The Generalized Fibonacci Oscillator as an Open Quantum System
We consider an open quantum system with Hamiltonian ₛ whose spectrum is given by a generalized Fibonacci sequence weakly coupled to a Boson reservoir in equilibrium at inverse temperature . We find the generator of the reduced system evolution and explicitly compute the stationary state of the syste...
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| author | Fagnola, Franco Ko, Chul Ki Yoo, Hyun Jae |
| author_facet | Fagnola, Franco Ko, Chul Ki Yoo, Hyun Jae |
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| description | We consider an open quantum system with Hamiltonian ₛ whose spectrum is given by a generalized Fibonacci sequence weakly coupled to a Boson reservoir in equilibrium at inverse temperature . We find the generator of the reduced system evolution and explicitly compute the stationary state of the system, which turns out to be unique and faithful, in terms of the parameters of the model. If the system Hamiltonian is generic, we show that convergence towards the invariant state is exponentially fast and compute explicitly the spectral gap for low temperatures, when quantum features of the system are more significant, under an additional assumption on the spectrum of ₛ.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 035, 19 pages
The Generalized Fibonacci Oscillator
as an Open Quantum System
Franco FAGNOLA a, Chul Ki KO b and Hyun Jae YOO c
a) Mathematics Department, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy
E-mail: franco.fagnola@polimi.it
URL: https://www.mate.polimi.it/qp/
b) University College, Yonsei University, 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, Korea
E-mail: kochulki@yonsei.ac.kr
c) School of Computer Engineering and Applied Mathematics,
Institute for Integrated Mathematical Sciences, Hankyong National University,
327 Jungang-ro, Anseong-si, Gyeonggi-do 17579, Korea
E-mail: yoohj@hknu.ac.kr
Received February 07, 2022, in final form April 19, 2022; Published online May 11, 2022
https://doi.org/10.3842/SIGMA.2022.035
Abstract. We consider an open quantum system with Hamiltonian HS whose spectrum
is given by a generalized Fibonacci sequence weakly coupled to a Boson reservoir in equi-
librium at inverse temperature β. We find the generator of the reduced system evolution
and explicitly compute the stationary state of the system, that turns out to be unique and
faithful, in terms of parameters of the model. If the system Hamiltonian is generic we show
that convergence towards the invariant state is exponentially fast and compute explicitly the
spectral gap for low temperatures, when quantum features of the system are more significant,
under an additional assumption on the spectrum of HS .
Key words: open quantum system; Fibonacci Hamiltonian; deformation of canonical com-
mutation relations; spectral gap; weak-coupling limit; quantum Markov semigroup
2020 Mathematics Subject Classification: 81S22; 81S05; 60J80
1 Introduction
Deformations of CCR and CAR have been extensively investigated in the literature. q-deformed
commutation relations are defined by means of a single parameter q in the interval [−1, 1] of
annihilation and creation operators a and a† satisfying aa† − qa†a = 1l and the CCR or CAR
are recovered in the limit as q → 1− or q → −1+ (see [7, 8, 9, 26] and the references therein).
Recently, the inclusion of two distinct deformation parameters r, q has been proposed to allow
more flexibility while retaining the good properties and the possibility of finding explicit formulas
as in the case of single parameter deformations (see [6, 14, 20, 25] and the references therein).
The two parameters deformed commutation relations become aa†− ra†a = qN , aa†− qa†a = rN
where N is the number operator (see Section 2 for precise definitions) in their one-mode Fock
space representation.
In this way one finds a quantum system with Hamiltonian HS = a†a which is a two parameter
deformation of the harmonic oscillator whose spectrum {(rn−qn)/(r−q)}n≥0 is a generalized Fi-
bonacci sequence (that turns out to be the well-known Fibonacci sequence for r =
(
1 +
√
5
)
/2,
q =
(
1 −
√
5
)
/2) and therefore is called Fibonacci Hamiltonian. Two parameters Hermite
This paper is a contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Ac-
tion. The full collection is available at https://www.emis.de/journals/SIGMA/non-commutative-probability.html
mailto:franco.fagnola@polimi.it
https://www.mate.polimi.it/qp/
mailto:kochulki@yonsei.ac.kr
mailto:yoohj@hknu.ac.kr
https://doi.org/10.3842/SIGMA.2022.035
https://www.emis.de/journals/SIGMA/non-commutative-probability.html
2 F. Fagnola, C.K. Ko and H.J. Yoo
polynomials have been computed and the energy spectrum has been studied showing that the
deformation is more effective in highly excited states (see [20, 25]). Deformed Fock spaces
and deformed Gaussian processes have been analyzed in connection with the single-parameter
deformation of the full Fock space of free probability [6]. Moreover, the arising quantum al-
gebra with two deformation parameters has been considered in applications to certain physical
models [14, 20].
In this paper we consider the r, q deformed oscillator with Hamiltonian HS weakly coupled to
a boson reservoir at inverse temperature β as an open quantum system. At first, we rigorously
deduce the reduced dynamics of the open system in the weak coupling limit [3] getting a quan-
tum Markov semigroup (QMS) which is a natural two-parameter deformation of the so-called
quantum Ornstein–Uhlenbeck semigroup [12, 13] and has a generator in the well-known Gorini–
Kossakowski–Lindblad–Sudharshan (GKLS) form, generalized to allow unbounded operators in
the case where one of the parameters is bigger than 1 that presents more difficulties (see [7]). We
emphasize that, as the reader may immediately note from the proof of Theorem 4.2, the choice
of a parameter bigger than 1 is necessary in order to find an equilibrium state for the dynamics
of the reduced system in order not to break the physical principle of thermal relaxation [1, 3].
In our analysis, we pay a special attention to the structure of the spectrum of HS whose order
plays an important role in determining the GKLS generator motivating the emergence of natural
inequalities among the two parameters r, q. In particular, the key conditions −1 ≤ q ≤ 1 < r
and r+q ≥ 1 that appear throughout the paper, are not for mere convenience because they affect
the order of the spectrum and, as a consequence, the QMS that emerges after the weak coupling
limit. However, they still allow us to analyze the behaviour of the system for parameters r → 1+
and q → 1−, when the r, q deformed commutation are “near” the CCR, for comparison with
the quantum Ornstein–Uhlenbeck semigroup [12, 13].
We then focus on the case where the spectrum of HS is generic (see last part of Section 3 for
the precise definition) in which the GKLS generator takes a simpler form (see [11, 19] and the
references therein). We emphasize that this happens for almost all choices of the deformation
parameters r, q (in the sense of Lebesgue measure on R2, for instance). We show that the arising
QMS has a normal invariant state, which is unique and faithful, and investigate the speed of
convergence towards the invariant state determining the spectral gap in the L2 space of the
invariant state.
Taking advantage of the structure of the GKLS generator we are able to compute explicitly the
spectral gap for low temperatures, when quantum features of the dynamics are more significant,
if r + q ≥ 2 (Theorem 5.6). We also provide evidence (see Remark 5.4) that the spectral
gap is strictly positive but it is not possible to obtain a simple closed-form expression for high
temperatures.
The case where a parameter r, q is strictly bigger than 1 is the most difficult when considering
deformations of the CCR (see [7]) not only because of unboundedness of creation and annihilation
operators, as in the boson case, but also because of additional pathologies that arise. It is well-
known, for instance, that field operators are not essentially self-adjoint on the domain of finite
particle vectors. However, our results complement those obtained for special cases q = r = 1
(Boson), r = 1, q = −1 (Fermi) and r = 1, q = 0 scattered in the literature.
In addition the computation of the spectral gap of a GKLS generator has its own interest
because of applications in the study of strong ergodicity of open quantum systems [5, 16, 27, 28]
and the explicit result is known only in few cases.
The paper is organized as follows. In Section 2 we discuss the structure of the spectrum
of generalized Fibonacci oscillators in order to justify the emergence of conditions on parame-
ters r, q that will be assumed in the paper. The deduction from the weak coupling limit of form
generators of QMSs for generalized Fibonacci Hamiltonians viewed as open quantum systems
is illustrated in Section 3 and the construction of QMSs from form generators by the minimal
The Generalized Fibonacci Oscillator as an Open Quantum System 3
semigroup method (see [17, Section 3]) is presented in Section 4. The spectral gap is computed
in Section 5 in a simple explicit formula for small temperatures of the reservoir and for param-
eters satisfying −1 < q ≤ 1 < r, r + q ≥ 2 also providing evidence that an explicit formula in
the general case cannot be achieved.
