Awesome and Mysterious Quantum Mechanics

While Ernest Rutherford, Hans Geiger and Ernest Marsden were bombarding Gold foils with Alpha particles in 1909 leading to the discovery of atomic nucleus, Chemists were finishing analysis of gas emission spectrum. When light obtained from heating up gases in a glass tube was passed through a prism, it formed distinct lines of light of different colors for example in case of Hydrogen, Red, Blue/Cyan and Violet colored lines were obtained. Niels Bohr, by putting together Einstein’s theory of light and J.J. Balmer's empirical formula came up with an explanation in 1913. He figured that Electrons absorb energy when gas is heated up and leap to higher energy orbits around the nucleus, but they only leap to certain orbits after absorbing specific amount of energy. When cooled down the Electrons emit energy in specific amounts termed as Quanta and leap in to lower energy orbits. We obtain distinct lines of different colored light because electrons absorb and emit energy in certain amounts. Electrons can only have certain amounts of energy, not just any value. This explanation also rescued Rutherford’s atomic model because now it was understood why Electrons did not get pulled into the nucleus given their opposite charges.

Balmer's formula

Here R_H is Rydberg constant for Hydrogen, n₁=2 and n₂ greater than n₁

Louis-Victor De Broglie after intensely studying wave particle duality of light pioneered by Max Planck and Albert Einstein, presented his famous hypothesis in 1924 stating not only light has both wave and particle properties but particles such as Electrons too should have both wave and particle like characteristics. De Broglie called the wave associated to a particle, matter wave. He showed that wave, characteristic to a particular particle should have a wavelength of

λ=h/p

Where h=Planck’s constant having a value of 6.626068*10^-34m²kg/s and p=momentum of the particle.

Werner Heisenberg et al presented matrix mechanics to describe wave nature of particles in 1925. In 1926 Erwin Schrodinger created wave mechanics in an attempt to describe matter waves. Latter Schrodinger showed that matrix mechanics and wave mechanics are equivalent. Max born showed a statistical version of wave function and pointed that solution to Schrodinger’s wave equation for particles gives probability of finding a subatomic particle like an Electron at a particular location, it doesn't describes a smeared out Electron as Schrodinger himself had thought. By 1927 George Paget Thomson in UK and Clinton Davisson and Lester Germer in US had experimental evidence of wave like properties of Electrons. They were conducting their experiments with Electron beams although with different purposes but found similar diffraction patterns indicating wave like nature of Electrons. These were very first experimental support for De Broglie’s matter waves, 3 years after they were presented. Latter while calculating probability of finding an Electron at different locations using Schrodinger’s wave equation, scientists found a definite probability of finding the Electron on other side of the detector screen. Further they found a certain probability of finding an Electron on other side of a thick wall and even a mountain. This surprising probability is described as Quantum Tunneling and it is real. It is thought that Electron borrows energy required to get through solid wall from future and returns that energy once it is on the other side. As bizarre as it may sound but most probably, that’s how it is. Scientists are harnessing this property of Electrons in creating faster and smaller transistors and Quantum Computers. Transistors utilizing Quantum Tunneling will be much smaller and much faster than the smallest transistors we have today like those 11nm CMOS transistors.

After the equivalence of Matrix and wave mechanics was established, Jordan in Gottingen University and Paul Dirac in Cambridge university merged these two different ways of representing a matter wave mathematically and came up with what became known as transformation theory. Heisenberg while pondering over the papers of Jordan and Dirac figured that more precisely the position of a subatomic particle is known, less precisely its momentum can be known. In 1927, He came up with a mathematical expression to quantize this uncertainty.

ΔxΔp ≥ h/4Π

Where Δx= Standard deviation of position, Δp= Standard deviation of momentum, h= Planck’s constant

Same uncertainty holds for certain other pair of variables too such as energy and time for which the particle can have that energy. In order to explain this result Heisenberg thought up an experiment using Gamma ray microscope. Gamma rays have higher frequency than Electrons and therefore when used to determine position of Electron it will significantly change the direction of motion of Electron upon impact. Even visible light when shone to find the location of Electron will alter its velocity by some amount as energy of photon is comparable to energy of an Electron, leading Heisenberg to conclude that uncertainty is inbuilt into nature itself. As a matter of fact, with Uncertainty principle the predictions of Quantum Mechanics became consistent. Of ‘course the principle has been verified in many experiments since 1927, most common of them being the slit experiment. 

