# Unified approach to the nonlinear Rabi models

Liwei Duan  Department of Physics, Zhejiang Normal University, Jinhua 321004,
China<EMAIL_ADDRESS>
###### Abstract

An analytical approach is proposed to study the two-photon, two-mode and
intensity-dependent Rabi models. By virtue of the su(1,1) Lie algebra, all of
them can be unified to the same Hamiltonian with $\mathcal{Z}_{2}$ symmetry.
There exist exact isolated solutions, which are located at the level crossings
between different parities and correspond to eigenstates with finite
dimension. Beyond the exact isolated solutions, the regular spectrum can be
achieved by finding the roots of the G-function. The corresponding eigenstates
are of infinite dimension. It is noteworthy that the expansion coefficients of
the eigenstates present an exponential decay behavior. The decay rate
decreases with increasing coupling strength. When the coupling strength tends
to the spectral collapse point $g\rightarrow\omega/2$, the decay rate tends to
zero which prevents the convergence of the wave functions. This work paves a
way for the analysis of novel physics in nonlinear quantum optics.

††: New J. Phys.

Keywords: nonlinear Rabi model, su(1,1) Lie algebra, exact solution

##  1 Introduction

As a paradigmatic model to study the light-matter interacting systems, the
Rabi model has been proposed for more than $80$ years [1, 2, 3]. Renewed
attention has been paid to the quantum Rabi model over the last decades, due
to the burst of experiments that push into ultrastrong and even deep strong
coupling regimes [4, 5, 6], the emergence of the quantum phase transition in
the finite component systems [7, 8, 9, 10, 11], as well as the breakthrough of
the analytical exact solutions obtained from the G-functions in the Bargmann
space [12] and Bogoliubov operator approach [13]. The quantum Rabi model
serves as a building block for the quantum information processing [4], and
forms a connecting link between mathematics, physics, and technology [3]. The
quantum Rabi model originally describes a two-level system linearly
interacting with a single bosonic mode [14]. Recently, a generalized Rabi
model has stepped into the spotlight which considers the nonlinear interaction
between the two-level system and the bosonic field. Among them, two-photon,
two-mode and intensity-dependent Rabi models are three typical ones which
introduce different forms of nonlinear interactions.

The two-photon and two-mode Rabi models describe the transitions of the two-
level system accompanied by emitting or absorbing two photons in single- and
two-mode bosonic field respectively. They can be used to describe the second-
order process with consequently small coupling strengths in different physical
setups [15]. Two-photon processes can generate high-order correlations between
the emitted photons which is of great significance in quantum optics and
quantum information science [16, 17]. Recently, implementations of the two-
photon Rabi models in the trapped ion [15, 18] and superconducting circuits
[19, 20] have been proposed which can reach the ultrastrong coupling regime.
The increase in the coupling strength prompts us to search for more accurate
methods beyond the rotating wave approximation (RWA) [21, 22, 23]. Based on
the numerical diagonalization in a truncated basis, Ng et alfound that there
exist significant differences in the energy spectra with and without RWA [24].
Emary and Bishop found exact isolated solutions for the two-photon Rabi model
based on the Bogoliubov transformations [25]. Furthermore, Chen et alfirst
proposed a G-function based on the Bogoliubov operator approach, with which
they achieved the exact isolated solutions and the complete regular spectrum
of the two-photon and two-mode Rabi models [13, 26, 27]. Braak provided a
rigorous proof of validity of Chen’s G-function based on the normalizability
of the wavefunctions in the Bargmann space [28].

Pioneered by Buck and Sukumar, they proposed an intensity-dependent Jaynes-
Cummings model, namely the Buck-Sukumar model, to study the collapse and
revival behavior of the two-level system [29]. The intensity-dependent Rabi
model can be regarded as a generalization of the Buck-Sukumar model which
introduces the counter-rotating wave terms and the Holstein-Primakoff
realization of the su(1,1) operators [24, 30, 31]. The trapped ion far away
from the Lamb-Dicke regime can be used to simulate the nonlinear Rabi model
[32], and it can be used to generate arbitrary $n$-phonon Fock states [33]. To
the best of our knowledge, neither the exact isolated solutions nor the
regular spectrum have been found in this model.

Although the Hamiltonians of the two-photon, two-mode and intensity-dependent
Rabi models are quite different, they share some common features: (i) One can
introduce the su(1,1) Lie algebra to describe the bosonic parts of three
Hamiltonians. The Bargmann index can be used to characterize different Hilbert
subspaces. (ii) All of them exist spectral collapse phenomena [24, 15, 26,
31]. When the coupling strength is large enough, the discrete energy levels
tend to form a continuous energy band except for some low-lying states [26].
Beyond the spectral collapse point, the nonlinear Rabi models become no longer
self-adjoint. In this paper, we employ the su(1,1) Lie algebra to unify three
models to a general Hamiltonians with $\mathcal{Z}_{2}$ symmetry. Then, the
analytical solutions to the general Hamiltonian are achieved by employing the
Bogoliubov operator approach.

The paper is structured as follows. In section 2, we revisit the su(1,1) Lie
algebra. In section 3, we introduce a general Hamiltonian which recovers three
nonlinear Rabi models by employing different realizations of su(1,1) algebra.
In section 4, we construct an ansatz for the general Hamiltonian by choosing
appropriate basis states. The asymptotic behavior of the expansion
coefficients of the ansatz is analyzed. The condition to achieve exact
isolated solutions is given. Beyond the exact isolated solutions, the regular
spectrum is achieved by solving the G-function. The energy spectrum and the
behavior of the eigenstates can be found in section 5. Finally, a brief
summary is given in section 6.

##  2 SU(1,1) group

The group theory has been employed in various branches in quantum optics [34,
35]. We begin by briefly reviewing the basic properties of the SU(1,1) group
and its associate su(1,1) algebra. The SU(1,1) group is non-compact. The
generators associated with SU(1,1) group satisfy

$\displaystyle\left[\hat{K}_{0},\hat{K}_{\pm}\right]=\pm\hat{K}_{\pm},\quad\left[\hat{K}_{+},\hat{K}_{-}\right]=-2\hat{K}_{0}.$
(1)

The corresponding Casimir $\hat{C}$ operator can be written as

$\displaystyle\hat{C}$ $\displaystyle=$
$\displaystyle\hat{K}_{0}^{2}-\frac{1}{2}\left(\hat{K}_{+}\hat{K}_{-}+\hat{K}_{-}\hat{K}_{+}\right),$
(2)

which commutes with all the elements of the su(1,1) Lie algebra. One can
choose the basis state $\left|{}k,m\right>$, which satisfies the following
relations,

$\displaystyle\hat{K}_{0}\left|{}k,m\right>$ $\displaystyle=$
$\displaystyle(k+m)\left|{}k,m\right>,$ (3a)
$\displaystyle\hat{K}_{+}\left|{}k,m\right>$ $\displaystyle=$
$\displaystyle\sqrt{(m+1)(m+2k)}\left|{}k,m+1\right>,$ (3b)
$\displaystyle\hat{K}_{-}\left|{}k,m\right>$ $\displaystyle=$
$\displaystyle\sqrt{m(m+2k-1)}\left|{}k,m-1\right>,$ (3c)
$\displaystyle\hat{C}\left|{}k,m\right>$ $\displaystyle=$ $\displaystyle
k(k-1)\left|{}k,m\right>,$ (3d)

with $m=0,1,2,\dots$ All states can be obtained from the lowest one
$\left|{}k,0\right>$ by successive actions of the raising operator
$\hat{K}_{+}$ according to

$\displaystyle\left|{}k,m\right>=\sqrt{\frac{\Gamma(2k)}{m!\Gamma(2k+m)}}\hat{K}_{+}^{m}\left|{}k,0\right>.$
(3d)

The number $k$ is known as the Bargmann index which separates different
irreducible representations.

