1 Introduction

In 2012, the long-sought Higgs boson is found [1, 2], which establishes the triumph of the Brout–Englert–Higgs mechanism [3, 4]. This mechanism tells us how no-mass gauge particles gain mass in the standard model (SM), while any gauge particle itself alone cannot have its mass due to gauge symmetry. That finding shows the Higgs-particle mass of 125GeV. This value is compatible with the electroweak scale (\(\sim 10^{2}\)GeV). When considering the interaction of the Higgs particle in the theory including the electroweak scale, the grand-unified-theory (GUT) scale, and the Planck scale, particle physicists normally need a special tuning to obtain the Higgs mass [5]. Since the GUT-scale mass (\(\sim 10^{16}\)GeV) and the Planck-scale mass (\(\sim 10^{19}\)GeV) are so much heavier than the Higgs mass, particle physicists usually employ the so-called fine-tuning in SM to cope with the mass gap with the enormous ratio (\(\sim 10^{14}\)GeV–\(10^{17}\)GeV); thus, they perform the unnatural, huge cancellation between the bare mass term and the quantum corrections to obtain the Higgs mass. This is a reduced paraphrase of the so-called hierarchy problem. Moreover, the Higgs mass of 125GeV could result in the possibility of the flat Higgs potential when the electroweak scale is together with the Planck scale [6,7,8,9,10,11]. It says that the Higgs quartic interaction may be invalidated. Removing this apprehension, we probably should need to find another mass-enhancement mechanism such as the radiative generation without the Higgs potential. Against these difficulties, supersymmetry (SUSY) is among the strong candidates for natural theories to solve the problems. It is expected that the gaps in mass are plugged by the SUSY breaking [12,13,14,15,16,17,18,19,20,21], a kind of spontaneous symmetry breaking.

In the light of relativistic quantum field theory, although Coleman and Mandula’s no-go theorem states that it is totally impossible to make a non-trivial theory including both the Poincaré symmetry and internal one [22], the Haag-Łopuszański-Sohnius theorem gives us a loophole in the Coleman-Mandula theorem, which says that a way nontrivially to mix the Poincaré and internal symmetries is through SUSY [23]. If SUSY is not a physical entity, the two theorems may show the theoretical limitation of relativistic quantum field theory.

Unfortunately, any superpartner (i.e., supersymmetric particle paired with an elementary particle) has not yet been found [24]; in fact, any fingerprint of SUSY and its spontaneous breaking had not been firmly, directly observed. We probably should study what parts of the theory of SUSY are realized as a physical entity, and clarify them one by one. In the first place, we should confirm the physically real existences of SUSY and its spontaneous breaking. Witten squeezes the minimal essence of SUSY and its spontaneous breaking, and develops it in quantum mechanics [25, 26]. This quantum mechanics is the supersymmetric quantum mechanics (SUSY QM) [27,28,29,30]. Although the verification of the full theory of SUSY needs a huge facility, that of SUSY QM requires the reasonable facility in a laboratory. Actually, Cai et al. report that they succeed in observing the signatures of SUSY and its spontaneous breaking in SUSY QM [31].

Some months before the Higgs-boson discovery, actually, the quantum simulation for the Brout–Englert–Higgs mechanism is succeeded [32]. Quantum simulation is for the study of quantum phenomena, and is implemented on a programmable quantum system consisting of quantum devices especially designed to realize those quantum phenomena. Thus, the quantum simulation is different from the virtual simulations by conventional computers. The original idea of quantum simulation is based on Feynman’s proposal [33] and has experimentally been developed [31, 32, 34,35,36,37,38,39,40]. Some theoretical models for quantum simulation of SUSY and its spontaneous breaking are proposed [41,42,43,44,45,46]. In particular, a simple prototype model is given, and it has the transition from the \({\mathcal {N}}=2\)  SUSY to its spontaneous breaking [41, 42]. It is based on the quantum Rabi model [47,48,49]. The quantum Rabi model is the 1-mode scalar boson version of the spin-boson model [50]. The success in an experimental observation of that transition in a trapped ion quantum simulator is reported [31]. In this transition, we cannot observe any mass enhancement because the Lagrangian of the prototype model does not include any mass-enhancement mechanism. Thus, we are interested in quantum simulation showing a mass enhancement in the fermionic states as the SUSY breaking occurs. One of the candidates for the mass enhancement is adding the quadratic term, often called ‘\(A^{2}\)-term’ [51, 52], for the bosonic states as an extra mass term to the quantum Rabi model. Since the quantum Rabi model describes the electromagnetic interaction basically, its ‘A’ corresponds to the photon gauge field. For the prototype model [41, 42], the strong coupling limit is used to obtain that transition. As shown in this paper, however, we can derive a no-go theorem for the SUSY breaking in the strong coupling limit if the prototype model has the \(A^{2}\)-term. On the other hand, Cai et al. propose another limit experimentally to obtain the transition for the prototype model [31]. We show that their limit makes our model avoid the no-go theorem. Employing their limit, therefore, we extend the prototype model such that we can make quantum simulation for the mass enhancement in SUSY breaking. Our model’s interaction has no Higgs potential; nonetheless, our model makes the mass enhancement.

In this paper, we consider scalar boson only. Thus, we call scalar boson merely ‘boson’ for short. The structure of this paper is as follows: In Sect. 2, we prove that the quantum Rabi model with the \(A^{2}\)-term meets the no-go theorem for the SUSY breaking in the strong coupling limit. On the other hand, we also prove that it can avoid the no-go theorem under the scheme by Cai et al. [31], and it has the transition from the \({\mathcal {N}}=2\)  SUSY to its spontaneous breaking. Then, the mass enhancement in the fermionic states takes place in the SUSY breaking. In Sect. 3, we explain what works for spontaneous symmetry breaking in the mass-enhancement process instead of the Higgs potential using the Lagrangian formalism. In Sect. 4, we discuss the experimental realization of our quantum simulation. We introduce some problems on the Goldstino (i.e., Nambu–Goldstone fermion) arising from the results of this paper.

2 Mass enhancement for quantum Rabi model in SUSY breaking

In this section, we explain the role of the quantum Rabi model for the transition from the \({\mathcal {N}}=2\)  SUSY to its spontaneous breaking. The quantum Rabi model has been coming in handy for quantum simulation lately [53,54,55,56,57], and it can be a powerful tool for our purpose. For the spontaneous symmetry breaking, we follow the definition in Ref. [58, 59]: Suppose that the total Hamiltonian H or Lagrangian L has the symmetry with respect to a transformation \({\mathfrak {G}}\), i.e., \([H, {\mathfrak {G}}]=0\) or \([L, {\mathfrak {G}}]=0\). We say that a spontaneous symmetry breaking occurs if the transformation \({\mathfrak {G}}\) has the multidimensional representation and it does not leave the ground state invariant. In other words, the spontaneous symmetry breaking emerges when \({\mathfrak {G}}\!\!\mid \!\!\textrm{g}\rangle \) becomes a ground state orthogonal with the ground state \(\mid \!\!\textrm{g}\rangle \) due to the degeneracy. The reason why we employ this direct definition instead of the definition with the order parameter is that we do not have to mind the divergence of \({\mathfrak {G}}\!\!\mid \!\!\textrm{g}\rangle \) because we consider our problem in quantum mechanics. We say that the symmetry is continuous if the group which the transformation belongs to is continuous, and it is discrete if the group is discrete. The former case, the continuity enables us to derive the Nambu–Goldstone theorem thanks to Noether’s theorem. Regarding the latter case, for example, the \(\phi ^{4}\)-theory Lagrangian for a ‘real’ scalar field \(\phi _{\textrm{rs}}\) with a quartic interaction has the discrete symmetry for the parity transformation, \(\phi _{\textrm{rs}}\longrightarrow -\phi _{\textrm{rs}}\), [60, 61], that is, the so-called \({\mathbb {Z}}_{2}\) symmetry.

