A Microscopic Derivation of Quantum Stochastic Differential Equations for A Non-Linear Damped Oscillator

(1)

A

Microscopic

Derivation

of

Quantum

Stochastic Differential

Equations

for

A Non-Linear Damped

Oscillator

Takeshi

SAITO

(斎藤健) and

Toshihico

ARIMITSU

(有光敏彦)

Institute of Physics, Universityof Tsukuba

Ibaraki 305, Japan

1 Introduction

For a dissipative non-linear quantum system, the effect of the non-linearity on its relaxation

wasconsidered $[1]-[5]$in deriving a quantummasterequation for the system within the damping

theory $[6, 7]$.

Let us consider the system of a non-linear damped oscilator where the Hamiltonian of the relevant system is given by

$H_{S}= \omega a^{\mathrm{t}}a+\frac{1}{2}ga^{\uparrow}a^{\uparrow}aa$, (1)

where$a$ and $a^{\mathrm{t}}$ are boson operators satisfying the commutation relations

$[a, a^{\uparrow}]=1$, $[a, a]=0$

.

(2)

In the non-conventiondtreatment $[1]-[5]$ of the damping theory, the effect of the non-linearity

within a relevant systemon its relaxation is taken into account, which ensuresthat the density

operator of the relevant system, $\rho_{S}(t)$, leads to the true final equilibrium state, i.e. $\rho_{S}(t)arrow$

$\mathrm{e}^{-\beta H_{S}}$ as _{$tarrow\infty$}

.

In the conventionaltreatment of the damping theory which ignores the

effect of the non-linearity within a relevant system on its relaxation, $\rho_{S}(t)$ converges to the

equilibrium state for a damped harmonic oscilator, i.e. $\rho_{S}(t)arrow \mathrm{e}^{-\beta wa}\mathrm{v}_{a}$ as $tarrow\infty$

.

This

shows that the effect of the non-linearity within a relevant system on its relaxation plays the

important role for its long time behavior. Haake et al. [5] derived the master equation for the

non-linear damped oscillator in thenon-conventional treatment within the damping theory.

Withinthe framework of Non-Equilibrium Thermo Field Dynamics (NETFD) $[8]-[12]$, a

uni-fied canonical operator formalism of quantum stochastic differential equations was constructed

including the quantum Langevin equation and the quantum stochastic Liouvile equation $[10]-$

[24]. Within this formalism, quantum stochastic differential equations for a non-linear damped

oscillator are constructed [20].

Accardiet al. $[25]-[29]$ gave a microscopic foundationtoquantum stochastic processes. They

considered a quantum system interacting with thermal reservoir which consists of boson fields.

Then, they showed that, in the weak coupling limit (the van Hove limit) [30], suitably chosen

(2)

and thetime-evolution equation ofawave function in the interaction representation toaquantum stochastic differential equation where the infinitesimal time-evolution generator contains the

increments of thequantum Wiener processes.

In this paper, we $\mathrm{w}\mathrm{i}\mathrm{l}$applythe procedure of Accardiet al. to a non-linear damped oscillator

within the formalism of NETFD, and give a microscopic foundation of quantum stochastic

differential equations for a non-linear damped oscilator where the effect of the non-linearity

within arelevant system on its relaxation is taken into account.

2 Microscopic Model

We consideranon-linearoscillator interactingwitha reservoirwhich is described by the following

Hamiltonian $H=H_{0}+H_{1}$, (3) where $H_{0}=Hs+H_{R}$, (4) and $H_{1}=i \lambda\sum_{k}(ab_{k}\dagger-b_{k}\dagger_{a})$

.

(5)

Here, $H_{S}$ is given by (1) and

$H_{R}= \sum_{k}\epsilon_{k}b_{k}\uparrow b_{k}$

.

(6)

The operators $a,$ $a\dagger$ and $b_{k},$ $b_{k}^{\dagger}$ are boson operators satisfying the commutation relations (2) and

$[b_{k}, b_{l}^{\dagger}]=\delta_{k}l$, _{$[b_{k}, b_{l}]=0$}

.

(7)

We introduce operators with tilde, $\tilde{a},\tilde{a}\dagger,\tilde{b}_{k},\tilde{b}_{k}^{\dagger}$. The tilde _{$conjugation\sim \mathrm{i}\mathrm{s}$} defined by

$(A_{1}A_{2})^{\sim}=\tilde{A}1\tilde{A}_{2}$, $(c_{1}A_{1}+c_{2}A_{2})^{\sim}=c_{1}^{*}\tilde{A}_{1}+c_{2}^{*}\tilde{A}_{2}$, (8) $(\tilde{A})^{\sim}=A$, $(A\dagger)^{\sim}=\tilde{A}\dagger$, (9)

where $A_{1},$ $A_{2}$ and $A$ are arbitrary operators and $c_{1}$ and $c_{2}$ are $\mathrm{c}$-numbers. The representation

space of $(a, a\tilde{a},\tilde{a})\uparrow,\dagger$ will be denoted by $\mathcal{H}_{S}$, while that of $(b_{k}, b_{k’ k}^{\uparrow\tilde{b}}k,\tilde{b}^{\uparrow})$ by $\Gamma_{R}$.

Thermal vacuums $|0_{R}\rangle$ and $\langle$_{$1_{R}|$} in $\Gamma_{R}$ are characterized by $\langle 1_{R}|b_{k}^{\uparrow}bl|\mathrm{o}_{R}\rangle=\overline{n}_{k}\delta_{kl}$ with the

Planck distribution $\overline{n}_{k}=\frac{1}{\mathrm{e}^{\epsilon_{k}/\tau}-1}$

.