##  3 Nonlinear Rabi model

A general nonlinear Rabi model with an su(1,1) coupling scheme can be written
as

$\displaystyle\hat{H}=\frac{\epsilon}{2}\hat{\sigma}_{z}+\omega\hat{K}_{0}+g\hat{\sigma}_{x}\left(\hat{K}_{+}+\hat{K}_{-}\right),$
(3e)

where $\epsilon$ and $\omega$ correspond to the frequency of the two-level
system and bosonic field respectively, $g$ is the coupling strength. Like the
linear Rabi model, the nonlinear one has $\mathcal{Z}_{2}$ symmetry. The
parity operator can be defined as $\hat{\Pi}=-\hat{\sigma}_{z}\otimes\hat{T}$
with $\hat{T}=\exp\left[\rmi\pi\left(\hat{K}_{0}-k\right)\right]$. We can
easily verify that

$\displaystyle\hat{\Pi}\hat{\sigma}_{z}\hat{\Pi}^{\dagger}=\hat{\sigma}_{z},\quad\hat{\Pi}\hat{\sigma}_{x}\hat{\Pi}^{\dagger}=-\hat{\sigma}_{x},\quad\hat{\Pi}\hat{K}_{0}\hat{\Pi}^{\dagger}=\hat{K}_{0},\quad\hat{\Pi}\hat{K}_{\pm}\hat{\Pi}^{\dagger}=-\hat{K}_{\pm},$

which leads to $\hat{\Pi}\hat{H}\hat{\Pi}^{\dagger}=\hat{H}$ and
$\left[\hat{H},\hat{\Pi}\right]=0$. The parity operator $\hat{\Pi}$ has
eigenvalues $\Pi=\pm 1$, and it can separate the whole Hilbert space into two
subspaces with even and odd parities respectively. Unlike the linear Rabi
model, the nonlinear one also commutes with the Casimir operator $\hat{C}$,
which separates the whole Hilbert space into different subspaces indexed by
the Bargmann index $k$.

Such a Hamiltonian has been studied by Penna et al[31] who mainly focused on
the two-mode and Holstein-Primakoff realizations of the su(1,1) algebra.
Depending on the choice of the realizations, $\hat{H}$ can be expressed in
different forms.

###  3.1 Two-photon Rabi model

In the one-mode bosonic realization, the generators can be expressed as

$\displaystyle\hat{K}_{0}=\frac{1}{2}\left(\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\right),\quad\hat{K}_{+}=\frac{1}{2}\left(\hat{a}^{\dagger}\right)^{2},\quad\hat{K}_{-}=\frac{1}{2}\hat{a}^{2},$
(3f)

where $\hat{a}$ ($\hat{a}^{\dagger}$) is the bosonic annihilation (creation)
operator. The corresponding Bargmann index is $k=\frac{1}{4}$ or
$\frac{3}{4}$. Given the Fock states $\left|{}n\right>_{a}$ which satisfies
$\hat{a}^{\dagger}\hat{a}\left|{}n\right>_{a}=n\left|{}n\right>_{a}$, the
basis state $\left|{}k,m\right>$ can be rewritten as

$\displaystyle\left|{}k,m\right>=\left|{}2\left(m+k-\frac{1}{4}\right)\right>_{a}.$
(3g)

Therefore, the number of bosons is even and odd for $k=\frac{1}{4}$ and
$\frac{3}{4}$ respectively.

One can obtain the well-known two-photon Rabi model [24, 25, 26] by
substituting $\hat{K}_{0}$ and $\hat{K}_{\pm}$ in (3e) with those in (3f),
which leads to

$\displaystyle\hat{H}_{\mathrm{2p}}$ $\displaystyle=$
$\displaystyle\hat{H}-\frac{\omega_{\mathrm{2p}}}{2},$ (3h) $\displaystyle=$
$\displaystyle\frac{\epsilon}{2}\hat{\sigma}_{z}+\omega_{\mathrm{2p}}\hat{a}^{\dagger}\hat{a}+g_{\mathrm{2p}}\hat{\sigma}_{x}\left(\left(\hat{a}^{\dagger}\right)^{2}+\hat{a}^{2}\right),$

with $\omega_{\mathrm{2p}}=\omega/2$, $g_{\mathrm{2p}}=g/2$.

###  3.2 Two-mode Rabi model

In the two-mode bosonic realization, the generators can be expressed as

$\displaystyle\hat{K}_{0}=\frac{1}{2}\left(\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}+1\right),\quad\hat{K}_{+}=\hat{a}^{\dagger}\hat{b}^{\dagger},\quad\hat{K}_{-}=\hat{a}\hat{b}.$
(3i)

The corresponding Bargmann index is
$k=\frac{1}{2},~{}1,~{}\frac{3}{2},~{}\dots$ Given the Fock states
$\left|{}n\right>_{s}$ ($s=a,b$) which satisfies
$\hat{s}^{\dagger}\hat{s}\left|{}n\right>_{s}=n\left|{}n\right>_{s}$, the
basis state $\left|{}k,m\right>$ can be rewritten as

$\displaystyle\left|{}k,m\right>=\left|{}m+2k-1\right>_{a}\otimes\left|{}m\right>_{b}.$
(3j)

Therefore, the Bargmann index are related with the number difference between
two modes.

One can obtain the two-mode Rabi model [27, 31, 36] by substituting
$\hat{K}_{0}$ and $\hat{K}_{\pm}$ in (3e) with those in (3i), which leads to

$\displaystyle\hat{H}_{\mathrm{2m}}$ $\displaystyle=$
$\displaystyle\hat{H}-\omega_{\mathrm{2m}}$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{2}\hat{\sigma}_{z}+\omega_{\mathrm{2m}}\left(\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}\right)+g_{\mathrm{2m}}\hat{\sigma}_{x}\left(\hat{a}^{\dagger}\hat{b}^{\dagger}+\hat{a}\hat{b}\right),$

with $\omega_{\mathrm{2m}}=\omega/2$ and $g_{\mathrm{2m}}=g$.

###  3.3 Intensity-dependent Rabi model

In the Holstein-Primakoff realization, the generators can be expressed as

$\displaystyle\hat{K}_{0}=\hat{a}^{\dagger}\hat{a}+k,\quad\hat{K}_{+}=\sqrt{\hat{a}^{\dagger}\hat{a}+2k-1}\hat{a}^{\dagger},\quad\hat{K}_{-}=\hat{a}\sqrt{\hat{a}^{\dagger}\hat{a}+2k-1},$
(3l)

where $k$ is the Bargmann index. In this case, the basis state
$\left|{}k,m\right>$ is nothing but the Fock state, namely,
$\left|{}k,m\right>=\left|{}m\right>_{a}$.

One can obtain the intensity-dependent Rabi model [31, 37] by substituting
$\hat{K}_{0}$ and $\hat{K}_{\pm}$ in (3e) with those in (3l), which leads to

$\displaystyle\hat{H}_{\mathrm{I}}$ $\displaystyle=$
$\displaystyle\hat{H}-k\omega_{\mathrm{I}}$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{2}\hat{\sigma}_{z}+\omega_{\mathrm{I}}\hat{a}^{\dagger}\hat{a}+g_{\mathrm{I}}\hat{\sigma}_{x}\left(\sqrt{\hat{a}^{\dagger}\hat{a}+2k-1}\hat{a}^{\dagger}+\hat{a}\sqrt{\hat{a}^{\dagger}\hat{a}+2k-1}\right),$

with $\omega_{\mathrm{I}}=\omega$ and $g_{\mathrm{I}}=g$. The well-known Buck-
Sukumar model [29] is recovered if we choose $k=\frac{1}{2}$ and perform the
RWA.

