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Andreev reflection, familiar in superconducting systems, shows a much more extensive set of behaviors in the p-wave condensate of superfluid 3He. We discuss the basic ideas of Andreev reflection in the superfluid and its various manifestations. The fact that the process displays almost perfect retroreflection allows us to exploit the remarkable properties of the phenomenon for characterizing the pure quantum turbulence that can exist in these condensates. Finally, we discuss Andreev-reflection “optics” as a means of visualizing this turbulence in real time through the vortex video camera.
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Supplemental Videos 1 and 2. Animation of the Basic Andreev Reflection Process
Supplemental Video 1. The basic process of Andreev reflection, an animated version of Figure 4. The group velocity (slope of the dispersion curve) falls as the gap increases. An incoming quasiparticle (red) decelerates as the energy gap increases, finally coming to rest. It then accelerates out again, retracing its inward trajectory, climbing the other side of the dispersion curve as a quasihole (blue).
Supplemental Video 2. The same process as in the previous video, but for an incoming quasihole (blue).
Supplemental Videos 3 and 4. Animation of the Feynman Picture
Supplemental Video 3. The Feynman-diagram representation of an incoming quasiparticle undergoing Andreev reflection. We represent the changing gap “potential” by the green “boson.” The incoming quasiparticle interacts with the potential to create a further quasiparticle–quasihole pair. The quasihole leaves and the new quasiparticle joins with the incoming one to create a new Cooper pair.
Supplemental Video 4. An incoming quasihole annihilates with one of the particles in a Cooper pair, with the destruction of the pair and the emission of the now unpaired particle. In the first process a Cooper pair is created, and in the second process a Cooper pair is destroyed.
Supplemental Videos 5 and 6. Animation of the Damping of an Object Moving in the Superfluid
Supplemental Video 5. An incoming quasiparticle (red), approaching the leading side of the moving wire, penetrates the region of liquid near the wire, is normally reflected at the wire surface, and escapes into the bulk liquid again. (The dispersion curves are drawn in the rest frame of the wire.)
Supplemental Video 6. Conversely, an incoming quasihole finds that there are no available states near the wire and is Andreev reflected. Thus it cannot reach the surface to exchange momentum with the wire.
The picture for excitations approaching the trailing side of the wire is similar except that the “flavor” of the excitations is reversed: Incoming quasiparticles are Andreev reflected without reaching the wire, whereas quasiparticles can reach the surface and exchange momentum. This has the rather counterintuitive effect that the excitations hitting the front of the wire have positive momenta and those hitting the rear, being quasiholes, have negative momentum, and both slow the motion of the wire.
Supplemental Videos 7 and 8. The Moving Objects We Use in the Superfluid
Supplemental Video 7. The vibrating wire resonator. As the wire is driven back and forth in the vertical magnetic field a voltage is generated across the ends of the wire that allows us to map out the mechanical resonance and thus calculate the damping. (The motion is highly exaggerated for clarity).
Supplemental Video 8. A quartz tuning fork. This is a much smaller device than the vibrating wire of the previous video and does not need an ambient magnetic field to operate. However, the geometry is more complex. These devices are now beginning to replace vibrating wires for many uses. (Again the motion is highly exaggerated.)
Supplemental Text. Some Further Experiments Demonstrating Andreev Reflection
We discuss briefly below a few further experiments, expanding on the list in Section 4 of the main text.
The Effect of Textural Direction
In the B phase with its quasi-isotropic gap the orientation of the texture (see Section 2 of the main text) plays essentially no role in the damping experienced by a wire moving through the liquid. Provided the magnetic field is small, the gap experienced by the quasiparticle/hole excitations is not direction-dependent. However, this is not the case for the A phase, which has nodes in the gap along the L vector direction (as seen in Figure 1 of the main text). As discussed in Section 3.3.2, the nodal direction must impinge normally on a boundary, meaning that if the nodal direction is very different out in the bulk then it must bend near the wall. An excitation in the bulk moving at an angle close to this direction will see a relatively low gap, but with the bending of the L vector near a boundary, the excitation will see an increasing gap and will be Andreev reflected, as shown in Supplemental Videos 9 and 10 below. In the case of a moving wire, this bending prevents a fraction of low-energy excitations in the bulk liquid from reaching the wire surface to undergo normal scattering. Furthermore, the fraction affected will clearly depend on the direction of the L vector with respect to the motion of the wire. The direction of the L vector in the bulk liquid will depend on the sample history, will be different on each cool down, and can be disturbed by thermally shocking the superfluid with a heat pulse to stir up the texture and change it.
