Not only is it normal, but it shows very well how the system operates.
Sleep requires your body to just "tick over", -so very little demand for oxygen
, and so the circulation is minimum. But not only is your pulse rate low but also the pressures generated by your heart drop very low too. If you measure your BP readings at the same time, you'll find they're about half your daytime maximum.
Now: the interesting bit.... Upon waking and on rising to start the day, you'll usually get an (automatic) swift "Kick" of adrenaline
. This boosts the pulse and makes the pressures rise quickly to make your cardiovascular system ready to spring into action. This phase last perhaps an hour or so in most cases, and gradually fades back to normality.
Sometimes .. it doesn't . Then you know something's amiss -illness,. impairment, old age, cardiac failure, -... whatever. So now you can see that if you DIDN'T exhibit the symptoms you describe, THAT would be abnormal.
You have no cause for concern.