# Can quantum Rabi model with $A^{2}$-term avoid no-go theorem for spontaneous
SUSY breaking?

Masao Hirokawa<EMAIL_ADDRESS>https://nvcspm.net/qstl/ Graduate
School of Information Science and Electrical Engineering, Kyushu University.

###### Abstract

The hierarchy problem asks why the mass of the Higgs particle is so much
lighter than the Planck-scale mass. Considering the interaction of the Higgs
particle and an elementary particle in the Planck-scale, to cope with that big
difference, the conventional calculation needs the help of an arbitrary,
excessive fine-tuning, that is, the huge cancellation between the bare mass
term and the quantum correction, without obeying a physical principle such as
symmetry. Thus, it is often said to be unnatural. On the other hand, the
theory of supersymmetry (SUSY) is a strong candidate naturally to solve the
hierarchy problem. However, any sign of SUSY even for the quantum mechanics
(QM) version [1, 2] had not been firmly, directly observed in the physical
reality until Cai et al. reported the observation of the $N=2$ SUSY and its
spontaneous breaking in a trapped ion quantum simulator [3] for the prototype
model for SUSY QM [4]. In this discussion, I derive a no-go theorem for the
spontaneous SUSY breaking in the strong coupling limit for the quantum Rabi
model [5] with the $A^{2}$-term, and at the same time, I show another limit
proposed in the scheme by Cai et al. [3] can avoid the no-go theorem and take
that model from the $N=2$ SUSY to its spontaneous breaking. I propose a
theoretical method to observe how the effect of $A^{2}$-term appears in the
spontaneous SUSY breaking.

††preprint: Commentary Discussion

In the discussion below, the annihilation and creation operators of a $1$-mode
boson are respectively denoted by $a$ and $a^{\dagger}$. The annihilation
operator $\sigma_{-}$ and the creation operator $\sigma_{+}$ of a $2$-level
atom (i.e., spin) are given by $\sigma_{\pm}:=\frac{1}{2}(\sigma_{x}\pm
i\sigma_{y})$. The standard notation $\sigma_{x}$, $\sigma_{y}$, and
$\sigma_{z}$ are used for the Pauli matrices. The Hamiltonian is defined by

$H(\omega_{\mathrm{a}},\omega_{\mathrm{c}},g,C)=\frac{\hbar\omega_{\mathrm{a}}}{2}\sigma_{z}+\hbar\omega_{\mathrm{c}}\left(a^{\dagger}a+\frac{1}{2}\right)+\hbar
g\sigma_{x}\left(a+a^{\dagger}\right)+\hbar
Cg^{2}\left(a+a^{\dagger}\right)^{2},$ (1)

where the first and second terms of RHS of equation (1) are respectively the
free energies of the $2$-level atom and the $1$-mode boson. The third term is
the linear interaction between the atom and boson with the parameter $g$
representing the coupling strength. The last term is the quadratic interaction
$\hbar C\\{g\sigma_{x}(a+a^{\dagger})\\}^{2}$ with the parameter $C$ which
controls the dimension and volume of the quadratic interaction energy. This is
often called $A^{2}$-term.

It is already proved that, tuning the parameters, $\omega_{\mathrm{a}}$,
$\omega_{\mathrm{c}}$, as $\omega_{\mathrm{a}}=\omega_{\mathrm{c}}=\omega$ for
a positive constant $\omega$, the Hamiltonian $H(\omega,\omega,0,0)$ without
the $A^{2}$-term has the $N=2$ SUSY [4]. As the coupling strength $g$ gets
stronger enough, the $A^{2}$-term may appear. Similarly to the case of the
superradiant phase transition [6, 7], a no-go theorem caused by the
$A^{2}$-term [8] should be minded, and its avoidance should be argued [9] also
for the prototype model for SUSY QM.

For every non-negative $C$, there exists a unitary operator $U_{A^{2}}$ such
that

