Evolution of decoherence in a degenerate parametric-amplifier system with thermal noise

Abstract

Quantum decoherence is considered a core obstacle in quantum computing; therefore, suppressing decoherence is a primary task. Based on the integral representation of the solution to the thermal master equation, we focus on the evolution of decoherence in a degenerate parametric amplifier system described by a known Hamiltonian in a thermal environment. We also investigate the evolution of the photon number distribution, Wigner distribution, and von Neumann entropy in a thermal environment. This work can provide a theoretical reference for experimental studies of degenerate parametric amplifier systems.

Graphical abstract

Keywords

degenerate parametric amplifier system / thermal noise / decoherence evolution / operator ordering method / von Neumann entropy

1 Introduction

Quantum decoherence is a central concept in quantum mechanics that describes the loss of quantum properties caused by interactions between quantum systems and their environments. In essence, it is the process in which a quantum system (such as a superposition state or an entangled state) undergoes irreversible interactions with the external environments (such as thermal fluctuations and electromagnetic radiation), resulting in the destruction of quantum phase relationships and ultimately exhibiting classical behaviors. In the context of quantum information processing and quantum computing, decoherence is regarded as a major obstacle, so understanding and suppressing decoherence is a primary task.

A degenerate parametric amplifier (DPA) is a quantum amplifier based on nonlinear optics or microwave circuits, whose core feature is the realization of phase-sensitive signal amplification through a parametric down-conversion process. Theoretically, the amplification process of a DPA can reach the quantum-noise limit (adding only vacuum noise), making it suitable for high-fidelity quantum signal processing [1–3]. Importantly, the DPA can enhance the quality of dispersive qubit measurements [4], quantum tomography [5], stochastic feedback cooling of a mechanical oscillator [6], and photonic entanglement and antibunching [7]. Consequently, many studies have recently focused on DPA systems, including generating mechanisms [8, 9], the frequency-detuning effect [10], entropy flow [11], optimization of nonlinearity and dissipation [12, 13], and the generation of spin squeezing, mechanical squeezing, and squeezed light [14–16]. However, to the best of our knowledge, investigations of the evolutions of DPA systems in decoherence channels have not been previously reported, mainly due to the lack of effective analytical methods.

Building on current theoretical progress on DPA systems, in this work we use the integral representation of the solution to the thermal master equation to investigate the evolution of decoherence in a DPA system with the Hamiltonian

(1){\mathcal{H}}_{\mathrm{d}}={\varrho }^{\ast }{a}^{†2}+\omega a^{†}a+\varrho a^2,

in a thermal environment, where \varrho is the interaction strength between the signal light and its surrounding nonlinear medium, and \omega is the frequency of the signal light. Our aims are twofold: first, to introduce the normalized density operator {\rho }_{\mathrm{d}} of the DPA system via the partition function and the operator-ordering method; and second, to discuss the evolutions of the density operator, quantum statistical functions, and von Neumann entropy.

The motivations for this work are threefold. First, the newly introduced density operator {\rho }_{\mathrm{d}} enables an analytical study of the system’s evolution, especially of the impacts of the parameters \varrho and \omega, which cannot be fully accessed using only the Hamiltonian {\mathcal{H}}_{\mathrm{d}} of the system. Second, the von Neumann entropy plays an important role in many areas of quantum theory; however, it is usually not easy to calculate because of the natural logarithmic function \ln {\rho }_{\mathrm{d}}, especially for the time-dependent density operator {\rho }_{\mathrm{d},t} in a thermal environment. Finally, the state {\rho }_{\mathrm{d}} is more general, since it includes squeezing, mixedness, vacuum, and related states, which makes the results of this work potentially applicable more broadly in quantum information theory and non-equilibrium statistical mechanics.

The paper is organized as follows. In Section 2, the corresponding density operator {\rho }_{\mathrm{d}} of the DPA system is presented. In Section 3, the analytical evolution of the operator {\rho }_{\mathrm{d}} in the presence of thermal noise is obtained via the integral representation of the solution to the thermal master equation. The evolutions of the photon-number distribution, Wigner distribution, and von Neumann entropy of the state {\rho }_{\mathrm{d}} in a thermal environment is investigated in Sections 4, 5, and 6, respectively. Finally, the main results of this work are summarized in Section 7.

2 Normalized density operator

For the DPA system with the given Hamiltonian {\mathcal{H}}_{\mathrm{d}}, its density operator is given by {\rho }_{\mathrm{d}}= e{}^{-\beta {\mathcal{H}}_{\mathrm{d}}}/\left(\mathtt{t}\mathtt{r}{\mathrm{e}}^{-\beta {\mathcal{H}}_{\mathrm{d}}}\right) in quantum statistical ensemble theory, where \beta ={\left(k_{B}T\right)}^{-1}, and k_B and T are, respectively, the Boltzmann constant and the temperature of the thermal field. Given that the Hamiltonian {\mathcal{H}}_{\mathrm{d}} and the natural exponential function e{}^{-\beta {\mathcal{H}}_{\mathrm{d}}} appear in the operator {\rho }_{\mathrm{d}}, we need to use the Baker−Hausdorff operator formula, that is, e{}^{a}be{}^{-b}=b+[a,b]+ [a,[a,b]]/2!+[a,[a,[a,b]]]/3!+\cdots, where [a,b]\ne 0, to calculate

