摘要:众所周知, 量子态的演化可用与其相应的Wigner函数演化来代替. 因为量子态的Wigner函数和量子态的密度矩阵一样, 都包含了概率分布和相位等信息, 因此对量子态的Wigner函数进行研究, 可以更加快速有效地获取量子态在演化过程的重要信息. 本文从经典扩散方程出发, 利用密度算符的P表示, 导出了量子态密度算符的扩散方程. 进一步通过引入量子算符的Weyl编序记号, 给出了其对应的Weyl量子化方案. 另外, 借助于密度算符的另一相空间表示—Wigner函数, 建立了Wigner算符在扩散通道中演化方程, 并给出了其Wigner算符解的形式. 本文推导出了Wigner算符在量子扩散通道中的演化规律, 即演化过程中任意时刻Wigner算符的形式. 在此结论的基础上, 讨论了相干态经过量子扩散通道的演化情况.
关键词: Wigner函数/
Weyl编序/
量子扩散通道/
演化规律

English Abstract

全文HTML

--> --> -->

近来, 量子调控已经成为研究微观世界的一个重要手段, 而用单光子实现量子操控尤为可行, 例如向光腔中逐个注入光子制备非高斯态, 理论上这属于量子扩散机制^[1,2]. 鉴于量子态的Wigner函数包含了量子态的概率分布和相位等信息, 量子态的演化可代之以研究相应的Wigner函数的演化^[3-5]. 本文旨在研究量子相空间的Wigner算符在量子扩散通道的时间演化规律, 它简洁而物理清晰, 展现了从点源函数$\dfrac{1}{2}\begin{array}{*{20}{c}} : \\ : \end{array}\delta \left( {{z^ * } - {a^\dagger }} \right)\delta \left( {z - a} \right)\begin{array}{*{20}{c}} : \\ : \end{array}$向$t$时刻高斯型函数$\dfrac{1}{{2 kt}}\begin{array}{*{20}{c}} : \\ : \end{array}\exp \left[ {\dfrac{{ - 1}}{{kt}}\left( {{a^\dagger } - {z^ * }} \right)\left( {a - z} \right)} \right]\begin{array}{*{20}{c}} : \\ : \end{array}$的演变, $k$是扩散系数, 这里$\begin{array}{*{20}{c}} : \\ : \end{array}\!\! \begin{array}{*{20}{c}} {} \\ {} \end{array}\begin{array}{*{20}{c}} : \\ : \end{array}$代表Weyl编序; ${a^\dagger }, a$是玻色产生和湮灭算符. 用有序算符内的积分方法也可进一步将Wigner算符的Weyl编序式转化为其他排序形式, 如正规乘积序等, 为计算量子态的Wigner函数演化规律带来便利. 本文安排如下, 先从经典扩散方程推导出量子扩散方程, 并以相干光场为例, 讨论其量子扩散. 鉴于初始相干光场的反正规乘积形式是Delta函数, 其演化就体现在从$\delta \left( {z - a} \right)\delta \left( {{z^ * } - {a^\dagger }} \right)$演化为

$\dfrac{1}{{kt}} \vdots \exp \left[ {\dfrac{{ - 1}}{{kt}}\left( {{z^ * } - {a^\dagger }} \right)\left( {z - a} \right)} \right] \vdots\;,$

这里$ \vdots \begin{array}{*{20}{c}} {} \\ {} \end{array} \vdots $代表反正规排序^[6,7]. 然后再讨论Wigner算符的量子扩散, 鉴于Wigner算符的Weyl排序形式是Delta函数,

$ \dfrac{1}{2}\begin{array}{*{20}{c}} : \\ : \end{array}\delta \left( {{\alpha ^ * } - {a^\dagger }} \right)\delta \left( {\alpha - a} \right)\begin{array}{*{20}{c}} : \\ : \end{array}, $

符号$\begin{array}{*{20}{c}} : \\ : \end{array}\begin{array}{*{20}{c}} {} \\ {} \end{array}\begin{array}{*{20}{c}} : \\ : \end{array}$代表Weyl排序, 为范洪义首创, 所以可以立即从扩散方程导出Wigner算符在扩散通道中的演化规律.

经典扩散方程是

$\frac{{\partial P\left( {z,t} \right)}}{{\partial t}} = - k\frac{{{\partial ^2}P\left( {z,t} \right)}}{{\partial z\partial {z^ * }}},$

其中$k$是扩散率, $P\left( {z, t} \right)$是系统的某种密度分布函数. 下面推导相应的量子扩散方程. 我们将密度算符$\rho $用相干态表象中的$P $-表示^[8,9]:

${\rho _t} = \int {\frac{{{{\rm{d}}^2}z}}{{\text{π}}}} P\left( {z,t} \right)\left| {\left. z \right\rangle } \right.\left\langle {\left. z \right|} \right.,$

其中

$\left| {\left. z \right\rangle } \right. = \exp \Big( { - \frac{{{{\left| z \right|}^2}}}{2} + z{a^\dagger }} \Big)\left| {\left. 0 \right\rangle } \right.$

是相干态, 则密度算符的时间演化满足方程为

$\frac{{{\rm{d}}{\rho _t}}}{{{\rm{d}}t}} = \int {\frac{{{{\rm{d}}^2}z}}{{\text{π}}}} \frac{{\partial P\left( {z,t} \right)}}{{\partial t}}\left| {\left. z \right\rangle } \right.\left\langle {\left. z \right|} \right..$

将经典扩散方程(1)式代入(4)式即有

$\frac{{{\rm{d}}{\rho _t}}}{{{\rm{d}}t}} = - k\int {\frac{{{{\rm{d}}^2}z}}{{\text{π}}}} \frac{{{\partial ^2}P\left( {z,t} \right)}}{{\partial z\partial {z^ * }}}\left| {\left. z \right\rangle } \right.\left\langle {\left. z \right|} \right.,$