##  4 Methods

Up to a global shift, three nonlinear Rabi models can be expressed in a united
form $\hat{H}$ (3e) in terms of the su(1,1) generators. Instead of solving
three nonlinear Rabi models individually, we only need to find the analytical
solutions to $\hat{H}$ itself.

In the basis state of $\\{\left|{}\pm\right>\\}$ which satisfy
$\hat{\sigma}_{x}\left|{}\pm\right>=\pm\left|{}\pm\right>$, Hamiltonian (3e)
can be written in a matrix form,

$\hat{H}=\left(\begin{array}[]{cc}\omega\hat{K}_{0}+g\left(\hat{K}_{+}+\hat{K}_{-}\right)&-\frac{\epsilon}{2}\\\
-\frac{\epsilon}{2}&\omega\hat{K}_{0}-g\left(\hat{K}_{+}+\hat{K}_{-}\right)\end{array}\right).$
(3n)

Since diagonal elements only consist of the linear combination of the su(1,1)
generators, one can easily achieve its eigenstates and eigenvalues which
satisfy

$\left[\omega\hat{K}_{0}\pm
g\left(\hat{K}_{+}+\hat{K}_{-}\right)\right]\left|{}k,m\right>_{\pm}=\beta\left(k+m\right)\left|{}k,m\right>_{\pm},$
(3o)

with $\beta=\sqrt{\omega^{2}-4g^{2}}$,
$r=\mathrm{arctanh}\left(\frac{2g}{\omega}\right)$ and

$\displaystyle\left|{}k,m\right>_{\pm}$ $\displaystyle=$
$\displaystyle\hat{S}(\mp r)\left|{}k,m\right>,$ (3p)
$\displaystyle\hat{S}(\mp r)$ $\displaystyle=$
$\displaystyle\exp\left[\mp\frac{r}{2}\left(\hat{K}_{+}-\hat{K}_{-}\right)\right].$
(3q)

For the one-mode and two-mode realizations, $\hat{S}(\mp r)$ corresponds to
the well-known Bogoliubov transformation or squeezing operator [2]. It should
be noted that when $g\rightarrow\omega/2$, $\beta$ tends to zero, which leads
to spectral collapse [15, 24, 26, 28, 31]. Beyond the spectral collapse point
($g>\omega/2$), the nonlinear Rabi model becomes no longer self-adjoint [28].
Therefore, we only focus on $0<g<\omega/2$ in this paper.

To achieve the eigenstates of $\hat{H}$ (3n), one can construct an ansatz
which is written as a superpostion of $\\{\left|{}k,m\right>_{+}\\}$, namely,

$\displaystyle\left|{}\psi\right>=\sum_{m=0}^{+\infty}\left(\begin{array}[]{c}c_{m}\left|{}k,m\right>_{+}\\\
d_{m}\left|{}k,m\right>_{+}\end{array}\right).$ (3t)

From the Schrödinger equation
$\hat{H}\left|{}\psi\right>=E\left|{}\psi\right>$, the expansion coefficients
$c_{m}$ and $d_{m}$ should satisfy

$\displaystyle\beta\left(k+m\right)c_{m}-\frac{\epsilon}{2}d_{m}=Ec_{m},$ (3u)
$\displaystyle-\frac{\epsilon}{2}c_{m}+\beta\left[\cosh
2r\left(k+m\right)d_{m}-\frac{\sinh
2r}{2}\left(\sqrt{m(m+2k-1)}d_{m-1}\right.\right.$
$\displaystyle\left.\left.+\sqrt{(m+1)(m+2k)}d_{m+1}\right)\right]=Ed_{m}.$
(3v)

Equation (3u) leads to

$\displaystyle c_{m}=\frac{\epsilon/2}{\beta\left(k+m\right)-E}d_{m}.$ (3w)

Substituting the above equation into (3v), we can obtain a three-term
recurrence relation for $d_{m}$, namely,

$\displaystyle d_{m+1}$ $\displaystyle=$ $\displaystyle
T_{m}d_{m}-R_{m-1}d_{m-1}.$ (3x)

with

$\displaystyle T_{m}$ $\displaystyle=$ $\displaystyle\frac{2\left(\beta\cosh
2r\left(k+m\right)-\frac{\epsilon^{2}/4}{\beta\left(k+m\right)-E}-E\right)}{\beta\sinh
2r\sqrt{(m+1)(m+2k)}},$ (3y) $\displaystyle R_{m}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(m+1)(m+2k)}{(m+2)(m+2k+1)}}.$ (3z)

In the limit of $m\rightarrow+\infty$, we find that

$\displaystyle\lim_{m\rightarrow+\infty}\frac{d_{m+1}}{d_{m}}=\left\\{\begin{array}[]{c}\tanh
r=\frac{2g}{\omega}<1\\\ \\\ \coth r=\frac{\omega}{2g}>1\end{array}\right..$
(3ad)

Therefore, equation (3x) has two linearly independent solutions $d_{m}^{(0)}$
and $d_{m}^{(1)}$ with the following limit behaviors respectively,

$\displaystyle\lim_{m\rightarrow+\infty}\frac{d_{m+1}^{(0)}}{d_{m}^{(0)}}=\frac{2g}{\omega},\quad\lim_{m\rightarrow+\infty}\frac{d_{m+1}^{(1)}}{d_{m}^{(1)}}=\frac{\omega}{2g}.$
(3ae)

Generally, $d_{m}$ can be written as

$\displaystyle d_{m}=A^{(0)}d_{m}^{(0)}+A^{(1)}d_{m}^{(1)}.$ (3af)

Due to $0<g<\omega/2$, we expect that the eigenstates correspond to
$A^{(1)}=0$, otherwise it will lead to divergence.

###  4.1 Exact isolated solutions

Special attention should be paid to (3u). When the energy satisfies
$E=\beta\left(k+M\right)$ with $M=1,2,3,\dots$, the expansion coefficient
$c_{m=M}$ and the corresponding wavefunction will not diverge only if
$d_{m=M}=0$. $E=\beta\left(k+M\right)$ is called the baseline energy [38, 39].
From the three-term recurrence relation (3x), $d_{M}=0$ corresponds to

$\displaystyle\left|\begin{array}[]{ccccc}-T_{0}&1\\\ R_{0}&-T_{1}&1\\\
&R_{1}&-T_{2}&1\\\ &&\ddots&\ddots&\ddots\\\
&&&R_{M-2}&-T_{M-1}\end{array}\right|=0,$ (3al)

which gives a relation between the system parameters $\epsilon$, $\omega$ and
$g$. When this relation is satisfied, $E=\beta\left(k+M\right)$ is the
eigenenergy, also known as the exact isolated solutions. The eigenstates
corresponding to the exact isolated solutions can be reduced to a closed form
as follows,

$\displaystyle\left|{}\psi\right>=\left(\begin{array}[]{c}\sum_{m=0}^{M}c_{m}\left|{}k,m\right>_{+}\\\
\sum_{m=0}^{M-1}d_{m}\left|{}k,m\right>_{+}\end{array}\right),$ (3ao)

where $d_{m<M}$ and $c_{m<M}$ are determined by (3x) and (3w) respectively.
$c_{M}$ is determined by (3v) with $d_{M}=d_{M+1}=0$, namely,

$\displaystyle-\frac{\epsilon}{2}c_{M}-\beta\frac{\sinh
2r}{2}\sqrt{M(M+2k-1)}d_{M-1}=0.$ (3ap)

Such kinds of exact isolated solutions were first discovered by Judd in the
Jahn-Teller model [38], which correspond to the level crossings in the energy
spectrum. Soon after that, Reik et alfound them in the linear Rabi model [39].
The exact isolated solutions for two-photon and two-mode Rabi models were also
brought into the spotlight [25, 27, 26, 28]. A new type of exact isolated
solutions, also known as the dark-like state, were found when generalizing it
to the multi-qubit cases [40, 41, 42]. However, less attention has been paid
to those in the intensity-dependent Rabi model.

