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

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Transcript of Lecture 2. Why BEC is linked with single particle quantum behaviour over macroscopic length scales...
Lecture 2. Why BEC is linked with single particle quantum
behaviour over macroscopic length scales
Interference between separately prepared condensates
of ultracold atoms
Quantised vortices in 4He and ultracold 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 nonzero 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 rrn < 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 ultracold 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
Nonuniform 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
PenroseOnsagerCriterion 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
Nonlinear 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 nonlinear
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 GrossPitaevski Equation in
weakly interacting system
GrossPitaevski 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 givesnonlinear 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
rr’
ρ1(rr’)
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