Enhanced electron-phonon coupling for a semiconductor charge qubit in a surface phonon cavity

Electron-phonon coupling is a major decoherence mechanism, which often causes scattering and energy dissipation in semiconductor electronic systems. However, this electron-phonon coupling may be used in a positive way for reaching the strong or ultra-strong coupling regime in an acoustic version of the cavity quantum electrodynamic system. Here we propose and demonstrate a phonon cavity for surface acoustic waves, which is made of periodic metal fingers that constitute Bragg reflectors on a GaAs/AlGaAs heterostructure. Phonon band gap and cavity phonon modes are identified by frequency, time and spatially resolved measurements of the piezoelectric potential. Tunneling spectroscopy on a double quantum dot indicates the enhancement of phonon assisted transitions in a charge qubit. This encourages studying of acoustic cavity quantum electrodynamics with surface phonons.

Coupling of electrons to intrinsic phonons has been known to cause scattering 1 or energy relaxation [2][3][4] in semiconductor electronic systems. Previous studies 2,3 showed that spontaneous emission of energies into the phonon bath is a major dissipation mechanism in charge and spin qubits in GaAs/AlGaAs systems. However, this negative effect can be applied in a constructive way to realise an acoustic analog of cavity quantum electrodynamic (cQED) system [5][6][7][8] for reaching the strong or ultra-strong coupling regime in cQED. It will be advantageous to fabricate an acoustic cQED system because it traps phonons with wavelengths in the nanometer regime, which is smaller than the microwave cavities used in conventional cQED systems. Smaller wavelengths allow for the fabrication of a smaller cavity with the potential of achieving a larger electron-cavity coupling 9 . Previously phonon cavities have been demonstrated on suspended nanowires [10][11][12][13] however it is challenging to confine intrinsic crystallographic phonons [6][7][8] . Acoustic waves in piezoelectric materials, such as GaAs, has a large coupling to quantum dots [14][15][16][17] or superconducting qubits 18,19 , and are advantageous for realising strong or ultra-strong coupling regimes with a small cavity 9 .
In this study we demonstrate phonon cavities for surface acoustic waves (SAW) on a GaAs/AlGaAs heterostructure. Brag reflectors (BR) deposited onto the surface of the heterostructure are used as mirrors to reflect and trap SAW. By observing the piezoelectric potential of the SAW through time, spatial and frequency measurements, we identify the phonon band gap and cavity modes of our phonon cavities. Furthermore coupling of the cavity mode to a charge qubit is demonstrated through phonon assisted transport measurements on a double quantum dot and observe to be enhanced at the resonant frequency of the cavity.
The coupling frequency g between a qubit and a cavity mode can be enhanced by reducing the mode volume V of the cavity. For standard cQED with electromagnetic waves, g normalized by the resonant frequency ω is bounded by the fine structure constant α ~ 1 137 , as seen in the form for an electric dipole of the length l and the wavelength λ 5,20 . Since geometric constraints require V to be greater than l 2 λ, ω g is practically limited by α . The phonon cavity is attractive for overcoming this limitation by taking advantage of its small mode volume associated with the slow speed v compared with the light speed c. Considering the electromechanical coupling K 2 (0.07% for GaAs SAW) 21,22 , the normalized coupling for acoustic cQED is bounded by the effective fine structure constant α α ε = / K c v eff 2 r (ε r being the relative dielectric constant). This can be greater than α in some piezoelectric materials (α α / eff ~ 10 for GaAs SAW and potentially greater than 100 for ZnO SAW and other piezoelectric materials 23 ), and thus attractive for reaching the ultra-strong coupling regime. We show that GaAs SAW phonons are successfully confined in a cavity for enhancing the electron-phonon coupling.
The coupling between SAW and a charge qubit in two-dimensional electron gas (2DEG) of a GaAs/ AlGaAs heterostructure is significant even without forming a cavity 24 . SAW is a Rayleigh wave localized near the surface within the penetration length (≈ 0.3 λ) comparable to the SAW wavelength λ, which provides a large piezoelectric field at the 2DEG located near the surface 25 . A charge qubit, in which an excess electron occupies one of the two dots, has an electric dipole (el) for the dot distance l 26 . The coupling can be maximized when l is comparable to λ/2 of SAW that resonate with the level spacing of the qubit (typically a few GHz) 2 . Typical GaAs DQDs approximately meet this condition (λ = 800 nm for the SAW frequency f = 3.2 GHz, 2DEG depth d = 95 nm and l = 240 nm in our device). We show that SAW can be confined in a cavity and the coupling can be further enhanced by designing a SAW cavity.