Table 1:  Exact isolated solutions for $k=\frac{1}{4}$ and $\frac{1}{2}$ with
$\epsilon=\omega=1$. The energy is located at the baseline
$E=\beta\left(k+M\right)$, while the corresponding coupling strength $g$ is
determined by (3al).

  * | $k=\frac{1}{4}$ | $k=\frac{1}{2}$
---|---|---
| $g$ | $E$ | $g$ | $E$
$M=1$ | 0.353 553 390 6 | 0.883 883 476 5 | 0.306 186 217 8 | 1.185 854 122 6
$M=2$ | 0.220 400 240 2 | 2.019 611 501 3 | 0.199 407 656 4 | 2.292 578 169 8
$M=2$ | 0.454 731 653 8 | 0.935 514 425 9 | 0.429 179 356 3 | 1.282 625 043 5
$M=3$ | 0.156 833 678 1 | 3.085 982 030 1 | 0.145 778 939 2 | 3.347 936 161 9
$M=3$ | 0.362 621 090 4 | 2.237 605 006 9 | 0.341 905 645 5 | 2.553 806 169 2
$M=3$ | 0.478 267 278 3 | 0.947 761 754 5 | 0.463 652 920 3 | 1.310 065 840 3

Table 1 gives the exact isolated solutions for $k=\frac{1}{4}$ and
$\frac{1}{2}$, where we have fixed $\epsilon=\omega=1$. The coupling strength
$g$ corresponding to $E=\beta(k+M)$ is determined by (3al). When
$k=\frac{1}{4}$, it corresponds to the two-photon Rabi model. We exactly
reproduce the exact isolated solutions in the two-photon Rabi model presented
in [25]. The results for $k=\frac{1}{2}$ is associated with the intensity-
dependent Rabi model or the Buck-Sukumar model, which is first discovered to
the best of our knowledge.

###  4.2 G-functions

Beyond the exact isolated solutions, one can set $d_{0}=1$ in general, while
$d_{m>0}$ as a function of $E$ can be achieved from the three-term recurrence
relation (3x) successively. Due to the parity symmetry, the eigenstates of
$\hat{H}$ should also be those of the parity operator $\hat{\Pi}$, namely,
$\hat{\Pi}\left|{}\psi\right>=\Pi\left|{}\psi\right>$. The left-hand side
corresponds to

$\displaystyle\hat{\Pi}\left|{}\psi\right>$ $\displaystyle=$
$\displaystyle-\hat{\sigma}_{z}\otimes\hat{T}\sum_{m=0}^{+\infty}\left(\begin{array}[]{c}c_{m}\left|{}k,m\right>_{+}\\\
d_{m}\left|{}k,m\right>_{+}\end{array}\right)=\sum_{m=0}^{+\infty}\left(\begin{array}[]{c}d_{m}(-1)^{m}\left|{}k,m\right>_{-}\\\
c_{m}(-1)^{m}\left|{}k,m\right>_{-}\end{array}\right),$ (3au)

while the right-hand side corresponds to

$\displaystyle\Pi\left|{}\psi\right>$ $\displaystyle=$
$\displaystyle\Pi\sum_{m=0}^{+\infty}\left(\begin{array}[]{c}c_{m}\left|{}k,m\right>_{+}\\\
d_{m}\left|{}k,m\right>_{+}\end{array}\right).$ (3ax)

Therefore,

$\displaystyle\sum_{m=0}^{+\infty}d_{m}(-1)^{m}\left|{}k,m\right>_{-}$
$\displaystyle=$
$\displaystyle\Pi\sum_{m=0}^{+\infty}c_{m}\left|{}k,m\right>_{+},$ (3ay)
$\displaystyle\sum_{m=0}^{+\infty}c_{m}(-1)^{m}\left|{}k,m\right>_{-}$
$\displaystyle=$
$\displaystyle\Pi\sum_{m=0}^{+\infty}d_{m}\left|{}k,m\right>_{+}.$ (3az)

It should be noted that (3ay) and (3az) are equivalent, since they can be
transformed to each other by the unitary transformation $\hat{T}$. Note that
[34, 35]

$\displaystyle\langle{}k,0|{}k,m\rangle_{+}$ $\displaystyle=$
$\displaystyle(-1)^{m}\langle{}k,0|{}k,m\rangle_{-}$ $\displaystyle=$
$\displaystyle\left(1-\xi^{2}\right)^{k}\sqrt{\frac{\Gamma(2k+m)}{m!\Gamma(2k)}}\xi^{m},$

with $\xi=\tanh\frac{r}{2}$. Projecting (3ay) onto $\left|{}k,0\right>$, we
achieve the G-function as follows,

$\displaystyle G_{k}^{\Pi}(E)$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{+\infty}d_{m}(-1)^{m}\langle{}k,0|{}k,m\rangle_{-}-\Pi
c_{m}\langle{}k,0|{}k,m\rangle_{+}$ (3bb) $\displaystyle=$
$\displaystyle\sum_{m=0}^{+\infty}\Xi_{k,m}^{\Pi}(E)\xi^{m},$

with

$\displaystyle\Xi_{k,m}^{\Pi}(E)=\left(d_{m}-\Pi
c_{m}\right)\left(1-\xi^{2}\right)^{k}\sqrt{\frac{\Gamma(2k+m)}{m!\Gamma(2k)}},$
(3bc)

where we have employed (4.2). $G_{k}^{\Pi}(E)$ can be regarded as a power
series in $\xi$. From (3ad), we can find that

$\displaystyle\lim_{m\rightarrow+\infty}\frac{\Xi_{k,m+1}^{\Pi}}{\Xi_{k,m}^{\Pi}}$
$\displaystyle\leq$ $\displaystyle\coth r,$ (3bd)

which indicates that the radius of convergence satisfies $R\geq\tanh r$.
Therefore, $G_{k}^{\Pi}(E)$ is well-defined and will always converge due to
$\xi<R$. The roots of $G_{k}^{\Pi}(E)=0$ determine the eigenenergies, with
which we can obtain the eigenstates according to (3w) and (3x).

The G-functions of the two-photon and two-mode Rabi models [27, 26, 28] have
been analyzed separately, whereas few studies focused on the intensity-
dependent Rabi model. In this paper, three nonlinear Rabi models can be
described by a unified G-function (3bb), and one only need to keep in mind
that they correspond to different realizations of su(1,1) algebra.

##  5 Results and discussions

Figure 1: Energy spectrum as a function of the coupling strength $g$ at
$\epsilon=\omega=1$ for (a) $k=\frac{1}{4}$ and (b) $k=\frac{1}{2}$. The blue
(red) solid lines correspond to the even (odd) parity. The dashed lines refer
to the baseline energies. Different symbols are related with the exact
isolated solutions given in Table 1. The circle, square and triangle
correspond to the exact isolated solutions with $M=1$, $2$ and $3$
respectively.

