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On the open Dicke-type model generated by an infinite-component vector spin

Ryota Kyokawa†, Hajime Moriya† and Hiroshi Tamura† 1

†Graduate School of the Natural Science and Technology, Kanazawa University, Kakuma-machi Kanazawa 920-1192, Japan

AbstractWe consider an open Dicke model comprising a single infinite-component vector spinand a single-mode harmonic oscillator which are connected by Jaynes–Cummings-typeinteraction between them. This open quantum model is referred to as the OISD (OpenInfinite-component Spin Dicke) model. The algebraic structure of the OISD Liouvillian isstudied in terms of superoperators acting on the space of density matrices. An explicit in-vertible superoperator (precisely, a completely positive trace-preserving map) is obtainedthat transforms the OISD Liouvillian into a sum of two independent Liouvillians, onegenerated by a dressed spin only, the other generated by a dressed harmonic oscillatoronly. The time evolution generated by the OISD Liouvillian is shown to be asymptoticallyequivalent to that generated by an adjusted decoupled Liouvillian with some synchronizedfrequencies of the spin and the harmonic oscillator. This asymptotic equivalence impliesthat the time evolution of the OISD model dissipates completely in the presence of any(tiny) dissipation.

Keywords : open quantum dynamics, Liouvillian, decoupling, open Dicke model, infinite-component vector spin, synchronization

1 Introduction

Investigating open quantum dynamical systems described by the Gorini–Kossakowski–Sudarshan–Lindblad–Davies-type master equation [1, 2, 3, 4] is fundamentally importantfor researching quantum optics [5]. In general, it is difficult to understand the dynamicalproperties of open quantum models, even qualitatively. However, some known modelshave been studied to date. The simplest but nevertheless important examples are openharmonic oscillator models whose dynamics can be analyzed by various methods; forexample, see [6, 7, 8]. Note that the (original) Dicke model [9] pertains to non-trivialinteraction between light and matter and has been subjected to intensive research; see[10] and the references therein. Needless to say, open-type Dicke models as in [11] giveanother prototype of open quantum models, thereby stimulating both theoretical andnumerical investigations; for example, see [12] and the references therein. However, open

1present address: Komatsu University, Shichoumachi nu 1-3, Komatsu 923-8511, Japan

1

Dicke models are difficult to analyze in general. The present paper gives a concrete openDicke model to which a rigorous approach is possible.

In order to explain our motivation in a broad context, we pose the following question.To understand a complicated quantum model, what should we do first? Not in every case,but sometimes it is useful to transform it into a simpler model. As a special realization ofthis simple idea, we recall Bogoljubov’s method, which has been used widely in condensedmatter physics and quantum field theory [13]. Bogoljubov’s method is tightly relatedto the concept of quasiparticles, and it is essential for many important physics modelssuch as Bardeen–Cooper–Schrieffer models and polaron models [14]. However, althoughBogoljubov’s method has many applications as mentioned above, it seems that its rolesin open quantum systems are yet to be explored fully: Let us mention a recent work [15]that treats the dynamics of open fermion models. On that basis, we seek concrete openquantum models to which a Bogoljubov-like method can apply. In fact, our open Dickemodel (mentioned above and specified later) is such an example, although it is somewhatartificial as a physics model.

In our Bogoljubov-like method applied to open dynamics, we make use of a completelypositive and trace preserving (CPTP) map for its similarity transformation. Althoughour argument is limited to this particular model only, we advocate a general scheme fordealing with open quantum dynamics as follows: To simplify (e.g., decouple, diagonalize)Lindblad operators of open quantum systems, we make use of CPTP maps for similaritytransformations rather than unitary maps used in closed quantum systems.

We specify our open-type Dicke model. First, its radiation field is given by a single-mode harmonic oscillator as in the original Dicke model [9]. Second, its matter is given bya single infinite-component vector spin. Third, we add dissipation in the following way:the harmonic oscillator is implicitly connected to a thermal bath, whereas the matter mayor may not be connected to a thermal bath. Finally, assuming a Jaynes–Cummings-typeinteraction between the harmonic oscillator and the spin, we obtain our open infinite-spinDicke model. Hereinafter for abbreviation, we refer to this model as the OISD model.

We provide some background to the OISD model. First, we note that infinite-componentvector spins can be extracted from n-compound quantum spin systems by a certain lim-iting procedure n → ∞. The infinite-component spin Dicke model, ISD for short, is aconserved (non-dissipative) quantum model that is generated by an infinite-componentvector spin and a single-mode harmonic oscillator as described above. It has been inves-tigated in [16, 17, 18].

Notably, the OISD model has both dissipation (induced by a hidden thermal bath)and non-trivial interaction (between the matter and the radiation). We investigate theirinterplay and effects upon the open quantum dynamics. Through the decoupling trans-formation, we delete the interaction producing new dissipation terms in both the spin andthe harmonic oscillator. Under this transformation, no dissipation term in the form of amixture of the spin and the harmonic oscillator appears.

This paper is divided into several parts as follows. In §2, we discuss the algebraicproperties of the open harmonic oscillator model in a self-contained manner. Althoughsome of our results in that section are either obvious or easily derived from known results

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[7, 19, 20, 21, 22], our reformulation based on the algebra of superoperators is usefulfor gaining information about the OISD model as well. In §3, we introduce the infinite-component spin system. We show that dissipation (even a tiny amount) added to theinfinite-component spin makes the system thoroughly unstable. The special algebraicstructure of the infinite-component spin renders the OISD model tractable by purelymathematical methods (without resorting to numerical computations). In §4, we definethe OISD model and investigate its Liouvillian based on §2 and §3. In Proposition 4.1, weshow that the OISD model is decomposed into a sum of two independent Liouvillians bya similarity transformation generated by a CPTP superoperator (and its inverse). Here,independence means that one is written by a dressed spin only, while the other is writtenby a dressed harmonic oscillator only. As a result of the decomposition, the dynamicsof the OISD model are examined. Proposition 4.2 shows that the time evolution of theOISD model is asymptotically equivalent to that of an decoupled Liouvillian. This de-coupled Liouvillian is adjusted so that it has synchronized frequencies of the spin andthe harmonic oscillator. See [23] for general information about synchronization for openquantum dynamics. Note that the decoupled Liouvillian in Proposition 4.1 and the ad-justed decoupled Liouvillian in Proposition 4.2 differ. Furthermore, in Proposition 4.2 itis shown that in the presence of any (tiny) dissipation, the time evolution of the OISDmodel dissipates completely as time tends to infinity.

