# Note on the chromatic number of Minkowski planes:
the regular polygon case

Panna Gehér  Eötvös Loránd University, Budapest<EMAIL_ADDRESS>
###### Abstract

The famous Hadwiger-Nelson problem asks for the minimum number of colors
needed to color the points of the Euclidean plane so that no two points unit
distance apart are assigned the same color. In this note we consider a variant
of the problem in Minkowski metric planes, where the unit circle is a regular
polygon of even and at most $22$ vertices. We present a simple
lattice–sublattice coloring scheme that uses $6$ colors, proving that the
chromatic number of the Minkowski planes above are at most 6. This result is
new for regular polygons having more than 8 vertices.

##  1 Introduction

In 1950, Nelson raised the following question: What is the minimum number of
colors that are needed to color the Euclidean plane so that no two points of
the same color determine unit distance? We refer to such a coloring with $k$
color classes as a proper $k$-coloring. Thus Nelson’s question asks for the
smallest $k$ value such that the plane can be properly $k$-colored. This value
is known as the chromatic number of the Euclidean plane, and is denoted by
$\chi(\mathbb{R}^{2})$. Immediately after the question was raised the
following easy-to-get bounds were established:

$4\leq\chi(\mathbb{R}^{2})\leq 7.$

The lower bound is due to Moser [17] who constructed a unit-distance graph
(that is a graph whose edges connect vertices unit distance apart) with
chromatic number $4$. The upper bound is due to Isbell [10] who considered a
tilting of the plane by translates of a regular hexagon with diameter slightly
less than one and defined a periodic proper $7$-coloring shown in Figure 1.

Figure 1: A proper $7$-coloring of the Euclidean plane shown by Isbell

Despite the numerous attempts to improve these bounds, only little progress
was made for more that $60$ years – for a historical survey on the problem see
[20]. However, in 2018 mathematicians were shaken when biologist de Grey [9]
constructed a $5$-chromatic unit-distance graph, proving that the chromatic
number of the plane is at least $5$. Shortly afterwards Exoo and Ismailescu
[7] independently published another proof.
The problem has regained a lot of attention since the breakthrough and a
Polymath project was launched with the main goal of creating a human-
verifiable proof of the new result. Although the proofs are still relying on
computers, quite some progress has been made: while the distance graph
published by de Grey had a total of $1581$ vertices, the current known
smallest example consists only $509$ [18].

As a consequence of the breakthrough many variations of the problem have
gained more attention in the last couple of years. One interesting research
area is generalizing the question to Minkowski planes: Let $C$ be a two-
dimensional centrally symmetric bounded convex domain centered at the origin
and let $(\mathbb{R}^{2},\text{ }C)$ denote the Minkowski plane where $C$
determines the unit circle; the $C$-norm of an $x\in\mathbb{R}^{2}$ point is
the value:

$\norm{x}_{C}:=\min\left\\{\lambda\in\mathbb{R}^{+}|x\in\lambda C\right\\}.$

The $C$-distance of two points $x$ and $y$ is defined by $\norm{x-y}_{C}$.
Naturally, the chromatic number of the Minkowski planes – denoted by
$\chi(\mathbb{R}^{2},C)$ – is the minimum number of colors needed to color the
points of $\mathbb{R}^{2}$ such that no monochromatic point pair determines a
unit $C$-distance. The main result concerning the chromatic number of
Minkowski planes is due to Chilakamarri [2]: by extending the arguments of
Moser and the construction of Isbell he proved that the bounds

$4\leq\chi(\mathbb{R}^{2},C)\leq 7$

hold for all centrally symmetric bounded convex domain $C$.

An interesting special case of the above problem is when the unit circle is a
regular polygon of even number of vertices. Or more generally we can consider
any affine image of a regular polygon since the problem itself is affine
invariant. The study of the chromatic number of such normed planes was also
initiated by Chilakamarri who considered the cases of regular polygons with
few vertices. He gave a tile-based proper $4$-coloring for the parallelogram
and the symmetric hexagon’s case, proving that the answer here is exactly $4$.
He also gave a proper $6$-coloring in case $C$ is a centrally symmetric
octagon: he considered a packing of $C/2$, that is a packing of circles of
radius half. He showed that the translates of $C/2$ can be colored using $4$
colors and two more colors can take care of the remaining squares (for details
see Figure 2). It is worth noting that we have to be careful with choosing the
colors of boundary points of the octagons as no antipodal point pair can have
the same color.

Figure 2: Chilakamarri’s proper $6$-coloring of the Minkowski plane equipped
with the regular octagon metric

Chilakamarri asked whether or not the chromatic number of a plane equipped
with the norm defined by the regular octagon or any centrally symmetric
octagon is exactly $4$. No progress was made until very recently when Exoo,
Fisher and Ismailescu [6] answered his question negatively: they constructed
$5$-chromatic unit-distance graphs in the cases of regular polygons with $8$,
$10$, and $12$ vertices. Together with the Euclidean case these are the only
known examples of a normed plane with chromatic number at least $5$. Table 1
summarizes the mentioned results:

Unit circle $C$ |  $\chi(\mathbb{R}^{2},C)$
---|---
Parallelogram, centrally symmetric hexagon (see [2]) Regular octagon (see [6], and [2]) Regular decagon, regular dodecagon (see [6] and [2]) Euclidean circle (see [9], [7] and [10]) Arbitrary symmetric convex domain (see [2]) |  $4$ $5$ or $6$ $5$, $6$ or $7$ $5$, $6$ or $7$ $4$, $5$, $6$ or $7$
Table 1: Possible values of the chromatic numbers of Minkowski planes

In this note we extend Chilakamarri’s result for regular octagons to regular
polygons with at most $22$ vertices by giving a simple lattice–sublattice
coloring scheme that uses only $6$ colors. It also slightly strengthens the
result of Chilakamarri as our colorings are regular: We call a proper
$k$-coloring of $\mathbb{R}^{n}$ with color classes $C_{1}$, $C_{2}$ …$C_{k}$
regular, if there exist vectors $v_{1}\dots$, $v_{k}$ such that
$C_{i}=C_{1}+v_{i}$ for $i=1\dots k$, that is the color classes are translates
of each other. Now we state our main theorem:

######  Theorem 1.

Let $C$ be a regular polygon with an even number of vertices. In case $C$ has
at most $22$ vertices then there exists a regular proper $6$-coloring of the
Minkowski plane equipped with the $C$-metric such that no points unit
$C$-distance apart are identically colored. Hence,

$\chi(\mathbb{R}^{2},C)\leq 6$

holds for any regular $2k$-gon $C$ where $k\leq 11$.

It follows that in the case of a regular decagon and dodecagon – similarly to
the octagon’s case – the chromatic number is either $5$ or $6$.

In Section 2 we describe the coloring scheme used in all cases and give some
details and figures about the proofs. As an example, in Section 3 all
computations are given in the regular dodecagon’s case and calculations for
the $22$-gon can be found in the Appendix. Finally, in Section 4 we describe a
consequence of Theorem 1 that considers a closely related asymmetric Ramsey-
type question raised by Szlam [21].

##  2 The coloring scheme

Let $C$ be a regular octagon first and define a symmetric convex hexagon $H$
inscribed in $C/2$ as follows: Choose two opposite sides of $C/2$ and form a
hexagon using their four endpoints and two additional boundary points of
$C/2$. The choice of the additional points can be made in various ways, here
we simply chose the ones that halve the boundary line of $C/2$ connecting the
chosen sides. Denote the vertices of $H$ by $A_{i}$ ($i=1\dots 6$) in a
clockwise order as shown in Figure 3.

Figure 3: The centrally symmetric hexagons inscribed in $C/2$ in the case of
the regular decagon

To avoid unit $C$-distance in $H$, remove the boundary points lying between
the points $A_{1}$ and $A_{4}$, including $A_{4}$ but not $A_{1}$. In this way
no antipodal point pair is monochromatic. For simplicity call the resulting
half-open hexagon still $H$. Now consider a tiling of the plane by translates
of $H$ and assign colors $1$ through $6$ periodically as shown in Figure 4. We
can assume that the centers of the hexagons form a lattice, that we denote by
$\mathcal{L}$.

Figure 4: A tiling of the plane with translates of hexagon $H$ defines a
periodic $6$-coloring

Note that the main difference between this coloring scheme and Chilakamarri’s
general $7$-coloring is that here we can take advantage of $C$ not being
strictly convex: in the direction perpendicular to the sides shared with $C/2$
the monochromatic hexagons can be placed such that they are separated by only
one differently colored hexagon.

Now we justify that our coloring is proper: as mentioned earlier a unit
$C$-distance is not realized within the hexagons. All is left is to check that
two hexagons of the same color are too far from each other to determine a
point pair unit $C$-distance apart. As the color classes are congruent, it is
enough to verify the statement for one specific color class, say the class of
red points. We can also assume that one of the red hexagons has the origin as
its center, thus the set of centers of red hexagons form a sublattice
$\mathcal{L^{\prime}}$. By the symmetry of $C$ it is enough to show that
polygons $\mathcal{L^{\prime}}+C/2\oplus H$ form a packing, where $\oplus$
denotes the Minkowski sum of the two polygons, that is:

$C/2\oplus H=\\{c+h\text{ }|\text{ }c\in C/2,\text{ }h\in H\\}.$

Straightforward calculations finish the proof. Without loss of generality we
can assume that $C$ has circumradius one. Let $v_{1}$ and $v_{2}$ denote the
basis vectors of the lattice $\mathcal{L^{\prime}}$ where we can assume that
$v_{1}$ is perpendicular to the sides shared with $C/2$. Then for any vector
$\lambda\in\mathcal{L^{\prime}}$, polygons $\lambda+C/2\oplus H$ and
$\lambda\pm v_{1}+C/2\oplus H$ are trivially disjoint. From definition
$H+C/2\subseteq C$ so $H+C/2$ also has circumradius at most one. Hence it is
enough to check that with the exception of $\underline{0}$ and $\pm v_{1}$ any
lattice vector has Euclidean length at least two. This obviously holds (see
Figure 5), thus the coloring is indeed proper.

Figure 5:  For any lattice vector $v$ of $\mathcal{L^{\prime}}\setminus\\{\pm
v_{1},\underline{0}\\}$ the Euclidean unit circle $B_{2}$ is disjoint from all
translates $B_{2}+v$

Now consider regular polygons with greater number of vertices. We show that
almost the same coloring scheme works for all the remaining cases: Two sides
of $H$ can always be two opposite sides of $C/2$, we only have to be careful
with the choice of the remaining two vertices. For the case of a regular
$10$\- and $12$-gons choosing the halving points on the boundary line of $C/2$
still works, we can simply define hexagon $H$ as shown in Figure 6.