The annihilation operators $(c_{k},\tilde{c}_{k})$ and creation operators

$(c_{k}^{*},\tilde{c}_{k}^{\mathrm{i}})$on $\Gamma_{R}$ satisfying the relations

$c_{k}|\mathrm{o}_{R}\rangle=\tilde{c}k|0R\rangle=0$, $\langle$$1_{R}|c_{k}*=\langle 1_{R}|\tilde{c}_{k}^{*}=0$, (10)

and the canonical commutation relations

$[c_{k}, c_{l}^{*}]=[\tilde{c}k,\tilde{c}_{l}^{*}]=\delta kl$, (11)

are

introduced by the Bogoliubov

_{transformation}

(3)

The space $\Gamma_{R}$ is spanned by the basic vectors introduced by cyclic operations of

$(c_{k}\tilde{c}_{k})*,*$ on $|0_{R}\rangle$

and $(c_{k},\tilde{c}_{k})$ on $\langle 1_{R}|$.

We introduce the time-evolution generator $\hat{U}_{\lambda}(t)$ in theinteraction picturedefined by

$\hat{U}_{\lambda}(t)=\mathrm{e}i\hat{H}\mathrm{o}tt\mathrm{e}^{-i\hat{H}}$

, (13)

where

$\hat{H}=H-\tilde{H}$, $\hat{H}_{0}=H_{0}-\tilde{H}0$

.

(14)

The generator $\hat{U}_{\lambda}(t)$ is the operator actingon the thermalspace $\mathcal{H}_{S}\otimes\Gamma_{R}$. The time-evolution

equation of $\hat{U}_{\lambda}(t)$ is given by

$\frac{\partial}{\partial t}\hat{U}_{\lambda}(t)=-i\hat{H}_{1}^{I}(t)\hat{U}_{\lambda}(t)$, (15)

with

$\hat{H}_{1}^{I}(t)=\mathrm{e}^{i\hat{H}_{\mathrm{O}}t}(H_{1}-\tilde{H}_{1})\mathrm{e}-i\hat{H}_{\mathrm{O}}t$

$=i \lambda\sum_{k}\mathrm{t}a^{\dagger 1}b_{k}\mathrm{e}^{-i}-k(\mathrm{t}y+g\alpha\dagger a)1t-\epsilon b\dagger 1^{\epsilon_{k}}-(\omega+ga\dagger_{a})]t\}k-\mathrm{e}^{i}a\mathrm{t}_{\mathrm{C}}..$, (16)

where$\mathrm{t}.\mathrm{c}$. indicates tilde conjugates of the previous term.

We introduce vacuum states $|0,\tilde{0}\rangle$ and $\langle$$0,\tilde{0}|$ by

$a|0,\tilde{0}\rangle=\tilde{a}|0,\tilde{0}\rangle=0$, $\langle$$0,\tilde{0}|a=\dagger\langle 0,\tilde{\mathrm{o}}|\tilde{a}^{\dagger}=0$, (17)

and define ket- and bra-vectors

$(a^{\uparrow})^{m}(\tilde{a}\dagger)^{n}$

$|m,\tilde{n}\rangle=\overline{\sqrt{m!}}\overline{\sqrt{n!}}^{1}0,\tilde{0}\rangle$,

$\langle$$m, \tilde{n}|=\langle 0,\tilde{\mathrm{o}}|\frac{a^{m}}{\sqrt{m!}}\frac{(\tilde{a})^{n}}{\sqrt{n!}}$ , (18)

whichsatisfy the orthonormalization condition

$\langle m,\tilde{n}|m’,\tilde{n}’\rangle=\delta_{mm^{\prime\delta}}nn’$

’ (19)

and the completeness relation

$\sum_{mn}|m,\tilde{n}\rangle\langle m,\tilde{n}|=I$

.

(20)

The representation space $\mathcal{H}_{S}$can be spanned by the basic vectors $|m,\tilde{n}\rangle$ and $\langle m,\tilde{n}|$.

Using the vectors $|m,\tilde{n}\rangle$ and $\langle$$m,\tilde{n}|,\hat{H}_{1}^{I}(t)$ can be expressed as

$\hat{H}_{1}^{I}(t)=i\lambda\sum_{mn}\sum_{k}\{\sqrt{m+1}|m+1,\tilde{n}\rangle\langle m,\tilde{n}|bk\mathrm{e}^{-i}(\epsilon k-\phi m)t$

$-b_{k}^{\uparrow_{e^{i(}}\epsilon_{k}}-\emptyset m)t_{\sqrt{m+1}\langle m}|m,\tilde{n}\rangle+1,\tilde{n}|\}-\mathrm{t}.\mathrm{c}.$ , (21)

(4)

3 Evaluation

of

$\hat{U}_{\lambda}(t)$

We introduce exponential vectors in $\Gamma_{R}$ defined by

$|e(z, w) \rangle_{R}=\exp[\sum_{k}z_{k^{C_{k}^{*}}}+w_{k}^{**}\tilde{c}_{k}]|0_{R}\rangle$, $R \langle e(z, W)|=\langle 1_{R}|\exp[\sum_{k}Z_{k^{Ck}}^{*}+wk^{\tilde{C}}k]$ , (22)

where$z_{k},$ $w_{k}$ are $\mathrm{c}$-numbers. The exponential vectors have the properties that the actions of$c_{k}$,

$c_{k}^{*}$ and their tilde conjugates on them are as folows:

$c_{k}|e(Z, w)\rangle_{R}=z_{k}|e(z, w)\rangle_{R}$, $\tilde{c}_{k}|e(_{Z}, W)\rangle_{R}=wk|*e(z, w)\rangle_{R}$, (23) $R\langle e(z, w)|c^{\}=R\langle ke(z, w)|z_{k}^{*}.$, $R\langle e(z, w)|\tilde{C}_{k}*=_{R}\langle e(z, w)|wk$, (24)

which indicates that the exponential vectors are the coherent states. Let us introduce the

exponential vectors (22) with $z_{k}$ and $w_{k}$ replaced with

$z_{k}= \lambda\sum\int_{s/}^{T}nnn/\lambda^{2}\lambda 2duZnk\mathrm{e}i(\epsilon_{k}-\phi_{n})u$ , $\langle$25)

and

$w_{k}= \lambda\sum\int_{s_{n}/\lambda}^{T}n/\lambda 2duw_{n}k\mathrm{e}^{i(\epsilon_{k\phi_{n}}}n2-)u$, (26)

respectively, and denote them by $|e_{\lambda(Z,w)\rangle_{R}}$ and$R\langle e_{\lambda}(z, w)|$. Here, $z_{nk}$ and $w_{nk}$ are c-numbers.

The exponential vectors $|e_{\lambda(Z,w)\rangle_{R}}$ and $R\langle e_{\lambda}(z, w)|$ arecalled the collective exponentialvectors.

Let $\hat{K}_{\lambda}(t)$ bedefined by

$\hat{K}_{\lambda}(t)=R\langle e_{\lambda(w_{1}}Z_{1},)|\hat{U}\lambda(t/\lambda^{2})|e_{\lambda(}z2, W2)\rangle R$

.

(27)

Using (15), we see that the equation of motion of$\hat{K}_{\lambda}(t)$ is given by

$\frac{d}{dt}\hat{K}_{\lambda}(t)=\frac{1}{\lambda^{2}}\frac{d}{d(t/\lambda^{2})}\hat{K}_{\lambda}(\iota)=R\langle e_{\lambda()|}Z_{1}, w_{1}\frac{-i}{\lambda^{2}}\hat{H}^{I}1(t/\lambda^{2})\hat{U}\lambda(t/\lambda 2)|e\lambda(_{Z,w_{2}}2)\rangle R$

.

(28)

Substituting (21) with $(b_{k}, b_{k}^{\uparrow},\tilde{b}_{k}, \tilde{b}_{k}^{\dagger})$expressed by $(c_{k,k}c_{k’ k}^{*\}\tilde{C},\tilde{C})$ into (28), we have

$\frac{d}{dt}\hat{K}_{\lambda}(t)=\hat{I}\lambda+II_{\lambda}\wedge$, (29)

where

$\hat{I}_{\lambda}=\frac{1}{\lambda}R\langle e_{\lambda}(_{Z_{1},w_{1}})|\sum_{nm}\sum_{k}\{\sqrt{m+1}|m+1,\tilde{n}\rangle\langle m,\tilde{n}|\overline{n}k\tilde{c}_{k}\_{\mathrm{e}}-i(\epsilon_{k}-\phi m)t/\lambda^{2}$

$-(\overline{n}_{k}+1)_{C_{k}^{*i}}\mathrm{e}-t/\lambda^{2}\sqrt{m+1}(\epsilon k\phi_{m})|m,\tilde{n}\rangle\langle m+1,\tilde{n}|+\mathrm{t}_{\mathrm{C}}..\}\hat{U}_{\lambda}(t/\lambda^{2})|e\lambda(z_{2}, W2)\rangle_{R},$ (30)

and

$I^{\wedge}I_{\lambda}= \frac{1}{\lambda}R\langle e_{\lambda}(z_{1}, w_{1})|\sum_{nm}\sum_{k}\{\sqrt{m+1}|m+1,\tilde{n}\rangle\langle m,\tilde{n}|c_{k}\mathrm{e}^{-i\mathrm{t}\epsilon}k-\phi m)t/\lambda^{2}$

(5)

Making use of(23) and (24) together with the relations

$c_{k}\hat{U}_{\lambda}(t/\lambda 2)=\hat{U}\lambda(t/\lambda 2)C_{k}+[ck,\hat{U}_{\lambda}(t/\lambda^{2})]$, (32)

$\tilde{c}_{k}\hat{U}_{\lambda(/\lambda)=\hat{U}}t2\lambda(t/\lambda 2)_{\tilde{C}}k+[\tilde{c}_{k},\hat{U}_{\lambda}(t/\lambda^{2})]$, (33)

we evaluate the limits of$\hat{I}_{\lambda}$ and $I^{\wedge}I_{\lambda}$ as $\lambdaarrow 0$, which gives

$\frac{d}{dt}\hat{K}(t)=\lim_{\lambdaarrow 0}\frac{d}{dt}\hat{K}_{\lambda}(t)(\hat{I}\lambda+II_{\lambda})\wedge$

$=-i \sum_{mn}\{i\sqrt{m+1}|m+1,\tilde{n}\rangle\langle m,\tilde{n}|2\kappa(\phi_{m})\overline{n}(\phi m)w_{1m}(\phi m)\chi[s\prime 1m’ m]\tau_{1}’(t)$

$-i2\kappa(\phi_{m})[\overline{n}(\phi m)+1|_{Z_{1m}}*(\phi m)x_{1^{s_{1}}]}m’ m(T_{1}t)^{\sqrt{m+1}\rangle}|m,\tilde{n}\langle m+1,\tilde{n}|$

$+i\sqrt{n+1}|m,n\overline{+}1\rangle\langle m,\tilde{n}|2\kappa(\phi_{n})\overline{n}(\phi n)z1n(*\phi_{n})\chi[s1n’ T1n1(t)$

