DSolve[{ca'[t] == -I h Exp[- I t] cb[t], cb'[t] == -I h Exp[ I t] ca[t], ca[0]==1, cb[0]==0}, {ca,cb},t]
ca[x_]=First[ ca /. %][x]
cb[x_]=First[ cb /. %%][x]
ca2[t_]=Simplify[ComplexExpand[ca[t] Conjugate[ca[t]] ], Element[h,Reals]]
cb2[t_]=Simplify[ComplexExpand[cb[t] Conjugate[cb[t]] ], Element[h,Reals]]

Simplify[ca2[t]+cb2[t], Element[h,Reals]]

Plot[Evaluate[{ca2[t],cb2[t]} /. h->.1],{t,0,10}]
Plot[Evaluate[{ca2[t],cb2[t]} /. h->.2],{t,0,10}]
Plot[Evaluate[{ca2[t],cb2[t]} /. h->.4],{t,0,10}]
Plot[Evaluate[{ca2[t],cb2[t]} /. h->.8],{t,0,10}]
Plot[Evaluate[{ca2[t],cb2[t]} /. h->1.6],{t,0,10}]
Plot[Evaluate[{ca2[t],cb2[t]} /. h->3.2],{t,0,10}]