Figure 6: The centrally symmetric hexagons inscribed in $C/2$ chosen in the
case of the regular $10$\- and $12$-gon

However, in the remaining cases we had to flatten the hexagons in order to get
a proper coloring. In the case of regular $14$-, $16$\- and $18$-gons some
other vertices of $C/2$ were chosen. But in the final two cases only non-
vertex points seemed to be working: for $n=20$ bisectors of some other sides
were chosen, and for $n=22$ we divided one of the sides in the ratio
$0.68:0.32$. For details see Figure 7.

Figure 7: The centrally symmetric hexagon $H$ inscribed in $C/2$ chosen for
the regular $14$-, $16$-, $18$-, $20$\- and $22$-gon

Proving that the colorings defined by the above hexagons are proper needs more
and more computations as the number of vertices grows. Since all cases are
similar, we will only give the details of the calculations in the regular
dodecagon’s case, in Section 3 and in the regular $22$-gon’s case in the
Appendix.

##  3 The regular dodecagon’s case

Now we present the details of the proof in the case of the regular dodecagon.
Consider the regular dodecagon centered at the origin with circumradius $2$,
whose vertices are:

$\big{(}\pm 1,\pm\sqrt{3}\big{)},\big{(}\pm\sqrt{3},\pm 1\big{)},\big{(}\pm
2,0\big{)},\big{(}0,\pm 2\big{)}.$

Let $H$ be the symmetric hexagon inscribed in $C/2$ as defined in Section 2:
take two opposite sides of $C/2$, for example the sides parallel to vector
$(2-\sqrt{3},1)$ and choose the two additional points such that they halve
parts of the boundary line of $C/2$ between the chosen sides. Denote these six
vertices by $A_{i}$ ($i=1\dots 6$) in a clockwise order as shown in Figure 8.
As before, let $H$ be the half-open hexagon defined by the points $A_{i}$ that
does not contain the line segment connecting the points $A_{1}$ and $A_{4}$
and the point $A_{1}$ itself.

Figure 8: $H$ hexagon inscribed in $C/2$

Therefore the hexagonal tiling of the plane with hexagon $H$ is the packing by
Voronoi regions of the lattice $\mathcal{L}$ spanned by vectors
$\Big{(}\frac{1-2\cdot\sqrt{3}}{4},\frac{4+\sqrt{3}}{4}\Big{)}$ and
$\Big{(}\frac{2+\sqrt{3}}{2},-\frac{1}{2}\Big{)}$. The basis vectors of the
sublattice $\mathcal{L^{\prime}}$ corresponding to the single color class
containing the hexagon centered at the origin are:

  * •

$v_{1}=\big{(}\frac{3-6\cdot\sqrt{3}}{4},\frac{12+3\cdot\sqrt{3}}{4}\big{)}$,

  * •

$v_{2}=\big{(}2+\sqrt{3},-1\big{)}$.

As mentioned in Section 2 what we need to show is that polygons
$\mathcal{L^{\prime}}+C/2\oplus H$ form a packing.

The vertices of $C/2\oplus H$ are:

$B_{1}=\Bigg{(}\frac{1}{4},\frac{6+\sqrt{3}}{4}\Bigg{)},\text{
}B_{2}=\Bigg{(}\frac{3}{4},\frac{2+3\cdot\sqrt{3}}{4}\Bigg{)},\text{
}B_{3}=\Bigg{(}\frac{1+2\cdot\sqrt{3}}{4},\frac{4+\sqrt{3}}{4}\Bigg{)},\text{
}B_{4}=\Bigg{(}\frac{2+\sqrt{3}}{2},\frac{1}{2}\Bigg{)},$
$B_{5}=\Big{(}2,0\Big{)},\text{ }B_{6}=\Bigg{(}\sqrt{3},-1\Bigg{)},\text{
}B_{7}=\Bigg{(}\frac{1+\sqrt{3}}{2},-\frac{1+\sqrt{3}}{2}\Bigg{)},\text{
}B_{8}=\Bigg{(}\frac{1}{4},-\frac{2+3\cdot\sqrt{3}}{4}\Bigg{)}\text{ etc.}$

The coordinates of the remaining vertices can be obtained by symmetry – see
Figure 9.

Figure 9: Minkowski sum of $H$ and $C/2$

As the coloring is regular, it is enough to pick hexagon $H$ centered at the
origin and show that $H\oplus 1/2C$ is disjoint from $\lambda^{\prime}+H\oplus
1/2C$ for all $\lambda^{\prime}\not\equiv 0$ in $\mathcal{L^{\prime}}$.

By definition $H\oplus C/2$ has circumradius $2$. Inside the circle of radius
$4$ centered at the origin there are $4$ lattice points of
$\mathcal{L^{\prime}}$ besides the origin and by symmetry we only have to
check $2$ of them and the corresponding hexagons, namely:

  * •

$H_{1}:=H+v_{2}$ and

  * •

$H_{2}:=H+v_{1}+v_{2}$.

$H$ and $H_{1}$ are separated by exactly one differently colored hexagon which
is enough as $v_{2}$ is perpendicular to the common sides of $H$ and $C/2$.
All is left is to give a line that separates $H$ from $H_{2}$. For example
consider the line $l$ defined by equation:

$y=-\frac{2+\sqrt{3}}{3}x+\frac{5+2\cdot\sqrt{3}}{3}.$

Figure 10: Line $l$ separates $H\oplus C/2$ and $H_{2}\oplus C/2$

It is straightforward to check that $l$ goes through two parallel sides of
$H\oplus C/2$ and $H_{2}\oplus C/2$ (which are on the opposite sides of their
centers) and the remaining vertex points of $H\oplus C/2$ are below line $l$,
while all of the remaining vertex points of $H_{2}\oplus C/2$ are above it
(see Figure 10). Therefore $H\oplus C/2$ and $H_{2}\oplus C/2$ are disjoint,
thus the coloring is proper.

We remark that in the presented example one can define hexagon $H$ in many
different ways as the coloring scheme is quite flexible in this case. However,
as the number of vertices increases, the range of possible choices narrows
down quickly. For example in the case of the regular $22$-gon we have to be
really carefull with the definition of hexagon $H$: as Figure 11 shows, our
coloring is almost rigid.

Figure 11: In the $6$-coloring of the Minkowski plane equipped with the
regular $22$-gon metric
monochromatic hexagons get dangerously close together

##  4 An asymmetric Ramsey-type problem

Another direction for generalizing the Hadwiger-Nelson problem is to replace
the pair of points at unit distance by another finite point configuration.
Moreover we can look for different configurations in each color class. Solving
a problem raised in [5] Juhász [12] showed that in any red-blue coloring of
the plane there are either two red points distance one apart or there is a
blue congruent copy of any configuration with at most $4$ points. In the
opposite direction Juhász also showed that her theorem does not remain true if
we replace $4$ by $12$. Csizmadia and Tóth [4] later improved the above
result, they proved that $4$ can not even be increased to $8$.

In the rest of the paper we are interested in the following variation of the
question above: is it true for a given point configuration
$K\subseteq\mathbb{R}^{n}$ that in any red-blue coloring of the
$n$-dimensional Euclidean space there are either two red points distance one
apart or there is a blue translate of configuration $K$? Let $k_{n}$ denote
the largest value $k$ such that the answer is ‘yes’ for all configuration $K$
of at most $k$ points. Note that the $2$-dimensional case is not the same as
it was in the original problem since ‘congruent copy’ has been replaced by
‘translate’.
This variant of the problem was first considered in [21] by Szlam who showed
that $k_{n}$ grows exponentially in the number of dimensions. More precisely,
Szlam’s theorem states that $\chi(\mathbb{R}^{n})$, the chromatic number of
the $n$-dimensional Euclidean space (i. e. the smallest number of colors that
are needed to color $\mathbb{R}^{n}$ so that no two points of the same color
determine unit distance) provides a lower bound on the value of $k_{n}$. For
the sake of completeness, we include his short proof as well.

######  Lemma 1 (Szlam I. [21]).

Assume that there exists a $k$-point configuration $K$ and a red-blue coloring
of $\mathbb{R}^{n}$ such that the red color class avoids unit distance and the
blue color class avoids all translates of $K$. Then $\mathbb{R}^{n}$ can be
properly $k$-colored that is none of the $k$ color classes contains unit
distance. Hence $k_{n}+1\geq\chi(\mathbb{R}^{n})$.

###### Proof.

Assume that we are given a configuration $K=\\{a_{1},\dots a_{k}\\}$ and a
red-blue coloring of $\mathbb{R}^{n}$ with the desired properties. Then, for
each $x\in\mathbb{R}^{n}$ there is at least one index $i$ such that $x+a_{i}$
is red. Now let us color the point $x$ with color number $i$: as there are no
red points unit distance apart, this indeed defines a proper $k$-coloring.
$\Box$

The famous theorem of Frankl and Wilson [8] gives an exponential lower bound
for the chromatic number of the Euclidean space: it states that in case $n$
tends to infinity

$\chi(\mathbb{R}^{n})\geq(1.2+o(1))^{n}.$

Hence, by Lemma 1 if $n$ tends to infinity, every red-blue coloring of
$\mathbb{R}^{n}$ contains a blue translate of all $k$ point configuration when
$k<(1.2+o(1))^{n}$.

Szlam also a gave a partial converse to Lemma 1 that considers only regular
colorings:

######  Lemma 2 (Szlam II. [21]).

Assume that $\mathbb{R}^{n}$ can be properly $k$-colored by a regular
coloring, with color classes $C_{i}=C_{1}+v_{i}$ (for $i=1\dots k$). Then
there exists a red-blue coloring of $\mathbb{R}^{n}$ and a $k$-point
configuration, namely $K=\\{v_{1},v_{2},\dots v_{k}\\}$ such that the red
color class avoids unit distance and the blue color class avoids all
translates of $K$.

Since Szlam’s original paper was published many applications and
generalizations of his work were considered (see for example [1, 11, 13, 16]).
Here we need the analogous result considering Minkowski spaces: Let $k_{n}(C)$
denote the largest $k$ value such that in any red-blue coloring of the
Minkowski space determined by $C$ there are either two red points distance one
apart or there is a blue translate of any configuration with at most $k$
points. An easy observation is that both Lemma 1 and Lemma 2 can be extended
to Minkowski spaces (as noted e.g. in [1]). As the $6$-colorings described in
Theorem 1 are a regular, we immediately get the following corollary:

######  Corollary 1.

Let $(\mathbb{R}^{2},C)$ be a Minkowski plane whose unit circle is a regular
polygon with an even number of vertices. In case $C$ has at most $22$ vertices
then there exists a red-blue coloring of the plane and a configuration $K$ of
$6$ points such that there is no red point pair unit $C$-distance apart, and
the blue color class avoids all translates of $K$.

