In our laboratory we were able to create the conditions to generate and observe both the a.c. and d.c. Josephson effects in a BEC of rubidium atoms. Here we present the experiment, its results, and some of the theory behind it.
When two superconductors are joined together, separated only by a thin insulating barrier, some new phenomena arise. First, if a voltage is applied across the junction, a small oscillating current starts flowing back and forth through the junction, without equilibrating the two sides. This is known as the a.c. Josephson effect. Second, if a constant current is made to flow through the junction, no voltage drop is detected across it, as long as the current stays below some critical value. This is known as the d.c. Josephson effect. Accordingly, the setup of a superconductor-insulator-superconductor is known as a Josephson junction.
The phenomena of the a.c. and d.c. Josephson effects manifest because of the quantum nature of each superconductor as a whole. Quantum mechanics usually apply to systems at the microscopic level. However, in superconductors, the many electrons in the metal “condense” into the same quantum state, creating a macroscopic quantum entity. B.D. Josephson originally predicted the effect in 1962 for superconductors relying on this fact1. However, it is possible in principle to observe the a.c. and d.c. Josephson effects in any two macroscopic quantum systems separated by a tunneling barrier.
In 1997, the a.c. Josephson effect was observed in superfluid 3He.2 Superfluid 3He is another example of a system where quantum phenomena manifest macroscopically. In 3He, it is the helium atoms that form the quantum state, and the current is not of electrical type, but the flow of atoms through a thin hole separating two helium reservoirs. The d.c. Josephson effect in superfluid 3He, and both a.c. and d.c. effects in superfluid 4He, have not yet been observed.
Bose-Einstein condensate (BEC) is a state of matter where, due to the laws of quantum mechanics, all the particles constituting the matter are coerced into a single quantum state of the lowest energy, reshaping themselves as a single entity.
This new form of matter was predicted by Satyendra Nath Bose and Albert Einstein in 1925. In the following years, it was established that superfluids and superconductors are a form of BECs, but an ‘ideal’ BEC of weakly-interacting atoms, corresponding to the original theory, required extremely low temperatures, less than millionth of a degree above the absolute zero. Only in 1995 were scientists able to produce a BEC from a gas of dilute atoms, 70 years after its prediction.
Bose-Einstein condensates provide a rich ground for the testing of fundamental physics, of which this research is an instance. For much more information on this exciting phenomenon, please see the Wikipedia article.
Bose-Einstein condensates comprise a new physical system for observing the a.c. and d.c. Josephson effects, Apart from superconductors and superfluids. Being a distinct quantum state, a BEC is a macroscopic entity of definite quantum phase. This property, also called coherence, is the key to Josephson-like behavior.
In a typical apparatus for studying BECs, a large number of atoms (several millions) are collected inside a magnetic trap that confines them. Subsequently the atoms are cooled by various means to a temperature below the critical temperature for the transition to condensate phase to occur, around 100 nano-Kelvin. After creating the condensate, a smaller number of atoms remain (in our apparatus, approximately 100,000 atoms). The BEC is further probed only with magnetic or optical forces, since any contact with material bodies will cause the temperature to quickly rise, destroying the condensate.
The a.c. and d.c. Jospephson effects can actually be described by three very simple equations that have direct mechanical analogs. Before introducing the equations we need to specify the quantities involved.
Each condensate is characterized by the number of atoms it contains, N, and by its chemical potential, μ. The chemical potential plays the role of potential in electrical systems. A difference in chemical potential between two systems means that the systems can equilibrate by flow of atoms from the higher-potential system to the lower-potential one. Because a condensate is a (macroscopic) quantum state, it has a defined phase φ.
We denote the two condensates 1 and 2, and define the relative population imbalance as η = (N1 − N2)/(N1 + N2), with −1 < η < 1. Also, the chemical potential differnce is defined as Δμ = μ1 − μ2. Finally, the phase difference between the condensates φ = φ1 − φ2 has a major role in the dynamics of the Josephson junction.
The physics of Josephson junction is governed by three simple equations.
is the current flow through the tunneling barrier,
is
Planck's constant, ηequil is the equilibrium value of η, and the ω's
are parameters that characterize the physical system.
The first of these equations conveys the sinusoidal nature of the Josephson current. The second relates the rate of change of phase difference to the chemical potential difference. The third connects potential difference with population imbalance.
The three equations can be combined into
with 
. This is the equation of
motion for the phase difference φ.
For the a.c. Josephson effect, consider the first two equations with an
applied constant chemical potential difference Δμ. Then
and
. The relative
population imbalance oscillates with frequency Δμ/h
with amplitude inversely proportional to Δμ.
More generally, if the external potential is static, then
and the equation of motion for φ
reduces to 
which describes the
motion of a classical pendulum with angle φ. The a.c. Josephson
effect corresponds to the case of the pendulum executing complete rotations with
always-increasing angle, with an oscillation period of (Δμ/h)−1.
In contrast, in a bulk macroscopic quantum system, the current is
proportional to the gradient of the phase, and we would expect the relation
.
Such a system would show no oscillating current because φ
increases at a constant rate. Note that the a.c. Josephson effect is very
different from the plasma oscillations, also indicated in the above figure, for which the
pendulum oscillates about zero with −π < φ < π.
