Системи, редуцируеми до системи с две и три нива: ...

Системи, редуцируеми до системи с две и три нива:

Декомпозиция на Морис-ШорХаусхолдер трансформация

Декомпозиция на Морис-Шор за системи с две нива

Трансформация на Морис-Шор (МШ) е възможна когато: всички взаимодействия са една и съща функция на

времето и когато имат еднакъв детунинг( )f t ( )t

a bN N

2 31

bN

( )f t

aN

bN

( )f t

Трансформация на МШ смесва състояния само от едно и също ниво:

1) Основните състояния на системата след трансформацията са суперпозиция само от основните състояния на системата преди МШ трансфомацията

2) Възбудените състояния на системата след трансформацията са суперпозиция само от възбудените състояния на системата преди МШ трансфомацията

Тъмни състояния

Светли състояния

-изроденост на основно ниво

-изроденост на възбудено ниво

Морис-Шор трансформация

( )t( )t


2J

1J

M

M 1 0 1

2 1 0 1 2



aN

bN

a bN NТъмни състояния

2 31

Светли състоянияbN

( )f t

( )f t

aN

bN

( )f t

( ) ( ) ( )d

i C t H t C tdt

†

0 ( )( )

( ) ( )

V tH t

V t t

Имаме уравнението на Шрьодингер

За Хамилтониана

e матрица,( )V t b aN xN

е матрица,†( )V t a bN xN

е матрица( )t b bN xN

a bN N

2 31

bN

( )f t

( ) ( )MSC t SC t0

0a

b

SS

S

Матрици на МШ трасформатцията

aS a aN xNе константна матрица,

( ) ( ) ( )MS MS MSdi C t H t C tdt

bS b bN xN


Тъмни състояния

Светли състояния



( )t

Уравнение на Шрьодингер в МШ базиса

е константна матрица,

††

0 ( )( ) ( )

( )MS M t

H t SH t SM t

Хамилтониана в МШ базиса

†( ) ( )a bM t S V t S

Разлагането на H на няколко независими системи с две нива изисква, след премахване на нулевите колони или редове, матрицата M да се редуцира (възможно е пренареждане на елементите) до диагонална матрица

† †( ) ( ) ( ) ( )a aM t M t S V t V t S† †( ) ( ) ( ) ( )b bM t M t S V t V t S

aS bS†( ) ( )V t V t †( ) ( )V t V t

и се дефенират така, че да диагонализират и

†( ) ( ) n n nV t V t u u † † †( ) ( )n n n nu V t V t u u u

† † † †( ( ) ) ( ( ) )n n n nV t u V t u u u† 0n nu u † † †( ( ) ) ( ( ) ) 0n nV t u V t u

0n 2n n

Но

Матрицата има същите собствени стойности като и допълнително нулеви собствени стойностиa bN N

†( ) ( )V t V t

Собствениете стойности на ( ) са неотрицателни†( ) ( )V t V t 2n

†( ) ( )V t V t

0 ( )( )

( )n

nn

tH t

t

Each of these two-state Hamiltonians has the same detuningBut they have different Rabi frequencies ( )n t

It is important for MS transformation that the couplings share the same time dependence and have the same detuning ( )f t

† † 2 †( ) ( ) ( ) ( ) ( ( ))a a a aM t M t S V t V t S f t S VV S

† † 2 †( ) ( ) ( ) ( ) ( ( ))b b b bM t M t S V t V t S f t S V VS

If all couplings share the same time dependence then( )f t

Thus and areno time dependent

aS bS

† † †( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )a a a aM t M t S V t V t S S t V t V t S t

† † †( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )b b b bM t M t S V t V t S S t V t V t S t

If some couplings share different time dependence then

Thus and aretime dependent

aS bS

Which means that the MS basis changes with time.

