Making hybrid quantum information systems
Different physical implementations of qubits—quantum bits—each have their pros and cons. An appealing idea is to combine them into hybrid architectures, taking advantage of their respective strengths. Tabuchi et al. placed a ferromagnetic sphere and a superconducting qubit in a cavity and used an electromagnetic mode of the cavity as the mediator between the two. They achieved strong coupling between a collective magnetic mode of the sphere and the qubit. Viennot et al. coupled a single spin in a double quantum dot to photons in a cavity. Both approaches hold promise for future applications.
Abstract
Rigidity of an ordered phase in condensed matter results in collective excitation modes spatially extending to macroscopic dimensions. A magnon is a quantum of such collective excitation modes in ordered spin systems. Here, we demonstrate the coherent coupling between a single-magnon excitation in a millimeter-sized ferromagnetic sphere and a superconducting qubit, with the interaction mediated by the virtual photon excitation in a microwave cavity. We obtain the coupling strength far exceeding the damping rates, thus bringing the hybrid system into the strong coupling regime. Furthermore, we use a parametric drive to realize a tunable magnon-qubit coupling scheme. Our approach provides a versatile tool for quantum control and measurement of the magnon excitations and may lead to advances in quantum information processing.
Low-dissipative magnon dynamics in ferromagnetic insulators have been extensively studied in the contexts of ferromagnetic resonance (1, 2), Bose-Einstein condensation (3), and spintronics (4, 5). Moreover, the coupling of magnons and microwave photons in a resonator has been investigated (6–9) with the aim of realizing hybrid quantum systems for quantum memories and transducers. However, coherent manipulation of a magnon at the single-quantum level has remained elusive because of the lack of anharmonicity in the system.
Single-electron spins, being a natural and genuine two-level system, play crucial roles in numerous applications in quantum information processing. However, they have intrinsic drawbacks, such as a small magnetic moment equal to 
(the Bohr magneton) and the limited spatial extension of the electron wave function, making coherent coupling with an electromagnetic field rather weak. To circumvent these problems, paramagnetic spin ensembles such as atoms (10), nitrogen vacancy centers (11, 12), and rare-earth ions in a crystal (13, 14) have been studied. Generally, the coupling strength is enhanced by the square root of the number of the spins involved. At the same time, a collective spin excitation mode, which matches the input electromagnetic-field mode, is spanned among the spatially and spectrally extended spin ensemble. However, with an increased spin density needed for a stronger coupling, the spin-spin interactions within the ensemble drastically degrade the coherence of the system, leading to a trade-off.
Here we take a different approach by introducing, counterintuitively, a strong exchange interaction between neighboring spins to make them fully ordered in the ferromagnetic state. Even though they typically have a spin density several orders of magnitude higher than those of paramagnetic spin crystals mentioned above, the strong exchange and dipolar interactions among the spins dominate their dynamics, leading to narrow-linewidth magnetostatic modes. The simplest mode, called the Kittel mode, has uniform spin precessions in the whole volume.
In this Report, we demonstrate a hybrid quantum system that combines two heterogeneous collective-excitation modes: i.e., the Kittel mode in a ferromagnetic crystal and a superconducting qubit. In the latter system, the nonlinearity of Josephson junctions plays a crucial role for the realization of the qubit: i.e., an effective two-level system. The progress in the past decade has made these qubits and their integrated circuits one of the most advanced technologies for quantum information processing (15–17). In the setups of circuit quantum electrodynamics, a qubit as an artificial atom is coupled strongly to a microwave resonator (18) or a waveguide (19). These setups allow precise control and readout of the qubit states, as well as synthesis and characterization of arbitrary quantum states in the microwave modes (20); these techniques can readily be applied to quantum engineering with magnon excitations. The anharmonicity contributed by the superconducting qubit is the critical element.
In our experimental setup (Fig. 1), a transmon-type superconducting qubit and a single-crystalline yttrium iron garnet (YIG) sphere are mounted in a microwave cavity. The qubit with a 0.7-mm-long dipole antenna has a resonant frequency 
of 
GHz. It strongly couples to the electric fields of the cavity modes; e.g., the coupling strength 
between the qubit and the TE
(transverse electric) mode at 
GHz is 
MHz (21). The YIG sphere with a diameter of 
mm is glued to an aluminum oxide rod and mounted near the antinode of the magnetic field of the TE
mode. We also apply a local static field 
T, which makes the YIG sphere a single-domain ferromagnet (fig. S1). The sphere now has an enormous magnetic dipole moment 
, which couples strongly to the magnetic field of the cavity mode. The large enhancement factor 
is the number of the net electron spins in the sphere; we take advantage of the high spin density compared to that of previously studied paramagnetic systems. We perform a series of spectroscopic measurements in a dilution refrigerator at 
= 10 mK. All the data are taken in the quantum regime, where very few thermally excited photons and magnons exist. The average probe-photon number in the cavity is also kept below one. To characterize the coupling between the magnon and the photon, we perform spectroscopy with the qubit frequency far detuned. Figure 1B shows the normal-mode splitting between the TE
mode and the Kittel mode in the YIG sphere. The pronounced anticrossing indicates the strong coupling between the two systems (7). We obtain the coupling strength, 
= 21.0 MHz, and the linewidths of the TE
and Kittel modes, 
= 2.5 MHz and 
= 1.4 MHz, from the fit. The additional splitting seen in the upper branch originates from another magnetostatic mode, which is detuned from the Kittel mode.
