Symmetry of asymmetric quantum Rabi models

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by  Masato Wakayama

Symmetry of asymmetric quantum Rabi models
Masato Wakayama
January 17, 2017
Abstract
7
1 The aim of this paper is a better understanding for the eigenstates of the asymmetric quantum
0 RabimodelbyLiealgebrarepresentationsofsl . Wedefineasecondorderelementoftheuniversal
2
2 enveloping algebra (sl ) of sl (R), which, through the action of a certain infinite dimensional
n representation of slU(R),2provid2es a picture of the asymmetric quantum Rabi model equivalent to
2
a
the one drawn by confluent Heun ordinary differential equations. Using this description, we prove
J
the existence of level crossingsin the spectral graphof the asymmetric quantum Rabi model when
4
the symmetry-breaking parameter ǫ is equal to 1, and conjecture a formula that ensures likewise
1 2
the presence of level crossings for general ǫ 1Z. This result on level crossings was demonstrated
∈ 2
] numerically by Li and Batchelor in 2015, investigating an earlier empirical observation by Braak
h
(2011). The first analysis of the degenerate spectrum was given for the symmetric quantum Rabi
p
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2
at partoftheexceptionalspectrumfromtheviewpointofinfinitedimensionalrepresentationsofsl2(R)
m having lowest weight vectors.
[ 2010 Mathematics Subject Classification: Primary 34L40, Secondary 81Q10,34M05, 81S05.

