数理解析研究所講究録
第 1904 巻 2014 年 175-188

175

A Calculable Model for a Cavity and an Atomic Beam
H. TAMURA
Institute of Science and EngineeringandGraduate School of Natural Science and Technology
Kanazawa University
ABSTRACT
We consider a simple model for the physical system consists of a cavity
and a beam of atoms which pass the cavity successively.
The Hamiltonian contains time-dependent (piecewise constant) term describing interaction between the cavity and the atom in the beam which is
passing the cavity at the prescribed moment. We deal with the model inwhich the radiation field inside the cavity and each atoms of the beam ismodeled by simple harmonic oscillators.
We calculate the time evolution of the density matrix of the system andthe asymptotic behavior of the cavity, both in the Hamiltonian dynamics andthe Markovian dynamics of Kossakowski-Lindblad-Davies type. We also discuss the entropies and evolution of the reduced density matrix for subsystems
near the cavity.
This talk is based on the joint work with Prof. V.A. Zagrebnov. The detailed description of the subject will be given in the forth coming paper.[TZ]

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1

The ModelLetbe the annihilation and the creation operators living in the one
mode Fock space :$a,$$a^{*}$$\mathscr{F}$$[a, a^{*}]=1, [a, a]=0, [a^{*}, a^{*}]=0.$$\mathscr{F}=\overline{\mathscr{F}fin}$

algebraic span offor a arbitrary but finite
Let $\mathscr{H}_{n}(n=0,1, \cdots, N)$ be copies ofOn the Hilbert space tensor product$\{\Omega, a^{*}\Omega, \cdots, a^{*k}\Omega, \}.$$\mathscr{F}fin=$$\mathscr{F}$$N\in \mathbb{N}.$$\mathscr{H}=\bigotimes_{n=0}^{N}\mathscr{H}_{n},$we define the operators$b_{0}=a\otimes 1\otimes\cdots\otimes 1, b_{0}^{*}=a^{*}\otimes 1\otimes\cdots\otimes 1,$$b_{1}=1\otimes a\otimes 1\otimes\cdots\otimes 1, b_{1}^{*}=1\otimes a^{*}\otimes 1\otimes\cdots\otimes 1,$$b_{2}=1\otimes 1\otimes a\otimes 1\otimes\cdots\otimes 1, b_{2}^{*}=1\otimes 1\otimes a^{*}\otimes 1\otimes\cdots\otimes 1,$

and so on. The operators$b_{j},$$b_{j}^{*}$$(j=0,1,2, \cdots, N)$ satisfy CCR$[b_{i}, b_{j}^{*}]=\delta_{ij}, [b_{i}, b_{j}]=[b_{i}^{*}, b_{j}^{*}]=0.$

as the state space for the photon inside the cavity
Remark: We regard
are creation andand $\mathscr{H}_{n}(n=1,2, \cdots)$ for internal states of atoms. So,
are raising and lowering operator
annihilation operators of photon andof the level of the j-th atom.
Remark: We consider the case $N<\infty$ , for simplicity.
defined byLet
be the self-adjoint Hamiltonian in$\mathscr{H}_{0}$$b_{0}^{*},$$b_{j}^{*},$$b_{0}$$b_{j}$$\mathscr{H}$$H_{n}$$H_{n}=Eb_{0}^{*}b_{0}+ \epsilon\sum_{k=1}^{N}b_{k}^{*}b_{k}+\eta b_{0}^{*}b_{n}+\eta b_{n}^{*}b_{0}, (n=1,2, \cdots, N)$

the energy level spacing of theatoms andthe interaction between the photon and atoms. We assume
is bounded below. (We understand thatthat is small enough so thatis theare taken to be closed.) We regard that
all operators like$[(n-1)\tau,n\tau$) when the n-th atom is
Hamiltonian during the time intervalpassing inside the cavity.

where $E>0$ is the photon energy,$\epsilon>0$$\eta>0$$H_{n}$$\eta$$H_{n}$$H_{n}$

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2

Hamiltonian Dynamics

In this section, we consider the time evolution of the system governed by thetime dependent (piecewise constant) Hamiltonian:$H(t)= \sum_{n=1}^{N}\chi_{[(n-1)\tau,n\tau)}(t)H_{n}.$The commutation relations$[H_{n}, b_{0}]=-Eb_{0}-\eta b_{n}, [H_{n}, b_{j}]=-\epsilon b_{j}-\delta_{jn}\eta b_{0},$$[H_{n}, b_{0}^{*}]=Eb_{0}^{*}+\eta b_{n}^{*}, [H_{n}, b_{j}^{*}]=\epsilon b_{j}^{*}+\delta_{jn}\eta b_{0}^{*}$

hold for $j=1,$$\cdots,$$N$

and yield the follgwing lemma.

