Evolution of decoherence in a two-mode entangled system under amplitude decay

Abstract

Based on our recently published work [Front. Phys. 21, 093201 (2026)], we extend the analysis to the density-operator evolution of a two-mode entangled quantum system subject to amplitude decay and show that the output state retains the same functional structure as the initial density operator throughout the decay. We further examine the evolutions of several quantum statistical distributions and the von Neumann entropy used to quantify entanglement during the process. This work offers a systematic approach to studying two-mode entangled open systems and yields results on the von Neumann entropy that may help facilitate entanglement measurement.

Graphical abstract

Keywords

two-mode entangled quantum system / operator ordering method / amplitude decay / quantum statistical distribution / von Neumann entropy

1 Introduction

Quantum decoherence, first discussed in pioneering work by Zeh and later developed extensively [1, 2], describes the loss of quantum coherence that drives a quantum system toward effectively classical behavior, i.e., a quantum-to-classical transition. This occurs because an open quantum system inevitably becomes entangled with its surrounding environment, leading to a gradual suppression of phase coherence [3–5]. Numerous experiments have confirmed decoherence effects. For example, in the double-slit experiment, interference fringes can disappear when which-path information becomes available through environmental coupling, reflecting the loss of phase coherence.

Decoherence is a rapid degradation of quantum coherence and a central obstacle to the development of quantum technology. In particular, it plays a critical role in quantum computing, where coherence loss directly limits computational fidelity, efficiency, and stability. Since the concept was introduced, decoherence has attracted broad attention across quantum physics. Reference [6] shows that decoherence can lead to environment-induced superselection (einselection), a process associated with the selective suppression of information in certain bases. The decoherence dynamics of continuous-variable two-qubit systems have also been studied under the simultaneous action of two different noise channels (e.g., amplitude and phase damping), which better reflect realistic experimental environments [7]. In addition, Ref. [8] reviews decoherence mechanisms, the evolution of geometric quantum correlations and quantum coherence in various noisy channels, and the adverse impacts of decoherence on quantum-information tasks. Overall, decoherence arising from coupling to a noisy environment is unavoidable, and mitigating its effects remains essential for executing quantum information protocols reliably.

For any open quantum system, quantum noise inevitably arises in its fundamental dynamical processes. Notably, amplitude decay is one of the most common causes of nonclassicality deterioration in open systems. Here, we consider a two-mode bosonic system coupled to a bosonic amplitude-decay environment (a Markovian process). In the interaction picture, the time-dependent density operator {\rho }_t of the system satisfies the following master equation [9–11]:

(1){\displaystyle \frac{\text{d}{\rho }_t}{\text{d}t}}=\kappa {\displaystyle \sum _{i=a,b}\left(2i{\rho }_t{i}^{†}-i^{†}i{\rho }_t-{\rho }_t{i}^{†}i\right),}

where the decay rate \kappa for the two modes is assumed to be the same, and a^{†} and b^{†} are the boson creation operators of the two-mode system, respectively. Unlike phase damping and thermal processes, amplitude decay represents energy transfer from the system to a zero-temperature environment. To solve Eq. (1), we review the thermal entangled state |\zeta ⟩ and its relevant properties, where |\zeta ⟩= e{}^{-{\left|\zeta \right|}^2/2+\zeta a^{†}-{\zeta }^{\ast }{\tilde{a}}^{†}+a^{†}{\tilde{a}}^{†}}|00⟩ describes the entanglement between the system and its surrounding environment, and {\tilde{a}}^{†} represents the creation operator of the environment. In the entangled state |\zeta =0⟩, there exist operator correspondence relations between the environment mode (\tilde{a},{\tilde{a}}^{†}) and the system mode (a,a^{†}), i.e., a\iff {\tilde{a}}^{†}, a^{†}\iff \tilde{a}and (a^{†}a{)}^n\iff ({\tilde{a}}^{†}\tilde{a}{)}^n [12, 13]. Using these operator relations, Eq. (1) can be successfully solved, and its explicit solution can be written in the operator-sum form [12, 14, 15], i.e.,

(2){\rho }_t={\displaystyle \sum _{l,m=0}^{\mathrm{\infty }}{\mathit{M}}_{l,m}{\rho }_0{\mathit{M}}_{l,m}^{†},}

where {\mathit{M}}_{l,m} is the Kraus operator, i.e.,

(3){\mathit{M}}_{l,m}=\sqrt{{\displaystyle \frac{{\mathcal{T}}^{l+m}}{l!m!}}}{\text{e}}^{-\kappa t\left(a^{†}a+b^{†}b\right)}{a}^l{b}^m,\mathcal{T}=1-{\text{e}}^{-2\kappa t}.

Using the operator ordering method and the operator identities

(4){\text{e}}^{\lambda i^{†}i}i{\text{e}}^{-\lambda i^{†}i}={\text{e}}^{-\lambda }i,{\text{e}}^{\lambda i^{†}i}=:\exp \left[\left({\text{e}}^{\lambda }-1\right)i^{†}i\right]:,

where the symbol :: means normal ordering, showing that all creation operators are on the left side of the annihilation operators [16–18], \lambda is an arbitrary parameter, we can check {\sum }_{l,m=0}^{\mathrm{\infty }}{\mathit{M}}_{l,m}^{†}{\mathit{M}}_{l,m}=1, such that tr{\rho }_t=1.

