Macroscopic quantum phenomena

[3] The concept of macroscopically occupied quantum states is introduced by Fritz London.

[4][5] In this section it will be explained what it means if a single state is occupied by a very large number of particles.

The wave function is normalized so that The physical interpretation of the quantity depends on the number of particles.

1 represents a container with a certain number of particles with a small control volume ΔV inside.

the vector potential; cc stands for the complex conjugate of the other term inside the brackets.

At temperatures below the lambda point, helium shows the unique property of superfluidity.

The fraction of the liquid that forms the superfluid component is a macroscopic quantum fluid.

(8) reduces to For an arbitrary loop in the liquid, this gives Due to the single-valued nature of the wave function with n integer, we have

For a circular motion with radius r In case of a single quantum (n = 1) When superfluid helium is put in rotation, Eq.

(13) will not be satisfied for all loops inside the liquid unless the rotation is organized around vortex lines (as depicted in Fig. 2).

These lines have a vacuum core with a diameter of about 1 Å (which is smaller than the average particle distance).

The number of vortex lines increases with the angular velocity (as shown in the upper half of the figure).

[7] In the original paper[8] Ginzburg and Landau observed the existence of two types of superconductors depending on the energy of the interface between the normal and superconducting states.

In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value Hc.

Depending on the geometry of the sample, one may obtain an intermediate state[9] consisting of a baroque pattern[10] of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field.

In Type II superconductors, raising the applied field past a critical value Hc1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large.

[11] For superconductors the bosons involved are the so-called Cooper pairs which are quasiparticles formed by two electrons.

The so-called fluxoid is defined by In general the values of κ and Φ depend on the choice of the loop.

So, for thick rings, the total magnetic flux in the loop is quantized according to Weak links play a very important role in modern superconductivity.

In these regions the velocity contribution to the total phase change in the loop is given by (with Eq.

So the value of the line integral is well-defined (e.g. independent of the choice of the end points).

The energy difference of a Cooper pair, moving from one side of the contact to the other, is ΔE = 2eV.

(32) can be written as ΔE = hν which is the relation for the energy of a photon with frequency ν.

It has a strong resemblance with the interference pattern generated by a laser beam behind a double slit.

The classical types of quantum systems, superconductors and superfluid helium, were discovered in the beginning of the 20th century.

[17] They are trapped using magnetic fields or optical dipole potentials in ultrahigh vacuum chambers.

Isotopes which have been used include rubidium (Rb-87 and Rb-85), strontium (Sr-87, Sr-86, and Sr-84) potassium (K-39 and K-40), sodium (Na-23), lithium (Li-7 and Li-6), and hydrogen (H-1).

A team of NIST and the University of Colorado has succeeded in creating and observing vortex quantization in these systems.

[18] The concentration of vortices increases with the angular velocity of the rotation, similar to the case of superfluid helium and superconductivity.

Fig. 1 Left: only one particle; usually the small box is empty. However, there is a nonzero probability that the particle is in the box. This chance is given by Eq. ( 3 ). Middle: a few particles. There are usually some particles in the box. We can define an average, but the actual number of particles in the box has large fluctuations around this average. Right: a very large number of particles. There is generally a large number of particles in the box. The fluctuations around the average are small compared to the number in the box.
Fig. 2 Lower part: vertical cross section of a column of superfluid helium rotating around a vertical axis. Upper part: Top view of the surface showing the pattern of vortex cores. From left to right the rotation speed is increased, resulting in an increasing vortex-line density.
Fig. 3. Two superconducting rings in an applied magnetic field
  1. thick superconducting ring. The integration loop is completely in the region with v s = 0 ;
  2. thick superconducting ring with a weak link. The integration loop is completely in the region with v s = 0 except for a small region near the weak link.
Fig. 4. Schematic of a weak link carrying a superconducting current i s . The voltage difference over the link is V . The phases of the superconducting wave functions at the left and right side are assumed to be constant (in space, not in time) with values of φ 1 and φ 2 respectively.
Fig. 5. Two superconductors connected by two weak links. A current and a magnetic field are applied.
Fig. 6. Dependence of the critical current of a DC-SQUID on the applied magnetic field
Fig. 7. Magnetic flux lines penetrating a type-II superconductor. The currents in the superconducting material generate a magnetic field which, together with the applied field, result in bundles of quantized flux.