BOSE-EINSTEIN CONDENSATTIONIncluding an introduction to Fermionic Condensates & Ultra-Slow light in a BEC

Aaron Flierl

SUNY BUFFALO

PHY 402

Spring 2016

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Statistical Distributions

• µ ≈ EF up to temperatures of 2000K

• F-D and B-E are limiting case where particle has wave

function with a λ comparable to interatomic spacing

• At high T B-E and F-D statistics converge to classical

regime and agree with M-B

• F-D always guarantees a 50% chance of finding a

Fermion at EF

• With increasing T F-D statistics shows increased chance of

exciting electrons into the conduction band

• At T ≈ 0K F-D statistics yields 0% chance of finding a

Fermion with E > EF “Fermi-Sea”

• Increase in T yields an increased chance of finding a

particle at a higher energy

• For Bosons, chemical potential must always be less than

the minimum allowed energy [Griffiths problem 5.30]

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Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016

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Statistical Distributions

• Plots of all 3 distributions with increasing T

• Note different behavior at low T, similar behavior at high T

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Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016

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What is a Bose-Einstein Condensate?• State of matter

• First predicted theoretically by Bose and Einstein in 1925

• First BEC created in 1995 using a gas of Rb cooled to 170nK

• Cool VERY dilute gas of non-interacting Bosons to near absolute

zero using combination of laser and evaporative cooling

• In a BEC Bosons macroscopically occupy the lowest energy state

• This “quantum degeneracy” occurs when de Broglie wavelength

becomes comparable to spacing between atoms

• BEC’s can have superfluid properties; behave as a fluid with zero

viscosity. Defy gravity and surface tension.

• BEC’s can have EXTREME optical properties

• In 1998 light was slowed to 17m/s in a BEC of Na atomsVelocity distribution [16]

Velocity distribution [1]

Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016

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What is a Bose-Einstein Condensate?

• At low T the de Broglie wavelength is large enough that

wave functions of individual atoms begin to overlap

• These particles can now be described by a single wave

function

• BEC forms when phase space density = 1 and at a

Temperature called the “critical temperature” Tc

λdB = ( 2πћ2/ mkBT )1/2

PSD = npkλ3dB

Npk is the peak number density of the sample

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Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016

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Critical Temperature

• Must occur when all of particles can barely

be accounted for in excited states

g(ϵ) is confining potential

• This means any further loss in KE will lead to

larger occupation of ground state

• At this temperature, chemical potential must

be zero (Griffiths problem 5.30)**

• Potential of trap approximated as 3d HO with

cylindrical symmetry

ρ2 = x2 + y2 + λ2z2 and

λ = ωz/ωr

(ratio of axial and radial trap

frequencies)

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BEC of 84Sr : T > Tc T ≈ Tc T < Tc

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[5]Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016

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Laser Cooling

Laser Cooling [15, 18] • Lasers are “detuned” to a frequency which corresponds to an energy BELOW a

desired energy level transition

• Due to Doppler effect, photon scattering occurs for atoms moving towards light

• Loss of momentum is in direction of motion

• After emission of photon, gained momentum is in random direction

• Repeat process many times, net loss of momentum

• Thermal energy related to KE, therefore a net loss in thermal energy

• Temperature limit due to mean squared velocity of random process.

• γ term is inverse of lifetime for excited state of the atom

Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016

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Evaporative Cooling“atom trap” [1]

• Combined with laser cooling the high phase space densities

required for BEC can be achieved

• Atoms trapped in “potential well”

• “hot” atoms with enough KE escape

• Slowly decrease well depth to achieve further cooling

• rf-induced “spin-flips” remove higher energy atoms

• Magnetic field vanishes at center of spherical quadrupole

potential where it changes direction rapidly. A hole in the trap!

[9] Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016

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• Magnetic moment of trapped atoms required to be

in opposite direction of magnetic field

• “Majorana flip” magnetic moment of some atoms

will flip without a field present

• Slightly “detuned” laser ultra-focused on this “hole”

creates a repulsive optical dipole force which acts

as a “plug”

• Optical dipole force arises due to coherent

interaction of inhomogenous EM field with induced

dipole moment of the atom

• These magneto-optic traps only permit study of

weak-field seeking states, whose spin degree of

freedom is frozen.

