Entropy evolution law of general quadratic state in a laser channel

Abstract

Using the integration within ordered products, we obtain the analytical density-operator evolution of the general quadratic state {\rho }_0 in a laser channel, which always maintains the same form as the initial density operator {\rho }_0. Introducing the differential representation of the von-Neumann entropy and two important resemble formulas, we analytically and numerically investigate the entropy evolution of the state {\rho }_0 in a laser channel, which is determined by not only the competition of the gain and loss of the laser channel, but also the coefficients \alpha, \beta of the initial quadratic state {\rho }_0.

Graphical abstract

Keywords

general quadratic state / laser channel / quantum entropy / operator ordering / integration within ordered product

1 Introduction

As one of the most important concepts in physics, entropy measures the disorder of a system. Quantum entropy is used in perturbation theory, central limit theorems, thermodynamics of spin systems, entropic uncertainty relations and optical communication, among others. The behaviors of entropy are governed by the competition of the density operators of systems and their surrounding environment [1, 2]. Studying the evolution of the entropy explains how the self-organization phenomenon happens between them. However, the evolution of quantum entropy with a time-dependent density operator {\rho }_t is difficult to investigate, since the formula for entropy in its von-Neumann form is given by [3, 4]

(1)S\left({\rho }_t\right)=-k_B \text{tr}\left({\rho }_t\ln {\rho }_t\right),

especially, the natural logarithm of the density operator {\rho }_t, i,e., \ln {\rho }_t, is hardly treated, and where k_B is the Boltzmann constant.

In this work, we focus on the entropy variation of a kind of systems during a laser channel process, without loss of generality, the system is described by the following quadratic density operator in normal ordering (indicated by the symbol ::), i.e.,

(2)\begin{array}{rl}{\rho }_0=& {\mathrm{e}}^{\delta }:\exp (\alpha a^{†2}+\beta a^{†}a\\ & +{\alpha }^{\ast }{a}^2+\gamma a^{†}+{\gamma }^{\ast }a):,\end{array}

where \alpha ,\beta and \gamma are the coefficients of the state {\rho }_0 and \delta is the normalization factor, a^{†} and a are respectively bosonic creation and annihilation operators. This density operator is general as it can describe squeezing, displacement, degenerate parametric amplification, and more.

It is also known that the laser channel is described by the master equation \mathrm{d}{\rho }_t/\mathrm{d}t=g\left([a^{†}\rho ,a]+[a^{†},\rho a]\right)+\kappa \left([a\rho ,a^{†}]+[a,\rho a^{†}]\right) [5-7], where a^{†}\rho a, a\rho a^{†} respectively refer to laser gain and loss, and g, \kappa respectively represent the gain and loss rate of laser cavity. The solution to this equation is in the form

(3){\rho }_t={\displaystyle \sum _{i,j=0}^{\mathrm{\infty }}{\mathcal{M}}_{ij}{\rho }_0{\mathcal{M}}_{ij}^{†},}

where {\mathcal{M}}_{ij} is the Kraus operator corresponding to the density operator {\rho }_t. In Refs. [8, 9], using the entangled state representation found to solve the master equation for the laser channel and obtain the Kraus operator {\mathcal{M}}_{ij}, i.e.,

(4){\mathcal{M}}_{ij}=\sqrt{{\displaystyle \frac{T_3{\kappa }^i{g}^j{T}_1^{i+j}}{i!j!T_2^{2j}}}}{\mathrm{e}}^{a^{†}a\ln T_2}{a}^{†j}{a}^i,

where T_1=\varrho \left[1-{\mathrm{e}}^{-2\left(\kappa -g\right)t}\right], T_2= \varrho \left(\kappa -g\right) {\mathrm{e}}^{-\left(\kappa -g\right)t} and T_3= \varrho \left(\kappa -g\right)=1-g{T}_1 with \varrho ={\left[\kappa -g{\mathrm{e}}^{-2\left(\kappa -g\right)t}\right]}^{-1}, and they obey the relations \left(1-g{T}_1\right)\left(1-\kappa T_1\right)= T_2^2 and 1-\kappa T_1=T_2{\mathrm{e}}^{-\left(\kappa -g\right)t}> 0. One can check \sum _{i,j=0}^{\mathrm{\infty }}{\mathcal{M}}_{ij}^{†}{\mathcal{M}}_{ij}=1, thus

(5)\text{tr}{\rho }_t= \text{tr}\left({\displaystyle \sum _{i,j=0}^{\mathrm{\infty }}{\mathcal{M}}_{ij}^{†}{\mathcal{M}}_{ij}{\rho }_0}\right)=\text{tr}{\rho }_0= 1.

Thus before deriving {\rho }_t for {\rho }_0, we should let tr{\rho }_0= 1, that is to determine \delta in terms of \alpha , \beta ,\gamma in Eq. (2). Here, the research motivation of this work needs to be emphasized in the two aspects. One is that the initial quadratic state and laser channel are general, so the entropy evolution law of general quadratic state is general sense, the other is that the differential representation of the von-Neumann entropy is a new method for calculating quantum entropy and suitable for solving quantum entropy problems in other systems.

This work is arranged as follows. In Section 2, we obtain the normalization factor of the general quadratic normally ordered density operator {\rho }_0. In Section 3, we further obtain the analytical evolution of the density operator {\rho }_0 in a laser channel. In Section 4, we first propose the differential form of the von-Neumann entropy and two important resemble formulas, and use them to analytically obtain the entropy evolution S\left({\rho }_t\right) of {\rho }_0 in a laser channel, and then numerically investigate the impact of the parameters \kappa , g on the entropy S\left({\rho }_t\right). Finally, Sections 5 and 6 are, respectively, devoted to discussing several special cases of the entropy S\left({\rho }_t\right) and summarizing some main results of this work.

