Strong Coupling in Nanocavities

• Strong coupling in nanocavities is defined by energy exchange rates that exceed dissipation, forming hybrid photon–spin states with observable Rabi oscillations.
• The criterion relies on balancing enhanced coupling in optimally sized nanomagnets against limitations from crystalline anisotropy and multi-domain effects.
• Practical designs require tuning nanomagnet dimensions to preserve the macrospin regime, enabling robust coherence for applications like quantum memory and magnonic transduction.

Strong coupling in nanocavities defines a regime in which the rate of energy exchange between two (or more) quantum subsystems—typically a confined optical cavity mode and another degree of freedom such as a collective spin (macrospin) in a nanomagnet—exceeds all dissipation and dephasing processes. In this regime, new hybridized eigenstates are formed, leading to phenomena such as Rabi oscillations, macroscopic coherence, and robust entanglement between constituent degrees of freedom. The strong coupling criterion is both quantitative (involving explicit inequalities on coupling and loss rates) and qualitative, depending crucially on system-specific interactions, size effects, and internal degrees of freedom. The criterion for strong coupling in photonic nanocavities coupled to nanomagnets is sharply determined by the interplay between coupling strength, nanomagnet size-dependent phenomena (macrospin validity, crystalline anisotropy), and cavity or material losses (Soykal et al., 2010).

1. Coupling Dynamics: Hamiltonian and Coupling Strength Scaling

The underlying system consists of a nanomagnet of $N$ exchange-locked spins (forming a macrospin $S$), placed in a photonic cavity supporting an appropriate electromagnetic mode. The light–matter interaction Hamiltonian is of the form

$$\mathcal{H}_{\text{int}} = -g \mu_B \Gamma \left(a S_+ + a^\dagger S_-\right),$$

where $a$ ($a^\dagger$) are the photon annihilation (creation) operators, $S_{\pm}$ are the collective spin raising/lowering operators, $g$ is the g-factor, $\mu_B$ the Bohr magneton, and $\Gamma$ an overall cavity-mode dependent factor. This term describes coherent energy exchange between cavity photons and nanomagnet spin-flips.

The coupling strength $\Gamma$ depends on the nanomagnet size:

• For small excitation numbers, the total spin $S$ scales as $S \approx \sqrt{N}$ where $N$ is the number of locked spins, i.e., proportional to nanomagnet volume $V$. Accordingly, the coupling scales as $\Gamma \propto \sqrt{V}$.
• In the superradiant regime (large photon or excitation number), the effective coupling scales as $V^{3/2}$ due to collective enhancement (Soykal et al., 2010).

This dynamical exchange of energy mediates the hybridization of photon–spin states; only when the coupling rate exceeds dissipative rates can reversible, coherent Rabi oscillations occur.

2. Quantum States: Entanglement, Coherence, and Eigenstate Structure

The coupled system conserves the total excitation number, so eigenstates are superpositions of joint photon number and spin projection basis states $|n, m_s\rangle$. The effective Hamiltonian can be written in a tight-binding form,

$$\mathcal{H} = \sum_n E_0 |n\rangle\langle n| - \tau(n)\left(|n+1\rangle\langle n| + |n\rangle\langle n+1|\right), \quad \tau(n) = \hbar \Gamma g\mu_B (n+1)\sqrt{2\xi - n},$$

where $\xi = N/2$.

The spread of the eigenstate (i.e., how many photonic and spin states are mixed) and the emergence of robust, oscillatory "coherent states" depend on the breadth of $\tau(n)$. As nanomagnet size increases, more photonic and spin states participate, enabling extended coherence and larger amplitude Rabi oscillations in photon and spin expectation values.

The ability to combine highly entangled photon–spin eigenstates into coherent oscillatory states is a haLLMark of the strong coupling regime.

3. Size-Dependent Regimes: Optimal Nanomagnet Dimensions

Nanomagnet size is a double-edged parameter for strong coupling. Larger nanomagnets:

• Increase the overall coupling strength and the number of active photon/spin states involved in coherent oscillations.
• Broaden the Hilbert space (states from $m_s = -S$ up to $S$), increasing the coherence "bandwidth."

However, there is a crucial trade-off:

• For small nanomagnets, crystalline anisotropy energy (CMA) is dominant (see below), localizing the eigenstates and suppressing coherent mixing.
• For large nanomagnets, the macrospin approximation fails; the material breaks into multiple domains each with its own local spin, which decouples the system and destroys the single, collective magnon–photon resonance.

