# Quantum Sensing of Antiferromagnetic Magnon Two-Mode Squeezed Vacuum

Anna-Luisa E. Römling<EMAIL_ADDRESS>Condensed Matter Physics
Center (IFIMAC) and Departamento de Física Teórica de la Materia Condensada,
Universidad Autónoma de Madrid, E-28049 Madrid, Spain  Akashdeep Kamra
Condensed Matter Physics Center (IFIMAC) and Departamento de Física Teórica de
la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain

###### Abstract

Néel ordered antiferromagnets exhibit two-mode squeezing such that their
ground state is a nonclassical superposition of magnon Fock states. Here we
theoretically demonstrate that antiferromagnets can couple to spin qubits via
direct dispersive interaction stemming from, e.g., interfacial exchange. We
demonstrate that this kind of coupling induces a magnon number dependent level
splitting of the excited state resulting in multiple system excitation
energies. This series of level splittings manifests itself as nontrivial
excitation peaks in qubit spectroscopy thereby revealing the underlying
nonclassical magnon composition of the antiferromagnetic quantum state. By
appropriately choosing the drive or excitation energy, the magnonic state can
be controlled via the qubit, suggesting that Fock states of magnon pairs can
be generated deterministically. This enables achieving states useful for
quantum computing and quantum information science protocols.

##  I Introduction

Antiferromagnets (AFMs) are materials with magnetic order and a vanishing net
macroscopic magnetization [1]. Due to their robustness against magnetic
fields, their fast THz dynamics and phenomena such as exchange bias and spin-
orbit effects, AFMs have been investigated especially for their potential in
spintronics [2, 3, 4, 5, 6, 7]. The classical AFM ground state can be
described by a Néel ordered state comprising two sublattices of oppositely
oriented spins [8]. Coherent excitations on the magnetic order generate a
collective precession of the magnetic moments around their equilibrium
position referred to as spin waves or magnons [9, 4]. Being two-sublattice
magnets, easy-axis collinear AFMs host two kinds of spin waves that are
distinguished by chirality [4]. While the semi-classical spin wave description
is successful in explaining many phenomena [4], it misses important physics
[10, 11, 12] as the true quantum ground state of the ordered AFM is
superposition of states with an equal number of spin-up and spin-down magnons
[10]. Therefore, the AFM ground state is a nonclassical state which may be
exploited for useful quantum information processing purposes [13, 14, 15, 16].

If two observables are non-commuting, their quantum fluctuations obey
Heisenberg’s uncertainty principle [17]. Squeezing is a phenomenon where the
quantum fluctuations of one observable are reduced beyond the standard quantum
limit at the expense of the other’s [18, 19]. In quantum optics, squeezed
states of light have been exploited in feats such as the detection of
gravitational waves due to reduced quantum noise [20]. While squeezed states
of light are nonquilibrium, magnets exhibit squeezing in equilibrium
suggesting to be a useful platform for quantum computing purposes [21, 10, 12,
22, 23]. If a ferromagnet possesses anisotropy, the ground state fluctuation
of the total spin $S_{x}$ and $S_{y}$ components are squeezed, adjusting to
the energy cost dictated by the magnet’s anisotropy. The ferromagnetic ground
state then becomes a one-mode squeezed-magnon vacuum which is composed of even
magnon number states making the ground state a nonclassical superposition [21,
19]. While ferromagnets exhibit squeezing only in the presence of
anisotropies, AFMs display (typically large) squeezing as an intrinsic
property due to strong exchange interaction between the sublattices [24, 10].
As a consequence, the fluctuations of the total spin of the two sublattices
become quantum correlated, such that the AFM exhibits two-mode squeezing of
the two sublattice modes. As a result, the ground state is a two-mode squeezed
vacuum – a superposition of entangled pairs of spin-up and spin-down
sublattice modes [10]. Two-mode squeezed states exhibit Bell nonlocality [25,
26], making them useful for manipulation of quantum information and study of
fundamental principles of quantum mechanics [27, 25, 28]. While there have
been experiments successfully realizing two-mode spin squeezing with coherent
drives [19, 10, 29, 30, 31], generating squeezed magnon Fock states poses a
bigger challenge. This, in part, motivates the present study to generate two-
mode squeezed states in AFMs and the design of experiments to study their
quantum properties.

However, the detection of AFM structure is challenging due to insensitivity to
magnetic fields and makes investigating the quantum nature of AFMs an
experimental challenge [32, 5]. A recently suggested theoretical protocol [33]
finds that the two-mode magnon entanglement exhibited by AFMs can be detected
via modulation of Rabi oscillations in a transmon qubit. While the number
state resolution of nonequilibrium states has been demonstrated experimentally
for photons [34, 35, 36] and magnons [37], a recent theoretical proposal
suggests that the equilibrium magnon composition of a ferromagnetic ground
state [21] can also be resolved via a qubit [38]. This proposal is based on a
unique direct dispersive magnon-qubit coupling stemming from, e.g.,
interfacial exchange interaction [39]. The coupling induces a magnon number
dependent excitation energy of the qubit, enabling to generate magnon number
states via controlled drive of the qubit and revealing the ground state
composition related to squeezing via qubit spectroscopy. Since AFMs
potentially also offer interfacial exchange interaction [40], we are motivated
to examine control and probe of the nonclassical magnon states in an AFM via a
spin qubit.

Here, we investigate a two-sublattice AFM. Its quantum ground state, a two-
mode squeezed magnon vacuum, is a superposition of states with an equal number
of spin-up and spin-down magnons [Fig. 1(a)] [10]. We theoretically
demonstrate that the AFM can be coupled to a spin qubit via direct dispersive
interaction [Fig. 1(b)] [38] and that the ground state of the coupled system
is a superposition of excited states with an equal number of spin-up and spin-
down squeezed-magnons. These levels are non-degenerate such that there are
multiple magnon-dependent qubit excitation energies, allowing to control the
magnonic state and probe the magnon composition of the AFM ground state via
the qubit. While the large frequency of AFM magnons makes the probe via a
qubit difficult, we find that excitation probabilities under the qubit drive
are exponentially enhanced via the squeezing present in the AFM [41, 42, 43,
44, 24]. We expect this squeezing-mediated enhancement to be strong due to the
large squeezing present in AFMs and therefore probing the AFM ground state
easier to achieve. We show that the direct dispersive coupling stems from
interfacial spin exchange interaction [39] and that the structure of the AFM
interface determines the coupling strength, i.e. if the interface is
compensated or uncompensated [Fig. 1(c)]. Our findings suggest that the qubit
probe and control only shows nontrivial effects if the AFM interface is
uncompensated.

We structure the paper as follows: In Sec. II we develop and analyze the
bosonic model for an AFM coupled to a spin qubit. In Sec. III we turn our
attention to realistic systems and derive the bosonic Hamiltonian for hematite
and the direct dispersive coupling from a spin model. Finally, we discuss and
conclude our paper in Secs. IV and V.

Figure 1: Schematic depiction of (a) two-mode squeezed magnon vacuum (b)
magnon-qubit interaction and (c) interfaces of the antiferromagnet. (a) The
two-mode squeeze operator $\hat{S}_{2}(r)$ applied to the Néel ordered state
$\ket{}$ is a superposition of states with equal number of delocalized spin
flips on the down-spin (blue) and up-spin (red) sublattices [10]. (b) The spin
qubit $\hat{\sigma}_{z}$ (green) couples to the antiferromagnetic sublattice
(spin flip) magnons $\hat{a}$ and $\hat{b}$ via direct dispersive interaction
[38] $\chi_{a}\hat{a}^{\dagger}\hat{a}-\chi_{b}\hat{b}^{\dagger}\hat{b}$
(orange). The eigenmodes of the antiferromagnet are two-mode squeezed-magnons
$\hat{\alpha}$ and $\hat{\beta}$ (purple). (c) A compensated interface
consists of equal number of up and down spins corresponding to equal direct
dispersive coupling strength $\chi_{a}=\chi_{b}$ (left). In a completely
uncompensated interface, $\chi_{a}$ is maximized while $\chi_{b}=0$ (right)
[39].

