Application of generalized operator representation in the time evolution of quantum systems

Abstract

We have systematically explored the application of generalized operator representation including P-, W-, and Husimi representation in the time evolution of quantum systems. In particular, by using the method of differentiation within an ordered product of operators, we give the normally and antinormally ordered forms of the integral kernels of Husimi operator representations and its corresponding formulations. By making use of the generalized operator representation, we transform exponentially complex operator equations into tractable phase–space equations. As a simple application, the time evolution equation of Husimi function in the amplitude dissipative channel is clearly obtained.

Appendix

$\Delta (\alpha ,\alpha ^{*})$’s antinormally ordered form

$$\begin{aligned} \Delta (\alpha ,\alpha ^{*})=\frac{-1}{\pi }\vdots \mathrm{e}^{2(\alpha ^{*}-a^{\dag })(\alpha -a)}\vdots . \end{aligned}$$

(45)

We can obtain through the procedure as follows:

Proof

$$\begin{aligned} \Delta (\alpha ,\alpha ^{*})=\frac{1}{\pi }:\mathrm{e}^{-2(\alpha ^{*}-a^{\dag })(\alpha -a)}:=\frac{\mathrm{e}^{-2|\alpha |^{2}}}{\pi }\mathrm{e}^{2\alpha a^{\dag }}:\mathrm{e}^{-2a^{\dag }a}:\mathrm{e}^{2\alpha ^{*}a}. \end{aligned}$$

(46)

Due to

$$\begin{aligned} \mathrm{e}^{\lambda a^{\dag }a}=:\mathrm{e}^{(\mathrm{e}^{\lambda }-1)a^{\dag }a}:, \end{aligned}$$

(47)

we have

$$\begin{aligned} (-)^{N}=:\mathrm{e}^{-2a^{\dag }a}:. \end{aligned}$$

(48)

Using

$$\begin{aligned} a^{\dag }(-)^{N}=-(-)^{N}a^{\dag };(-)^{N}a=-a(-)^{N}, \end{aligned}$$

(49)

we can obtain

$$\begin{aligned} \Delta (\alpha ,\alpha ^{*})=\frac{\mathrm{e}^{-2|\alpha |^{2}}}{\pi }(-)^{N}\mathrm{e}^{-2\alpha a^{\dag }}\mathrm{e}^{2\alpha ^{*}a}=\frac{\mathrm{e}^{2|\alpha |^{2}}}{\pi }\mathrm{e}^{2\alpha ^{*}a}(-)^{N}\mathrm{e}^{-2\alpha a^{\dag }}. \end{aligned}$$

(50)

Because

$$\begin{aligned} (-)^{N}=:\mathrm{e}^{-2a^{\dag }a}:=\underset{n=0}{\sum }\frac{(-2)^{n}}{n!}a^{\dag n}a^{n} \end{aligned}$$

(51)

and using the formulation

$$\begin{aligned} a^{\dag n}a^{n}=\vdots H_{n,n}(a,a^{\dag })\vdots , \end{aligned}$$

(52)

here, $H_{m,n}(x,y)$ is Hermite polynomial, we can derive

$$\begin{aligned} (-)^{N}=\underset{n=0}{\sum }\frac{(-2)^{n}}{n!}\vdots H_{n,n}(a,a^{\dag })\vdots . \end{aligned}$$

(53)

Using

$$\begin{aligned} L_{n}(x\cdot y)=H_{n,n}(x,y)\frac{(-1)^{n}}{n!} \end{aligned}$$

(54)

and using

$$\begin{aligned} \frac{1}{1-z}\mathrm{e}^{\frac{xz}{z-1}}=\underset{n=0}{\sum }z^{n}L_{n}(x), \end{aligned}$$

(55)

where $L_{n}$ is Laguerre polynomial, we can obtain

$$\begin{aligned} (-)^{N}=\underset{n=0}{\sum }2^{n}\vdots L_{n}(aa^{\dag })\vdots =-\vdots \mathrm{e}^{2a^{\dag }a}\vdots . \end{aligned}$$

(56)

From this, we have

$$\begin{aligned} \Delta (\alpha ,\alpha ^{*})=\frac{-1}{\pi }\vdots \mathrm{e}^{2(\alpha ^{*}-a^{\dag })(\alpha -a)}\vdots . \end{aligned}$$

(57)

$\square$