Universal Quantum Transducers Based on Surface Acoustic Waves

We propose a universal, on-chip quantum transducer based on surface acoustic waves in piezoactive materials. Because of the intrinsic piezoelectric (and/or magnetostrictive) properties of the material, our approach provides a universal platform capable of coherently linking a broad array of qubits, including quantum dots, trapped ions, nitrogen-vacancy centers, or superconducting qubits. The quantized modes of surface acoustic waves lie in the gigahertz range and can be strongly confined close to the surface in phononic cavities and guided in acoustic waveguides. We show that this type of surface acoustic excitation can be utilized efficiently as a quantum bus, serving as an on-chip, mechanical cavity-QED equivalent of microwave photons and enabling long-range coupling of a wide range of qubits.

Popular Summary

The realization of long-range interactions between remote qubits is arguably one of the greatest challenges in developing a scalable, solid-state quantum information architecture. Here, we propose and analyze quantum sound in the form of surface-acoustic-wave phonons in piezoactive materials as a universal mediator for long-range spin-spin couplings instead of photons. Surface acoustic waves occupy a middle ground between previously investigated electromagnetic (transmission lines) and mechanical (fixed resonators) coupling mechanisms and naturally combine the advantageous properties of both systems.

Because of the plethora of physical properties associated with surface acoustic waves, our approach is accessible to a broad class of systems such as quantum dots, trapped ions, nitrogen-vacancy centers, or superconducting qubits. We show that our proposed system also bears striking similarities to the established fields of cavity (circuit) quantum electrodynamics, opening up the possibility to implement the on-chip many-quantum communication protocols well known from the context of optical quantum networks. Furthermore, typical surface-acoustic-wave frequencies lie in the gigahertz range, closely matching the transition frequencies of artificial atoms and enabling ground-state cooling by conventional cryogenic techniques. Our theoretical predictions suggest that our proposed surface acoustic wave-based quantum-state-transfer protocol—for coupling qubits over large distances—can be realized using existing experimental technology and device dimensions on the order of micrometers.

We believe that the techniques and concepts of quantum optics and quantum information, in conjunction with the technological expertise of surface acoustic wave devices, are likely to lead to rapid theoretical and experimental progress in the field of quantum acoustics. In particular, hybrid surface-acoustic-wave architectures containing quantum dots, nitrogen-vacancy centers, and/or superconducting qubits may prove to be particularly robust.

Figure 1

SAW as a universal quantum transducer. Distributed Bragg reflectors made of grooves form an acoustic cavity for surface acoustic waves. The resonant frequency of the cavity is determined by the pitch $p$, $f_c=v_s/2p$. Reflection occurs effectively at some distance inside the grating; the fictitious mirrors above the surface are not part of the actual experimental setup, but are shown for illustrative purposes only. Red arrows indicate the relevant decay channels for the cavity mode: leakage through the mirrors, internal losses due to, for example, surface imperfections, and conversion into bulk modes. Qubits inside and outside of the solid can be coupled to the cavity mode. In more complex structures, the elastic medium can consist of multiple layers on top of some substrate.

Figure 2

Characterization of a groove-based SAW cavity. (a) Quality factor $Q$ for $N=100$ (dashed blue curve) and $N=300$ (red solid curve) grooves as a function of the normalized grove depth $h/{\lambda }_c$. For shallow grooves, $Q$ is limited by leakage losses due to imperfect acoustic mirrors ($Q_r$ regime, gray area), whereas for deep grooves conversion to bulk modes dominates ($Q_b$ regime); compare asymptotics (dash-dotted lines). (b) Ratio of desired to undesired decay rates ${\kappa }_{\mathrm{gd}}/{\kappa }_{\mathrm{bd}}$. The stronger $Q$ is dominated by $Q_r$, the higher ${\kappa }_{\mathrm{gd}}/{\kappa }_{\mathrm{bd}}$. Here, $w/p=0.5$, $D=5.25{\lambda }_c$, and $f_c=3\mathrm{GHz}$; typical material parameters for ${\mathrm{LiNbO}}_3$ are used (cf. Appendix pp5).

