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Coherent coupling between a ferromagnetic magnon and a superconducting qubit

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Science  24 Jul 2015:
Vol. 349, Issue 6246, pp. 405-408
DOI: 10.1126/science.aaa3693

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.

Science, this issue pp. 405 and 408

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 (69) 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 Embedded Image (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 (1517). 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 Embedded Image of Embedded Image GHz. It strongly couples to the electric fields of the cavity modes; e.g., the coupling strength Embedded Image between the qubit and the TEEmbedded Image (transverse electric) mode at Embedded Image GHz is Embedded Image MHz (21). The YIG sphere with a diameter of Embedded Image mm is glued to an aluminum oxide rod and mounted near the antinode of the magnetic field of the TEEmbedded Image mode. We also apply a local static field Embedded Image T, which makes the YIG sphere a single-domain ferromagnet (fig. S1). The sphere now has an enormous magnetic dipole moment Embedded Image, which couples strongly to the magnetic field of the cavity mode. The large enhancement factor Embedded Image 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 Embedded Image = 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 TEEmbedded Image 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, Embedded Image= 21.0 MHz, and the linewidths of the TEEmbedded Image and Kittel modes, Embedded Image= 2.5 MHz and Embedded Image = 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.

Fig. 1 Qubit-magnon hybrid quantum system in a microwave cavity.

(A) Simulated microwave field distribution of the TEEmbedded Image (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 Embedded Image is applied locally to the sphere (see fig. S1). (B) Magnon-photon normal-mode splitting. Amplitude of the microwave transmission coefficient, Re(Embedded Image), is measured through the TEEmbedded Image mode as a function of the probe frequency and the static magnetic field. The field is represented by the relative coil current Embedded Image, which is defined to be zero at the anticrossing. The dashed lines show the TEEmbedded Image-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, Embedded Image and Embedded Image, while both are far detuned from the cavity frequency Embedded Image). When Embedded Image, 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 Embedded Image (1)where Embedded Image is the effective qubit-magnon coupling strength, and Embedded Image and Embedded Image are annihilation operators of the qubit excitation and the magnon, respectively. The detuning Embedded Image 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.

Fig. 2 Magnon-vacuum–induced Rabi splitting of the superconducting qubit.

(A) Cavity-mediated coupling scheme. The energy diagram illustrates the frequencies of the cavity TEEmbedded Image and TEEmbedded Image modes, Embedded Image (Embedded Image) and Embedded Image; the Kittel-mode frequency Embedded Image; and the qubit frequency Embedded Image. The qubit couples to the electric field of the TEEmbedded Image mode with a coupling strength Embedded Image, whereas the Kittel mode in the YIG sphere magnetically couples to the same mode with a coupling strength Embedded Image. When Embedded Image, a qubit excitation is transformed to a Kittel-mode magnon and vice versa via the virtual-photon excitation in the TEEmbedded Image mode. The qubit also couples to the TEEmbedded Image 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 Embedded Image at Embedded Image is measured as a function of the qubit excitation frequency and the static magnetic field represented by the relative coil current Embedded Image, which is defined to be zero at the anticrossing. An additional splitting around Embedded Image 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 TEEmbedded Image mode. Despite the large detuning, the TEEmbedded Image mode at Embedded Image GHz is subject to a dispersive frequency shift: The change in the cavity reflection coefficient, Re(Embedded Image), at Embedded Image reflects the qubit state. Figure 2B shows Re(Embedded Image) 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 Embedded Image = Embedded Image MHz is obtained. The coupling strength far exceeds the linewidths of the qubit and the magnon, Embedded Image MHz and Embedded Image MHz, respectively. The value is also in a good agreement with the calculated value of Embedded Image 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 (2325) (Fig. 3A). To suppress the static coupling, a large detuning of Embedded Image MHz between the qubit and the Kittel mode (Embedded Image 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., Embedded Image—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 Embedded Image and Embedded Image. The interaction Hamiltonian for the parametrically induced coupling is written as Embedded Image (2)where the effective coupling strength Embedded Image is proportional to the drive microwave power Embedded Image (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 Embedded Image and the Kittel-mode magnon-number states Embedded Image. The two-photon drive induces the Rabi splitting Embedded Image between the states Embedded Image and Embedded Image. With a probe microwave at frequencies near Embedded Image, 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 Embedded Image iii). The shift from 8.410 to 8.405 GHz is identified as a qubit-state–dependent shift, corresponding to the transition Embedded Image for small Embedded Image (i, ii) and Embedded Image for large Embedded Image (iii to vi). This is caused by the population transfer from Embedded Image to Embedded Image at large Embedded Image, as well as the residual static coupling between the qubit and the Kittel mode (21). The anticrossing observed in the Embedded Image 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 Embedded Image, 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 Embedded Image 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 Embedded Image and do not show any notable broadening in the presence of the microwave drive.

Fig. 3 Tunable coupling between a superconducting qubit and the Kittel mode.

(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 Embedded Image. The inset shows energy levels labeled with the qubit state (g or e) and the magnon number (Embedded Image). The parametric drive (red dashed arrows) induces the two-photon transitions, e.g., Embedded Image, resulting in the Rabi splitting. The blue arrows depict the Kittel-mode decay, and the red circle at Embedded Image illustrates the dominant steady-state population at higher drive powers. (B) Kittel-mode spectra for various drive powers Embedded Image indicating tunable coupling with the qubit. (C) Cross sections of (B) at the drive frequencies corresponding to Embedded Image [yellow dashed lines in (B)]. Note that the frequency depends on Embedded Image through the Stark shift of the qubit. For clarity, the individual curves i to v are offset vertically by Embedded Image from each other from top to bottom. Inset: Coupling strength Embedded Image as a function of the drive power Embedded Image. 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 (2729). 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

www.sciencemag.org/content/349/6246/405/suppl/DC1

Materials and Methods

Figs. S1 to S4

References (30, 31)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. 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).

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