1 Keywords and phrases: quantumRabimodels,levelcrossings,confluentHeundifferentialequa-
v tions, exceptional spectrum, irreducible representations.
8
8
8
1 Introduction
3
0
. ThequantumRabimodel(QRM)is thefully quantizedversion[16]ofthe originalRabimodel(for
1
areviewsee[8])andisknowntobethesimplestmodelusedinquantumopticstodescribetheinter-
0
7 action of light and matter. As such, it appears ubiquitously in various quantum systems including
1 cavity and circuit quantum electrodynamics, quantum dots etc. (see e.g [9]). The Hamiltonian of
: the quantum Rabi model (~=1) reads
v
i
X H =ωa†a+∆σ +gσ (a†+a), (1.1)
Rabi z x
r
a
0 1 1 0
with a† =(x ∂ )/√2,a=(x+∂ )/√2(∂ := d ) and σ = ,σ = are the Pauli
− x x x dx x 1 0 z 0 1
(cid:20) (cid:21) (cid:20) − (cid:21)
matrices, 2∆ is the energy difference between the two levels, and g denotes the coupling strength
between the two-level system and the bosonic mode with frequency ω. It is well-known that the
Jaynes-Cummings model [16] (see [9] for a comprehensive overview and recent applications), the
RWA (rotating-wave approximation) of the quantum Rabi model, has a U(1)-symmetry and is
known to be integrable. Although the quantum Rabi model has no such U(1)-symmetry, it has
a Z -symmetry (parity). Using this Z -symmetry, D. Braak [3] has shown the integrability of
2 2
the quantum Rabi model in 2011. In the present paper, we study the spectrum of the following
asymmetricquantumRabimodel[34](called“generalized”quantumRabimodelin[3,19],“biased”
in [2] and “driven” in [21]) with broken Z -symmetry. This asymmetric model provides actually a
2
more realistic description of circuit QED experiments employing flux qubits than the QRM itself
[24]:
Hǫ =ωa†a+∆σ +gσ (a†+a)+ǫσ . (1.2)
Rabi z x x
1
2 M. Wakayama
Wesetω =1inthefollowing. Thereareseveralstudiesonthespectrumoftheasymmetricquantum
Rabi model [5, 7, 2, 19, 20]. Here we are particularly interested in the level crossingsvisible in the
spectral graph, i.e. the set of discrete eigenvalues of Hǫ as function of the coupling strength g.
Rabi
Thepresenceoflevelcrossingsintheasymmetricmodelishighlynon-trivialbecausetheadditional
term ǫσ breaks the Z -symmetry which couples the bosonic mode and the two-level system by
x 2
allowing spontaneous tunneling between the two atomic states. Although the Z -symmetry may
2
be recovered by “integrable embedding” [3], so that the same solution method as used for the
“symmetric” QRM can be applied to the asymmetric model as well [5], the broken Z -symmetry
2
removesalllevelcrossingsfromthespectralgraphforgeneralǫ,renderingthemodelnon-integrable
according to the criterion for quantum integrability proposed in [3]. Without the symmetry, there
seem to be no invariant subspaces anymore whose respective spectral graphs may intersect to
create “accidental” degeneracies in the spectrum for specific values of the coupling. Surprisingly,
thecrossingshavebeennumericallyconfirmedin[19]whentheparameterǫisahalf-integer,hinting
at a hidden symmetry present in this case.
In general, these level crossings(or degeneracies of eigenstates) have been studied also in pure
mathematics for different reasons [11, 12, 13]. Especially,the “gamma factor”(a terminology used
innumbertheory)ofthespectralzetafunction[30]forthequantumRabimodelmaycorrespondto
the so-calledexceptional spectrum of the model [3, 31]. It wouldbe very interesting if this gamma
factor (exceptionaleigenstates)could be interpretedas, e.g. a topologicalpropertyof the states as
in the case of the Selberg zeta functions for discontinuous groups (see, e.g [10]). Also, we expect
that representation theoretic treatment of the lever crossings developed here may help the future
study of monodoromy problems for the confluent Heun ODE.
In the following we shall employ the theory worked out previously [33]. Using this description,
we prove the presence of level crossings of the asymmetric quantum Rabi model for ǫ = 1. This
2
result has been demonstrated numerically by Li and Batchelor [19]. As a subsequent observation,
we give a representation theoretic description of such crossings for general ǫ 1Z by means of
finite dimensional representations of sl . Thus we may observe a certain rec∈ipr2ocity (or a Z -
2 2
symmetry) behind this structure. We leave the proof of the crossings for general ǫ 1Z and a
∈ 2
detailed mathematical investigation for future study but discuss briefly the non-degenerate part
of the exceptional spectrum of the model from the viewpoint of infinite dimensional irreducible
submodules(orsubquotients)ofthenon-unitaryprincipalseriessuchasthe(holomorphic)discrete
series representations of sl (R).
2
2 Confluent Heun’s picture of the model
In this section, we recall the confluent Heun picture of the asymmetric quantum Rabi model. We
may assume ω = 1 without loss of generality in the Hamiltonian Hǫ = Hǫ (g,∆,ω) of the
Rabi Rabi
asymmetricquantumRabimodel. Inwhatfollows,wewilluse theBargmannrepresentationofthe
boson operators [1]. Here a† and a are realized as the multiplication and differentiation operators
over the complex variable: a† = (x ∂ )/√2 z and a = (x+∂ )/√2 ∂ := d . These
− x → x → z dz
operators act on the Hilbert space of entire functions equipped with the inner product
B
1
(f g)= f(z)g(z)e−|z|2d(Re(z))d(Im(z)).
| π C
Z
In this Bargmann picture, the Hamiltonian
d d
Hǫ H˜ǫ =z I +g z+ σ +∆σ +ǫσ .
Rabi → Rabi dz dz x z x
(cid:0) (cid:1)
Therefore, by the standard procedure (see e.g. [5, 19]), we observe that the Schr¨odinger equation
Hǫ ϕ=λϕ is equivalent to the system of first order differential equations
Rabi
ψ (z)
H˜ǫ ψ =λψ, ψ = 1 .
Rabi ψ (z)
2
(cid:20) (cid:21)
Hence, in order to have an eigenstate of Hǫ , it is sufficient to obtain an eigenstate ψ , that
is, BI: (ψ ψ ) < , and BII: ψ are holoRmaboirphic everywhere in the whole complex pla∈nBe C for
i i i
| ∞
i=1,2.
Symmetry of asymmetric quantum Rabi models 3
Let W be the eigenspace of Hǫ , whence of H˜ǫ , corresponding to the eigenvalue λ. It is
λ Rabi Rabi