Recently, solving various master equations that describe quantum decoherence processes and investigating the time evolutions of quantum states in these processes have received great attention. A comprehensive review is given in Ref. [12], which used the entangled-state approach to derive the evolutions of quantum states in phase diffusion, damping, amplification, and phase-sensitive processes, and revealed the decoherence mechanism in an intuitive way. Other studies typically involve at most two types of simple quantum states or decoherence processes [7, 19, 20]. In view of the current progress in theoretical research, investigations of the evolutions of two-mode entangled states governed by known Hamiltonians have not, to the best of our knowledge, been reported. So, different from the previous work [6–8, 12, 19–22], here we make full use of the operator ordering method to find the entangled density operator describing the two-mode entangled quantum system (e.g., a non-degenerate optical parametric amplifier system) with the Hamiltonian

(5)H_0=\varrho a^{†}{b}^{†}+\omega \left(a^{†}a+b^{†}b\right)+{\varrho }^{\ast }ab,

and investigate its decoherence evolution in the amplitude decay process, where \varrho is the entanglement coefficient of the system and \omega refers to the frequency of input signal light [23]. The reason for choosing this topic is twofold. One is that the entangled density operator corresponding to the Hamiltonian H_0, especially its normal ordering product, and the amplitude decay model provide a more general and realistic framework for investigating the evolutions of two-mode open systems. The other is that the operator ordering method and the integration within ordered products offer a powerful tool for investigating the evolutions of two-mode entangled systems and the von Neumann entropy describing quantum entanglement, and this method can also be applied to other two-mode or multi-mode open quantum systems.

The rest of the work is arranged as follows. In Section 2, we obtain the two-mode entangled state {\rho }_0belonging to Hamiltonian H_0. In Section 3, we further obtain the evolution of the density operator {\rho }_0 in the amplitude decay process. Sections 4, 5 and 6 are, respectively, devoted to discussing the evolutions of the average photon number, photon number distribution and Wigner distribution function in this process. The evolved von Neumann entropy of the state {\rho }_0 in this process is investigated in Section 7. Finally, we summarize some main results of this work in Section 8.

2 Entangled density operator

To investigate the evolution of the entangled system with the Hamiltonian H_0 under amplitude decay, we first obtain the normalized density operator of the entangled system. Using the theory of statistical ensembles, the normalized density operator {\rho }_0 of the entangled system is represented as

(6){\rho }_0={\text{e}}^{-\beta H_0}/\left(\mathtt{t}\mathtt{r}{\text{e}}^{-\beta H_0}\right),

where \beta =1/\left(kT\right), k is the Boltzmann constant and T is the temperature of the thermal field. Given that the operators a^{†}{b}^{†},a^{†}a+b^{†}b+1, and ab yield the standard su(1,1) Lie algebra, we obtain the following decomposition [24]

(7)\begin{array}{rl}& \exp \left[m{a}^{†}{b}^{†}+g\left(a^{†}a+b^{†}b\right)+nab\right]\\ & ={\text{e}}^{-g}\exp \left(\frac{m}{E\coth E-g}{a}^{†}{b}^{†}\right)\\ & \times \exp \left[\left(a^{†}a+b^{†}b+1\right)\ln \frac{E}{E\cosh E-g\sinh E}\right]\\ & \times \exp \left(\frac{n}{E\coth E-g}{a}^{†}{b}^{†}\right)\end{array}

with E^2=g^2-mn. Using Eqs. (5)−(7), we obtain the normal ordering product of e{}^{-\beta H_0} as

(8){\text{e}}^{-\beta H_0}=\varkappa :\exp \left[\alpha a^{†}{b}^{†}+\sigma \left(a^{†}a+b^{†}b\right)+{\alpha }^{\ast }ab\right]:,

where the parameters are, respectively,

(9)\begin{array}{rl}& \varkappa ={\text{e}}^{-{\omega }^{\mathrm{\prime }}}\left(\sigma +1\right),\alpha =\frac{-{\varrho }^{\mathrm{\prime }}}{\mathrm{\Lambda }\coth \mathrm{\Lambda }-{\omega }^{\mathrm{\prime }}},\\ & \sigma =\frac{\mathrm{\Lambda }}{\mathrm{\Lambda }\cosh \mathrm{\Lambda }-{\omega }^{\mathrm{\prime }}\sinh \mathrm{\Lambda }}-1,\\ & \mathrm{\Lambda }={\left({\omega }^{\mathrm{\prime }2}-{\left|{\varrho }^{\mathrm{\prime }}\right|}^2\right)}^{1/2},{\omega }^{\mathrm{\prime }}=-\beta \omega ,{\varrho }^{\mathrm{\prime }}=-\beta \varrho .\end{array}

On the other hand, using the completeness relation of two-mode coherent states |z_1{z}_2⟩, i.e., {\pi }^{-2}\intd{}^2{z}_1d{}^2{z}_2|z_1{z}_2⟩⟨z_1{z}_2|=1 and the integral formula

(10)

we obtain

(11)\mathtt{t}\mathtt{r}{\text{e}}^{-\beta H_0}={\displaystyle \frac{\varkappa }{{\sigma }^2-{\left|\alpha \right|}^2}}.

Therefore, the normalized entangled density operator {\rho }_0 is given by

(12){\rho }_0={\text{e}}^{\mathrm{\Delta }}:\exp \left[\alpha a^{†}{b}^{†}+\sigma \left(a^{†}a+b^{†}b\right)+{\alpha }^{\ast }ab\right]:,

where e{}^{\mathrm{\Delta }}={\sigma }^2-{\left|\alpha \right|}^2. Eqs. (10) and (12) imply that, in order to guarantee that the state {\rho }_0 is physically valid, the parameters \alpha ,\sigma must satisfy the conditions: and {\sigma }^2-{\left|\alpha \right|}^2>0 or equivalently, and .

Obviously, the density operator {\rho }_0 is in general since it can involve squeezing, vacuum, mixedness, etc. For examples, for \alpha =0, {\rho }_0 becomes the thermal state with the density operator {\sigma }^2e {}^{\left(a^{†}a+b^{†}b\right)\ln \left(\sigma +1\right)}\equiv {\rho }_{\text{th}} in terms of Eq. (4), and for \alpha =0 and \sigma =-1, {\rho }_0 reduces to vacuum (|00⟩⟨00|) via the normal ordering product of two-mode vacuum projection operator, i.e.,

(13)|00⟩⟨00|=:\exp \left[-\left(a^{†}a+b^{†}b\right)\right]:.

However, for \alpha \ne 0, {\rho }_0 refers to the two-mode squeezed vacuum or thermal state.

