Strong Field Interactions between a Nanomagnet and a Photonic Cavity

We analyze the interaction of a nanomagnet (ferromagnetic) with a single photonic mode of a cavity in a fully quantum-mechanical treatment and find that exceptionally large quantum-coherent magnet-photon coupling can be achieved. Coupling terms in excess of several THz are predicted to be achievable in a spherical cavity of $\sim 1\mathrm{mm}$ radius with a nanomagnet of $\sim 100\mathrm{nm}$ radius and ferromagnetic resonance frequency of $\sim 200\mathrm{GHz}$. Eigenstates of the magnet-photon system correspond to entangled states of spin orientation and photon number, in which over ${10}^5$ values of each quantum number are represented; conversely, initial (coherent) states of definite spin and photon number evolve dynamically to produce large oscillations in the microwave power (and nanomagnet spin orientation), and are characterized by exceptionally long dephasing times.

Figure 1

Schematic of the nanomagnet-cavity system with a spherical nanomagnet of radius $r_m$ placed at a distance of $d$ from the center of a cavity of radius $R$. A uniform magnetic field, ${\mathit{B}}_0$, applied along the $\mathbf{z}$ axis causes precession of the nanomagnet macrospin, $\mathit{S}$, with frequency of $\omega$, in resonance with TM mode of the cavity.

Figure 2

Wave functions of the nanomagnet-cavity system as a function of photon number, $n$, centered about $n_0=4\xi /3=6.66667\times {10}^8$ for $N={10}^9$ spins: (a) ground state with a width of roughly $5\times {10}^4$ photons (or equivalently spin quantum numbers $m_s$), (b) 1st, (c) 2nd, and (d) 150th excited states.

Figure 3

(a) Amplitude of a coherent state of nanomagnet/photon system shown as a function of photon number $n$. The large oscillations of this coherent state about $n_0=6.667\times {10}^8$ occur between photon numbers $-267000$ (Filled line), and $+267000$ (Dashed line) with a period of $T=4.74\mu \mathrm{s}$. (b) Time evolution of the Zeeman energy of the nanomagnet (Red line), and transverse magnetic mode of the cavity field (Green line) at $z=d$ in this coherent state representation.

Figure 4

Dephasing time of the coherent state obtained by a Gaussian fit to the peak values of the correlation function (inset) at successive time intervals.