obvious that dimW 2. Upon writing f :=ψ ψ , we have
λ ± 1 2
≤ ±
(z+g) d f +(gz+ǫ λ)f +∆f =0,
dz + − + − (2.1)
((z−g)ddzf−−(gz+ǫ+λ)f−+∆f+ =0.
It is elementary to see that this system of first order equations is regular singular at z = g,
±
irregular singular at , and no other singularities. However, since the irregular singular point at
∞
has rank 1 [15], we observe that the asymptotic expansions of f (i = 1,2) for z reads
±
∞ → ∞
f (z)=eczzρ(c +c /z+ ) with some constants c ,ρ ,c ,c ,..., whence the condition BI
± 0 1 ± ± ±,1 ±,2
···
is always satisfied, because all functions growing like ecz for z satisfy BI. This fact implies
| |→∞
thattheanalyticityofψ,resp. f attheregularsingularpointsz = g,correspondingtocondition
±
±
BII, is sufficient for f . We leave the detailed discussion to [5].
±
∈B
Now, from these equations, we can get two sets of solutions for f (z) and f (z) as follows:
+ −
Substitute φ (z) := egzf (z) into the equations and eliminate φ (z) from these equations.
1,± ± 1,−
•
Further, put φ (x)=φ (z), where x:=(g+z)/2g. Then we have ǫ(λ)φ (x)=0, where
1 1,+ H1 1
d2 1 (λ+g2)+ǫ 1 (λ+g2+1) ǫ d
ǫ(λ):= + 4g2+ − + − −
H1 dx2 − x x 1 dx
n 4g2(λ+g2 ǫ)x+µ+4ǫg2 ǫ2 − o
+ − − .
x(x 1)
−
Substitute φ (z) := e−gzf (z) into the equations and eliminate φ (z) from these equations.
2,± ± 2,−
•
Further, put φ (x)=φ (z), where x:=(g z)/2g. Then we have ǫ(λ)φ (x)=0, where
2 2,+ − H2 2
d2 1 (λ+g2+1) ǫ 1 (λ+g2)+ǫ d
ǫ(λ):= + 4g2+ − − + −
H2 dx2 − x x 1 dx
n 4g2(λ+g2 1+ǫ)x+µ 4ǫg2 ǫ2 − o
+ − − − .
x(x 1)
−
Here the constant µ is defined by
µ:=(λ+g2)2 4g2(λ+g2) ∆2. (2.2)
− −
and contributes the accessory parameter of each equation ǫ(λ)φ (x) = 0 (j = 1,2) (e.g. [29]).
Hj j
Notice that these ǫ(λ)(j = 1,2) are the confluent Heun differential operators and give regular
Hj
singularities at x = 0,1 and irregular one at x = to the corresponding equations (the Heun
∞
pictureofasymmetricquantumRabimodels). Since the procedureofderivationofthese equations
for φ (x)(j =1,2) respectively is standard,we leave its details to the reader. We close the section
j
by observing the exponents of these equations.
Lemma2.1. Theexponentsρofeachregularsingularpointsfortheequations ǫ(λ)φ (x)=0(j =
Hj j
1,2) are respectively given as follows:
ǫ(λ) :ρ=0,λ+g2 ǫ(x=0), ρ=0,λ+g2+1+ǫ(x=1),
H1 −
ǫ(λ) :ρ=0,λ+g2+1+ǫ(x=0), ρ=0,λ+g2 ǫ(x=1).
H2 −
In particular, both exponents at x = 0,1 of ǫ(λ) (resp. ǫ(λ)) are integers iff one of two cases,
i.e. either λ+g2,ǫ Z or λ+g2,ǫ Z+ 1Hh1olds. H2
∈ ∈ 2
sl
3 Representations of
2
We recall briefly the representation theoretic setting for the Lie algebra sl in order to elucidate
2
the symmetry behind the asymmetric quantum Rabi model.
4 M. Wakayama
Let H,E and F be the standard generators of sl defined by
2
1 0 0 1 0 0
H = , E = , F = .
0 1 0 0 1 0
(cid:20) − (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
These satisfy the commutation relations
[H, E]=2E, [H, F]= 2F, [E, F]=H.
−
Let a C. Put V1 := x−14C[x,x−1] and V2 := x14C[x,x−1]. Consider the following algebraic
actions∈of sl on V , (j =1,2) defined by
2 j
1 1 1 1 1
̟ (H):=2x∂ + , ̟ (E):=x2∂ + (a+ )x, ̟ (F):= ∂ + (a )x−1.
a x a x a x
2 2 2 − 2 − 2
These operators indeed act on the spaces V , (j = 1,2) and define infinite dimensional repre-
j
sentations (non-unitary principal series representations) of sl2. Write ̟j,a = ̟a|Vj and put
e1,n :=xn−41, e2,n :=xn+41. Then we have
the spherical principal series: on V1,a :=V1 = n∈ZC e1,n
• ⊕ ·
̟ (H)e =2ne ,
1,a 1,n 1,n
̟ (E)e = n+ a e ,
 1,a 1,n 2 1,n+1
̟1,a(F)e1,n =(cid:0)−n+(cid:1)a2 e1,n−1.
and  (cid:0) (cid:1)
the non-spherical principal series: on V2,a :=V2 = n∈ZC e2,n
• ⊕ ·
̟ (H)e =(2n+1)e ,
2,a 2,n 2,n
̟ (E)e = n+ a+1 e ,
 2,a 2,n 2 2,n+1
̟2,a(F)e2,n =(cid:0)−n+ a(cid:1)−21 e2,n−1.
Note that (̟ ,V ) (resp. (̟ ,V ) is irred(cid:0)ucible when(cid:1) a 2Z (resp. a 2Z 1) and there
1,a 1 2,a 2
6∈ 6∈ −
is an equivalence between ̟ and ̟j,2 a under the same condition.
j,a
For a non-negative integer m, define su−bspaces D± ,F of V (=V ), and D± ,F
2m 2m−1 1,2m 1 2m+1 2m
of V (=V ) respectively by
2,2m+1 2
D± := C e , F := C e ,
2m · 1,±n 2m−1 · 1,n
n≥m −m+1≤n≤m−1
M M
D− := C e , D+ := C e , F := C e .
2m+1 · 2,−n 2m+1 · 2,n 2m · 2,n
n≥m+1 n≥m −m≤n≤m−1
M M M
The spaces D± (resp. D± ) are invariant under the action ̟ (X) (resp. ̟ (X))
2m 2m+1 1,2m 1,2m+1
(X sl ), anddefine irreducible representationsknownto be equivalentto (holomorphicand anti-
2
holo∈morphic)discreteseriesform>0ofsl (R)(seee.g. [22,14]). Moreover,thefinitedimensional
2
space Fm (dimCFm =m), is invariant and defines irreducible representationof sl2 for a=2 2m
−
when j =1 and a=1 2m when j =2, respectively.
−
D−2m F2m−1 D+2m
❏ ✡ ❏ ✡
V1,2m(m∈Z) : ❏ ✡ ❏ ✡
• • • • •
2m 2m+2 0 2m 2 2m
− − −
Fora Z,summarizingtheactions̟ ofsl ,werecallthefollowingirreducibledecomposition
j,a 2
of (̟ , V∈ )(a=m Z).
a j,a
∈
Lemma 3.1. Let m Z .
≥0
∈
Symmetry of asymmetric quantum Rabi models 5
D− F D+
2m+1 2m 2m+1
❏ ✡ ❏ ✡
V2,2m+1(m∈Z): ❏ ✡ ❏ ✡
• • • • •
2m 1 2m+1 1 2m 1 2m+1
− − − −
Figure 1: Weight decompositions of V for m Z
j,m
∈
1. The subspaces D± are irreducible submodules of V under the action ̟ and F
2m 1,2m 1,2m 2m−1
is an irreducible submodule of V under ̟ . In the former case, the finite dimen-
1,2−2m 1,2−2m
sional irreducible representation F can be obtained as the subquotient as V /D−
D+ = F . In the latter case,2mth−e1discrete series D± can be realized as the1,2imrredu2cmib⊕le
2m ∼ 2m−1 2m
components of the subquotient representation as V /F =D− D+ .
1,2−2m 2m−1 ∼ 2m⊕ 2m
2. ThesubspacesD± areirreduciblesubmoduleofV undertheaction̟ andF
2m+1 2,2m+1 2,2m+1 2m
is an irreducible submodule of V under ̟ . In the former case, the finite dimen-
2,1−2m 2,1−2m
sional irreducible representation F can beobtained as thesubquotientas V /D−
2m 2,2m+1 2m+1⊕
D+ =F , while in the latter case, the discrete series D± can be realized as the irre-
2m+1 ∼ 2m 2m+1
ducible components of the subquotient representation as V /F =D− D+ .
2,1−2m 2m ∼ 2m+1⊕ 2m+1
3. The space V is decomposed as the irreducible sum: V =D− D+.