3 Evolution of {\rho }_0 under amplitude decay

In this section, we investigate how the two-mode entangled state {\rho }_0 evolves when it undergoes amplitude decay. By observing the bosonic operator-ordering structure in the Kraus operator-sum representation of {\rho }_t in Eq. (2), we find that the antinormally ordered form of {\rho }_0 is particularly convenient for discussing the evolution of {\rho }_0 under amplitude decay. Therefore, inserting Eq. (12) into the formula that rewrites any two-mode boson operator in antinormal ordering [25, 26], namely,

(14)\begin{array}{rl}\rho =& \int \frac{{\text{d}}^2{z}_1{\text{d}}^2{z}_2}{{\pi }^2}⟨-z_1,-z_2|\rho |z_1,z_2⟩\\ & ⋮{\text{e}}^{{\left|z_1\right|}^2+{\left|z_2\right|}^2+z_1^{\ast }a-z_1{a}^{†}+z_2^{\ast }b-z_2{b}^{†}+a^{†}a+b^{†}b}⋮,\end{array}

where ⋮ ⋮ denotes antinormal ordering, whose ordering rules are opposite to those of normal ordering, and using the integration formula in Eq. (10) twice, we obtain the antinormal ordering of {\rho }_0 as

(15)\begin{array}{rl}\rho =& {\text{e}}^{\mathrm{\Delta }}\int \frac{{\text{d}}^2{z}_1{\text{d}}^2{z}_2}{{\pi }^2}{\text{e}}^{\alpha z_1^{\ast }{z}_2^{\ast }+{\alpha }^{\ast }{z}_1{z}_2}\\ & ⋮{\text{e}}^{-\left(1+\sigma \right)\left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)+z_1^{\ast }a-z_1{a}^{†}+z_2^{\ast }b-z_2{b}^{†}+a^{†}a+b^{†}b}⋮\\ =& {\text{e}}^{\mathrm{\Delta }}\mathcal{G}⋮\exp \{\mathcal{G}\alpha a^{†}{b}^{†}+\left[1-\mathcal{G}\left(1+\sigma \right)\right]\\ & \times \left(a^{†}a+b^{†}b\right)+\mathcal{G}{\alpha }^{\ast }ab\}⋮,\end{array}

where we have used the inner product ⟨-z_1,-z_2|z_1,z_2⟩= e{}^{-2\left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)}, \mathcal{G}={\left[{\left(1+\sigma \right)}^2-{\left|\alpha \right|}^2\right]}^{-1}. Further, inserting Eq. (15) into Eq. (2) and using the completeness relation of two-mode coherent states |z_1,z_2⟩, we have

(16)\begin{array}{rl}{\rho }_t=& {\text{e}}^{\mathrm{\Delta }}\mathcal{G}\sum _{l,m=0}^{\mathrm{\infty }}\frac{{\mathcal{T}}^{l+m}}{l!m!}{\text{e}}^{-\kappa t\left(a^{†}a+b^{†}b\right)}⋮a^l{b}^m\exp \{\mathcal{G}\alpha a^{†}{b}^{†}\\ & +\left[1-\mathcal{G}\left(1+\sigma \right)\right]\left(a^{†}a+b^{†}b\right)+\mathcal{G}{\alpha }^{\ast }ab\}\\ & \cdot b^{†m}{a}^{†l}⋮{\text{e}}^{-\kappa t\left(a^{†}a+b^{†}b\right)}\\ =& {\text{e}}^{\mathrm{\Delta }}\mathcal{G}\int \frac{{\text{d}}^2{z}_1{\text{d}}^2{z}_2}{{\pi }^2}{\text{e}}^{\mathcal{T}\left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)}{\text{e}}^{-\kappa t\left(a^{†}a+b^{†}b\right)}|z_1,z_2⟩\\ & \cdot ⟨z_1,z_2|{\text{e}}^{-\kappa t\left(a^{†}a+b^{†}b\right)}\exp \{\mathcal{G}\alpha z_1^{\ast }{z}_2^{\ast }\\ & +\left[1-\mathcal{G}\left(\sigma +1\right)\right]\left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)+\mathcal{G}{\alpha }^{\ast }{z}_1{z}_2\}.\end{array}

Using the identity e{}^{-\lambda a^{†}a}|\alpha ⟩=\exp [-\frac{1}{2}(1-{\text{e}}^{-2\lambda }){\left|\alpha \right|}^2]|\alpha {\text{e}}^{-\lambda }⟩, we can rewrite {\rho }_t as

(17)\begin{array}{rl}{\rho }_t=& {\text{e}}^{\mathrm{\Delta }}\mathcal{G}\int \frac{{\text{d}}^2{z}_1{\text{d}}^2{z}_2}{{\pi }^2}|z_1{\text{e}}^{-\kappa t},z_2{\text{e}}^{-\kappa t}⟩\\ & \cdot ⟨z_1{\text{e}}^{-\kappa t},z_2{\text{e}}^{-\kappa t}|\exp \{\mathcal{G}\alpha z_1^{\ast }{z}_2^{\ast }+\\ & \left[1-\mathcal{G}\left(1+\sigma \right)\right]\left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)+\mathcal{G}{\alpha }^{\ast }{z}_1{z}_2\}.\end{array}

Finally, using the integral formula in Eq. (10) twice, we obtain

(18){\rho }_t={\text{e}}^{{\mathrm{\Delta }}_t}:\exp \left[{\alpha }_t{a}^{†}{b}^{†}+{\sigma }_t\left(a^{†}a+b^{†}b\right)+{\alpha }_t^{\ast }ab\right]:,

where \mathcal{H}={\left\{{\left[\left(1+\sigma \right)\mathcal{G}-\mathcal{T}\right]}^2-{\mathcal{G}}^2{\left|\alpha \right|}^2\right\}}^{-1}, and the parameters {\alpha }_t, {\sigma }_t and {\mathrm{\Delta }}_t are, respectively,

(19)\begin{array}{rl}& {\alpha }_t=\mathcal{H}\mathcal{G}\alpha {\text{e}}^{-2\kappa t},\\ & {\sigma }_t=\mathcal{H}\left[\mathcal{G}\left(1+\sigma \right)-\mathcal{T}\right]{\text{e}}^{-2\kappa t}-1,\\ & {\text{e}}^{{\mathrm{\Delta }}_t}={\text{e}}^{\mathrm{\Delta }}\mathcal{G}\mathcal{H}\equiv {\sigma }_t^2-{\left|{\alpha }_t\right|}^2.\end{array}

Obviously, Eq. (18) shows that the time-dependent density operator {\rho }_t always has the same form as the initial density operator {\rho }_0. However, under the influence of amplitude-decay noise, {\rho }_t is always a mixed quantum state regardless of whether the initial state {\rho }_0 is pure or mixed, which results from the loss of coherence in the initial state {\rho }_0.

