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.

Introduction

A future quantum processor will benefit from the advantages of different qubit implementations¹. Two prominent workhorses of solid state qubit implementations are spin- and superconducting qubits. While spin qubits have a high anharmonicity, a small footprint² and promise long coherence times^3,4,5, superconducting qubits allow fast and high fidelity read-out and control^6,7. A coherent link, which couples both qubit systems controllably over distances exceeding the physical size of the spin qubit, typically hundreds of nanometers, by several orders of magnitude is required to create an integrated scalable quantum device. An architecture to provide such a link is circuit quantum electrodynamics (circuit QED)⁸, where microwave photons confined in a superconducting resonator couple coherently to the qubits. Circuit QED was initially developed for superconducting qubits⁹, where long-distance coupling^10,11 enables two-qubit gate operations¹². Recently, coherent qubit-photon coupling was demonstrated for spin qubits^13,14,15 in few electron quantum dots. However, coupling a spin qubit to another distant qubit^16,17 has not yet been shown. One major challenge for an interface between spin and superconducting qubits^{\(| {S}_{11}|\) reflectance spectrum shown in Fig. 2a and extract a spin qubit-photon coupling strength of ${g}_{{\rm{RX}}}/2\pi =52\ {\rm{MHz}}$ from a fit to the vacuum Rabi mode splitting shown in black in Fig. 2c. This coupling strength enhancement compared to earlier work in ref. 15 is an important ingredient to realize coherent virtual-photon-mediated qubit–qubit interaction, which is weaker compared to resonant qubit-photon interaction. The enhancement is related to an enhanced characteristic impedance of the resonator, to the position of the qubit at the open end of the resonator, where the voltage vacuum fluctuations are maximal, as well as to an optimized TQD gate design with an increased overlap of the resonator gate with the underlying quantum dot. The spin qubit and the coupling resonator photons are strongly coupled since ${g}_{{\rm{RX}}}\, > \, {\kappa }_{{\rm{C}}},{\gamma }_{2,\rm{RX}}$, with the RX qubit decoherence rate ${\gamma }_{{\rm{2,RX}}}/2\pi =11\ {\rm{MHz}}$ and the bare coupling resonator linewidth ${\kappa }_{{\rm{C}}}/2\pi =4.6\ {\rm{MHz}}$. The decoherence rate is determined independently with power dependent two-tone spectrosopy. We dispersively detune the coupling resonator with ${\it{\Phi} }_{{\rm{C}}}$ from the RX qubit and extrapolate the width of the peak observed in the two-tone spectroscopy response (c.f. Fig. 1e) to zero drive power33.}

Fig. 2

Resonant interaction. The schematics at the top of the graphs indicate the energy levels of the RX qubit (${\nu }_{{\rm{RX}}}$), coupling resonator (${\nu }_{{\rm{C}}}$) and transmon (${\nu }_{{\rm{T}}}$). Theory curves in the absence (presence) of coupling are shown as dashed black (red) lines. a Reflected amplitude $| {S}_{11}|$ as a function of RX detuning $\it \Delta$ and probe frequency ${\nu }_{{\rm{p}}}$ for RX qubit tunnel coupling configuration 2. b Reflected amplitude $| {S}_{11}|$ as a function of relative transmon (Tmon) flux ${\it{\Phi} }_{{\rm{T}}}/{\it{\Phi} }_{0}$ and ${\nu }_{{\rm{p}}}$. The states $\left|\pm \right\rangle$ are discussed in the main text. c Cuts from panel a at ${\it{\Delta}} /h\simeq -7.6\ {\rm{GHz}}$ (black) and from panel b at ${\it{\Phi} }_{{\rm{T}}}/{\it{\Phi} }_{0}\simeq 0.3$ (green) as marked with arrows in the respective panels. The black trace is offset in $| {S}_{11}|$ by 0.1. Theory fits are shown as red dashed lines. d $| {S}_{11}|$ as a function of ${\it{\Delta}}$ and ${\nu }_{{\rm{p}}}$ for RX qubit tunnel coupling configuration 2. The states $\left|-,\pm \right\rangle$ and $\left|+,\pm \right\rangle$ 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 $| {S}_{11}|$ are scaled to the experimental data range in d. f $| {S}_{11}({\nu }_{{\rm{p}}})|$ at ${\it{\Delta}} /h\simeq -9.8\ {\rm{GHz}}$ and ${\it{\Delta}} /h\simeq -5.6\ {\rm{GHz}}$ as marked with the corresponding colored arrows in panels d, e. The blue trace is offset in $| {S}_{11}|$ by 0.2

