Download 2 - IS MU

Cold atoms
15th
Lecture 5.
November, 2006
Preliminary plan/reality in the fall term
Lecture 1
…
Something about everything (see next slide)
The textbook version of BEC in extended systems
Sep 22
Lecture 2
…
thermodynamics, grand canonical ensemble, extended
gas: ODLRO, nature of the BE phase transition
Oct 4
Lecture 3
…
atomic clouds in the traps – independent bosons, what
is BEC?, "thermodynamic limit", properties of OPDM
Oct 18
Lecture 4
…
atomic clouds in the traps – interactions, GP equation at
zero temperature, variational prop., chem. potential
Nov 1
Lecture 5
…
Infinite systems: Bogolyubov theory, BEC and symmetry
breaking, coherent states
Nov 15
Lecture 6
…
Time dependent GP theory. Finite systems: BEC theory
preserving the particle number
2
Previous class:
Interacting atoms
L4: Scattering length, pseudopotential
Beyond the potential radius, say
propagates in free space
3 , the scattered wave
For small energies, the scattering is purely isotropic , the s-wave
scattering. The outside wave is
sin(kr   0 )

r
For very small energies the radial part becomes just
r  as , as ... the scattering length
This may be extrapolated also into the interaction sphere
(we are not interested in the short range details)
Equivalent potential ("pseudopotential")
U (r )  g   (r )
4 as
g
m
2
4
Previous class:
Mean-field treatment of interacting atoms
L4: Many-body Hamiltonian and the Hartree approximation
1 2
1
ˆ
H 
pa  V (ra ) 
2
a 2m
 U (ra  rb )
a  b
We start from the mean field approximation.
This is an educated way, similar to (almost identical with) the
HARTREE APPROXIMATION we know for many electron systems.
Most of the interactions is indeed absorbed into the mean field and
what remains are explicit quantum correlation corrections
1 2
ˆ
H GP  
pa  V (ra )  VH (ra )
a 2m
VH (ra )   drbU (ra  rb ) n(rb )  g  n(ra )
n(r )   n   r 
2
self-consistent
system

 1 2

p

V
(
r
)

V
(
r
)
H

  r   E   r 
 2m

6
L4:
Gross-Pitaevskii equation at zero temperature
Consider a condensate. Then all occupied orbitals are the same and
we have a single self-consistent equation for a single orbital
2
 1 2
p  V (r )  gN 0  r  0  r   E00  r 

 2m

Putting
 (r )  N  0 (r )
we obtain a closed equation for the order parameter:
The lowest level
coincides with the
chemical potential
2
 1 2
p  V (r )  g   r    r     r 

 2m

This is the celebrated Gross-Pitaevskii equation.
For a static condensate, the order parameter has ZERO PHASE.
Then
 ( r )  N  0 ( r )  n( r )
N [ n]  N   d 3 r  ( r )   d 3 r  n( r )  N
2
7
Gross-Pitaevskii equation – homogeneous gas
The GP equation simplifies
2

2
  g   r    r     r 

 2m

For periodic boundary conditions in a box with V  Lx  Ly  Lz
1
0 (r ) 
V
 ( r )  N  0 ( r ) 
N
 n
V
g   r    r     r 
2
... GP equation
  g   r   gn
2
 1
E
1 3  2
1
2
2
 d r 
( n )  V (r )n  g n   g n
N
N
2
 2m
 2
8
Field theoretic reformulation
(second quantization)
Field operator for spin-less bosons
Definition by commutation relations
 (r ), † (r' )    (r  r' ),  (r ), (r' )   0,  † ( r ), † ( r' )   0
basis of single-particle states (  complete set of quantum numbers)
 
    
r     r 
      ,  ... single particle state
r  r   
decomposition of the field operator
  r     r  a , a  "   "   d3 *  r   r 
 †  r   *  r  a†
commutation relations
 a , a†     ,  a , a   0,  a† , a†   0
10
Field operator for spin-less bosons – cont'd
Plane wave representation (BK normalization)
  r   V 1/ 2  ei kr ak , ak  V 1/ 2  d3 r e  i kr   r 
 †  r   V 1/ 2  e i kr ak†  V 1/ 2  ei kr a-k†
 ak , ak'†    kk' ,  ak , ak'   0,  ak† , ak'†   0
11
Operators
Additive observable
X Xj
 X    d3 r d3 r'  † (r ) r X r'  (r' )
General definition of the OPDM
X    d 3 r d 3 r'  † (r ) r X r'  (r' )    d 3 r d 3 r' r X r'  † (r ) (r' )
   d 3 r d 3 r' r X r' r'  r  Tr X 
r'  r
Particle number
N  1OP, j

