CROSS-REFERENCE TO RELATED APPLICATIONS [0001] This application claims the benefit of U.S. Provisional Application No.

63/119,228, filed November 30, 2020, which is hereby incorporated by reference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR

DEVELOPMENT

[0002] This invention was made with government support under grant N00014-15-1- 2761 awarded by the Department of Defense/Office of Naval Research. The government has certain rights in the invention.

BACKGROUND

[0003] Embodiments of the present disclosure relate to acoustic waveguides, and more specifically, to electrical control of on-chip traveling acoustic waves.

BRIEF SUMMARY

[0004] According to embodiments of the present disclosure, an acoustic waveguide is provided. The acoustic waveguide comprises: a piezoelectric substrate having a first acoustic velocity; a film disposed on the piezoelectric substrate, the film having a slot extending therethrough to the piezoelectric substrate, the film having a second acoustic velocity greater than the first acoustic velocity; at least one pair of electrodes configured to apply an electric field to the piezoelectric substrate at the slot.

[0005] According to embodiments of the present disclosure, a method of electro-acoustic modulation is provided. A sinusoidal signal is applied to at least one pair of electrodes. The at least one pair of electrodes is configured to apply an electric field to a piezoelectric substrate. The piezoelectric substrate has a first acoustic velocity. A film is disposed on the piezoelectric substrate. The film has a slot extending therethrough to the piezoelectric substrate. The film has a second acoustic velocity greater than the first acoustic velocity. The at least one pair of electrodes is configured to apply the electric field to the piezoelectric substrate at the slot. An acoustic wave is supplied to the slot. [0006] According to embodiments of the present disclosure, a method of acoustic frequency shifting is provided. A sawtooth signal is applied to at least one pair of electrodes. The at least one pair of electrodes is configured to apply an electric field to a piezoelectric substrate. The piezoelectric substrate has a first acoustic velocity. A film is disposed on the piezoelectric substrate. The film has a slot extending therethrough to the piezoelectric substrate. The film has a second acoustic velocity greater than the first acoustic velocity. The at least one pair of electrodes is configured to apply the electric field to the piezoelectric substrate at the slot. An acoustic wave is supplied to the slot. [0007] According to embodiments of the present disclosure, a Mach-Zehnder interferometer is provided. The Mach-Zehnder interferometer comprises: a piezoelectric substrate having a first acoustic velocity; a film disposed on the piezoelectric substrate, the film having a first and second slot extending therethrough to the piezoelectric substrate, the film having a second acoustic velocity greater than the first acoustic velocity; at least three electrodes, the electrodes configured to apply a first electric field to the piezoelectric substrate at the first slot and a second electric field at the second slot, the first and second electric fields having opposite directions.

[0008] In some embodiments, the piezoelectric substrate comprises lithium niobate. In some embodiments, the piezoelectric substrate comprises single-crystal lithium niobate.

In some embodiments, the film is disposed on an X-cut surface of the single-crystal lithium niobate.

[0009] In some embodiments, the film comprises silicon nitride.

[0010] In some embodiments, the at least one pair of electrodes is disposed on the film at opposing sides of the slot.

[0011] In some embodiments, the at least one pair of electrodes comprise aluminum. [0012] In some embodiments, the slot has a width of about 10 pm.

[0013] In some embodiments, the acoustic propagation loss is at most 1.7 dB/mm.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0014] Figs. 1A-D illustrate a Lithium niobate (LN) electro-acoustic modulator according to embodiments of the present disclosure.

[0015] Figs. 2A-F illustrate electro-acoustic phase modulation according to embodiments of the present disclosure. [0016] Figs. 3A-C illustrate electro-acoustic amplitude modulation according to embodiments of the present disclosure.

[0017] Figs. 4A-D illustrate nonreciprocal phase modulation of acoustic waves according to embodiments of the present disclosure.

[0018] Fig. 5 illustrates simulated acoustic phase velocities at various directions on X-cut LN according to embodiments of the present disclosure.

[0019] Fig. 6 illustrates acoustic phase change on different modulation amplitudes at room and cryogenic temperatures according to embodiments of the present disclosure.

[0020] Fig. 7 illustrates phase matching between the traveling acoustic wave and the quasi-traveling modulation according to embodiments of the present disclosure.

[0021] Figs. 8A-F illustrate Electro-acoustic phase modulation according to embodiments of the present disclosure.

[0022] Fig. 9 is a graph of measured transmission spectra of an acoustic modulator according to embodiments of the present disclosure at temperatures of 300 and 1.3 K.

[0023] Fig. 10 is a graph of sideband power showing modulation bandwidth of a 1-cm- long electro-acoustic phase modulator according to embodiments of the present disclosure.

[0024] Fig. 11 is a schematic view of a fifty millikelvin measurement setup according to embodiments of the present disclosure.

