This is an interesting question and I suspect historians of physics would debate the answer a bit. I'll just say that from the beginning there is a thread in quantum theory that connected it to things like Eastern religions and mystical notions, even by its creators. Niels Bohr considered complementarity, for example, to be related to the yin and yang idea, and many of the early quantum pioneers explicitly tied their work to notions in Hinduism (see e.g. Oppenheimer) and Buddhism (see e.g. Schrödinger). They of course took these things more as inspirational than literal; they liked the gnomic aspects of Eastern thought, aspects that could tolerate ambiguity better than many of the aspects of Western thought. But I bring this up because while of course these things were dramatically increased in later years, they were there in the beginning.

During the mid-20th century much of this kind of speculative, philosophical approach to quantum theory was stamped out. This is sometimes called the "just calculate!" period of quantum physics, and was marked by the huge increases in physics budgets, and physics class sizes, post-World War II and post-Sputnik. These are not the cigarette-filled seminars of Oppenheimer or the meandering seminars of Bohr, but auditoriums full of people learning how to use Feynman diagrams en masse.

The return of the mysticism, and its popularization, comes just after this, in the 1970s, when the physics market crashed. This meant a huge number of people with very intensive educations suddenly found themselves unemployable and they began to seek non-academic and non-governmental uses of their talents. A few of them ended up getting very much into the woo scene, pushing very strange approaches to quantum mechanics (a few of which actually became useful), and looking for "far out" connections between the counterculture and science. This is what a few scholars have called "groovy science", and this is probably what you are looking for. This is where you start to see a lot of popularization efforts and an increasingly explicit tying of quantum concepts into religious and woo approaches. Interestingly, this did lead to a re-engagement in the deeper questions of quantum mechanics itself, and some of these "woo" ideas became somewhat influential within physics — multiverse theory and Bell's inequality both come to mind, both of which were conceived in non-woo contexts but heavily taken up by the woo-ers.

For more on this, see David Kaiser and Patrick McCray's edited volume, Groovy Science, as well as Kaiser's How the Hippies Saved Physics: Science, Counterculture, and the Quantum Revival (the latter of which I helped as a researcher and illustrator).