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BEC AND MATTER WAVESan overviewAllan Griffin, University of Toronto

The discovery of Bose-Einstein condensation ( BEC ) in 1995 in dilute, ultracold trapped atomic gases is one of the most exciting developments in recent physics research. In 2002, there are over 40 labs around the world which can routinely produce these atomic condensates.My goal is to give you some understanding of these atomicBose condensates and why they are causing so much excitement in the world of physics. The discovery of BEC has led to a whole new field of ultracold atomic gases, and opened the door to a new magic kingdom .

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Nobel prize in Physics in 2001

Eric Cornell and Carl Wieman, NIST and University of Colorado, Boulder.Wolfgang Ketterle, MIT

1925 →1995“This discovery must be viewed as one of the most beautiful physics experiments of the 20th century” - Lev Pitaevskii

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Atomic Bose condensates can involve millions of atoms all occupying the identical single-particle quantum state.This is because the atoms are Bosons, and the exclusion principle does not hold. As a result these millions of atoms are now described by a macroscopic single particle wavefunction Φ(r).They are a macroscopic de Broglie matter wave.

These matter waves are best thought of as a macroscopic quantum order parameter, describing a new kind of condensed matter, like a solid or a liquid. A Bose-condensed gas is not a gas! Using external fields, these condensates can move, change shape, oscillate, and scatter off each other, producing spectacular interferencepatterns. They allow one to see quantum effects with our eyes!

A new kind of condensed matter

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Bosons and FermionsOrdinary matter is composed of elementary particles like

protons, neutrons and electrons and of “composite” particles such as atoms and molecules.

Neutrons, protons and electrons are all Fermions, since their spin S is an odd multiple of . Fermions obey the Pauli exclusion principle, which means that only one Fermion(of a given type) can occupy a single particle quantum state. This is the origin of the periodic table, since as one adds more electrons, they have to occupy different states or orbitals.

However, atoms may be Bosons if their net spin F is an even multiple of .The net spin is the total spin of all the Fermions (electrons, protons and neutrons) composing the atom. About 80% of the atoms in the periodic table are Bosons.

h2

h2

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4He ( 2 protons, 2 neutrons and 2 electrons) → Boson3He ( 2 protons, 1 neutron, 2 electrons) → FermionMost recent ultracold atom studies have used alkali atoms,since their electronic structure involves one s-electron outside a closed shell. Net spin of an alkali atom is F = I ±1/2 .

Bose atoms of choice: 87Rb , 85Rb 23Na , 7Li , 133CsFermi atoms of choice: 6Li , 40K

Atoms, atoms and more atoms

In quantum liquids, we only have 4He, 3He and 3He -4He mixtures.Ultracold gases are much richer! One can produce multicomponent quantum gases using (2F + 1)different hyperfine atomic states (mF).

A BEC of 85Rb- 85Rb molecules was recently produced byWieman’s group at JILA. A very hot topic now.

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How do we produce atomic matter waves?

Quantum effects are smeared out at high temperatures dueto thermal motion. This is why physicists have been on a long quest to go to lower and lower temperatures. Life is moreinteresting as T → 0, where more delicate phases of matter can become stabilized. Superfluidity only lives at low T.

BEC and ultracold atomic gases are just the latest spectacular discovery in this quest for absolute zero over the last century.

What do we mean by low temperatures?milli micro nano

273 K 1K 10-3 K 10-6 K 10-9 K( < 1875) (1910 -1960) ~ 1970 ~ 1980 > 1995

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What is the physics behind BEC in a gas?

Atoms in a gas are described by a quantum wavefunction, with a wavelength λ related to the atom’s momentum p by

Average atomic kinetic energy =

Thus we see in a gas of atoms in thermal equilibrium at a temperature T, the de Broglie relation gives

As T decreases , the wavelength λ increases (Einstein , 1925)

r p = m

r v =

hλ

λ ≈h

mkT

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mv 2 =p 2

2m≈ kBT

(de Broglie , 1924)

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When does a gas become quantum?

