This document discusses the application of dynamical groups and coherent states in quantum optics and molecular spectroscopy. It provides an introduction to using Lie groups and algebras to describe quantum systems and defines coherent states. Specific applications discussed include using dynamical symmetries to calculate energy levels of systems like the harmonic oscillator and hydrogen atom. Coherent states are used to derive classical equations of motion and represent open quantum systems. Examples of coherent state dynamics are shown for two-level and three-level atoms interacting with laser fields.

1. Dynamical Groups, Coherent States and
Some of their Applications in Quantum
Optics and Molecular Spectroscopy
Alexander V. Gorokhov,
Samara State University, Russia
gorokhov@samsu.ru
8 August, 2014

2. Samara 2

3. 3
Contents
Ö Introduction
Ö Dynamical algebras and groups in
Quantum Physics
Ö Coherent States (CS), Path Integrals and
Classical Equations
Ö Open Systems & Fokker – Planck
equations in CS representation
Ö Summary

4. 4
Introduction
“In the thirties, under the demoralizing influence
of quantum – theoretic perturbation theory, the
mathematics required of a theoretical physicist
was reduced to a rudimentary knowledge of the
Latin and Greek alphabets.” R. Iost
Lie groups and Lie algebras have been successfully applied in quantum
mechanics since its inception.

5. Wigner’s approach to quantum physics
5
, { } , 1.
ei ei
θ θ
Ψ Ψ Ψ ⊂Η Ψ =
→ ˆ( ) : Ψ ˆ +
ˆ Φ = Ψ Φ
,
g T g T T
ˆ ˆ
+ ∗
T T
Ψ Φ = Ψ Φ
∼
SO(3 ) ~ SU(2)/Z2
SU(2)
Spin 0 Spin ½ Spin 1, ….
Scalar Spinor Vector …
Poincare group
ISL(2,C)
Trivial
reps.
Massive
states
Massless
states
Vacuum electron neutrinos ?
quarks photon,
… graviton
G
Jˆ 2 = j( j+1)Iˆ, j = 0, 1 2, 1, ...

6. 6
Unitary representations of the Lorentz group
Tˆ(g )Tˆ(g )=ω(g , g )Tˆ(g ⋅ g ), |ω(g , g ) |=1
1 2 1 2 1 2 1 2
Ray representations, covering groups; central extensions
, { } , 1.
ei ei
θ θ
Ψ Ψ Ψ ⊂Η Ψ =
→ ˆ( ) : Ψ ˆ +
ˆ Φ = Ψ Φ
,
g T g T T
ˆ ˆ
+ ∗
T T
Ψ Φ = Ψ Φ
∼
V. Bargmann

7. 7
Symmetry group applications
Elementary particles, atoms, molecules,
crystals
(classification of states, selection rules,
"hidden" symmetry of the hydrogen atom,
the Lorentz group, Poincaré and
classification of particles)
J.L. Birman,
CUNY, USA
Ya.A. Smorodinsky, JIRN,
Dudna
V.A. Fock, Leningrad
University

8. 8
Dynamical Symmetries, 1964
• Classification of hadrons:
SU (3), SU (6) symmetry
of the flavors, the quarks,
the mass formulas
• Dynamical symmetries of
quantum systems
• Spectrum generating
algebra
• Coherent states
(1963, Glauber;
1972 Perelomov,
Gilmore,…) M. Gell-Mann
Y. Neeman
Asim Orhan Barut
1926-1994

9. Eightfold way and dynamical (spectrum generating) groups
The baryon octet
Eightfold Way, classification of hadrons into groups on the basis of their
symmetrical properties, the number of members of each group being 1, 8 (most
frequently), 10, or 27. The system was proposed in 1961 by M. Gell-Mann and
Y. Neʾeman. It is based on the mathematical symmetry group SU(3); however, the
name of the system was suggested by analogy with the Eightfold Path of
Buddhism because of the centrality of the number eight..
9
SU(3) … SUI(2)
8 = 1 Δ 2 Δ 2 Δ 3

