research article the rabi oscillation in subdynamic system...

Research ArticleThe Rabi Oscillation in Subdynamic System forQuantum Computing

Bi Qiao and Gu Jiayin

Department of Physics Science School Wuhan University of Technology Wuhan 430070 China

Correspondence should be addressed to Bi Qiao biqiaogmailcom

Received 16 August 2015 Revised 3 October 2015 Accepted 5 October 2015

Academic Editor Pavel Kurasov

Copyright copy 2015 B Qiao and G JiayinThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A quantum computation for the Rabi oscillation based on quantum dots in the subdynamic system is presentedThe working statesof the original Rabi oscillation are transformed to the eigenvectors of subdynamic system Then the dissipation and decoherenceof the system are only shown in the change of the eigenvalues as phase errors since the eigenvectors are fixed This allows bothdissipation and decoherence controlling to be easier by only correcting relevant phase errors This method can be extended togeneral quantum computation systems

1 Introduction

The Rabi oscillation based on the quantum dot system isan efficient way to realize quantum logic operation [1] Theprojected formalism of control rotation gate (CROT) for twoqubits is expressed as negative CNOT operation [2] Thephysical observation of the Rabi oscillation with 10120587 periodhas been reported and the detail analysis of quantum com-putation based on the Rabi oscillation and relevant mech-anism of decoherence and dissipation have been discussedby several works with various methods [3ndash12] Howeverthe controlling dissipation and decoherence in the Rabioscillation system is still an important problem to challengeapplication of the Rabi oscillation in quantum computation

On the other hand subdynamics theory rooted from theold Brussels-Austin school [13ndash15] and followed by some up-to-date works [16 17] has been found that there is a potentialadvantage to cancel both decoherence and dissipation inquantum information systemsThis is so because the relevantsubdynamic kinetic equation (SKE) intertwines with theoriginal Schrodinger equation or Liouville equation by a sim-ilarity transformation which allows to construct the relativeideal subdynamic system in which the eigenvectors are fixedwhen decoherence or dissipation happens Therefore theerrors will only exist in the phase change of the evolution ofprojected states whichmakes easier to cancel the decoherenceand dissipation

Considering this background in this work we present asubdynamic proposal [13ndash20] to realize quantum computingbased on the Rabi oscillation This proposal can provide arelative ideal Rabi oscillation used in quantum computingby controlling both dissipation and decoherence Below theformalism is firstly introduced

2 Subdynamic Formalism

Let us consider a two-level system described by a Hamilto-nian

(119905) = 120596012059000

+ 120596112059011

+

1

2

119892 (12059010119890minus119894120596119871119905

+ 12059001119890+119894120596119871119905

) (1)

where 1205960and 120596

1is an eigen-frequency of states |0⟩ and

|1⟩ respectively 12059000

= |0⟩⟨0| 12059011

= |1⟩⟨1| 12059001

=

|0⟩⟨1| and 12059010

= |1⟩⟨0| 120596119871is a frequency of exciting

optical field 119892 = minus120583119864(0) is a Rabi frequency betweenstates |0⟩ and |1⟩ jumping 120583 is a dipole moment between|0⟩ and |1⟩ jumping and 119864(119905) is an envelope of excitingoptical field 119864(119905) = 119864(0) cos120596

119871119905 = (12)119864(0)(119890

minus119894120596119871119905+

119890+119894120596119871119905

) Notice the so-called rotation wave approximationhas been used to produce above Hamiltonian Then forsimplicity here the explicitly time-dependent Hamiltoniancan be transformed to the time-independent one by a time-dependent unitary transformation 119880

00(119905) = exp(minus119894120596

11987111990512059000)

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 151690 7 pageshttpdxdoiorg1011552015151690

2 Advances in Mathematical Physics

where 12059000

= |0⟩⟨0| is a projector [21] So we obtain a time-independent Hamiltonian as

119867 = 119880minus1

00(119905) (119905) 11988000 (

119905)

= 120596012059000

+ 120596112059011

+

1

2

119892 (12059001

+ 12059010) = 119867

0+ 1198671

(2)

where defining

1198670= 120596012059000

+ 120596112059011

1198671=

1

2

119892 (12059001

+ 12059010)

(3)

When the exciting optical field takes off 11988000(119905) = 1 we have

(119905) = 119867 So the corresponding Schrodinger equation forthe Hamiltonian in (2) is

119894

120597120595

120597119905

= 119867120595 (4)

which can reflect the same dynamics described by theHamiltonian in (1)

