1. 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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2. 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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3. 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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4. 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 atoms
Velocity distribution
[16]
Velocity distribution [1]
Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016
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5. 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 = npkλ3
dB
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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6. 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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Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016
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7. 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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8. 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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9. [7]
• 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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10. 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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11. 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 velocity
Dr Lene Hau - Harvard [17]
Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016
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12. References
[1] Bose-Einstein Condensation in a Gas of Sodium Atoms
K. 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 condensation
of atomic gases
James R. Anglin & Wolfgang Ketterle
Research 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 light
Zachary Dutton, Naomi S Ginsberg, Christopher Slower, and Lene Hau
Lyman Laboratory, Harvard University, Cambridge MA 02138
[4] Fermi Condensates
Markus Greiner, Cindy A. Regal, and Deborah S. Jin
JILA, 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 Universität
Heidelberg 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,b
aDepartment 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
Aaron Flierl PHY 402 SUNY BUFFALO Spring 2016
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