PhD Student : Publish Paper with Wife?

Nobody will notice or care, unless you share a last name with your wife, in which case the strongest reaction is likely to be, "aw how cute, a husband and wife published a paper together."

I know several instances where spouses/couples are co-authors, though often it is not obvious to an outsider if the co-authors are spouses. (There are also many examples of parent-child and sibling co-authors, so it is not obvious even if the spouses have the same last name.) Thus it is not a strange thing to happen, particular if non-academic co-authors are common in your field.

There is only one case where I even thought one spouse may possibly be going out of their way to bring up the research profile of the other (academic) spouse, but that is due to some specifics of the situation, and I don't regard them poorly because of this (which may not be true anyway), just am unclear on one of the spouse's contributions. In your case, presumably your wife has little to gain career-wise from coauthorship, so there is not even reason for people to think this.

If you personally feel weird about it, you can always explain in the paper what each author's role was. (And, of course, you should never deny a deserving person co-authorship, especially someone you (hopefully) like.)

In terms of relationship with the external world, it need not be any problem. If you begin collaborating scientifically on a regular basis, however, it is likely to affect your relationship, by making you colleagues as well as spouses.

For some couples, this can be a good thing, as the shared interest and partnership can add a new dimension to the relationship.

For other couples, the stresses of scientific collaboration outweigh the benefits. Some people also find that they lose a place of refuge from work, as their work life and home life become more entangled.

Bottom line: have fun and publish together if you want, but make sure to have an explicit discussion about how you want to relate to collaboration as a couple, and revisit it from time to time if necessary.