2 Normalization of general quadratic normally ordered density operator

The trace of {\rho }_0 must be one, using the completeness relation of coherent states |z⟩, i.e., {\pi }^{-1}\int {\mathrm{d}}^{2}z |z⟩⟨z|=1, where |z⟩= {\mathrm{e}}^{-|z{|}^2/2}{\mathrm{e}}^{z{a}^{†}}|0⟩ [10-15], and the integral formula

(6)\begin{array}{rl}& \int \frac{{\mathrm{d}}^2\alpha }{\pi }{\mathrm{e}}^{h{\left|\alpha \right|}^2+s\alpha +\eta {\alpha }^{\ast }+f{\alpha }^2+g{\alpha }^{\ast 2}}\\ & ={\left(h^2-4fg\right)}^{-1/2}\exp \left(\frac{-hs\eta +s^{2}g+{\eta }^{2}f}{h^2-4fg}\right),\end{array}

which is valid for the convergence condition Re, we have

(7)\begin{array}{rl}\text{tr}{\rho }_0=& {\mathrm{e}}^{\delta }\text{tr}[\int \frac{{\mathrm{d}}^{2}z}{\pi }|z⟩⟨z|:\exp (\alpha a^{†2}\\ & +\beta a^{†}a+{\alpha }^{\ast }{a}^2+\gamma a^{†}+{\gamma }^{\ast }a):]\\ =& {\mathrm{e}}^{\delta }\int \frac{{\mathrm{d}}^{2}z}{\pi }{\mathrm{e}}^{\beta {\left|z\right|}^2+\alpha z^{\ast 2}+{\alpha }^{\ast }{z}^2+\gamma z^{\ast }+{\gamma }^{\ast }z}\\ =& {\mathrm{e}}^{\delta }{\left({\beta }^2-4{\left|\alpha \right|}^2\right)}^{-1/2}{\mathrm{e}}^{\mathrm{\Delta }}.\\ =& 1,\end{array}

where \mathrm{\Delta }=(\alpha {\gamma }^{\ast 2}-\beta {\left|\gamma \right|}^2+ {\alpha }^{\ast }{\gamma }^2)/({\beta }^2-4{\left|\alpha \right|}^2). So the unique of \delta is

(8)\delta ={\displaystyle \frac{1}{2}}\ln \left({\beta }^2-4{\left|\alpha \right|}^2\right)-\mathrm{\Delta } .

The following conditions on the parameters \alpha ,\beta must be satisfied for {\rho }_0 to be a valid state in physics: and {\beta }^2-4{\left|\alpha \right|}^2> 0 or equivalently, and , which is actually the convergent condition for the integral in Eq. (6).

3 Evolution of ρ₀

Let us suppose that, for the initial state {\rho }_0, the output state at time t is {\rho }_t. Inserting the initial state {\rho }_0 and the completeness relations of coherent states of two modes |z_1⟩ and |z_2⟩ into Eq. (3), and using the operator identity {\mathrm{e}}^{a^{†}a\ln T_2}=:{\mathrm{e}}^{a^{†}a\left(T_2-1\right)}: [16, 17], we therefore see the evolution of {\rho }_0 in a laser channel, that is

(9)\begin{array}{rl}{\rho }_t=& {\mathrm{e}}^{\delta }\sum _{i,j=0}^{\mathrm{\infty }}\frac{{\kappa }^i{g}^j{T}_3{T}_1^{i+j}}{i!j!}{a}^{†j}{\mathrm{e}}^{a^{†}a\ln T_2}{a}^i:\exp (\alpha a^{†2}\\ & +\beta a^{†}a+{\alpha }^{\ast }{a}^2+\gamma a^{†}+{\gamma }^{\ast }a):a^{†i}{\mathrm{e}}^{a^{†}a\ln T_2}{a}^j\\ =& {\mathrm{e}}^{\delta }{T}_3\sum _{i,j=0}^{\mathrm{\infty }}\frac{{\kappa }^i{g}^j{T}_1^{i+j}}{i!j!}\int \frac{{\mathrm{d}}^2{z}_1{\mathrm{d}}^2{z}_2}{{\pi }^2}{a}^{†j}:{\mathrm{e}}^{a^{†}a\left(T_2-1\right)}:\\ & \times a^i|z_1⟩⟨z_1|:\exp (\alpha a^{†2}+\beta a^{†}a+{\alpha }^{\ast }{a}^2\\ & +\gamma a^{†}+{\gamma }^{\ast }a):|z_2⟩⟨z_2|a^{†i}:{\mathrm{e}}^{a^{†}a\left(T_2-1\right)}:a^j.\end{array}

Noting that, for two different coherent states |z_1⟩ and |z_2⟩, the inner product reads ⟨z_1|z_2⟩={\mathrm{e}}^{-{\left|z_1\right|}^2/2-{\left|z_2\right|}^2/2+z_1^{\ast }{z}_2}, and the corresponding quantum mechanical operator is