(2)\begin{array}{rl}& \exp \left({\varrho }^{\ast }{a}^{†2}+\omega a^{†}a+\varrho a^2\right)\\ & =\sqrt{\mathcal{A}\mathrm{\Delta }}{\mathrm{e}}^{-\omega /2}\exp \left(\mathrm{\Lambda }{\varrho }^{\ast }{a}^{†2}\right)\\ & \times \exp \left(a^{†}a\ln \left(\mathcal{A}\mathrm{\Delta }\right)\right)\exp \left(\mathrm{\Lambda }\varrho a^2\right),\end{array}

where \mathrm{\Lambda }={\left(\mathcal{A}\coth \mathcal{A}-\omega \right)}^{-1}, \mathrm{\Delta }={\left(\mathcal{A}\cosh \mathcal{A}-\omega \sinh \mathcal{A}\right)}^{-1}, and \mathcal{A}={\left({\omega }^2-4{\left|\varrho \right|}^2\right)}^{1/2}. Combining Eqs. (1) and (2) leads to

(3){\mathrm{e}}^{-\beta {\mathcal{H}}_{\mathrm{d}}}=\sqrt{\mathcal{B}{\mathrm{e}}^{\omega \beta }}{\mathrm{e}}^{{\mathcal{C}}^{\ast }{a}^{†2}}{\mathrm{e}}^{a^{†}a\ln \mathcal{B}}{\mathrm{e}}^{\mathcal{C}{a}^2},

where \mathcal{B}=\mathcal{A}/\left[\omega \sinh \left(\beta \mathcal{A}\right)+\mathcal{A}\cosh \left(\beta \mathcal{A}\right)\right]and \mathcal{C}=-\left[\mathcal{B}\varrho \sinh \left(\beta \mathcal{A}\right)\right]/\mathcal{A}. Using the normal ordering product (indicated by the symbol :: [17, 18], that is, all creation operators are located to the left of the annihilation operators) of the boson exponential operator e{}^{\varsigma a^{†}a} with an arbitrary parameter \varsigma, i.e.,

(4){\mathrm{e}}^{\varsigma a^{†}a}=:\exp \left[\left({\mathrm{e}}^{\varsigma }-1\right)a^{†}a\right]:,

we write the operator e{}^{-\beta {\mathcal{H}}_{\mathrm{d}}} in Eq. (3) in its normal-ordered form, i.e.,

(5){\mathrm{e}}^{-\beta {\mathcal{H}}_{\mathrm{d}}}=\sqrt{\mathcal{B}{\mathrm{e}}^{\omega \beta }}:{\mathrm{e}}^{{\mathcal{C}}^{\ast }{a}^{†2}+\left(\mathcal{B}-1\right)a^{†}a+\mathcal{C}{a}^2}:.

On the other hand, using the completeness relation of coherent states |\alpha ⟩, where |\alpha ⟩= e {}^{-{\left|\alpha \right|}^2/2}e{}^{\alpha a^{†}}|0⟩, and the mathematical integral formula [17]

(6)

we obtain the trace tre{}^{-\beta {\mathcal{H}}_{\mathrm{d}}} as

(7)\mathtt{t}\mathtt{r}{\mathrm{e}}^{-\beta {\mathcal{H}}_{\mathrm{d}}}=\sqrt{\mathrm{\Gamma }\mathcal{B}{\mathrm{e}}^{\omega \beta }},

where \mathrm{\Gamma }={\left[{\left(\mathcal{B}-1\right)}^2-4{\left|\mathcal{C}\right|}^2\right]}^{-1}. Therefore, using Eqs. (4), (5) and (7), we show that

(8)\begin{array}{rl}{\rho }_{\mathrm{d}}=& \sqrt{{\mathrm{\Gamma }}^{-1}}:{\mathrm{e}}^{{\mathcal{C}}^{\ast }{a}^{†2}+\left(\mathcal{B}-1\right)a^{†}a+\mathcal{C}{a}^2}:\\ =& \sqrt{{\mathrm{\Gamma }}^{-1}}{\mathrm{e}}^{{\mathcal{C}}^{\ast }{a}^{†2}}{\mathrm{e}}^{a^{†}a\ln \mathcal{B}}{\mathrm{e}}^{\mathcal{C}{a}^2},\end{array}

which is exactly the normalized density operator (Gaussian-enhanced quantum state) describing the DPA system with the Hamiltonian {\mathcal{H}}_{\mathrm{d}}.

Special cases are discussed below. For \varrho =0, we have \mathcal{A}=\omega, \mathcal{B}= e{}^{-\beta \omega }, \mathcal{C}=0, and \mathrm{\Gamma }={\left({\mathrm{e}}^{-\beta \omega }-1\right)}^{-2}, so the density operator {\rho }_{\mathrm{d}} becomes {\rho }_{\mathrm{t}\mathrm{h}}=\left(1-{\mathrm{e}}^{-\beta \omega }\right)e{}^{-\beta \omega a^{†}a}, which describes the ordinary thermal field. However, for \omega =2\left|\varrho \right|, \mathcal{A}=\mathcal{B}=0, \mathcal{C}=-\varrho, and \mathrm{\Gamma }={\left(1-4{\left|\varrho \right|}^2\right)}^{-1}, thus {\rho }_{\mathrm{s}\mathrm{q}\mathrm{u}}=\sqrt{1-4{\left|\varrho \right|}^2}e{}^{-{\varrho }^{\ast }{a}^{†2}}|0⟩⟨0|e{}^{-\varrho a^2}, describing the usual squeezed field, where |0⟩⟨0|=:e{}^{-a^{†}a}: refers to the normal ordering of the vacuum projector.

