Lecture 2. Why BEC is linked with single particle quantum behaviour over macroscopic length scales...

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Lecture 2. Why BEC is linked with single particle quantum

behaviour over macroscopic length scales

Interference between separately prepared condensates

of ultra-cold atoms

Quantised vortices in 4He and ultra-cold trapped gases

http://cua.mit.edu/ketterle_group/

Single particle behaviour over macroscopic length scalesis a consequence of the delocalisation of Ψ(r1,r2…rN)

This delocalisation is a necessary consequence of BEC.

Delocalisation leads to;

A new thermodynamic quantity – the order parameter.

Factorisation of Ψ over macroscopic length scales

A macroscopic single particle Schrödinger equation.

Macroscopic single particle behaviour

Outline of Lecture

Ground state of 4He.

ψS(r) is MPWF Ψ(r,s), normalised over r

ψS(r) is non-zero within volume

of at least fV

(7% of total volume in 4He)

ψS(r)

ψS(r) occupies the spaces between

particles at s

ψS(r) = 0 if |r-rn| < a

ψS(r) = constant otherwise

f = volume of white regions

Feynman Model

192atoms

2)(1

)0( rrSS dV

nf

rrS d)(has same value for allpossible s inmacroscopic system

Probability distribution for fS becomes

narrower and more Gaussian for large N

f

f

N/1

Δf

Width of Gaussian is ~1/√√N

J. Mayers PRL 84 314, (2000) PRB64 224521,(2001)

24 atoms

Feynman model

The independence of s of the integral is a general

property of the ground state wave function of any Bose condensed system

Other physically relevant integrals over ψS(r) are also independent of s to ~1/√N

Due to delocalisation of the wave function in presence of BEC

Leads to single particle behaviour over macroscopic length scales

rrS d)(

3. Why independence of s ?

Volume of spaces between particlessimilarly becomes independent of s

rrS d)( independent of s

Similar to physical reason why numberof particles in large volume Ω of fluid isindependent of s.

NN

N 1~

Rigorous result of liquids theory

Basic Assumptions

3. Pair correlations extend only over distances of a few interatomic spacings. • definition of a fluid

4. Interactions between particles extend only over a few interatomic spacings

• true for atoms - i.e. liquid helium and ultra-cold trapped gases.

• implicit in assumption 3.

2. Fluid of uniform macroscopic density.

1. ψS(r) is a delocalised function of r- Necessary consequence of BEC

,

i

igg )()( ss

sss dgPG )()( sss dGgPG22 )()(

Vdg rrs S )()( How does vary with s?

Divide V intoN cells ofvolume V/N

Average of1 atom/cell

Define ii dg rsrs )|()(

Integral over single cell

Uniform density - all cells give the same average contribution

gNGdgP i /)()( sss

ggg ii )()( ssCell fluctuation with s

i

j

Arrangement of atoms near cell i is not correlated with that near widely separated cell j.

Short range interactions

+

Form of ψS(r) within cell i not

correlated with that within cell jΔgi(s), Δgj(s) uncorrelated

For cells of size V/N, NΩ~1ggi ~ NNN ~~ ggi ~)(s

])([)()( i

ii

i gggg sss

NGG /~

gNgN ~

Random walk

Gaussian distributionfor g(s)

Sign of Δgi(s) varies randomly with i gNgN

ii ~)( s

Argument fails if ψS(r) is localised function of r

Only ~1 cell contributes to integrals

No cancellation of fluctuations from large number of cells

Consequence of delocalisation of ψS(r)

sss dGgPG22 ))()(

sss dggPN

ii

2

1

)()(

iigg )()( ss

gNG

ssss dggPji

ji,

)()()(

sss dgPi

i

2)]([)( No correlations in fluctuations

of widely separated cells i ≠ j

ggi ~)(sNGgNG /~ 222

Second Demonstration

g(s) = G ± ~G/√N

Potential energy

sss dPvv )()( rrrrs S dvvm

m

2

1

)()()(

All n make same contribution

n

NNnm

mn dddv rrrrrrrr ...),...,()( 21

2

21

NNm

m dddvv rrrrrrrr ...),...,()( 21

2

211

1

22)()(),( rssr SPNrrsrr .., 21

<v> is mean potential energy of each particle=N<v>

Nv /1~1Independentof s

Kinetic Energy

n

V N

n

NN ddd

mrrr

r

rrrrrr ..