$\displaystyle
U_{A^{2}}^{*}H(\omega_{\mathrm{a}},\omega_{\mathrm{c}},g,C)U_{A^{2}}$
$\displaystyle=$ $\displaystyle
H(\omega_{\mathrm{a}},\omega(g),\widetilde{g},0)$ $\displaystyle=$
$\displaystyle\frac{\hbar\omega_{\mathrm{a}}}{2}\sigma_{z}+\hbar\omega(g)\left(a^{\dagger}a+\frac{1}{2}\right)+\hbar\widetilde{g}\sigma_{x}\left(a+a^{\dagger}\right),$

where $\omega(g)=\sqrt{\omega_{\mathrm{c}}^{2}+4C\omega_{\mathrm{c}}g^{2}\,}$
and $\widetilde{g}=g\sqrt{\omega_{\mathrm{c}}/\omega(g)}$. This unitary
operator $U_{A^{2}}$ is obtained in the meson pair theory of nuclear physics
[10, 11]. Equation (Can quantum Rabi model with $A^{2}$-term avoid no-go
theorem for spontaneous SUSY breaking? ) is often called the Hopfield-
Bogoliubov transformation [9] of
$H(\omega_{\mathrm{a}},\omega_{\mathrm{c}},g,C)$. The effect of the
$A^{2}$-term is stuffed into $\omega(g)$ and $\widetilde{g}$. We note that,
for $C=0$, the parameters satisfy $\omega(g)=\omega_{\mathrm{c}}$,
$\widetilde{g}=g$, and then, the unitary operator is $U_{A^{2}}=I$ (i.e., the
identity operator).

For the displacement operator
$D(g/\omega_{\mathrm{c}}):=\exp\left[g(a^{\dagger}-a)/\omega_{\mathrm{c}}\right]$,
a unitary operator is defined by
$U(g/\omega_{\mathrm{c}}):=\frac{1}{\sqrt{2}}\left\\{\left(\sigma_{-}-1\right)\sigma_{+}D(g/\omega_{\mathrm{c}})+\left(\sigma_{+}+1\right)\sigma_{-}D(-g/\omega_{\mathrm{c}})\right\\}$.
Then, it makes the equation,

$\displaystyle
U(g/\omega_{\mathrm{c}})^{*}\left\\{H(\omega_{\mathrm{a}},\omega_{\mathrm{c}},g,0)+\hbar\frac{g^{2}}{\omega_{\mathrm{c}}}\right\\}U(g/\omega_{\mathrm{c}})$
(3) $\displaystyle=$
$\displaystyle\hbar\omega_{\mathrm{c}}\left(a^{\dagger}a+\frac{1}{2}\right)-\frac{\hbar\omega_{\mathrm{a}}}{2}\left\\{\sigma_{+}D(g/\omega_{\mathrm{c}})^{2}+\sigma_{-}D(-g/\omega_{\mathrm{c}})^{2}\right\\}.$

Since the arguments on the limit of Hamiltonians used below are already
established as a mathematical method [4, 10, 11]. Thus, for simplicity,
mathematically naive arguments are made in this discussion to explain the no-
go theorem and its avoidance.

Now we consider the strong coupling limit for the quantum Rabi model with the
$A^{2}$-term. For instance, the strong coupling limit is experimentally
realized for the quantum Rabi model in circuit QED [12] as deep-strong
coupling regime [13]. The parameters, $\omega_{\mathrm{a}}$,
$\omega_{\mathrm{c}}$, $g$, are set as
$\omega_{\mathrm{a}}=\omega_{\mathrm{c}}=\omega$ and $g=\mathrm{g}$ for a non-
negative parameter $\mathrm{g}$. The Hamiltonian
$H(\omega,\omega,\mathrm{g},0)$ is for the quantum Rabi model, and denoted by
$H_{\mbox{\tiny Rabi}}(\mathrm{g})$ for simplicity. In the renormalization for
$A^{2}$-term, the quantities, defined by
$\omega(\mathrm{g}):=\sqrt{\omega^{2}+4C\omega\mathrm{g}^{2}\,}$ and
$\widetilde{\mathrm{g}}:=\mathrm{g}\sqrt{\omega/\omega(\mathrm{g})}$, are
used.

In case $C=0$, according to the mathematical results [4, 10], the
approximation,

$H_{\mbox{\tiny Rabi}}(\mathrm{g})+\hbar\frac{\mathrm{g}^{2}}{\omega}\approx
U(\mathrm{g}/\omega)\left[\hbar\omega\left(a^{\dagger}a+\frac{1}{2}\right)\right]U(\mathrm{g}/\omega)^{*},$
(4)

is obtained as $\mathrm{g}\to\infty$. Since the mechanism for spontaneous SUSY
breaking for the quantum Rabi model [4] works by the appearance of only the
boson free energy in RHS of equation (4), the $N=2$ SUSY of $H_{\mbox{\tiny
Rabi}}(\mathrm{g})$ is spontaneously broken in the strong coupling limit
$\mathrm{g}\to\infty$ (Fig.1a). Naively, how to obtain equation (4) is
explained in the following. Due to the violent vibrations in the displacement
operator $D(\pm\mathrm{g}/\omega)$ as $\mathrm{g}/\omega$ grows larger, the
displacement operator decays. Thus, the second term of RHS of equation (3)
disappears in the limit $\mathrm{g}\to\infty$.