If considering the influence of environment the totalHamiltonian can be supposed as 119867 = 119867

0+ 1198671+ Δ119867

where Δ119867 comes from the interaction between the systemand environment which may introduce dissipation anddecoherence in the system Then a main frame to constructthe subdynamic systems is firstly to establish a subdynamicequation [13ndash19] as

119894

120597

120597119905

= Θ (5)

where is a projected state 119867 + Δ119867 = ΩΘΩminus1 and Ω is

a similarity operator These parameters and relations can beconstructed by the following several rules

(1) Propose projectors as

119875119899119896

=1003816100381610038161003816119899 otimes 120593119896⟩ ⟨119899 otimes 120593

119896

1003816100381610038161003816

for 119899 = 0 1 119896 = 0 1 2

(6)

and 119876119899119896

= 1 minus 119875119899119896 where 119899 corresponds to the index

of eigenvectors for Hamiltonian in system 119896 denotes theindex to the Hamiltonian in environment and 120593

119896is the 119896th

eigenvector of diagonal part of the interaction Hamiltonianbetween the system and environment

(2) Introduce the creation (destruction) operatorexpressed as

119862119899119896

= 119875119899119896119867119875119899119896

+ 119875119899119896Δ119867119876119899119896

1

119864119899119896

minus 119876119899119896 (

119867 + Δ119867)119876119899119896

119876119899119896Δ119867119875119899119896

119862dagger

119899119896= 119863119899119896

(7)

(3) Introduce the projected state in the subdynamicsystem as

= Ωminus1120595 (8)

where120595 is a solution of the original Schrodinger equation andthe similarity operatorΩ is defined by

Ω119899119896

= (119862119899119896

+ 119875119899119896)

Ω =

1

sum

119899=0

sum

119896

Ω119899119896

Ωminus1

=

1

sum

119899=0

sum

119896

Ωminus1

119899119896

=

1

sum

119899=0

sum

119896

(119875119899119896

+ 119863119899119896119862119899119896)minus1

(119875119899119896

+ 119863119899119896)

(9)

(4)The ruling equation is subdynamic Liouville equation

119894

120597

120597119905

120588 = [Θ 120588] (10)

where Θ operator is introduced by

Θ =

1

sum

119899=0

sum

119896

Θ119899119896

=

1

sum

119899=0

sum

119896

(119875119899119896119867119875119899119896

+ 119875119899119896Δ119867119876119899119896119862119899119896119875119899119896)

(11)

and the relationship of the aboveΘ operator with the original119867+Δ119867 can be proven by a kind of similarity transformation

Θ = Ωminus1

(119867 + Δ119867)Ω (12)

So it is obvious to use (12) and consider (8) one can obtainSKE (5) from the Schrodinger equation or the Liouvilleequation

3 Rabi Oscillation in Subdynamic Space

Using the Rabi oscillation between |1⟩ (or |1⟩⟨1|) and |0⟩

(or |0⟩⟨0|) as the working states of quantum computationis a reasonable approach of quantum computing based onquantum dot system [11 22] In the ideal situation Δ119867 = 0the Liouville equation for this two-level system with the Rabioscillation is expressed by

119894119880minus1

11(119905)

120597120588

120597119905

11988011 (

119905) = 119880minus1

11(119905) [119867 120588]119880

11 (119905)

= 119880minus1

11(119905)119867120588119880

11(119905) minus 119880

minus1

11(119905) 120588119867119880

11(119905)

= 119880minus1

11(119905) 11988011 (

119905) (119905) 119880minus1

11(119905) 12058811988011 (

119905)

minus 119880minus1

11(119905) 120588119880

11(119905) (119905) 119880

minus1

11(119905) 11988011(119905)

= (119905) 119880minus1

11(119905) 12058811988011 (

119905) minus 119880minus1

11(119905) 12058811988011 (

119905) (119905)

= [ (119905) 120588 (119905)] = 119894

120597119880minus1

11(119905) 120588119880

11(119905)

120597119905

= 119894

120597120588 (119905)

120597119905

(13)

Advances in Mathematical Physics 3

where one notices that using (13) one has

120597119880minus1

11(119905)

120597119905

12058811988011(119905) + 119880

minus1

11(119905) 120588

12059711988011 (

119905)

120597119905

= 0

120588 = 11988011 (

119905) 120588 (119905) 119880minus1

11(119905)

= 12058811 |

1⟩ ⟨1| + 12058800 |

0⟩ ⟨0| + 12058810 |

1⟩ ⟨0|

+ 12058801 |

0⟩ ⟨1|

(14)

