# Analyzing the Influence of Geometrical Deformation on Photon Sphere and
Shadow Radius: A New Analytical Approach -Stationary, and Axisymmetric
Spacetime

Vitalii Vertogradov<EMAIL_ADDRESS>Physics department, Herzen state
Pedagogical University of Russia, 48 Moika Emb., Saint Petersburg 191186,
Russia  SPB branch of SAO RAS, 65 Pulkovskoe Rd, Saint Petersburg 196140,
Russia  Ali Övgün<EMAIL_ADDRESS>Physics Department, Eastern
Mediterranean University, Famagusta, 99628 North Cyprus, via Mersin 10,
Turkiye  Reggie C. Pantig<EMAIL_ADDRESS>Physics Department, Mapúa
University, 658 Muralla St., Intramuros, Manila 1002, Philippines

###### Abstract

Black hole shadows and photon spheres offer valuable tools for investigating
black hole properties. Recent observations by the Event Horizon Telescope
Collaboration have confirmed the existence of rotating black holes. Black hole
parameters influence the observed shadow size. This paper explores the impact
of geometrical deformations on black hole shadow size using gravitational
decoupling applied to axially-symmetric spacetime. We find the results are
more complex than the spherically-symmetric case. We compare shadows in well-
known models with those of an Kerr black hole. Our approach suggests that the
influence of an accretion disc on the observed shadow shape can be accurately
described despite negligible impact on the black hole geometry itself.

Rotating Black hole; Shadow; Photon sphere; Gravitational decoupling;
Deformation

###### pacs:

95.30.Sf, 04.70.-s, 97.60.Lf, 04.50.Kd

##  I Introduction

Black hole shadow is one of the observable quantities of a black hole. The
spacetime is curved so strong near these objects that light can move along
circular orbit which forms a black spot on the observer’s sky $\sim 2.5$ times
bigger than Euclidean size111Here, under Euclidean size we mean light
propagation along straight lines. In this case the visible black spot is the
size of the event horizon .. Recent observation by Event horizon Collaboration
Telescope bib:eht1 ; bib:eht2  revealed the black hole in the centre of
galaxies M87 and Milky Way. There are many papers exploring properties of
static black holes bib:tsupko_plasma ; bib:tsupko_deflection ;
bib:tsupko_lensing ; bib:tsupko_kerr ; bib:klaudel ; bib:hod ;
Virbhadra:2024pru ; Adler:2022qtb ; Virbhadra:2022iiy ; Virbhadra:2007kw ;
Virbhadra:1999nm ; Virbhadra:1998dy ; bib:khoo ; bib:decaniny ; bib:shoom ;
bib:sederbaum ; bib:jokhannsen ; bib:teo ; Okyay:2021nnh ; Pantig:2022gih ;
Kuang:2022xjp ; Zakharov:2023lib ; Zakharov:2021gbg ; Qiao:2022hfv ;
Hod:2020pim ; Lu:2019zxb ; Paithankar:2023ofw ; Aratore:2024bro ;
Tsupko:2022kwi ; bib:rezzolla ; bib:rezzolla2 ; bib:expanding ; bib:vaidya1 ;
bib:vaidya2 ; bib:ver_shadow ; bib:tsupko_first ; bib:understanding ;
bib:joshi_shadow ; Tsukamoto:2014tja ; Tsukamoto:2017fxq ; Gomez:2024ack ;
Adair:2020vso ; Ghosh:2023kge . (See also a recent review bib:tsupko_review ).
The black hole shadow can serve as a cosmological ruler bib:tsupko_ruller ;
bib:vanozi_ruller  and as a means to test various black hole models bib:vanozi
. Real astrophysical black hole should be surrounded by an accretion disc
which has an impact on visible size of a shadow. Indeed, one approach involves
considering a black hole surrounded by plasma and examining how light
propagation within it affects the angular size of the shadow. Alternatively,
the black hole-accretion disk system can be treated as a solution of the
Einstein field equations, allowing for the analysis of light trajectories
within the obtained spacetime.

One can describe a black hole surrounded by an accretion disc by using
gravitational decoupling method bib:gd1 ; bib:gd2 ; Contreras:2021yxe . It is
a novel tool which allows, under some conditions, consider a solution of
Einstein field equation as a straight superposition of two solutions. One can
consider the deformation of $g_{rr}$ component of the metrec (minimal
geometrical deformation) or one can deform both $g_{rr}$ and $g_{tt}$ metric
components (extended gravitational decoupling). Using gravitational
decoupling, several new solutions of the Einstein field equations have been
discovered, depicting deformed versions of well-known black hole solutions
bib:bh1 ; bib:bh2 ; bib:ovalle_regular ; bib:max ; bib:max_regular ;
bib:kiselev ; bib:rotating ; bib:varmhole . These deformations arise due to
the presence of an additional anisotropic matter field surrounding the black
hole. In recent paper Vertogradov:2024qpf , the effect of minimal geometrical
deformations, which can be caused by the extra matter field surrounding a
black hole, on the radius of a photon sphere and visible angular size has been
explored. One more important fact regarding real astrophysical black holes is
that they should be rotational ones. The comparison of obtained images with
Kerr spacetime surrounded by an accretion disc revealed angular momentum of a
black hole around $a\sim 0.7$ bib:doc .

In this paper, we investigate the influence of minimal geometrical
deformations on apparent shape of a shadow in a general axially-symmetric
black hole. As opposed to spherical-symmetric black hole, which casts a shadow
as a circle, in rotational case the shadow is deformed and flattened on one
side. The apparent shape of a shadow depends on the angle of observation. In
order to describe the shadow of rotational black hole, one needs to introduce
two impact parameters $\xi$ and $\eta$. If one considers the geometrical
deformations of the seed spacetime then the impact parameters and apparent
shape changes. Here, we describe the influence of these deformations on the
apparent shape and compare results with a shadow of a seed spacetime.

The paper organized as follows: In sec.II we describe the method of
calculation of a shadow in general axially-symmetric spacetime. In sec. III we
elaborate the influence of minimal geometrical deformations on the apparent
shape of a black hole shadow. We give several examples in sec. IV and discuss
obtained results in sec. V. The system of units $G=c=1$ and signature $-+++$
will be used throughout the paper.

##  II The influence of deformation on shadow of axially symmetric and
rotating black holes

Without loss of generality, we can assume Vertogradov:2024qpf

$\displaystyle
ds^{2}=-\left(1-\frac{M(r)}{r}\right)e^{\alpha\gamma(r)}dt^{2}+\left(1-\frac{M(r)}{r}\right)^{-1}e^{-\alpha\gamma(r)}dr^{2}+r^{2}d\Omega^{2},$
(1)

and for the axially symmetric case one can write it in this form
Contreras:2021yxe :

$ds^{2}=\left[1-\frac{2\,r\,{m}(r)}{\rho^{2}}\right]dt^{2}+\frac{4\,a\,r\,{m}(r)\sin^{2}\theta}{\rho^{2}}\,dt\,d\phi-\frac{\rho^{2}}{\Delta}\,dr^{2}-\rho^{2}d\theta^{2}-\frac{\Sigma\sin^{2}\theta}{\rho^{2}}d\phi^{2},$
(2)

where the expressions for $\rho$, $\Sigma$ and $\Delta$ are

$\displaystyle m(r)=\frac{1}{2}\left[e^{\alpha\gamma(r)}(2M-r)+r\right],$ (3)

with

$\displaystyle\rho^{2}$ $\displaystyle=$ $\displaystyle
r^{2}+a^{2}\cos^{2}\theta,$ (4) $\displaystyle\tilde{\Delta}$ $\displaystyle=$
$\displaystyle r^{2}-2\,r\,m(r)+a^{2},$ (5) $\displaystyle\Sigma$
$\displaystyle=$
$\displaystyle\left(r^{2}+a^{2}\right)^{2}-a^{2}\,\Delta\,\sin^{2}\theta,$ (6)

and

$a\,=\,J/M\ ,$ (7)

where $J$ is the angular momentum and $M$ the total mass of the system. Note
that the line-element (2) reduces to the Kerr solution when the metric
function $m=M$. Moreover, when $a=0$, we obtain the Schwarzschild-like metric.