3 Density-operator evolution for thermal noise

3.1 Solution of thermal master equation

Given a quantum state {\rho }_0 as an initial state immersed in the thermal environment, its density-operator evolution obeys the following master equation [19, 20]:

(9)\begin{array}{rl}\frac{\mathrm{d}{\rho }_t}{\mathrm{d}t}=& \kappa \overline{\mathfrak{n}}\left([a^{†}\rho ,a]+[a^{†},\rho a]\right)\\ & +\kappa \left(\overline{\mathfrak{n}}+1\right)\left([a\rho ,a^{†}]+[a,\rho a^{†}]\right),\end{array}

where \kappa is the dissipation factor and \overline{\mathfrak{n}} is the mean photon number of the thermal environment. Clearly, Eq. (9) describes the amplitude-decay model when \kappa is finite and \overline{\mathfrak{n}}\to 0 (\kappa \overline{\mathfrak{n}}\to 0).

To solve the thermal master equation in Eq. (9), we first introduce the thermal entangled state |\varkappa ⟩= e{}^{-\frac{1}{2}{\left|\varkappa \right|}^2+\varkappa a^{†}-{\varkappa }^{\ast }{\tilde{a}}^{†}+a^{†}{\tilde{a}}^{†}}|0\tilde{0}⟩ [21, 22], where {\tilde{a}}^{†} is a fictitious mode representing the thermal environment as a counterpart to the real system mode a^{†}, and |0\tilde{0}⟩ is the two-mode (real and fictitious) vacuum state. It is worth emphasizing that, in the entangled state |\varkappa =0⟩, the well-defined operator correspondence relations read

(10)a\iff {\tilde{a}}^{†},a^{†}\iff \tilde{a},a^{†}a\iff {\tilde{a}}^{†}\tilde{a}.

Accordingly, we rewrite the master equation for the density operator in Eq. (9) as the evolution equation of the state |{\rho }_t⟩\equiv {\rho }_t|\varkappa =0⟩ and obtain its Kraus operator-sum solution as follows:

(11){\rho }_t={\displaystyle \sum _{m,n=0}^{\mathrm{\infty }}{\mathcal{M}}_{m,n}{\rho }_0{\mathcal{M}}_{m,n}^{†},}

where {\mathcal{M}}_{m,n} is the Kraus operator corresponding to {\rho }_t, i.e.,

(12)\begin{array}{rl}& {\mathcal{M}}_{m,n}=\sqrt{\frac{{\mathcal{T}}_1^{m+n}{\left(\overline{\mathfrak{n}}+1\right)}^n{\overline{\mathfrak{n}}}^m}{m!n!\left(\overline{\mathfrak{n}}\mathcal{T}+1\right)}}{a}^{†m}{\mathrm{e}}^{a^{†}a\ln {\mathcal{T}}_2}{a}^n,\\ & {\mathcal{T}}_1=\frac{\mathcal{T}}{\overline{\mathfrak{n}}\mathcal{T}+1},{\mathcal{T}}_2=\frac{{\mathrm{e}}^{-\kappa t}}{\overline{\mathfrak{n}}\mathcal{T}+1},\\ & \mathcal{T}=1-{\mathrm{e}}^{-2\kappa t},\end{array}

where \kappa t refers to the decay time. Based on the operator-ordering method, we prove the completeness relation of the Kraus operators, which takes the form

(13){\displaystyle \sum _{m,n=0}^{\mathrm{\infty }}{\mathcal{M}}_{m,n}^{†}{\mathcal{M}}_{m,n}=1,}

that is, the evolved density operator {\rho }_t in the thermal environment is always normalized. Consequently, the trace is preserved, that is,

(14)\mathtt{t}\mathtt{r}{\rho }_t=\mathtt{t}\mathtt{r}\left({\displaystyle \sum _{m,n=0}^{\mathrm{\infty }}{\mathcal{M}}_{m,n}^{†}{\mathcal{M}}_{m,n}{\rho }_0}\right)=\mathtt{t}\mathtt{r}{\rho }_0=1.

3.2 Evolution of {\rho }_{\mathrm{d}}

In this section, we aim to investigate how the state {\rho }_{\mathrm{d}} evolves in a thermal environment. For convenience, we first derive the integral form of the solution to the thermal master equation, instead of using the Kraus operator-sum solution in Eq. (11). Inserting the P-representation, i.e., {\rho }_0={\pi }^{-1}\intd{}^2\alphaP(\alpha ,0)|\alpha ⟩⟨\alpha |, of any initial state {\rho }_0 into Eq. (11), we obtain

(15)\begin{array}{rl}{\rho }_t=& \int \frac{{\mathrm{d}}^2\alpha }{\pi }\mathrm{P}(\alpha ,0)\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\mathcal{T}}_1^{m+n}{\left(\overline{\mathfrak{n}}+1\right)}^n{\overline{\mathfrak{n}}}^m}{m!n!\left(\overline{\mathfrak{n}}\mathcal{T}+1\right)}\\ & \times {\left|\alpha \right|}^{2n}{a}^{†m}{\mathrm{e}}^{a^{†}a\ln {\mathcal{T}}_2}|\alpha ⟩⟨\alpha |{\mathrm{e}}^{a^{†}a\ln {\mathcal{T}}_2}{a}^m.\end{array}

Using the two operator identities |0⟩⟨0|=:e{}^{-a^{†}a}: and e{}^{a^{†}a\ln {\mathcal{T}}_2}|\alpha ⟩= e{}^{-{\left|\alpha \right|}^2/2+{\mathcal{T}}_2\alpha a^{†}}|0⟩, and summing over the indices m and n, respectively, we obtain