)..,()..,(

2 21221

*2

21*

2

VdP sss )()(

Vd

mr

r

rrs S

S 2

2*

2 )()(

2)(

=Nκκ is mean kinetic energy/atom

Kinetic and potential energy can be accurately calculated in macroscopicsystem by calculating single particle integral for any possible s

N

Independent of s

Non-uniform particle density

.r

Cell of volume Ω centred at r

assume density varies sufficiently slowly that it is constant within cell

N

)(r within single cell

Contains on average NΩ >>1 atoms

ΔNΩ/ NΩ ~1/√NΩ

Integrals over cell at can be treated in same way as integrals over total volume V at constant density

1/√NΩ fluctuations in NΩ do not change this

Nd /1~1)()(1

)(rrr

r S Same for all possibles if NΩ is large

Nd /1~1)()(1

)(

2rrr

r S Same for all possibles if NΩ is large

Coarse grained average of potential energy

)()()'()(

1)(

2rrrrr

r r S vdvm

m

)(

2)()(

r S rrr d Normalisation factor

Coarse grained average of kinetic energy

Mean potential energyof particle in Ω(r)

)( 2

2*

2

)()(

)()(

1

2 r

SS rr

r

rr

r totdm

Mean kinetic energyof particle in Ω(r)

Localised ψS(r)

X

Integrals over Ω are notindependent of s if ψS(r) is localised

S

X

0)]([ rrS dF

S'0)]([ rrS dF

The order parameter

ssrsrrr d),(),(),( *1Single particle

density matrix

)()(),( *1 rrrr rrif

Penrose-OnsagerCriterion for BEC

α(r) is the “order parameter”

)()()(),( *1 rrssrr SS dP

)()(),( rssr SP

Coarse grained average of SPDM

)(

)(1

)(r S rrr d

)()(1 ),(11

),(rr

rrrrrr dd

)(

*

)(

* )(1

)(1

)(r Sr S rrrrss dddP

Means equal to withinterms ~ 1/√NΩ

srrsrr dP )()()(),(1

1)( ss dP

)()(),(1 rrrr

•Order parameter is coarse grained average of ψS(r).

•Valid for averages over macroscopic regions of space

•New thermodynamic variable created by BEC

)()(),(1 rrrr

)(

)(1

)(r S rrr d

srsssrr S dPd22

)()(),()( Microscopic density

)(

2

)()(

1)()(

1)(

r Srrrssrrr ddPd Macroscopic density

N

d1

~)()(1

)(

2

r S rrr

1)( ss dP

)(

2)(

1r S rr d Same for all s to ~1/√NΩ

Macroscopic density is integral of ψS(r) over Ω(r)

for any possible s

)(

2

21)( 2)( 1

2

21 ..,(..1

..,(21 N

NNNN dddrrr

rrrrrrrrr

Coarse grained average of many particle wave function

Integral of each coordinate over cube of volume Ω

)( 1

2

1)( 2)(21

2

2112

)(1

)...(..1

..,(r Sr r

rrrrrrrrr dPdd NNNNN

)()..,()()..,(..,( 321232

2

21 rrrrrrrrrrr NNN PP

N

nnN

1

2

21 )(..,( rrrr

)()..( 12 rrr NP

Same is true for r2, r3 etc

Single particle behaviourover macroscopic length scales

tiv

m

N

n nmmnn

n

12

22

)()(2

rrrr

ti n

nn

nmm

)()()(

rrr

N

nnt

i

1

)(2

r

Coarse grained average of N particle Schrödinger equation

tiv

m

N

n nmmnn

n

*

1

22

2

2*

2

)()(2

rrrr

2)()( nn rr

1 2 3 4

t

i

ti

][

2

2*

Ψ is real

4

N

n nmmnv

1

2)( rr

Consider term n=12

11 )(

mmv rr

)( 1

2

11

1)( 2)(2112

)()(1

)...(..1

r Sr rrrrrrrrr dvPdd

mmNNN

N

)()()()()(11

2nn

N

n nmnm

N

n nmmn vv rrrrrr

Potential energy of interactionbetween particles.

3

)()()...( 112 rrrr vP N

Kinetic energy

N

n nm12

2*

2

2 r

)()()()(1

nn

N

n nmnm rrrr

0)(

)()()()()()(1

t

iTv nnnnn

N

n nmnm

rrrrrrr

Every particle satisfies

t

iTv

)(

)()()()(r

rrrr

1

2)( nr )()()()(

1nn

N

n nmnm T rrrr

3

t

iTv

)(

)()()()(r

rrrr

Derivation neglects contribution to kinetic energy due to long range

variation in particle density.

t

ivm

)(

)()()()()(

2 2

22 rrrrr

r

r

This contributes extra term 2

22 )(

2 r

r

m

)()()( rrr vveff

Microscopic kinetic and potential energy gives effective single particle potential

)()()( rrr vveff

)(r is mean kinetic energy/particle at uniform macroscopic density )(r

)(rv is mean potential energy/particle

Both depend upon 2)()( rr

Non-linear single particle Schrödinger equation

tiv

m eff

)(

)(])([)()()(

2

2

2

22 rrrrr

r

r

Limits of validity

tiv

m eff

)(

)(])([)()()(

2

2

2

22 rrrrr

r

r

Accurate to within ~1/√NΩ where NΩ is number of atoms within resolution vol. Ω

Describes time evolution of particle density if this is a meaningful concept

Valid providing Ψ(r,s) is delocalised over macroscopic length scales.