A SAW cavity can be formed by two BRs made of periodic metal arrays with a period a, shown by the schematic in Fig. 1a. Metallization of the semiconductor surface decreases the SAW velocity by about 10% and causes a small reflection with a coefficient r (≈ 1%) at the metal edges 27 . With a large number N BR ( 1/r) of the array, the total reflection coefficient approaches unity for a specific wavelength λ BR close to 2a. When two BRs are separated by a gap distance D which equals an odd integer multiples of λ BR /2, the fundamental cavity mode exhibits a node at the center, which maximizes the coupling to the DQD, as shown schematically in Fig. 1a.
The important parameter of the cavity for cQED is the finesse F, which determines the number of times the wave is reflected inside the cavity before decaying 20 . Therefore, the coupling is enhanced by the factor F. We experimentally evaluate the enhancement by the following schemes. The time-resolved reflectometry reveals each reflected wavepacket to characterize the phonon band gap inside the BR. The finesse is evaluated as a ratio of the band gap to the resonance width of the cavity mode. The obtained F ~ 80 is promising for cQED applications. Moreover, the SAW in the cavity mode induces phonon-assisted qubit transitions more efficiently than off-resonant SAW, which manifests the enhancement of electron-phonon coupling in the system. Periodic arrays serve as BRs for forming a phonon cavity and IDT for generating SAWs. Fine metal fingers in the middle of the cavity are used to define point contacts for the phase-sensitive local potential measurements or a DQD for studying phonon assisted tunneling.

Results and Discussions
In our experiments, SAW phonons are excited by applying an RF voltage to an interdigital transducer (IDT) sharing the metal array with the BR, and the enhancement of the piezoelectric potential is evaluated with electronic devices inside the cavity. Device A illustrated in Fig. 2a consists of two metal gratings with a = 0.4 μm (λ BR = 0.8 μm) which serve as two BRs and an IDT shared with the left BR. A short SAW packet launched from the IDT bounces between the two BRs with a large distance  D 160 μm. The time resolved phase-sensitive local potential measurements were performed with a point contact (PC), where i-th PC (i = 1−5) is selectively activated by using gates G 0 and G i 28,29 . The measurement was done by applying two synchronized burst waves, V ex for generating a wave packet and V det for detecting the potential wave φ i PC at i-th PC, and by measuring the resultant dc current I det through the PC as a function of the delay time t d between the two bursts [See pulse sequence in Fig. 2b and Methods]. The PCs separated by 0.12 μm (0.15λ) allow us to identify the phase evolution and thus the direction of the travelling waves. Figure 2c shows the derivative of the detection current ( / ) dI dt det d with PC2 as a function of t d , which oscillates with a period (1/f ~ 0.3 ns) of the carrier frequency f = 3.2 GHz as seen in the insets. The time evolution of the envelope in the main trace involves the electromagnetic (EM) crosstalk with a triangular shape centered at t d = 0, direct SAW packet at around t d = 31 ns (d-SAW), and other reflected waves (r1and r2-SAW). Their travelling-wave nature is confirmed from the phase evolution of the signals at different PCs. For the direct SAW packet in the left inset of Fig. 2c, the phase of the oscillations measured at the rightmost PC4 is seen to be delayed with respect to that at the leftmost PC1, indicating that the wave packet is propagating to the right. Similarly, examining the phase evolutions around t d ≈ 99 ns (the central inset) and t d ≈ 142 ns (the right inset) allows us to assign the oscillations in these time domains to be the primary reflected waves moving to the left (r1-SAW) and the secondary reflected waves moving to the right (r2-SAW), respectively. Observation of SAW packets bouncing between BRs suggests high coherency of SAW even in the presence of electronic devices with fine gate patterns G 0−5 .
