谱隙界与哈密顿-指针相互作用中纯度损失的时间尺度

中文摘要 AI 辅助

我们研究了通过冯·诺依曼型相互作用哈密顿量$\hat H_{\mathrm{int}}=g\\,\hat H\otimes\hat p$与连续变量指针耦合的量子系统的纯度动力学。对于初始为高斯态的指针，相互作用产生依赖于能量的条件平移，其相互重叠由系统哈密顿量的布居谱间隔显式确定。在迹除指针自由度后，我们获得约化密度算符，并推导出时间依赖纯度的精确解析表达式。利用该表达式，我们建立了由布居谱支撑上的最小和最大非零能量间隙控制的双边纯度界。这些界提供了纯度损失的态依赖谱表征，并且对于二能级系统是精确的。我们进一步证明，纯度的短时间下降由初始态的哈密顿量方差控制，即$\mathcal P''(0)=-(g^2/\sigma^2) \operatorname{Var}_{\psi_S}(\hat H)$。此外，推导了一个明确的充分时间尺度，使得纯度在给定容差内接近其渐近值，揭示了标度$t_{\varepsilon}\propto\sigma/(|g|\Delta_{\min})$。最后，以具有等间距谱的等权重$N$能级系统为例说明了一般结果，其渐近纯度为$1/N$。该分析阐明了谱间隔、能量方差、耦合强度和指针宽度在哈密顿条件纯度损失中的不同作用。

英文摘要

We investigate the purity dynamics of a quantum system coupled to a continuous-variable pointer through a von Neumann-type interaction Hamiltonian of the form $\hat H_{\mathrm{int}}=g\,\hat H\otimes\hat p$. For an initially Gaussian pointer state, the interaction generates energy-dependent conditional translations whose mutual overlaps are determined explicitly by the populated spectral separations of the system Hamiltonian. After tracing out the pointer degrees of freedom, we obtain the reduced density operator and derive an exact analytical expression for the time-dependent purity. Using this expression, we establish two-sided purity bounds governed by the minimum and maximum nonzero energy gaps on the populated spectral support. These bounds provide a state-dependent spectral characterization of the loss of purity and become exact for two-level systems. We further show that the short-time decrease of purity is controlled by the Hamiltonian variance of the initial state, with $\mathcal P''(0)=-(g^2/σ^2) \operatorname{Var}_{ψ_S}(\hat H)$. In addition, an explicit sufficient timescale is derived for the purity to approach its asymptotic value within a prescribed tolerance, revealing the scaling $t_{\varepsilon}\proptoσ/(|g|Δ_{\min})$. Finally, the general results are illustrated for an equally weighted $N$-level system with an equally spaced spectrum, for which the asymptotic purity is $1/N$. The analysis clarifies the distinct roles of spectral separation, energy variance, coupling strength, and pointer width in Hamiltonian-conditioned purity loss.

发表机构

• Atatürk University（阿塔图尔克大学）
• Gümüşhane University（居姆沙汉大学）

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