2 Fibonacci oscillators
Let q, r be two real numbers with q ̸= r. (q, r)-integers are defined by
ε0 = 0, ε1 = 1, εn =
rn − qn
r − q
for n ≥ 2,
where we can assume r > q without loss of generality. In the case where q → 1− and r → 1+ one
finds the natural numbers and, if r = 1 the usual q-integers (also allowing q > 1). It is worth
noticing that, in the special case where r =
(√
5+ 1
)
/2, q =
(
1−
√
5
)
/2, one finds the sequence
of Fibonacci numbers. For this reason (εn)n≥0 is called generalized Fibonacci sequence. Note
that εn ≥ 0 for all n ≥ 0.
Let h = Γ(C) be the one-mode Fock space with canonical orthonormal basis (en)n≥0. The
Fibonacci oscillator is the quantum system with Hamiltonian
HS =
∑
n≥0
εn|en⟩⟨en|, (2.1)
whose spectrum is the generalized Fibonacci sequence. Defining the (q, r) annihilation and
creation operators
Dom(a) =
{
u ∈ h
∣∣∣∣ ∑
n≥0
εn|un|2 < ∞
}
, aen =
√
εnen−1, (2.2)
Dom
(
a†
)
=
{
u ∈ h
∣∣∣∣ ∑
n≥0
εn+1|un|2 < ∞
}
, a†en =
√
εn+1en+1,
one can write HS = a†a on the domain F of finite linear combinations of vectors of the canonical
orthonormal basis, also called finite particle vectors.
One immediately checks that a and a† are bounded operators if and only if −1 ≤ q < r ≤ 1,
they are mutually adjoint and satisfy the commutation relations
aa† − ra†a = qN , aa† − qa†a = rN , (2.3)
where N is the usual number operator defined by
Dom(N) =
{
u ∈ h
∣∣∣∣ ∑
n≥0
n2|un|2 < ∞
}
, Nu =
∑
n≥1
nunen.
Paying attention to the operator domains these properties can be extended to the general case
q < r. In particular, for r = 1, we have the q-commutation relations aa† − qa†a = 1l. These
commutation relations can be found also considering creations and annihilations on interacting
Fock spaces (see [2, 21] and the references therein) but it is more convenient to consider the usual
one-mode Fock space for our analysis. Moreover, we would like to mention that two parameter
deformed commutation relations lead to remarkable combinatorial formulas (e.g., for moments
of field operators) as those of canonical commutation and anti-commutation relations (see [22]).
Since we are interested in the generalized Fibonacci oscillator as an open quantum system
weakly coupled with a reservoir and the weak coupling crucially depends on ordering of the
4 F. Fagnola, C.K. Ko and H.J. Yoo
spectrum of HS , throughout the paper assume that eigenvalues εn of HS form an increasing
sequence. Clearly, this is not the case, for example, if 0 < q < r < 1 because rn+1−qn+1 < rn−qn
for big n. Moreover, in order to exclude high oscillatory behaviours also for reasons that will
be clear in the next section, we are mostly interested in the case −1 ≤ q ≤ 1 < r therefore this
inequality will also be assumed throughout the paper.
Note that ε2 ≥ ε1 if and only if r2 − q2 ≥ r − q, i.e., r + q − 1 ≥ 0, therefore we need at
least this additional condition. Once it holds, the sequence (εn)n≥0 is obviously increasing if
r > 1 > q ≥ 0 because
rn+1 − qn+1 − (rn − qn) = rn(r − 1) + qn(1− q) > 0.
The case −1 ≤ q < 0 < 1 < r needs a slightly more detailed analysis. First of all note that,
since r, q solve the equation 0 = (x − r)(x − q) = x2 − (r + q)x + rq and ε0 = 0, ε1 = 1 then
εn+2 − (r + q)εn+1 + rqεn = 0. Therefore, since r + q − 1 ≥ 0 and −rq > 0, the identity
εn+2 − εn+1 = (r + q − 1)εn+1 − rq εn
shows that the sequence (εn)n≥0 is non decreasing whenever r + q − 1 ≥ 0.
The above discussion is summarized by the following
Lemma 2.1. Assume −1 ≤ q ≤ 1 < r. The sequence (εn)n≥0 is non-decreasing (resp. strictly
increasing) if and only if r + q − 1 ≥ 0 (resp. r + q − 1 > 0).
Remark 2.2. The “free” r = 1, q = 0 and Fibonacci r =
(√
5 + 1
)
/2, q =
(
1 −
√
5
)
/2 cases
lie on the boundary of the region. Note that the q = r = 1 (Bose) and q = 1 = −r (Fermi)
cases, are formally excluded, but can arise as limiting cases. However, the spectrum of system
Hamiltonian is no more generic and the study of QMS arising from the weak coupling limit has
been carried on separately [13].
3 QMS of weak coupling limit type
Let HS be a Hamiltonian with spectral decomposition
HS =
∑
m≥0
εmPεm ,
where εm, with εm < εn for m < n, are the eigenvalues of HS and Pεm are the corresponding
eigenspaces. QMSs of weak coupling limit type (WCLT), associated with the Hamiltonian HS
(see [3, 18] and the references therein), have generators of the form L =
∑
ω∈B+
Lω where B+
is the set of all Bohr frequencies (Arveson spectrum)
B+ := {εn − εm : εn − εm > 0}.
Given a system operator D, whose domain contains ranges of projections Pεm , depending on the
interaction of the system with a reservoir, for every Bohr frequency ω, consider a generator Lω
with the Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) structure
Lω(x) = i[Hω, x]−
Γ−
ω
2
(D∗
ωDωx− 2D∗
ωxDω + xD∗
ωDω)
− Γ+
ω
2
(DωD
∗
ωx− 2DωxD
∗
ω + xDωD
∗
ω) (3.1)
The Generalized Fibonacci Oscillator as an Open Quantum System 5
for all x ∈ B(h), with Kraus operators Dω defined by
Dω =
∑
(εn,εm)∈B+,ω
PεmDPεn ,
where B+,ω = {(εn, εm) | εn−εm = ω}, Γ−
ω , Γ
+
ω are non-negative real constants with Γ−
ω +Γ+
ω > 0
and Hω is a bounded self-adjoint operator on h commuting with HS .
In the case when the set of Bohr frequencies is infinite, for L to be the generator of a norm
continuous QMS the series∑
ω∈B+
(
Γ−
ωD
∗
ωDω + Γ+
ωDωD
∗
ω
)
must be strongly convergent in B(h), the von Neumann algebra of all bounded operators on h
(see [29, Corollary 30.13, p. 268 and Theorem 30.16, p. 271]).
QMSs generators in this form arise in the weak coupling limit of a system with Hamilto-
nian HS coupled to a Boson reservoir in equilibrium at inverse temperature β with coupling
D ⊗A†(ϕ) +D∗ ⊗A(ϕ),
where A†(ϕ), A(ϕ) are the creation and annihilation operator of the reservoir with test function ϕ.
Constants Γ±
ω = fωγ
±
ω are given explicitly by
γ−ω =
eβω
eβω − 1
, γ+ω =
1
eβω − 1
, fω =
∫
{y∈R3 | |y|=ω}
|ϕ(y)|2dsy, (3.2)
where ds denotes the surface integral and the cut-off ϕ is a square-integrable function on R3.
A realistic cut-off could be a function which is constant in some bounded “big” region and slowly
vanishes as ω goes to infinity. For this reason, slightly modifying generators Lω after the weak
coupling limit, if necessary, it looks reasonable to assume throughout the paper fω constant and
fix fω = 1.
From the above discussion it is clear that the spectral decomposition of the Hamiltonian HS
plays a key role. In particular, if we consider the case r = 1, q = 0 (in which (2.3) are the
well-known free commutation relations) the Hamiltonian HS becomes
HS = a†a =
∑
n≥1
|en⟩⟨en| = P1
with P1 projection. Therefore there is only the Bohr frequency ω = 1,
D1 = |e0⟩⟨e0|DP1 = |e0⟩ ⟨P1D
∗e0|
is determined by the vector v = P1D
∗e0 orthogonal to e0 so that
D1 = |e0⟩ ⟨v| , D∗
1 = |v⟩ ⟨e0| , D∗
1D1 = |v⟩ ⟨v| , D1D
∗
1 = ∥v∥2 |e0⟩ ⟨e0|
and H1 is a multiple of P1 up to addition of a multiple of the identity operator. In this way,
calling e the normalized vector v, we get the GKLS generator
Lω(x) = iκ[ |e⟩⟨e|, x]− Γ−
1 ∥v∥2
2
(|e⟩ ⟨e|x− 2 |e⟩ ⟨e0|x |e0⟩ ⟨e|+ x |e⟩ ⟨e|)
− Γ+
1 ∥v∥2
2
(|e0⟩ ⟨e0|x− 2 |e0⟩ ⟨e|x |e⟩ ⟨e0|+ x |e0⟩ ⟨e0|) .
