Transcript Slide 1

Lecture 2.
Why BEC is linked with single particle quantum
behaviour over macroscopic length scales
http://cua.mit.edu/ketterle_group/
Quantised vortices in 4He and
ultra-cold trapped gases
Interference between
separately prepared condensates
of ultra-cold atoms
Outline of Lecture
Single particle behaviour over macroscopic length scales
is 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
Ground state of
4He.
ψS(r)
ψS(r) is MPWF Ψ(r,s), normalised over r
ψS(r) occupies the spaces between
particles at s
ψS(r) is non-zero within volume
Feynman Model
of at least fV
ψS(r) = 0 if |r-rn| < a
(7% of total volume in 4He)
ψS(r) = constant otherwise
f = volume of white regions
J. Mayers PRL 84 314, (2000)
PRB64 224521,(2001)

1
f S  n(0)   S (r )dr
V
24
atoms

2
Feynman model
Δf
Probability distribution for fS becomes
narrower and more Gaussian for large N
Width of Gaussian is ~1/√√N
192
atoms
f
f

1/ N
S
(r )dr
has same value for all
possible s in
macroscopic system
The independence of s of the integral

S
(r )dr
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
3. Why independence of s ?
Similar to physical reason why number
of particles in large volume Ω of fluid is
independent of s.
Rigorous result of liquids theory
N 
~
N
1
N
Volume of spaces between particles
similarly becomes independent of s


S
(r ) d r
independent of s
Basic Assumptions
1.
ψS(r) is a delocalised function of r- Necessary consequence of BEC
2. Fluid of uniform macroscopic density.
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.
How does
g (s)    S (r )dr
vary with s?
V
G   P (s) g (s)ds
Define
,
g i (s)   (r | s)dr
i
Integral over single cell
Divide V into
N cells of
volume V/N
Average of
1 atom/cell
g (s)   gi (s)
i


G   P(s) g (s)  G ds
2
2
Uniform density - all cells give the same average contribution
 P(s) g (s)ds  G / N  g
i
i
Cell fluctuation with s
gi (s)  gi (s)  g
Arrangement of atoms near cell i is not
correlated with that near widely separated cell j.
+
j
Short range interactions
Form of ψS(r) within cell i not
correlated with that within cell j
Δgi(s), Δgj(s) uncorrelated
gi ~ g
For cells of size
V/N, NΩ~1
N ~ N ~ N
gi (s) ~ g
N
Sign of Δgi(s) varies randomly with i
 g (s) ~
i
Ng
i
g (s)   gi (s)  [ g  gi (s)]
i
Random walk
i
 Ng  ~ N g
 G ~ G / N
Gaussian
distribution
for g(s)
Consequence of delocalisation of ψS(r)
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
Second Demonstration


G 2   P(s) g (s)  G ) ds
2
2


  P(s)  g i (s)  g  ds
 i 1

N
g (s)   gi (s)
i
G  Ng
  P(s) gi (s)g j (s)ds
i, j
  P(s)[gi (s)] ds
2
i
G 2 ~ Ng 2  G 2 / N
g(s) = G ± ~G/√N
No correlations in fluctuations
of widely separated cells i ≠ j
gi (s) ~ g
Potential energy
  v(rn  rm ) (r1, r2 ,...rN ) dr1dr2 ...drN
2
n
=N<v>
m n
<v> is mean potential
energy of each particle
All n make same contribution
 v    v(r1  rm ) (r1 , r2 ,...rN ) dr1dr2 ...drN
2
m1
r  r1 , s  r2 ..rN
 (r, s)  P (s) S (r )
 v   v(s) P (s) ds
2
2
v(s)    v(r  rm ) S (r) dr
2
m1

 v 1 ~ 1 / N

Independent
of s
Kinetic Energy
 2 * (r1 , r2 ..rN )
2
*

dr1dr2 ..drN
  (r1, r2 ..rN )
2
2m n V
rn
    (s) P(s)ds
V
=Nκ
κ is mean kinetic
energy/atom
 2 S (r )
2
*
 (s)  
 S (r)
dr
2

V
2m
r
 

N
Independent
of s
Kinetic and potential energy can be accurately calculated in macroscopic
system by calculating single particle integral for any possible s
Non-uniform particle density
Cell of volume Ω centred at r
Contains on average NΩ >>1 atoms
ΔNΩ/ NΩ ~1/√NΩ
assume density varies sufficiently
slowly that it is constant within cell
N
 (r ) 

within single cell
.r
Integrals over cell at can be treated in same way
as integrals over total volume V at constant density

1
 S (r)dr   (r ) 1 ~ 1 / N 

  (r )

Same for all possible
s if NΩ is large
1/√NΩ fluctuations in NΩ do not change this

1
2

 S (r ) dr   (r ) 1 ~ 1 / N 

 (r )