10. A simple example - Harmonic Oscillator
( )
( ) ( ) ( )
3 ˆ ˆ ˆ ˆ ˆ ˆ.
= − − = − I =− I
10
ˆ 1 ˆ 2 ˆ 2
, ( 1)
H p x m
= + = = =
=
2 2 2 2
K p x K p x K px xp
= + = − = +
0 1 2
⎧⎪⎪ ⎪⎪⎫ = ± ⎨ = = ⎬ ⎩⎪⎪ ⎪⎪⎭
K K 1 iK 2
K aa K aa
⎡ K K ⎤ =± K ⎡ K K ⎤ ⎢⎣ 0 ⎥⎦ ⎢⎣ ⎥⎦
=− K
0
SU SL R Sp R SO
k
+
2
ˆ 1 ˆ ˆ , ˆ 1 ˆ ˆ , ˆ 1 ˆ ˆ ˆˆ ;
4 4 4
ˆ ˆ ˆ , ˆ 1 ˆ ˆ , ˆ 1 ˆ ˆ ,
2 2
ˆ , ˆ ˆ , ˆ , ˆ 2 ˆ .
(1,1) (2, ) (2, ) (2,1).
T , 1 , 3 ;
4 4
k
ω
+ +
± + −
± ± + −
= = ≈
=
2 2 2 2
K K K
k k
K ( 1)
0 1 2
N
dimensionaloscill
ator: W ^
1
)
6
N Sp (2 N
,
R
−

11. 11
Dynamical symmetries:
condensed matter, quantum optics
Murray Gell-Mann,
lecturing in 2007,
Ennackal Chandy
George Sudarshan
SU(2), SU(3), SU(n), SO(4,2), SU(m,n), Sp(2N,R), W(N)^Sp(2N,R),…

12. 12

13. 13
Dynamical symmetry in action
R. Feynman’s method of operator exponent
disentanglement
In a linear case it is possible to find exact solution:
• Energy levels and corresponding wave functions (time independent
Hamiltonian);
• Transition probabilities (time dependent Hamiltonian);
• Quasi energy and quasi energy states (periodic Hamiltonian).
Aharonov - Anandan geometric phase

14. Lie groups and an energy levels calculation
V.A. Fock, (1935) V. Bargmann, (1936)
A.O. Barut, H. Kleinert, Yu.B. Rumer, A.I. Fet (1971)
14
H-atom, SO(4),
SO(4,2)
SU(1,1), SU(N,1), Sp(2n,R), quantum optics, superfluidity
ˆ ( ˆ ), ˆ
k k H f A A
= ∈
A ⊃ A ' ⊃ A "
⊃
...,
A A A
A,
( ( )) ( ) ( ) 1 12 2 2
ˆ ' " ...
= + ⎜⎛ ⎟⎞+ ⎜⎛ ⎟⎞+
H α f C α f C α f C
⎝ ⎠ ⎝ ⎠
Molecular rotational- vibrational levels
F. Iachello, R.D. Levine
u(4)⊃so(4)⊃so(3) ⊃so(2)

15. -group and Franck – Condon overlap Integrals,
- Born-Oppenheimer app. & harmonic approximation for vibrations)
G G G G G G
β α
exp 1 [ ] [ ]
β α = ⎡⎢− β + α ⎤⎥ × ⎣ ⎦ ⋅ Σ
Vˆ Vˆ
15
Molecular spectra & vibronic transitions
Wn ∧ Sp(2N,R)
ˆ ˆˆ ˆ
VH
+
G G G G
ˆ , , ,..., ;
( )
'
N q Rq q q q q q
R L L q
= +Δ =
1 2
= Δ = Δ Δ = −
;
− 1 −
1 ( ) G ( )
G G G G
'
0 0
([ ] ) [ ] [ ] [ ]
ˆ ˆ ' ˆ, ˆ ' ,
G G
ˆ V=T( )ˆ D( )ˆ U(ˆ R)
I , ' |
'
ˆ
VH
L r r r r
m n
m n m n
V
δ ρ
⎡⎣ ⎤⎦ = ⎡⎣ ⎤⎦
′ =
< >=
( ) 2 *[ ] [ ] 2
2 [ ],[ ]
[ !] [ !]
v n
v n
v n
v n
I m V n I H I x
ˆ , ' , ( ') (2 ') exp ' ,
00 ( ) ( )2
' 00
! ! 2( ')
' 2 , ' 2 ' ; ( 1).
' '
mn mn
m n
x x M
ωω
ω ω ωω
ω ω
ω ω ω ω
ω ω ω ω
⎡ ⎤
= = Δ Δ = + ⎢− Δ ⎥ ⎣ + ⎦
Δ = Δ Δ = − Δ = =
+ +
=

