Not sure the historical evidence suggests that they show up flat for games after playoff bye weeks.
Since 2001, the Patriots are 11-1 in the postseason after receiving a first round bye, and 7-0 since the one loss in 2010.
They have outscored their opponent in those 11 wins by a score of 344 to 191.
Only three of those eleven opponents had fewer than 10 wins (the 2011 Broncos, 2016 Texans and 2017 Titans) and six of those eleven opponents had 11 or more wins. Combined record of those eleven opponents was 115-61
The only loss in twelve tries after a postseason first round bye was to the Jets in January of 2011.
Conversely, in the three playoff games they have played since 2001 in the wild card round, they are 2-1, the only defeat being that ugly loss to the Ravens in January 2010.
Brady has been historically good after a playoff bye week. 3,117 passing yards in those eleven wins, with 24 TDs against only 4 INTs.
After a regular season bye, maybe they come out flat. I'm not entirely sure that is the case but it feels that way without looking up the data. But in the postseason with a bye, they are pretty dominant.