The state space of the 1-mode boson is given by the boson Fock space \({\mathcal {F}}_{\textrm{b}}\), which is spanned by the boson Fock states. The boson Fock state with n bosons is denoted by \(\mid \!\!n\rangle \); thus, \(\mid \!\!0\rangle \) is the Fock vacuum in particular. The 2-level atom in our model is represented by spin. We denote the up-spin state by \(\mid \uparrow \rangle =\bigl ( {\begin{matrix} 1\\ 0 \end{matrix}} \bigl )\), and the down-spin state by \(\mid \downarrow \rangle =\bigl ( {\begin{matrix} 0 \\ 1 \end{matrix}} \bigl )\). We denote by \({\mathbb {C}}\) the set of all the complex numbers. Then, \({\mathbb {C}}^{2}\) is the 2-dimensional unitary space with the natural inner product. We use the Hilbert space \({\mathbb {C}}^{2}\otimes {\mathcal {F}}_{\textrm{b}}\) for the total state space of our model. The orthonormal basis of \({\mathbb {C}}^{2}\otimes {\mathcal {F}}_{\textrm{b}}\) is given by the set of all the vectors \(\mid \downarrow \rangle \otimes \mid \!\!n\rangle \) and \(\mid \uparrow \rangle \otimes \mid \!\!n'\rangle \) for \(n, n'=0, 1, 2, \cdots \). We often omit the symbol ‘\(\otimes \)’ in the vectors of \({\mathbb {C}}^{2}\otimes {\mathcal {F}}_{\textrm{b}}\) throughout this paper. 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, that is, spin or qubit, are given by \(\sigma _{\pm }=(1/2)(\sigma _{x}\pm i\sigma _{y})\). Thus, \(\sigma _{-}\) and \(\sigma _{+}\) are, respectively, the spin-annihilation operator and spin-creation operator. Here, the standard notations, \(\sigma _{x}\), \(\sigma _{y}\), and \(\sigma _{z}\), are used for the Pauli matrices: \(\sigma _{x}=\bigl ( {\begin{matrix} 0 &{} 1 \\ 1 &{} 0\\ \end{matrix}} \bigl )\), \(\sigma _{y}=\bigl ( {\begin{matrix} 0 &{} -i \\ i &{} 0 \end{matrix}} \bigl )\), and \(\sigma _{z}=\bigl ( {\begin{matrix} 1 &{} 0 \\ 0 &{} -1 \end{matrix}} \bigl )\). We use the notation ‘1’ for the 2-by-2 identity matrix, i.e., \(1=\bigl ( {\begin{matrix} 1 &{} 0 \\ 0 &{} 1 \end{matrix}} \bigl )\), and for the identity operator acting in \({\mathcal {F}}_{\textrm{b}}\) as well as the numerical character 1. We often omit the symbols, ‘\(1\otimes \)’ and ‘\(\otimes 1\),’ in operators throughout this paper.

2.1 Our problems

We consider the physical system consisting of the 2-level atom and 1-mode boson. The two ideal, free Hamiltonians, \(H(0, \Omega _{\textrm{b}}, 0, 0)\) and \(H(\Omega _{\textrm{a}}, \Omega _{\textrm{b}}, 0, 0)\), are defined by

$$\begin{aligned} H(0, \Omega _{\textrm{b}}, 0, 0)&= 1\otimes \hbar \Omega _{\textrm{b}}\left( a^{\dagger }a+\frac{1}{2}\right) , \\ H(\Omega _{\textrm{a}}, \Omega _{\textrm{b}}, 0, 0)&=\frac{\hbar \Omega _{\textrm{a}}}{2}\sigma _{z}\otimes 1+H(0, \Omega _{\textrm{b}}, 0, 0), \end{aligned}$$

where \(\Omega _{\textrm{b}}\) denotes the frequency of the 1-mode boson, and \(\Omega _{\textrm{a}}\) is the atom transition frequency.

It is easy to check that, for a common constant \(\Omega >0\), the Hamiltonian \(H(\Omega , \Omega , 0, 0)\) has the \({\mathcal {N}}=2\)  SUSY, and the Hamiltonian \(H(0, \Omega , 0, 0)\) makes its spontaneous breaking [41, 42]. The individual algebraic structures are given in the following.

For the Hamiltonian \(H(\Omega , \Omega , 0, 0)\), its real supercharges, \(q_{1}\) and \(q_{2}\), are given by

$$\begin{aligned} q_{1} =\sqrt{\frac{\hbar \Omega }{2}}\left( \sigma _{+}a+\sigma _{-}a^{\dagger }\right) ,\qquad q_{2} =i\sqrt{\frac{\hbar \Omega }{2}}\left( \sigma _{-}a^{\dagger }-\sigma _{+}a\right) . \end{aligned}$$

Then, they satisfy

$$\begin{aligned}&\left\{ q_{k}, q_{\ell }\right\} =\delta _{k\ell }H(\Omega , \Omega , 0, 0), \\&\quad \left[ q_{k}, H(\Omega , \Omega , 0, 0)\right] =0, \\&\quad \left\{ q_{k}, N_{ \text{ F }}\right\} =0, \end{aligned}$$

where \(N_{ \text{ F }}\) is the grading operator defined by \(N_{ \text{ F }}=-\sigma _{z}\). The ground state (i.e., vacuum) \({\mid \downarrow \rangle }\otimes {\mid \!\!0\rangle }\) of \(H(\Omega , \Omega , 0, 0)\) is a bosonic state since \(N_{ \text{ F }}{\mid \downarrow \rangle }\otimes {\mid \!\!0\rangle } =\, {\mid \downarrow \rangle }\otimes {\mid \!\!0\rangle }\), and it satisfies \(q_{k}\mid \downarrow \rangle \otimes \mid \!\!0\rangle =0\), \(k=1, 2\). The complex supercharges, \(q^{+}\) and \(q^{-}\), are given by

$$\begin{aligned} q^{+}=\frac{1}{\sqrt{2}}\left( q_{1}+iq_{2}\right) =\sqrt{\hbar \Omega }\, \sigma _{+}a,\qquad q^{-}=\frac{1}{\sqrt{2}}\left( q_{1}-iq_{2}\right) =\sqrt{\hbar \Omega }\, \sigma _{-}a^{\dagger }, \end{aligned}$$

such that

$$\begin{aligned}&H(\Omega , \Omega , 0, 0)=\left\{ q^{+}, q^{-}\right\} , \\&\quad \left\{ q^{\pm }, q^{\pm }\right\} =0, \\&\quad \left[ H(\Omega , \Omega , 0, 0), q^{\pm }\right] =0. \end{aligned}$$

These complex supercharges make the connection between the bosonic and fermionic states:

$$\begin{aligned}&q^{-}\mid \downarrow \rangle \otimes \mid \!\!n\rangle =q^{+}\mid \uparrow \rangle \otimes \mid \!\!n\rangle =0, \\&\quad \mid \uparrow \rangle \otimes \mid \!\!n\rangle =\frac{1}{\sqrt{(n+1)\hbar \omega \,}} q^{+}\mid \downarrow \rangle \otimes \mid \!\!n+1\rangle , \\&\quad \mid \downarrow \rangle \otimes \mid \!\!n+1\rangle =\frac{1}{\sqrt{(n+1)\hbar \omega \,}} q^{-}\mid \uparrow \rangle \otimes \mid \!\!n\rangle . \end{aligned}$$

We immediately have \(q^{\pm }{\mid \downarrow \rangle }\otimes {\mid \!\!0\rangle }=0\) for the vacuum \({\mid \downarrow \rangle }\otimes {\mid \!\!0\rangle }\). Since this vacuum is a unique ground state of the Hamiltonian \(H(\Omega , \Omega , 0, 0)\), the Witten index is 1.