16. 16

17. 17
Model Hamiltonians,
dynamical groups and coherent states
Klauder, Glauber, Sudarshan, Perelomov, Berezin, Gilmore, …

18. Heisenberg – Weyl group W1 and coherent states
18
J.R. Klauder
R. Glauber, 2005, October 5
1963
Coherence properties of
quantum electromagnetic
fields, lasers

19. 19
CS and holomorphic function representations
G-invariant Kähler’s
2-form

20. 20
Path Integrals in CS - representation
ˆ U ( t , t 0 ) ≡ ˆ U ( t ˆ N , t 0 ) = U ( t N , t N − 1 ) ˆ U ( t N − 1 , t ˆ N − 2 ) ⋅⋅⋅
U ( t 1 , t 0
)

21. 21
“Classical” Equations
2
i zz Kzzz
( , ) ln ( , ) ,
( , ) ln ( , ) .
∂ ∂
Σ
Σ
z z z
1
2
=
i zz Kzzz
1
n
n
z z z
β
α α β
β
β
α α β
β
=
=
∂ ∂ ∂
∂ ∂
− =
∂ ∂ ∂
=
=
(
(
∂
2 g = ln K(z, z ) , g ⋅ g = , g ⋅ g
=
. z z
αβ α α β β
αβ α β κβ κ αγ γ δ δ
∂ ∂
For linear systems classical dynamics
of coherent states is exact!

22. 22
Quantum oscillator in a field of external force
Oscillator coherent states, R. Glauber, 1963

23. P(x,t) = |x|z(t)|2 23

24. 24

25. 25
n(t)
W0(t)

26. 26
-1 0 1
x
0
t
-1 0 1
0
2 Ψ(x,t) ← a quantumcarpets

27. 27
F(t) = F0 sin(ω t)

28. N-level atoms in classical fields. G = SU(N), ( |kl| oe u(N) )
28

29. 29
G= SU(2), (2j+1)-level atom
Coherent state dynamics. (a) – trajectory, (b) – Upper level probability P(t).
(z(0) = 1+i, ω0 = 1. ω = 2/3, A = 2)
j=1/2

30. 30
Complex plane Bloch’s sphere
ϑ z = eiφ tan
⎛⎜⎞⎟
2
⎝ ⎠

31. 31
SU(2) CS generation: (a) – trajectory, (b) Upper level probability P(t).
(z(0) = 0, ω0 = 1. ω = 2, A = 1.5, t0 = 5, t = (3/5)1/2)

32. 32
Three-level atom, G = SU(3)

33. We have not here a funny
pictures for CS trajectories as
in SU(2) case…
33
coherent population trapping
Case of V – atom transitions and dynamics of the level populations.