The energy spectra for $k=\frac{1}{4}$ and $\frac{1}{2}$ are illustrated in
figure 1. The two-photon Rabi model is associated with $k=\frac{1}{4}$, as
shown in figure 1(a). It should be noted that the eigenenergies of the two-
photon Rabi model should be
$E_{\mathrm{2p}}=E-\frac{\omega_{\mathrm{2p}}}{2}$, as indicated in (3h).
Figure 1(b) depicts the energy spectrum at $k=\frac{1}{2}$, which is
associated with the two-mode and intensity-dependent Rabi models. The
eigenenergies of the two-mode and intensity-dependent Rabi models should be
$E_{\mathrm{2m}}=E-\omega_{\mathrm{2m}}$ and
$E_{\mathrm{I}}=E-k\omega_{\mathrm{I}}$ respectively, as indicated in (3.2)
and (3.3). Especially, different symbols located at the baseline correspond to
the exact isolated solutions given in Table 1, which can be achieved by
solving (3al). Clearly, the exact isolated solution corresponds to the level
crossing between even and odd parities, and its number on each baseline is
given by $M$.

Figure 2: $G_{k}^{\Pi}$ as function of $E$ for $\epsilon=\omega=1$, $g=0.4$,
and $k=\frac{1}{2}$. The blue (red) solid lines correspond to the even (odd)
parity. The dashed lines correspond to the baseline energies. The black
circles refer to the eigenvalues obtained from the numerical diagonalization.

Beyond the exact isolated solutions, the regular spectrum is determined by
finding the roots of the G-function. As an example, figure 2 shows the
G-function at $\epsilon=\omega=1$, $g=0.4$ and $k=\frac{1}{4}$. It can be used
to describe either the two-mode Rabi model with
$\omega_{\mathrm{2m}}=\omega/2=0.5$ and $g_{\mathrm{2m}}=g=0.4$, or the
intensity-dependent Rabi model with $\omega_{\mathrm{I}}=\omega=1$ and
$g_{\mathrm{I}}=g=0.4$. The eigenenergies obtained from the numerical
diagonalization are also depicted as a benchmark. The roots of
$G_{k}^{\Pi}(E)=0$ correspond to the eigenenergies, which fit well with those
obtained from the numerical diagonalization.

Figure 3: $\ln|d_{m}|$ for $\epsilon=\omega=1$ and $k=\frac{1}{2}$. The solid
lines are plotted as a benchmark, which correspond to the asymptotic behavior
$|d_{m}|\propto\exp\left(-\gamma_{\mathrm{d}}m\right)$ with
$\gamma_{\mathrm{d}}=\ln(\omega/2g)$. (a) The coupling strength is fixed at
$g=0.4$. Different $E$ are marked with different symbols, and the
corresponding $G_{k}^{\Pi}(E)$ are also shown in the inset. Note that
$E=0.38991138$ marked with green circle corresponds to the second black circle
from left to right in figure 2. (b) $\ln|d_{m}|$ for different coupling
strength $g$. $E$ is fixed at the corresponding lowest eigenenergy for
$\Pi=-1$.

Figure 3(a) shows the expansion coefficients $d_{m}$ of wavefunctions near the
eigenenergy, and the asymptotic behaviors are also depicted. The exact
eigenenergy corresponds to the root of the G-function, which is given in the
inset. As demonstrated in (3ad)-(3af), there exist two linearly independent
solutions for $d_{m}$ at $0<g<\omega/2$: in the limit of
$m\rightarrow+\infty$,
$|d_{m}^{(0)}|\propto\left(2g/\omega\right)^{m}=\exp\left(-\gamma_{\mathrm{d}}m\right)$,
$|d_{m}^{(1)}|\propto\left(\omega/2g\right)^{m}=\exp\left(\gamma_{\mathrm{d}}m\right)$,
with the decay rate $\gamma_{\mathrm{d}}=\ln(\omega/2g)$. The decay rate
$\gamma_{\mathrm{d}}$ only depends on $g$ and $\omega$ rather than $k$. Due to
$\gamma_{\mathrm{d}}>0$, $d_{m}^{(0)}$ tends to exponentially decay, whereas
$|d_{m}^{(1)}|$ tends to exponentially increase. As indicated by the red dots
in figure 3(a), when $E$ is far away from the eigenenergy, $d_{m}$ obtained
from (3x) is dominated by $d_{m}^{(1)}$. When $E$ is closer to the
eigenenergy, the weight of $d_{m}^{(0)}$ increases. For small $m$, $d_{m}$
shows an exponential decay behavior which is governed by $d_{m}^{(0)}$. On the
contrary, an exponential increase behavior emerges for large $m$ which is
governed by $d_{m}^{(1)}$. When $E$ is exactly the eigenenergy, we expect that
$d_{m}=d_{m}^{(0)}$ which leads to exponential decay of the expansion
coefficients of the wavefunction, as illustrated by the green circles in
figure 3(a).

The influences of the coupling strength $g$ on $d_{m}$ are shown in figure
3(b), where we have chosen $E$ the exact eigenenergies. The asymptotic
behavior in the limit of $m\rightarrow+\infty$ is well described by
$|d_{m}^{(0)}|\propto\exp\left(-\gamma_{\mathrm{d}}m\right)$. As indicated by
the blue crossing, the expansion coefficients decay very fast for weak
coupling strength, and one can easily obtain the convergent eigenstates.
Increasing the coupling strength $g/\omega$ leads to the decease of
$\gamma_{\mathrm{d}}$, which indicates that one need more bases to describe
the corresponding eigenstate. When the coupling strength tends to the spectral
collapse point $g\rightarrow\omega/2$, the decay rate $\gamma_{\mathrm{d}}$
tends to zero. Therefore, one can hardly describe the properties near the
spectral collapse point with a truncated Hilbert space.

##  6 Summary

In the last decades, exploring the strong and nonlinear coupling between light
and matter has achieved great processes. The interest in the nonlinear Rabi
models has blossomed both experimentally and theoretically. In this paper, we
focus on three typical nonlinear Rabi models: two-photon, two-mode and
intensity-dependent Rabi models, and propose a unified analytical approach.

Previous studies mainly dealt with three models individually, and their common
behaviors didn’t receive sufficient attention. By virtue of different
realizations of the su(1,1) Lie algebra, three models can be described by the
same Hamiltonian with $\mathcal{Z}_{2}$ symmetry. By choosing appropriate
basis states, we construct an ansatz to describe the eigenstates, whose
expansion coefficients satisfy a three-term recurrence relation. Of special
significance is the baseline energy identifying the exact isolated solutions
at the level crossings between different parities, for which the eigenstates
can be reduced to a closed form. We reproduce the exact isolated solutions in
the two-photon and two-mode Rabi models, whereas those in the intensity-
dependent Rabi models are first achieved.

Beyond the exact isolated solutions, we propose a unified G-function based on
the $\mathcal{Z}_{2}$ symmetry, whose roots give the regular spectrum. The
expansion coefficients of the eigenstates present an exponential decay
behavior in the limit of $m\rightarrow+\infty$, and the decay rate
$\gamma_{\mathrm{d}}$ can be achieved analytically. With increasing coupling
strength $g/\omega$, the decay rate $\gamma_{\mathrm{d}}$ decreases and it
tends to zero in the spectral collapse point $g\rightarrow\omega/2$.

With the unified analytical approach, we achieve the eigenstates and
eigenenergies of three nonlinear Rabi models, and their common behaviors are
addressed. The nonlinear Rabi model introduces new physical mechanisms which
cannot be captured by the linear one. The squeezing effect is inherent in the
nonlinear Rabi model, compared to the linear one which is obvious only if the
frequency of the two-level system is large enough [43]. The exotic nonlinear
phenomena, the squeezing effect, and their applications in quantum information
deserve further consideration, which are left to future research.