Meanwhile, the algebraic structure for the SUSY breaking of \(H(0, \Omega , 0, 0)\) is determined as follows: Its real supercharges, \(Q_{1}\) and \(Q_{2}\), are given by

$$\begin{aligned} Q_{1}=\sqrt{\frac{\hbar \Omega }{2}}\,\sigma _{x} \sqrt{a^{\dagger }a+\frac{1}{2}},\qquad Q_{2}=\sqrt{\frac{\hbar \Omega }{2}}\,\sigma _{y} \sqrt{a^{\dagger }a+\frac{1}{2}}. \end{aligned}$$

Then, they satisfy

$$\begin{aligned}&\left\{ Q_{k}, Q_{\ell }\right\} =\delta _{k\ell }H(0, \Omega , 0, 0), \\&\quad \left[ Q_{k}, H(0, \Omega , 0, 0)\right] =0, \\&\quad \left\{ Q_{k}, N_{ \text{ F }}\right\} =0, \end{aligned}$$

where \(N_{ \text{ F }}\) is the grading operator defined by \(N_{ \text{ F }}=-\sigma _{z}\). The ground sates (i.e., vacuums) \({\mid \!\!\sharp \rangle }\otimes {\mid \!\!0\rangle }\), \(\sharp =\downarrow , \uparrow \), of \(H(0, \Omega , 0, 0)\) have the lowest, strictly positive eigenvalue \(\hbar \Omega /2>0\). We have \(Q_{k}{\mid \!\!\sharp \rangle }\otimes {\mid \!\!0\rangle }\ne 0\), \(k=1, 2\). The complex supercharges, \(Q^{+}\) and \(Q^{-}\), are given by

$$\begin{aligned} Q^{\pm }=\frac{1}{\sqrt{2}}\left( Q_{1}\pm iQ_{2}\right) =\sigma _{\pm }\sqrt{\hbar \Omega \left( a^{\dagger }a+\frac{1}{2}\right) } \end{aligned}$$

such that

$$\begin{aligned}&H(0, \Omega , 0, 0)=\left\{ Q^{+}, Q^{-}\right\} , \\&\quad \left\{ Q^{\pm }, Q^{\pm }\right\} =0, \\&\quad \left[ H(0, \Omega , 0, 0), Q^{\pm }\right] =0. \end{aligned}$$

These complex supercharges have the relations, \( Q^{-}\mid \downarrow \rangle \otimes \mid \!\!n\rangle =Q^{+}{\mid \uparrow \rangle }\otimes \mid \!\!n\rangle =0 \). They do cut the connection with the boson annihilation and creation but the connection between the bosonic and fermionic states as

$$\begin{aligned}&\mid \uparrow \rangle \otimes \mid \!\!n\rangle =\frac{1}{\sqrt{(n+\frac{1}{2})\hbar \Omega \,}} Q^{+}\mid \downarrow \rangle \otimes \mid \!\!n\rangle , \\&\quad \mid \downarrow \rangle \otimes \mid \!\!n\rangle =\frac{1}{\sqrt{(n+\frac{1}{2})\hbar \Omega \,}} Q^{-}\mid \uparrow \rangle \otimes \mid \!\!n\rangle , \end{aligned}$$

in particular, \(Q^{+}\mid \downarrow \rangle \otimes \mid \!\!0\rangle \ne 0\) and \(Q^{-}\mid \uparrow \rangle \otimes \mid \!\!0\rangle \ne 0\) for the vacuums \(\mid \!\!\sharp \rangle \otimes \mid \!\!0\rangle \), \(\sharp =\downarrow , \uparrow \). In terms of the grading operator \(N_{ \text{ F }}\), since \(N_{ \text{ F }}{\mid \downarrow \rangle }\otimes {\mid \!\!n\rangle }= {\mid \downarrow \rangle }\otimes {\mid \!\!n\rangle }\) and \(N_{ \text{ F }}{\mid \uparrow \rangle }\otimes {\mid \!\!n\rangle }= -{\mid \uparrow \rangle }\otimes {\mid \!\!n\rangle }\), the vacuum \({\mid \downarrow \rangle }\otimes {\mid \!\!0\rangle }\) is a bosonic state and the vacuum \({\mid \uparrow \rangle }\otimes {\mid \!\!0\rangle }\) is a fermionic state. Thus, the Witten index is 0, and the SUSY is spontaneously broken. In this model, actually, the \({\mathbb {Z}}_{2}\) symmetry is spontaneously broken: \([ H(0, \Omega , 0, 0), \sigma _{x}]=0\) for the transformation \({\mathfrak {G}}=\sigma _{x}\), however, 2-fold degenerate vacuums, \(\mid \downarrow \rangle \otimes {\mid \!\!0}\rangle \) and \(\mid \uparrow \rangle \otimes \mid {\!\!0}\rangle \), break the \({\mathfrak {G}}=\sigma _{x}\) invariance, \(\mid \downarrow \rangle \otimes {\mid \!\!0}\rangle \ne {\mathfrak {G}}\mid \downarrow \rangle \otimes {\mid \!\!0}\rangle =\mid \uparrow \rangle \otimes {\mid \!\!0}\rangle \). We note that the mathematical structure of our spontaneous symmetry breaking is somewhat similar to that of Susskind’s lattice fermions (see Section V of Ref. [62]).

The collaboration by the supercharges, \(Q^{\pm }\), can make the oscillation between \(\mid \downarrow \rangle \otimes \mid \!\!0\rangle \) and \(\mid \uparrow \rangle \otimes \mid \!\!0\rangle \), the degenerate ground states. This may emerge a fingerprint of the Goldstino mode [12, 26, 27, 29, 30, 63,64,65,66,67] even though the symmetry is discrete. Briefly, the states, \({Q^{+}\!\!\mid \downarrow \rangle \otimes \mid \!\!0\rangle }\) and \({Q^{-}\!\!\mid \uparrow \rangle \otimes \mid \!\!0\rangle }\), are eigenstates of \(H(0, \Omega , 0, 0)\), and they are made up of the excitation by the supercharges \(Q^{\pm }\). Therefore, there might be proper particles on the vacuums, \(\mid \downarrow \rangle \otimes \mid \!\!0\rangle \) and \(\mid \uparrow \rangle \otimes \mid \!\!0\rangle \). Since there is no energy increment between the individual vacuum and the corresponding excited state, the particles might be Goldstinos.

Our problems are described in the following.

Problem 1. How can we introduce an interaction \(H_{\textrm{int}}\) between the 2-level atom and 1-mode boson to make the transition from the \({\mathcal {N}}=2\)  SUSY Hamiltonian \(H(\Omega , \Omega , 0, 0)\) to its spontaneous-breaking Hamiltonian unitarily equivalent to the Hamiltonian \(H(0, \Omega , 0, 0)\)?

Problem 2. How can we make a mass term in the interaction \(H_{\textrm{int}}\) which causes the mass enhancement in the SUSY breaking?

The prototype model in [41, 42] is proposed for a partial solution to Problem 1. Thus, we extend it in this paper such that the extended model gives a solution to Problem 2.

Our model is based on the quantum Rabi model whose Hamiltonian is given by

$$\begin{aligned} H_{ \text{ Rabi }}(\Omega _{\textrm{a}}, \Omega _{\textrm{b}}, G) =H(\Omega _{\textrm{a}}, \Omega _{\textrm{b}}, 0, 0) +\hbar G\sigma _{x} \left( a+a^{\dagger }\right) , \end{aligned}$$

where the last term is the linear interaction between the atom and boson with the parameter G representing the coupling strength. For our candidate of the interaction \(H_{\textrm{int}}\), we add the quadratic interaction in addition to the linear one, and thus, our total Hamiltonian reads

$$\begin{aligned} H(\Omega _{\textrm{a}}, \Omega _{\textrm{b}}, G, C) =&H_{ \text{ Rabi }}(\Omega _{\textrm{a}}, \Omega _{\textrm{b}}, G) +\hbar CG^{2}\left( a+a^{\dagger }\right) ^{2}, \end{aligned}$$
(1)

where the last term of Eq. (1) 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 quadratic term is often called ‘\(A^{2}\)-term’ [51, 52].