Next, we characterize the interaction between the transmon and the coupling resonator. We tune the transmon through the resonator resonance by swee** ${\it{\Phi} }_{{\rm{T}}}$. For this measurement, the RX qubit is far detuned in energy. We resolve the hybridized states of the transmon and the resonator photons in the measured $| {S}_{11}|$ spectrum in Fig. 2b. They are separated in energy by the vacuum Rabi mode splitting $2{g}_{{\rm{T}}}/2\pi =360\ {\rm{MHz}}$ illustrated in Fig. 2c in green. We perform power dependent two-tone spectroscopy to extract the transmon linewidth by probing the read-out resonator. We obtain ${\gamma }_{2,{\rm{T}}}/2\pi =0.7\ {\rm{MHz}}$, which we estimate to be limited by Purcell decay^42,43. Consequently, the strong coupling limit ${g}_{{\rm{T}}}\, > \, {\kappa }_{{\rm{C}}},{\gamma }_{2,\rm{T}}$ is also realized for transmon and coupling resonator.

We now demonstrate that the two qubits interact coherently via resonant interaction with the coupling resonator. For this purpose, we first tune the transmon and the coupling resonator into resonance, where the hybrid system forms the superposition states $\left|\pm \right\rangle =(\left|{0}_{{\rm{T}}},{1}_{{\rm{C}}}\right\rangle \pm \left|{1}_{{\rm{T}}},{0}_{{\rm{C}}}\right\rangle )/\sqrt{2}$ of a single excitation in either the resonator or the qubit. Then, we sweep ${\it{\Delta}}$ to tune the RX qubit through a resonance with both the lower energy state $\left|-\right\rangle$ and the higher energy state $\left|+\right\rangle$. In the $| {S}_{11}|$ spectrum in Fig. 2d, avoided crossings are visible at both resonance points. This indicates the coherent interaction of the three quantum systems which form the states $\left|-,\pm \right\rangle$ and $\left|+,\pm \right\rangle$, where the second label indicates a symmetric or antisymmetric superposition of the RX qubit state with the transmon-resonator $\left|\pm \right\rangle$ states. The splitting $2{g}_{{{\mp }}}$ between $\left|-,\pm \right\rangle$ and $\left|+,\pm \right\rangle$ is extracted from the $| {S}_{11}|$ reflection measurements in Fig. 2f. We obtain $2{g}_{+}/2\pi =84\ {\rm{MHz}}$ at ${\it{\Delta}} /h\simeq -5.6\ {\rm{GHz}}$ and $2{g}_{-}/2\pi =63\ {\rm{MHz}}$ at ${\it{\Delta}} /h\simeq -9.8\ {\rm{GHz}}$ from the fits in Fig. 2f. The smaller ${g}_{-}$ compared to ${g}_{+}$ is due the decrease of the RX qubit dipole moment with more negative ${\it{\Delta}}$. The RX qubit, the transmon and the resonator are on resonance (${\nu }_{{\rm{RX}}}={\nu }_{{\rm{T}}}={\nu }_{{\rm{C}}}$) between the avoided crossings in Fig. 2d at ${\it{\Delta}} /h\simeq -7.8\ {\rm{GHz}}$ (see purple arrow). There, the splitting of the dips in the reflection spectrum is enhanced by $\approx 16\ {\rm{MHz}}$ compared to the off-resonant splitting of $\left|\pm \right\rangle$ at ${\it{\Delta}} /h\approx -11.4\ {\rm{GHz}}$ in Fig. 2d (see Supplementary Note 1). This enhancement is an experimental signature of the coherent resonant interaction of all three quantum systems in good agreement with the theoretical value $(2{g}_{{\rm{T}}}-2\sqrt{{g}_{{\rm{T}}}^{2}+{g}_{{{\rm{RX}}}}^{2}})/2\pi \approx 15\ {\rm{MHz}}$ calculated from independently extracted parameters.

The experimental observation in Fig. 2d is well reproduced by a quantum master equation simulation shown in Fig. 2e and further discussed in Supplementary Note 2.