(10)|z_1⟩⟨z_2|={\mathrm{e}}^{-({\left|z_1\right|}^2+{\left|z_2\right|}^2)/2}:{\mathrm{e}}^{z_1{a}^{†}+z_2^{\ast }a-a^{†}a}:,

we thus have

(11)\begin{array}{rl}{\rho }_t=& {\mathrm{e}}^{\delta }{T}_3\sum _{i,j=0}^{\mathrm{\infty }}\frac{{\kappa }^i{g}^j{T}_1^{i+j}}{i!j!}\int \frac{{\mathrm{d}}^2{z}_1{\mathrm{d}}^2{z}_2}{{\pi }^2}:a^{†j}\\ & \times {\mathrm{e}}^{a^{†}{z}_1\left(T_2-1\right)}{z}_1^i{\mathrm{e}}^{-{\left|z_1\right|}^2-{\left|z_2\right|}^2+z_1^{\ast }{z}_2+a^{†}{z}_1+z_2^{\ast }a-a^{†}a}\\ & \times z_2^{\ast i}{\mathrm{e}}^{z_2^{\ast }a\left(T_2-1\right)}{a}^j:{\mathrm{e}}^{\alpha z_1^{\ast 2}+\beta z_1^{\ast }{z}_2+{\alpha }^{\ast }{z}_2^2+\gamma z_1^{\ast }+{\gamma }^{\ast }{z}_2}\\ =& {\mathrm{e}}^{\delta }{T}_3\int \frac{d^2{z}_1{d}^2{z}_2}{{\pi }^2}{\mathrm{e}}^{-{\left|z_1\right|}^2-{\left|z_2\right|}^2+z_1^{\ast }{z}_2+\kappa T_1{z}_1{z}_2^{\ast }}\\ & \times {\mathrm{e}}^{\alpha z_1^{\ast 2}+\beta z_1^{\ast }{z}_2+{\alpha }^{\ast }{z}_2^2+\gamma z_1^{\ast }+{\gamma }^{\ast }{z}_2}\\ & \times :\exp [\left(g{T}_1-1\right)a^{†}a+a^{†}{z}_1+z_2^{\ast }a\\ & +a^{†}{z}_1\left(T_2-1\right)+z_2^{\ast }a\left(T_2-1\right)]:.\end{array}

After rearranging the similar terms, we have

(12)\begin{array}{rl}{\rho }_t=& {\mathrm{e}}^{\delta }{T}_3\int \frac{{\mathrm{d}}^2{z}_1{\mathrm{d}}^2{z}_2}{{\pi }^2}:\exp [\left(g{T}_1-1\right)a^{†}a\\ & -{\left|z_1\right|}^2+\alpha z_1^{\ast 2}+\gamma z_1^{\ast }+T_2{a}^{†}{z}_1\\ & -{\left|z_2\right|}^2+{\alpha }^{\ast }{z}_2^2+\kappa T_1{z}_1{z}_2^{\ast }\\ & +\left(\beta +1\right)z_1^{\ast }{z}_2+T_{2}a{z}_2^{\ast }+{\gamma }^{\ast }{z}_2]:,\end{array}

which is in normal ordering, and thus first integrating over z_2and then over z_1 by using Eq. (6), we finally obtain the analytical evolution of {\rho }_0 for laser noise as

(13)\begin{array}{rl}{\rho }_t=& {\mathrm{e}}^{\delta }{T}_3\int \frac{{\mathrm{d}}^2{z}_1}{\pi }:\exp \{\left(g{T}_1-1\right)a^{†}a-{\left|z_1\right|}^2\\ & +\alpha z_1^{\ast 2}+\gamma z_1^{\ast }+T_2{a}^{†}{z}_1+[\left(\beta +1\right)z_1^{\ast }+{\gamma }^{\ast }]\\ & \times \left(\kappa T_1{z}_1+a{T}_2\right)+{\alpha }^{\ast }{\left(\kappa T_1{z}_1+a{T}_2\right)}^2\}:\\ =& {\mathrm{e}}^{{\delta }_t}:\exp ({\alpha }_t{a}^{†2}+{\beta }_t{a}^{†}a\\ & +{\alpha }_t^{\ast }{a}^2+{\gamma }_t{a}^{†}+{\gamma }_t^{\ast }a):,\end{array}

where the time-dependent parameters {\alpha }_t, {\beta }_t, {\gamma }_t and {\delta }_t are, respectively,

(14)\begin{array}{rl}{\alpha }_t=& \alpha \varpi T_2^2,\\ {\beta }_t=& \varpi [\left(\beta +1\right)-\kappa T_1{\left(\beta +1\right)}^2\\ & +4{\left|\alpha \right|}^2\kappa T_1]T_2^2+g{T}_1-1,\\ {\gamma }_t=& \varpi \left[\gamma -\kappa T_1\left(\beta +1\right)\gamma +2\alpha {\gamma }^{\ast }\kappa T_1\right]T_2,\\ {\mathrm{e}}^{{\delta }_t}=& {\mathrm{e}}^{\delta }{\varpi }^{1/2}{T}_3\exp \{\varpi \kappa T_1\left[1-\left(\beta +1\right)\kappa T_1\right]{\left|\gamma \right|}^2\\ & +\varpi {\kappa }^2{T}_1^2\left(\alpha {\gamma }^{\ast 2}+{\alpha }^{\ast }{\gamma }^2\right)\}\end{array}

with \varpi ={\left\{{\left[1-\left(\beta +1\right)\kappa T_1\right]}^2-4{\left|\alpha \right|}^2{\kappa }^2{T}_1^2\right\}}^{-1}, so we find that the evolved density operator {\rho }_t still takes the same form as the initial density operator {\rho }_0. Note that due to Eq. (14), we know that the unique of the parameter {\delta }_t guarantees tr{\rho }_t= 1. Obviously, this also requires us to check if the final state {\rho }_t satisfies the condition . Since T_1= \varrho \left[1-{\mathrm{e}}^{-2\left(\kappa -g\right)t}\right]increases monotonically with time t, and \underset{t\to +\mathrm{\infty }}{lim}\kappa T_1=1 when \kappa ⩾ g, and \underset{t\to +\mathrm{\infty }}{lim}\kappa T_1=\kappa /g when , we therefore have 0⩽\kappa T_1⩽ 1, and we also find that

