Download Bogolyubov transformation

Cold atoms
21st
Lecture 7.
February, 2007
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
Nov 15
Lecture 6
…
BEC and symmetry breaking, coherent states
Nov 29
Lecture 7
…
Time dependent GP theory. Finite systems: BEC theory
preserving the particle number
2
Recapitulation
Offering many new details and
alternative angles of view
BEC in atomic clouds
Nobelists I. Laser cooling and trapping of atoms
The Nobel Prize in Physics 1997
"for development of methods to cool and trap atoms with
laser light"
Steven Chu
Claude CohenTannoudji
William D. Phillips
1/3 of the prize
1/3 of the prize
1/3 of the prize
USA
France
USA
Stanford University
Stanford, CA, USA
Collège de France; École
Normale Supérieure
Paris, France
National Institute of
Standards and
Technology
Gaithersburg, MD, USA
b. 1948
b. 1933
(in Constantine, Algeria)
b. 1948
5
Doppler cooling in the Chu lab
6
Doppler cooling in the Chu lab
atomic cloud
7
Nobelists II. BEC in atomic clouds
The Nobel Prize in Physics 2001
"for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms,
and for early fundamental studies of the properties of the condensates"
Eric A. Cornell
Wolfgang Ketterle
Carl E. Wieman
1/3 of the prize
1/3 of the prize
1/3 of the prize
USA
Federal Republic of Germany
USA
University of Colorado, JILA
Boulder, CO, USA
Massachusetts Institute of
Technology (MIT)
Cambridge, MA, USA
University of Colorado, JILA
Boulder, CO, USA
b. 1961
b. 1957
b. 1951
8
Trap potential
evaporation
cooling
Typical profile
?
coordinate/ microns

This is just one direction
Presently, the traps are mostly 3D
The trap is clearly from the real world, the
atomic cloud is visible almost by a naked eye
9
Ground state orbital and the trap potential
400 nK
level
number
200 nK
87 Rb
a0  1 m
 =10 nK
x / a0 x
 0 ( x, y, z )  0 x  x 0 y  y 0 z  z 
0 (u ) 
1
a0

e
u2
2 a 02
,
N ~ 106 at.

2
2
1
1
1
a0 
, E0    
 
2
m
2
2 ma0 2 Mum a02
 u 
1
1
2 2
V (u )  m u    
2
2
 a0 
2
• characteristic energy
• characteristic length10
BEC observed by TOF in the velocity distribution
11
Ketterle explains BEC to the King of Sweden
12
Simple estimate of TC (following the Ketterle slide)
The quantum breakdown sets on when
the wave clouds of the atoms start overlapping
mean
interatomic
distance
V 
 
 N
1
3
thermal
de Broglie
wavelength
h
mk BTc
Critical temperature
2
ESTIMATE
TRUE EXPRESSION
Tc
h
mk B
N
 
V
h2
Tc 
4 mk B
2
3
 N 


2,612
V


2
3
13
Interference of atoms
Two coherent condensates are interpenetrating and interfering.
Vertical stripe width …. 15 m
Horizontal extension of the cloud …. 1,5mm
14
Today, we will be mostly concerned with
the extended (" infinite" ) BE gas/liquid
Microscopic theory well developed
over nearly 60 past years
Interacting atoms
Importance of the interaction – synopsis
Without interaction, the
condensate would occupy the
ground state of the oscillator
(dashed - - - - -)
In fact, there is a significant
broadening of the condensate of
80 000 sodium atoms in the
experiment by Hau et al. (1998),
perfectly reproduced by the
solution of the GP equation
17
Many-body Hamiltonian
1 2
1
ˆ
H 
pa  V (ra ) 
2
a 2m
 U (ra  rb )
a  b
True interaction potential at low energies (micro-kelvin range)
replaced by an effective potential, Fermi pseudopotential
U (r )  g   (r )
4 as
g
m
2
,
as ... the scattering length
Experimental data
18
Mean-field treatment of interacting atoms
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

20
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
21
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
22
Field theoretic reformulation
(second quantization)
Purpose: go beyond the GP
approximation, treat also the excitations
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
24
Action of the field operators in the Fock space
basis of single-particle states
 
      ,  ... single particle state
    
r  r   
r     r 
FOCK SPACE F
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,
25
Action of the field operators in the Fock space
Average values of the field operators in the Fock states
Off-diagonal elements only!!! The diagonal elements vanish:
n1 , n2 , n3 ,
, np ,
n1 , n2 , n3 ,
, np ,
a p n1 , n2 , n3 ,
, np ,

, n p  1,
n p n1 , n2 , n3 ,
0
Creating a Fock state from the vacuum:
n1 , n2 , n3 ,
, np ,

p
1
np
a 

!
†
p
np
vac
26
Field operator for spin-less bosons – cont'd
Important special case – an extended homogeneous system
Translational invariance suggests to use the
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†
The other form is  made possible by the inversion symmetry (parity)
 important, because the combination
u  ak + v  a-k†
corresponds to the momentum transfer by k
Commutation rules do not involve a  - function, because the BK momentum
is discrete, albeit quasi-continuous:
 ak , ak'†    kk' ,  ak , ak'   0,  ak† , ak'†   0
27
Operators
Additive observable
X Xj
 X    d3 r d3 r'  † (r ) r X r'  (r' )
General definition of the one particle density matrix – 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   a† a
Momentum
P   pj
N   d3 r  † (r ) (r )

P   d3 r  † (r )   i   (r )
P   k  a† a
28
Hamiltonian
1 2
H 
pa  V (ra ) single-particle additive
a 2m
1
  U (ra  rb ) two-particle binary
2 a b


  d 3 r  † (r )  2 m   V (r )  ( r )
2
 12  d 3 r d 3 r'  † (r ) † (r' )U (r  r' ) (r' ) (r )
29
Hamiltonian
1 2
H 
pa  V (ra ) single-particle additive
a 2m
acts in the N -particle sub-space H N  F
1
  U (ra  rb ) two-particle binary
2 a b


  d 3 r  † (r )  2 m   V (r )  ( r )
2
acts in the whole Fock space F
 12  d 3 r d 3 r'  † (r ) † (r' )U (r  r' ) (r' ) (r )
30
Hamiltonian
1 2
H 
pa  V (ra ) single-particle additive
a 2m
acts in the N -particle sub-space H N  F
1
  U (ra  rb ) two-particle binary
2 a b


  d 3 r  † (r )  2 m   V (r )  ( r )
2
acts in the whole Fock space F
 12  d 3 r d 3 r'  † (r ) † (r' )U (r  r' ) (r' ) (r )
but
Particle number conservation
H
,N 0
Equilibrium density operators and the ground state (ergodic property)
P P (H ),
N , P   0
31
On symmetries and conservation laws
Hamiltonian is conserving the particle number
Particle number conservation
H
,N 0
Equilibrium density operators and the ground state (ergodic property)
P P (H ),
N , P   0
Typical selection rule
 (r )  Tr (r )P  0
is a consequence:
(similarly   0,  †  0,
)
Proof:
0  Tr  [N , P ]  Tr P [ , N ]  Tr P 
Tr A[ B, C ]  Tr C[ A, B ]
[ ( x),  d x ' † ( x ') ( x ')]   d x '( † ( x ')[ ( x), ( x ')]  [ ( x), † ( x ')] ( x '))   ( x)
QED
Deeper insight: gauge invariance of the 1st kind
33
Gauge invariance of the 1st kind
Particle number conservation
H
,N 0
Equilibrium density operators and the ground state (ergodic property)
P P (H ),
N , P   0
Gauge invariance of the 1st kind
H
,N 0