(A) Simulated microwave field distribution of the TE
(transverse electric) mode in the cavity. The upper half of the sketch displays the electric field lines (red arrows); the electric field couples to a transmon-type superconducting qubit (right insets: optical image of the qubit’s antenna pads and false-color scanning-electron micrograph of the Josephson junction bridging them). The lower half of the sketch shows the magnetic field (blue arrows), which couples to the spins in a single-crystalline sphere of yttrium iron garnet (YIG; photo in the left inset). A static magnetic field 
is applied locally to the sphere (see fig. S1). (B) Magnon-photon normal-mode splitting. Amplitude of the microwave transmission coefficient, Re(
), is measured through the TE
mode as a function of the probe frequency and the static magnetic field. The field is represented by the relative coil current 
, which is defined to be zero at the anticrossing. The dashed lines show the TE
-mode (white) and Kittel-mode (green) frequencies obtained from fitting.
Whereas the qubit and the magnon electrically and magnetically couple to the cavity mode, respectively, they have a negligibly small direct interaction in between. Therefore, we first establish a static coupling scheme between the magnon and the qubit by using the presence of the cavity mode (Fig. 2A) (22). We tune the qubit and the magnon frequencies, 
and 
, while both are far detuned from the cavity frequency 
). When 
, coherent exchange of the qubit excitation and a magnon is mediated by the virtual-photon excitation in the cavity mode. The interaction is described by a Jaynes-Cummings-type Hamiltonian, which is written as
(1)where 
is the effective qubit-magnon coupling strength, and 
and 
are annihilation operators of the qubit excitation and the magnon, respectively. The detuning 
is the difference between the bare frequencies of the qubit and the cavity mode (21). The first and the second excited states of the hybridized system are bonding and antibonding states between the qubit and the magnon excitation.
(A) Cavity-mediated coupling scheme. The energy diagram illustrates the frequencies of the cavity TE
and TE
modes, 
(
) and 
; the Kittel-mode frequency 
; and the qubit frequency 
. The qubit couples to the electric field of the TE
mode with a coupling strength 
, whereas the Kittel mode in the YIG sphere magnetically couples to the same mode with a coupling strength 
. When 
, a qubit excitation is transformed to a Kittel-mode magnon and vice versa via the virtual-photon excitation in the TE
mode. The qubit also couples to the TE
mode weakly, which results in the dispersive frequency shift of the cavity mode depending on the states of the qubit. (B) Magnon-vacuum–induced Rabi splitting. Change in the cavity reflection coefficient 
at 
is measured as a function of the qubit excitation frequency and the static magnetic field represented by the relative coil current 
, which is defined to be zero at the anticrossing. An additional splitting around 
GHz is attributed to another magnetostatic mode (21). (C) Cross sections of (B) at various static magnetic fields. For clarity, the individual curves are offset vertically by 0.06 each from bottom to top.
To demonstrate the qubit-magnon coupling, we perform qubit excitation spectroscopy by using the qubit readout through the cavity TE
mode. Despite the large detuning, the TE
mode at 
GHz is subject to a dispersive frequency shift: The change in the cavity reflection coefficient, Re(
), at 
reflects the qubit state. Figure 2B shows Re(
) as a function of the excitation microwave frequency and the static magnetic field. The anticrossing is a manifestation of the magnon-vacuum–induced Rabi splitting, which indicates coherent coupling between the qubit and the Kittel mode. From the size of the splitting, the effective coupling strength 
= 
MHz is obtained. The coupling strength far exceeds the linewidths of the qubit and the magnon, 
MHz and 
MHz, respectively. The value is also in a good agreement with the calculated value of 
MHz based on the lowest-order approximation (21).
