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Assume for a moment that the US has switched to a nation-wide proportional voting system for Senate elections. During the election the Democrats get 45%, the Republicans get 45% and Independents get 10%. Now the question becomes - which candidate becomes responsible for which state district?
In the current FPTP system it's extremely simple - whoever wins the district gets the assignment. But how would it work in proportional voting?
In most systems with proportional representation, each party has a numbered list of candidates. The list is determined by the party, usually through a party-internal process, and published before the election.
The seats gained by a party are then filled with the people from the party list in the stated order. For example, when Party A got 25 seats, they go to the 25 first people they put on their list. The people on these seats represent the party as a whole, not any particular district.
One property of this system which you might consider a disadvantage is that it puts a lot of power into the hands of the parties. The top list-candidates of a popular party are almost guaranteed to get into the parliament, no matter how unpopular they might be among party-sympathizers who are not party-members.
Another problem is that list-candidates don't represent any particular district.
That's why there are many variations of this system which give some power to nominate specific people back to the voters.
In German parliament elections, for example, people have two votes. One for the proportional representation and one for the local candidate. When a local candidate wins a district, that candidate must gain a seat owned by their party, no matter if they are on the party list or not. There are twice as many seats as voting districts. So only half the seats represent districts while the other half just represent the party as a whole. This system is well explained in the video Mixed-Member Proportional Representation by CGP Grey.
Other countries simplify this by unifying those two votes. Your vote for your local candidate is also a vote for the national party list. So when your local candidate loses, your vote still counts for the representation of the national party of that candidate.
Now the question becomes - which candidate becomes responsible for which state district?
They wouldn't. You specified:
Assume for a moment that the US has switched to a nation-wide proportional voting system for Senate elections.
It wouldn't have state districts. It would have a single nationwide district. Presumably that would be why people would switch, because they think that the state districts give undue power to small states relative to large states (by population).
If you don't insist on the national aspect, an alternative would be to have state districts where both Senators were up at the same time. In many states, this would result in one Senator from each major party. In a few states (e.g. Utah, Hawaii, etc.), both Senators might be from the same party. In some states, this might give independents a serious chance.
Changing Senate voting to be proportional would require a constitutional amendment. That's part of why it's more common to talk about making the House proportional, since that's possible with only legal changes. Given a constitutional amendment, they could change the system however they wanted. Presumably the new system would have the same kind of tradeoffs as the old system.
It's worth noting that there are single office systems with ranked voting. They aren't proportional, but they are ranked rather that first-past-the-post. IRV, Condorcet-compliant, even Range, Approval, etc.
You simply make larger districts, usually with more senators per district.
The excessive version is that the whole country is one single district, so with 100 fictional seats you'd get 45 democrats, 45 republicans and 10 independents for the single district "US". Usually a less excessive middle ground is chosen.
Another system is to have a couple of extra non-district party slots in a system where you chose a candidate and a party during the vote, and the non-district slots get allocated as necessary to "correct" rounding errors of the district votes so the party distribution in the senate matches the party votes of the nation. Example: You have 90 seats bound to districts and 10 non-district seats. As a result of the vote 45 district seats go to Republicans, 37 district seats go to Democrats and 8 seats go to Independents. Because the vote distribution is 45%-45%-10%, of the remaining non-district seats 8 go to Democrats and 2 to Independents, giving the parties 45, 45, and 10, seats respectively.