(16)\begin{array}{rl}{\rho }_t=& \int \frac{{\mathrm{d}}^2\alpha }{\pi }{\mathrm{e}}^{-{\left|\alpha \right|}^2}\mathrm{P}(\alpha ,0)\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\mathcal{T}}_1^{m+n}{\left(\overline{\mathfrak{n}}+1\right)}^n}{m!n!\left(\overline{\mathfrak{n}}\mathcal{T}+1\right)}\\ & \times {\overline{\mathfrak{n}}}^m{\left|\alpha \right|}^{2n}:a^{†m}{\mathrm{e}}^{{\mathcal{T}}_2\left(\alpha a^{†}+{\alpha }^{\ast }a\right)-a^{†}a}{a}^m:\\ =& {\mathcal{T}}_2{\mathrm{e}}^{\kappa t}\int \frac{{\mathrm{d}}^2\alpha }{\pi }\mathrm{P}(\alpha ,0)\\ & \times :{\mathrm{e}}^{{\mathcal{T}}_2\left(-{\mathrm{e}}^{-\kappa t}{\left|\alpha \right|}^2+\alpha a^{†}+{\alpha }^{\ast }a-{\mathrm{e}}^{\kappa t}{a}^{†}a\right)}:,\end{array}

which corresponds to the integral representation of the solution to the thermal master equation. It is clear that, once the P-representation P(\alpha ,0) of the initial state {\rho }_0 is known, the evolution of {\rho }_0 in a thermal environment can be obtained directly by integrating Eq. (16).

To this end, we need the P-representation of the state {\rho }_{\mathrm{d}}. Using the formula for calculating the antinormal-ordering product (denoted by ⋮ ⋮, that is, all annihilation operators are on the left of the creation operators) of a bosonic operator \rho, namely \rho ={\pi }^{-1}\int {\mathrm{d}}^2\gamma ⟨-\gamma |\rho \left|\gamma \right|⋮{\mathrm{e}}^{{\left|\gamma \right|}^2+{\gamma }^{\ast }a-\gamma a^{†}+a^{†}a}⋮, where |\gamma ⟩ is the coherent state, we obtain the antinormal-ordering product of {\rho }_{\mathrm{d}} as

(17)\begin{array}{rl}{\rho }_{\mathrm{d}}=& {\pi }^{-1}\sqrt{{\mathrm{\Gamma }}^{-1}}\int {\mathrm{d}}^2\gamma {\mathrm{e}}^{{\mathcal{C}}^{\ast }{\gamma }^{\ast 2}}\\ & \times {\mathrm{e}}^{-\mathcal{B}{\left|\gamma \right|}^2+\mathcal{C}{\gamma }^2}⋮{\mathrm{e}}^{{\gamma }^{\ast }a-\gamma a^{†}+a^{†}a}⋮\\ =& \sqrt{\mathcal{D}{\mathrm{\Gamma }}^{-1}}⋮{\mathrm{e}}^{{\mathcal{D}\mathcal{C}}^{\ast }{a}^{†2}}{\mathrm{e}}^{\left(1-\mathcal{D}\mathcal{B}\right)a^{†}a}{\mathrm{e}}^{\mathcal{D}\mathcal{C}{a}^2}⋮,\end{array}

where \mathcal{D}=1/\left({\mathcal{B}}^2-4{\left|\mathcal{C}\right|}^2\right). Using Eq. (17), we find that the P-representation of {\rho }_{\mathrm{d}} reads

(18){\mathrm{P}}_{\mathrm{d}}(\alpha ,0)=\sqrt{\mathcal{D}{\mathrm{\Gamma }}^{-1}}{\mathrm{e}}^{{\mathcal{D}\mathcal{C}}^{\ast }{\alpha }^{\ast 2}}{\mathrm{e}}^{\left(1-\mathcal{D}\mathcal{B}\right){\left|\alpha \right|}^2}{\mathrm{e}}^{\mathcal{D}\mathcal{C}{\alpha }^2}.

Inserting Eq. (18) into Eq. (16) and using Eq. (6), we obtain

(19){\rho }_{\mathrm{d},t}=c:{\mathrm{e}}^{c_1^{\ast }{a}^{†2}+\left(\mathrm{\Pi }-1\right)a^{†}a+c_1{a}^2}:,

where c=\sqrt{\mathcal{D}\mathcal{E}{\mathrm{\Gamma }}^{-1}}{\mathcal{T}}_2e{}^{\kappa t}, c_1=\mathcal{D}\mathcal{C}{\mathcal{E}\mathcal{T}}_2^2, \mathcal{E}=[(1-\mathcal{D}\mathcal{B}-{\mathcal{T}}_2{\mathrm{e}}^{-\kappa t}{)}^2-4{\mathcal{D}}^2{\left|\mathcal{C}\right|}^2{]}^{-1} and \mathrm{\Pi }=-{\mathcal{T}}_2[{\mathrm{e}}^{\kappa t}+(1-\mathcal{D}\mathcal{B}-{\mathcal{T}}_2{\mathrm{e}}^{-\kappa t}){\mathcal{E}\mathcal{T}}_2]+1. It follows from Eqs. (4) and (19) that

(20){\rho }_{\mathrm{d},t}=c{\mathrm{e}}^{c_1^{\ast }{a}^{†2}}{\mathrm{e}}^{a^{†}a\ln \mathrm{\Pi }}{\mathrm{e}}^{c_1{a}^2}.

Clearly, by comparing Eqs. (8) and (20), we see that the time-dependent operator {\rho }_{\mathrm{d},t} in a thermal environment retains the same exponential functional form as the initial operator {\rho }_{\mathrm{d}}, but becomes a genuinely mixed quantum state determined by the dissipation factor \kappa and the thermal mean photon number \overline{n}. This means that the initial state {\rho }_{\mathrm{d}} completely loses its coherence in a thermal environment.

