a review of atomic-gas bose-einstein condensation experiments for workshop: “hawking radiation in...

A review of atomic-gas Bose-Einstein condensation experiments for

Workshop: “Hawking Radiation in condensed-matter systems”

Eric Cornell JILABoulder, 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.

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 trajectoriesthat 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 completedwithin limited time.

Bang!

Three-body molecular-formation processcauses 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 recombinationcan 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. ArraysObservables. 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=1m=1

m=-1m=0

m=2

m=-2Energy

Typical single-atom energy level diagramin 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 atomsrest at equilibriumhere?

Gravity (real gravity!) often is important force in experiments

U(z)

z

m=-1

m=0

m=+1

Why do atomsrest at equilibriumhere?

UB(z)+mgz

z

m=-1

m=0

m=+1

Gravity (real gravity!) often is important force in experiments

Shape of magnetic potential.

U= x2 + y2 + z2 Why?

D

L

R electromagnet coilstypically much larger, farther apart, than size of atomcloud

D, L >> R

so, order x3 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 structuredmagnetic potentials.

If electromagnets are based on “Atom chip”design, one can have sharper, more structuredmagnetic potentials.But there is a way to escape entirelyfrom 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 providedissipative (heating,cooling) forces

Interaction oflight and atoms.

One pair of beams: standing wave in intensity

Can provide tight confinement in 1-D with almostfree 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 system3.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.

Rayleighrange

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-noisetypically good, but sampleis destroyed.

Off-resonant imaging:index of refraction mapping. Signal-noise typically less good, but (mostly) nondestructive..

Watching a shock coming into existence110 ms

using a bigger beam

Absorption imaging: one usuallyinterprets images as of a continuousdensity 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= (N0 +/- N0

1/2) /A

The N01/2 term is the”atom

shot noise”. and it can dominatetechnical noise in the image.

Absorption depth observed in a given box of area A is OD= (N0 +/- N0

1/2) /AThe N0

1/2 term is the”atomshot noise”. and it can dominatetechnical noise in the image.

Atom shot noise will likely be an important backgroundeffect which can obscure Hawking radiation unlessexperiment 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

rErVm2 extern

2

BEC: many weakly interacting particles Gross-Pitaevskii equation

rrrm

a4V

m22

2

extern

2

Why are BECs so interesting?

The condensate isself-interacting (usually self-repulsive)

BEC: many weakly interacting particles Gross-Pitaevskii equation

t

rirr

m

aV

m extern

222 4

2

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

t

rirr

m

aV

m extern

222 4

2

Can be solved in various approximations.

The Thomas-Fermi approximation:ignore KE term, look for stationary states

externVrm

a 2

24

The Thomas-Fermi approximation:ignore KE term, look for stationary states

externVrm

a 2

24

V(x)

x

n(x)

Self-interacting condensate expands to fill confining potential to height

BEC: many weakly interacting particles Gross-Pitaevskii equation

t

rirr

m

aV

m extern

222 4

2

Can be solved in various approximations.

Ignore external potential, look for plane-waveexcitations

BEC: many weakly interacting particles Gross-Pitaevskii equation

t

rirr

m

aV

m extern

222 4

2

Can be solved in various approximations.

Ignore external potential, look for plane-waveexcitations

Data from Nir Davidson

c = (/m)1/2

= (hbar2/m)1/2

speed of sound:

Healing length:

Chemical potential:

= 4 hbar2 a n /m

n(x)

(x)

Long wavelengthexcitations(k << 1/)

relatively little density fluctuation,large phase fluctuation(which we can’t directly image).

n(x)

(x) But, if we turn offinteractions suddenly ( goes to zero), and wait a little bit:

n(x)

(x)

n(x)

(x)

m goes to zero, gets largenow (k > 1/)

the same excitation nowevolves much larger densityfluctuations.

But, if we turn offinteractions 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>

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)