Appendix: Exchange-dipolar magnons - Important note To cite this publication, please use the fi

5

The option to shrink the cavity and optical volume is limited by the wavelength 𝜆 /𝑛 . For 𝜆 = 1.3 𝜇m and 𝑛 = 2.2, a cavity with an optical volume of 𝜆 /𝑛 gives an upper limit ∼ 2𝜋 × 50 kHz for pure YIG. In a Bi:YIG sphere of radius ∼ 𝜆 /𝑛 , the optical first Mie resonance may strongly couple with the Kittel mode [4].

The coupling can be enhanced by the ellipticity angle𝜃 of the magnetization, which is controlled by crystalline anisotropy, saturation magnetization, and geom-etry. Linear polarization𝜃 → 0 or 𝜃 → 𝜋/2 would lead to a unphysical diverging coupling, because in practice magnons are close to circularly polarized, 𝜃 ≈ 𝜋/4.

For YIG spheres the weak ellipticity even suppresses the coupling, 𝑀 < 1 in Eq.

(5.48).

In purely dipolar theory, the surface magnons are chiral, i.e. only modes with 𝑚 > 0 exist, implying a complete suppression of the red sideband that hinders magnon cooling [3]. This is not necessarily the case when the exchange interaction kicks in [35]. An analysis similar to the one above indeed indicates that exchange-dipolar magnons are only partially chiral, since modes with 𝑚 < 0 acquire finite amplitude¹.

We find that light may efficiently pump or cool certain surface (low wavelength) magnons that do not couple easily to microwaves. This could be used to ma-nipulate macroscopically coherent magnons, raising hopes of accessing interesting non-classical dynamics in the foreseeable future.

5.6. Appendix: Exchange-dipolar magnons

Here, we solve Eqs. (5.23)-(5.26) with Maxwell boundary conditions, Eq. (5.27), and pinned surface magnetization𝑚_±(𝑅) = 0. The magnetization in the linearized LL equation, Eq. (5.24), can be eliminated in favor of the scalar potential𝜓, Eq.

(5.23) [28],

[(𝒪 − 𝜔 ) ∇ + 𝜔 𝒪 (∇ − 𝜕

𝜕𝑧 )] 𝜓 = 0, (5.55)

where 𝒪 = 𝜔 − 𝐷_ex∇ with 𝐷_ex = 𝜔 /𝑘_ex. The general solution for a sphere is complicated because the magnetization breaks the rotational symmetry, but it can be simplified for the surface magnons near the equator. The ansatz

𝜓(r) = 𝑌 (𝜃, 𝜙)Ψ(𝑟), (5.56) are spherical harmonic functions with associated Legendre polynomials

𝑃 (𝑥) =(−1)

2 𝑙! (1 − 𝑥 ) ^/ 𝑑

𝑑𝑥 (𝑥 − 1) , (5.58)

1Our calculations not discussed here

5

have spherical Bessel functions of order𝑙 as eigenfunctions. The surface magnons with large angular momentum 𝑙 are localized near the equator. They have a large

“kinetic energy” along the equator. The confinement along the 𝜃-direction is not so strong, however, so the magnon amplitude looks like a flat tire. A posteriori, we find 𝑘 ∝ √𝑙, while 𝑘 ∝ 𝑙. For large 𝑙, the terms 𝜕 ≈ 𝑅 𝜕 near the equator, may therefore be disregarded in Eq. (5.55). This gives a cubic in ̂𝑂 , similar to a magnetic cylinder [31], i.e. waves propagating along the equator [see Sec. 5.4].

Consider the eigenvalue equation ̂𝑂 Ψ = −𝜇 Ψ with reciprocal “length scales”

𝜇 ∈ {0, 𝑘, 𝑖𝜅}. Its two linearly independent solutions are spherical Bessel functions of first and second kind, which in the limit𝑙 ≫ 1 are proportional to Bessel functions of first [𝐽 (𝜇𝑟)] and second [𝑌 (𝜇𝑟), not to be confused with the spherical harmonic 𝑌 ] kind, respectively. 𝑌 (𝜇𝑟) diverges at 𝑟 = 0, so inside the sphere Ψ = 𝐽 (𝜇𝑟).