6 F. Fagnola, C.K. Ko and H.J. Yoo
This GKLS generator essentially describes the dynamics of a 2-level system than can be explicitly
computed. One finds the same dramatic simplification for r = 1, q = −1 and so we are not
interested in these special cases.
It is worth noticing that the GKLS generator written down just by analogy with the Boson
case, namely
L(x) = iκ[ a†a, x]− Γ−
1
2
(
aa†x− 2axa† + xaa†x
)
− Γ+
1
2
(
a†ax− 2a†xa+ xa†a
)
has another structure.
It is well-known that, when the system Hamiltonian HS is generic namely:
(i) Its spectrum is pure point and each eigenvector has multiplicity one,
(ii) For all ω ∈ B+ there exists a unique pair (n,m) of energy levels such that εn − εm = ω,
the structure of the generator is very simple (see, e.g., [19]) because operators Dω are multiples
of rank one operators |em⟩⟨en|, where n, m are determined by the unique pair of such that
εn − εm = ω. In particular each off-diagonal rank one operator |ei⟩⟨ej | is an eigenvector for L
and the action of maps Tt of the QMS (Tt)t≥0 generated by L (see Section 4 for details) on
|ei⟩⟨ej | is explicit.
The Fibonacci type Hamiltonian HS as in (2.1) is clearly generic for almost all choices of
parameters r, q. However, in other cases, the WCLT generator might be more complex because
of the structure of B+ (see [15] for a detailed analysis of the structure of norm-continuous QMSs).
Indeed, if (εn)n≥0 is the Fibonacci sequence, then ε0 = 0, ε1 = 1 and εn+1 = εn+ εn−1 for n ≥ 1
so that B+ = {εn |n ≥ 1} because, for all n ≥ 1
εn+1 − εn = εn − εn−2 = εn−1
and, for k ≥ 3,
εn+k − εn = εn+k−1 + εn+k−2 − εn = 2εn+k−2 + εn+k−3 − εn ≥ 2εn+k−2 > εn′ − εm′
for all n′ < n+ k − 2 and so, in particular, for all n′ < n. In this case, as a consequence,
Dε1 = D1 = ⟨e0, De1⟩|e0⟩⟨e1|+ ⟨e0, De2⟩|e0⟩⟨e2|
+ ⟨e1, De3⟩|e1⟩⟨e3|+ ⟨e2, De3⟩|e2⟩⟨e3|+ ⟨e3, De4⟩|e3⟩⟨e4|.
Clearly, the operator D also plays a key role in the GKLS generator L because transitions
between levels εn and εm (εn − εm = ω > 0) can be forbidden if ⟨em, Den⟩ is zero even if Γ±
ω is
strictly positive. The most natural choice for D is the annihilator D = a defined by (2.2). With
this choice of D and the Fibonacci sequence as (εn)n≥0 we find
Dε1 = |e0⟩⟨e1|+
√
2|e2⟩⟨e3|+
√
3|e3⟩⟨e4|.
However, with other choices of the operator D, Kraus operators Dω can be rank one also in the
case where (εn)n≥0 is the Fibonacci sequence.
4 Generic open Fibonacci type oscillators
From now on we consider the GKLS form generator
L =
∑
n≥1
Ln, (4.1)
The Generalized Fibonacci Oscillator as an Open Quantum System 7
where
Ln(x) = i
[
κ−n |en⟩⟨en|+ κ+n |en−1⟩⟨en−1|, x
]
− Γ−
n
2
rn − qn
r − q
(|en⟩⟨en|x− 2|en⟩⟨en−1|x|en−1⟩⟨en|+ x|en⟩⟨en|)
− Γ+
n
2
rn − qn
r − q
(|en−1⟩⟨en−1|x− 2|en−1⟩⟨en|x|en⟩⟨en−1|+ x|en−1⟩⟨en−1|) ,
Γ±
n are as in (3.2) and κ±n are real constants. As explained in the previous section, these are
generators (3.1) where all operators Dω are rank-one because either HS is generic or by suitable
choice of the operator D. The set of ω with terms contributing to the generator is in one-to-one
correspondence with N∗ = {1, 2, 3, . . . } and transitions from level εn (n > 0) can occur only to
levels εn+1 and εn−1 for n > 0 and from level ε0 = 0 to ε1 = 1 so that the graph of the process
is as follows:
ε0 = 0
&&
ε1 = 1
$$
ff
ε2
ee
##
ε3
cc
""
. . .
dd
Graph of the nearest neighbour jump process.
The definition of L is only formal because the sum on n in (4.1) is infinite, therefore some
clarifications are now in order. First of all note that the operator
G =
∑
n≥0
(
i(κ−n + κ+n+1)−
Γ−
n
2
rn − qn
r − q
−
Γ+
n+1
2
rn+1 − qn+1
r − q
)
|en⟩⟨en|
is well defined as a normal operator on the domain Dom(G) of vectors u =
∑
n unen ∈ h, i.e.,
such that
∑
n≥0 |un|2 < ∞ for which
∑
n≥0
∣∣κ−n + κ+n+1
∣∣2 + ∣∣∣∣∣Γ−
n
2
rn − qn
r − q
+
Γ+
n+1
2
rn+1 − qn+1
r − q
∣∣∣∣∣
2
|un|2 < ∞.
In particular, if sequences (κ−n + κ+n )n≥1 and ((Γ−
n + Γ+
n )(r
n − qn))n≥1 are bounded, then G is
bounded, L is a bounded operator on B(h) and generates a norm continuous QMS on B(h) with
Kraus operators Lℓ (ℓ ≥ 1) which are rank-one and given by
L2ℓ =
(
Γ−
ℓ (r
ℓ − qℓ)
r − q
)1/2
|eℓ−1⟩⟨eℓ|, L2ℓ+1 =
(
Γ+
ℓ (r
ℓ − qℓ)
r − q
)1/2
|eℓ⟩⟨eℓ−1|.
However, even if G is unbounded as in typical cases with r > 1, it is possible to construct
a uniquely determined QMS on B(h) by the minimal semigroup method (see, e.g., [17, Sec-
tions 3.3 and 3.4] and also [23] and the references therein). Indeed, G generates a strongly
continuous semigroup (Pt)t≥0 on h and the explicit form of the operators Pt is immediately
written. For x ∈ B(h) let £(x) be the quadratic form with domain Dom(G)×Dom(G)
£(x)[v, u] = ⟨Gv, xu⟩+
∑
ℓ≥1
⟨Lℓv, xLℓu⟩+ ⟨v, xGu⟩ . (4.2)
The minimal semigroup associated with operators G, Lℓ is constructed, on elements x of B(h),
by means of the non decreasing sequence of positive maps (T (n)
t )n≥0 defined, by recurrence, as
8 F. Fagnola, C.K. Ko and H.J. Yoo
follows
T (0)
t (x) = P ∗
t xPt, (4.3)〈
v, T (n+1)
t (x)u
〉
= ⟨Ptv, xPtu⟩+
∑
ℓ≥1
∫ t
0
〈
LℓPt−sv, T (n)
s (x)LℓPt−su
〉
ds
for all x ∈ B(h), t ≥ 0, v, u ∈ Dom(G). Indeed, we have
Tt(x) = sup
n≥0
T (n)
t (x)
for all positive x ∈ B(h) and all t ≥ 0. The definition of positive maps Tt is then extended to
all the elements of B(h) by linearity. The minimal semigroup associated with G, Lℓ satisfies the
integral equation
⟨v, Tt(x)u⟩ = ⟨v, xu⟩+
∫ t
0
£(Ts(x))[v, u] ds (4.4)
for all x ∈ B(h), t ≥ 0, v, u ∈ Dom(G). Moreover, it is the unique solution to the above
equation if and only if it is conservative (or Markov), i.e., Tt(1l) = 1l for all t ≥ 0 (see, e.g., [17,
Corollary 3.23]).
