The coupling between acoustic vibrations in a lithium niobate bulk acoustic wave resonator and microwave photons of a re-entrant microwave cavity was investigated at a temperature close to 4 K. Coupling was achieved by placing the acoustic resonator in the location of the re-entrant cavity electric field maxima in a symmetric “split-post” configuration with a large overlap between the microwave field and the acoustic mode, allowing acoustic modulations of the microwave frequency. We show that the acoustic modes in this setup retain large inherent quality factors of greater than 106. A maximum optomechanical coupling rate was determined to be g0 = 0.014 mHz, four orders of magnitude larger than previous results obtained using a quartz BAW at 4 K in a similar experimental setup but using a single post-re-entrant cavity resonator.
Investigating the fundamental interaction between photonic and phononic systems is crucial for advancing various applications within the domain of quantum technology and quantum electrodynamics.1,2 This understanding has led to the emergence of numerous associated predicted phenomena. Resolved sideband cooling,3,4 parametric amplification,5–7 optomechanically induced transparency/absorption,6,8 and long-range entanglement9,10 are just a few of these phenomena that have had important ramifications for testing fundamental quantum limits,11–15 dark matter detection,16–18 quantum communication, information processing, and storage, along with generation and manipulation of quantum states.19–21 Importantly, for these systems, highly coherent quantum state readouts and increased coupling rates are integral for parametric detection techniques. This is particularly relevant in the search for gravitons22,23 and in the development of parametric transducers for achieving low noise readouts at microwave frequencies.3,24
Ground state preparation of truly macroscopic (gram-scale) mechanical resonators is a highly sought after experimental demonstration.12,25,26 For one, there still remain questions regarding the quantum-to-classical crossover point that will be revealed by such systems, but in addition, the precision readout of such devices may offer insights into quantum gravity and high frequency gravitational wave detection.13,27–31
Studies on these photon–phonon interactions employ various types of resonant photonic device architectures, such as superconducting circuits,32 co-planar resonators,33 3-D microwave cavity resonators,34 and whispering gallery resonators,21,35,36 combined with various types of mechanical architectures displaying high coupling rates, such as membrane-in-the-middle systems,6 trapped ion particles,37 and acoustic-mechanical modes such as bulk acoustic wave (BAW) resonators.6,25,26,35 BAW resonators are low-loss mechanical resonators that exhibit long mechanical coherence times due to their specifically engineered convex surface, which helps trap the majority of phonons in the center of the resonator.27
In this work, a macroscopic centimeter size-scale lithium niobate (LiNbO3) crystal with its axis cut in the z-direction was manufactured with a diameter of 30 mm, a center thickness of 2 mm, and a convex radius of curvature of 100 mm. The crystal exhibited coherent phonon modes trapped in the center of the crystal with frequencies of order MHz. Figure 1 shows a photo of the LiNbO3 crystal along with the two halves of a “split-post” microwave cavity resonator,38 which sandwiches the BAW in the post gap, allowing the maximum electric field to be applied to the region of maximum phonon trapping as shown in Fig. 2. This means the acoustic phonons will be parametrically upconverted to microwave frequencies. The motivation was drawn from similar experiments conducted with quartz BAW resonators.26,39 However, the cavity design differs from before as we have used a symmetric design in which the gap is located in the center of a coaxial cavity. This particular design results in a more highly confined electric field distribution focused into the center of the BAW, and we term this design the “split-post re-entrant cavity.” Moreover, the reason for the emerging popularity of LiNbO3 over quartz is that it exhibits a stronger piezoelectric coefficient in addition to a low loss tangent. For example, LiNbO3-BAW has demonstrated a high quality factor × frequency product of Hz at room temperature.40
The convex lithium niobate crystal BAW resonator, along with the two halves of the re-entrant split-post microwave cavity, fabricated from oxygen-free copper.
The convex lithium niobate crystal BAW resonator, along with the two halves of the re-entrant split-post microwave cavity, fabricated from oxygen-free copper.
