# Principles of a phase slip gyroscope

This section describes the principles of a 4He phase slip gyroscope which can be viewed as an analog to a superconducting rf SQUID (Superconducting Quantum Interference Device). Whereas SQUIDs are the most sensitive detectors of magnetic flux, the 4He SQUID has the potential to be a very sensitive detector of absolute rotation. We have used this device to measure the Earth's rotation and are working on scaling up the device to make it even more sensitive.

# Rotation induced aperture flow

 To understand the effects of rotation on a superfluid-filled container, we perform a series of gedanken (thought) experiments. We begin by choosing the superfluid filled container to be a torus sitting motionless. (For a basic introduction to superfluid 4He, click here.)

Step 1
We can first ask the question of what will happen when a torus filled with superfluid is slowly rotated (starting from rest) at a rate given by the angular velocity W.

Answer: It turns out that the superfluid remains motionless since it doesn't feel the wall (no viscosity).

Step 2
This time, a partition is inserted in the torus before it is slowly rotated from rest. What happens?

Answer: The superfluid is forced to rotate with average velocity vbulk = WRave. Nevertheless, the circulation of the fluid around any complete loop within the superfluid (such as the dotted line) remains zero.

Step 3

Next, a very small aperture is made in the partition, and the experiment is repeated. What happens?

Answer: The superfluid still wants to remain in a zero circulation state, but with the hole in the wall, we can now draw a loop which goes all the way around the torus. The flow throughout the bulk of the loop is still given by the velocity vbulk = WRave, but in order for the circulation to remain zero around the dotted yellow line, an enormous backflow is generated in the small hole.

How large is this backflow? Since circulation is defined as the velocity times path length around a loop, we find that vaplap+ vbulklbulk=0 (here bulk refers to the main loop, and ap refers to the aperture).Therefore, setting the length of the torus loop lbulk equal to its circumference 2pRave, we find that the velocity of the flow in the aperture is given by vap= -vbulk (2pRave/lap). There are two important points:

• First, the negative sign indicates that the flow through the aperture is in the opposite direction to rotation (hence the term backflow).
• Secondly, since Rave is macroscopic (i.e., on the order of centimeters) and the length of the microaperture lap is microscopic (i.e., ~100nm), the velocity of the aperture vap can be over a million times as large as the velocity of the superfluid through the bulk vbulk!

In other words, a torus with a submicron aperture is a rotational velocity amplifier. If we can find some way to measure the velocity in the aperture, we will have an extremely sensitive rotation device. The phase slip critical velocity will allow us to measure this backflow.

Mathematical note: If we define the rotational flux Krot as Krot = 2WA cos(a) (where a is the angle between the rotation vector W and the vector A defining the area of the torus) then the velocity in the aperture due to rotation can also be expressed as vap= -Krot/lap. Therefore, for fixed W and A, the rotational flux can be modified by changing the angle a.

# A Diaphragm-Aperture Oscillator (DAO)

In order to understand how we observe critical velocities, it may help to know a little about our experimental setup. In general, we study superfluid flow through small apertures with the aid of Diaphragm Aperture Oscillators (DAO).

The simplest variant of a DAO is shown below. It consists of a box filled with superfluid. Inside that box is another container which has a soft flexible plastic membrane on one side, and a small aperture (represented by the X) in the wall on the opposite side. The membrane is coated with a thin metallic film. By applying voltages between this metallic film and a nearby electrode, one can move the membrane --thereby displacing fluid through the small hole. The flip side of the membrane is coated with another metallic film which is superconducting. A nearby superconducting wire wound in a spiral is connected to a SQUID. This spiral coil has a current in it which produces a magnetic field in the region between the coil and the membrane. Since a superconductor expells magnetic flux, when the membrane moves, the changing magnetic flux (between the coil and the membrane) induces an additional current in the spiral coil. The SQUID then produces a voltage in proportion to this small change in current. Therefore, the SQUID based displacement sensor essentially converts membrane motion into voltage which we can easily read out. We are able to measure displacements as small as 10-15 meters (the size of a proton) in one second.