Lemma 2.0.1 For $j=0$ , 1, 2,$\cdots,$$N$

and $n=1$ , 2,$\cdots,$$N,$$e^{-i\tau H_{n}}b_{j}e^{i\tau H_{n}}= \sum_{k=0}^{N}(U_{n})_{jk}b_{k}, e^{-i\tau H_{n}}b_{j}^{*}e^{i\tau H_{n}}=\sum_{k=0}^{N}\overline{(U_{n})_{jk}}b_{k}^{*},$$e^{i\tau H_{n}}b_{j}e^{-i\tau H_{n}}= \sum_{k=0}^{N}(U_{n}^{*})_{jk}b_{k}, e^{i\tau H_{n}}b_{j}^{*}e^{-i\tau H_{n}}=\sum_{k=0}^{N}\overline{(U_{n}^{*})_{jk}}b_{k}^{*}$

hold. Hereand

$U_{n}$

and$V_{n}$

are $(N+1)\cross(N+1)$ matrices given by$U_{n}=e^{i_{\mathcal{T}}\epsilon}V_{n}$$(V_{n})_{jk}=\{\begin{array}{ll}gz\delta_{k0}+gw\delta_{kn} (j=0)-g\overline{w}\delta_{k0}+g\overline{z}\delta_{kn} (j=n)\delta_{jk} (otherwise)\end{array}$with$g=e^{i\tau(E-\epsilon)/2}, w= \frac{2i\eta}{\sqrt{(E-\epsilon)^{2}+4\eta^{2}}}\sin\tau\sqrt{\frac{(E-\epsilon)^{2}}{4}+\eta^{2}}$$z= \cos\tau\sqrt{\frac{(E-\epsilon)^{2}}{4}+\eta^{2}}+\frac{i(E-\epsilon)}{\sqrt{(E-\epsilon)^{2}+4\eta^{2}}}\sin\tau\sqrt{\frac{(E-\epsilon)^{2}}{4}+\eta^{2}}.$

178
Note that $|z|^{2}+|w|^{2}=1$ and that$(\begin{array}{ll}z w-\overline{w} \overline{z}\end{array})$

is a unitary matrix. And so are$V_{n}$’s and$U_{n}’ s.$$e.g.$$U_{1}=e^{i\tau\epsilon}V_{1}=e^{i\tau\epsilon}(\begin{array}{llllll}gz gw 0 [0 0 \cdots-g\overline{w} g\overline{z} 0 0 0 \cdots 0 0 1 0 0 \cdots 0 0 0 1 0 \cdots 0 0 0 0 1 \end{array})$$U_{2}=e^{i\tau\epsilon}V_{2}=e^{i\tau\epsilon}(-g\overline{w}gz000 00001gwg\overline{z}0000000100001 )$

2.1
For

Time evolution of Product states$y\in \mathbb{C},$$w(y)=e^{i(\overline{y}a+ya^{*})}$

denotes the Weyl operator over . We consider the Weyl algebra
overwhich isand the algebra
generated byovergenerated by$\mathscr{F}$$\mathscr{F}$$\{w(y)\}_{y\in \mathbb{C}}$$\mathscr{A}(\mathscr{F})$$\mathscr{A}(\mathscr{H})$$W( \zeta)=\bigotimes_{k=0}^{N}w(\zeta_{k}) , (\zeta=\{\zeta_{k}\}_{k=0}^{N})$.