BEC implies that all particles satisfy the same non-linear

Schrödinger equation on macroscopic length scales

)()( rrS

fdV

rr2

)( 1)(2 V r

Single particle Schrödinger equation is valid for any Bose

condensed system irrespective of size of condensate fraction

The order parameter is not

the condensate wave function

Calculations in a dilute Bose gas give

2)()( rr effv

tiA

m

)(

)()()()()(

2

2

2

22 rrrrr

r

r

Reduces to Gross-Pitaevski Equation in

weakly interacting system

Gross-Pitaevski equation My Equation

Can only be derived in weakly Derivation valid for anyInteracting system strength of interaction

Requires presence of BEC Requires delocalisation (implied by BEC)

Valid only in weakly Valid for anyInteracting system strength of interaction

Existing derivations assume Derivation valid for fixed particle number is not fixed or variable N

Delocalisation is necessary consequence of BEC

Order parameter is integral of ψS(r) over macroscopic region of space.

Delocalisation implies that integrals over r of quantities involving ψS(r) are independent of s

Coarse grained average of single particle wave function factorises

Coarse grained average of many particle Schrödinger equation givesnon-linear single particle equation

Summary

BEC implies single particle behaviour over macroscopic length scales

True for any size of condensate fraction

Division of Κtot(r)

Contribution due to short range structure in ψS(r)

Contribution due to long range variation in average density over V

)(/)()( rrr SS )()( rr

2

2

2

2

2

2 )()(

)()(2

)()(

)(

r

rr

r

r

r

r

r

rr

r

rS

SSS

)( 2

2*

2 )()(

)(

1

2)(

r

SS r

r

rr

rr d

mtot

rr

rrr

r r

SS

d

M )( 2

2*

2 )()()(

)(

1

2

Mean kinetic energyof particle in gnd stateat constant density

)(r

const within Ω(r)

rr

r

r

rr

r r

SS

d

M )(

*2 )()(

2)()(

1

2

Proportional to mean momentum of atomsin gnd state at constantdensity =0 ±~1/√NΩ

rr /)(

const within Ω(r)

rrr

r

rr r S

d

M )(

2

2

22

)()(

)(

1

)(

1

2

22 /)( rr

const within Ω(r)2

22 )(

)(

1

2 r

r

r

M

Mean kinetic energy of particle in system at constant density

)()(

)(

1

2)(

2

22

rr

r

rr

mtot

Kinetic energy due to structure of density on macroscopic scales

SSSSS rrrrss )()()()()( ** dP

Denote average over s as < >S

Standard Theory T=0

)(ˆ)(ˆ),( *1 rrrr

Quantum average overfield operator

SSS rrrr )()(),( *

1

Here

Quantum average of ψS overpossible particle positions s

)(ˆ rOrderParameter

SS r)(OrderParameter

suggests

SSS

SS

SS rrrrrr )()()()(),( **1 d rr

Integrate over r,r/

22

SS

SS

Penrose criterion for BEC

If Ω is sufficiently large d~rr Makes negligible contribution

must be independent of s SHence this definition is consistent with proven properties of ψS(r)in ground state

|r-r’|

ρ1(r-r’)

f

1

d

02

SSS

)()()()()()( ** rrrrss SSSS jjjjj

j dPTB

Finite T

Notation

Standard Theory

)(ˆ rOrderParameter

Quantum and thermalaverage over field operator

)(rSjOrderParameter

suggests

Here

Quantum average over s given j Thermal average over states j

rrr ss djj )(

1)( must be the same for all j and s

Finite T

ψjS(r) = √ρSexp[iφj(s)] ψS(r) + ψSR(r)

• ψS(r) is phase coherent ground state• ψSR(r) is phase incoherent in r

Ne jiSj /1~1)()( )( rr s

S

Phase φj(s) must be the same for all j and s

Physical interpretation

When BEC first occurs particular N particle state j is occupied with a random value of φj(s)

Delocalisation of wave function implies thermally induced transition to states with different phase must occur simultaneously over macroscopic volume

Therefore very unlikely – like transition to different direction of M in ferromagnet.

Hence broken symmetry – states of different phase are degenerate but only one particular phase is accessible

Not broken gauge symmetry. Particle number is fixed.

Interference between condensates

Not necessary to assume that interferencefringes are created by observation

Total number of particles is fixed, but necessaryto assume that condensates exchange particlesΔN1 = ΔN2~√N

Only superfluid component contributes to interference effects

)/.(cos)(2)()( 2 rvMrrr SN

New testable prediction