##  II Antiferromagnetic magnons dispersively coupled to a qubit

In this section, we analyse the quantum model of a two-sublattice AFM coupled
to a spin qubit via direct dispersive interaction [38]. We discuss the
uncoupled AFM and introduce useful notation and functions. We then determine
the eigenmodes and eigenenergies of the coupled system via projection onto the
qubit ground ($\ket{g}$) and excited ($\ket{e}$) states. The resulting reduced
Hamiltonians reveal the system ground state and a set of excited states.
Finally, we discuss probe and control of the system state via controlled drive
of the qubit $\ket{g}\rightarrow\ket{e}$.

###  II.1 Uncoupled antiferromagnet

We consider an AFM with Heisenberg exchange and Dzyaloshinskii–Moriya
interaction between neighboring spins $\hat{\bm{S}}_{i}$ and
$\hat{\bm{S}}_{j}$, easy-axis anisotropy in $\hat{\bm{z}}$-direction and an
applied magnetic field along the easy-axis. In Sec. III.1, we present the
corresponding full spin Hamiltonian. Here, starting from Néel ordering, we
define two sublattices: Sublattice A is the sublattice of all spins pointing
along $-\hat{\bm{z}}$ and sublattice $B$ is the sublattice of all spins
pointing along $+\hat{\bm{z}}$ [10, 45, 46] [see Fig. 1(a)]. We map spin
operators $\hat{\bm{S}}_{i}$ and $\hat{\bm{S}}_{j}$ onto boson operators
$\hat{a}_{i}$ and $\hat{b}_{j}$ via Holstein-Primakoff transformations [47]
and switch to Fourier space [see Eqs. (34)–(39) in Sec. III.1 for details].
Considering a small magnet, we only take the uniform $\bm{k}=\bm{0}$ mode into
account [44]. We define bosonic excitations on the two sublattices via
$\hat{S}_{i\in A}^{+}\propto\hat{a}_{\bm{k}=\bm{0}}^{\dagger}$ and
$\hat{S}_{j\in B}^{-}\propto\hat{b}_{\bm{k}=\bm{0}}^{\dagger}$ interpreting
them as delocalized spin flips on sublattice A and B respectively [see Fig.
1(a)]. We drop the index $\bm{k}=\bm{0}$ and refer to them as spin-up
($\hat{a}$) and spin-down ($\hat{b}$) sublattice-magnons. We obtain the
following Hamiltonian [10, 45, 46]

$\displaystyle\hat{\mathcal{H}}_{\mathrm{AFM}}=A\hat{a}^{\dagger}\hat{a}+$
$\displaystyle
B\hat{b}^{\dagger}\hat{b}+C^{*}\hat{a}\hat{b}+C\hat{a}^{\dagger}\hat{b}^{\dagger},$
(1)

where the parameters $A$ and $B$ are evaluated to be real from the spin model.
The parameter $C$ can be complex and quantifies the exchange between the two
sublattices [see Sec. III.1 for the full derivation]. Our goal is to
diagonalize the Hamiltonian $\hat{\mathcal{H}}_{\text{AFM}}$ [Eq. (1)] to
understand the eigenmodes of the AFM and its quantum ground state.

In the following, it will be convenient to define the functions

$\displaystyle\mathcal{E}\left(x,y\right)$
$\displaystyle=\sqrt{\left(x+y\right)^{2}/4-\left|C\right|^{2}}$ (2)
$\displaystyle\Delta\left(x,y\right)$ $\displaystyle=\left(x-y\right)/2,$ (3)
$\displaystyle\mathcal{U}\left(x,y\right)$
$\displaystyle=\frac{1}{\sqrt{2}}\sqrt{\frac{x+y}{2\mathcal{E}\left(x,y\right)}+1},$
(4) $\displaystyle\mathcal{V}\left(x,y\right)$
$\displaystyle=\frac{e^{i\phi}}{\sqrt{2}}\sqrt{\frac{x+y}{2\mathcal{E}\left(x,y\right)}-1},$
(5)

with the phase $e^{i\phi}=C/\left|C\right|$. We diagonalize the Hamiltonian
$\hat{\mathcal{H}}_{\mathrm{AFM}}$ [Eq. (1)] via the Bogoliubov
transformations [48, 46] $\hat{\alpha}=u\hat{a}+v\hat{b}^{\dagger}$ and
$\hat{\beta}=u\hat{b}+v\hat{a}^{\dagger}$, where the Bogoliubov coefficients
are given by $u=\mathcal{U}\left(A,B\right)$ and
$v=\mathcal{V}\left(A,B\right)$ [Eqs. (4) and (5)]. We obtain the diagonalized
Hamiltonian
$\hat{\mathcal{H}}_{\text{AFM}}=\omega_{\alpha}\hat{\alpha}^{\dagger}\hat{\alpha}+\omega_{\beta}\hat{\beta}^{\dagger}\hat{\beta}+E_{0}$
with the eigenfrequencies $\omega_{\alpha/\beta}$ and a zero-point energy
$E_{0}=\left(\omega_{\alpha}+\omega_{\beta}-A-B\right)/2$. The
eigenfrequencies explicitly read

$\displaystyle\omega_{\alpha}$
$\displaystyle=\mathcal{E}\left(A,B\right)+\Delta\left(A,B\right),$ (6)
$\displaystyle\omega_{\beta}$
$\displaystyle=\mathcal{E}\left(A,B\right)-\Delta\left(A,B\right),$ (7)

where we used the functions $\mathcal{E}\left(x,y\right)$ and
$\Delta\left(x,y\right)$ [Eqs. (2) and (3)]. It is convenient to denote the
average of the eigenfrequencies as
$\varepsilon=\left(\omega_{\alpha}+\omega_{\beta}\right)/2$ which corresponds
to the AFM resonance without an applied magnetic field.

The eigenmodes ($\hat{\alpha}$ and $\hat{\beta}$) are related to the
sublattice-magnons ($\hat{a}$ and $\hat{b}$) via the two-mode squeeze
transformations
$\hat{\alpha}=\hat{S}_{2}\left(\xi\right)\hat{a}\hat{S}_{2}^{\dagger}\left(\xi\right)$
and
$\hat{\beta}=\hat{S}_{2}\left(\xi\right)\hat{b}\hat{S}_{2}^{\dagger}\left(\xi\right)$
[10, 49, 19]. The two-mode squeeze operator $\hat{S}_{2}\left(\xi\right)$ is
defined by [50, 19, 49]

$\hat{S}_{2}\left(\xi\right)=\exp\left(\xi^{*}\hat{a}\hat{b}-\xi\hat{a}^{\dagger}\hat{b}^{\dagger}\right),$
(8)

with the two-mode squeeze factor $\xi=re^{i\phi}$. Its absolute value is given
by

$\tanh\left(r\right)=\left|\mathcal{V}\left(A,B\right)\right|/\mathcal{U}\left(A,B\right),$
(9)

and the phase $\phi$ is the same as in Eq. (5). Note that per our definition,
$\mathcal{U}\left(A,B\right)$ [Eq. (4)] is real and positive, such that $r$
[Eq. (9)] is real and positive as well. We refer to the AFM eigenexcitations
as spin-up ($\hat{\alpha}$) and spin-down ($\hat{\beta}$) two-mode squeezed-
magnons (TMSMs).