Using sesquilinear form notation$\langle\zeta, b\rangle=\sum_{j=0}^{N}\overline{\zeta}_{j}b_{j}, \langle b, \zeta\rangle=\sum_{j=0}^{N}\zeta_{j}b_{j}^{*},$$\mathscr{H}$

(2.1)

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$W(\zeta)$

can be written as$W(\zeta)=\exp[i(\langle\zeta, b\rangle+\langle b, \zeta$

Letbe a normalized self-adjoint non-negative trace class operator on$N$ . It can be regarded as a state on
for $k=0$ , 1, 2,. We use
the notation$\rho_{k}$$\mathscr{F}$$\mathscr{A}(\mathscr{F})$$\cdots,$$C_{k}(y)=Tx_{\mathscr{F}}[w(y)\rho_{k}].$Similarly, we consider the operator$\rho=\bigotimes_{k=0}^{N}\rho_{k}$as a state on$\mathscr{A}(\mathscr{H})$:$\omega_{\rho}(W(\zeta))=$$Tr$ $\mathscr{H}[W(\zeta)\rho]=\prod_{k=0}^{N}C_{k}(\zeta_{k})$Let us consider the time evolution of$\rho$

by $H(t)$.

$(0\leq t\leq N\tau)$:$\rho(N\tau):=e^{-i\tau H_{N}}\cdots e^{-i\tau H_{1}}\rho e^{i\tau H_{1}}\cdots e^{i\tau H_{N}}.$

Lemma 2.1.1$\omega_{\rho(N\tau)}(W(\zeta))=\omega_{\rho}(W(U_{1}\cdots U_{N}\zeta))=\prod_{k=0}^{N}C_{k}((U_{1}\cdots U_{N}\zeta)_{k})$$holds_{f}$where$(U_{1} \cdots U_{N}\zeta)_{0}=e^{iN\tau\epsilon}((gz)^{N}\zeta_{0}+\sum_{j=1}^{N}9^{w(g_{Z})^{j-1}\zeta_{j})}$and$(U_{1}\cdots U_{N}\zeta)_{k}=e^{iN\tau\epsilon}(-g\overline{w}(gz)^{N-k}\zeta_{0}+9^{\overline{Z}\zeta_{k}-\sum_{j=k+1}^{N}g^{2}|w|^{2}(gz)^{j-k-1}\zeta_{j})}$for $k>0.$

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2.2

The product Gibbs state

For the product Gibbs density matrix

with$\rho=\otimes_{k=0}^{N}\rho_{k}$$\rho_{0}=e^{-\beta_{0}a^{*}a}/Z(\beta_{0}) , \rho_{j}=e^{-\beta a^{*}a}/Z(\beta) (j=1, \cdots, N)$

where$\beta,$$\beta_{0}>0$

,

(2.2)

and $Z(\beta)=(1-e^{-\beta})^{-1}$ , we have

Lemma 2.2.1 The$\mathcal{S}late$corresponding to the density matrix (2.4) satisfies$\omega_{\rho}(W(\zeta))=Tr_{\mathscr{H}}[W(\zeta)\rho]=\exp[-\frac{|\zeta_{0}|^{2}}{2}(\frac{1+e^{-\beta_{0}}}{1-e^{-\beta_{0}}}-\frac{1+e^{-\beta}}{1-e^{-\beta}})-\frac{\langle\zeta,\zeta\rangle}{2}\frac{1+e^{-\beta}}{1-e^{-\beta}}]$and$S(\rho)=-Tr_{\mathscr{H}}[\rho\log\rho]=Ns(\beta)+s(\beta_{0})$,where $s(\beta):=\beta(e^{\beta}-1)^{-1}-\log(1-e^{-\beta})$ .

The time evolution of the density matrix$\rho(N\tau)=e^{-i\tau H_{N}}e^{-i\tau H_{N-1}}\cdots e^{-i\tau H_{1}}\rho e^{i\tau H_{1}}\cdots e^{i\tau H_{N}}.$satisfies the following properties:
Lemma 2.2.2$\omega_{\rho(N\tau)}(W(\zeta))=\omega_{\rho}(W(U_{1}\cdots U_{N}\zeta))=$$\exp[-\frac{|(U_{1}\cdots U_{N}\zeta)_{0}|^{2}}{2}(\frac{1+e^{-\beta_{0}}}{1-e^{-\beta_{0}}}-\frac{1+e^{-\beta}}{1-e^{-\beta}})-\frac{\langle\zeta,\zeta\rangle}{2}\frac{1+e^{-\beta}}{1-e^{-\beta}}],$

and$S(\rho(N\tau))=Ns(\beta)+s(\beta_{0})=S(\rho)$

The relative entropy of$\rho(N\tau)$w.r.t.

