high $Q$-factor phonon modes in the focus of an incident laser beam. This interaction can be engineered in practically any transparent crystal and can be viewed as a new type of ultra-sensitive Brillouin-like materials spectroscopy. Bulk crystalline optomechanics provides a broadly tunable platform for the study of new parametric processes as well as basic material properties.

Acknowledgments

Primary support for this work was provided by NSF MRSEC DMR-1119826. This work was supported in part by the Packard Fellowship for Science and Engineering and Yale University. The authors thank Paul Fleury, Yiwen Chu, Eric Kittlaus, Nils Otterstrom, Jack Harris, Kale Johnson, Alexey Shkarin, Anna Kashkanova, and Glen Harris for valuable feedback and discussions.

Author contributions

W.H.R. and P.T.R. conceived the device and spectroscopy approach. W.H.R. conducted experiments to produce the initial results. W.H.R. and P.K. jointly advanced these techniques to produce the final results under the guidance of P.T.R. W.H.R. and P.K. developed simulation methods with input from R.O.B. and P.T.R. P.K. and R.O.B. developed the analytical theory with guidance from W.H.R. and P.T.R. All authors participated in the writing of this manuscript.

Additional information

The authors declare no competing financial interests.