We briefly review how to study null geodesics in a rotating space-time like
the one in Eq. (2), and find the celestial coordinates describing the BH
shadow. The Hamilton-Jacobi equation

$\frac{\partial
S}{\partial\lambda}=\frac{1}{2}\,g^{\mu\nu}\,\partial_{\mu}S\,\partial_{\nu}S\
,$ (8)

where $\lambda$ is a parameter along the curve and $S$ the Jacobi action.
Given the symmetries of the space-time (2), Eq. (8) is separable and one has

$S=-E\,t+\Phi\,\phi+S_{r}(r)+S_{\theta}(\theta)\ ,$ (9)

with $E$ and $\Phi$ being the conserved energy and angular momentum,
respectively. Replacing (9) in Eq. (8), we obtain

$\displaystyle S_{r}$ $\displaystyle=$
$\displaystyle\int\frac{\sqrt{R(r)}}{\Delta}\,dr$ $\displaystyle S_{\theta}$
$\displaystyle=$ $\displaystyle\int\sqrt{\Theta(\theta)}\,d\theta\ ,$

where

$\displaystyle R$ $\displaystyle=$
$\displaystyle\left[(r^{2}+a^{2})\,E-a\,\Phi\right]^{2}-\Delta\left[Q+(\Phi-a\,E)^{2}\right]$
$\displaystyle\Theta$ $\displaystyle=$ $\displaystyle
Q-(\Phi^{2}\,\csc^{2}\theta-a^{2}\,E^{2})\,\cos^{2}\theta\ ,$

with $Q$ the Carter constant.

The (unstable) circular photon orbits are determined by $R=R^{\prime}=0$,
namely

$\displaystyle\left(a^{2}-a\,\xi+r^{2}\right)^{2}-\left(a^{2}+r^{2}F\right)\left[(a-\xi)^{2}+\eta\right]=0$
(12) $\displaystyle
4\left(a^{2}-a\,\xi+r^{2}\right)-\left[(a-\xi)^{2}+\eta\right]\left(r\,F^{\prime}+2\,F\right)=0\
,$

where $\xi=\Phi/E$ and $\eta=Q/E^{2}$ are the impact parameters. Accordingly,

$\displaystyle\xi$ $\displaystyle=$ $\displaystyle
a+\frac{r^{2}}{a}-\frac{4\left(a^{2}+r^{2}\,F\right)}{a\left(r\,F^{\prime}+2\,F\right)}\
,$ $\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{r^{2}\left[r^{2}+2\,a\left(a-\xi\right)-\left(a-\xi\right)^{2}F\right]}{a^{2}+r^{2}\,F}\
,$ (13)

where $r$ is the radius of the unstable photon orbit.

The apparent shape of the shadow is finally described by the celestial
coordinates

$\displaystyle\sigma$ $\displaystyle\equiv$
$\displaystyle\lim\limits_{r_{0}\to\infty}\left(-r_{0}^{2}\,\sin\theta_{0}\frac{d\phi}{dr}\bigg{|}_{(r_{0,\theta_{0}})}\right)$
(14) $\displaystyle=$ $\displaystyle-\frac{\xi}{\sin\theta_{0}}$

and

$\displaystyle\beta$ $\displaystyle\equiv$
$\displaystyle\lim\limits_{r_{0}\to\infty}\left(r_{0}^{2}\,\frac{d\theta}{dr}\bigg{|}_{(r_{0},\theta_{0})}\right)$
(15) $\displaystyle=$
$\displaystyle\sqrt{\eta-\xi^{2}\,\cot^{2}\theta_{0}+a^{2}\,\cos^{2}\theta_{0}}\
,$

where $(r_{0},\theta_{0})$ are the coordinates of the observer.

##  III The influence of geometrical deformation on apparent shape of a shadow

The dimensionless parameter $\alpha$ is supposed to be small $\alpha\ll 1$, so
one can consider its influence on the apparent shape of a shadow which depends
on impact parameters $\xi$ and $\eta$ (II) by assuming

$\displaystyle\xi$ $\displaystyle=$ $\displaystyle\xi_{0}+\alpha\xi_{1},$
$\displaystyle\eta$ $\displaystyle=$ $\displaystyle\eta_{0}+\alpha\eta_{1},$
(16)

here $\xi_{0}$ and $\eta_{0}$ are impact parameters along null geodesics in
seed spacetime, $\xi_{1}$ and $\eta_{1}$ are small changes caused by presence
of an extra matter field. We can write $\xi$ and $\eta$ as functions of
$\alpha$ and expand with respect to coupling constant. By notesing that
$\xi(0)=\xi_{0}$ and $\eta(0)=\eta_{0}$, substituting (III) into (II) and
expanding, we find $\xi_{1}$ and $\eta_{1}$ as

$\displaystyle\xi_{1}$ $\displaystyle=$
$\displaystyle\frac{r\left(r(r-2M)\left(a^{2}+r(r-2M)\right)\gamma^{\prime}(r)+2a^{2}(r-M)\gamma(r)\right)}{a(M-r)^{2}},$
$\displaystyle\eta_{1}$ $\displaystyle=$
$\displaystyle-\frac{2r^{3}\left(r(r-3M)(r-2M)\left(a^{2}+r(r-2M)\right)\gamma^{\prime}(r)+2a^{2}M(M-r)\gamma(r)\right)}{a^{2}(M-r)^{3}}.$
(17)

where for the seed Schwarzschild spacetime impact parameters are:

$\displaystyle\xi_{0}=\frac{a^{2}(M+r)+r^{2}(r-3M)}{a(M-r)}$ (18)

and

$\displaystyle\eta_{0}=-\frac{r^{3}\left(r(r-3M)^{2}-4a^{2}M\right)}{a^{2}(M-r)^{2}}.$
(19)

As one can notice to estimate the influence of geometrical deformation, one
should know the impact parameters in seed spacetime and the value of $\gamma$
and $\gamma^{\prime}$.By substituting (III)into (14) and (15) with $\xi_{1}$
and $\eta_{1}$ from (III), we find the apparent shape of deformed axially-
symmetric black hole

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle-\frac{\xi_{0}+\alpha\xi_{1}}{\sin\theta_{0}},$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\sqrt{\eta_{0}+\alpha\eta_{1}-\xi_{0}(\xi_{0}+2\alpha\xi_{1})\cot^{2}\theta_{0}+a^{2}\cos^{2}\theta_{0}}.$
(20)

The celestial coordinates $\sigma$ and $\beta$ show the apparent perpendicular
distances of the image around the black hole. At $\theta_{0}=\pi/2$, the
celestial coordinates become

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle-\xi_{0}-\alpha\xi_{1},$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\sqrt{\eta_{0}+\alpha\eta_{1}}.$ (21)

The photonsphere radius $r_{\rm ph}$ can then be determined by solving
$\beta=0$ for r. After substituting all parameters we find

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle\frac{\alpha
r^{2}(2M-r)\left(a^{2}+r(r-2M)\right)\gamma^{\prime}(r)-(M-r)\left(a^{2}(M+r)-2\alpha
a^{2}r\gamma(r)+r^{2}(r-3M)\right)}{a(M-r)^{2}},$ $\displaystyle\beta$
$\displaystyle=$ $\displaystyle\sqrt{\frac{r^{3}\left((r-M)\left(4\alpha
a^{2}M\gamma(r)-4a^{2}M+r(r-3M)^{2}\right)-2\alpha
r(r-3M)(r-2M)\left(a^{2}+r(r-2M)\right)\gamma^{\prime}(r)\right)}{a^{2}(M-r)^{3}}}.$
(22)

Note that one can also define Feng:2019zzn

$R_{\mathrm{sh}}=\frac{1}{2}\left[\sigma\left(r_{\mathrm{ph}}^{+}\right)-\sigma\left(r_{\mathrm{ph}}^{-}\right)\right].$
(23)