$-i2\kappa(\phi_{n})[\overline{n}(\phi m)+1]w_{1}n(\phi n)x_{1s\prime}\tau_{1}^{r}11n’ n(t)\sqrt{n+1}|m,\tilde{n}\rangle\langle m,$ $n\overline{+}1|\}\hat{K}(t)$

$-i \sum_{mn}\{i\sqrt{m+1}|m+1,\tilde{n}\rangle\langle m,\tilde{n}|2\kappa(\phi_{m})z2m(\phi m)\chi[S2m’ T2m](t)$

$-i\sqrt{m+1}|m,\tilde{n}\rangle\langle m+1,\tilde{n}|2\kappa(\phi_{m})w2*m(\phi_{m})x_{1}s’\tau’]2m’ 2m(t)$

$+i\sqrt{n+1}|m,n\overline{+}1\rangle\langle m,\tilde{n}|2\kappa(\phi_{n})w2*n(\phi_{n})\chi_{1s\tau’}\prime 12n’ 2n(t)$

$-i\sqrt{n+1}|m,\tilde{n}\rangle\langle m,$ $n\overline{+}1|2\kappa(\phi n)z_{2n}(\phi_{n})\chi 1^{S_{2}}n’ T_{2}n](t)\}\hat{K}(t)$

$-i(\hat{\Delta}+i\Pi\wedge)\hat{K}(i)$, (34)

where $\hat{K}(t)=\lim_{\lambdaarrow 0^{\hat{K}_{\lambda}(}}t)^{*}.$ Here, $\chi_{[s,\eta}(t)=\theta(t-s)\theta(\tau-t)$, with the step function $\theta(t)$

defined by

$\theta(t)=\{$ 1, for

$t\geq 0$,

(35)

$0$, for $t\leq 0$,

and we introduced the operators $\hat{\Delta}$

and $\hat{\Pi}$ as

$\hat{\Delta}=P\int d\epsilon\sum_{mn}\{[\overline{n}(\epsilon)+1]\frac{\rho(\epsilon)}{\phi_{m}-\epsilon}(m+1)|m+1,\tilde{n}\rangle\langle m+1,\tilde{n}|$

$-(m+1)|m, \tilde{n}\rangle\langle m,\tilde{n}|\overline{n}(\epsilon)\frac{\rho(\epsilon)}{\phi_{m}-\epsilon}-\mathrm{t}.\mathrm{c}.\}$ , (36)

and

$\hat{\Pi}=-\sum_{mn}\{\kappa(\phi_{m})[\overline{n}(\phi_{m})+1](m+1)|m+1,\tilde{n}\rangle\langle m+1,\tilde{n}|$

$+(m+1)|m,\tilde{n}\rangle\langle m,\tilde{n}|\kappa(\phi m)\overline{n}(\phi m)+\mathrm{t}.\mathrm{c}.\}$

$+2 \sum_{m}\{(m+1)|m+1,$$m\overline{+}1\rangle\langle m,\tilde{m}|\kappa(\phi m)\overline{n}(\phi m)$

$+\kappa(\phi_{m})[\overline{n}(\phi m)+1](m+1)|m,\tilde{m}\rangle\langle m+1,$$m\overline{+}1|\}$

.

(37)

(6)

In deriving (34), wechanged the summation with respect to $k$ to the integral with respect to $\epsilon$

with adensity ofstates $\rho(\epsilon)$ defined by

$\sum_{k}\delta(\epsilon-\epsilon_{k})=\rho(\epsilon)$, (38)

and

_used.

the relation

$\int_{-\infty}^{\infty}dv\mathrm{e}-\phi_{n})v=(\epsilon 2\pm i\delta(\epsilon-\phi_{n})\pi$

.

(39)

4 Quantum Wiener Processes

In this section, weconstruct the quantum Wiener processes affected by the non-linearity within

a relevant system.

We introduce boson operators $c_{t,k}(\phi_{n}),$ $C_{t,k}^{*}(\phi_{n})$ and their tilde conjugates satisfying the

commutation relations

$[c_{t,k}(\phi_{n}), c_{tk’}^{*},,(\phi_{n}r)]=2\pi\delta(\epsilon_{k}-\phi n)\mathit{6}(t-t/)\delta_{k}k’\delta nn’$, (40) $[\tilde{c}_{t,k}(\phi_{n}),\tilde{C}_{tk’}^{*},,(\phi_{n}’)]=2\pi\delta(\epsilon k-\phi_{n})\delta(t-t’)\delta_{k},k’\delta_{n}n’$

’ (41)

and define the vacuums $|\rangle$ and $\langle$$|$ by

$c_{t,k}(\phi_{n})|\rangle=\tilde{c}_{t},k(\phi_{n})|\rangle=0$,

{

$|c_{t},(\_{kn})\phi=\langle|\tilde{c}t,k\phi_{n})=0$

.

(42)

Let the Fock space builtonthe basic ket- and br -vectorsmade bycyclic operations of$(c_{t,k}^{*}(\phi n),\tilde{c}t\#,k(\phi_{n}))$

on $|\rangle$ and of_{$(c_{t,k}(\phi_{n}),\tilde{c}t,k(\phi n))$} on $\langle$$|$ be denoted by

$\Gamma^{\beta}$

.