$\rho$.

is:Lemma 2.2.3$S(\rho(N\tau)|\rho)=$-Tr $[\rho(N\tau)(\log\rho(N\tau)-\log\rho)]=-H[\rho(\log\rho-\log\rho(-N\tau))]$$=- \frac{(\beta_{0}-\beta)(e^{\beta_{0}}-e^{\beta})}{(e^{\beta_{0}}-1)(e^{\beta}-1)}(1-|z|^{2N})$.

The relative entropy is non-positive generally. In this case, it
decreases monotonically as $Narrow\infty$ and converges to the limit:
Remark$\lim_{Narrow\infty}S(\rho(N\tau)|\rho)=-\frac{(\beta_{0}-\beta)(e^{\beta_{0}}-e^{\beta})}{(e^{\beta 0}-1)(e^{\beta}-1)}.$

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2.3

Subsystems

In this section, we devide the system into 2 subsystems. At time $t=k\tau$ , theobjects are ordered as follows:

the first atom,$\cdots$, the k-th atom, the cavity, the $k+1$ -th atom,

We regard the cavity and the$n$$\cdots$, the N-th atom.

atoms ahead the cavity as the subsystem:$\mathscr{H}=\mathscr{H}_{s}\otimes \mathscr{H}_{r},$where$\mathscr{H}_{s}=\mathscr{H}_{0}\otimes\bigotimes_{j=1}^{n}\mathscr{H}_{k-j+1},\mathscr{H}_{r}=\bigotimes_{j=1}^{k-n}\mathscr{H}_{j}\otimes\bigotimes_{j=n+1}^{N}\mathscr{H}_{j}$We want to re-number the atoms in the subsystem:takeFor$arrow 0$$(\theta_{0}0\}$$arrow 1$-th-th$\theta=(\begin{array}{l}\theta_{0}\theta_{1}\vdots\theta_{n}\end{array})\in \mathbb{C}^{n+1},$$arrow k-n$ -th$arrow k-n+1$ -th$arrow k-n+2$ -th$\in \mathbb{C}^{N+1}$$\zeta_{\theta}^{(k)}=\ovalbox{\tt\small REJECT}\theta_{n-1}\theta_{2}\theta_{n}\theta_{1}0/00$$arrow k-1$ -th$arrow k$-th-th$arrow k+1$$\mathfrak{s}$$arrow N$

-th

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to get$\theta_{0}b_{0}^{*}+\overline{\theta}_{0}b_{0}+\sum_{j=1}^{n}(\theta_{j}b_{k-j+1}^{*}+\overline{\theta}_{j}b_{k-j+1})=\sum_{j=0}^{N}(\zeta_{\theta,j}^{(k)}b_{j}^{*}+\overline{\zeta}_{\theta,j}^{(k)}b_{j})$And consider the Weil operator on.

$\mathscr{H}_{s}$$W_{s}( \theta) = \exp[i(\theta_{0}b_{0}^{*}+\overline{\theta}_{0}b_{0}+\sum_{j=1}^{n}(\theta_{j}b_{k-j+1}^{*}+\overline{\theta}_{j}b_{k-j+1}))]$$= \exp[i(\theta_{0}\tilde{b}_{0}^{*}+\overline{\theta}_{0}\tilde{b}_{0}+\sum_{j=1}^{n}(\theta_{j}\tilde{b}_{j}^{*}+\overline{\theta}_{j}\tilde{b}_{j}$,where,. (We used abused notations: e.g.,
is not an
operator in, etc. ), whileinbut inFor the density matrix , let
be the reduced density matrix of the
sub-system i.e.,.
(2.3)$\tilde{b}_{0}=b_{0},$$\mathscr{H}_{s}$$\tilde{b}_{j}=b_{k-j+1}$$b_{0}$$\tilde{b}_{0}$$\mathscr{H}$$\mathscr{H}_{s}$$\rho$$\rho_{s}$$\rho_{s}=Tr_{\mathscr{H}_{r}}\rho$Then, we get$\omega_{\rho_{s}}(W_{s}(\theta))=\omega_{\rho}(W(\zeta_{\theta}))$.