###  III.1 Example 1: Hairy Kerr spacetime by gravitational decoupling

The hairy Kerr spacetime by gravitational decoupling has been found in the
paper bib:rotating . The $\gamma$ and $M(r$ for this spacetime are given by

$\displaystyle M(r)$ $\displaystyle\equiv$ $\displaystyle M=\text{const.},$
$\displaystyle\alpha\gamma(r)$ $\displaystyle=$
$\displaystyle\ln\left|1-\alpha\frac{l-re^{-\frac{r}{M}}}{r-2M}\right|,$ (24)

The photonsphere radius $r_{\rm ph}$ is found numerically by $\beta=0$ and
then from Eq. 23, shadow radius is obtained as shown in Table 1 and in Fig. 1.
Clearly, increasing the parameter l results in an increase in both the photon
sphere size and the shadow radius.

| $r_{\mathrm{ph}}^{-}$ | $r_{\mathrm{ph}}^{+}$ | $R_{\mathrm{sh}}$
---|---|---|---
l=0.1 | 2.85 | 3.09 | 5.28
l=0.3 | 2.89 | 3.12 | 5.37
l=0.5 | 2.91 | 3.16 | 5.64
l=0.7 | 2.93 | 3.20 | 6.16
l=0.9 | 2.95 | 3.26 | 7.16
Table 1: Table illustrating the behavior of photon spheres and shadow radius
with varying values of the parameter l with constants $M=1$, $\alpha=0.1$, and
$a=0.1$.

Figure 1: Hairy Kerr black hole shadow ($a=0.95M$, $\theta=\pi/2$). Upper left
panel: $\alpha=0.35$. Upper right panel: $\alpha=-0.35$. Lower left panel:
$\alpha=0.10$. Lower right panel: $\alpha=-0.35$.

###  III.2 Example 2: Rotating Kiselev solution

In rotating Kiselev solution bib:kiselev , we have

$\displaystyle M$ $\displaystyle=$ $\displaystyle\text{const}.,$
$\displaystyle\alpha\gamma(r)$ $\displaystyle=$
$\displaystyle\ln\left|1-\frac{N}{r^{3\omega}(r-2M)}\right|.$ (25)

Here $N$ is cosmological parameter and $\omega$ is a parameter of the equation
of state. In order to satisfy energy condition $N$ and $\omega$ should have
opposite signs. The photonsphere radius $r_{\rm ph}$ is found numerically by
$\beta=0$ and then from Eq. 23, shadow radius is obtained as shown in Table 2
and in Fig. 2. Clearly, increasing the parameter $\omega$ results in an
decrease in both the photon sphere size and the shadow radius.

| $r_{\mathrm{ph}}^{-}$ | $r_{\mathrm{ph}}^{+}$ | $R_{\mathrm{sh}}$
---|---|---|---
$\omega$=0.1 | 2.894 | 3.125 | 5.240
$\omega$=0.2 | 2.891 | 3.123 | 5.224
$\omega$=0.3 | 2.889 | 3.121 | 5.213
$\omega$=0.4 | 2.887 | 3.118 | 5.206
$\omega$=0.5 | 2.886 | 3.117 | 5.202
Table 2: Table illustrating the behavior of photon spheres and shadow radius
with varying values of the parameter $\omega$ with constants $M=1$, $N=0.01$,
$\alpha=0.1$, and $a=0.1$..

Figure 2: Kiselev shadow ($a=0.95M$, $\theta=\pi/2$). Upper left panel:
$\omega=0.5$. Upper right panel: $\omega=-0.15$. Lower left panel: $N=0.5M$.
Lower right panel: $N=-0.09M$.

###  III.3 Example 3: Kerr spacetime

Our last example is the Kerr spacetime, wherein:

$\alpha\gamma(r)=\ln\left[\frac{2Mr-r^{2}}{r(2M-r)}\right].$ (26)

The rotating spacetime also possesses an event horizon, located at $r=r_{+}$
such that $g^{rr}\left(r_{+}\right)=0$, which turns out to be equivalent to
the condition

$g_{tt}g_{\phi\phi}-g_{t\phi}^{2}=0.$ (27)

Solving this equation using the location of the horizon in the Kerr metric and
with the ansatz $r_{+}^{\alpha^{\prime}}=r_{+}^{SK}+\alpha\mu_{1}$, we found
that the location of the event horizon for this spacetime is

$r_{+}=2M-\frac{a^{2}}{2M}+\frac{\alpha}{6M}+\mathcal{O}(a^{2}\alpha,\alpha^{2}).$
(28)

Using the condition $g_{tt}(r)=0$ to find the location of the ergosphere, one
finds

$r_{erg}^{\alpha^{\prime}}=2M-\frac{a^{2}}{2M}\cos^{2}\theta-\dfrac{\alpha}{12M^{2}}+\mathcal{O}(a^{2}\alpha,\alpha^{2}).$
(29)

Note that the $\alpha^{\prime}$-corrected ergosphere is smaller than the
ergosphere of Kerr, namely
$r^{\alpha^{\prime}}_{\mathrm{erg}}<r_{\mathrm{erg}}^{SK}$, while the location
of the event horizon of the $\alpha^{\prime}$-corrected geometry is larger
than the location of the Kerr event horizon

The photonsphere radius $r_{\rm ph}$ is found numerically by $\beta=0$ and
then from Eq. 23, shadow radius is obtained as shown in Table 3 and in Fig.3.
Clearly, increasing the parameter $M$ results in an inreases in both the
photon sphere size and the shadow radius.

| $r_{\mathrm{ph}}^{-}$ | $r_{\mathrm{ph}}^{+}$ | $R_{\mathrm{sh}}$
---|---|---|---
$M$=1.0 | 2.882 | 3.113 | 5.193
$M$=1.2 | 3.483 | 3.714 | 6.233
$M$=1.4 | 4.083 | 4.314 | 7.273
$M$=1.6 | 4.683 | 4.914 | 8.312
$M$=1.8 | 5.283 | 5.514 | 9.352
Table 3: Table illustrating the behavior of photon spheres and shadow radius
with varying values of the parameter $M$ with constants $\alpha=0.1$, and
$a=0.1$.

Figure 3: Kerr shadow. Left panel: $\theta=\pi/2$. Right panel:
$\theta=\pi/4$.

##  IV Conclusions

A black hole’s shadow and photon sphere are critical tools for understanding
its properties. Recent observations by the Event Horizon Telescope
Collaboration have confirmed that real astrophysical black holes are likely
rotating, with a spin of around $0.7$. These black holes are typically
surrounded by an accretion disc, which affects the visible size of their
shadows. Light propagation in Kerr spacetime, accounting for light deflection
due to plasma presence, offers insights into these phenomena. By treating the
black hole and accretion disc as a single system, we can better understand
light propagation in their combined spacetime.

Gravitational decoupling is a powerful method for analyzing these systems. Our
recent work on spherically-symmetric spacetimes has provided clear insights
into how geometrical deformations impact black hole shadows and the radius of
photon spheres. Building on this, our current study focuses on axially-
symmetric spacetimes, where we have found that the influence of geometrical
deformations on the apparent size of a black hole image is more intricate.

In this paper, we have analyzed several well-known models and calculated the
shadows in these spacetimes using a novel method. Comparing our results with
those of an undistorted Kerr spacetime, we have shown that the influence of an
accretion disc on black hole geometry is likely negligible. Our approach
accurately describes the disc’s influence on the apparent shape of the black
hole image, providing valuable insights into these complex astrophysical
systems.

###### Acknowledgements.

V. Vertogradov thanks the Basis Foundation (grant number 23-1-3-33-1) for the
financial support. A. Ö. and R. P. would like to acknowledge networking
support by the COST Action CA18108 - Quantum gravity phenomenology in the
multi-messenger approach (QG-MM). A. Ö. would like to acknowledge networking
support by the CA22113 - Fundamental challenges in theoretical physics
(THEORY-CHALLENGES) and by the COST Action CA21106 - COSMIC WISPers in the
Dark Universe: Theory, astrophysics and experiments (CosmicWISPers).We also
thank TUBITAK and SCOAP3 for their support.