Then from (13) one gets the time-independent equations of⟨119899|120588|119898⟩ = 120588

119899119898as

119894

119889

119889119905

12058811

=

1

2

119892 (12058801

minus 12058810)

119894

119889

119889119905

12058800

= minus119894

119889

119889119905

12058811

= minus

1

2

119892 (12058801

minus 12058810)

119894

119889

119889119905

12058810

=

1

2

119892 (12058800

minus 12058811)

(15)

Then by introducing 119880 119881119882 as a Bloch vector that is

= 1198801198901+ 1198811198902+1198821198903

(16)

with

119880 = 12058810

+ 12058801

119881 = 119894 (12058810

minus 12058801)

119882 = 12058811

minus 12058800

(17)

where 1198901 1198902 1198903is an unit vector respectively and 119890

119894sdot 119890119895= 120575119894119895

thus (15) can be changed as

119889

119889119905

(

119880

119881

119882

) = (

0 0 0

0 0 119892

0 minus119892 0

)(

119880

119881

119882

) (18)

Then using the subdynamic similarity operatorΩ above onegets

Ωminus1(

(

(

119889119880

119889119905

119889119881

119889119905

119889119882

119889119905

))

)

= Ωminus1

(

0 0 0

0 minus0 119892

0 minus119892 0

)ΩΩminus1

(

119880

119881

119882

) (19)

119889

119889119905

(

) = (

Θ1

0 0

0 Θ2

0

0 0 Θ3

)(

) (20)

The solution of (20) is given by

Ω(

(119905)

(119905)

(119905)

) = Ω(

minus119894119892 0 0

0 119894119892 0

0 0 0

)Ωminus1Ω(

(0)

(0)

(0)

) (21)

where

(

119880

119881

119882

) = Ω(

) = (

0 0 1

119894 minus119894 0

1 1 0

)(

) (22)

so (20) can also give a solution based on the Bloch ball

(

119880(119905)

119881 (119905)

119882 (119905)

) = (

1 0 0

0 cos 120579 minus sin 120579

0 sin 120579 cos 120579)(

119880(0)

119881 (0)

119882 (0)

) (23)

where 120579 is defined as a radial rotation angle In the initialcondition (by considering (17)which corresponds to an initialpure state of the Rabi oscillation 120588

00= 1 and else three states

12058811 12058801 and 120588

10as zero) one gets

(

119880(0)

119881 (0)

119882 (0)

) = (

0

0

minus1

) (24)

This allows one from (23) to get

(

119880(119905)

119881 (119905)

119882 (119905)

) = (

0

minus sin 120579 (119905)

minus cos 120579 (119905)) (25)

which reflects the Rabi oscillation in the ideal situationwithout the interaction from environment

4 Controlling Dissipation and Decoherence

Now let us consider there is an interaction between the systemand environment This can introduce some dissipation anddecoherence For example an interaction of Liouvillian forthe Rabi oscillation system can be specified by

Δ119871 (120588) =

1

2

12057410(21205900112058812059010

minus 12059011120588 minus 120588120590

11) (26)

where 12059010

or 12059001

expressed the same meaning as in (1) and12057410

is a coupling number reflecting dissipation velocity Sothe corresponding master equation for the pseudovector 120588 =(12058800 12058811 12058801 12058810) is expressed as

119889

119889119905

12058811

= minus119894

1

2

119892 (12058801

minus 12058810) minus 1205741012058811

119889

119889119905

12058800

= minus

119889

119889119905

12058811

= 119894

1

2

119892 (12058801

minus 12058810) + 1205741012058811

119889

119889119905

12058810

= minus11989412057512058810

minus 119894

1

2

119892 (12058800

minus 12058811) minus

1

2

1205741012058810

(27)


then through the subdynamic similarity operator Ω oneobtains

119894

119889

119889119905

Ωminus1

(

119880

119881

119882

)

= Ωminus1

(

minus

1

1198792

minus120575 0

120575 minus

1

1198792

119892

0 minus119892 minus

1

1198791

)ΩΩminus1

(

119880

119881

119882

)

+Ωminus1

(

0

0

minus

1

1198791

)

(28)

where we notice that 120575 = 1205961minus1205960minus120596119871 1198792= 12119879