In 1801 Thomas Young took up the task of measuring light’s wavelength. He used a paper card to split a single pinhole beam of sunlight into two to carry out his experiment. Latter the practice of using two narrowly separated slits caught on. Italian physicists Pier Giorgio Merli, Gian Franco Missiroli and Giulio Pozzi did the double slit experiment with a single Electron in 1974. They found an interference pattern on the detector screen, kind of pattern which is created by waves, proving the predictions of Quantum Mechanics. Latter on Scientists discovered that when they look at the Electron fired from Electron gun, it formed a particle like pattern on detector screen and when they did not look, it formed a wave interference pattern. But, how a single Electron can form wave interference pattern? Although the predictions of Quantum Mechanics were verified, nobody had any idea, how. This mystery is still without a good explanation. In more recent versions of this experiment, Scientists observed the Electron after it crossed the slits. To their surprise they found that when they look, the Electron formed a particle like pattern and when they didn’t, it again formed a wave like pattern. How is that possible? Nobody knows the answer even today, but we do know now that Bohr was right. That, act of observation does really changes behavior of a particle. In addition, these observations also gave support to Heisenberg’s Uncertainty Principle. These experiments supported probability wave nature of Electron as predicted by Schrodinger’s wave equation and supported De Broglie’s matter waves or wave particle duality. We know now that the Electron can be in many positions at the same time or wave like if we are not looking at it, a Quantum Mechanical phenomena known as Quantum Superposition, and takes a particular position or behaves particle like if we do look at it.

Time-dependent Schrödinger equation- General

Ψ is wave function of Quantum system, Ĥ is Hamiltonian operator, ћ is reduced Planck’s constant, i=sqrt(-1), r is 3d position vector

Albert Einstein had other ideas about Quantum Mechanics. He and two of his colleagues Boris Podolsky and Nathan Rosen came up with a prediction of Quantum Mechanics in 1935, which he thought cannot be true, leading to his declaration that Quantum Mechanics is not wrong but it is incomplete. That prediction was Quantum Entanglement. Einstein together with his colleagues said that according to Schrodinger’s wave equation and other principles of Quantum Mechanics if two fundamental particles share their source of origin then their properties will have to be linked in such a way that sum of measurements of their Quantum properties give the Quantum property of source particle no matter how far apart these two particles are. This can be possible only if an act of measurement on one of these entangled particles simultaneously affects the Quantum state of other particle thus keeping the sum total same as it would be for the Quantum state of the source particle. This indicates some form of communication between the two particles happening at speeds greater than speed of light, leading Einstein to conclude that the theory is incomplete and requires modifications. Bohr and his colleagues remained unmoved. In his 1964 paper, Irish Physicist John Stewart Bell came with an inequality to settle this dispute. According to the inequality if Einstein was right then Quantum Mechanics would not be just incomplete but it would be wrong. When experiment was done in 1972 by John Clauser and Stuart Freedman, Einstein was found wrong. Results of this experiment supported immediate communication between entangled particles. Latter in 1981-82 a more precise experiment was conducted by Alain Aspect which corroborated faster than light communication between entangled particles when an act of measurement is performed. Bell’s inequality indicated that if Quantum Entanglement was real then entangled particles will have higher correlation than classical physics would allow. Results obtained from experiments based on Bell’s inequality unequivocally indicated a higher correlation between entangled particles. Now scientists around world are harnessing this phenomenon for creating Quantum Computers and to achieve Teleportation.

Because of the uncertain nature of reality, first indicated by Heisenberg, we can’t say with certainty that empty space is without any energy-mass density. According to Quantum Mechanics there is a finite probability that empty space may have certain amount of energy as long as nobody observes it. More specifically, particles and antiparticles may originate from empty space in pairs and exist for a certain while before annihilating each other. Amount of time they exist for depends on their energy, as Quantum Mechanics predicts. More the energy, shorter the duration they exist for and vice versa. This prediction came to be known as quantum vacuum fluctuations of electromagnetic field and is inherent to space.