ei N  H e i N  H
unitary transform
The equilibrium states have then the same invariance property:
N , P   0 
e i N P ei N   P
Selection rule rederived:
Tr P  Tr e  i  N P ei  N  Tr ei  N  e  i  N P  ei  Tr P
(1  ei  ) Tr P  0  Tr ( r )P   ( r )  0
34
Hamiltonian of a homogeneous gas
H 
a
1 2
1
pa  V 
2m
2

 U (r
a  b
a
 rb ),
V  const.

  d 3 r  † (r )  2 m   V  (r )  12  d 3 r d 3 r'  † (r ) † (r' )U ( r  r' ) ( r' ) ( r )
2
ToTranslationally
study the symmetry
properties
Hamiltonian
invariant system
… howof
to the
formalize
(and to learn more about the
gauge invariance)
Proceed
in three steps …
T † (a )HT (a )  H,
a  R3 ... translation vector
in the direction reverse to that for the gauge invariance
Constructing the unitary operatorT
T (a ) (r1 , r2 , r3 ,
 e
i p a /
 (r1 ,
rp ,
(a )
rN )  (r1  a, r2  a, r3  a,
rN )  e
i  p a /
 (r1 ,
T (a )  e  i P a /
rp  a ,
rN )  e  iP a /  (r1 ,
rN  a )
rN )
... compare O ( )  e  i N
[N ,H ]=0
Infinitesimal translation
H  T † (a )HT (a )  e  iP a / H e  iP a /  H  i/ P aH  i/ HP a  O( a 2 )

[H , P ]a  0

[H , P x, y , z ]  0
... momentum conservation
35
Hamiltonian of the homogeneous gas
H 
a
1 2
1
pa  V 
2m
2

 U (r
a  b
a
 rb ),
V  const.

  d 3 r  † (r )  2 m   V  (r )  12  d 3 r d 3 r'  † (r ) † (r' )U ( r  r' ) ( r' ) ( r )
2
Translationally invariant system … how to formalize (and to learn more about the
gauge invariance)
†
T (a )H T (a )  H,
a  R3 ... translation vector
Constructing the unitary operatorT
T (a ) (r1 , r2 , r3 ,
 e
i p a /
 (r1 ,
rp ,
(a )
rN )  (r1  a, r2  a, r3  a,
rN )  e
i  p a /
 (r1 ,
T (a )  e  i P a /
rp  a ,
rN )  e  iP a /  (r1 ,
rN  a )
rN )
... compare O ( )  e  i N
[N ,H ]=0
Infinitesimal translation
H  T † (a )HT (a )  e  iP a / H e  iP a /  H  i/ P aH  i/ HP a  O( a 2 )

[H , P ]a  0

[H , P x, y , z ]  0
... momentum conservation
36
Hamiltonian of the homogeneous gas
H 
a
1 2
1
pa  V 
2m
2

 U (r
a  b
a
 rb ),
V  const.

  d 3 r  † (r )  2 m   V  (r )  12  d 3 r d 3 r'  † (r ) † (r' )U ( r  r' ) ( r' ) ( r )
2
Translationally invariant system … how to formalize (and to learn more about the
gauge invariance)
†
T (a )H T (a )  H,
a  R3 ... translation vector
Constructing the unitary operatorT
(a )
T (a ) (r1 ,in
r2 ,the
r3 , one-particle
rp , rN )  orbital
(r1  a, rspace
Translation
2  a , r3  a ,
i p a /
rp  a ,
rN  a )
i  p a /
 iP an/
 e
 (r1 , rN ) 1e

(
r
,
r
)

e
rN /)
1

i
pa   ( r1 ,  i pa
1
N

n
T (a ) (r )   (r  a )   (a )  (r )   
  (r )  e  i N  ( r )

i
P
a
/
n ! compare
 O ( )  e
Tn !(a )  e
...
[N ,H ]=0
Infinitesimal translation
H  T † (a )HT (a )  e  iP a / H e  iP a /  H  i/ P aH  i/ HP a  O( a 2 )

[H , P ]a  0

[H , P x, y , z ]  0
... momentum conservation
37
Hamiltonian of the homogeneous gas
H 
a
1 2
1
pa  V 
2m
2

 U (r
a  b
a
 rb ),
V  const.

  d 3 r  † (r )  2 m   V  (r )  12  d 3 r d 3 r'  † (r ) † (r' )U ( r  r' ) ( r' ) ( r )
2
Translationally invariant system … how to formalize (and to learn more about the
gauge invariance)
†
T (a )H T (a )  H,
a  R3 ... translation vector
Constructing the unitary operatorT
T (a ) (r1 , r2 , r3 ,
 e
i p a /
 (r1 ,
rp ,
(a )
rN )  (r1  a, r2  a, r3  a,
rN )  e
i  p a /
 (r1 ,
T (a )  e  i P a /
rp  a ,
rN )  e  iP a /  (r1 ,
rN  a )
rN )
... compare O ( )  e  i N
[N ,H ]=0
Infinitesimal translation
H  T † (a )HT (a )  e  iP a / H e  iP a /  H  i/ P aH  i/ HP a  O( a 2 )

[H , P ]a  0

[H , P x, y , z ]  0
... momentum conservation
38
Hamiltonian of the homogeneous gas
H 
a
1 2
1
pa  V 
2m
2

 U (r
a  b
a
 rb ),
V  const.

  d 3 r  † (r )  2 m   V  (r )  12  d 3 r d 3 r'  † (r ) † (r' )U ( r  r' ) ( r' ) ( r )
2
Translationally invariant system … how to formalize (and to learn more about the
gauge invariance)
†
T (a )H T (a )  H,
a  R3 ... translation vector
Constructing the unitary operatorT
T (a ) (r1 , r2 , r3 ,
 e
i p a /
 (r1 ,
rp ,
(a )
rN )  (r1  a, r2  a, r3  a,
rN )  e
i  p a /
 (r1 ,
T (a )  e  i P a /
rp  a ,
rN )  e  iP a /  (r1 ,
rN  a )
rN )
... compare O ( )  e  i N
[N ,H ]=0
Infinitesimal translation
H  T † (a )HT (a )  e  iP a / H e  iP a /  H  i/ P aH  i/ HP a  O( a 2 )

[H , P ]a  0

[H , P x, y , z ]  0
... momentum conservation
39
Hamiltonian of the homogeneous gas
H 
a
1 2
1
pa  V 
2m
2