As explained above, tuning the parameters \(\Omega _{\textrm{a}}\) and \(\Omega _{\textrm{b}}\) as \(\Omega _{\textrm{a}}=\Omega _{\textrm{b}}=\omega \) for a positive, common constant \(\omega \), the Hamiltonian \(H(\omega , \omega , 0, 0)\) has the \({\mathcal {N}}=2\)  SUSY. In our model, as the coupling strength G gets stronger enough, the \(A^{2}\)-term may appear, i.e., \(C\ne 0\). Then, similar to the case of the superradiant phase transition [68, 69], a no-go theorem caused by the \(A^{2}\)-term [51] should be minded for our target transition. In that case, its avoidance should be argued for our model described by Eq. (1) in SUSY QM as well as for the superradiant-phase-transition model [52]. We investigate this problem in Sect. 2.2.

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

$$\begin{aligned} U_{A^{2}}^{*}H(\Omega _{\textrm{a}}, \Omega _{\textrm{b}}, G, C)U_{A^{2}}= & {} H(\Omega _{\textrm{a}}, \Omega (G), {\widetilde{G}}, 0) =H_{ \text{ Rabi }}(\Omega _{\textrm{a}}, \Omega (G), {\widetilde{G}}), \end{aligned}$$
(2)

where \(\Omega (G)=\sqrt{\Omega _{\textrm{b}}^{2}+4C\Omega _{\textrm{b}}G^{2}\,}\) and \({\widetilde{G}}=G\sqrt{\Omega _{\textrm{b}}/\Omega (G)}\). This unitary operator \(U_{A^{2}}\) is obtained using the (meson) pair theory of nuclear physics [70, 71]. Equation (2) means a renormalization of the \(A^{2}\)-term, and is often called the Hopfield-Bogoliubov transformation [52, 72] of \(H(\Omega _{\textrm{a}}, \Omega _{\textrm{b}}, G, C)\). The effect of the \(A^{2}\)-term is stuffed into \(\Omega (G)\) and \({\widetilde{G}}\). We note that, for \(C=0\), the parameters satisfy \(\Omega (G)=\Omega _{\textrm{b}}\), \({\widetilde{G}}=G\), and then, the unitary operator \(U_{A^{2}}\) is 1, the identity operator.

For the displacement operator \(D(G/\Omega _{\textrm{b}}) = \exp \left[ G(a^{\dagger }-a)/\Omega _{\textrm{b}}\right] \), a unitary operator \(U(G/\Omega _{\textrm{b}})\) is defined by

$$\begin{aligned} U(G/\Omega _{\textrm{b}})=\frac{1}{\sqrt{2}} \left\{ \left( \sigma _{-}-1\right) \sigma _{+}D(G/\Omega _{\textrm{b}}) +\left( \sigma _{+}+1\right) \sigma _{-}D(-G/\Omega _{\textrm{b}})\right\} . \end{aligned}$$

Then, it makes the equation,

$$\begin{aligned}{} & {} U(G/\Omega _{\textrm{b}})^{*} \left\{ H(\Omega _{\textrm{a}}, \Omega _{\textrm{b}}, G, 0) +\hbar \frac{G^{2}}{\Omega _{\textrm{b}}} \right\} U(G/\Omega _{\textrm{b}}) \nonumber \\{} & {} \quad = H(0, \Omega _{\textrm{b}}, 0, 0) -\frac{\hbar \Omega _{\textrm{a}}}{2} \left\{ \sigma _{+}D(G/\Omega _{\textrm{b}})^{2} +\sigma _{-}D(-G/\Omega _{\textrm{b}})^{2}\right\} . \end{aligned}$$
(3)

The arguments on the limit of Hamiltonians used below are already established as a mathematical method [42, 70, 71]. Thus, for simplicity, mathematically naive arguments are made in this section to explain the no-go theorem and its avoidance.

2.2 No–go theorem in strong coupling limit

Now we consider the strong coupling limit for the quantum Rabi model without and with the \(A^{2}\)-term. This limit is approximately realized in experiments of the deep-strong coupling regime [73], for instance, in circuit QED [54]. The parameters, \(\Omega _{\textrm{a}}\), \(\Omega _{\textrm{b}}\), and G, are set as \(\Omega _{\textrm{a}}=\Omega _{\textrm{b}}=\omega \) and \(G=\textrm{g}\) for a non-negative parameter \(\textrm{g}\). The Hamiltonian \(H(\omega , \omega , \textrm{g}, 0) =H_{ \text{ Rabi }}(\omega , \omega , \textrm{g})\) is for the quantum Rabi model, and denoted by \(H_{ \text{ Rabi }}(\textrm{g})\) for simplicity. In the renormalization for the \(A^{2}\)-term, the quantities \(\Omega (\textrm{g})\) and \(\widetilde{\textrm{g}}\) are defined by \(\Omega (\textrm{g})= \sqrt{\omega ^{2}+4C\omega \textrm{g}^{2}\,}\) and \(\widetilde{\textrm{g}}=\textrm{g}\sqrt{\omega /\Omega (\textrm{g})}\).

In case \(C=0\), the mathematical results [42, 70] say that the approximation,

$$\begin{aligned} H_{ \text{ Rabi }}(\textrm{g})+\hbar \frac{\textrm{g}^{2}}{\omega } \approx U(\textrm{g}/\omega )H(0, \omega , 0, 0)U(\textrm{g}/\omega )^{*}, \end{aligned}$$
(4)

is obtained as \(\textrm{g}\rightarrow \infty \). Due to the appearance of the Hamiltonian \(H(0, \omega , 0, 0)\) in Eq. (4), the \({\mathcal {N}}=2\)  SUSY is spontaneously broken in the strong coupling limit \(\textrm{g}\rightarrow \infty \). This is completely characterized with the energy-spectrum property, for instance, as in the left graph of Fig. 1. We here note that the ground states of \(U(\textrm{g}/\omega )H(0, \omega , 0, 0)U(\textrm{g}/\omega )^{*}\) become the Schrödinger-cat-like states. Naively, how to obtain Eq. (4) is explained in the following. Since the vibration in the displacement operator, \(D(\pm \textrm{g}/\omega ) =\exp \left[ \pm i\textrm{g}\left\{ i\left( a-a^{\dagger }\right) \right\} /\omega \right] \), becomes violent as \(\textrm{g}/\omega \) grows larger, the displacement operator decays. Thus, the second term of RHS of Eq. (3) disappears in the limit \(\textrm{g}\rightarrow \infty \).

In the case \(C>0\), on the other hand, the mathematical result [71] says that

$$\begin{aligned}{} & {} H_{ \text{ Rabi }}(\textrm{g}) +\hbar C\textrm{g}^{2}\left( a+a^{\dagger }\right) ^{2} +\hbar \frac{\widetilde{\textrm{g}}^{2}}{\Omega (\textrm{g})} \nonumber \\{} & {} \quad \approx U_{A^{2}} U(\widetilde{\textrm{g}}/\Omega (\textrm{g})) \left[ H(0, \Omega (\textrm{g}), 0, 0) -\, \frac{\hbar \omega }{2}\sigma _{x} \right] U(\widetilde{\textrm{g}}/\Omega (\textrm{g}))^{*} U_{A^{2}}^{*} \end{aligned}$$
(5)

as \(\textrm{g}\rightarrow \infty \). The atomic term \(\hbar \omega \sigma _{x}/2\) appears in addition to the Hamiltonian \(H(0, \Omega (\textrm{g}), 0, 0)\) in RHS of Eq. (5). This appearance interferes with the transition to the SUSY breaking then. Moreover, the divergence of \(\Omega (\textrm{g})\) as \(\textrm{g}\rightarrow \infty \), together with the atomic term, rudely crushes and explicitly breaks that SUSY. We can see this explicit SUSY breaking in the energy spectrum, for instance, as in the right graph of Fig. 1. Thus, the above quantum Rabi model with the \(A^{2}\)-term cannot go to the SUSY breaking as \(\textrm{g}\) changes from \(\textrm{g}=0\) to \(\textrm{g}\approx \infty \). This is the ‘no-go theorem’ caused by the \(A^{2}\)-term for the SUSY breaking in the strong coupling limit. The reason why \(\hbar \omega \sigma _{x}/2\) appears in RHS of Eq. (5) is naively explained as follows: Since \(\lim _{\textrm{g}\rightarrow \infty } \widetilde{\textrm{g}}/\Omega (\textrm{g})=0\) for \(C>0\) though \(\lim _{\textrm{g}\rightarrow \infty } \widetilde{\textrm{g}}/\Omega (\textrm{g})= \lim _{\textrm{g}\rightarrow \infty }\textrm{g}/\omega =\infty \) for \(C=0\), the displacement operator \(D(\pm \widetilde{\textrm{g}}/\Omega (\textrm{g}))\) in Eq. (3) remains as \(D(0)=1\), the identity operator, in the limit \(\textrm{g}\rightarrow \infty \).