Further, we need to verify whether the output state {\rho }_t satisfies the condition in order to show that the state {\rho }_t is always physically valid in the amplitude decay process. Since \mathcal{T}=1- e{}^{-2\kappa t}increases monotonically with time t, and \underset{t\to 0}{lim}\mathcal{T}\to 0, \underset{t\to +\mathrm{\infty }}{lim}\mathcal{T}\to 1, we have . Further, using , we obtain the inequality or equivalently, \left(\sigma -\left|\alpha \right|+1\right)\mathcal{G}-\mathcal{T}>0. Hence, we have

(20)\begin{array}{rl}{\mathcal{H}}^{-1}=& {\left[\left(\sigma +1\right)\mathcal{G}-\mathcal{T}\right]}^2-{\mathcal{G}}^2{\left|\alpha \right|}^2\\ =& \left[\left(\sigma +\left|\alpha \right|+1\right)\mathcal{G}-\mathcal{T}\right]\\ & \times \left[\left(\sigma -\left|\alpha \right|+1\right)\mathcal{G}-\mathcal{T}\right]\\ >& 0,\end{array}

which leads to , that is, {\rho }_t is a physically valid state.

We now consider several special cases. For the case of \alpha =0, e{}^{\mathrm{\Delta }}={\sigma }^2, \mathcal{G}={\left(\sigma +1\right)}^{-2}, \mathcal{H}={\left[{\left(\sigma +1\right)}^{-1}-\mathcal{T}\right]}^{-2}, we therefore have

(21)\begin{array}{rl}& {\alpha }_t=0,\\ & {\sigma }_t=\frac{{\text{e}}^{-2\kappa t}}{{\left(\sigma +1\right)}^{-1}-\mathcal{T}}-1,\\ & {\text{e}}^{{\mathrm{\Delta }}_t}=\frac{{\sigma }^2}{{\left[1-\left(\sigma +1\right)\mathcal{T}\right]}^2},\end{array}

thus {\rho }_t reduces to the evolution of the two-mode thermal state {\rho }_{\text{th}} under amplitude decay, where we have used the identity in Eq. (4). For \sigma =-1, owing to e{}^{\mathrm{\Delta }}=1-{\left|\alpha \right|}^2, \mathcal{G}=-{\left|\alpha \right|}^{-2}, \mathcal{H}={\left({\mathcal{T}}^2-{\left|\alpha \right|}^{-2}\right)}^{-1}, thus

(22)\begin{array}{rl}& {\alpha }_t=\frac{\alpha {\text{e}}^{-2\kappa t}}{1-{\mathcal{T}}^2{\left|\alpha \right|}^2},\\ & {\sigma }_t=\frac{\mathcal{T}{\left|\alpha \right|}^2{\text{e}}^{-2\kappa t}}{1-{\mathcal{T}}^2{\left|\alpha \right|}^2}-1,\\ & {\text{e}}^{{\mathrm{\Delta }}_t}=\frac{1-{\left|\alpha \right|}^2}{1-{\mathcal{T}}^2{\left|\alpha \right|}^2},\end{array}

so {\rho }_t becomes the evolution of the two-mode squeezed vacuum with the density operator {\rho }_{\text{squ}}=\left(1-{\left|\alpha \right|}^2\right)e{}^{\alpha a^{†}{b}^{†}}|00⟩⟨00|e{}^{{\alpha }^{\ast }ab} in the amplitude decay process. In the limiting case, \kappa t=0, \mathcal{T}=0, \mathcal{H}=G^{-1}, \mathcal{G}\mathcal{H}=1, {\alpha }_t\to \alpha, {\sigma }_t\to \sigma, e{}^{{\mathrm{\Delta }}_t}\to e{}^{\mathrm{\Delta }}, thus Eq. (18) naturally becomes the initial density operator {\rho }_0. In the long-time limit, \kappa t\to \mathrm{\infty }, \mathcal{T}\to 1, \mathcal{G}\mathcal{H}\to{\left({\sigma }^2-{\left|\alpha \right|}^2\right)}^{-1}, {\alpha }_t\to 0, {\sigma }_t\to -1, e{}^{{\mathrm{\Delta }}_t}\to 1, so {\rho }_{t\to \mathrm{\infty }}\to |00⟩⟨00|.

4 Average photon number

The average photon number is one of the fundamental parameters that characterizes the amount of light radiation, and its measurement is of great significance in the continuous-variable quantum key distribution. For the state {\rho }_t, the average photon number in the a mode reads

(23)\begin{array}{rl}⟨n_a⟩=& {\text{e}}^{{\mathrm{\Delta }}_t}\mathtt{t}\mathtt{r}\{:a^{†}\exp [{\alpha }_t{a}^{†}{b}^{†}\\ & +{\sigma }_t\left(a^{†}a+b^{†}b\right)+{\alpha }_t^{\ast }ab]a:\}-1.\end{array}

Inserting the completeness relation of two-mode coherent states |z_1,z_2⟩ and using the mathematical integral formula