34. ⎛ ⎞
H t J J J V t
= ⎜ + + ⎟+ =
I I I
⎝ ⎠
=
⎛ ⎞ ⎛ ⎞
J J J J J J J ( ) ˆ,
G G G
( ) 1 2 B(t) = B(t) e cosωt + e sinωt .
z z t z H t z
= =
⎛ + + + ⎞ + + − = ⎪⎧ ⎨ 2 J ⎜ a (2 J − 1) z z + b 4 Jzz 2 Jz z 1 + c J (1 z z ) / 2 (1 J )
zz
⎪⎫ ⎟ ⎬− ⎩⎪ ⎝ + + ⎠ + ⎪⎭
34
Molecular rotator in external magnetic field
ˆ 2 ˆ 2 2
ˆ ˆ ( ) 2 1 2 3
ˆ ( )
2 2 2
1 2 3
( ) ( ) 2
2 1 1 ˆ ˆ 2 1 1 ˆ ˆ ˆ ˆ =
ˆ2
8 8 8 8 2
γ B t J + − + − − + − + + + +
= 2 2
⎜ ⎟ + ⎜ ⎟ −
I I I I I
⎝ ⎠ ⎝ ⎠
A symbol of the hamiltonian:
3
1 2 1 2 3
G G
= = =
( , , ) ˆ ( )
2 2 2 2 2 2
2 2 2
zz zz zz
(1 ) (1 ) (1 )
i t i t
ω ω
JB t ze ze
( ) .
1
zz
γ
−
+
−
+
=
=
(
a I I b I I c
, , 1 .
− +
= = = = = =
2 1 2 1
1 II 2 1 II 2 I
3
8 8 2

35. −
+ ∂
z − i − i t i ω t
ω
35
( ) ( 3 ) 2 2 1
2 ( ) (2 ) (1 ) ( ).
1
j
a z z b c z zz ih e z e
zz
− + − − + −
+
=
(1 )2 ( , ; ) .
2
z i zz z z t
j z
= −
∂
=
(
h =γ B/ 2

36. 36
36
Superposition of CS
ψ t j j eiφ t j n j n ± = ⎡⎣ ± + + ± + ⎤⎦
( ) 1 ( 0 ( ) .
( 1 ( ) 2 ( ) )
ψ t = eiϕ t z t ± eiϕ t z t
1 2
2
z i c j z z z ih e z e
(2 1) 1 ( ),
( )
1 1 2
1 1 1 1
1 1
2 2 2
2 2 2 2
2 2
1
2 1 1 ( ).
1
i t i t
i t i t
z z
z i c j z z z ih e z e
z z
ω ω
ω ω
−
−
−
= − + −
+
−
= − + −
+
j,n ( ) 1 , ( ) , ( )
2
Sang Jae Yun , Chang Hee Nam, Molecular quantum rotors in gyroscopic motion with
a nonspreading rotational wave packet // Physical Review A 87, 032520, 2013.

37. 37
37
CS trajectory of z1 CS trajectory of z2

38. 38

39. 39
Dynamical chaos of CS parameters
for two-level models
ω
0 ω
G =W1 ⊗SU(2), CS = α ⊗ ζ
j = N / 2; N =1,2,...
The Dicke model (N à1)
m =1, 2.
One-atom maser j = 1/2

40. “Bee-like” trajectory for
atomic SU(2)
g=λ=0.2; ν=1.1, ω=1. J = 1
40

41. 41
Mean values and chaos

42. 42

43. 43
The Dicke – model with parametric interaction
(1) (2)

44. 44
Lyapunov’s maximal coefficient

45. 45

46. 46

47. 47
Chaos and entanglement
The quantum entanglement criteria
Closed (pure) quantum systems; (“atoms” + field)
A F Ψ ≠ Ψ ⊗ Ψ
( ) ˆ ( ) log ˆ ( ) , A F A F A F S = − Sp ⎡⎣ρ ρ ⎤⎦
( ) ( ) ˆ , A F A F ρ = Sp Ψ Ψ . A F A F S − S ≤ S ≤ S + S
- “Purity”, and linear (atomic) entropy
Open quantum systems, environment
- Separable “mixed” states
Entangled mixed states:

48. 48
The Dicke Model (j = N/2), N=9), Poincare sections:

49. d d d d j d d α α ς ς
π π ς
Q-corrections
∂
Ψ = Ψ
∂
i (t) Hˆ (t)
t
Ψ(0) = |α (0) ⊗|ζ (0) . Ε = E = Ψ(0) Hˆ Ψ(0) = Ψ(t) Hˆ Ψ(t) ;
= ∼ ⎡− ⋅ − ⎤ = Σ ⎣ ⎦
2 ( ; ) ( ); ( ; ) exp ( ) ( ) ; ( ) ,
f f α t ψ ζ f α t κ t α α t ψ ζ ζ j m
The initial condition:
ˆ + | = ∂ ∂ + (1 + ) | ; ˆ − | = − ⋅ ∂ ∂ + (1 + ) |
;
ˆ | j (1 ) (1 ) | .
ζ ζ ζ ζζ ζ ζ ζ ζ ζ ζζ ζ
ζ ζ ζ ζζ ζζ ζ
1 () . T
49
Ψ (t) d 2
=∫∫
α d(μ ) ς f Ε (,α ζ ;α0 ,ζ 0 | t) α ,ζ . Re Im , ( ) 2 1 Re Im .
α μς
( )
2
2 2
1 | |
+
= =
+
j j
E m m m m cl m
f α ζ α ζ t δ α α δ ζ ζ → Ε
lim ( , | , ; ) = ( − ) ( −
)
t
0 0 0 2 0 2 0
j
m =−
j
ˆ | | ; ˆ | 2 | ;
ˆ ˆ| 2| .
a a
a a
+
α α α α α α α
α α α αα α
J j J j
J
= ⋅ ∂ ∂ + − +
( ) 1 ˆ ( )2 , A F A E t = − Sp ⎡⎣ρ t ⎤⎦ { } max max ( ) , t A
E = E t ( ) 0
A A E = T ⋅ ∫ E t dt
We have to calculate:
( )
( )
+
= = ∂ ∂ +
= ⋅ ∂ ∂ +
( ) ( 2
)
( )
0 2
=

50. 50
Dynamical chaos of three-level models
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ
ˆ ( ˆ ˆ ˆ ˆ ) ( ˆ ˆ ˆ ) ( ˆ ˆ ˆ ˆ ) ( ˆ ˆ ˆ ˆ ), 1,
H H V
H H H aa aa
V g a J aJ f a J aJ g a L a L f a L a L
f
= 0
+
= +Ω + ω + +
+
ω ω = + + − + + + + =
⎧
=⎨⎩
0 0 1 0 2 11 1 2 2 2
1 1 1 1 1 1 2 2 2 1 2 2
0,
i
i
g
+ + + +
− +− + − − + + −
=
- Regular model
- Irregular model, chaos
2 12 1 2 G =W ⊗SU(3), CS = α ,α ⊗ z , z
i g z z
⎧ + ⎪ = + ⎪ + +
⎪ +
⎪ = + + + ⎨⎪
1 1
,
α ωα μ
1
,
α ω α μ
1
( ) ( )
( ) ( )( 1),
( )( 1) ( ) .
= +Ω − + + + −
( )
1 1 1 1
1 1 21 2
2 2
2 2 2 2
1 1 21 2
1
1 0 0 1 1 1 1 12 2 2 2 1
0 2
2 0 2 1 1 1 2 2 2 2 1 2
2
z z z z
i g z z
z z z z
i z z g z z g z
i z z g z g z z
ω μ α α α α
ω
μ α α α α
⎪⎪
= ⎛ +Ω ⎞ + + − + + ⎪ ⎜ ⎟
⎩ ⎝ ⎠
1 ω
2 ω
Irregular case

51. 51
OPEN SYSTEMS
Coherent relaxation (the Markovian approximation)
Finite number of “atomic” levels, SU(N)

52. 52
Glauber – Sudarshan P-representation for density operator

53. 53
G=SU(2), (2j+1)-level atom ( j=1/2, 1; j •)
The Fokker-Planck (FP) equation:

54. 54
FP-equation in the case of squeezed bath:
Method on spherical functions expansion:

55. FP-equation propagator for a qubit in a squeezed bath
55

56. 56
Emission line contour for j=1/2 “atom” in a squeezed bath
No squeezing
squeezing
The ratio of the emission lines width is
independent on the bath temperature, r is a
squeezing parameter