The reflection spectrum of a BR is obtained by plotting the amplitude A det of the oscillation in / dI dt det d around some specific delay time t d as a function of f in Fig. 2e. The broad spectrum of the direct SAW at t d = 35 ns, which is determined by the Fourier spectrum of the SAW burst wave. This is regarded as an incident spectrum to the right BR. The reflected spectrum measured at a later time t d > 60 ns is narrower and the width is independent of t d . This is a manifestation of the phonon band gap in which SAW cannot propagate through the BR. The observed spectra as well as the multiple reflections are consistent with the one-dimensional coupling-of-mode (1D COM) theory 27  A well-defined cavity mode can be formed in device B (the same λ BR = 0.8 μm) with a small D of 4.4 μm shown in Fig. 3a. Inherent cavity spectrum is obtained by using the time-resolved measurement to exclude the EM crosstalk. A short IDT of 10 pairs is used to generate SAW in a broadband, and a long burst wave of about 2500 cycles is used to minimize the spectral broadening of the burst. As shown in Fig. 3b, the spectrum shows sharp peaks, each of which is much narrower (Δ f cav = 1.2 MHz for peak I) than the band gap (Δ f BR = 100 MHz). Both peak I and II exhibit standing wave nature of the cavity mode in the phase evolution at different PCs, as shown in the right inset for peak I. An approximately π phase shift is observed between PC1 to PC3 and PC4 to PC5. This means that the piezoelectric potential has a node located close to the central gate, which proves the realization of a cavity mode [See the left inset].
The above measurements of phonon band gap in the BR and the line width of the cavity mode suggests the cavity finesse, F = Δ f BR /Δ f cav ≈ 80, by considering the ratio of the cavity decay time (∆ ) − f cav 1 and the time required to reflect SAWs in a BR (∆ ) − f BR 1 . For a given input power or a number of phonons, the cavity is expected to enhance the spontaneous emission as well as stimulated processes by the factor F, which is examined with a DQD.
We demonstrate phonon assisted tunneling in a DQD by using the cavity mode. For the time-gated scheme described below, the transport path of Device B is rerouted as shown in Fig. 4a, and the ohmic contact on the left is used as an in-plane gate (IPG). A weakly coupled DQD is formed with G 0−5 at 30 mK, where thermal excitation of SAW phonons can be neglected. When the detuning ε between the ground levels in the left and right dots is varied, a sharp resonant tunneling peak with a width of 16 μeV appears at ε = 0, as shown by the bottom trace of Fig. 4b. The spectrum can be fitted with a Lorentzian profile shown by a solid line with a negligible deviation associated with the cotunneling processes at ε µ < + e 50 V. Current associated with the spontaneous phonon emission could not be observed because it was too small in this weak coupling limit. When a continuous wave (CW) at the resonant frequency of 3222 MHz is applied to the IDT, the peak splits into multiple peaks with fine structures in Fig. 4b. In this CW experiment, both photon (the EM crosstalk) and phonon (SAW) contribute to the splitting, as the DQD feels the sum of the oscillating detuning potential ε . The current spectrum can be explained by phonon and/or photon assisted tunneling with absorption (n < 0) or emission (n > 0) of n phonons/ photons (See insets) 24,30,31 . They are seen as small current peaks at ε = nhf (highlighted by vertical lines) in Fig. 4b even though the linewidth of 16 μeV is slightly larger than the spacing hf = 13 μeV. The theory for CW excitation suggests that each peak current should be proportional to the square of the Bessel function of the first kind α ( )  J n of a normalized amplitude α ε = /   hf . Using a Lorentzian function fitted to the original peak profile at V ex = 0, all current profiles can be reproduced accurately by varying α  in proportional to V ex , as shown by solid lines.
Contribution of the phonons can be extracted by employing the time-gated scheme with a burst RF wave V ex for exciting phonons and a square voltage pulse V IPG for defining the measurement period just after the EM crosstalk has disappeared, as shown in the inset of Fig. 4c and Methods. The left peak, labeled SAW, in the current profile of Fig. 4c measures the phonon-assisted tunneling spectrum driven by phonons accumulated in the cavity. When the carrier frequency f is varied as shown, a clear splitting into peaks at ε ±  hf is observed only at the cavity's resonant frequency (the thick trace at zero offset ∆ = ) f 0 . This demonstrates that SAW phonons can be used as an excitation source for a charge qubit in a DQD. The splitting vanishes with a small offset ∆f  1 MHz, consistent with the narrow peak in Fig. 3b. This shows the enhancement of piezoelectric field in the cavity by noting that almost same RF power is introduced to the IDT in the frequency range.
