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Dynamical Groups, Coherent States and Some of their Applications in Quantum
Optics and Molecular Spectroscopy
Alexander V. Gorokhov,Samara State University, [email protected]
8 August, 2014
3
Contents
IntroductionDynamical algebras and groups in Quantum PhysicsCoherent States (CS), Path Integrals and Classical EquationsOpen Systems & Fokker – Planck equations in CS representation Summary
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{ }, , 1.
ˆ ˆ ˆ( ) : ,ˆ ˆ
i ie e
g T g T T
T T
θ θ
+
∗+
Ψ Ψ Ψ ⊂Η Ψ =
→ Ψ Φ = Ψ Φ
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
statesVacuum electron neutrinos ?
quarks photon,
… graviton 2ˆ ˆ( 1) , 0, 1 2 , 1, ...J j j I j= + =
6
Unitary representations of the Lorentz group
{ }, , 1.
ˆ ˆ ˆ( ) : ,ˆ ˆ
i ie e
g T g T T
T T
θ θ
+
∗+
Ψ Ψ Ψ ⊂Η Ψ =
→ Ψ Φ = Ψ Φ
Ψ Φ = Ψ Φ
∼
1 2 1 2 1 2 1 2ˆ ˆ ˆ( ) ( ) ( , ) ( ), | ( , ) | 1T g T g g g T g g g gω ω= ⋅ =
V. Bargmann
Ray representations, covering groups; central extensions
7
Symmetry group applications
Elementary particles, atoms, molecules, Elementary particles, atoms, molecules, crystals crystals (classification of states, selection rules, (classification of states, selection rules, "hidden" symmetry of the hydrogen atom, "hidden" symmetry of the hydrogen atom, the Lorentz group, Poincarthe Lorentz group, Poincaréé and and classification of particles)classification of particles)
J.L. Birman, CUNY, USA
Ya.A. Smorodinsky, JIRN, Dudna
V.A. Fock, Leningrad University
8
Dynamical Symmetries, 1964Dynamical 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..
SU(3) … SUI(2)8 = 1 ∆ 2 ∆ 2 ∆ 3
10
A simple example - Harmonic Oscillator
( )
( ) ( ) ( )
2 2
2 2 2 20 1 2
1 2
0 0
k+
1ˆ ˆ ˆ , ( 1)21 1 1ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆˆ, , ;4 4 4
1 1ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ, , ,2 2
ˆ ˆ ˆ ˆ ˆ ˆ, , , 2 .
(1,1) (2, ) (2, ) (2,1).1 3T , , ;4 4
H p x m
K p x K p x K px xp
K K i K K a a K aa
K K K K K K
SU SL R Sp R SO
k
ω
+ +± + −
± ± + −
= + = = =
= + = − = +
⎧ ⎫⎪ ⎪⎪ ⎪= ± = =⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭⎡ ⎤ ⎡ ⎤=± =−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
= = ≈
= 2 2 2 20 1 2
N
3ˆ ˆ ˆ ˆ ˆ .̂
dimensionaloscill
K ( 1)
ator: W ^1
)6
(2 ,N
K K K
Sp N
k k
R
I I= − − = − =−
−
11
Dynamical symmetries:Dynamical symmetries:condensed mattercondensed matter, , quantum opticsquantum 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),…
13
Dynamical symmetry in action
In a linear case it is possible to find exact solution:• Energy levels and corresponding wave functions (time independentHamiltonian);
• Transition probabilities (time dependent Hamiltonian);
• Quasi energy and quasi energy states (periodic Hamiltonian).Aharonov - Anandan geometric phase
R. Feynman’s method of operator exponent disentanglement
14
( )( ) ( ) ( )1 1 2 2 2
ˆ ˆˆ ( ),' "
' "ˆ ...