Hendrik Casimir while investigating the discrepancies in measurements done on colloids, using the theory of Fritz London concerning Van Der Walls forces, the forces that govern colloids, thought up an interesting event. In his 1948 paper he indicated that if two neutral conducting plates are placed about a micron apart in vacuum, then vacuum between plates could only create virtual photon pairs with very small wavelengths compared to vacuum around as kind of wavelengths that can exist between plates becomes limited because of the very short gap. This will cause a higher pressure due to impact of colliding virtual photons from vacuum around compared to pressure caused by virtual photons contained in between, causing plates to get pushed together. This became known as Casimir effect. Until recently, scientists were not able to create experimental conditions required to measure the force predicted by Casimir’s equation.

F=(Πhc/480L⁴)A

F= Force experienced by the plates, h= Planck’s constant, c= Speed of light, L= Gap between plates, A= Area of plates

In 1996, Steven Lamoreaux finally succeeded in creating lab conditions required for the experiment and found the experimental results to be within 5% of the value predicted by Casimir. By 2011, more accurate experimental confirmations have been achieved. Particles other than Photons also affect the plates but in magnitudes too small to be measurable by current technology. Effect of Bosons is attractive on the other hand effect of Fermions is repulsive. Existence of Casimir effect reveals a lot about nature of reality. For one it entails a broken Super symmetry, because it indicates absence of Fermionic photinos. Super symmetry has not been detected yet. It has other implications such as quantum gravity and about nature of space-time itself. Dynamic Casimir effect is a direct derivation and predicts that if the two plates move to and fro at speeds close to speed of light then virtual particles of vacuum may become real by converting energy of motion of plates.

References:

1) https://www.nobelprize.org/nobel_prizes/physics/laureates/1922/bohr-lecture.pdf

2) http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/quantum_theory_ origins/
3) http://math.ucr.edu/home/baez/physics/Quantum/bells_inequality.html
4) http://www.casimir-network.org/IMG/pdf/Casimir_20effect.pdf

Image credits goes to respective sources.

Quantum entanglement- The advent of Quantum computing and Teleportation

In Quantum computing, iterations are done using Quantum phenomena such as Superposition and Entanglement. Analogous to bits of classical computing we have Qubits in Quantum computing. In some cases spin of an Electron is used as Qubit as it is an inherent property and it fulfills the requirements for Qubit, apart from this certain properties of atom, ion and even a photon which is nothing but quanta of energy, can be used as Qubit. Electron spin can be assigned a value of up or down, but we can only calculate the probability of either state. Until measured, it could have both spin values at same time.

Scientists are constantly looking for new candidates for Qubit. A Qubit does not have absolute state of either 0 or 1 like a classical bit rather it is a superposition of both states. Qubit can be 0, 1 and any value in between at the same time, until an act of measurement is performed which forces it to relinquish all possible states except for one. Act of measurement turns the Qubit into a classical bit and all the Quantum mechanical advantage is lost. This is where property of Entanglement helps out. Quantum Entanglement is the phenomena under which properties of two particles that have previously interacted are inextricably linked in such a way that any change in the state of one particle simultaneously changes the state of other, and this holds true even if the two particles are at opposite ends of universe. Superposition of Qubit is maintained while figuring out its state by performing the measurement on its Entangled pair from which state of Qubit under consideration is inferred. Quantum Entangled bits have higher correlation than two classical correlated bits as established by Bell's inequality. While performing measurements through Entanglement, high correlation between the two Qubits is desirable for faithful results. Correlation giving a fidelity of 96-97% has been achieved by Professor Andrea Morello and his team.

