Analogous optical coherent state sourced by a partially coherent beam

Jinqi Song; Fengqi Liu; Mingli Sun; Bingsong Cao; Zhifang Qiu; Naichen Zhang; Xiangyu Tong; Wenzhe Wang; Yuanxing Liu; Kaikai Huang; Xian Zhang; Xuanhui Lu

doi:10.1117/1.APN.4.4.046009

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Abstract

Based on the first-order correlation function of light, we propose analogous optical coherent states (AOCSs) sourced by partially coherent beams, which can nondiffractively propagate with sinusoidal oscillation in the harmonic potential when the nondiffraction propagation matching condition (NPMC) is met. Unlike the traditional quantum coherent state, the minimum uncertainty of AOCS is related to the coherence of light, and only when the NPMC is met, its uncertainty is the least. Furthermore, based on the mathematical similarity between the Schrödinger and the Helmholtz equations, we find that our proposed AOCSs correspond to the partially coherent steady states of the harmonic oscillator. Our research not only increases the understanding of the coherence of light and enriches the types of nondiffraction beams but also increases the understanding of the quantum coherence regulating the evolution of probability waves.

Keywords

coherent state harmonic potential nondiffraction partially coherent beam

1 Introduction

In 1926, Schrödinger1 proposed, when searching for the closest solution to the classical state of the quantum harmonic oscillator, that the coherent state is a Gaussian wave packet that oscillates over time and does not diffuse. It corresponds directly to the classical phase space and has the minimum orthogonal component uncertainty. Subsequently, in 1963, when studying the quantum coherence of light, Glauber2 pointed out that the eigenstates of the non-Hermitian annihilation operator of an electromagnetic field were the coherent states, and benefiting from the characteristic that the eigenvalues of coherent states were continuous and forming a super-complete set, they have been used as a mathematical tool to describe any quantum state,3 which led to a deeper study and understanding of the coherence of light fields.4 At the same time, Klauder5^,6 introduced the basic ideas of coherent states for arbitrary Lie groups and developed a method to generate a set of continuous states; then, after ten years of development, Perelomov7 and Gilmore8^,9 realized the direct construction of coherent states for any physical system using group pairs with dynamic symmetry. Since then, coherent states have become an essential component of modern physics and are widely applied in many regions.10^–16 In addition to the traditional coherent state of bosons in Fock space, the concept of coherent state has also been extended to fermion systems and atomic ensembles. Unlike boson systems, the generation and annihilation operators of fermions have an anti-commutative relationship, so the eigenstate of the fermion annihilation operator—coherent state—is defined in a super Hilbert space over the Graßmann algebra, i.e., an enlarged Hilbert space spanned by linear combinations of vectors with Graßmann number coefficients.17^,18 For atomic ensembles, Gilmore and Feng17 proposed the spin coherent state in angular momentum space, which described the collective properties of atomic systems very well.

In this paper, we extend the concept of coherent states to classical optics and propose the analogous optical coherent state (AOCS). As coherence is an inherent statistical degree of freedom in the light field and has outstanding applications in many fields,20^–28 the AOCS we proposed is sourced by a partially coherent beam. When the nondiffraction matching condition (NPMC) is met, it can nondiffractively propagate with sine oscillations in the harmonic potential, just like the dynamic evolution of traditional quantum coherent states; except that the minimum value of its uncertainty is related to the coherence of the light source, and it only reaches the minimum value when the NPMC is met. Furthermore, due to the mathematical similarity between the Schrödinger and the Helmholtz equations, we found that the AOCSs we proposed correspond to the partially coherent steady states of the harmonic oscillator. So, our research has potential values for controlling the dynamic evolution of probability waves by regulating coherence.