In the case $C>0$, on the other hand, the mathematical results [11] says that

$\displaystyle H_{\mbox{\tiny Rabi}}(\mathrm{g})+\hbar
C\mathrm{g}^{2}\left(a+a^{\dagger}\right)^{2}+\hbar\frac{\widetilde{\mathrm{g}}^{2}}{\omega(\mathrm{g})}$
(5) $\displaystyle\approx$ $\displaystyle
U_{A^{2}}U(\widetilde{\mathrm{g}}/\omega(\mathrm{g}))\left[\hbar\omega(\mathrm{g})\left(a^{\dagger}a+\frac{1}{2}\right)-\,\frac{\hbar\omega}{2}\sigma_{x}\right]U(\widetilde{\mathrm{g}}/\omega(\mathrm{g}))^{*}U_{A^{2}}^{*}$

as $\mathrm{g}\to\infty$. The atomic term $\hbar\omega\sigma_{x}/2$ appears in
addition to the boson free energy in RHS of equation (5), and then, this
appearance interferes with the conversion to the spontaneous SUSY breaking,
and moreover, the divergnce of $\omega(\mathrm{g})$, together with the atomic
term, rudely crushes that SUSY (Fig.1b). Thus, the above quantum Rabi model
with the $A^{2}$-term cannot go to its spontaneous breaking as $\mathrm{g}$
chages from $\mathrm{g}=0$ to $\mathrm{g}\approx\infty$. This is the no-go
theorem for the spontaneous SUSY breaking in the strong coupling limit caused
by the $A^{2}$-term. The reason why $\hbar\omega\sigma_{x}/2$ appears in RHS
of equation (5) is naively explained as follows: Since
$\lim_{\mathrm{g}\to\infty}\widetilde{\mathrm{g}}/\omega(\mathrm{g})=0$ for
$C>0$ though
$\lim_{\mathrm{g}\to\infty}\widetilde{\mathrm{g}}/\omega(\mathrm{g})=\lim_{\mathrm{g}\to\infty}\mathrm{g}/\omega=\infty$
for $C=0$, the displacement operator
$D(\pm\widetilde{\mathrm{g}}/\omega(\mathrm{g}))$ in equation (3) remains as
the identity operator $I$ in the limit $\mathrm{g}\to\infty$.

The approximations given by equations (4) and (5) are mathematically
established in the norm resolvent sense, and the limit is valid over the
energy spectrum [14]. Thus, the limit energy spectrum is obtained by those
approximations. Whether $N=2$ SUSY of $H(\omega,\omega,0,0)$ is taken to its
spontaneous breaking is checked by the energy degeneracy between the bosonic
and the fermionic states. The energy spectrum by the numerical analysis with
QuTiP [15] is obtained, for instance, as in Fig.1. (a) (b)

Figure 1: Energy Spectrum of $H_{\mbox{\tiny Rabi}}(\mathrm{g})+\hbar
C\mathrm{g}^{2}\left(a+a^{\dagger}\right)^{2}$ with $\omega=6.2832$: a shows
the energy spectrum for $C=0$. The right graph is for $C=0.3770$. The quantum
Rabi model (without $A^{2}$-term) has the transition from the $N=2$ SUSY to
its spontaneous breaking. On the other hand, b shows the loss of the
spontaneous breaking. Here, it should be noted
$\lim_{\mathrm{g}\to\infty}\hbar\omega(\mathrm{g})=\infty$ and
$\lim_{\mathrm{g}\to\infty}\hbar\widetilde{\mathrm{g}}^{2}/\omega(\mathrm{g})=\hbar/(4C)$.

As stated above, the Rabi model with the $A^{2}$-term is faced with the no-go
theorem in the strong coupling limit, which is caused by the effect coming
from the $A^{2}$-term. Indeed the no-go theorem appears in the strong coupling
limit, but there is another limit in the scheme which is proposed by Cai et
al. [3]. From now on, it is proved that the limit has the advantage over the
strong coupling limit in order that the Rabi model with the $A^{2}$-term avoid
the no-go theorem and has the spontaneous SUSY breaking.