Now let us consider time evolution. The time evoluted density
the initial Gibbs state$\rho=\exp[-\beta_{0}b_{0}^{*}b_{0}-\beta\sum_{j=1}^{N}b_{j}^{*}b_{j}]/(Z(\beta_{0})Z(\beta)^{N})$$\rho(k\tau)$,

has the reduced density matrix given by

Lemma 2.3.1$\omega_{\rho(k\tau)_{s}}(W_{s}(\theta))=\omega_{\rho(k\tau)}(W(\zeta_{\theta}))$$= \exp[-\frac{|(U_{1}\cdots U_{k}\zeta_{\theta})_{0}|^{2}}{2}(\frac{1+e^{-\beta_{0}}}{1-e^{-\beta_{0}}}-\frac{1+e^{-\beta}}{1-e^{-\beta}})-\frac{\langle\theta,\theta\rangle}{2}\frac{1+e^{-\beta}}{1-e^{-\beta}}],$

Now consider the limit $karrow\infty(Narrow\infty)$ with$n$fixed. We get

of

(2.4)

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Proposition 2.3.2$\rho(k\tau)_{s}$converge to$\rho^{(\beta)}$and$\lim_{karrow\infty}S(\rho(k\tau)_{s})=S(\rho^{(\beta)})$

,

where
$\rho^{(\beta)}=\exp[-\beta b_{0}^{*}b_{0}-\beta\sum_{j=1}^{n}b_{N-j+1}^{*}b_{N-j+1}]/Z(\beta)^{n+1}$

Remark The local entropy decreases or increases accordong to, respectively.

$\beta>\beta_{0}$or$\beta<\beta_{0}$2.4

A scaling limit for product states

Here, we mention an asymptotic behavior of the state of the cavity under
the influence of the beam where the state for the atoms is product of generaltype.
We assume that
(1) $\rho_{1}=\rho_{2}=\cdots=\rho_{N}$ ;(2) $H[a\rho_{1}]=\ulcorner b[a^{2}\rho_{1}]=H[a^{*}\rho_{1}]=n[a^{*2}\rho_{1}]=0$ ;(3) $h[(a^{*}a)^{2}\rho_{1}]<\infty.$

Proposition 2.4.1 Under the limit$\tauarrow 0$

and $Narrow\infty$ subject to$\tau^{2}Narrow\infty$

and $\tau^{3}Narrow 0(e.g., \tau=O(N^{-0.4}))$ ,$\lim\omega_{\rho(N\tau)_{s}}(w(\theta))=\lim\omega_{\rho(N\tau)}(W(\zeta_{\theta}))=e^{-Tr[(a^{*}a+aa^{*})\rho_{1}]|\theta|^{2}/2}$

holds for3$\theta\in \mathbb{C}^{0+1}$

Markovian Evolution

We consider here the evolution of the system under the Kossakowski-LindbladDavies equation, which yields a behavior the system in a large reservoir:$\partial_{t}\rho(t)=L_{\sigma}(t)(\rho(t)) , \rho=\rho(t)|_{t=0}\in \mathfrak{C}_{1}(\mathscr{H})$

,

where
$L_{\sigma}(t)( \rho(t)) :=-i[H(t), \rho(t)]+\sigma_{-}b_{0}\rho(t)b_{0}^{*}-\frac{\sigma_{-}}{2}\{b_{0}^{*}b_{0}, \rho(t)\}$$+ \sigma+b_{0}^{*}\rho(t)b_{0}-\frac{\sigma_{+}}{2}\{b_{0}b_{0}^{*}, \rho(t)\}$

.

(3.1)

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To satisfy the complete positivity-preserving the parameters of non-Hamiltonianpart of dynamics must satisfy inequality. We also impose condition-for the boundedness of expectations in the state, see [NVZ].
We introduce the familyof trace-preserving and completepositive evolution mappings:$\sigma_{\mp}\geq 0$$\sigma_{+}\leq\sigma$$\{T_{t,t’}\}_{0\leq t’\leq t}$$T_{t,0}^{\sigma}:\rho\mapsto\rho_{\sigma}(t)=T_{t,0}^{\sigma}(\rho(O))$

with$T_{t_{)}0}^{\sigma}=T_{t}^{\sigma}{}_{t}T_{t,0}^{\sigma},$$(0\leq t’\leq t)$

.