2 Theory

The cross-spectral density function of a one-dimensional AOCS can be expressed as follows: $W (x_{1}, x_{2}, 0) = \exp (- \frac{x_{1}^{2} + x_{2}^{2}}{w^{2}}) \exp [i c (x_{1} - x_{2})] \times \exp [- (\frac{x_{1} - x_{2}}{δ})^{2}] .$ (1)

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Among them, $w$ is the waist, $c$ is a linear chirp parameter, and $δ$ is the coherent width. The AOCS proposed here is the same beam as a linearly chirped modulated Gaussian Schell mode.29 However, as we prove in Appendix B, it has the same mathematical expression of the partially coherent quantum mixed state obtained by superimposing coherent states incoherently, and to highlight its dynamic transmission characteristics in the harmonic potential, we chose to use AOCS to call it in this paper.

Under the paraxial approximation conditions, the evolution of the AOCS in harmonic potential is governed by29 $i \frac{\partial}{\partial z} W = \frac{- 1}{2 k} (\frac{\partial^{2} W}{\partial x_{1}^{2}} - \frac{\partial^{2} W}{\partial x_{2}^{2}}) + [V (x_{1}) - V^{*} (x_{2})] W,$ (2)where $V (x) = b \cdot x^{2}$ is the harmonic potential, which can correspond to the GRIN fiber30 or curved surfaces31^,32 in real physical systems, $k$ is the wavenumber, $b$ is the depth of the harmonic potential, and * represents the complex conjugation.

For Eqs. (1) and (2), the symmetric split Fourier transform can be used to simulate the propagation of cross-spectral density. Taking the simulation of a one-step symmetric split Fourier transform as an example, the calculation process can be expressed as ${\begin{matrix} W (x_{1}, x_{2}, z + h) = I F {F [B] \cdot \exp [\frac{i h}{4 k} (k_{x_{1}}^{2} - k_{x_{2}}^{2})]} \\ B = I F [A] \cdot \exp {- i h [V (x_{1}) - V^{*} (x_{2})]} \\ A = F [W (x_{1}, x_{2}, z)] \cdot \exp [\frac{i h}{4 k} (k_{x_{1}}^{2} - k_{x_{2}}^{2})] . \end{matrix}$ (3)

Here, $I F$ is the inverse Fourier transform, $F$ is the Fourier transform, and $h$ is the step size.

For a partially coherent beam, the degree of coherence is defined as29 $μ (x_{1}, x_{2}, z) = \frac{W (x_{1}, x_{2}, z)}{\sqrt{W (x_{1}, x_{1}, z) \cdot W (x_{2}, x_{2}, z)}} .$ (4)

We theoretically derive that the AOCS can propagate nondiffractively in the harmonic potential when the following NPMC is met (a detailed proof is provided in Appendix A): $b = \frac{1}{2 k} (\frac{4}{w^{4}} + \frac{8}{w^{2} δ^{2}}) .$ (5)

As AOCS represents a quantum mixed state, the NPMC here can also be regarded as the steady state condition, and when the NPMC [Eq. (5)] is met, the AOCS’s analytical propagation formula in harmonic potential can be expressed as $W (x_{1}, x_{2}, z) = \exp {- \frac{[x_{1} + a \sin (d z)]^{2} + [x_{2} + a \sin (d z)]^{2}}{w^{2}}} \times \exp [i c \cos (d z) (x_{1} - x_{2})] \exp [- \frac{(x_{1} - x_{2})^{2}}{δ^{2}}] .$ (6)

Among them, $d^{2} = \frac{2 b}{k}$ and $a = - \frac{c}{d k} = - \frac{c}{\sqrt{2 b k}}$ . From Eq. (6), it can be seen that when the AOCS propagates in the harmonic potential, in addition to no-diffraction, it will also sinusoidally oscillate around the center of harmonic potential. The oscillation frequency $\frac{d}{2 π}$ is determined by the depth of parabolic potential $b$ and the wavenumber of the beam $k$ . The oscillation amplitude $a$ is proportional to the linear chirp parameter $c$ and inversely proportional to the depth of parabolic potential $b$ and the wavenumber of the beam $k$ .

