Try:

   $Version
   (*"13.2.0 for Microsoft Windows (64-bit) (November 18, 2022)"*)


   timeRange =  2 Pi;
   \[CapitalDelta] = 10^-25; \[CapitalGamma] = 10^25;
   solution = 
     NDSolve[{I x'[
          t] == (Exp[-I \[CapitalDelta] t] + 
           Exp[-I \[CapitalGamma] t]) y[t], 
       I y'[t] == (Exp[I \[CapitalDelta] t] + 
           Exp[I \[CapitalGamma] t]) x[t], x[0] == 1, y[0] == 0}, {x, 
       y}, {t, 0, timeRange}, MaxSteps -> 50000, AccuracyGoal -> 4];
   Plot[{Evaluate[Norm[x[t] /. solution]^2], 
     Evaluate[Norm[y[t] /. solution]^2]}, {t, 0, timeRange}, 
    PlotTheme -> "Scientific", 
    FrameLabel -> {"t(s)", "|a(t)\!\(\*SuperscriptBox[\(|\), \(2\)]\)"}]

you get a nice plot.

Regards M.I.