RX qubit optimal working point

While ${\gamma }_{{\rm{2,T}}}$ is limited by Purcell decay and therefore does not depend on ${\it{\Phi} }_{{\rm{T}}}$, ${\gamma }_{{\rm{2,RX}}}$ changes with ${\it{\Delta}}$¹⁵. For obtaining the data shown in Fig. 3a we use power dependent two-tone spectroscopy via the coupling resonator to measure ${\gamma }_{{\rm{2,RX}}}$ as a function of ${\it{\Delta}}$. We observe an increase of ${\gamma }_{{\rm{2,RX}}}$ as the charge character of the qubit is increased with ${\it{\Delta}}$. Compared to ref. ¹⁵, the data in Fig. 3a covers a larger range in ${\it{\Delta}}$, in particular for $| {\it{\Delta}} | \gg {t}_{{\rm{l,r}}}$. The data suggests a lower limit of ${\gamma }_{{\rm{2,RX}}}/2\pi \simeq 6.5\ {\rm{MHz}}$ for ${\it{\Delta}} \ll 0$. This is in agreement with refs. ⁴⁴ and ¹⁵, where the RX qubit was operated at a finite magnetic field of a few hundred mT. Hence, our experiment indicates that the RX qubit can be operated near zero magnetic field without reducing its optimal coherence. In our experiment, the maximum external magnetic field determined by ${\it{\Phi} }_{{\rm{C}}}$ is of the order of $1\ {\rm{mT}}$. To model the RX qubit decoherence in Fig. 3a, we consider an ohmic spectral density for the charge noise as well as the hyperfine field of the qubit host material that acts on the spin part of the qubit (see Supplementary Notes 2 and 3). Theory and experiment in Fig. 3a match for a width ${\sigma }_{\rm{B}}=3.48\ {\rm{mT}}$ of the hyperfine fluctuations in agreement with other work on spin in GaAs^45,46,47. This suggests that ${\gamma }_{{\rm{2,RX}}}$ is limited by hyperfine interactions.

Fig. 3

RX qubit working points and virtual-photon-mediated interaction. a RX qubit decoherence rate ${\gamma }_{{\rm{2,RX}}}$ as a function of detuning ${\it{\Delta}}$. 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 ${\nu }_{{\rm{C}}}-{\nu }_{{\rm{RX}}}\simeq (13.7,8.0,5.1,4.4){g}_{{\rm{RX}}}$ for ${\it{\Delta}} /h\simeq (-9.9,-3.3,3.4,10.2)\ {\rm{GHz}}$ and the RX qubit tunnel coupling configurations 3 (circle) and 4 (triangle). For the black data points, ${\nu }_{{\rm{C}}}-{\nu }_{{\rm{RX}}}\ge 9.7\ {g}_{{\rm{RX}}}$ 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 $50\ {\rm{MHz}}$. b Ratio of ${g}_{{\rm{RX}}}$, as obtained from theory, and ${\gamma }_{{\rm{2,RX}}}$ 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 (${\left|0/1\right\rangle }_{{\rm{C}}}$ are the two lowest photon number states). The RX qubit is driven at frequency ${\nu }_{{\rm{dRX}}}$, the transmon is probed via the read-out resonator at frequency ${\nu }_{{\rm{R}}}$. d–g Two-tone spectroscopy at ${\nu }_{{\rm{p}}}={\nu }_{{\rm{R}}}\simeq 5.6\ {\rm{GHz}}$ as a function of ${\it{\Delta}}$ and drive frequency ${\nu }_{{\rm{dRX}}}$. 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 ${\it{\Delta}}$ as specified with arrows in the corresponding panels. The cuts are centered around zero by accounting for a frequency offset ${\nu }_{{\rm{dRX,0}}}\equiv {\nu }_{{\rm{dRX}}}-{\it{\Delta}} {\nu }_{{\rm{dRX}}}$. The dashed lines show the corresponding theory results

The colored data points in Fig. 3a were measured for a smaller RX qubit-coupling resonator detuning compared to the black data points (numbers are given in Fig. 3 caption). The smaller detuning is used for the virtual interaction measurements discussed below. We observe an increase of ${\gamma }_{{\rm{2,RX}}}$ for small qubit-resonator detuning compared to large detuning. This increase is about one order of magnitude larger than our estimated difference of Purcell decay and measurement induced dephasing for those different data sets (see Supplementary Note 4). In contrast, for the transmon that is insensitive to charge noise, we do not observe this effect. This suggests that the effect is due to charge noise induced by the coupling resonator.