We now introduce the mathematical notation that is used throughout this paper. LetH be an infinite-dimensional Hilbert space. Let B(H ) be the set of all bounded linearoperators on H and C1(H ) be the Banach algebra of the trace class operators on H withtrace norm ‖ · ‖1. For linear operators A, B, and C acting in H , we define superoperatorsKA and DB◦C acting in C1(H ) by

KA(ρ) := [A, ρ], DB◦C(ρ) := 2BρC − {CB, ρ} for ρ ∈ C1(H ). (1)

We can verify the following relations by straightforward calculations:

[KA,KB] = K[A,B], [KA,DB◦C ] = D[A,B]◦C +DB◦[A,C], (2)

[DA◦B,DC◦D] = D[A,C]◦{B,D} −D{A,C}◦[B,D] +D[DC,A]◦B

+DA◦[B,DC] −D[BA,C]◦D −DC◦[D,BA] +K[BA,DC], (3)

DA◦1 = KA, D1◦B = −KB. (4)

We investigate the open quantum dynamics on H determined by the following typeof master equation for ρ(t) ∈ C1(H ):

d

dtρ(t) = −iKH(ρ(t)) +

∑

j

DAj◦A†j(ρ(t)) , (5)

where KH represents the infinitesimal change induced by a Hamiltonian H of the corre-sponding closed quantum system, and the term

∑

j DAj◦A†jwith a set of operators {Aj,

A†j} represents the effect of dissipation.

3

We make some remarks about the paper. We assume (implicitly) that superoperatorsact on the space of trace class operators rather than on the space of Hilbert–Schmidtoperators; in some literature of mathematical physics, the Hilbert–Schmidt class seemsto be more common. Throughout this paper, linear operators in a Hilbert space andsuperoperators are generically unbounded. From a mathematics standpoint, our resultis a realization of a particular open quantum dynamics with unbounded generators. Werefer to [24], which suggests general investigation of unbounded open quantum dynamicalsystems. Nevertheless, we put more emphasis on explicit calculations of this particularmodel rather than on general arguments based on functional analysis.

2 Algebraic structure of open harmonic oscillator

Because of the fundamental importance of the open harmonic oscillator in open quantumsystems, it is worth considering yet another formalism that is easier to handle. We givesuch in terms of the algebra of superoperators on the space of trace class operators on theone-mode Fock space.

Let a and a† denote the annihilation and creation operators, respectively, acting in theone-mode Fock space F . The number operator is denoted as N = a†a. It is known thatthe Fock space F has the following complete orthonormal system { |n 〉 }n∈N∪{0}, suchthat

N |n〉 = n|n〉 (n ∈ N ∪ {0}), (6)

and

a|0〉 = 0, a†|0〉 = |1〉, a |n〉 =√n |n− 1〉, a† |n〉 =

√n + 1 |n+ 1〉 (n ∈ N). (7)

Let us consider the master equation

d

dtρ(t) = Lph(ρ(t)) for ρ(t) ∈ C1(F ) (8)

with its Liouvillian

Lph := −iωKN + γ(

(J + 1)Da◦a† + JDa†◦a

)

, (9)

where ω > 0 denotes the angular frequency of the oscillator, γ > 0 denotes the strength ofdissipation, and J > 0 is a parameter related to the temperature (of the hidden thermalbath) [5]. The subscript of Lph signifies “photon.”

We consider the algebraic structure of the Liouvillian Lph and its constituents. From(2-4) and [a, a†] = 1, we have

[Ka,Ka† ] = 0 , [Ka,KN ] = Ka , [Ka† ,KN ] = −Ka† , (10)

[Ka,Da◦a† ] = Ka , [Ka† ,Da◦a† ] = Ka† , (11)

[Ka,Da†◦a] = −Ka , [Ka† ,Da†◦a] = −Ka† (12)

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and[KN ,Da◦a† ] = [KN ,Da†◦a] = 0 , (13)

[Da◦a† ,Da†◦a] = −2(

Da◦a† +Da†◦a

)

. (14)

We then investigate the eigenvalue problem of the Liouvillian (9) using the algebraicrelations listed above. For convenience, we define Φn,m = Kn

a†Kma (|0〉〈0|) for n,m =

0, 1, 2, · · · . We see that the set {Φn,m}∞n,m=0 is total in C1(F ), namely its linear spanis a dense subspace of C1(F ). In fact, for each non-negative integer k, the identityLin.Span{Φn,m |n > 0, k > m > 0} =Lin.Span{ |n〉〈m| |n > 0, k > m > 0} holds.

We can show this set of identities by induction on k noting Φn,0 =√n! |n〉〈0| and√

m+ 1 |n〉〈m + 1| =√n |n − 1〉〈m| − Ka(|n〉〈m|). Using (10) and (11) inductively,

we have

KN(Φn,m) = (n−m)Φn,m , Da◦a†(Φn,m) = −(n+m)Φn,m (n,m ∈ N∪{0} ) . (15)

Thus all Φn,m are common eigenvectors for both KN and Da◦a† , but not for Da†◦a. In thenext paragraph, we focus on Da†◦a.

We investigate the semigroup generated by Da†◦a. Set

St(ρ) =∞∑

n=0

(1− e−2t)n

n!a†ne−taa†ρ e−taa†an (16)

for t > 0 and ρ ∈ C1(F ). It is straightforward to derive the following properties:

d

dtSt(ρ) = Da†◦a(St(ρ)) , St1+t2(ρ) = St1

(

St2(ρ))

, S0(ρ) = ρ . (17)

The semigroup St defined above is CPTP because by definition (16) it has the followingform of CPTP maps (e.g., see [25] §§8.2.4, [26]):

St(ρ) =

∞∑

n=0

En(t)ρE†n(t) (18)

with bounded operators En(t) satisfying the normalization condition

∞∑

n=0

E†n(t)En(t) = 1 (19)

for every t > 0. Precisely, conditions (18) and (19) are satisfied by

En(t) =(1− e−2t)n/2√

n!a†ne−taa† ∈ B(F ) . (20)

Similarly Da◦a† generates a one-parameter semigroup of CPTP maps on C1(F ), as thefollowing formula holds:

etDa◦a† (ρ) =∞∑

n=0

(e2t − 1)n

n!ane−ta†aρ e−ta†aa†n . (21)

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We show the following identity of superoperators:

eτDa†◦aDa◦a† =(

e2τDa◦a† + (e2τ − 1)Da†◦a

)

eτDa†◦a (t > 0) . (22)

Denote the right-hand side of (22) by Xτ . Then we immediately see that X0 = Da◦a† andddτXτ = Da†◦aXτ because of (14).Setting τ > 0 by

e2τ = J + 1 (23)

together with (13), we have

eτDa†◦a

(

− iωKN + γDa◦a†)

= Lph eτD

a†◦a . (24)

Because of (15), (22), and (24), we obtain(

(J+1)Da◦a†+JDa†◦a

)

eτDa†◦a(Φn,m) = −(n+m)eτDa†◦a(Φn,m) (n,m ∈ N∪{0}) (25)

and

LpheτD

a†◦a(Φn,m) =(

− iω(n−m)− γ(n+m))

eτDa†◦a(Φn,m) (n,m ∈ N ∪ {0}) . (26)

To see that this gives the complete solution of the eigenvalue problem for Lph, it isenough to show that the range of eτDa†◦a is dense in C1(F ), because {Φn,m}∞n,m=0 is totalin C1(F ). We note that the right-hand side of (16) is still well defined for t < 0 andgives a densely defined operator as the (left) inverse of e|t|Da†◦a when |t| is small (i.e.,|1 − e−2t| < 1). Hence the range of eτDa†◦a is dense in C1(F ) for small τ > 0 . Thebounded semigroup property ensures the same for arbitrary τ > 0. We refer to previouswork [7, 19, 20, 21] for the information regarding the eigenvalue problem of Lph.