In particular, for \varrho =0, thus \mathcal{D}= e{}^{2\beta \omega }, \mathcal{E}=(1-{\mathrm{e}}^{\beta \omega }-{\mathcal{T}}_2{\mathrm{e}}^{-\kappa t}{)}^{-2}, and \mathrm{\Pi }=1-{\mathcal{T}}_2e{}^{\kappa t}-{\left(1-{\mathrm{e}}^{\beta \omega }-{\mathcal{T}}_2{\mathrm{e}}^{-\kappa t}\right)}^{-1}{\mathcal{T}}_2^2\equiv {\mathcal{F}}_0, thus we obtain the evolution of the thermal state {\rho }_{\mathrm{t}\mathrm{h}} for thermal noise, i.e.,

(21){\rho }_{\mathrm{d},t}={\displaystyle \frac{\left(1-{\mathrm{e}}^{\beta \omega }\right){\mathcal{T}}_2{\mathrm{e}}^{\kappa t}}{1-{\mathrm{e}}^{\beta \omega }-{\mathcal{T}}_2{\mathrm{e}}^{-\kappa t}}}{\mathrm{e}}^{a^{†}a\ln {\mathcal{F}}_0}.

However, for \omega =2\left|\varrho \right|, thus \mathcal{D}={\left(-4{\left|\varrho \right|}^2\right)}^{-1}, \mathcal{E}={\left[{\left(1-{\mathcal{T}}_2{\mathrm{e}}^{-\kappa t}\right)}^2-{\left(4{\left|\varrho \right|}^2\right)}^{-1}\right]}^{-1}\equiv \mathcal{F}, and \mathrm{\Pi }=1-{\mathcal{T}}_2e{}^{\kappa t}-\left(1-{\mathcal{T}}_2{\mathrm{e}}^{-\kappa t}\right){\mathcal{F}\mathcal{T}}_2^2\equiv {\mathcal{F}}_1, we obtain the evolution of the squeezed state {\rho }_{\mathrm{s}\mathrm{q}\mathrm{u}} in a thermal environment, that is,

(22)\begin{array}{rl}{\rho }_{\mathrm{d},t}=& \sqrt{\mathcal{F}\left[1-{\left(4{\left|\varrho \right|}^2\right)}^{-1}\right]}{\mathcal{T}}_2{\mathrm{e}}^{\kappa t}\\ & \times {\mathrm{e}}^{\frac{{\mathcal{F}\mathcal{T}}_2^2}{4\varrho }{a}^{†2}}{\mathrm{e}}^{a^{†}a\ln {\mathcal{F}}_1}{\mathrm{e}}^{\frac{{\mathcal{F}\mathcal{T}}_2^2}{4{\varrho }^{\ast }}{a}^2}.\end{array}

Moreover, for t=0, {\mathcal{T}}_1=0, {\mathcal{T}}_2=1, \mathcal{T}=0, thus the operator {\rho }_{\mathrm{d},t} naturally reduces to the initial operator {\rho }_{\mathrm{d}} in Eq. (8). For \kappa t\to \mathrm{\infty }, {\mathcal{T}}_1\to {\left(\overline{\mathfrak{n}}+1\right)}^{-1}, {\mathcal{T}}_2\to 0, \mathcal{T}\to 1, thus {\rho }_{\mathrm{d},t} reduces to the density operator \frac{1}{\overline{\mathfrak{n}}+1}\exp \left(a^{†}a\ln \frac{\overline{\mathfrak{n}}}{\overline{\mathfrak{n}}+1}\right)\equiv {\rho }_{\mathrm{t}\mathrm{h}}^{\mathrm{\prime }} describing a thermal field. This result means that, when \kappa t\to \mathrm{\infty }, the initial state {\rho }_{\mathrm{d}} completely loses its non-classicality and finally becomes a mixed thermal state that depends only on the thermal mean photon number \overline{\mathfrak{n}}.

4 Photon number distribution evolution

The photon number distribution represents the probability distribution of detecting n photons in the optical field and is a core statistical property for describing quantum states of light in quantum optics [23, 24]. Therefore, studying it is of crucial significance for many fields, such as quantum communication and quantum computing.

For any quantum state \rho, its photon number distribution is usually defined as \mathcal{P}(m)= tr\left(\rho |m⟩⟨m|\right), where |m⟩ is the number state. To investigate the evolution of the photon number distribution of the state {\rho }_{\mathrm{d}} in the thermal environment, we first use Eq. (19) and the relation \sqrt{m!}|m⟩={\mathrm{\partial }}_i^m{\Vert i⟩|}_{i=0} (\Vert i⟩ is the unnormalized coherent state, i=\alpha ,\gamma) to obtain

(23)\begin{array}{rl}\mathcal{P}(m,t)=& \frac{c}{m!}{\mathrm{\partial }}_{\alpha }^m{\mathrm{\partial }}_{{\gamma }^{\ast }}^m⟨\gamma \Vert :{\mathrm{e}}^{c_1^{\ast }{a}^{†2}}\\ & \times {\mathrm{e}}^{\left(\mathrm{\Pi }-1\right)a^{†}a+c_1{a}^2}:{\Vert \alpha ⟩|}_{\alpha ,{\gamma }^{\ast }=0}\\ =& \frac{c}{m!}{\mathrm{\partial }}_{\alpha }^m{\mathrm{\partial }}_{{\gamma }^{\ast }}^m{{\mathrm{e}}^{c_1^{\ast }{\gamma }^{\ast 2}+\mathrm{\Pi }{\gamma }^{\ast }\alpha +c_1{\alpha }^2}|}_{\alpha ,{\gamma }^{\ast }=0}.\end{array}