The three-level Morris-Shore (MS) transformation

A. A. Rangleov, N. V. Vitanov, B. W. Shore Phys. Rev. A 7, (2006)

(1) (2) (3) (4) (5)

Morris-Shoretransformation

The quasi-two-level case

( )f t

( )f t

c

b d

a

a

( )f t

( )f t

( )f t

c

b

d

1) All couplings share the same time dependence 2) The two-photon resonances a-c and b-d are fulfilled Then one can carry out the MS factorization on the new degenerate two-level system, as displayed in the figure

( )f t

The three-level Morris-Shore transformation


c

2 ( )f t

1( )f t (3)1

(2)1

(1)1

(2)2(1)

2

b

c

a

cc

b

2 ( )f t

1( )f t

b

a

bN

aN

cN

We search for a transformation that mixes only sublevels of a given level. Reducethe system of three degenerate levels to:several three level problems, several two state problems and possibleseveral dark states.

1

†1 2

†2

0 0

( )

0

b

c

V

H t V V

V

1

† †1 2

†2

0 0

0

MSb

c

M

H SHS M M

M

The Hamiltonian in MS basis:

0 0

0 0

0 0

a

b

c

S

S S

S

Transformation matrix:

The Hamiltonian:

, and are constant square unitary matricesof dimensions , and respectively

aS bS cS

aN cNbN( ) ( )MSC t SC t

( ) ( ) ( )d

i C t H t C tdt

Schrödinger Equation:

Schrödinger Equation in MS basis:

( ) ( ) ( )MS MS MSdi C t H t C tdt

†1 1( ) ( )a bM t S V t S

†2 2( ) ( )b cM t S V t S

The matrices and may have null rows; these correspond to dark states. The desired decomposition of into a set of independent two- or three-state systems requires that, after removing the null rows, and reduce (possibly after an appropriate relabeling) to diagonal matrices.

1M 2M

1M 2MH

† † †1 1 1 1( ) ( )a aM M S V t V t S diag † † †

1 1 1 1( ) ( )b bM M S V t V t S diag

† † †2 2 2 2( ) ( )c cM M S V t V t S diag

† † †2 2 2 2( ) ( )b bM M S V t V t S diag

Hence and are defined by the condition that they diagonalize and respectively. The matrix must, by definition, diagonalize both and . This can only occur if these two matrixes commute,

bSaS

†2 2( ) ( )V t V t

cS

1W 2W

1 2, 0W W 1 2 2 1WW W W

†1 1( ) ( )V t V t

†1 1( ) ( )a bM t S V t S †

2 2( ) ( )b cM t S V t S

We define the matrix products

†1 1 1( ) ( ) ( )W t V t V t

†2 2 2( ) ( ) ( )W t V t V t

Special case: Single intermediate state


cc

b

a

b

1( )f t

2 ( )f t

c

a

1( )f t

2 ( )f t

1

2

b

c

b

In case of a , and reduce to scalars

and condition hold automatically. Hence for a single

intermediate state the MS transformation is always possible.

1bN

1 2, 0W W

1W 2W

Extension to N levels

The results for the three-level MS transformation are readily extended to N degenerate levels. For each transition n we form the matrixes when n is a odd number and when n is even number then the N-level MS transformation exists if and only if:

†n n nW V V

1 2

2 3

3 4

2 1

, 0

, 0

, 0

.......................

, 0N N

W W

W W

W W

W W

When the left system is full filled then MS transformation will produce sets of independent nondegenerate N-state systems, (N -1)-state systems, and so on, and a numberof uncoupled states, depending on the particular system.

†n n nW V V

STIRAP between degenerate levels

Well known three levels lambda and ladder systems. One cane make effective population transfer in them using the STIRAP method.

b

a

b

1( )f t2 ( )f t

c c

c

a

1( )f t

2 ( )f t

1

2

b

c

b


cc

b

a

b

1( )f t

2 ( )f t

c

a

1( )f t

2 ( )f t

1

2

b

c

b

Well known three levels lambda and ladder systems. One cane make effective population transfer in them using the STIRAP method.

b

a

b

1( )f t2 ( )f t

c c

c

a

1( )f t

2 ( )f t

1

2

b

c

b

c c

b

1( )f t

2 ( )f t

a

b


a

b

2 ( )f t

1( )f t

b

c

(2)2 (2)

2

(2)1

(1)1

c


f

e

1( )p t

1( )s t

e

f

( )p t

( )s t( )sN ft

( )pNgt

is degeneracy of ground level gN

is degeneracy of final level fN

-bright state of ground level1

1| |

Ng

g pk gkp

b k

1

1| |

N f

f sk fks

b k

-bright state of final level

2

1

Ng

p pkk

2

1

N f

s skk

|1g | ggN | 2g | gb

| fb |1 f | f fN

....................