In our framework it is not difficult to show that conservativity is equivalent to the Karlin–
McGregor condition for non-explosion of Markov jump processes. Indeed, one immediately
checks that the diagonal algebra generated by projections |en⟩⟨en| is invariant for maps T (n)
t
(for all n ≥ 0) defined recursively by (4.3) because each vector en is an eigenvector of G so
that P ∗
t |ej⟩⟨ej |Pt = ezjt|ej⟩⟨ej | for some zj ∈ C and, looking at iterations (4.3), if T (n)
s (|ej⟩⟨ej |)
belongs to the diagonal algebra, then
P ∗
t−sL
∗
ℓT (n)
s (|ej⟩⟨ej |)LℓPt−s
belongs to the diagonal algebra as well for all 0 ≤ s ≤ t, and so also T (n+1)
s (|ej⟩⟨ej |) belongs
to the diagonal algebra. It follows that Ts(|ej⟩⟨ej |) belongs to the diagonal algebra which is
invariant for the QMS T , as expected from the quadratic form computation
£(f(a†a)) =
∑
n≥1
Γ−
n ϵn (f(ϵn−1)− f(ϵn)) |en⟩⟨en|+
∑
n≥0
Γ+
n+1ϵn+1 (f(ϵn+1)− f(ϵn)) |en⟩⟨en|
for all bounded function f on the spectrum of HS . In this way, we see that the restriction of
maps Tt to the diagonal algebra coincides with the minimal semigroup of the classical birth-and-
death process with birth (resp. death) rates λn (resp. µn)
λn = Γ+
n+1
rn+1 − qn+1
r − q
= Γ+
n+1εn+1, µn = Γ−
n
rn − qn
r − q
= Γ−
n εn. (4.5)
Moreover, writing (4.4) for x = |ej⟩⟨ej |, u = v = ei and recalling that ⟨ei, Tt(|ej⟩⟨ej |)ei⟩ is the
probability of visiting j at time t starting from i at time 0, we find the backward Kolmogorov
equations of the birth-and-death process. Therefore the minimal semigroup is Markov if and
only if the minimal semigroup of the classical birth-and-death process with the above rates is
conservative.
Let π0 = 1 and, for n ≥ 1,
πn = (λ0λ1 · · ·λn−1) / (µ1µ2 · · ·µn) = e−βεn .
The Generalized Fibonacci Oscillator as an Open Quantum System 9
It is well-known [4, Theorem 2.2, p. 100] that the minimal semigroup of the classical birth-and-
death process with these rates is conservative (more precisely, the minimal semigroup is identity
preserving and it is the unique solution of the backward Kolmogorov equations) if and only if
∑
n≥1
1
λnπn
n∑
k=1
πk = +∞.
Theorem 4.1. The minimal semigroup associated with the above G, Lℓ is Markov for all −1 ≤
q ≤ 1 < r with r + q ≥ 1.
Proof. Notice that ωn = ϵn − ϵn−1 and so, by (3.2), we have
Γ+
n =
1
eβωn − 1
, Γ−
n =
eβωn
eβωn − 1
.
Thus we get πn = e−βεn for all n ≥ 0 and
1
λnπn
=
eβεn+1 − eβεn
εn+1
=
eβεn
εn+1
(
eβ(r
n(r−1)+qn(1−q))/(r−q) − 1
)
.
Since the sequence (1/λnπn)n≥0 diverges as n goes to +∞ and
∞∑
n=1
1
λnπn
n∑
k=1
πk > π1
∞∑
n=1
1
λnπn
= +∞.
This completes the proof. ■
In the sequel we will assume that parameters −1 ≤ q ≤ 1 < r satisfy r+ q ≥ 1. Theorem 4.1
also implies that [17, Proposition 3.33] the domain of the generator of T is the space of x ∈ B(h)
for which the quadratic form £(x) (4.2) with domain Dom(G) × Dom(G) is bounded. This
happens, in particular, for all off diagonal rank-one operators |ej⟩⟨ek| (j ̸= k)
L(|ej⟩⟨ek|) =
(
i(κ−j − κ−k + κ+j+1 − κ+k+1)
−
Γ−
j
2
εj −
Γ+
j+1
2
εj+1 −
Γ−
k
2
εk −
Γ+
k+1
2
εk+1
)
|ej⟩⟨ek|, (4.6)
that are eigenvectors of L with nonzero eigenvalue. This remark allows us to prove in a simple
way existence and uniqueness of a normal invariant state.
Theorem 4.2. Suppose that −1 ≤ q ≤ 1 < r and r+q ≥ 1. The QMS admits a unique invariant
state ρ
ρ =
1
Zβ
∑
n≥0
e−βεn |en⟩⟨en|, Zβ =
∑
n≥0
e−βεn . (4.7)
Proof. First of all note that
∑
n>0 e
−βεn < +∞. Let ρ be a normal invariant state. Since rank
one operators |ej⟩⟨ek| (j ̸= k) belong to the domain of the generator L and are eigenvectors with
nonzero eigenvalue ξjk, say, differentiating the identity tr(ρTt(|ej⟩⟨ek|)) = tr(ρ|ej⟩⟨ek|) at t = 0,
we get
0 = tr (ρL(|ej⟩⟨ek|)) = ξjk⟨ek, ρej⟩
10 F. Fagnola, C.K. Ko and H.J. Yoo
and so ρ is diagonal in the same basis as HS , i.e., ρ =
∑
n≥0 ρn|en⟩⟨en|. Now a simple computa-
tion shows that the probability density (ρn)n≥0 on N is an invariant measure for the associated
classical birth-and-death process. Therefore (see, e.g., [4, Example 4.2, p. 197]) the state ρ
given by (4.7) is invariant because ρn := e−βεn/Zβ defines an invariant density for classical
birth-and-death process.
Uniqueness follows immediately because we proved that a normal invariant state is diagonal
and it determines an invariant density for the associated classical birth-and-death process which
is unique because the birth-and-death process has strictly positive transition rates whence it is
irreducible. ■
Remark 4.3. Note that the invariant state (4.7) is faithful.
The rest of the paper is dedicated to the study of the speed of convergence of T towards the
invariant state.
5 Spectral gap
Strong ergodic properties, such as the speed of convergence towards the invariant state, are
a natural problem on the behaviour of an open quantum system with a unique faithful normal
invariant state. In this section we discuss the spectral gap of the generator in (4.1) that solves
this problem in suitable norm.
Given a QMS with a faitful normal invariant state we may embed B(h) into L2(h), the space
of Hilbert-Schmidt operators on h with inner product ⟨x, y⟩ = tr(x∗y), in the following way:
ι : B(h) → L2(h), ι(x) = ρ1/4xρ1/4.
Let T = (Tt)t≥0 be the strongly continuous contraction semigroup on L2(h) defined by
Tt(ι(x)) = ι(Tt(x))
and let L be the generator of the semigroup (Tt)t≥0. We can check that
L
(
ρ1/4xρ1/4
)
= ρ1/4L(x)ρ1/4, for x ∈ Dom(L).
The Dirichlet form, defined for ξ ∈ Dom(L), is the quadratic form E
E(ξ) = −Re⟨ξ, L(ξ)⟩.
The spectral gap of the operator L is the nonnegative number
gap(L) := inf
{
E(ξ) | ∥ξ∥ = 1, ξ ∈ (Ker(L))⊥
}
.
Since rank-one operators |ej⟩⟨ek| (j ̸= k) are eigenvectors for L (as for all generic QMSs [19]), and
the diagonal algebra is invariant we have the same properties also for the induced semigroup T
on L2(h). LetD be diagonal operators
∑
n≥0 ξn|en⟩⟨en| in L2(h), i.e., such that
∑
n≥0 |ξn|2 < +∞
and let Djk be the linear space generated by |ej⟩⟨ek| j, k ≥ 0, j ̸= k. One can easily check that
L2(h) = D ⊕j,k≥0, j ̸=k Djk
and, for all ξ =
∑
j,k≥0 ξjk|ej⟩⟨ek| in Dom(L), defining ξ0 =
∑
j≥0 |ξjj |2ej⟩⟨ej |, it turns out that
E(ξ) = E(ξ0) +
∑
j,k≥0, j ̸=k
|ξjk|2 E(|ej⟩⟨ek|).
As a consequence we have the following
The Generalized Fibonacci Oscillator as an Open Quantum System 11
Proposition 5.1. Let g0 = inf
{
E(ξ) | ∥ξ∥ = 1, ξ ∈ D, ξ ⊥ ρ1/2
}
. The spectral gap of L is
gap(L) = min
{
g0, inf
j,k≥0, j ̸=k
E(|ej⟩⟨ek|)
}
.
We will now study separately the off-diagonal minimum and the diagonal spectral gap be-
ginning by the former that we can compute explicitly.