Same for all possible
s if NΩ is large
Coarse grained average of potential energy
1
2

v(r 'rm ) S (r ) dr  v (r )



(
r
)
 (r )
m
 (r)  
(r )
 S (r) dr
2
Mean potential energy
of particle in Ω(r)
Normalisation factor
Coarse grained average of kinetic energy
 2 S (r)
2
1
*

 S (r)
dr   tot (r)
2


(
r
)
2m  (r)
r
Mean kinetic energy
of particle in Ω(r)
Localised ψS(r)
X


F [ S (r )]dr  0
S
Integrals over Ω are not
independent of s
if ψS(r) is localised


F [ S (r )]dr 0
S'
X
The order parameter
1 (r, r)   * (r) (r) if
r r  
Penrose-Onsager
Criterion for BEC
α(r) is the “order parameter”
Single particle
density matrix
1 (r, r)    * (r, s) (r, s)ds
(r, s)  P(s) S (r)
*

1 (r, r )   P(s)ds S (r ) S (r)
Coarse grained average of SPDM
1 (r, r) 
1
1


d
r
dr (r, r)




(
r
)

(
r
)


  P(s)ds
 (r ) 
1
1
*
*




 

(
r
)
d
r

S
S (r ) dr




(
r
)

(
r
)


1
 S (r)dr


(
r
)


Means equal to within
terms ~ 1/√NΩ
1 (r, r)   P(s) (r ) (r)ds
 P(s)ds  1
1 (r, r)   (r) (r)
1 (r, r)   (r) (r)
1
 (r )    S (r)dr
 (r )
•Order parameter is coarse grained average of ψS(r).
•Valid for averages over macroscopic regions of space
•New thermodynamic variable created by BEC
 (r)    (r, s) ds   P(s) S (r ) ds
2
2
Microscopic density
1
2
1



 (r)    (r )dr   P(s)ds    S (r) dr
 (r )
  (r )

1
2
 S (r) dr


(
r
)

Macroscopic density
Same for all s to ~1/√NΩ
 P(s)ds  1
1
2


(
r
)
dr   (r) ~
S


(
r
)

1
N
Macroscopic density is
integral of ψS(r) over Ω(r)
for any possible s
Coarse grained average of many particle wave function
 (r1 , r2 ..rN 
2
1
N

 ( r1 )
dr1
 ( r2 )
dr2 ..
 ( rN )
 (r1, r2 ..rN drN
2
Integral of each coordinate over cube of volume Ω
2
2
 1
 1

(r1 , r2 ..rN   N 1  dr2 ..
drN P(r2 ...rN )     S (r1) dr1
 ( r2 )
 ( rN )

   (r1 )

 P (r2 ..rN ) (r1 )
Same is true for r2, r3 etc
(r1 , r2 ..rN  P (r2 , r3 ..rN )  (r2 )  P (r1 , r2 ..rN )  (r3 )
2
N
 (r1 , r2 ..rN    (rn )
2
n 1
Single particle behaviour
over macroscopic length scales
Coarse grained average of N particle Schrödinger equation
2  2




(
r
)


v
(
r

r
)


i



n
n
m
2
2
m
t
rn
n 1
m n
N
  2 *  2
2
2
* 

 (rn )    v(rn  rm )    i


2
t
rn
n 1  2m
m n

N
1
2
3
4
Ψ is real
4
i [ 2 ]
* 
i 

t
2 t
i  N

 (rn )

2 t n1
 (rn )   (rn )2

   (rn ) 
   (rm ) (rn )  i


t


 n m n

3
Potential energy of interaction
between particles.
Consider term n=1

2
  v (rn  rm ) 

n 1  m  n

N
2
v
(
r

r
)

 1 m
m1

2
 1
 1
  N 1  dr2..
drN P(r2...rN )     v(r1  rm ) S (r1) dr1
 ( r2 )
 ( rN )

   (r1 ) m1

 P (r2 ...rN )   (r1 )v (r1 )
N


2
 v(rn  rm )     (rm ) (rn )  v (rn ) (rn )

n 1 m  n
  n 1 m n

N
Kinetic energy
 2 *  2



2
2m
rn
n 1
N
1
3
N

   (rm ) (rn )   (rn ) (rn )
 n1 m n

N


(rn )     (rm ) (rn )  T (rn ) (rn )


 n1 m n

2
N
 
 (rn ) 
  (rm ) (rn )   (rn )  v (rn )  T (rn ) (rn )  i t   0

 n1 m n
 
Every particle satisfies
 (r )
 (r)  v (r)  T (r) (r)  i
t
 (r )
 (r)  v (r)  T (r) (r)  i
t
Derivation neglects contribution to kinetic energy due to long range
variation in particle density.
This contributes extra term
 2  2 (r)

2m r 2
 2  2 (r )
 (r )

