Single-photon quantum regime of artificial radiation pressure on a surface acoustic wave resonator

Electromagnetic fields carry momentum, which upon reflection on matter gives rise to the radiation pressure of photons. The radiation pressure has recently been utilized in cavity optomechanics for controlling mechanical motions of macroscopic objects at the quantum limit. However, because of the weakness of the interaction, attempts so far had to use a strong coherent drive to reach the quantum limit. Therefore, the single-photon quantum regime, where even the presence of a totally off-resonant single photon alters the quantum state of the mechanical mode significantly, is one of the next milestones in cavity optomechanics. Here we demonstrate an artificial realization of the radiation pressure of microwave photons acting on phonons in a surface acoustic wave resonator. The order-of-magnitude enhancement of the interaction strength originates in the well-tailored, strong, second-order nonlinearity of a superconducting Josephson junction circuit. The synthetic radiation pressure interaction adds a key element to the quantum optomechanical toolbox and can be applied to quantum information interfaces between electromagnetic and mechanical degrees of freedom.


SUPPLEMENTARY NOTE 1: SAMPLE
The circuit was fabricated on a 500-µm-thick ST-X cut quartz substrate. The Bragg mirrors, the interdigitated transducers (IDTs), the nonlinear microwave (MW) resonator, and the coplanar waveguides for the external feed lines were simultaneously patterned in a wet-etching process from a 50-nm-thick evaporated aluminum film. The Bragg mirrors have 750 fingers each. The IDT for the external coupling has a pair of four fingers, and the IDT connected to the MW resonator has a pair of ten fingers (Fig. 1e in the main text). All those fingers have a width and a spacing of 1 µm. The length of the surface-acoustic-wave (SAW) resonator, the inner distance between the Bragg mirrors, is 240 µm. The widths of the Bragg mirrors and the IDTs are 500 µm. The Josephson junctions for the SNAIL are made from Al/AlO x /Al junctions, which are simultaneously fabricated by the shadow evaporation technique with the bridgeless resist mask. The size of the junctions are 150 × 150 nm for the small one and 300 × 300 nm for the large ones.
Supplementary Figure 1 shows the resonance frequency and the loss rates of the MW resonator as a function of the magnetic flux in the SNAIL loop. Note that the loss rates are periodically fluctuating depending on the flux bias. The periodic modulation is presumably caused by the resonant acoustic radiation from the MW resonator. The internal loss rate κ in is divided into the electric loss κ e and the acoustic radiation loss κ a from the MW resonator. The part of the acoustic radiation is picked up by the IDT electrode of the SAW resonator, and thus κ a = κ cross + κ rad , where κ cross is the external coupling rate of the MW resonator though acoustic waves to the SAW input port (port 3 in Fig. 1b) and κ rad is the acoustic radiation rate to the environment. Supplementary Figure 1d shows the acoustic external coupling rate κ cross of the MW resonator to the SAW input port (port 3 in Fig. 1b).

SUPPLEMENTARY NOTE 2: NONLINEAR RESONATOR WITH SNAIL
Our SNAIL has a single small junction and two large junctions. It is shunted with a large capacitor whose singleelectron charging energy E C is estimated to be h × 35 MHz. To determine the Josephson energies in the device, we fit the flux-dependent spectrum in Fig. 1g and obtain E ′ J = h × 47.5 GHz and E J = h × 163.5 GHz, respectively. Supplementary Figure 2 shows the inductive energy U (θ) of the SNAIL, given by Eq.(1) in the main text, in units E J . For Φ ̸ = 0, the parity symmetry is broken and the Pockels nonlinearity appears. The inductive energy is expanded around the minimum at θ 0 in a power series ofθ ≡ θ − θ 0 as The Hamiltonian of this nonlinear resonator in the transmon limit ( whereN is the number operator of the excess Cooper pairs in the superconducting electrode connected to the ground via the SNAIL. For the phase operatorθ, we omit the tilde for simplicity. This Hamiltonian can be rewritten with

the creation and annihilation operators aŝ
The Pockels (second-order) and self-Kerr (third-order) nonlinearities appear in Eq.(S3). This relates the circuit parameters to the coefficients of the nonlinear tems in Eq.
(2) of the main text as

SUPPLEMENTARY NOTE 3: ARTIFICIAL OPTOMECHANICAL COUPLING
The total Hamiltonian of the hybrid system consisting of a nonlinear MW resonator and a SAW resonator piezoelectrically coupled to each other is described without rotating wave approximation aŝ whereĤ The parameters and the operators are defined in the main text. g p is the piezoelectric coupling coefficient of the nonlinear MW resonator and SAW resonator.
By treatingV 0 as a perturbation, we find an effective Hamiltonian aŝ where δ = ω m −ω s is the detuning between the MW and SAW resonators. This calculation is valid when {ω m , ω s , δ} ≫ {|α 0 |, |β|, |g p |} is satisfied. While the second term on the right-hand side gives the self-Kerr nonlinearity, the third term leads to the radiation pressure interaction, and the fourth term introduces dynamical Casimir effect. When α 0 = 3β 2 /ω m , the self-Kerr nonlinearity vanishes, and the effective Hamiltonian is rewritten aŝ with the rotating wave approximation and a large detuning (δ ∼ ω m ).

SUPPLEMENTARY NOTE 4: LINEARIZED HAMILTONIAN
We irradiate the MW drive at frequency ω d , the annihilation operator of the MW resonator becomeŝ a → e −iω d t Ω +â (S13) and the interaction term becomesV where Ω is the complex amplitude of the MW drive. On the rotating frame with a unitary operator the Hamiltonian becomesĤ When the bandwidth of the MW resonator κ and the strength of the radiation pressure interaction g 0 are both smaller than ω s , we can apply the rotating approximation to eliminate the third term and obtain the linearized Hamiltonian.

SUPPLEMENTARY NOTE 5: SELF-KERR NONLINEARITY
We characterize the amount of the self-Kerr nonlinearity of the MW resonator by measuring the frequency shift as a function of the probe power. Supplementary Figures 4a and 4b show the frequency shift and the saturation of the absorption in the MW resonator at zero flux bias, respectively. To analyze the result, we solve the master equation of the resonator with the third-order nonlinearity and fit the experimental data. In the steady state, it fulfills whereρ is the density-matrix operator of the MW resonator,L is the Lindblad superoperator, and Here, A m is the attenuation through the input line of the MW feedline, and P m is the probe power at the input port outside the refrigerator. The saturation effect is highly nonlinear so that we can calibrate the absolute internal photon number with respect to the applied MW power. The strength of the self-Kerr nonlinearity and the attenuation in the input lines are determined from the fits as α 0 /2π = −13.0 MHz and −57.3 dB, respectively.

SUPPLEMENTARY NOTE 6: STARK SHIFT BY THE SAW EXCITATION
To calibrate the SAW input power, we measure the Stark shift of the MW resonator induced by the SAW excitation. Supplementary Figure 5 shows the Stark shift as a function of the phonon number in the SAW resonator at zero flux