We finish the paper with a small remark on Szlam’s results: We noticed that
although the proof of Lemma 2 is short and straightforward, it can be a bit
misleading. To see its inconvenience notice how the proper coloring defined in
Lemma 1 is not regular: color classes are generated by covering the plane with
translates of the unit distance avoiding red set. Call a coloring with such
structure subregular. More precisely we call a proper $k$-coloring with color
classes $C_{1}$ $\dots$ $C_{k}$ subregular if there exist vectors $v_{1}$
$\dots$ $v_{k}$ such that $C_{i}$ is a subset of $C_{1}+v_{i}$. We show that
Lemma 2 can be extended to subregular colorings in a very natural way:

######  Theorem 2.

Let $(\mathbb{R}^{n},C)$ be an $n$-dimensional Minkowski space. Assume
$\mathbb{R}^{n}$ can properly be $k$-colored by a subregular coloring defined
by a $C$-unit distance avoiding set $C_{1}$ and vectors $v_{1}$, $v_{2}$
$\dots$ $v_{k}$. Then there exists a red-blue coloring of $\mathbb{R}^{n}$ and
a $k$-point configuration, namely $K=\\{-v_{1},-v_{2},\dots-v_{k}\\}$ such
that the red color class avoids unit $C$-distance and the blue color class
avoids all translates of $K$.

###### Proof.

Let the points of $C_{1}$ be colored red, and color all the remaining points
blue. As promised, let us consider the configuration
$K=\\{-v_{1},-v_{2},\dots,-v_{k}\\}$. We wish to show that for an arbitrary
vector $m$ color class $C_{1}$ contains at least one point of $K+m$. Without
loss of generality we can assume $v_{1}\equiv 0$. Hence if $m\in C_{1}$ there
is nothing to prove. Assume that $m\notin C_{1}$. In this case there exists an
index $i$ such that $m\in C_{1}+v_{i}$ which leads to $-v_{i}+m\in C_{1}$.
$\Box$

It follows that for all $n$ and $C$ the value $k_{n}(C)$ is exactly the
smallest number $k$ such that there exists a subregular $k$-coloring of
$(\mathbb{R}^{n},C)$. As we have seen, known proper colorings of Minkowski
planes are usually regular. In the $3$-dimensional Euclidean a proper
$15$-coloring was defined in [19] and independently in [3] which give the best
upper bound for the chromatic number of $\mathbb{R}^{3}$. Although these
colorings are not rigorously regular, they can be turned into regular
colorings in a trivial way. However, in higher dimensions proper colorings are
typically only subregular. The best known upper bound on the chromatic number
of the Euclidean space was established by Larman and Rogers [15] who
identified a subregular proper coloring of $\mathbb{R}^{n}$ with
$(3+o(1))^{n}$ color classes, meaning $k_{n}<(3+o(1))^{n}$. By Lemma 2 this
result is tight up to the constant factor. For Minkowski spaces the analogous
theorem with $(4+o(1))^{n}$ color classes was proved by Kupavskii [14],
meaning $k_{n}(C)<(4+o(1))^{n}$.

## Acknowledgement

I would like to express my gratitude to Géza Tóth for his great help and for
his many helpful remarks. I am also thankful to Dan Ismailescu for informing
us about the current state of the problem.

## References

  * [1] J. Burkert, P. Johnson _Szlam’s lemma: mutant offspring of a Euclidean Ramsey problem from 1973, with numerous applications_ , Ramsey theory (pp. $97-113$), Birkhäuser, Boston, MA., 2011.
  * [2] K. B. Chilakamarri _Unit-distance graphs in Minkowski metric spaces_ , Geometriae Dedicata (1991), $37.3:345-356$.
  * [3] D. Coulson _A 15-colouring of 3-space omitting distance one_ , Discrete Mathematics (2001), $256$, $83-90$.
  * [4] Gy. Csizmadia, G. Tóth _Note on a Ramsey-type problem in geometry_ , Journal of Combinatorial Theory, Series A (1994), $65(2)$, $302-306$.
  * [5] P. Erdős, R.L. Graham, P. Montgomery, B.L. Rothschild, J.H. Spencer, E.G. Straus _Euclidean Ramsey Theorems II._ , Colloq. Math. Soc. J. Bolyai vol. 10, Infinite and Finite Sets: $529-557$, North-Holland, Amsterdam, 1975.
  * [6] G. Exoo, D. Fisher, D. Ismailescu _The chromatic number of the Minkowski plane – the regular polygon case_ , Geombinatorics (2021), $31$, $49$–$67$.
  * [7] G. Exoo, D. Ismailescu _The chromatic number of the plane is at least 5: A new proof_ , Discrete $\&$ Computational Geometry (2020), $1-11$.
  * [8] P. Frankl, R. M. Wilson _Intersection theorems with geometric consequences_ , Combinatorica (1981), $1(4)$, $357-368$.
  * [9] A. de Grey _The Chromatic Number of the Plane Is at least 5_ , Geombinatorics (2018), $28$, $5-18$.
  * [10] H. Hadwiger, H. DeBrunner, V. Klee _Combinatorial Geometry in the Plane_ , Holt, Rinehart $6$ Winston, New York, 1964.
  * [11] P. Johnson A. D. Szlam _A new connection between two kinds of Euclidean coloring problems_ Geombinatorics (2001), $10(4)$, $172-178$.
  * [12] R. Juhász _Ramsey type theorems in the plane_ , Journal of Combinatorial Theory, Series A (1979), $27(2)$, $152-160$.
  * [13] C. Krizan _Euclidean Szlam Numbers_ PhD Dissertation, Auburn University, 2016.
  * [14] A. Kupavskii _On the chromatic number of $\mathbb{R}^{n}$ with an arbitrary norm_, Discrete Math. (2011), $311(6):437-440$.
  * [15] D. G. Larman, C. A. Rogers _The realization of distances within sets in Euclidean space_ , Mathematika (1972), $19(1)$, $1-24$.
  * [16] A. Li, M. Plummer, A. Shapiro, K. Weatherspoon _An Assortment of Euclidean Coloring Problems_ , 2021.
  * [17] L. Moser, W. Moser _Solution to problem $10$_, Can. Math. Bull. (1961), $4$, $187-189$.
  * [18] J. Parts _Graph minimization, focusing on the example of 5-chromatic unitdistance graphs in the plane_ , Geombinatorics (2020), $29$, no. 3, $137$-$166$.
  * [19] R. Radoičić, G. Tóth _Note on the chromatic number of the space_ , Discrete and computational geometry (2003), Springer, Berlin, Heidelberg, $695-698$.
  * [20] A. Soifer _The mathematical coloring book: Mathematics of coloring and the colorful life of its creators_ , Springer Science & Business Media 2008.
  * [21] A. D. Szlam _Monochromatic translates of configurations in the plane_ , Journal of combinatorial theory. Series A (2001), $93(1)$, $173-176$.
  * [22]

## Appendix

For the sake of completeness, we give the calculations for the regular
$22$-gon’s case: Consider the regular $22$-gon centered at the origin with
circumradius $2$ so that the coordinates of the vertices of $C/2$ are:

$\left(\pm\cos(\frac{2t\pi}{11}),\pm\sin(\frac{2t\pi}{11})\right)$

where $t=0,1,2,3,4,5$.

Let $H$ be the symmetric hexagon inscribed in $C/2$ as defined in Section 2:
take the sides of $C/2$ parallel to vector
$\left(1+\cos(\frac{20\pi}{22}),\sin(\frac{20\pi}{22})\right)$ and choose the
two additional points such that they divide the sides parallel to vector
$\left(\cos(\frac{6\pi}{22})-\cos(\frac{4\pi}{22}),\sin(\frac{6\pi}{22})-\sin(\frac{4\pi}{22})\right)$
in the ratio $0.68:0.32$.
Coordinates of the vertices of $H$ are:

  * •

$A_{1}=-A_{4}=\left(0.32\cos(\frac{4\pi}{22})+0.68\cos(\frac{6\pi}{22}),0.32\sin(\frac{4\pi}{22})+0.68\sin(\frac{6\pi}{22})\right)$,

  * •

$A_{2}=-A_{5}=\left(1,0\right)$,

  * •

$A_{3}=-A_{6}=\left(\cos(\frac{2\pi}{22}),\sin(\frac{-2\pi}{22})\right)$.

As before, let $H$ be the half-open hexagon defined by the points $A_{i}$ that
does not contain the line segment connecting the points $A_{1}$ and $A_{4}$
and the point $A_{1}$ itself. For $i=1\dots 6$ let $a^{i}$ denote the position
vector of point $A_{i}$. Then the hexagonal tiling of the plane with hexagon
$H$ is the packing by Voronoi regions of the lattice $\mathcal{L}$ spanned by
vectors $\left(a^{1}+a^{6}\right)$ and $\left(a^{2}+a^{3}\right)$. The basis
vectors of the sublattice $\mathcal{L^{\prime}}$ corresponding to the single
color class containing the hexagon centered at the origin are:

  * •

$v_{1}=\left(3(a^{1}+a^{6})\right)$,

  * •

$v_{2}=\left(2(a^{2}+a^{3})\right)$.

Once again, we need to show is that polygons $\mathcal{L^{\prime}}+C/2\oplus
H$ form a packing. The vertices of $C/2\oplus H$ are:

  * •

$\pm\left(\cos(\frac{12\pi}{22})+a^{1}_{x},\text{
}\sin(\frac{12\pi}{22})+a^{1}_{y}\right)$

  * •

$\pm\left(\cos(\frac{10\pi}{22})+a^{1}_{x},\text{
}\sin(\frac{10\pi}{22})+a^{1}_{y}\right)$

  * •

$\pm\left(\cos(\frac{8\pi}{22})+a^{1}_{x},\text{
}\sin(\frac{8\pi}{22})+a^{1}_{y}\right)$

  * •

$\pm\left(\cos(\frac{6\pi}{22})+a^{1}_{x},\text{
}\sin(\frac{6\pi}{22})+a^{1}_{y}\right)$

  * •

$\pm\left(\cos(\frac{4\pi}{22})+a^{1}_{x},\text{
}\sin(\frac{4\pi}{22})+a^{1}_{y}\right)$

  * •

$\pm\left(\cos(\frac{2\pi}{22})+a^{1}_{x},\text{
}\sin(\frac{2\pi}{22})+a^{1}_{y}\right)$

  * •

$\pm\left(\cos(\frac{2\pi}{22})+a^{2}_{x},\text{
}\sin(\frac{2\pi}{22})+a^{2}_{y}\right)$

  * •

$\pm\left(1+a^{2}_{x},\text{ }a^{2}_{y}\right)=(2,0)$

  * •

$\pm\left(\cos(\frac{2\pi}{22})+a^{3}_{x},\text{
}\sin(\frac{-2\pi}{22})+a^{3}_{y}\right)$

  * •

$\pm\left(\cos(\frac{4\pi}{22})+a^{3}_{x},\text{
}\sin(\frac{-4\pi}{22})+a^{3}_{y}\right)$

  * •

$\pm\left(\cos(\frac{6\pi}{22})+a^{3}_{x},\text{
}\sin(\frac{-6\pi}{22})+a^{3}_{y}\right)$

  * •

$\pm\left(\cos(\frac{8\pi}{22})+a^{3}_{x},\text{
}\sin(\frac{-8\pi}{22})+a^{3}_{y}\right)$

  * •

$\pm\left(\cos(\frac{10\pi}{22})+a^{3}_{x},\text{
}\sin(\frac{-10\pi}{22})+a^{3}_{y}\right)$

Since $H\oplus C/2$ has circumradius $2$, it is enough to consider lattice
points of $\mathcal{L^{\prime}}$ with length at most 4. Besides the origin
there are $6$ such points and by symmetry we only have to check $3$ of them
and the corresponding hexagons, namely:
$H_{1}:=H+v_{2}$, $H_{2}:=H+v_{1}$ and $H_{3}:=H+v_{1}+v_{2}$.

