Experiment description

The experiment is designed to detect an acoustic signal generated by vortex slips in superfluid 4He flowing through an aperture. The acoustic spectrum is expected to have a Fourier component at a frequency proportional to the chemical potential (pressure) difference between the regions on either side of the aperture. The coefficient of proportionality is independent of the experimental details and can be expressed through fundamental constants only, providing a possible pressure standard.

Simplified experimental setup

The experimental setup consists of two reservoirs filled with superfluid helium, separated by a small aperture. A pressure differential is applied by means of a "piston". Flow rate of a normal fluid in such system would be uniform, limited by viscosity in the aperture.

Superfluid flow

One of the remarkable properties of liquid helium is its ability under certain circumstances to flow without dissipation of energy. For example, superfluid flow will persist in a ring-shaped pipe indefinitely. The viscosity associated with such flow is exactly zero. The flow is potential: the curl of the velocity field is zero and the velocity field itself can be described by a gradient of a scalar potential. It is easy to show that in a quantum mechanical description this potential is the phase of the superfluid wave function. The quantum mechanical description applies an additional restriction on the potential. For the wave function to remain single-valued, the phase variation integral along any path originating and ending at the same point can only be a multiple of . Since the phase gradient is proportional to flow velocity, the path integral of the velocity field is also quantized. This condition is called circulation quantization, and the unit of such quantization is , where is Plank's constant and is the mass of a helium atom.

Vortex generation

Consider a system of two reservoirs connected with a small aperture filled with superfluid helium. The implication of viscosity-free flow is that there can be a constant non-zero flow from one side of the system into the other without any pressure difference between them.

When a pressure difference is applied, the liquid in the flow channel accelerates until some critical flow is reached. At this point, the flow becomes dissipative. Although Landau theory predicts breakdown of superfluidity via creation of elementary excitations when the relative velocity of the superfluid and normal components reaches a critical velocity (of the order of 55 meters per second), in macroscopic systems the observed critical velocities are typically lower and appear to depend on the flow channel geometry. This dissipation is believed to be due to formation of quantized vortices. Superfluidity is broken near the center (core) of such vortex. The circulation for any path that has the vortex inside is exactly . Vortices with mutiple quanta are unstable and will break up into several vortices with one quantum of circulation each.

The vortex is most likely to spawn from the surface in the spot with the highest local velocity, probably near a protrusion on the inside surface of the aperture. Once the vortex is formed, it is driven across the flow by Magnus force and collapses after crossing the aperture. The typical time to cross the aperture is estimated to be several to tens of microseconds.

Slip size quantization

The crossing of the vortex results in a flow velocity drop near the aperture. The phase difference between velocity path integrals taken along Path 1 and Path 2 differ by one unit of circulation . (only a single vortex is inside the loop formed by the two paths). If points A and B are taken to be in the regions where the velocity does not change significantly during the phase slip, the velocity path integral before and after the slip occurred differ by one unit of circulation, or . Assuming that velocities along any given path scale with the velocity at some point on that path inside the aperture , the jump in during the slip is , where is the effective length of the aperture, i.e., the characteristic length over which the velocity varies. In a long channel, the effective length is equivalent to the length of the channel; in case of an aperture the length is the thickness of the membrane plus a length associated with edge effects.

Phase evolution; Josephson frequency

In the time interval between the slips, the liquid accelerates due to the pressure differential. In the absence of energy loss, the effect of the pressure gradient is to accelerate helium uniformly, with acceleration proportional to the pressure gradient. The difference in phases of the superfluid wave function between points A and B evolves according to the Josephson-Anderson phase evolution formula: . Indeed, if the points A and B are in regions of space little perturbed by the flow through the aperture, the amplitude of the wave function remains constant and the phase evolution is linear in time, with the rate of evolution proportional to the local potential. Although the potential is only defined up to an additive constant, the potential difference is a well defined quantity as long as the pressure in each of the volumes is well defined. If the phase difference (which is proportional to the helium flow velocity), at which the slips occur, is the same for all slips, and the size of the phase slip is exactly , the slip rate (frequency) is .

There are several important points regarding this frequency dependence:

The velocity step size is a function of the aperture size, with larger steps accuring in smaller apertures.

Detection Method

The idea of the experiment is to detect sound waves generated by periodic velocity variations in the aperture. A microphone mounted in one of the helium reservoirs can detect pressure oscillations associated with the periodic flow. In addition, the reservoir itself acts as a resonant cavity, amplifying the weak signal.

A constant pressure drop across the aperture is maintained by a feedback loop which includes a capacitive pressure gauge and a piezoelectric driver (piston). The signal is detected at the resonance frequencies of the cavity as a function of the applied pressure differential.

Stochastic vortex generation

In the discussion above the assumption was that all the slips will happen at exactly the same velocity. This assumption, however, is not justified. In fact, critical flow velocities in an aperture typically have some finite distribution width. Moreover, critical flow velocities show a characteristic temperature dependence, increasing linearly as the temperature is lowered from the superfluid transition temperature. The average critical flow velocity also appears to increase logarithmically with the pressure head. Phenomenologically, both of these effects can be explained by a model (Packard and Vitale 1992) in which the vortex nucleation process is stochastic with probability described by Arrhenius equation

where is the attempt frequency and is the activation energy.

Instead of the regular sawtooth velocity evolution described above, one expects to find an irregular pattern. The extent of the irregularity can be quantified by the ratio of the critical velocity distribution width to the slip size. The fourier components of the velocity spectrum get less pronounced and the signal appears more noise-like as this ratio increases. The noise is similar to shot noise due to finite electron charge in electronics.

Computer simulations

The effect of the stochastic nature of phase slip generation is illustrated in the results of a computer simulation below. For a set of different distribution widths, the velocity evolution is simulated. The fourier spectrum is then computed. For each set, an ideal (zero width) distribution and 1/f noise of the same RMS amplitude are shown. You can also listen to the generated waveform

Narrow distribution
Time evolution (.wav format sound):

Fourier transform:

Medium distribution
Time evolution (.wav format sound):

Fourier transform:

Wide distribution
Time evolution (.wav format sound):

Fourier transform:

Past results and future research directions

Shot-like noise generated by superfluid flow through an aperture was detected and characterized in this laboratory (Backhaus and Packard 1998). The acoustic noise amplitude was measured as a function of the pressure head for two different cell resonances. However, the signal did not appear to contain features corresponding to the Josephson frequency. The size of the apertures used in the experiment was such that the critical velocity distribution width was larger than the slip size, and the stochastic spectrum dominated.

To see the signal at the Josephson frequency one has to use apertures of smaller size. The acoustic amplitude of a signal generated by a smaller aperture is reduced. However, if the detection limit is determined by the signal to noise ratio rather than by the signal intensity, this would not be a problem. In the past experiments, a capacitance microphone was used to detect the acoustic signal. In addition to the acoustic noise generated by the phase slips, the microphone was sensitive to electric noise from elsewhere. A SQUID-based displacement transducer will provide a less noisy and a more sensitive method of detection. Other possible directions of research include the use of aperture arrays and attempts to approach the superfluid transition temperature where the coherence length approaches the aperture size.

Further reading