(a) Not to scale microwave-BAW cavity setup with a frequency discriminator output. The synthesizer was tuned to the microwave cavity and measured at the mixer output with the FFT to detect the upconverted BAW frequency. (b) Microwave transmission measurements (S21) with a vector network analyzer between ports 1 and 2, comparing 300 K with 4 K, showing an increase in microwave Q-factor. (c) To scale electric flux density and magnetic field intensity density plots inside the split-post microwave cavity. The electric flux density is concentrated in the center of the BAW in the vicinity of the MHz phonon modes, which are indicated in (a).
(a) Not to scale microwave-BAW cavity setup with a frequency discriminator output. The synthesizer was tuned to the microwave cavity and measured at the mixer output with the FFT to detect the upconverted BAW frequency. (b) Microwave transmission measurements (S21) with a vector network analyzer between ports 1 and 2, comparing 300 K with 4 K, showing an increase in microwave Q-factor. (c) To scale electric flux density and magnetic field intensity density plots inside the split-post microwave cavity. The electric flux density is concentrated in the center of the BAW in the vicinity of the MHz phonon modes, which are indicated in (a).
Much smaller LiNbO3 nanostructures have been used previously in various optical applications, such as quantum sensors41,42 and optical transducers.43–48 Our study investigated LiNbO3 microwave-BAW coupling rates, with the goal of creating strongly coupled coherent microwave readouts of large macroscopic phonon modes for fundamental physics applications that require larger masses. Here upconversion opens up possibilities for harnessing effects such as parametric amplification, cooling, and squeezing, while also providing a method for reading out displacement in large non-piezoelectric crystals.
Figure 2(a) illustrates the experimental setup for measurements conducted at 4 K. The experiment utilizes the split post-microwave cavity, which functions as a frequency discriminator. This phase bridge detects frequency shifts of the cavity in response to mechanical modes by ensuring that the LO and RF signals are in quadrature, making the system insensitive to amplitude fluctuations. The power splitting ratio is set by a −6 dB directional coupler. The signal generator is an analog synthesizer (Keysight E8663D), covering the frequency range of 100 kHz–9 GHz. Therefore, the experiment measures the frequency modulation of the BAW resonator up-converted to microwave frequencies. The output of the mixer is observed on the FFT spectrum analyzer.
The setup in Fig. 2 also shows the two posts of the microwave cavity, one of which was anodized to provide two electrically isolated posts, allowing them to function as electrodes to piezoelectrically excite the BAW mechanical modes with an arbitrary waveform generator (AWG). Since the applied voltage to the electrodes is known, we can estimate the piezoelectric-induced crystal displacement using known LiNbO3 electrostatic material parameters, allowing us to determine the change in displacement dx. The measured change in microwave cavity frequency, dωc, is then used to determine G.
The operational cavity mode was the fundamental split-post-mode [Fig. 2(c)], with a frequency of 6.075 GHz at 4 K. The loaded quality factor, QL, improved significantly from 300 to 4 K to QL = 330–2500, respectively, due to reduced resistive losses from the copper cavity walls as shown in Fig. 2(b). The coupling coefficients were determined from reflection measurements from ports 1 and 2 and calculated to be β1 = 0.8 and β2 = 0.136. Therefore, the unloaded microwave Q-factor can thus be determined to be Q0 = QL(1 + β1 + β2) = 4250. The increased Q-factor helped achieve better discriminator sensitivity for the phase bridge setup. The incident input power to the microwave cavity was 0.01 mW.
Figures 3(c) and 3(d) show the mixer output voltages for the longitudinal A5,0,0 mode and the shear B7,0,0 modes, respectively. Normalized reflection coefficients S11 are measured by driving the BAW through the anodized post and measuring the reflected power on an impedance analyzer, which is shown as the green trace. Figure 3 illustrates that the mixer output voltage is indeed a result of the mechanical mode being driven.
FEM simulations of the displacement field of the (a) 8.3 MHz longitudinal and (b) 5.8 MHz shear modes, with dashed lines indicating the effective mode volumes. (c) and (d) show the corresponding mixer output voltage in blue and S11 reflection measurements in green using an impedance analyzer across the split-post. Loaded Q-factors are determined by curve fits, as shown in orange.