$H$ and $H_{1}$ are separated by exactly one differently colored hexagon which
is enough as $v_{2}$ is perpendicular to the common sides of $H$ and $C/2$.
All is left is to give a line $l_{1}$ that separates $H$ from $H_{2}$ and a
line $l_{2}$ that separates $H$ from $H_{3}$. For example consider the lines
$l_{1}$ and $l_{2}$ defined by the following equations:

$\displaystyle y-\sin(\frac{12\pi}{22})-a^{6}_{x}-\frac{1}{300}$
$\displaystyle=\frac{a^{6}_{y}-a^{1}_{y}}{a^{6}_{x}-a^{1}_{x}}\left(x-\cos(\frac{12\pi}{22})-a^{6}_{x}\right)$
$\displaystyle y-\sin(\frac{4\pi}{22})-a^{1}_{y}-\frac{1}{50}$
$\displaystyle=\frac{\sin(\frac{4\pi}{22})-\sin(\frac{2\pi}{22})}{\cos(\frac{4\pi}{22})-\cos(\frac{2\pi}{22})}\left(x-\cos(\frac{4\pi}{22})-a^{1}_{x}\right)$

Figure 12: Lines $l_{1}$ and $l_{2}$ separate $H$ from $H_{2}\oplus C/2$ and
from $H_{3}\oplus C/2$

It is straightforward to check that all vertex points of $H\oplus C/2$ are
below line $l_{1}$ and line $l_{2}$, while all vertex points of $H_{2}\oplus
C/2$ are above $l_{1}$, and all vertex points of $H_{3}\oplus C/2$ are above
$l_{2}$ (see Figure 12). Thus the coloring is indeed proper.


aainstitutetext: Ernest Orlando Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USAbbinstitutetext: Department of Physics, University of
California, Berkeley, CA 94720, USAccinstitutetext: Kavli Institute for the
Physics and Mathematics of the Universe (WPI), University of Tokyo Institutes
for Advanced Study, University of Tokyo, Kashiwa 277-8583,
Japanddinstitutetext: Institute for Solid State Physics, University of Tokyo,
Kashiwa, Chiba 277-8581, Japan

# Axion detection via superfluid 3He ferromagnetic phase and quantum
measurement techniques

So Chigusa  0000-0001-6005-4447<EMAIL_ADDRESS>c Dan Kondo
<EMAIL_ADDRESS>a,b,c,1 Hitoshi Murayama 111Hamamatsu Professor
0000-0001-5769-9471<EMAIL_ADDRESS>c Risshin Okabe  0000-0002-5351-174X
<EMAIL_ADDRESS>d and Hiroyuki Sudo  0000-0003-4744-3100
<EMAIL_ADDRESS>
###### Abstract

We propose to use the nuclear spin excitation in the ferromagnetic A1 phase of
the superfluid 3He for the axion dark matter detection. This approach is
striking in that it is sensitive to the axion-nucleon coupling, one of the
most important features of the QCD axion introduced to solve the strong CP
problem. We review a quantum mechanical description of the nuclear spin
excitation and apply it to the estimation of the axion-induced spin excitation
rate. We also describe a possible detection method of the spin excitation in
detail and show that the combination of the squeezing of the final state with
the Josephson parametric amplifier and the homodyne measurement can enhance
the sensitivity. It turns out that this approach gives good sensitivity to the
axion dark matter with the mass of
$\mathcal{O}(1)\,$\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$ depending
on the size of the external magnetic field. We estimate the parameters of
experimental setups, e.g., the detector volume and the amplitude of squeezing,
required to reach the QCD axion parameter space.

##  1 Introduction

Axion Peccei:1977hh  is a proposed solution to the strong CP problem, namely
to explain why the quantum chromodynamics (QCD) does not violate the time-
reversal symmetry. The experimental upper limit on the neutron electric dipole
moment $d_{n}<1.8\times 10^{-26}\,e\,$\mathrm{c}\mathrm{m}$$ Abel:2020pzs
implies that the so-called vacuum angle of QCD to be extremely small
$\bar{\theta}<10^{-10}$. The theory assumes a new global $\mathrm{U}(1)$
Peccei–Quinn symmetry broken spontaneously at the energy scale called the
axion decay constant $f_{a}$ as well as explicitly by the QCD anomaly. The
effective operator of the axion coupling to gluons is

$\displaystyle{\cal
L}_{a}=\frac{g_{s}^{2}}{64\pi^{2}}\left(\bar{\theta}+\frac{a}{f_{a}}\right)\epsilon^{\mu\nu\rho\sigma}G^{b}_{\mu\nu}G^{b}_{\rho\sigma}\
.$ (1)

Switching to the chiral Lagrangian, it can be shown that the axion settles to
the ground state where $\bar{\theta}$ is dynamically canceled.

Interestingly, it was pointed out that the axion can also comprise the dark
matter of the Universe from misalignment mechanism or emission from axion
strings Preskill:1982cy . The initial version of the theory assumed $f_{a}=v$
(electroweak scale) and was excluded by beam dump experiments. It was later
proposed to take $f_{a}\gg v$ dubbed “invisible axion” Kim:1979if ;
Shifman:1979if ; Dine:1981rt ; Zhitnitsky:1980tq . The axion abundance is
higher for higher $f_{a}$, and
$f_{a}\simeq${10}^{12}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ is typically
regarded as a preferred range. It translates to
$m_{a}\simeq$\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$ scale.

Many direct detection experiments for the dark matter axion, such as
asztalos2010squid ; ADMX:2018gho ; ADMX:2019uok ; ADMX:2021nhd ;
HAYSTAC:2018rwy ; HAYSTAC:2020kwv ; HAYSTAC:2023cam ; McAllister:2017lkb ;
Quiskamp:2022pks ; Alesini:2019ajt ; Alesini:2020vny ; Alesini:2022lnp ;
Lee:2020cfj ; Jeong:2020cwz ; CAPP:2020utb ; Lee:2022mnc ; Kim:2022hmg ;
Yi:2022fmn , rely on the axion coupling to photons
$aF_{\mu\nu}\tilde{F}^{\mu\nu}$. Their prospect in the near future is becoming
exciting. Yet the axion coupling to photons is highly model-dependent. To
fully verify that the axion solves the strong CP problem, measuring its
coupling to hadrons would be crucial. In particular, the axion couples to the
nucleon spins $\vec{\nabla}a\cdot\vec{s}_{N}$ with relatively little model
dependence. Search for dark matter axion using the nuclear spins, or
confirming detected axion signal with nuclear signs, would be crucial to
enhance our understanding of both the strong CP problem as well as the nature
of dark matter. In spite of its importance, there are relatively few
experiments and proposals including JacksonKimball:2017elr ; Wu:2019exd ;
Garcon:2019inh ; Bloch:2021vnn ; Bloch:2019lcy ; Lee:2022vvb ; Graham:2020kai
; Gao:2022nuq ; Dror:2022xpi ; Brandenstein:2022eif ; Wei:2023rzs ;
Chigusa:2023hmz  in this direction.

In this paper, we propose a new experimental technique to detect dark matter
axions using their coupling to nuclear spins. Interactions among the nuclear
spins are very weak because their magnetic moments are suppressed by the
nucleon mass $\mu_{N}=e/m_{N}$ rather than the electron mass
$\mu_{B}=e/m_{e}$. One needs to identify material where nuclear spins play a
major role at very low temperatures.

We point out that the A1 phase of superfluid 3He is a unique material that has
an ordering of nuclear spins without relying on their coupling to electron
spins. This is because the Cooper pairs of 3He atoms are in the $p$-wave
(anti-symmetric) with total spin $S=1$ (symmetric) as required by Fermi
statistics. In a high magnetic field, it becomes basically a ferromagnet of
nuclear spins. The corresponding nuclear magnon is gapped due to the external
magnetic field and the gap can be tuned to the axion mass. It is quite
remarkable that the gap happens to be in the range of the preferred axion mass
for dark matter with an achievable magnetic field. Then the magnon can be
converted to a cavity photon resonantly due to the polariton mixing between
the magnon and photon. Again the size of the cavity is such that it can be
fitted in a laboratory. Note that our setup is distinct from other proposals
to use superfluid 3He for axion dark matter search Gao:2022nuq ; Dror:2022xpi
in the superfluid phase used and/or the signal detection method.

Because our experiments are performed at such low temperatures
$T\lesssim$3\text{\,}\mathrm{m}\mathrm{K}$$ that the target 3He shows
superfluidity, the quantum noise caves1982quantum  becomes non-negligible.
These days several applications of quantum measurement techniques to axion
detections have been studied in order to circumvent the quantum noise
Malnou:2018dxn ; HAYSTAC:2020kwv ; Wurtz:2021cnm ; Zheng:2016qjv ;
Ikeda:2021mlv ; Sushkov:2023fjw ; Dixit:2020ymh ; Shi:2022wpf ;
Lamoreaux:2013koa . In this paper, we apply the squeezing technique, which has
been discussed in Malnou:2018dxn ; HAYSTAC:2020kwv , and evaluate the
improvement in the sensitivity of our experiment.

This paper is organized as follows. In section 2, we review the properties of
3He. We analyze superfluid phases of 3He using the spinor BEC formalism and
understand the properties of nuclear magnons in the ferromagnetic A1 phase. In
LABEL:sec:Axiondetection, we discuss how the axion dark matter signal can be
detected using superfluid 3He; we use a nuclear magnon mode, which is
converted into a cavity photon through the polariton mixing. We also discuss
how noise reduction is realized by using squeezing and the homodyne
measurement. We show sensitivities for several different setups in
LABEL:sec:sensitivity and conclude in LABEL:sec:conclusion. A detailed
description of our noise estimate and statistical treatment is summarized in
LABEL:sec:SNR. Finally, we review the Josephson parametric amplifier (JPA),
which is a representative apparatus for squeezing, in LABEL:sec:JPA.