(15)

implies that

(16)

Thus, we give

(17)\begin{array}{rl}& {\left[1-\left(\beta +1\right)\kappa T_1\right]}^2-4{\left|\alpha \right|}^2{\kappa }^2{T}_1^2\\ & =\left[1-\left(2\left|\alpha \right|+\beta +1\right)\kappa T_1\right]\\ & \times \left[1-\left(\beta +1\right)\kappa T_1+2\left|\alpha \right|\kappa T_1\right]\\ & >0,\end{array}

which leads to

(18)

thus {\rho }_t is guaranteed to be a valid state in physics.

4 Evolution of entropy

In order to calculate the von-Neumann entropy S\left({\rho }_t\right) of the initial state {\rho }_0 in a laser channel, we first rewrite S\left({\rho }_t\right) in Eq. (1) as its differential form

(19)S\left({\rho }_t\right)/k_B=-{{\displaystyle \frac{\mathrm{\partial }\text{tr}\left({\rho }_t^s\right)}{\mathrm{\partial }s}}|}_{s=1}.

Actually, the proof of Eq. (19) is as follows. Noting that

(20){\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}{y}^x= {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}{\mathrm{e}}^{x\ln y}={\mathrm{e}}^{x\ln y}\ln y=y^x\ln y,

so we have

(21){\rho }_t\ln {\rho }_t=\underset{s\to 1}{lim}{\rho }_t^s\ln {\rho }_t=\underset{s\to 1}{lim} {\displaystyle \frac{\mathrm{d}}{\mathrm{d}s}}{\rho }_t^s,

that is, the differential representation of the entropy S\left({\rho }_t\right) in Eq. (19) can be obtained by using the Lopital rule. For simplifiying the calculation of tr\left({\rho }_t^s\right), we notice that if performing a unitary transformation W for \rho, that is W\rho W^{-1}={\rho }^{\mathrm{\prime }}, then

(22)\begin{array}{rl}\text{tr}\left({\rho }^s\right)=& \text{tr}\left[{\left(W^{-1}{\rho }^{\mathrm{\prime }}W\right)}^s\right]\\ =& \text{tr}\left[W^{-1}{\rho }^{\mathrm{\prime }s}W\right]=\text{tr}\left({\rho }^{\mathrm{\prime }s}\right),\end{array}

so in Eq. (2) we make the unitary translational transformation for {\rho }^s, i.e., a=b+ϵand a^{†}=b^{†}+{ϵ}^{\ast }, we can write

(23)\begin{array}{rl}& :\exp \left(\alpha a^{†2}+\beta a^{†}a+{\alpha }^{\ast }{a}^2+\gamma a^{†}+{\gamma }^{\ast }a\right):\\ & =:\exp [\alpha \left(b^{†2}+2{ϵ}^{\ast }{b}^{†}+{ϵ}^{\ast 2}\right)+\beta \left(b^{†}+{ϵ}^{\ast }\right)\left(b+ϵ\right)\\ & +{\alpha }^{\ast }\left(b^2+2ϵb+{ϵ}^2\right)+\gamma \left(b^{†}+{ϵ}^{\ast }\right)+{\gamma }^{\ast }\left(b+ϵ\right)]:\\ & =:\exp [\alpha b^{†2}+\beta b^{†}b+{\alpha }^{\ast }{b}^2+\left(2\alpha {ϵ}^{\ast }+\beta ϵ+\gamma \right)b^{†}\\ & +(2{\alpha }^{\ast }ϵ+\beta {ϵ}^{\ast }+{\gamma }^{\ast })b+\alpha {ϵ}^{\ast 2}\\ & +{\alpha }^{\ast }{ϵ}^2+\beta {\left|ϵ\right|}^2+\gamma {ϵ}^{\ast }+{\gamma }^{\ast }ϵ]:.\end{array}

If we choose ϵ=\left(2\alpha {\gamma }^{\ast }-\beta \gamma \right)/({\beta }^2-4{\left|\alpha \right|}^2), then we find

(24)\begin{array}{rl}& 2\alpha {ϵ}^{\ast }+\beta ϵ+\gamma =0,\\ & 2{\alpha }^{\ast }ϵ+\beta {ϵ}^{\ast }+{\gamma }^{\ast }=0,\\ & \alpha {ϵ}^{\ast 2}+{\alpha }^{\ast }{ϵ}^2+\beta {\left|ϵ\right|}^2+\gamma {ϵ}^{\ast }+{\gamma }^{\ast }ϵ=\mathrm{\Delta }.\end{array}

So, the linear terms in Eq. (2) disappear, and Eq. (2) becomes

(25)\begin{array}{rl}& :\exp \left(\alpha a^{†2}+\beta a^{†}a+{\alpha }^{\ast }{a}^2+\gamma a^{†}+{\gamma }^{\ast }a\right):\\ & =:\exp \left(\alpha b^{†2}+\beta b^{†}b+{\alpha }^{\ast }{b}^2+\mathrm{\Delta }\right):\end{array}

again, we do not need to calculate the c-number factor \mathrm{\Delta } either. Note that this unitary translational transformation does not change the values of \alpha and \beta, so if we only care about the entropy, we do not need to calculate the coefficient {\gamma }_t either.