We wish to thank Daniel Braak and Qing-Hu Chen for stimulating discussions.

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# A tight closure approach
to a result of G. Faltings

Tirdad Sharif  Tirdad Sharif, School of Mathematics, Institute for Studies in
Theoretical Physics and Mathematics, P. O. Box 19395-5746, Tehran, Iran
<EMAIL_ADDRESS>http://www.ipm.ac.ir/IPM/people/personalinfo.jsp?PeopleCode=IP0400060

###### Abstract.

Using a result of M. Hochster and C. Huneke on $F$-rational rings a criterion
for complete intersection rings of characteristic $p>0$ is presented. As an
application, we give a completely different proof for an algebraic result of
G. Faltings that was used by Taylor and Wiles in [9] for a simplification of
the proof of the minimal deformation problem.

###### Key words and phrases:

Complete Intersection Algebras, Deformation Algebras, Hecke Algebras, Tight
Closure

###### 2000 Mathematics Subject Classification:

13A35, 13H10

The author was supported by a grant from IPM, (No. 83130311).

##  1\. Introduction

The theory of tight closure was created by Melvin Hochster and Craig Huneke.
An important notion in the theory of tight closure is the notion of _$F$
-rationality_. There is a close connection between this notion and _rational
singularity_ , see [8].

The main purpose of this note is to use a result of M. Hochster and C. Huneke
in [3] on $F$-rational rings, to give a criterion for complete intersection
rings of characteristic $p>0$, see Theorem 2.5 for the precise statement.

In [9] G. Faltings proved an algebraic result that was used by Taylor and
Wiles to reprove the _minimal deformation problem_ by a simpler method. As an
application of Theorem 2.5 we prove Proposition 2.8, which is an extension of
Faltings result. Our approach to prove this proposition is fundamentally
different from Faltings method where he simplifies Taylor and Wiles argument,
see Remark 2.6.

In order to give a view for the reader we recall the minimal deformation
problem very briefly.

The modularity conjecture for semistable elliptic curves depends on a critical
conjecture of Wiles that was proved in [9, 10], see Conjecture (2.16) of [10].

Let $\mathcal{D}$ be a deformation theory and let $R_{\mathcal{D}}$ and
$T_{\mathcal{D}}$ be the universal deformation and Hecke algebras associated
to $\mathcal{D}$, respectively. The universal property of $R_{\mathcal{D}}$
implies that there is a homomorphism
$\varphi_{\mathcal{D}}:R_{\mathcal{D}}\rightarrow T_{\mathcal{D}}$. Wiles
conjecture asserts that $\varphi_{\mathcal{D}}$ is an isomorphism. In the
special case, if $\mathcal{D}$ is a minimal deformation theory, then the above
conjecture is called the minimal deformation problem. For different types of
deformation theories and their relations see Chapter 1 and 2 of [10].

Wiles method in [10] to prove the above conjecture, was closely related to
show that certain Hecke algebras are complete intersection that was shown in
[9].

There is no need for the reader to know the number theoretic materials of
Wiles proof in detail. We refer the interested readers to [6] for an elegant
exposition of the above matters.

##  2\. Definitions, Notations and The Main Theorem

Throughout this note all rings are commutative and Noetherian of
characteristic $p>0$ and all of modules are finite (that is, finitely
generated).

In this note we use the notations $\nu_{R}(M)$ and $\ell_{R}(M)$ respectively
for the minimal number of generators of $M$ and the length of $M$ over $R$.

######  Definition 2.1.

Let $(R,\mathfrak{m},k)$ be a local ring and let $R[[X]]=R[[X_{1},...,X_{n}]]$
be the formal power series of $n$ variables over $R$. Write $A=R[[X]]/J$ for
an ideal $J$ of $R[[X]]$. If the local ring $A\otimes_{R}k$ is complete
intersection, then $A$ is called a complete intersection $R$-algebra.

######  Definition 2.2.

Let $R$ be a ring and let $J=(a_{1},...,a_{t})$ be an ideal of $R$. An element
$x\in R$ is said to be in the tight closure of $J$ and write $x\in J^{*}$ if
there is an element $c\in R^{0}$ such that for all large $q=p^{n}$ we have
$cx^{q}\in J^{[q]}$, wherein $R^{0}$ is the complement of the union of all
minimal primes of $R$ and $J^{[q]}=(a_{1}^{q},...,a_{t}^{q})$.

An ideal $J$ is called tightly closed if $J=J^{*}$. The ring $R$ is called
$F$-rational if every parameter ideal of $R$ is tightly closed and in
particular, if every ideal of $R$ is tightly closed, then it is called
$F$-regular.

######  Fact 2.3.

Let $Q$ be a regular local ring, then it is $F$-regular, see [2, (4.4)].

To state the main theorem of this note we will make use of the following
simple lemma the proof of which is omitted.

######  Lemma 2.4.

Let I be an ideal of R, then $(I^{*})^{n}\subseteq(I^{n})^{*}$ for $n\geqslant
1$.

######  Theorem 2.5.

Let $(A,\mathfrak{n},k)$ be a local ring of dimension $t$ and let
$\mathfrak{q}$ be an ideal of $A$, generated by a system of parameters. Let
$\eta\colon A\to Q$ and $\theta\colon Q\to B$ be local ring homomorphisms such
that $Q$ is regular of dimension $t$, $\theta$ is surjective, and $\eta$
induces an isomorphism between the residue fields of $A$ and $Q$. Set
$\delta=\theta\eta\colon A\to B$ and $\ell(B/\mathfrak{q}^{*}B)=d$. Assume for
an integer $q=p^{n}>d^{t}t^{t-1}$ we have
$\ell(B/{\mathfrak{q}}^{[q]}B)\geqslant\ell(A/{\mathfrak{q}}^{[q]})$. If the
ideal ${\mathfrak{q}}^{[q]}$ is tightly closed, then the canonical
homomorphism $\pi:Q/q^{*}Q\rightarrow B/\mathfrak{q}^{*}B$ is an isomorphism.

In particular, if $A$ is equidimensional homomorphic image of a Cohen-Macaulay
ring, then $\pi$ is an isomorphism between complete intersection rings.

###### Proof.

Write $\mathfrak{m}$ for the maximal ideal of $Q$ and
$J=\operatorname{Ker}\theta.$ We may assume that $B=Q/J$. From the assumptions
it follows that $\mathfrak{m}^{d}\subseteq J+{\mathfrak{q}}^{*}Q.$ Therefore
we have the next inclusions
$\mathfrak{m}^{tqd}\subseteq(J+{\mathfrak{q}}^{*}Q)^{tq}\subseteq(J+(\mathfrak{q}^{*})^{tq}Q)$.
Lemma 2.4 implies that $\mathfrak{m}^{tqd}\subseteq
J+(\mathfrak{q}^{tq})^{*}Q$. Since $\mathfrak{q}$ is generated by $t$ elements
it is easy to see that ${\mathfrak{q}}^{tq}\subseteq{\mathfrak{q}}^{[q]}$.
This yields that $\mathfrak{m}^{tqd}\subseteq J+(\mathfrak{q}^{[q]})^{*}Q$ and
from our assumptions we have $\mathfrak{m}^{tqd}\subseteq
J+\mathfrak{q}^{[q]}Q$.