##  2 Understanding 3He via spinor BEC

In this section, we will describe the phase structure of the superfluid 3He
using Ginzburg-Landau formalism and simplified spinor BEC formalism. We
summarize the phase structure in the following tree diagram. The important
thing in this paper is that we will use the fact that the A1 phase has a
ferromagnetic property.

for tree= grow’=0, child anchor=west, parent anchor=south, anchor=west,
calign=first, edge path= [draw, edge] (!u.south west) +(7.5pt,0) —-
node[fill,inner sep=1.25pt] (.child anchor)edge label; , before typesetting
nodes= if n=1 insert before=[,phantom] , fit=band, before computing xy=l=15pt,
[Superfluid phases of 3He [External magnetic field $H\neq 0$ [A1 phase
Ferromagnetic$\rightarrow$ magnon] [A2 phase (A phase in a magnetic field)]
[B2 phase (B phase in a magnetic field)] ] [No external magnetic field [A
phase Anti-Ferromagnetic] [B phase Spin-Orbit Coupling] ] ]

In section 2.1, we will sketch the phase structure of 3He with Ginzburg–Landau
formalism. In LABEL:subsec:HespinorBEC, we will describe the A phase of 3He
with an external magnetic field with spinor BEC formalism. In
LABEL:subsec:nuclearmagnon, we will study how to utilize the ferromagnetic
property of the A1 phase of 3He as magnon.

###  2.1 Phases of superfluid 3He

Historically, after the success of the BCS theory PhysRev.108.1175 , people
tried to look for the description of the superfluid 3He because it is liquid
and has no lattice structure inside. Some people thought about the pairing
states which are not $s$-wave. One is about the general anisotropic case by
Anderson and Morel PhysRev.123.1911 . This model has a peculiar feature that
the nodes exist on the Fermi surface for the axial $p$-wave state (ABM state
named after Anderson-Brinkman-Morel). It turned out that this theory describes
what is called the A phase nowadays. The number of the substates for $p$-wave
Cooper pair is three. Later, it was shown that the mixing of all these states
is favored energetically PhysRev.131.1553 , which turned out to be the B phase
nowadays. At early stages, the thermodynamic properties are explored by
Leggett using the framework of renormalized quasi-particles PhysRev.140.A1869
; PhysRevLett.14.536 . Experimentally, the A and B phases were discovered at
$2.6\text{\,}\mathrm{m}\mathrm{K}$ and $1.8\text{\,}\mathrm{m}\mathrm{K}$
respectively PhysRevLett.29.920 , which confirmed the existence of the phase
structure of the superfluid 3He.

The nucleus of a 3He atom consists of two protons and one neutron. The proton
spins are aligned anti-parallel with each other, while the neutron spin is
isolated, making the total spin angular momentum to be $I=1/2$. In the
superfluid phase, two 3He atoms form a Cooper pair, whose ground state is a
spin-triplet $p$-wave condensate Vollhardt1990TheSP . The corresponding order
parameter is expressed in terms of annihilation operators of nuclei
$\hat{a}_{\vec{k}\alpha}$ as

$\expectationvalue{\hat{a}_{-\vec{k}\beta}\hat{a}_{\vec{k}\alpha}}\propto\Delta_{\vec{k}\alpha\beta}\equiv\sum_{\mu=1}^{3}d_{\mu}(\vec{k})(\sigma_{\mu}i\sigma_{2})_{\alpha\beta}\,,$
(2)

where $\vec{k}$ and $\alpha$ ($\beta$) are the momentum and the spin of a 3He
nucleus, respectively, and $\sigma_{\mu}$ is the Pauli matrix. Since a Cooper
pair forms a spin-triplet $L=1$ relative angular momentum state, the vector
$d_{\mu}(\vec{k})$ can be represented as a linear combination of spherical
harmonics $Y_{1m}(\vec{k}/|\vec{k}|)\propto\vec{k}/|\vec{k}|$,

$d_{\mu}(\vec{k})=\sqrt{3}\sum_{j=1}^{3}A_{\mu
j}\frac{\vec{k}_{j}}{|\vec{k}_{j}|}.$ (3)

The phenomenological Lagrangian of the Cooper pairs, i.e., Ginzburg–Landau
Lagrangian, can be expressed in terms of the $3\times 3$ order parameter
matrix $A_{\mu j}$ PhysRevA.8.2732 ; PhysRevLett.30.1135 . The index
$\mu=1,2,3$ refers to the $S=1$ states while $j=1,2,3$ to the $L=1$ state both
in the Cartesian basis. Namely $A_{\mu j}$ transforms as a bi-vector under
$\mathrm{SO}(3)_{L}\times\mathrm{SO}(3)_{S}$. Note that $A_{\mu j}$ is complex
as its phase $\mathrm{U}(1)_{\phi}$ corresponds to the conserved number
operator of the Cooper pairs. Because the Lagrangian has to be Hermitian and
invariant under the global
$\mathrm{SO}(3)_{L}\times\mathrm{SO}(3)_{S}\times\mathrm{U}(1)_{\phi}$
symmetry, we have only one second-order term of $A_{\mu j}$

$I_{0}=\tr(AA^{\dagger}),$ (4)

and five fourth-order terms

$\displaystyle I_{1}$ $\displaystyle=\absolutevalue{\tr(AA^{T})}^{2},$ (5)
$\displaystyle I_{2}$ $\displaystyle=\quantity[\tr(AA^{\dagger})]^{2},$ (6)
$\displaystyle I_{3}$ $\displaystyle=\tr[(AA^{T})(AA^{T})^{*}],$ (7)
$\displaystyle I_{4}$ $\displaystyle=\tr[(AA^{\dagger})^{2}],$ (8)
$\displaystyle I_{5}$ $\displaystyle=\tr[(AA^{\dagger})(AA^{\dagger})^{*}],$
(9)

in the effective potential. As a result, in the absence of any external
fields, the effective potential per volume is given by

$V_{0}=\alpha(T)I_{0}+\frac{1}{2}\sum_{i=1}^{5}\beta_{i}I_{i}\,,$ (10)

where we neglect higher-order terms of $A_{\mu j}$, which can be justified
when we consider the phenomenology of a system sufficiently close to the phase
transition, and the numerical values of $|A_{\mu j}|$ are small. The
coefficients $\alpha$ and $\beta_{i}$ are determined by the microscopic
theory. For example, they have been calculated in the weak-coupling theory
Vollhardt1990TheSP , and their numerical values are

$\displaystyle\alpha(T)\sim-10^{-3}\quantity(1-\frac{T}{T_{c}})\;$\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}^{-1}\mathrm{\textup{\AA}}^{-3}$\,,$
(11)
$\displaystyle(\beta_{1}^{\mathrm{WC}},\beta_{2}^{\mathrm{WC}},\beta_{3}^{\mathrm{WC}},\beta_{4}^{\mathrm{WC}},\beta_{5}^{\mathrm{WC}})=\frac{6}{5}\beta_{0}\quantity(-\frac{1}{2},1,1,1,-1)\,,$
(12) $\displaystyle\beta_{0}\sim
10^{-3}\;$\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}^{-3}\mathrm{\textup{\AA}}^{-3}$\,,$
(13)

where $T_{c}$ is the transition temperature
$\sim$2.6\text{\,}\mathrm{m}\mathrm{K}$$ in the absence of external magnetic
fields. The values of $\beta_{i}$ can differ from those of
$\beta^{\mathrm{WC}}_{i}$ depending on pressure. Nevertheless, we will use the
numerical values in eqs. 12 and 13 for $\beta_{i}$ below since the
experimentally measured values differ from $\beta^{\mathrm{WC}}_{i}$ by only
$\order{1}$ factors,
$(\beta_{i}-\beta_{i}^{\mathrm{WC}})/\beta_{0}\simeq\mathcal{O}(1)$
choi2007strong .

As noted above, the effective Lagrangian has a global symmetry
$\mathrm{SO}(3)_{L}\times\mathrm{SO}(3)_{S}\times\mathrm{U}(1)_{\phi}$, which
corresponds to the rotation in the momentum space, the rotation in the spin
space, and the overall phase rotation, respectively. It is known that,
depending on the values of coefficients in eq. 10, the matrix $A$ acquires a
non-zero expectation value in the ground state, which spontaneously breaks the
global symmetry and leads to different phases. Without an external magnetic
field, there are two superfluid phases for 3He, the A and B phases. Their
expectation values are expressed as

(14)

where $\phi$ is an overall phase, and $R_{\mu j}$ is a relative rotation of
the spin and orbital spaces, represented by a rotation axis $\vec{n}$ and a
rotation angle $\theta$. Note that there are more than one choice of the order
parameter in the A phase corresponding to the choices of particular directions
of spin and orbital spaces, both of which are assumed to be the $z$-axis in
the above expression.

When we turn on an external magnetic field $\vec{B}$, the potential $V$ has
two more invariant terms

$\displaystyle F^{(1)}$ $\displaystyle=i\eta\sum_{\mu\nu\lambda
j}\epsilon_{\mu\nu\lambda}B_{\mu}A_{\nu j}^{*}A_{\lambda j}\,,$ (15)
$\displaystyle F^{(2)}$ $\displaystyle\propto\sum_{\mu\nu j}B_{\mu}A_{\mu
j}B_{\nu}A_{\nu j}^{*}\,.$ (16)

The term $F^{(1)}$ is there only for the indices $\mu,\nu$ because the 3He
atoms are electrical neutral and their orbital angular momentum does not have
a magnetic moment, while their spins do. Assuming that $\vec{B}$ is along the
$z$-direction, one can see that $F^{(1)}$ and $F^{(2)}$ break the global
symmetry to
$\mathrm{SO}(3)_{L}\times\mathrm{U}(1)_{S_{z}}\times\mathrm{U}(1)_{\phi}$.
Because these interaction terms $F^{(1)}$ and $F^{(2)}$ bring three types of
spontaneous symmetry breaking depending on the coefficients, there are three
corresponding phases: Similarly to LABEL:eq:C.10, The Hamiltonian describing
the resonator part of FJPA is

$\displaystyle H_{\text{sys}}$
$\displaystyle=\frac{(2en)^{2}}{2C_{t}}-E_{J}\cos\vartheta_{1}-E_{J}\cos\vartheta_{2}$
$\displaystyle=4E_{C}n^{2}-E_{J}^{\text{eff}}(\Phi_{\text{ext}})\cos\vartheta,$
(153)

where $E_{C}=e^{2}/(2C_{t})$,
$E_{J}^{\text{eff}}(\Phi_{\text{ext}})=2E_{J}\cos(e\Phi_{\text{ext}})=2E_{J}\cos(\pi\Phi_{\text{ext}}/\Phi_{0})$.
We set the DC part of $\Phi_{\text{ext}}$ to the quarter of magnetic flux
quantum, i.e., bias the ampliﬁer at $\Phi_{DC}=\Phi_{0}/4$. In the absence of
a pump,

$\displaystyle H_{\text{sys}}$
$\displaystyle=4E_{C}n^{2}-\sqrt{2}E_{J}\cos\vartheta.$ (154)

Expanding $\cos\vartheta$ to order $\vartheta^{2}$, we can write the
Hamiltonian by the ladder operator.