Let us consider the semigroup etLph . Because of the commutativity (13), we have

etLph = e−itωKN etγ(

(J+1)Da◦a†

+JDa†◦a

)

. (27)

We have also

etγ(

(J+1)Da◦a†

+JDa†◦a

)

= eτ1(tγ)Da†◦ae(tγ+τ1(tγ))Da◦a† , (28)

where

τ1(s) :=1

2log(J + 1− Je−2s) for s > 0 . (29)

To see (28), we set its right-hand side as Y (t). Then Y (0) = 1 holds and its derivativesatisfies

dY (t)

dt=γτ ′1(tγ)Da†◦ae

τ1(tγ)Da†◦ae(tγ+τ1(tγ))Da◦a†

+ eτ1(tγ)Da†◦aγ(

1 + τ ′1(tγ))

Da◦a†e(tγ+τ1(tγ))Da◦a†

=γ(

τ ′1(tγ)Da†◦a +(

1 + τ ′1(tγ))

(e2τ1(tγ)Da◦a† + (e2τ1(tγ) − 1)Da†◦a))

Y (t)

=γ(

(J + 1)Da◦a† + JDa†◦a

)

Y (t) ,

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where we have used (22). With (27), (28), and (29), we obtain the following decompositionformula of etLph :

etLph = e−itωKN eτ1(tγ)Da†◦ae(tγ+τ1(tγ))Da◦a† . (30)

Because e−itωKN (which generates a unitary evolution) is obviously a CPTP semigroupon C1(F ), and both etDa†◦a and etDa◦a† are CPTP maps as we have seen, etLph given asthe composition of these CPTP maps in (30) is also a CPTP map.

In the following, we discuss the asymptotic behavior of the dynamical semigroup{etLph}t>0. From (16) and (23), we have

eτDa†◦a (Φ0,0) = Sτ (Φ00) =∞∑

n=0

e−βωn

Z|n〉〈n | ≡ ρph,G, β = ω−1 log(1 + J−1). (31)

The right-hand side of the above equality is the Gibbs state with respect to the Hamil-tonian ωN at the inverse temperature β. It follows from (26) that ρ(t) approaches theGibbs state (31) in trace norm as t → ∞ from arbitrary initial state ρ(0). To see thisasymptotic property, we note that the trace-preserving property of eτDa†◦a and the traceproperty yield

T[eτDa†◦a(Φn,m)] = T[Φn,m] = δn,0δm,0 (n,m ∈ N ∪ {0}) . (32)

Because the trace-preserving map eτDa†◦a : C1(F ) → C1(F ) has dense range with respectto the trace norm, any density matrix ρ0 is approximated by the finite linear combination

∑

n,m>0

cn,meτD

a†◦a(Φn,m), cn,m ∈ C

in C1(F ). Note that c0,0 = 1 because of T(ρ0) = 1 and (32). From (26) and (31), weobtain

limt→∞

etLph(

ρ0)

=∑

n,m>0

cn,m limt→∞

etLph(

eτDa†◦a(Φn,m))

= eτDa†◦a(Φ0,0) = ρph,G . (33)

The above heuristic limiting procedure can be made rigorous by using the uniform bound-edness of CPTP semigroup { etLph }t>0. From (33) it follows that for any t > 0,

etLph(

ρph,G)

= ρph,G . (34)

Recall that the Gibbs state ρph,G is with respect to the Hamiltonian ωN , so it is alsoinvariant under the unitary evolution generated by KN : For any t ∈ R, we have

e−itωKN (ρph,G) = ρph,G . (35)

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3 Infinite-component vector spin system

For the matter, we consider an infinite-component spin system [17]. We introduce thissystem by employing the algebraic formulation as in §2.

Let G be a Hilbert space and { |n)}n∈Z be a complete orthonormal system of G . Asoperators on G , we consider l± and M defined by

M |n) = n |n), l±|n) = |n± 1) (n ∈ Z) . (36)

The following fundamental relations hold:

[M, l±] = ±l±, l+l− = l−l+ = 1. (37)

Remark 3.1. In the Dicke model, the n-compound system of two-component spins isresolved into a set of irreducible representations. Picking a representation whose totalspin is ℓ from the set, we realize it in G as

J±,ℓ |m) =√

(ℓ∓m)(ℓ±m+ 1) |m± 1), J3,ℓ |m) = m |m) ( |m| 6 ℓ ),

J±,ℓ |m) = J3,ℓ |m) = 0 ( |m| > ℓ ) .

Then we have operators l± as

s- limℓ→∞

J±,ℓ

ℓ= l±,

where s-lim stands for the strong limit of operators on the Hilbert space G , that is,J±,ℓ|ψ〉/ℓ → l±|ψ〉 in the norm of G for arbitrary |ψ〉 ∈ G . The convergence J3,ℓ → Malso holds in norm on the domain of M . Moreover in [17], it has been shown that theHamiltonian

Hℓ = ω 1⊗ a†a+ µ J3,ℓ ⊗ 1+λ

ℓ

(

J+,ℓ ⊗ a+ J−,ℓ ⊗ a†)

(38)

of the (2ℓ+ 1)-component Dicke model acting in G⊗

F converges to the Hamiltonian

H = ω 1⊗ a†a+ µM ⊗ 1+ λ(

l+ ⊗ a + l− ⊗ a†)

(39)

of the ISD model in the strong generalized sense, i.e.,

s- limℓ→∞

e−itHℓ = e−itH

on G⊗

F . Intuitively, those strong limits give the idealization of the property of cor-responding operators that hold for the action only on the vectors consisting of linearcombinations of |m) satisfying |m| ≪ ℓ for large but finite ℓ. In the rest of this section,we deal with the dissipative infinite-component spin system described by l± and M . Weconsider the open system on G

⊗

F given by the Hamiltonian (39) and dissipation termsin the next section.

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We consider the master equation

d

dtρ(t) = Lsp(ρ(t)) for ρ(t) ∈ C1(G ) , (40)

where the Liouvillian is defined by

Lsp := −iµKM + α−Dl−◦l+ + α+Dl+◦l− (41)

with constants µ > 0, α± > 0. The subscript of Lsp signifies “spin.” We consider mainlythe case of α± = 0, i.e., no dissipation for the spin as a component of the OISD model in§4. However, it will become clear that treating both cases with or without dissipation onan equal footing is helpful for our discussion.