A,A A A ...,
A A A
k kH f A A
H f C f C f Cα α α
= ∈
⎛ ⎞ ⎛ ⎞= + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⊃ ⊃ ⊃
Lie groups and an energy levels calculation
Molecular rotational- vibrational levels
F. Iachello, R.D. Levine
(4) (4) (3) (2)u so so so⊃ ⊃ ⊃
H-atom, SO(4), SO(4,2)
V.A. Fock, (1935) V. Bargmann, (1936)
A.O. Barut, H. Kleinert, Yu.B. Rumer, A.I. Fet (1971)
SU(1,1), SU(N,1), Sp(2n,R), quantum optics, superfluidity
15
Molecular spectra & vibronic transitions(2 , )nW Sp N∧ R
( )
( ) ( )
[ ]( ) [ ] [ ] [ ]
'1 2
1 1 '0 0
' '
;ˆ , , ,..., ;
ˆ ˆ ˆ ˆ' , ' ,
, |
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆV =T( )D( ) U(R)ˆI V
Nq R q q q q q q
R L L q
H
L r r r r
m n
VH
m n m n
V
δ ρ
− −
+
= +Δ =
= Δ = Δ Δ = −
⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦
′ =
< >=
( ) *[ ] [ ]2 2
[ ],[ ]
1exp [ ] [ ]2 [ !] [ !]
ˆ ˆV Vv n
v nv n
v nβ αβ α β α⎡ ⎤= − + ×⎢ ⎥ ⋅⎣ ⎦
∑
-group and Franck – Condon overlap Integrals,
- Born-Oppenheimer app. & harmonic approximation for vibrations)
( ) ( )200' 00
'ˆ , ' , ( ') (2 ') exp ,2( ')! !
' 2 2 ', ' ; ( 1).' '
mn mnII m V n H I xm n
x x M
ωωω ω ωωω ω
ω ω ω ωω ω ω ω
⎡ ⎤= = Δ Δ = + − Δ⎢ ⎥+⎣ ⎦
Δ = Δ Δ = − Δ = =+ +
17
Model Hamiltonians, dynamical groups and coherent states
Klauder, Glauber, Sudarshan, Perelomov, Berezin, Gilmore, …
18
J.R. Klauder
R. Glauber, 2005, October 5
Heisenberg – Weyl group W1 and coherent states
Coherence properties of quantum electromagnetic fields, lasers
1963
20
Path Integrals in CS - representation
0 0 1 1 2 1 0ˆ ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , ) ( , )N N N N NU t t U t t U t t U t t U t t− − −≡ = ⋅⋅⋅
21
“Classical” Equations
2
1
2
1
( , ) ln ( , ) ,
( , ) ln ( , ) .
n
n
i z z K z z zz z z
i z z K z z zz z z
βα α β
β
βα α β
β
=
=
∂ ∂=
∂ ∂ ∂
∂ ∂− =
∂ ∂ ∂
∑
∑
2 ln ( , ) , , .K z zg g g g gz z
α β α α β βκ αγ γαβ κβα β δ δ∂
= ⋅ = ⋅ =∂ ∂
For linear systems classical dynamics of coherent states is exact!
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
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)
33Case of V – atom transitions and dynamics of the level populations.
We have not here a funny pictures for CS trajectories as in SU(2) case…
coherent population trapping
34
Molecular rotator in external magnetic field
A symbol of the hamiltonian:
( ) ( )2
2 23
1 2 1 2 3
22 22 31 2
1 2 3
2 2 21 1 1 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ8 8 8 8 2
ˆˆ ˆˆ ˆ( ) ( )2 2 2
ˆ( ) ,J J J J J J JI I I I I
JJ JH t V tI I I
B t Jγ+ − + − − +− + + + +
⎛ ⎞= + + + =⎜ ⎟
⎝ ⎠⎛ ⎞ ⎛ ⎞
= + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
2 2 2 2 2 2
2 2 2
ˆ( , , ) ( )
4 2 1 (1 ) / 2 (1 )2 (2 1)(1 ) (1 ) (1 )
( ) .1
i t i t
z z t z H t z
z z Jzz Jz z J z z J zzJ a J b czz zz zz
ze zeJB tzz
ω ω
γ−
= =
⎧ ⎫⎛ ⎞+ + + + + −⎪ ⎪= − + + −⎨ ⎬⎜ ⎟+ + +⎪ ⎪⎝ ⎠⎩ ⎭+
−+
2 1 2 1
1 2 1 2 3
1, , .8 8 2I I I Ia b c
I I I I I− +
= = =
( )1 2( ) ( ) cos sin .B t B t e t e tω ω= +
35
( ) ( )3 22 12 ( ) (2 ) (1 ) ( ).