The average distance between the atoms in the gas = d(n)• The average thermal de Broglie wavelength = λ(T)The average density n ~ (the volume per atom) -1 ~ 1/ d3

At high T → λ(T) << d(n)At low T → λ(T) ≈ d(n) ← quantum featuresAt ultralow T → λ(T) >> d(n)

Clearly the wavelike nature of the atoms in the gas becomes crucial. The wavefunctions of different atoms now overlap, and the atoms move in a highly correlated collective manner.

When the λ ~ d , many Bosons start to occupy the lowest energy single particle state. At T = 0, all the Bose atoms in a gas occupy the zero momentum state (Einstein, 1925).

9From the Ketterle MIT homepage

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How do we trick nature to obtain a dilute gas at ultracold temperatures?

To get a BEC in a gas of a given density, we must go to low T toincresase λ ~ d . How do we prevent the interacting atoms from forming a solid? The solution is that in a very dilute gas,collisions between atoms are very infrequent. Thus it takes a long time for a cluster of atoms (precusor to a solid ) to nucleate.

This gives BEC a chance to be born - if we can make it happenfast enough (typical numbers are seconds). This was first achieved in a gas of 87Rb atoms in 1995 by Cornell and Wieman. Such gases are only metastable, but this is not a problem.

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How do we cool a dilute gas?Laser cooling gets us down to 10-5-10-6 K using a

magnetoptical trap or MOT:

Laser frequency is off - resonance , so only atoms moving towards laser source absorb photons ( Doppler effect) .

Total momentum of system = ( pat + pph ) ~ 0

After a photon is absorbed, the atom velocity decreases - coolersince temperature is proportional to (velocity)2. With 6 laser beams along ±x , ±y, ±z directions, no atom can escape!(1997 Nobel Physics Prize)

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What does a BEC apparatus look like ?

Here is one at the MPI in Munich. The U of T apparatus in Rm 054looks similar! The hope is that eventually, these will be available to be purchased off the shelf, just like a Helium liquifier!

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Evaporative cooling gets us to 10-8 -10-9 K

- The laser-cooled atoms are transferred to a magnetic trap, using the fact that the atoms are in a selected Zeemanlevel of the atomic hyperfine state being used. Other atomsare expelled by the magnetic trap.

- One then removes the high energy atoms from the trappedgas, using a rf em field.

- Remaining low energy atoms quickly re-thermalize and go to lower energy levels and to a lower temperature. Note that this requires that atoms have a large collision cross-section.

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Simple harmonic well trapping potentials

In most current BEC experiments, the magnetic field holding the ultracold gas corresponds to a simple anisotropic harmonic potential trap,

The trap frequencies ωo used are typically 10 ~ 200 Hz .pancake traps , ωox = ωoy << ωoz (2D in xy plane)cigar shaped traps , ωox = ωoy >> ωoz (1D along z axis)

The energy levels for atoms in an isotropic harmonic wellare separated by . In a non-interacting Bose gas of N atoms trapped in such a potential , the BEC transition temperatureis

TBEC , N ~ 105 - 106 atoms

V(

r r ) =

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m[ωox2 x2 +ωoy

2 y 2 + ωoz2 z2 ]

hωo

≅ N1/3hωo

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At temperatures just above TBEC , all the the lowest lyingenergy states are occupied according to the Bose distributionfunction

N E j( )=

1eβ E j −µ( ) − 1

, E j = jhωo , j = 0,1,2, ..

Below TBEC , the number of atoms in the lowest energy levelof the harmonic trap abruptly starts to increase. All the atoms occupy this state when T << TBEC .These atoms are the Bosecondensate of a trapped non-interacting gas , with N ~ 106 atoms in the same single-particle state.This is the atomic matter wave, given by the Gaussian ground state wavefunction of a SHO.

hω0←

A Bose- condensed gas is not a gas!!

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The density of atoms in a trap as the temperaturegoes from above TBEC to below

T > Tc T= Tc T << Tc

The peak on the right is almost a pure condensate(from MPI, Munich)

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Gross -Pitaevskii equation for matter wavesThese ultracold gases are extremely dilute, ten thousand

times more dilute than the air in this room! Yet, it turns out that the interatomic interactions between atoms absolutely controls their behaviour. This is the key to why they are so interesting.