5.1 The off-diagonal minimum
Note that, for j ̸= k, L(|ej⟩⟨ek|) is given by (4.6) and the action of the generator L of the
semigroup in L2(h) of the invariant state L
(
ρ1/4xρ1/4
)
= ρ1/4L(x)ρ1/4 is the same by
L(|ej⟩⟨ek|) = ρ1/4L
(
ρ−1/4|ej⟩⟨ek|ρ−1/4
)
ρ1/4 = L(|ej⟩⟨ek|).
Therefore, by (4.6), it suffices to find the minimum on j ̸= k of
1
2
(
Γ−
j εj + Γ−
k εk + Γ+
j+1εj+1 + Γ+
k+1εk+1
)
=
1
2
(
eβωj εj
eβωj − 1
+
eβωk εk
eβωk − 1
+
εj+1
eβωj+1 − 1
+
εk+1
eβωk+1 − 1
)
. (5.1)
As a result, we can compute the off-diagonal minimum after the following preliminary
Lemma 5.2. The following hold:
1. The sequence (ωk)k≥1 is non-decreasing if and only if r + q ≥ 2.
2. If r + q ≥ 2, for any c ≥ 0 we have
εk
ωk
≥ 1 +
1
r + q + c
(5.2)
for all k ≥ 2 if and only if (1 + c)(r + q) + rq ≥ 0. In particular, if we fix c = 1, the
inequality (5.2) holds for all r > 1 and −2/3 ≤ q ≤ 1.
Proof. 1. Write ωk =
(
rk−1(r − 1) + qk−1(1− q)
)
/(r − q) and note that, for all k ≥ 1,
(r − q)(ωk+1 − ωk) = rk−1(r − 1)2 − qk−1(1− q)2
≥ (r − 1)2 − (1− q)2 = (r − q)(r + q − 2).
2. The claimed inequality is equivalent to ωk/εk ≤ 1 − 1/(r + q + c + 1) which is, in turn,
equivalent to
εk
εk−1
=
rk − qk
rk−1 − qk−1
≤ r + q + 1 + c. (5.3)
We show that it is equivalent to (1 + c)(r + q) + rq ≥ 0 distinguishing two cases according to
the sign of q.
If 0 ≤ q ≤ 1, then the sequence (εk/εk−1)k≥2 is non-increasing. Indeed defining f : [2,+∞[ →
]0,+∞[ by
f(x) =
(
rx − qx
)
/
(
rx−1 − qx−1
)
12 F. Fagnola, C.K. Ko and H.J. Yoo
we have
f ′(x) =
(rx log(r)− qx log(q))
(
rx−1 − qx−1
)
−
(
rx−1 log(r)− qx−1 log(q)
)
(rx − qx)
(rx−1 − qx−1)2
=
−(rq)x−1(r − q)(log(r)− log(q))(
rx−1 − qx−1
)2 ≤ 0.
Hence sup
k≥2
εk/εk−1 = ε2/ε1 = r+ q and (5.3) is obviously true as well as (1 + c)(r+ q) + rq ≥ 0.
If −1 ≤ q < 0, then considering g(x) =
(
r2x+1 + |q|2x+1
)
/
(
r2x − |q|2x
)
and h(x) =
(
r2x −
|q|2x
)
/
(
r2x−1 + |q|2x−1
)
instead of f(x), we immediately get
sup
k≥1
ε2k
ε2k−1
= lim
k→∞
ε2k
ε2k−1
= r, sup
k≥1
ε2k+1
ε2k
=
ε3
ε2
=
r2 + rq + q2
r + q
.
We easily see that r ≤
(
r2 + rq + q2
)
/(r + q) and so, in the case −1 ≤ q < 0, the supremum of
εk/εk−1 for k ≥ 2 is smaller than r+ q+1+ c if and only if
(
r2+ rq+ q2
)
/(r+ q) ≤ r+ q+1+ c
which is equivalent to (1 + c)(r + q) + rq ≥ 0.
Finally, if we fix c = 1, then (1 + c)(r + q) + rq ≥ 0 if and only if q ≥ −2r/(2 + r), and from
r ≥ 1, we find q ≥ −2/3. ■
Theorem 5.3. For −1 ≤ q ≤ 1 < r such that r + q > 1 there is a pair (j0, k0) (j0 ̸= k0) such
that gapoff-diag(L) = E(|ej0⟩⟨ek0 |). In particular, if r + q ≥ 2, −2/3 ≤ q ≤ 1 then the pair is
(0, 1) or (1, 0) and the off-diagonal minimum is given by
gapoff-diag(L) =
1
2
(
eβ + 1
eβ − 1
+
r + q
eβ(r+q−1) − 1
)
.
Proof. The first claim is an immediate consequence of lim
j→+∞
εj = +∞ and lim
j→+∞
ωj = +∞ so
that the first two terms in (5.1) diverge as j and k go to infinity.
Suppose now r + q ≥ 2. In order to find the minimum for j ̸= k of (5.1) we first note that,
for j = 1, k = 2 we have
eβωjεj
eβωj − 1
+
eβωkεk
eβωk − 1
+
εj+1
eβωj+1 − 1
+
εk+1
eβωk+1 − 1
∣∣∣
j=1,k=2
=
eβ
eβ − 1
+
eβω2ε2
eβω2 − 1
+
ε2
eβω2 − 1
+
ε3
eβω3 − 1
=
eβ
eβ − 1
+
eβ(r+q−1) (r + q)
eβ(r+q−1) − 1
+
r + q
eβ(r+q−1) − 1
+
ε3
eβω3 − 1
≥ eβ
eβ − 1
+
eβ(r+q−1)(r + q)
eβ(r+q−1) − 1
+
r + q
eβ(r+q−1) − 1
.
The right-hand side will be bigger or equal than
eβ + 1
eβ − 1
+
r + q
eβ(r+q−1) − 1
if the second term satisfies
eβ(r+q−1)(r + q)
eβ(r+q−1) − 1
≥ 1
eβ − 1
,
i.e., taking inverses, 1− e−β(r+q−1) ≤ (r + q)(eβ − 1) which holds true because
1− e−β(r+q−1) ≤ β(r + q − 1) < β(r + q) < (r + q)
(
eβ − 1
)
.
The Generalized Fibonacci Oscillator as an Open Quantum System 13
If 2 ≤ j < k, first recall that the sequence (ωk)k≥1 is non-decreasing by Lemma 5.2(1). for
r + q ≥ 2. Then note that functions on ]0,+∞[
x 7→ x eβx
eβx − 1
, x 7→
x
(
eβx + 1
)
eβx − 1
(5.4)
are increasing because
d
dx
xeβx
eβx − 1
=
eβx
(
eβx − 1− βx
)(
eβx − 1
)2 ≥ 0,
d
dx
x
(
eβx + 1
)
eβx − 1
=
sinh(βx)− βx
2(sinh(βx))2
≥ 0
by the elementary inequalities eβx ≥ 1+βx and sinh(βx) ≥ βx. Therefore we have the inequality
eβωj εj
eβωj − 1
+
εj+1
eβωj+1 − 1
+
eβωk εk
eβωk − 1
+
εk+1
eβωk+1 − 1
≥ εj
ωj
eβωj ωj
eβωj − 1
+
εj+1
eβωj+1 − 1
+
εk
ωk
eβωk ωk
eβωk − 1
dropping the last term in the left-hand side, multiplying and dividing the first (resp. third)
by ωj (resp. ωk). Now, by monotonicity of (5.4), multiplying and dividing the second term in
the right-hand side by ωj+1, we find
eβωjεj
eβωj − 1
+
εj+1
eβωj+1 − 1
+
eβωkεk
eβωk − 1
+
εk+1
eβωk+1 − 1
≥ εj
ωj
eβωjωj
eβωj − 1
+
εj+1
ωj+1
ωj+1
eβωj+1 − 1
+
εk
ωk
eβωj+1ωj+1
eβωj+1 − 1
.
Finally, by Lemma 5.2(2). and monotonicity of (5.4)
eβωjεj
eβωj − 1
+
εj+1
eβωj+1 − 1
+
eβωkεk
eβωk − 1
+
εk+1
eβωk+1 − 1
≥ r + q + 2
r + q + 1
(
eβωjωj
eβωj − 1
+
(
eβωj+1 + 1
)
ωj+1
eβωj+1 − 1
)
≥ r + q + 2
r + q + 1
(
eβω2ω2
eβω2 − 1
+
eβ + 1
eβ − 1
)
.
