Tensor product notation as a power I chance upon a notation while reading a paper which I do not quite understand.
Suppose that $\hat{J}$ is a operator in a tensor product of two N$ dimensional Hilbert space. Explicitly, it is given by 


$\hat{J} = \frac{1}{\sqrt{2}}(\hat{I}^{\otimes N} + i\hat{\sigma}_{x}^{\otimes N})$


How should the two terms enclosed in the parentheses be understood? 

In general, when dealing with any product-like operation $\star$, you should understand $a^{\star n}$ as $$\underbrace{a \star a \star \dots \star a}_{n \text { times}}.$$ This might be written as simply $a^n$ in case where it's completely obvious which product is meant, but putting $\star$ in the exponent specifies which product it is.
I personally encounter this notation most often in graph theory, where (for example) $G^{\boxtimes n}$ is the $n$-fold strong product of graph $G$ with itself. But it shows up in many places, and tensor products are a common example.