(24)\begin{array}{rl}& \int \frac{{\text{d}}^2\gamma }{\pi }{\gamma }^n{\gamma }^{\ast m}{\text{e}}^{h{\left|\gamma \right|}^2+s\gamma +\eta {\gamma }^{\ast }}\\ & ={\text{e}}^{-\frac{s\eta }{h}}\sum _{l=0}^{min(n,m)}\left(\begin{array}{c}m\\ l\end{array}\right)\left(\begin{array}{c}n\\ l\end{array}\right)\frac{l!s^{m-l}{\eta }^{n-l}}{{\left(-h\right)}^{m+n+1-l}}\end{array}

and

(25){\displaystyle \int {\displaystyle \frac{{\text{d}}^2\gamma }{\pi }}{\gamma }^n{\gamma }^{\ast m}{\text{e}}^{h{\left|\gamma \right|}^2}={\delta }_{m,n}n!{\left(-{\displaystyle \frac{1}{h}}\right)}^{n+1},}

where Re, we thus obtain the evolved average photon number of a-mode under amplitude decay, i.e.,

(26)\begin{array}{rl}⟨n_a⟩=& {\text{e}}^{{\mathrm{\Delta }}_t}\int \frac{{\text{d}}^2{z}_1{\text{d}}^2{z}_2}{{\pi }^2}{\left|z_1\right|}^2\exp [{\alpha }_t{z}_1^{\ast }{z}_2^{\ast }\\ & +{\sigma }_t\left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)+{\alpha }_t^{\ast }{z}_1{z}_2]-1\\ =& -\frac{{\sigma }_t}{{\sigma }_t^2-{\left|{\alpha }_t\right|}^2}-1.\end{array}

Clearly, ⟨n_a⟩=⟨n_b⟩. In particular, for \alpha =0 or \sigma =-1, Eq. (26) reduces to the evolved average photon number of the a mode in the state {\rho }_{\text{th}} or {\rho }_{\text{squ}} in the amplitude decay process, i.e., {⟨n_a⟩}_{\text{th}}=-\left(1+{\sigma }^{-1}\right)e{}^{-2\kappa t} or {⟨n_a⟩}_{\text{squ}}={\left|{\alpha }_t\right|}^2/\left(1-{\left|{\alpha }_t\right|}^2\right), where {\alpha }_t is given in Eq. (22). In the limit, \kappa t\to \mathrm{\infty }, ⟨n_a⟩\to 0.

Figure 1 shows the evolved average photon number ⟨n_a⟩ as a function of the decay time \kappa t for the parameters \beta, \omega and \varrho. For any \beta, \omega and \varrho, ⟨n_a⟩ decreases gradually with increasing \kappa t. Especially, ⟨n_a⟩ has different initial values as \kappa t\to 0 and approaches zero when \kappa t is sufficiently large. This is consistent with the analysis in Sec. 3, as a result of the decoherence effect caused by amplitude-decay noise, which confirms that amplitude decay leads to a decrease in the average photon number. For a fixed \kappa t, ⟨n_a⟩ increases with increasing \varrho but decreases with increasing \beta and \omega. For small \kappa t, the changes of ⟨n_a⟩ with the parameters \beta, \omega and \varrho are more pronounced.

5 Photon number distribution

The photon number distribution describes the statistical probability features of the photon quantity in a quantum light field and often used to characterize the non-classicality of the light field [27, 28].

Using the relation between the number state |n_a,n_b⟩ and the unnormalized coherent state \Vert z_1,z_2⟩, i.e., |n_a,n_b⟩=(n_a!n_b!{)}^{-1/2}{\mathrm{\partial }}_{z_1}^{n_a}{\mathrm{\partial }}_{z_2}^{n_b}{\Vert z_1,z_2⟩|}_{z_1,z_2=0} and the inner product ⟨z^{\mathrm{\prime }}|z⟩= e{}^{-{\left|z^{\mathrm{\prime }}\right|}^2/2-{\left|z\right|}^2/2+z^{\mathrm{\prime }\ast }z}, we obtain the photon number distribution of {\rho }_t as

(27)\begin{array}{rl}& P(n_a,n_b,t)\\ & ={\text{e}}^{{\mathrm{\Delta }}_t}⟨n_a,n_b|:{\text{e}}^{{\alpha }_t{a}^{†}{b}^{†}+{\sigma }_t\left(a^{†}a+b^{†}b\right)+{\alpha }_t^{\ast }ab}:|n_a,n_b⟩\\ & ={\text{e}}^{{\mathrm{\Delta }}_t}\sum _{l=0}^{min(n_a,n_b)}\left(\begin{array}{c}{n}_a\\ l\end{array}\right)\left(\begin{array}{c}{n}_b\\ l\end{array}\right)\frac{{\left|{\alpha }_t\right|}^{2l}{\left({\sigma }_t+1\right)}^{n_a+n_b}}{{\left({\sigma }_t+1\right)}^{2l}}.\end{array}

In general, letting n_a⩽n_b, and using the standard definition of Jocobi polynomials P{}_m^{\left(x,y\right)}(\cdot ), we can rewrite P(n_a,n_b,t) as

(28)\begin{array}{rl}P(n_a,n_b,t)=& {\text{e}}^{{\mathrm{\Delta }}_t}{\left({\sigma }_t+1\right)}^{n_b-n_a}{\left({\left({\sigma }_t+1\right)}^2-{\left|{\alpha }_t\right|}^2\right)}^{n_a}\\ & \times {\text{P}}_{n_a}^{\left(0,n_b-n_a\right)}\left(\frac{{\left({\sigma }_t+1\right)}^2+{\left|{\alpha }_t\right|}^2}{{\left({\sigma }_t+1\right)}^2-{\left|{\alpha }_t\right|}^2}\right).\end{array}

Especially, when \alpha =0, P(n_a,n_b,t) represents the photon number distribution evolution of the state {\rho }_{\text{th}} under amplitude decay, i.e.,

(29)P_{\text{th}}(n_a,n_b,t)={\displaystyle \frac{{\sigma }^2}{{\left[1-\mathcal{T}\left(\sigma +1\right)\right]}^2}}{\left[{\displaystyle \frac{\left(\sigma +1\right){\text{e}}^{-2\kappa t}}{1-\mathcal{T}\left(\sigma +1\right)}}\right]}^{n_a+n_b},

and when \sigma =-1, Eq. (27) reduces to the evolved photon number distribution of the state {\rho }_{\text{squ}} under amplitude decay, i.e., P_{\text{squ}}(n_a,n_b,t)=(1-{\left|{\alpha }_t\right|}^2)n_a!n_b!{\left|{\alpha }_t\right|}^{n_a+n_b}, where {\alpha }_t can be found in Eq. (22).