We have proposed and demonstrated a SAW phonon cavity, in which electron-phonon coupling is enhanced. The enhancement should also be applied to the spontaneous emission process, known as the Purcell effect. The spontaneous phonon emission of a charge qubit in a similar GaAs DQD yields the rate Γ spon ( Γ = . − ħ 0 02 1 spon μeV) at the energy spacing of ε = 10-30 μeV 2,26 . If the DQD emits all phonons into the SAW cavity mode, the ultra-strong coupling regime with Γ ε ≈ spon can be reached by enhancing the phonon emission by a factor of 30-500. Such crude estimate encourages further study of cQED with SAW phonons.

Methods
The time-resolved phase-sensitive local potential measurement was performed with a PC as a local potential probe. Its conductance is adjusted in the tunneling regime (≈ 0.6 e 2 /h), where it is sensitive to the potential of the barrier. As shown in Fig. 2b, a burst RF wave V ex at the carrier frequency f is applied to the IDT to generate a burst SAW, which propagates along the sample surface and induces piezoelectric potential φ PC at i-th PC. This time-dependent potential is demodulated by applying another burst RF wave V det at the same f on the bias voltage of the PC. The resultant dc current (I det ) through the PC is proportional to the convolution of φ PC and V det . Monitoring I det while varying the time delay t d between V det and V ex provides a phase-sensitive potential measurement 28 . Generally a burst wave of N B cycles repeated for a period of N R cycles (N B and N R in the unit of 1/f ) involves spurious spectra of the spacing f/N R in the frequency range of ± / f f N B . We have carefully chosen N B and N R such that the spurious spectra does not affect the spectrum of interest. For data in Fig. 2c,e, a SAW packet of about 150 cycles is generated by applying V ex with N B = 50 to the left IDT with 100 metal pairs. The spatial spreading (≈ 120 μm) of the packet is shorter than the gap distance D = 160 μm to discriminate the right-and left-moving packets. For data in Fig. 3b, a long burst wave of N B = 2500 cycles is applied for accumulating phonons in the cavity mode, and is repeated for N R = 6000 cycles. The slowly-decaying SAW after the excitation is demodulated with a detection burst of 2500 cycles with delay time t d of 3000 cycles of 1/f. The time-gated phonon-assisted tunneling measurement is performed with an IPG for time gating. A burst RF wave of 1000 cycles is applied to the 10-pair IDT to accumulate phonons in the cavity. The square wave V IDT in the inset of Fig. 3c is designed to measure purely the phonon-assisted tunneling excited by the cavity phonons for the period labeled 'SAW' after the excitation burst is turned off. This was done by shifting the spectrum by 100 μeV in detuning with the square wave V IDT . Corresponding spectrum appears in the current profiles labeled 'SAW' . The additional spectrum in the profiles labeled 'EM+SAW' measure the excitation spectrum from both photons (EM) and phonons (SAW). The 1D COM simulation is performed to deduce the phonon band gap and the device parameters from the time-domain reflectometry in Fig. 2c and the reflection spectrum in Fig. 2e. In this model, the right-and left-moving SAWs are coupled by the scattering matrix with r, and the SAWs are coupled to the RF voltage and current in IDT with K 2 . A simple reflection coefficient for a continuous-wave SAW can be obtained by the standard frequency domain analysis for the total scattering matrix of the BR. Since a burst wave is employed in this work, the actual spectrum as well as the detector response involves the Fourier series of the burst wave. We took this effect into account in the simulations of Fig. 2d,f. We determined r = 0.01 and the damping (25% energy loss at each etching step) to reproduce the experimental data in Fig. 2c,e. The corresponding phonon band gap in an infinite metal array should be π  rf 100 MHz as shown by the arrow labeled Δ f BG . The multiple peaks in Device B. Similar 1D COM simulation for Device B suggests only one cavity mode in the band gap. The multiple peaks in Fig. 3b might be caused by the complicated gate patterns with inhomogeneous metal coverage. Since the velocity in the metalized region is reduced by 10% in our device, the resonant frequency should be affected by the wider gate patterns extended from the central region [See SEM micrograph with an identical design in Fig. 1b]. The excess metallized regions of 0.6 μm along the SAW propagation direction are sufficient to explain the spacing between peaks I and II. Two-dimensional mode analysis is desirable to identify multiple modes as well as to design a phononic crystal cavity.