We discuss the behavior of the open quantum dynamics generated by the LiouvillianLsp. Its qualitative picture may be described as follows. Because its Hamiltonian partgenerated byM is not lower bounded (36), it is considered to be unstable. One may guessthat if we add dissipation as in the Liouvillian (41), then any density matrix made bythe eigenstates { |n)}n∈Z for M will be no longer stable under the dissipative dynamicsgenerated by Lsp. Actually, we verify this naive picture in the following.

We consider the algebraic structure of the Liouvillian Lsp and its constituents. From(2), (3), and (37), we have

[KM ,Dl−◦l+ ] = [KM ,Dl+◦l− ] = [Dl−◦l+ ,Dl+◦l−] = 0 . (42)

We have alsoe−iµtKM (ρ) = e−iµtMρ eiµtM , (43)

etα−Dl−◦l+ (ρ) =

∞∑

n=0

(2tα−)n

n!e−2tα− ln−ρl

n+, (44)

and

etα+Dl+◦l− (ρ) =∞∑

m=0

(2tα+)m

m!e−2tα+ lm+ρl

m− (45)

for ρ ∈ C1(G ). From these expressions combined with the commutativity relations (42),the solution of the master equation is given as

etLsp(ρ) =etα−Dl−◦l+etα+Dl+◦l−e−iµtKM (ρ)

=

∞∑

n,m=0

(2tα−)n(2tα+)

m

n!m!e−2t(α−+α+)ln−m

− e−iµtMρ eiµtM ln−m+

=

∞∑

k=−∞

ck(t)lk−e

−iµtMρ eiµtM lk+, (46)

where

ck(t) ≡∞∑

n,m=0

δn−m,k(2tα−)

n(2tα+)m

n!m!e−2t(α−+α+) . (47)

9

The set of coefficients { ck(t) }k∈Z given in (47) may be considered as a time-dependentprobability distribution in the sense

ck(t) > 0,

∞∑

k=−∞

ck(t) = 1 , (48)

from which it follows that the evolution (46) has the CPTP property. On the other hand,the behavior of its mean and variance, namely

∞∑

k=−∞

kck(t) = 2(α− − α+)t,

∞∑

k=−∞

k2ck(t)−(

∞∑

k=−∞

kck(t))2

= 2(α− + α+)t, (49)

exhibits the floating and diffusive nature of the evolution. Moreover, it can be shown that

s- limt→∞

etLsp(ρ) = 0 (50)

holds for arbitrary initial state ρ unless (α+, α−) = (0, 0). In particular, with such non-trivial dissipation, there exists no eigenstate for Lsp and there exists no steady (i.e.,temporally invariant) state for the dynamical semigroup {etLsp}t>0. As an approximationto the (2ℓ + 1)-component spin system, the evolution formula (46) should not be usedbeyond the restriction t≪ ℓ/2(α+ + α−).

4 Open infinite-spin Dicke model

In this section, we investigate the OISD model, an open quantum model generated by theinfinite-component spin in §3 and the open harmonic oscillator in §2 with the Jaynes–Cummings interaction between them.

We provide the precise formulation of the OISD model in the following. The Hilbertspace of the system is H = G

⊗

F . Operators acting in H such as M ⊗1, 1⊗a†a, andl+ ⊗ a are denoted simply as M , a†a, and l+a, respectively, by obvious embedding. Weintroduce the shorthand notation

Ksp ≡ KM , Kint−+ ≡ Kl−a† , Kint

+− ≡ Kl+a, Kint ≡ Kint+− +Kint

−+, Kph ≡ KN , (51)

and for nonnegative constant J ,

Dsp ≡(J + 1)Dl−◦l+ + JDl+◦l− , Dph ≡ (J + 1)Da◦a† + JDa†◦a,

Dint−+ ≡(J + 1)Dl−◦a† + JDa†◦l−, Dint

+− ≡ (J + 1)Da◦l+ + JDl+◦a. (52)

With this notation, the Liouvillian of the OISD model is defined by

LOISD := −iµKsp − iλKint − iωKph + γDph (53)

with positive constants ω, µ, and γ. As we have anticipated, Kint is the Jaynes–Cummingsinteraction between the spin and the oscillator, and the constant λ ∈ R denotes the

10

strength of this interaction. The time evolution on the composed system is governed bythe above Liouvillian as

d

dtρ(t) = LOISD(ρ(t)) for ρ(t) ∈ C1(H ) . (54)

Note that in the Liouvillian (53) of the OISD model, dissipation is induced onlythrough the harmonic oscillator (not through the spin). Later, we discuss a more generalLiouvillian that has dissipation terms both for the harmonic oscillator and the infinite-component spin. It turns out that the analysis for such a general model is essentiallyreduced to the simple case (53). Hence we focus on this special setup for our OISDmodel.

Let us analyze the algebraic structure of the Liouvillian LOISD in (53). We can straight-forwardly check the following commutation relations:

[Kph,Kint−+] = −[Ksp,Kint

−+] = [Dph,Dint−+] = +Kint

∓±,

[Kph,Kint+−] = −[Ksp,Kint

+−] = [Dph,Dint+−] = −Kint

∓±,

[Dph,Kint−+] = +Dint

−+, [Dph,Kint+−] = −Dint

+−,

[Kint−+,Dint

+−] = −Dsp, [Kint+−,Dint

−+] = +Dsp, (55)

and

[Kph,Dph] = [Ksp,Dph] = [Kint−+,Kint

+−] = [Kint−+,Dint

−+] = [Kint+−,Dint

+−] = 0,

[Dsp,Kint−+] = [Dsp,Kint

+−] = [Dsp,Dph] = 0. (56)

For η ∈ C, defineW (η) := eηK

int−+−ηKint

+−. (57)

We see that for any η ∈ C, W (η) is a CPTP map with its bounded inverse W (−η). Thisfact will be important later. From (55) and (56), we have

W (η)

Ksp

Kint−+

Kint+−

Kph

Dsp

Dint−+

Dint+−

Dph

W (−η) =

Ksp + ηKint−+ + ηKint

+−

Kint−+

Kint+−

Kph − ηKint−+ − ηKint

+−

Dsp

Dint−+ − ηDsp

Dint+− − ηDsp

Dph − ηDint−+ − ηDint

+− + |η|2Dsp

, (58)

where the adjoint action Ad(W (η)) acts on each component. Note that higher terms of ηand η vanish in the right-hand side because of commutativity (56). For any σ, t > 0,

eσDph

Kint−+

Kint+−

Dint−+

Dint+−

=

Kint−+ cosh σ +Dint

−+ sinh σKint

+− cosh σ −Dint+− sinh σ

Dint−+ cosh σ +Kint

−+ sinh σDint

+− cosh σ −Kint+− sinh σ

eσDph (59)

11

and

e−it(µKsp+ωKph)

(

Kint−+

Kint+−

)

=

(

e−it(ω−µ)Kint−+

eit(ω−µ)Kint+−

)

e−it(µKsp+ωKph) (60)

hold. These identities can be verified by repeating an argument similar to that used in(22).