We denote the eigenstates of $\hat{\mathcal{H}}_{\text{AFM}}$ [Eq. (1)] by
$\ket{n_{\alpha},n_{\beta}}_{\mathrm{sq}}$ which is a TMSM Fock state with
$n_{\alpha}$ spin-up and $n_{\beta}$ spin-down TMSM excitations. The AFM
ground state is the TMSM vacuum denoted by $\ket{0,0}_{\mathrm{sq}}$ [10]. The
TMSM vacuum can be obtained from the sublattice-magnon vacuum
$\ket{0,0}_{\mathrm{sub}}$ (absence of spin flips) via
$\ket{0,0}_{\mathrm{sq}}=\hat{S}_{2}\left(\xi\right)\ket{0,0}_{\mathrm{sub}}$.
Denoting $\ket{n_{a},n_{b}}_{\mathrm{sub}}$ as a Fock state with $n_{a}$ spin-
up and $n_{b}$ spin-down sublattice-magnons, our TMSM vacuum can be expanded
as [10, 49, 19]

$\ket{0,0}_{\mathrm{sq}}=\sum_{n}\frac{\left[-e^{i\phi}\tanh\left(r\right)\right]^{n}}{\cosh\left(r\right)}\ket{n,n}_{\mathrm{sub}}.$
(10)

The AFM ground state is therefore a superposition of states with equal number
of delocalized spin flips on both sublattices [see Fig. 1(a)]. Due to
Heisenberg uncertainty relation [51], the total spin on sublattice A and B
fluctuate. Because of the strong exchange interaction between the sublattices,
these fluctuations are quantum correlated such that the ground state exhibits
squeezed quantum fluctuations [52, 10] [see Fig. 2(a)].

###  II.2 Eigenmodes of the coupled system

Figure 2: Schemetic depiction of (a) quantum fluctuations of the two-mode
squeezed magnon vacuum (b) energy levels of the ground state and the first 4
excited states and (c) qubit spectroscopy protocol. (a) We depict the phase
space of $S_{A_{x}}+S_{B_{x}}$ and $S_{A_{y}}-S_{B_{y}}$, corresponding to to
the sum of $x$ -and difference of $y$-components of the total spin on
sublattice A and B and compare the quantum fluctuation of the two-mode
squeezed magnon vacuum
$\ket{0,0}_{\mathrm{sq}}=\hat{S}_{2}\ket{0,0}_{\mathrm{sub}}$ with an
elliptical shape (red) to the isotropic quantum fluctuations of the
sublattice-magnon vacuum $\ket{0,0}_{\mathrm{sub}}$ (grey) [52]. (b) We depict
the two subspaces resulting from projection onto the qubit states $\ket{g}$
and $\ket{e}$, with squeezing $r_{g}$ (green) and $r_{e}$ (blue). The ground
state $\ket{0,0}_{\mathrm{sq}}^{g}$ can be excited into states with an equal
number of spin-up and spin-down two-mode squeezed magnons
$\ket{n,n}_{\mathrm{sq}}^{e}$ by driving the qubit with frequency
$\omega_{d}=\omega_{n}$ [Eq. (29)]. (c) We depict a lossy qubit that is driven
by a monochromatic microwave drive with frequency $\omega_{d}$. The qubit
evolves under the drive until a steady state is reached. Sweeping over drive
frequency $\omega_{d}$ and measuring steady state qubit excitation for each
$\omega_{d}$ results in the qubit spectrum with excitation peaks at
frequencies $\omega_{0}$, $\omega_{1},\dots$ [Eq. (29)].

We consider an AFM as discussed in Sec. II.1, described by Hamiltonian
$\hat{\mathcal{H}}_{\text{AFM}}$ [Eq. (1)], and couple it to a spin qubit
$\hat{\sigma}_{z}$. As depicted in Fig. 1(b) and (c), the qubit couples to the
spins in the antiferromagnetic interface via spin exchange interaction
$\propto\hat{\bm{S}}\cdot\hat{\bm{\sigma}}$. As detailed in Sec. III.2, the
dominating contribution comes from the $z$-component of the interfacial
exchange if the bare AFM eigenfrequencies $\omega_{\alpha}$ and
$\omega_{\beta}$ [Eqs. (6) and (7)] are far detuned from the bare qubit level
splitting $\omega_{q}$. From this, we obtain a direct dispersive coupling
between the spin qubit and sublattice-magnons $\hat{a}$ and $\hat{b}$ and find
that the coupling strength depends on the size of the magnet and the structure
of the interface. Assuming a large detuning between AFM and spin qubit, we
neglect coherent exchange [49, 44]. As demonstrated in Sec. III.2, we obtain
the following Hamiltonian describing the coupled magnet-qubit system

$\displaystyle\hat{\mathcal{H}}_{0}=A\hat{a}^{\dagger}\hat{a}+$ $\displaystyle
B\hat{b}^{\dagger}\hat{b}+C^{*}\hat{a}\hat{b}+C\hat{a}^{\dagger}\hat{b}^{\dagger}+\frac{\omega_{q}}{2}\hat{\sigma}_{z}$
$\displaystyle+\chi_{a}\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z}-\chi_{b}\hat{b}^{\dagger}\hat{b}\hat{\sigma}_{z},$
(11)

where the parameters $\chi_{a}$ and $\chi_{b}$ quantify the direct dispersive
coupling strength and are assumed to be real and positive for simplicity.

We start to diagonalize the Hamiltonian $\hat{\mathcal{H}}_{0}$ [Eq. (11)] by
projecting onto the qubit ground ($\ket{g}$) and excited ($\ket{e}$) states.
We find that the reduced Hamiltonian
$\hat{\mathcal{H}}_{g}=\bra{g}\hat{\mathcal{H}}_{0}\ket{g}$ is given by

$\displaystyle\hat{\mathcal{H}}_{g}=$
$\displaystyle\left(A-\chi_{a}\right)\hat{a}^{\dagger}\hat{a}+\left(B+\chi_{b}\right)\hat{b}^{\dagger}\hat{b}$
$\displaystyle+C^{*}\hat{a}\hat{b}+C\hat{a}^{\dagger}\hat{b}^{\dagger}-\frac{\omega_{q}}{2}.$
(12)

The excited state projection
$\hat{\mathcal{H}}_{e}=\bra{e}\hat{\mathcal{H}}_{0}\ket{e}$ can be obtained
from the expression for $\hat{\mathcal{H}}_{g}$ [Eq. (12)] by substituting all
minus signs by plus signs $(-\rightarrow+)$.

We diagonalize the reduced Hamiltonians $\hat{\mathcal{H}}_{g}$ and
$\hat{\mathcal{H}}_{e}$ [Eq. (12)] via the following Bogoliubov
transformations [48],

$\displaystyle\hat{\alpha}_{g/e}$
$\displaystyle=u_{g/e}\hat{a}+v_{g/e}\hat{b}^{\dagger},$ (13)
$\displaystyle\hat{\beta}_{g/e}$
$\displaystyle=u_{g/e}\hat{b}+v_{g/e}\hat{a}^{\dagger},$ (14)

with Bogoliubov coefficients

$\displaystyle u_{g}$
$\displaystyle=\mathcal{U}\left(A-\chi_{a},B+\chi_{b}\right),$ (15)
$\displaystyle v_{g}$
$\displaystyle=\mathcal{V}\left(A-\chi_{a},B+\chi_{b}\right),$ (16)
$\displaystyle u_{e}$
$\displaystyle=\mathcal{U}\left(A+\chi_{a},B-\chi_{b}\right),$ (17)
$\displaystyle v_{e}$
$\displaystyle=\mathcal{V}\left(A+\chi_{a},B-\chi_{b}\right).$ (18)

Note that we use the labels $g$ and $e$ to denote the state of the qubit.
Applying the Bogoliubov transformations [Eqs. (13) and (14)], we obtain the
following diagonalized Hamiltonians