In Fig. 2, the evolved photon number distribution P(n_a,n_b,t) of the state {\rho }_0 under amplitude decay is shown for different parameters \beta, \omega, \varrho, and \kappa t. With the increase of \beta and \omega, the distribution P(n_a,n_b,t) is obvious only for a few small photon number states, especially for the number states with n_a=n_b, and is almost zero for others, more and more photons occur in vacuum. Also, when \beta and \omega are sufficiently large, almost all photons will appear in vacuum. So, we say that the parameters \beta and \omega have the same role in the distribution P(n_a,n_b,t), however the impacts of the parameters \varrho and \kappa t on the distribution P(n_a,n_b,t) are exactly the opposite.

6 Wigner distribution function

The Wigner distribution function is a phase-space quasi-probability distribution, which is often used to assess the fidelity of logical qubits in quantum error correction experiments and is also applied to the study of non-commutative space quantum mechanics.

Using the two-mode Wigner operator \mathrm{\Delta }(z_1,z_2)=\mathrm{\Delta }(z_1)\otimes \mathrm{\Delta }(z_2) in the coherent state representation, where \mathrm{\Delta }(z_j)= {\pi }^{-2}e{}^{{\left|z_j\right|}^2}\intd{}^2{z}_j^{\mathrm{\prime }}|z_j^{\mathrm{\prime }}⟩⟨-z_j^{\mathrm{\prime }}|e{}^{2(z_j{z}_j^{\mathrm{\prime }\ast }-z_j^{\ast }{z}_j^{\mathrm{\prime }})}, (j=1,2) [29–34], together with Eq. (18) and the integral formula in Eq. (10), we obtain the Wigner distribution function for the state {\rho }_t as

(30)\begin{array}{rl}w(z_1,z_2,t)=& \frac{{\text{e}}^{{\left|z_1\right|}^2+{\left|z_2\right|}^2}{\text{e}}^{{\mathrm{\Delta }}_t}}{{\pi }^2}\int \frac{{\text{d}}^2{z}_1^{\mathrm{\prime }}{\text{d}}^2{z}_2^{\mathrm{\prime }}}{{\pi }^2}\\ & {\text{e}}^{-\left({\sigma }_t+2\right)\left({\left|z_1^{\mathrm{\prime }}\right|}^2+{\left|z_2^{\mathrm{\prime }}\right|}^2\right)+{\alpha }_t^{\ast }{z}_1^{\mathrm{\prime }}{z}_2^{\mathrm{\prime }}+{\alpha }_t{z}_1^{\mathrm{\prime }\ast }{z}_2^{\mathrm{\prime }\ast }}\\ & \times {\text{e}}^{2(z_1{z}_1^{\mathrm{\prime }\ast }-z_1^{\ast }{z}_1^{\mathrm{\prime }})+2(z_2{z}_2^{\mathrm{\prime }\ast }-z_2^{\ast }{z}_2^{\mathrm{\prime }})}\\ =& \frac{\mathrm{\Gamma }{\text{e}}^{{\mathrm{\Delta }}_t}}{{\pi }^2}\exp \{\left[1-4\mathrm{\Gamma }\left({\sigma }_t+2\right)\right]\left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)\\ & +4\mathrm{\Gamma }{\alpha }_t^{\ast }{z}_1{z}_2+4\mathrm{\Gamma }{\alpha }_t{z}_1^{\ast }{z}_2^{\ast }\},\end{array}

where \mathrm{\Gamma }={\left[{\left({\sigma }_t+2\right)}^2-{\left|{\alpha }_t\right|}^2\right]}^{-1}. Equation (30) shows that the evolved Wigner distribution function w(z_1,z_2,t) remains Gaussian under amplitude decay.

In particular, for \alpha =0, using Eq. (21), w(z_1,z_2,t) reduces to the evolved Wigner distribution function for the state {\rho }_{\text{th}} in the amplitude decay process, that is

(31)w_{\text{th}}(z_1,z_2,t)={\displaystyle \frac{{\sigma }^2{\mathcal{A}}_1^2}{{\pi }^2}}\exp \left[{\mathcal{A}}_1{\mathcal{A}}_2\left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)\right],

where {\mathcal{A}}_1={\left[\left(\sigma +1\right)\left(1-2\mathcal{T}\right)+1\right]}^{-1}, and {\mathcal{A}}_2=\left(\sigma +1\right)(1+2\mathcal{T})-3. However, for \sigma =-1, using Eq. (22), we obtain the evolution of the Wigner distribution function for the state {\rho }_{\text{squ}} under amplitude decay. In the long-time limit (\kappa t\to \mathrm{\infty }), w(z_1,z_2,t)\to {\pi }^{-2}e{}^{-2\left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)}, which corresponds to the Wigner distribution function for vacuum.

Figure 3 clearly shows that, for arbitrary values of \beta ,\omega ,\varrho, and \kappa t, the function w(z_1,z_2,t) remains Gaussian, consistent with the analytical result above. With increasing \beta and \varrho, the function w(z_1,z_2,t) exhibits stronger squeezing in phase space, whereas the frequency \omega and the decay time \kappa t evidently weaken the squeezing of the state {\rho }_0. Especially, when \kappa t is sufficiently large, the function w(z_1,z_2,t) evolves into the Gaussian wave packet corresponding to vacuum in phase space, which indicates that the initial state {\rho }_0 eventually loses nonclassicality and decays to vacuum under the long-term influence of amplitude-decay noise.

Besides, the peak value of the function w(z_1,z_2,t) increases with increasing \beta and \omega, but decreases with increasing entanglement coefficient \varrho and decay time \kappa t.