2,1 2,1 1 ⊕ 1
Moreover, the spaces of irreducible submodules D±( V ), F ( V ) and the direct sum
m ⊂ j,m m ⊂ j,1−m
D+ D−( V ) above are the only non-trivial invariant subspaces of V for j = 1 (resp.
j =m2⊕) whmen⊂m ijs,meven (resp. odd) under the action of (sl ), the universal ejn,mveloping algebra of
2
sl . U
2
In the lemma above, we notice that each of the following short exact sequences of sl -modules
2
for m>0 is not split.
0 D+ D− V F 0, (3.1)
−→ 2m⊕ 2m −→ 1,2m −→ 2m−1 −→
0 D+ D− V F 0. (3.2)
−→ 2m+1⊕ 2m+1 −→ 2,2m+1 −→ 2m −→
Remark 3.1. The irreducible representations D−, D+ are called the (infinitsimal version of) limit
1 1
of discrete series of sl (R).
2
4 Lie theoretical approach to the eigenspectrum
We recall first the result in [33]. Let (α,β,γ,C) R4. Define a second order element K =
K(α,β,γ;C) (sl ) and a constant λ = λ (α,β∈,γ) depending on the representation ̟ as
2 a a a
∈ U
follows:
1 1
K(α,β,γ;C):= H E+α (F +β)+γ H +C,
2 − − 2
(cid:20) (cid:21) (cid:20) (cid:21)
1 1
λ (α,β,γ):=β a+α +γ a .
a
2 − 2
(cid:18) (cid:19) (cid:18) (cid:19)
By the definition of ̟ we observe (∂ := d ) that
a x dx
1 1 1 1 1
̟ (K)= x∂ + x2∂ + (a+ )x +α ∂ + (a )x−1+β +2γx∂ +C.
a x x x x
4 − 2 2 − 2 − 2
n (cid:0) (cid:1) on o
Noticing x−21(a−12)x∂xx12(a−21) =x∂x+ 21(a− 12), we have the following lemma ([33]).
Lemma 4.1. We have the following expression.
x−12(a−21)̟a(K(α,β,γ;C))x12(a−12)
x(x 1)
−
d2 1a+α 1a+2γ α d aβx+λ (α,β,γ)+C
= + β+ 2 + 2 − + − a .
dx2 − x x 1 dx x(x 1)
n − o −
6 M. Wakayama
We construct two second order elements and ˜ (sl ) from K=K(α,β,γ;C) by suitable
2
K K ∈ U
choices of the parameters (α,β,γ;C). As in the case of the quantum Rabi model in [33], we show
that these elements provide the confluent Heun picture of the asymmetric quantum Rabi model
under the image of the representations (̟ ,V )(j =1,2).
j,a j,a
We now capture the confluent Heun operators Rabi(λ)(j = 1,2) through K (sl ). The
Hj ∈ U 2
proof of the following proposition is simple and done by the same way in [33].
Proposition 4.2. Let λ be the eigenvalue of Hǫ . We have the following expressions.
Rabi
1. Suppose a= (λ+g2 ǫ). Define
− −
λ+g2 ǫ 1 λ+g2+ǫ
:=K 1 − ,4g2, ; µ+4ǫg2 ǫ2 (sl ),
2
K − 2 2 − 2 − ∈U
(cid:16) λ+g2 ǫ 1 λ+g2+ǫ (cid:17)
Λ :=λ 1 − ,4g2, .
a a
− 2 2 − 2
(cid:16) (cid:17)
Then
x(x−1)H1ǫ(λ)=x−21(a−12)(̟a(K)−Λa)x12(a−12). (4.1)
2. Suppose a= (λ+g2 1+ǫ). Define
− −
1 λ+g2+ǫ λ+g2 ǫ
˜ :=K ,4g2, − ; µ 4ǫg2 ǫ2 (sl ),
2
K − 2 − 2 − 2 − − ∈U
(cid:16) 1 λ+g2+ǫ λ+g2 ǫ (cid:17)
Λ˜ :=λ ,4g2, − .
a a
− 2 − 2 − 2
(cid:16) (cid:17)
Then
x(x−1)H2ǫ(λ)=x−12(a−12)(̟a(K˜)−Λ˜a)x21(a−12). (4.2)
Remark 4.1. The choice of β as β = 4g2 is necessary (and unique) if we identify the operator
x−21(a−12)̟a(K(α,β,γ;C))x21(a−21) with ǫ(λ).
x(x−1) Hj
Remark 4.2. In Proposition4.2, the rangeofa is bounded (looksalmostnegative). However,since
there is an equivalence between ̟ and ̟ when a Z, no such restriction applies. Actually,
a 2−a
6∈
e.g. for the spherical case, the linear isomorphism A:=Diag( ,c , ,c , ,c , ), where
−n 0 n
··· ··· ··· ···
|n| k a
c =c − 2 (c =0),
n 0 k 1+ a 0 6
k=1 − 2
Y
intertwines two representations (̟ ,V ) and (̟ ,V ), i.e. A̟ (X) = ̟ (X)A holds
1,a 1 1,2−a 1 1,a 1,2−a
for any X sl . However, when a 2Z, since there is no such intertwiner (because there is no
2
∈ ∈
equivalent irreducible submodules between V and V (see Lemma 3.1)), it is neccessary
1,2m 1,2−2m
to consider another approach to relating a representation ̟ (N > 0). The investigation of the
N
relationbetweendiscreteseriesrepresentationsandthespectrumoftheasymmetricquantumRabi
models for a general ǫ 1Z, we will leave the details to another occasion.
∈ 2
Remark 4.3. We have a mathematical model called the non-commutative harmonic oscillator
(NcHO) introduced in [26, 27] (see [25] in detail) as
α 0 1 d2 1 0 1 d 1
Q=Qα,β = 0 β −2dx2 + 2x2 + 1 −0 xdx + 2 , (αβ >1,α,β >0).
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
which is a parity preserving (i.e. Z -symmetry) self-adjoint ordinary differential operator and
2
a generalization of the quantum harmonic oscillator having an interaction term. We observed
in [32] the close connection between NcHO and the quantum Rabi model (QRM). The QRM is
obtainedby a secondorder element (sl ), whichis arisingfrom NcHO throughthe oscillator
2
R∈U
representation. Precisely, the Heun picture of the QRM is obtained by a confluent Heun equation
derived from the Heun operator defined by under a non-unitary principal series representation
of sl and confluent procedure by a finite regRular singular point with [32]:
2
∞
non-unitaryprincipalseries Confluent Heun picture
NcHO Heun ODE
←R̟→a −→ conflue−n→tprocess of the QRM
Symmetry of asymmetric quantum Rabi models 7
The element (sl ) is given as
2
R∈U
√a2+1 ν
= E F H + (H ν)+(εν)2, (4.3)
R − − a a −
h i
dependingonnon-zeroparameters(a,ν,ε) R3. Forasuitablechoiceoftheseparametersdepend-
ing on α,β, (sl ) gives the NcHO th∈rough the oscillator representations ([31]). It would be
2
interesting toRfi∈ndUa Z -symmetry broken generalization of NcHO, which can give the asymmetric
2
quantum Rabi model by the same confluent procedure. Additionally, it would be desirable if there
is a similar symmetry, as we will see in the subsequent sections on the asymmetric quantum Rabi
model, especially in relation with the monodoromy problems for Heun ordinary equations. More-
over,we may construct a “G-function” for NcHO similar to the one defined by Braak[3]. It would
be quite interesting to examine the relation between those two G-functions.
5 Degeneracy of eigenstates of the asymmetric quantum
Rabi model for ǫ 1Z
∈ 2
In this section, we study the level crossing of the asymmetric quantum Rabi model when ǫ 1Z
in terms of the finite dimensional irreducible representations of sl . We first consider the ge∈ne2ral
2
case.
5.1 λ = 2m g2 +ǫ (F : spherical ̟ ( )-eigenproblems)
2m+1 1,−2m
− K
Let a= (λ+g2 ǫ)= 2m(m Z ). Recall the notation in Proposition 4.2. We take
>0
− − − ∈
α=1 λ+g2−ǫ =1 m,
− 2 −
β =4g2,