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# Particle Physics with the Cosmic Microwave Background with SPT-3G

J. S. Avva1 P. A. R. Ade2 Z. Ahmed3,4 A. J. Anderson5,6 K. Aylor7 R. Basu
Thakur6 A. N. Bender8,6 B. A. Benson5,6,9 L. E. Bleem8,6 S. Bocquet10,8 L.
Bryant11 J. E. Carlstrom6,11,12,8,9 F. W. Carter8,6 T. W. Cecil8 C. L.
Chang8,6,9 T. M. Crawford6,9 A. Cukierman1 T. de Haan1 J. Ding14 M. A.
Dobbs13,15 S. Dodelson16 D. Dutcher6,12 W. Everett17 K. R. Ferguson18 A.
Foster19 J. Gallicchio6,20 A. E. Gambrel6 R. W. Gardner11 J. C. Groh1 S. Guns1
N. W. Halverson17,21 A. H. Harke-Hosemann8,22 N. L. Harrington1 J. W.
Henning8,6 G. P. Holder15,23 W. L. Holzapfel1 D. Howe24 N. Huang1 K. D.
Irwin3,25,4 O. B. Jeong1 M. Jonas5 A. Jones24 T. S. Khaire14 L. Knox7 A. M.
Kofman22 M. Korman19 D. L. Kubik5 S. Kuhlmann8 C.-L. Kuo3,25,4 A. T. Lee1,26
A. E. Lowitz6 S. S. Meyer6,11,12,9 D. Michalik24 J. Montgomery13 A. Nadolski22
T. Natoli6,9 H. Nguyen5 G. I. Noble13 V. Novosad14 S. Padin6 Z. Pan6,12 P.
Paschos11 J. Pearson14 C. M. Posada14 W. Quan6,12 S. Raghunathan18 A.
Rahlin5,6 C. L. Reichardt27 D. Riebel24 J. E. Ruhl19 J.T. Sayre17 E.
Shirokoff6,9 G. Smecher28 J. A. Sobrin6,12 A. A. Stark29 J. Stephen11 K. T.
Story3,25 A. Suzuki26 K. L. Thompson3,25,4 C. Tucker2 K. Vanderlinde30,31 J.
D. Vieira22,23 G. Wang8 N. Whitehorn18 W. L. K. Wu6 V. Yefremenko8 K. W.
Yoon3,25,4 M. R. Young31<EMAIL_ADDRESS>
###### Abstract

The cosmic microwave background (CMB) encodes information about the content
and evolution of the universe. The presence of light, weakly interacting
particles impacts the expansion history of the early universe, which alters
the temperature and polarization anisotropies of the CMB. In this way, current
measurements of the CMB place interesting constraints on the neutrino energy
density and mass, as well as on the abundance of other possible light
relativistic particle species. We present the status of an on-going 1500 sq.
deg. survey with the SPT-3G receiver, a new mm-wavelength camera on the 10-m
diameter South Pole Telescope (SPT). The SPT-3G camera consists of 16,000
superconducting transition edge sensors, a 10x increase over the previous
generation camera, which allows it to map the CMB with an unprecedented
combination of sensitivity and angular resolution. We highlight projected
constraints on the abundance of sterile neutrinos and the sum of the neutrino
masses for the SPT-3G survey, which could help determine the neutrino mass
hierarchy.

1Department of Physics, University of California, Berkeley, CA, 94720, USA
2School of Physics and Astronomy, Cardiff University, Cardiff CF24 3YB, United
Kingdom
3Kavli Institute for Particle Astrophysics and Cosmology, Stanford University,
452 Lomita Mall, Stanford, CA, 94305, USA
4SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA,
94025, USA
5Fermi National Accelerator Laboratory, MS209, P.O. Box 500, Batavia, IL,
60510, USA
6Kavli Institute for Cosmological Physics, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL, 60637, USA
7Deptartment of Physics, University of California, One Shields Avenue, Davis,
CA 95616, USA
8High-Energy Physics Division, Argonne National Laboratory, 9700 South Cass
Avenue., Argonne, IL, 60439, USA
9Department of Astronomy and Astrophysics, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL, 60637, USA
10Ludwig-Maximilians-Universität, Scheiner- str. 1, 81679 Munich, Germany
11Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL, 60637, USA
12Department of Physics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL, 60637, USA
13Department of Physics and McGill Space Institute, McGill University, 3600
Rue University, Montreal, Quebec H3A 2T8, Canada
14Materials Sciences Division, Argonne National Laboratory, 9700 South Cass
Avenue, Argonne, IL, 60439, USA
15Canadian Institute for Advanced Research, CIFAR Program in Gravity and the
Extreme Universe, Toronto, ON, M5G 1Z8, Canada
16Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania,
15312, USA
17CASA, Department of Astrophysical and Planetary Sciences, University of
Colorado, Boulder, CO, 80309, USA
18Department of Physics and Astronomy, University of California, Los Angeles,
CA, 90095, USA
19Department of Physics, Center for Education and Research in Cosmology and
Astrophysics, Case Western Reserve University, Cleveland, OH, 44106, USA
20Harvey Mudd College, 301 Platt Boulevard., Claremont, CA, 91711, USA
21Department of Physics, University of Colorado, Boulder, CO, 80309, USA
22Department of Astronomy, University of Illinois at Urbana-Champaign, 1002
West Green Street, Urbana, IL, 61801, USA
23Department of Physics, University of Illinois Urbana-Champaign, 1110 West
Green Street, Urbana, IL, 61801, USA
24University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA
25Deptartment of Physics, Stanford University, 382 Via Pueblo Mall, Stanford,
CA, 94305, USA
26Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA,
94720, USA
27School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
28Three-Speed Logic, Inc., Vancouver, B.C., V6A 2J8, Canada
29Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge,
MA, 02138, USA
30Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St.
George Street, Toronto, ON, M5S 3H4, Canada
31Department of Astronomy & Astrophysics, University of Toronto, 50 St. George
Street, Toronto, ON, M5S 3H4, Canada

##  1 Introduction

The South Pole Telescope (SPT) is a 10-meter telescope located at the
Amundsen-Scott South Pole Station designed to make high-resolution millimeter-
wavelength measurements of the cosmic microwave background (CMB). The SPT-3G
camera on the SPT was deployed in the 2016-2017 austral season. It consists of
over 16,000 transition-edge sensors, a 10x increase in detectors over the
previous-generation camera on the SPT [1] [2]. These detectors are
polarization-sensitive and span three frequencies – 90, 150, and 220 GHz.
These significant improvements will allow the SPT-3G camera to map the
polarization and temperature anisotropies of the CMB with unprecedented depth
($\sim$ 2 $\mu$K in temperature at 150 GHz), resolution (1 arcmin), and sky
coverage (1500 deg2).

The CMB provides a unique way to probe particle physics since it was emitted
in the early universe. First, the anisotropies provide a snapshot at
$\sim$380,000 years after the Big Bang. In this period, neutrinos and other
light relics comprised around 10$\%$ of the energy density of the universe, as
opposed to $<$ 1$\%$ now. Small-scale anisotropies are particularly affected
by this radiation because small-scale structure is exponentially damped by an
increased energy-to-mass ratio in the early universe. This allows for a
sensitive measurement of $N_{\textup{eff}}$, the parameter that refers to the
effective number of neutrino species [3]. The measurement of
$N_{\textup{eff}}$ is described in Section 2. Second, CMB photons have been
traveling through the universe for billions of years. The intervening matter
between the surface of last scattering and the telescopes that observe the CMB
affects the CMB photons, for example by scattering and gravitational lensing.
The change to large-scale structure over time is thus imprinted in the CMB
measurements that are made today. In this way, the large-scale structure, and
therefore the effect of the sum of the neutrino masses on the expansion
history of the universe, can be studied [4] [5]. The measurement of $\sum
m_{\nu}$ is described in Section 3.

