A review of atomic-gas Bose-Einstein condensation experiments for Workshop: “Hawking Radiation in condensed-matter systems” Eric Cornell JILA Boulder, Colorado

The basic loop. Cooling. Minimum temperature. Stray heating. Confinement. Magnetic. Optical. Reduced dimensions. Arrays Observables. Images. Shot noise. Atom counting. Interactions. The G-P equation. Speed of sound. Time-varying interactions. feshbach resonance. Reduced dimensions. Thermal fluctuations. A range of numbers.

The basic loop. (once every minute) Cooling. Minimum temperature. Stray heating. Confinement. Magnetic. Optical. Reduced dimensions. Arrays Observables. Images. Shot noise. Atom counting. Interactions. The G-P equation. Speed of sound. Time-varying interactions. feshbach resonance. Reduced dimensions. Thermal fluctuations. A range of numbers.

MOT 2.5 cm Laser cooling

Basic cooling idea leads to Basic cooling limit. Heating happens. The basic loop. Cooling. Minimum temperature. Stray heating. Confinement. Magnetic. Optical. Reduced dimensions. Arrays Observables. Images. Shot noise. Atom counting. Interactions. The G-P equation. Speed of sound. Time-varying interactions. feshbach resonance. Reduced dimensions. Thermal fluctuations. A range of numbers.

V(x) x  Self-interacting condensate expands to fill confining potential to height 

V(x) x n(x)  Self-interacting condensate expands to fill confining potential to height 

V(x) x n(x) kT Cloud of thermal excitations made up of atoms on trajectories that go roughly to where the confining potential reaches kT

V(x) x n(x)  When kT <  then there are very few thermal excitations extending outside of condensate. Thus evaporation cooling power is small. kT

BEC experiments must be completed within limited time. Bang! Three-body molecular-formation process causes condensate to decay, and heat! Lifetime longer at lower density, but physics goes more slowly at lower density!

Dominant source of heat: Bang! Decay products from three-body recombination can collide “as they depart”, leaving behind excess energy in still-trapped atoms.

Bose-Einstein Condensation in a Dilute Gas: Measurement of Energy and Ground-State Occupation, J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 77, 4984 (1996)

The basic loop. Cooling. Minimum temperature. Stray heating. Confinement. Magnetic. Optical. Reduced dimensions. Arrays Observables. Images. Shot noise. Atom counting. Interactions. The G-P equation. Speed of sound. Time-varying interactions. feshbach resonance. Reduced dimensions. Thermal fluctuations. A range of numbers.

F=2 F=1 m=1 m=-1 m=0 m=2 m=-2 Energy Typical single-atom energy level diagram in the presence of a magnetic field. Note Zeeman splitting.

U(B) B m=-1 m=0 m=+1 B(x) x

U(B) B m=-1 m=0 m=+1 B(x) x U(x) x m=-1 m=0 m=+1

U(z) z m=-1 m=0 m=+1 Why do atoms rest at equilibrium here? Gravity (real gravity!) often is important force in experiments

U(z) z m=-1 m=0 m=+1 Why do atoms rest at equilibrium here? U B (z)+mgz z m=-1 m=0 m=+1 Gravity (real gravity!) often is important force in experiments

Shape of magnetic potential. U=  x 2 +  y 2 +  z 2 Why? D L R electromagnet coils typically much larger, farther apart, than size of atom cloud D, L >> R so, order x 3 terms are small. Magnetic confining potential typically quadratic only, except…

Atom chip substrate Cross-section of tiny wire patterned on chip If electromagnets are based on “Atom chip” design, one can have sharper, more structured magnetic potentials.

If electromagnets are based on “Atom chip” design, one can have sharper, more structured magnetic potentials. But there is a way to escape entirely from the boring rules of magnetostatics….

Laser beam laser Laser too red +atom diffracts light +light provides conservative, attractive potential Laser too blue +atom diffracts light +light provides conservative, repulsive potential Laser quasi-resonant +atom absorbs light +light can provide dissipative (heating, cooling) forces Interaction of light and atoms.

One pair of beams: standing wave in intensity Can provide tight confinement in 1-D with almost free motion in 2-d. (pancakes) This is a one-D array of quasi-2d condensates

Two pairs of beams: Can provide tight confinement in 2-D, almost-free motion in 1-d (tubes) Can have a two-D array of quasi-1d condensates

Why K-T on a lattice? 1.Ease of quantitative comparison. 2. Makes it easier to get a quasi-2d system 3.This is the Frontier in Lattices Session!!! (come on!) Our aspect ratio, (2.8:1), is modest, but... addition of 2-d lattice makes phase fluctuations much “cheaper” in 2 of 3 dimensions. z, the “thin dimension” x, y, the two “large dimemsions”.

Three pairs of beams: Can provide tight confinement in 3-D, (dots) Can have a three-D array of quasi-0d condensates

Single laser beam brought to a focus. Rayleigh range Can provide a single potential (not an array of potentials) with extreme, quasi-1D aspect ratio. Additional, weaker beams can change free-particle dispersion relation.