The difference of the right-hand side and twice the claimed lower bound is
ω2 + 3
ω2 + 2
(
eβω2 ω2
eβω2 − 1
+
eβ + 1
eβ − 1
)
−
(
ω2 + 1
eβω2 − 1
+
eβ + 1
eβ − 1
)
=
1
ω2 + 2
(
eβ + 1
eβ − 1
+
eβω2 ω2(ω2 + 3)− (ω2 + 1)(ω2 + 2)
eβω2 − 1
)
≥ 1
ω2 + 2
(
eβ + 1
eβ − 1
+
ω2(ω2 + 3)− (ω2 + 1)(ω2 + 2)
eβω2 − 1
)
=
1
ω2 + 2
(
1 +
2
eβ − 1
− 2
eβω2 − 1
)
≥ 1
ω2 + 2
(
1 +
2
eβ − 1
− 2
eβ − 1
)
=
1
ω2 + 2
≥ 0.
This shows the desired inequality. ■
Remark 5.4. It is worth noticing that the lower bound −2/3 on q can be relaxed to q ≥ q0
for some q0 ∈ [−1,−2/3[ at the price of a stronger lower bound the other parameter r > r0 for
some r0 ∈ ]1, 2]. It suffices to consider r0 such that q0 = −2r0/(2 + r0) so that, in Lemma 5.2
for c = 1 we have 2(r + q) + rq ≥ 0.
Moreover, for β small, it is not difficult to find values r > 1 and q near −1 for which the
off-diagonal minimum is attained at some (j0, j0 + 1) with j0 > 0.
In the following subsections we separately investigate the diagonal spectral gap for different
regions of the parameters.
14 F. Fagnola, C.K. Ko and H.J. Yoo
5.2 Diagonal spectral gap
As in the analysis of off-diagonal minima, parameters move in the region −1 ≤ q ≤ 1 < r with
a restriction r + q − 2 ≥ 0 so that the sequence (ωn)n≥1 is monotone increasing.
5.2.1 Lower bound
We already noted that, when restricted to the diagonal subalgebra our QMS reduces to the
Markov semigroup of a classical birth and death process with birth rates (λn)n≥0 and death
rates (µn)n≥1 given respectively by (4.5), namely
λn =
1
eβωn+1 − 1
rn+1 − qn+1
r − q
, µn =
eβωn
eβωn − 1
rn − qn
r − q
.
In detail, let ȷ be a map defined on the subspace of L2(h) consisting of the images of the
diagonal elements under the embedding ι into the sequence space defined as follows: for each
x =
∑
n≥0 xn|en⟩⟨en| ∈ B(h),
ȷ : ι(x) 7→ ξ = (xn)n≥0 ∈ ℓ2(π̃),
where π̃ denotes the probability density of the invariant measure of the aforementioned birth and
death process, i.e., π̃n = πn/Zβ = e−βεn/Zβ. We easily check that ȷ is a unitary isomorphism,
namely ∥ι(x)∥L2(h) = ∥ξ∥ℓ2(π̃). Let A be the generator of the classical birth and death process
with birth rates (λn) and death rates (µn) defined by
(Af)n =
{
µn(fn−1 − fn) + λn(fn+1 − fn), for n ≥ 1,
λ0(f1 − f0), for n = 0
for f = (fn)n≥0 ∈ ℓ2(π̃). For each diagonal element x =
∑
n≥0 xn|en⟩⟨en| ∈ B(h),
ȷ ◦ L(ι(x)) = A ◦ ȷ(ι(x)).
Therefore, the diagonal spectral gap of the generator of the QMS is equal to
inf
−⟨f,Af⟩ | ∥f∥2 = 1,
∑
n≥0
fnπ̃n = 0
,
where ⟨·, ·⟩ and ∥ · ∥ denote the inner product and the induced norm of ℓ2(π̃):
⟨f, g⟩ =
∑
n≥0
fngnπ̃n.
It is easy to see that
−⟨f,Af⟩ =
∑
n≥0
π̃nλn(fn+1 − fn)
2.
In order to compute the diagonal spectral gap we adopt the method described in [24] as in
[10, 16] and proceed as follows. or any f ∈ ℓ2(π̃) with
∑
n fnπ̃n = 0, by the Schwarz inequality,
∥f∥2 =
∑
y<x
(fy − fx)
2π̃xπ̃y ≤ Z−2
β
∑
x<y
(
y−1∑
u=x
(fu+1 − fu)
2
)
(y − x)πxπy
= Z−2
β
∑
u>0
(fu+1 − fu)
2πuλu
(∑
y>u πy
πuλu
) u−1∑
x=0
(u− x)πx
+ Z−2
β
∑
u≥0
(fu+1 − fu)
2πuλu
(∑
y>u(y − u)πy
πuλu
) u∑
x=0
πx. (5.5)
The Generalized Fibonacci Oscillator as an Open Quantum System 15
From the estimation (5.5), by using Lemmas A.1, A.2, A.3 and recalling that πu+1/πu = e−βωu+1 ,
we get
∥f∥2 ≤
∑
u>0
(fu+1 − fu)
2π̃uλuZ
−1
β
1
εu+1
(
u
1− e−β
−
e−β
(
1− e−βu
)(
1− e−β
)2
)
+
∑
u≥0
(fu+1 − fu)
2π̃uλuZ
−1
β
1
εu+1
1
1− e−βωu+1
1− e−β(u+1)
1− e−β
.
Noting that, since r + q ≥ 2 and so the sequence (ωk)k≥1 is non-decreasing,
εu+1 =
u+1∑
k=1
ωk ≥
u+1∑
k=1
ω1 = u+ 1 (5.6)
for all u ≥ 1 we have
Z−1
β
1
εu+1
(
u
1− e−β
−
e−β
(
1− e−βu
)(
1− e−β
)2 +
1
1− e−βωu+1
1− e−β(u+1)
1− e−β
)
= Z−1
β
1
εu+1
(
u+ 1
1− e−β
)
≤
Z−1
β
1− e−β
.
In addition, for u = 0,
Z−1
β
1
εu+1
1
1− e−βωu+1
1− e−β(u+1)
1− e−β
=
Z−1
β
1− e−β
,
and so
∥f∥2 ≤ −
Z−1
β
1− e−β
⟨f,Af⟩.
Finally, from the trivial inequality Zβ ≥ 1 + e−β, get the following result.
Theorem 5.5. Suppose that −1 ≤ q ≤ 1 < r with r + q − 2 ≥ 0. Then for all β > 0,
gap(A) ≥ Zβ
(
1− e−β
)
≥ 1− e−2β.
It turns out that, as the parameters change, in certain region the diagonal gap dominates
and in some other region the off-diagonal minimum dominates. For example, let us compare
the diagonal gap and off-diagonal minimum with fixed q = 1. If r > 1 is sufficiently large, then
the diagonal gap dominates the off-diagonal minimum (see Figure 1). On the other hand, when
r > 1 is near to 1 and β > 0 is sufficiently small, then the off-diagonal minimum dominates (see
Figure 2).
In order to better understand which one among the off-diagonal minimum and the diagonal
spectral gap is bigger and convince ourselves that, for β small, the spectral gap of the generator L
is actually given by the spectral gap of A, not just because of a poor estimate of the lower bound
of Theorem 5.5, we can study the upper bound of the diagonal spectral gap.
In Appendix B, by choosing a special f and evaluating −⟨f,Af⟩/∥f∥2, we found
gap(A) ≤ 1/
((
eβ − 1
)(
1− Z−1
β
))
showing that if β is sufficiently small and r > 1, q < 1 are sufficiently near to 1, then the
spectral gap of the generator L coincides with the one of A the diagonal subalgebra. Moreover,
we showed that the lower bound of Theorem 5.5 is near the optimal one for big β.
Summing up, from Proposition 5.1, Theorems 5.3 and 5.5, we get the following
Theorem 5.6. Suppose that −1 ≤ q ≤ 1 < r with r + q − 2 ≥ 0. Then for all β > 0,
gap(L) = min
{
1− e−2β,
1
2
(
eβ + 1
eβ − 1
+
r + q
eβ(r+q−1) − 1
)}
.
16 F. Fagnola, C.K. Ko and H.J. Yoo
Figure 1. Diagonal lower bound and off-diagonal minimum, β = 1.5, q = 1.
Figure 2. Diagonal lower bound and off-diagonal minimum, r = 2.
A Inequalities for partial sums
We collect here estimates on partial sums of series needed in the evaluation of the spectral gap.
Lemma A.1. For all u ∈ N we have∑
y>u(y − u)πy
πuλu
≤ 1
εu+1
(
1− e−βωu+1
) .