[0025] Fig. 12 a graph of noise measurement of an electro-acoustic modulator according to embodiments of the present disclosure at 50 mK.

DETAILED DESCRIPTION

[0026] Acoustic waves in solids are the basis for numerous signal processing applications including microwave filters, delay lines, and sensors, and are emerging as universal interfaces between quantum systems such as superconducting circuits, defect centers, and optical devices. Compared to electromagnetic waves, acoustic waves feature five-orders- of-magnitude slower phase velocity and do not suffer radiation into free -space. They therefore allow coherent information processing and manipulation in ultra-compact footprint with negligible crosstalk between devices and to the environment, which are limiting factors in scaling-up today’s superconducting microwave circuits. For this reason, on-chip phononic systems have been proposed as a promising candidate for quantum computing and storage. An integrated phononic system requires a few essential functionalities — efficient transduction, low-loss waveguiding and routing, and active modulation and control. Further, nonreciprocity is also crucial to achieve isolation and circulation for qubit protection and noise mitigation. However, some of these basic features, especially active modulation, have not been realized, and thus hinders the development of fully integrated functional phononic circuits.

[0027] Various approaches may be taken towards the realization of phononic integrated circuits. Microwave acoustic waveguides on chip may be implemented using suspended structures, two-dimensional phononic crystals, and high acoustic velocity substrates. The use of nonlinear interactions for controlling acoustic waves may be applied in acoustic four-wave mixing and resonance tuning, but due to the high linearity in elastic response of most materials, these methods require high acoustic powers which are not feasible for signals at the quantum level. On the other hand, optomechanical coupling permits control of individual phonons, yet mechanical bandwidths are limited to the sub-MHz linewidth of the mechanical resonance, and these systems are challenging to scale up. To achieve acoustic nonreciprocity in particular, approaches based on nonlinear materials, circulating fluids, water-submerged phononic crystals, deformed water-air interfaces, and optomechanical systems are limited to acoustic frequencies below a few megahertz, which restrict functionality for high bandwidth or quantum processing. Approaches at microwave frequencies using electric circuitry and semiconductor acoustoelectric effects are not compatible with sub-Kelvin temperatures required for quantum processing, while ones using ferromagnetic materials require magnetic field and thus restrict compatibility with many solid-state systems, such as superconductors.

[0028] The present disclosure provides electrical control of the fundamental properties of an acoustic microwave — its phase and amplitude — using an integrated lithium niobate (LN) electro-acoustic platform. These methods are illustrated by direct frequency modulation of guided microwave phonons and reconfigurable nonreciprocal phase modulation using quasi-traveling electric fields. Modulation of the acoustic waves are enabled by the electro-acoustic effect, which is the change in the elasticity of a solid due to an applied electric field, and results in a variation of the phase velocity of traveling acoustic waves. The longer the acoustic wave propagates in the modulated region, the more phase difference will be accumulated at the output. The electro-acoustic effect is characterized by the third-order piezoelectric coefficients d describing the change of elastic constants Ac due to the applied electric field E.ACi _j = d _kij E _k . where i,j, k can take values of 1 to 3, corresponding to crystal X, Y and Z directions, and d _kij is dependent on material symmetry. However, the electro-acoustic effect observed on bulk devices is too weak to build piratical devices, for example, achieving a p-phase shift, until we confine the acoustic waves in an on-chip waveguide. We achieve the acoustic waveguiding by creating acoustic index contrast by patching micro-structured, thin -film silicon nitride (SiN) on top of LN substrate.

[0029] In view of the above discussion, a “piezoelectric” material, component, or substrate as referred to herein is one that has an electrically-controllable elastic constant.

[0030] Figs. 1A-D illustrate a Lithium niobate (LN) electro-acoustic modulator according to embodiments of the present disclosure. Fig. 1A is an optical micrograph of a fabricated device. Bright regions (e.g., 101) are aluminum (Al), the darker thin lines (e.g., 102) are the edges of the silicon nitride (SiN) thin film. Interdigital transducers (IDT) are used to excite and collect acoustic waves. The device is on an X-cut LN substrate, and the acoustic waveguide is at an angle of 30° respect to the crystal Z-axis (coordinates indicated). That out-of-focus darker are remarks on the back side of the chip, which do not affect the device. Fig. IB is a cross-section of the acoustic waveguide. The normalized displacement field intensity (shading) shows the simulated fundamental Rayleigh-type acoustic mode and arrows indicate the simulated electric field direction and magnitude due to a voltage on the modulation electrodes. Fig. 1C is a false-colored scanning electron microscope image of the acoustic waveguide. Fig. ID shows the measured propagation loss of the acoustic waveguide with varied temperature.