We introduce the exponentialvectors defined by

$|e(z, w) \rangle=\exp[\sum_{n}\sum_{k}\{\int_{s_{n}}\tau_{n_{duz}}nkc(u,k\phi_{n}*)+\int S’*duwk\tilde{c}tau_{n}’\}n]nu,k(\phi n)|\rangle$, (43)

and

$\langle$_{$e(z, w)|= \langle|\exp[\sum_{n}\sum_{k}\{\int_{S_{n}}^{T_{n}}duz^{*}nku,k(C\phi n)+\int^{T_{n_{dw}}}s’\}n]\prime unk\tilde{c}\mathrm{u},k(\phi_{n})$}

.

(44)

Introducing the operators\dagger

$c_{t}( \phi_{n})=\sum ckt,k(\phi_{n})$, $c_{t}^{*}( \phi n)=\sum_{k}c_{t,k}*(\phi n)$, (45)

and their tilde conjugates, we have

$c_{t}(\phi_{n})|e(z, w)\rangle=x1^{s_{n}},Tn](t)2\kappa(\phi n)z_{n}(\phi_{n})|e(_{Z}, w)\rangle$ , (46) $\tilde{c}_{t}(\phi n)|e(z, w)\rangle=\chi[s’,\tau\prime 1nn(t)2\kappa(\phi n)w^{*}(n\phi n)|e(z, w)\rangle,$ (47)

$\langle$$e(z, w)|c_{\iota}(phi n)=\langle e(_{Z}, w)|x_{[1}s_{n},T_{n}(t)2\kappa(\phi n)z*(n\phi_{n})$ , (48)

and

$\langle$$e(z, w)|\tilde{C}^{*}\iota(\phi_{n})=\langle e(z, w)|x\iota S^{\prime\tau_{n}}’]n’(t)2\kappa(\phi_{n})wn(\phi n)$

.

(49)

$\overline{|\mathrm{T}\mathrm{h}\mathrm{e}}$_operators$c_{t}(\phi_{n}),$ $c_{t}^{\}(\emptyset n)$andtheir tilde conjugates correspond to the annihilation and creation operators

(7)

From the commutation relations (40) and (41), we seethat

$[_{Ct}(\phi_{n}),$ $c_{t}^{*\prime},(\phi_{n}’)\mathrm{I}=2\kappa(\phi_{n})\delta(t-t)\delta_{nn}’$, (50)

$[\tilde{c}_{t}(\phi_{n}),\tilde{c}_{t}^{*},(\emptyset n’)]=2\kappa(\phi_{n})\delta(t-t/)\delta nn’$

.

(51)

Here, we changed the summation with respect to $k$ to the

integral..with

respect to $\epsilon$ with the

density of states (38).

We introduce thequantum Wiener processes defined by

$C_{t}( \phi_{n})=\int_{0}^{t}dsCs(\phi_{n})$, $C_{t}^{*}( \phi_{n})=\int_{0}^{t}dsc_{S}^{*}(\phi_{n})$, (52)

and their tilde conjugates. We now investigate the product rule of the increments $dC_{t}(\phi_{n})$,

$dC_{\iota}^{*}(\phi n),$ $d\tilde{c}_{t}(\phi_{n}),$ $d\tilde{C}_{t(\emptyset}*n)$ defined by

$dC_{t}( \phi_{n})=C_{t+}dx(\phi_{n})-C_{t}(\phi_{n})=\int_{t}^{t+d}\iota_{dsc}s(\phi_{n})$, (53)

$dC_{t} \#(\phi_{n})=C_{tdt}+\phi_{n})-C^{\}t(\phi n)=\int_{t}^{t+dl}dsc_{\theta}^{*}(\phi_{n})$, (54)

and their tilde conjugates. It can be done by evaluating the matrix elements of the products

such as $dc_{t}^{*}(\phi n)dCt(\phi_{n^{\prime)}}$ with respect to the exponential vectors. By making use of

(46)$-(49)$,

we then have

$dC_{t}(\phi_{n})dct\phi n^{\prime)}=2\kappa(\phi_{n})\delta_{n}n’dt, d\tilde{C}_{t}(\phi_{n})d\tilde{c}_{t(}*\phi_{n}’)=2\kappa(\phi_{n})\delta_{nn}rdt$, (55)

and other products vanish\ddagger .

We introduce the quantum Wiener processes $B_{t}(\phi_{n}),$ $Bt\dagger(\phi_{n}),\tilde{B}_{t(\phi_{n}}),\tilde{B}_{t}^{\uparrow}(\phi_{n})$ defined by

$B_{t}(\phi_{n})=C_{t}(\phi n)+\overline{n}(\phi_{n})\tilde{c}t\phi_{n})$, $B_{t}^{\dagger}(\phi n)=\tilde{C}t(\phi_{n})+[\overline{n}(\phi_{n})+1]c_{t}\phi n)$, (56)

and their tilde conjugates. The definitions (56)of$B_{t}(\phi_{n})$ and $B_{t}\dagger(\phi_{n})$together with the product

rules (55) giveus thefollowingproduct rules of the increments$dB_{t}(\phi_{n}),$ $dBt\uparrow_{(}\phi_{n})$ and their tilde

conjugates:

$dB_{t}(\phi_{n})dBt\dagger(\phi_{n}’)=2\kappa(\phi n)[\overline{n}(\phi n)+1]\delta_{n}n’dt$_, $dB_{t}(\phi_{n})d\tilde{B}_{t}(\phi_{n}’)=2\kappa(\phi n)\overline{n}(\phi_{n})\delta_{nn}\prime dt$, (57) $dB_{t}^{\dagger}(\phi_{n})dB_{t(}\phi n)’=2\kappa(\phi_{n})\overline{n}(\phi n)\delta_{nn};dt$, $dB_{t}\dagger(\phi_{n})d\tilde{B}_{t}\uparrow(\phi n’)=2\kappa(\phi_{n})[\overline{n}(\phi n)+1]\delta_{n}n’dt$ , (58)

