Spin-Phonon-Photon Strong Coupling in a Piezoelectric Nanocavity

Solid-state quantum systems based on superconductors and spins are leading platforms that offer complementary advantages in quantum computing and networking. Superconducting quantum processors enable fast and high-fidelity entangling gates [1, 2], but challenges remain in quantum memory time and long-distance networking. Conversely, atom-like emitters in solid-state have demonstrated long spin coherence time, efficient spin-photon interfaces for long-distance entanglement, and high readout fidelity [3, 4, 5, 6, 7, 8]. Coupling these modalities is therefore an exciting direction in quantum information science.

Previous studies using magnetic coupling between microwave (MW) photons and spins have been limited to multi-spin ensemble interactions [9, 10, 11, 12, 13, 14, 15] due to low spin-magnetic susceptibility and the low magnetic energy density of MW resonators [16, 17, 18]. Alternate experiments and proposals rely on coupling via intermediate acoustic modes [19, 20, 21], which have experimentally demonstrated large coupling to superconducting circuits [22, 23, 24, 25, 26] and are predicted to have large coupling to diamond quantum emitters [27, 28, 29, 30, 31, 32, 33]. However, designing a device that strongly couples one phonon to both one MW photon and to one spin – enabling an efficient MW photon-to-spin interface – remains an outstanding challenge.

Here we address this problem through the co-design of a scandium-doped aluminum nitride (ScAlN) Lamb wave resonator with a heterogeneously-integrated diamond thin film. This structure piezoelectrically couples a MW photon and acoustic phonon while concentrating strain at the location of a diamond quantum emitter. Through finite-element modeling, we predict photon-phonon coupling $\sim 10$ MHz concurrent with phonon-spin coupling $\sim 3$ MHz. These rates yield photon-phonon and phonon-spin cooperativities of order $10^{4}$ assuming demonstrated lifetimes of spins, mechanical resonators, and superconducting circuits [34, 35]. We explore state transfer protocols via quantum master equation (QME) simulations and show that this device can achieve photon-to-spin transduction fidelity $F>0.97$ with conservative hardware parameters. We find that performance of these schemes is likely limited by two-level system (TLS) loss in current piezoelectrics. An improvement in piezoelectric TLS loss rates to that of silicon will pave the way towards SC-spin state transduction with $F>0.99$.

We consider a coupled tripartite system consisting of a superconducting circuit (SC), acoustic phonon, and Group-IV electron spin with the Hamiltonian (see [36] for detailed derivation)

Here, the SC frequency $\omega_{sc}$ is defined by the transmon Josephson and shunt capacitances, the spin frequency $\omega_{e}$ is given by the Zeeman splitting of the electron spin states, and the acoustic frequency $\omega_{p}$ is defined by the acoustic resonator geometry. The first three terms of this equation describe the energies of the uncoupled modes of the devices (Fig. 1(a-c)) while the fourth and fifth terms describe the interaction dynamics. Generally, SCs feature $\omega_{sc}\sim 4-6$ GHz [37]. Electron spin resonant frequencies can be arbitrarily set by an external magnetic field; to match this frequency range, fields $\sim 0.1$ T are required [38]. The coupling coefficient $g_{sc,p}$ is physically governed by the piezoelectric effect, whereby a strain field produces an electric response and vice versa (Fig. 1(d)). This interaction is described by the strain-charge equations

where $s_{ijkl}$ and $d_{ijk}$ are the elastic and piezoelectric coefficient tensors of the resonator’s piezoelectric material, $S_{ij}$ and $T_{ij}$ are the stress and strain fields, and $E_{i}$ and $D_{i}$ are the electric and displacement fields. At single quantum levels, a MW photon in the SC will generate an electric field in the volume of the piezoelectric resonator described by

where $\textbf{E}_{IDT}(\textbf{r})$ is the electric field profile of the IDT for an arbitrary applied voltage $V_{app}$, and the capacitances are indicated in Fig. 1. Since $C_{S}$ is typically much larger than $C_{IDT}$ and $C_{J}$ for transmon qubit configurations, the MW photon energy is largely contained in the shunt capacitor. Similarly, a phonon in the piezoelectric resonator will produce a strain field described by

where $\textbf{T}_{p}(\textbf{r})$ is the strain profile of the acoustic mode for an arbitrary mechanical displacement. Following (5), $\textbf{t}_{p}(\textbf{r})$ will produce an electric displacement field given by $\textbf{d}\cdot\textbf{t}_{p}(\textbf{r})$, where d is the piezoelectric coefficient tensor. Then the coupling $g_{sc,p}$ will be determined by the overlap integral between $\textbf{e}_{sc}(\textbf{r})$ and $\textbf{d}\cdot\textbf{t}_{p}(\textbf{r})$ [39],