利用分部积分法, 并注意到在无穷远处$P\left( {z, t} \right)$消失, 则有

$\begin{aligned} & \int {\frac{{{{\rm{d}}^2}z}}{{\text{π}}}} \frac{{{\partial ^2}P\left( {z, t} \right)}}{{\partial z\partial {z^ * }}}\left| {\left. z \right\rangle } \right.\left\langle {\left. z \right|} \right. \\=\; &\int {\frac{{{{\rm{d}}^2}z}}{{\text{π}}}} P\left( {z, t} \right)\frac{{{\partial ^2}}}{{\partial z\partial {z^ * }}}\left| {\left. z \right\rangle } \right.\left\langle {\left. z \right|} \right.,\end{aligned}$

所以

$\frac{{{\rm{d}}{\rho _t}}}{{{\rm{d}}t}} = - k\int {\frac{{{{\rm{d}}^2}z}}{{\text{π}}}} P\left( {z,t} \right)\frac{{{\partial ^2}}}{{\partial z\partial {z^ * }}}\left| {\left. z \right\rangle } \right.\left\langle {\left. z \right|} \right..$

现在利用相干态投影算符的正规乘积表示

$\left| {\left. z \right\rangle \left\langle {\left. z \right|} \right.} \right. = {:^{ - {{\left| z \right|}^2} + z{a^\dagger } + {z^ * }a - {a^\dagger }a}}:$

来分析$\dfrac{{{\partial ^2}}}{{\partial z\partial {z^ * }}}\left| {\left. z \right\rangle } \right.\left\langle {\left. z \right|} \right.$, 注意到

$\begin{split}{a^\dagger }\left| {\left. z \right\rangle \left\langle {\left. z \right|} \right.} \right.\; & = {a^\dagger }{:^{ - {{\left| z \right|}^2} + z{a^\dagger } + {z^ * }a - {a^\dagger }a}}:\\ &= \left( {{z^ * } + \frac{\partial }{{\partial z}}} \right)\left| {\left. z \right\rangle \left\langle {\left. z \right|} \right.} \right., \end{split}$

$\left| {\left. z \right\rangle \left\langle {\left. z \right|} \right.} \right.a = \left( {{z^ * } + \frac{\partial }{{\partial z}}} \right)\left| {\left. z \right\rangle \left\langle {\left. z \right|} \right.} \right.,$

则有

$\begin{split} & - \frac{\partial^2}{\partial z\partial z^*}|z\rangle \langle z|\\=\; & z\Big(z^* + \frac{\partial }{\partial z} \Big) |z \rangle \langle z| \!-\! \Big(z^* \!+\! \frac{\partial }{\partial z} \Big) \Big(z \!+\! \frac{\partial }{\partial z^*}\Big) |z\rangle \langle z | \\ & - |z|^2|z \rangle \langle z|+ \Big(z + \frac{\partial} {\partial z^*} \Big)(z^*|z\rangle \langle z|) \\ =\; & z a^\dagger |z\rangle \langle z| - \Big( z^* + \frac{\partial }{\partial z} \Big)|z\rangle \langle z|a-|z|^2 |z \rangle \langle z | \\ & + \Big(z + \frac{\partial }{\partial z^*}\Big)|z \rangle \langle z|a^\dagger \\= \; & a^\dagger a|z \rangle \langle z|- a^\dagger |z \rangle \langle z| a - a|z \rangle \langle z|a^\dagger +|z \rangle \langle z|a a^\dagger.\end{split} $

将(10)式代入(6)式得到

$\begin{split}\; & \frac{{{\rm{d}}{\rho _t}}}{{{\rm{d}}t}} = k\int {\frac{{{{\rm{d}}^2}z}}{{\text{π}}}} P(z,t) \big(a^\dagger a| z \rangle \langle z | - a^\dagger | z \rangle \langle z| a \\ & \qquad~ - a | z \rangle \langle z | a^\dagger + | z \rangle \langle z | a a^\dagger \big) \\=\; & k \bigg[ a^\dagger a \int \frac{{\rm d}^2 z}{\text{π}} P( z,t)| z \rangle\langle z | - a^\dagger \int \frac{ {\rm d}^2 z} {\text{π}} P(z,t) | z \rangle \langle z| a \\&-\! a\! \int \frac{{\rm d}^2 z}{\text{π}} P(z,t)| z \rangle \langle z | a^\dagger \!+\! \int \!\frac{{\rm d}^2 z} {\text{π}} P(z,t) |z\rangle \langle z | a {a^\dagger} \bigg].\end{split} $

这说明量子扩散方程为

$\frac{{{\rm{d}}{\rho _t}}}{{{\rm{d}}t}} = k\left( {{a^\dagger }a\rho - {a^\dagger }\rho a - a\rho {a^\dagger } + \rho {a^\dagger }a} \right), $

这是从经典扩散方程过渡到量子扩散方程的捷径.

当初态是纯相干光场时,

${\rho _0} = | z\rangle \langle z |, $

它的正规排序是

${\rho _0} =:\exp \left[ { - \left( {{z^ * } - {a^\dagger }} \right)\left( {z - a} \right)} \right]:, $

利用范洪义等^[10]给出的把正规乘积排序变为反正规乘积排序的公式

$\begin{aligned} \left| {\left. z \right\rangle } \right.\left\langle {\left. z \right|} \right. \; & =:\exp \left[ { - \left( {{z^ * } - {a^\dagger }} \right)\left( {z - a} \right)} \right]: \\ & = {\text{π}} \vdots \delta \left( {z - a} \right)\delta \left( {{z^ * } - {a^\dagger }} \right) \vdots ,\end{aligned}$

所以初态是纯相干光场时的反正规乘积排序是

${\rho _0} = {\text{π}}\, \vdots\, \delta \left( {z - a} \right)\delta \left( {{z^ * } - {a^\dagger }} \right) \vdots, $

故而它的$P - $表示为

${P_0} = {\text{π}}\delta \left( {{z^ * } - {\alpha ^ * }} \right)\delta \left( {z - \alpha } \right).$

