推导二能级原子系综自旋压缩的一般理论, 给出了计算自旋压缩系数的普遍方法. 作为例子, 在推广的单轴扭曲模型中研究失谐量δ和初始自旋相干态|θ0,Φ0>的极角θ0对自旋压缩效应的影响. 结果表明, 对任意δ和Φ0, 当Φ0=Π/2时, 回到理想单轴扭曲模型的结果. 但当θ0略微偏离Π/2时, 涨落(V_)min≈N2/3, 大于理想情况下的N1/3. 最大压缩时间满足标度率N-5/6. 得到了(V_)min和tmin的解析表达式, 结果与数值模拟符合.
We investigate the dependence of spin squeezing on the polar angle of the initial coherent spin state |θ 0 , ϕ 0 ⟩ in a generalized one-axis twisting model, where the detuning δ is taken into account. We show explicitly that regardless of δ and ϕ 0 , previous results of the ideal one-axis twisting are recovered as long as θ 0 =π/2. For a small departure of θ 0 from π/2, however, the achievable variance ( V − ) min ∼ N 2/3 , which is larger than the ideal case N 1/3 . We also find that the maximal squeezing time t min scales as N −5/6 . Analytic expressions of ( V − ) min and t min are presented and they agree with numerical simulations.
由于极限方法上的不同,存在着波函数在经典极限下可以描述单个点粒子和只能描述系综两种相左的结论.本文比较了不同极限方法的优缺点,并采用n→∞,h→0但保持E(n,h)为能量实际测量值的综合极限.在这种极限下不仅证明一个波函数只能统计性地描述系综,而且能说出这系综是由具有何种力学性质的粒子所构成.从计算中还可看出,造成量子力学不能描述单个粒子的原因不是来自运动方程,而是来自状态的定义本身.
How does the classical notion of ``phase'' apply to a quantum harmonic oscillator H=1/2(q${\mathrm{^}}^{2}$+p${\mathrm{^}}^{2}$), [q^,p^]=i\ensuremath{\Elzxh}, which cannot have sharp position and momentum? A quantum state \ensuremath{\rho}^ can be assigned a definite classical phase only if it is a large-amplitude localized state. Our only demand, therefore, on a (Hermitian) phase operator \ensuremath{\varphi}^ is that the phase distribution P(\ensuremath{\Phi})= Tr{\ensuremath{\delta}(\ensuremath{\varphi}^-\ensuremath{\Phi})\ensuremath{\rho}^} attribute the correct sharp phase to any such ``classical phase'' state. This requires that the Weyl symbol [\ensuremath{\varphi}^
]w(q,p) of \ensuremath{\varphi}^ tend to \ensuremath{\theta} mod2\ensuremath{\pi} as r\ensuremath{\rightarrow}\ensuremath{\infty}, where \ensuremath{\theta}=${\mathrm{tan}}^{\mathrm{\ensuremath{-}}1}$(p/q) and r=(
q2+
p2)1/2. There are infinitely many such phase operators. Each is expressible as \ensuremath{\varphi}^=[${\mathrm{tan}}^{\mathrm{\ensuremath{-}}1}$(p^/q^)${]}_{\mathrm{\ensuremath{\Omega}}}$, where \ensuremath{\Omega} specifies an ordering rule for q^ and p^. The commutator -i[H^,\ensuremath{\varphi}^]=1-2\ensuremath{\pi}[\ensuremath{\delta}(${\mathrm{tan}}^{\mathrm{\ensuremath{-}}1}$p^/q^)${]}_{\mathrm{\ensuremath{\Omega}}}$ corresponds to the Poisson bracket {H,${\mathrm{\ensuremath{\varphi}}}_{\mathrm{cl}}$${\mathrm{}}}_{\mathrm{}\mathrm{PB}}$=1-2\ensuremath{\pi}\ensuremath{\delta}(\ensuremath{\theta}) for the single-valued classical phase ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{cl}}$=\ensuremath{\theta} mod2\ensuremath{\pi}. Phase states \ensuremath{\Gamma}^(\ensuremath{\Phi}) are defined by the condition that their Weyl symbols [\ensuremath{\Gamma}^(\ensuremath{\Phi})
]w(r,\ensuremath{\theta})\ensuremath{\rightarrow}\ensuremath{\delta}(\ensuremath{\theta}-\ensuremath{\Phi}) as r\ensuremath{\rightarrow}\ensuremath{\infty}. If moreover ${\mathrm{\ensuremath{\int}}}_{0}^{2\mathrm{\ensuremath{\pi}}}$d\ensuremath{\Phi}\ensuremath{\Gamma}^(\ensuremath{\Phi})=1^, then \ensuremath{\Gamma}^(\ensuremath{\Phi}) is a phase probability operator measure (POM). In particular, \ensuremath{\delta}(\ensuremath{\varphi}^-\ensuremath{\Phi}) is a phase POM.Normalizable approximate phase states \ensuremath{\Gamma}${\mathrm{^}}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi}) are defined by [\ensuremath{\Gamma}${\mathrm{^}}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi})
]w(r,\ensuremath{\theta}) =2\ensuremath{\pi}${\mathrm{\ensuremath{\varepsilon}}}^{2}$${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\varepsilon}}\mathit{r}}$[\ensuremath{\Gamma}^(\ensuremath{\Phi})