N   d3 r  † (r ) (r )
N   a† a
12
Hamiltonian
H 
a
1 2
1
pa  V (ra ) 
2m
2

 U (r
a
a  b
 rb )

  d r  (r )  2 m   V (r )  (r )  12  d 3 r d 3 r'  † (r ) † (r' )U (r  r' ) (r' ) (r )
3
†
2
Particle number conservation
H
,N 0
Equilibrium density operators and the ground state
P P (H ),
N , P   0
Typical selection rule
 (r )  Tr ( r )P  0
is a consequence of the gauge invariance of the 1st kind:
Tr P  Tr ei N P e i N  Tr e i N  ei N P  e  i Tr P
13
Hamiltonian of the homogeneous gas
H   2m k 2 ak† ak  12 V 1 U q ak†  q , ak'† q ak' ak ,
2
kk'q
U k   d3 r e i kr U  r 
k q
k
k'  q
k'
14
Action of the field operators in the Fock space
basis of single-particle states
 
    
      ,  ... single particle state
r  r   
r     r 
FOCK SPACE space of many particle states
basis states … symmetrized products of single-particle states for bosons
specified by the set of occupation numbers 0, 1, 2, 3, …
 , , ,
, p ,
 n   n1 , n2 , n3 ,
, np ,
1
2
3

n -particle state n  Σn p
a †p n1 , n2 , n3 ,
, np ,
 n p  1 n1 , n2 , n3 ,
a p n1 , n2 , n3 ,
, np ,
 n p n1 , n2 , n3 ,
, n p  1,
, n p  1,
15
Bogolyubov method
Basic idea
Bogolyubov method
is devised for boson quantum fluids with weak interactions – at T=0 now
no interaction
g 0
weak interaction
g 0
1 N  N BE   ak† ak  N BE
N  N BE  a0† a0
1
k 0
The condensate dominates.
Strange idea
N 0  a0† a0
1  a0† a0
a0† a0  a0 a0†  like c-numbers
a0  N 0 , a0†  N 0
N  N 0   ak† ak
... mixture of c-numbers and q -numbers
k 0
17
Approximate Hamiltonian
Keep at most two particles out of the condensate
H   2m k 2 ak† ak  12 V 1 U q ak†  q , ak'† q ak' ak
2
kk'q
  2m k a a
UN
 2V0
 a a
 4 a a  ak a  k 
  2m k a a
 UN
2V
 a a
 2 a a  ak a  k 
2
2
k
2
2
†
k k
†
k k
k
† †
k k
†
k k
k
† †
k k
†
k k
k
condensate particle
UN 2
 2V
k
k
k
UN 02
 2V
k
k
k
18
Bogolyubov transformation
Last rearrangement
H 
1
2
  2m k
2
2

 gn a a  a a k  
†
k k
†
k
gn
2
a a
† †
k k
 ak a k 
k
UN 2
 2V
anomalous
mean field
Conservation properties: momentum … YES, particle number … NO
NEW FIELD OPERATORS notice momentum conservation!!
bk  uk ak  vk a† k
b†k  vk ak  uk a† k
ak  uk bk  vk b†k
a† k  vk bk  uk b†k
requirements

New operators should satisfy the boson commutation rules
bk , bk'†    kk' , bk , bk'   0, bk† , bk'†   0
iff uk2  vk2  1

When introduced into the Hamiltonian, the anomalous terms have to
vanish
19
Bogolyubov transformation – result
Without quoting the transformation matrix
H
1
2
 (k ) b b  b b

†
k k
†
k k

UN 2
 2V
 higher order constant
independent quasiparticles
 (k ) 

2

 (k )

2
2 m k  gn   gn   2 m k
2
2
2
asymptotically
merge
2
k
2m
 (k )  c  k
k

c
gn
m
2
2
2
2
k
 2 gn
2m
high energy region
quasi-particles are
nearly just particles
sound region
quasi-particles are
collective excitations
20
More about the sound part of the dispersion law
Entirely dependent on the interactions, both the magnitude of the velocity
and the linear frequency range determined by g
Can be shown to really be a sound:
V VV E

c "
"
,

mn
2
UN
E

2V
 (k )  c  k
c
gn
m
Even a weakly interacting gas exhibits superfluidity; the ideal gas does not.
The phonons are actually Goldstone modes corresponding to a broken
symmetry
The dispersion law has no roton region, contrary to the reality
The dispersion law bends upwards  quasi-particles are unstable, can
decay
21
Particles and quasi-particles
At zero temperature, there are no quasi-particles, just the condensate.
Things are different with the true particles. Not all particles are in the
condensate, but they are not thermally agitated in an incoherent way, they
are a part of the fully coherent ground state
ak† ak   vk bk  uk b†k   uk b k  vk bk†   vk2  0
The total amount of the particles outside of the condensate is
N  N0
8 3/ 2 1/ 2

as n
N
3 
as3 n
the gas parameter
is
the expansion variable
22
Coherent ground state
Reformulation of the Bogolyubov requirements
Looks like he wanted
a0    ,   N , so that
a0 
This is in contradiction with the rule derived above,
a0  0
The above equation is known and defines the ground state to be a coherent
state with the parameter 
For a coherent state, there is no problem with the particle number
conservation. It has a rather uncertain particle number, but a well defined
phase:
2
†
  e