The approximations given by Eqs. (4) and (5) are mathematically established in the norm resolvent sense, and the limit is valid over the energy spectrum (see Theorem VIII.24 of [74]). Thus, the limit energy spectra are obtained by those approximations. Whether the \({\mathcal {N}}=2\)  SUSY of \(H(\omega , \omega , 0, 0)\) is taken to its spontaneous breaking is checked by seeing the energy degeneracy and measuring each interval between adjacent energy levels. Each energy spectrum by the numerical computations with QuTiP [75, 76] is obtained, for instance, as in Fig. 1.

Fig. 1
figure 1

Energy Spectrum of \(H_{ \text{ Rabi }}(\textrm{g})+\hbar C\textrm{g}^{2} \left( a+a^{\dagger }\right) ^{2}+\hbar \widetilde{\textrm{g}}^{2}/\Omega (\textrm{g})\) with \(\omega =6.2832\). A ground state energy and six excited state energies from the bottom are shown in each graph. The left graph shows the energy spectrum for \(C=0\). The right graph is for \(C=0.0377\). The left graph says that the quantum Rabi model (without \(A^{2}\)-term) has the transition from the \({\mathcal {N}}=2\)  SUSY to its spontaneous breaking. On the other hand, the right graph shows the loss of the spontaneous breaking, and reveals the explicit breaking instead. Here, it should be noted \(\lim _{\textrm{g}\rightarrow \infty }\hbar \Omega (\textrm{g})=\infty \) and \(\lim _{\textrm{g}\rightarrow \infty }\hbar \widetilde{\textrm{g}}^{2}/\Omega (\textrm{g}) =\hbar /(4C)\)

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2.3 Limit for avoidance of no-go theorem and mass enhancement

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 used for the scheme by Cai et al. [31]. They employ their own limit experimentally to realize the transition for the prototype in the case without \(A^{2}\)-term. Their limit is more direct to make the transition from the \({\mathcal {N}}=2\)  SUSY Hamiltonian \(H(\omega , \omega , 0, 0)\) to its spontaneous-breaking Hamiltonian unitarily equivalent to the Hamiltonian \(H(0, \omega , 0, 0)\) than the strong coupling limit is. It is based on the following idea. They prepare a continuous function \(\omega [r]\) of 1-variable r, \(0\le r\le 1\), such that \(\omega [0]=\omega \) and \(\omega [1]=0\). Then, the Hamiltonian \(H(\omega [0], \omega , 0, 0) =H(\omega , \omega , 0, 0)\) has the \({\mathcal {N}}=2\)  SUSY, and the Hamiltonian \(H(\omega [1], \omega , 0, 0) =H(0, \omega , 0, 0)\) makes its spontaneous breaking. Cai et al. have the trapped-ion technology to realize this limit in the case \(C=0\). Indeed, the linear interaction cannot, alone, do anything to enhance the mass, but it works for the mass enhancement not only in the bosonic states but also in the fermionic states with the help of the \(A^{2}\)-term. We explain this in Sect. 3.2.

From now on, it is proved that the limit, \(r\rightarrow 1\), has the advantage over the strong coupling limit in order that the Rabi model with the \(A^{2}\)-term avoids the no-go theorem and has the SUSY breaking. Let g(r) be a continuous function of 1-variable r, \(0\le r \le 1\), satisfying \(g(0)=0\) and \(g(1)=\textrm{g}\). The parameters, \(\Omega _{\textrm{a}}\), \(\Omega _{\textrm{b}}\), and G, are given by \(\Omega _{\textrm{a}}=\omega [r]\), \(\Omega _{\textrm{b}}=\omega \), and \(G=g(r)\). The Hamiltonian \(H(\omega [r], \omega , g(r), 0)=H_{ \text{ Rabi }}(\omega [r], \omega , g(r))\) for the quantum Rabi model is denoted by \(H_{ \text{ Rabi }}[r]\) for simplicity. The renormalized quantities \({\widetilde{\omega }}[r]\) and \({\widetilde{g}}[r]\) are given by \({\widetilde{\omega }}[r]= \sqrt{\omega ^{2}+4C\omega g(r)^{2}\,}\) and \({\widetilde{g}}[r]=g(r)\sqrt{\omega /{\widetilde{\omega }}[r]}\). Then, the same argument as in [42, 70, 71] gives

$$\begin{aligned}{} & {} H_{ \text{ Rabi }}[r] +\hbar Cg(r)^{2}\left( a+a^{\dagger }\right) ^{2} +\hbar \frac{{\widetilde{g}}[r]^{2}}{{\widetilde{\omega }}[r]} \nonumber \\{} & {} \quad \longrightarrow U_{A^{2}} U(\widetilde{\textrm{g}}[1]/{\widetilde{\omega }}[1]) H(0, {\widetilde{\omega }}[1], 0, 0) U(\widetilde{\textrm{g}}[1]/{\widetilde{\omega }}[1])^{*} U_{A^{2}}^{*} \end{aligned}$$
(6)

in the norm resolvent sense [74] as \(r\rightarrow 1\). It is worthy to note that the ground states of \(U_{A^{2}} U(\widetilde{\textrm{g}}[1]/{\widetilde{\omega }}[1]) H(0, {\widetilde{\omega }}[1], 0, 0) U(\widetilde{\textrm{g}}[1]/{\widetilde{\omega }}[1])^{*} U_{A^{2}}^{*}\) are obtained as the unitary transformation of Schrödinger-cat-like states. The naive reason why this limit is obtained is because the limit, \(\omega [r]\rightarrow \omega [1]=0\), eliminates the second term of RHS of Eq. (3).

Equation (6) says that the Hamiltonian \(H(0, {\widetilde{\omega }}[1], 0, 0)\) appears in the limit, and therefore, the Rabi model with \(A^{2}\)-term, described by \(H_{ \text{ Rabi }}[r] +\hbar Cg(r)^{2}\left( a+a^{\dagger }\right) ^{2} +\hbar {\widetilde{g}}[r]^{2}/{\widetilde{\omega }}[r]\), yields the SUSY breaking in the limit \(r\rightarrow 1\). The limit in the norm resolvent sense guarantees the convergence of each energy level (see Theorem VIII.24 of [74]). Thus, it is worthy to note that how the transition from the \({\mathcal {N}}=2\)  SUSY to its spontaneous breaking occurs, and how the energy gap is produced in that transition. The energy gap is governed by the parameter C of the \(A^{2}\)-term. The energy spectrum is checked with QuTiP [75, 76], for instance, as in Fig. 2. In particular, the comparison of the two graphs of Fig. 2 shows the energy gap caused by the \(A^{2}\)-term.

