We solve for the statistics of the first detection of a quantum system in a particular desired state, when the system is subject to a projective measurement at independent identically distributed random time intervals. We present formulas for the probability of detection in the $n\mathrm{th}$ attempt. We calculate as well the mean and mean square, both of the number of the first successful detection attempt and the time until first detection. We present explicit results for a particle initially localized at a site on a ring of size $L$, probed at some arbitrary given site, in the case when the detection intervals are distributed exponentially. We prove that, for all interval distributions and finite-dimensional Hamiltonians, the mean detection time is equal to the mean attempt number times the mean time interval between attempts. We further prove that for the return problem when the initial and target state are identical, the total detection probability is unity and the mean attempts until detection is an integer, which is the size of the Hilbert space (symmetrized about the target state). We study an interpolation between the fixed time interval case to an exponential distribution of time intervals via the Gamma distribution with constant mean and varying width. The mean arrival time as a function of the mean interval changes qualitatively as we tune the interarrival time distribution from very narrow ($\delta$ peaked) to exponential, as resonances are wiped out by the randomness of the sampling.

中文翻译：

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随机探测下量子态的首次检测时间

当系统在独立的均匀分布的随机时间间隔内进行投射测量时，我们将解决在特定期望状态下首次检测量子系统的统计信息。我们提出了用于检测概率的公式$ñ日$试图。我们还计算了第一次成功检测尝试的次数和直到第一次检测的时间的均值和均方。我们为最初定位在一个大小为一个环的位置上的粒子提供了明确的结果$\mathrm{大号}$，在检测间隔呈指数分布的情况下，在任意给定位置进行探测。我们证明，对于所有间隔分布和有限维哈密顿量，平均检测时间等于平均尝试次数乘以两次尝试之间的平均时间间隔。我们进一步证明，对于初始状态和目标状态相同时的返回问题，总检测概率为1，并且直到检测为止的平均尝试次数为整数，该整数是希尔伯特空间的大小（对称于目标状态）。我们研究了固定时间间隔情况与时间间隔的指数分布之间的插值，该插值是通过均值恒定且宽度变化的Gamma分布进行的。$\delta$ 峰值）到指数级，因为采样的随机性消除了共振。