Fact that Qubit can be in two state at same time allows for performing millions of iterations simultaneously making Quantum computers astonishingly superior than classical ones. Quantum computers use sequence of Qubits. A Qubit can be 0, 1 and all points in between at same time. Quantum computer with two Qubits can be in 4 different states at same time. Quantum computer with n Qubits can be in 2^n different states at the same time on the other hand a classical computer can only be in one of these 2^n states at any given time. Thus with every additional Qubit computing power grows exponentially. A Qubit has 2 key states denoted as 0=(1 0) and 1=(0 1) known as basis states. Quantum computer uses these states to perform iterations  according to unitary matrix transformation. Theory and logic of Qubit computations is getting developed and numerous methods have been submitted to realize Quantum computing. One such method could be Quantum Annealing as indicated by T Lanting et al. Scientists so far have achieved successful multiplication of two integers using a Quantum computing method.

Quantum computers are based on behavior of  matter at Quantum level. Manner in which the spin, energy or speed of these particles changes on interaction is used to create the logic based on which Quantum logic gates are created which are then used to operate on a set of Quantum inputs so called Qubits to yield an output. Qubits are Input to Quantum computers but output comes in form of classical bits because act of getting an output forces particles to relinquish all states they can be in except for one. Binary computer uses binary code as input which is then operated upon using logic circuits made of binary logic gates such as AND, OR, XOR. A binary logic gate is a realization of binary operators such as AND or NOT operator and are made using transistors such as a CMOS transistor. Architecture of Quantum computer is different from architecture of binary computers. In a Quantum computer, Quantum transistors made of controlled Qubits are used to create Quantum gates in order to realize basic Quantum logical operators as in unitary matrix which are then used to create Quantum logic circuits designed to solve real world problems. Code corresponding to such circuits is developed and used for programming. Quantum algorithm is used for creating codes for solving problems using Quantum computers. Based on the definition of Qubit, Quantum transistors have been created such as single atom transistor wherein by controlling the state of Qubit, conduction path can be opened or closed. Using single electron Qubit, Quantum logic gates have been created such as the CNOT gate, a two Qubit gate wherein target Qubit flips its spin when control Qubit is pointing down and maintains its spin when control Qubit is pointing up. Here spin of electron serves as Qubit and control is exercised through microwaves. This 2 Qubit gate alongwith single Qubit operations can be used to create any other gate set. It gives us a way of creating Quantum computers with 100s of Qubits. For sheer processing power, Quantum computer with 300 Qubits will have more computing ability than all binary computers on Earth combined. Quantum computer with 300 Qubits in Entanglement will have processing power equivalent to 2³⁰⁰ bit conventional computer. 2³⁰⁰ is about the number of particles in observable universe.

Vector representation is used to present Quantum states mathematically. Mathematically a Qubit can be presented in terms of its basis vector states as
              
2 Qubits in terms of their 4 basis vector states can be shown as

Quantum logic gate operation represents the multiplication between matrix representing it and vector representing Quantum state of Qubits. Quantum logic gate acting on k Qubits is represented as 2^k x 2^k  unitary matrix. Qubits are input and output depends on type of  Quantum logic gate used. With clear understanding of Quantum state mathematics, logic circuits can be designed to realize operations such as addition, multiplication, division, encoding, decoding, multiplexing et al and to form registers to store information on Qubits. Quantum transistors are used to construct Quantum logic gates. Quantum computer components such as processor, memory, I/O devices are created using Logic circuits made of Quantum logic gates. Communication protocols for allowing communication between the various components has been developed and is getting improved. Toffoli, Feynman, CNOT, Pauli X, Pauli Y, Pauli Z, Fredkin are some of the Quantum logic gates we have. Pauli X gate corresponds to rotation of Bloch sphere around X axis by π radian. It flips the state of input Qubit. Other gates have their own effects. We have the math and the logic, work is going on to figure out ways of realizing it to build working Quantum computers.