$\displaystyle H_{\text{sys}}$
$\displaystyle=\omega_{c}\hat{a}^{\dagger}\hat{a},$ (155)

where

$\displaystyle\vartheta$
$\displaystyle=\left(\frac{\sqrt{2}E_{C}}{E_{J}}\right)^{1/4}(\hat{a}^{\dagger}+\hat{a}),$
(156) $\displaystyle n$
$\displaystyle=\frac{i}{2}\left(\frac{E_{J}}{\sqrt{2}E_{C}}\right)^{1/4}(\hat{a}^{\dagger}-\hat{a}),$
(157) $\displaystyle\omega_{c}$ $\displaystyle=2\sqrt{2\sqrt{2}E_{C}E_{J}}.$
(158)

Next, we consider including the AC part of the external field
$\Phi_{\text{ext}}$ due to the pumping

$\displaystyle\Phi_{\text{ext}}$
$\displaystyle=\Phi_{\text{DC}}+\Phi_{\text{AC}}\cos(\alpha\omega_{c}t).$
(159)

We set the AC part of $\Phi_{\text{ext}}$ smaller than the DC part
$\Phi_{\text{AC}}\ll\Phi_{\text{DC}}$, and evaluate
$E_{J}^{\text{eff}}(\Phi_{\text{ext}})$ as

$\displaystyle E_{J}^{\text{eff}}(\Phi_{\text{ext}})$ $\displaystyle\simeq
E_{J}^{\text{eff}}(\Phi_{\text{DC}})+\left.\frac{\partial
E_{J}^{\text{eff}}(\Phi)}{\partial\Phi}\right|_{\Phi=\Phi_{\text{DC}}}\Phi_{\text{AC}}\cos(\alpha\omega_{c}t)$
$\displaystyle=\sqrt{2}E_{J}-\left(\frac{\pi\Phi_{\text{AC}}}{\Phi_{0}}\right)\sqrt{2}E_{J}\cos(\alpha\omega_{c}t).$
(160)

Thus, Hamiltonian $H_{\text{sys}}$ becomes

$\displaystyle H_{\text{sys}}$
$\displaystyle\simeq\omega_{c}\hat{a}^{\dagger}\hat{a}+\mu_{r}\cos(\alpha\omega_{c}t)(\hat{a}^{\dagger}+\hat{a})^{2},$
(161)

where
$\mu_{r}=\frac{\pi\Phi_{\text{AC}}}{\Phi_{0}}\left(\frac{E_{C}E_{J}}{\sqrt{2}}\right)^{\frac{1}{2}}$.
We focus on the parametric amplifier region ($\alpha\simeq 2$). Applying
rotating wave approximation, we can estimate $H_{\text{sys}}$ as

$\displaystyle H_{\text{sys}}$
$\displaystyle\simeq\omega_{c}\hat{a}^{\dagger}\hat{a}+\frac{\mu_{r}}{2}e^{i\alpha\omega_{c}t}\hat{a}^{2}+\frac{\mu_{r}}{2}e^{-i\alpha\omega_{c}t}\hat{a}^{\dagger
2}.$ (162)

Figure 7: Schematic of the parametric amplifier. Here we use opto-mechanical
analogy (resonator consisting of the cavity) instead of Josephson parametric
amplifier.

We assume the resonator has a semi-infinite waveguide mode (the annihilation
operator of which is denoted as $\hat{b}_{k}$) connected as an input/output
port and also has internal losses in the resonator (the annihilation operator
of which is denoted as $\hat{c}_{k}$). The schematic of this parametric
amplifier using opto-mechanical analogy is fig. 7. The total Hamiltonian
describing this is

$\displaystyle H_{\text{tot}}$
$\displaystyle=H_{\text{sys}}+H_{\text{sig}}+H_{\text{loss}},$ (163)
$\displaystyle H_{\text{sys}}$
$\displaystyle=\omega_{c}\hat{a}^{\dagger}\hat{a}+\frac{\mu_{r}}{2}e^{i\alpha\omega_{c}t}\hat{a}^{2}+\frac{\mu_{r}}{2}e^{-i\alpha\omega_{c}t}\hat{a}^{\dagger
2},$ (164) $\displaystyle H_{\text{sig}}$
$\displaystyle=\int\mathrm{d}\omega\left[\omega\hat{b}^{\dagger}(\omega)\hat{b}(\omega)+i\sqrt{\frac{\kappa_{e}}{2\pi}}(\hat{a}^{\dagger}\hat{b}(\omega)-\hat{b}^{\dagger}(\omega)\hat{a})\right],$
(165) $\displaystyle H_{\text{loss}}$
$\displaystyle=\int\mathrm{d}\omega\left[\omega\hat{c}^{\dagger}(\omega)\hat{c}(\omega)+i\sqrt{\frac{\kappa_{i}}{2\pi}}(\hat{a}^{\dagger}\hat{c}(\omega)-\hat{c}^{\dagger}(\omega)\hat{a})\right].$
(166)

Here, $\kappa_{e}$ is the external loss rate of the resonator, and
$\kappa_{i}$ is the internal loss rate of the resonator. As we did in
LABEL:sec:SNR, we get Heisenberg equations for the resonator mode $\hat{a}$
and the input-output relationship of the waveguide:

$\displaystyle\derivative{\hat{a}(t)}{t}$
$\displaystyle=\left(-i\omega_{c}-\frac{\kappa}{2}\right)\hat{a}(t)-i\mu_{r}e^{-i\alpha\omega_{c}t}\hat{a}^{\dagger}(t)+\sqrt{\kappa_{e}}\hat{b}_{\text{in}}(t)+\sqrt{\kappa_{i}}\hat{c}_{\text{in}}(t),$
(167) $\displaystyle\hat{b}_{\text{out}}(t)$
$\displaystyle=\hat{b}_{\text{in}}(t)-\sqrt{\kappa_{e}}\hat{a}(t),$ (168)

where $\kappa\equiv\kappa_{i}+\kappa_{e}$.

###  B.3 Resonator equation

In this subsection, we neglect the internal loss ($\kappa=\kappa_{e}$) and
switch to a frame rotating at the angular frequency $\alpha\omega_{c}/2$, and
deﬁne the following operators:

$\displaystyle\hat{A}(t)$
$\displaystyle=e^{i\frac{\alpha}{2}\omega_{c}t}\hat{a}(t),$ (169)
$\displaystyle\hat{B}_{\text{in}\,(\text{out})}(t)$
$\displaystyle=e^{i\frac{\alpha}{2}\omega_{c}t}\hat{b}_{\text{in}\,(\text{out})}(t).$
(170)

Assuming $\alpha=2$ for simplicity, the resonator equation (167) and the
input-output relation become

$\displaystyle\derivative{\hat{A}(t)}{t}$
$\displaystyle=-\frac{\kappa}{2}\hat{A}(t)-i\mu_{r}\hat{A}^{\dagger}(t)+\sqrt{\kappa}\hat{B}_{\mathrm{in}}(t),$
$\displaystyle\hat{B}_{\text{out}}(t)$
$\displaystyle=\hat{B}_{\text{in}}(t)-\sqrt{\kappa}\hat{A}(t),$ (171)

We consider the case with monochromatic incident light, i.e.,

$\hat{B}_{\mathrm{in}}(t)=\hat{B}_{\mathrm{in}}(0)e^{-i\Delta\omega},$ (172)

where $\Delta\omega\equiv\omega-\omega_{c}$. In this case, the stationary
solution of $\hat{A}(t)$ has only two Fourier components $e^{\pm i\Delta\omega
t}$. The resonator equations for these components are

$-i\Delta\omega\matrixquantity(\hat{A}(\Delta\omega)\\\
\hat{A}^{\dagger}(-\Delta\omega))=\matrixquantity(-\kappa/2&-i\mu_{r}\\\
+i\mu_{r}&-\kappa/2)\matrixquantity(\hat{A}(\Delta\omega)\\\
\hat{A}^{\dagger}(-\Delta\omega))+\sqrt{\kappa}\matrixquantity(\hat{B}_{\mathrm{in}}(0)\\\
0),$ (173)

and

$+i\Delta\omega\matrixquantity(\hat{A}(-\Delta\omega)\\\
\hat{A}^{\dagger}(\Delta\omega))=\matrixquantity(-\kappa/2&-i\mu_{r}\\\
+i\mu_{r}&-\kappa/2)\matrixquantity(\hat{A}(-\Delta\omega)\\\
\hat{A}^{\dagger}(\Delta\omega))+\sqrt{\kappa}\matrixquantity(0\\\
\hat{B}^{\dagger}_{\mathrm{in}}(0)).$ (174)

Solving these equations, we obtain

$\displaystyle\hat{A}(t)=\frac{\frac{\kappa}{2}-i\Delta\omega}{\quantity(\frac{\kappa}{2}-i\Delta\omega)^{2}-\mu_{r}^{2}}\sqrt{\kappa}\hat{B}_{\mathrm{in}}(0)e^{-i\Delta\omega
t}+\frac{-i\mu_{r}}{\quantity(\frac{\kappa}{2}+i\Delta\omega)^{2}-\mu_{r}^{2}}\sqrt{\kappa}\hat{B}^{\dagger}_{\mathrm{in}}(0)e^{+i\Delta\omega
t}.$ (175)

The output field is derived using eq. 171 as

$\displaystyle\hat{B}_{\mathrm{out}}(t)=$
$\displaystyle\quantity[1-\frac{\quantity(\frac{\kappa}{2}-i\Delta\omega)\kappa}{\quantity(\frac{\kappa}{2}-i\Delta\omega)^{2}-\mu_{r}^{2}}]\hat{B}_{\mathrm{in}}(0)e^{-i\Delta\omega
t}$
$\displaystyle\qquad+\frac{-i\mu_{r}\kappa}{\quantity(\frac{\kappa}{2}+i\Delta\omega)^{2}-\mu_{r}^{2}}\hat{B}^{\dagger}_{\mathrm{in}}(0)e^{+i\Delta\omega
t}.$ (176)

The first term represents the signal component, and the second term represents
the idler component.