 U (r
a  b
a
 rb ),
V  const.

  d 3 r  † (r )  2 m   V  (r )  12  d 3 r d 3 r'  † (r ) † (r' )U ( r  r' ) ( r' ) ( r )
2
Translationally invariant system … how to formalize (and to learn more about the
gauge invariance)
†
T (a )H T (a )  H,
a  R3 ... translation vector
Constructing the unitary operatorT
T (a ) (r1 , r2 , r3 ,
 e
i p a /
 (r1 ,
rp ,
(a )
rN )  (r1  a, r2  a, r3  a,
rN )  e
i  p a /
 (r1 ,
T (a )  e  i P a /
rp  a ,
rN )  e  iP a /  (r1 ,
rN  a )
rN )
... compare O ( )  e  i N
[H ,N ]=0
Infinitesimal translation
H  T † (a )HT (a )  e  iP a / H e  iP a /  H  i/ P aH  i/ HP a  O( a 2 )

[H , P ]a  0

[H , P x, y , z ]  0
... momentum conservation
40
Summary: two symmetries compared
Gauge invariance of the 1st kind
Translational invariance
universal for atomic systems
specific for homogeneous systems
O † ( )HO ( )  H,   0, 2
T † (a )HT (a )  H, a  R3
O ( )  e i N
T (a )  e  i P a /
global phase shift of the wave
function
global shift in the configuration space
[H , P x , y , z ]  0
[H ,N ]=0
particle number conservation
N , P   0
 e
iN 
Pe
iN 
P
total momentum conservation
P , P   0
 e
i
 Pa
Pe
i
Pa
for equilibrium states
for equilibrium states
selection rules
selection rules

† 0
unless there are as many
ak ak'
†
as .
P
†
ak''
0
unless the total momentum transfer
41
k  k '  k '' is0zero
Hamiltonian of the homogeneous gas
In the momentum representation
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 q
k'
k
Momentum conservation
 k + q  +  k'  q   k  k' = 0
Particle number conservation
a † a †a a
42
Hamiltonian of the homogeneous gas
For the Fermi pseudopotential
In the momentum representation
U q  U 0  U ( g )
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 q
k'
k
Momentum conservation
 k + q  +  k'  q   k  k' = 0
Particle number conservation
a † a †a a
43
Bogolyubov method
Originally, intended and conceived for
extended (rather infinite) homogeneous
system.
Reflects the 'Paradoxien der Unendlichen'
Basic idea
Bogolyubov method
is devised for boson quantum fluids with weak interactions – at T=0 now
no interaction
weak interaction
g 0
g0
N  N BE  a0† a0
1
N  N BE   ak† ak  N BE
1
k 0
The condensate dominates,
but some particles are kicked out
by the interaction (not thermally)
Strange idea introduced by Bogolyubov
N 0  a0† a0
1  a0† a0
a0† a0  a0 a0†  1  like c-numbers
a0  N 0 , a0†  N 0
N  N 0   ak† ak
... mixture of c-numbers and q -numbers
k 0
45
Basic idea
Bogolyubov method
is devised for boson quantum fluids with weak interactions – at T=0 now
no interaction
weak interaction
g 0
g0
N  N BE  a0† a0
1 N  N BE   ak† ak  N BE
1
k 0
The condensate dominates,
but some particles are kicked out
by the interaction (not thermally)
Strange idea introduced by Bogolyubov
N 0  a0† a0
1  a0† a0
a0† a0  a0 a0†  1  like c-numbers
a0  N 0 , a0†  N 0
N  N 0   ak† ak
... mixture of c-numbers and q -numbers
k 0
46
Approximate Hamiltonian
Keep at most two particles out of the condensate, use
a0  N0 , a0†  N0
H   2m k 2 ak† ak  12 V 1 U q ak†  q , ak'† q ak' ak
2
kk'q
†
k k
UN
 2V0
 a a
 4a a  ak a k 
UN 02
 2V
†
k k
 UN
2V
 a a
 2a a  ak a k 
UN 2
 2V
  2m k a a
2

k
2
 2m k
2
2
aa
k
k 0
k 0
† †
k k
† †
k k
†
k k
†
k k
k
k
k
k
k
k
condensate particle
47
Approximate Hamiltonian
Keep at most two particles out of the condensate, use
a0  N0 , a0†  N0
H   2m k 2 ak† ak  12 V 1 U qq ak†  q , ak'† q ak' ak
2
kk'q
†
k k
UN
 2V0
 a a
 4a a  ak a k 
UN 02
 2V
†
k k
 UN
2V
 a a
 2a a  ak a k 
UN 2
 2V
  2m k a a
2

k
2
 2m k
2
2
aa
k
k 0
k 0
† †
k k
† †
k k
†
k k
†
k k
k
k
k
k
k
k
condensate particle
48
Approximate Hamiltonian
Keep at most two particles out of the condensate, use
a0  N0 , a0†  N0
H   2m k 2 ak† ak  12 V 1 U q ak†  q , ak'† q ak' ak
2
kk'q
†
k k
UN
 2V0
 a a
 4a a  ak a k 
UN 02
 2V
†
k k
 UN
2V
 a a
 2a a  ak a k 
UN 2
 2V
  2m k a a
2

k
2
 2m k
2
2
aa
k
k 0
k 0
† †
k k
† †
k k
†
k k
†
k k
k
k
k
k
k
k
condensate particle
49
Approximate Hamiltonian
Keep at most two particles out of the condensate, use
a0  N0 , a0†  N0
H   2m k 2 ak† ak  12 V 1 U q ak†  q , ak'† q ak' ak
2
kk'q
†
k k
UN
 2V0
 a a
 4a a  ak a k 
UN 02
 2V
†
k k
 UN
2V
 a a
 2a a  ak a k 
UN 2
 2V
0th order
condensate
  2m k a a
2

2
 2m k
2
2
aa
3rd & 4th order
neglected
k
k
k 0
k 0
† †
k k
† †
k k
†
k k
†
k k
2nd order
pair excitations
k
1st order
is zero
violates k-conservation
k
k
k
k
k
condensate particle
50
Approximate Hamiltonian
Keep at most two particles out of the condensate, use
a0  N0 , a0†  N0
H   2m k 2 ak† ak  12 V 1 U q ak†  q , ak'† q ak' ak
2
kk'q
†
k k
UN
 2V0
 a a
 4a a  ak a k 
UN 02
 2V
†
k k
 UN
2V
 a a
 2a a  ak a k 
UN 2
 2V
  2m k a a
2

k
2
 2m k
2
2
aa
k
k 0
k 0
† †
k k
† †
k k
†
k k
†
k k
k
k
k
k
k
k
condensate particle
51
Approximate Hamiltonian
Keep at most two particles out of the condensate, use
a0  N0 , a0†  N0
H   2m k 2 ak† ak  12 V 1 U q ak†  q , ak'† q ak' ak
2
use
N 0  N   ak† ak
k 0
kk'q
†
k k
UN
 2V0
 a a
 4a a  ak a k 
UN 02
 2V
†
k k
 UN
2V
 a a
 2a a  ak a k 
UN 2
 2V
  2m k a a
2