57. Tree-level systems. FP-equation, V-atom SU(3)
57

58. 58
Spherical functions on SU(3)/U(2)
Σ
f F t
=
∇ = +
∇ ∇ ∇
nl l m nl l m
1 2 1 2
nl l m
1 2
n n
M
1 2 1 2
2
2 2 2
1 2
( )Y ,
Y nl l m ( 3)Y nl l m
.
, , , ˆ

59. Coherent relaxation of large number of “atoms”
59

60. Line radiation contour in the first order
~ 1/N (N = 10)
60

61. 61
Line radiation contour in the first order ~ 1/N
(N = 100)

62. 62
Dipole – dipole interacting atoms

63. ↑ ↑
= ↑ ↓ + ↓ ↑
↓ ↓
= ↑ ↓ − ↓ ↑
| 1 | , | , ,
( 1 2 1 2
)
s
| 1 | , | , .
( )
, ,
1 2
, ;
1 2
1 2 1 2
2
2
a
Time Dependence of symmetric and antisymmetric states Probabilities 63
generation

64. 64
Two - level system in external stochastic
fields
ς ς ς −
Ω = ∫ Ω Ω = ∫
K t t t dt K t t t eiω 0 ( t t 1 )
dt
t t
( ) ( ) ( ) , ( ) ( ) ( )
1 1 1 1
0 0

65. 65

66. Probability to find atom in the upper level P↑ for Kubo – Andersen processes
66
Exact soluble model
Perturbation theory
Relaxation without stochastic field

67. 67
Longitudinal T1 and transverse T2 relaxation times
White
noise

68. 68
Line Radiation Contour for Dichotomic Process
(strong field)
Line contour without stochastic field

69. Parametric Amplifier in a thermal squeezed bath
69
ˆ ( ) ( ˆ ˆ 1) ( ˆ ˆ ˆ ˆ )
H t =ω a+a + + g aae 2 it ω+ a+a+e− 2
it
ω0
2
Uˆ ( , ) exp 1 ˆ ˆ 1 ˆ ˆ
I t + Δ ⎛ t t ≈ a+a+ −
ξ ξ aa⎞
2 2 ⎜ ⎟
⎝ ⎠
ξ = −2igΔt exp[−2i(ω −ω0 )t]
( ) ( ) ( ) 0 0 P t =Uˆ t,t P t
( 0 ) ( )
ˆ , ˆ ˆ ˆ ˆ , , .
0
U t t
LU U t t I
t
∂
= =
∂
( ) ( )
2 i t 2 i t r 0 r
0
− ω ω − −
ω ω
a ad be ge a ad be
ge
2 , 2 ,
+ + = + + =
2 1 ,
⎛ ⎞ + ⎜ ⎟
⎝ ⎠
= =
b + d + d b + ae r 2
ae r
=
N
re 2 2 ( 0 ω ) t ω , re 2 2 ( 0 ω ) t
ω ,
d d
2
1 .
2
d d d d
d d d d
d d i d d i
ge ge
γ
γ
− −
− −
− − − − −
= =

70. 70
Evolution operator of FP-equation

71. 71
The case of zero detuning

72. 72
Nonzero detuning

73. Summary and some problems
ƒ We have presented a mathematical formalism for
describing the dynamics and relaxation of quantum
systems.
ƒ Group theoretical method and the CS technique are
naturally used in quantum optics, quantum information
theory, condensed matter and so on.
ƒ Search for quantum corrections to the semi-classical
dynamics CS in this approach in the general case has not
been solved to date.
ƒ One of the main problems here is the inclusion of non-
Markovian effects into consideration.
ƒ Possible generalizations of the concept of dynamical
symmetries (super-algebras, associative algebras, …) to
more complicated and realistic systems also a worthy of
special consideration.
73

74. 74
Acknowledgements
I’m sincerely grateful to my former students and
colleagues (Ilya Sinayskiy, Vitaly Semin, Elena
Rogacheva, Viktor Mikhailov, Slava Ruchkov,
Vadim Asnin, Dmitriy Umov,…). The collaboration
with them was very helpful for me!
I am also very grateful to Professor
Francesco Petruccione for his kind
invitation and for the opportunity to
spend the amazing month in Durban!

75. 75
Thank You!
z