Thus, Eq. (5.60) has three linearly independent solutions, {Ψ , Ψ , Ψ } and the general solution is

The spatial distribution of the three components are discussed in more detail in the main text [see Sec. 5.3].

The derivative introduced in Sec. 5.2

𝜕_±𝜓 = 𝑌 𝑒^± ∑ 𝛼

𝐽 (𝜇 𝑅)(𝐽 (𝜇 𝑟) ∓ 𝑚𝐽 (𝜇 𝑟)

𝜇 𝜌 ) , (5.65)

5.6.Appendix: Exchange-dipolar magnons

5

where𝜕_±= 𝜕 ± 𝑖𝜕 . Close to the equator, 𝜌 ≈ 𝑟 and using 𝑙 ≫ |𝑙 − 𝑚|,

𝜕_±𝜓 ≈ ∓𝑌_±^± ∑ 𝐽_± (𝜇 𝑟)

𝐽 (𝜇 𝑅), (5.66)

where we used the recursion relations [10]

𝐽 _± (𝑥) = 𝛼

𝑥𝐽 (𝑥) ∓ 𝐽 (𝑥) (5.67)

and 𝑌_±^± ≈ 𝑒^± 𝑌 that holds for 𝑙 ≫ 1, |𝑙 − 𝑚|. Solving Eq. (5.24) for magneti-zation,

𝑚_±(r) = 𝑌_±^± ∑ 𝜁_,±𝐽_± (𝜇 𝑟)

𝐽 (𝜇 𝑅), (5.68)

with coefficients

𝜁_,±= 𝜔 𝛼

𝜔 ± ̃𝜔 , (5.69)

and𝜔 = 𝜔 + 𝐷̃ _ex𝜇 .

Outside the magnet, 𝜓 satisfies a Laplace equation Eq. (5.26). Using the continuity of magnetic potential and𝜓 → 0 at 𝑟 → ∞,

𝜓 = 𝑌 (𝜃, 𝜙) (𝑅

𝑟) ∑ 𝛼 𝐽 (𝜇 𝑅)

𝜇 𝐽 (𝜇 𝑅). (5.70)

The integration constants𝛼 are governed by the boundary conditions: Maxwell boundary conditions, Eq. (5.27), and pinned magnetization boundary condition for the LL equation𝑚_± = 0, which we justified a posteriori in Sec. 5.3. Demanding 𝑚 (𝑟 = 𝑅) = 0 and 𝜕 (𝜓 − 𝜓 )| = 0 gives

∑ 𝜔 𝛼

𝜔 − ̃𝜔 = 0 = ∑ 𝛼 , (5.71)

which is solved by

𝛼 = 𝑚 (𝜔 − ̃𝜔 )( ̃𝜔 − ̃𝜔 )

𝜔 , (5.72)

𝛼 = 𝑚 (𝜔 − ̃𝜔 )( ̃𝜔 − ̃𝜔 )

𝜔 , (5.73)

𝛼 = 𝑚 (𝜔 − ̃𝜔 )( ̃𝜔 − ̃𝜔 )

𝜔 , (5.74)

where𝑚 is a normalization constant.

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We now arrive at the solution discussed in the main text, Sec. 5.3. With {𝜇 , 𝜇 , 𝜇 } = {0, 𝑘, 𝑖𝜅}

lim→ 𝐽 (𝜇 𝑟) ≈ 1 𝑙!(𝜇 𝑟

2 ) ; 𝐽 (𝑖𝜅𝑟) = 𝑖 𝐼 (𝜅𝑟), (5.75) where𝐼 is the modified Bessel function. The above holds also for 𝑙 → 𝑙 ± 1. Substi-tuting into Eq. (5.68),

The Bessel function ratios in the third terms are real even though𝐽 (𝑖𝜅𝑟) need not be.

According to Eq. (5.69) the polarization does not depend on the coefficients𝛼 . With{ ̃𝜔 , ̃𝜔 , ̃𝜔 } = {𝜔 , 𝜔_sq− 𝜔 /2, −𝜔_sq− 𝜔 /2}, 𝜔_sq= 𝜔 + 𝜔 /4 the form Eq. (5.30) in the main text.