Using the following partial-derivative relation

(24)\begin{array}{rl}& {\mathrm{\partial }}_{\alpha }^m{\mathrm{\partial }}_{{\gamma }^{\ast }}^m{{\mathrm{e}}^{-{\alpha }^2-{\gamma }^{\ast }{}^2+2x\alpha {\gamma }^{\ast }}|}_{\alpha ,{\gamma }^{\ast }=0}\\ & =\sum _{n,l,k=0}^{\mathrm{\infty }}{\frac{(-1{)}^{n+l}(2x{)}^k}{n!l!k!}{\mathrm{\partial }}_{\alpha }^m{\mathrm{\partial }}_{{\gamma }^{\ast }}^m{\alpha }^{2l+k}{\gamma }^{\ast }{}^{2n+k}|}_{\alpha ,{\gamma }^{\ast }=0}\\ & =2^{m}m!\sum _{n=0}^{[m/2]}\frac{m!}{2^{2n}(n!{)}^2(m-2n)!}{x}^{m-2n},\end{array}

we obtain the time evolution of the photon number distribution of the state {\rho }_{\mathrm{d}} in the presence of thermal noise, namely,

(25)\mathcal{P}(m,t)={\displaystyle \sum _{n=0}^{\left[m/2\right]}{\displaystyle \frac{m!c{\mathrm{\Pi }}^m}{2^{2n}(n!{)}^2\left(m-2n\right)!}}{\left({\displaystyle \frac{2\left|c_1\right|}{\mathrm{\Pi }}}\right)}^{2n}.}

Furthermore, using the new form of Legendre functions in Ref. [25], we rewrite the evolved distribution \mathcal{P}(m,t) as

(26)\mathcal{P}(m,t)=c{\mathrm{{\rm Y}}}^m{\mathrm{P}}_m\left({\displaystyle \frac{\mathrm{\Pi }}{\mathrm{{\rm Y}}}}\right),

where {\mathrm{{\rm Y}}}^2={\mathrm{\Pi }}^2-4{\left|c_1\right|}^2, and which is related to the Legendre function P{}_m(\cdot ) of order m. In particular, for \varrho =0, thus \mathrm{\Pi }=\mathrm{{\rm Y}}\equiv {\mathcal{F}}_0, and hence

(27)\mathcal{P}(m,t)={\displaystyle \frac{\left(1-{\mathrm{e}}^{\beta \omega }\right){\mathcal{T}}_2{\mathrm{e}}^{\kappa t}}{1-{\mathrm{e}}^{\beta \omega }-{\mathcal{T}}_2{\mathrm{e}}^{-\kappa t}}}{\mathcal{F}}_0^m,

which gives the evolution of the photon number distribution of {\rho }_{\mathrm{t}\mathrm{h}} under thermal noise. In the case of \omega =2\left|\varrho \right|, we have {\mathrm{{\rm Y}}}^2={\mathcal{F}}_1^2-{\left(4{\left|\varrho \right|}^2\right)}^{-1}{\mathcal{F}}^2{\mathcal{T}}_2^4, so \mathcal{P}(m,t) represents the time evolution of the photon number distribution of {\rho }_{\mathrm{s}\mathrm{q}\mathrm{u}} in a thermal environment.

In Fig. 1, the photon number distribution \mathcal{P}(m,t) of the state {\rho }_{\mathrm{d},t} is shown for different parameters \omega, \varrho, \beta, \overline{\mathfrak{n}} and \kappa t. For relatively small values of \omega ,\beta and \kappa t, the probabilities of finding photons are spread over many photon-number states. As \omega ,\beta and \kappa t increase, the probabilities become concentrated on only a few low photon-number states and approach zero for the other states. In particular, for sufficiently large \omega ,\beta and \kappa t, almost all photons appear in the vacuum state. These results indicate that the parameters \omega ,\beta and \kappa t play almost the same role in influencing the photon number distribution \mathcal{P}(m,t). However, the behavior of the distribution \mathcal{P}(m,t) changes in the opposite way as \varrho and \overline{\mathfrak{n}} increase.

5 Wigner distribution evolution

The Wigner distribution is defined by the phase-space autocorrelation function of the wave function and can intuitively demonstrate the nonclassical properties of quantum states by mapping the states to phase space [26, 27]. In particular, its region of negative values directly reflects the strength of quantum correlations and is commonly used in quantum-optics experiments to analyze the entanglement characteristics of photon pairs.

For any single-mode quantum state \rho, its Wigner distribution reads w(\alpha )= tr\left[\rho \mathrm{\Delta }(\alpha )\right], where \mathrm{\Delta }(\alpha ) is the Wigner operator in the coherent-state representation. Using Eq. (10), the time evolution of the Wigner distribution of the state {\rho }_{\mathrm{d}} is obtained as

(28)w(\alpha ,t)=c\mathtt{t}\mathtt{r}\left(:{\mathrm{e}}^{c_1^{\ast }{a}^{†2}}{\mathrm{e}}^{\left(\mathrm{\Pi }-1\right)a^{†}a}{\mathrm{e}}^{c_1{a}^2}:\mathrm{\Delta }(\alpha )\right).

Furthermore, using the integration formula in Eq. (8), we obtain

(29)\begin{array}{rl}w(\alpha ,t)=& {\pi }^{-1}c\sqrt{\mathcal{G}}\exp \{4\mathcal{G}{c}_1^{\ast }{\alpha }^{\ast 2}\\ & +4\mathcal{G}{c}_1{\alpha }^2+\left[1-4\mathcal{G}\left(\mathrm{\Pi }+1\right)\right]{\left|\alpha \right|}^2\},\end{array}

where \mathcal{G}={\left[{\left(\mathrm{\Pi }+1\right)}^2-4{\left|c_1\right|}^2\right]}^{-1}. Clearly, the Wigner distribution w(\alpha ,t) remains Gaussian in phase space. In the case of \varrho =0, \mathcal{G}={\left({\mathcal{F}}_0+1\right)}^{-2}, and w(\alpha ,t) becomes the Wigner distribution evolution of {\rho }_{\mathrm{t}\mathrm{h}} in a thermal environment, i.e.,

