Virtual-photon-mediated spin-qubit-transmon coupling

Abstract

Spin qubits and superconducting qubits are among the promising candidates for realizing a solid state quantum computer. For the implementation of a hybrid architecture which can profit from the advantages of either approach, a coherent link is necessary that integrates and controllably couples both qubit types on the same chip over a distance that is several orders of magnitude longer than the physical size of the spin qubit. We realize such a link with a frequency-tunable high impedance SQUID array resonator. The spin qubit is a resonant exchange qubit hosted in a GaAs triple quantum dot. It can be operated at zero magnetic field, allowing it to coexist with superconducting qubits on the same chip. We spectroscopically observe coherent interaction between the resonant exchange qubit and a transmon qubit in both resonant and dispersive regimes, where the interaction is mediated either by real or virtual resonator photons.

Fig. 1

Sample and qubit dispersions. a Schematic of sample and measurement scheme. The signals at frequencies ${\nu }_{\mathrm{p}}$ (probe) and ${\nu }_{\mathrm{d}}$ (drive) are routed with circulators as indicated by arrows. The reflected signal $I+\mathrm{i}Q$ at ${\nu }_{\mathrm{p}}$ is measured. The sample (dashed line) contains four quantum systems with transition frequencies ${\nu }_i$: a coupling resonator that consists of an array of SQUID loops (${\nu }_{\mathrm{C}}$, blue), an RX qubit (${\nu }_{\mathrm{RX}}$, red), a transmon (${\nu }_{\mathrm{T}}$, green) and a read-out resonator (${\nu }_{\mathrm{R}}$, gray). Empty black double-rectangles indicate electron tunnel barriers separating the three quantum dots (red circles) as well as the source (S) and drain (D) electron reservoirs. A drive tone at frequency ${\nu }_{\mathrm{dRX}}$ can be applied to one of the dots. Crossed squares denote the Josephson junctions of SQUIDs. Yellow arrows indicate the coupling between the quantum systems with coupling strengths $g_i$. ${\mathrm{\Phi }}_{\mathrm{C}}$ and ${\mathrm{\Phi }}_{\mathrm{T}}$ denote coupling resonator and transmon flux, respectively. b Two-tone spectroscopy of the transmon, with the RX qubit energetically far detuned. We plot the complex amplitude change $\mid \mathit{A}-{\mathit{A}}_0\mid$ (see main text) as a function of drive frequency ${\nu }_{\mathrm{d}}$ and ${\mathrm{\Phi }}_{\mathrm{T}}∕{\mathrm{\Phi }}_0$. The dashed line indicates ${\nu }_{\mathrm{T}}$ as obtained from the system Hamiltonian. c Scanning electron micrograph of the TQD and quantum point contact (QPC) region of the sample. Unused gate lines are grayed out. The gate line extending to the coupling resonator is highlighted in blue. d TQD energy level diagram indicating the tunnel couplings $t_{\mathrm{l}}$ and $t_{\mathrm{r}}$ and the electrochemical potentials, parametrized by $\mathrm{\Delta }$, of the relevant RX qubit states ($N_{\mathrm{l}}$,$N_{\mathrm{m}}$,$N_{\mathrm{r}}$) with $N_{\mathrm{l}}$ electrons in the left, $N_{\mathrm{m}}$ electrons in the middle and $N_{\mathrm{r}}$ electrons in the right quantum dot. e Two-tone spectroscopy of the RX qubit, with the transmon energetically far detuned for ${\nu }_{\mathrm{p}}\simeq {\nu }_{\mathrm{C}}=4.84\mathrm{GHz}$ as a function of $\mathrm{\Delta }$ and ${\nu }_{\mathrm{dRX}}$. The dashed line shows the expected qubit energy obtained from the Hamiltonian of the system