According to Eq. (19), our next target is to calculate the exponential function {\rho }_t^s and then its trace tr{\rho }_t^s for

(26)\begin{array}{rl}{\rho }_t=& {\mathrm{e}}^{{\delta }_t}:\exp ({\alpha }_t{a}^{†2}+{\beta }_t{a}^{†}a\\ & +{\alpha }_t^{\ast }{a}^2+{\gamma }_t{a}^{†}+{\gamma }_t^{\ast }a):\\ =& {\mathrm{e}}^{{\delta }_t}:\exp \left({\alpha }_t{b}^{†2}+{\beta }_t{b}^{†}b+{\alpha }_t^{\ast }{b}^2+{\mathrm{\Delta }}_t\right)\\ =& {\mathrm{e}}^{{\tilde{\delta }}_t}:\exp \left({\alpha }_t{b}^{†2}+{\beta }_t{b}^{†}b+{\alpha }_t^{\ast }{b}^2\right):,\end{array}

where {\tilde{\delta }}_t= {\delta }_t{\mathrm{\Delta }}_t. For this purpose, we need to write the normally ordered product {\mathrm{e}}^{{\tilde{\delta }}_t}:\exp \left({\alpha }_t{b}^{†2}+{\beta }_t{b}^{†}b+{\alpha }_t^{\ast }{b}^2\right): into a single exponential form \exp \left[A{b}^{†2}+\frac{B}{2}\left(b^{†}b+b{b}^{†}\right)+A^{\ast }{b}^2\right] by using the following two important resemble formulas.

Formula 1:

(27)\begin{array}{rl}& \exp \left[A{b}^{†2}+\frac{B}{2}\left(b^{†}b+b{b}^{†}\right)+A^{\ast }{b}^2\right]\\ & ={\omega }^{1/2}:\exp [A\omega {\theta }^{-1}\sinh \theta b^{†2}\\ & +b^{†}b\left(\omega -1\right)+A^{\ast }\omega {\theta }^{-1}\sinh \theta b^2]:,\end{array}

where

(28)\begin{array}{rl}\omega =& {\left(\cosh \theta -B{\theta }^{-1}\sinh \theta \right)}^{-1},\\ \theta =& \sqrt{B^2-4{\left|A\right|}^2}.\end{array}

Proof:

Let \mathrm{\Pi }=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right), and

(29)\begin{array}{rl}& \exp \left[A{a}^{†2}+\frac{B}{2}\left(a^{†}a+a{a}^{†}\right)+A^{\ast }{a}^2\right]\\ & \equiv \exp \left[\left(a^{†},a\right)\mathrm{\Gamma }(\genfrac{}{}{0ex}{}{a^{†}}{a})\right],\end{array}

we thus have

(30)\begin{array}{rl}\mathrm{\Gamma }=& \left(\begin{array}{cc}2A& B\\ B& 2A^{\ast }\end{array}\right),\\ \mathrm{\Gamma }\mathrm{\Pi }=& \left(\begin{array}{cc}2A& B\\ B& 2A^{\ast }\end{array}\right)\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)=\left(\begin{array}{cc}B& -2A\\ 2A^{\ast }& -B\end{array}\right),\end{array}

and

(31)\begin{array}{rl}& \left(\begin{array}{cc}Q& L\\ N& P\end{array}\right)\\ & \equiv \exp \left(\mathrm{\Gamma }\mathrm{\Pi }\right)\\ & =\left(\begin{array}{cc}\cosh \theta +B{\theta }^{-1}\sinh \theta & -2A{\theta }^{-1}\sinh \theta \\ 2A^{\ast }{\theta }^{-1}\sinh \theta & \cosh \theta -B{\theta }^{-1}\sinh \theta \end{array}\right),\end{array}

therefore using Eq. (31), we have

(32)\begin{array}{rl}& \exp \left[A{b}^{†2}+\frac{B}{2}\left(b^{†}b+b{b}^{†}\right)+A^{\ast }{b}^2\right]\\ & =\frac{1}{\sqrt{\cosh \theta -B{\theta }^{-1}\sinh \theta }}:\exp [\frac{A\sinh \theta }{\theta \cosh \theta -B\sinh \theta }{b}^{†2}\\ & +b^{†}b\left(\frac{1}{\cosh \theta -B{\theta }^{-1}\sinh \theta }-1\right)\\ & +\frac{A^{\ast }\sinh \theta }{\theta \cosh \theta -B\sinh \theta }{b}^2]:.\end{array}

Formula 2:

We can show

(33)\begin{array}{rl}& :\exp \left(\alpha b^{†2}+\beta b^{†}b+{\alpha }^{\ast }{b}^2\right):\\ & =\frac{1}{\sqrt{1+\beta }}\exp [\frac{\alpha \theta }{\left(1+\beta \right)\sinh \theta }{b}^{†2}\\ & +\frac{\theta \left(\cosh \theta -{\displaystyle \frac{1}{1+\beta }}\right)}{2\sinh \theta }\left(b^{†}b+b{b}^{†}\right)+\frac{{\alpha }^{\ast }\theta }{\left(1+\beta \right)\sinh \theta }{b}^2],\end{array}

where

(34)\cosh \theta ={\displaystyle \frac{{\left(1+\beta \right)}^2-4{\left|\alpha \right|}^2+1}{2\left(1+\beta \right)}}.

