As a summary of the mechanism…
Consider electrons as the driver, and consider a fixed position through the plasma chamber: Position X. The driver bunch comes screaming into the plasma from the left. As it approaches Position X, it begins pushing the free electrons near X out of the way, primarily sideways. (It’s was previously doing this to electrons upstream of X, but we’ll just look at X since the same thing happens everywhere, just at different times.) The driver pulse soon passes X and continues on its way, and the electrons that were displaced find themselves far away from the positive charges left near the core of the plasma. (The positive ions are too heavy to move a significant distance on this timescale, so consider them frozen in place.) The electrons also find themselves in an over-density of electrons, and all told, they are pulled/pushed very strongly back to the middle. Just as with a mass on a spring, there is no reason for them to instantly stop when they get back to where they started. They will arrive at the neutral position with significant momentum and will thus overshoot. This recreates the original pushed-out situation, only now with each electron on the opposite side from before. This will repeat until it damps out. Note that this oscillation phenomenon is not part of the wakefield acceleration itself. It’s a side effect that can be used for significant engineering advantages (in particular, multiple bunches can be accelerated from a single driver bunch).
For a dual-bunch scheme, the acceleration of the so-called “witness” bunch is all about timing. If you send in this second bunch of electrons and time it such that the plasma’s electrons are all out of the way at X right before the incoming bunch gets to X, the incoming bunch will see ahead of it a nice attractive positive ion soup at X, and it will be accelerated toward it. As the witness bunch passes X, the plasma electrons have begun crashing back down behind it toward, and then through, their “neutral” position.
Since the driver bunch is continually creating a bubble of positive charge around it by pushing away electrons, and since the witness bunch is continually seeing that positive charge bubble out in front of it, you get a continuous acceleration through the plasma cavity.
The technology is at an early proof-of-principle stage. There is a long road ahead, and no one yet knows if work-horse accelerators are possible. To give a flavor of the complexities:
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A high-energy accelerator needs not only to produce high energy particles but also a lot of them, and getting a lot of particles through a PWA is non-trivial. For small bunches the plasma behaves linearly, but as you increase the bunch size (i.e., number of particles being accelerated), you get strong non-linear phenomena that are hard to control. The whole task is one of subtle balance even before non-linearities kick in, with details like the density of the plasma along the length of the chamber needing to be controlled carefully.
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An accelerator should provide particles with a narrow range of energies (within 1% of each other, say). Because the wakefield time scales are so short, the front of a bunch and the back of it will end up seeing different net accelerations as the plasma distributions respond and evolve from the passing of the driver, leading to a wide spread in energies. Using lasers for the driving together with very narrow witness bunches helps, but lasers have a ceiling on their usefulness due to diffraction limits for real-world-sized cavities and due to overall power consumption. (One quickly surpasses multi-terawatt power consumption for a competitive laser-based PWA.)
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Electrons are convenient drivers, but they can only boost other electrons by a factor of two in energy, which is hardly worth it. Using protons as the driver could solve this, but protons are not readily used for this purpose. For instance, you need very short (length-wise) bunches to drive the wakefield since the restoring timescales are very short. Today’s proton bunch lengths are typically tens of centimeters, but you need tenths-of-millimeter lengths (>1000x smaller) for PWA.
PWA is in a similar position to fusion power thirty years ago. It should be possible in principle, but only the next step or two out of hundreds is clearly visible. We might have something competitive and workable in 60-90 years, or we might not. As with fusion, I’m a big fan of supporting this sort of forward-looking R&D since the one thing that is certain is that we will hit practical limits with existing technologies.