Supplemental Video 9. Here we represent the gap following the texture by a steep-sided valley, but remember the gap actually varies with angle not with position. An excitation approaching a boundary (checkered) can travel along the textural direction in low gap, but as the texture bends near the boundary the gap increases with the changing angle of the texture and the excitation is Andreev reflected and does not reach the boundary surface.
Supplemental Video 10. Of course, the picture of the previous video also implies that excitations that can reach the boundary are trapped in the surface layer and cannot escape into the bulk liquid.
Because the A phase has nodes where the gap is much lower than that in the B phase, the A phase has a much higher concentration of excitations at the same temperature [because Boltzmann factor, exp(-Δ/kT), is much larger]. Thus, we would expect the damping in the A phase to be much higher than that in the B phase, but very irreproducible, and also susceptible to change by simply thermally shocking the system.
Supplemental Figure 1. In a simple cell where there is a magnetic field gradient stabilized B phase at one end and A phase at the other, a straightforward comparison could be made of the A-phase and B-phase damping on two similar vibrating wire resonators at the same temperature (Fisher SN, Guénault AM, Kennedy CJ, Pickett GR. 1991. Phys. Rev. Lett. 67:3788–91). Typical results are shown in the B part of the figure, where the A-phase and B-phase damping are plotted against each other. As can be seen the A-phase damping can be a factor of ten greater, the measurement is different for each run, and thermally shocking the system gives a different result again.
This result represents the first confirmation, albeit in a different form, of the experiment suggested by Greaves & Leggett (Greaves NA, Leggett AJ. 1983. J. Phys. C: Solid State Phys.16:4383--404), as discussed in Section 3.3.2.
Energy-Gap Magnitude and Spin
The easiest way to realize the classic Andreev scenario with a static step in the gap in the superfluid 3He system is to set up a phase boundary between the A and B phases of the superfluid, because the gaps for the two phases are different (see Figure 1). We can achieve this by applying a magnetic field gradient to the superfluid. Because the A phase has a higher magnetic susceptibility than the B phase, the field gradient will result in the A phase occupying the high field region and the B phase the low field region, with an A-to-B phase boundary in between.
This configuration has been used to probe the Andreev reflection of a beam of excitations emitted from a blackbody radiator in furnace mode. The field gradient is provided by a very small solenoid placed immediately outside the radiator orifice in the path of excitations flowing out of the radiator.
When heat is applied to the radiator container, we can measure an effective impedance of the orifice from the equilibrium temperature of the helium inside. As the ambient magnetic field is increased nothing much happens until the transition field is reached, whereupon a section of the A phase is stabilized in the coil and the emitted excitations now experience a substantial step in the static gap as the A-phase gap is higher than the B-phase gap parallel to a magnetic field. This step means that a larger fraction of the excitations is Andreev reflected back into the radiator volume and the effective impedance of the orifice increases, indicating clearly that the phase boundary has appeared. The actual situation is somewhat more complicated because the quasiparticle/hole excitations are associated with a spin thus giving rise to a splitting between the up- and down-spin excitations, which adds an extra increase or decrease in the effective gap as discussed further in (Cousins DJ, Enrico MP, Fisher SN, Phillipson SL, Pickett GR, et al. 1996. Phys. Rev. Lett. 77:5245-48).
Supplemental Figure 2. (A) A blackbody radiator with a small coil placed around the orifice, in which a slab of A phase will be stabilized in high field. The coil inner diameter is so small (~1 mm) that the tightly wound superconducting wire was not fully superconducting and a silver heat link had to encircle the coil to keep it at microkelvin temperatures. (B) The normalized impedance of the orifice. Nothing much happens until the B-to-A transition field is reached, at which point there is a step in the gap, the beam is partially reflected back into the BBR, and the impedance rises rapidly.
Supplemental Figures 3 and 4 and Supplemental Videos 11 and 12. The Blackbody Radiator with Paddle
The following figures and animations illustrate the amazing behavior of the blackbody radiator when an oscillating paddle is placed in front of the orifice.
Supplemental Figure 3. The blackbody radiator with a paddle in front of the orifice as shown in Figure 11.
Supplemental Figure 4. When we heat the excitation gas inside the radiator container a thermal beam of excitations is emitted. The excitations simply bounce off the paddle by normal scattering and are lost in the bulk surrounding liquid.
Supplemental Video 11. We can oscillate the paddle in the same way as we oscillate a vibrating wire. This sets up flow fields around the paddle as it moves through the liquid.