1+120574ph 1198791 =

112057410expresses the life time of state |1⟩ and 120574ph is a coupling

number reflecting pure phase decoherenceThen considering(18) one can define

(L + ΔL)(119880

119881

119882

) = (

minus

1

1198792

minus120575 0

120575 minus

1

1198792

119892

0 minus119892 minus

1

1198791

)(

119880

119881

119882

)

+(

0

0

minus

1

1198791

)

(29)

This allows us to introduce the subdynamics for (119880 119881119882) rarr

120588 = (12058800 12058811 12058822 12058801 12058810) or = (120588

00 12058811 12058822 12058801 12058810) by the

above formalism

119889

119889119905

Ωminus1120588 (119905) = Ω

minus1(L + ΔL) ΩΩ

minus1120588 (119905) = Θ120588 (119905)

=

119889

119889119905

120588 (119905)

(30)

with

Θ = Ωminus1

(L + ΔL) Ω

= 119875 (L + ΔL) 119875 + 119875 (L + ΔL) 119876119862119875

120588 (119905) = Ω120588 (119905) = (119875 + 119862) 120588 (119905) = sum

119899119898

(119875119899119898

+ 119862119899119898

) 120588 (119905)

(31)

by defining the projector

119875 = sum

119899119898

119875119899119898

= sum

119899119898

1003816100381610038161003816120588119899119898

) (120588119899119898

1003816100381610038161003816

for 119899119898 = 00 11 01 10

(32)

with119876 = 1 minus 119875 (33)

so that 120588 = (12058800 12058811 12058801 12058810) is expressed by the subdynamic

formalism1003816100381610038161003816120588119899119898

) = (119875119899119898

+ 119862119899119898

)1003816100381610038161003816120588119899119898

) (34)

This 120588119899119898

can be measured in the practical system withenergy term 119897

119899119898 and then using 120588

119899119898 119897119899119898

we can constructthe projected states 120588

119899119898in the subdynamic space by the

subdynamic formalism1003816100381610038161003816120588119899119898

) =1003816100381610038161003816120588119899119898

) + 119876119899119898

1

119872119899119899

minus 119876119899119898

(L + ΔL) 119876119899119898

sdot 119876119899119899 (

L + ΔL) 1003816100381610038161003816120588119899119899)

(35)

where it is marvelous that whatever the system has Lor L + ΔL the above formalism gives the same =

(12058800 12058811 12058822 12058801 12058810) rarr ( ) in the subdynamic space

namely if L of the computing system changes to L + ΔL thenthe similarity operatorΩ in (22) changes to

Ω = (

minus

1

120575

119892

120575

119892

120575

119892

0

1

119892

radicminus1205752minus 1198922

minus

1

119892


1 1 1

) (36)

which from the changed (119880 119881119882) still gives the same( ) as

(

)

=

((((

(

minus120575

119892

1205752+ 1198922

0

1205752

1205752+ 1198922

120575

119892

21205752+ 21198922

1

2

119892


1198922

21205752+ 21198922

1

2

120575

119892

1205752+ 1198922

minus

1

2

119892


1

2

1198922

1205752+ 1198922

))))

)

(

119880

119881

119882

)

(37)

Hence this equation group allows one from the measure-ment of states 120588 = (120588

00 12058811 12058801 12058810) and energy 119897

119899119898 for

119899119898 = 00 11 01 10 to determine the projected states =

(12058800 12058811 12058801 12058810) These projected states are invariant in the

subdynamic space in the sense

Θ1003816100381610038161003816120588119899119898

) = (120588119899119898

1003816100381610038161003816119867

1003816100381610038161003816120588119899119898

) + (120588119899119898

1003816100381610038161003816119867119876119899119898

119862119899119898

1003816100381610038161003816120588119899119898

)

= 120579119899119898

1003816100381610038161003816120588119899119898

)

Θ1003816100381610038161003816120588119899119898

) = (120588119899119898

1003816100381610038161003816(119867 + Δ119867)

1003816100381610038161003816120588119899119898

)

+ (120588119899119898

1003816100381610038161003816(119867 + Δ119867)119876

119899119898119862119899119898

1003816100381610038161003816120588119899119898

)

= 1205791015840

119899119898

1003816100381610038161003816120588119899119898

)

(38)

in which both cases have the same eigenvectors 120588119899119898

rarr

( ) although the eigenvalues are changed from 120579119899119898

to1205791015840

119899119898 So from (21) one can obtain


(

minus119894119892 0 0

0 119894119892 0

0 0 0

) 997888rarr (

(

minus

1

1198792

minus radicminus1205752minus 1198922

0 0

0 minus

1

1198792

+ radicminus1205752minus 1198922

0

0 0 minus

1

2

1198922

1198792(1198922+ 1205752)

