N维空间量子态的层析重建
Tomographic reconstruction of quantum states in N spatial dimensions
摘要：
大多数量子层析成像方法只能用于一维问题。我们展示了如何通过简单的逆Radon变换来推断非相对论N维谐振子系统的量子态。该方法同样适用于寻找不同谐振子势中几个可区分粒子的联合量子态。该过程的一个要求是N个谐波势的角频率是不可忽略的。我们讨论了什么样的信息可以找到，如果不满足的要求的不可见性，也在什么条件下，可以从有限时间测量重建的状态。作为N维量子态重构的另一个例子，我们考虑了具有周期边界条件的N维自由粒子和N维盒子中的粒子这两种相关情况，其中我们发现一维系统具有相似的不变性和有限递归时间条件。
英文摘要：
Most quantum tomographic methods can only be used for one-dimensional problems. We show how to infer the quantum state of a non-relativistic N-dimensional harmonic oscillator system by simple inverse Radon transforms. The procedure is equally applicable to finding the joint quantum state of several distinguishable particles in different harmonic oscillator potentials. A requirement of the procedure is that the angular frequencies of the N harmonic potentials are incommensurable. We discuss what kind of information can be found if the requirement of incommensurability is not fulfilled and also under what conditions the state can be reconstructed from finite time measurements. As a further example of quantum state reconstruction in N dimensions we consider the two related cases of an N-dimensional free particle with periodic boundary conditions and a particle in an N-dimensional box, where we find a similar condition of incommensurability and finite recurrence time for the one-dimensional system.
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