Generally, for AOCS beams, we can obtain the evolution of the position and momentum of them: ${\begin{matrix} ⟨ x ⟩ = \frac{\int x \times W (x, x, z) d x}{\int W (x, x, z) d x} = - a \cdot \sin (d z) \\ S (k_{x_{1}}, k_{x_{2}}, z) = \int \int W (x_{1}, x_{2}, z) \exp [i (k_{x_{1}} x_{1} - k_{x_{2}} x_{2})] d x_{1} d x_{2} \\ ⟨ k_{x} ⟩ = \frac{\int k_{x} \times S (k_{x}, k_{x}, z) d k_{x}}{\int S (k_{x}, k_{x}, z) d k_{x}} = - c \cdot \cos (d z), \end{matrix}$ (7)where $S$ is the cross-spectral density function in the momentum space. It can be seen from Eq. (7) that the position and momentum of AOCS are harmoniously oscillating with propagation. The uncertainty of the position and momentum of AOCS can be given as ${\begin{matrix} ⟨ x^{2} ⟩ = \frac{\int x^{2} \times W (x, x, z) d x}{\int W (x, x, z) d x} = \frac{4 a^{2} \sin^{2} (d z) + w^{2}}{4} \\ Δ x = ⟨ x^{2} ⟩ - ⟨ x ⟩^{2} = \frac{w^{2}}{4} \\ ⟨ k_{x}^{2} ⟩ = \frac{\int k_{x}^{2} \times S (k_{x}, k_{x}, z) d k_{x}}{\int S (k_{x}, k_{x}, z) d k_{x}} = \frac{4 c^{2} \cos^{2} (d z)}{4} + \frac{(δ^{2} + 2 w^{2})}{w^{2} δ^{2}} \\ Δ p = ⟨ k_{x}^{2} ⟩ - ⟨ k_{x} ⟩^{2} = \frac{δ^{2} + 2 w^{2}}{w^{2} δ^{2}} . \end{matrix}$ (8)

It can be seen from Eq. (8) that the position uncertainty of AOCS is only related to the beam waist. By contrast, the momentum uncertainty is related to both beam waist and coherence width. The smaller the coherence width, the greater the momentum uncertainty. So the principle of uncertainty of AOCS is related to the coherence of the light source: $Δ p Δ x = \frac{1}{4} \frac{δ^{2} + 2 w^{2}}{δ^{2}} \geq \frac{1}{4} .$ (9)

As shown in Eq. (9), the uncertainty principle’s value will decrease as the coherence width increases, and it will equal the limit $\frac{1}{4}$ when the coherence width is infinite.

As mentioned at the beginning, the AOCS represents a quantum mixed state obtained by incoherently superimposing coherent states. This implies that the position and momentum uncertainties of such a mixed state are related to its coherence and will decrease as the coherence width increases (a detailed proof can be found in Appendix B).

3 Simulation and Discussion

In our simulation, the wavelength, coherent width, linear chirp parameter, and waist of the beam are set to $λ = 1064 nm$ , $δ = 0.174 mm$ , $c = 3000 {mm}^{- 1}$ , and $w = 1.5 mm$ , respectively, and all transmission diagrams in this paper are drawn using symmetric split Fourier transform [Eq. (3)].

The NPMC is the most critical condition in this paper, so it was first demonstrated. As shown in Fig. 1, the waist of the beam is inversely proportional to the coherence width when the harmonic potential depth is fixed. The NPMC can be well understood as follows: as stated in the principle of laser, the divergence intensity of the beam during free space propagation is inversely proportional to the waist and coherence width. By contrast, the harmonic potential can play a converging role similar to a lens.33^,34 So, when the harmonic potential’s convergence intensity cancels out the beam’s divergence propagating in free space, the beam will propagate unchanged.