Fig. 2
figure 2

Energy Spectrum of \(H_{ \text{ Rabi }}[r]+\hbar Cg(r)^{2} \left( a+a^{\dagger }\right) ^{2}+\hbar {\widetilde{g}}[r]^{2}/{\widetilde{\omega }}[r]\) with \(\omega =6.2832\) and \(\textrm{g}=6.2832\). A ground state energy and six excited state energies from the bottom are shown in each graph. The left graph shows the energy spectrum for \(C=0\). The right graph is for \(C=0.2513\). 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 \({\mathcal {N}}=2\)  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. In these numerical computations, we employ \(\omega [r]=(1-r)\omega \) and \(g(r)=r\textrm{g}\)

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We can see the growth of the energy gap for the parameter C in the numerical analyses. The larger the parameter C becomes, the more the energy gap grows as in Fig. 3. This growth follows our result then.

Fig. 3
figure 3

Energy Spectrum of \(H_{ \text{ Rabi }}[r]+\hbar Cg(r)^{2}\left( a+a^{\dagger }\right) ^{2} +\hbar {\widetilde{g}}[r]^{2}/{\widetilde{\omega }}[r]\). All the parameter setting is the same as in Fig. 2 but the parameter C. Here, the parameter C is set as \(C=0.628\) and \(C=1.257\) from the left

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3 Revisiting of quantum Rabi model’s mass-enhancement in SUSY breaking: Lagrangian formalism

In this section, we make sure that the increasing energy gap is totally caused by the so-called mass term of boson in the Lagrangian formalism. Then, we describe the reason why the mass enhancement takes place in fermionic states as well as in bosonic ones even if our model does not have the Higgs potential.

3.1 Lagrangian for quantum simulation

We consider the position operator X and the momentum operator P acting in the boson Fock space \({\mathcal {F}}_{\textrm{b}}\), and identify them with \(1\otimes X\) and \(1\otimes P\) acting in the state space \({\mathbb {C}}^{2}\otimes {\mathcal {F}}_{\textrm{b}}\), respectively. For these identified position and momentum operators, X and P, we give the Hamiltonian H of a harmonic oscillator. This describes the energy operator of a 1-mode massive boson. It is given by

$$\begin{aligned} H = \left( \frac{1}{2}P^{2}+\frac{\omega _{\textrm{g}}^{2}}{2}X^{2}\right) \end{aligned}$$
(7)

acting in the state space \({\mathbb {C}}^{2}\otimes {\mathcal {F}}_{\textrm{b}}\), where \(\hbar \omega _{\textrm{g}}\) is the boson energy. We call this 1-mode massive boson the ‘heavy boson.’

We arbitrarily give a positive parameter \(\omega \), a non-negative parameter C, and a positive constant \(\textrm{g}\) such that \(\omega _{\textrm{g}}^{2}=\omega ^{2}+4C\omega \textrm{g}^{2}\). We consider another Hamiltonian \(H_{ \text{ SS }}\) for the position operator x and the momentum operator p acting in another boson Fock space \({\mathcal {F}}_{\textrm{b}}\). The Hamiltonian \(H_{ \text{ SS }}\) is popular in SUSY QM [27, 28] and given by

$$\begin{aligned} H_{ \text{ SS }} =1\otimes \frac{1}{2}\left( p^{2}+W^{2}\right) +\frac{\hbar }{2}\sigma _{z}\otimes \frac{d W}{dx}, \end{aligned}$$
(8)

where W is the superpotential defined by \(W(x)=\omega x\). It is worthy to note that Nambu had given the advice on the linearization of the superpotential [77]. We omit ‘\(\otimes \)’, and then, \(H_{ \text{ SS }}=(1/2)\left( p^{2}+W^{2}+\hbar \sigma _{z}(dW/dx)\right) \).

Our spin–boson interaction is based on \(\sigma _{x}x\), the position operator with the X-gate. It should be pointed out that the Pauli matrix \(\sigma _{x}\), i.e., X-gate, plays a role of the swap between the bosonic and fermionic states. We suppose that an extra second-order term \((2C/\omega )g(r)^{2}(\sigma _{x}W)^{2}=(2C/\omega )g(r)^{2}W^{2}\), different from the second-order term \(W^{2}/2\) in Eq. (8), is introduced in our interaction in addition to the first-order term \(g(r)\sqrt{2\hbar /\omega }\, \sigma _{x}W\). We note here that the X-gate disappears from the second-order term since \(\sigma _{x}^{2}=1\). We finally prepare an interaction,

$$\begin{aligned} H_{\textrm{int}}(r) =g(r)\sqrt{\frac{2\hbar }{\omega }}\, \sigma _{x}W +\frac{2C}{\omega }g(r)^{2}W^{2} +\frac{\hbar g(r)^{2}}{4Cg(r)^{2}+\omega } +\frac{\hbar }{2}\sigma _{z}\frac{dW_{\textrm{a}}(r)}{dx}, \end{aligned}$$
(9)

for r, \(0\le r\le 1\), with functions of r: g(r), \(W_{\textrm{a}}(r)=\left( \omega _{\textrm{a}}(r)-\omega \right) x\), \(\omega _{\textrm{a}}(r)\). The concrete definition of these functions are given below. This interaction \(H_{\textrm{int}}(r)\) is introduced such that it causes a SUSY breaking for the SUSY Hamiltonian \(H_{ \text{ SS }}\). Unlike Nambu and Jona-Lasinio’s case [78] and Goldstone’s [79], the interaction \(H_{\textrm{int}}(r)\) has no Higgs potential. In terms of oscillator, the second-order term, \((2C/\omega )g(r)^{2}W^{2}\), means that the oscillator is coupled not only to its nearest neighbor but also to itself at the equilibrium points, and induces a mass (see Chapter 3 of [80]). It is eventually expected that the extra second-order term in \(H_{\textrm{int}}(r)\) plays a role of radiatively making the mass enhancement. Therefore, our total Hamiltonian reads

$$\begin{aligned} H(r)=H_{ \text{ SS }}+H_{\textrm{int}}(r). \end{aligned}$$

We control the interaction appearance using the functions g(r) and \(\omega _{a}(r)\). Here, g(r) is a continuous function satisfying \(g(0)=0\) and \(g(1)=\textrm{g}\). The function \(\omega _{\textrm{a}}(r)\) is also continuous and satisfies \(\omega _{\textrm{a}}(0)=\omega \) and \(\omega _{\textrm{a}}(1)=0\). Then, the total Hamiltonian attains the SUSY Hamiltonian at \(r=0\): \(H(0)=H_{ \text{ SS }}\).

We bring up the parameter r from \(r=0\) to \(r=1\) in the total Hamiltonian H(r). Following the mathematical methods [42, 70, 71], we can show \(H(r)\rightarrow H(1)\) as \(r\rightarrow 1\) in the norm resolvent sense [74]. Actually, \(H(1)=H\). In the case \(C=0\), it can mathematically be proved that this limit produces the transition from the \({\mathcal {N}}=2\)  SUSY at \(r=0\) to its spontaneous breaking at \(r=1\) in the same way as in [42]. The condition \(C=0\) means that there is no mass-enhancement tool in the interaction \(H_{\textrm{int}}(r)\), and there is no possibility that the SUSY breaking can yields a mass enhancement. In the case \(C>0\), however, there is that possibility. We check it below.