]w(r,\ensuremath{\theta}), \ensuremath{\varepsilon}\ensuremath{\ll}1. Phase states are not, in general, ``pure orthogonal'' in the sense \ensuremath{\Gamma}^(\ensuremath{\Phi})\ensuremath{\Gamma}^(\ensuremath{\Phi}\ensuremath{'})=\ensuremath{\delta}(\ensuremath{\Phi}-\ensuremath{\Phi}\ensuremath{'})\ensuremath{\Gamma}^(\ensuremath{\Phi}), unless they are of the form \ensuremath{\delta}(\ensuremath{\varphi}^-\ensuremath{\Phi}). However, any phase state \ensuremath{\Gamma}${\mathrm{^}}_{1}$(\ensuremath{\Phi}) is trace orthogonal to any phase POM \ensuremath{\Gamma}${\mathrm{^}}_{2}$(\ensuremath{\Phi}), in the sense that Tr{\ensuremath{\Gamma}${\mathrm{^}}_{2}$(\ensuremath{\Phi}\ensuremath{'})\ensuremath{\Gamma}${\mathrm{^}}_{1}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi})}\ensuremath{\rightarrow}\ensuremath{\delta}(\ensuremath{\Phi}-\ensuremath{\Phi}\ensuremath{'}) as \ensuremath{\varepsilon}\ensuremath{\rightarrow}0. This implies that measurement of the POM \ensuremath{\Gamma}${\mathrm{^}}_{2}$(\ensuremath{\Phi}\ensuremath{'}) on the state \ensuremath{\Gamma}${\mathrm{^}}_{1}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi}) yields the outcome \ensuremath{\Phi} with probability 1 as \ensuremath{\varepsilon}\ensuremath{\rightarrow}0. Phase measurements of the first kind are possible in principle; they would allow one to prepare (approximate) phase states and monitor their phase evolution in a quantum (phase) nondemolishing manner. Cases of special interest are the Susskind-Glogower and the Cahill-Glauber ordered phase states and operators. The energy-phase (or number-phase) uncertainty relation is \ensuremath{\Delta}H\ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\ge}0, the lower limit \ensuremath{\Delta}H\ensuremath{\Delta}\ensuremath{\varphi}=0 being realized by pure number states; however, for states whose Wigner functions are localized away from the origin and from the extremities of the (single-valued) phase window (0,2\ensuremath{\pi}), the uncertainty relation is effectively \ensuremath{\Delta}H\ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\ge}1/2. \textcopyright{} 1996 The American Physical Society.
摘要 量子力学与经典物理的范式割裂、逻辑冲突是近代物理学百年核心困境,两大体系底层本体与数学框架无法自洽兼容,始终未能实现全尺度物理规律统一。本文基于 W≡0 全域拓扑极性守恒统一场论 0~第六阶段永久锁档成果,提出 \\ 一体拓扑(全域统一场 Φᵤ)、两相表象(量子 / 经典)\\ 的全新范式,论证量子与经典现象实为同一拓扑本体在不同能标、尺度下的差异化表现,全域统一场 Φᵤ无需额外量子化规则,其高能极限天然呈现量子行为、低能极限自然回归经典稳态。本文精准锚定第六阶段锁档场方程与拓扑不变量,给出量子 - 经典三相临界切换条件,厘清与主流量子 - 经典过渡理论的范式差异,细化体 - 相 - 用三线研究产出与论文绑定关系,刚性划定研究边界与范式不兼容范畴,明确 8 篇论文层级依赖与实验接口传递逻辑,预判高精度实验适配等核心挑战并给出应对方案,构建 “量子基础重构→数学统一证明→前沿疑难消解→体系锁档” 的完整研究路径,旨在彻底消解量子 - 经典鸿沟,为第八阶段判决性实验预言、第九阶段元物理理论闭环奠定标准化理论纲领。 关键词 W≡0 公理;全域拓扑极性守恒;量子 - 经典统一;一体拓扑两相表象;全域统一拓扑场 Φᵤ;拓扑闭合度;量子 - 经典过渡;研究边界规范
In this paper, we explore how to estimate the phase damping parameter γ and the tunneling amplitude parameter Δ from a spin-boson dephasing quantum model by periodical projective measurements. The preparation of initial states is accomplished by performing the period measurements in our scheme. The parameter γ can be always estimated when projective measurement bases are chosen as θ = π /2 and ϕ = 0. Based on the estimated value of γ and the interval information of Δ, we can select another measurement bases ( θ = π /4 and ϕ = π /2) to obtain the estimated value of Δ . A coherent control is indispensable to estimate Δ if γ is in the interval of Δ; whereas the control is not necessary if γ is out of the known interval of Δ . We establish the relation between the optimal period time and the parameter γ or Δ in terms of Fisher information. Although the optimal measurement period cannot be selected beforehand, the aforementioned relation can be utilized to adjust the measurement period to approach the optimal one.