$d\tilde{B}_{t}(\phi_{n})dBt(\phi_{n}’)=2\kappa(\phi_{n})\overline{n}(\phi n)\delta_{n}n’dt$,

$d\tilde{B}_{t}(\phi_{n})\mathrm{L}d\tilde{B}^{\dagger}t(\phi n)’=2\kappa(\phi n)[\overline{n}(\phi_{n}).+.1]\delta_{n}nt|\prime d$, (59)

$d\tilde{B}_{\mathrm{r}}^{\dagger}(\phi n)dB_{t}\dagger(\phi n’)=2\kappa(\phi_{n})[\overline{n}(\phi_{n})+1]\delta nn’dt$, $d\tilde{B}_{t}^{\uparrow}(\phi_{n})d\tilde{B}_{t}(\phi n’)=2\kappa(\phi_{n})\overline{n}(\phi n)\delta_{n}n’dt$, (60)

and other products vanish.

(8)

5 Stochastic

Time-Evolution

Generator

We define the operator $\hat{U}(t)$ such that

$\hat{K}(t)=\langle e(z_{1,w_{1}})|\hat{U}(t)|e(Z_{2}, w2)\rangle$

.

(61)

Using the properties (46)$-(49)$, we see from (34) that $\hat{U}(t)$ satisfies the quantum stochastic

differentialequation

$d\hat{U}(t)=-i\{(\hat{\Delta}+i\Pi\wedge)\hat{U}(t)dt$

$+i \sum_{mn}[\sqrt{m+1}|m+1,\tilde{n}\rangle\langle m,\tilde{n}|\hat{U}(t)\circ dct(\phi_{m})-\sqrt{m+1}|m,\tilde{n}\rangle\langle m+1,\tilde{n}|\hat{U}(t)\circ d\tilde{c}t(\phi_{m})$

$+\sqrt{n+1}|m,$$n\overline{+}1\rangle\langle m,\tilde{n}|\hat{U}(t)\circ d\tilde{c}_{t}(\phi n)-\sqrt{n+1}|m,\tilde{n}\rangle\langle m,$ $n\overline{+}1|\hat{U}(t)\circ dc_{t}(\phi_{n})]$

$+i \sum_{mn}[d\tilde{C}t(phi_{m})\sqrt{m+1}|m+1,\tilde{n}\rangle\langle m,\tilde{n}|\overline{n}(\phi m)-dc_{t}\phi_{m})\sqrt{m+1}|m,\tilde{n}\rangle\langle m+1,\tilde{n}|$

$\mathrm{x}[\overline{n}(\phi m)+1]$

$+dC_{t}^{\}(\phi_{n})\sqrt{n+1}|m,$$n\overline{+}1\rangle\langle m,\tilde{n}|\overline{n}(\phi_{n})-d\tilde{C}t\phi n)^{\sqrt{n+1}\rangle\langle}|m,\tilde{n}m,$ $n\overline{+}1|[\overline{n}(\phi_{n})+1]]$

$\mathrm{o}\hat{U}(t)\}$, (62)

where the symbol$\circ$indicates theStratonovichproduct. Here,weinterpreted (62) asthe

stochas-tic differential equation of the

Stratonovich

type, because it was derived from the ordinary

operator-valued differential equation (15) where the ordinary calculus rulecan be applied.

Using the relations between the

Stratonovich

and the Ito products

$X_{t} \circ dct(\phi_{n})=x_{t}dC_{t}(\phi_{n})+\frac{1}{2}dX_{t}dC_{t}(\phi n)$ , etc., (63)

wefind that (62) becomes

$d\hat{U}(t)=-i(\hat{\Delta}dt+d\hat{M}_{t}^{I})\circ\hat{U}(t)$, (64)

with $d\hat{M}_{t}^{I}$ defined by

$d \hat{M}_{t}^{I}=i\sum_{mn}[\sqrt{m+1}|m+1,\tilde{n}\rangle\langle m,\tilde{n}|dB_{t}(\phi_{m})-dB\mathrm{t}(\phi_{m})\sqrt{m+1}|m,\tilde{n}\rangle\langle m+1,\tilde{n}|]t-\mathrm{t}.\mathrm{C}.$

.

(65)

6 Quantum Stochastic Differential Equations

6.1

Quantum

Stochastic Liouville

Equation

Let us introduce the stochastic time-evolution generator $\hat{V}_{f}(t)$ defined by

$\hat{V}_{f}(t)=\mathrm{e}^{-}\hat{U}i\hat{H}st(t)$, (66)

with $\hat{H}_{S}=H_{s-}\tilde{H}_{S}$. The time-evolution equation of$\hat{V}_{f}(t)$ isgiven by

$d\hat{V}_{f}(t)=-i\hat{H}_{f,tf^{()}}dt\circ\hat{V}i$, (67)

where

(9)

with

$d\hat{M}_{t}=\mathrm{e}^{-i\hat{H}tI}s_{d}\hat{M}_{t}\mathrm{e}^{i\hat{H}s\iota}$

$=i \sum_{mn}[\sqrt{m+1}|m+1,\tilde{n}\rangle\langle m,\tilde{n}|\mathrm{e}^{-i}\phi mdB_{t(}\phi m)-dB^{|i\phi}t(\phi_{m})\mathrm{e}m\sqrt{m+1}|m,\tilde{n}\rangle\langle m+1,\tilde{n}|]$

$-\mathrm{t}.\mathrm{c}.$

.