Models of Quantum computing include Adiabatic, one way, Quantum gate array and Topological. Many methods are available for implementing a Quantum computer such as Nuclear magnetic resonance, Fullerine based ESR, Linear optical, Trapped ion and Quantum dot. Logic of Quantum computing is far more complex than classical binary logic and requires simplification. Apart from this Scientists have to deal with Decoherence while designing these Quantum computers. Decoherence is the characteristic of getting into disorderly and unorganized state due to external interference or internal causes. Quantum state of Qubit changes with slightest of disturbance. To do calulations its essential to maintain their state that is up spin or down spin state for example, for entire duration of calculation. Scientists try to work around this problem by keeping Qubits in super cooled, ultra vacuum environment causing them to get in Quantum mechanical ground state. Liquid Nitrogen or liquid Helium is used for cooling. Preservation of Quantum state has been achieved for a maximum period of upto 2-3 hours so far. With more and more research in Quantum error correction, factors that can affect the state of Qubits are being recognized and ways of neutralizing those factors are getting created by research teams around the world. We only have a certain probability of getting the expected result when computing using quantum mechanical properties because state of particles at Quantum level cannot be measured with absolute certainty, we can only have a probability of one result or other determined according to Heisenberg uncertainty principle. This is why Quantum algorithms have to be run several times in succession to get result expected as per Quantum logical operators (unitary matrix transformations).

Researchers at the National Institute of Standards and Technology (NIST) have now managed a significant breakthrough by ‘teleporting’ or transferring, quantum information from one photon to another over a distance of 100 km of optical fiber. In this experiment as shown in their infographic they created a Photon and then split it in two using a special crystal generating a pair of entangled photons whose states are identical. One of the Entangled Photon is transported to receiving end through a spool of Optical Fiber. Then they generate the input Photon and select its state either early, late or a superposition of both. Input and helper Photon are made to meet at a beam splitter with a 50/50 chance of getting straight through or reflecting at an angle. Detectors developed at NIST and based on superconducting nanowires made of molybdenum silicide are placed suitably to detect the arrival of Photons. When one detector clicks early and the other late, it means Photons are out of phase. Detectors at receiving station measure the state of output Photons from which the state of input photon can be inferred. Thus teleportation of Quantum information is achieved. NIST’s Marty Stevens says “Only about 1 percent of photons make it all the way through 100 km of fiber”.

This is nothing like the general idea of Teleportation but it’s a great progress towards achieving the same. After all this is how the first conventional computers were developed and over 50 years of constant improvements have given us the computer and internet as we know it. Just remember the early experiments on Silicon and Germanium crystals in order to develop the very first transistors. We know that teleportation in principle is achievable because of the insight provided by the team of C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. Wootters. They pointed that complete teleportation of quantum information can be achieved in theory at least. Say you want to teleport object A. You can do this by scanning the quantum state of A and another object B together. The unscanned part is transferred to another object C through B which is Entangled with C. Using the scanned data of original Quantum state of object A, one of many treatments can be applied to C in order to recreate the complete original Quantum state of A, transforming C into A. This theory has been used by many scientists to demonstrate quantum teleportation at numerous occasions.

Anton Zeilinger and his team are carrying out Photon teleportation between the Islands of La Palma and Tenerife, over a distance of 143 km through open space. In their experiments they create two identical Photons (Heralded Single Photon), one of which is transported to Tenerife over a high energy Laser. A third Photon which they are going to teleport is brought close to the Photon at La Palma and their interactions are observed. Due to entanglement the state of distant Photon changes with the state of the Photon at the sending station. The observations made at sending station are used to convert the photon at Tenerife into an exact copy of the third Photon. A number of Photons have already been teleported using this method.

These experiments are very early steps towards making Quantum communication possible and creating Quantum Internet. Since a qubit cannot be copied, as any attempt to do so will alter the information due to uncertainty principle, information can be sent and received securely over Quantum Internet. Work is going on for the development of Quantum computer at institutes like MIT, IQC and corporations like IBM, Google. With Quantum Internet all the Quantum computers in world can be instantaneously connected and with help of Quantum Entanglement  may be we could create a universal network one day, enabling us to communicate from anywhere in the universe instantaneously.

References:
1) https://arxiv.org/pdf/1501.00011.pdf

2) http://newsroom.unsw.edu.au/news/quantum-computing-taps-nucleus-single-atom

3) http://researcher.watson.ibm.com/researcher/files/us-bennetc/BBCJPW.pdf
4) https://arxiv.org/pdf/quant-ph/9511027.pdf

Image credits goes to respective sources.