7 Von Neumann entropy

Von Neumann entropy unifies the concepts of thermodynamic entropy and information entropy through density matrices, and it is a core tool in quantum information theory because it can be used to quantify the degree of entanglement in a system [35–37]. Its main applications include quantum computing, black hole physics, and many-body physics. To obtain the time-evolved von Neumann entropy of the state {\rho }_t in the amplitude decay process, we first use Eq. (12) to derive the following operator identity:

(32)\begin{array}{rl}& :\exp \left[\alpha a^{†}{b}^{†}+\sigma \left(a^{†}a+b^{†}b\right)+{\alpha }^{\ast }ab\right]:\\ & =\frac{{\text{e}}^{{\omega }^{\mathrm{\prime }}}}{\sigma +1}\text{exp}\left[{\varrho }^{\mathrm{\prime }}{a}^{†}{b}^{†}+{\omega }^{\mathrm{\prime }}\left(a^{†}a+b^{†}b\right)+{\varrho }^{\mathrm{\prime }\ast }ab\right],\end{array}

where

(33)\begin{array}{rl}& {\omega }^{\mathrm{\prime }}=\frac{\mathrm{\Lambda }\left[\left(\sigma +1\right)\cosh \mathrm{\Lambda }-1\right]}{\left(\sigma +1\right)\sinh \mathrm{\Lambda }},\\ & {\varrho }^{\mathrm{\prime }}=-\frac{\alpha \mathrm{\Lambda }}{\left(\sigma +1\right)\sinh \mathrm{\Lambda }},\end{array}

and

(34)\cosh \mathrm{\Lambda }={\displaystyle \frac{\sigma +1}{2}}+{\displaystyle \frac{1-{\left|\alpha \right|}^2}{2\left(\sigma +1\right)}}.

The proof of Eq. (32) is as follows. Using Eq. (9), we have

(35)\begin{array}{rl}& {\omega }^{\mathrm{\prime }}{\mathrm{\Lambda }}^{-1}\sinh \mathrm{\Lambda }=\cosh \mathrm{\Lambda }-\frac{1}{\sigma +1},\\ & {\varrho }^{\mathrm{\prime }}{\mathrm{\Lambda }}^{-1}\sinh \mathrm{\Lambda }=-\frac{\alpha }{\sigma +1},\end{array}

which leads to the identity

(36)\begin{array}{rl}& {\left({\mathrm{\Lambda }}^{-1}\sinh \mathrm{\Lambda }\right)}^2\left({\omega }^{\mathrm{\prime }2}-{\left|{\varrho }^{\mathrm{\prime }}\right|}^2\right)\\ & ={\left(\cosh \mathrm{\Lambda }-\frac{1}{\sigma +1}\right)}^2-\frac{{\left|\alpha \right|}^2}{{\left(\sigma +1\right)}^2},\\ & {\sinh }^2\mathrm{\Lambda }={\cosh }^2\mathrm{\Lambda }-\frac{2\cosh \mathrm{\Lambda }}{\sigma +1}+\frac{1-{\left|\alpha \right|}^2}{{\left(\sigma +1\right)}^2},\\ & \cosh \mathrm{\Lambda }=\frac{\sigma +1}{2}+\frac{1-{\left|\alpha \right|}^2}{2\left(\sigma +1\right)}.\end{array}

Thus, Eq. (32) follows from Eqs. (8), (9), (35) and (36).

Therefore, using Eq. (32), we can rewrite Eq. (18) as

(37)\begin{array}{rl}{\rho }_t=& {\text{e}}^{{\mathrm{\Delta }}_t}:\exp \left[{\alpha }_t{a}^{†}{b}^{†}+{\sigma }_t\left(a^{†}a+b^{†}b\right)+{\alpha }_t^{\ast }ab\right]:\\ =& \frac{{\text{e}}^{{\mathrm{\Delta }}_t}{\text{e}}^{{\omega }_t^{\mathrm{\prime }}}}{{\sigma }_t+1}\text{exp}\left[{\varrho }_t^{\mathrm{\prime }}{a}^{†}{b}^{†}+{\omega }_t^{\mathrm{\prime }}\left(a^{†}a+b^{†}b\right)+{\varrho }_t^{\mathrm{\prime }\ast }ab\right],\end{array}

where {\omega }_t^{\mathrm{\prime }}, {\varrho }_t^{\mathrm{\prime }} and {\mathrm{\Lambda }}_t are obtained by making the substitutions \alpha \to {\alpha }_t, \sigma \to {\sigma }_t in Eqs. (33) and (34), and {\alpha }_t, {\sigma }_t can be found in Eq. (19).

Using the von Neumann entropy, S\left({\rho }_t\right)=-k_B\mathrm{t}\mathrm{r}\left({\rho }_t\ln {\rho }_t\right) [19], together with Eqs. (18) and (37), we have

(38)\begin{array}{rl}& S\left({\rho }_t\right)/k_B\\ & =-\text{tr}\left[{\rho }_t\ln \left(\frac{{\text{e}}^{{\mathrm{\Delta }}_t}{\text{e}}^{{\omega }_t^{\mathrm{\prime }}}}{{\sigma }_t+1}{\text{e}}^{{\varrho }_t^{\mathrm{\prime }}{a}^{†}{b}^{†}+{\omega }_t^{\mathrm{\prime }}\left(a^{†}a+b^{†}b\right)+{\varrho }_t^{\mathrm{\prime }\ast }ab}\right)\right]\\ & =-\ln \frac{{\text{e}}^{{\mathrm{\Delta }}_t}{\text{e}}^{{\omega }_t^{\mathrm{\prime }}}}{{\sigma }_t+1}-{\text{e}}^{{\mathrm{\Delta }}_t}\text{tr}(:{\text{e}}^{{\alpha }_t{a}^{†}{b}^{†}+{\sigma }_t\left(a^{†}a+b^{†}b\right)+{\alpha }_t^{\ast }ab}:\\ & \times \left[{\varrho }_t^{\mathrm{\prime }}{a}^{†}{b}^{†}+{\omega }_t^{\mathrm{\prime }}\left(a^{†}a+b^{†}b\right)+{\varrho }_t^{\mathrm{\prime }\ast }ab\right]).\end{array}