由此可以直接验证扩散方程$\dfrac{{\partial P\left( {z, t} \right)}}{{\partial t}} \!=\!$$ - k\dfrac{{{\partial ^2}P\left( {z, t} \right)}}{{\partial z\partial {z^ * }}} $的解是

${P_t} = \frac{1}{{kt}}\exp \left[ {\frac{{ - 1}}{{kt}}\left( {{z^ * } - {\alpha ^ * }} \right)\left( {z - \alpha } \right)} \right],$

此解满足初始条件, 即:

$\begin{split} &\mathop {\lim }\limits_{t \to 0} \frac{1}{{kt}}\exp \left[ {\frac{{ - 1}}{{kt}}\left( {{z^ * } - {\alpha ^ * }} \right)\left( {z - \alpha } \right)} \right] \\=\; & {\text{π}}\delta \left( {{z^ * } - {\alpha ^ * }} \right)\delta \left( {z - \alpha } \right),\end{split}$

这是$\delta \left( x \right) = \mathop {\lim }\limits_{t \to 0} \dfrac{1}{{\sqrt {\text{π}} t}}{{\rm{e}}^{ - {{{x^2}} / t}}}$的推广.
${P_t}$是密度算符${\rho _t}$在相干态表象中的表示, 所以便可得到${\rho _t}$的反正规乘积形式为

${P_t} = \frac{1}{{kt}} \vdots \exp \left[ {\frac{{ - 1}}{{kt}}\left( {{z^ * } - {a^\dagger }} \right)\left( {z - a} \right)} \right] \vdots ,$

这就是相干态在扩散通道中的演化公式. 再用相干态表象^[11]和有序算符内的积分技术^[12-14]可以将它化为正规乘积,

$\begin{split} {P_t} =\;& \frac{1}{{kt}} \vdots \exp \left[ {\frac{{ - 1}}{{kt}}\left( {{z^ * } - {a^\dagger }} \right)\left( {z - a} \right)} \right] \vdots \int {\frac{{{{\rm{d}}^2}\alpha }}{{\text{π}}}} \left| {\left. \alpha \right\rangle } \right.\left\langle {\left. \alpha \right|} \right. \\ =\;& \frac{1}{{kt}}\int {\frac{{{{\rm{d}}^2}\alpha }}{{\text{π}}}} \left| {\left. \alpha \right\rangle } \right.\left\langle {\left. \alpha \right|} \right.\exp \left[ {\frac{{ - 1}}{{kt}}\left( {{z^ * } - {\alpha ^ * }} \right)\left( {z - \alpha } \right)} \right] \\ =\;& \frac{1}{{kt}}\int {\frac{{{{\rm{d}}^2}\alpha }}{{\text{π}}}}:\exp \left[\frac{{ - 1}}{{kt}}\left( {{z^ * } - {\alpha ^ * }} \right)\left( {z - \alpha } \right)\right. \\ & \left.- {{\left| \alpha \right|}^2} + \alpha a + {\alpha ^ * }a - {a^\dagger }a \right]: \\ =\;& \frac{1}{{1 + kt}}{{\rm{e}}^{\frac{z}{{1 + kt}}{a^\dagger }}}:{{\rm{e}}^{\left( {\frac{{kt}}{{1 + kt}} - 1} \right){a^\dagger }a}}:{{\rm{e}}^{\frac{{{z^ * }}}{{1 + kt}}a}}{{\rm{e}}^{ - \frac{{{{\left| z \right|}^2}}}{{1 + kt}}}} \\ =\;& \frac{1}{{1 + kt}}{{\rm{e}}^{\frac{z}{{1 + kt}}{a^\dagger }}}{{\rm{e}}^{{a^\dagger }a\ln \frac{{kt}}{{1 + kt}}}}{{\rm{e}}^{\frac{{{z^ * }}}{{1 + kt}}a}}{{\rm{e}}^{ - \frac{{{{\left| z \right|}^2}}}{{1 + kt}}}}.\\[-15pt]\end{split} $

通过(20)式可发现它不再是纯态. 同时可验证${\rm{tr}}{\rho _t} = 1$, 故而${\rho _t}$是一个新光场密度算符, 代表一个广义的混沌光场^[15,16]. 计算$t$时刻的光子数:

${\rm{tr}}\left( {{a^\dagger }a{\rho _t}} \right) = {\left| z \right|^2} + kt.$

比较初始时刻的光子数${\rm{tr}}\left( {{a^\dagger }a{\rho _0}} \right) = \left\langle {\left. z \right|{a^\dagger }a\left| {\left. z \right\rangle } \right.} \right.=$${\left| z \right|^2} $, 可见光子数增加了$kt$, 这是扩散的结果.

现在讨论Wigner函数在扩散通道中的演化. 鉴于$t$时刻的密度算符$\rho \left( t \right)$的Wigner函数是^[17,18]

$\begin{split} W\left( {p,q,t} \right) \;& = 2{\rm{{\text{π}} tr}}\left[ {\rho \left( t \right)\varDelta \left( {p,q,0} \right)} \right] \\ & = 2{\rm{{\text{π}} tr}}\left[ {\rho \left( 0 \right)\varDelta \left( {p,q,t} \right)} \right],\end{split}$

这里$\varDelta \left( {p, q} \right)$是Wigner算符, 所以也可转而讨论Wigner算符在扩散通道中的时间演化, 即从$\varDelta \left( {p, q, 0} \right)$演化为$\varDelta \left( {p, q, t} \right)$. 历史上, Wigner算符最早是在坐标表象中定义的^[19],

$\varDelta \left( {q,p} \right) = \frac{1}{{2{\text{π}}}}\int_{ - \infty }^\infty {{\rm{d}}v{{\rm{e}}^{{\rm{i}}pv}}\left| {q + \frac{v}{2}} \right\rangle \left\langle {q - \frac{v}{2}} \right|} ,$