γ = 21 − λ+g22+ǫ = 21 −m−ǫ,
C =µ+4ǫg2 ǫ2.
−
We have the following lemma.
Lemma 5.1. Let ν = ν := m a e F . Then the equation (̟ ( )
2m+1 n=−m n 1,n ∈ 2m+1 1,−2m K −
Λ )ν =0 is equivalent to the following
−2m
P
β+a +α+a +γ+a =0,
n n+1 n n n n−1
where
α+ =(m n)2 4g2(m n) ∆2+2ǫ(m n),
n − − − − −
β+ =(m+n+1)(m n 1),
 n − −
γ+ =4g2(m n+1).
n −
Proof. The proof can be obtained in a similar way to that in [33]. In fact, we observe
̟ ( )e =(m+n)(m n)e
1,−2m 1,n 1,n−1
K −
1 1
+ (m+n)(n 1 m)+4g2(n+1 m)+( m ǫ)(2n )+µ+4ǫg2 ǫ2) e
1,n
− − − 2 − − − 2 −
+4ng2(m n)e . o
1,n+1
−
It follows that
m−1
̟ ( )ν = (m+n+1)(m n 1)a e
1,−2m n+1 1.n
K − −
n=−m
X
m
1 1
+ (m+n)(m n+1) 4g2(m n 1)+( m ǫ)(2n )+µ+4ǫg2 ǫ2 a e
n 1,n
− − − − − 2 − − − 2 −
n=X−mn o
m
+4g2 (m n+1)a e F .
n−1 1,n 2m+1
− ∈
n=−m+1
X
8 M. Wakayama
Since C =µ+4ǫg2 ǫ2, where µ=(λ+g2)2 4g2(λ+g2) ∆2 and
− − −
1 1
Λ =Λ =β( a+α)+γ(a )
a −2m
2 − 2
1 1
=4g2( 2m+1)+( m ǫ)( 2m ),
− 2 − − − − 2
we observe that the eigenvalue equation (̟ ( ) Λ )ν = 0 is equivalent to the stated
1,−2m −2m
K −
recurrence equation.
Define now a matrix M(2m,ǫ) =M(2m,ǫ)((2g)2),∆) (k =0,1,2, ,2m) by
k k ···
α+ γ+ 0 0 0 0
β+m α+m γ+ 0 ··· ···
 m0−1 βm+−1 αm+−1 γ+ ·0·· ··· · · 
m−2 m−2 m−2 ··· · ·
M(2m,ǫ) = · ··· ... ... ... ··· 0 · .
k  ... ... γ+ 0 
 · ··· ··· m−k+3 · 
 0 β+ α+ γ+ 0 
 · ··· ··· m−k+2 m−k+2 m−k+2 
 0 0 β+ α+ γ+ 
 ··· ··· ··· m−k+1 m−k+1 m−k+1
 0 0 0 β+ α+ 
 ··· ··· ··· m−k m−k 
 