Substituting 𝛼 for the pinned boundary conditions, Eqs. (5.72-5.74), into Eq.

(5.69)

The above solutions satisfy Maxwell’s boundary conditions, Eq. (5.27), and 𝑚 (𝑅) = 0 by design [see Eq. (5.71)]. The last condition 𝑚 (𝑅) = 0 gives the The roots of the above equation are counted by 𝜈 ≥ 0. For 𝑘 > 0, the lowest root 𝜈 = 0 occurs near 𝑘 ≈ 0 at frequency 𝜔 ≈ √𝜔 + 𝜔 𝜔 . The next and higher roots occurs only around 𝑘𝑅 ≳ 𝑙 as plotted in Fig. 5.4 [the root 𝜈 = 0 is to the

References

5 0 1 2 3 4 5 6 7

(ω − ω

)/ω

[ ×10

⁻³

]

−4

−2 0 2

4

1(ω) R²(ω)

Figure 5.4: The resonance conditionℛ ℛ gives the allowed magnon frequencies when the magne-tization is pinned at the surface. is the frequency at which .

far left of the origin]. ℛ is a rapidly varying function, while ℛ ≈ 1.2 is nearly constant. Sufficiently far from the zeroes of 𝐽 (𝑘𝑅), ℛ < 0 and at the crossing with ℛ , ℛ ≈ 1.2. This implies that at magnon resonances, 𝐽 (𝑘𝑅) ≈ 0 or 𝑘𝑅 ≈ 𝑙 + 𝛽 (𝑙/2) ^/ , while𝜔(𝑘) is given by Eq. (5.61). Their explicit values are discussed in Sec. 5.3

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[3] S. Sharma, Y. M. Blanter, and G. E. W. Bauer, Optical cooling of magnons, Phys. Rev. Lett. 121, 087205 (2018).

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Enhancing the modulation of near-infrared light by spin waves,Phys. Rev. B 97, 184406 (2018).

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Curriculum Vitæ

Sanchar SHARMA

11-09-1992 Born in Jaipur, Rajasthan, India.

Education

2010 High School

Vidya Mandir School, Kota, Rajasthan, India

2010–2015 Dual Degree (Bachelors & Masters) in Electrical Engineering Indian Institute of Technology, Mumbai, India

Thesis: Self-oscillations in domain wall magnets Supervisors: Prof. dr. Bhaskaran Muralidharan and Prof. dr.

Ashwin Tulapurkar 2015–2019 PhD in Applied Physics

Delft University of Technology, Delft, the Netherlands Thesis: Cavity Optomagnonics

Promotors: Prof. dr. ir. Gerrit E. W. Bauer and Prof. dr.

Yaroslav M. Blanter

List of Publications

6. S. Sharma, B. Z. Rameshti, Y. M. Blanter, and G. E. W. Bauer, Optimal mode matching in cavity optomagnonics,arXiv:1903.01718

5. T. Yu, S. Sharma, Y. M. Blanter, and G. E. W. Bauer, Surface dynamics of rough magnetic films,Phys. Rev. B 99, 174402 (2019)

4. S. Sharma, Y. M. Blanter, and G. E. W. Bauer, Optical cooling of magnons, Phys. Rev. Lett. 121, 087205 (2018)

3. J. A. Haigh, N. J. Lambert, S. Sharma, Y. M. Blanter, G. E. W. Bauer, and A.

J. Ramsay, Selection rules for cavity-enhanced Brillouin light scattering from magnetostatic modes,Phys. Rev. B. 97, 214423 (2018)

2. S. Sharma, Y. M. Blanter, and G. E. W. Bauer, Light scattering by magnons in whispering gallery mode cavities,Phys. Rev. B. 96, 094412 (2017)

1. S. Sharma, B. Muralidharan, and A. Tulapurkar, Proposal for a Domain Wall Nano-Oscillator driven by Non-uniform Spin Currents, Sci. Rep. 5, 14647 (2015)

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