The equation of motion for the condensatewavefunction was written down, independently, by Pitaevskii and Gross in 1961 in the theory of superfluid 4He ( where it is not valid).

The interactions enter by the Hartree mean-field , which one condensate atom feels from all the other condensate atoms:

VH (r,t) = dr'v(r − r')∫ nc(r',t)

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The condensate atoms are all in the state described by Φ(r,t), satisfying the “Schrodinger-like” wave equation

where the condensate local density nc(r,t) = |Φ(r,t)|2. Note that the GP equation is a closed (but non-linear) equation for Φ(r,t).

ih

∂Φ∂t

= −h2∇2

2m+Vext (r) + gnC (r,t)

Φ

Gross-Pitaevskii(GP) equation

At ultralow temperatures, a tremendous simplification occurs - only the l = 0 partial wave scattering contribution to the interatomic potential between two atoms is not frozen out. Thus we need only can keep the s-wave scattering length a and can use an effective pseudopotential (this is rigorous!)

v(r-r´) = g δ(r-r´) , g ≡ 4πa/m ≡ interaction strength

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The role of the s-wave scattering length a

Until 1995, values of a for alkali atoms were not well known. In the last few years, there have been hundreds of papers on this topic, since the value of a is the only parameter needed to describe the interactions in ultracold Bose and Fermi gases.

Moreover, one can tune the value of a by working close to a Feshbach resonance, where there is a molecular bound stateclose the total energy of two colliding atoms. One can then make a very large, decrease it to zero, or make it negative!

Repulsive interactions , a > 0 Attractive interactions , a < 0

This ability to tune the strength and sign of the interatomic interaction is a unique feature of ultracold atoms - for a theorist, they are the ideal interacting many body system.

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Dynamics of the macroscopic wavefunction

Solutions of the GP equation describe the behaviour of Φ(r,t). In equilibrium, Φ0(r) is time-independent. One can induce changes from equilibrium using external laser fieldsand excite collective oscillations of the condensate. These are like phonons in a crystal. Here is a centre of mass andquadrupole oscillation of 23Na atoms in a cigar-shaped trap:

From Ketterle MIT group

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Superfluid velocity vs(r,t)The dynamics of the condensate matter wave is that of superfluid, with no dissipation since we are dealing with “millions” of atoms all in one single quantumstate. This is the same physics as in superconductors and in superfluid 4He.

Φ(r,t) = (nc)1/2 eiS : Amplitude and Phase variables

The condensate density is nc(r, t ) = Φ(r, t)2

The superfluid velocity is .

It is the phase of Φ(r,t) which is the origin of all thewavelike and superfluid properties of the condensate.The motion is irrotational (curl vs = 0 ).

vs(r, t) =

h

m∇S(r,t)

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Condensate matter wave interferenceThe most direct evidence of the long range coherence

properties of matter waves are interference experimentsbetween different condensates. This year, the group at MPI in Munich produced a spectacular example of this.

They trapped their ultracold atoms in a periodic latticeassociated with the standing waves produced by intersecting laser beams ( Zeilinger calls this a crystal of light ).The atoms are trapped in the potential minima.

One can work out the Bosonic energy band structure of these Bloch Bosons moving in a periodic lattice. One gets Brillouin zones , Bragg scattering, etc. The big difference from electrons is that now each Bloch state can be occupied by thousands of atoms and are easily manipulated by lasers.

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Superfluidity of atoms in optical lattices Each lattice site can have a Bose condensate in it, and

its phase is coherently coupled to those on other lattice sites by (Josephson) tunneling if the potential barrier is less than a certain critical value. If the “crystal of light” is turned off,each condensate ballistically expands and constructivelyinterfers with condensate waves from other lattice sites.

from Henk Stoof, Nature, 2002

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Superfluid-Mott insulator phase transitionEvidence after optical lattice is turned off

Superfluid phase - all lattice site matter wave phases were coherent with each other.

Potential barriers are too high. Phases of different lattice site condensates were incoherent.

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What is the connection to superfluid 4He ?

First application of BEC concept was to liquid 4He in 1938 by Fritz London, where the superfluid transition is at Tc = 2.17K .

However, in liquid 4He with strong interactions, even at T = 0, only 9% of the 4He atoms are in the condensate.