利用$| {q \!+\! {v}/{2}}\rangle = {{\rm e}^{{{ - {\rm{i}}Pv} / 2}}} |q\rangle $和$\left| q \right\rangle \left\langle q \right| = \delta \left( {q - Q} \right)$, $Q$是坐标算符, P 是动量算符, (23)式可化为

$\begin{split} \varDelta \left( {q,p} \right) \; &= \frac{1}{{2{\text{π}}}}\int_{ - \infty }^\infty {{\rm{d}}v{{\rm{e}}^{{\rm{i}}pv}}{{\rm{e}}^{{{ - {\rm{i}}Pv} / 2}}}\delta \left( {q - Q} \right){{\rm{e}}^{{{ - {\rm{i}}Pv} / 2}}}} \\ &= \frac{1}{{4{{\text{π}}^2}}}\int\!\!\! {\int_{ - \infty }^\infty {{\rm{d}}u{\rm{d}}v{{\rm{e}}^{{\rm{i}}pv}}{{\rm{e}}^{{{ - {\rm{i}}Pv} / 2}}}{{\rm{e}}^{{\rm{i}}u\left( {q - Q} \right)}}{{\rm{e}}^{{{ - {\rm{i}}Pv} / 2}}}} } \\ &= \frac{1}{{4{{\text{π}}^2}}}\int \!\!\!{\int_{ - \infty }^\infty {{\rm{d}}u{\rm{d}}v{{\rm{e}}^{{\rm{i}}v\left( {p - P} \right)}}^{ + {\rm{i}}u\left( {q - Q} \right)}} } .\\[-18pt]\end{split} $

为了对(24)式进行积分, 我们引入Weyl编序$\begin{array}{*{20}{c}} : \\ : \end{array}\begin{array}{*{20}{c}} {} \\ {} \end{array}\begin{array}{*{20}{c}} : \\ : \end{array}$来标志已经Weyl排序好了的算符^[20-22]. 例如Weyl编序可以将经典量${q^m}{p^r}$通过积分变换(积分核就是Wigner算符$\varDelta \left( {q, p} \right)$)量子化为

$\begin{split}& {q^m}{p^r} \to {\iint_{ - \infty }^\infty {{\rm{d}}p{\rm{d}}q\varDelta \left( {p,q} \right)} } {q^m}{p^r} \\ =\; & {\left( {\frac{1}{2}} \right)^m}\sum\limits_{l = 0}^\infty {\left( {\begin{array}{*{20}{c}} m \\ l \end{array}} \right)} {Q^{m - l}}{P^r}{Q^l},\end{split}$

它是Weyl排序好的, 以$\begin{array}{*{20}{c}} : \\ : \end{array}\begin{array}{*{20}{c}} {} \\ {} \end{array}\begin{array}{*{20}{c}} : \\ : \end{array}$标识,

$\begin{split} & {\left( {\frac{1}{2}} \right)^m}\sum\limits_{l = 0}^\infty {\left( {\begin{array}{*{20}{c}} m \\ l \end{array}} \right)} {Q^{m - l}}{P^r}{Q^l} \\=\; & \begin{array}{*{20}{c}} : \\ : \end{array}{\left( {\frac{1}{2}} \right)^m}\sum\limits_{l = 0}^\infty {\left( {\begin{array}{*{20}{c}} m \\ l \end{array}} \right)} {Q^{m - l}}{P^r}{Q^l}\begin{array}{*{20}{c}} : \\ : \end{array},\end{split}$

再注意到在记号$\begin{array}{*{20}{c}} : \\ : \end{array}\begin{array}{*{20}{c}} {} \\ {} \end{array}\begin{array}{*{20}{c}} : \\ : \end{array}$内部玻色算符是可交换的, 所以有

$\begin{split}& \begin{array}{*{20}{c}} : \\ : \end{array}{\left( {\frac{1}{2}} \right)^m}\sum\limits_{l = 0}^\infty {\left( {\begin{array}{*{20}{c}} m \\ l \end{array}} \right)} {Q^{m - l}}{P^r}{Q^l}\begin{array}{*{20}{c}} : \\ : \end{array} \! =\! \begin{array}{*{20}{c}} : \\ : \end{array}{Q^m}{P^r}\begin{array}{*{20}{c}} : \\ : \end{array} \\\; & = {\iint_{ - \infty }^\infty {{\rm d}\,p{\rm d}\,q} } {q^m}{p^r}\begin{array}{*{20}{c}} : \\ : \end{array}\delta \left( {q - Q} \right)\delta \left( {p - P} \right)\begin{array}{*{20}{c}} : \\ : \end{array}. \\[-20pt]\end{split} $

与一般算符$H\left( {P, Q} \right)$及其经典对应$h\left( {p, q} \right)$的Weyl对应式为^[23]

$H\left( {P,Q} \right) = {\iint_{ - \infty }^\infty {{\rm{d}}q{\rm{d}}p\varDelta \left( {p,q} \right)} } h\left( {p,q} \right),$

通过比较可见Wigner算符的Weyl排序形式是

$\varDelta \left( {p,q} \right) = \begin{array}{*{20}{c}} : \\ : \end{array}\delta \left( {q - Q} \right)\delta \left( {p - P} \right)\begin{array}{*{20}{c}} : \\ : \end{array},$

从而

$H\left( {P,Q} \right) = \begin{array}{*{20}{c}} : \\ : \end{array}h\left( {P,Q} \right)\begin{array}{*{20}{c}} : \\ : \end{array}.$

可见$H\left( {P, Q} \right)$的Weyl排序形式, 可以直接将$h\left( {p, q} \right)$中的$p \to P, q \to Q$, 并放入$\begin{array}{*{20}{c}} : \\ : \end{array}\begin{array}{*{20}{c}} {} \\ {} \end{array}\begin{array}{*{20}{c}} : \\ : \end{array}$内得到. 例如, $\begin{array}{*{20}{c}} : \\ : \end{array}{{\rm{e}}^{{\rm{i}}Pv + {\rm{i}}Qu}}\begin{array}{*{20}{c}} : \\ : \end{array}$的经典对应是${{\rm{e}}^{{\rm{i}}pv + {\rm{i}}qu}}$:

$\begin{split} & {\iint_{ - \infty }^\infty {{\rm{d}}p{\rm{d}}q\varDelta \left( {p,q} \right)} } {{\rm{e}}^{{\rm{i}}pv + {\rm{i}}qu}} \\ =\; & {\iint_{ - \infty }^\infty {{\rm{d}}p{\rm{d}}q\begin{array}{*{20}{c}} : \\ : \end{array}\delta \left( {q - Q} \right)\delta \left( {p - P} \right)\begin{array}{*{20}{c}} : \\ : \end{array}} } {{\rm{e}}^{{\rm{i}}pv + {\rm{i}}qu}} \\ =\; & \begin{array}{*{20}{c}} : \\ : \end{array}{{\rm{e}}^{{\rm{i}}Pv + {\rm{i}}Qu}}\begin{array}{*{20}{c}} : \\ : \end{array}, \\[-20pt] \end{split} $

进一步令

$\alpha = ({{q + {\rm{i}}p}})/{{\sqrt 2 }},~~a = ({{Q + {\rm{i}}P}})/{{\sqrt 2 }},$

可得Wigner算符的Weyl排序形式是

$\varDelta (p,q) \!\to\! \varDelta ( {\alpha,{\alpha ^ * }} ) \!=\! \frac{1}{2}\!\begin{array}{*{20}{c}} : \\ : \end{array}\!\delta ({{\alpha ^ * } \!-\! {a^\dagger }})\delta ({\alpha \!-\! a} )\!\begin{array}{*{20}{c}} : \\ : \end{array}\!.$

由于Wigner算符满足:

$ {\iint_{ - \infty }^\infty {{\rm{d}}p{\rm{d}}q\varDelta \left( {p,q} \right)} } = 1$

或

$2\int {{{\rm{d}}^2}\alpha \varDelta \left( {\alpha,{\alpha ^ * }} \right) = 1} ,$

Wigner算符本身看作是一个混合态的密度算符, 根据(12)式, 它所满足的扩散方程是

$\begin{split} \; & \frac{{{\rm{d}}\varDelta \left( {\alpha,{\alpha ^ * }} \right)}}{{{\rm{d}}t}} = k\left[ {a^\dagger }a\varDelta \left( {\alpha,{\alpha ^ * }} \right) - {a^\dagger }\varDelta \left( {\alpha,{\alpha ^ * }} \right)a \right. \\ & \qquad - \left.a\varDelta \left( {\alpha,{\alpha ^ * }} \right){a^\dagger } + \varDelta \left( {\alpha,{\alpha ^ * }} \right)a{a^\dagger } \right], \end{split}$

此方程也可从Wigner算符的正规乘积形式方程(37)直接导出(具体详见附录A).

$\begin{split}& \varDelta \left( {\alpha,{\alpha ^ * }} \right) = \frac{1}{2}\begin{array}{*{20}{c}} : \\ : \end{array}\delta \left( {{\alpha ^ * } - {a^\dagger }} \right)\delta \left( {\alpha - a} \right)\begin{array}{*{20}{c}} : \\ : \end{array} \\ =\; & \int {\frac{{{{\rm{d}}^2}\beta }}{{2{{\text{π}}^2}}}} \begin{array}{*{20}{c}} : \\ : \end{array}\exp \left\{ {{\rm{i}}\left. {\left[ {{\beta ^ * }\left( {\alpha - a} \right) + \beta \left( {{\alpha ^ * } - {a^\dagger }} \right)} \right]} \right\}} \right.\begin{array}{*{20}{c}} : \\ : \end{array} \\ = \; &\int {\frac{{{{\rm{d}}^2}\beta }}{{2{{\text{π}}^2}}}} \exp \left\{ {{\rm{i}}\left. {\left[ {{\beta ^ * }\left( {\alpha - a} \right) + \beta \left( {{\alpha ^ * } - {a^\dagger }} \right)} \right]} \right\}} \right. \\ =\; & \int {\frac{{{{\rm{d}}^2}\beta }}{{2{{\text{π}}^2}}}}:\exp \left[ {{\rm{i}}\left. {\left( {{\beta ^ * }a - {\beta ^ * }\alpha + \beta {a^\dagger } - \beta {\alpha ^ * }} \right)} \right]} \right.{{\rm{e}}^{{{ - {{\left| \beta \right|}^2}} / 2}}}: \\ =\; & \frac{1}{{\text{π}}}:{{\rm{e}}^{ - 2\left( {{\alpha ^ * } - {a^\dagger }} \right)\left( {\alpha - a} \right)}}:. \\[-12pt]\end{split} $

对照本文第2节内容可见此扩散方程的经典对应是

$\frac{{\partial W}}{{\partial t}} = - k\frac{{{\partial ^2}}}{{\partial \alpha \partial {\alpha ^ * }}}W, $

此方程即为Wigner函数$W$应该满足的扩散方程.

初始的Wigner算符$\varDelta \left( {\alpha, {\alpha ^ * }, 0} \right)$在Weyl编序下是^[24-26]

$\varDelta \left( {\alpha,{\alpha ^ * },0} \right) = \frac{1}{2}\begin{array}{*{20}{c}} : \\ : \end{array}\delta \left( {{\alpha ^ * } - {a^\dagger }} \right)\delta \left( {\alpha - a} \right)\begin{array}{*{20}{c}} : \\ : \end{array},$

那么类比于本文第2节的结果可知:

$\varDelta \left( {\alpha,{\alpha ^ * },t} \right) = \frac{1}{{2kt}}\begin{array}{*{20}{c}} : \\ : \end{array}\exp \left[ {\frac{{ - 1}}{{kt}}\left( {{a^\dagger } - {\alpha ^ * }} \right)\left( {a - \alpha } \right)} \right]\begin{array}{*{20}{c}} : \\ : \end{array}, $

