WEAK LINK HISTORY
Although Josephson's first theoretical results for weak links were verified in superconductors shortly after his discovery, they have lain dormant waiting for verification by another type of phase coherent system. There were two candidates that could most likely fulfill this prophecy, superfluid 3He and 4He. Superfluids are phase coherent systems that can be described by a "macroscopic" quantum wave function (fulfilling the first requirement of the theory). However, is it possible to create a weak link separating the superfluid? And secondly, how does one measure such small mass currents through the weak link?
In order to weakly couple two volumes of superfluid it is necessary to separate them by a strong thin wall containing a small hole. In addition, the dimensions of the thin wall and the hole must be on the order of the healing length for the superfluid. The superfluid healing length is the length over which the order parameter or wave function can vary while still minimizing the free energy (similar to the coherence length for superconductors). The healing length is given by:
Where Tc is the onset temperature for superfluidity and xo is the T = 0 value of the healing length, which differs considerably for 3He and 4He. For 3He this value is ~ 65 nm while for 4He it is ~ 0.1 nm. It is precisely this difference which makes the likelihood of producing a weak link suitable for 3He much more favorable than 4He. This length scale is also one strong reason for the difficulty in discovering the Josephson effect in a superfluid.
Our weak link efforts were enhanced by novel techniques of micro-fabrication which have allowed the construction of a square array of 4,225 holes each with a diameter equal to 100 nm and separated by 3 mm etched into a 50 nm thick SiN membrane. The array of holes allows the passage of a large coherent flow of superfluid that is thousands of times larger than the flow through just one hole.
In order to detect Josephson oscillations in a superfluid one must use a SQUID-based displacement transducer (which was developed for the pursuit of gravity wave detection). The SQUID (Superconducting Quantum Interference Device) uses two superconducting Josephson junctions in parallel to measure very slight changes in magnetic fields. It enables one to detect the motion of a flexible Kapton diaphragm with a precision of 10-15 m/ÖHz.
Many attempts to see Josephson effects in
superfluids have been made. It has taken many years and many different minds tackling this
problem to learn enough information to successfully observe these effects directly. Many
improvements along the way in conjuction with our new ideas have led to recent success for
Superfluid Weak Link Physics.
In 1962, Brian Josephson made a remarkable discovery when
he considered the possibility of cooper pairs tunneling their way through a thin
insulating barrier separating two superconductors. He found that when there was a fixed
chemical potential difference, cooper pairs would tunnel back and forth through the
barrier. These oscillations, known as Josephson oscillations or the Josephon effect, still
bares his name and has nearly formed its own field of physics. Although Josephson's
original results were predicted for superconductors, he later reformulated his ideas along
with Feynman and Anderson to describe a more general arrangement where two phase coherent
systems are separated by a thin barrier called a "weak link". Phase coherence
requires that one be able to describe a system by a wave function that has a single
definite overall phase (i.e. all constituent particles in the fluid have the same phase).
If a thin barrier allows only small interactions between two wave functions then it is a
weak link. Recent advances in technology have made it possible to create a weak link and
detect extremely small currents. Measurements have now been made verifying Josephson's
ideas in a more general framework.
A cylindrical pillbox containing an "outer" volume of superfluid houses within it a smaller pillbox which contains an "inner" volume of superfluid. The smaller pillbox is formed by an Al cylinder, 140-mm in length, which has been capped on the top by a soft, flexible circular Kapton diaphragm, which acts as a piston to move the fluid. The bottom of this Al cylinder has a cap made from a stiff circular Kapton membrane that contains a hole over which a Si chip is glued in place. The chip contains the SiN membrane that has etched into it a micro-aperture array of 4,225 holes and acts as a weak link. The upper Kapton diaphragm has a 100-nm layer of Pb and 20-nm layer of In evaporated on its surface to 1) expel magnet field lines for detection by a SQUID-based displacement transducer, and 2) act as an electrode. In the outer volume, about 100-mm above the diaphragm, a second upper electrode is mounted which is used to electro-statically activate the diaphragm/piston, applying pressure differences across the micro-aperture array. About 10-mm behind this upper electrode lies the input coil of the d.c. SQUID, which will detect flux changes when the diaphragm moves. The pillboxes are connected to a nuclear demagnitization refrigerator capable of cooling the liquid helium to temperatures as low as 0.30-mK.