We claim that $J\subseteq\mathfrak{m}^{d+1}$, otherwise choose $u$ in $J$ not
in $\mathfrak{m}^{d+1}$ and consider the following exact sequence

$0\longrightarrow\operatorname{Ker}(\alpha)\longrightarrow
Q/\mathfrak{m}^{c}\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}Q/\mathfrak{m}^{c}\longrightarrow\operatorname{Coker}(\alpha)\longrightarrow
0\hskip 72.26999pt(2.5.1)$

wherein the map $\alpha$ is the homothety by element $u$ and $c=tdq$.
Obviously, there is a surjective map as the following

$\operatorname{Coker}(\alpha)=Q/(uQ+\mathfrak{m}^{c})\longrightarrow
Q/(J+{\mathfrak{q}}^{[q]}Q)=B/\mathfrak{q}^{[q]}B.$

Therefore
$\ell_{Q}(\operatorname{Coker}(\alpha))\geqslant\ell_{Q}(B/{\mathfrak{q}}^{[q]}B)$.
It is clear that $B/{\mathfrak{q}}^{[q]}B$ has finite length over $Q$. The
ideal $\mathfrak{q}$ is an $\mathfrak{n}$-primary ideal of $A$, therefore
$\ell_{A}(B/{\mathfrak{q}}^{[q]}B)$ is finite and since
$k=A/\mathfrak{n}\simeq Q/\mathfrak{m}$, clearly the length of
$B/{\mathfrak{q}}^{[q]}B$ over both of $Q$ and $A$ are equal. By assumptions
$\ell(B/{\mathfrak{q}}^{[q]}B)\geqslant\ell(A/{\mathfrak{q}}^{[q]})$,
therefore we have
$\ell_{Q}(\operatorname{Coker}(\alpha))\geqslant\ell(B/{\mathfrak{q}}^{[q]}B)\geqslant\ell(A/{\mathfrak{q}}^{[q]})$
and from [5, (14.10)] we find that $\ell(A/\mathfrak{q}^{[q]})\geqslant
q^{t}e(A)\geqslant q^{t}$ wherein the symbol $e(A)$ is the multiplicity of
$A$. Thus $\ell_{Q}(\operatorname{Coker}(\alpha))\geqslant q^{t}$.

Let $x_{1},x_{2},...,x_{t}$ be a minimal generator for $\mathfrak{m}.$ The
local ring $Q$ is regular, so the associated graded ring
$gr_{\mathfrak{m}}(Q)$ is a polynomial ring in $t$ variables over $k$, see [7,
Page 76,Theorem 9(d)]. This fact implies that $x_{j}$ for $1\leqslant
j\leqslant t$ are analytically independent in $\mathfrak{m}$ and so that
$\nu_{Q}(\mathfrak{m}^{c-i})={c-i+t-1\choose t-1}$ for $i\geqslant 1$. Since
$u$ is not an element of $\mathfrak{m}^{d+1}$ therefore
$\operatorname{Ker}(\alpha)\subseteq\mathfrak{m}^{c-d}/\mathfrak{m}^{c}$ and
hence
$\ell_{Q}(\operatorname{Ker}(\alpha))\leqslant\ell_{Q}(\mathfrak{m}^{c-d}/\mathfrak{m}^{c})$.

The following equality is clear

$\ell_{Q}(\mathfrak{m}^{c-d}/\mathfrak{m}^{c})=\displaystyle{\sum_{i=1}^{d}}\nu_{Q}(\mathfrak{m}^{c-i}).\hskip
72.26999pt(2.5.2)$

Now from (2.5.1) we have
$\ell_{Q}(\operatorname{Ker}\alpha)=\ell_{Q}(\operatorname{Coker}\alpha)$ and
from (2.5.2) we find that
$\ell_{Q}(\mathfrak{m}^{c-d}/\mathfrak{m}^{c})\leqslant d{c+t-2\choose
t-1}\leqslant dc^{t-1}=d(tdq)^{t-1}.$ Therefore $q^{t}\leqslant
d^{t}t^{t-1}q^{t-1}$ and this contradicts the choice of $q$. Hence we must
have $J\subseteq\mathfrak{m}^{d+1}$ and consequently
$\mathfrak{m}^{d}\subseteq
J+{\mathfrak{q}}^{*}Q\subseteq\mathfrak{m}^{d+1}+{\mathfrak{q}}^{*}Q.$ Thus
$\mathfrak{m}^{d}+{\mathfrak{q}}^{*}Q=\mathfrak{m}^{d+1}+{\mathfrak{q}}^{*}Q$
and by using Nakayama’s Lemma
$\mathfrak{m}^{d+1}\subseteq{\mathfrak{q}}^{*}Q$. This yields that
$J\subseteq{\mathfrak{q}}^{*}Q$ and this shows that the canonical homomorphism
$\pi:Q/{\mathfrak{q}}^{*}Q\rightarrow B/{\mathfrak{q}}^{*}B$ is an
isomorphism.

Now let $A$ be an equidimensional ring which is homomorphic image of a Cohen-
Macaulay ring. In this case, since ${\mathfrak{q}}^{[q]}$ is generated by a
system of parameters and it is tightly closed it follows from [4, (4.2.e)]
that $A$ is $F$-rational and hence $\mathfrak{q}={\mathfrak{q}}^{*}$. Now it
is easy to see that $\pi$ is an isomorphism between complete intersection
rings. ∎

######  Remark 2.6.

Let $(\mathcal{O},\mathfrak{m},k)$ be a complete discrete valuation ring with
finite residue field $k$ and let $\mathcal{D}$ be a minimal deformation
theory. Let $R$ and $T$ be the universal deformation and Hecke
$\mathcal{O}$-algebras associated to $\mathcal{D}$, respectively and let
$\varphi=\varphi_{\mathcal{D}}$ be the homomorphism described in Section 1.
Taylor and Wiles in the appendix of [9] for a simplification of some arguments
in Section 3 of [9] and Chapter 3 of [10] use a commutative algebraic result
of G. Faltings to reprove that $\varphi$ is an isomorphism between complete
intersection algebras. In Proposition 2.8 and Corollary 2.11 we give an
extension of corresponding results due to Taylor-Wiles and Faltings without
the assumption of finiteness of $k$. Our method to prove the above results is
totally different from their methods.

In the following we refine the definition of a _(level) n-structure_ due to
Wiles and Faltings in [9].

######  Definition 2.7.

Let $(\mathcal{O},\mathfrak{m},k)$ be a local ring and let
$\mathcal{O}[[Y]]=\mathcal{O}[[Y_{1},...,Y_{t}]]$. Set
$\mathfrak{q}=(Y_{1},...,Y_{t})$ and assume that for an integer $n\geqslant 1$
we have a commutative diagram of local $\mathcal{O}$-algebras as the following
in which $\varphi$ is surjective and $T$ is a finite free
$\mathcal{O}$-module.

$\begin{array}[]{llll}\mathcal{O}[[Y]]\stackrel{{\scriptstyle\psi_{n}}}{{\longrightarrow}}&R_{n}&\stackrel{{\scriptstyle\varphi_{n}}}{{\longrightarrow}}&T_{n}\\\
&\downarrow&&\downarrow\\\
&R&\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}&T\\\ \end{array}$

By a level $n$-structure we mean the above diagram with the following
properties:

$(1)$ There is a surjective homomorphism
$\lambda_{n}:\mathcal{O}[[X_{1},X_{2},...,X_{t}]]\longrightarrow R_{n}$.

$(2)$ The ring homomorphism $R_{n}\longrightarrow T_{n}$ is surjective.

$(3)$ From the vertical homomorphisms we get the isomorphisms
$R_{n}/{\mathfrak{q}}R_{n}\rightarrow R$ and
$T_{n}/{\mathfrak{q}}T_{n}\rightarrow T$.