其满足的初始条件为

$\begin{split} & \mathop {\lim }\limits_{t \to 0} \varDelta \left( {\alpha,{\alpha ^ * },t} \right) \\=\; & \mathop {\lim }\limits_{t \to 0} \frac{1}{{2kt}}\begin{array}{*{20}{c}} : \\ : \end{array}\exp \left[ {\frac{{ - 1}}{{kt}}\left( {{a^\dagger } - {\alpha ^ * }} \right)\left( {a - \alpha } \right)} \right]\begin{array}{*{20}{c}} : \\ : \end{array}\\=\; & \frac{1}{2}\begin{array}{*{20}{c}} : \\ : \end{array}\delta \left( {{\alpha ^ * } - {a^\dagger }} \right)\delta \left( {\alpha - a} \right)\begin{array}{*{20}{c}} : \\ : \end{array}.\end{split}$

(41)式就是量子扩散通道中Wigner算符的演化律公式, 可以看出, 它简洁明了, 同时展现了从点源函数$\dfrac{1}{2}\begin{array}{*{20}{c}} : \\ : \end{array}\delta \left( {{z^ * } - {a^\dagger }} \right)\delta \left( {z - a} \right)\begin{array}{*{20}{c}} : \\ : \end{array}$向高斯型函数$\dfrac{1}{{2 kt}}\begin{array}{*{20}{c}} : \\ : \end{array}$$\exp \left[ {\dfrac{{ - 1}}{{kt}}\left( {{a^\dagger } - {z^ * }} \right)\left( {a - z} \right)} \right]\begin{array}{*{20}{c}} : \\ : \end{array} $的演变, 所以此数学表达式的物理意义十分明晰.
从(40)式可知$\varDelta \left( {\alpha, {\alpha ^ * }, t} \right)$的经典对应是

$\varDelta \left( {\alpha,{\alpha ^ * },t} \right) \to \frac{1}{{2kt}}\exp \left[ {\frac{{ - 1}}{{kt}}\left( {{\alpha ^ * } - {z^ * }} \right)\left( {\alpha - z} \right)} \right], $

根据(28)式可知

$\begin{split}\varDelta \left( {\alpha,{\alpha ^ * },t} \right) =\; & \int {{{\rm{d}}^2}z} \frac{1}{{2kt}}\exp \left[ {\frac{{ - 1}}{{kt}}\left( {{\alpha ^ * } - {z^ * }} \right)\left( {\alpha - z} \right)} \right]\\ &\times\varDelta \left( {\alpha,{\alpha ^ * },0} \right).\\[-10pt]\end{split}$

而对(24)式做积分可得:

$\varDelta \left( {\alpha,{\alpha ^ * },0} \right) = \frac{1}{{\text{π}}}:{{\rm{e}}^{ - 2\left( {{\alpha ^ * } - {a^\dagger }} \right)\left( {\alpha - a} \right)}}:,$

所以$\varDelta \left( {\alpha, {\alpha ^ * }, t} \right)$的正规乘积是

$\begin{split} &\varDelta \left( {\alpha,{\alpha ^ * },t} \right) \\ =\;& \int {{{\rm{d}}^2}z} \frac{1}{{kt}}\exp \left[ {\frac{{ - 1}}{{kt}}\left( {{\alpha ^ * } - {z^ * }} \right)\left( {\alpha - z} \right)} \right]\\ &\times \frac{1}{{\text{π}}}: {{\rm{e}}^{ - 2\left( {{\alpha ^ * } - {a^\dagger }} \right)\left( {\alpha - a} \right)}}: \\ = \;&\frac{1}{{{\text{π}}\left( {2kt{\rm{ + 1}}} \right)}}:\exp \left[ {\frac{{ - 2}}{{2kt{\rm{ + 1}}}}\left( {{a^\dagger } - {\alpha ^ * }} \right)\left( {a - \alpha } \right)} \right]:. \\ \end{split} $

这样就从(40)式的Weyl编序形式导出了其正规乘积形式. 另外, (45)式还可进一步得到验证(具体详见附录B), 即将正规乘积形式转化为Weyl编序形式. 举例, 当初态是纯相干态, 那么经过扩散通道后, 从(40)式可知它的Wigner函数为

$\begin{split} \; & {W_{\rm{F}}} = \left\langle z \right|\varDelta \left( {\alpha,{\alpha ^ * },t} \right)\left| z \right\rangle \\={}& \frac{1}{{{\text{π}}\left( {2kt{\rm{ + 1}}} \right)}} \exp \left[ {\frac{{ - 2}}{{2kt{\rm{ + 1}}}}\left( {{\alpha ^ * } - {z^ * }} \right)\left( {\alpha - z} \right)} \right], \\& \qquad z = q + {\rm i} p.\end{split}$

图1所示为Wigner算符的演化规律, 图1(a)描绘了相干态初始的Wigner函数, 尖峰象征Delta函数; 图1(b) 描绘了$kt = 0.8$时高斯型Wigner函数. 对比两图中Wigner函数的峰值及形状, 可以看出相干态在耗散通道的演化情况.
图 1 (a) $kt = 0$, (b) $kt = 0.8$时, 相干态下Wigner算符的演化规律($\alpha = 1/2\left( {1 + i} \right)$)
Figure1. Evolution law of Wigner operator for the coherent state with $\alpha = 1/2\left( {1 + i} \right)$ for (a) $kt = 0$; (b) $kt = 0.8$

本文引入算符的Weyl编序记号, 导出量子扩散通道中Wigner算符的演化律公式, 它简洁而物理清晰, 展现了从点源函数

$\frac{1}{2}\begin{array}{*{20}{c}} : \\ : \end{array}\delta \left( {{z^ * } - {a^\dagger }} \right)\delta \left( {z - a} \right)\begin{array}{*{20}{c}} : \\ : \end{array}$

向高斯型函数

$\frac{1}{{2 kt}}\begin{array}{*{20}{c}} : \\ : \end{array}\exp \left[ {\frac{{ - 1}}{{kt}}\left( {{a^\dagger } - {z^ * }} \right)\left( {a - z} \right)} \right]\begin{array}{*{20}{c}} : \\ : \end{array}$

的演变, $k$是扩散系数. 由此也可转化为Wigner算符的其他排序形式, 如正规乘积序. 值得指出, 对于相干态的演化用了反正规乘积来讨论, 而对Wigner算符的演化用Weyl排序来讨论, 这两者的演化在数学形式上是一样的.