We consider the limit, \(H(r)\rightarrow H(1)\) as \(r\rightarrow 1\). Defining the 1-mode boson annihilation operator B by

$$\begin{aligned} B=\sqrt{\frac{\omega _{\textrm{g}}}{2\hbar }}\, X+i\sqrt{\frac{1}{2\hbar \omega _{\textrm{g}}}}\, P, \end{aligned}$$

the Hamiltonian H in Eq. (7) of the heavy boson can be rewritten as

$$\begin{aligned} H=\hbar \omega _{\textrm{g}}\left( B^{\dagger }B+\frac{1}{2}\right) . \end{aligned}$$

Meanwhile, we define the 1-mode boson annihilation operator b by

$$\begin{aligned} b=\sqrt{\frac{\omega }{2\hbar }}\, x+i\sqrt{\frac{1}{2\hbar \omega }}\, p. \end{aligned}$$

We call this 1-mode boson the ‘light boson’ compared with the heavy boson. Then, we can rewrite the total Hamiltonian H(r) of the light boson as

$$\begin{aligned} H(r) =H_{ \text{ Rabi }}(r) +\hbar Cg(r)^{2}( b+b^{\dagger })^{2} +\frac{\hbar g(r)^{2}}{4Cg(r)^{2}+\omega }, \end{aligned}$$

where \(H_{ \text{ Rabi }}(r)\) is the Hamiltonian of the quantum Rabi model [47,48,49] given by

$$\begin{aligned} H_{ \text{ Rabi }}(r)=\hbar \omega \left( b^{\dagger }b+\frac{1}{2}\right) +\hbar g(r)\sigma _{x}\left( b+b^{\dagger }\right) +\frac{\hbar \omega _{\textrm{a}}(r)}{2}\sigma _{z}. \end{aligned}$$

The total Hamiltonian H(r) is unitarily equivalent to the Hamiltonian \(H(\omega _{\textrm{a}}(r), \omega , g(r), C)+\hbar g(r)^{2}/(4Cg(r)^{2}+\omega )\), where the definition of \(H(\omega _{\textrm{a}}(r), \omega , g(r), C)\) is given in Sect. 2. The second-order term \(\hbar Cg(r)^{2}( b+b^{\dagger })^{2}\) is the \(A^{2}\)-term [51, 52]. The \(A^{2}\)-term naturally appears in quantum electrodynamics (QED) and cavity QED when the coupling strength g(r) is not so small. Moreover, it may be controlled in circuit QED (see [52] and Methods of [54]).

For every r with \(0\le r\le 1\), we prepare functions, \(\omega _{\textrm{g}}(r)\) and \({\widetilde{g}}(r)\), of 1-variable r by \(\omega _{\textrm{g}}(r)=\sqrt{\omega ^{2}+4C\omega g(r)^{2}}\) and \({\widetilde{g}}(r)=g(r)\sqrt{\omega /\omega _{\textrm{g}}(r)}\). Replacing \(\Omega _{\textrm{a}}\), \(\Omega _{\textrm{b}}\), G, \(\Omega (G)\), and \({\widetilde{G}}\) in Eqs. (2) and (3) by \(\omega _{\textrm{a}}(r)\), \(\omega \), g(r), \(\omega _{\textrm{g}}(r)\), and \({\widetilde{g}}(r)\), respectively, we can make the unitary operator \(U_{r}\), and define a boson annihilation operator \(B_{r}\) and the spin operators \({\mathcal {D}}_{\pm }\) by

$$\begin{aligned} B_{r}&=U_{r}bU_{r}^{*}=(c_{1}+c_{2})b+(c_{1}-c_{2})b^{\dagger } +\frac{{\widetilde{g}}(r)}{\omega _{\textrm{g}}(r)}\, \sigma _{x}, \end{aligned}$$
(10)
$$\begin{aligned} {\mathcal {D}}_{\pm }&=U_{r}\sigma _{\pm } \exp \left[ \pm 2\frac{{\widetilde{g}}(r)}{\omega _{\textrm{g}}(r)} \left( b^{\dagger }-b\right) \right] U_{r}^{*} =-\, \frac{1}{2}\left( \sigma _{z}\mp i\sigma _{y}\right) , \end{aligned}$$
(11)

where \(c_{1}=(1/2)\sqrt{\omega _{\textrm{g}}(r)/\omega }\) and \(c_{2}=(1/2)\sqrt{\omega /\omega _{\textrm{g}}(r)}\). Then, \(B_{1}\) is unitarily equivalent to B since \(\omega _{\textrm{a}}(1)=0\), \(g(1)=\textrm{g}\), \(\omega _{\textrm{g}}(1)=\omega _{\textrm{g}}\), and \({\widetilde{g}}(1)=\widetilde{\textrm{g}}\equiv \textrm{g}\sqrt{\omega /\omega _{\textrm{g}}}\). Thus, we identify \(B_{1}\) with B, i.e., \(B_{1}=B\), from now on.

We note that the canonical commutation relation and canonical anticommutation relation respectively hold:

$$\begin{aligned} \left[ B_{r}, B_{r}^{\dagger }\right] =\left[ b, b^{\dagger }\right] =1,\qquad \left\{ {\mathcal {D}}_{-}, {\mathcal {D}}_{+}\right\} =1,\qquad \left\{ {\mathcal {D}}_{\pm }, {\mathcal {D}}_{\pm }\right\} =0. \end{aligned}$$

In addition to these, we realize the spin-chiral symmetry,

$$\begin{aligned}{}[\sigma _{x}, B_{r}]=[\sigma _{x}, B_{r}^{\dagger }]=0. \end{aligned}$$

In other words, it is the symmetry with respect to the swap between the bosonic and fermionic states. Equation (10) says that the boson annihilation operator \(B_{r}\) consists of the pair of the annihilation and creation of the light boson with the X-gate. This pair is produced following the (meson) pair theory [70, 71, 80]. Since \(B_{1}=B\) in particular, we can think that the heavy boson is a quasi-particle of the annihilation and creation of the light bosons. The quasi-particle eats the swap \(\sigma _{x}\), or rather the vibration, between the fermionic and boson states then. Equation (11) says that the heavy boson cannot see the displacement by the light boson directly in the spin.

Then, we have the equation between the Hamiltonian described by the light boson coupled with the spin and the Hamiltonian described by the heavy boson coupled with the spin,

$$\begin{aligned} \hbar \omega _{\textrm{g}}(r)\left( B_{r}^{\dagger }B_{r}+\frac{1}{2}\right) -\, \frac{\hbar \omega _{\textrm{a}}(r)}{2} \left( {\mathcal {D}}_{-}+{\mathcal {D}}_{+}\right) = H(r). \end{aligned}$$
(12)

We have \(\omega _{\textrm{a}}(1)=0\), and \(\omega _{\textrm{g}}(1)=\omega _{\textrm{g}}\) because \(g(1)=\textrm{g}\). Thus, we obtain the limit

$$\begin{aligned} H(r)&= H_{ \text{ Rabi }}(r)+\hbar Cg(r)^{2}( b+ b^{\dagger })^{2} +\frac{\hbar g(r)^{2}}{4Cg(r)^{2}+\omega } \nonumber \\&\longrightarrow H=\hbar \omega _{\textrm{g}}\left( B^{\dagger }B+\frac{1}{2}\right) \end{aligned}$$
(13)

as \(r\rightarrow 1\). This limit is consistent with Eq. (6) and its rephrasing in the present case.

3.2 Mechanism of radiative mass-enhancement

Following the Nambu and Jona-Lasinio’s theory [78], and Goldstone’s [79], a Mexican-hat potential implies the spontaneous symmetry breaking. In the Brout–Englert–Higgs mechanism [3, 4], the Higgs potential among the Mexican-hat potentials causes the mass generation. The interaction of our model does not have the Higgs potential. However, a certain kind of its approximation works. We call this approximation the two-level system approximation in this paper. It is based on the idea of building an infinitely tall barrier between the two wells and adopting the tunnel effect between the two wells. For further details, see Section III B of Ref. [81]. When seeking the representation of the ground-state energy of the Schrödinger particle in a double-well potential, we use this idea in the WKB approximation with the help of the instanton solution for the double-well potential [82]. We can obtain a similar representation of the ground-state energy of the spin-boson model and the quantum Rabi model [41, 83].

We think that the potential, \(X^{2}\), in Eq. (7) makes a 2-level-system approximation of the Higgs potential in the following sense. Let us consider the Higgs potential as a double-well potential for the real scalar field now. The real scalar field \(\phi _{\textrm{rs}}\) of the Higgs potential, \(V_{ \text{ H }}(\phi _{\textrm{rs}})\), has the parity symmetry, \(\phi _{\mathrm{{rs}}}\longleftrightarrow -\phi _{\textrm{rs}}\). The \(X^{2}\)-potential on each level corresponds to one of the wells of the Higgs potential, and the two \(X^{2}\)-potentials on the two levels give an approximation of the double-well. Although the two \(X^{2}\)-potentials have a kind of the infinitely tall barrier, the swap by the X-gate, \(\sigma _{x}\), makes the role of the tunnel effect between the two wells. Thus, the two-level-system approximation plays a role as a substitute for the Higgs potential for the real scalar field in our story. The X-gate, \(\sigma _{x}\), makes the \({\mathbb {Z}}_{2}\)-symmetry instead of the the parity transformation then.