2
 2m k
2
2
aa
k 0
k 0
† †
k k
† †
k k
†
k k
†
k k
The idea: replace the unknown condensate occupation by the known particle number
neglecting again higher than pair excitations
k
k
k
k
k
k
k
k
condensate particle
52
Approximate Hamiltonian
Keep at most two particles out of the condensate, use
a0  N0 , a0†  N0
H   2m k 2 ak† ak  12 V 1 U q ak†  q , ak'† q ak' ak
2
use
N 0  N   ak† ak
k 0
kk'q
†
k k
UN
 2V0
 a a
 4a a  ak a k 
UN 02
 2V
†
k k
 UN
2V
 a a
 2a a  ak a k 
UN 2
 2V
  2m k a a
2

2
 2m k
2
2
aa
k 0
k 0
† †
k k
† †
k k
†
k k
†
k k
The idea: replace the unknown condensate occupation by the known particle number
neglecting again higher than pair excitations
k
k
k
k
k
k
k
k
condensate particle
53
Approximate Hamiltonian
Keep at most two particles out of the condensate, use
a0  N0 , a0†  N0
H   2m k 2 ak† ak  12 V 1 U q ak†  q , ak'† q ak' ak
2
use
N 0  N   ak† ak
k 0
kk'q
†
k k
UN
 2V0
 a a
 4a a  ak a k 
UN 02
 2V
†
k k
 UN
2V
 a a
 2a a  ak a k 
UN 2
 2V
  2m k a a
2

k
2
 2m k
2
2
aa
k
k 0
k 0
† †
k k
† †
k k
†
k k
†
k k
k
k
k
k
k
k
condensate particle
54
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
gN 2
2V
anomalous
mean field
Conservation properties: momentum … YES, particle number … NO
NEW FIELD OPERATORS notice momentum conservation!!
bk  uk ak  vk a† k
ak  uk bk  vk b†k
b†k  vk ak  uk a† 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

In terms of the new operators, the anomalous terms in the Hamiltonian
have to vanish
55
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
gN 2
2V
anomalous
mean field
Conservation properties: momentum … YES, particle number … NO
NEW FIELD OPERATORS notice momentum conservation!!
bk  uk ak  vk a† k
ak  uk bk  vk b†k
b†k  vk ak  uk a† 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

In terms of the new operators, the anomalous terms in the Hamiltonian
have to vanish
56
Bogolyubov transformation – result
Without quoting the transformation matrix
 (k ) bk†bk  b†k b k 
H  12 

independent quasiparticles
 (k ) 

2
k  gn
2m
2

2
gN 2
 higher order constant
2V
ground state energy E
  gn   2m k 2 2m k 2  2 gn
2
2
2
57
Bogolyubov transformation – result
Without quoting the transformation matrix
H 
1
2

 (k )b b
gN 2
2V

†
k k
ind. quasi-particles
 (k ) 


2
k  gn
2m
2

2
ground state energy E
  gn   2m k
2
asymptotically
merge
2
2m
 (k )
2
2
k
2m
k 
k
 higher order constant
2
2
2
2
k
 2 gn
2m
high energy region
k  gn
2
quasi-particles are
nearly just particles
GP
 (k )  c  k
c
gn
m
cross-over
k 
sound region
4mgn
2
defines scale for k
quasi-particles are
collective excitations
58
More about the sound part of the dispersion law
Entirely dependent on the interactions, both the magnitude of the velocity
 (k )  c  k
and the linear frequency range determined by g
Can be shown to really be a sound:
V VV E

c "
"
,

mn
2
gN
E

2V
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 in 4He
The dispersion law bends upwards  quasi-particles are unstable, can
decay
59
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 fraction of particles outside of the condensate is
N  N0
8 3/ 2 1/ 2

as n
N
3 
as3 n
square root of the
gas parameter
is
the expansion
variable
60
What is the Bogolyubov approximation about
The results for various quantities are
8


N 0  N  1 
as3/ 2 n1/ 2 
3 


gn
128 3/ 2 1/ 2 

E
N  1 
as n 
2
15 


  gn  1 


as3/ 2 n1/ 2 
3 


[BG]  [GP]  1 


as3/ 2 n1/ 2 



general
pattern
32
square root of the
gas parameter
as3 n
is the expansion
variable
The Bogolyubov theory is the lowest order correction to the mean field (GrossPitaevskii) approximation
It provides thus the criterion for the validity of the mean field results
It is the simplest genuine field theory for quantum liquids with a condensate
61
Trying to understand the Bogolyubov
method
Notes to the contents of the Bogolyubov theory
The first consistent microsopic theory of the ground state and elementary
excitations (quasi-particles) for a quantum liquid (1947)
The quantum condensate turns into the classical order parameter in the
thermodynamic limit N  , V  , N / V  n  const.
The Bogolyubov transformation became one of the standard technical means for
treatment of "anomalous terms" in many body Hamiltonians (…de Gennes)
Central point of the theory is the assumption
a0  N0 , a0†  N0
Its introduction and justification intuitive, surprisingly lacks mathematical rigor.
Two related problems:
lowering operator
a0 G, N  H N 1
 ? 
gauge symmetry, s. rule
G, N a0 G, N  N 0
a0  0
Additional assumptions: something of a crutch/bar to study of finite systems
 homogeneous system
 infinite system
Infinity as a problem: philosophical, mathematical, physical
63
ODLRO in the Bogolyubov theory
Basic expressions for the OPDM for a homogeneous system
r'  r   † (r ) (r' )  V 1  e  i kr ak†   ei k'r' ak'
 V 1  ei k'r' e i kr ak† ak'  V 1  ei k'r' e  i kr ak† ak  kk'
k ,k '
by definition
transl. invariance
k ,k '
64
Off-diagonal long range order
ODLRO in the Bogolyubov theory
One particle density matrix
Basic expressions for the OPDM for a homogeneous system
r'  r   † (r ) (r' )  V 1  e  i kr ak†   ei k'r' ak'
 V 1  ei k'r' e i kr ak† ak'  V 1  ei k'r' e  i kr ak† ak  kk'
k ,k '
by definition
transl. invariance
k ,k '
65
ODLRO in the Bogolyubov theory
Basic expressions for the OPDM for a homogeneous system
r  r'   † (r' ) (r )  V 1  e  i k'r' ak'†  ak ei kr
 V 1  ei kr e  i k'r' ak'† ak  V 1  ei kr e  i k'r' ak† ak  kk'
k ,k '
by definition
transl. invariance
k ,k '
General expression for the one particle density matrix with condensate
 (r , r ')  V 1  ei k ( r  r ') nk
k0  0, n0  N 0
k
 V 1 ei k0 ( r  r ') n0  V 1  ei k ( r  r ') nk
coherent across
the sample
 (r )  (r' )
dyadic
k  k0
FT of a smooth function
has a finite range
 V 1  ei k ( r  r ') nk
N 0 i i k0 r
 (r ) 
e e
V