本节验证Wigner算符所满足的扩散方程. 由${\rm{Wigner}}$算符的正规乘积表达式(44)式可算出:

$\begin{split}\frac{\partial }{{\partial \alpha }}\varDelta \left( {\alpha,{\alpha ^ * }} \right)\; &= \frac{{\rm{1}}}{{\text{π}}}\frac{\partial }{{\partial \alpha }}:{{\rm{e}}^{ - 2\left( {{\alpha ^ * } - {a^\dagger }} \right)\left( {\alpha - a} \right)}}: \\ &= 2\left( {{a^\dagger } - {\alpha ^ * }} \right)\varDelta \left( {\alpha,{\alpha ^ * }} \right),\end{split}\tag{A1}$

$\begin{split}\frac{\partial }{{\partial {\alpha ^ * }}}\varDelta \left( {\alpha,{\alpha ^ * }} \right)\; & = \frac{1}{{\text{π}}}\frac{\partial }{{\partial {\alpha ^ * }}}:{{\rm{e}}^{ - 2\left( {{\alpha ^ * } - {a^\dagger }} \right)\left( {\alpha - a} \right)}}:\\ & = \varDelta \left( {\alpha,{\alpha ^ * }} \right)2\left( {a - \alpha } \right),\end{split}\tag{A2}$

因此有

${a^\dagger }\varDelta \left( {\alpha,{\alpha ^ * }} \right) = \left( {\frac{\partial }{{2\partial \alpha }} + {\alpha ^ * }} \right)\varDelta \left( {\alpha,{\alpha ^ * }} \right),\tag{A3}$

$\varDelta \left( {\alpha,{\alpha ^ * }} \right)a = \left( {\frac{\partial }{{2\partial {\alpha ^ * }}} + \alpha } \right)\varDelta \left( {\alpha,{\alpha ^ * }} \right).\tag{A4}$

另一方面, 从Wigner算符的反正规乘积表达式

$\varDelta \left( {\alpha,{\alpha ^ * }} \right) = - \frac{1}{{\text{π}}} \vdots {{\rm{e}}^{2\left( {{\alpha ^ * } - {a^\dagger }} \right)\left( {\alpha - a} \right)}} \vdots ,\tag{A5}$

可以推导出:

$\begin{split}\frac{\partial }{{\partial {\alpha ^ * }}}\varDelta \left( {\alpha,{\alpha ^ * }} \right) ={}& \varDelta \left( {\alpha,{\alpha ^ * }} \right)2\left( {a - \alpha } \right)\\={}& 2\left( {\alpha - a} \right)\varDelta \left( {\alpha,{\alpha ^ * }} \right),\end{split}\tag{A6}$

$\begin{split}\frac{\partial }{{\partial \alpha }}\varDelta \left( {\alpha,{\alpha ^ * }} \right)\; & = 2\left( {{a^\dagger } - {\alpha ^ * }} \right)\varDelta \left( {\alpha,{\alpha ^ * }} \right) \\ &= 2\varDelta \left( {\alpha,{\alpha ^ * }} \right)\left( {{\alpha ^ * } - {a^\dagger }} \right),\end{split}\tag{A7}$

所以

$\left( {\alpha - \frac{\partial }{{2\partial {\alpha ^ * }}}} \right)\varDelta \left( {\alpha,{\alpha ^ * }} \right) = a\varDelta \left( {\alpha,{\alpha ^ * }} \right),\tag{A8}$

$\left( {{\alpha ^ * } - \frac{\partial }{{2\partial \alpha }}} \right)\varDelta \left( {\alpha,{\alpha ^ * }} \right) = \varDelta \left( {\alpha,{\alpha ^ * }} \right){a^\dagger },\tag{A9}$

$\begin{split}& \quad \varDelta \left( {\alpha,{\alpha ^ * }} \right){a^\dagger }a \\\;& = \left( {{\alpha ^ * } - \frac{\partial }{{2\partial \alpha }}} \right)\varDelta \left( {\alpha,{\alpha ^ * }} \right)a\\ & = \left( {{\alpha ^ * } - \frac{\partial }{{2\partial \alpha }}} \right)\left( {\frac{\partial }{{2\partial {\alpha ^ * }}} + \alpha } \right)\varDelta\left( {\alpha,{\alpha ^ * }} \right),\end{split}\tag{A10}$

由以上这些关系式就能导出(36)式.

相干态$\left| z \right\rangle \left\langle z \right|$的经典对应是

$\begin{split}& \quad 2{\rm{{\text{π}} tr}}\left[ {\left| z \right\rangle \left\langle z \right|\varDelta\left( {q,p} \right)} \right] \\\; & = 2{\text{π}}\frac{1}{{\text{π}}}\left\langle z \right|:{{\rm{e}}^{ - 2\left( {a - \alpha } \right)\left( {{a^\dagger } - {\alpha ^ * }} \right)}}:\left| z \right\rangle \\ &= 2{{\rm{e}}^{ - 2\left( {{z^ * } - {\alpha ^ * }} \right)\left( {z - \alpha } \right)}},\end{split}\tag{B1} $

其Weyl对应式则为

$\left| z \right\rangle \left\langle z \right| = 4\int {{{\rm{d}}^2}\alpha } {{\rm{e}}^{ - 2\left( {{z^ * } - {\alpha ^ * }} \right)\left( {z - \alpha } \right)}}\varDelta \left( {\alpha,{\alpha ^ * }} \right),\tag{B2}$