$\displaystyle\hat{\mathcal{H}}_{g}=$
$\displaystyle\omega_{\alpha}^{g}\hat{\alpha}_{g}^{\dagger}\hat{\alpha}_{g}+\omega_{\beta}^{g}\hat{\beta}_{g}^{\dagger}\hat{\beta}_{g}$
$\displaystyle+\frac{1}{2}\left(\omega_{\alpha}^{g}+\omega_{\beta}^{g}-A-B+\chi_{a}-\chi_{b}-\omega_{q}\right),$
(19) $\displaystyle\hat{\mathcal{H}}_{e}=$
$\displaystyle\omega_{\alpha}^{e}\hat{\alpha}_{e}^{\dagger}\hat{\alpha}_{e}+\omega_{\beta}^{e}\hat{\beta}_{e}^{\dagger}\hat{\beta}_{e}$
$\displaystyle+\frac{1}{2}\left(\omega_{\alpha}^{e}+\omega_{\beta}^{e}-A-B-\chi_{a}+\chi_{b}+\omega_{q}\right),$
(20)

with eigenfrequencies

$\displaystyle\omega_{\alpha}^{g}$
$\displaystyle=\mathcal{E}\left(A-\chi_{a},B+\chi_{b}\right)+\Delta\left(A-\chi_{a},B+\chi_{b}\right),$
(21) $\displaystyle\omega_{\beta}^{g}$
$\displaystyle=\mathcal{E}\left(A-\chi_{a},B+\chi_{b}\right)-\Delta\left(A-\chi_{a},B+\chi_{b}\right),$
(22) $\displaystyle\omega_{\alpha}^{e}$
$\displaystyle=\mathcal{E}\left(A+\chi_{a},B-\chi_{b}\right)+\Delta\left(A+\chi_{a},B-\chi_{b}\right),$
(23) $\displaystyle\omega_{\beta}^{e}$
$\displaystyle=\mathcal{E}\left(A+\chi_{a},B-\chi_{b}\right)-\Delta\left(A+\chi_{a},B-\chi_{b}\right).$
(24)

For later, it will be convenient to define the average frequencies
$\varepsilon_{g/e}=\left(\omega_{\alpha}^{g/e}+\omega_{\beta}^{g/e}\right)/2$.

Similar to $\hat{\alpha}$ and $\hat{\beta}$ discussed in Sec. II.1, the
eigenmodes $\hat{\alpha}_{g/e}$ and $\hat{\beta}_{g/e}$ [Eqs. (13) and (14)]
are spin-up and spin-down TMSM. We obtain that their squeeze factors are given
by $\xi_{g/e}=r_{g/e}e^{i\phi}$, having the same phase $\phi$ as $\xi$ for the
uncoupled AFM in Sec. II.1. We find that their respective absolute values
$r_{g/e}$ are given by the relations

$\tanh\left(r_{g/e}\right)=\frac{\left|v_{g/e}\right|}{u_{g/e}},$ (25)

where we used Bogoliubov coefficients [Eqs. (15)–(18)]. All of the Bogoliubov
coefficients $u_{g/e}$ and $v_{g/e}$ [Eqs. (15)–(18)] depend explicitly on the
difference between the direct dispersive coupling strengths
$(\chi_{a}-\chi_{b})$. If the direct dispersive coupling strengths are the
same $\chi_{a}=\chi_{b}$ then $u_{g}=u_{e}=u$ and $v_{v}=v_{e}=v$. This
follows from the invariances
$\mathcal{U}\left(x+\lambda,y-\lambda\right)=\mathcal{U}\left(x,y\right)$ and
$\mathcal{V}\left(x+\lambda,y-\lambda\right)=\mathcal{V}\left(x,y\right)$
[Eqs. (4) and (5)]. Inserting the relations
$(A+B)=(\omega_{\alpha}+\omega_{\beta})\cosh(2r)$ and
$\left|C\right|=(\omega_{\alpha}+\omega_{\beta})\left|\sinh(2r)\right|/2$
[obtained from Eqs. (6), (7) and (9)], one can show that $u_{g/e}$ and
$\left|v_{g/e}\right|$ [Eqs. (15)–(18)] only depend on squeezing $r$ and the
ratio
$\left(\chi_{a}-\chi_{b}\right)/\left(\omega_{\alpha}+\omega_{\beta}\right)$.
Therefore, $r_{g}$ and $r_{e}$ [Eq. (25)] are functions of $r$ and
$\left(\chi_{a}-\chi_{b}\right)/\left(\omega_{\alpha}+\omega_{\beta}\right)$
and have different values if the direct dispersive coupling strengths fulfil
$\chi_{a}\neq\chi_{b}$.

We denote $\hat{\alpha}_{g}$ and $\hat{\beta}_{g}$ as ground state TMSM with
squeeze factor $\xi_{g}$ and correspondingly the excited state TMSM
$\hat{\alpha}_{e}$ and $\hat{\beta}_{e}$ with $\xi_{e}$. We conclude that the
eigenstates of $\hat{\mathcal{H}}_{g}$ and $\hat{\mathcal{H}}_{e}$ [Eqs. (19)
and (20)] are given by TMSM Fock states. We denote them by
$\ket{n_{\alpha}^{g},n_{\beta}^{g}}_{\mathrm{sq}}^{g}$ and
$\ket{n_{\alpha}^{e},n_{\beta}^{e}}_{\mathrm{sq}}^{e}$, where
$n_{\alpha(\beta)}^{g}$ is the number of spin-up (spin-down) ground state TMSM
excitations and $n_{\alpha(\beta)}^{e}$ the number of spin-up (spin-down)
excited state TMSM excitations. From this, one can see that the system ground
state is given by the ground state TMSM vacuum $\ket{0,0}_{\mathrm{sq}}^{g}$.
Finally, we remark that the stability of the ground state and excited state
TMSM requires
$2\left|C\right|\leq\min\left[A+B-\left(\chi_{a}-\chi_{b}\right),A+B+\left(\chi_{a}-\chi_{b}\right)\right]$
and $\omega_{\alpha}^{g/e}$ and $\omega_{\beta}^{g/e}$ have to remain
positive. As a consequence the Bogoliubov coefficients $u_{g/e}$ [Eqs. (15)
and (17)] are real and positive, such that $r_{g/e}$ [Eq. (25)] are real and
positive as well.

###  II.3 Quantum state-dependent excitation frequencies

Figure 3: Plot of (a) factors $r_{g}$, $r_{e}$ and $r_{\mathrm{eff}}$ and (b)
contrast vs. direct dispersive coupling in units of magnon frequency
$(\chi_{a}-\chi_{b})/(\omega_{\alpha}+\omega_{\beta})$. (a) Ground state
squeezing $r_{g}$ (solid line), excited state squeezing $r_{e}$ (dashed) and
the difference $r_{\mathrm{eff}}=r_{g}-r_{e}$ (dashed-dotted) [Eq. (25)] for a
bare magnon squeezing of $r=1$ (blue) and $r=1.5$ (red). (b) Contrast $c$ [Eq.
(30)] for several values of $r$.

Our goal is to detect the equilibrium superposition that comprises the
nonclassical AFM ground state [Eq. (10)] via qubit spectroscopy as depicted in
Fig 2(c). In other works such as [34, 53, 36], a qubit is coupled dispersively
to the eigenmode of a quantum system hosting bosons, like a cavity or a
magnet, while a nonequilibrium state is injected into the quantum system. The
dispersive coupling induces boson-number-dependent frequency shifts in the
qubit, matching the boson number $n$ conserving transitions
$\ket{g}\ket{n}\rightarrow\ket{e}\ket{n}$. In the qubit spectrum, the injected
state reveals itself as excitation peaks at the shifted qubit frequency
representing each eigenmode number state contribution to the superposition
comprising the injected state. Here, the idea is that the qubit couples
dispersively to noneigenmodes – the sublattice magnons $\hat{a}$ and $\hat{b}$
– via
$\left(\chi_{a}\hat{a}^{\dagger}\hat{a}-\chi_{b}\hat{b}^{\dagger}\hat{b}\right)\hat{\sigma}_{z}$
[Eq. (11)] in order to resolve the noneigenmode composition of the AFM ground
state. As shown for ferromagnets [38], the dispersive coupling to the
noneigenmode leads to a eigenmode-number-dependent frequency shift in the
system resonance frequencies. Qubit excitation $\ket{g}\rightarrow\ket{e}$ is
not eigenmode-number-conserving [see Fig. 2(b)]. In this procedure, the qubit
is driven by a weak monochromatic drive, reading out the qubit excitation in
the steady state under the drive. This measurement is performed for a range of
frequencies in order to take the qubit spectrum [Fig. 2(c)]. If squeezing is
present in the magnet, the qubit spectrum contains nontrivial peaks [38]. In
the following, we demonstrate theoretically that this protocol can also be
used for AFMs and predict under which conditions nontrivial peaks arise.