For dynamical control of the magnon quantum state, fast on-off switching of the interaction with the qubit would be useful. In the static coupling scheme, however, it is technically challenging to sweep the magnetic field in a time scale faster than that of the coupling strength. To obtain a dynamically tunable coupling between the qubit and the Kittel mode, we adopt a parametrically induced interaction that has been proposed and demonstrated in superconducting circuits (23–25) (Fig. 3A). To suppress the static coupling, a large detuning of 
MHz between the qubit and the Kittel mode (
GHz) is chosen. Next, we introduce a microwave drive at a frequency equal to the average of the qubit and the Kittel-mode frequencies—i.e., 
—to induce the parametric coupling. This is basically a third-order nonlinear process; the system absorbs two drive photons and excites the qubit and a magnon simultaneously. The substantial third-order nonlinearity stems from the anharmonicity of the qubit, as well as the large coupling strengths 
and 
. The interaction Hamiltonian for the parametrically induced coupling is written as
(2)where the effective coupling strength 
is proportional to the drive microwave power 
(21). To understand how the continuous-wave spectroscopy reveals the parametrically induced coupling, we consider the inset of Fig. 3A, which illustrates the energy levels of the driven hybrid system in the bases of the qubit states 
and the Kittel-mode magnon-number states 
. The two-photon drive induces the Rabi splitting 
between the states 
and 
. With a probe microwave at frequencies near 
, the change of the magnon excitation spectrum in the driven system is monitored. The color intensity plots in Fig. 3B show the reflection coefficient as a function of the probe frequency and the parametric drive frequency for several drive powers. When the drive frequency hits the two-photon resonance condition (yellow dashed lines), the dip in the spectrum splits into two. As the drive power increases, the anticrossing feature grows in the spectra, and an abrupt shift of the magnon frequency occurs (ii 
iii). The shift from 8.410 to 8.405 GHz is identified as a qubit-state–dependent shift, corresponding to the transition 
for small 
(i, ii) and 
for large 
(iii to vi). This is caused by the population transfer from 
to 
at large 
, as well as the residual static coupling between the qubit and the Kittel mode (21). The anticrossing observed in the 
transition manifests parametrically induced coupling between the qubit and the magnon. As seen in the cross sections presented in Fig. 3C, the spacing between the dips, corresponding to 
, increases linearly with the drive power. This indicates the capability of dynamically tunable coupling between the qubit and the Kittel mode via tailoring the parametric drive. The maximum coupling 
obtained is 3.4 MHz, which exceeds the decoherence rates of the qubit and the magnon and ensures coherent coupling between them. Importantly, the linewidths of the dips are consistent with the expected value 
and do not show any notable broadening in the presence of the microwave drive.
(A) Parametrically induced coupling scheme. The coupling between the qubit and the Kittel mode detuned from each other is parametrically induced by a drive microwave at the frequency 
. The inset shows energy levels labeled with the qubit state (g or e) and the magnon number (
). The parametric drive (red dashed arrows) induces the two-photon transitions, e.g., 
, resulting in the Rabi splitting. The blue arrows depict the Kittel-mode decay, and the red circle at 
illustrates the dominant steady-state population at higher drive powers. (B) Kittel-mode spectra for various drive powers 
indicating tunable coupling with the qubit. (C) Cross sections of (B) at the drive frequencies corresponding to 
[yellow dashed lines in (B)]. Note that the frequency depends on 
through the Stark shift of the qubit. For clarity, the individual curves i to v are offset vertically by 
from each other from top to bottom. Inset: Coupling strength 
as a function of the drive power 
. Also plotted are the linear fit (red dashed line) and the theoretical expectation (green dashed line) (21).
Magnons in a macroscopic-scale ferromagnetic crystal are now ready to be controlled in a quantum manner, enabling the investigation of the ultimate limit of spintronics and magnonics at the single-quantum level. It would be of particular interest to consider an analogy with recent advances in optoelectromechanics (26): Phonons in nanomechanical devices, yet another example of spatially extended collective excitations in solids, coherently interact both with microwave and optical degrees of freedom and thus are studied as a candidate for realizing quantum transducers between two spectrally distant frequency domains (27–29). Given the demonstrated strong coupling to microwave and the anticipated magneto-optical coupling, magnons in ferromagnetic insulators may provide an alternative route toward that goal.
Supplementary Materials
References and Notes
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- Acknowledgments: We acknowledge P.-M. Billangeon for fabricating the transmon qubit. This work was partly supported by the Project for Developing Innovation System of the Ministry of Education, Culture, Sports, Science and Technology, Japan Society for the Promotion of Science KAKENHI (grant no. 26600071, 26220601), the Murata Science Foundation, Research Foundation for Opto-Science and Technology, and National Institute of Information and Communications Technology (NICT).