##  2 Light Relics and Neutrinos in the Early Universe

A particle species in chemical equilibrium with the Standard Model particles
at any point in the early universe contributes to the universe’s total energy
density. The fractional amounts of the energy density of the early universe
that are composed of matter, dark energy, or radiation all change the
expansion rate of the universe. The net effect of this is observed in the CMB
temperature and polarization power spectra. The power spectra of the CMB have
a series of peaks and troughs that decrease in amplitude at small angular
scales and are the products of acoustic oscillations and photon diffusion
damping [6] [7]. The locations of the peaks and the shape of the damping tail
determine the parameter $N_{\textup{eff}}$, the effective number of neutrino
species [8]. The Standard Model predicts a value of $N_{\textup{eff}}$ =
3.046. The difference between the Standard Model prediction and the naive
prediction of 3 stems from entropy transferred to neutrinos during electron-
positron annihilation due to neutrinos not being fully decoupled from the
early universe plasma. Additional light relativistic particles contribute a
change to $N_{\textup{eff}}$ related to the point at which they decoupled from
the primordial plasma (related to their masses and cross-sections). This
potential extra energy contribution to $N_{\textup{eff}}$ could be due to a
sterile neutrino, axion-like particles, a matter/antimatter asymmetry in the
neutrino sector, or any other possible light relic in the early universe [9].
In Figure 1, contributions from additional light relics are shown as functions
of their spin and freeze-out temperature. As seen in the figure, for a
$\Lambda$CDM + $Y_{\textup{p}}$ (the primordial helium abundance) +
$N_{\textup{eff}}$ cosmology, Planck has measured the current best limit, a
change in $N_{\textup{eff}}$ of $\Delta N_{\textup{eff}}$ = 0.19 at 1$\sigma$
[10], which rules out particles that decoupled after quarks became confined to
hadrons (the QCD phase transition). This corresponds to many more degrees of
freedom in the early universe plasma, and consequentially smaller changes in
$N_{\textup{eff}}$ for additional particle species. With SPT-3G’s sensitivity
of $\Delta N_{\textup{eff}}$ = 0.1 at 1$\sigma$, it is projected to be one of
the first experiments which will measure beyond the QCD phase transition,
placing interesting constraints on new physics.

##  3 Measuring Neutrino Mass from Clusters and Lensing

Figure 1: Contributions to $N_{\textup{eff}}$ from a single, massless,
particle that froze-out from the Standard Model particles at temperature
$T_{F}$. Figure adapted from Figure 10 of [11]. The black dashed line shows
the current best constraint on $\Delta N_{\textup{eff}}$ from $Planck$ [10],
and the magenta line is the forecast for SPT-3G, showing that SPT-3G could go
beyond the QCD phase transition.

Figure 2: Projected 68 and 95$\%$ confidence intervals for $S8$ vs. $\sum
m_{\nu}$ determined from: $Planck$ TT+TE+EE+lowE CMB [10] (green), the
forecasted improvement adding first SPT-3G CMB (gray) and then DESI BAO data
[12] (purple), and the forecasted constraints from $Planck$ TT+lowE with the
SPT-3G cluster data (red).

In addition to being a snapshot of the early universe, the CMB also provides a
probe of the evolution of cosmic structure over time. As neutrinos free-
stream, they suppress small-scale structure as described in the previous
section. This suppression is caused by the particles being ultra-relativistic.
As the universe ages and they cool, they slow, changing their effect on
structure. Combining the measurement of early structure from the preceding
section with any of a number of measurements of more recent times thus
provides an opportunity to measure the neutrinos’ velocity as a function of
energy, and thus their mass [4] [5]. This is a complementary approach to
terrestrial neutrino experiments – the CMB measures $\sum m_{\nu}$, as opposed
to the differences between the neutrino flavors. Due to SPT-3G’s high angular
resolution, the data will actually be able to provide two separate constraints
of $\sum m_{\nu}$: one from galaxy clusters and one from CMB lensing. These
two methods have different systematic uncertainties, so this provides a
compelling cross-check for $\sum m_{\nu}$ constraints from cosmological data.

First, the abundance of galaxy clusters over time traces large-scale
structure. As CMB photons pass through the hot gaseous regions of galaxy
clusters, they can scatter off high-energy electrons. This imprints a
characteristic spectral distortion called the Sunyaev-Zeldovich effect [13] in
CMB maps at the location of galaxy clusters. In this way, large-scale
structure over time can be characterized by the abundance of galaxy clusters
as a function of mass and redshift.

Second, lensing also probes large-scale structure. The intervening matter
between us and the surface of last scattering distorts the CMB spectrum
through lensing. The amplitude of the lensing signal is proportional to the
amount of large-scale structure in the universe and as a result can be used to
characterize the effect of massive neutrinos.

These two measurements constrain $\sum m_{\nu}$ on a similar order of
magnitude as displayed in Figure 2. This figure shows the joint confidence
intervals for the sum of the neutrino masses and a parameter describing the
combination of the amplitude of the matter power spectrum and the matter
energy density. The quantity $S8$ is defined by the amplitude of the matter
power spectrum ($\sigma_{8}$) and the sum of the cold dark matter, baryon, and
massive neutrino energy densities ($\Omega_{m}$) via the relationship
$S8=\sigma_{8}\sqrt{\Omega_{m}/0.3}$. The constraint from SPT-3G galaxy
clusters is forecast to be $\sigma(\sum m_{\nu})\simeq$ 0.055 eV and
$\sigma(\sum m_{\nu})\simeq$ 0.06 eV from CMB lensing (SPT-3G + $Planck$).
Adding in the Dark Energy Spectroscopic Instrument (DESI) baryon acoustic
oscillations (BAO) data [12], the SPT-3G forecast improves to $\sigma(\sum
m_{\nu})\simeq$ 0.038 eV. Currently, neutrino oscillation experiments limit
$\sum m_{\nu}\geq$ 0.067 eV. The inverted hierarchy dictates that there are
two neutrinos with $m_{\nu}\geq$ 0.049 eV, which would make $\sum m_{\nu}\geq$
0.107 eV [14]. Assuming the normal hierarchy and the minimum value of $\sum
m_{\nu}=$ 0.067 eV, SPT-3G could provide evidence against the inverted
hierarchy. Conversely, assuming the inverted hierarchy, SPT-3G could rule out
$\sum m_{\nu}=$ 0 at 2$\sigma$.

##  4 Summary

Using the polarization and temperature power spectra, SPT-3G will provide a
1$\sigma$ constraint of $\Delta N_{\textup{eff}}$ = 0.1, probing new parameter
space for light relic particles in the early universe. Between the two
systematically independent metrics of galaxy clusters and CMB lensing, SPT-3G
will set a constraint of $\sigma(\sum m_{\nu})\simeq$ 0.038 eV, a 30$\%$
improvement over $Planck$. Studying the CMB using SPT-3G data provides a
cutting-edge and complementary window into the Standard Model and beyond.

The South Pole Telescope is supported by the National Science Foundation (NSF)
through grant PLR-1248097. The author acknowledges support from the National
Science Foundation Graduate Research Fellowship under Grant No. DGE 1752814.
Partial support is also provided by the NSF Physics Frontier Center grant
PHY1125897 to the Kavli Institute of Cosmological Physics at the University of
Chicago, and the Kavli Foundation and the Gordon and Betty Moore Foundation
grant GBMF 947. Work at Argonne National Laboratory, including Laboratory
Directed Research and Development support and use of the Center for Nanoscale
Materials, a U.S. Department of Energy, Office of Science (DOE-OS) user
facility, was supported under Contract No. DE-AC02-06CH11357. We acknowledge
R. Divan, L. Stan, C.S. Miller, and V. Kutepova for supporting our work in the
Argonne Center for Nanoscale Materials. Work at Fermi National Accelerator
Laboratory, a DOE-OS, HEP User Facility managed by the Fermi Research
Alliance, LLC, was supported under Contract No. DE-AC02-07CH11359. NWH
acknowledges support from NSF CAREER grant AST-0956135. The McGill authors
acknowledge funding from the Natural Sciences and Engineering Research Council
of Canada, Canadian Institute for Advanced Research, and the Fonds de
recherche du Qubec Nature et technologies. JV acknowledges support from the
Sloan Foundation.