[0031] Electro-acoustic modulators are fabricated on an X-cut LN substrate (Fig. 1A), where the large coefficient d ₂ n of LN is employed by using Rayleigh-type guided acoustic modes and applying electric fields mainly in the Y direction. Inter-digital transducers (IDT) are used to electrically generate and detect microwave acoustic waves. The pitch of the IDT finger electrodes is 650 nm and equal to the half wavelength of the most efficiently transduced acoustic waves at 2.5 GHz. To optimize transduction efficiency, the width of the IDT (75 um) is designed to be larger than the acoustic waveguide (10 um). The generated acoustic wave is then tapered to an acoustic waveguide and propagates along an angle of 30° to the crystal Z axis, as this direction features the smallest acoustic velocity on the X-cut surface and thus has better confinement (Fig. 5). The acoustic waveguide is defined by an opened slot within a SiN thin film on top of LN (Figs. 1-C). Since the acoustic velocity (index) in SiN is greater (less) than that in LN, the Rayleigh-type acoustic mode is thus confined by the SiN thin film (Fig. IB). Aluminum electrodes on the SiN layer are used to apply the electric field for the modulation. The width of the waveguide is chosen to be 10 pm to achieve low propagation loss.

[0032] The propagation loss of the acoustic waveguide is characterized with varied temperature (Fig. ID). The propagation loss is determined by comparing the transmission of two devices that differ in waveguide length. At room temperature (300 K), the acoustic propagation loss is a=1.7 dB/mm The propagation loss reduces with lowered temperature to a=0.5 dB/mm at liquid nitrogen temperatures (77 K) and <0.1 dB/mm at 1.3 K. The acoustic propagation is consistent with that measured using acoustic cavities on LN. Thermoelastic dissipation is the likely source of loss at temperatures above 1 K.

[0033] Figs. 2A-F illustrate electro-acoustic phase modulation according to embodiments of the present disclosure. Fig. 2A shows an experimental setup for characterizing the electro-acoustic phase modulator. A signal generator 201 is used to generate the acoustic wave via an IDT, a low-frequency signal is applied to the modulation electrodes, and a real-time spectrum analyzer (RSA) 202 detects the phase and amplitude of the modulated acoustic wave. Fig. 2B shows transmission spectra of the acoustic modulator temperatures of 300 and 1.3 K, showing a 25 dB improvement in peak transmission at low temperature. Fig. 2C shows acoustic phase modulation of 180° by a 10 kHz sine wave signal with peak-to-peak voltage V _pp of 53 V. Fig. 2D shows a spectrum of a 2.483 GHz acoustic wave modulated by a 10 kHz sine wave with V _p = 2.3 V _% . Fig. 2E shows modulation bandwidth indicated by the first sideband power due to the phase modulation. Fig. 2F shows frequency shift of an acoustic wave by 1 kHz by applying a repeating linear voltage ramp (serrodyning) with V = 2 V _% . The spectral powers in Figs. 2D-F are normalized to the unmodulated signal received by the RSA. The results shown in this figure are measured from the same device with a modulation length of 10 mm.

[0034] Next, we measure the performance of the electro-acoustic phase modulator with a modulation length of 10 mm. While the transmission is measured by a network analyzer, the phase and amplitude of the acoustic wave modulation is detected by a real-time spectrum analyzer (RSA) when signals are applied to the modulation electrodes (Fig. 2A). The transmission spectra of the phase modulator are measured at temperatures of 300 (lower curve 201 with fill) and 1.3 K (upper curve 202 without fill) (Fig. 2B). Although the peak transmission increases up to 25 dB with lowered temperature as expected, the transmission spectrum detunes due to temperature dependent elasticity of LN. The insertion loss of the device, measured from a microwave signal applied to one IDT and detected after the other, is 10 dB at 1.3 K, which is mainly caused by the acoustic loss from the symmetric geometry of the IDTs (6 dB) and the tapers that guide acoustic waves to the waveguide.

[0035] When a sinusoidal signal with peak-peak voltage (V _pp ) of 53 V is applied on the phase modulator electrode, a p-phase change of the received acoustic wave is observed at room temperature (Fig. 2C), inferring a 1 = 53 V at carrier acoustic frequency of 2.483 GHz. The phase change increases linearly with V of the modulation signal (Fig. 6). At 1.3 K, an increased V _% of 133 V is measured at the acoustic frequency of 2.532 GHz, corresponding to maximum transmission (Fig. 6). Although the V _% increases by a factor of 3 at cryogenic temperature, an important figure of merit for the modulator, the product of half-wave length voltage and propagation loss V aL, reduces by a factor of 7 (from 90 V dB to 12 V dB) at 1.3 K owing from reduced propagation loss compared to room temperature. The V of the phase modulator could be further reduced by using narrower acoustic waveguides, which is possible using materials with higher acoustic contrast, such as LN on the sapphire.