(2m,ǫ)
Then the continuant detM (i.e., a multivariate polynomial representing the deter-
{ k }0≤k≤2m
minant of a tridiagonal matrix) satisfies the recurrence relation
detM(2m,ǫ) =α+ detM(2m,ǫ) γ+ β+ detM(2m,ǫ)
k m−k k−1 − m−k+1 m−k k−2
with initial values
detM(2m,ǫ) =α+ = ∆2,
0 m −
(detM1(2m,ǫ) =α+mα+m−1 =−∆2(1−4g2−∆2+2ǫ).
Noticing that
α+ =k2 4g2k ∆2+2ǫk,
m−k − −
β+ =(2m k+1)(k 1)(=(N k+1)(k 1)) ifN =2m,
 m−k − − − −
γ+ =4kg2.
m−k+1

5.2 λ = 2m g2 1+ǫ (F : non-spherical ̟ ( )-eigenproblems)
2m 2,1−2m
− − K
Let a= (λ+g2 ǫ)=1 2m(m Z ). We take
>0
− − − ∈
α=1 λ+g2−ǫ = 3 m,
− 2 2 −
β =4g2,

γ = 21 − λ+g22+ǫ =1−m−ǫ
C =µ+4ǫg2 ǫ2.
−
Lemma5.2. Letν =ν := m−1 b e F . Then theequation [(̟ ( ) Λ ]ν =
2m n=−m n 2,n ∈ 2m 2,1−2m K − 1−2m
0 is equivalent to the following
P
β−b +α−b +γ−b =0,
n n+1 n n n n−1
where
α− =(m n 1)2 4g2(m n 1 ∆2 2ǫ(m n 1),
n − − − − − − − − −
β− =(m+n+1)(m n 2),
 n − −
γ− =4g2(m n).
n −