When $\Delta\omega=0$, these two modes degenerate. In this case, the output
gain shows the phase-sensitivity. In order to verify this, we define the
following quadratures:

$\displaystyle\hat{X}_{\theta}$
$\displaystyle\equiv\frac{\hat{B}e^{-i\theta}+\hat{B}^{\dagger}e^{i\theta}}{\sqrt{2}},$
(177) $\displaystyle\hat{Y}_{\theta}$
$\displaystyle\equiv\frac{\hat{B}e^{-i\theta}-\hat{B}^{\dagger}e^{i\theta}}{\sqrt{2}i}.$
(178)

From eq. 176 with $\Delta\omega=0$, we find

$\displaystyle\hat{X}_{\theta,\,\text{out}}$
$\displaystyle=\left[1-\frac{\frac{\kappa^{2}}{2}}{\frac{\kappa^{2}}{4}-\mu_{r}^{2}}-\frac{\mu_{r}\kappa\sin(2\theta)}{\frac{\kappa^{2}}{4}-\mu_{r}^{2}}\right]\hat{X}_{\theta,\,\text{in}}-\frac{\mu_{r}\kappa\cos(2\theta)}{\frac{\kappa^{2}}{4}-\mu_{r}^{2}}\hat{Y}_{\theta,\,\text{in}},$
(179) $\displaystyle\hat{Y}_{\theta,\,\text{out}}$
$\displaystyle=\left[1-\frac{\frac{\kappa^{2}}{2}}{\frac{\kappa^{2}}{4}-\mu_{r}^{2}}+\frac{\mu_{r}\kappa\sin(2\theta)}{\frac{\kappa^{2}}{4}-\mu_{r}^{2}}\right]\hat{Y}_{\theta,\,\text{in}}-\frac{\mu_{r}\kappa\cos(2\theta)}{\frac{\kappa^{2}}{4}-\mu_{r}^{2}}\hat{X}_{\theta,\,\text{in}}.$
(180)

When $\theta=(1/4+n)\pi$ ($n\in\mathbb{Z}$) in particular, they take the
following form:

$\displaystyle\hat{X}_{\theta,\text{out}}=\sqrt{G}\hat{X}_{\theta,\text{in}},\quad\hat{Y}_{\theta,\text{out}}=\frac{1}{\sqrt{G}}\hat{Y}_{\theta,\text{in}},$
(181)

where the parameter $G$ is

$G=\quantity(\frac{\mu_{r}+\frac{\kappa}{2}}{\mu_{r}-\frac{\kappa}{2}})^{2}.$
(182)

Equation 181 represents the squeezing by a JPA and is what we used in
LABEL:eq:squeeze_SQ and LABEL:eq:squeeze_AMP.

