How do soul sleep adherents explain Lk 23:43? Luke 23:43 says:


  And he said to him, “Truly, I say to you, today you will be with me in paradise.”


Jesus told the thief on the cross beside him that he'll be in paradise with Jesus that day. Being in paradise amounts to some kind of conscious experience. And as per soul-sleep doctrine, there is no conscious experience after death until resurrection.
Jehovah’s Witnesses believe that the soul dies when the body dies.
The condition of the dead is like sleep.
Without a resurrection there would be no hope for a soul to live again.
That may be enough to qualify us as soul sleepers. At any rate we do have an explanation for why Luke 23:43 is not in conflict with our view on what happened to the repentant evildoer upon his death.
The Insight on the Scriptures book
Vol 2 on the subject of Paradise talks about this very topic

Luke’s account shows that an evildoer, being executed alongside Jesus Christ, spoke words in Jesus’ defense and requested that Jesus remember him when he ‘got into his kingdom.’ Jesus’ reply was: “Truly I tell you today, You will be with me in Paradise.” (Lu 23:39-43) The punctuation shown in the rendering of these words must, of course, depend on the translator’s understanding of the sense of Jesus’ words, since no punctuation was used in the original Greek text. Punctuation in the modern style did not become common until about the ninth century C.E. Whereas many translations place a comma before the word “today” and thereby give the impression that the evildoer entered Paradise that same day, there is nothing in the rest of the Scriptures to support this. Jesus himself was dead and in the tomb until the third day and was then resurrected as “the firstfruits” of the resurrection. (Ac 10:40; 1Co 15:20; Col 1:18) He ascended to heaven 40 days later.—Joh 20:17; Ac 1:1-3, 9.
The evidence is, therefore, that Jesus’ use of the word “today” was not to give the time of the evildoer’s being in Paradise but, rather, to call attention to the time in which the promise was being made and during which the evildoer had shown a measure of faith in Jesus. It was a day when Jesus had been rejected and condemned by the highest-ranking religious leaders of his own people and was thereafter sentenced to die by Roman authority. He had become an object of scorn and ridicule. So the wrongdoer alongside him had shown a notable quality and commendable heart attitude in not going along with the crowd but, rather, speaking out in Jesus’ behalf and expressing belief in his coming Kingship. Recognizing that the emphasis is correctly placed on the time of the promise’s being made rather than on the time of its fulfillment, other translations, such as those in English by Rotherham and Lamsa, those in German by Reinhardt and W. Michaelis, as well as the Curetonian Syriac of the fifth century C.E., rendered the text in a form similar to the reading of the New World Translation, quoted herein.

So our belief is that the evil doer did not go to heaven on the day he died but his soul died. His hopes for living again rested on God’s power and promise to resurrect humans back to the living. Jesus’ promise was to resurrect him to life in paradise at some future date. The same unknown date spoken of in Acts 24:15 .