Symmetry of asymmetric quantum Rabi models 9
Similarly to the case of M(2m,ǫ), we define a matrix M(2m−1,ǫ) = M(2m−1,ǫ)((2g)2),∆) (k =
k k k
0,1, ,2m 1) by
··· −
α− γ− 0 0 0 0
m−1 m−1 ··· ···
β− α− γ− 0
 m0−2 βm−−2 αm−−2 γ− ·0·· ··· · · 
m−3 m−3 m−3 ··· · ·
 .. .. .. 
M(2m−1,ǫ) = · ··· . . . ··· 0 · .
k  ... ... γ− 0 
 · ··· ··· m−k+2 · 
 0 β− α− γ− 0 
 · ··· ··· m−k+1 m−k+1 m−k+1 
 0 0 β− α− γ− 
 ··· ··· ··· m−k m−k m−k 
 0 0 0 β− α− 
 ··· ··· ··· m−k−1 m−k−1
 
Then the continuant detM(2m−1,ǫ) satisfies the recurrence relation
{ k }0≤k≤2m−1
detM(2m−1,ǫ) =α− detM(2m−1,ǫ) γ− β− detM(2m−1,ǫ)
k m−k−1 k−1 − m−k m−k−1 k−2
with initial values
detM(2m−1,ǫ) =α− = ∆2,
0 m−1 −
(detM1(2m−1,ǫ) =α−m−1α−m−2 =−∆2(1−4g2−∆2+2ǫ).
Noticing that
α− =k2 4g2k ∆2+2ǫk,
m−k−1 − −
β− =(2m k)(k 1)(=(N k+1)(k 1)) ifN =2m 1,
 m−k−1 − − − − −
γ− =4kg2.
m−k