We can see that etLOISD is a CPTP map on C1(H ) for each t > 0 as follows. Notethat etγDph is identical to 1⊗ exp

(

tγ(J + 1)Da◦a† + tγJDa†◦a

)

. Because its second factoris completely positive on C1(F ) as we have seen in §2, etγDph is a completely positive mapon C1(H ) = C1(G )

⊗

C1(F ). The trace-preserving property follows from the tensor-product structure as well. Similarly, etγ

′Dsp and thereby et(γDph+γ′Dsp) = etγDphetγ′Dsp are

CPTP maps for γ′ > 0. We have the following formula of the semigroup generated byLOISD:

etLOISD = e−it(µKsp+ωKph)W(

η1(t))

etγDph+τ2(t)DspW(

η2(t))

, (61)

where η1(t), η2(t), and τ2(t) are the solutions of the differential equations

η′2(t) sinh tγ − γη1(t) = 0, (62)(

η′1(t) + γη1(t) coth tγ)

e−it(ω−µ) + iλ = 0, (63)

τ ′2(t)− γ|η1(t)|2 = 0 (64)

with initial condition η1(0) = η2(0) = τ2(0) = 0. The solution for η1(t) is given explicitlyby

η1(t) =iλ

(ω − µ)2 + γ2

(

i(ω − µ)ei(ω−µ)t +γ(1− ei(ω−µ)t cosh γt)

sinh γt

)

, (65)

and η2(t) and τ2(t) are obtained readily from (62) and (64). The identity (61) can beshown as in (28). Namely by putting the right hand-side by Z(t), we have

Z(0) = 1,dZ(t)

dt= LOISD(Z(t)) (66)

with the help of (58), (59), and (60). Because each factor in the right-hand side of (61)is CPTP, etLOISD is also CPTP.

Next we derive the decomposition formula of the Liouvillian LOISD. For this purpose,we need a map that gives rise to the decomposition. By using Dph and W (η) defined in(57), we set

V (σ) := W (ζ1)eσDphW (ζ2), σ > 0, (67)

where the constants are defined by

δ :=λ

γ2 + (ω − µ)2, ζ1 := −(ω − µ+ iγ coth σ)δ , ζ2 :=

iγδ

sinh σ. (68)

From (58) and (59), we obtain(

−iµKsp − iλKint − iωKph + γDph

)

V (σ) = V (σ)(

−iµKsp + λγδDsp − iωKph + γDph

)

.

Thus, we arrive at our first main result as follows.

12

Proposition 4.1. The Liouvillian LOISD of the OISD model is decomposed into the de-coupled Liouvillian Ldecoupled under the similarity transformation induced by V (σ) as

LOISDV (σ) = V (σ)Ldecoupled,

Ldecoupled := Lsp + Lph, Lsp := −iµKsp + λγδDsp, Lph := −iωKph + γDph. (69)

The transformation V (σ) used above is an invertible CPTP map on C1(H ).

Note that Lph is the Liouvillian (9) acting in C1(F ) (imbedded into C1(H )), and Lsp isthe Liouvillian (41) acting in C1(G ) (imbedded into C1(H )) with values α− = λγδ(J+1),α+ = λγδJ . The formula (69) tells us that the similarity transformation by V (σ) erasesthe Jaynes–Cummings interaction −iλKint in the original LOISD but generates the newdissipation term λγδDsp in the new decoupled Liouvillian Ldecoupled.

To complete the proof of Proposition 4.1, it remains to show that V (σ) is an invertibleCPTP map. Note that W (ζ1) and W (ζ2) are CPTP maps with their bounded inversesW (−ζ1) and W (−ζ2), respectively.

Furthermore, note that eσDph is a CPTP map, and that it has its dense range and(unbounded) inverse because of the spectrum of Dph given in (25). Thus V (σ) is aninvertible CPTP map on C1(H ) with its dense range.

In the following, we discuss the asymptotic behavior of the dynamical semigroup{etLOISD}t>0. To this end, let us rewrite the decomposition formula (69) in tensor-productformat as

LOISDV (σ) = V (σ)Ldecoupled, Ldecoupled := Lsp ⊗ 1+ 1⊗ Lph . (70)

The solution for the master equation (54) of the OISD model is written formally as

ρ(t) ≡ etLOISD(

ρ(0))

= V (σ)(

etLsp ⊗ etLph

)

V (σ)−1(

ρ(0))

. (71)

By applying an argument similar to that leading to (33) at the end of §2 to the secondtensor component of ρ(t) in the right-hand side of (71), we obtain the convergence

(

1⊗ etLph

)

V (σ)−1(

ρ(0))

→ ρ∗ ⊗ ρph,G (72)

in norm as t → ∞, where ρ∗ is some uniquely determined density matrix in C1(G ) and

ρph,G ∈ C1(F ) is the Gibbs state as in (31).Let us modify the decoupled Liouvillian Ldecoupled = −iµKsp+λγδDsp− iωKph+γDph

of (69) by setting ω = µ and deleting the term γDph as

Lsyn.dec. := −iµ(Ksp +Kph) + λγδDsp. (73)

Let us call it the synchronized decoupled Liouvillian. Note that it differs from Ldecoupled

with ω = µ (and γ = 0). We show the following proposition.

13

Proposition 4.2. Let ρ(0) be any density matrix of C1(H ), and let ρ(t) (t > 0) denoteits time evolution under the OISD Liouvillian LOISD. Define the density matrix

ρ(0) := V (σ)(

ρ∗ ⊗ ρph,G)

∈ C1(F ) (74)

and consider its time evolution under the synchronized decoupled Liouvillian Lsyn.dec. givenas

ρ(t) := etLsyn.dec.(

ρ(0))

. (75)

Then

limt→∞

‖ρ(t)− ρ(t)‖1 = 0. (76)

Namely, the time evolution of any state ρ(0) under the OISD Liouvillian is asymptoti-cally equivalent to the time evolution of the state ρ(0) under the synchronized decoupledLiouvillian. If λγδ 6= 0, then

s- limt→∞

ρ(t) = 0. (77)

Proof. We first note the following remarkable fact. For ρ(0), which has the specific formas defined in (74), its time evolution under Lsyn.dec. is identical to the time evolution underthe genuine OISD Liovillian LOISD, namely for all t > 0, we have

ρ(t) ≡ etLsyn.dec.(

ρ(0))

= etLOISD (ρ(0)) . (78)

We show this identity in the following. We compute the right-hand side of (78) as

etLOISD (ρ(0)) = etLOISD(

V (σ)(

ρ∗ ⊗ ρph,G))

= V (σ)etLdecoupled(

ρ∗ ⊗ ρph,G)

(79)

= V (σ)(

etLsp ⊗ 1)(

1⊗ etLph

)

(

ρ∗ ⊗ ρph,G)

= V (σ)(

etLsp ⊗ 1)

