Do not telegraph a jump by dropping the arms or bending the knees. Remember, the arms are used for blocking and attacking, not for balance or to add momentum to a spin or jump. The knees should already be bent if you have a good fighting stance. To jump, just suddenly jump. Try to minimize all other body movements. This requires explosive leg strength, so good jumpers have powerful legs.
Jump as if you were trying to touch your knees to your shoulders. Do not bring the feet straight up under the body and do not lift the feet behind you; rather, draw the knees upward in front of you. The height of a jump kick is judged by how high the foot kicks, not how high the body is off the floor. By drawing the knees up in front, the feet have maximum height off the floor giving you more "hang time" to perform the kick, greater kicking height, and it lets you quickly get the feet back down for a safe landing. Also, by drawing the knees upward in front, you are chambering the leg for the kick that follows.
After a jump, the lead leg or arm may attack or you may close the range by rotating body 180° forward on trailing leg side and then using trailing leg or arm to attack. To close the range during a jump, jerk non-kicking knee high in front of body and then jump and kick with other leg. For example, jerk training knee high in front of body and then jump and kick with lead leg.