Quantum coherence and coherence length of correlated Gaussian states

Abstract

The superposition principle is at the heart of quantum mechanical interference phenomena. For instance, Quantum coherence, developed by all systems that develop quantum correlations, is deeply rooted in the superposition principle. On the other hand, the coherence length in position and momentum has been shown to encode the interference extension over which phase space correlations are developed. In this work, using partially coherent correlated Gaussian states, we show how the quantum coherence measured by relative entropy can be related with the coherence length.

Appendix A: Parameters of the density matrices and second moments

The parameters of the density matrix in position and in momentum are given by

$$\begin{aligned} A_{1} & = {} \frac{m^{2}}{8\hbar ^{2}t^{2}\sigma _{0}^{2}B^{2}},\;\;A_{2}=\frac{m^{2}}{8\hbar ^{2}t^{2}\ell _{0}^{2}B^{2}},\end{aligned}$$

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$$\begin{aligned} A_{3} & = {} \frac{m}{4 \hbar t\sigma _{0}^{2}B^{2}}\left( \frac{1}{2\sigma _{0}^{2}}+\frac{1}{\ell _{0}^{2}}\right) \nonumber \\{} &\quad {} +\frac{m\gamma }{8\hbar t \sigma _{0}^{2}B^{2}}\left( \frac{m}{\hbar t}+\frac{\gamma }{\sigma _{0}^{2}}\right) ,\end{aligned}$$

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$$\begin{aligned} B^{2} & = {} \frac{1}{4\sigma _{0}^{4}}+\frac{1}{2\sigma _{0}^{2}\ell _{0}^{2}}+\left( \frac{m}{2\hbar t}+\frac{\gamma }{2\sigma _{0}^{2}}\right) ^{2},\end{aligned}$$

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$$\begin{aligned} C_1 & = {} \frac{A_1}{4\hbar ^{2}(A_{1}^{2}+2A_1A_2+A_{3}^{2})},C_2=\frac{A_2}{A_{1}}C_1, \end{aligned}$$

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and

$$\begin{aligned} C_3=\frac{A_3}{A_{1}}C_1. \end{aligned}$$

(26)

The dimensionless second moments are defined by

$$\begin{aligned} \sigma _{11}= & {} \frac{\langle \hat{x}^{2}\rangle }{\sigma _{0}^{2}}=\frac{1}{\sigma _{0}^{2}}Tr[\rho (x,x^{\prime },t)\hat{x}^{2}],\end{aligned}$$

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$$\begin{aligned} \sigma _{22}= & {} \frac{\sigma _{0}^{2}}{\hbar ^{2}} \langle \hat{p}^{2}\rangle =\frac{\sigma _{0}^{2}}{\hbar ^{2}}Tr[\rho (p,p^{\prime },t)\hat{p}^{2}], \end{aligned}$$

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and

$$\begin{aligned} \sigma _{12}=\frac{1}{2\hbar } \langle (\hat{x}\hat{p}+\hat{p}\hat{x})\rangle =\frac{1}{2\hbar }Tr[\rho (x,x^{\prime },t)(\hat{x}\hat{p}+\hat{p}\hat{x})]. \end{aligned}$$

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