## References

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††thanks: Authors to whom correspondence should be addressed<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Authors to whom correspondence should be
addressed<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
# Model for 1/f Flux noise in Superconducting Aluminum Devices: Impact of
External Magnetic Fields

José Alberto Nava Aquino  Rogério de Sousa  Department of Physics and
Astronomy, University of Victoria, Victoria, British Columbia V8W 2Y2, Canada
Centre for Advanced Materials and Related Technology, University of Victoria,
Victoria, British Columbia V8W 2Y2, Canada

###### Abstract

Superconducting quantum interference devices (SQUIDs) and related circuits
made of aluminum are known to display $1/\omega$ flux noise, where $\omega$ is
frequency. A recent experiment showed that the application of an external
magnetic field in the $10-100$ G range changed the noise to a single
Lorentzian peaked at $\omega=0$. Here it is shown that a model based on
independent impurity spin flips with coexisting cross and direct mechanisms of
spin relaxation may explain these experiments. The model shows that
application of an external magnetic field can be used to reduce the impact of
flux noise in qubits.

Superconducting qubits are a promising candidate for building practical
quantum computers due to their relatively low fabrication costs, low
decoherence and scalability compared to other qubit technologies.Kjaergaard
_et al._ (2019); You and Nori (2005) Frequency tunability is required to scale
up to thousands of qubits and prevent errors associated to frequency crowding.
This requires the addition of SQUIDs to the circuits, making them sensitive to
flux noise. Hutchings  _et al._ (2017) Flux noise can alter the energy levels
of the qubits and lead to errors in quantum operations, making it a crucial
factor in the development of practical quantum computers. Oliver and Welander
(2013); Wellstood and Urbina (1987)

Despite extensive research, the underlying mechanism of flux noise remains
unclear, making it difficult to mitigate its impact. However, flux noise is
widely believed to arise from the dynamics of magnetic impurities near the
superconducting wires. de Sousa (2007); Koch, DiVincenzo, and Clarke (2007);
Sendelbach  _et al._ (2008); Faoro and Ioffe (2008); Kumar  _et al._ (2016);
de Graaf  _et al._ (2017); Quintana  _et al._ (2017)

Several experiments Bylander  _et al._ (2011); Anton  _et al._ (2013); Lanting
_et al._ (2014); Quintana  _et al._ (2017); Zaborniak and de Sousa (2021) show
that the frequency dependence of flux noise follows a power law,
$\tilde{S}_{\Phi}(\omega)\propto 1/\omega^{\alpha}$, with exponent $\alpha$
independent of frequency $\omega$ over several decades. Experimental
measurements on Nb devices show $\alpha(T)$ going from $0.8$ to $0.4$ with
increasing temperature $T$. Anton  _et al._ (2013) On the other hand, in Al
devices, $\alpha=0.96-1.05$.Quintana  _et al._ (2017) Recently, we showed that
the $\alpha(T)$ measured in Nb devices can be explained by a model that
assumes the impurity spins interact via ferromagnetic exchange interactions.
Nava Aquino and de Sousa (2022) In contrast, the exponent $\alpha$ measured in
Al devices could not be explained by spin-spin exchange; it required instead
the assumption of an extra individual spin relaxation channel, presumably due
to the interaction between each spin and other non-spin degrees of freedom
such as phonons, electron gas, or amorphous two-level systems. Nava Aquino and
de Sousa (2022)

A recent experiment, focused in Al superconducting qubits, presented for the
first time measurements of flux noise in the presence of an external magnetic
field $B$. Rower  _et al._ While at $B=0$ the flux noise showed the expected
$1/\omega$ frequency dependence Quintana  _et al._ (2017) consistent with our
model for spin relaxation disorder, Nava Aquino and de Sousa (2022) the
measurements at weak fields ($B=10-100$ G) were quite puzzling. At $B>0$ the
noise transitioned smoothly into a Lorentzian in frequency, suggesting
dramatic reduction in spin disorder.

Here we propose a model for these observations, and show that fitting to
experimental data will shed light on the microscopic mechanism responsible for
flux noise in Al devices.

Consider a set of spin impurities spatially distributed on the surfaces and
interfaces of the wires forming the superconducting device. Each impurity is
located at position $\bm{R}_{j}$ and has spin described by the dimensionless
operator $\hat{\bm{s}}_{j}$. Their magnetic moment $-g\mu_{B}\hat{\bm{s}}_{j}$
imprints a flux on the device LaForest and de Sousa (2015)

$\hat{\Phi}=-\sum_{j}\bm{F}_{j}\cdot\hat{\bm{s}}_{j},$ (1)

where the flux vector $\bm{F}_{j}=g\mu_{B}\bm{B}_{I}(\bm{R}_{j})/I$ accounts
for the dependence of the impurity-generated flux on different spin
orientations. Here $g$ is the impurity’s $g$-factor, $\mu_{B}$ is the Bohr
magneton, $\bm{B}_{I}(\bm{R}_{j})$ is the magnetic field generated by the wire
at the spin site, and $I$ is the total current flowing through the wire.

The spins themselves are coupled to the wire’s current according to the
Hamiltonian,

${\cal H}=g\mu_{B}\sum_{j}\bm{B}_{j}\cdot\hat{\bm{s}}_{j},$ (2)

where $\bm{B}_{j}=\bm{B}_{I}(\bm{R}_{j})+\bm{B}_{{\rm ext}}$ is the spin’s
local field, including an externally applied $\bm{B}_{{\rm ext}}$. For typical
SQUIDs, $I\lesssim 1$ $\mu$A, leading to peak $B_{I}<\mu_{0}I/b\sim 0.1$ G
where $b\sim 0.1$ $\mu$m is the thickness of the superconducting wire. As a
result it is safe to approximate $\bm{B}_{j}\approx\bm{B}_{{\rm ext}}$ when
$B_{{\rm ext}}>1$ G.

Flux noise arises from time-dependent correlations of the flux fluctuation
operator $\delta\hat{\Phi}(t)=\hat{\Phi}(t)-\langle\hat{\Phi}\rangle$,

$\displaystyle\tilde{S}_{\Phi}(\omega)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dt\textrm{e}^{i\omega
t}\left\langle\delta\hat{\Phi}(t)\delta\hat{\Phi}(0)\right\rangle$ (3)
$\displaystyle=$
$\displaystyle\sum_{j,k,a,b}F^{a}_{j}\tilde{S}^{ab}_{jk}(\omega)F^{b}_{k},$

where the superscripts $a,b=x,y,z$ denote the components of the flux vector,
and the spin noise is defined as

$\tilde{S}^{ab}_{jk}(\omega)=\int_{-\infty}^{\infty}dte^{i\omega
t}\langle[\hat{s}_{j}^{a}(t)-\langle\hat{s}_{j}^{a}\rangle][\hat{s}_{k}^{b}(0)-\langle\hat{s}_{k}^{b}\rangle]\rangle.$
(4)

In Ref. Nava Aquino and de Sousa, 2022 a numerical method to compute Eq. (4)
for a general Hamiltonian of interacting spins was described. While it was
shown that spin-spin interactions lead to the dominant mechanism of flux noise
in Nb devices, the so called spin diffusion mechanism, comparison with
experiments in Al devices suggested a quite different picture. Namely, that
the spins near Al wires were fluctuating independently with spin-spin
interaction and spin diffusion playing a minor role.

In the current letter, it is assumed that independent spin flips dominate
$\omega>0$ noise, so that $\tilde{S}_{jk}(\omega)\ll\tilde{S}_{jj}(\omega)$
when $j\neq k$. Together with
$\tilde{S}^{ab}_{jj}(\omega)=-\tilde{S}^{ba}_{jj}(\omega)$ when $a\neq b$, Not
the flux noise simplifies to

$\tilde{S}_{\Phi}(\omega)\approx\sum_{j}\left[\left|\bm{F}_{j}\cdot\hat{\bm{B}}_{j}\right|^{2}\tilde{S}^{\parallel}_{jj}(\omega)+\left|\bm{F}_{j}\times\hat{\bm{B}}_{j}\right|^{2}\tilde{S}^{\perp}_{jj}(\omega)\right].$
(5)

Here $\hat{\bm{B}}_{j}$ is the unit vector pointing along the local magnetic
field acting on the spin,
$\tilde{S}^{\parallel}_{jj}(\omega)\equiv\tilde{S}^{\hat{B}_{j}\hat{B}_{j}}_{jj}(\omega)$,
and
$\tilde{S}^{\perp}_{jj}(\omega)\equiv\tilde{S}^{\hat{P}_{j}\hat{P}_{j}}_{jj}(\omega)$
were $\hat{\bm{P}}_{j}$ is any direction perpendicular to $\bm{B}_{j}$.

