Lecture: April 12, 2021

Macroscopic Quantum Phenomenon

• (1) Quantization of Resistance : Quantum Hall Effect

• (2) Superconductivity

• (3) Bose Einstein Condensation

—————————————————————————————————————————

QUANTUM HALL EFFECT

What do we know about Electrical Resistance of a material The electrical resistance of an object is a measure of its opposition to the flow of electric current. The reciprocal quantity is electrical conductance, and is the ease with which an electric current passes. The resistance of an object depends in large part on the material it is made of. Objects made of electrical insulators like rubber tend to have very high resistance and low conductivity, while objects made of electrical conductors like metals tend to have very low resistance and high conductivity. This relationship is quantified by resistivity or conductivity. The nature of a material is not the only factor in resistance and conductance, however; it also depends on the size and shape of an object because these properties are extensive rather than intensive. For example, a wire’s resistance is higher if it is long and thin, and lower if it is short and thick. All objects resist electrical current, except for superconductors, which have a resistance of zero. Finally, impurities will change the resistance. Therefore, quantization of resistance as described above is very very surprising and unexpected phenomena.

Resistance R of some sheets of material, in a magnetic field assumes quantized values that depend on charge of the electron and Planck constant.

1 ne2 Conductance = Resistance = h , n = 1, 2, 3....

1 • (1) Why is the resistance quantized? ( Recall what causes quantization in Bohr model).

• (2) Why is this quantization observed with extreme precision , better than one part in billion

• (3) Why is the conductivity independent of geometry of the sample and impurities in the sample

In Bohr model, quantization comes from integer number of wave lengths that fit into a Bohr orbit.

The quantization of resistance is lot more subtle: it depends on the “number of holes” in the wave function of the the electrons. Since we cannot have fractional number of holes, we have quantized resistance.

Mathematically, we say that quantization is the “topological” property of the wave function.

What is Topology ??? In mathematics, topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. This concept that topology determines the properties of some materials have led to

2 revolutionary discoveries of new class of materials that may have applications in quantum computers and many other novel devices.

FIG. 1: Cartoon diagram: It is the number of holes in the wave function that determine quantization.

FIG. 2: From the book: “ Butterfly in the Quantum World” :Theoretical model, known as Hofstadter butterfly that predicts all possible quantization of resistance. Horizontal axis is magnetic field and vertical axis is energy of electrons in a crystal.

3 Nobel Prizes: (1) Von Klitzing in 1980 who was at that time Heisenberg fellow at universiity of Wrzburg (2) Robert Laughlin, 1990 ( For theoretical work on fractional quantum Hall effect ) (3) David Thouless for theoretical explanation of Quantum Hall Effect. Answers to above three questions lies in the fact that the effect is topological. This mysterious phenomenon was explained by David Thouless described theoretically, using TOPOLOGY and was awarded Nobel prize in 2016.

4 Bose Einstein Condensate

Video:

Macroscopic Number of Bosons at very low temperature form a new kind of quantum state that exists in lowest possible quantum state. This state was first predicted, generally, in 1924-25 by Satyendra Nath Bose and Albert Einstein. On June 5, 1995 the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NIST-JILA lab, in a gas of rubidium atoms cooled to 170 nanokelvin. Shortly thereafter, Wolfgang Ketterle at MIT demonstrated important BEC properties. For their achievements Cornell, Wieman, and Ketterle received the 2001 Nobel Prize in Physics.

BEC is a new state of matter where particles loose their identity as de-Broglie wave length of different particles overlap. Such a state of matter is a quantum mechanical state whose properties

5 6 can be tamed. Nobel Prizes:

The Nobel Prize in Physics 2001 was awarded jointly to Eric A. Cornell, Wolfgang Ketterle and Carl E. Wieman ”for the achievement of Bose-Einstein condensation. APPLICATIONS: They have applications in clock precision, new type of lasers and sensors and also exploring new phenomena in physics.

7