Bunching & Anti-Bunching of Quantum Particles: Applications to Cold Atoms in Optical Lattices

1. Bunching & Anti-Bunching of Quantum Particles:Applications to Cold Atoms in Optical Lattices Indu Satija George Mason University Collaborators: Ana Maria: Harvard Univ Charles Clark, NIST

3. Fermions have half units of spin, and tend to shy away from each other, like people who always stay in single rooms at the fermion motel. Bosons have zero or integer units of spin, and like to be with each other, like people who stay in shared dormitories at the boson inn. There are two types of particles in nature: fermions and bosons.

4. Ultra-cold Matter Research The nucleus of an atom is a fermion or boson depending on whether the total number of its protons and neutrons is odd or even, respectively Fermions: 3He , 40K Bosons: 4He , 87Rb In atomic gases at ultra-low temperatures (10-1000 nK), the physics of the gas is no longer classical and can only be described by the laws of quantum mechanics. When a system is made of identical indistinguishable particles, quantum mechanics predicts that their wave functions have to obey following properties: Two identical Bosons : f = (f1 f2 + f2 f1) Two identical Fermions : f = (f1 f2 - f2 f1)

5. Simultaneous Detection: Bosons , Fermions & Tennis Balls (1)Particles at both detectors come from source 1 Amplitude equal to A1. (2) Particles at both detectors come from source 2 Amplitude equal to A2, (3) Paricle at detector 1 comes from source 1 and at detector 2 comes from 2 Amplitude equal to A3 (4) Paricle at detector 2 comes from source 1 and at detector 2 comes from 1 Amplitude equal to A4 Classically, all paths are distinguishable and total probability of simultaneous detector is the sum of the 4 amplitude squares However, if particles are bosons or fermions, the last two processes are indistinguishable I= (A1)2+(A2)2 +(A3 A4)2where respectively refers to bosons and fermions. 1 2 1 2 special case, A1=A2=A3=A4=A P= 4(A)2for classical particles P= 6(A)2 for bosons particles P= 2(A)2 for fermions particles

6. Hanbury-Brown and Twiss (HBT) effect

8. News and Views Nature 418, 377-379 (25 July 2002) Quantum physics: Spaced-out electrons John C. H. Spence Top of page Abstract In a stream of photons, the particles tend to bunch together, but electrons in a beam do the opposite. At last, this quantum effect for free electrons — the Hanbury Brown–Twiss anticorrelation — has been seen experimentally. Like the gentle patter of raindrops, we expect photons, the quanta of sunlight, to arrive at Earth at random intervals, their arrival times distributed in just the same, natural way that customers arrive at a box office to buy tickets for a play. A histogram of the number of people or photons arriving per unit time follows what is known as the Poisson distribution. But in 1909, in the first clear evidence for wave–particle duality, Einstein pointed out1 that the width of this distribution for sunlight contains both the Poisson contribution of random arrival times, and a second 'wave-noise' contribution, which causes the photons to arrive in bunches. An experiment performed by Kiesel et al.2, reported on page 392 of this issue, shows that the same principle applies to a beam of coherent electrons — but with the opposite effect. In contrast to the bunching of photons3, this experiment confirms the theoretical prediction4 that a stream of coherent electrons will 'anti-bunch', tending to become more equally spaced than the classical Poisson prediction (Fig. 1). In the quantum regime, electrons have an innate tendency to avoid each other, thereby demonstrating a fundamental difference in the way light and electrons interfere with themselves. Figure 1: Random and bunched distributions. Figure 1 : Random and bunched distributions. Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com a, The vertical lines represent a random sequence, such as the times at which theatre-goers arrive at the box office or photons might arrive at Earth from a distant star. Mathematically, this distribution was described by Poisson (1781–1840) and bears his name. b, But in fact the case for photons is not quite so straightforward. Quantum correlations cause the photon arrival times to bunch together — an effect exploited by Hanbury Brown and Twiss to measure the angular size of stars. c, For electrons, the reverse is true: quantum effects cause free electrons to 'anti-bunch', or spread out. Kiesel et al.2 have now measured this 'Hanbury Brown–Twiss anticorrelation'.

9. Physics in Action Oct 2, 2002 Electron antibunching finally made beautiful The Hanbury Brown and Twiss effect - one of the classic experiments in quantum optics - has been observed with free electrons for the first time Interference is a classic property of waves. It can be seen most clearly when a coherent wave is split into two partial waves that are then recombined to produce a pattern of bright and dark fringes on a screen. The classic interferometer set-up is the Young’s double-slit experiment: a small light source emitting in a narrow frequency band is placed on one side of the slits, and an interference pattern is observed at the screen on the other side. The number of fringes that can be seen in the interference pattern is a measure of the coherence of the source. What I have just described is well known and widely taught to undergraduates. Unfortunately, it is less well known that coherence can also be probed by simply placing two detectors behind the two slits and recording the respective light intensities as a function of time. This second approach to measuring coherence is a variant of an experimental set-up that was invented half a century ago by the astronomers Robert Hanbury Brown and Richard Twiss to measure the size of Sirius by determining the coherence of light from the star with two spatially separated telescopes.