由于是Delta函数型, 所以$\left| z \right\rangle \left\langle z \right|$的Weyl排序形式为

$\left| z \right\rangle \left\langle z \right| = 2\begin{array}{*{20}{c}} : \\ : \end{array}{{\rm{e}}^{ - 2\left( {{z^ * } - {a^\dagger }} \right)\left( {z - a} \right)}}\begin{array}{*{20}{c}} : \\ : \end{array},\tag{B3}$

代入(2)式得到

$\rho = 2\int {\frac{{{{\rm{d}}^2}z}}{{\text{π}}}} P\left( z \right)\begin{array}{*{20}{c}} : \\ : \end{array}{{\rm{e}}^{ - 2\left( {{z^ * } - {a^\dagger }} \right)\left( {z - a} \right)}}\begin{array}{*{20}{c}} : \\ : \end{array},\tag{B4}$

鉴于

$P\left( z \right) = {{\rm{e}}^{{{\left| z \right|}^2}}}\int {\frac{{{\rm d^2}\beta }}{{\text{π}}}} \left\langle { - \beta } \right|\rho \left| \beta \right\rangle {{\rm{e}}^{{{\left| \beta \right|}^2}}}{{\rm{e}}^{{\beta ^ * }z - \beta {z^ * }}},\tag{B5}$

这里$\left| \beta \right\rangle $为相干态, $\left| \beta \right\rangle = \exp \left[ {{{ - {{\left| \beta \right|}^2}} / 2} + \beta {a^\dagger }} \right]\left| 0 \right\rangle $, 所以

$\begin{split}\rho =\; & 2\int {\frac{{{{\rm{d}}^2}z}}{{\text{π}}}} {{\rm{e}}^{{{\left| z \right|}^2}}}\int {\frac{{{{\rm{d}}^2}\beta }}{{\text{π}}}} \left\langle { - \beta } \right|\rho \left| \beta \right\rangle {{\rm{e}}^{{{\left| \beta \right|}^2}}}{{\rm{e}}^{{\beta ^ * }z - \beta {z^ * }}}\\ & \times \begin{array}{*{20}{c}}:\\:\end{array}{{\rm{e}}^{ - 2\left( {{z^ * } - {a^\dagger }} \right)\left( {z - a} \right)}}\begin{array}{*{20}{c}}:\\:\end{array}\\ =\; & 2\int {\frac{{{{\rm{d}}^2}\beta }}{{\text{π}}}} \left\langle { - \beta } \right|\rho \left| \beta \right\rangle \begin{array}{*{20}{c}}:\\:\end{array}{{\rm{e}}^{2\left( {{\beta ^ * }a - \beta {a^\dagger } + {a^\dagger }a} \right)}}\begin{array}{*{20}{c}}:\\:\end{array},\\[-15pt]\end{split}\tag{B6}$

这就是将算符转化为Weyl编序的形式. 当取$\rho $为(45)式时,

$\varDelta \left( {\alpha,{\alpha ^ * },t} \right)\! =\! \frac{1}{{{\text{π}}\left( {2kt \!+\! 1} \right)}}:\exp \left[ {\frac{{ - 2}}{{2kt\! +\! 1}}\left( {{a^\dagger } \!-\! {\alpha ^ * }} \right)\left( {a\! -\! \alpha } \right)} \right]\!:,\tag{B7}$

此为正规乘积形式, 则代入(B6)式便可得到Weyl编序形式.

$\begin{split} & \varDelta \left( {\alpha,{\alpha ^ * },t} \right) \\=\; & 2\begin{array}{*{20}{c}} : \\ : \end{array}\int {\frac{{{{\rm{d}}^2}\beta }}{{{\text{π}}\left( {2kt + 1} \right)}}} \exp \Big[- 2{{\left| \beta \right|}^2} \\ & \left.- \frac{2}{{2kt + 1}}\left( { - {\beta ^ * } - {\alpha ^ * }} \right)\left( {\beta - \alpha } \right) + 2{a^\dagger }a + 2a{\beta ^ * } - 2\beta {a^\dagger }\right]\!\!\begin{array}{*{20}{c}} : \\ : \end{array} \\ = \; &2\int \!{\frac{{{{\rm{d}}^2}\beta }}{{{\text{π}}\left( {2kt + 1} \right)}}} \!\!\begin{array}{*{20}{c}} : \\ : \end{array}\!\!\exp \left[ { - {{\left| \beta \right|}^2}\frac{{4kt}}{{2kt \!+\! 1}} \!+\! 2\beta \left( {\frac{{{\alpha ^ * }}}{{2kt \!+\! 1}}\! -\! {a^\dagger }} \right)} \right. \\ &\left. { - 2{\beta ^ * }\left( {\frac{\alpha }{{2kt + 1}} - a} \right)- \frac{{2{{\left| \alpha \right|}^2}}}{{2kt + 1}} + 2{a^\dagger }a} \right]\begin{array}{*{20}{c}} : \\ : \end{array} \\ = \; &\frac{1}{{2kt}}\begin{array}{*{20}{c}} : \\ : \end{array}\exp \left[ - \frac{{2kt + 1}}{{kt}}\left( {\frac{{{\alpha ^ * }}}{{2kt + 1}} - {a^\dagger }} \right)\left( {\frac{\alpha }{{2kt + 1}} - a} \right)\right. \\ &\left.- \frac{{2{{\left| \alpha \right|}^2}}}{{2kt + 1}} + 2{a^\dagger }a \right]\begin{array}{*{20}{c}} : \\ : \end{array} \\ = \; &\frac{1}{{2kt}}\begin{array}{*{20}{c}} : \\ : \end{array}\exp \left[ { - \frac{1}{{kt}}\left( {{\alpha ^ * } - {a^\dagger }} \right)\left( {\alpha - a} \right)} \right]\begin{array}{*{20}{c}} : \\ : \end{array}.\\[-15pt]\end{split}\tag{B8}$

此即验证了(40)式.