Cavity magnon-polaritons in cuprate parent compounds

Cavity control of quantum matter may offer new ways to study and manipulate many-body systems. A particularly appealing idea is to use cavities to enhance superconductivity, especially in unconventional or high-$T_c$ systems. Motivated by this, we propose a scheme for coupling terahertz resonators to the antiferromagnetic fluctuations in a cuprate parent compound, which are believed to provide the glue for Cooper pairs in the superconducting phase. First, we derive the interaction between magnon excitations of the Neél order and polar phonons associated with the planar oxygens. This mode also couples to the cavity electric field, and in the presence of spin-orbit interactions mediates a linear coupling between the cavity and magnons, forming hybridized magnon-polaritons. This hybridization vanishes linearly with photon momentum, implying the need for near-field optical methods, which we analyze within a simple model. We then derive a higher-order coupling between the cavity and magnons, which is only present in bilayer systems, but does not rely on spin-orbit coupling. This interaction is found to be large, but only couples to the bimagnon operator. As a result, we find a strong, but heavily damped, bimagnon-cavity interaction which produces highly asymmetric cavity line shapes in the strong-coupling regime. To conclude, we outline several interesting extensions of our theory, including applications to carrier-doped cuprates and other strongly correlated systems with terahertz-scale magnetic excitations.

Figure 1

Overview of our proposal for forming cavity magnon-polaritons in insulating cuprates. The paper is structured into two parts. In part one (Sec. 4), we study a mechanism based on the dynamical Dzyaloshinskii-Moriya interaction. This interaction originates from spin-orbit effects, and the relevant displacement of the phonon mode is shown in (a). We illustrate the energy scales and interaction mechanisms for the cavity-phonon and magnon-phonon interactions, which results in an effective linear magnon-cavity coupling, shown in (b). In part two (Sec. 5), we consider a mechanism which is based on a bilayer-squeezing interaction. This interaction doesn't require spin-orbit effects and involves a bilayer-odd perturbation to the scalar exchange. The relevant phonon displacement is shown in (c). We illustrate the energy scales and interaction mechanisms for the cavity-phonon and magnon-phonon interactions, which results in an effective quadratic magnon-cavity coupling, shown in (d).

Figure 2

Dispersion of the two spin-wave bands for easy plane toy model with dispersion given in Eq. (15). This is shown for an exaggerated anisotropy of $\mathrm{\Gamma }\sim .1J_0$, which results in a large spin-wave gap for one of the modes (blue, upper). In reality, the anisotropy is expected to be much smaller, since the gap scales with the square root of the anisotropy.

Figure 3

Spin-wave spectrum of the bilayer system for realistic parameters of $J_0=150\mathrm{meV},\lambda /t=0.017,J_{ud}/J_0=0.0018$ for the case of Néel order along the $a$ axis ($\theta =0$). We see that is very close to a doubling of the easy-plane dispersion, except for one of the Goldstone modes is rendered massive by the bilayer coupling. Inset: Expanded view of the behavior near the origin, illustrating the masses of various modes. The bands are classified according to their parity under $\mathrm{\Pi }={\zeta }_1{l}_1$, with the upper and lower bands (red) being odd parity and the two middle bands (blue) being even parity.

Figure 4

Magnon contribution to the electromagnetic absorption $\Im {\chi }_{zz}^{\mathrm{diel}}(\omega ,\mathbf{q})$ for the easy-plane model, from Eq. (22). We take characteristic values of $J_0=150\mathrm{meV}, \mathrm{\Gamma }=0.01\mathrm{meV}$, and oscillator strength $F=6.4\times {10}^4{\mathrm{meV}}^2$, and choose to consider $\mathbf{q}\parallel \widehat{\mathbf{N}}\parallel {\widehat{\mathbf{e}}}_x$. We note that the effective oscillator strength vanishes as $\mathbf{q}\to 0$, due to the hybridization form factor ${\gamma }_{\mathbf{q}}^{\parallel }$. We also note that the dispersion of the spin wave sets in fairly rapidly.

Figure 5

Cavity spectral function including capacitive coupling to magnons obtained from Eq. (25). We take $V_{\mathrm{eff}}={\left(40\mathrm{nm}\right)}^3, d\sim R\sim 300a, h\sim 10a$, and parameters of spin-wave system from main text. We assume a cavity linewidth of $\kappa =1\mathrm{meV}$. We see a clear avoided crossing in the cavity spectral function as the cavity comes in to resonance with the spin-wave.

Figure 6

Size of effective coupling constant, characterized by $K\left(0\right)$, as per Eq. (26), as a function of distance $d$ to the resonator dipole, for three different values of the size of the dipole radius $R$ (for details on the specific model, see Appendix pp5). Other than varying $R$, we use the same parameters as in Fig. 5. In general, this depends on $R$ and $d$ as ${(R/d)}^6$.

Figure 7

Magnon contribution to the $\Im {\chi }_{zz}^{\mathrm{diel}}(\omega ,\mathbf{q})$ for the Rashba bilayer model, from Eq. (29). We again observe the vanishing oscillator strength at long wavelengths. In this case, the dispersion relation of the magnon is less trivial and we illustrate the susceptibility for three different configurations of the in-plane momentum and Neel order, illustrated in the lower part of the figure. This is shown specifically for three scenarios of Neel order at angles relative to the $a$ axis of (a) $\theta =0$, (b) $\theta ={45}^{\circ }$, and (c) $\theta ={75}^{\circ }$, respectively. We for simplicity only show $\mathbf{q}\parallel a$. While it is not visible on the chosen scale, there is a small contribution for a higher-lying spin-wave band which behaves similarly as the contribution from the dominant band.