As ${\gamma }_{{\rm{2,RX}}}$ increases with ${\it{\Delta}}$ in Fig. 3a, the RX qubit coupling strength ${g}_{{\rm{RX}}}$ to the coupling resonator increases. This implies the existence of an optimal working point for the RX qubit, where ${g}_{{\rm{RX}}}/{\gamma }_{{\rm{2,RX}}}$ is maximal. While a distinct optimal point is not discernible for the black data points in Fig. 3b, the averaged value of ${g}_{{\rm{RX}}}/{\gamma }_{{\rm{2,RX}}}\simeq 9$ in the spin dominated regime for $-6\, <\, {\it{\Delta}} /h\, <\, 0\ {\rm{GHz}}$ is about a factor of $1.7$ larger than values reported so far for Si spin qubits^13,14. In contrast, for the colored data points we observe an optimal working point at ${\it{\Delta}} /h\simeq -3.3\ {\rm{GHz}}$ since ${g}_{{\rm{RX}}}/{\gamma }_{{\rm{2,RX}}}$ is reduced at small qubit-resonator detuning in Fig. 3b compared to the black data points at large detuning due to the influence of the coupling resonator on ${\gamma }_{{\rm{2,RX}}}$ discussed above.

Virtual photon coupling

In the following, we investigate the RX qubit-transmon interaction mediated by virtual photons in the coupling resonator at the RX qubit working points marked in color in Fig. 3a. The two qubits are resonant while the coupling resonator is energetically detuned, such that the photon excitation is not dominant in the superposed two-qubit eigenstates. This coupling scheme, illustrated in Fig. 3c, is typically used for superconducting qubits to realize two-qubit operations¹². We measure the virtual coupling at the optimal working point (${\it{\Delta}} /h\simeq -3.3\ {\rm{GHz}}$), at ${\it{\Delta}} /h\simeq -9.9\ {\rm{GHz}}$ and at ${\it{\Delta}} /h\simeq 10.2\ {\rm{GHz}}$, where ${\gamma }_{{\rm{2,RX}}}$ in Fig. 3a saturates, as well as in the intermediate regime at ${\it{\Delta}} /h\simeq 3.4\ {\rm{GHz}}$. While the RX qubit is tuned through a resonance with the transmon by changing ${\it{\Delta}}$, they are both detuned by ${\it{\Delta} }_{{\rm{C}}}\equiv {\nu }_{{\rm{C}}}-{\nu }_{{\rm{T}}}\simeq 3{g}_{{\rm{T}}}$ from the coupling resonator. To realize this detuning for every working point, we adjust the qubit and resonator energies with ${\it{\Phi} }_{{\rm{T}}}$, ${t}_{{\rm{l,r}}}$ and ${\it{\Phi} }_{{\rm{C}}}$. We drive the RX qubit at frequency ${\nu }_{{\rm{dRX}}}$ (see Fig. 1a) and investigate its coupling to the transmon by probing the dispersively coupled read-out resonator at its resonance frequency (${\nu }_{{\rm{p}}}={\nu }_{{\rm{R}}}\simeq 5.6\ {\rm{GHz}}$). This measurement is shown in Fig. 3d for the working point at ${\it{\Delta}} /h\simeq -9.9\ {\rm{GHz}}$. For large transmon-spin qubit detuning (${\it{\Delta}} /h\ll -10\ {\rm{GHz}}$), the spectroscopic signal of the transmon is barely visible as the drive mainly excites the bare RX qubit. The signal increases with ${\it{\Delta}}$ as the RX qubit approaches resonance with the transmon, such that driving the RX qubit also excites the transmon due to their increasing mutual hybridization. On resonance, we resolve the two hybridized spin-qubit-transmon states ${\left|\pm \right\rangle }_{{\rm{e}}}\simeq (\left|{0}_{{\rm{RX}}},{1}_{{\rm{T}}}\right\rangle \pm \left|{1}_{{\rm{RX}}},{0}_{{\rm{T}}}\right\rangle )/\sqrt{2}$ by about a line width. These states are separated in energy by the virtual-photon-mediated exchange splitting $2J\simeq 2{g}_{{\rm{RX}}}{g}_{{\rm{T}}}/{\it{\Delta} }_{{\rm{C}}}$. The splitting is enhanced at the other working points in Fig. 3e–g, for which the RX qubit control parameter ${\it{\Delta}}$ and consequently ${g}_{{\rm{RX}}}$ is larger. The result of a master equation simulation shown in Fig. 3e agrees well with the experimental observation. The influence of the RX qubit decoherence rate ${\gamma }_{{\rm{2,RX}}}$ on the virtual interaction measurement is quantified in Fig. 3h, where we show averaged measurements of the two-tone spectroscopy signal from Fig. 3d–g at ${\it{\Delta}}$ as indicated by arrows in the corresponding panels. The fits of a master equation model in Fig. 3h show excellent quantitative agreement with the experimental curves. As discussed in detail in Supplementary Note 3, fit parameters previously obtained from Fig. 2 were adjusted to account for significant power broadening in these measurements. The exchange splitting is best resolved at the optimal working point, corresponding to the solid green curve in Fig. 3h, where we obtain $2J/2\pi \simeq 32\ {\rm{MHz}}$ from the fit.