(3.2)

As in the Hamiltonian evolution, we consider tuned repeated interactions,when the Hamiltonian part of dynamics is piecewise constant. Then for$t\in[(k-1)\tau, k\tau)$ , the generator has the form:$L_{\sigma,k}( \rho(t)) :=-i[H_{k}, \rho(t)]+\sigma_{-}b_{0}\rho(t)b_{0}^{*}-\frac{\sigma_{-}}{2}\{b_{0}^{*}b_{0}, \rho(t)\}$$+ \sigma+b_{0}^{*}\rho(t)b_{0}-\frac{\sigma+}{2}\{b_{0}b_{0}^{*}, \rho(t)\}(k\geq 1)$

(3.3)

.

The solution of the corresponding Cauchy problem$\partial_{t}\rho(t)=L_{\sigma}(t)(\rho(t)) , \rho(t)|_{t=0}=\rho_{0}\otimes\bigotimes_{k=1}^{N}\rho_{k}$

(3.4)

,

has a form:$\rho(N\tau)=T_{N\tau,0}^{\sigma}(\rho(0))=e^{\tau L_{\sigma,N}} e^{\tau L_{\sigma,2}}e^{\tau L_{\sigma,1}}(\rho(0))$.

Let us use the notation:$T_{k}^{\sigma}:=T_{k\tau,(k-1)\tau}^{\sigma}=e^{\tau L_{\sigma,k}}$

.

(3.5)

And we consider evolution of the Weyl operators, which is dual to the evolution of states$Tr_{\mathscr{H}}[T_{N\tau,0}^{\sigma}(\rho)W(\zeta)]=Tr_{\mathscr{H}}[\rho T_{N\tau,0}^{\sigma*}(W(\zeta))]$

.

(3.6)

Note that$T_{N\tau,0}^{\sigma}=e^{\tau L_{\sigma,N}} e^{\tau L_{\sigma,2}}e^{\tau L_{\sigma,1}}$and its dual evolution$T_{N\tau,0}^{\sigma*}=e^{\tau L_{\sigma,1}^{*}}e^{\tau L_{\sigma,2}^{*}} e^{\tau L_{\sigma,N}^{*}}$

(3.7)

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3.1

Evolution of Open System

First we establish a formula for the one-step mappings in (3.7) of the Weyl
operators.
Lemma 3.1.1 Let $l=0$ , 1, 2, . . . , $N$ and $n=1$ , 2, . . . , N. Let vector
be as in (2.1). Then we obtain$k,$$\zeta=$$\{\zeta_{k}\}_{k=0}^{N}\in \mathbb{C}^{N+1}$$T_{n}^{\sigma*}(W(\zeta)) :=e^{tL_{\sigma,n}^{*}}(W(\zeta))=\Omega_{t}^{\sigma,n}(\zeta)W(U_{n}^{\sigma}(t)\overline{\zeta})$

(3.8)

,

where

,

$\Omega_{t}^{\sigma,n}(\zeta)=\exp[-\frac{1}{4}\frac{\sigma_{+}+\sigma_{-}}{\sigma_{+}-\sigma_{-}}(\langle U_{n}^{\sigma}(t)\zeta, U_{n}^{\sigma}(t)\zeta\rangle-\langle\zeta, \zeta$$U_{n}^{\sigma}(t)= \exp[it(Y_{n}-i\frac{\sigma_{+}-\sigma_{-}}{2}P_{0})],$$(P_{0})_{kl}=\delta_{k0}\delta_{l0}$

.

(3.9)
(3.10)

Remark The main diﬀerence between the mapping forand (3.8),(3.10) is that the energy parameter (Lemma 2.0.1) has the shift:$\sigma_{\mp}=0$$E arrow E_{\sigma}:=E-i\frac{\sigma_{+}-\sigma_{-}}{2}$Note that${\rm Im}(E_{\sigma})>0$, if$\sigma+<\sigma_{-}.$

Corollary 3.1.2$T_{N\tau,0}^{\sigma*}(W( \zeta))=\exp[-\frac{\sigma_{+}+\sigma_{-}}{4(\sigma_{+}-\sigma_{-})}(\langle U_{1}^{\sigma}(\tau)\ldots U_{N}^{\sigma}(\tau)\zeta,$$\cross W(U_{1}^{\sigma}(\tau)\ldots U_{N}^{\sigma}(\tau)\zeta)$$U_{1}^{\sigma}(\tau)\ldots U_{N}^{\sigma}(\tau)\zeta\rangle-\langle\zeta,$$\zeta$.