Figure 8

Cavity spectral function including capacitive coupling to magnons obtained from Eq. (25) and using ${\chi }_{zz}^{\mathrm{diel}}$ as obtained in Eq. (29). We take $V_{\mathrm{eff}}={\left(40\mathrm{nm}\right)}^3, d\sim R\sim 300a, h\sim 10a$, and parameters of spin-wave system from main text. We assume a cavity linewidth of $\kappa =1\mathrm{meV}$. We see a clear avoided crossing in the cavity spectral function as the cavity comes in to resonance with the spin wave.

Figure 9

Magnon bands in the bilayer system: (a) The set of four magnon bands for $J_0=128 \mathrm{meV}, J_{ud}=0.5 \mathrm{meV}$, and $\mathrm{\Gamma }=0.016 \mathrm{meV}$; (b) Feynman diagram illustrating the bimagnon contribution to the dielectric susceptibility; (c) expanded view of the magnon band dispersion near the center of the Brillouin zone, with the interband pairs of magnons with opposite momenta shown schematically.

Figure 10

Contribution to the dielectric susceptibility due to bimagnon coupling, corresponding to the Feynman diagram in the inset of Fig. 9. We show the real part (solid blue) and imaginary part (dashed red) of the relative susceptibility ${\chi }_{zz}^{\mathrm{diel}}(\omega ,\mathbf{q}=0)/{\epsilon }_0$ as a function of the electric field frequency $\omega$. We use scalar exchange $J_0=128\mathrm{meV}$ and take interlayer exchange $J_{ud}=0.5\mathrm{meV}$, compatible with our estimates from Sec. 3b. (a) Response when easy-plane anisotropy = 0, corresponding to the case of two doubly degenerate magnon bands. The response sets in once the single-magnon threshold is passed, since in this case the bimagnon is actually a single gapped magnon, along with a soft Goldstone mode. (b) Response when $\mathrm{\Gamma }=0.16\mathrm{meV}$, the value obtained in previous sections. The dominant feature corresponds to a bimagnon excitation gap ${\mathrm{\Omega }}_{\mathrm{bimag}}\sim 33\mathrm{meV}$. In contrast to (a), we find a strong response at this frequency driven by the Van Hove singularity, which in two dimensions is logarithmic for the real part and steplike for the imaginary part.

Figure 11

Cavity spectral functions include contribution to the dielectric susceptibility due to bimagnon coupling illustrated in Fig. 7. (a) Density plot illustrating the behavior of the cavity spectral function $A_{\mathrm{cav}}\left(\omega \right)$ as the bare cavity resonance frequency ${\mathrm{\Omega }}_{\mathrm{cav}}$ is tuned across the bimagnon resonance for the case of $\mathrm{\Gamma }=0$, which corresponds to ${\chi }_{zz}$ from Fig. 7. Upon crossing the threshold for pair production, the imaginary part of ${\chi }^{\mathrm{diel}}$ sets in rapidly, leading to a mostly dissipative response and resulting diffuse line shape. (b) Density plot for cavity spectral function with finite $\mathrm{\Gamma }=0.16\mathrm{meV}$. At the edge of the bimagnon continuum, located around $\omega =33\mathrm{meV}$, the logarithmic singularity manifests as a visible feature in the cavity spectral function. Though not a true avoided crossing due to the onset of strong damping by the continuum, we do see some type of level repulsion as we attempt to tune the bare cavity frequency ${\mathrm{\Omega }}_{\mathrm{cav}}$ through the resonance. Note the cavity spectral function is renormalized such that the resonance occurs at ${\omega }^2={\mathrm{\Omega }}_{\mathrm{cav}}^2$ as ${\mathrm{\Omega }}_{\mathrm{cav}}^2\to 0$.

Figure 12

Line cuts of cavity spectral function from Fig. 11, with finite $\mathrm{\Gamma }=0.16\mathrm{meV}$ for constant ${\mathrm{\Omega }}_{\mathrm{cav}}$. (a) Line cut along ${\mathrm{\Omega }}_{\mathrm{cav}}=30\mathrm{meV}$. We see a small secondary shoulderlike feature appears in the cavity spectral function at the bimagnon resonance, but it is largely overshadowed by the bare resonance of the cavity. (b) As the cavity is tuned through the resonance, such that now ${\mathrm{\Omega }}_{\mathrm{cav}}=50\mathrm{meV}$, the shoulderlike feature becomes washed out be a broad overdamped continuum. In this case, the dominant effect of the magnons is to provide an asymmetric damping to the cavity line shape, which essentially just reflects this contribution.

Figure 13

Schematic of the proposed near-field coupling scheme: A metal object (grey sphere) is placed inside the resonator, which we model as an RLC circuit. The cavity electric field ${\mathbf{E}}_{\text{cav}}\left(t\right)$ induces a dipole moment ${\mathbf{P}}_{\text{ind}}\left(t\right)$ in the metal object. The sample (of thickness $h)$ is placed a distance $d$ below the induced dipole and experiences a dipole field ${\mathbf{E}}_{\text{ind}}\left(t\right)$, which samples momenta of order $\sim 1/d$.