Figure 3

(a) Schematic illustration for coupling to a NV center via a piezomagnetic (PM) material (see text for details); surface grooves (not shown) can be used to provide SAW phonon confinement. (b) SAWs can be generated electrically based on standard interdigital transducers (IDTs) deposited on the surface. Typically, an IDT consists of two thin-film electrodes on a piezoelectric material, each formed by interdigitated fingers. When an ac voltage is applied to the IDT on resonance (defined by the periodicity of the fingers as ${\omega }_{\mathrm{IDT}}/2\pi =v_s/p_{\mathrm{IDT}}$, where $v_s$ is the SAW propagation speed), it launches a SAW across the substrate surface in the two directions perpendicular to IDT fingers [14, 15, 17].

Figure 4

(a) Average fidelity $\overline{\mathcal{F}}$ of the state transfer protocol for a coherent superposition $|\psi ⟩=(|0⟩-|1⟩)/\sqrt{2}$ in the presence of quasistatic (non-Markovian) Overhauser noise, as a function of the root-mean-square fluctuations ${\sigma }_{\mathrm{nuc}}$ in the detuning parameters ${\delta }_i(i=1,2)$, for ${\kappa }_{\mathrm{bd}}/{\kappa }_{\mathrm{gd}}=0$ (solid line, circles) and ${\kappa }_{\mathrm{bd}}/{\kappa }_{\mathrm{gd}}=10\%$ (dash-dotted line, squares). (b) After $n=100$ runs with random values for ${\delta }_i$, $\overline{\mathcal{F}}$ approximately reaches convergence. The curves refer to ${\sigma }_{\mathrm{nuc}}/{\kappa }_{\mathrm{gd}}=(0,2,\dots ,10)\%$ (from top to bottom) for ${\kappa }_{\mathrm{bd}}/{\kappa }_{\mathrm{gd}}=10\%$. (c) Pulse shape $g_1(t)$ for first node.

Figure 5

Dispersion relation ${\omega }_n=v_{n}k$ of the three $(n=1,2,3)$ Rayleigh-type SAW modes for propagation along ${\widehat{x}}^{\prime }\parallel [110]$. If not stated otherwise, we refer to the lowest frequency solution as the SAW mode (solid line).

Figure 6

Depth dependence of the (normalized) vertical displacement $u_z/U$ along ${\widehat{x}}^{\prime }\parallel [110]$ for a Rayleigh surface acoustic wave propagating on a (001) GaAs crystal. The acoustic amplitude decays away from the surface into the bulk on a characteristic length scale approximately given by the SAW wavelength $\lambda =2\pi /k\approx 1\mu \mathrm{m}$.

Figure 7

The dimensionless function $\mathcal{F}(z)$ determines the decay of the electrical potential away from the surface into the bulk; the characterisitc length scale is approximately set by the SAW wavelength $\lambda =2\pi /k\approx 1\mu \mathrm{m}$. Inset: Density plot of the (normalized) electric potential $\mathrm{Re}[\varphi ]/{\varphi }_0=-\mathcal{F}(kz)\sin (k{x}^{\prime }-\omega t)$ along ${\widehat{x}}^{\prime }\parallel [110]$ for a Rayleigh surface acoustic wave propagating on a (001) GaAs crystal at $t=0$.

Figure 8

Total reflection coefficient $|R|$ as a function of the normalized groove depth $h/{\lambda }_c$ for $N=100$ (blue dashed curve) and $N=300$ (red solid curve). Here, $w/p=0.5$, and material parameters for ${\mathrm{LiNbO}}_3$ are used (see text).

Figure 9

Illustration of the relevant electronic levels under consideration. The triplet levels with $S_{\mathrm{tot}}^z\ne 1$ can be tuned off resonance by applying a sufficiently large homogeneous magnetic field.

Figure 10

Spectrum of $H_0$ for $t_c=5\mathrm{\Delta }$. The three eigenstates $|l⟩$ are displayed in green dotted $(l=2)$, red solid $(l=1)$, and blue dashed $(l=0)$ lines, respectively. The triplets $|T_{\pm }⟩$ are assumed to be far detuned by a large external field and are not shown. For large negative detuning $\epsilon >-t_c$, the hybridized levels $\{|0⟩,|1⟩\}$ can be used as qubit.