Combining the above Corollary and Lemma, we get the following theorem.

Theorem 3.1.3 Let$\rho$

be the Gibbs density matrix (2.2). Then, we get$\omega_{T_{N_{\mathcal{T}},0}^{\sigma}\rho}(W(\zeta))=\exp[-\frac{1}{4}\langle\zeta, X^{\sigma}(N\tau)\zeta\rangle],$where$X^{\sigma}(N\tau)$is the $(N+1)\cross(N+1)$ matrix given by$X^{\sigma}(N \tau)=U_{N}^{\sigma}(\tau)^{*}\ldots U_{1}^{\sigma}(\tau)^{*}[(\frac{\sigma_{+}+\sigma_{-}}{\sigma_{+}-\sigma_{-}}+\frac{1+e^{-\beta}}{1-e^{-\beta}})I+(\frac{1+e^{-\beta_{0}}}{1-e^{-\beta_{0}}}-\frac{1+e^{-\beta}}{1-e^{-\beta}})P_{0}]$$\cross U_{1}^{\sigma}(\tau)\ldots U_{N}^{\sigma}(\tau)-\frac{\sigma_{+}+\sigma_{-}}{\sigma_{+}-\sigma_{-}}I.$

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3.2

Limit of reduced density for the cavity

Hereafter, we use the notations:and$U_{n}^{\sigma}(t)=e^{it\epsilon}V_{n}^{\sigma}(t)$$(V_{n}^{\sigma}(t))_{jk}=\{\begin{array}{ll}g^{\sigma}(t)z^{\sigma}(t)\delta_{k0}+g^{\sigma}(t)w^{\sigma}(t)\delta_{kn} (j=0)g^{\sigma}(t)w^{\sigma}(t)\delta_{k0}+g^{\sigma}(t)z^{\sigma}(-t)\delta_{kn} (j=n)\delta_{jk} (otherwise)\end{array}$with$g^{\sigma}(t)=e^{it(E_{\sigma}-\epsilon)/2},$$w^{\sigma}(t)= \frac{2i\eta}{\sqrt{(E_{\sigma}-\epsilon)^{2}+4\eta^{2}}}\sin t\sqrt{\frac{(E_{\sigma}-\epsilon)^{2}}{4}+\eta^{2}},$(3.11)$z^{\sigma}(t)= \cos\iota\sqrt{\frac{(E_{\sigma}-\epsilon)^{2}}{4}+\eta^{2}}+\frac{i(E_{\sigma}-\epsilon)}{\sqrt{(E_{\sigma}-\epsilon)^{2}+4\eta^{2}}}\sin\iota\sqrt{\frac{(E_{\sigma}-\epsilon)^{2}}{4}+\eta^{2}}.$

Note the relation$z^{\sigma}(t)z^{\sigma}(-t)-w^{\sigma}(t)^{2}=1$

holds, but(3.12)
for$z^{\sigma}(-t)\neq\overline{z^{\sigma}(t)}$$\sigma+\neq\sigma-.$

We consider the system with initial product state$\rho:=\bigotimes_{k=0}^{N}\rho_{k}$whereFor fixed$\rho_{0},$with$\rho_{1}=\rho_{2}=\cdots=\rho_{N}$(3.13),are density matrices on . We assume that is gaude invariant.
by, we define one step evolution of the cavity state$\mathscr{F}$$\rho_{1}$$\rho_{1}$$\rho_{1}$$\rho_{0}$$\mathcal{T}(\rho_{0})=(T_{\tau,0}^{\sigma}\rho)_{0}$(3.14),

in the righthand side represents thewhere is (3.13) and the subscript
densitycorresponding
subsystem
reducedto theconsists of the cavity only.
The application of can be expressed explicitly by the use of the expectationof the Weyl operator:$()_{0}$$\rho$$\mathcal{T}$$\omega_{\mathcal{T}(\rho 0)}(\hat{w}(\theta))=\exp[-\frac{|\theta|^{2}}{4}\frac{\sigma_{-}+\sigma_{;}}{\sigma_{-}-\sigma+}(1-|g^{\sigma}(\tau)z^{\sigma}(\tau)|^{2}-|g^{\sigma}(\tau)w^{\sigma}(\tau)|^{2})]$$\cross\omega_{\rho 0}(\hat{w}(e^{i\tau\epsilon}g^{\sigma}(\tau)z^{\sigma}(\tau)\theta)\omega_{\rho_{1}}(\hat{w}(e^{i\tau\epsilon}g^{\sigma}(\tau)w^{\sigma}(\tau)\theta))$

Then we get:.