View image of
micro-aperture array
By knowing the position of the soft membrane as a function of time, it is possible to measure the pressure across the micro-aperture array using Hook's law. The flexable Kapton membrane essentually acts as a spring with a spring constant K. As the position of the membrane changes with time, a volume of fluid is either swept in or out of the inner cell through the aperture array. By knowing the rate of change of this volume, dV/dt, it is possible to calculate the current through the array. By combining these relationships with Josephson's AC and DC equations we arrive at the equation for a physical pendulum. The quantum phase f takes the place of the angular displacement of the physical pendulum. The low amplitude frequency of oscillations or the "pendulum mode" frequency wp is directly related to the critical current Ic of the weak link.
THE PENDULUM EQUATION FOR THE QUANTUM PHASE
The physical pendulum equation for the phase describes two different regions of quantum phase behavior:
The "Quantum Oscillations" mode
When the initial pressure in the system is quite high the system exhibits "phase winding". The initial pressure stays constant (if there is no dissipation) and the phase continues to wind up linearly with time. This is the case for a physical pendulum hit very hard so that it spins round and round its axis, with an angular velocity proportional to the initial kick. This causes the current through the weak link to oscillate with a frequency given by the solution to the AC Josephson Relation.
We see that this frequency is linearly proportional to the pressure across the weak link.
The "Pendulum" mode
For small initial pressures we provoke oscillations whose phase varies between -p and p with an average pressure of zero across the weak. In this case, the force delivered to the pendulum initially is not enough to send it over the top. The phase simply oscillates back and forth like a swinging pendulum. When the phase oscillates in this manner we say that the weak link is in the "pendulum mode" (the low amplitude oscillations occur at the pendulum frequency wp)
In deriving the pendulum equation for the phase we have ignored any dissipation effects. Our experimental system does have dissipation. This allows us to access both regions of quantum phase behavior with one method of excitation. By applying a step voltage drive between the upper electrode and the soft diaphragm we create a large initial pressure head which will relax due to the dissipation. The result is a "transient" behavior. The system begins at large pressure, with phase winding, performing "quantum oscillations" of the current. As the pressure across the weak link relaxes the frequency of these oscillations falls. The system relaxes towards equilibrium. When the pressure is close to zero the system then undergoes "pendulum mode" oscillations. The analogy would be a physical pendulum hit very hard so that it spins round and round its axis, but in the presence of dissipation (friction) it slows down to the point where it can no longer make another revolution and so begins to swing back and forth. This is clearly seen if we look at the pressure stored in the diaphragm as a function of time. This is done by monitoring the position of the diaphragm a function of time using the SQUID.
VERIFYING THE AC JOSEPHSON EFFECT
We can verify Josephson's AC equation by measuring the frequency-pressure relationship directly. We begin by taking a transient and slicing up the time-axis into bins. For each bin, the frequency information is acquired through a fourier transform of the data while the corresponding pressure is related to the average position of the diaphragm. A calibration done prior to the measurement allows us to convert the SQUID signal as a function of time into position, x(t). It is then possible to plot the largest frequency peak in the fourier transform vs. the averaged pressure for each bin. We find that the Josephson-Anderson phase evolution relation or the AC Josephson equation is clearly satisfied. The fit-line has a slope of 194 Hz/mPa at all temperatures. We take this to be a clear verification of Josephson's AC relation for a superfluid 3He weak link.
Using Transient data we can extract the current through the
weak link as well as the phase difference across it as a function of time. The current is
calculated by differentiating the position of the membrane as a function of time and using
the conservation of mass principle. We calculate the phase by integrating the AC Josephson
equation directly using the information we've measured for x(t). By eliminating
time we arrive at the current-phase relationship I(f).
