Prof. Robert B. Laughlin, Department of Physics, Stanford University

Josephson Whistling in Liquid Helium: The Music of the Spheres

Loren Alegria
March 22, 2009

(Submitted as coursework for Physics 376, Stanford University, Winter 2009)

Fig. 1: Schematic diagram of the apparatus of Hoskinson, Packard and Haard for detecting the Josephson effect in superfluid Helium.

In a Berkeley physics laboratory, Professor Richard Packard and his colleagues are making liquid helium sing with a tone whose purity is unmatched in the natural world. When the experimenters push superfluid helium through a tiny constriction, it does not flow continuously, but with a superimposed oscillatory motion that has no analogue on human-sized length scales. The frequency at which the flow wiggles depends only on fundamental constants and how hard the experimenters push. The equivalent of this effect using electrons has revolutionized the sensitive measurement of magnetic fields, and Packard's group hopes to similarly revolutionize the measurement of physical rotations using the techniques they have developed.

The greatest challenge in realizing this phenomenon, known as the Josephson effect, is creating a constriction small enough. In order to create the oscillation, the helium must be constricted to a passage just narrow enough that the superfluid state can only barely exist. This critical width is known as the "coherence length" of the superfluid, and depends upon the fluid's temperature. Assuming you can make such a hole, subtler still is detecting the tiny flows involved. In fact, to create an observable flow it is necessary to make many such holes, and the detection scheme must nevertheless be fairly sophisticated. Let us see how these hurdles are overcome.

The experimental set up is shown schematically in Fig. 1. A chamber is immersed in superfluid liquid helium-4 (the common isotope of helium). At the bottom of this chamber is a silicon nitride membrane, only 50 nm thick, that has been patterned with an array of 65x65 = 4225 holes. The holes are each 70 nm in diameter and are produced by electron beam lithography. Now, as mentioned, the coherence length of the superfluid depends upon temperature. As it so happens, this length becomes large at the superfluid transition temperature. So, in addition to making the holes small, we can make the superfluid coherence length large by bringing the temperature very near to the transition temperature. For the size of the holes in Packard's experiment, the helium needed to be held just 2 mK below the transition temperature for the coherence length to become long enough.

With conditions ripe for observing the Josephson effect, all that is left is to apply a pressure difference across the porous membrane and measure the resulting flow. To this end, the roof of the chamber is a kapton diaphragm, the top of which is plated in metal (yellow). Opposite this metal plating is an electrode. When a voltage is applied to the electrode, charges of opposite sign collect on the metal plating and their attraction flexes the diaphragm outward. In this way we can electronically control the pressure inside the chamber. Once a pressure is applied, flow is measured by monitoring the subsequent motion of the diaphragm. A coil (blue, in cross section) above the electrode is connected to a device that monitors the mutual inductance between the coil and the diaphragm. [1] Based on the mutual induction, displacements of the diaphragm can be measured to within a fraction of a picometer. Such extreme sensitivity is necessary, however, to detect the tiny reverberations from the oscillating flow through the membrane.

In a typical experiment, a sudden increase in voltage is applied to the electrodes, producing a sudden change in pressure in the chamber. The experimenters, listening to the output of the displacement detector with headphones, hear a distinct "whistle," at first high in pitch and becoming lower and lower as the pressure equalizes. At equilibrium, the whistle dies out entirely. When they extract the frequency of the whistle from the data, they find that it matches what would be expected from the Josephson effect: it is proportional to the pressure. Moreover, the constant of proportionality they find is that given theoretically for the Josephson effect. It is this constant of nature that makes the process so fundamental and so we will now discuss its origin.

Schrödinger's equation, which forms the basis of quantum mechanics, says that a mechanical state of energy E will carry a sort of clock (a phase), which ticks around at a rate of exactly E/h revolutions per second (h being Planck's constant). Now what is so very remarkable about superfluid helium is the fact that a whole body of it can act like a state with a specific energy. Consequently, if two separate bodies of helium have different energies, they will have clocks that tick at a rate

relative to one another. As it turns out, spatial variations of phase give rise to motion. If the two bodies of helium are just barely connected, a flow of helium will be created whose velocity will be due to the difference in the position of the hands of the two clocks. As the two clocks go around at their respective rates, their hands will sometime line up, sometimes be quite different, and in short, will have a difference that will oscillate back and forth over time at a rate of precisely fj. What we witness physically is a flow of liquid back and forth through the linkage at a frequency fj. This rather counterintuitive motion is the Josephson effect.

Now that we have hammered down the Josephson effect, what is left is to apply it to the experiment at hand. [2] All that we must really know is the relevant energy difference Δ E between the two superfluids. Although I will not prove it here, in our scenario this is given simply by the pressure difference times the volume per atom in the liquid. With this knowledge, the formula for the Josephson frequency becomes

By running the experiment at many Δ P's, Packard confirmed that the constant of proportionality with the frequency of the whistling observed was indeed Vatom/h.

Apart from elegantly demonstrating a quantum mechanical principle, Packard's work has implications for the development of rotation sensors that make use of this effect. He and others have long since demonstrated the Josephson effect with He-3. But using He-4, as the experiment described above does, would be more suited to applications because (a) less sophisticated cooling techniques are necessary, and (b) He-4 is a far more readily available than He-3. In fact, despite the sophisticated techniques used in the experiment above, the set-up is simpler by far than many experiments in the field, and this is its general appeal. [3]

© 2009 Lauren Alegria. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


[1] More explicitly, the coil is connected to another coil, which is placed next to a SQUID (a magnetic field sensor that is itself based on the Josephson effect). When the metal plating on the diaphragm moves relative to the first coil, it creates a current through both coils, and the SQUID measures the magnetic field produced in the second coil.

[2] R. P. Feynman, The Feynman Lectures on Physics, vol. III (Addison-Wesley, 1963).

[3] E. Hoskinson, R. E. Packard and T. M. Haard, "Oscillatory Motion: Quantum Whistling in Superfluid Helium-4," Nature 433, 376 (2005).