In Sec. II.2, we theoretically demonstrate that the ground state of
$\hat{\mathcal{H}}_{0}$ [Eq. (11)] is the ground state TMSM vacuum
$\ket{0,0}_{\mathrm{sq}}^{g}$. Our goal is to find out into which states the
ground state $\ket{0,0}_{\mathrm{sq}}^{g}$ can be excited by driving the qubit
$\ket{g}\rightarrow\ket{e}$ with a monochromatic microwave drive. We model the
contribution of the qubit drive to the Hamiltonian via
$\hat{\mathcal{H}}_{d}=\Omega_{d}\cos(\omega_{d})\left(\hat{\sigma}_{+}+\hat{\sigma}_{-}\right)$
where $\Omega_{d}$ denotes the drive amplitude or Rabi frequency and
$\omega_{d}$ the drive frequency. The Hamiltonian describing the full driven
system now reads
$\hat{\mathcal{H}}_{\mathrm{full}}=\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{d}$.

As discussed in Sec. II.2, there are two types of two-mode squeezing if direct
dispersive coupling strengths have different values $\chi_{a}\neq\chi_{b}$. If
the qubit is in the ground state $\ket{g}(\ket{e})$ the two-mode squeezing is
$\xi_{g}(\xi_{e})$ [see Eq. (25) and the schematic in Fig. 2(b)]. As a
consequence, there exist two subspaces with a set of Fock states each, namely
the ground state (excited sate) TMSM Fock states
$\left\\{\ket{n_{\alpha}^{g(e)},n_{\beta}^{g(e)}}_{\mathrm{sq}}^{g(e)}\right\\}$.
We find that, by making use of the Baker-Campbell-Hausdorff formula [54, 38],
the vacuum states $\ket{0,0}_{\mathrm{sq}}^{g}$ and
$\ket{0,0}_{\mathrm{sq}}^{e}$ are connected by a two-mode squeeze operation
[Eq. (8)] via
$\ket{0,0}_{\mathrm{sq}}^{g}=\hat{S}_{2}\left(\xi_{\text{eff}}\right)\ket{0,0}_{\mathrm{sq}}^{e}$.
Here, the factor $\xi_{\mathrm{eff}}$ denotes the relative squeezing between
the vacuum states $\ket{0,0}_{\mathrm{sq}}^{g}$ and
$\ket{0,0}_{\mathrm{sq}}^{e}$. We find that it reads
$\xi_{\mathrm{eff}}=r_{\mathrm{eff}}e^{i\phi}$ with the absolute value given
by $r_{\text{eff}}=r_{g}-r_{e}$. Choosing the convention $\chi_{a}>\chi_{b}$
results in $r_{g}>r_{e}$, such that the difference between the absolute values
of the squeeze factors is positive $r_{\text{eff}}>0$. For small
$\left|\chi_{a}-\chi_{b}\right|\ll\min[\left|2\varepsilon^{2}/(A+B)\right|,\left|8\varepsilon^{2}/(A+B+2\varepsilon)\right|]$
this parameter becomes approximately

$r_{\text{eff}}=\frac{\sinh(2r)}{2}\frac{\chi_{a}-\chi_{b}}{\omega_{\alpha}+\omega_{\beta}},$
(26)

and is an odd function of $\chi_{a}-\chi_{b}$. In Fig. 3(a) we plot the full
expression of $r_{\text{eff}}=r_{g}-r_{e}$ [Eq. 25] as a function of
$\left(\chi_{a}-\chi_{b}\right)/\left(\omega_{\alpha}+\omega_{\beta}\right)$
comparing it with $r_{g}$ and $r_{e}$.

Here we note, that the relatively simple relation
$\ket{0,0}_{\mathrm{sq}}^{g}=\hat{S}_{2}\left(\xi_{\text{eff}}\right)\ket{0,0}_{\mathrm{sq}}^{e}$
is a result of the fact that the phase $\phi$ is the same for $\xi_{g}$ and
$\xi_{e}$ which simplifies the Baker-Campbell-Hausdorff formula [54, 38]. In
Sec. II.1, we discuss that the vacuum state of the uncoupled AFM can be
expanded in the sublattice-magnon basis [10, 19, 49], as presented in Eq.
(10). We find that a similar expansion of the ground state TMSM vacuum
$\ket{0,0}_{\mathrm{sq}}^{g}$ is possible in terms of excited state TMSM Fock
states $\ket{n_{\alpha}^{e},n_{\beta}^{e}}_{\mathrm{sq}}^{e}$, following from
the relation
$\ket{0,0}_{\mathrm{sq}}^{g}=\hat{S}_{2}\left(\xi_{\text{eff}}\right)\ket{0,0}_{\mathrm{sq}}^{e}$.
We obtain the following expression

$\ket{0,0}_{\mathrm{sq}}^{g}=\sum_{n}c_{n}\ket{n,n}_{\mathrm{sq}}^{e},$ (27)

with the factors

$c_{n}=\frac{\left[-e^{i\phi}\tanh\left(r_{\text{eff}}\right)\right]^{n}}{\cosh\left(r_{\text{eff}}\right)}.$
(28)

If $r_{\text{eff}}=0$ then $c_{n}=0$ for $n\geq 1$, trivializing the
superposition [Eq. (27)] into
$\ket{0,0}_{\mathrm{sq}}^{g}=\ket{0,0}_{\mathrm{sq}}^{e}$. In conclusion, a
nontrivial superposition will only occur if $\chi_{a}\neq\chi_{b}$ is
fulfilled.

As one can see from Eq. (27), there is a nonzero overlap between the system
ground state $\ket{0,0}_{\mathrm{sq}}^{g}$ and the excited state TMSM Fock
states with an equal number of spin-up and spin-down magnons
$\ket{n,n}_{\mathrm{sq}}^{e}$, given by
$c_{n}=\vphantom{}_{\mathrm{sq}}^{~{}e}\\!\innerproduct{n,n}{0,0}_{\mathrm{sq}}^{g}$
[Eq. (28)]. This allows transitions
$\ket{0,0}_{\mathrm{sq}}^{g}\rightarrow\ket{n,n}_{\mathrm{sq}}^{e}$ from the
ground state into an excited state, if the qubit is driven
$\ket{g}\rightarrow\ket{e}$ and the drive frequency $\omega_{d}$ is matching
the energy difference between the states $\ket{0,0}_{\mathrm{sq}}^{g}$ and
$\ket{n,n}_{\mathrm{sq}}^{e}$ [see the schematic in Fig. 2(b)]. Denoting the
energy difference as $\omega_{n}$, we use
$\omega_{n}=\vphantom{}_{\mathrm{sq}}^{~{}e}\\!\bra{n,n}\hat{\mathcal{H}}_{e}\ket{n,n}_{\mathrm{sq}}^{e}-\vphantom{}_{\mathrm{sq}}^{~{}g}\\!\bra{0,0}\hat{\mathcal{H}}_{g}\ket{0,0}_{\mathrm{sq}}^{g}$
with $\hat{\mathcal{H}}_{g/e}$ [Eqs. (19) and (20)] and obtain the expression

$\omega_{n}=2n\varepsilon_{e}+\left(\varepsilon_{e}-\varepsilon_{g}\right)-\left(\chi_{a}-\chi_{b}\right)+\omega_{q}.$
(29)

