arXiv quant-ph/05110462005-11quant-ph

精确可解的含时耦合量子态约化模型

Exactly solvable quantum state reduction models with time-dependent coupling

Dorje C. Brody, Irene C. Constantinou, James D. C. Dear, Lane P. Hughston

备注：: 27 pages
发表信息：: Journal of Physics A39, 11029-11051 (2006)
DOI：: 10.1088/0305-4470/39/35/006

摘要：

得到了含时耦合的基于能量的随机薛定谔方程的封闭解。解是代数性质的，并且直接用独立的随机数据表示。数据包括（i）具有分布P（H=E_i）= pi_i的随机变量H，其中pi_i是从初始状态到具有能量E_i的Luders状态的转移概率;和（ii）独立的P-布朗运动，其中P是与还原过程的动力学相关的物理概率度量。当耦合是时间无关的，它是已知的，状态减少发生渐近-也就是说，在一个无限的时间范围。在依赖于时间的耦合的情况下，我们表明，如果耦合的幅度足够迅速地减少，那么在动力学下的能量方差将减少，但状态不需要达到能量本征态。这种情况对应于能量的"部分“或"不完全”测量的情况。我们还构建了一个模型的例子，其中相反的情况下，在一个有限的时间段的通过后，实现了完整的状态还原。

英文摘要：

A closed-form solution to the energy-based stochastic Schrodinger equation with a time-dependent coupling is obtained. The solution is algebraic in character, and is expressed directly in terms of independent random data. The data consist of (i) a random variable H which has the distribution P(H=E_i) = pi_i, where pi_i is the transition probability from the initial state to the Luders state with energy E_i; and (ii) an independent P-Brownian motion, where P is the physical probability measure associated with the dynamics of the reduction process. When the coupling is time-independent, it is known that state reduction occurs asymptotically--that is to say, over an infinite time horizon. In the case of a time-dependent coupling, we show that if the magnitude of the coupling decreases sufficiently rapidly, then the energy variance will be reduced under the dynamics, but the state need not reach an energy eigenstate. This situation corresponds to the case of a ``partial'' or ``incomplete'' measurement of the energy. We also construct an example of a model where the opposite situation prevails, in which complete state reduction is achieved after the passage of a finite period of time.

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