Off-resonance coupling between a cavity mode and an ensemble of driven spins

We study the interaction between a superconducting cavity and a spin ensemble. The response of a cavity mode is monitored while simultaneously the spins are driven at a frequency close to their Larmor frequency, which is tuned to a value much higher than the cavity resonance. We experimentally find that the effective damping rate of the cavity mode is shifted by the driven spins. The measured shift in the damping rate is attributed to the retarded response of the cavity mode to the driven spins. The experimental results are compared with theoretical predictions and fair agreement is found.

Figure 1

The device is made of two $40\times 40\times 0.5$ mm sapphire wafers carrying the radio-frequency omega resonator, and a $5\times 5\times 0.5$ mm silicon wafer carrying the microwave spiral resonator. The DPPH powder is placed between the omega inductor and the spiral. The three wafers are vertically shifted in the sketch for clarity. In the assembled device, both the top sapphire wafer and the silicon wafer are placed directly on top of the bottom sapphire wafer. The three wafers and a loop antenna are assembled together inside a package made of high-conductivity oxygen-free copper. Both omega and spiral resonators are made by dc-magnetron sputtering of a 200-nm-thick niobium layer. The radius of the omega inductor is $500\mu m$ and the linewidth is $40\mu m$. The spiral dimensions are inner radius $500\mu m$, outer radius $580\mu m$, linewidth $10\mu m$, and number of turns 4. The measured frequency of the omega (spiral) resonator is ${\omega }_{\mathrm{a}}/2\pi =0.173GHz$ (${\omega }_{\mathrm{b}}/2\pi =2.00GHz$), whereas the value obtained from numerically simulating the structure is $0.176GHz$ ($2.07GHz$).

Figure 2

The experimental setup. A power combiner (PC) is employed for combining the injected signals of a signal generator (SG) and a network analyzer (NA). The combined injected signal is transmitted through an amplifier (A) and a coupler (C), and feeds the loop antenna (LA), which is positioned above the device under study (DUS). The back-reflected signal is split by a power splitter (PS) and measured by both a NA and a spectrum analyzer (SA).

Figure 3

ESR of (a) the omega and (b) the spiral resonators. The color-coded plots display the measured reflectivity coefficient ${\left|S_{11}\right|}^2$ vs ${\omega }_{\mathrm{L}}$ (i.e., vs static magnetic field) and the probing frequency ${\omega }_{\mathrm{NA}}$. Measurements are performed by a network analyzer at a temperature of $T=3.1K$, for which the polarization coefficient $p_0$ [see Eq. (A20)] is given by $p_0=-1.4\times {10}^{-3}$ ($p_0=-1.6\times {10}^{-2}$) for the omega (spiral) resonator.

Figure 4

The (a) measured and (b) calculated change in the damping rate $-Im{\mathrm{{\rm Y}}}_{\mathrm{a}}$ vs ${\omega }_{\mathrm{L}}$ and the pump frequency ${\omega }_{\mathrm{p}}$. The experimental value is obtained from the line shape of the omega resonance. The Larmor frequency ${\omega }_{\mathrm{L}}$ is tuned by applying a static magnetic field in a direction parallel to the wafers. The pump power is set to the value $17\mathrm{dBm}$, which corresponds to a driving amplitude of ${\omega }_1/2\pi =12MHz$. (b) The calculated shift is based on Eqs. (1) and (2). The following parameters are used in the calculation: ${\gamma }_{\mathrm{b}}=0.4MHz$ and ${\gamma }_2=8.3MHz$ (other parameters are specified above).

Figure 5

The contribution to cavity mode damping rate ${\gamma }_{\mathrm{aL}}$ due to coupling to the driven spins. (a) The normalized contribution ${\gamma }_{\mathrm{aL}}/{\omega }_{\mathrm{a}}$ vs normalized detuning ${\mathrm{\Delta }}_{\mathrm{pL}}/{\omega }_{\mathrm{a}}$ and normalized driving amplitude ${\omega }_1/{\omega }_{\mathrm{a}}$. The calculation is based on Eq. (1) with the following assumed parameters: ${\gamma }_1/{\omega }_{\mathrm{a}}=2{\gamma }_2/{\omega }_{\mathrm{a}}=0.05,g_{\mathrm{a}}/{\omega }_{\mathrm{a}}=0.1$, and $p_0=-0.1$. (b) Normalized spin polarization $-p_z/p_0$ vs cavity mode amplitude $x_{\mathrm{a}}$ for the case of blue-detuned driving. The black solid line represents the steady-state normalized spin polarization $-p_{z0}/p_0$. Retardation in the response of the spins to periodic oscillation of $x_{\mathrm{a}}$ is illustrated by the blue closed curve.