(

ρ∗ ⊗ ρph,G)

. (80)

For the derivation of (79) the relation (70) is used, and for (80) the invariance of the Gibbs

state ρph,G under etLph given as (34) is used. Substituting the identity e−itµKph(ρph,G) =ρph,G (which obviously holds as in (35)) into (80), we obtain

etLOISD (ρ(0)) = V (σ)(

etLsp ⊗ 1)(

1⊗ e−itµKph

)

(

ρ∗ ⊗ ρph,G)

= V (σ)et(−iµKsp+λγδDsp−iµKph)(

ρ∗ ⊗ ρph,G)

= V (σ)et(−iµ(Ksp+Kph)+λγδDsp)(

ρ∗ ⊗ ρph,G)

(81)

= et(−iµ(Ksp+Kph)+λγδDsp)V (σ)(

ρ∗ ⊗ ρph,G)

(82)

= etLsyn.dec.[µ=ω]V (σ)(

ρ∗ ⊗ ρph,G)

= etLsyn.dec. (ρ(0)) ,

14

which gives our desired (78). To show (82), we note the commutativity relations [Dsp,Kint−+] =

[Dsp,Kint+−] = [Dsp,Dph] = [Ksp + Kph,Dph] = [Ksp + Kph,Kint

−+] = [Ksp + Kph,Kint+−] = 0,

which follow from (55) and (56), and then recall the definition of V (σ) given in (57) and(67). All the ingredients Kint

−+, Kint+−, and Dph generating V (σ) commute with each of Dsp

and Ksp +Kph.Noting (71) and (80), we have the estimate

‖ρ(t)− ρ(t)‖1=∥

∥

∥V (σ)

(

etLsp ⊗ 1)(

1⊗ etLph

)

V (σ)−1(

ρ(0))

− V (σ)(

etLsp ⊗ 1)

(

ρ∗ ⊗ ρph,G)

∥

∥

∥

1

=∥

∥

∥V (σ)

(

etLsp ⊗ 1)[(

1⊗ etLph

)

V (σ)−1(

ρ(0))

− ρ∗ ⊗ ρph,G

]∥

∥

∥

1

6

∥

∥

∥

[(

1⊗ etLph

)

V (σ)−1(

ρ(0))

− ρ∗ ⊗ ρph,G

]∥

∥

∥

1. (83)

The last inequality is a consequence of the fact that CPTP maps act in a contractivemanner on self-adjoint elements of C1(H ). By combining (72) with (83), we have (76).

We show (77). By (78) and (80), we have

ρ(t) = V (σ)(

etLsp(ρ∗)⊗ ρph,G

)

. (84)

If λγδ 6= 0, equivalently, the dissipation part of Lsp = −iµKsp + λγδDsp is non-zero, thenaccording to the argument in the last part of §3, we have

s- limt→∞

etLsp(ρ∗) = 0,

therefore

s- limt→∞

ρ(t) = 0 (85)

holds. Because the trace norm convergence implies the strong convergence, we obtains-limt→∞ ρ(t) = 0 from (76) and (85).

Remark 4.3. From the above argument, we see that the OISD model demonstrates atypical mechanism of synchronization, namely, the separation of multiple dissipative timescales as described in [23]. The essential point lies in the different decay rates betweenthe two systems: the dressed photon decays into the Gibbs state with the positive decayrate γ, while the dressed spin does not decay into a certain state in a finite time scale.

Remark 4.4. We recall [27], which derives the master equation of the reduced densitymatrix for the spin sector of an open Dicke model based on approximate dissipation of theharmonic oscillator into the Gibbs state. Its (reduced) Liouvillian has dissipative termshaving a coefficient corresponding to λγδ of our case.

Remark 4.5. Note that V (σ) for any σ > 0 works for the same transformation identity(69). This freedom of choice of σ reflects the fact that −iµKsp+λγδDsp−iωKph+γDph inthe right-hand side of (69) commutes with Dsp and Dph. From (68), if γ (i.e., the strengthof the dissipation of the harmonic oscillator) is large, then the strength of the dissipationof the dressed spin in the decoupled Liouvillian (i.e., λγδ) is suppressed.

15

In the rest of this section, we suggest a straightforward generalization of our results.So far, we have discussed the OISD model whose dissipation is assigned to the harmonicoscillator only as in (53). As we have anticipated in §1, we can similarly treat a moregeneral OISD model, where each of the harmonic oscillator and the infinite-componentspin has its dissipation term. Namely, let

LOISD(γ, γ) := −iµKsp − iλKint − iωKph + γDph + γDsp (86)

with positive constants ω, µ, γ, γ and real λ. Correspondingly, we define the decoupledLiouvillian as

Ldecoupled(γ, γ) := −iµKsp − iωKph + γDph + γDsp. (87)

Because Dsp commutes with any of Ksp, Kint, Kph, Dsp, Dint−+, Dint

+−, and Dph by (56), byusing the same V (σ) as in (67) we obtain

LOISD(γ, γ)V (σ) = V (σ)Ldecoupled(γ, λγδ + γ) . (88)

From (88), we see that the Jaynes–Cummings interaction in the original Liouvillian iserased, whereas extra dissipation is added to the infinite-component spin but not to theharmonic oscillator.

We can discuss the asymptotic behavior of the dynamical semigroup generated byLOISD(γ, γ). Because of (88), we can easily show a similar statement to that given inProposition 4.2.

So far, we have always assumed γ > 0. Finally, let us briefly discuss the case of γ = 0(and γ > 0). By recalling the argument that yields (88), for any ω 6= µ we see

LOISD(0, γ)V (σ) = V (σ)Ldecoupled(0, γ) (89)

by taking V (σ) = W (λ/(µ − ω))eσDph. Thus, by applying the similarity transformationgenerated by this V (σ) to LOISD(0, γ), the interaction term is erased as before, whereas nodissipation appears in the harmonic oscillator. Hence we do not expect a similar statementto that in Proposition 4.2. If the frequencies are identical, that is, ω = µ, then the abovedecoupling method is no longer valid. However, assuming ω = µ from the outset does notseem to be a natural setup.

AcknowledgmentsThis work was supported by JSPS KAKENHI Grant Number JP17K05272. The authorsare grateful to Professor Valentin A. Zagrebnov for useful discussions and suggestions.This work was done as part of the project of the Rigaku Yugo 1 Group of KanazawaUniversity.

A Appendix

In the text of the note, there are some identities whose derivations are tedious. We showhere the details of such derivations to facilitate the reader’s understanding.