[0036] Since the frequency of a wave is determined by the dynamics of its phase, the electro-acoustic phase modulator that can achieve 2p phase shift also allows for frequency control of acoustic waves. By driving the phase modulator using a 10 kHz sinusoidal signal with Tp _P =2.3 V V, an electro-acoustic frequency comb is generated with 19 comb lines separated by 10 kHz and centered around the input acoustic frequency of 2.483 GHz (Fig. 2D). To investigate the modulation bandwidth, which is a critical performance parameter, a weak signal modulation (Tp _P = 0.2 n ₎ is applied with varied frequency and the modulation efficiency is extracted from the first sideband power (Fig. 2E). The 3-dB bandwidth is measured to be 110 kHz for the 10-mm phase modulator. Because of the slow group velocity of the acoustic wave compared to the electrical modulation signals, full oscillations of the sideband power are observed as a function of modulation frequency. The modulation sideband power approaches zero at every modulation frequency with period of D/ ₀ =336 kHz, when the traveling acoustic wave acquires an integer number of 2p phase. The zero-modulation frequency is related to the modulation length L and acoustic group velocity by D/ ₀ = — , infers an acoustic group velocity of vg v _g = 3.36 km/s, which is consistent with simulated velocity of 3.38 km/s (Fig. 5), and confirms that the acoustic wave is being directly modulated. Beyond symmetric sideband generation, single-sideband modulation, or frequency shifting, is also demonstrated using a serrodyne approach. Specifically, by applying a repeating linear voltage ramp (sawtooth) signal at the frequency of 1 kHz with V _PP of 2 n _p , the modulated acoustic wave experiences an approximately linear phase ramp in time and results in a near-ideal frequency shift of the acoustic wave by 1 kHz with a measured efficiency of 92% (Fig. 2F). The efficiency is defined as the detected acoustic power at the shifted frequency 204 compared to that of an unmodulated acoustic wave 203. The carrier suppression of the frequency shifting is 21 dB, and the additional harmonics are due to the limited bandwidth of the voltage amplifier used to drive the phase modulator.

[0037] Figs. 3A-C illustrate electro-acoustic amplitude modulation according to embodiments of the present disclosure. Fig. 3A is a schematic of the acoustic Mach- Zehnder modulator. The modulation signal is applied at the central electrode 301 while two outer electrodes (302, 303) are ground, and thus two acoustic waveguides experience applied electric fields with opposite direction. Fig. 3B shows measured acoustic amplitude with slowly varying modulation voltage to determine the V _% of the modulator. Fig. 3C shows measured acoustic amplitude with a weak modulation signal biased at 0.5 n _.

[0038] Amplitude modulation of an acoustic wave is realized by using an acoustic Mach- Zehnder interferometer with electro-acoustic phase modulators (Fig. 3A). Accordingly, the acoustic wave is split into two waveguides that, by design, each experiences undergo phase shifts of opposite sign due to the opposite directions of the applied electric fields from the coplanar electrode, and acoustic interference occurs by recombining the waveguides to yield an amplitude modulation. Maximum output amplitude occurs when the difference in phase acquired by the acoustic waves traveling in each arm is zero or an even integer number of p, while the minimum amplitude occurs when the phase difference is an odd integer number of p. The phase difference can be adjusted by an applied DC bias voltage to the electrode. V =29 V is measured at quasi-DC frequency for such an acoustic Mach-Zehnder modulator that is 8 mm long, while the extinction ratio between the maximum and minimum output intensity is over 15 dB (Fig. 3B).

When the modulator is biased at 0.514, the amplitude of the resultant acoustic waves accordingly varies proportional to a weak modulation voltage (Fig. 3C).

[0039] Figs. 4A-D illustrate nonreciprocal phase modulation of acoustic waves according to embodiments of the present disclosure. Fig. 4A is a schematic of the device used for nonreciprocal phase modulation. Fig. 4B is an illustration of nonreciprocal modulation of forward and backward propagating acoustic waves. Fig. 4C shows measured acoustic phases at the output port in both forward and backward directions. The modulation signal on each electrode is delayed by 120°. Fig. 4D shows measured modulation sideband power of the forward and backward propagating acoustic waves for varying phase delays between the voltages applied to the electrodes. The modulation frequency is 336 kHz in Figs. 4C-D.