/2
 e a0 vac
 a0  
 a0† a0   
2
 a a a a   
†
†
0 0 0 0
 N0  
4
2
24
The end
ADDITIONAL NOTES
On the way to the mean-field Hamiltonian
26
ADDITIONAL NOTES
On the way to the mean-field Hamiltonian
 First, the following exact transformations are performed
ˆ
ˆ
W
V
1
1
2
ˆ
H 
pa  V (ra )
2
a 2m
a

Uˆ
U (ra  rb )
a  b
Vˆ  V (ra )   d 3 r V (r )  (r  ra )   d 3r V (r )  nˆ (r )
a
1
ˆ
U
2
a

U (ra  rb ) 
a  b
1
2
3
3
d
r
d
r ' U  r  r '

particle
density operator
    r  ra   r ' rb 
a  b
TRICK!! 



  d 3 r d 3 r 'U  r  r '    r  ra     r ' rb     r  r ' 
2
a
b
SI
eliminates
nˆ (r' )
nˆ (r ) 
1
(self-interaction)
1
Hˆ  Wˆ   d3 r V (r )  nˆ (r )   d 3 r d 3 r 'U  r  r '  nˆ (r ) nˆ (r' )    r  r ' 
2
27
ADDITIONAL NOTES
On the way to the mean-field Hamiltonian
 Second, a specific many-body state is chosen, which defines
the mean field:
  n(r )  nˆ (r )   nˆ (r ) 
Then, the operator of the (quantum) density fluctuation is defined:
nˆ  r   n  r    nˆ  r 
nˆ  r  nˆ  r'   nˆ  r  n  r'   n  r  nˆ  r'    nˆ  r   nˆ  r'   n  r  n  r' 
The Hamiltonian, still exactly, becomes


Hˆ  Wˆ   d 3 r V (r )   d 3 r' U  r  r '  n(r' )  nˆ (r )


1
2
1
2
3
3
d
r
d
r 'U  r  r '  n( r ) n( r' )

3
3
d
r
d
r 'U  r  r '  nˆ (r ) nˆ ( r' )  nˆ ( r )  r  r ' 

28
ADDITIONAL NOTES
On the way to the mean-field Hamiltonian
 In the last step, the third line containing exchange, correlation
and the self-interaction correction is neglected. The mean-field
Hamiltonian of the main lecture results:


Hˆ  Wˆ   d 3 r V (r )   d 3r' U  r  r '  n(r' )  nˆ (r )


1
d r d r 'U  r  r '  n( r ) n( r' )

2
1
2
3
3
VH  r 
substitute back
nˆ (r )    (r  ra )
a
and integrate
3
3
d
r
d
r 'U  r  r '  nˆ (r ) nˆ (r' )  nˆ (r )  r  r ' 

REMARKS
• Second line … an additive constant compensation for doublecounting of the Hartree interaction energy
• In the original (variational) Hartree approximation, the self-interaction
is not left out, leading to non-orthogonal Hartree orbitals
BACK
29
ADDITIONAL NOTES
Variational approach
to the condensate ground state
30
ADDITIONAL NOTES
Variational estimate of the condensate properties
 VARIATIONAL PRINCIPLE OF QUANTUM MECHANICS
The ground state and energy are uniquely defined by
E   Hˆ    ' Hˆ  ' for all  '  H NS ,  '  '  1
In words,  ' is a normalized symmetrical wave function of N particles. The
minimum condition in the variational form is
  Hˆ   0 equivalent with the SR Hˆ   E 
 HARTREE VARIATIONAL ANSATZ FOR THE CONDENSATE WAVE F.
For our many-particle Hamiltonian,
1 2
1
ˆ
H 
pa  V (ra ) 
2
a 2m
 U (ra  rb ),
U (r )  g   (r )
a  b
the true ground state is approximated by the condensate for non-interacting
particles (Hartree Ansatz, here identical with the symmetrized Hartree-Fock)
  r1 , r2 ,
, rp ,

 
, rN  0  r1 0  r2  0 rp
0  rN 
31
ADDITIONAL NOTES
Variational estimate of the condensate properties
Here,  0 is a normalized real spinless orbital. It is a functional variable to be
found from the variational condition
 E 0 ]   0 ] Hˆ 0 ]  0 with 0 ] 0 ]  1  0 0  1
Explicit calculation yields
E 0 ]
2
2m
N  d 3 r 0  r   N  d 3 r V  r  0  r   
2
2
4
1
N  N  1 g  d 3r 0  r  
2
Variation of energy with the use of a Lagrange multiplier:

 N 1E 0 ]   0 0

0  0  r  , 0  0  r 
BY PARTS
2 2 3
4
3
3
3

d
r





2
d
r


V
r




N

1
g
d
r











0
0
0
0
0
0


2m 
2
This results into the GP equation derived here in the variational way:
2
 1 2
p  V (r )   N -1 g 0  r  0  r   0  r 

 2m

eliminates self-interaction
BACK
32
ADDITIONAL NOTES
Variational estimate of the condensate properties
 ANNEX Interpretation of the Lagrange multiplier 
The idea is to identify it with the chemical potential. First, we modify the notation
to express the particle number dependence
1 2
1


E N  ] N  
p    V    N  1 g  d3r  4 
2
 2m

2
 1 2
EN  E N 0 N ], 
p  V ( r )   N - 1 g 0 N  r   0  r    N 0 N  r 
 2m

The first result is that  is not the average energy per particle:
1 2
1
p 0 N  0 N V 0 N   N  1 g  d 3r 0 N 4
2m
2
1 2
 N  0 N
p 0 N  0 N V 0 N   N  1 g  d3r 0 N 4
2m
EN / N  E N 0 N ]/ N  0 N
from the GPE
33
ADDITIONAL NOTES
Variational estimate of the condensate properties
Compare now systems with N and N -1 particles:
EN  E N 0 N ]  E N 1 0 N ]   N  E N 1 0, N 1 ]   N  EN 1   N
N … energy to remove a particle
use of the variational
principle for GPE
without relaxation of the condensate
In the "thermodynamic"asymptotics of large N, the inequality tends to equality.
This only makes sense, and can be proved, for g > 0.
Reminescent of the
Derivation:
E N  ]
BACK
theorem in the HF theory of atoms.
N  V   2 N  N  1 g  d3 r  4
N
1
2m
p2  
E N 1  ]  N  1 
1
2m
p 2   N  1  V   2  N  1 N  2  g  d 3r  4
E N  E N 1 
1
2m
1
1
p 2    V   2  N  N  1   N  1 N  2   g  d 3r  4
1
 N for 
0 N
34
ADDITIONAL NOTES
Variational estimate of the condensate properties
 SCALING ANSATZ FOR A SPHERICAL PARABOLIC TRAP
The potential energy has the form

V  r   12 m02  r 2  12 m02 x 2  y 2  z 2

Without interactions, the GPE reduces to the SE for isotropic oscillator
 1 2 1
2
2
3   r
p

m


r

r



0
0 0 

 0
2
2
 2m

The solution (for the ground state orbital) is
1 r2
  2
2 a0
3
00  r   A0 e
, a0 
m0
,
0 
2
A0 
ma02
 
1/ 4
2
a0 
We (have used and) will need two integrals:
I1   




du e
u2

2
   , I 2   




du e
u2
2
u 2  12  3 
35
ADDITIONAL NOTES
Variational estimate of the condensate properties
SCALING ANSATZ
The condensate orbital will be taken in the form
1 r2
  2
2b
3
0  r   A e
,
 
A b 
2
1/ 4
It is just like the ground state orbital for the isotropic oscillator, but with a
rescaled size. This is reminescent of the well-known scaling for the ground
state of the helium atom.
Next, the total energy is calculated for this orbital
E 0 ]
2
2m
N  d 3 r 0  r   N  d 3 r V  r  0  r   
The solution (for)
2 is
1
6  a0
 0 NA  4  d3 r e
2
b

2
r2
 2
b
r2 
4
1
N  N  1 g  d 3r 0  r  
2
2r 2 
2
 2
ma

r 2   N  1 A6 20 g  d3 r e b 


2
1
3
d
re
2 
a0
r2
 2
b
36
ADDITIONAL NOTES
Variational estimate of the condensate properties
For an explicit evaluation, we (have used and) will use the identities:
2
m
E 0 ]

0 a02 ,
m02
0
4 2 as
1
1
 2 , A 

, g
I1  b  b 
m
a0
2
The integrals, by the Fubini theorem, are a product of three:

2
I
b
 3I 2  b   1   
 0 N 
2
2
b

I
b

 1  

 a02
 4
1
 2    N  1
a0 
2b 
b
ma02
1
3 3/ 2
2
4
2
m
as
 
I1 b / 2

 I1  b  
3
3





 3  a02 b 2   N  1 as a03 
E 0 ]  0 N   2  2  
 3   0 N  E  
a0 
2 a0 b 
 4  b
3 1
1
b dimension-less

dimension-less
E     2   2     3


orbital size
energy per particle
4 

a0

Finally,
This expression is plotted in the figures in the main lecture.
BACK
37