(3.15)

187
Proposition 3.2.1 Suppose that$E( \theta):=\prod_{k=0}^{\infty}\omega_{\rho_{1}}(\hat{w}(e^{i(k+1)\tau\epsilon}g^{\sigma}(\tau)^{(k+1)}z^{\sigma}(\tau)^{k}w^{\sigma}(\tau)\theta))$$= \prod_{k=0}^{\infty}C_{1}(e^{i(k+1)\tau\epsilon}g^{\sigma}(\tau)^{(k+1)}z^{\sigma}(\tau)^{k}w^{\sigma}(\tau)\theta)$

is convergent and continuous for allonwhich atisfies$\mathscr{A}(\mathscr{F})$$\theta\in \mathbb{C}$. Then there is a unique state$\rho_{*}$$\mathcal{S}$

( )
( )
$i$

$\mathcal{T}(\rho_{*})=\rho_{*}$

$ii$

(iii)

:

$\lim_{karrow\infty}\mathcal{T}^{k}(\rho_{0})=\rho_{*}$$\lim_{Narrow\infty}(T_{N\tau,0}^{\sigma}\rho)_{s}=(T_{n\tau,0}^{\sigma}\rho_{(*)})_{s}$where in the third item,
is (3.13) withfor the reduced density to the subsystemnear the cavity, see (2.3).
Moreoverhave the expectation$\rho_{(*)}$(3.16)(3.17),$\forall\rho_{0}\in \mathfrak{C}_{1}(\mathscr{F})$,(3.18),and the subscript stands
theof cavity and the atoms$s$$\rho_{0}=\rho_{*}$$consist_{\mathcal{S}}$$n$$\rho_{*}$$\omega_{\rho_{*}}(\hat{w}(\theta))=\exp[-\frac{|\theta|^{2}}{4}\frac{\sigma_{-}+\sigma_{;}}{\sigma_{-}-\sigma+}\backslash (1-\frac{|_{9^{\sigma}}(\tau)w^{\sigma}(\tau)|^{2}}{1-|g^{\sigma}(\tau)z^{\sigma}(\tau)|^{2}})]E(\theta)$4.

(3.19)

Summary

As a simple mathematical model for atomic beam passing through a cavity,
we considered a system consists of harmonic oscillators.
We have studied the Hamiltonian evolution of the system by calculating
the expectation values of Weyl operators, explicitly. For Gibbs initial states,
we consider a relaxation phenomena of the sub-system arround the cavity.
For initial product states, we saw the convergence to the Gibbsian density
matrix in a certain scaling limit.
We also studied the Markovian evolution of the model. We gave a formulafor the dual evolution of the Weyl operators, explicitly. For a certain initial
product states, we gave the asysmptotic behavior of the states for subsystems
around the cavity.
The detailed presentation of the subject and their proofs will be given in
[TZ].

188

References[AJPI]

Open Quantum Systems I, The Hamiltonian Approach, S. Attal,
A. Joye, C.-A. Pillet (Eds.), Lecture Notes in Mathematics 1880,
Springer-Verlag, Berlin-Heidelberg 2006.

[AJPII]

Open Quantum Systems II, The Markovian Approach, S. Attal,
A. Joye, C.-A. Pillet (Eds.), Lecture Notes in Mathematics 1881,
Springer-Verlag, Berlin-Heidelberg 2006.

[AJP3]

Open Quantum Systems III, Recent Developements, S. Attal,
A. Joye, C.-A. Pillet (Eds.), Lecture Notes in Mathematics 1882,
Springer-Verlag, Berlin-Heidelberg 2006.

[BJM]

L.Bruneau, A.Joye, and M.Merkli, Repeated interactions in open
quantum systems, (May 14, 2013). Submitted to J.Math.Phys.

[NVZ]

B. Nachtergaele, A. Vershynina, and V. A. Zagrebnov,
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