 (r )   (r )  v (r ) (r )  i
2
2m r
t
Microscopic kinetic and potential energy gives effective single particle potential
veff (r)   (r)  v (r)
veff (r)   (r)  v (r)
 (r )
is mean kinetic energy/particle at uniform macroscopic density
v (r)
is mean potential energy/particle
Both depend upon
 (r)   (r)2
 2  2 (r )
 (r)
2

 (r) (r)  veff [  (r) ] (r)  i
2
2m r
t
Non-linear single particle Schrödinger equation
 (r )
Limits of validity
 2  2 (r )
 (r)
2



(
r
)

(
r
)

v
[

(
r
)
]

(
r
)

i

eff
2m r 2
t
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
 S (r)   (r)

V
 (r ) dr  f
2
The order parameter is not
the condensate wave function
  (r)
V
2
1
Single particle Schrödinger equation is valid for any Bose
condensed system irrespective of size of condensate fraction
Calculations in a dilute Bose gas give
veff (r )   (r )
2
 2  2 (r )
 (r )
2



(
r
)

(
r
)

A

(
r
)

(
r
)

i

2m r 2
t
Reduces to Gross-Pitaevski Equation in
weakly interacting system
Gross-Pitaevski equation
My Equation
Requires presence of BEC
Requires delocalisation
(implied by BEC)
Can only be derived in weakly
Interacting system
Derivation valid for any
strength of interaction
Existing derivations assume
particle number is not fixed
Derivation valid for fixed
or variable N
Valid only in weakly
Interacting system
Valid for any
strength of interaction
Summary
Delocalisation is necessary consequence of BEC
Delocalisation implies that integrals over r of quantities
involving ψS(r) are independent of s
Order parameter is integral of ψS(r) over macroscopic region of space.
Coarse grained average of single particle wave function factorises
Coarse grained average of many particle Schrödinger equation gives
non-linear single particle equation
BEC implies single particle behaviour over macroscopic length scales
True for any size of condensate fraction
Division of Κtot(r)
2

 S (r)
 2 1
*
 tot (r) 
 S (r)
dr
2


(
r
)
2m  (r)
r
Contribution due to short range structure in ψS(r)
Contribution due to long range variation in average density over V
S (r)   S (r) / (r)
 (r)   (r)
 2 S (r)
 2  S (r)
 S (r)  (r)
 2 (r)
  (r)
2
  S (r)
2
2
r
r
r
r
r2
 2  S (r)
2
1
*

 S (r) (r)
dr
2


(
r
)

2M  (r)
r
2
1
 (r)  S (r)
*



(
r
)
2
dr
S


(
r
)
2M  (r)
r
r


1
1   (r)
2


(
r
)
dr
S
2


(
r
)
2M  (r)  (r) r
2
2
 (r)
const within Ω(r)
 (r) / r
const within Ω(r)
 2 (r) / r2
const within Ω(r)
Mean kinetic energy
of particle in gnd state
at constant density
Proportional to mean
momentum of atoms
in gnd state at constant
density =0 ±~1/√NΩ
 2 1  2 (r)

2M  (r) r 2
 2 1  2 (r)
 tot (r)  
  (r)
2
2m  (r) r
Kinetic energy due to structure of
density on macroscopic scales
Mean kinetic energy of particle in
system at constant density
Denote average over s as <
>S
 P(s)ds S (r) S (r)   S (r) S (r)
*
1 (r, r)   S* (r) S (r)
1 (r, r)  ˆ (r )ˆ (r)
*
S
Quantum average of ψS over
possible particle positions s
Quantum average over
field operator
 ˆ (r)
S
Here
Standard Theory T=0
Order
Parameter
*
suggests
Order
Parameter
  S (r)
S
Penrose criterion for BEC
1 (r, r)   S* (r) S (r)   S* (r)  S (r)
S
S
S
r r  d
Integrate over r,r/
If Ω is sufficiently large
S
2
 S
S

S
 S
ρ1(r-r’)
1
r  r ~ d Makes negligible
contribution
2
d
S
f
|r-r’|

2
0
S
S
must be independent of s
Hence this definition is consistent
with proven properties of ψS(r)
in ground state
Finite T
 B j (T ) P(s)ds jS (r) jS (r)   jS (r) jS (r)
*
*
Notation
j
Standard Theory
Order
Parameter
 ˆ (r)
Here
suggests
Quantum and thermal
average over field operator
1
 js (r )    js (r )dr
 
Order
  jS (r )
Parameter
Quantum average over s given j
Thermal average over states j
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
 jS (r)   S e
i j ( s )
 (r)1 ~ 1/
N

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
Only superfluid component contributes to interference effects




2
 (r )   N (r )  2 S (r ) cos (Mv.r / )
New testable prediction
Not necessary to assume that interference
fringes are created by observation
Total number of particles is fixed, but necessary
to assume that condensates exchange particles
ΔN1 = ΔN2~√N