1i t i ti
ja z z b c z zz ih e z e
zzz ω ω−−
−− + − − + −
+=
2(1 ) ( , ; ) .2
zz z z tz ij z
+ ∂= −
∂
/ 2h Bγ=
3636
Superposition of CS
( )1 2( ) ( )1 2
1( ) ( 0 ( ) .2
i t i tt e z t e z tϕ ϕψ = ±
( )
21 11 1 1 1
1 1
22 22 2 2 2
2 2
1(2 1) ( ),112 1 ( ).1
i t i t
i t i t
z zz i c j z ih e z ez zz zz i c j z ih e z ez z
ω ω
ω ω
−
−
−= − + −
+−
= − + −+
( )j,n ( )1( ) , ,2
i tt j j e j n j nφψ ± ⎡ ⎤= ± + + ± +⎣ ⎦
Sang Jae Yun , Chang Hee Nam, Molecular quantum rotors in gyroscopic motion witha nonspreading rotational wave packet // Physical Review A 87, 032520, 2013.
39
Dynamical chaos of CS parameters for two-level models
ω0ω
1 (2),G W SU CS α ζ= ⊗ = ⊗/ 2; 1,2,...j N N= =
( 1)N àThe Dicke model
m =1, 2.
One-atom maser j = 1/2
47
Chaos and entanglement
A FΨ ≠ Ψ ⊗ Ψ
The quantum entanglement criteria
( ) ( ) ( )ˆ ˆlog ,A F A F A FS Sp ρ ρ⎡ ⎤=− ⎣ ⎦
( ) ( )ˆ ,A F A FSpρ = Ψ Ψ .A F A FS S S S S− ≤ ≤ +
- “Purity”, and linear (atomic) entropy
Closed (pure) quantum systems; (“atoms” + field)
- Separable “mixed” states
Entangled mixed states:
Open quantum systems, environment
49
20 0( ) ( ) ( , ; , | ) , .t d d f tα μ ς α ζ α ζ α ζΕΨ = ∫∫
( )2
22
Re Im 2 1 Re Im, ( ) .1 | |
d d j d dd dα α ς ςα μ ςπ π ς
+= =
+
0 0 2 0 2 00lim ( , | , ; ) ( ) ( )t
f tα ζ α ζ δ α α δ ζ ζΕ→= − −
2ˆ( ) 1 ( ) ,A F AE t Sp tρ⎡ ⎤= − ⎣ ⎦ { }max max ( ) ,AtE E t= ( )
01 ( ) .
T
A AE T E t dt= ⋅ ∫
We have to calculate:
2( ; ) ( ); ( ; ) exp ( ) ( ) ; ( ) ,j
j jE m m m m cl m
m j
f f t f t t t j mα ψ ζ α κ α α ψ ζ ζ=−
⎡ ⎤= ∼ − ⋅ − =⎣ ⎦∑
( )( )
ˆ ˆ| | ; | 2 | ;
ˆ ˆ | 2 | .