an arbitrary phase
k  k0
66
ODLRO in the Bogolyubov theory
Basic expressions for the OPDM for a homogeneous system
r  r'   † (r' ) (r )  V 1  e  i k'r' ak'†  ak ei kr
 V 1  ei kr e  i k'r' ak'† ak  V 1  ei kr e  i k'r' ak† ak  kk'
k ,k '
by definition
transl. invariance
k ,k '
General expression for the one particle density matrix with condensate
 (r , r ')  V 1  ei k ( r  r ') nk
k0  0, n0  N 0
k
 V 1 ei k0 ( r  r ') n0  V 1  ei k ( r  r ') nk
coherent across
the sample
 (r )  (r' )
dyadic
k  k0
FT of a smooth function
has a finite range
 V 1  ei k ( r  r ') nk
N 0 i
 (r ) 
e
V

an arbitrary phase
k  k0
67
ODLRO in the Bogolyubov theory
Basic expressions for the OPDM for a homogeneous system
r  r'   † (r' ) (r )  V 1  e  i k'r' ak'†  ak ei kr
by definition
 V 1  ei kr e  i k'r' ak'† ak  V 1  ei kr e  i k'r' ak† ak  kk'
k ,k '
transl. invariance
k ,k '
General expression for the one particle density matrix with condensate
 (r , r ')  V 1  ei k ( r  r ') nk
k0  0, n0  N 0
k
 V 1 ei k0 ( r  r ') n0  V 1  ei k ( r  r ') nk
coherent across
the sample
 (r )  (r' )
dyadic
N 0 i
 (r ) 
e
V

an arbitrary phase
k  k0
FT of a smooth function
has a finite range
 V 1  ei k ( r  r ') nk
k  k0
Interpretation in the Bogolyubov theory – at zero temperature
 (r , r ')  V 1/ 2 a0  V 1/ 2 a0†  V 1  ei k ( r  r ') vk2
k  k0
Rich microscopic content hinging on the Bogolyubov assumption
68
Three methods of reformulating the Bogolyubov theory
In the original BEC theory … no need for non-zero averages of linear field operators
Why so important? … microscopic view of the condensate phase
quasi-particles and superfluidity
basis for a perturbation treatment of Bose fluids
We shall explore three approaches having a common basic idea:
 relax the particle number conservation  work in the thermodynamic limit
69
Three methods of reformulating the Bogolyubov theory
In the original BEC theory … no need for non-zero averages of linear field operators
Why so important? … microscopic view of the condensate phase
quasi-particles and superfluidity
basis for a perturbation treatment of Bose fluids
We shall explore three approaches having a common basic idea:
 relax the particle number conservation  work in the thermodynamic limit
I
explicit construction of the classical
part of the field operators
Pitaevski in LL IX (1978)
II
the condensate represented by a
coherent state
Cummings & Johnston (1966)
Langer, Fisher & Ambegaokar
(1967 – 1969)
III
spontaneous symmetry breakdown,
particle number conservation violated
Bogolyubov (1960)
Hohenberg&Martin (1965)
P W Anderson (1983 – book)
70
I.
explicit construction of the classical part of
the field operators
Quotation from Landau-Lifshitz IX
72
Quotation from Landau-Lifshitz IX
… that part of the  operators, which changes the condensate particle number by 1,
we have, then, by definition
the symbols
denoting two "identical"
states, differing only by the number of the particles in the system, and
is a complex number.
These statements are strictly valid
in the limit
The definition of the quantity
has thus to be written in the form
the limiting transition is to be performed at a given fixed value of the
liquid density
73
Quotation from Landau-Lifshitz IX
… that part of the  operators, which changes the condensate particle number by 1,
we have, then, by definition
the symbols
denoting two "identical"
states, differing only by the number of the particles in the system, and
is a complex number.
These statements are strictly valid
in the limit
The definition of the quantity
has thus to be written in the form
the limiting transition is to be performed at a given fixed value of the
liquid density
If the  operators are represented in the form
then their remaining ("supercondensate") parts transform the state
to states which are orthogonal to it, that is, the matrix
elements are
In the limit
, the difference between the states
vanishes entirely and in this sense the quantity
comes the mean value of the operator
over this state.
and
be-
74
Quotation from Landau-Lifshitz IX
… that part of the  operators, which changes the condensate particle number by 1,
we have, then, by definition
the symbols
denoting two "identical"
states, differing only by the number of the particles in the system, and
is a complex number.
These statements are strictly valid
in the limit
The definition of the quantity
has thus to be written in the form
the limiting transition is to be performed at a given fixed value of the
liquid density
If the  operators are represented in the form
then their remaining ("supercondensate") parts transform the state
to states which are orthogonal to it, that is, the matrix
elements are
In the limit
, the difference between the states
vanishes entirely and in this sense the quantity
comes the mean value of the operator
over this state.
and
be-
75
II.
the condensate represented by
a coherent state
Reformulation of the Bogolyubov requirements
Bogolyubov himself and his faithful followers never speak of the many
particle wave function. Looks like he wanted
a0     ,   N 0 ei , so that
a0  
The ground state
This is in contradiction with the selection rule,
a0  0
The above eigenvalue equation is known and defines the ground state to be
a coherent state with the parameter 
77
Reformulation of the Bogolyubov requirements
Bogolyubov himself and his faithful followers never speak of the many
particle wave function. Looks like he wanted
a0     ,   N 0 ei , so that
a0  
The ground state
This is in contradiction with the selection rule,
a0  0
The above eigenvalue equation is known and defines the ground state to be
a coherent state with the parameter 
HISTORICAL REMARK
 The coherent states (not their name) discovered by Schrödinger as the
minimum uncertainty wave packets, obtained by shifting the ground state
of a harmonic oscillator.
 These states were introduced into the quantum theory of the
coherence of light by Roy Glauber (NP 2005). Hence the name.
 The uses of the coherent states in the many body theory and quantum
field theory have been manifold.
78
About the coherent states
OUR BASIC DEFINITION
a0     ,   N 0 ei ,
a0  
If a particle is removed from a coherent state, it remains unchanged (cf. the
Pitaevskii requirement above). It has a rather uncertain particle number, but
a reasonably well defined phase
General coherent state
 e
2
  /2
e
a0†
Condensate
 
vac
 N 0 ei 
 a0   
 a0† a0   
n0  N 0
2
 a0† a0 a0† a0     
4
 n0  
2
n02  N 02  N 0
 n0  N 0
N0
79
New vacuum and the shifted field operators
Does all that make sense? Try to work in the full Fock space F rather in its
fixed N sub-space HN This implies using the "grand Hamiltonian"
H  N
Let us define the shifted field operator
b0  a0  , b0†  a0†  *
b0 , b0†   1, b0   0 ... new vacuum
What next? Example: BE system without interactions – ideal Bose gas
H   N