• Single spin state BEC “scalar” BEC

• Optical traps allow for study of states with non-zero

quantum number m

• Spinor BEC, spin-f BEC has 2f + 1 space/time varying

compoonents

U = -µ·B

Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016

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Fermionic Condensates

BEC “Fermi Sea”

• Pairs of Fermions have integer

spin and can form

condensates

• Major breakthrough was the

ability to control interactions

• Favor pairing such as cooper

pairs of electrons

• Current experiments aim to

study connection between

BEC’s, superfluidity, and

superconductivity

• BSC-BEC Crossover theory

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Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016

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Ultra-Slow Light in a BEC!

• BEC is illuminated with a coupling laser

• Optical properties of atoms can be dramatically altered

• Becomes a coupled atomic-light medium

• Coupling laser couples state |2 › and |3 ›

• lower level unoccupied, coupling laser splits higher level into two

symmetric energy levels

• Energy gap proportional to E of coupling laser

• Probe laser tuned to the |1 › |3 › transition is “injected” into the BEC

• It is this laser pulse which travels at extremely low group velocityDr Lene Hau - Harvard [17]

Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016

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References

[1] Bose-Einstein Condensation in a Gas of Sodium AtomsK. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle

Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139

[2] Bose–Einstein condensationof atomic gasesJames R. Anglin & Wolfgang KetterleResearch Laboratory for Electronics, MIT-Harvard Center for Ultracold Atoms, and Department of Physics, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139, USA

[3] The art of taming light: ultra-slow and stopped lightZachary Dutton, Naomi S Ginsberg, Christopher Slower, and Lene HauLyman Laboratory, Harvard University, Cambridge MA 02138

[4] Fermi CondensatesMarkus Greiner, Cindy A. Regal, and Deborah S. JinJILA, National Institute of Standards and Technology and University of Colorado,and Department of Physics, University of Colorado, Boulder, CO 80309-0440

[5] Bose-Einstein Condensate : http://massey.dur.ac.uk/resources/mlharris/Chapter2.pdf

[6] Creating new states of matter: Selim Jochim MPI für Kernphysik and UniversitätHeidelberg Experiments with ultra-cold Fermi gases Henning Moritz ETH Zürich.

[7] “Plugging the hole” : http://cua.mit.edu/ketterle_group/Projects_1995/Plugged_trap/Plugged_trap.htm

[8] “The Strontium Project” : http://www.strontiumbec.com/

[9] “Cooling and Trapping Techniques With Ultra-cold Atoms” : http://large.stanford.edu/courses/2009/ph376/amet1/

[10] Spinor Bose-Einstein condensates Yuki Kawaguchi 1a, Masahito Uedaa,baDepartment of Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan bERATO Macroscopic Quantum Control Project, JST, Tokyo 113-8656,

Japan

[11] Chapter 5 lecture slides Dr. Hao Zeng : UBLearns.buffalo.edu

[12] https://commons.wikimedia.org/wiki/File:Mplwp_Fermi_Boltzmann_Bose.svg

[13] "Bose-Einstein, Fermi-Dirac, and Maxwell-Boltzmann Statistics" from the Wolfram

Demonstrations Project

http://demonstrations.wolfram.com/BoseEinsteinFermiDiracAndMaxwellBoltzmannStatist

ics/

[14] Introduction to Quantum Mechanics, 2nd Edition by Griffiths, David J., Pearson Education 2005

[15] “Laser Cooling” : https://en.wikipedia.org/wiki/Laser_cooling

[16] “Bose-Einstein Condensate” : https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate

[17] Q+A with Dr Lene Hau : http://www.physicscentral.com/explore/people/hau.cfm

[18] “Doppler Cooling” : https://en.wikipedia.org/wiki/Doppler_cooling

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