16

Derivation of (2):

[KA,KB](ρ) = KAKB(ρ)−KBKA(ρ) = [A, [B, ρ]]− [B, [A, ρ]]

= A(Bρ− ρB)− (Bρ− ρB)A− B(Aρ− ρA) + (Aρ− ρA)B

= ABρ+ ρBA− BAρ− ρAB = [A,B]ρ− ρ[A,B]

= K[A,B](ρ) ,

[KA,DB◦C ](ρ) = KADB◦C(ρ)−DB◦CKA(ρ)

= A(2BρC − CBρ− ρCB)− (2BρC − CBρ− ρCB)A

− 2B(Aρ− ρA)C + CB(Aρ− ρA) + (Aρ− ρA)CB

= 2[A,B]ρC + 2Bρ[A,C]− ACBρ− AρCB + CBρA

+ ρCBA + CBAρ− CBρA + AρCB − ρACB

= 2[A,B]ρC + 2Bρ[A,C]− (ACB − CBA)ρ− ρ(ACB − CBA)

= 2[A,B]ρC + 2Bρ[A,C]− ([A,C]B + C[A,B])ρ− ρ([A,C]B + C[A,B])

= D[A,B]◦C(ρ) +DB◦[A,C](ρ) .

Derivation of (3)Applying each side of the equality to ρ ∈ C1(H ), we have

[DA◦B,DC◦D](ρ)

= DA◦B

(

DC◦D(ρ))

−DC◦D

(

DA◦B(ρ))

= 2A(

2CρD − {DC, ρ})

B − {BA,(

2CρD − {DC, ρ})

}− 2C

(

2AρB − {BA, ρ})

D + {DC,(

2AρB − {BA, ρ})

}= 4ACρDB − 2A{DC, ρ}B − 2{BA,CρD}+ {BA, {DC, ρ}}− 4CAρBD + 2C{BA, ρ}D + 2{DC,AρB} − {DC, {BA, ρ}}= ({A,C}+ [A,C])ρ({B,D} − [B,D])

− ({A,C} − [A,C])ρ({B,D}+ [B,D])

+ 2[DC,A]ρB + 2Aρ[B,DC]− 2[BA,C]ρD − 2Cρ[D,BA]

+BADCρ+ ρDCBA−DCBAρ− ρBADC

= 2[A,C]ρ{B,D} − 2{A,C}ρ[B,D]

+ 2[DC,A]ρB + 2Aρ[B,DC]− 2[BA,C]ρD − 2Cρ[D,BA]

+ [[BA,DC], ρ] , (90)

where the underlined terms in the fourth expression correspond to those in the fifth

17

expression, and

(

D[A,C]◦{B,D} −D{A,C}◦[B,D]

+D[DC,A]◦B +DA◦[B,DC]

−D[BA,C]◦D −DC◦[D,BA]

+K[BA,DC]

)

(ρ)

= 2[A,C]ρ{B,D} − {{B,D}[A,C], ρ} − 2{A,C}ρ[B,D] + {[B,D]{A,C}, ρ}+ 2[DC,A]ρB − {B[DC,A], ρ}+ 2Aρ[B,DC]− {[B,DC]A, ρ}− 2[BA,C]ρD + {D[BA,C], ρ} − 2Cρ[D,BA] + {[D,BA]C, ρ}+ [[BA,DC], ρ] . (91)

Then the difference of both sides is

(91) − (90)

= −{{B,D}[A,C], ρ}+ {[B,D]{A,C}, ρ}− {B[DC,A], ρ} − {[B,DC]A, ρ}+ {D[BA,C], ρ}+ {[D,BA]C, ρ}= {−{B,D}[A,C] + [B,D]{A,C}− B[DC,A]− [B,DC]A+D[BA,C] + [D,BA]C, ρ}= {2BDCA− 2DBAC − BDCA+BADC − BDCA+DCBA

+DBAC −DCBA+DBAC − BADC, ρ}= {+BADC +DCBA−DCBA−BADC, ρ} = 0 . (92)

Thus we have shown (3).

Derivation of the first equation in (17)

18

By differentiating (16) with respect to t, we have

d

dtSt(ρ) =

∞∑

n=1

(1− e−2t)n−1

(n− 1)!2e−2ta†ne−taa†ρ e−taa†an

+

∞∑

n=0

(1− e−2t)n

n!a†n

{

− aa†, e−taa†ρ e−taa†}

an

=

∞∑

n=1

(1− e−2t)n−1

(n− 1)!2e−2ta†ne−taa†ρ e−taa†an

+∞∑

n=0

(1− e−2t)n

n!2na†ne−taa†ρ e−taa†an

+∞∑

n=0

(1− e−2t)n

n!

{

− aa†, a†ne−taa†ρ e−taa†an}

=∞∑

n=1

(1− e−2t)n−1

(n− 1)!

(

2e−2t + 2(1− e−2t))

a†ne−taa†ρ e−taa†an

+

∞∑

n=0

(1− e−2t)n

n!

{

− aa†, a†ne−taa†ρ e−taa†an}

=2a†(

∞∑

n=0

(1− e−2t)n

n!a†ne−taa†ρ e−taa†an

)

a

−{

aa†,

∞∑

n=0

(1− e−2t)n

n!a†ne−taa†ρ e−taa†an

}

=Da†◦a(St(ρ)) .

Derivation of the second equation in (17):By the use of (16) in the right-hand side, we have

St1

(

St2(ρ))

=

∞∑

n=0

(1− e−2t1)n

n!a†ne−t1aa†

(

∞∑

m=0

(1− e−2t2)m

m!a†me−t2aa†ρ e−t2aa†am

)

e−t1aa†an

=∞∑

n=0

∞∑

m=0

(1− e−2t1)n(1− e−2t2)m

n!m!a†(n+m)e−t1(aa†+m)e−t2aa†ρ e−t2aa†e−t1(aa†+m)an+m

=∞∑

n,m=0

(1− e−2t1)n(1− e−2t2)me−2t1m

n!m!a†(n+m)e−(t1+t2)aa†ρ e−(t1+t2)aa†an+m

=

∞∑

k=0

k∑

n=0

1

k!

k!(1− e−2t1)n(

e−2t1 − e−2(t1+t2))k−n

n!(k − n)!a†ke−(t1+t2)aa†ρ e−(t1+t2)aa†ak

=∞∑

k=0

(

1− e−2(t1+t2))k

k!a†ke−(t1+t2)aa†ρ e−(t1+t2)aa†ak = St1+t2(ρ) .

19

Derivation of (19):Making use of (7) and (20), we see the action of the left-hand side of (19) on |m〉.

∞∑

n=0

En(t)†En(t)|m〉 =

∞∑

n=0

(1− e−2t)n

n!e−taa†ana†ne−taa† |k〉

=

∞∑

n=0

(n+m)!

n!m!(1− e−2t)ne−2t(m+1)|m〉

=e−2t(m+1)

(1− 1 + e−2t)m+1|m〉 = |m〉 .

Here we have used the Maclaurin expansion formula

1

(1− x)m+1=

∞∑

n=0

(n +m)!

n!m!xn (|x| < 1) (93)

in the fourth equality. Because { |m〉}m∈N∪{0} forms C.O.N.S. in F , we obtain (19).