10. Intensity Interferometry ( HBT Interferometry) Jeltes et al, Nature 2007

11. Young’s Double-Slit Experiment Bragg Diffraction

12. Electrons emitted from a source are sent to the electron biprism. The electrons are attracted toward the central filament and overlap in the electrons arrived lower detector plane at the detector are displayed as bright spots on the monitor. Even when the electron arrival rate is as low as 10 electrons/sec, the accumulation of single electrons forms a biprism interference pattern MOST BEAUTIFUL EXPERIMENT, 2002

13. 1 Young's double-slit experiment applied to the interference of single electrons 2 Galileo's experiment on falling bodies (1600s) 3 Millikan's oil-drop experiment (1910s) 4 Newton's decomposition of sunlight with a prism (1665-1666) 5 Young's light-interference experiment (1801) 6 Cavendish's torsion-bar experiment (1798) 7 Eratosthenes' measurement of the Earth's circumference (3rd century BC) 8 Galileo's experiments with rolling balls down inclined planes (1600s) 9 Rutherford's discovery of the nucleus (1911) 10 Foucault's pendulum (1851) Others experiments that were cited included: Archimedes' experiment on hydrostatics Roemer's observations of the speed of light Joule's paddle-wheel heat experiments Reynolds's pipe flow experiment Mach & Salcher's acoustic shock wave Michelson-Morley measurement of the null effect of the ether Röntgen's detection of Maxwell's displacement current Oersted's discovery of electromagnetism The Braggs' X-ray diffraction of salt crystals Eddington's measurement of the bending of starlight Stern-Gerlach demonstration of space quantization Schrödinger's cat thought experiment Trinity test of nuclear chain reaction Wu et al.'s measurement of parity violation Goldhaber's study of neutrino helicity Feynman dipping an O-ring in water

14.  Folling et al Nature, 2005 Intrinsic quantum noise: Noise Correlations n=n(x)-<n(x)> <n(x+d/2)n(x-d/2)>

16. HBT Interferometry Each absorption image conatins large fluctuations related to intrinsic quantum noise, and their HBT correlations contain information about the spatial order in the system. Noise correlations (HBT) not only complement, but provide lot more useful informaton than the momentum distribution ( Bragg ).

17. Metal-Insulator ( superfluid-insulator ) Transition in Two-Color Lattice a/b = irrational a b • Hard core bosons: fermionic and bosonic characteristics, fractional Mott • Electrons • Superfluid-Mott Transtion in Period-2 bosonic Lattice • Rotating Bosonic Ring: Entanglement and Mott transition *Entanglement Measure

18. Optical lattices Imagine you could produce an artificial crystal for quantum matter, defect free and with complete control over the periodic crystal potential. The shape of the periodic potential, its depth and the interactions between the underlying particles could be changed at will and the particles could be moved around in a highly controlled way, at essentially zero temperature. This sounds almost too good to be true, but it is in fact what optical lattices have made possible for cold and ultracold atoms.

30. Peak to Dip Transition in HCB

34. Period-2 Bosonic Systems No metal-insulator transition: Mott transition for finite U Fermionic character for large U Fractional Mott Constructing Phase Diagram using Noise Correlation

38. Entanglement: Quantum Correlations with NO classical Counterpart Fluctuation driven zero temperature Quantum Phase transitions are characterized by long range correlations. It has been suggested that the critical point may correspond to maximally entangled state. In other other words, property responsible for long range correlations is the entanglement Therefore, it is important to quantify entanglement of systems exhibiting quantum phase transition . Conjecture: Entanglement can be quantified by Noise Correlations At present, there is no consensus as to the best method to define an entanglement measure for all possible multipartile states. However, there is an unambiguous way to construct suitable measures in two cases

40. I think I finally understand atoms

41. CONCLUSIONS (The Tragedy of Hamlet, by Shakespeare): There are more thing inheaven and earth,Horatio, than are dreamt of inyour philosophy.

42. (1)Metal-Insulator Transition Revisited for Cold atoms in Non-Abelian Gauge Potentials", Indubala I Satija, Daniel K Dakin and Charles W Clark, Phys Rev Letter, 97, 216401, 2006 (2)Quantum Coherence of Hard Core Bosons and Fermions: Extended, Glassy and Mott Phases", Ana Maria Rey, Indubala I Satija and Charles W Clark, Phys Rev A, 73, 063610, 2006. (3)”Noise Correlations of Hard-Core Bosons: Quantum Coherence and Symmetry Breaking, Ana Maria Rey, Indubala I Satija and Charles W Clark, J. Phys B, 39, S177, 2006 (4)Hanbury Brown -Twiss Interometry for Fractional and Integer Mott Phases", Ana Maria Rey, Indubala I Satija and Charles W Clark, New Journal of Physics, Special Issue on Cold Atoms, 8, 2006, 155 (5)” Rotating bosonic ring lattice: Entanglement and Phase Transition", Ana Maria Rey, Keith Burnett, Indubala I Satija, and Charles W Clark, Phys rev A, 75, 063616 (2007) (6) " Noise Correlations of Fermions and Hard Core Bosons in a Quasiperiodic Potential", Ana Maria Rey, Indubala I Satija and Charles W Clark, Laser Physics, 2006