There are several variations of the simple DAO shown above. For instance, the single aperture can be replaced with an array of apertures. Another variant is the use of a single aperture in parallel with a larger tube (shown below without the driving electrode and SQUID coil). This changes the topology of the device and allows for trapped circulation in the loop traversing the small hole and the parallel path. It turns out that this latter design is exactly what one needs to make a superfluid gyroscope, as we will see later.

# Critical velocity and phase slips

Superfluid flows with essentially zero dissipation through very small holes, long narrow paths, and powders up to some well defined velocity, known as the critical velocity. The value of the critical velocity depends on the details of a particular experimental cell, but once determined, it is stable for a given set of conditions (i.e., constant temperature and pressure). The source of dissipation for the critical velocity is the creation and movement of quantized vortex lines, in a process known as phase slippage.

## Phase Slips

 A phase slip is the process whereby a quantized vortex (similar to a whirlpool, hurricane, or tornado) moves across a channel, and in doing so, reduces the energy of the flow by a discrete (quantum) amount. One could just as easily have called this phenomena a velocity slip, since a superfluid phase difference is directly proportional to its velocity. In this drawing, the blue lines represent the flow through an aperture, and the red lines correspond to a vortex (looking down through its eye).

## Critical Velocity

In our experiements, we drive these diaphragm-aperture oscillators on resonance, slowly building up the velocities that flow through the apertures until the critical velocity is reached and phase slips occur. A typical measurement is shown below.

 This figure displays the amplitude of the diaphragm as a function of time as the membrane is oscillated at a fixed driving force. The oscillator rings up (with essentially zero dissipation) until it reaches its critical velocity (the velocity of the flow through the aperture is directly proportional to the amplitude of the membrane). The sharp drops in amplitude are single phase slips as illustrated above.

# Detection of rotation induced flow with phase slips

 In order to detect the rotation induced flow, we use a DAO that is configured with a small aperture and a parallel loop in the shape of a torus.

We begin with the torus at rest. If an oscillating drive is applied to the membrane, superflow will slosh back and forth through the small aperture and the rest of the torus. One particular half cycle of the oscillation is represented in the figure below (the black arrows represent the motion of the fluid).

As we drive the membrane harder, the velocity of the flow through the aperture will eventually exceed its critical velocity . The critcal velocity will always be reached in the small aperture first (and not anywhere else in the loop) as long as the aperture is sufficiently small.

When the critcal velocity is reached, a phase slip occurs. Since the critical velocity is a constant, the onset of phase slippage tells us precisely what the total flow velocity through the aperture is.

To see how this allows us to measure rotation, let's first assume that there is no rotation (the fluid is only moving because we are pushing it around with our diaphragm).

The first important point is that we know exactly how much fluid we are pushing with the diaphragm (when we move the diaphragm, the displaced volume of fluid must go through the torus loop or the hole; the ratio of how much goes through the loop vs. the hole is easily determined by the geometry of the two paths). Since the critical velocity is fixed, so too is the total amount of current we need to push through the hole to reach this critical velocity. If, however, some additional current were added to the hole (which we don't know about), then we would need to push less current ourselves in order to reach the critical velocity (i.e., the amount of current we contribute + the amount contributed by other sources must equal the critical flow) . Therefore, from our point of view, when we see a lower critical velocity than normal, it signifies that some other mechanism has contributed some current through the hole. In our case, this additional current is the backflow produced by rotations.

Note: if some current were subtracted, then the critical velocity would indeed appear higher, so we can also tell which direction the torus is rotating!.

To summarize:

• Quantization of circulation stipulates that there must be a backflow through the aperture any time the torus is rotated.
• This rotation-induced velocity increases in direct proportion with the rotation rate(W), the size of the torus (area A), and inversely with the aperture size (small apertures imply rotation-induced velocities that can be over 1 million times larger than the rotational velocity of the torus).
• The magnitude of the change in the apparent critical velocity (due to phase slips) reveals the magnitude of this rotation-induced flow (and hence the rotation rate).