   2 m k 2   ak† ak 
   a0† a0   0
2
for   0
Here,  is a true eigenstate,  coincides with the previous result for the
particle number conserving state B  N
. 0 , 0 ,0 , ,0 ,
Two different, but macroscopically equivalent possibilities.
80
L1: Thermodynamics: which environment to choose?
THE ENVIRONMENT IN THE THEORY SHOULD CORRESPOND
TO THE EXPERIMENTAL CONDITIONS
… a truism difficult to satisfy
 For large systems, this is not so sensitive for two reasons
•
System serves as a thermal bath or particle reservoir all by itself
•
Relative fluctuations (distinguishing mark) are negligible

Adiabatic system
SB heat exchange – the slowest
S
B
Real system
medium fast
process
• temperature lag
• interface layer
Isothermal system
the fastest
S
B
 Atoms in a trap: ideal model … isolated. In fact: unceasing energy exchange
(laser cooling). A small number of atoms may be kept (one to, say, 40). With
107, they form a bath already. Besides, they are cooled by evaporation and
they form an open (albeit non-equilibrium) system.
 Sometime, N =const. crucial (persistent currents in non-SC mesoscopic rings)
81
L1: Thermodynamics: which environment to choose?
THE ENVIRONMENT IN THE THEORY SHOULD CORRESPOND
TO THE EXPERIMENTAL CONDITIONS
… a truism difficult to satisfy
 For large systems, this is not so sensitive for two reasons
•
System serves as a thermal bath or particle reservoir all by itself
•
Relative fluctuations (distinguishing mark) are negligible

Adiabatic system
SB heat exchange – the slowest
S
B
Real system
medium fast
process
• temperature lag
• interface layer
Isothermal system
the fastest
S
B
 Atoms in a trap: ideal model … isolated. In fact: unceasing energy exchange
(laser cooling). A small number of atoms may be kept (one to, say, 40). With
107, they form a bath already. Besides, they are cooled by evaporation and
they form an open (albeit non-equilibrium) system.
 Sometime, N =const. crucial (persistent currents in non-SC mesoscopic rings)
82
L1: Thermodynamics: which environment to choose?
THE ENVIRONMENT IN THE THEORY SHOULD CORRESPOND
TO THE EXPERIMENTAL CONDITIONS
… a truism difficult to satisfy
 For large systems, this is not so sensitive for two reasons
•
System serves as a thermal bath or particle reservoir all by itself
•
Relative fluctuations (distinguishing mark) are negligible

Adiabatic system
SB heat exchange – the slowest
S
B
Real system
medium fast
process
• temperature lag
• interface layer
Isothermal system
the fastest
S
B
 Atoms in a trap: ideal model … isolated. In fact: unceasing energy exchange
(laser cooling). A small number of atoms may be kept (one to, say, 40). With
107, they form a bath already. Besides, they are cooled by evaporation and
they form an open (albeit non-equilibrium) system.
 Sometime, N =const. crucial (persistent currents in non-SC mesoscopic rings)
83
L1: Homogeneous one component phase:
boundary conditions (environment) and state variables
T P  dual variables, intensities
"intensive"
SV
S V N isolated, conservative
open S V 
S P N isobaric
isothermal T V N
S P  not in use
grand T V 
T P N isothermal-isobaric
not in use T P 
84
L1: Homogeneous one component phase:
boundary conditions (environment) and state variables
T P  dual variables, intensities
"intensive"
SV
S V N isolated, conservativ
conservativee
sharp boundary isolated/open
open S V 
At zero temperature the
two coincide
TVN
T=0 isothermal
S=0
(Third
Principle)
S P N isobaric
S P  not in use
grand T V 
T P N isothermal-isobaric
not in use T P 
85
New vacuum and the shifted field operators
Does all that make sense? Try to work in the full Fock space F rather in its
fixed N sub-space HN This implies using the "grand Hamiltonian"
H  N
Let us define the shifted field operator
b0  a0  , b0†  a0†  *
b0 , b0†   1, b0   0 ... new vacuum
What next? Example: BE system without interactions – ideal Bose gas
H   N



   2 m k 2   ak† ak 
   a0† a0   0
2
for   0
Here,  is a true eigenstate,  coincides with the previous result for the
particle number conserving state B  N
. 0 , 0 ,0 , ,0 ,
Two different, but macroscopically equivalent possibilities.
86
New vacuum and the shifted field operators
Does all that make sense? Yes: work in the full Fock space F rather than in its
fixed N sub-space HN This implies using the "grand Hamiltonian"
H  N
Let us define the shifted field operator
b0  a0  , b0†  a0†  *
b0 , b0†   1, b0   0 ... new vacuum
What next? Example: BE system without interactions – ideal Bose gas
H   N



   2 m k 2   ak† ak 
   a0† a0   0
2
for   0
Here,  is a true eigenstate,  coincides with the previous result for the
particle number conserving state B  N
. 0 , 0 ,0 , ,0 ,
Two different, but macroscopically equivalent possibilities.
87
New vacuum and the shifted field operators
Does all that make sense? Yes: work in the full Fock space F rather than in its
fixed N sub-space HN This implies using the "grand Hamiltonian"
H  N
Let us define the shifted field operator
b0  a0  , b0†  a0†  *
b0 , b0†   1, b0   0 ... new vacuum
What next? … is this coherent state able to represent the condensate?
Test example: ideal Bose gas – limit of a BE system without interactions
H   N  


  2 m k 2   ak† ak 
   a0† a0   0
2
for   0
Here,  is a true eigenstate,  coincides with the previous result for the
particle number conserving state B  N. 0 , 0 ,0 , ,0 ,
Two different, but macroscopically equivalent possibilities.
88
General case: the approximate vacuum


H   d3 r  † (r )  2m   V (r )  (r )  12  d3r d3 r'  † (r ) † (r' )U (r  r' ) (r' ) (r )
2
● Non-zero interaction, repulsive forces make the
lowest energies N dependent
g  r  r ' 
● Inhomogeneous, possibly finite, system with a
confinement potential
● There is no privileged symmetry related basis
of one-particle orbitals
89
General case: the approximate vacuum


H   d3 r  † (r )  2m   V (r )  (r )  12  d3r d3 r'  † (r ) † (r' )U (r  r' ) (r' ) (r )

2
Trial function … a coherent state a marked generalization!!
  r     r  
g  r  r ' 
We should minimize the average grand energy
 H   N    d r  (r )     V (r )    (r )
3
*
2
2m
 12  d 3r d 3r'  *(r ) (r )U (r  r' ) * (r' ) ( r' )
This is precisely the energy functional of the Hartree type we met already

and the Euler-Lagrange equation is the good old Gross-Pitaevski equation
 2
2
  V (r )  g   r    r     r 

 2m

with the normalization condition
N [n]  N   d3 r  (r )
2
90
General case: the approximate vacuum


H   d3 r  † (r )  2m   V (r )  (r )  12  d3r d3 r'  † (r ) † (r' )U (r  r' ) (r' ) (r )

2
Trial function … a coherent state a marked generalization!!
  r     r  
g  r  r ' 
We should minimize the average grand energy
 H   N    d r  (r )     V (r )    (r )
PROPERTIES OF THE
3
2
*
2m