If the resonance condition $\omega_{d}=\omega_{n}$ is fulfilled, the
transition $\ket{0,0}_{\mathrm{sq}}^{g}\rightarrow\ket{n,n}_{\mathrm{sq}}^{e}$
occurs with probability $p_{n}=\left|c_{n}\right|^{2}$ [Eq. (28)], allowing to
generate TMSM Fock states $\ket{n,n}_{\mathrm{sq}}^{e}$. Since
$\tanh(r_{\mathrm{eff}})<1$, the probabilities $p_{n}$ are decreasing with
increasing $n$. If the qubit is driven continuously at
$\omega_{d}=\omega_{n}$, the qubit excitation reaches a steady state
$\langle\hat{\sigma}_{+}\hat{\sigma}_{-}\rangle_{\mathrm{ss}}$ which is
proportional to $p_{n}$. Sweeping over the qubit frequency $\omega_{d}$
therefore results in peaks at $\omega_{n}$ whose heights are proportional to
the corresponding transition probability $p_{n}$ [see schematic in Fig.
(2)(c)]. We refer to the described procedure as qubit spectroscopy [34, 36,
38] and conclude that the superposition [Eq. (27)] reveals itself as
nontrivial excitation peaks at transition frequencies
$\omega_{0},\omega_{1},\dots$ [Eq. (29)]. To quantify the visibility of the
first nontrivial peak, we define contrast as
$c=\left|c_{1}\right|^{2}/\left|c_{0}\right|^{2}$ (with factors $c_{n}$ [Eq.
(28)]) which reads explicitly

$c=\tanh\left(r_{\text{eff}}\right)^{2}.$ (30)

We plot contrast [Eq. (30)] against
$\left(\chi_{a}-\chi_{b}\right)/\left(\omega_{\alpha}+\omega_{\beta}\right)$
for several values of squeezing $r$ in Fig. 3(b). For small
$\left|\chi_{a}-\chi_{b}\right|\ll\min[\left|2\varepsilon^{2}/(A+B)\right|,\left|8\varepsilon^{2}/(A+B+2\varepsilon)\right|]$,
the contrast [Eq. (30)] can be expanded resulting in

$c\approx\frac{\sinh(2r)}{4}^{2}\frac{\left(\chi_{a}-\chi_{b}\right)^{2}}{\left(\omega_{\alpha}+\omega_{\beta}\right)^{2}}.$
(31)

This expression [Eq. (31)] shows explicitly the dependence of contrast on
$\chi_{a}-\chi_{b}$. This confirms that $\chi_{a}$ and $\chi_{b}$ have to be
finite but different ($\chi_{a}\neq\chi_{b}$) in order to observe nontrivial
peaks in qubit spectroscopy. While the ratio
$(\chi_{a}-\chi_{b})/(\omega_{\alpha}+\omega_{\beta})$ is expected to be small
due to large AFM frequencies, the factor $\sinh(2r)^{2}$ in Eq. (31) provides
a squeezing-mediated amplification expected to be large due to typically large
AFM squeezing [41, 24].

Under the condition
$\left|\chi_{a}-\chi_{b}\right|\ll\left|2\varepsilon^{2}/(A+B)\right|$ the
expansion of system excitation energies $\omega_{n}$ [Eq. (29)] reads

$\displaystyle\omega_{n}\approx n$
$\displaystyle\left[2\varepsilon+\cosh\left(2r\right)\left(\chi_{a}-\chi_{b}\right)\right]$
$\displaystyle+2\sinh^{2}r\left(\chi_{a}-\chi_{b}\right)+\omega_{q}.$ (32)

This expression [Eq. (32)] shows that the separation of two peaks in qubit
spectroscopy [Fig. 2(c)] is given by the average TMSM frequency
$\sim\varepsilon$ of the uncoupled antiferromagnet.

To conclude this section, we demonstrated that the qubit state can control the
magnon squeezing if $\chi_{a}\neq\chi_{b}$, leading to nontrivial overlaps
between the system ground state $\ket{0,0}_{\mathrm{sq}}^{g}$ and excited
states $\ket{n,n}_{\mathrm{sq}}^{e}$. Driving the qubit at frequencies
$\omega_{d}=\omega_{n}$ [Eq. (29)] enables the deterministic generation of
TMSM Fock states $\ket{n,n}_{\mathrm{sq}}^{e}$, where a selection rule imposes
that only states with an equal number of spin-up and spin-down magnons can be
generated from the ground state. Finally, we demonstrated that nontrivial
excitation peaks emerge in qubit spectroscopy if $\chi_{a}\neq\chi_{b}$
enabling to resolve the nonclassical AFM ground state.

##  III Physical realizations

In this section, we start by deriving the AFM Hamiltonian
$\hat{\mathcal{H}}_{\mathrm{AFM}}$ [Eq. (1)] from a spin model, taking
hematite ($\alpha-\mathrm{Fe}_{2}\mathrm{O}_{3}$) as an experimentally
available example. We then continue to derive the dispersive coupling [see Eq.
(11)] from an interfacial spin exchange interaction and discuss the role of
the interface.

###  III.1 Hematite

Hematite is antiferromagnetic insulator. Its magnetic properties stem from the
iron ions while the oxygen ions mediate the exchange between the iron ions
[55]. Below the Morin temperature $T_{M}\approx 250~{}\text{K}$ hematite
crystallizes in the easy-axis antiferromagnetic phase [56] which is also
referred to as rhombohedral phase.

We consider the spin model with Heisenberg exchange interaction,
Dzyaloshinskii-Moriya interaction (DMI), easy-axis single ion anisotropy in
second and fourth order and an external magnetic field along the easy-axis.
For simplicity, we only take into account the dominating terms in the
isotropic Heisenberg exchange interaction. Choosing the $z$-axis as the easy
axis, the spin Hamiltonian reads ($\hbar=1$) [55, 57, 58, 59]

$\displaystyle\mathcal{H}_{\mathrm{hem}}$ $\displaystyle=\sum_{l\neq
m}\left[J_{lm}\hat{\bm{S}}_{l}\cdot\hat{\bm{S}}_{m}+\mathcal{D}_{lm}\hat{\bm{z}}\cdot\left(\hat{\bm{S}}_{l}\times\hat{\bm{S}}_{m}\right)\right]$
$\displaystyle-\sum_{l}\left[K_{2}\left(\hat{S}_{l}^{z}\right)^{2}+K_{4}\left(\hat{S}_{l}^{z}\right)^{4}+\gamma\mu_{0}H_{0}\hat{S}_{l}^{z}\right],$
(33)

where $\hat{\bm{S}}_{l}$ denotes the spin operator at site $l$ in the magnetic
lattice, $S$ the total spin on one lattice site, $J_{lm}$ the Heisenberg
exchange integral and $\mathcal{D}_{lm}$ the DMI coupling strength between
lattice sites $l$ and $m$, $K_{2}$ and $K_{4}$ denote the second and fourth
order single-ion anisotropy constants, $H_{0}$ is a constant magnetic field in
the $z$-direction and $\gamma<0$ denotes the gyromagnetic ratio.

We transform the spin Hamiltonian $\hat{\mathcal{H}}_{\mathrm{hem}}$ [Eq.
(33)] into a bosonic form by using linearized Holstein-Primakoff
transformations [47, 58]. We also switch into momentum space via Fourier
transformation. Considering lattice sites $i\in$A and $j\in$B, the spin
operators read

$\displaystyle\hat{S}_{i}^{+}$
$\displaystyle=\sqrt{\frac{2S}{N}}\sum_{\bm{k}}e^{-i\bm{k}\cdot\bm{r}_{i}}\hat{a}_{\bm{k}}^{\dagger},$
(34) $\displaystyle\hat{S}_{i}^{-}$
$\displaystyle=\sqrt{\frac{2S}{N}}\sum_{\bm{k}}e^{i\bm{k}\cdot\bm{r}_{i}}\hat{a}_{\bm{k}},$
(35) $\displaystyle\hat{S}_{i}^{z}$
$\displaystyle=-S+\frac{1}{N}\sum_{\bm{k},\bm{k}^{\prime}}e^{-i\left(\bm{k}-\bm{k}^{\prime}\right)\cdot\bm{r}_{i}}\hat{a}_{\bm{k}}^{\dagger}\hat{a}_{\bm{k}^{\prime}},$
(36) $\displaystyle\hat{S}_{j}^{+}$
$\displaystyle=\sqrt{\frac{2S}{N}}\sum_{\bm{k}}e^{i\bm{k}\cdot\bm{r}_{j}}\hat{b}_{\bm{k}},$
(37) $\displaystyle\hat{S}_{j}^{-}$
$\displaystyle=\sqrt{\frac{2S}{N}}\sum_{\bm{k}}e^{-i\bm{k}\cdot\bm{r}_{j}}\hat{b}_{\bm{k}}^{\dagger},$
(38) $\displaystyle\hat{S}_{j}^{z}$
$\displaystyle=S-\frac{1}{N}\sum_{\bm{k},\bm{k}^{\prime}}e^{-i\left(\bm{k}-\bm{k}^{\prime}\right)\cdot\bm{r}_{j}}\hat{b}_{\bm{k}}^{\dagger}\hat{b}_{\bm{k}^{\prime}},$
(39)