Proof:

We write

(35)\begin{array}{rl}& \frac{A\sinh \theta }{\theta \cosh \theta -B\sinh \theta }=\alpha ,\\ & \frac{1}{\cosh \theta -B{\theta }^{-1}\sinh \theta }-1=\beta \end{array}

in Eq. (32), then we obtain

(36)\begin{array}{rl}B{\theta }^{-1}\sinh \theta =& \cosh \theta -\frac{1}{1+\beta },\\ A{\theta }^{-1}\sinh \theta =& \frac{\alpha }{1+\beta }\end{array}

and

(37)\begin{array}{rl}& {\left({\theta }^{-1}\sinh \theta \right)}^2\left(B^2-4{\left|A\right|}^2\right)\\ & ={\left(\cosh \theta -\frac{1}{1+\beta }\right)}^2-\frac{4{\left|\alpha \right|}^2}{{\left(1+\beta \right)}^2},\\ & {\sinh }^2\theta ={\cosh }^2\theta -\frac{2\cosh \theta }{1+\beta }+\frac{1-4{\left|\alpha \right|}^2}{{\left(1+\beta \right)}^2},\\ & \cosh \theta =\frac{1+\beta }{2}+\frac{1-4{\left|\alpha \right|}^2}{2\left(1+\beta \right)},\end{array}

thus Eq. (27) becomes Eq. (33).

Using Eqs. (27) and (33), we can rewrite the s-power of {\mathrm{e}}^{\delta }:\exp \left(\alpha b^{†2}+\beta b^{†}b+{\alpha }^{\ast }{b}^2\right):\equiv \rho as

(38)\begin{array}{rl}{\rho }^s=& {\left[{\mathrm{e}}^{\delta }:\exp \left(\alpha b^{†2}+\beta b^{†}b+{\alpha }^{\ast }{b}^2\right):\right]}^s\\ =& {\mathrm{e}}^{s\delta }(\exp [A{b}^{†2}+\frac{B}{2}\left(b^{†}b+b{b}^{†}\right)\\ & {+A^{\ast }{b}^2-\frac{\ln \left(1+\beta \right)}{2}])}^s\\ =& {\mathrm{e}}^{s\delta -s\ln \left(1+\beta \right)/2}\exp \left[sA{b}^{†2}+\frac{sB}{2}\left(b^{†}b+b{b}^{†}\right)+s{A}^{\ast }{b}^2\right]\\ =& \frac{{\mathrm{e}}^{s\delta -s/2\ln \left(1+\beta \right)}}{\sqrt{\cosh s\theta -sB{\displaystyle \frac{\sinh s\theta }{s\theta }}}}:\exp [\frac{sA\sinh s\theta }{s\theta \cosh s\theta -B\sinh s\theta }{b}^{†2}\\ & +b^{†}b\left(\frac{1}{\cosh s\theta -sB{\displaystyle \frac{\sinh s\theta }{s\theta }}}-1\right)\\ & +\frac{s{A}^{\ast }\sinh s\theta }{s\theta \cosh s\theta -sB\sinh s\theta }{b}^2]:.\end{array}

Substituting Eq. (36) into Eq. (38) leads to

(39)\begin{array}{rl}{\rho }^s=& {\mathrm{e}}^{s\delta -s/2\ln \left(1+\beta \right)}{\mathrm{\Lambda }}^{1/2}\sqrt{\left(1+\beta \right)\sinh \theta }\\ & :\exp [b^{†}b\left(\left(1+\beta \right)\mathrm{\Lambda }\sinh \theta -1\right)\\ & +\alpha \mathrm{\Lambda }\sinh s\theta b^{†2}+{\alpha }^{\ast }\mathrm{\Lambda }\sinh s\theta b^2]:,\end{array}

where \mathrm{\Lambda }={\left[\sinh s\theta -\left(1+\beta \right)\sinh \left(s-1\right)\theta \right]}^{-1}. So, we obtain

(40)\begin{array}{rl}{\rho }^s=& {\mathrm{e}}^{s\delta -s/2\ln \left(1+\beta \right)}\exp [sA{b}^{†2}+s{A}^{\ast }{b}^2+\frac{sB}{2}\left(b^{†}b+b{b}^{†}\right)]\\ =& {\mathrm{e}}^{s\delta -s/2\ln \left(1+\beta \right)}{\mathrm{\Lambda }}^{1/2}\sqrt{\left(1+\beta \right)\sinh \theta }\\ & \times :\exp [\alpha \mathrm{\Lambda }\sinh s\theta b^{†2}+{\alpha }^{\ast }\mathrm{\Lambda }\sinh s\theta b^2\\ & +b^{†}b\left(\left(1+\beta \right)\mathrm{\Lambda }\sinh \theta -1\right)]:,\end{array}

and

(41)\begin{array}{rl}\text{tr}{\rho }^s=& \int \frac{{\mathrm{d}}^{2}z}{\pi }⟨z|{\rho }^s|z⟩\\ =& {\left\{{\left[\left(1+\beta \right)\mathrm{\Lambda }\sinh \theta -1\right]}^2-4{\left|\alpha \right|}^2{\mathrm{\Lambda }}^2{\sinh }^{2}s\theta \right\}}^{-1/2}\\ & \times {\mathrm{e}}^{s\delta -s/2\ln \left(1+\beta \right)}{\mathrm{\Lambda }}^{1/2}\sqrt{\left(1+\beta \right)\sinh \theta }.\end{array}