(119890minus119894(1199051198792)

minus 1)

)

)

(39)

where one notices that the evolution is given by

119894

119889

119889119905

= minus

1

1198792

minus

1

2

1198922

1198792(1198922+ 1205752)

(119905) = minus119890minusint(11198792)119889119905

[int

1

2

1198922

1198792(1198922+ 1205752)

119890int(11198792)119889119905

+ 119862]

= minus119890minus119894(1199051198792)

int

119905

0

119894

2

1198922

1198792(1198922+ 1205752)

119890119894(11198792)119905

119889119905

=

1

2

1198922

1198792(1198922+ 1205752)

(119890minus119894(1199051198792)

minus 1)

(40)

Hence based on the above subdynamic formalism the phys-ical realizing states in the subdynamic space are related to( ) which can be determined by the states (119880 119881119882)

The remaining change of the eigenvalue from 120579119899119898

to 1205791015840

119899119898

which includes two parts of influence of the dissipation anddecoherence may introduce the relative phase error How-ever since the eigenvector in the subdynamic space is fixedthe correction procedure only needs to consider phase errorthis makes the procedure easier for instance suppose theworking states in quantum computing process are designedas (12058800 12058811 12058801 12058810) then the phase error correction in the

evolution of ((119905) (119905) (119905)) is only necessarily consideredThis can be given by using (39)

(

(119905)

(119905)

(119905)

) = (

119890119894(11198792+radicminus120575

2minus1198922)

0 0

0 119890119894(11198792minusradicminus120575

2minus1198922)

0

0 0 119890minus119894((12)(119892

21198792(119892

2+1205752))(119890minus119894(1199051198792)minus1))

)(

) (41)

so the correction is to replace (minus11198792∓ radicminus120575

2minus 1198922) by ∓119892

2

and replace ((12)(11989221198792(1198922+ 1205752))(119890minus119894(1199051198792)

minus 1)) by 0 whichenables (41) to return (21) For example using the quantumthree-editing-code method [23] if there are phase errors in for a quantum computing tolerance time Δ119905 then onecan correct the errors by using

119909119910119911 119909 oplus 119911 119910 oplus 119911 error-qubit

0

1015840 1

1015840 2

1015840

3

(42)

where means 1015840oplus = because

1015840 has a phaseerror Thus by means of one can conclude which qubit haserror This process is described by an error point formula|119909119910119911⟩ rarr |119909119910119911 119909 oplus 119911 119910 oplus 119911⟩ Then the correction is toreplace 1015840 by through replacing the relevant eigenvaluesafter considering change of the eigenvalues This process cankeep the working states to be corrected by canceling thetime evolution phase error of the states during every errortolerance time interval Δ119905 in the subdynamic spaces

Although the above Rabi oscillation seems to deal withone qubit system it is not problem to generalize to multi-qubits systems For example it can be applied into a two-qubitsystem based on a quantum dot with two excitons in whichthere are four degenerate energy levels where |00⟩ is vacuumstate of excitons and |10⟩ and |01⟩ are 119909 or 119910 polarization ofexciton states respectively while |11⟩ is states of two excitonsThus using 119909 and 119910 polarization light the CROT for twoqubits can be realized as

119880crot (| 00⟩ | 01⟩ | 10⟩ | 11⟩ )

= (|00⟩ | 01⟩ minus |11⟩ | 10⟩ )

(43)

with

119880crot = (

1 0 0 0

0 1 0 0

0 0 0 minus1

0 0 1 0

) (44)

which gives an operation on the states based on (17) as

119880crot (

119880119880

119881119881

119882119882

) = (

119880119880

119881119881

1198821198821015840

) (45)


so that 119882119882 changes to 1198821198821015840 Therefore in the subdynamic

space one can perform the CROT operation for two-qubitsystem ideally that is

119880crot (

)

= 119880crot (

0 minus

1

2

119894

1

2

0

1

2

119894

1

2

1 0 0

)(

119880

119881

119882

)(

0 minus

1

2

119894

1

2

0

1

2

119894

1

2

1 0 0

)(

119880

119881

1198821015840

)

= (

1015840

)

(46)

The relevant errors can be corrected using the same approachas the previous subdynamic treatment as in (41) and Table(42)