COHERENT
 12 d 3GROUND
r d 3r'  *(STATE
r ) (r )U (r
 r' ) * (r' ) ( r' )
● energy
Groundfunctional
state in the
mean
field type we met already
This is precisely the
of the
Hartree
approximation
and the Euler-Lagrange equation is the good old Gross-Pitaevski equation

 ● 2All particles at zero temperature
2
in the
V (condensate
r )  g   r    r     r 
 are 
2
 2m

 
r 
N [n]  N   d
● condition
r n
with the normalization
3
r  (r )
2
91
General case: perturbative expansion


Define the shifted field operators and the condensate as the new vacuum
  r     r    r  ,  †  r    †  r   * r 
  r  , †  r'      r  r'  etc.,   r    0
Expansion parameter … concentration of the supra-condensate particles
deviation from the MF condensate = -operators

If we keep only the terms not more than quadratic in the new operators, the
resulting approximate Hamiltonian becomes
92
General case: perturbative expansion


Define the shifted field operators and the condensate as the new vacuum
  r     r    r  ,  †  r    †  r   * r 
  r  , †  r'      r  r'  etc.,   r    0
Expansion parameter … concentration of the supra-condensate particles
deviation from the MF condensate = -operators

If we keep only the terms not more than quadratic in the new operators, the
resulting approximate Hamiltonian becomes

  d r  (r )   2 m   V (r )    gn
  d r  (r )   2 m   V (r )    (r )

H   N   d 3r  *(r )  2 m   V (r )    12 gnBE (r )  ( r )
2
3
†
2
BE
3
†

(r )  ( r )  h.c.
2
 2  d 3r nBE (r ) † (r ) † (r )  4 † (r ) ( r )   ( r ) ( r )}
g
Here (see in a moment )
nBE r     r 
2
93
General case: perturbative expansion

  d r  (r )   2 m   V (r )    gn
  d r  (r )   2 m   V (r )    (r )

H   N   d 3r  *(r )  2 m   V (r )    12 gnBE (r )  ( r )
2
3
†
2
BE
3
†

(r )  ( r )  h.c.
2
 2  d 3r nBE (r ) † (r ) † (r )  4 † (r ) ( r )   ( r ) ( r )}
g
94
General case: perturbative expansion

  d r  (r )   2 m   V (r )    gn
  d r  (r )   2 m   V (r )    (r )

H   N   d 3r  *(r )  2 m   V (r )    12 gnBE (r )  (r )
2
3
†
2
BE
3
†

(r )  (r )  h.c.
2
 2  d 3r nBE (r ) † (r ) † (r )  4 † (r ) (r )   (r ) (r )}
g
1.
The linear part must vanish to have minimum at   0 . This is identical
with the Gross-Pitaevskii equation and justifies the identification
nBE r     r 
2
95
General case: perturbative expansion
2
H   N   d 3r   12  gnBE
(r )

  d r  (r )   2 m   V (r )    (r )

  d 3r  †(r )  2 m   V (r )    gnBE (r )  (r )  h.c.
2
3
†
2
 2  d 3r nBE (r )  † (r ) † (r )  4 † (r ) (r )   (r ) (r )}
g
1.
The linear part must vanish to have minimum at   0 . This is identical
with the Gross-Pitaevskii equation and justifies the identification
nBE r     r 
2.
2
The zero-th order part simplifies – substitute from the GPE
96
General case: perturbative expansion
 
2
H   N   d 3 r  12 gnBE
(r )

  d r  (r )  

  d 3 r  †(r )  2 m   V (r )    gnBE (r )  ( r )  h.c.
3
†
2
2
2m
  V (r )    (r )
 2  d 3 r nBE (r )  † (r ) † (r )  4 † (r ) ( r )   ( r ) (r )}
g
1.
The linear part must vanish to have minimum at   0 . This is identical
with the Gross-Pitaevskii equation and justifies the identification
nBE r     r 
2
2.
The zero-th order part simplifies – substitute from the GPE
3.
There remains to eliminate the anomalous terms from the quadratic part:
Bogolyubov transformation
97
General case: the Bogolyubov transformation
A simple method: use EOM for the field operators
i t (r , t )   (r , t ),H   N



  2 m   V (r )    2 gnBE (r )  ( r , t )  gnBE ( r ) †( r , t )
2
i t †(r , t )   †( r , t ),H   N 


  2 m   V (r )    2 gnBE (r )  †(r , t )  gnBE ( r ) ( r , t )
2
These are linear eqs. To find their mode structure, make a linear ansatz
iEk t
 bk†vk* ( r )e
 iEk t
iEk t
 bk†uk* ( r )e
 iEk t
 (r , t )   bk uk (r )e
 † (r , t )   bk vk (r )e
Reminescent of the old u and v for infinite system.
Substitute into the EOM and separate individual frequencies.
Bogolyubov – de Gennes eqs. are obtained.
98
General case: the Bogolyubov transformation
Bogolyubov – de Gennes eqs.

 

v
Ek uk   2 m   V    2 gnBE uk  gnBE vk*
 Ek vk
2
2 m   V    2 gnBE
2
k
 gnBE uk*
Strange coupled “Schrödinger” equations.
Strange orthogonality relations:
3
*
*
d
r
u
u

v

k
k v  (r )   k

It is our choice to normalize to unity
3
d
 r  uk v  vk u  (r )  0
The definition of the QP field operators can be inverted
 (r )   bk uk (r )  bk† vk* (r )
 † (r )   bk vk (r )  bk†uk* (r )


  d r  v   u  †  ( r )
bk   d 3 r uk*  vk* † ( r )
bk†
3
k
k
99
General case: the Bogolyubov transformation
These field operators satisfy the correct commutation rules
bk , b†    k , bk , b   0, bk† , b†   0
Finally,
H  N   d r
3
  gn
 12
2
BE

(r )   E b b   d r vk (r )
†
k k k
3
2

a neat QP form of the grand Hamiltonian.
100
Detail: the mean-field for a homogeneous system
Before: minimize the energy functional with fixed particle number N, find the
chemical potential  afterwards
Now: minimize the grand energy functional with fixed chemical potential, find
the average particle number in the process
101
Detail: the mean-field for a homogeneous system
Before: minimize the energy functional with fixed particle number N, find the
chemical potential  afterwards
Now: minimize the grand energy functional with fixed chemical potential, find
the average particle number in the process
Homogeneous system:
order parameter
 ( r )   const.  N 0 /V  n


 H   N    d r  (r )  2 m   V (r )    (r )
3
*
2
 12  d 3 r d 3 r'  *(r ) ( r ) g  r  r '  * ( r' ) ( r )