The optimal nanomagnet radius is thus found just below the threshold for domain formation or macrospin breakdown—typically in the $10$–$50\,\text{nm}$ range (Soykal et al., 2010).

4. Crystalline Anisotropy: Localization and the Low-Size Limit

For nanomagnets with small volume, the crystalline magnetic anisotropy (CMA) dominates. The anisotropy energy is given by

$$E_{\text{CMA}} = U_1\left(\kappa_1^2 \kappa_2^2 + \kappa_2^2 \kappa_3^2 + \kappa_1^2 \kappa_3^2\right) + U_2 \kappa_1^2 \kappa_2^2 \kappa_3^2,$$

with $U_1, U_2$ as material-dependent constants and $\kappa_{i}$ denoting directional cosines.

When the CMA energy dispersion across spin orientations $>$ magnet–cavity coupling, the system eigenstates are pinned in spin number: the admixture of photon and spin states is energetically suppressed, and coherent state formation is quenched. Therefore, no matter how high the quality factor or external drive, sufficiently small nanomagnets cannot reach the strong coupling regime due to this intrinsic localization effect.

5. Macrospin Validity: Domain Effects and Upper Size Bound

The coupling model presumes the macrospin approximation—that all $N$ microscopic spins are exchange-locked into a single rigid entity. This approximation strictly fails once the nanomagnet exceeds the critical size for single-domain stability, at which point:

• Internal spin dynamics (e.g., domain wall formation, noncollinear spin waves) become relevant.
• Different spatial domains couple independently, leading to decoherence and suppression of the collective magnon–photon resonance.
• The simple collective spin–photon coupling Hamiltonian becomes invalid.

Thus, the strong coupling criterion requires not only that the magnet–photon coupling rate be large but also that the nanomagnet be below the single-domain threshold size.

6. Quantitative Criterion and Summary Table

The emergence of strong coupling is encapsulated by both dynamical and structural criteria. These can be summarized as follows:

Parameter	Weak Coupling (WC)	Strong Coupling (SC)	Limiting Factors
Coupling strength	$\ll$ loss rates	$>$ loss rates, Rabi frequency exists	-
Eigenstate participation	Localized (few states)	Extended (many photon/spin states)
Coherence (oscillations)	Damped, nonoscillatory	Macroscopic, coherent Rabi oscillations	CMA/localization (small size); Macrospin breakdown (large size)
Size regime	$r<10\,\text{nm}$ or $r>50\,\text{nm}$	$r\sim$10–50 nm	CMA, domain formation

In explicit form, the core strong coupling condition is: $$\tau(n)/\hbar \gg \kappa_{\text{cavity}},\,\gamma_{\text{magnet}},\ \text{and single-domain macrospin},$$ where $\kappa_{\text{cavity}}$ is the cavity photon loss rate, $\gamma_{\text{magnet}}$ the spin relaxation rate, and $\tau(n)$ is the $n$-dependent coupling matrix element.

The practical recipe is:

• Choose nanomagnet sizes $r\approx10$–$50\,\text{nm}$ to maximize coupling but avoid multi-domain formation.
• Ensure the loss rates—both cavity and magnetic—lie well below the characteristic coupling-induced oscillation rate $\tau(n)/\hbar$.
• Verify that the crystalline anisotropy energy at the relevant field is insufficient to localize the eigenstates.

7. Broader Implications and Applicability

The strong coupling criterion for nanocavities is central for engineering entangled light–matter states and creating macroscopic quantum phenomena such as coherent state oscillations and superradiance in nanomagnet–cavity systems. These conditions are also vital for potential applications in quantum memory, magnonic quantum transduction, and cavity-enhanced magnetic resonance, where robustness against both environmental decoherence and intrinsic localization/demagnetization is required.

This rigorous synthesis, as elaborated in (Soykal et al., 2010), reveals that achieving strong coupling in nanocavity–nanomagnet systems is governed by a delicate balance between enhanced photon–spin coupling (favoring larger nanomagnets), suppression of anisotropy-induced localization (favoring intermediate sizes), and preservation of the macrospin regime (imposing an upper size limit). For well-engineered systems, this window enables realization of highly coherent hybridized states with prospects for quantum technology integration.