$(4)$ The ring $T_{n}/{\mathfrak{q}^{[p^{n}]}}T_{n}$ is finite and free as a
module over $\mathcal{O}[[Y]]/{\mathfrak{q}^{[p^{n}]}}$.

For convenience we represent a level $n$-structure by the notation
$L_{n}(\mathcal{O},\varphi,\varphi_{n},\psi_{n},\lambda_{n})$.

######  Proposition 2.8.

Let $m\geqslant 1$ be an integer such that for every $n\geqslant m$, the level
$n$-structure $L_{n}(\mathcal{O},\varphi,\varphi_{n},\psi_{n},\lambda_{n})$
has the following additional property:

$(5)$ For each $x\in\mathfrak{m}$ we have $\psi_{n}(x)\in\mathfrak{m}R_{n}$.

Then $\varphi$ is an isomorphism between complete intersection
$\mathcal{O}$-algebras.

###### Proof.

Set $\widehat{\square}=\square\otimes_{\mathcal{O}}k$,
$\mathfrak{q}_{n}=\mathfrak{q}^{[p^{n}]}$ and
$\mathcal{O}[[X]]=\mathcal{O}[[X_{1},X_{2},...,X_{t}]]$. From (1) there is a
surjective homomorphism $k[[X]]\rightarrow\widehat{R_{n}}$. The fifth property
implies that $\psi_{n}(\mathfrak{m}[[Y]])\subseteq\mathfrak{m}R_{n}$. Thus
$\psi_{n}$ induces a local homomorphism
$\widehat{\psi_{n}}:k[[Y]]\longrightarrow\widehat{R_{n}}$. Since $k[[X]]$ is a
complete local ring, using [1, (7.16)] we can lift the homomorphism
$k[[Y]]\rightarrow\widehat{R_{n}}$ to a unique homomorphism $k[[Y]]\rightarrow
k[[X]]$. Thus from (1), (2) and (3) we find that there is a surjective
homomorphism as the following

$\alpha:k[[X]]/{\mathfrak{q}}k[[X]]\longrightarrow\widehat{T_{n}}/{\mathfrak{q}}\widehat{T_{n}}\simeq\widehat{T}.$

From (3) it follows that $\widehat{T_{n}}/{\mathfrak{q}}\widehat{T_{n}}$ is a
finite vector space over $k$. We may assume that the length of
$\widehat{T_{n}}/{\mathfrak{q}}\widehat{T_{n}}$ over $k[[X]]$ is equal $d$.
Now choose $n\geqslant m$ such that $p^{n}>t^{t-1}d^{t}$. From the fourth
property it is clear that the length of
$\widehat{T_{n}}/{\mathfrak{q}_{n}}{\widehat{T_{n}}}$ over $k[[Y]]$ is equal
or greater than the length of $k[[Y]]/{\mathfrak{q}_{n}}k[[Y]]$.

Now by writing $A=k[[Y]]$, $Q=k[[X]]$ and $B=\widehat{T_{n}}$ from Theorem 2.5
and Fact 2.3 we get that $\alpha$ is an isomorphism between complete
intersection rings.

On the other hand, $\varphi_{n}$ is surjective, from this and the third
property we have the following surjective homomorphisms

$\beta:k[[X]]/{\mathfrak{q}}k[[X]]\longrightarrow\widehat{R_{n}}/{\mathfrak{q}}\widehat{R_{n}}$
$\gamma:\widehat{R_{n}}/{\mathfrak{q}}\widehat{R_{n}}\longrightarrow\widehat{T_{n}}/{\mathfrak{q}}\widehat{T_{n}}$

such that $\alpha=\gamma\beta$. It was shown that $\alpha$ is an isomorphism
thus from (3) it is clear that $\widehat{\varphi}$ is an isomorphism between
$\widehat{R}$ and $\widehat{T}$ as complete intersection rings. Since $T$ is a
finite free module and $\varphi$ is surjective, Nakayama’s Lemma implies that
$\varphi$ is an isomorphism. It is obvious that the local ring $T$ is a
homomorphic image of $\mathcal{O}[[X]]$. Since $\widehat{T}$ is a complete
intersection ring from Definition 2.1 it follows that $T$ and so $R$ are
complete intersection $\mathcal{O}$-algebras. ∎

######  Remark 2.9.

In the proof of the above proposition the fifth property enables us to reduce
the level $n$-structure
$L_{n}(\mathcal{O},\varphi,\varphi_{n},\psi_{n},\lambda_{n})$ to the level
$n$-structure
$L_{n}(k,\widehat{\varphi},\widehat{\varphi_{n}},\widehat{\psi_{n}},\widehat{\lambda_{n}})$
and then by lifting property which is based on [1, (7.16)] we can prove the
proposition as a simple corollary of our main theorem. If we assume that
$\mathcal{O}$ is $\mathfrak{m}$-adic complete, then $\mathcal{O}[[X]]$ is
$(\mathfrak{m}[[X]]+(X))$-adic complete. From (1) we have a surjective
homomorphism $\mathcal{O}[[X]]\longrightarrow R_{n}$. Now again by using [1,
(7.16)] we can lift the local homomorphism $\psi_{n}$ to a unique local
homomorphism $\mathcal{O}[[Y]]\longrightarrow\mathcal{O}[[X]]$. Now by a
similar method to our proof we can show that $\varphi$ is an isomorphism
between complete intersection $\mathcal{O}$-algebras. In this case, we don’t
need to use the fifth property directly. However, in the following corollary
as a simple application of the level $n$-structures, we make clear the
relation between the completeness of $\mathcal{O}$ and the fifth property. In
a sense, we can say that the assumption of completeness of our base ring
implies the fifth property. We use the next simple lemma the proof of which is
omitted.

######  Lemma 2.10.

Let $L_{n}(\mathcal{O},\varphi,\varphi_{n},\psi_{n},\lambda_{n})$ be a level
$n$-structure, then there is a level $n$-structure
$L^{\prime}_{n}(\mathcal{O},\varphi,\varphi^{\prime}_{n},\psi^{\prime}_{n},\lambda^{\prime}_{n})$
such that $\lambda^{\prime}_{n}$ is the canonical epimorphism.

######  Corollary 2.11.

Let $(\mathcal{O},\mathfrak{m},k)$ be an $\mathfrak{m}$-adic complete local
ring, then $\varphi$ is an isomorphism between complete intersection
$\mathcal{O}$-algebras.

###### Proof.

From Lemma 2.10 it follows that there is a level $n$-structure such that the
homomorphism
$\lambda_{n}:\mathcal{O}[[X]]\longrightarrow\mathcal{O}[[X]]/J_{n}=R_{n}$ is a
canonical epimorphism. Therefore for each $g\in\mathcal{O}[[X]]$ we have
$\lambda_{n}(g)=g+J_{n}$, where $J_{n}=\operatorname{Ker}\lambda_{n}$. Since
$\mathcal{O}$ is $\mathfrak{m}$-adic complete hence $\mathcal{O}[[X]]$ is
$(\mathfrak{m}[[X]]+(X))$-adic complete. Thus from [1, (7.16)] we can lift
$\psi_{n}$ to a unique homomorphism
$\xi:\mathcal{O}[[Y]]\longrightarrow\mathcal{O}[[X]]$ such that $\xi(x)=x$ for
each $x\in\mathfrak{m}$. On the other hand, we have $\lambda_{n}\xi=\psi_{n}$.
Hence $\psi_{n}(x)=x+J_{n}$ and this shows that
$\psi_{n}(x)\in\mathfrak{m}R_{n}$. Now from Proposition 2.8 assertion holds. ∎

## Acknowledgments

The author is grateful to Sean Sather-Wagstaff and Irena Swanson for their
useful comments on this note.

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