(30)\begin{array}{rl}w(\alpha ,t)=& \frac{\left(1-{\mathrm{e}}^{\beta \omega }\right){\mathcal{T}}_2{\mathrm{e}}^{\kappa t}}{\pi \left(1-{\mathrm{e}}^{\beta \omega }+{\mathcal{T}}_2{\mathrm{e}}^{-\kappa t}\right)\left({\mathcal{F}}_0+1\right)}\\ & \times \exp \left\{\left[1-4{\left({\mathcal{F}}_0+1\right)}^{-1}\right]{\left|\alpha \right|}^2\right\},\end{array}

whereas for \omega =2\left|\varrho \right|, \mathcal{G}={\left[{\left({\mathcal{F}}_1+1\right)}^2-{\left(4{\left|\varrho \right|}^2\right)}^{-1}{\mathcal{F}}^2{\mathcal{T}}_2^4\right]}^{-1}, and thus w(\alpha ,t) gives the Wigner distribution evolution of {\rho }_{\mathrm{s}\mathrm{q}\mathrm{u}} under thermal noise.

In Fig. 2, we present the Wigner distribution w(\alpha ,t) of the state {\rho }_{\mathrm{d},t} for different parameters \omega, \varrho, \overline{\mathfrak{n}}, \beta and \kappa t. As expected from the analytical result above, the distribution w(\alpha ,t) always exhibits a Gaussian form for arbitrary \omega, \varrho, \overline{\mathfrak{n}}, \beta and \kappa t. Moreover, the parameters \varrho and \beta significantly enhance the squeezing of the state {\rho }_{\mathrm{d}}, whereas the parameters \omega, \overline{\mathfrak{n}} and \kappa t weaken it. For a sufficiently large \kappa t, the distribution w(\alpha ,t) approaches the Gaussian form corresponding to the thermal state {\rho }_{\mathrm{t}\mathrm{h}}^{\mathrm{\prime }} in phase space. In addition, the peak height of the distribution w(\alpha ,t) increases as the parameters \omega ,\beta and \kappa t increase, but decreases as \varrho and \overline{\mathfrak{n}} increase.

6 Von Neumann entropy evolution

The von Neumann entropy is a core concept in quantum mechanics that describes the information entropy of quantum states [28–31]. Its mathematical definition is S(\rho )/k_B=-tr\left(\rho \ln \rho \right), where \rho is the density matrix. In quantum information theory, entanglement entropy (such as the reduced entropy of subsystems) is a key tool for quantifying entanglement strength and can be calculated from the von Neumann entropy of the reduced density matrix [32–36].

In this section, we calculate the time evolution of the von Neumann entropy of the state {\rho }_{\mathrm{d}} under thermal noise. Since the original definition of the von Neumann entropy involves the natural logarithm \ln {\rho }_{\mathrm{d},t}, we first need to get rid of the normal-ordering symbol :: in Eq. (19) and rewrite {\rho }_{\mathrm{d},t} in a single-exponential form. To achieve this, we use the following operator identity [28]

(31)\begin{array}{rl}& :\exp \left({\epsilon }^{\ast }{a}^{†2}+ϵa^{†}a+\epsilon a^2\right):\\ & =\frac{1}{\sqrt{1+ϵ}}\exp [\frac{{\epsilon }^{\ast }\theta }{\left(1+ϵ\right)\sinh \theta }{a}^{†2}\\ & +\frac{\theta \left[\left(1+ϵ\right)\cosh \theta -1\right]}{2\left(1+ϵ\right)\sinh \theta }\left(2a^{†}a+1\right)\\ & +\frac{\epsilon \theta }{\left(1+ϵ\right)\sinh \theta }{a}^2],\end{array}

where \cosh \theta =\left[{\left(1+ϵ\right)}^2-4{\left|\epsilon \right|}^2+1\right]/2\left(1+ϵ\right) (\epsilon ,ϵ are arbitrary parameters), and rewrite Eq. (19) as

(32){\rho }_{\mathrm{d},t}=d\exp \left(d_1^{\ast }{a}^{†2}+d_2{a}^{†}a+d_1{a}^2\right),

where

(33)\begin{array}{rl}& d=c{\mathrm{\Pi }}^{-1/2}{\mathrm{e}}^{d_2/2},d_1=\frac{c_1\theta }{\mathrm{\Pi }\sinh \theta },\\ & d_2=\frac{\theta \left(\mathrm{\Pi }\cosh \theta -1\right)}{\mathrm{\Pi }\sinh \theta },\\ & \cosh \theta =\frac{{\mathrm{\Pi }}^2-4{\left|c_1\right|}^2+1}{2\mathrm{\Pi }}.\end{array}

Thus, combining Eqs. (19) and (32), we obtain

(34)\begin{array}{rl}\mathcal{S}({\rho }_{\mathrm{d},t})/k_B=& -\mathrm{t}\mathrm{r}[c\ln d:{\mathrm{e}}^{c_1^{\ast }{a}^{†2}+\left(\mathrm{\Pi }-1\right)a^{†}a+c_1{a}^2}:\\ & +c:{\mathrm{e}}^{c_1^{\ast }{a}^{†2}+\left(\mathrm{\Pi }-1\right)a^{†}a+c_1{a}^2}:\\ & \times \left(d_1^{\ast }{a}^{†2}+d_2{a}^{†}a+d_1{a}^2\right)].\end{array}

To evaluate the entropy S({\rho }_{\mathrm{d},t})/k_B, we first use the integral formula in Eq. (6) and the differential relation in Eq. (24) to calculate the expectation values of the operators a^2,a^{†2} and a^{†}a in the state {\rho }_{\mathrm{d},t}, i.e.,