The spin-phonon coupling $g_{p,e}$ results from the spin-strain susceptibility $\mathbf{\chi}_{spin}$ of quantum emitters in a strain field [38, 40, 41]. For a single-phonon strain profile $\textbf{t}_{p}$, the resulting coupling is $g_{spin}(\textbf{r})=\mathbf{\chi}_{spin}\cdot\textbf{t}_{p}(\textbf{r})$. In Group IV emitters in diamond, $\chi_{spin}$ depends heavily on the spin-orbit mixing enabled by an off-axis magnetic field (see [36]) and primarily interacts with transverse strain in the emitter frame [38]. Therefore, for the rest of this analysis, we set this expression to be

where $\textbf{t}^{\prime}(\textbf{r})$ is the single-phonon strain profile in the coordinate system of the emitter and $\chi_{eff}\approx 0.28$ PHz/strain [41].

To implement the device in Fig.1, we require a platform with (i) superconductivity, (ii) piezoelectricity, (iii) acoustic cavities, and (iv) strain transfer to diamond emitters. To address (i-ii), we propose a silicon-on-insulator (SOI) platform with a thin-film deposition of scandium-doped aluminum nitride (ScAlN). This material system allows for superconducting qubits and piezoelectrics to co-inhabit one chip [42, 43]. To answer (iii-iv), we co-design a Nb-on-Sc_0.32Al_0.68N-on-SOI piezoelectric resonator with a heterogeneously integrated diamond thin membrane. We propose Niobium (Nb) as a well-characterized superconductor with high $H_{c1}=0.18$ T and $H_{c2}=2$ T [44, 45, 46], as required for operation with the spin. SOI platforms have previously been used for piezoelectric resonators [47, 48], and diamond-AlN interfaces have been used to acoustically drive emitters in diamond [49, 50, 51]. ScAlN further boosts the piezoelectric coefficient of AlN, allowing us to achieve a stronger interaction [52, 53].

We present the resonator design in Fig. 2. Our device is based on Lamb wave resonators, which produce standing acoustic waves dependent on electrode periodicity $\lambda$ and material thickness [54, 55, 56]. We localize the strain in the diamond thin film using a central “defect cell” (Fig. 2a inset) with a suspended taper. To maintain high quality factors, we tether the Lamb wave resonator via phononic crystal tethers placed at displacement nodes of the box. [57]. We further propose an angled ScAlN sidewall in the transducer (15^∘ from normal) that allows the electrodes to ”climb” on top of the ScAlN film, rather than requiring a continuous piezoelectric layer over the phononic tethers. This both facilitates the design of wide-bandgap phononic tethers and is compatible with current fabrication techniques.

To calculate device performance, we simulate the architecture using the finite element method (FEM) in COMSOL to produce the phononic tether band structures and mode profiles (Fig. 2b-e). The tether band structure exhibits a 500 MHz bandgap around the device’s $\approx$ 4.11 GHz resonant mode. This frequency is desirable as it falls near the central operating range of most superconducting qubits [37]. Additionally, the 4.11 GHz resonant mode is itself isolated from other acoustic modes of the system by $\sim$56 MHz, which is enough to neglect parastic couplings and treat the transducer in the single-mode approximation (see SI). Fig. 2(d-e) show the mechanical and electrical displacement fields of this mode, from which we derive $\mathbf{e}_{sc}(\mathbf{r})$ and $\mathbf{t}_{p}(\mathbf{r})$, respectively. We calculate a $g_{sc,p}\approx 7.0-20.5$ MHz (for a shunt capacitance of 65-190 fF, corresponding to $100\text{ MHz}<E_{C}/h<300\text{ MHz}$[37]) and a maximum $g_{p,e}\approx 3.2$ MHz according to Equations (8) and (9). The strain maximum occurs at the edges of the central diamond taper, which maximizes $g_{p,e}$ (Fig. 2f).