# Measuring the Earth's rotation

## Description of the cell

Our prototype cell was microfabricated out of a silicon wafer and is shown here adjacent to the schematic representation. The chip is 1.5cm on a side and is 0.5mm thick. The central depression as well as the loop are recessed by 80mm from the surface. A single small aperture (in this case, 0.19mm x 1.0mm) is made in the central depression, and a much larger window (1mm square) is made at the end of the loop. The raised surface is subsequently covered with a thin plastic membrane and sealed using epoxy. The whole chip is then placed inside a superfluid-filled box.

The similarity of the two devices becomes apparent when tracing the flow path of the superfluid. Let's begin in the central depression (this corresponds to sitting on one side of the diaphragm). The superfluid can either go through the small aperture, or through the peripheral loop. Following the outer loop, the fluid will continue through the window, around the backside of the chip and then in through the small aperture to arrive at its starting point. The plastic membrane which is glued to the chip's surface acts as the soft diaphragm used to push the fluid back and forth.

## Performing the Experiment

As shown above, the rotation induced backflow is directly proportional to the rotational flux through the torus. Since our aim was to measure the Earth's rotation (and since the Earth rotates at a constant rate of once per day), the only way to vary the amount of backflow was to change the orientation of the chip with respect to the rotation axis of the Earth (i.e., by changing the angle a, we changed the rotational flux through the chip). The figure below illustrates how one would do the experiment if our lab were located near the equator.

 The chip sitting at the equator (with the orientation shown) would not sense any of the Earth's rotational flux, and so there would be no rotation induced backflow. If that same chip were rotated by 90 degrees, as shown here, then it would pick up the full effect of the Earth's rotation (in this case, the vectors A and W are lined up parallel to each other). Therefore, if one continuously re-oriented the chip, one would expect the rotational flux (and hence the rotational induced flow and the apparent critical velocity) to vary sinusoidally (see the actual data below). Since our lab is in Berkeley at 38 degrees latitude, our setup looks more like this. Being in Berkeley (vs near the equator) simply decreases the magnitude of the rotational coupling by cos(38) as compared to the lab at the equator.

## Data

 This figure shows the expected sinusoidal modulation of the critical amplitude (or critical velocity) when the gyroscope chip is slowly rotated about the vertical axis in our lab. The angle a is taken as zero when the chip has zero sensitivity to the Earth's rotational flux (i.e., similar to the orientation of the chip located at the equator in the figure above).

# References

The first successful attempts to see single phase slips were performed by Avenel and Varoquaux using a low frequency diaphragm-aperture oscillator.

• Avenel, O., Varoquaux, E. Observation of singly quantized dissipation events obeying the Josephson frequency relation in the critical flow of superfluid 4He through an aperture. Physical Review Letters, vol.55, (no.24), 2704-2707 (1985).
• These references describe the successful attempts (in Berkeley and in Orsay/Saclay, France) to measure the Earth's rotation.

• Avenel, O., Hakonen, P., Varoquaux, E. Detection of the rotation of the Earth with a superfluid gyrometer. Physical Review Letters, vol.78, (no.19), 3602-3605 (1997).
• Schwab, K., Bruckner, N., Packard, R.E. Detection of the Earth's rotation using superfluid phase coherence. Nature, vol.386, (no.6625), 585-587 (1997).
• Schwab, K., Bruckner, N., Packard, R. The superfluid 4He analog of the RF SQUID. Journal of Low Temperature Physics, vol.110, (no.5-6),1043-1104 (1998).
• A comprehensive treatment of the principles of superfluid helium gyroscopes can be found in:

• Packard, R.E. and Vitale, S. Principles of superfluid helium gyroscopes. Phys. Rev. B 46, 3540-3549 (1992).

Updated October 25, 1999

Send comments or questions to: Richard Packard