For the semigroup property of etDa◦a† , we put

St(ρ) =∞∑

n=0

En(t)ρEn(t)† , (94)

where

En(t) =(e2t − 1)n/2√

n!ane−ta†a . (95)

20

Derivation ofd

dtSt = Da◦a† St:

d

dtSt(ρ) =

∞∑

n=1

(e2t − 1)n−1

(n− 1)!2e2tane−ta†aρ e−ta†aa†n

+∞∑

n=0

(e2t − 1)n

n!an{−a†a, e−ta†aρ e−ta†a}a†n

=∞∑

n=1

(e2t − 1)n−1

(n− 1)!2e2tane−ta†aρ e−ta†aa†n

−∞∑

n=0

(e2t − 1)n

n!2nane−ta†aρ e−ta†aa†n

+

∞∑

n=0

(e2t − 1)n

n!{−a†a, ane−ta†aρ e−ta†aa†n}

=

∞∑

n=1

(e2t − 1)n−1

(n− 1)!

(

2e2t − 2(e2t − 1))

ane−ta†aρ e−ta†aa†n

+∞∑

n=0

(e2t − 1)n

n!{−a†a, ane−ta†aρ e−ta†aa†n}

=2a(

∞∑

n=0

(e2t − 1)n

n!ane−ta†aρ e−ta†aa†n

)

a†

− {a†a,∞∑

n=0

(e2t − 1)n

n!ane−ta†aρ e−ta†aa†n}

=Da◦a† St(ρ) .

Derivation of St1 St2 = St1+t2 :

St1

(

St2(ρ))

=

∞∑

n=0

∞∑

m=0

(e2t1 − 1)n

n!

(e2t2 − 1)m

m!ane−t1a†aame−t2a†aρ e−t2a†aa†me−t1a†aa†n

=∞∑

n=0

∞∑

m=0

(e2t1 − 1)n(e2t2 − 1)m

n!m!an+me−t1(a†a−m)e−t2a†aρ e−t2a†ae−t1(a†a−m)a†(n+m)

=∞∑

n,m=0

(e2t1 − 1)n(e2t2 − 1)me2t1m

n!m!an+me−(t1+t2)a†aρ e−(t1+t2)a†aa†(n+m)

=

∞∑

k=0

k∑

n=0

1

k!

k!(e2t1 − 1)n(

e2(t1+t2) − e2t1)k−n

n!(k − n)!ake−(t1+t2)a†aρ e−(t1+t2)a†aa†k

=∞∑

k=0

(

e2(t1+t2) − 1)k

k!ake−(t1+t2)a†aρ e−(t1+t2)a†aa†k = St1+t2(ρ) .

21

Derivation of∑

n EnEn†= 1 by applying it on each |m〉 (m ∈ N ∪ {0}):

∞∑

n=0

En(t)†En(t)|m〉 =

∞∑

n=0

(e2t − 1)n

n!e−ta†aa†nane−ta†a|m〉

=m∑

n=0

(e2t − 1)n

n!

m!

(m− n)!e−2tm|m〉

= (1 + e2t − 1)me−2tm|m〉 = |m〉 .

Derivation of (69): We show

eσDphW (ζ2)(

−iµKsp + λγδDsp − iωKph + γDph

)

W (−ζ2) (96)

−W (−ζ1)(

−iµKsp − iλKint − iωKph + γDph

)

W (ζ1)eσDph = 0 . (97)

By the use of (58), (59), and part of (56):

[Dph,Kph] = [Dph,Ksp] = [Dph,Dsp] = 0 , (98)

we have

eσDphW (ζ2)[

−iµKsp + λγδDsp − iωKph + γDph

]

W (−ζ2)

−W (−ζ1)[

−iµKsp − iλKint − iωKph + γDph

]

W (ζ1)eσDph

= eσDph

[

−iµ(

Ksp + ζ2Kint−+ + ζ2Kint

+−

)

+ λγδDsp − iω(

Kph − ζ2Kint−+ − ζ2Kint

+−

)

+ γ(

Dph − ζ2Dint−+ − ζ2Dint

+− + |ζ2|2Dsp

)

]

−[

−iµ(

Ksp − ζ1Kint−+ − ζ1Kint

+−

)

− iλ(Kint−+ +Kint

+−)− iω(

Kph + ζ1Kint−+ + ζ1Kint

+−

)

+ γ(

Dph + ζ1Dint−+ + ζ1Dint

+− + |ζ1|2Dsp

)

]

eσDph

= eσDph

[

−iµKsp − i(µ− ω)ζ2Kint−+ − i(µ− ω)ζ2Kint

+− + λγδDsp − iωKph

+ γ(

Dph − ζ2Dint−+ − ζ2Dint

+− + |ζ2|2Dsp

)

]

−[

−iµKsp + i(

(µ− ω)ζ1 − λ)

Kint−+ + i

(

(µ− ω)ζ1 − λ)

Kint+− − iωKph

+ γ(

Dph + ζ1Dint−+ + ζ1Dint

+− + |ζ1|2Dsp

)

]

eσDph

=[

−iµKsp − i(µ− ω)ζ2(

Kint−+ cosh σ +Dint

−+ sinh σ)

− i(µ− ω)ζ2(

Kint+− cosh σ −Dint

+− sinh σ)

+ λγδDsp − iωKph

+ γ(

Dph − ζ2(

Dint−+ cosh σ +Kint

−+ sinh σ)

− ζ2(

Dint+− cosh σ −Kint

+− sinh σ)

+ |ζ2|2Dsp

)

]

eσDph

−[

−iµKsp + i(

(µ− ω)ζ1 − λ)

Kint−+ + i

(

(µ− ω)ζ1 − λ)

Kint+− − iωKph

+ γ(

Dph + ζ1Dint−+ + ζ1Dint

+− + |ζ1|2Dsp

)

]

eσDph

22

=[(

− i(µ− ω)ζ2 cosh σ − γζ2 sinh σ − i(

(µ− ω)ζ1 − λ)

)

Kint−+

+(

− i(µ− ω)ζ2 cosh σ + γζ2 sinh σ − i(

(µ− ω)ζ1 − λ)

)

Kint+−

+(

− i(µ− ω)ζ2 sinh σ − γζ2 cosh σ − γζ1

)

Dint−+

+(

i(µ− ω)ζ2 sinh σ − γζ2 cosh σ − γζ1

)

Dint+−

+(

λγδ + γ|ζ2|2 − γ|ζ1|2)

Dsp

]

eσDph

=[(

− i(µ− ω)(ζ1 + ζ2 cosh σ)− iγ2δ + iλ)

Kint−+

+(

− i(µ− ω)(ζ1 + ζ2 cosh σ)− iγ2δ + iλ)

Kint+−

+(

(µ− ω)γδ − iγ2δ coth σ − γζ1

)

Dint−+

+(

(µ− ω)γδ + iγ2δ coth σ − γζ1

)

Dint+−

+(

λγδ + γ|ζ2|2 − γ|ζ1|2)

Dsp

]

eσDph = 0 .

We have used (68) in the last two steps.

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