[0040] When the modulation electrode is separated into three segments, a quasi-traveling electric field can be realized by adjusting the relative phase of modulation signals applied to each electrode (Fig. 4A). With this design, a nonreciprocal acoustic phase modulation can be achieved when the quasi-traveling modulation signal is phase matched with the traveling acoustic wave in one direction but mismatched with that in the opposite direction. Maximum nonreciprocity occurs when the signal to each succeeding modulation electrode are phase delayed by 120° and the modulation frequency matches the traveling time of the acoustic wave. In this case, the forward propagating acoustic wave acquires the same segment-dependent phase as the modulation signal, while the backward propagating wave experience three different segment-dependent phases that result in a (integer number of) 2p phase shift of the modulation signal and thus shows no phase modulation at the output (Fig. 4B). Such a nonreciprocal acoustic modulator of 10 mm length is provided and the required 336 kHz modulation frequency is applied for maximum nonreciprocity. At this condition, (an absence of) acoustic phase modulations in the forward 402 (backward 401) propagation direction are observed (Fig. 4C). The acoustic nonreciprocity can be modified by adjusting the relative phase of the signals applied to the modulation electrodes 411...413 (Fig. 4D, forward 403, backward 404). A maximum nonreciprocity of over 40 dB of the modulation sideband power is observed. While sweeping the modulation frequency and the phase delay between the applied voltages on the electrodes, maximum modulation sideband powers are clearly observed when the traveling acoustic wave and the modulation signal are phase-matched, and shows zero modulation when they are phase mismatched by any positive integer number of 2p phases (Fig. 7). Segmenting the modulation could also be used to extend the modulation bandwidth of the acoustic phase modulator by a factor equal to the number of segments.

[0041] An on-chip phononic quantum network may be realized using the demonstrated electro-acoustic platform. This approach addresses challenges faced by electromagnetic quantum processors including the cross-talks between routing waveguides and high- quality resonators with small footprints. The ability to modulate the phase and frequency of phonons could enable entanglement among multiple solid-state qubits such as defect centers, which are usually mismatched in frequency due to their mesoscopic environment. In applications of acoustic signal processing, the ability to tune the phase of acoustic waves may be used to compensate for any unwanted drift caused by the operating environment such as temperatures, and develop tunable acoustic elements to reduce the number of acoustic components for the next-generation telecommunication.

[0042] Exemplary Materials and Methods [0043] Design of the Electro-acoustic Modulators

[0044] The interdigital transducer (IDT) is optimized for maximum transduction at about 2.5 GHz between acoustic waves and electrical signals. The aperture of IDT is 75 pm, the pitch of finger electrode is 650 nm, and the number of finger electrodes pair is 25.

The measured peak transmission between an IDT pair at 2.5 GHz is -8 dB at room temperature, where 6 dB is due to the symmetric design of both IDTs.

[0045] The adiabatic taper that connects the IDT and the acoustic waveguide is 400 pm in length. The insertion loss for the acoustic taper is about 5 dB per taper at room temperature, extracted from the measured transmission of devices with and without the waveguide structure.

[0046] The X-cut lithium niobate (LN) substrate is chosen such that the majority of the strain field of the Rayleigh-type guided acoustic mode is in XX component and the modulation electric field can be applied on the crystal Y and Z axes. This configuration features a non-zero electro-acoustic modulation coefficient. The direction of the acoustic waveguide on the X-cut waveguide is at an angle of 30° to the crystal Z axis (Fig. 5, free surface 501, shorted surface 502). This direction features the slowest acoustic wave phase velocity on the surface of X-cut LN and thus leads to a well-confined acoustic mode for the waveguide.

[0047] Fig. 5 illustrates simulated acoustic phase velocities at various directions on X-cut LN according to embodiments of the present disclosure. The direction is defined by the angle respective to the crystal Z axis. The electromechanical coupling coefficient k ² =

2 (v ₀ — v _m )/v ₀ . where v ₀ and v _m are the phase velocities when the top surface are free and electrically shorted, respectively. The direction of the waveguide used in our device is 30°, indicated by the dash line.

[0048] Device Fabrication

[0049] Electro-acoustic modulators are fabricated on an X-cut LN substrate. A silicon nitride (SiN) layer of about 400 nm in thickness is deposited by chemical vapor deposition on the LN substrate. The SiN layer is pattern by a direct write lithography tool (Heidelberg Instruments MLA150) and etched by reactive ion etching using carbon tetrafluoride (CF4), sulfur hexafluoride (SF6) and hydrogen (H2) gases. The metal layer is pattern by an electron lithography tool (Elionix ELS-F125) using PMMA resist. A 115- nm-thick aluminum is deposited by an electron beam evaporation tool, and lift-off in an NMP (1 -methyl -2 -pyrrolidone) solvent for more than 3 hours at 80 °C.

[0050] Device Measurements

[0051] Exemplary devices are mounted and wire-bonded to a printed circuit board (PCB). The transmission spectra are measured by a vector network analyzer (Keysight N5224A). In the modulation experiments, a microwave signal generator providing a single frequency source at the maximum transmission frequency near 2.5 GHz is applied to one IDT, and a real-time spectrum analyzer (RSA) is connected to the other IDT. The RSA not only measures the spectrum of the acoustic wave received by the IDT, but also demodulates the signal providing real-time I/Q data, which are converted to the phase and amplitude of the received signal. The microwave signal generator and the RSA are synchronized by the 10 MHz clock. A function generator is used to provide the low- frequency modulation signal, and a voltage amplifier (Falco Systems, WMA-005) is used when necessary to provide a 20 times amplification in voltage up to ±75 V, which has a bandwidth of 20 kHz. In the nonreciprocal measurement, a four-channel arbitrary function generator (Tabor WS8104A-DST) is used to generate the three synchronized modulation signals with various phase delays. [0052] Low Temperature Setup