It then follows that

(42)\begin{array}{rl}& S\left(\rho \right)/k_B\\ & =-{\frac{\mathrm{\partial }(\text{tr}{\rho }^s)}{\mathrm{\partial }s}|}_{s=1}\\ & =-\frac{\mathrm{\partial }}{\mathrm{\partial }s}\{{\mathrm{e}}^{s\delta -s/2\ln \left(1+\beta \right)}{\mathrm{\Lambda }}^{1/2}\sqrt{\left(1+\beta \right)\sinh \theta }\\ & \times {{\left\{{\left[\left(1+\beta \right)\mathrm{\Lambda }\sinh \theta -1\right]}^2-4{\left|\alpha \right|}^2{\mathrm{\Lambda }}^2{\sinh }^{2}s\theta \right\}}^{-1/2}\}|}_{s=1}\\ & =\frac{1}{2}\ln \frac{1+\beta }{{\beta }^2-4{\left|\alpha \right|}^2}+\frac{\theta }{\sinh \theta }\frac{{\left(2+\beta \right)}^2-4{\left|\alpha \right|}^2}{4\left(1+\beta \right)},\end{array}

where \cosh \theta =(1+{\mathrm{e}}^{2\beta }-4{\left|\alpha \right|}^2){\mathrm{e}}^{-\beta }/2=\cosh \beta -2 {\left|\alpha \right|}^2{\mathrm{e}}^{-\beta } and we have used the fact that

(43){\displaystyle \frac{{\mathrm{e}}^{\delta }}{{\left({\beta }^2-4{\left|\alpha \right|}^2\right)}^{1/2}}}=\text{tr}\rho =1,

thus the above expression (42) gives the entropy of a quadratic state with the density operator \rho ={\mathrm{e}}^{\delta }:\exp \left(\alpha b^{†2}+\beta b^{†}b+{\alpha }^{\ast }{b}^2\right):.

Making the following substitutions

(44)\begin{array}{rl}\alpha \to {\alpha }_t=& \alpha \varpi T_2^2,\\ \beta \to {\beta }_t=& \varpi [\left(\beta +1\right)-\kappa T_1{\left(\beta +1\right)}^2\\ & +4{\left|\alpha \right|}^2\kappa T_1]T_2^2+g{T}_1-1,\end{array}

we will find the evovled von-Neumann entropy S\left({\rho }_t\right) of general quadratic state {\rho }_0 in a laser channel, a very long expression. It is remarkable that the entropy S\left({\rho }_t\right) is completely independent of the coefficient \gamma of the initial density operator {\rho }_0 in Eq. (2). This is why the entropy S\left({\rho }_t\right) is completely independent of the amplitude z when {\rho }_0=|z⟩⟨z|=:{\mathrm{e}}^{-{\left|z\right|}^2+z{a}^{†}+z^{\ast }a-a^{†}a}:.

To observe the influence of the laser cavity gain and loss rate g, \kappa on the von-Neumann entropy S\left({\rho }_t\right) more clearly, we plotted the variation law of S\left({\rho }_t\right)/k_B as a function of time t for different values of \kappa, g. It is pointed out that the parameters \alpha and \beta only take some special values (e.g., \alpha =0.2 and \beta = -1 here) because of the limited conditions and given by the normalization in Section 2. Obviously, the entropy S\left({\rho }_t\right) always monotonically increases with time t, and this trend is independent of other parameters. This is because the high-speed noise generated by laser light sources is a good random entropy source that can lead to a rapid increase in quantum entropy. Also, the choice of the parameters \alpha, \beta can affect the entropy at initial time t=0, but cannot affect the variation law of the entropy with time t.

On the other hand, when the gain rate g is less than the loss rate \kappa, the entropy S\left({\rho }_t\right) quickly approaches constant values with time t. However, when g is larger than \kappa, the entropy S\left({\rho }_t\right) increases linearly with time t. For a fixed time t, the entropy S\left({\rho }_t\right) always decreases as \kappa is increased or g is decreased. These results show that a laser beam can generate and output only for the sufficiently high gain rate g.

5 Some special cases

In particularly, for the case of \alpha =0, we therefore have

(45){\alpha }_t=0, {\beta }_t={\displaystyle \frac{\left(\beta +1\right)T_2^2}{1-\left(\beta +1\right)\kappa T_1}}+g T_1-1,

which leads to

(46)\cosh {\theta }_t= {\displaystyle \frac{1+{\beta }_t}{2}}+{\displaystyle \frac{1}{2\left(1+{\beta }_t\right)}},

that is, {\theta }_t=\ln \left(1+{\beta }_t\right), and thus making the substitutions {\alpha }_t\to \alpha, {\beta }_t\to \beta and \cosh {\theta }_t\to \cosh \theta in Eq. (42), we can give

(47)\begin{array}{rl}S\left({\rho }_t\right)/k_B=& \frac{1+{\beta }_t}{{\beta }_t}\ln \left(1+{\beta }_t\right)-\ln \left|{\beta }_t\right|\\ =& \frac{\left(\beta +1\right)\left(1-\kappa T_1\right)-\beta g{T}_1}{\beta \left(1-g{T}_1\right)}\\ & \times \ln \left[\frac{\left(\beta +1\right)\left(1-\kappa T_1\right)-\beta g{T}_1}{1-\left(\beta +1\right)\kappa T_1}\right]\\ & -\ln \left|\frac{\beta \left(1-g{T}_1\right)}{1-\left(\beta +1\right)\kappa T_1}\right|,\end{array}

which corresponds to the entropy evolution of the mixture of thermal states, i.e., {\rho }_0={\mathrm{e}}^{\delta }{\mathrm{e}}^{\gamma a^{†}}:{\mathrm{e}}^{\beta a^{†}a}:{\mathrm{e}}^{{\gamma }^{\ast }a} for laser noise, and is only related to the parameter \beta. Further, taking \beta = -1, we thus give