where the spin ladder operators are defined by
$\hat{S}_{i}^{\pm}=\hat{S}_{i}^{x}\pm i\hat{S}_{i}^{y}$ and
$\hat{S}_{j}^{\pm}=\hat{S}_{j}^{x}\pm i\hat{S}_{j}^{y}$ and $N$ denotes the
number of sites in one sublattice. Here we assume that both sublattices have
an equal number of sites. Note that $\hat{a}_{\bm{k}}$ and $\hat{b}_{\bm{k}}$
denote the sublattice-magnon modes (as discussed in Sec. II.1) with momentum
$\bm{k}$ and $\bm{r}_{i}$ denotes the position of lattice site $i$. Inserting
the transformations Eqs. (34)–(39) into Eq. (33) results in

$\displaystyle\hat{\mathcal{H}}=\sum_{\bm{k}}A\hat{a}^{\dagger}_{\bm{k}}\hat{a}_{\bm{k}}+B\hat{b}^{\dagger}_{\bm{k}}\hat{b}_{\bm{k}}+C_{\bm{k}}^{*}\hat{a}_{\bm{k}}\hat{b}_{-\bm{k}}+C_{\bm{k}}\hat{a}^{\dagger}_{\bm{k}}\hat{b}^{\dagger}_{-\bm{k}},$
(40)

with the parameters

$\displaystyle A$
$\displaystyle=SzJ+Sz^{\prime}J^{\prime}+2SK_{2}+4S^{3}K_{4}-\gamma\mu_{0}H_{0},$
(41) $\displaystyle B$
$\displaystyle=SzJ+Sz^{\prime}J^{\prime}+2SK_{2}+4S^{3}K_{4}+\gamma\mu_{0}H_{0},$
(42) $\displaystyle C$
$\displaystyle=\gamma_{\bm{k}}Sz\left(J+i\mathcal{D}\right)+\gamma_{\bm{k}}^{\prime}Sz^{\prime}\left(J^{\prime}+i\mathcal{D}\right),$
(43)

and with dominant exchange couplings $J$ and $J^{\prime}$ from third and
fourth nearest neighbors [57], the DMI coupling strength $\mathcal{D}$,
$z$($z^{\prime}$) being the number of coupled third(fourth) neighbors and
structure factors $\gamma_{\bm{k}}$($\gamma_{\bm{k}}^{\prime}$) from
third(fourth) neighbors.

Here, we consider a small magnet. Due to the confinement and its resulting
boundary conditions, the magnon modes exhibit standing waves with discrete
spectrum and modes with a finite $\bm{k}$ are well separated from the
$\bm{k}=\bm{0}$ by a few GHz [44]. We will therefore keep only $\bm{k}=\bm{0}$
in the sum over $\bm{k}$ in the transformed Hamiltonian $\hat{\mathcal{H}}$
[Eq. (40)]. As a consequence, the structure factors simplify to
$\gamma_{\bm{0}}=1$ and $\gamma_{\bm{0}}^{\prime}=1$. We define the
sublattice-magnon modes as $\hat{a}\equiv\hat{a}_{\bm{k}=\bm{0}}$ and
$\hat{b}\equiv\hat{b}_{\bm{k}=\bm{0}}$. This way, we end up with a Hamiltonian
in the form of $\hat{\mathcal{H}}_{\text{AFM}}$ [Eq. (1)]. Using theoretical
estimates on the exchange coupling and single ion anisotropies from Mazurenko
and Anisimov [57] ($z=3$, $z^{\prime}=6$, $J=25.2$ meV, $J^{\prime}=17.5$ meV,
$S=2$, $K_{2}=112.8\,\mu$eV, $K_{4}=1.1\,\mu$eV) and the DMI strength
$\mathcal{D}\left(z+z^{\prime}\right)=\gamma\times 2.2\,$T from [60], we
estimate that the parameters [Eqs. (41)– (43)] approximately read
$(A+B)/4\pi=87.46\,$THz and $\left|C\right|/2\pi=87.34$ THz. These values
result in an average resonance frequency of
$(\omega_{\alpha}+\omega_{\beta})/4\pi=4.53$ THz and a squeezing of $r=1.826$
for the uncoupled AFM.

In conclusion, we were able to demonstrate that hematite below the Morin
temperature realizes the Hamiltonian $\hat{\mathcal{H}}_{\mathrm{AFM}}$ [Eq.
(1)].

###  III.2 Role of the interface

In this section, we derive the dispersive interaction between the AFM and spin
qubit from a spin model. We consider an exchange coupling between the
interfacial spins of the AFM and the spin qubit. The Hamiltonian of the
interfacial exchange reads [44]

$\hat{\mathcal{H}}_{\text{int}}=J_{\text{int},A}\sum_{l\in
A}\hat{\bm{S}}_{l}\cdot\hat{\bm{s}}_{l}+J_{\text{int},B}\sum_{m\in
B}\hat{\bm{S}}_{m}\cdot\hat{\bm{s}}_{m},$ (44)

where $\hat{\bm{S}}_{l(m)}$ denotes the spin operator at interfacial lattice
site $l(m)$, $\hat{\bm{s}}_{l(m)}$ the spin operator of the qubit at
interfacial lattice site $l(m)$ and $J_{\mathrm{int},A(B)}$ are the exchange
coupling strength [61, 62, 63, 64] to the qubit with interfacial lattice site
$l\in\mathrm{A}(m\in\mathrm{B})$.

The product between spin operators can be written as
$2\hat{\bm{S}}_{l}\cdot\hat{\bm{s}}_{l}=\hat{S}^{+}_{l}\hat{s}^{-}_{l}+\hat{S}^{-}_{l}\hat{s}^{+}_{l}+2\hat{S}^{z}_{l}\hat{s}^{z}_{l}$.
Skogvoll  _et al._ [44] demonstrate that terms
$\propto\hat{S}^{+}_{l}\hat{s}^{-}_{l}$ and
$\propto\hat{S}^{-}_{l}\hat{s}^{+}_{l}$ result in a coherent interaction. Here
we consider a large detuning between the AFM and qubit such that coherent
interaction is suppressed [49, 38] and will be neglected in the further
analysis.

Using the transformations [Eqs. (34)–(39)], the terms
$\propto\hat{\bm{S}}^{z}\cdot\hat{\bm{s}}^{z}$ provide two terms up to second
order in the sublattice magnon operators $\hat{a}$ and $\hat{b}$. As discussed
in [44] and [38], one of these contributions renormalizes the qubit frequency
$\omega_{q}$, whereas the other is proportional to the sublattice-magnon
number operators $\hat{a}^{\dagger}\hat{a}$ and $\hat{b}^{\dagger}\hat{b}$.
Assuming a constant qubit wave function over the interface, we obtain that the
magnon number dependent part leads to the direct dispersive interaction [44,
38]
$\hat{\mathcal{H}}_{\text{dis}}=\left(\chi_{a}\hat{a}^{\dagger}\hat{a}-\chi_{b}\hat{b}^{\dagger}\hat{b}\right)\hat{\sigma}_{z},$
with the coupling strengths

$\displaystyle\chi_{a}$
$\displaystyle=\frac{J_{\text{int},A}\left|\psi\right|^{2}}{2N}N_{\text{int},A},$
(45) $\displaystyle\chi_{b}$
$\displaystyle=\frac{J_{\text{int},B}\left|\psi\right|^{2}}{2N}N_{\text{int},B}.$
(46)

Here $\left|\psi\right|^{2}=1/(2N)$ denotes the average qubit wave function
over the interface [44] and $N_{\mathrm{int},A(B)}$ the number of interfacial
lattice sites belonging to sublattice A(B).