## References

  * (1) R.D. Peccei and H.R. Quinn, _CP Conservation in the Presence of Instantons_ , _Phys. Rev. Lett._ 38 (1977) 1440.
  * (2) C. Abel et al., _Measurement of the Permanent Electric Dipole Moment of the Neutron_ , _Phys. Rev. Lett._ 124 (2020) 081803 [2001.11966].
  * (3) J. Preskill, M.B. Wise and F. Wilczek, _Cosmology of the Invisible Axion_ , _Phys. Lett. B_ 120 (1983) 127.
  * (4) J.E. Kim, _Weak Interaction Singlet and Strong CP Invariance_ , _Phys. Rev. Lett._ 43 (1979) 103.
  * (5) M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, _Can Confinement Ensure Natural CP Invariance of Strong Interactions?_ , _Nucl. Phys. B_ 166 (1980) 493.
  * (6) M. Dine, W. Fischler and M. Srednicki, _A Simple Solution to the Strong CP Problem with a Harmless Axion_ , _Phys. Lett. B_ 104 (1981) 199.
  * (7) A.R. Zhitnitsky, _On Possible Suppression of the Axion Hadron Interactions. (In Russian)_ , _Sov. J. Nucl. Phys._ 31 (1980) 260.
  * (8) S.J. Asztalos, G. Carosi, C. Hagmann, D. Kinion, K. Van Bibber, M. Hotz et al., _Squid-based microwave cavity search for dark-matter axions_ , _Physical review letters_ 104 (2010) 041301.
  * (9) ADMX collaboration, _A Search for Invisible Axion Dark Matter with the Axion Dark Matter Experiment_ , _Phys. Rev. Lett._ 120 (2018) 151301 [1804.05750].
  * (10) ADMX collaboration, _Extended Search for the Invisible Axion with the Axion Dark Matter Experiment_ , _Phys. Rev. Lett._ 124 (2020) 101303 [1910.08638].
  * (11) ADMX collaboration, _Search for Invisible Axion Dark Matter in the 3.3–4.2 $\mu$eV Mass Range_, _Phys. Rev. Lett._ 127 (2021) 261803 [2110.06096].
  * (12) HAYSTAC collaboration, _Results from phase 1 of the HAYSTAC microwave cavity axion experiment_ , _Phys. Rev. D_ 97 (2018) 092001 [1803.03690].
  * (13) HAYSTAC collaboration, _A quantum-enhanced search for dark matter axions_ , _Nature_ 590 (2021) 238 [2008.01853].
  * (14) HAYSTAC collaboration, _New results from HAYSTAC’s phase II operation with a squeezed state receiver_ , _Phys. Rev. D_ 107 (2023) 072007 [2301.09721].
  * (15) B.T. McAllister, G. Flower, J. Kruger, E.N. Ivanov, M. Goryachev, J. Bourhill et al., _The ORGAN Experiment: An axion haloscope above 15 GHz_ , _Phys. Dark Univ._ 18 (2017) 67 [1706.00209].
  * (16) A.P. Quiskamp, B.T. McAllister, P. Altin, E.N. Ivanov, M. Goryachev and M.E. Tobar, _Direct search for dark matter axions excluding ALP cogenesis in the 63- to 67- $\mu$eV range with the ORGAN experiment_, _Sci. Adv._ 8 (2022) abq3765 [2203.12152].
  * (17) D. Alesini et al., _Galactic axions search with a superconducting resonant cavity_ , _Phys. Rev. D_ 99 (2019) 101101 [1903.06547].
  * (18) D. Alesini et al., _Search for invisible axion dark matter of mass m ${}_{a}=43~{}\mu$eV with the QUAX–$a\gamma$ experiment_, _Phys. Rev. D_ 103 (2021) 102004 [2012.09498].
  * (19) D. Alesini et al., _Search for Galactic axions with a high-Q dielectric cavity_ , _Phys. Rev. D_ 106 (2022) 052007 [2208.12670].
  * (20) S. Lee, S. Ahn, J. Choi, B.R. Ko and Y.K. Semertzidis, _Axion Dark Matter Search around 6.7 $\mu$eV_, _Phys. Rev. Lett._ 124 (2020) 101802 [2001.05102].
  * (21) J. Jeong, S. Youn, S. Bae, J. Kim, T. Seong, J.E. Kim et al., _Search for Invisible Axion Dark Matter with a Multiple-Cell Haloscope_ , _Phys. Rev. Lett._ 125 (2020) 221302 [2008.10141].
  * (22) CAPP collaboration, _First Results from an Axion Haloscope at CAPP around 10.7 $\mu$eV_, _Phys. Rev. Lett._ 126 (2021) 191802 [2012.10764].
  * (23) Y. Lee, B. Yang, H. Yoon, M. Ahn, H. Park, B. Min et al., _Searching for Invisible Axion Dark Matter with an 18 T Magnet Haloscope_ , _Phys. Rev. Lett._ 128 (2022) 241805 [2206.08845].
  * (24) J. Kim et al., _Near-Quantum-Noise Axion Dark Matter Search at CAPP around 9.5 $\mu$eV_, _Phys. Rev. Lett._ 130 (2023) 091602 [2207.13597].
  * (25) A.K. Yi et al., _Axion Dark Matter Search around 4.55 $\mu$eV with Dine-Fischler-Srednicki-Zhitnitskii Sensitivity_, _Phys. Rev. Lett._ 130 (2023) 071002 [2210.10961].
  * (26) D.F. Jackson Kimball et al., _Overview of the Cosmic Axion Spin Precession Experiment (CASPEr)_ , _Springer Proc. Phys._ 245 (2020) 105 [1711.08999].
  * (27) T. Wu et al., _Search for Axionlike Dark Matter with a Liquid-State Nuclear Spin Comagnetometer_ , _Phys. Rev. Lett._ 122 (2019) 191302 [1901.10843].
  * (28) A. Garcon et al., _Constraints on bosonic dark matter from ultralow-field nuclear magnetic resonance_ , _Sci. Adv._ 5 (2019) eaax4539 [1902.04644].
  * (29) NASDUCK collaboration, _New constraints on axion-like dark matter using a Floquet quantum detector_ , _Sci. Adv._ 8 (2022) abl8919 [2105.04603].
  * (30) I.M. Bloch, Y. Hochberg, E. Kuflik and T. Volansky, _Axion-like Relics: New Constraints from Old Comagnetometer Data_ , _JHEP_ 01 (2020) 167 [1907.03767].
  * (31) J. Lee, M. Lisanti, W.A. Terrano and M. Romalis, _Laboratory Constraints on the Neutron-Spin Coupling of feV-Scale Axions_ , _Phys. Rev. X_ 13 (2023) 011050 [2209.03289].
  * (32) P.W. Graham, S. Hacıömeroğlu, D.E. Kaplan, Z. Omarov, S. Rajendran and Y.K. Semertzidis, _Storage ring probes of dark matter and dark energy_ , _Phys. Rev. D_ 103 (2021) 055010 [2005.11867].
  * (33) C. Gao, W. Halperin, Y. Kahn, M. Nguyen, J. Schütte-Engel and J.W. Scott, _Axion Wind Detection with the Homogeneous Precession Domain of Superfluid Helium-3_ , _Phys. Rev. Lett._ 129 (2022) 211801 [2208.14454].
  * (34) J.A. Dror, S. Gori, J.M. Leedom and N.L. Rodd, _On the Sensitivity of Spin-Precession Axion Experiments_ , 2210.06481.
  * (35) C. Brandenstein, S. Stelzl, E. Gutsmiedl, W. Schott, A. Weiler and P. Fierlinger, _Towards an electrostatic storage ring for fundamental physics measurements_ , _EPJ Web Conf._ 282 (2023) 01017 [2211.08439].
  * (36) K. Wei et al., _Dark matter search with a strongly-coupled hybrid spin system_ , 2306.08039.
  * (37) S. Chigusa, T. Moroi, K. Nakayama and T. Sichanugrist, _Dark matter detection using nuclear magnetization in magnet with hyperfine interaction_ , 2307.08577.
  * (38) C.M. Caves, _Quantum limits on noise in linear amplifiers_ , _Physical Review D_ 26 (1982) 1817.
  * (39) M. Malnou, D.A. Palken, B.M. Brubaker, L.R. Vale, G.C. Hilton and K.W. Lehnert, _Squeezed vacuum used to accelerate the search for a weak classical signal_ , _Phys. Rev. X_ 9 (2019) 021023 [1809.06470].
  * (40) K. Wurtz, B.M. Brubaker, Y. Jiang, E.P. Ruddy, D.A. Palken and K.W. Lehnert, _Cavity Entanglement and State Swapping to Accelerate the Search for Axion Dark Matter_ , _PRX Quantum_ 2 (2021) 040350 [2107.04147].
  * (41) H. Zheng, M. Silveri, R.T. Brierley, S.M. Girvin and K.W. Lehnert, _Accelerating dark-matter axion searches with quantum measurement technology_ , 1607.02529.
  * (42) T. Ikeda, A. Ito, K. Miuchi, J. Soda, H. Kurashige and Y. Shikano, _Axion search with quantum nondemolition detection of magnons_ , _Phys. Rev. D_ 105 (2022) 102004 [2102.08764].
  * (43) A.O. Sushkov, _Quantum Science and the Search for Axion Dark Matter_ , _PRX Quantum_ 4 (2023) 020101 [2304.11797].
  * (44) A.V. Dixit, S. Chakram, K. He, A. Agrawal, R.K. Naik, D.I. Schuster et al., _Searching for Dark Matter with a Superconducting Qubit_ , _Phys. Rev. Lett._ 126 (2021) 141302 [2008.12231].
  * (45) H. Shi and Q. Zhuang, _Ultimate precision limit of noise sensing and dark matter search_ , _npj Quantum Inf._ 9 (2023) 27 [2208.13712].
  * (46) S.K. Lamoreaux, K.A. van Bibber, K.W. Lehnert and G. Carosi, _Analysis of single-photon and linear amplifier detectors for microwave cavity dark matter axion searches_ , _Phys. Rev. D_ 88 (2013) 035020 [1306.3591].
  * (47) J. Bardeen, L.N. Cooper and J.R. Schrieffer, _Theory of superconductivity_ , _Phys. Rev._ 108 (1957) 1175.
  * (48) P.W. Anderson and P. Morel, _Generalized bardeen-cooper-schrieffer states and the proposed low-temperature phase of liquid ${\mathrm{he}}^{3}$_, _Phys. Rev._ 123 (1961) 1911.
  * (49) R. Balian and N.R. Werthamer, _Superconductivity with pairs in a relative $p$ wave_, _Phys. Rev._ 131 (1963) 1553.
  * (50) A.J. Leggett, _Theory of a superfluid fermi liquid. i. general formalism and static properties_ , _Phys. Rev._ 140 (1965) A1869.
  * (51) A.J. Leggett, _Spin susceptibility of a superfluid fermi liquid_ , _Phys. Rev. Lett._ 14 (1965) 536.
  * (52) D.D. Osheroff, W.J. Gully, R.C. Richardson and D.M. Lee, _New magnetic phenomena in liquid ${\mathrm{he}}^{3}$ below 3 mk_, _Phys. Rev. Lett._ 29 (1972) 920.
  * (53) D. Vollhardt, P. Wölfle and R.B. Hallock, _The Superfluid Phases of Helium 3_ , Taylor & Francis, London (1990).
  * (54) W.F. Brinkman and P.W. Anderson, _Anisotropic superfluidity in ${}^{3}\mathrm{He}$: Consequences of the spin-fluctuation model_, _Phys. Rev. A_ 8 (1973) 2732.
  * (55) N.D. Mermin and C. Stare, _Ginzburg-landau approach to $l\neq 0$ pairing_, _Phys. Rev. Lett._ 30 (1973) 1135.
  * (56) H. Choi, J. Davis, J. Pollanen, T. Haard and W. Halperin, _Strong coupling corrections to the ginzburg-landau theory of superfluid he 3_ , _Physical Review B_ 75 (2007) 174503.
  * (57) Y. Kawaguchi and M. Ueda, _Spinor bose–einstein condensates_ , _Physics Reports_ 520 (2012) 253.
  * (58) D.M. Stamper-Kurn and M. Ueda, _Spinor bose gases: Symmetries, magnetism, and quantum dynamics_ , _Rev. Mod. Phys._ 85 (2013) 1191.
  * (59) H. Godfrin and E. Krotscheck, _The dynamics of quantum fluids_ , _arXiv preprint arXiv:2206.06039_ (2022) .
  * (60) M.A. Lee, K.E. Schmidt, M.H. Kalos and G.V. Chester, _Green’s function monte carlo method for liquid ${}^{3}\mathrm{He}$_, _Phys. Rev. Lett._ 46 (1981) 728.
  * (61) R.M. Panoff and J. Carlson, _Fermion monte carlo algorithms and liquid ${}^{3}\mathrm{He}$_, _Phys. Rev. Lett._ 62 (1989) 1130.
  * (62) V.R. Pandharipande and R.B. Wiringa, _Variations on a theme of nuclear matter_ , _Rev. Mod. Phys._ 51 (1979) 821.
  * (63) E. Manousakis, S. Fantoni, V.R. Pandharipande and Q.N. Usmani, _Microscopic calculations for normal and polarized liquid ${}^{3}\mathrm{He}$_, _Phys. Rev. B_ 28 (1983) 3770.
  * (64) A. Schneider et al., _Direct measurement of the 3He+ magnetic moments_, _Nature_ 606 (2022) 878 [2206.05943].
  * (65) H. Watanabe and H. Murayama, _Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance_ , _Phys. Rev. Lett._ 108 (2012) 251602 [1203.0609].
  * (66) Y. Hidaka, _Counting rule for Nambu-Goldstone modes in nonrelativistic systems_ , _Phys. Rev. Lett._ 110 (2013) 091601 [1203.1494].
  * (67) D.S. Greywall, _${}^{3}\mathrm{He}$ specific heat and thermometry at millikelvin temperatures_, _Phys. Rev. B_ 33 (1986) 7520.
  * (68) M. Gorghetto and G. Villadoro, _Topological Susceptibility and QCD Axion Mass: QED and NNLO corrections_ , _JHEP_ 03 (2019) 033 [1812.01008].
  * (69) L.F. Abbott and P. Sikivie, _A Cosmological Bound on the Invisible Axion_ , _Phys. Lett. B_ 120 (1983) 133.
  * (70) M. Dine and W. Fischler, _The Not So Harmless Axion_ , _Phys. Lett. B_ 120 (1983) 137.
  * (71) P.D. Group, R. Workman, V. Burkert, V. Crede, E. Klempt, U. Thoma et al., _Review of particle physics_ , _Progress of theoretical and experimental physics_ 2022 (2022) 083C01.
  * (72) S. Chigusa, T. Moroi and K. Nakayama, _Detecting light boson dark matter through conversion into a magnon_ , _Phys. Rev. D_ 101 (2020) 096013 [2001.10666].
  * (73) D. Einzel, _Spin dependent transport and relaxation parameters of 3he-b_ , _Physica B+C_ 108 (1981) 1143.
  * (74) X. Zhang, C.-L. Zou, L. Jiang and H.X. Tang, _Strongly coupled magnons and cavity microwave photons_ , _Phys. Rev. Lett._ 113 (2014) 156401.
  * (75) Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami and Y. Nakamura, _Hybridizing ferromagnetic magnons and microwave photons in the quantum limit_ , _Physical review letters_ 113 (2014) 083603.
  * (76) Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami et al., _Quantum magnonics: The magnon meets the superconducting qubit_ , _Comptes Rendus Physique_ 17 (2016) 729.
  * (77) K. van Bibber, K. Lehnert and A. Chou, _Putting the squeeze on the axion_ , _Physics today_ (2020) .
  * (78) A.A. Clerk, M.H. Devoret, S.M. Girvin, F. Marquardt and R.J. Schoelkopf, _Introduction to quantum noise, measurement, and amplification_ , _Rev. Mod. Phys._ 82 (2010) 1155.
  * (79) Y.K. Semertzidis et al., _Axion Dark Matter Research with IBS/CAPP_ , 1910.11591.
  * (80) M.D. Bird, _Ultra-high field solenoids and axion detection_ , in _Microwave Cavities and Detectors for Axion Research_ , G. Carosi and G. Rybka, eds., (Cham), pp. 9–16, Springer International Publishing, 2020.
  * (81) J. Lough et al., _First Demonstration of 6 dB Quantum Noise Reduction in a Kilometer Scale Gravitational Wave Observatory_ , _Phys. Rev. Lett._ 126 (2021) 041102 [2005.10292].
  * (82) R. Dassonneville, R. Assouly, T. Peronnin, A.A. Clerk, A. Bienfait and B. Huard, _Dissipative Stabilization of Squeezing Beyond 3 dB in a Microwave Mode_ , _PRX Quantum_ 2 (2021) 020323 [2102.02863].
  * (83) T. Kashiwazaki, T. Yamashima, K. Enbutsu, T. Kazama, A. Inoue, K. Fukui et al., _Over-8-dB squeezed light generation by a broadband waveguide optical parametric amplifier toward fault-tolerant ultra-fast quantum computers_ , 2301.12658.
  * (84) H. Vahlbruch, M. Mehmet, K. Danzmann and R. Schnabel, _Detection of 15 db squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency_ , _Phys. Rev. Lett._ 117 (2016) 110801.
  * (85) P. Carenza, T. Fischer, M. Giannotti, G. Guo, G. Martínez-Pinedo and A. Mirizzi, _Improved axion emissivity from a supernova via nucleon-nucleon bremsstrahlung_ , _JCAP_ 10 (2019) 016 [1906.11844].
  * (86) M.V. Beznogov, E. Rrapaj, D. Page and S. Reddy, _Constraints on Axion-like Particles and Nucleon Pairing in Dense Matter from the Hot Neutron Star in HESS J1731-347_ , _Phys. Rev. C_ 98 (2018) 035802 [1806.07991].
  * (87) C. O’HARE, _cajohare/axionlimits: Axionlimits_ , July, 2020.  10.5281/zenodo.3932430.
  * (88) M. Martinello, D. Bice, C. Boffo, S. Chandrasekeran, G. Eremeev, F. Furuta et al., _Q-factor optimization for high-beta 650 mhz cavities for pip-ii_ , _Journal of Applied Physics_ 130 (2021) .
  * (89) T. Kashiwazaki, T. Yamashima, N. Takanashi, A. Inoue, T. Umeki and A. Furusawa, _Fabrication of low-loss quasi-single-mode PPLN waveguide and its application to a modularized broadband high-level squeezer_ , _Appl. Phys. Lett._ 119 (2021) 251104 [2201.01457].
  * (90) J.W. Foster, N.L. Rodd and B.R. Safdi, _Revealing the Dark Matter Halo with Axion Direct Detection_ , _Phys. Rev. D_ 97 (2018) 123006 [1711.10489].