(69)

Using the relation (63), we can transform the equation (67) of the Stratonovich type into

that of the Ito type

$d\hat{V}_{f}(t)=-i\hat{\mathcal{H}}_{f^{t}},dt\hat{V}_{f}(t)$, (70)

where

$\hat{\mathcal{H}}_{f^{t}},dt=\hat{H}_{f,t}dt-i\frac{1}{2}\hat{H}f,tdt\hat{H}f,tdt$

.

(71)

Evaluating $\hat{H}_{f,t}dt\hat{H}_{f},tdt$in terms of the product rules (57)$-(60)$, we have

$\hat{H}_{f^{t}},dt\hat{H}_{f,t}dt=d\hat{M}_{t}d\hat{M}_{t}=-2\hat{\Pi}dtr$

.

(72)

Therefore, we find that $\hat{\mathcal{H}}_{ft},dt$is given by

$\hat{\mathcal{H}}_{f_{1}^{t}}dt=(\hat{H}_{S}+\hat{\Delta}+i\hat{\Pi})dt+d\hat{M}$_t. (73)

We define the thermalvacuum

$|0_{f}(t)\rangle=\hat{V}_{f}(t)|0_{f}(\mathrm{o})\rangle$

.

(74)

In terms of the time-evolution equation (70),weobtain the quantum stochastic Liouville equation

ofthe Ito type

$d|0_{f}(t)\rangle=-i\hat{\mathcal{H}}_{f,t}dt|0f(t)\rangle$, (75)

where$\hat{\mathcal{H}}_{f,t}dt$ isgiven by (73).

Applying athermal$\mathrm{b}\mathrm{r}\mathrm{a}$-vacuum $\langle$$|$ in

$\Gamma^{\beta}$ to the stochastic Liouville equation (75) of the Ito

type, we see that

$d\langle|0_{f}(t)\rangle=-i\langle|\hat{\mathcal{H}}_{f},tdt|0_{f}(t)\rangle=-i\hat{H}dt\langle|0f^{(t)}\rangle$ , (76)

wherewe defined $\hat{H}$

by

$\hat{H}=\tilde{H}_{S}+\hat{\Delta}+\cdot.i\Pi\wedge$

.

(77)

Here, under the $\mathrm{a}_{\mathfrak{l}}\mathfrak{B}\mathrm{u}\mathrm{m}\mathrm{p}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$ that _{$|0_{f}(0)\rangle=|\mathrm{o}_{S}\rangle|\rangle$} with the thermal vacuum $|0_{S}\rangle$ of relevant

system at $t=0$, we evaluated as $\langle$$|d\hat{M}_{t}|0f^{(t)\rangle}=\langle|d\hat{M}_{t}\hat{V}f(t)|0_{f}(\mathrm{o})\rangle=0$ with the help of the

properties of the Ito type

$\langle|dB_{t}(\phi_{n})\hat{V}_{f}(t)|\rangle=0$,

. $\langle|dB_{t}^{\dagger}(\phi_{n})\hat{V}_{f}(t)|\rangle=0$, (78)

$\langle|d\tilde{B}_{t}(\phi n)\hat{V}f(t)|\rangle=0$, $\langle|d\tilde{B}_{t}^{\dagger}(\emptyset n)\hat{V}_{f}(t)|\rangle=0$

.

(79)

Therefore, putting $|0(t)\rangle=\langle|0_{f}(t)\rangle$, we obtain thequantum master equation

$\frac{\partial}{\partial t}|0(t)\rangle=-i\hat{H}|\mathrm{o}(t)\rangle$, (80)

(10)

6.2 Quantum

Langevin Equation

For any relevant system operator $A$, we define the Heisenberg operator by

$A(t)=\hat{V}_{f}^{-1}(t)A\hat{V}_{f(t})$

.

(81)

With the help of the calculus rule of the Ito type together with (70) and the equation of$\hat{V}_{f}^{-}1(t)$

of the Ito type

$d\hat{V}_{ff}^{-1}(t)=i\hat{V}^{-1}(t)\hat{\mathcal{H}}_{f},dt(-t1)$, (82)

with

$\hat{\mathcal{H}}_{f,t}^{(-1)}dt=\hat{\mathcal{H}}_{f1t}dt+id\hat{M}_{t}d\hat{M}_{t}$ , (83)

we have the quantum Langevin equation of the Ito type

$dA(i)=d\hat{V}_{f}^{-1}(t)A\hat{V}_{f(t})+\hat{V}_{ff}^{-1}(t)Ad\hat{V}(t)+d\hat{V}^{-1}(ft)Ad\hat{V}f(t)$

$=i[\hat{\mathcal{H}}_{f}(t)dt,$ $A(t)]-d\hat{M}(t)[d\hat{M}(t),$ $A(t)]$ , (84)

where$\hat{\mathcal{H}}_{f}(t)dt=\hat{V}_{f}^{-1}(t)\hat{\mathcal{H}}f,tdt\hat{V}_{f(t)}$ and $d\hat{M}(t)=\hat{V}^{-1}(f)td\hat{M}_{t}\hat{V}f(t)$

.

Applying $\langle\langle$$1|=\langle|\langle 1_{S}|$ to the equation (84), we have

$d\langle\langle 1|A(t)=-i\langle\langle 1|A(t)[\hat{H}_{S}(t)dt+i\hat{\Pi}(t)dt+d\hat{M}(t)]$ , (85)

where we used the property $\langle\langle 1|\hat{\mathcal{H}}_{f}(t)=0$ and $\langle\langle 1|d\hat{M}(t)=0$

.