FEM simulations of the displacement field of the (a) 8.3 MHz longitudinal and (b) 5.8 MHz shear modes, with dashed lines indicating the effective mode volumes. (c) and (d) show the corresponding mixer output voltage in blue and S11 reflection measurements in green using an impedance analyzer across the split-post. Loaded Q-factors are determined by curve fits, as shown in orange.
We can determine the linewidth of the mechanical modes from the mixer output voltage and obtain Q-factors. These values closely agree with the previously measured Q at 4 K using a parallel cathode plate.40 Figure 4 shows measured mechanical Q-factors at 4 K for various A and B modes.
Q values for the A3,0,0 and A5,0,0 longitudinal and B5,0,0, B7,0,0, and B9,0,0 shear modes, obtained from mixer trace output. The Q-factors demonstrate insignificant power dependence until the onset of duffing nonlinearity. The Q × f product of the best modes is of order 1014.
Q values for the A3,0,0 and A5,0,0 longitudinal and B5,0,0, B7,0,0, and B9,0,0 shear modes, obtained from mixer trace output. The Q-factors demonstrate insignificant power dependence until the onset of duffing nonlinearity. The Q × f product of the best modes is of order 1014.
The effective mode mass for various A and B modes of the BAW vary slightly from each other, depending on the value of their lumped circuit parameters. These values were calculated using COMSOL finite element modeling, similar to the analysis for quartz, as detailed in the works of,12,26 and related analytical effective mass calculations,50,51 with details given in the Appendix. Various determined parameters for each of the mechanical modes are presented in Table I, where co-operativity C0 is defined as , Γm is the linewidth of the mechanical resonance and κc is the linewidth of the microwave cavity.1 As observed, different modes show different coupling rates due to different displacement polarizations and different overlaps between the re-entrant post-mode and the mechanical modes.
Comparison of the parameters for some BAW modes. Frequencies, fm, in MHz;40 Meff,m in gram, evaluated from finite element modeling; g0, the coupling rate in Hz; and the single photon cooperativity, C0.
| Xn,m,p | fm | Meff,m | g0 | C0 |
|---|---|---|---|---|
| B5,0,0 | 4.20 | 3.53 × 10−4 | 2.67 × 10−7 | 2.40 × 10−20 |
| A3,0,0 | 4.70 | 4.15 × 10−4 | 1.44 × 10−5 | 2.40 × 10−17 |
| B7,0,0 | 5.80 | 3.14 × 10−4 | 2.12 × 10−8 | 2.96 × 10−23 |
| B9,0,0 | 7.40 | 2.77 × 10−4 | 4.79 × 10−8 | 7.62 × 10−22 |
| A5,0,0 | 8.30 | 3.52 × 10−4 | 2.38 × 10−6 | 1.96 × 10−18 |
| Xn,m,p | fm | Meff,m | g0 | C0 |
|---|---|---|---|---|
| B5,0,0 | 4.20 | 3.53 × 10−4 | 2.67 × 10−7 | 2.40 × 10−20 |
| A3,0,0 | 4.70 | 4.15 × 10−4 | 1.44 × 10−5 | 2.40 × 10−17 |
| B7,0,0 | 5.80 | 3.14 × 10−4 | 2.12 × 10−8 | 2.96 × 10−23 |
| B9,0,0 | 7.40 | 2.77 × 10−4 | 4.79 × 10−8 | 7.62 × 10−22 |
| A5,0,0 | 8.30 | 3.52 × 10−4 | 2.38 × 10−6 | 1.96 × 10−18 |
The displacement of the crystal structure is calculated from the electrical current arising from the applied piezoelectric voltage using Eqs. (4) and (5). The frequency shift from the phase bridge measurement divided by the calculated piezoelectric induced displacement provides the frequency pull factor, as detailed by Eqs. (6) and (7). By measuring Δfc for different input voltages to the piezoelectric BAW (converted to displacements), coupling rates at 4 K for different modes were calculated from the gradients of dfc/dx. Figure 5 shows coupling rates for A and B modes of LiNbO3-BAW, compared to other mechanical resonators. The inset figure shows dfc/dx for the A5,0,0 mode and is equal to 4.4 × 1013 Hz/m. The coupling rate g0 is derived from this value by multiplying it with xzpf.