Similar to the derivation of Eq. (26), we have

(39)\begin{array}{rl}⟨ab⟩=& {\text{e}}^{{\mathrm{\Delta }}_t}\mathtt{t}\mathtt{r}\left[:{\text{e}}^{{\alpha }_t{a}^{†}{b}^{†}+{\sigma }_t\left(a^{†}a+b^{†}b\right)+{\alpha }_t^{\ast }ab}ab:\right]\\ =& {\text{e}}^{{\mathrm{\Delta }}_t}\int \frac{{\text{d}}^2{z}_1{\text{d}}^2{z}_2}{{\pi }^2}{z}_1{z}_2{\text{e}}^{{\alpha }_t{z}_1^{\ast }{z}_2^{\ast }++{\sigma }_t\left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)+{\alpha }_t^{\ast }{z}_1{z}_2}\\ =& \frac{{\alpha }_t}{{\sigma }_t^2-{\left|{\alpha }_t\right|}^2}={⟨a^{†}{b}^{†}⟩}^{\ast },\end{array}

and using

(40)⟨n_a⟩=⟨n_b⟩=-{\displaystyle \frac{{\sigma }_t}{{\sigma }_t^2-{\left|{\alpha }_t\right|}^2}}-1,

we finally obtain the von Neumann entropy of the state {\rho }_t, i.e.,

(41)\begin{array}{rl}S\left({\rho }_t\right)/k_B=& -\ln \frac{\left({\sigma }_t^2-{\left|{\alpha }_t\right|}^2\right){\text{e}}^{{\omega }_t^{\mathrm{\prime }}}}{{\sigma }_t+1}-\frac{{\varrho }_t^{\mathrm{\prime }}{\alpha }_t^{\ast }+{\varrho }_t^{\mathrm{\prime }\ast }{\alpha }_t}{{\sigma }_t^2-{\left|{\alpha }_t\right|}^2}\\ & +2{\omega }_t^{\mathrm{\prime }}\left(\frac{{\sigma }_t}{{\sigma }_t^2-{\left|{\alpha }_t\right|}^2}+1\right).\end{array}

As a special case, setting \alpha =0, we obtain the evolved von Neumann entropy of the state {\rho }_{\text{th}} under amplitude decay, that is

(42)S\left({\rho }_t\right)/k_B=-\ln {\displaystyle \frac{{\sigma }_t^2{\text{e}}^{{\omega }_t^{\mathrm{\prime }\mathrm{\prime }}}}{{\sigma }_t+1}}+{\displaystyle \frac{2{\omega }_t^{\mathrm{\prime }\mathrm{\prime }}\left({\sigma }_t+1\right)}{{\sigma }_t}},

where {\sigma }_t can be found in Eq. (21), and

(43){\omega }_t^{\mathrm{\prime }\mathrm{\prime }}={\displaystyle \frac{\left[\left({\sigma }_t+1\right)\cosh \ln \left({\sigma }_t+1\right)-1\right]\ln \left({\sigma }_t+1\right)}{\left({\sigma }_t+1\right)\sinh \ln \left({\sigma }_t+1\right)}}.

However, for \sigma =-1, using Eq. (22), Eq. (41) becomes the entropy evolution of the state {\rho }_{\text{squ}} in the amplitude decay process.

To illustrate the effects of the parameters \omega, \varrho, and \beta on the von Neumann entropy S\left({\rho }_t\right) in the amplitude decay process, we plot S\left({\rho }_t\right)/k_B as a function of the decay time \kappa t for different values of \omega, \varrho, and \beta. Clearly, the initial value of the entropy S\left({\rho }_t\right) is determined by \omega, \varrho, and \beta. As the decay time \kappa t increases, the entropy S\left({\rho }_t\right) gradually decreases and eventually approaches zero, regardless of the values of \omega, \varrho, and \beta. Moreover, when \varrho is larger or when \omega and \beta are smaller, the initial value of S\left({\rho }_t\right) is larger and the entropy approaches zero more slowly as \kappa t increases.

In addition, for a fixed decay time \kappa t, the entropy S\left({\rho }_t\right) increases with increasing \varrho, but decreases with increasing \omega and \beta. Since the von Neumann entropy can quantify the degree of entanglement of the system, a larger \varrho or smaller \omega and \beta enhances the entanglement of the initial state {\rho }_0 and leads to a slower reduction of entanglement over the decay time \kappa t.

8 Conclusions

In summary, using the operator ordering method, we obtained the normalized entangled density operator {\rho }_0 corresponding to the Hamiltonian H_0 and found that the conditions and ensure that {\rho }_0 is physically valid. Starting from the initial state {\rho }_0, we derived the time evolution of the density operator {\rho }_0 in the amplitude decay process using the Kraus-operator representation of the master equation for amplitude decay, and found that the density operator {\rho }_t retains the same functional form as the initial density operator {\rho }_0. We further showed , which guarantees that the evolved density operator {\rho }_t is physically valid. Clearly, the decoherence evolution is fully determined by the decay rate \kappa and the system parameters \omegaand \varrho. In the long-time limit \kappa t\to \mathrm{\infty }, the initial state {\rho }_0 decays to vacuum due to amplitude decay.

In addition, the evolution of {\rho }_0 under amplitude decay was characterized by examining its average photon number, photon number distribution, and Wigner distribution function. We found that larger \beta and \varrho enhance the nonclassicality of the initial state {\rho }_0, whereas larger \kappa and \omega lead to stronger decoherence. Moreover, it is worth emphasizing that larger \varrho or smaller \omega and \beta yields stronger entanglement of the state {\rho }_0. These results suggest that examining the decoherence evolution of a two-mode entangled quantum system under amplitude decay is broadly relevant and may have applications in studies of decoherence models in quantum optics and quantum information.

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