From the expressions for $\chi_{a}$ and $\chi_{b}$ [Eqs. (45) and (46)], we
deduce that the direct dispersive coupling strength depends on the size of the
AFM ($\propto 1/N$) and the structure of the interface. Assuming
$J_{A}=J_{B}$, for a compensated interface
($N_{\text{int},A}=N_{\text{int},B}$) the direct dispersive coupling strengths
are the same $\chi_{a}=\chi_{b}$ whereas for uncompensated interfaces
($N_{\text{int},A}\neq N_{\text{int},B}$) the coupling strengths are different
$\chi_{a}\neq\chi_{b}$. For perfectly uncompensated interfaces, one coupling
strength is maximized while the other is zero [see Fig. 1(c) for
illustration].

Semiconductor qubits implement the required exchange interaction [Eq. (44)]
[65]. We therefore want to estimate $\chi_{a}$ and $\chi_{b}$ for an AFM with
100 interfacial sites and 6 monolayers coupled to such a qubit. We use an
interfacial coupling strength of $J_{A}=J_{B}\approx 10\,\text{meV}$ which has
been extracted from spin-pumping experiments [66, 67, 68]. For a compensated
interface, we calculate a direct dispersive coupling strengths of
$\chi_{a}=\chi_{b}\approx 2.11\,\text{GHz}$. For a perfectly uncompensated
interface with $N_{\text{int},B}=0$ and assuming that the layers are perfectly
composed of either sublattice A or B, we find $\chi_{a}=4.22\,\text{GHz}$ and
$\chi_{b}=0$.

In conclusion, the size of the AFM and structure of the interface are crucial
factors in determining the direct dispersive coupling strengths $\chi_{a}$ and
$\chi_{b}$ [Eqs. (45) and (46)]. We also conclude that the protocol to resolve
the nonclassical ground state composition of an AFM requires that the
interface is uncompensated ($\chi_{a}\neq\chi_{b}$). Finally, our estimation
resulted in a direct dispersive coupling strengths in the GHz regime.

##  IV Discussion

Having derived the composition of the ground state
$\ket{0,0}_{\mathrm{sq}}^{g}$ in terms of excited state TMSM Fock states
$\ket{n,n}_{\mathrm{sq}}^{e}$ [Eq. (27)] and the excitation energies
$\omega_{n}$ [Eq. (29)], we predict that qubit spectroscopy can reveal
nontrivial excitation peaks. Now, we want to discuss the resolvability of
spectroscopy peaks and requirements for realistic systems.

As discussed in [38], the direct dispersive coupling can be realized with spin
qubits, such as semiconducting quantum dots [65] or NV centers. The latter
interact with other spins via dipole-dipole interaction [69, 70]. However, it
has been shown in [38] that the direct dispersive coupling strength stemming
from dipole-dipole interaction is expected to be vanishingly small. As a
consequence, NV centers are not promising candidates for resolving the
nonclassical ground state composition of an AFM. We also want to mention that
coherent interaction in the dispersive limit [49], as provided by
superconducting qubits [34, 35, 71], is not able to resolve nonclassical
equilibrium states [38] and therefore not suitable for resolving the AFM
ground state composition.

In Sec. III.2, we estimated that the direct dispersive coupling strengths
$\chi_{a}$ and $\chi_{b}$ between an AFM and a semiconductor qubit is in the
GHz regime. This competes with the resonance frequencies of AFMs up to THz
[3]. The difference of around 3 orders of magnitude between the coupling
strength and the AFM frequency lowers the contrast in qubit spectroscopy. The
expectedly low $(\chi_{a}-\chi_{b})/(\omega_{\alpha}+\omega_{\beta})$ can be
compensated by large two-mode squeezing $r$ via $\sinh(2r)^{2}$ [see Eq.
(31)]. For instance the estimated squeezing factor of $r=1.826$ in hematite
results in $\sinh(2r)^{2}\approx 371.00$. However, this is a given property
and depends on the material.

Even for magnets with low squeezing $r$ and low
$(\chi_{a}-\chi_{b})/(\omega_{\alpha}+\omega_{\beta})$, there is the
possibility of amplifying the peak height of the nontrivial peaks at
$\omega_{1}$, $\omega_{2},\dots$ by increasing the drive amplitude
$\Omega_{d}$ [38]. The excitation process into states
$\ket{n,n}_{\mathrm{sq}}^{e}$ for $n>0$ is suppressed by a factor
$\left|c_{n}\right|^{2}$ allowing for larger $\Omega_{d}$ while still being in
the linear response regime. Since the peaks at $\omega_{n}$ and $\omega_{n+1}$
are well separated by the AFM resonance $\sim\varepsilon$ ($\sim$THz), we
conclude that indeed large $\Omega_{d}$ can be chosen without affecting the
resolvability with peak overlaps.

Lastly we want to discuss the requirement of the uncompensated interface [see
Fig.1(a)], as concluded from the ground state composition [Eqs. (27) and
(28)]. The coupling strength is maximized by a perfectly uncompensated
interface. AFMs, can be grown with uncompensated interfaces [72]. Some van der
Waals materials, such as bilayer $\mathrm{CrI}_{3}$ [73, 74], consist of
ferromagnetic layers that are antiferromagnetically coupled and therefore
exhibit a maximized $\chi_{a}$ while $\chi_{b}=0$. Also hematite, discussed in
Sec. III.1 can be grown in layers having uncompensated spins at the surface
[75]. We conclude that the condition $\chi_{a}\neq\chi_{b}$ can be fulfilled
by real systems. Since the nontrivial excitation peaks depend on
$\chi_{a}-\chi_{b}$, qubit spectroscopy provides a measure of how compensated
the interface is. This is complementary to already existing techniques of
probing spin surface structures, such as spin-polarized scanning tunneling
microscopy [76, 77].

##  V Conclusions

We have shown that the recently suggested protocol for resolving the
nonclassical ground state composition of a single-mode squeezed vacuum in a
ferromagnet [38] can be adequately adapted for a similar probing of AFMs
exhibiting two-mode squeezing [10]. We have shown that the qubit excitation
peaks, that serve as the experimental probe of the quantum superposition, are
well separated by the typically large AFM resonance frequency
$\sim\varepsilon$. Furthermore, hematite below the Morin temperature [56, 57,
55] realizes the bosonic two-mode squeezing Hamiltonian [Eq. (1)] [19, 10] for
the $\bm{k}=\bm{0}$ mode. We estimated that the direct dispersive coupling
strength is achievable to be in the range of $\sim\,$GHz suggesting small
contrast in qubit spectroscopy. However large squeezing of the AFM ground
state and optimizing the qubit drive can amplify the qubit excitation peaks
making them experimentally detectable. We therefore conclude that this
protocol has the potential to resolve the ground state composition of AFMs.

The direct dispersive coupling to a spin qubit also enables the deterministic
generation of non-equilibrium Fock states $\ket{n,n}_{\mathrm{sq}}^{e}$ by
driving the qubit at $\omega_{d}=\omega_{n}$ [Eq. (29)]. The deterministic
generation of magnon pairs has the potential to furthermore enable use of the
intrinsic entanglement. This could, for instance, be utilized to realize
magnonic Hong-Ou-Mandel effect [78, 79].

###### Acknowledgements.

We acknowledge financial support from the Spanish Ministry for Science and
Innovation – AEI Grant CEX2018-000805-M (through the “Maria de Maeztu”
Programme for Units of Excellence in R&D) and grant RYC2021-031063-I funded by
MCIN/AEI/10.13039/501100011033 and “European Union Next Generation EU/PRTR”.
A. E. R. acknowledges that the project that gave rise to these results
received the support of a fellowship from “la Caixa” Foundation (ID 100010434)
with the fellowship code LCF/BQ/DI22/11940029.

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