Figure 1.Variation of coherence width versus the waist of AOCS under the NPMC. The units for $w$ and $δ$ are mm and the unit for $b$ is ${mm}^{- 3}$ .

To prove the AOCS’s nondiffraction and the sinusoidal oscillation properties like the quantum coherent state when the NPMC is satisfied, we draw the propagation diagrams of the AOCS with different linear chirp parameters in Fig. 2 and depict the motion of the beam centroid. As shown in Figs. 2(a)–2(c), when the NPMC is satisfied, the beam’s intensity and degree of coherence will not diffuse with the propagation, and the light intensity will oscillate sinusoidally around the center of the potential well when the linear chirp is not 0; moreover, as shown in Fig. 2(d), the center of mass of the beam is consistent with the theoretical prediction [Eqs. (6) and (7)].

Figure 2.Propagation properties of AOCS in the harmonic potential with $b = 10^{7} {mm}^{- 3}$ when the NPMC is met. (a) The intensity of AOCS with $c = 0$ . (b) The intensity of AOCS with $c = 3000 {mm}^{- 1}$ . (c) The coherence degree of AOCS with different $c$ is the same. (d) The center of mass of AOCS. The unit of $x$ -axis is mm and the unit of $z$ -axis is m.

Due to the similar mathematical form of the Helmholtz and the Schrödinger equations, so this means the quantum mixed state corresponding to the AOCS should also exhibit nondiffusion and sinusoidal oscillation characteristics as it evolves in the harmonic potential and can be seen as a steady state (see Appendix B for detailed definitions). This also means that controlling the coherence of quantum states can achieve certain unique evolution characteristics of quantum states and indicate that the coherence of quantum states is also wave propagated. From this perspective, the NPMC we proposed has rich physical significance and potential applications in classical optics and quantum mechanics. Moreover, as the coherence can be continuously changed, compared with the completely coherent eigenstates for harmonic oscillator potentials with discrete eigenvalues,35^–37 the partially coherent AOCSs have continuously changing characteristic parameters [eigenstates: $| δ, c ⟩ = W (x_{1}, x_{2}, δ, w, c)$ ] and may have potential applications in laser scanning technology,30 optical communication, particle manipulation, and microscopy.38

Next, we study the transmission of AOCS in the harmonic potential when the NPMC is unsatisfied. The unsatisfied condition is divided into two cases: the depth of the harmonic potential is greater than or less than the matching condition. The propagation of the AOCS in the harmonic potential in the first cases is first shown in Fig. 3.

Figure 3.Propagation properties of AOCS in the harmonic potential with $b = 2 \times 10^{7} {mm}^{- 3}$ when the NPMC is not met. (a) The intensity of AOCS with $c = 0$ . (b) The intensity of AOCS with $c = 3000 {mm}^{- 1}$ . (c) The coherence degree of AOCS with different $c$ is the same. (d) The center of mass of AOCS. The unit of $x$ -axis is mm and the unit of $z$ -axis is m.

As shown in Figs. 3(a) and 3(b), when the depth of the harmonic potential is greater than the matching condition, the intensity of the AOCS will periodically converge to a maximum and then diverge to the initial state. In addition, the coherence width of AOCS also shows a similar trend: it first decreases to a minimum and then increases to the initial state [see Fig. 3(c)]. However, the motion of the beam centroid is not affected by the matching conditions; it is still a sinusoidal oscillation, and the oscillation frequency and amplitude are consistent with Eq. (7) [see Fig. 3(d)].

Furthermore, we study the second case, as shown in Figs. 4(a) and 4(b), contrary to situations $b$ greater than NPMC, when the depth of the harmonic potential is less than the matching condition, the intensity of the AOCS will periodically diverge to a minimum and then converge to the initial state. The coherence width of AOCS also has the opposite performance: it first increases to a maximum and then decreases to the initial state [see Fig. 4(c)]. However, the motion of the beam centroid is also not affected by the matching conditions and behaves as described in Eq. (7) [see Fig. 4(d)].