[0053] The low-temperature performance of the device is measured in a closed-cycle cyrostat (ICE Oxford) that can reach abase temperature of -0.8 K. The transmission (S21) and half-wave voltage of the devices are monitored continuously (with tempertaure recorded at the same time) as the cryostat cools down from room temperature to the base temperature. The measurements are repeated during warm up. Cable losses are independently calibrated from a separate cooldown. To characterize the propagation loss of the SAW, we measured the transmissions of two acoustic waveguides with different lengths, fabricated on the same chip, packaged on the same PCB, and connected using identical cables. The difference in transmissions over the difference in lengths gives the propagation loss of the acoustic waveguides.

[0054] Electro-acoustic Effect

[0055] Hooke’s law relates the force on a spring to its displacement by a spring constant. The resonant frequency of a mass-spring system depends on the spring constant and the mass. The ability to tune the spring constant would result in the control of the resonant frequency.

[0056] For acoustic waves in solids, the generalized Hooke’s law relates stress s and strain e by an elasticity matrix, also named stiffness matrix, C, which is a 6-by-6 matrix in Voigt notation. It is subject to the symmetry of the material. LN is of point group 3m, which has a three-fold rotation symmetry about its Z axis and mirror symmetry on its X axis.

[0057] In LN, the elasticity matrix can be tuned by the applied electric field, Ac _i; - = d- _kij E _k , where i,j= 1..6, k= 1. 2, 3, E is the applied electric field, and D (c/ky) is the third- order piezoelectric tensor. Subject to the symmetry of the material, D of LN is in the following form,

[0058] Coupling between the Traveling Acoustic Wave and the Electric Field [0059] The electric field affects the traveling acoustic wave by tuning the elasticity of the material. When the applied electric field is small, such tuning in elasticity can be treated by the perturbation theory. The wave equation for the acoustic mode is

— rw ² u = V (C e), where u is the displacement field, w is the angular frequency of the eigenmode at given wavenumber k.

[0060] For such a guided acoustic wave mode, the first-order shift in the frequency due to the change of elasticity AC is given by

[0061] Further the change of wavenumber \k at certain mode frequency w is calculated by the dispersion relation of the acoustic mode,

Ak Aw v _p k w V _g where v _p and v _g are the phase and group velocity of the guided acoustic mode. The overall acoustic phase change due to the applied electric field is Ak L.

[0062] Fig. 6 illustrates acoustic phase change on different modulation amplitudes at room and cryogenic temperatures according to embodiments of the present disclosure. The modulation signals are sine waves with various peak-peak voltage at a frequency of 10 kHz. The linear fittings show V _% of 53 V at room temperature (300 K) and 135 V at 1.3 K.

[0063] Fig. 7 illustrates phase matching between the traveling acoustic wave and the quasi-traveling modulation according to embodiments of the present disclosure. The modulation sideband power is measured with various modulation frequency and phase delay between the electrodes. The peak modulations showing the phase matching condition are indicated by the red line. The point for the maximum nonreciprocity in phase modulation is indicated by the black dots, as the counter propagating acoustic waves see opposite phase delays. The sideband power is normalized to the unmodulated carrier acoustic wave power.

[0064] Referring to Fig. 8A, an experimental setup for characterizing the electro-acoustic phase modulator at room temperature is provided. A signal generator 801 is used to excite the acoustic wave via an IDT at carrier frequency f _c . A modulating signal 802 with frequency fmoi is applied to the modulation electrodes 811, 812, and a real-time spectrum analyzer 802 is used to detect the phase and amplitude of transmitted acoustic signal detected by an identical IDT. In the transmission-mode microwave impedance microscopy (TMIM), a scanning probe 803 is used to detect the acoustic field.

[0065] In Fig. 8B, TMIM images show phase shift of the acoustic wave due to an applied bias voltage on the modulation electrodes. The scanning region is located at the center of the waveguide near the output of the modulator.

[0066] The acoustic-wave profiles are directly imaged using the transmission-mode microwave impedance microscopy (TMIM), which is implemented on a commercial atomic-force microscopy (AFM) platform ParkAFM XE-70. The IDT is driven by a continuous microwave input signal (Anritsu MG 3692A), which launches the propagating surface acoustic wave. During the AFM scanning, the customized cantilever probe from PrimeNano Inc. picks up the GHz piezoelectric potential accompanying the acoustic wave. By using the same excitation frequency as the reference, the TMIM electronics demodulate the tip signal into a time-independent spatial pattern that is shown in Fig. 8B. Note that the TMIM image contains information on the phase of the propagating wave.