V    
energy per unit volume
2
*
average particle density

 
 N    d r  (r ) (r )  V  
3
4
 12 g 
2
n  
102
Detail: the mean-field for a homogeneous system
The GP equation reduces from differential to an algebraic one:

 x   0,   x
x
2 x  12 g  4 x3  0, 
 n
2
min

max
 0, 
min


g
,
min   12 g 
4
min

2
2g


g
103
Detail: the mean-field for a homogeneous system
The GP equation reduces from differential to an algebraic one:

 x   0,   x
x
2 x  12 g  4 x3  0, 
 n
2
min

max
 0, 
min

2g

     ref
4
min   12 g 
2
g
choose  ref ;  ref   ref / g
    g  ref
g
,
4

Plot in relative units
     ref
min




locus of the
minima


104
III.
broken symmetry and quasi-averages
Zero temperature limit of the grand canonical ensemble


1

Z
e
P
  E  N
 Z 1   N e   N   N
  H  N
 Z 1  0 N e

  E0 N   N

0N   0N 0N
Picks up the correct ground state energy,
all ground states are taken with equal statistical weight
NORMAL
NON-DEGENERATE GROUND STATE
SYSTEMS
"ANOMALOUS"
DEGENERATE GROUND STATE
106
Degenerate ground state
NORMAL
NON-DEGENERATE GROUND STATE
SYSTEMS
"ANOMALOUS"
DEGENERATE GROUND STATE
Characterized by a classical order parameter … macroscopic quantity
Typical cause: a symmetry degeneracy
Everything depends on the system characteristic parameters
Ginsburg – Landau phenomenological model
E    a 2  b 4
a0
a0
stable equilibrium
metastable equilibrium
non-degenerate
degenerate
107
Degenerate ground state
NORMAL
NON-DEGENERATE GROUND STATE
SYSTEMS
"ANOMALOUS"
DEGENERATE GROUND STATE
Characterized by a classical order parameter … macroscopic quantity
Typical cause: a symmetry degeneracy
Everything depends on the system characteristic parameters
Ginsburg – Landau phenomenological model
E    a 2  b 4
spontaneous
symmetry breaking
a0
a0
stable equilibrium
metastable equilibrium
non-degenerate
degenerate
108
Rovnovážná struktura molekul AB3
BF3
NH3
F
N
H
B
H
F
H
F
N
U
U
rovina F
h
rovina H
h
U adiabatická potenciální energie
109
Equilibrium structure of the AB3 molekules
BF3
NH3
F
H
B
F
N
H
H
F
N
U
U
F plane
h
H plane
h
U adiabatic potential energy
stable equilibrium
non-degenerate
ground state
metastable equilibrium
degenerate
ground state
110
Equilibrium structure of the AB3 molekules
BF3
NH3
F
H
B
F
N
H
H
F
N
U
U
F plane
h
H plane
h
U adiabatic potential energy
stable equilibrium
non-degenerate
ground state
metastable equilibrium
degenerate
ground state
111
Equilibrium structure of the AB3 molekules
F
BF3
Ammonia molecule
pyramidal molecule.
B
two minima
of
F potential energy
F
separated by a barrier.
U Different from a typical
extended system:
 Small system:
quantum barrier & tunneling
rovina F
h
NH3
N
H
H
H
N
U
H plane
 Discrete symmetry
broken:
U adiabatická potenciální energie
discrete set of equivalent
equilibria states
h
112
Broken continouous symmetries in extended systems
Three popular cases
System
Isotropic
ferromagnet
Atomic crystal
lattice
Bosonic gas/liquid
Hamiltonian
Heisenberg spin
Hamiltonian
Distinguishable
atoms with int.
Bosons with short
range interactions
Symmetry
3D rotational in
spin space
Translational
Global gauge
invariance
Order parameter
homogeneous
magnetization
periodic
particle density
macroscopic
wave function
Symmetry
breaking field
external magnetic
field
"empty lattice"
potential
particle
source/drain
Goldstone modes
magnons
acoustic phonons
sound waves
For a nearly exhaustive list see the PWA book of 1983
113
Bose condensate - degeneracy of the ground state
The coherent ground state
mean field energy
order parameter
mf ground state

E      
2
 
N 0  ei 
 e

2
1

2
e
4
 12 g 

any from  0,2 
N0 ei a0
degeneracy
genuinely different
for different 
vac
Selection rule
a0

E 
  ei   0
a0   d  a0


0
average over all degenerate states


"Mexican hat"
114
Symmetry breaking – removal of the degeneracy
The coherent ground state
mean field energy
order parameter
mf ground state

E      
2
 
N 0  ei 
 e

2
1

2
e
4
 12 g 
any from  0,2 
N0 ei a0

H   N -  a0† ei  a0 e  i
E 
degeneracy
genuinely different
for different 
vac
Symmetry broken by a small
perturbation picking up one 
H  N 



particle number NOT conserved


"Mexican hat"
115
Symmetry breaking – removal of the degeneracy
The coherent ground state
mean field energy
order parameter
mf ground state

E      
2
 
N 0  ei 
 e

2
1

2
e
4
 12 g 
any from  0,2 
N0 ei a0

H   N -  a0† ei  a0 e  i
E 
degeneracy
genuinely different
for different 
vac
Symmetry broken by a small
perturbation picking up one 
H  N 




particle number NOT conserved
For
 0
one particular phase selected


"Mexican hat"
116
How the symmetry breaking works – ideal BE gas
Without interactions, the ground level is uncoupled from the excited levels:

H   N -  a0† ei  a0 e i

i
   a a -  a e  a0 e
†
0 0
†
0

 i


   a0† a0 -  a0† ei  a0 e i
  2m  k 2    ak† ak
2

k 0
The control parameter is the chemical potential , but it will be adjusted to yield
a fixed average particle number in the condensate.
Transformation:




  a0† a0   a0† ei  a0 e  i     a0† a0  ei a0†  e  i a0 







    a0† a0  a0†  * a0      a0†  *  a0     * 

    b0†b0  *  
117
How the symmetry breaking works – ideal BE gas
Now we determine the many-body ground state
  b0†b0  *     E  ,
b0  a0  ,    1 ei ,   0
The lowest energy corresponds to
b0   0, i.e. a0    
... coherent state
*    a0† a0   N 0 ,   N 0 ei
E  *   N 0
   / N 0
The control parameter is the chemical potential , but it will be adjusted to yield
a fixed average particle number in the condensate.
Infinitesimal symmetry breaking field   0
  0 with N 0 fixed:
  00
E 0
  N 0 ei  ,

fixed
118
How the symmetry breaking works – ideal BE gas
Now we determine the many-body ground state
  b0†b0  *     E  ,
b0  a0  ,    1 ei ,   0
The lowest energy corresponds to
b0   0, i.e. a0    
... coherent state
*    a0† a0   N 0 ,   N 0 ei
E  *   N 0
   / N 0
The control parameter is the chemical potential , but it will be adjusted to yield
a fixed average particle number in the condensate.
Infinitesimal symmetry breaking field   0
  0 with N 0 fixed:
  00
E 0
  N 0 ei  ,

fixed
• The coherent state is the exact
ground state for the ideal BE gas
• The order parameter picks up the
phase from the perturbing field
• The order of limits: first   0, only
then the thermodynamic limit N0 
.
119
The end