Figure 4.Propagation properties of AOCS in the harmonic potential with $b = 5 \times 10^{6} {mm}^{- 3}$ when the NPMC is not met. (a) The intensity of AOCS with $c = 0$ . (b) The intensity of AOCS with $c = 3000 {mm}^{- 1}$ . (c) The coherence degree of AOCS with different $c$ is the same. (d) The center of mass of AOCS. The unit of $x$ -axis is mm and the unit of $z$ -axis is m.

When the NPMC is satisfied, the partially coherent mixed state represented by the AOCS is a steady state, so its position uncertainty $Δ x$ , momentum uncertainty $Δ p$ , and the product of position uncertainty and momentum uncertainty $Δ x Δ p$ are all fixed values [see Eqs. (8) and (9)]. However, when the NPMC is not satisfied, the beam profile and coherence of the AOCS will change periodically with the propagation, which means that $Δ x$ , $Δ p$ , and $Δ x Δ p$ of the mixed state will also change periodically. Taking the value of $Δ x = Δ x_{s}$ , $Δ p = Δ p_{s}$ , and $Δ x Δ p = Δ x_{s} Δ p_{s}$ as a unit under steady state condition (NPMC) [see Eqs. (8) and (9)], we normalize the evolution of the position uncertainty, momentum uncertainty, and uncertainty principle with propagation under two unmatched conditions.

As shown in Fig. 5(a), when the depth of the harmonic potential is larger than the NPMC, the $\frac{Δ x}{Δ x_{s}}$ will periodically decrease to the minimum and then increase to the steady state value. At the same time, the $\frac{Δ p}{Δ p_{s}}$ will periodically increase to the maximum and then decrease to the steady state value. The $\frac{Δ x Δ p}{Δ x_{s} Δ p_{s}}$ will also periodically increase to the maximum and then decrease to the steady state value but its frequency is twice that of $\frac{Δ x}{Δ x_{s}}$ and $\frac{Δ p}{Δ p_{s}}$ . Although the depth of the harmonic potential is smaller than the NPMC, the evolution of $\frac{Δ x}{Δ x_{s}}$ and $\frac{Δ p}{Δ p_{s}}$ is opposite to that when the depth is larger than the NPMC. However, the change of the $\frac{Δ x Δ p}{Δ x_{s} Δ p_{s}}$ is consistent in two cases [see Fig. 5(b)]. This means that the value of the uncertainty principle satisfied by the mixed state is the smallest only when it is the steady state (the NPMC is satisfied).

Figure 5.Normalized uncertainty of AOCS as functions of propagation distance $z$ (unit: m) when the NPMC is not met. (a) $b = 2 \times 10^{7} {mm}^{- 3}$ and (b) $b = 5 \times 10^{6} {mm}^{- 3}$ .

4 Conclusion

To sum up, in this paper, focusing on the coherence of the light field, we propose a partially coherent beam (AOCS) that can nondiffractively propagate with sinusoidal oscillation like a coherent state in the harmonic potential and give the corresponding NPMC, analytical propagation formula, and numerical simulation method. Moreover, the propagation characteristics and uncertainty of the beam when the NPMC is not satisfied are studied. Due to the mathematical similarity between the paraxial Helmholtz and the Schrödinger equations, the cross-spectral density function of AOCS mathematically corresponds to the density matrix of a quantum mixed state. Hence, the transmission of cross-spectral density function of AOCS in the harmonic potential corresponds to the evolution of the density matrix of the mixed state in the same potential. Therefore, the transmission dynamics properties of AOCS can be directly extended to the evolution dynamics properties of mixed states. This correspondence suggests potential applications for regulating probability wave evolution through coherence control. Although our research is based on a one-dimensional paraxial wave equation, when the polarization of the beam is not considered, the conclusions can be directly extended to two dimensions using the separation of variables method. When the beam has a nonuniform polarization state (e.g., radial polarization and angular polarization), a vector cross-spectral density matrix is required to describe its transmission characteristics.