The basic loop. Cooling. Minimum temperature. Stray heating. Confinement. Magnetic. Optical. Reduced dimensions. Arrays Observables. Images. Shot noise. Atom counting. Interactions. The G-P equation. Speed of sound. Time-varying interactions. feshbach resonance. Reduced dimensions. Thermal fluctuations. A range of numbers.

Observables: It’s all about images. Temperature, pressure, viscosity, all these quantities are inferred from images of the atomic density.

Near-resonant imaging: absorption. signal-noise typically good, but sample is destroyed. Off-resonant imaging: index of refraction mapping. Signal-noise typically less good, but (mostly) nondestructive..

V , 110 ms expansion Watching a shock coming into existence 110 ms

Bigger beam now 12.4 pixel, 7.2 mW using a bigger beam

Absorption imaging: one usually interprets images as of a continuous density distribution, but density distribution actually os made up discrete quanta of mass known, (in my subdiscipline of physics) as “atoms”.. Hard to see an individual atom, but can see effects of the discrete nature. Absorption depth observed in a given box of area A is OD= (N 0 +/- N 0 1/2 )  /A The N 0 1/2 term is the”atom shot noise”. and it can dominate technical noise in the image.

Absorption depth observed in a given box of area A is OD= (N 0 +/- N 0 1/2 )  /A The N 0 1/2 term is the”atom shot noise”. and it can dominate technical noise in the image. Atom shot noise will likely be an important background effect which can obscure Hawking radiation unless experiment is designed well.

Atom shot noise limited imaging Data from lab of Debbie Jin.

N.B: imaging atoms with light is not the only way to detect them: I think we will hear from Chris Westbrook about detecting individual metastable atoms.

The basic loop. Cooling. Minimum temperature. Stray heating. Confinement. Magnetic. Optical. Reduced dimensions. Arrays Observables. Images. Shot noise. Atom counting. Interactions. The G-P equation. Speed of sound. Time-varying interactions. feshbach resonance. Reduced dimensions. Thermal fluctuations. A range of numbers.

QM: Particle described by Schrödinger equation BEC: many weakly interacting particles  Gross-Pitaevskii equation Why are BECs so interesting? The condensate is self-interacting (usually self-repulsive)

BEC: many weakly interacting particles  Gross-Pitaevskii equation Can be solved in various approximations. The Thomas-Fermi approximation: ignore KE term, look for stationary states

BEC: many weakly interacting particles  Gross-Pitaevskii equation Can be solved in various approximations. The Thomas-Fermi approximation: ignore KE term, look for stationary states 

The Thomas-Fermi approximation: ignore KE term, look for stationary states V(x) x n(x)  Self-interacting condensate expands to fill confining potential to height 

BEC: many weakly interacting particles  Gross-Pitaevskii equation Can be solved in various approximations. Ignore external potential, look for plane-wave excitations

BEC: many weakly interacting particles  Gross-Pitaevskii equation Can be solved in various approximations. Ignore external potential, look for plane-wave excitations

Data from Nir Davidson c = (  /m) 1/2  = (hbar 2 /m) 1/2 speed of sound: Healing length: Chemical potential:  = 4  hbar 2 a n /m

n(x)  (x) Long wavelength excitations (k << 1/  ) relatively little density fluctuation, large phase fluctuation (which we can’t directly image).

n(x)  (x) But, if we turn off interactions suddenly (  goes to zero), and wait a little bit:

n(x)  (x) n(x)  (x) m goes to zero,  gets large now (k > 1/  ) the same excitation now evolves much larger density fluctuations. But, if we turn off interactions suddenly (  goes to zero), and wait a little bit: Can you DO that?

The basic loop. Cooling. Minimum temperature. Stray heating. Confinement. Magnetic. Optical. Reduced dimensions. Arrays Observables. Images. Shot noise. Atom counting. Interactions. The G-P equation. Speed of sound. Time-varying interactions. feshbach resonance. Reduced dimensions. Thermal fluctuations. A range of numbers.

Magnetic-field Feshbach resonance molecules → ← attractive repulsive BB > free atoms

Single laser beam brought to a focus. Suddenly turn off laser beam…. Data example from e.g. Ertmer.

The basic loop. Cooling. Minimum temperature. Stray heating. Confinement. Magnetic. Optical. Reduced dimensions. Arrays Observables. Images. Shot noise. Atom counting. Interactions. The G-P equation. Speed of sound. Time-varying interactions. feshbach resonance. Reduced dimensions. Thermal fluctuations. (a serious problem) A range of numbers.

The basic loop. Cooling. Minimum temperature. Stray heating. Confinement. Magnetic. Optical. Reduced dimensions. Arrays Observables. Images. Shot noise. Atom counting. Interactions. The G-P equation. Speed of sound. Time-varying interactions. feshbach resonance. Reduced dimensions. Thermal fluctuations. A range of numbers. (coming later)