( ) | 1gb ( ) | 1fb

REFLECTION for students

the vector is the reflection of over the plane P with a normal vector :

unitary operator

1 2ˆ M

ˆ ˆ 2 M 11 †ˆ ˆ M M

2 M I

2 1 †2 2 M M MI I MI

1 2 1 2 1 2 2 1, 4 0 M M

A.S. Householder, J. ACM 5, 339 (1958)

Householder Reflection: action on arbitrary matrix

produces an upper (lower) triangular matrix

by only N1 operations

N(N-1)/2 steps needed with Givens SU(2) rotations

1

0

0

0

0

0

* * * * * * * * * * * *

* * * * * * * * * * *

* * * * * * * * * * *( )

* * * * * * * * * * *

* * * * * * * * * * *

* * * * * * * * * * *

M 2

0 0

0 0 0

0 0 0

0 0 0

0 0

* * * * * * * * * * * *

* * * * * * * * * *

* * * * * * * * *( )

* * * * * * * * *

* * * * * * *

0

* *

* * * * * * * * *

M 3

0 0

0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0

* * * * * * * * * * * *

* * * * * * * * * *

* * * * * * * *( )

* * * * * * *

* * * * * * *

* * * * 0 0 * * *

M 4

0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0

* * * * * * * * * * * *

* * * * * * * * * *

* * * * * * * *( )

* * * * * *

* * * * *

* * * * *0 0 0

M 5

0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

* * * * * * * * * * * *

* * * * * * * * * *

* * * * * * * *( )

* * * * * *

* * * *

* * *

M

2 M I

1 2 1( ) ( ) ( )N M M M

Householder Reflection: action on unitary matrix

turns a unitary matrix into a diagonal matrix!!! by only N1 operations

a recipe for synthesis of arbitrary preselected unitaries!

N(N-1)/2 steps needed with Givens SU(2) rotationsM Reck, A Zeilinger, HJ Bernstein, P Bertani, Phys. Rev. Lett. 73, 58 (1994)

2 M I

1

0 0 0 0 0

0

0

0

0

0

* * * * * * *

* * * * * * * * * * *

* * * * * * * * * * *( )

* * * * * * * * * * *

* * * * * * * * * * *

* * * * * * * * * * *

M 2

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0

0 0 0

0 0

* *

* * * * * *

* * * * * * * * *( )

* * * * * * * * *

* * * * * * * * *

* * * * * * * * *0

M 3

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0

* *

* *

* * * * *( )

* * * * * * *

* * * * * * *

* * * * * * *0 0

M 5

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0

* *

* *

0 0 0 0

* *( )

* *

* * *

* * *

M 4

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0

* *

* *

* *( )

* * * *

* * * * *

* 0 0* * * *0

M

1 2 1( ) ( ) ( )N U M M M D

Householder Reflection: action on Hermitian matrix

turns a Hermitian matrix into a tri-diagonal matrix!!!

reduces any interaction linkage patternto a nearest-neighbour (chain) linkage

2n n n M I

0 0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0

* * * * * * * *

* * * * * * * * *

* * * * * * * * *QHR* * * * * * * * *

* * * * * * * * *

* * * * * 0 * *0* 0

1n n nn

x x e

norm

2 2 1 1 2 2( ) ( ) ( ) ( ) ( ) ( )N N HM M M M M M

AA Rangelov, NVV, BW Shore, PRA 77, 033404 (2008)

Householder Reflection on a Hamiltonian

turns a Hermitian matrix into a tri-diagonal matrix!!!reduces any interaction linkage pattern to a nearest-neighbour (chain)

linkage

* * * * * 0QHR

* * * * * *

* * * 0 * *


Householder Reflection: Three-state loop


Householder Reflection: Three-state loop


Applications chain breaking and two-state subspaces analytic solutions creation of superpositions hidden spectator states

when 3 0

example:

Системи, редуцируеми до системи с две и три нива: ...

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