5 Appendix A: The Proof of NPMC

Assuming that the analytical propagation formula of AOCS has the expression described in Eq. (6), then there is $\frac{\partial^{2} W}{\partial x_{1}^{2}} - \frac{\partial^{2} W}{\partial x_{2}^{2}} = [(\frac{4}{w^{4}} + \frac{8}{w^{2} δ^{2}}) (x_{1}^{2} - x_{2}^{2}) + (\frac{8}{w^{4}} + \frac{16}{w^{2} δ^{2}}) (x_{1} - x_{2}) \cdot a \cdot \sin (d z) + \frac{- 4 i c [x_{1} + x_{2} + 2 a \cdot \sin (d z)] \cos (d z)}{w^{2}}] W, i \frac{\partial W}{\partial z} = i [- i c (x_{1} - x_{2}) d \sin (d z) + \frac{- 2 [x_{1} + x_{2} + 2 a \cdot \sin (d z)] a d \cdot \cos (d z)}{w^{2}}] W .$ (10)

If Eq. (2) is to be satisfied, then there are $\frac{1}{2 k} (\frac{4}{w^{4}} + \frac{8}{w^{2} δ^{2}}) = b, d^{2} = \frac{2 b}{k}, a = \frac{- c}{d k} .$ (11)

Hence, when the NPMC is met, the AOCS will nondiffractively propagate with sinusoidal oscillation in the harmonic potential, and its analytical propagation formula is Eq. (6).

6 Appendix B: The Mixed State Corresponding to AOCS

The Hamiltonian of one particle in harmonic potential can be expressed as $\begin{matrix} H_{a} = \frac{P_{x}^{2}}{2 m} + \frac{1}{2} m ω_{a}^{2} x^{2} \end{matrix},$ (12)where $P_{x} = - i ℏ \frac{\partial}{\partial x}$ . The expression for its corresponding vacuum state (ground state) in coordinate representation is35 $| 0 ⟩ = \exp (\frac{- β^{2} x^{2}}{2}) = \exp (\frac{- x^{2}}{w_{1}^{2}}),$ (13)where $β = \sqrt{\frac{m ω_{a}}{ℏ}}$ and $w_{1} = \sqrt{\frac{2 ℏ}{m ω_{a}}}$ . The translation operator acting on a vacuum state can yield a coherent state, namely35 $| α ⟩ = \exp (α a^{+} - α^{*} a) | 0 ⟩ .$ (14)

From $a^{+} = \sqrt{\frac{1}{2 m ℏ ω_{a}}} (m ω_{a} x - i P_{x})$ , $a = \sqrt{\frac{1}{2 m ℏ ω_{a}}} (m ω_{a} x + i P_{x})$ , it can be inferred that $α a^{+} - α^{*} a = (α - α^{*}) \sqrt{\frac{m ω_{a}}{2 ℏ}} x + (- i α - i α^{*}) \sqrt{\frac{1}{2 m ℏ ω_{a}}} P_{x} .$ (15)

When $α = i \frac{v}{2}$ , there is $| i \frac{v}{2} ⟩ = \exp (i \frac{v}{2} a^{+} + i \frac{v}{2} a) | 0 ⟩ = φ_{c} (v, x), φ_{c} (v, x) = \exp (- \frac{x^{2}}{w_{1}^{2}}) \exp (i v \sqrt{\frac{m ω_{a}}{2 ℏ}} x),$ (16)where $φ_{c} (v, x)$ is the mathematical expression of the coherent state of the harmonic oscillator ( $ω_{a}$ ) in the coordinate representation. The cross-spectral density function of AOCS can also be expressed as $W (x_{1}, x_{2}, 0) = \exp (- \frac{x_{1}^{2} + x_{2}^{2}}{w_{1}^{2}}) \times \exp [i c (x_{1} - x_{2})] \times \exp [(\frac{x_{1} - x_{2}}{δ})^{2}] = \int g (v) \cdot κ^{*} (x_{1}, v) \cdot κ (x_{2}, v) d v, κ (x_{j}, v) = \exp (- \frac{x_{j}^{2}}{w_{1}^{2}}) \exp [i (v + c) x_{j}], g (v) = \frac{δ}{2 \sqrt{π}} \exp (- \frac{δ^{2} v^{2}}{4}) .$ (17)