In Fig. 3, we explore different protocols for quantum transduction from an initialized SC to a spin. The time evolution of the system when initialized in the $\rho_{0}=\ket{100}\bra{100}$ state (where the indices consecutively refer to the state of the SC, the Fock state of the phonon, and the z-projection of the spin) is calculated using the Lindblad master equation,

where the Hamiltonian in a frame rotating at rate $\omega_{p}$ is

Here, $\Delta_{sc}(t)\equiv\omega_{sc}(t)-\omega_{p}$ is the superconducting qubit detuning and $\Delta_{e}(t)\equiv\omega_{e}(t)-\omega_{p}$ is the spin detuning at time $t$. The use of time-varying detuning can be easily implemented, e.g. via on-chip flux bias lines [58, 59, 60], unlike time-varying coupling rates explored in previous works [20]. We account for dephasing in each mode with conservative estimates on decoherence rates $\frac{\kappa_{sc}}{2\pi}=100$ kHz, $\frac{\kappa_{p}}{2\pi}=\frac{\omega_{p}}{2\pi Q}\approx 40$ kHz, and $\frac{\kappa_{e}}{2\pi}=1$ MHz [35, 34, 61, 62, 63]. As cryogenic operation of ScAlN-on-SOI acoustic resonators–as well as diamond hybrid intergration on said devices–has not been previously explored, we further discuss prospects for $Q_{mech}$ below.

Fig. 3(b,e,h) plot the state transfer fidelity $F_{i}\equiv\bra{\psi_{i}}\rho(t)\ket{\psi_{i}}$ to the target state $\ket{\psi_{i}}=\ket{1_{i}}$ under different conditions. In Fig. 3a where the modes are all resonant ($\omega_{sc}=\omega_{p}=\omega_{e}=4.11$ GHz), and $g_{sc,p}/2\pi=10$ MHz, $F_{e}$ is poor due to the mismatch $\Delta g(g_{p,e})=g_{sc,p}-g_{p,e}$ (Fig. 3c). Assuming one reduces $g_{sc,p}$ or $g_{p,e}$, for example by increase the qubit shunt capacitance $C_{S}$ or reducing the transverse magnetic field, $F_{e}$ may increase at the cost of maximum coupling rates.

In Fig. 3b we detune the phonon mode by $\Delta_{p}\equiv\omega_{p}-\omega_{sc}$ where $\omega_{sc}=\omega_{e}$ and keep the coupling rates matched at 3.0 MHz. In this case, $F_{e}\sim 0.95$ via virtual excitation of the phonon mode, if the phonon mode is detuned by $30$ MHz. This protocol generates very low population in the phonon mode, primarily exchanging states between the superconducting qubit and spin. If the phonon mode is lossy, this transduction method is then preferred. However, while this protocol features wider efficiency peaks in time, which may require less stringent pulse control (see Fig. 3e), it does not overcome the issue of coupling imbalance and additionally suffers from decoherence of the superconducting qubit and spin modes over a longer protocol time (Fig. 3f).

Fig. 3g shows the optimal solution, assuming control over $\Delta_{sc}(t)$ and $\Delta_{e}(t)$, in a double Rabi-flop protocol. During this protocol, it is assumed that $g_{sc,p}/2\pi=10$ MHz (which overcomes losses during the Rabi flop while still allowing mode isolation during the next flop) and $g_{p,e}/2\pi=3.0$ MHz. We also assume $\Delta_{sc}(t)=0$ and $0\text{ MHz}\leq\Delta_{e}(t)\leq 1$ GHz for $t\in\{0,\pi/(2g_{sc,p})\}$–the duration of a Rabi flop between the SC and phonon. Then, $\Delta_{sc}(t)=\Delta_{e}(t=0)$ MHz and $\Delta_{e}(t)=0$ for $t\in\{\pi/(2g_{sc,p}),\pi/(2g_{sc,p})+\pi/(2g_{p,e})\}$–the duration of a Rabi flop between the phonon and spin. This sequentially transfers states between the modes (Fig. 3h), and for $\Delta_{e}(t=0)>500$ MHz, can achieve $F_{e}>0.97$ (Fig. 3i; for $\Delta_{i}=1.0$ GHz, $F_{e}=0.971$). In this protocol, we have neglected the losses that can occur when varying $\Delta_{sc}$ and $\Delta_{e}$. In reality, one has to select a pair of $\Delta_{sc}$ and $\Delta_{e}$ that do not fall on resonance with another acoustic mode of the system to prevent Rabi oscillations between the SC qubit or electron spin and an undesired acoustic mode (see SI for more details).