Applying $|0\rangle\rangle=|\mathrm{o}_{S}\rangle|\rangle$ to the

equation (85), we obtain the equation of motion ofexpectation value

$\frac{d}{dt}\langle\langle 1|A(t)|\mathrm{o}\rangle\rangle=i\langle\langle 1|[Hs(t), A(t)]|0\rangle\rangle+\langle\langle 1|A(t)\hat{\Pi}(t)|0\rangle\rangle$, (86)

where we used the thermal state condition

_{

$1|\tilde{A}\dagger(t)=\langle 1|A(t)$ for any operator $A$ of relevant

system, and the properties (78) and (79). The equation (86) can be also derived from the

quantum master equation (80).

7 Summary and

Discussion

In this paper, applying the procedure of Accardi et al. to anon-linear oscillatorinteractingwith

thermal reservoir, we obtained the quantum stochastic differential equations for the non-linear

damped oscillator.

We showed that, in the weak coupling limit, the equation of motion of the matrix element

ofthe time-evolution generator with respect to collective exponential vectors in reservoir space

converges to the equation of motion of the matrix element of the stochastic time-evolution

generator with respect to exponential vectors in the space of quantum Wiener processes. In

the senseofthe matrix elements, wefound that the stochastic time-evolution generator satisfies

a quantum stochastic differential equation. This indicates that the convergence of the time

evolution equation is the weak convergencein the sense of the matrix elements, in other words,

the change of the equation for the time-evolution generator to the one for the stochastic

time-evolution generator can be interpreted as the change of arepresentation space.

Taking account of the effect of the non-linearity within a relevant system, we constructed

quantum Wiener processes together with their representation space. The effect of thequantum

Wiener processes on the time-evolution equation is appeared in the expression of quantum

master equation and the equation of motion of expectation value of an observable. A further

(11)

References

[1] T. Arimitsu, Y. Takahashi and F. Shibata, PhysicaA100 (1980) 507.

[2] T. Arimitsu, PhysicaA104 (1980) 126.

[3] T. Arimitsu, J. Phys. Soc. Japan 51 (1982) 1054. [4] M. Ban and T. Arimitsu, PhysicaA129 (1985) 455.

[5] F. Haake, H. Risken, C. Savage and D. Walls, Phys. Rev. A34 (1986) 3969.

[6] F. Shibataand T. Arimitsu, J. Phys. Soc.Japan. 49 (1980) 891, and references therein.

[7] T. Arimitsu, J. Phys. Soc. Japan 51 (1982) 1720.

[8] T. Arimitsu and H. Umezawa, Prog. Theor. Phys. 74 (1985) 429. [9] T. Arimitsu and H. Umezawa, Prog. Theor. Phys. 77 (1987) 32.

[10] T. Arimitsu, in Thermal Fidd Theories, eds. H. Ezawa, T. Arimitsu and Y. Hashimoto

(North-Holland, 1991) 207.

[11] T. Arimitsu, Lecture Note of the Summer School

_for

YoungerPhysicistsinCondensedMauerPhysics

[published in ”Bussei Kenkyu” (Kyoto) 60 (1993) 491-526, written inEnglish], and the references

therein.

[12] T. Arimitsu, CondensedMatter Physics (Lviv, Ukraine) 4 (1994) 26. [13] T. Arimitsu, Phys. Lett. A153 (1991) 163.

[14] T. Saitoand T. Arimitsu, Modern Phys. Lett. B6 (1992) 1319.

[15] T. Arimitsu and T. Saito, Bussei Kenkyu59-2 (1992) 213, in Japanese.

[16] T. Arimitsu and T. Saito, A

_Unified

Framework

_of

Quantum Stochastic

_Differential

Equations, in Proceedings of the Conferenceon Field Theory and $\mathrm{C}\mathrm{o}\mathrm{i}\mathrm{l}\mathrm{e}\mathbb{C}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{e}}$

Phenomena(1993) inpress. [17] T. Arimitsu and T. Saito, Vistasin Astronomy37 (1993) 99.

[18] T. Arimitsu, M. Ban and T. Saito, PhysicaA177 (1991) 329.

[19] T. Arimitsu, M. Ban and T. Saito, in Structure:

_from

Physics to General Systems, e&. M. Marinaro and G. Scarpetta (WorldScientific, 1991) 163.

[22] T. Arimitsu and T. Saito, Quantum Stochastic

_Differential

Equationsin Phase-SpaceMethods, Mod. Phys. Lett. B (1994) submitted.

[23] T. Saito andT. Arimitsu, Bussei Kenkyu62-1 (1994) 215, in Japanese.

[24] T.Arimitsu and T. Saito, General Structure

_of

the Time-Evolution Generator

_for

Quantum

Stochas-. tic Liouville Equation(1995) in preparationtosubmit.

[25] L. Accardi, A. bigerio and Y. G, Lu, Lect. Notes inMath. 1396 (Springer 1989) 20.

[26] L. Accardi, A. Rigerio and Y. G, Lu, Commun. Math. Phys. 131 (1990) 537.

[27] L. Accardi and Y. G, Lu, Ann. Inst. Henri Poincar\’e 54 (1991) 435.

[28] L. Accardi and L. Y. Gang, Quantum Measurements in Optics, eds. P. Tombesi and D. F. Walls,

(PlenumPress, NewYork 1992) 247.

[29] L. Accardi, J. Gough and Y. G, Lu, Rep. Math. Phys. 36 (1995) 155.

[30] L. van Hove, Physica21 (1955) 617.

[31] T. SaitoandT. Arimitsu, A System

_of

Quantum Stochastic

_Differential

Equationsin tems

_of

Non-Equibbrium Therrno FieldDynamics, J. Phys. A (1997) submitted.