a a
a a
α α α α α α α
α α α αα α
+
+
>= > >= ∂ ∂ + >
>= ⋅∂ ∂ + >
( ) ( )( )
2
0 2
ˆ ˆ| (1 ) | ; | (1 ) | ;
ˆ | (1 ) (1 ) | .j
J j J j
J
ζ ζ ζ ζζ ζ ζ ζ ζ ζ ζζ ζ
ζ ζ ζ ζζ ζζ ζ
+ −>= ∂ ∂ + + > >= − ⋅∂ ∂ + + >
>= ⋅∂ ∂ + − + >
The initial condition:
(0) (0) (0) .α ζΨ = | > ⊗ | > ˆ ˆ(0) (0) ( ) ( ) ;E H t H tΕ = = Ψ Ψ = Ψ Ψ
ˆ( ) ( )i t H tt∂
Ψ = Ψ∂
Q-corrections
50
Dynamical chaos of three-level models0
0 0 1 0 2 1 1 1 2 2 2
1 1 1 1 1 1 2 2 2 1 2 2
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ), 1,
0,i
i
H H V
H H H a a a a
V g a J a J f a J a J g a L a L f a L a L
fg
ω ω ω+ +
+ + + +− +− + − − + + −
= +
= +Ω + +
= + + − + + + + =
⎧= ⎨⎩
2 1 2 1 2(3), , ,G W SU CS z zα α= ⊗ = ⊗
( ) ( )
( )
1 11 1 1 1
1 1 21 2
2 22 2 2 2
1 1 21 2
11 0 0 1 1 1 1 1 2 2 2 2 1
202 0 2 1 1 1 2 2 2 2 1 2
,1
,1
( ) ( )( 1) ,
( )( 1) ( ) .2
z zi gz z z z
z zi gz z z z
i z z g z z g z
i z z g z g z z
α ωα μ
α ω α μ
ω μ α α α α
ω μ α α α α
+⎧ = +⎪ + +⎪+⎪
= +⎪ + +⎨⎪ = +Ω − + + + −⎪⎪ ⎛ ⎞= +Ω + + − + +⎪ ⎜ ⎟
⎝ ⎠⎩
- Regular model
- Irregular model, chaos
1ω
2ω
Irregular case
51
OPEN SYSTEMSCoherent relaxation (the Markovian approximation)
Finite number of “atomic” levels, SU(N)
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
58
Spherical functions on SU(3)/U(2)
1 2 1 2
1 2
1 2 1 2
2
2 2 21 2
( )Y ,
Y ( 3)Y .ˆ, , ,
nl l m nl l mnl l m
nl l m nl l m
f F t
n n
M
=
∇ = +
∇ ∇ ∇
∑
63Time Dependence of symmetric and antisymmetric states Probabilities generation
( )
( )
1 2
1 2 1 2
1 2
1 2 1 2
, ,
1| | , | , ,2
, ;
1| | , | , .2
s
a
↑ ↑
>= ↑ ↓ > + ↓ ↑ >
↓ ↓
>= ↑ ↓ > − ↓ ↑ >
64
Two - level system in external stochastic fields
0 1( )1 1 1 1
0 0
( ) ( ) ( ) , ( ) ( ) ( )t t
i t tK t t t dt K t t t e dtως ς ς −
Ω = Ω Ω =∫ ∫
66
Probability to find atom in the upper level P↑ for Kubo – Andersen processes
Exact soluble modelPerturbation theoryRelaxation without stochastic field
68
Line Radiation Contour for Dichotomic Process
(strong field)
Line contour without stochastic field
69
Parametric Amplifier in a thermal squeezed bath2 2
01ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )2
i t i tH t a a g aae a a e+ + + −= + + +ω ωω1 1ˆ ˆ ˆ ˆ ˆU ( , ) exp2 2I t t t a a aaξ ξ+ +⎛ ⎞
⎜ ⎟⎝ ⎠
+Δ ≈ −02 exp[ 2 ( ) ]ig t i tξ ω ω= − Δ − −
( ) ( ) ( )0 0ˆ ,P t U t t P t=
( ) ( )00
ˆ , ˆ ˆ ˆ ˆ, , .U t t
LU U t t It
∂= =
∂
( ) ( )
0 02 ( ) t 2 ( ) t
0 02 2
2
2 , 2 ,
2 , 2 ,12 ,2
1 .2
d d d d
d d d d
i id d d d
i t i tr r
r ae r
ge ge
a ad be ge a ad be ge
b d d b ae N
re re d dω ω ω ω
ω ω ω ω
γ
γ
− −
− −
− − −− −
− − −
⎛ ⎞ +⎜ ⎟⎝ ⎠
= =
+ + = + + =
+ + + =
= =
73
Summary and some problemsWe have presented a mathematical formalism for describing the dynamics and relaxation of quantum systems. Group theoretical method and the CS technique are naturally used in quantum optics, quantum information theory, condensed matter and so on. Search for quantum corrections to the semi-classical dynamics CS in this approach in the general case has not been solved to date.One of the main problems here is the inclusion of non-Markovian effects into consideration.Possible generalizations of the concept of dynamical symmetries (super-algebras, associative algebras, …) to more complicated and realistic systems also a worthy of special consideration.
74
AcknowledgementsI’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!