These plasma oscillations exist even for small-angle oscillations, for which also
,
as in a bulk macroscopic quantum system. The a.c. Josephson effect has no analogue in a bulk system.
The equation of motion for φ can be rewritten as
with
The mechanical analogy here is a point particle with “mass”
and “position” φ moving in
an external “washboard” potential U. See the figure below.
Mechanically,
controls the tilt of the washboard. Since atoms flow
in order to reach equilibrium value ηequil, this tilt is
analogous to an applied current.
As long as the applied current
is smaller than ωJ, the particle sits in a
local minimum of the washboard: the phase difference φ remains constant and there is no chemical potential
difference Δμ. When the applied current reaches the critical
value ωJ, the tilt is too high and the particle is free to fall down the slope – the phase difference
φ starts increasing, resulting in a non-zero chemical potential difference.
The Josephson junction thus exhibits two different states, depending on current: a “supercurrent” state, where current flows with no potential difference across the junction, and a “normal” state, with normal current flowing alongside supercurrent, causing a finite potential difference. This behavior is usually illustrated in a V-I (or Δμ-I) curve, for voltage vs. current:
We have successfully observed both the a.c. and d.c. Josephson effects in a BEC of 87Rb atoms by containing two BECs in a double-well potential. The barrier is sufficiently thin to allow coupling (tunneling) between the two condensates.
In order to realize the double-well potential, atoms are first gathered in a magnetic trap. Then a thin “sheet” of laser light divides the trap into two regions. The laser is specially frequency-tuned to repel the atoms, effectively creating a potential barrier. The atoms are now evaporatively-cooled to the stage of forming two Bose-Einstein condensates in the two wells.
More information on our experimental setup can be found in the research page.
In the a.c. Josephson effect, a constant average chemical potential difference is maintained between the two BECs. By the Josephson equations, a sinusodial current through the barrier is established, with atoms flowing back and forth between the condensates. The chemical potentials do not equilibrate – this is known as macroscopic quantum self-trapping (MQST).
We create an initial chemical potential difference by starting from an asymmetric double-well, and then shifting the barrier to the middle (see figure), resulting in a symmetric double-well configuration with different number of atoms in each well. The shift is done very quickly so as not to allow the atoms to ‘pass’ through the barrier.
We observe the Josephson oscillations in-situ by using non-destructive phase-contrast imaging. This allows taking multiple images without destroying the cloud. Imaging pulses are sent through the atomic cloud to the camera once per frame. The imaging process integrates the atom density of the cloud along the imaging direction. Thus the resulting image is a map of the column density of the cloud.
The imaging camera is a CCD array of 1024×256 pixels. The CCD array is logically divided into 16 strips of 64×256 pixels each, all covered except the first one. After taking each image, the entire array is shifted one strip, thus exposing the next strip for the next image. After 16 (or so) images are taken, the entire CCD array is read out and stored in an image file.
Below we see an example image. There are 14 frames with 0.33 msec between each frame. The left cloud is bigger than the right one because of the chemical potential difference. Seen below the image is a plot of an integration of the image along its y axis. This is the second integration performed on the cloud.
From these plots, we can extract the chemical potential difference Δμ and the relative population imbalance η as a function of time. The a.c. Jospehson effect is manifest through oscillations in η in a frequency equal to Δμ (in units of Planck's constant, h). Below we see an example of η vs. time. The upper curve is for Δμ = 750 Hz × h, and the lower curve is for Δμ = 450 Hz × h. The curves show an oscillating behavior, an it can be seen that the curve with the higher chemical potential difference is also of higher frequency.
We can extract the frequencies of the plots with varying delta mu by inspecting their Fourier transform. We can then plot the frequency as a function of Δμ and verrify that the relation is as predicted. The plot below shows our main result for the a.c. Josephson effect in a BEC medium. We can see that measuring Δμ and the frequency in the same units excellently agrees with the theoretical prediction.
In the d.c. Josephson effect, current flows through the barrier with no chemical potential difference between the two sides. The d.c. Josephson effect has so far been observed only in superconducting systems. Observing it a BEC medium requires a method to create flow of atoms from one side of the junction to the other, in various magnitudes, and to measure the chemical potentials.
We do this by slowly shifting the barrier from an off-center position to the center of the trap, ending with a symmetric potential. In the d.c. Josephson effect, the atoms do not ‘feel’ the barrier and are allowed to tunnel through it without resistance. The magnitude of the atom current is proportional to the velocity of the barrier. If this current should not rise above the critical current, the flow of atoms will not incur a difference in the chemical potentials, which will remain at its initial value of zero (equilbrium).
At the end of the experiment we reach a symmetric trap configuration, where condensates of equal chemical potential have the same size. We can measure the chemical potential difference by comparing the sizes of the condensates, and repeat the experiment at various atomic currents by moving the barrier with different velocity each time.
The d.c. Josephson effect is demonstrated by plotting the characteristic I-Δμ curve of the system — the dependence of the chemical potential difference on the current.
We can see the behavior predicted by the Josephson equations, with a critical current value of about 30 sec−1.