Proof. Recalling that the sequence (ωk)k≥1 is non-decreasing, by r + q − 2 ≥ 0, we can write∑
y>u(y − u)πy
πuλu
=
eβωu+1 − 1
εu+1
∑
y>u
(y − u)e−β(εy−εu) =
eβωu+1 − 1
εu+1
∑
y>u
(y − u)e−β
∑y
j=u+1 ωj
≤ eβωu+1 − 1
εu+1
∑
y>u
(y − u)e−β(y−u)ωu+1 =
1
εu+1 (1− e−βωu+1)
. ■
Lemma A.2. For all u ∈ N, u ≥ 1, we have
u−1∑
x=0
(u− x)πx ≤ u
1− e−β
−
e−β
(
1− e−βu
)(
1− e−β
)2 ,
u∑
x=0
πx ≤ 1− e−β(u+1)
1− e−β
.
Proof. Notice that εx =
∑x
j=1 ωj ≥ xω1 = x (inequality (5.6)). Then both follow from the
explicit summation formulae. ■
The Generalized Fibonacci Oscillator as an Open Quantum System 17
Lemma A.3. For the tail of the invariant measure we have the bound.
πu+1 ≤
∑
y>u
πy ≤ πu+1
1− e−βωu+1
.
Proof. The lower bound is obvious. For the upper bound,∑
y>u
πy =
∑
y>u
e−βεy = e−βεu
∑
y>u
e−β(εy−εu) = e−βεu
∑
y>u
e−β
∑y
j=u+1 ωj
≤ e−βεu
∑
y>u
e−β(y−u)ωu+1 = e−βεu e−βωu+1
1− e−βωu+1
=
e−βεu+1
1− e−βωu+1
=
πu+1
1− e−βωu+1
. ■
B An upper bound for the diagonal spectral gap
We first consider the limiting case: r → 1+, q → 1−. In that case the jump rates become
λn(β, r, q) −→
r→1+, q→1−
λn(β, 1, 1) =
n+ 1
eβ − 1
, µn(β, r, q) −→
r→1+, q→1−
µn(β, 1, 1) =
n eβ
eβ − 1
and, for small β (i.e., high temperatures), the diagonal spectral gap is near 1 as expected by
[10, Section 5] or [13, Section 7].
Next, we find an upper bound with a simple (fn)n≥0. Define for u ≥ 0, Zβ(u) :=
∑
x≥u πx;
Zβ := Zβ(0). Put f = (fn)n≥0 as
f0 = 1, fn = −c, for n ≥ 1,
where c is chosen so that
∑
x≥0 fxπx = 0, and hence c = π0/Zβ(1) = 1/(Zβ − 1),
E(f) =
∞∑
x=0
π̃xλx(fx+1 − fx)
2 = Z−1
β π0λ0(1 + c)2 =
Zβ λ0
Zβ(1)2
,
∥f∥2 =
∞∑
x=0
π̃xf
2
x = Z−1
β
(
π0 + c2Zβ(1)
)
=
1
Zβ(1)
.
Therefore,
gap(A) ≤
Zβ λ0
Zβ(1)
=
1(
eβ − 1
)(
1− Z−1
β
) := α(β, r, q).
Noticing that α(β, r, q) is continuous with respect to both β > 0, r > 1 and 0 < q < 1
α(β, 1, 1) := lim
r→1+, q→1−
α(β, r, q) =
1
1− e−β
.
On the other hand, comparing with the off-diagonal minimum for r = q = 1, which is
(
1 +
3e−β
)
/2
(
1− e−β
)
, we see that when 0 < β < log 3,
1
1− e−β
<
1 + 3e−β
2
(
1− e−β
) ,
which says, together with the continuity of both α(β, r, q) and the formula of off-diagonal mini-
mum, that if β is sufficiently small and r > 1, q < 1 are sufficiently near to 1, then the spectral
gap of the QMS occurs at the diagonal subalgebra.
18 F. Fagnola, C.K. Ko and H.J. Yoo
Fix q = 1 and let’s find regions in the r-β plane to see which gap would dominates. In
Figure 3, the upper line is the level curve found by equating the off-diagonal minimum and
the diagonal lower bound, 1 − e−2β. In the above the curve, the off-diagonal minimum is less
than the lower bound of the diagonal gap, and hence in that region the spectral gap occurs
in the off-diagonal subspace. Similarly, the lower line in the figure is the level curve obtained
by equating the off-diagonal minimum and the upper bound of the diagonal gap, α(β, r, 1). In
the region below the curve, the off-diagonal minimum dominates the diagonal upper bound and
therefore the spectral gap occurs in the diagonal subspace.
Figure 3. Gap dominators: above the upper curve, the spectral gap occurs at off-diagonal subspaces
and below the lower curve, the spectral gap occurs at the diagonal subspace (q = 1, horizontal axis for r
and vertical axis for β).
Acknowledgements
The work of HJY was supported by the National Research Foundation of Korea (NRF) grant
funded by the Korean government (MSIT) (no. 2020R1F1A101075). FF is a member of GNAM-
PA-INdAM Italy. FF first met Michael Schürmann at the conference Quantum Probability and
Applications III held in Oberwolfach, January 25–31, 1987, organized by their respective advisors
Professors Luigi Accardi and Wilhelm von Waldenfels [30]. Over most of these years he has had
the pleasure of meeting him at the annual QP conferences, that nowadays reached the number 42,
visiting him in Greifswald, exchange views and follow reports on his scientific work. He would
like to congratulate Michael on his retirement and wish him endless happy days with his friends
and family.
References
[1] Accardi L., Garćıa-Corte J.C., Guerrero-Poblete F., Quezada R., Breaking of the similarity principle in
Markov generators of low density limit type and the role of degeneracies in the landscape of invariant states,
Open Syst. Inf. Dyn. 27 (2020), 2050018, 40 pages.
[2] Accardi L., Lu Y.G., The Wigner semi-circle law in quantum electrodynamics, Comm. Math. Phys. 180
(1996), 605–632.
[3] Accardi L., Lu Y.G., Volovich I., Quantum theory and its stochastic limit, Springer-Verlag, Berlin, 2002.
[4] Anderson W.J., Continuous-time Markov chains: an applications-oriented approach, Springer Series in
Statistics: Probability and its Applications, Springer-Verlag, New York, 1991.
[5] Becker S., Datta N., Salzmann R., Quantum Zeno effect in open quantum systems, Ann. Henri Poincaré
22 (2021), 3795–3840, arXiv:2010.04121.
[6] Blitvić N., The (q, t)-Gaussian process, J. Funct. Anal. 263 (2012), 3270–3305.
https://doi.org/10.1142/S1230161220500183
http://projecteuclid.org/euclid.cmp/1104287457
https://doi.org/10.1007/978-3-662-04929-7
https://doi.org/10.1007/978-1-4612-3038-0
https://doi.org/10.1007/s00023-021-01075-8
https://arxiv.org/abs/2010.04121
https://doi.org/10.1016/j.jfa.2012.08.006
The Generalized Fibonacci Oscillator as an Open Quantum System 19
[7] Bożejko M., Remarks on q-CCR relations for |q| > 1, in Noncommutative Harmonic Analysis with Applica-
tions to Probability, Banach Center Publ., Vol. 78, Polish Acad. Sci. Inst. Math., Warsaw, 2007, 59–67.
[8] Bożejko M., Kümmerer B., Speicher R., q-Gaussian processes: non-commutative and classical aspects,
Comm. Math. Phys. 185 (1997), 129–154, arXiv:funct-an/9604010.
[9] Bożejko M., Lytvynov E., Wysoczański J., Fock representations of Q-deformed commutation relations,
J. Math. Phys. 58 (2017), 073501, 19 pages, arXiv:1603.03075.
[10] Carbone R., Fagnola F., Exponential L2-convergence of quantum Markov semigroups on B(h), Math. Notes
68 (2000), 452–463.
[11] Carbone R., Sasso E., Umanità V., Structure of generic quantum Markov semigroup, Infin. Dimens. Anal.
Quantum Probab. Relat. Top. 20 (2017), 1750012, 19 pages.
[12] Carlen E.A., Maas J., Gradient flow and entropy inequalities for quantum Markov semigroups with detailed
balance, J. Funct. Anal. 273 (2017), 1810–1869, arXiv:1609.01254.
[13] Cipriani F., Fagnola F., Lindsay J.M., Spectral analysis and Feller property for quantum Ornstein–Uhlenbeck
semigroups, Comm. Math. Phys. 210 (2000), 85–105.