It is obvious that $κ (x, v)$ and $φ_{c} (v, x)$ have the same mathematical expression. So, it is clear that the AOCS corresponds to a mixed state obtained by superimposing coherent states incoherently,39 and its density matrix can be expressed as follows: $ρ_{d} = \frac{δ}{2 \sqrt{π}} \sqrt{\frac{m ω_{a}}{2 ℏ}} \int \exp (- \frac{m ω_{a} δ^{2} v^{2}}{8 ℏ}) \times φ_{c}^{*} (v + c, x_{1}) φ_{c} (v + c, x_{2}) d v = \exp (- \frac{x_{1}^{2} + x_{2}^{2}}{w_{1}^{2}}) \times \exp [i c \sqrt{\frac{m ω_{a}}{2 ℏ}} (x_{1} - x_{2})] \times \exp [(\frac{x_{1} - x_{2}}{δ})^{2}] .$ (18)

So far, we have proven that the cross-spectral density function of AOCS corresponds mathematically to the density matrix of the mixed state obtained by superimposing coherent states incoherently. Next, we will discuss the dynamic evolution characteristics of this mixed state.

The evolution of this mixed state in a new harmonic potential $(H_{b} = \frac{P_{x}^{2}}{2 m} + \frac{1}{2} m ω_{b}^{2} x^{2})$ is described by the following equation: $i ℏ \frac{\partial}{\partial t} ρ_{d} = [H_{b}, ρ_{d}] = H_{b} ρ_{d} - ρ_{d} H_{b} = {\frac{P_{x_{1}}^{2} - P_{x_{2}}^{2}}{2 m} + \frac{1}{2} m ω_{b}^{2} (x_{1}^{2} - x_{2}^{2})} ρ_{d},$ (19)where $P_{x_{1}} = - i ℏ \frac{\partial}{\partial x_{1}}$ and $P_{x_{2}} = - i ℏ \frac{\partial}{\partial x_{2}}$ . It is clear that Eqs. (2) and (19) have similar mathematical forms so that the mixed state will have similar dynamic evolution characteristics as AOCS, and corresponding to the NPMC, as shown in Eq. (5), when the following condition is met, the mixed state will not diffuse with evolution, $\frac{ℏ}{2 m} (\frac{4}{w_{1}^{4}} + \frac{8}{w_{1}^{2} δ^{2}}) = \frac{m ω_{b}^{2}}{2 ℏ}, \frac{m ω_{a}^{2}}{2 ℏ} + \frac{2 ω_{a}}{δ^{2}} = \frac{m ω_{b}^{2}}{2 ℏ} .$ (20)