Figure 1: Flux noise produced by a single spin in a magnetic field $\bm{B}$.
Spin fluctuations along $\bm{B}$ ($\tilde{S}^{\parallel}(\omega)$) remain
peaked at $\omega=0$. In contrast, spin fluctuations perpendicular to $\bm{B}$
($\tilde{S}^{\perp}(\omega)$) give rise to “spin precession peaks” centered at
$\omega=\pm g\mu_{B}B/\hbar$. As a result the flux noise contribution due to
the component of the flux vector perpendicular to $\bm{B}$ is expelled from
the low frequency range (See Eq. (5)). Both contributions are Lorentzian with
linewidth set by the corresponding spin decay rate
$\Gamma_{\parallel},\Gamma_{\perp}$. The figure assumes
$\Gamma_{\parallel}=2\Gamma_{\perp}=0.1\;g\mu_{B}B/\hbar$.  Figure 2: Effect
of external magnetic field $\bm{B}$ on flux noise with
$\bm{F}_{j}=\bm{F}\parallel\bm{B}$. The spin relaxation rates are assumed to
be distributed according to Eq. (8), with exponent $n=4$ representing direct
phonon emission due to spin-orbit admixture. Flux noise transitions from
$1/\omega$ “disorder” noise to an “undisordered” Lorentzian with increasing
$B$. The noise is normalized by $\tilde{S}_{\Phi}^{0}(\omega_{{\rm min}})$,
its value at $B=0$ and $\hbar\omega_{{\rm min}}/k_{B}T=10^{-10}$. Parameters
used in the calculation: $\hbar\Gamma_{0}/k_{B}T=1,\hbar\Gamma_{1}=5\times
10^{-6}(k_{B}T)^{-4},\lambda_{{\rm max}}=30$.

It is assumed that each spin evolves independently of the others according to
the equation of motion

$\displaystyle\frac{d\bm{s}}{dt}$ $\displaystyle=$
$\displaystyle\frac{g\mu_{B}}{\hbar}\bm{B}\times\bm{s}\color[rgb]{0,0,0}-\Gamma_{\parallel}[(\bm{s}-\bm{s}^{{\rm
inst~{}eq}})\cdot\hat{\bm{B}}]\hat{\bm{B}}\color[rgb]{0,0,0}$ (6)
$\displaystyle\color[rgb]{0,0,0}-\Gamma_{\perp}[\hat{\bm{B}}\times(\bm{s}-\bm{s}^{{\rm
inst~{}eq}})]\times\hat{\bm{B}}\color[rgb]{0,0,0},$

where $\bm{s}=\langle\hat{\bm{s}}_{j}\rangle$ represents a quantum average
(the subscript $j$ is now omitted for simplicity). In addition to spin
precession, this equation includes _spin energy relaxation rate_
$\Gamma_{\parallel}=\frac{1}{T_{1}}$ and _spin decoherence rate_
$\Gamma_{\perp}=\frac{1}{T_{2}}\geq\Gamma_{\parallel}/2$. These drive $\bm{s}$
towards its “instantaneous equilibrium” value $\bm{s}^{{\rm
inst~{}eq}}=\bm{s}^{{\rm
eq}}-g\mu_{B}[\tilde{\chi}_{\parallel}(0)\delta\bm{B}_{\parallel}(t)+\tilde{\chi}_{\perp}(0)\delta\bm{B}_{\perp}(t)]$,
where $\tilde{\chi}_{\parallel}(0)$ and $\tilde{\chi}_{\perp}(0)$ are the
$\omega=0$ longitudinal and transverse susceptibilities, respectively. For
small deviations from equilibrium we write
$\bm{B}\rightarrow\bm{B}+\delta\bm{B}(t)$, and $\bm{s}\rightarrow\bm{s}^{{\rm
eq}}+\delta\bm{s}(t)$, where both $\delta\bm{B}(t)$ and $\delta\bm{s}(t)$ are
small time-dependent perturbations. Plugging these into Eq. (6) and dropping
non-linear terms enables the calculation of the dynamical susceptibility and
the spin noise through the fluctuation-dissipation theorem (See Nava Aquino
and de Sousa, 2022 for details). In the current independent spin model the
spin noise can be calculated analytically:

$\displaystyle\tilde{S}^{\parallel}(\omega)$ $\displaystyle=$
$\displaystyle\frac{2\hbar\omega}{1-e^{-\hbar\omega/k_{B}T}}\frac{\Gamma_{\parallel}\tilde{\chi}_{\parallel}(0)}{\omega^{2}+\Gamma_{\parallel}^{2}},$
(7a)
$\displaystyle\color[rgb]{0,0,0}\tilde{S}^{\perp}(\omega)\color[rgb]{0,0,0}$
$\displaystyle\color[rgb]{0,0,0}=\color[rgb]{0,0,0}$
$\displaystyle\color[rgb]{0,0,0}\frac{\hbar\omega\Gamma_{\perp}\tilde{\chi}_{\perp}(0)}{1-e^{-\hbar\omega/k_{B}T}}\left[\frac{1}{(\omega-g\mu_{B}B/\hbar)^{2}+\Gamma_{\perp}^{2}}\right.\color[rgb]{0,0,0}$
(7b)
$\displaystyle\color[rgb]{0,0,0}+\left.\frac{1}{(\omega+g\mu_{B}B/\hbar)^{2}+\Gamma_{\perp}^{2}}\right]\color[rgb]{0,0,0},$

where $s^{{\rm eq}}=-\frac{1}{2}{\rm tanh}(g\mu_{B}B/2k_{B}T)$ and
$\tilde{\chi}_{\parallel}(0)=1/[4k_{B}T\cosh^{2}(g\mu_{B}B/2k_{B}T)]$,
$\tilde{\chi}_{\perp}(0)={\rm tanh}(g\mu_{B}B/2k_{B}T)/(2g\mu_{B}B)$.

In the presence of an external magnetic field, $\tilde{S}^{\parallel}(\omega)$
is drastically different from $\tilde{S}^{\perp}(\omega)$, as illustrated in
Fig. 1. From Eq. (5) it follows that at each spin location, the flux vector’s
component parallel to the external field will produce Lorentzian flux noise
peaked at $\omega=0$. In contrast, the component of the flux vector
perpendicular to $\bm{B}$ will produce instead two Lorentzians peaked at
$\omega=\pm g\mu_{B}B/\hbar=\pm 2\pi(B/1~{}{\rm G})\times 2.8~{}{\rm MHz}$.
This shows that even fields of a few Gauss have huge impact on flux noise:
They shift the perpendicular flux vector contribution to a much higher
frequency range.