[14] Daoud M., Kibler M., Statistical mechanics of qp-bosons in D dimensions, Phys. Lett. A 206 (1995), 13–17,
arXiv:quant-ph/9512006.
[15] Deschamps J., Fagnola F., Sasso E., Umanità V., Structure of uniformly continuous quantum Markov
semigroups, Rev. Math. Phys. 28 (2016), 1650003, 32 pages, arXiv:1412.3239.
[16] Dhahri A., Fagnola F., Yoo H.J., Quadratic open quantum harmonic oscillator, Lett. Math. Phys. 110
(2020), 1759–1782, arXiv:1905.09965.
[17] Fagnola F., Quantum Markov semigroups and quantum flows, Proyecciones 18 (1999), 95–134.
[18] Fagnola F., Quezada R., A characterization of quantum Markov semigroups of weak coupling limit type,
Infin. Dimens. Anal. Quantum Probab. Relat. Top. 22 (2019), 1950008, 10 pages.
[19] Fagnola F., Umanità V., Generic quantum Markov semigroups, cycle decomposition and deviation from
equilibrium, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 (2012), 1250016, 17 pages.
[20] Gavrilik A.M., Kachurik I.I., Nonstandard deformed oscillators from q- and p, q-deformations of Heisenberg
algebra, SIGMA 12 (2016), 047, 12 pages, arXiv:1602.07927.
[21] Gerhold M., Skeide M., Interacting Fock spaces and subproduct systems, Infin. Dimens. Anal. Quantum
Probab. Relat. Top. 23 (2020), 2050017, 53 pages, arXiv:1808.07037.
[22] Gough J., Kupsch J., Quantum fields and processes: a combinatorial approach, Cambridge Studies in
Advanced Mathematics, Vol. 171, Cambridge University Press, Cambridge, 2018.
[23] Kumar D., Sinha K.B., Srivastava S., Stability of the Markov (conservativity) property under perturbations,
Infin. Dimens. Anal. Quantum Probab. Relat. Top. 23 (2020), 2050009, 20 pages.
[24] Liggett T.M., Exponential L2 convergence of attractive reversible nearest particle systems, Ann. Probab. 17
(1989), 403–432.
[25] Marinho A.A., Brito F.A., Hermite polynomials and Fibonacci oscillators, J. Math. Phys. 60 (2019), 012101,
9 pages, arXiv:1805.03229.
[26] Mouayn Z., El Moize O., A generalized Euler probability distribution, Rep. Math. Phys. 87 (2021), 291–311.
[27] Mukhamedov F., Al-Rawashdeh A., Generalized Dobrushin ergodicity coefficient and uniform ergodicities
of Markov operators, Positivity 24 (2020), 855–890, arXiv:2001.06266.
[28] Mukhamedov F., Al-Rawashdeh A., Generalized Dobrushin ergodicity coefficient and ergodicities of non-
homogeneous Markov chains, Banach J. Math. Anal. 16 (2022), 18, 37 pages, arXiv:2001.07703.
[29] Parthasarathy K.R., An introduction to quantum stochastic calculus, Modern Birkhäuser Classics,
Birkhäuser/Springer Basel AG, Basel, 1992.
[30] Schürmann M., von Waldenfels W., A central limit theorem on the free Lie group, in Quantum Probability
and Applications, III (Oberwolfach, 1987), Lecture Notes in Math., Vol. 1303, Springer, Berlin, 1988, 300–
318.
https://doi.org/10.4064/bc78-0-4
https://doi.org/10.1007/s002200050084
https://arxiv.org/abs/funct-an/9604010
https://doi.org/10.1063/1.4991671
https://arxiv.org/abs/1603.03075
https://doi.org/10.1007/BF02676724
https://doi.org/10.1142/S0219025717500126
https://doi.org/10.1142/S0219025717500126
https://doi.org/10.1016/j.jfa.2017.05.003
https://arxiv.org/abs/1609.01254
https://doi.org/10.1007/s002200050773
https://doi.org/10.1016/0375-9601(95)00580-V
https://arxiv.org/abs/quant-ph/9512006
https://doi.org/10.1142/S0129055X16500033
https://arxiv.org/abs/1412.3239
https://doi.org/10.1007/s11005-020-01274-0
https://arxiv.org/abs/1905.09965
https://doi.org/10.22199/s07160917.1999.0003.00006
https://doi.org/10.1142/S0219025719500085
https://doi.org/10.1142/S0219025712500166
https://doi.org/10.3842/SIGMA.2016.047
https://arxiv.org/abs/1602.07927
https://doi.org/10.1142/S0219025720500174
https://doi.org/10.1142/S0219025720500174
https://arxiv.org/abs/1808.07037
https://doi.org/10.1017/9781108241885
https://doi.org/10.1142/S0219025720500095
https://doi.org/10.1214/aop/1176991408
https://doi.org/10.1063/1.5040016
https://arxiv.org/abs/1805.03229
https://doi.org/10.1016/S0034-4877(21)00038-0
https://doi.org/10.1007/s11117-019-00713-0
https://arxiv.org/abs/2001.06266
https://doi.org/10.1007/s43037-021-00173-3
https://arxiv.org/abs/2001.07703
https://doi.org/10.1007/978-3-0348-0566-7
https://doi.org/10.1007/BFb0078071
1 Introduction
2 Fibonacci oscillators
3 QMS of weak coupling limit type
4 Generic open Fibonacci type oscillators
5 Spectral gap
5.1 The off-diagonal minimum
5.2 Diagonal spectral gap
5.2.1 Lower bound
A Inequalities for partial sums
B An upper bound for the diagonal spectral gap
References
|
| id | nasplib_isofts_kiev_ua-123456789-211633 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T10:37:46Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Fagnola, Franco Ko, Chul Ki Yoo, Hyun Jae 2026-01-07T13:42:24Z 2022 The Generalized Fibonacci Oscillator as an Open Quantum System. Franco Fagnola, Chul Ki Ko and Hyun Jae Yoo. SIGMA 18 (2022), 035, 19 pages 1815-0659 2020 Mathematics Subject Classification: 81S22; 81S05; 60J80 arXiv:2202.02196 https://nasplib.isofts.kiev.ua/handle/123456789/211633 https://doi.org/10.3842/SIGMA.2022.035 We consider an open quantum system with Hamiltonian ₛ whose spectrum is given by a generalized Fibonacci sequence weakly coupled to a Boson reservoir in equilibrium at inverse temperature . We find the generator of the reduced system evolution and explicitly compute the stationary state of the system, which turns out to be unique and faithful, in terms of the parameters of the model. If the system Hamiltonian is generic, we show that convergence towards the invariant state is exponentially fast and compute explicitly the spectral gap for low temperatures, when quantum features of the system are more significant, under an additional assumption on the spectrum of ₛ. The work of HJY was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (no. 2020R1F1A101075). FF is a member of GNAMPA-INdAM Italy. FF first met Michael Schurmann at the conference Quantum Probability and Applications III held in Oberwolfach, January 25–31, 1987, organized by their respective advisors, Professors Luigi Accardi and Wilhelm von Waldenfels [30]. Over most of these years, he has had the pleasure of meeting him at the annual QP conferences, which nowadays reach the number 42, visiting him in Greifswald, exchanging views, and following reports on his scientific work. He would like to congratulate Michael on his retirement and wish him endless happy days with his friends and family. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Generalized Fibonacci Oscillator as an Open Quantum System Article published earlier |
| spellingShingle | The Generalized Fibonacci Oscillator as an Open Quantum System Fagnola, Franco Ko, Chul Ki Yoo, Hyun Jae |
| title | The Generalized Fibonacci Oscillator as an Open Quantum System |
| title_full | The Generalized Fibonacci Oscillator as an Open Quantum System |
| title_fullStr | The Generalized Fibonacci Oscillator as an Open Quantum System |
| title_full_unstemmed | The Generalized Fibonacci Oscillator as an Open Quantum System |
| title_short | The Generalized Fibonacci Oscillator as an Open Quantum System |
| title_sort | generalized fibonacci oscillator as an open quantum system |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211633 |
| work_keys_str_mv | AT fagnolafranco thegeneralizedfibonaccioscillatorasanopenquantumsystem AT kochulki thegeneralizedfibonaccioscillatorasanopenquantumsystem AT yoohyunjae thegeneralizedfibonaccioscillatorasanopenquantumsystem AT fagnolafranco generalizedfibonaccioscillatorasanopenquantumsystem AT kochulki generalizedfibonaccioscillatorasanopenquantumsystem AT yoohyunjae generalizedfibonaccioscillatorasanopenquantumsystem |