In the United States towards the middle of World War II on October 28th, 1943. During this experiment, the United States Naval Destroyer battleship escort known as the USS Eldridge was aimed to be rendered entirely optically invisible to any and all human observers for a short period of time. 

For a long time, this experiment was largely thought to be a hoax because the thought of making something optically invisible was something that had previously been born entirely out of science fiction literature. The United States Navy still maintains that there was never an experiment to try and make something invisible. The details of this story are heavily contradicted by the well-established facts about the USS Eldridge. Though, despite the government’s assurances that the Philadelphia Experiment never occurred, it has still raised some serious conspiracy theories around the United States as well as in some venues around the globe that specialize in the validation or study of certain thought conspiracies. Though the Philadelphia Experiment may never be confirmed, it is still the subject of much speculation and criticism by numerous groups.

The story has several different accounts since no record of its existence has ever been confirmed by documentation, but the major belief is this; the experiment was supposed to be based on a certain aspect of the unified field theory, a term that was originally coined by Albert Einstein. This theory aims primarily to describe physically and mathematically the interrelated nature of the natural forces that comprise both gravity and electromagnetic radiation. However, this theory remains just that, a theory, and it has never been fully proven to be an absolute truth.
According to the accounts of the story, scientists believed that some version of the unified field theory could enable the Navy to utilize huge electronic generators that could bend the light around a specific object so that it would no longer reflect light, but refract it, causing it to appear invisible. The Navy was thought to be attempting these experiments because optical invisibility would be an invaluable tool against the United States’ opponents in the war. Thus, it allegedly sponsored this experiment.

According to most accounts, the naval experiments succeeded to some degree during the summer tests of 1943. During a test on July 22 of 1943, it is thought that the Navy was able to render an object almost entirely invisible, though it was reported to emit some kind of green fog around the area.
However, once the ship was brought back into optical sight, there were numerous anomalies surrounding the ship including some ship mates being embedded in the metal sides and structures of the ship. One sailor even reportedly ended up a full deck level below where he was before the experiment.

Because of this controversial occurrence after the Philadelphia Experiment, it was thought to be kept silent for liability reasons by the government. Furthermore, the scientific community states that there is still no confirmed applied variant of the unified field theory, thus supposedly disproving the existence of the Philadelphia Experiment at all. Though there have been some experiments stating that certain “cloaks” or single person fields may be worn that apply some aspects of the unified field theory, there is no evidence that anything larger than a human could be rendered entirely invisible no matter how much displacement of light was achieved. Though it has never been proven, there are still beliefs that the Philadelphia Experiment existed.

source: http://www.unsolved-mysteries.com/unexplained_events/philadelphia_experiment.html