Discussion

In conclusion, we have implemented a coherent link between an RX qubit and a transmon. The link either utilizes real or virtual microwave photons for the qubit–qubit interaction. The RX qubit was operated in both spin and charge dominated regimes. We found an optimal working point at which the ratio between its resonator coupling and its decoherence rate is maximal and comparable to state of the art values achieved with spin qubits in Si. We also reported that the coupling resonator potentially introduces charge noise that can have significant impact on the RX qubit coherence. The performance of the quantum link in this work is limited by the minimum decoherence rate of the qubit, which is determined by hyperfine interaction in the GaAs host material. Once the spin coherence is enhanced by using hyperfine free material systems such as graphene^48,49 or isotopically purified silicon⁵⁰, the spin could be used as a memory that can be coupled on-demand to the transmon by pulsing the qubit control parameter. While three-electron spin qubits have already been implemented in Si^51,52, a sufficiently high transition frequency allowing for circuit QED experiments to be performed, which requires large inter-dot tunnel couplings, has not yet been realized. Compared to GaAs, electrostatic control of single electrons can be more challenging related to overlap** fine gate structures for silicon and graphene quantum dots as well as due to the smaller quantum dot size in silicon. An additional potential challenge in both materials is the valley degree of freedom³⁵. As the coherence properties of the RX qubit are retained at zero magnetic field in contrast to other spin qubit implementations, the quantum device architecture used in this work is compatible for realizing a high fidelity transmon–spin-qubit and spin-qubit–spin-qubit interface in a future quantum processor.

Methods

RX qubit tunnel coupling configurations

Throughout this work, we use the four RX qubit tunnel coupling configurations listed in Table 1.

Table 1 Tunnel couplings

The measurements to extract the tunnel couplings are explained in Supplementary Note 1. Different configurations were necessary for two reasons. To realize the virtual interaction scheme in Fig. 3c for different RX qubit working points while kee** the same transmon flux, the tunnel couplings had to be adjusted. The current in the transmon flux line was kept below a level at which an increase in the refrigerator temperature was observed. This ensured that the device operation took place at the lowest accessible measurement temperature. Second, when readjusting the RX qubit after a random charge rearrangement occurred in the host material, which was observed on the time scale of days, identical tunnel coupling configurations could not be achieved.

Details of sample and measurement scheme

In Fig. 4a we show a false-colored optical micrograph of the part of the sample that was illustrated schematically in Fig. 1a. The microwave read-out scheme is also indicated. The sample is measured in a dilution refrigerator with a base temperature of $10\ {\rm{mK}}$. The four quantum systems are highlighted in different colors. A magnified image of the transmon is shown in Fig. 4b. One side of the SQUID loop is grounded, the other is connected to a big shunt capacitor (highlighted in green). We control the transmon transition frequency with a current $I$ though an inductively coupled flux line. The transmon is capacitively coupled to one end of a $\lambda /2$ $50\ {{\Omega }}$ (read-out) resonator, which is shown to the full extent in Fig. 4c. It is capacitively coupled to a transmission line that is used for resonator read-out.

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 $50\ \Omega$ (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 $50\ \Omega$ resonator and its microwave read-out line. The location of the region shown in a is outlined with a black rectangle