From now on, we use the rough correspondence [77] between the position (resp. momentum) operator and the field (resp. conjugate field). We introduce the 1-mode scalar field \(\Phi _{r}\) and its conjugate field \(\Pi _{r}\) of the heavy boson by

$$\begin{aligned} \Phi _{r}=\sqrt{\frac{\hbar }{2\omega _{\textrm{g}}(r)}}\left( B_{r}+B_{r}^{\dagger }\right) ,\qquad \Pi _{r}=-i\sqrt{\frac{\hbar \omega _{\textrm{g}}(r)}{2}}\left( B_{r}-B_{r}^{\dagger }\right) , \end{aligned}$$
(14)

for \(0\le r\le 1\). We denote \(\Phi _{1}\) and \(\Pi _{1}\) by \(\Phi \) and \(\Pi \), respectively, because \(B_{1}=B\). Then, we have \([\Phi _{r}, \Pi _{r}]=i\hbar \). The Lagrangian \(L_{r}\) corresponding to H(r) is given by

$$\begin{aligned} L_{r}=\frac{1}{2}\Pi _{r}^{2}-\frac{\omega _{\textrm{g}}(r)^{2}}{2}\Phi _{r}^{2} +\frac{\hbar \omega _{\textrm{a}}(r)}{2} \left( {\mathcal {D}}_{-}+{\mathcal {D}}_{+}\right) . \end{aligned}$$

In particular, we have

$$\begin{aligned} L_{1}=\frac{1}{2}\Pi ^{2}-\frac{\omega _{\textrm{g}}^{2}}{2}\Phi ^{2} \end{aligned}$$
(15)

since \(\omega _{\textrm{a}}(1)=0\). The Lagrangian \(L_{1}\) corresponds to the Hamiltonian H since \(B=B_{1}\).

We introduce a scalar field \(\phi \) and its conjugate field \(\pi \) of the light boson by

$$\begin{aligned} \phi =\sqrt{\frac{\hbar }{2\omega }}\left( b+b^{\dagger }\right) ,\qquad \pi =-i\sqrt{\frac{\hbar \omega }{2}}\left( b-b^{\dagger }\right) . \end{aligned}$$
(16)

We use the fields, \(\phi \) and \(\pi \), as auxiliary fields for the fields, \(\Phi _{r}\) and \(\Pi _{r}\). Taking the limit \(r\rightarrow 1\), we have \(L_{r}\rightarrow L_{1}\). Thus, using Eqs. (10), (14), and (16), we can rewrite \(L_{r}\) and obtain the limit,

$$\begin{aligned} L_{r}&=\,\,\frac{1}{2}\pi ^{2}-\frac{\omega ^{2}}{2}\phi ^{2} - g(r)\sqrt{2\hbar \omega }\, \sigma _{x}\phi -2C\omega g(r)^{2}\phi ^{2} \nonumber \\&\quad -\frac{\hbar g(r)^{2}}{4Cg(r)^{2}+\omega } -\,\frac{\hbar \omega _{\textrm{a}}(r)}{2}\sigma _{z} \nonumber \\&\quad \mathop {\longrightarrow }_{r\rightarrow 1}\,\, L_{1}=\frac{1}{2}\pi ^{2}-\frac{\omega _{\textrm{g}}^{2}}{2}\phi ^{2} - \textrm{g}\sqrt{2\hbar \omega }\, \sigma _{x}\phi -\frac{\hbar \textrm{g}^{2}}{4C\textrm{g}^{2}+\omega }. \end{aligned}$$
(17)

In the Lagrangian \(L_{r}\), an extra second-order term, \(2C\omega g(r)^{2}\phi ^{2}\), appears. Indeed an effect of \(\sigma _{x}\) is invisible in it since \(\sigma _{x}^{2}=1\), but the interaction in the Lagrangian \(L_{r}\) is basically constructed with \(\sigma _{x}\phi \) which makes the swap between the bosonic and fermionic states. The increment of the mass enhancement is included in the factor, \(4C\omega \textrm{g}^{2}\), in the renormalized frequency \(\omega _{\textrm{g}}\). Considering the dimension, the mass increment \(\Delta m\) is given by \(\omega _{\textrm{g}}=\sqrt{\omega ^{2}+(\Delta m)^{2}/\hbar ^{2}}\), that is, \(\Delta m=2\sqrt{C\omega }\,\hbar \textrm{g}\).

We here summarize the above results.

  1. (1)

    The two-level-system approximation works instead of the Higgs potential for the real scalar field, and then, the transition from \({\mathcal {N}}=2\) SUSY to its spontaneous breaking occurs.

  2. (2)

    The transition changes the free field \(\phi \) of the light boson to the free field \(\Phi \) of heavy boson. The heavy boson acquires a part of its mass, caused by the \(A^{2}\)-term, from the excitation of the light boson. This makes the mass enhancement for the fermionic states as well as for the bosonic ones.

  3. (3)

    The Lagrangian \(L_{1}\) has the spin-chiral symmetry, \([\sigma _{x}, L_{1}]=0\), though the Lagrangian \(L_{r}\) does not have it, \([\sigma _{x}, L_{r}]\ne 0\), for \(0\le r<1\) because of the existence of the spin term, \(-\hbar \omega _{\textrm{a}}(r)\sigma _{z}/2\).

4 Conclusion and discussion

We have proposed a mathematical model, though very simple, for quantum simulation of a mass enhancement in the SUSY breaking. This model is based on the quantum Rabi model with the \(A^{2}\)-term, and reveals a transition from the \({\mathcal {N}}=2\)  SUSY to its spontaneous breaking. We have proved that the \(A^{2}\)-term works for the mass enhancement in the fermionic states as well as in the bosonic ones. We have shown that, in the process of the transition, the quasi-particle of the light bosons eats the swap effect of the X-gate and becomes the heavy boson. Our results state that we need a strict parameter-setting for \(\omega [r]\) and g(r) of the model to have SUSY and its spontaneous breaking. We can manage to make SUSY for \(\Omega _{\textrm{a}}=\omega [0]=\omega \), \(G=g(0)=0\) and its spontaneous breaking for \(\Omega _{\textrm{a}}=\omega [1]=0\), \(G=g(1)=\textrm{g}\). On the other hand, we realize that SUSY is ‘explicitly’ broken for any other parameter-setting at \(r=0, 1\) and for all the case \(0<r<1\).

We have explained that the qubit system (i.e., the two-level system) coupled with boson might be good at simulating the double-well potential such as the Higgs potential. For another example, we know that the quantum Rabi model has some properties similar to the instanton [84, 85] as well as the spin-boson model has (see [83, Theorem 1.5] and [41, Appendix B]).

In the case without the \(A^{2}\)-term, it is reported that the transition is experimentally observed in a trapped ion quantum simulator by Cai et al. [31]. Thus, a future experimental problem would be whether \(A^{2}\)-term can be added to their experimental set-ups in a quantum simulator, and an experimental observation of the energy spectrum can be performed.

The results in this paper raise the following issues: Can we see a fingerprint of the mode of the so-called Goldstino (i.e., Nambu–Goldstone fermion) [12, 26, 27, 29, 30, 63,64,65,66,67] in the SUSY breaking for our quantum-mechanical model? From this point of view, it is worthy to note that Cai et al. have been developing the technology to observe the supercharges [31]. In our SUSY breaking, the oscillation between the bosonic and fermionic states is that between qubits, the down-spin and the up-spin states. Is there any relation between the Goldstino and the Rabi oscillation or any relation between the Goldstino and the instanton? For further details of these questions, see Ref. [81].