Each of these scenarios achieves transduction to the spin with high fidelity. The third scenario allows the quantum state to persist in the spin without continued interaction with the acoustic or SC modes. While in this state, the electron spin can access other degrees of freedom (e.g. ${}^{13}C$ spins [64, 65]).

Since acoustic losses and therefore the total mechanical quality factor $Q_{mech}$ are difficult to predict from first principles, we evaluate the transduction fidelity $F_{e}$ of each protocol in different regimes of $Q_{mech}$ in Fig. 4. Here, protocol 1 is the resonant protocol with $g_{sc,p}=g_{p,e}=3$ MHz; protocol 2 is the virtual excitation protocol with identical $g_{sc,p}$ at a detuning of $30$ MHz; and protocol 3 is the Rabi protocol with $g_{sc,p}=10$ MHz, $g_{p,e}=3$ MHz, and $\Delta_{e}(t=0)=1$ GHz. $Q_{mech}$ is the inverse sum of three components,

Here, $Q_{c}$ is the mechanical clamping loss, which we can engineer to be non-limiting (see [36]), and $Q_{A}$ is the Akhieser loss-related $Q$, which at millikelvin temperatures is negligible [66]. These two losses are well-described for analogous systems; in contrast, $Q_{TLS,i}$–the dielectric loss-related $Q$–is harder to predict. These $Qs$ depends on the number of quasi-particles or TLSs trapped in each of the device’s material interfaces and are weighted by the electric field participation $p_{i}$ in each interface. Given this uncertainty in $Q_{TLS_{i}}$, we lay out the protocol hierarchy as a function of the overall $Q_{mech}$:

• •

  If $Q_{mech}\lesssim 2\times 10^{3}$, protocol 2 is superior.

• •

  If $2\times 10^{3}\lesssim Q_{mech}\lesssim 5\times 10^{5}$, protocol 1 is superior.

• •

  If $5\times 10^{5}\lesssim Q_{mech}$, protocol 3 is superior.

In existing hardware, the largest challenge to reach the high-fidelity regime ($F\gtrsim 0.99$) is reducing dielectric loss in the thin-film piezoelectric, as indicated by published intrinsic quality factors of, e.g., monolithic aluminum nitride or lithium niobate resonators [67, 68]. So, while current hardware may encourage us to utilize the virtual coupling protocol for coupling through a lossy intermediary phononic mode, future iterations of this scheme with improved materials and interfaces can expect to break the 0.99 transduction fidelity barrier using a resonant protocol. At this fidelity, SC-spin transduction would surpass the 1% error correction thresholds of common codes and thus be compatible with scalable quantum information processing schemes [69, 70, 71].

An open question remains in the bonding strength between the diamond thin film and underlying resonator, which, if poor, can incur additional losses. However, for single-phonon occupation, the Van der Waals static frictional force exceeds the strain-generated force on the resonator.

Ultimately, we have proposed a resonator architecture capable of simultaneously coupling a microwave photonic mode from a superconducting circuit and an electronic spin from a solid state color center to a single phonon. For our calculated coupling parameters and conservatively assumed $Q$s across the three modes, we expect SC-phonon cooperativity $C_{sc,p}=\frac{4g_{sc,p}^{2}}{\kappa_{sc}\kappa_{p}}\sim 4\times 10^{3}$ and similarly, spin-phonon cooperativity $C_{p,e}=\frac{4g_{p,e}^{2}}{\kappa_{p}\kappa_{e}}\sim 10^{2}$. This doubly strongly-coupled architecture has a number of uses. Firstly, it can provide superconducting circuit qubits access to a long-lived quantum memory in the form of a nuclear spin register surrounding the electron spin. Secondly, this resonator can grant superconducting circuit qubits a spin-photon interface for efficient coupling to fiber optical quantum networks. Finally, by multiplexing each SC with several acoustic resonators and each acoustic resonator with several spins, this architecture can yield a memory bank of quantum memories for computational superconducting circuits (see [36] for a more detailed discussion). We believe introducing this quantum transducer into existing superconducting circuits is a large step towards developing a specialized hybrid quantum computer with fast superconducting qubits for processing and slow, long-lived memory qubits in the solid state for storage and communication.