Research ArticleQUANTUM MAGNETISM

Resolving quanta of collective spin excitations in a millimeter-sized ferromagnet

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Science Advances  05 Jul 2017:
Vol. 3, no. 7, e1603150
DOI: 10.1126/sciadv.1603150
  • Fig. 1 Hybrid system and qubit-magnon coherent interaction.

    (A) Schematic illustration of a ferromagnetic YIG sphere and a superconducting transmon qubit inside a three-dimensional microwave cavity. A magnetic field B0 is applied to the YIG sphere using permanent magnets and a coil. The magnetostatic mode in which spins uniformly precess in the ferromagnetic sphere, or the Kittel mode, couples to the magnetic field of the cavity modes. The qubit and the Kittel mode interact through virtual excitations in the cavity modes at a rate gq-m. (B) The spectrum of the qubit is measured by probing the change of the reflection coefficient Re(Δr) of a microwave excitation resonant, with the probe mode at frequency ωp as a function of the spectroscopy frequency ωs and the coil current I, changing the magnetic field at the ferromagnet. The avoided crossing indicates a coherent interaction between the qubit and the Kittel mode. Vertical dashed lines indicate that I = −4.25 mA, where the qubit and the Kittel mode are hybridized (Fig. 1D), and that I = −5.02 mA, where the qubit-magnon interaction is in the dispersive regime (Figs. 2 to 4). (C) Magnon-vacuum Rabi splitting of the qubit on resonance, with the Kittel mode at I = −4.25 mA. From the fit, we extract the qubit-magnon coupling rate gq-m/2π = 7.79 MHz.

  • Fig. 2 Dispersive qubit-magnon interaction.

    (A) Schematic illustration of the hybrid system in the strong dispersive regime. A microwave excitation at frequency ωmw is used to create a magnon coherent state in the Kittel mode. The excitation is detuned from the magnon frequency, with the qubit in the ground state, ωmg, by Δmw = ωmg − ωmw. In the strong dispersive regime, magnon number states |nm〉 (of probability distribution Embedded Image) are mapped into the qubit spectrum as peaks at frequencies Embedded Image, separated by 2χq-m + Δmw and with a spectral weight closely related to Embedded Image. (B) Measurement of the qubit spectrum for a coil current I = −5.02 mA as a function of the Kittel mode excitation frequency ωmw and the spectroscopy frequency ωs. The excitation frequency producing the maximum magnon-induced ac Stark shift of the qubit from ωq (horizontal dashed line) yields an estimation of ωmg/2π ≈ 7.95 GHz (vertical dashed line). The Kittel mode spectrum, measured via its dispersive interaction with the probe mode, appears as a faint vertical line at ~7.95 GHz. The signature corresponding to the two-photon transition involving both the spectroscopy and the excitation photons and exciting both the qubit and a magnon (fig. S1) is indicated by the diagonal dashed line given by Embedded Image, calculated with χq-m/2π = 1.5 MHz at ωmg/2π = 7.95 GHz.

  • Fig. 3 Resolving magnon number states.

    (A) Measurements of the qubit spectrum at different Kittel mode excitation powers Pmw for a coil current of −5.02 mA and Kittel mode excitation frequency of 7.95 GHz. Black lines show fits of the data to the spectrum of a qubit dispersively coupled to a harmonic oscillator. Color-coded shaded areas show components of the spectrum corresponding to different photon number states |np〉 of the probe cavity mode and magnon number states |nm〉. Vertical offsets are shown by horizontal dashed lines. (B) Measured qubit spectra as a function of Pmw. For clarity, after subtracting a power-dependent offset, Re(Δr) is normalized relative to its maximum for each drive power. For both (A) and (B), vertical dashed lines indicate the frequencies of the qubit |g〉 ↔ |e〉 transitions corresponding to the first four magnon number states, neglecting a power-dependent ac Stark shift, which is small relative to the dispersive shift per magnon for this range of Pmw.

  • Fig. 4 Magnon occupancy and probability distribution.

    (A) Magnon occupancy Embedded Image as a function of the excitation power Pmw. Dashed black line and solid orange line show fits to a driven linear and nonlinear Kittel mode, respectively. Inset: Deviations Embedded Image from the linear fit, indicating a significant magnon Kerr nonlinearity. (B) Probability Embedded Image of the first four magnon number states as a function of Pmw. Poisson distributions are shown as solid lines. Inset: Embedded Image for Pmw = 3.1 fW. For both (A) and (B), error bars correspond to 95% confidence intervals.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/7/e1603150/DC1

    section S1. Hamiltonian of the hybrid system

    section S2. Cavity-magnon coupling

    section S3. Qubit spectrum in the dispersive regime

    section S4. Qubit spectroscopy—Magnon vacuum state

    section S5. Qubit spectroscopy—Magnon coherent state

    section S6. Magnon Kerr nonlinearity

    fig. S1. Qubit-magnon hybrid system.

    fig. S2. Experimental setup.

    fig. S3. Cavity-magnon coupling.

    fig. S4. Power broadening of the qubit spectrum.

    fig. S5. Dispersive qubit-magnon interaction.

    fig. S6. Probability distributions.

    fig. S7. Magnon Kerr nonlinearity.

    fig. S8. Effect of the finite Kerr nonlinearity on the magnon probability distribution.

    table S1. Parameters of the hybrid system.

    table S2. Comparison between experimental and theoretical values.

    table S3. Linewidths of the hybrid system.

    table S4. Experimental parameters of the measurements.

    References (3134)

  • Supplementary Materials

    This PDF file includes:

    • section S1. Hamiltonian of the hybrid system
    • section S2. Cavity-magnon coupling
    • section S3. Qubit spectrum in the dispersive regime
    • section S4. Qubit spectroscopy—Magnon vacuum state
    • section S5. Qubit spectroscopy—Magnon coherent state
    • section S6. Magnon Kerr nonlinearity
    • fig. S1. Qubit-magnon hybrid system.
    • fig. S2. Experimental setup.
    • fig. S3. Cavity-magnon coupling.
    • fig. S4. Power broadening of the qubit spectrum.
    • fig. S5. Dispersive qubit-magnon interaction.
    • fig. S6. Probability distributions.
    • fig. S7. Magnon Kerr nonlinearity.
    • fig. S8. Effect of the finite Kerr nonlinearity on the magnon probability
      distribution.
    • table S1. Parameters of the hybrid system.
    • table S2. Comparison between experimental and theoretical values.
    • table S3. Linewidths of the hybrid system.
    • table S4. Experimental parameters of the measurements.
    • References (31–34)

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