(35)\begin{array}{rl}⟨a^2⟩=& c\mathrm{t}\mathrm{r}\left(:{\mathrm{e}}^{c_1^{\ast }{a}^{†2}+\left(\mathrm{\Pi }-1\right)a^{†}a+c_1{a}^2}:a^2\right)\\ =& 2c^{-2}{c}_1^{\ast }={⟨a^{†2}⟩}^{\ast },\end{array}

and

(36)\begin{array}{rl}⟨a^{†}a⟩=& \mathrm{t}\mathrm{r}\left({\rho }_{\mathrm{d},t}a{a}^{†}\right)-1\\ =& c\mathrm{t}\mathrm{r}\left(:a^{†}{\mathrm{e}}^{c_1^{\ast }{a}^{†2}+\left(\mathrm{\Pi }-1\right)a^{†}a+c_1{a}^2}a:\right)-1\\ =& c^{-2}\left(1-\mathrm{\Pi }\right)-1,\end{array}

and thus we have

(37)\begin{array}{rl}\mathcal{S}({\rho }_{\mathrm{d},t})/k_B=& d_2{c}^{-2}\left(\mathrm{\Pi }-1\right)-\ln d+d_2\\ & -2c^{-2}\left(d_1^{\ast }{c}_1+d_1{c}_1^{\ast }\right).\end{array}

In particular, for \varrho =0, we have d=c{\mathcal{F}}_0^{-1/2}e{}^{d_2/2}, d_1=0, d_2=\theta \left({\mathcal{F}}_0\cosh \theta -1\right)/{\mathcal{F}}_0\sinh \theta, \cosh \theta =\left({\mathcal{F}}_0^2+1\right)/2{\mathcal{F}}_0, c=\left(1-{\mathrm{e}}^{\beta \omega }\right){\left(1-{\mathrm{e}}^{\beta \omega }-{\mathcal{T}}_2{\mathrm{e}}^{-\kappa t}\right)}^{-1}{\mathcal{T}}_2e{}^{\kappa t}, and c_1=0, and hence

(38)\mathcal{S}({\rho }_{\mathrm{d},t})/k_B=d_2{c}^{-2}\left({\mathcal{F}}_0-1\right)-\ln d+d_2.

In Fig. 3, the von Neumann entropy S\left({\rho }_{\mathrm{d},t}\right) of the state {\rho }_{\mathrm{d},t} is plotted as a function of the decay time \kappa t for different parameters \omega, \varrho, \overline{\mathfrak{n}} and \beta. It is clearly seen from Fig. 3 that, for arbitrary \omega, \varrho, \overline{\mathfrak{n}} and \beta, the entropy S\left({\rho }_{\mathrm{d},t}\right) always decreases slowly as \kappa t increases and approaches a constant value when \kappa t is sufficiently large. This is easy to understand, because for sufficiently large \kappa t the initial state {\rho }_0 reduces to the usual thermal state {\rho }_{\mathrm{t}\mathrm{h}}^{\mathrm{\prime }}, which depends only on the thermal mean photon number \overline{\mathfrak{n}}.

Moreover, for a fixed \kappa t, the entropy S\left({\rho }_{\mathrm{d},t}\right) increases with the parameters \varrho and \overline{\mathfrak{n}}, but decreases as the parameters \omega and \beta increase. In addition, the parameters \omega, \varrho and \beta have a significant impact on the initial value of the entropy S\left({\rho }_{\mathrm{d},t}\right), whereas the parameter \overline{\mathfrak{n}} has almost no effect on it. More importantly, the parameters \omega, \varrho, \overline{\mathfrak{n}} and \beta have qualitatively different effects on the entropy evolution over the decay time \kappa t. In summary, the larger the parameters \varrho and \overline{\mathfrak{n}}, the more slowly the entropy S\left({\rho }_{\mathrm{d},t}\right) decreases, whereas the effects of the parameters \omega and \beta on S\left({\rho }_{\mathrm{d},t}\right) are exactly the opposite.

7 Conclusions

In summary, using partition-function theory and the operator-ordering method, we obtain the normalized density operator {\rho }_{\mathrm{d}} describing the DPA system with the known Hamiltonian {\mathcal{H}}_{\mathrm{d}}, which is exactly a Gaussian-enhanced quantum state. In addition, we present the integral representation of the solution to the thermal master equation and use it to obtain the analytical evolution of the DPA system in a thermal environment. The analytical results show that the evolved density operator {\rho }_{\mathrm{d},t} describing the system under thermal noise retains the same functional form as the initial state {\rho }_{\mathrm{d}}, but becomes a genuinely mixed state as a result of thermal noise.

Moreover, we investigate the evolution of the photon number distribution and the Wigner distribution of the state {\rho }_{\mathrm{d}} in a thermal environment. The results show that the evolved photon number distribution is always related to the Legendre function, and the evolved Wigner distribution is always Gaussian. In addition, the nonclassicality of the state {\rho }_{\mathrm{d}} can be enhanced as \varrho and \beta increase, but is weakened as \omega, \overline{\mathfrak{n}} and \kappa t increase. In particular, we handle the von Neumann entropy evolution of the state {\rho }_{\mathrm{d}} under thermal noise by means of a meaningful operator identity and numerically study its evolution behavior. We find that the entropy always decreases gradually as the decay time \kappa t increases for any values of \omega, \varrho, \overline{\mathfrak{n}} and \beta, and that, for a given \kappa t, the entropy increases as \varrho and \overline{\mathfrak{n}} increase, but decreases as \omega and \beta increase. Given the generality of the newly prepared Gaussian-enhanced quantum state and the quantum thermal environment considered here, this work provides an effective approach for studying quantum entropy evolution in realistic environments, and the main results can help guide experimental measurements of quantum entropy.

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