Now we define
P(N,ǫ) =P(N,ǫ)((2g)2,∆2)):=( 1)kdetM(N,ǫ)/( ∆2).
k k − k −
Then the following lemma is obvious from the recurrence equations for M(2m,ǫ) and M(2m−1,ǫ).
k k
Proposition 5.3. Let x = (2g)2. Then P(N,ǫ)(x) = P(N,ǫ)(x,∆2) (k = 0,1, ,N) satisfies the
k k ···
following recursion formula.
P(N,ǫ) =1, P(N,ǫ) =x+∆2 1 2ǫ,
0 1 − − (5.1)
(P(N,ǫ) =[kx+∆2 k2 2kǫ]P(N,ǫ) k(k 1)(N k+1)xP(N,ǫ).
k − − k−1 − − − k−2
In particular, degP(N,ǫ) =k as a polynomial in x.
k
WecallP(N,ǫ)(x)theconstraintpolynomial. Itisobviousthatifx>0isarootoftheconstraint
N
polynomial P(N,ǫ)(x), then there is a non-zero eigenvector ν( F ) of the eigenvalue equation
N ∈ N
(̟ ( )ν = Λ ν, that is, ν gives rise an eigenvector of the asymmetric quantum Rabi model
1,a a
K
through Proposition 4.2.
5.3 λ = 2m g2 ǫ (F : non-spherical ̟ (˜)-eigenproblems)
2m 2,1−2m
− − K
Let a= (λ+g2 1+ǫ)=1 2m(m Z ). We take
>0
− − − ∈
α= 1 λ+g2+ǫ = 1 m,
−2 − 2 −2 −
β =4g2,

γ =−λ+g22−ǫ =−m+ǫ
C =µ 4ǫg2 ǫ2.
− −

10 M. Wakayama
Lemma 5.4. Let ν˜=ν˜ := m−1 c e (c =0) F . Then the equation (̟ (˜)
2m n=−m n 2,n m−1 6 ∈ 2m 2,1−2m K −
Λ˜ )ν˜=0 is equivalent to the following
1−2m P
β˜−c +α˜−c +γ˜−c =0,
n n+1 n n n n−1
where
α˜− =(m n)2 4g2(m n) ∆2 2ǫ(m n),
n − − − − − −
β˜− =(m+n+1)(m n),
 n −
γ˜− =4g2(m n).
n −
Since cm+1 =cm =0 and cm−1 6=0 at the equation
β˜−c +α˜−c +γ˜−c =0,
m m+1 m m m m−1
we find that γ˜− = 0 and may take α˜− = ∆2 without loss of generality. We define a matrix
m m −
M˜(2m,ǫ) =M˜(2m,ǫ)((2g)2),∆) (k =0,1,2, ,2m) by
k k ···
α˜− γ˜−(=0) 0 0 0 0
m m ··· ···
β˜− α˜− γ˜− 0
 m−1 m−1 m−1 ··· ··· · · 
0 β˜− α˜− γ˜− 0
m−2 m−2 m−2 ··· · ·
 ... ... ... 0 
M˜(2m,ǫ) = · ··· ··· · .
k  ... ... γ˜− 0 
 0·· ······ ······ 0 β˜m−−0k+2 αβ˜˜m−m−−−kk++32 αγ˜˜m−−−k+2 γ˜−0· 
 0 ··· ··· ··· 0 m−0k+1 βm˜−−k+1 αm˜−−k+1
 ··· ··· ··· m−k m−k 
 
Then the continuant detM˜(2m,ǫ) satisfies the recurrence relation
{ k }0≤k≤2m
detM˜(2m,ǫ) =α˜− detM˜(2m,ǫ) γ˜− β˜− detM˜(2m,ǫ)
k m−k k−1 − m−k+1 m−k k−2
with initial values
detM˜(2m,ǫ) =α˜− = ∆2,
0 m −
(detM˜1(2m,ǫ) =α˜−mα˜−m−1 =−∆2(1−4g2−∆2−2ǫ).
Noticing that
α˜− =k2 4g2k ∆2 2ǫk,
m−k − − −
β˜− =(2m k+1)k (=(N k+1)k) ifN =2m,
 m−k − −
γ˜− =4(k 1)g2.
m−k+1 −

5.4 λ = 2m g2 +1 ǫ (F : spherical ̟ (˜)-eigenproblems)
2m+1 1,−2m
− − K
Let a= (λ+g2 1+ǫ)= 2m(m Z ). We take
>0
− − − ∈
α= 1 λ+g2+ǫ = 1 m,
−2 − 2 − −
β =4g2,

γ =−λ+g22−ǫ =−12 −m+ǫ
C =µ 4ǫg2 ǫ2.
− −
Lemma5.5. Letν˜=ν˜ := m d e (d =0) F . Thentheequation(̟ (˜)
2m+1 n=−m n 1,n n 6 ∈ 2m+1 1,−2m K −
Λ˜ )ν˜=0 is equivalent to the following
−2m P
β˜+d +α˜+d +γ˜+d =0,
n n+1 n n n n−1
where
α˜+ =(m n+1)2 4g2(m n+1) ∆2 2ǫ(m n+1),
n − − − − − −
β˜+ =(m+n+1)(m n+1),
 n −
γ˜+ =4g2(m n+1).
n −