Therefore, when the matching conditions, as shown in Eq. (20), are met, the analytical evolution formula for the mixed state, as well as the coordinate uncertainty and momentum uncertainty, is $ρ_{d} (x_{1}, x_{2}, t) = \exp {- \frac{[x_{1} + a_{q} \sin (d_{q} t)]^{2} + [x_{2} + a_{q} \sin (d_{q} t)]^{2}}{w_{1}^{2}}} \times \exp [i c_{q} \cos (d_{q} t) (x_{1} - x_{2})] \exp [- \frac{(x_{1} - x_{2})^{2}}{δ^{2}}],$ (21) ${\begin{matrix} ⟨ x ⟩ = \frac{\int x \times ρ_{d} (x, x, t) d x}{\int ρ_{d} (x, x, t) d x} = - a_{q} \cdot \sin (d_{q} t) \\ S_{q} (p_{x_{1}}, p_{x_{2}}, t) = \int \int ρ_{d} (x_{1}, x_{2}, t) \exp [\frac{i}{ℏ} (p_{x_{1}} x_{1} - p_{x_{2}} x_{2})] d x_{1} d x_{2} \\ ⟨ p_{x} ⟩ = \frac{\int p_{x} \times S_{q} (p_{x}, p_{x}, t) d p_{x}}{\int S_{q} (p_{x}, p_{x}, t) d p_{x}} = - ℏ c_{q} \cos (d_{q} t), \end{matrix}$ (22) ${\begin{matrix} ⟨ x^{2} ⟩ = \frac{\int x^{2} \times ρ_{d} (x, x, t) d x}{\int ρ_{d} (x, x, t) d x} = \frac{4 a_{q}^{2} \sin^{2} (d_{q} t) + w_{1}^{2}}{4} \\ Δ x = ⟨ x^{2} ⟩ - ⟨ x ⟩^{2} = \frac{w_{1}^{2}}{4} \\ ⟨ p_{x}^{2} ⟩ = \frac{\int p_{x}^{2} \times S_{q} (p_{x}, p_{x}, t) d p_{x}}{\int S_{q} (p_{x}, p_{x}, t) d p_{x}} = \frac{4 c_{q}^{2} \cos^{2} (d_{q} t)}{4} ℏ^{2} + \frac{(δ^{2} + 2 w_{1}^{2})}{w_{1}^{2} δ^{2}} ℏ^{2} \\ Δ p = ⟨ p_{x}^{2} ⟩ - ⟨ p_{x} ⟩^{2} = \frac{δ^{2} + 2 w_{1}^{2}}{w_{1}^{2} δ^{2}} ℏ^{2}, \end{matrix}$ (23) $Δ p Δ x = \frac{ℏ^{2}}{4} \frac{δ^{2} + 2 w_{1}^{2}}{δ^{2}} \geq \frac{ℏ^{2}}{4},$ (24)where $d_{q} = ω_{b}$ , $a_{q} = \frac{- c_{p} ℏ}{ω_{b} m}$ , $c_{q} = c \sqrt{\frac{m ω_{a}}{2 ℏ}}$ , and $S_{q}$ is the density matrix of the mixed state in momentum space. From Eqs. (22)–(24), it is clear that the uncertainty of this mixed state is similar to that of AOCS.

In 1926, Schrödinger defined the harmonic oscillation Gaussian wave packet as a coherent state that does not diffuse in the harmonic potential. Considering the original definition of coherent states, when Eq. (20) is satisfied, the mixed state corresponding to AOCS is also a harmonic oscillation Gaussian wave packet that does not diffuse over time, so it should be regarded as a generalized coherent state. In addition, because it is nondiffusive (wave shape remains unchanged), it should also be considered a steady state of the harmonic potential.

For matter waves, the first-order spatial correlation function can be expressed as40^,41 ${\begin{matrix} G^{1} (x_{1}, x_{2}, t) = ⟨ ψ^{+} (x_{1}, t) ψ (x_{2}, t) ⟩ = ρ (x_{1}, x_{2}, t) \\ g^{1} (x_{1}, x_{2}, t) = \frac{G^{1} (x_{1}, x_{2}, t)}{\sqrt{G^{1} (x_{1}, x_{1}, t)} \sqrt{G^{1} (x_{2}, x_{2}, t)}}, \end{matrix}$ (25)where $ρ (x_{1}, x_{2}, t)$ is the density matrix of the quantum state. It is obvious that Eqs. (4) and (25) have the same mathematical form. Hence, the evolution of the first-order spatial correlation function of the mixed state over time is also consistent with the variation of AOCS spatial coherence with propagation.