Let $\omega[r]$ be a continuous function of variable $r$, $0\leq r\leq 1$,
satisfying $\omega[0]=\omega$ and $\omega[1]=0$. The parameters
$\omega_{\mathrm{a}}$, $\omega_{\mathrm{a}}$, $g$ are given by
$\omega_{\mathrm{a}}=\omega[r]$, $\omega_{\mathrm{c}}=\omega$,
$g=r\mathrm{g}_{0}$ for a positive constant $\mathrm{g}_{0}$. The Hamiltonian
$H(\omega[r],\omega,r\mathrm{g}_{0},0)$ for the quantum Rabi model is denoted
by $H_{\mbox{\tiny Rabi}}[r]$ for simplicity. The quantities are given by
$\widetilde{\omega}[r]:=\sqrt{\omega^{2}+4C\omega r^{2}\mathrm{g}_{0}^{2}\,}$
and
$\widetilde{\mathrm{g}}[r]:=r\mathrm{g}_{0}\sqrt{\omega/\widetilde{\omega}[r]}$.
Then, the same argument as in [4, 10, 11] gives

$\displaystyle H_{\mbox{\tiny Rabi}}[r]+\hbar
Cr^{2}\mathrm{g}_{0}^{2}\left(a+a^{\dagger}\right)^{2}+\hbar\frac{\widetilde{\mathrm{g}}[r]^{2}}{\widetilde{\omega}[r]}$
(6) $\displaystyle\longrightarrow$ $\displaystyle
U_{A^{2}}U(\widetilde{\mathrm{g}}[1]/\widetilde{\omega}[1])\left[\hbar\widetilde{\omega}[1]\left(a^{\dagger}a+\frac{1}{2}\right)\right]U(\widetilde{\mathrm{g}}[1]/\widetilde{\omega}[1])^{*}U_{A^{2}}^{*}$

in the norm resolvent sense [14] as $r\to 1$. The naive reason why this limit
is obtained is because the limit $\omega[r]\to\omega[1]=0$ eliminates the
second term of RHS of equation (3).

Equation (6) says that the mechanism for spontaneous SUSY breaking for the
quantum Rabi model [4] works, and therefore, the Rabi model with $A^{2}$-term,
described by $H_{\mbox{\tiny Rabi}}[r]+\hbar
Cr^{2}\mathrm{g}_{0}^{2}\left(a+a^{\dagger}\right)^{2}+\hbar\widetilde{\mathrm{g}}[r]^{2}/\widetilde{\omega}[r]$,
yields the spontaneous SUSY breaking in the limit $r\to 1$. The Lagrangian for
this Hamiltonian obtains the extra second-order field,
$-(m[r]^{2}/(2\hbar^{2}))\phi^{2}$, from the $A^{2}$-term, where $m[r]$ is a
mass and $\phi=\sqrt{\hbar/(2\omega)}(a+a^{\dagger})$ the $1$-mode Bose field.
We have $m[r]^{2}=4\hbar^{2}C\omega r^{2}\mathrm{g}_{0}^{2}$ then. The limit
in the norm resolvent sense guarantees the convergence of each energy level
[14]. Thus, it is worthy to note that the energy gap is produced in the
process from the $N=2$ SUSY to its spontaneous breaking. The energy gap is
governed by the parameter $C$ of the $A^{2}$-term. Whether $N=2$ SUSY of
$H(\omega,\omega,0,0)$ goes to its spontaneous breaking can be shown by the
energy degeneracy between the bosonic and the fermionic states. The energy
spectrum is checked with QuTiP [15], for instance, as in Fig.2. In particular,
the comparison of Fig.2a and Fig.2b shows the energy gap caused by the
$A^{2}$-term.

(a) (b)

Figure 2: Energy Spectrum of $H_{\mbox{\tiny Rabi}}[r]+\hbar
Cr^{2}\mathrm{g}_{0}^{2}\left(a+a^{\dagger}\right)^{2}$ with $\omega=6.2832$
and $\mathrm{g}_{0}=6.2832$: a shows the energy spectrum for $C=0$. b is for
$C=0.3770$. The quantum Rabi models without $A^{2}$-term (i.e., $C=0$) and
with $A^{2}$-term (i.e., $C>0$) have the transition from the $N=2$ SUSY to its
spontaneous breaking. In particular, the energy gap by the $A^{2}$-term
appears in $\hbar\widetilde{\omega}[1]$ of the right graph.

###### Acknowledgements.

The author acknowledges the support from JSPS Grant-in-Aid for Scientific
Researchers (C) 20K03768. He would like to dedicate this study to Hiroshi
Ezawa and Elliott H. Lieb on the occasions of their 90th birthdays.

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