This effect is the spin analogue of high-frequency charge noise produced by
quantum coherent two-level systems in amorphous materials.Shnirman  _et al._
(2005)

While a single spin produces Lorentzian noise, a system of spins with a wide
distribution of relaxation rates $\Gamma_{\parallel}$ can give rise to
$1/\omega$ noise in a large interval of frequencies. de Sousa (2007); Belli,
Fanciulli, and de Sousa (2020); Nava Aquino and de Sousa (2022)

Motivated by recent flux noise measurements in Al devices under weak magnetic
fields, Rower  _et al._ we propose the following relaxation rates for a spin
$\bm{s}_{j}$ placed on the surface or interface of a superconducting wire,

$\Gamma_{\parallel,j}=\Gamma_{0}e^{-\lambda_{j}}+\gamma(k_{B}T)(g\mu_{B}B)^{n}.$
(8)

The first contribution to Eq. (8) models _cross spin relaxation_ due to the
impurity spin interaction with one or more amorphous two-level systems (TLSs),
where $\lambda_{j}$ is a random variable uniformly distributed in the interval
$[0,\lambda_{{\rm max}}]$. de Sousa (2007); Belli, Fanciulli, and de Sousa
(2020) Cross relaxation occurs when a TLS switches due to thermal fluctuation,
causing a simultaneous impurity spin flip. Since the majority of these
processes occur for TLSs with energy splitting of the order of $k_{B}T$, cross
relaxation is independent of magnetic field when $g\mu_{B}B<k_{B}T$. de Sousa
(2007) The second contribution to Eq. (8) models _direct spin relaxation_ ,
whereby the spin emits a phonon, either due to modulation of its hyperfine
interaction ($\Gamma_{\parallel,j}\propto T\;B^{2}$), or due to the spin-orbit
admixture mechanism ($\Gamma_{\parallel,j}\propto T\;B^{4}$). In both cases
the rates scale linearly in $T$ because they are proportional to the phonon
occupation at the Zeeman frequency; they also scale as $B^{n}$ because the
direct mechanism requires the breaking of time-reversal symmetry. Van Vleck
(1940); de Sousa and Das Sarma (2003) The constants $\Gamma_{0}$, $\gamma$,
and $n$ are to be determined by fitting to experiments.

The $\Gamma_{\parallel,j}$ are distributed according to probability density
$p(\Gamma)=1/(\lambda_{{\rm max}}|\frac{d\Gamma}{d\lambda}|)$, leading to the
disorder-averaged single spin noise,

$\displaystyle\tilde{S}_{{\rm avg}}^{\parallel}(\omega)$ $\displaystyle=$
$\displaystyle\int_{\Gamma_{{\rm min}}}^{\Gamma_{{\rm
max}}}\tilde{S}^{\parallel}(\omega)p(\Gamma)d\Gamma=\color[rgb]{0,0,0}\frac{\hbar\omega\tilde{\chi}_{\parallel}(0)}{1-e^{-\hbar\omega/k_{B}T}}\left\\{\frac{1}{\lambda_{{\rm
max}}}\left(\frac{\omega^{2}}{\omega^{2}+\Gamma_{B}^{2}}\right)\frac{{\rm
arctan}(\Gamma_{{\rm max}}/\omega)-{\rm arctan}(\Gamma_{{\rm
min}}/\omega)}{\omega}\right.$ (9)
$\displaystyle\color[rgb]{0,0,0}\left.+\left[1-\frac{1}{2\lambda_{{\rm
max}}}{{\rm log}}\left(\frac{\Gamma_{{\rm max}}^{2}+\omega^{2}}{\Gamma_{{\rm
min}}^{2}+\omega^{2}}\right)\right]\frac{\Gamma_{B}}{\omega^{2}+\Gamma_{B}^{2}}\right\\}\color[rgb]{0,0,0},$

where $\Gamma_{B}=\gamma(k_{B}T)(g\mu_{B}B)^{n}$ and the minimum and maximum
rates are $\Gamma_{{\rm min}}=\Gamma_{0}e^{-\lambda_{{\rm max}}}+\Gamma_{B}$
and $\Gamma_{{\rm max}}=\Gamma_{0}+\Gamma_{B}$.

When $B=0$, $\Gamma_{B}=0$ and the first term in the curly bracket of Eq. (9)
gives rise to $1/|\omega|$ noise for a wide range of frequencies $\Gamma_{{\rm
min}}\ll|\omega|\ll\Gamma_{{\rm max}}$. As $B$ is increased from zero, there
will be a range of frequencies satisfying
$|\omega|<\Gamma_{B}\approx\Gamma_{{\rm min}}$. In this range the $1/|\omega|$
contribution to flux noise is suppressed, and the second term in the curly
bracket of Eq. (9) becomes important, with the square bracket approximately
independent of $\omega$. As $B$ increases further $\Gamma_{{\rm min}}$ gets
close to $\Gamma_{{\rm max}}$ and the low frequency noise is converted into a
simple “reduced disorder” Lorentzian, centered at $\omega=0$ with width
$\Gamma_{B}$.

Figure 2 shows the predicted transition from $1/\omega$ to Lorentzian flux
noise with increasing magnetic field. The behavior is in qualitative agreement
to recent experiments in Al devices (Fig. 3(a) of Ref. Rower  _et al._ , ).
Our Eqs. (5) and (9) can explain the experimental data if we assume
$\lambda_{{\rm max}}=10-30$ and spin areal density $3-8\times 10^{16}/{\rm
m}^{2}$.

The spin precession peaks at $\omega=\pm g\mu_{B}B/\hbar$, the ones that occur
due to the nonzero components of $\bm{F}_{j}\perp\bm{B}$, may also broaden due
to disorder. If these are dominated by spin energy relaxation, we get
$\Gamma_{\perp,j}\approx\frac{1}{2}\Gamma_{\parallel,j}$ and the peaks
transition from $\propto 1/|\omega\pm g\mu_{B}B/\hbar|$ at low $B$ to
Lorentzians centered at $\pm g\mu_{B}B/\hbar$ with width $\Gamma_{B}$ at
higher $B$. Measuring the shape of these spin precession peaks will yield
information about the distribution of spin decoherence rates
$\Gamma_{\perp,j}$.

In conclusion, a model for spin noise is proposed to explain flux noise
experiments in superconducting Al devices. The model is based on the
assumption that the interaction between each impurity spin and the vibrational
modes of the lattice, formed by amorphous TLSs and phonons, dominate finite
frequency fluctuations. Within this scenario, the impact of spin-spin
interactions such as dipolar and exchange can still be observed as a Curie-
Weiss temperature $T_{{\rm CW}}$ in the zero-frequency susceptibility, because
decay rates $\Gamma_{\parallel},\Gamma_{\perp}$ do not impact
$\tilde{\chi}_{\parallel}(0)$. Thus one can use
$\tilde{\chi}_{\parallel}(0)=1/[4k_{B}(T-T_{{\rm
CW}})\cosh^{2}(g\mu_{B}B/2k_{B}T)]$ in Eq. (9) to model spin-spin
interactions.

An analytical expression for the flux noise in the presence of an external
magnetic field $B$ is obtained, revealing a transition from $1/\omega$ flux
noise at $B=0$ to Lorentzian flux noise in the frequency range where direct
spin relaxation by phonon emission is stronger than cross relaxation due to
amorphous TLSs. The transition is similar to what is observed in recent
experiments.Rower  _et al._

Fitting the current theory to experimental data will yield the characteristic
exponent $n$ for direct phonon emission, elucidating whether the mechanism is
mediated by spin-orbit coupling or hyperfine interaction. This will in turn
yield valuable information on the identity of the spins causing flux noise.

The model predicts that application of a $B$ field reduces low frequency noise
arising from $\bm{F}_{j}\parallel\bm{B}$ ($\bm{F}_{j}\perp\bm{B}$) by a factor
of $1/\cosh^{2}{(g\mu_{B}B/2k_{B}T)}$ (${\rm
tanh}(g\mu_{B}B/2k_{B}T)/(2g\mu_{B}B)$). It also shifts the contribution of
$\bm{F}_{j}\perp\bm{B}$ out of the low frequency range, transferring noise to
spin precession peaks centered at $\omega=\pm g\mu_{B}B/\hbar$. For $T=10$ mK
and $B=100$ G, $1/\cosh^{2}{(g\mu_{B}B/2k_{B}T)}\approx 2/3$, and
approximately one half of the total noise power will be transferred to a
higher frequency band. Therefore, a total reduction of $\approx 1/3$ is made
to the noise power in the low frequency band. Further reduction can be
achieved for $B>100$ G. These results show that an external magnetic field can
be used to reduce the impact of flux noise in qubits.

###### Acknowledgements.

The authors thank M. Amin, R. Harris, and T. Lanting for several insights into
flux noise, and L. Ateshian and D. A. Rower for discussions and sharing
experimental results before publication. This work was supported by NSERC
(Canada) through its Discovery program (Grant number RGPIN-2020-04328).

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