Coupling rates for various A (blue) and B (orange) modes of the LiNbO3-BAW at 4 K, compared to other acoustic resonator coupling rates.6,26,39 The inset figure shows the change in microwave frequency Δfc (Hz) corresponding to displacement Δx for A5,0,0 mode, the slope of which multiplied by xzpf gives g0.
Coupling rates for various A (blue) and B (orange) modes of the LiNbO3-BAW at 4 K, compared to other acoustic resonator coupling rates.6,26,39 The inset figure shows the change in microwave frequency Δfc (Hz) corresponding to displacement Δx for A5,0,0 mode, the slope of which multiplied by xzpf gives g0.
The value of these coupling rates is benchmarked against similar experiments using a quartz BAW at 4 K with a single post-re-entrant resonator, as described in Refs. 26 and 39. The coupling rates g0 for LiNbO3-BAW in the split-post re-entrant cavity are observed to be up to four orders of magnitude greater compared to quartz for the longitudinal mode at A3,0,0, demonstrating higher coupling rates and making it a suitable candidate for further studies. Additionally, we compared our results with a Si3N4 membrane inside a re-entrant cavity, which achieved coupling rates of 25 mHz.6 These findings motivate further investigation into the coupling of LiNbO3-BAW with photonic systems.
The degree to which the electromagnetic and mechanical mode volumes overlap plays a key role in determining the optomechanical coupling. The improvement in coupling rates resulting from the use of a split-post vs a single-post cavity can be estimated by the ratios of a so-called overlap factor. This factor can be determined from the percentage of electromagnetic mode that exists within the mechanical mode volume determined by the dashed lines in Fig. 3. The A5,0,0 mechanical mode, for example, has all of its displacement field within a diameter of ∼6.0 mm. Inside this volume, the electromagnetic field distribution for both the split- and single-post cavities is primarily determined by the Ez field, which makes up of the electrical energy, which is only 5% in the case of a single post. The amount of electrical field within the mechanical mode volume is 20 times larger for a split-post design compared to the single-post design, explaining one of the contributions to the observed coupling rate increase of these results compared to the results in quartz.26,39 The remaining contribution stems from the material properties.
In conclusion, we investigated the acoustic modes of a LiNbO3-BAW resonator by directly exciting them with a radio frequency external voltage at mechanical resonance through piezoelectricity at 4 K. Coupling rates of order 0.014 mHz were obtained for the A3,0,0 longitudinal mode, four orders of magnitude larger than previous results obtained using quartz. Therefore, with further development, it could be utilized as an alternate material in quantum sensing, communication and testing of fundamental physics.
ACKNOWLEDGMENTS
This research was supported by the ARC Center of Excellence for Engineered Quantum Systems (EQUS, Grant No. CE170100009) along with support from the Defense Science and Technology Group (DSTG) as part of the EQUS Quantum Clock Flagship program. Additional support was provided by the ARC Center of Excellence for Dark Matter Particle Physics (CDM, Grant No. CE200100008).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
S. Parashar: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). W. M. Campbell: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). J. Bourhill: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). E. Ivanov: Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). M. Goryachev: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal); M.E. Tobar: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.
APPENDIX: SOME ACOUSTIC MODE PROPERTIES
1. Effective mass of the mechanical modes
2. Power dependence of the mechanical mode
Figure 6 shows the density plot for A(5,0,0) mode where the input voltage to the cavity is varied in the range of 0.1–3.0 V, and the output of the mixer is plotted for different input power. As the power is increased, the amplitude of the sideband increases; however, there is no significant change in linewidth, which indicates no variation in the Q-factor.