Supplemental Video 12. When the paddle is moving and we then turn on the beam of excitations through the orifice, the flow fields Andreev reflect part of the beam, but since these processes are almost perfect retro-reflections the reflected part of the beam travels back into the radiator container raising the temperature inside, allowing us to calculate the fraction of excitations Andreev reflected.
Supplemental Videos 13 and 14. The Effect of Turbulence on the Damping of a Moving Wire
Below are two animation of the effect on the dynamics of a vibrating wire of a region of flow in the near neighborhood (for example, from a nearby vortex).
Supplemental Video 13. Here we show how the excitation dispersion curve evolves as we pass through a region of flow near a vibrating wire. We see that an incoming quasiparticle (red) cannot penetrate this region of moving liquid but is Andreev reflected and thus never reaches the wire to exchange momentum with it.
Supplemental Video 14. This shows that an incoming quasihole (blue), on the other hand, can pass through the region of flow and can therefore reach the wire to be normally reflected. However, on reversing its path it is now approaching the external region of flow from the opposite direction and cannot pass; thus it is Andreev reflected back to the wire, where it is normally reflected again and is then able to finally escape.
In each of the above cases, the incoming and outgoing excitations have the opposite flavor and thus the net momentum exchange with the wire is zero. That has the surprising effect that when we measure the damping of a vibrating wire in the superfluid, the damping falls when we introduce vorticity, because the flow fields of the vortices shield the wire from exchanging momentum with the surrounding excitation gas. Because the wire damping falls, it looks as if the temperature of the liquid has fallen. Normally when we induce vorticity, we put heat into the liquid, so to see an associated apparent temperature fall is a problem. Not surprisingly, it took us a long time to understand this mechanism, as it appears, at first sight, to violate the second law. The animations just show one direction of fluid flow. However, whether it is an incoming quasiparticle or quasihole, or whether the flow is positive or negative, the result is similar. A large fraction of the excitations that would otherwise have reached the wire do not do so. Once we understood the mechanism, this provided us with a most incredibly useful and sensitive turbulence detector in the superfluid.
Supplemental Videos 15 and 16. Ring Coalescence
The interesting thing about the experiment in Section 5.1.3 in the main text is that it shows us how the moving grid first creates a shower of microscopic vortex rings that, if the shower is dense enough, will collide and reconnect to generate turbulence. This is most entertainingly portrayed in simulations below by Shoji Fujiyama and Makoto Tsubota of Osaka City University. This is an unusual situation in turbulence in that large structures are built up from microscopic precursor vortex rings. Classical turbulence is generally associated with larger structures decaying into smaller-scale eddies.
Supplemental Video 15. A simulation of the creation of vorticity by a moving grid. Vortex loops trapped on the grid are exposed to an oscillating flow field as the grid is oscillated to and fro. This excites loops of a particular size that expand and recombine to release vortex rings with a very narrow range of diameters. In this simulation the left-hand side of the “box” represents the grid surface producing rings that are emitted into the bulk liquid. In this case the density of the rings is low (the grid is only oscillated with low amplitude), and thus individual rings have a low probability of colliding with other rings and they simply propagate as an independent gas of noninteracting particles.
Supplemental Video 16. Here the grid is oscillated with higher amplitude producing a higher density of rings that do collide, reconnect, and form an incoherent block in the bulk liquid that traps further rings from the grid and gradually builds up to form a vortex tangle. Because the rings are created with a narrow range of sizes the changeover from the independent gas behavior to the creation of a tangle also happens over a narrow range of grid velocities.
Supplemental Videos 17 and 18. A Sample of the Output of the Vortex Video Camera
Supplemental Video 17. The animation shows the quasiparticle flux detected by the camera pixels when the vibrating wire resonator in front of the camera changes velocity in 13 steps (not quite linearly) from 10 mm/s to 13mm/s. The camera pixels colors show the width parameter for the appropriate camera pixel (with backgrounds subtracted) normalized by the highest value in the whole video. The rate is one frame per second. The increasing width parameter for each pixel indicates the increasing quasiparticle flux as the velocity of the generating vibrating-wire resonator (VWR) is ramped up. The image yields valuable information on the angular dependence of the pair-breaking processes taking place at the moving boundary of the VWR.
Supplemental Video 18. An animated camera image when an incoming beam from the radiator is illuminating a vortex tangle between the beam orifice and the camera. As can be seen, the image detects a lot of activity, but the resolution is just too low to detect the motion of individual vortices in the tangle in the field of view. From the experience gained with these and similar images, we now know how to increase the resolution and can move forward to developing a camera with single-vortex sensitivity. (Observant viewers will have noticed that in this video, one row of pixels was not operating owing to a touch somewhere in the leads for this bank of tuning forks.)