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April 3, 2017 Quantum Physics becomes visible in the cold Quantum Effects in Macroscopic World at Low

Superconductivity Quantum

Bose Einstein

Deep down, all is governed by the laws of quantum physics. , and are the usual phases of matter, in which quantum effects are often hidden by random atomic movements. But in extreme cold, close to absolute zero temperature, matter assumes strange new phases and behaves in unexpected ways. Quantum physics, which otherwise only works in the micro-scale world, suddenly becomes visible.

1 The following phenomena describe quantum effects that exist in macroscopic world at low .

• Superconductivity is a phenomenon of exactly zero electrical resistance and expulsion of magnetic fields occurring in certain materials when cooled below a characteristic critical temperature. It was discovered by Dutch physicist Heike Kamerlingh Onnes on April 8, 1911 in Leiden.

Two important properties of superconductors: (1) Zero Resistance, so they conduct without heating the wires (2) Repel

• Quantum Hall Effect: Resistance R of some two-dimensional sheets of material, in a magnetic field assumes quantized values that depend on charge of the electron and .1/R = ne2/h, n = 1, 2, 3....

• Bose-Einstein Condensate (BEC): Macroscopic Number of Bosons at very low temperature form a new kind of that exists in lowest possible quantum state. This state was first predicted, generally, in 1924-25 by 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 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 .

BEC is a new 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 can be tamed. They have applications in clock precision, new type of lasers and sensors and also exploring new phenomena in physics.

2 I. OF RESISTANCE– TOPOLOGICAL STATES OF MATTER

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Why is the resistance quantized and why is this quantization observed with extreme precision and is independent of geometry of the sample and impurities in the sample ?? This mysterious phenomenon was explained by David Thouless described theoretically, using topology. Ordinary phases of matter transition between each other when the temperature changes. For example, such a transition occurs when ice, which consists of well-ordered crystals, is heated and melts into water, a more chaotic phase of matter. The transition in QHE is a topological transition.

The topological is not an ordinary phase transition, like that between ice and water. In physics, it is not uncommon for drastic things to happen when the temperature is lowered; for example, many materials become magnetic. This happens because all the small atomic magnets in the material suddenly point in the same direction, giving rise to a strong magnetic field, which can also be measured. However, the quantum Hall effect is more difficult to understand; the electrical conductance in the layer appears to only be able to assume particular values, which are also extremely precise, something that is unusual in physics. Measurements provide precisely the same results even if the temperature, magnetic field or the amount of impurities in the vary. When the mag- netic field changes enough, the conductance of the layer also changes, but only in steps; reducing the strength of the magnetic field makes electrical conductance first exactly twice as big, then it triples, quadruples, and so on. These integer steps could not be explained by the physics known at the time, but David Thouless found the solution to this riddle using topology. Answered by Topology

Topology describes the properties that remain intact when an object is stretched, twisted or deformed, but not if it is torn apart. Topologically, a sphere and a bowl belong to the same category,

3 because a spherical lump of clay can be transformed into a bowl. However, a bagel with a hole in the middle and a coffee cup with a hole in the handle belong to another category; they can also be remodelled to form each others shapes. Topological objects can thus contain one hole, or two, or three, or four... but this number has to be an integer. This turned out to be useful in describing the electrical conductance found in the quantum Hall effect, which only changes in steps that are exact multiples of an integer. In the quantum Hall effect, electrons move relatively freely in the layer between the semi-conduc- tors and form something called a topological quantum fluid. In the same way as

4 new properties often appear when many particles come together, electrons in the topological quantum fluid also display surprising characteristics. Just as it cant be ascertained whether there is a hole in a coffee cup by only looking at a small part of it, it is impossible to determine whether electrons have formed a topological quantum fluid if you only observe what is happening to some of them. However, con- ductance describes the electrons collective and, because of topology, it varies in steps; it is quantized. Another characteristic of the topological quantum fluid is that its borders have unusual properties. These were predicted by the theory and were later confirmed experimentally.

Butterfly Graph and Quantization of Resistance

Theoretical model used by Thouless was extremely simple. Figure (??) shows the graph of energies of electrons ( or electron density) as a function of magnetic field. The graph is known as Hofstadter butterfly. The colored parts show the regions that give different quantum Hall states, each color representing a unique value of the integer.

5 FIG. 1: Hofstadter butterfly graph discovered by Douglas Hofstadter in 1976

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