Figure 11

The product ${\kappa }_0{\kappa }_1$ directly affects the effective single-phonon Rabi frequency $g_{\mathrm{QD}}/g_0={\kappa }_0{\kappa }_1$ [69], while the difference $|{\kappa }_1^2-{\kappa }_0^2|$ determines the robustness of the qubit against charge noise. Here, $t_c=5\mathrm{\Delta }$.

Figure 12

Exact numerical simulations of the full (blue, solid line) and approximate (cyan circles) master equations as given in Eqs. (g14) and (g15), respectively. Plots are shown for (a) the electronic inversion ${⟨S^z⟩}_t$, (b) the cavity occupation ${⟨n⟩}_t$, and (c) the error $\mathrm{Tr}[\rho |2⟩⟨2|]$ quantifying the leakage to $|2⟩$. The latter is found to be negligibly small $\sim {10}^{-5}$. We set $\overline{\delta }={\omega }_0-{\omega }_c=0$. Numerical parameters are $t_c=10\mu \mathrm{eV}$, $\epsilon =-7\mu \mathrm{eV}$, $\mathrm{\Delta }=1\mu \mathrm{eV}$, ${\eta }_{\mathrm{geo}}e{\phi }_0=5.2\times {10}^{-2}\mu \mathrm{eV}$, such that $g_{\mathrm{QD}}=4\times {10}^{-3}\mu \mathrm{eV}\approx 6\mathrm{MHz}$. The cavity decay rate is $\kappa =g_{\mathrm{QD}}/2$, corresponding to $Q\approx {10}^3$.

Figure 13

Transition frequency ${\omega }_0$ (blue solid line) and its sensitivity against charge-noise-induced fluctuations in $\epsilon$ for (a) intermediate and (b) large negative detuning; for large negative detuning, ${\omega }_0\approx 2\mathrm{\Delta }$, and the sensitivity $\partial {\omega }_0/\partial \epsilon$ practically vanishes, leaving nuclear noise as the dominant dephasing process. By occasional refocusing of the spin states, this regime can be used for long-term storage of quantum information [69]. Other numerical parameters: $t_c=5\mathrm{\Delta }$.

Figure 14

Success probability $p_{\mathrm{suc}}$ for a qubit excitation to leak through the mirror, as a function of the cooperativity $C$ for $ϵ={\kappa }_{\mathrm{bd}}/{\kappa }_{\mathrm{gd}}=0$ (black solid line) and $ϵ=5\%$ (blue dashed line).

Figure 15

Numerical simulation of state transfer for two different initial states ${|1⟩}_1$ (red solid line) and $({|0⟩}_1-{|1⟩}_1)/\sqrt{2}$ (blue dashed line) in the presence of Markovian noise. Black (dash-dotted) curves refer to the ideal, noise-free scenario $({\mathcal{L}}_{\text{noise}}=0)$, where perfect transfer is achieved, while colored curves take into account decoherence processes. (a) Transfer fidelity $\mathcal{F}$. (b) Excited-state occupation ${⟨S_i^{+}{S}_i^{-}⟩}_t$ for first and second qubit. Numerical parameters are $g_1(t\ge 0)={\kappa }_{\mathrm{gd}}$, ${\kappa }_{\mathrm{bd}}/{\kappa }_{\mathrm{gd}}=5\%$, and ${\mathrm{\Gamma }}_{\mathrm{deph}}/{\kappa }_{\mathrm{gd}}=5\%$.

Figure 16

Quantum state transfer protocol in the presence of Markovian noise. State transfer fidelity $\mathcal{F}$ for a coherent superposition $|\psi ⟩=(|0⟩-|1⟩)/\sqrt{2}$ as a function of the qubit’s dephasing rate ${\mathrm{\Gamma }}_{\mathrm{deph}}$ for different values of intrinsic phonon losses ${\kappa }_{\mathrm{bd}}/{\kappa }_{\mathrm{gd}}=0$ (black solid line), ${\kappa }_{\mathrm{bd}}/{\kappa }_{\mathrm{gd}}=5\%$ (blue dashed line), and ${\kappa }_{\mathrm{bd}}/{\kappa }_{\mathrm{gd}}=10\%$ (red dash-dotted line). Inset: Pulse shape $g_1(t)$ for first node.