Fig. 2

Resonant interaction. The schematics at the top of the graphs indicate the energy levels of the RX qubit (${\nu }_{\mathrm{RX}}$), coupling resonator (${\nu }_{\mathrm{C}}$) and transmon (${\nu }_{\mathrm{T}}$). Theory curves in the absence (presence) of coupling are shown as dashed black (red) lines. a Reflected amplitude $\mid S_{11}\mid$ as a function of RX detuning $\mathrm{\Delta }$ and probe frequency ${\nu }_{\mathrm{p}}$ for RX qubit tunnel coupling configuration 2. b Reflected amplitude $\mid S_{11}\mid$ as a function of relative transmon (Tmon) flux ${\mathrm{\Phi }}_{\mathrm{T}}∕{\mathrm{\Phi }}_0$ and ${\nu }_{\mathrm{p}}$. The states $∣\pm ⟩$ are discussed in the main text. c Cuts from panel a at $\mathrm{\Delta }∕h\simeq -7.6\mathrm{GHz}$ (black) and from panel b at ${\mathrm{\Phi }}_{\mathrm{T}}∕{\mathrm{\Phi }}_0\simeq 0.3$ (green) as marked with arrows in the respective panels. The black trace is offset in $\mid S_{11}\mid$ by 0.1. Theory fits are shown as red dashed lines. d $\mid S_{11}\mid$ as a function of $\mathrm{\Delta }$ and ${\nu }_{\mathrm{p}}$ for RX qubit tunnel coupling configuration 2. The states $∣-,\pm ⟩$ and $∣+,\pm ⟩$ are explained in the main text. The black and blue arrows are referred to in f, the purple arrow is discussed in the text. e Result of master equation simulation for parameters as in d. The values for $\mid S_{11}\mid$ are scaled to the experimental data range in d. f $\mid S_{11}\left({\nu }_{\mathrm{p}}\right)\mid$ at $\mathrm{\Delta }∕h\simeq -9.8\mathrm{GHz}$ and $\mathrm{\Delta }∕h\simeq -5.6\mathrm{GHz}$ as marked with the corresponding colored arrows in panels d, e. The blue trace is offset in $\mid S_{11}\mid$ by 0.2

Fig. 3

RX qubit working points and virtual-photon-mediated interaction. a RX qubit decoherence rate ${\gamma }_{\mathrm{2,RX}}$ as a function of detuning $\mathrm{\Delta }$. The dotted vertical lines specify the four working points used in d–g. The corresponding colored data points were obtained for a coupling resonator-RX qubit detuning of ${\nu }_{\mathrm{C}}-{\nu }_{\mathrm{RX}}\simeq \left(13.7,8.0,5.1,4.4\right)g_{\mathrm{RX}}$ for $\mathrm{\Delta }∕h\simeq \left(-9.9,-3.3,3.4,10.2\right)\mathrm{GHz}$ and the RX qubit tunnel coupling configurations 3 (circle) and 4 (triangle). For the black data points, ${\nu }_{\mathrm{C}}-{\nu }_{\mathrm{RX}}\ge 9.7g_{\mathrm{RX}}$ with qubit tunnel coupling configuration 1 (circle) and 2 (triangle). The dashed red line is a fit of a model (see main text) to the black data points. Error bars indicate the standard error of fits and an estimated uncertainty of the RX qubit energy of $50\mathrm{MHz}$. b Ratio of $g_{\mathrm{RX}}$, as obtained from theory, and ${\gamma }_{\mathrm{2,RX}}$ as shown in a. The color and shape code of the data points is the same as in a. c Schematic of the measurement scheme. Bare qubit transitions (black arrows) are coupled by virtual photon excitations (red arrows) in the detuned coupling resonator (${∣0∕1⟩}_{\mathrm{C}}$ are the two lowest photon number states). The RX qubit is driven at frequency ${\nu }_{\mathrm{dRX}}$, the transmon is probed via the read-out resonator at frequency ${\nu }_{\mathrm{R}}$. d–g Two-tone spectroscopy at ${\nu }_{\mathrm{p}}={\nu }_{\mathrm{R}}\simeq 5.6\mathrm{GHz}$ as a function of $\mathrm{\Delta }$ and drive frequency ${\nu }_{\mathrm{dRX}}$. Dashed black (red) lines indicate transmon (Tmon) and RX qubit energies in the absence (presence) of coupling. The frame color refers to the RX qubit working points as specified in a. The inset in e shows the result from a master equation simulation with the same axes as the main graph. h Two-tone spectroscopy response from panels d–g at $\mathrm{\Delta }$ as specified with arrows in the corresponding panels. The cuts are centered around zero by accounting for a frequency offset ${\nu }_{\mathrm{dRX,0}}\equiv {\nu }_{\mathrm{dRX}}-\mathrm{\Delta }{\nu }_{\mathrm{dRX}}$. The dashed lines show the corresponding theory results

Fig. 4

Sample details. False-colored optical micrographs. a Relevant part of the sample (dashed region) false colored as follows: SQUID array (coupling) resonator in orange with corresponding drive and probe port in yellow, transmon in green, one end of the $50\mathrm{\Omega }$ (read-out) resonator in gray, GaAs in black and grounded Al in white. The position of the TQD (see Fig. 1c) is outlined with a red rectangle. The microwave lines that route probe and drive tones to the sample are shown. The frequencies ${\nu }_i$ are labeled as in Fig. 1a. b Magnified optical image of the transmon region in a. The positions of transmon SQUID and flux line with current $I$ are marked. c Optical micrograph of the $50\mathrm{\Omega }$ resonator and its microwave read-out line. The location of the region shown in a is outlined with a black rectangle