As a result, a lateral shift of the wave pattern indicates that the acoustic wave is modulated by the DC bias electric field. Due to the charging effects at interfaces between layers, a higher DC bias voltage is required to achieve the same phase shift as that of a modulating signal at /mod = 10 kHz.

[0067] In Fig. 8C, a graph of normalized power showing the modulation of an acoustic wave by a 10 kHz sine wave with V _PP = 2.3 V _% .

[0068] Fig. 8D is a graph of normalized power showing the serrodyne frequency shift of an acoustic wave by 1 kHz. This is achieved by applying a repeating linear voltage ramp (serrodyne) with V _pp = 2 V _% (Left inset). Right inset plots the acoustic powers in a linear scale, showing an efficiency of 92%. The spectral powers in Figs. 8C-D are normalized to the unmodulated signal received by the spectrum analyzer. The measurements in Figs. 8B-D were conducted at room temperature and usingyc= 2.483 GHz.

[0069] Fig. 8E is a graph of mean intra-waveguide phonon number showing modulation of an acoustic wave by a 10 Hz sine wave with V = 140 V. Unmodulated signal 821 is also shown. The input microwave signal is attenuated such that the mean phonon number of the acoustic wave in the waveguide is much less than 1. Fig. 8F is a graph of mean intra- waveguide phonon number showing serrodyne frequency shift of 10 Hz at the single-phonon level with an efficiency of 94.8%. This is achieved by applying a repeating linear voltage ramp (serrodyne) with V = 2 V _% . Figs. 8E-F are measured at 50 mK temperature with fc = 2.528 GHz. All results presented in this figure are measured from the same device with a modulation length of 1 cm.

[0070] The performance of a 1-cm-long electro-acoustic phase modulator is measured by inputting an acoustic wave at carrier frequency fc ~ 2.5 GHz and detecting phase and amplitude of the modulated acoustic wave (Fig. 8A). The insertion loss of the device, measured from a microwave signal applied to one IDT and detected after the other, is 10 dB at cryogenic temperature. This loss is dominated by the symmetric IDTs that excite and collect acoustic waves bidirectionally, which result in a 3 dB loss at each IDT, and the tapers that guide acoustic waves to the waveguide. At room temperature, the insertion loss increases by 25 dB due to higher propagation loss and a lower efficiency of IDTs (Fig. 9). Transmission-mode microwave impedance microscopy (TMIM), a scanning probe technique that coherently measures the profiles of traveling acoustic waves near the output of the modulator waveguide, is used. A ~p/2 phase shift of the acoustic wave is observed when a DC bias voltage is applied on the modulation electrode (Fig. 8B). This measurement of the traveling-wave profile confirms that the acoustic wave is being directly modulated. [0071] The modulation bandwidth of the 1-cm-long phase modulator (Fig. 8) is measured. A weak signal modulation (V _pp = 0.2 n ₎ is applied with varied frequencies and measure the modulation efficiency. The modulation efficiency is indicated by the first sideband power generated by the phase modulation. Due to the phase mismatch between the slowly propagating acoustic wave and the fast-varying electrical modulating signals, we measure the 3-dB bandwidth to be 110 kHz for the 1-cm phase modulator and observe periodic variations of the sideband power as a function of modulation frequency (Fig. 10). The modulation efficiency approaches zero every time when the modulation frequency is an integer multiple (N) of / _mod = 336 kHz. At these frequencies, the electric field oscillates exactly N full cycles as the acoustic wave travels through the modulator, resulting in a vanishing cumulative modulation effect. The zero-modulation frequency is related to the modulation length L and the acoustic group velocity v _g by

/mod ⁼ —· Using this relationship, an acoustic group velocity of v _g = 3.36 km/s is inferred, consistent with the simulated velocity of 3.38 km/s.

[0072] Despite the increased V _% at the cryogenic temperature, an important figure of merit for the modulator, the product between its half-wave voltage V _% , length L, and propagation loss a, i.e., V _% L a, is reduced by a factor of 7 (from 900 V dB to 120 V dB) at 1.3 K owing to reduced propagation loss compared to room temperature. The V _% of the phase modulator could be further reduced by a factor of 2 by using narrower acoustic waveguides, which is possible using materials with higher acoustic contrast. In addition, the 3-dB bandwidth is measured to be 110 kHz and observe zero modulation at /mod =336 kHz when the period of the modulating signal matches the traveling time of the acoustic wave (Fig. 10).