The analogy between superfluid He and a dilute gas only appears at finite temperatures, where there are a lot of atoms thermally excited out of the condensate. If the collisions are strong enough to produce local equilibrium , one can derive the famous Landau two-fluid hydrodynamic equations

( Zaremba, Nikuni and Griffin, 1999)

Superfluid component = condensate atomsNormal fluid component = non-condensate atoms

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Rotating condensates and quantized vortices

One can have quantized atomic currents in a trapped gas in equilibrium - corresponds to another topologicalstate of the gas

Φo(r) = nc(r,θ) e iφ , 0 < φ < 2π ; phase S = φ

This atomic current around the z-axis moves with thesuperfluid velocity given by

vs(r, t) =

h

m∇S (r, t)

=

h

mφ∧ 1

rsinθ∂∂φ

(φ) =h

mρφ∧

ρ = (x2 + y2)1/2;

We note that vs ∝ 1/ρ → ∞ as ρ→ 0. This leads to the condensate density nc vanishing at ρ = 0. The vortex has a hole at centre.

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Rotating condensatesVortices can be produced by rotating a slightly anisotropic trapwith an angular speed greater than a certain critical value Ωc .Recent work has shown vortices are nucleated when a surface quadrupole mode = 2 (of frequency , where ω0 is the trap frequency ) becomes unstable and starts to grow. This critical rotation speed is given by Landau’s criterion Ωc = 0.7ω0 .

ωl= lω0 l

Note how densityvanishes at vortex core.

From Dalibard’s group, ENS, Paris (2001)

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Many vortices form the usual vortex lattice

Same triangular Abrikosov lattice as found in rotating superfluid Helium and in type II-superconductors in a magnetic field.

Current experiments produce up to 200 vortices

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Ultracold Fermi gases and BCS Cooper pairs!

One can also laser-cool Fermion atoms, often using Bosonsas a coolant. At ENS in Paris, for example, they have produced a degenerate Fermi gas of gas 6Li atoms ( Fermi temperature is TF = 4.6µK ) in a Bose gas of 7Li atoms ( TBEC =1.4µK ) .

The big race is now on to produce Cooper pairing between fermions in different atomic hyperfine states ( analogue of spin up and spin down in superconductors). One can use a Feshbach resonance to produce a large attractive interaction between the atoms. Evidence in 6Li ( F =1/2, mF = ±1/2) just appeared in Science.

All this activity is getting more and more CMP theoristsinvolved in these “ highly correlated atomic Fermion systems”

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BEC recently achieved in metastable He4He atoms in ground state has zero spin - cannot be trapped.

However, 4He in first excited (2s) state is metastable(lifetime is about an hour) . In 2001, BEC was produced in *He by two groups in Paris.

Thus 4He atoms can form a superfluidliquid as well as a superfluid gas.

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Ultracold atom research

Besides the USA, Germany, France, Italy and Austriaare probably the strongest groups -both theory and experiment.

Many young women physicists are playing leading roles.

BEC in Canada?Laser cooling in done at York and several other universities.

World-class theoretical work on ultracold atoms is being done at Queen’s (Eugene Zaremba) and Calgary (David Feder).

Aephraim Steinberg’s BEC group ( Jofre, Siercke, and Maone ) are getting very close. Visit their lab in Rm 054.Joesph Thywissen (from Harvard and Orsay) arrives at U of T in Jan., 2003, as a new faculty member in AMO.

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Conclusion

A beautiful new field of research has been born,a new superposition of:

atomic physics, laser physics, condensed matter physics, quantum optics, statistical physics, many body theory low temperature physics.

If you are in any of these sub-fields, you can jump in and have a lot of fun in the next decade!

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More information on BEC and quantum gases

1. An excellent text book has just been published,

BOSE-EINSTEIN CONDENSATION IN DILUTE GASES, by Pethick and Smith(Cambridge UP , 2002), also in paperback.

2. BEC World Homepage - has everything!http://amo.phy.gasou.edu/bec.html.

3. For more details on my own research , seehttp://www.physics.utoronto.ca/~griffin

4. The Nobel prize site http://www.nobel.se/physics/laureates/2001/