WHAT DO WE EXPECT FOR THE CURRENT-PHASE RELATION
From Josephson's theory we know that the current-phase
relation should look sine-like when the microaperture array provides sufficiently weak
coupling between the two reservoirs of superfluid. The amount of coupling these two
reservoirs feel is a function of the superfluid density near the apertures. In considering
the suppression of superfluid in each hole, we see that the superfluid healing length
determines an effective diameter for each hole. This effective diameter appears to
increase as the temperature falls, so that the weak link coupling increases. At
temperatures close to Tc, we expect weak coupling. As the temperature decreases
the weak link will soon become stronger than Josephson's criteria and the current-phase
relation should change. At the low temperatures, we expect a linear current-phase
relationship indicative of an array of just simple holes with some hydrodynamic
inductance.
VERIFYING
THE DC JOSEPHSON EFFECT
Here we show the extracted current-phase relation of the weak link array for temperatures close to and below Tc. At temperatures close to Tc, the Current-Phase relation is clearly sine-like as predicted by the DC Josephson relation. The maximum amplitude gives a measurement of the critical current for the weak link.
As temperature is changed we find the
behavior of these curves qualitatively matches our predictions. However, a closer look
reveals that our method of excitation (a step voltage producing a transient) would not
allow the system to access certain regions of phase near |f| = p. We decided
it was important to investigate these regions directly by trying to excite the system to
these higher values of phase. We simply needed to "swing" the pendulum mode to
larger amplitudes at these low temperatures.
A) A low amplitude sine wave burst is applied to the soft diaphragm. The frequency of the burst is chosen to closely match the frequency of the pendulum mode at a given temperature.
B) The diaphragm's motion "rings up" in reaction to the applied wave train. These oscillations strongly persist even after the excitation has passed since there is very little damping.
C) As we increase the amplitude of the excitation we see the diaphragm's oscillation amplitude increase as expected.
D) When the excitation is increased to some
critical level something remarkable happens! Suddenly, the amplitude drops and the
frequency of oscillations is reduced. At some later time the system spontaneously returns
to its original energy state.
PHASE
PORTRAITS OF THE p-STATE
By making a phase portrait of the diaphragms response to the sine burst (i.e. using curves like D), one gets a better idea of what is happening to the system. As the system rings up, the trajectory assumes an increasing orbit about f = 0. At some phase near p the system makes a transition to a smaller orbit centered about p. The system resides in this "metastable p-state" for a limited time before spontaneously making a transition to an orbit centered about 2p with its original amplitude.
When we extract the current-phase relation for the weak link array, we find that a new branch centered about f = p has formed. It was this branch that wasn't accessed in the previous curves for the current-phase relation. Notice that the slope of this new branch is noticeably less than the major branch. This clearly shows the difference in the frequency of pendulum oscillations of the two states.
We further extract the Free energy stored in
the weak link array, F(f), from the current-phase relation. These curves
clearly show that some energy is stored while in the p-state. The minimum in the
energy at f = p is indicative of a metastable state.
POSSIBLE THEORIES
1) Measurements of Current-Pressure curves using Feedback
2) The "Fiske Effect"
3) Non-Linear Dynamics of the Pendulum Mode
4) The "Shapiro Effect"
5) More Interesting Characteristics of the Weak Link
6) Superfluid 3He Gyroscope
QUESTIONS ABOUT THIS RESEARCH?
Contact the following:
Prof. Richard Packard (packard@socrates.berkeley.edu) (web site)
Prof. J.C. (Séamus) Davis (jcdavis@socrates.berkeley.edu) (web site)
...along with questions about this site
Ray Simmonds
(simmonds@physics.berkeley.edu) (web site)
The research is supported in part by grants from the NSF.
Send comments to Richard Packard
Last modified 3/00.