[0073] To assess electro-acoustic modulators for quantum applications, coherent modulation of single-phonon-level acoustic waves is demonstrates at 50 mK (see below and Fig. 11). The input microwave signal is attenuated so that the mean phonon number of the field in the acoustic waveguide is much less than one. Basic functionalities of the electro-acoustic modulator are preserved at low temperature: symmetric sidebands are observed when applying a 10 Hz sinusoidal modulation signal (Fig. 8E) and serrodyne frequency shifting, with an efficiency of 94.8%, is realized by applying a repeating linear voltage ramp at 10 Hz using F _PP of 2 n _p (Fig. 8F). As shown, the modulation process adds less than one noise phonon (Fig. 12). [0074] For data shown in Figs. 8E-F, and Fig. 12, a 50 mK measurement setup is used (Fig. 11). The device is mounted on the mixing chamber plate of a dilution fridge (Bluefors) with a base temperature below 50 mK. The microwave input signal to the IDT of the electro-acoustic modulator is attenuated such that the mean phonon number on the 1-cm-long acoustic waveguide is less than one. The readout line includes a circulator, a high-electron-mobility transistor (HEMT) amplifier at 4 K, and two room-temperature low -noise amplifiers. The output signal is recorded using a real-time spectrum analyzer (RSA). The detection bandwidth of the RSA is set to 78 mHz to realize a high signal-to- noise ratio, but this detection bandwidth limits the span to 50 Hz. The modulation signal of 10 Hz is supplied by a function generator followed by a high-voltage amplifier (Trek 2210). A controlled and isolated thermal source, consisting of a heater, temperature sensor, and a 30 dB attenuator, is installed on the input line before the electro-acoustic modulator to calibrate the readout gain and added noise. The temperature of this thermal source is varied from 100 mK to 6.7 K and the output noise spectrum density is measured. By comparing the measured noise spectrum density on the RSA against the calibrated thermal noise source, a total readout gain of 91.49 dB is extracted, consistent with the specifications of the amplifiers and cables used.

[0075] The noise performance of t electro-acoustic modulator is investigated at 50 mK (Fig. 12) by measuring the noise floor near the carrier frequency in three situations: (A) no signal is applied to the electro-acoustic modulator, (B) only the carrier microwave signal is applied, and (C) both carrier microwave signal and modulation signal are applied. When no signal is applied (Situation A), we measure a noise phonon number of N _tot,A = 64.35 + 0.39 quanta/s/Hz, which is mainly from the HEMT amplifier. When applying the carrier microwave signal (Situation B), we measure N _{tot B} = 64.23 + 0.36 quanta/s/Hz. The modulation signal is further applied (Situation C) and observe N _{tot c} = 64.32 + 0.37 quanta/s/Hz. As no observable difference (within error) in noise are measured, we conclude that the electro-acoustic modulation adds negligible noise and is thus suitable for quantum applications. This result agrees with the fact that the electro acoustic modulation is a parametric process.

[0076] Referring to Fig. 9, a graph of measured transmission spectra of the acoustic modulator at temperatures of 300 and 1.3 K is provided. The results indicate a 25 dB improvement in peak transmission at low temperature. The frequency shift of the spectrum is due to the temperature dependent elasticity of LN. [0077] Referring to Fig. 10, a graph of sideband power is provided showing modulation bandwidth of the 1-cm-long electro-acoustic phase modulator. The modulation efficiency is indicated by the first sideband power due to the phase modulation. The measured 3-dB bandwidth is 110 kHz. The modulation approaches zero at /mod = 336 kHz when the acoustic traveling time through the modulator equals l/ mod. The same device is measured as that in Fig. 8.

[0078] Referring to Fig. 11, a schematic view is provided of a fifty millikelvin measurement setup. The electro-acoustic modulator is mounted on a mixing plate of a dilution fridge with a base temperature below 50 mK. The input microwave signal is provided by a signal generator and, to ensure negligible contribution of thermal noise, passed through attenuators at various temperature stages of the fridge before going into our device. The output microwave signal from the modulator passes through a circulator, a high-electron-mobility transistor (HEMT) amplifier at 4 K, two low-noise amplifiers at room temperature, and is finally detected by a real-time spectrum analyzer (RSA). The modulation signal is provided by a function generator. A controlled and thermally isolated thermal source, which consists of a heater, temperature sensor, and a 30 dB attenuator, is installed in the microwave line before the electro-acoustic modulator to calibrate the gain and added noise in the output/readout line.

[0079] Referring to Fig. 12, a graph of noise measurement of the electro-acoustic modulator at 50 mK is provided. Total noise power spectrum density near f _c when (1) no signal is applied to the electro-acoustic modulator, (2) only the carrier microwave signal is applied, and (3) both the carrier microwave and modulation signals are applied. The total noise power N _tot = N _dev + N _add , where N _dev is the noise of the electro-acoustic modulator and N _add is the added noise from the readout chain. N _add is mainly determined by the high-electron-mobility transistor (HEMT) at the 4 K stage. The electro-acoustic modulation adds negligible noise and is thus suitable for quantum phononics. The same device is measured as that in Fig. 8.

[0080] The descriptions of the various embodiments of the present disclosure have been presented for purposes of illustration, but are not intended to be exhaustive or limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen to best explain the principles of the embodiments, the practical application or technical improvement over technologies found in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments disclosed herein.