In reading this article, I come across this paragraph:
The pink marked place is where I can't understand, why can we use direct product of the former but not the later?
This is may be a basic quantum mechanics question, but I don't remember when can we use the direct product of the subsystem states to represent state of the whole system.
I can only remembered that when $H_A=H_B+H_C$, one can write $|\psi_A>=|\psi_B>\otimes |\psi_C>$. I don't know how this is related to the above situation.
My basic understanding is this: if the system $B$ has a different phase of that of $C$, then $B$ and $C$ will interact through josephson effect. The particle number will flow from $B$ to $C$(or $C$ to $B$), depending on their relative phase. However, if their phase is same, they have no interaction, thus the whole system can be considered as two independent part.
However, why can we choose a $N_B$ that the phase $\phi_B=\phi_C$? Even if the initial phase is not same, and the Josephson effect happens; after sometime, their phase will be the same, because the current cannot flow forever. At this time, we count the number of B and C, why can't we write A as a direct product of them.