(48){\beta }_t=g{T}_1-1

and

(49)\begin{array}{rl}S\left({\rho }_t\right)/k_B=& \frac{1+{\beta }_t}{{\beta }_t}\ln \left(1+{\beta }_t\right)-\ln \left|{\beta }_t\right|\\ =& -\frac{g{T}_1\ln g{T}_1}{1-g{T}_1}-\ln \left(1-g{T}_1\right),\end{array}

which is exactly the entropy evolution of the coherent state {\rho }_0= {\mathrm{e}}^{\delta }{\mathrm{e}}^{\gamma a^{†}}|0⟩⟨0| {\mathrm{e}}^{{\gamma }^{\ast }a} in the laser channel, and not related to the amplitude \gamma of the state {\rho }_0.

For a squeezed coherent state

(50)\begin{array}{rl}& |\psi \left(\lambda ,z\right)⟩\\ & =s\left(\lambda \right)|z⟩\\ & ={\mathcal{A}}^{1/2}:\exp \left[-\mathcal{B}{a}^{†2}+\left(\mathcal{A}-1\right)a^{†}a+\mathcal{B}{a}^2\right]:|z⟩\\ & ={\mathcal{A}}^{1/2}\exp \left[-\mathcal{B}{a}^{†2}+\left(\mathcal{A}-1\right)a^{†}z+\mathcal{B}{z}^2\right]|z⟩\\ & ={\mathcal{A}}^{1/2}{\mathrm{e}}^{-{\left|z\right|}^2/2}\exp \left(-\mathcal{B}{a}^{†2}+\mathcal{A}{a}^{†}z+\mathcal{B}{z}^2\right)|0⟩,\end{array}

where s \left(\lambda \right) is the single-mode squeezing operator, A=1/\cosh \lambda and B=\left(\tanh \lambda \right)/2, then we have

(51)\begin{array}{rl}{\rho }_0=& |\psi \left(\lambda ,z\right)⟩⟨\psi \left(\lambda ,z\right)|\\ =& \mathcal{A}{\mathrm{e}}^{-{\left|z\right|}^2}:\exp [-\mathcal{B}\left(a^{†2}+a^2\right)-a^{†}a\\ & +\mathcal{A}\left(a^{†}z+a{z}^{\ast }\right)+\mathcal{B}\left(z^2+z^{\ast 2}\right)]:.\end{array}

Noting that

(52)\alpha =-\mathcal{B}, \beta =-1 ,

we thus obtain

(53)\begin{array}{rl}& {\alpha }_t=\frac{-\mathcal{B}{T}_2^2}{1-4{\kappa }^2{T}_1^2{\mathcal{B}}^2},\\ & {\beta }_t=\frac{4\kappa T_1{T}_2^2{\mathcal{B}}^2}{1-4{\kappa }^2{T}_1^2{\mathcal{B}}^2}+g{T}_1-1,\\ & \cosh {\theta }_t=\frac{1+{\beta }_t}{2}+\frac{1-4{\left|{\alpha }_t\right|}^2}{2\left(1+{\beta }_t\right)}.\end{array}

So, we can obtain the entropy evolution of the squeezed coherent state |\psi \left(\lambda ,z\right)⟩ in a laser channel in terms of Eqs. (42) and (53). Obviously, the entropy evolution of the state |\psi \left(\lambda ,z\right)⟩ in a laser channel only depends on the gain and loss rate g, \kappa and the squeezing parameter \lambda, and is also independent of the amplitude z of the coherent state |z⟩.

6 Conclusions

In summary, using the IWOP method, we obtained the normalization of the general quadratic state {\rho }_0 and found that the conditions and guarantee that the state {\rho }_0 is valid in physics. Taking the state {\rho }_0 as an initial state, we obtained the evolved density operator {\rho }_t in a laser channel via the Kraus operator of master equation for laser noise, and found that the operator {\rho }_t is exactly the same form as the initial general quadratic normally ordered density operator {\rho }_0. On the basis of this, via introducing the differential representation of the von-Neumann entropy S\left(\rho \right) and two important resemble formulas, we obtained the analytical entropy evolution S\left({\rho }_t\right) of the state {\rho }_0 in a laser channel, which is decided by the gain and loss of the laser channel and the coefficients \alpha, \beta of the state {\rho }_0, but has noting to do with the the codfficient \gamma of the state {\rho }_0.

On the other hand, the numerical results indicate that the entropy S \left({\rho }_t\right) always increases monotonically as time t is increased, and the initial entropy at t=0 is determined by the values of the parameters \alpha and \beta. With the increase of t, the entropy S\left({\rho }_t\right) gradually approaches certain constant values for the case that g is less than \kappa, but increases linearly when g is larger than \kappa. This happens to be the reason for the generation of laser light sources.

Besides, as several special cases, the entropies of the coherent state, the mixture of thermal states and squeezed coherent state are also obtained, the results show that they are related to the parameters \kappa ,g of laser channel, and all independent of the paramter \gamma, a meaningful result. Given that the generality of the initial quadratic state and quantum laser process, this work will provide a new approach for calculating the evolved quantum von-Neumann entropy in open systems, and the main results can provide theoretical reference for measuring quantum entropy experimentally.

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