5 Conclusions and Remarks

The subdynamic system for the quantum computation basedon the Rabi oscillation is introduced The working states ofsubdynamic system are the eigenvectors of Θ = Ω119867Ω

minus1

which is similarity operator of the original Hamiltonian orLiouvillian These eigenvectors constructed are invariant inthe subdynamic space so they are independent on the inter-action part of Hamiltonian or Liouvillian between systemand environment While the dissipation and decoherenceinduced by the interaction part of Hamiltonian or Liouvillianexist in the change of the eigenvalues there are only phaseerrors in the subdynamic space This makes easier for oneto find a procedure to cancel the phase error since theeigenvectors are fixed such as the quantum three-editing-code method listed in this work Moreover these correctionsof phase errors contain not only an error induced by decoher-ence but also an error induced by dissipation therefore thismethod has an advantage compared with other approacheswhich only deal with the decoherence Furthermore thismethod principally can be extended to general quantumcomputing system without restriction For example if thesystem uses laser pulses instead of a continuous laser it stillcan find the subsystem dynamics where the eigenstates areconstant This is generally true for subdynamics

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors thank the support of this work from WuhanUniversity of Technology

References

[1] M V Gurudev Dutt Y W Wu X Q Li D C Steel DGammon and L J Sham CP772 Physics of Semiconductors27th International Conference on the Physics of Semiconductorsvol 32 2005

[2] X Q Li YWWuD C Steel et al ldquoAn all-optical quantum gatein a semiconductor quantumdotrdquo Science vol 301 no 5634 pp809ndash811 2003

[3] J M Villas-Boas E Ulloa Sergio and A O Govorov ldquoDeco-herence of Rabi oscillations in a single quantum dotrdquo PhysicalReview Letters vol 94 no 5 Article ID 057404 2005

[4] P Borri W Langbein S Schneider et al ldquoUltralong dephasingtime in InGaAs quantum dotsrdquo Physical Review Letters vol 87no 15 Article ID 157401 2001

[5] H Htoon T Takagahara D Kulik O Baklenov A L HolmesJr and C K Shih ldquoInterplay of rabi oscillations and quantuminterference in semiconductor quantum dotsrdquo Physical ReviewLetters vol 88 no 8 Article ID 087401 2002

[6] A Zrenner E Beham S Stufler F Findeis M Bichler and GAbstreiter ldquoCoherent properties of a two-level system based ona quantum-dot photodioderdquoNature vol 418 no 6898 pp 612ndash614 2002

[7] H Kamada H Gotoh J Temmyo T Takagahara and H AndoldquoExciton Rabi oscillation in a single quantum dotrdquo PhysicalReview Letters vol 87 no 24 Article ID 246401 2001

[8] P Borri W Langbein S Schneider et al ldquoRabi oscillations inthe excitonic ground-state transition of InGaAs quantum dotsrdquoPhysical Review B vol 66 no 8 Article ID 081306 2002

[9] L Besombes J J Baumberg and J Motohisa ldquoCoherentspectroscopy of optically gated charged single InGaAs quantumdotsrdquo Physical Review Letters vol 90 no 25 Article ID 2574022003

[10] H J Zhou S D Liu Q Q Wang M S Zhan and Q KXue ldquoRabi flopping in a V-type three-level system with twoorthogonal eigenstatesrdquo Acta Physica Sinica vol 54 no 2 pp710ndash714 2005

[11] Q Q Wang A Muller P Bianucci et al ldquoDecoherenceprocesses during optical manipulation of excitonic qubits insemiconductor quantum dotsrdquo Physical Review BmdashCondensedMatter and Materials Physics vol 72 no 3 Article ID 0353062005

[12] L Allen and J H Eberly Optical Resonance and Two-LevelAtoms Dover Publications New York NY USA 1987

[13] I Antoniou and S Tasaki ldquoGeneralized spectral decomposi-tions of mixing dynamical systemsrdquo International Journal ofQuantum Chemistry vol 46 no 3 pp 425ndash474 1993

[14] T Petrosky and I Prigogine ldquoAlternative formulation of classi-cal and quantum dynamics for nonintegrable systemsrdquo PhysicaA vol 175 no 1 pp 146ndash209 1991

[15] I Antoniou Y Melnikov and B Qiao ldquoMaster equation fora quantum system driven by a strong periodic field in thequasienergy representationrdquo Physica A Statistical Mechanicsand its Applications vol 246 no 1-2 pp 97ndash114 1997

[16] B Qiao H E Ruda M S Zhan and X H Zeng ldquoKineticequation non-perturbative approach and decoherence freesubspace for quantum open systemrdquo Physica A StatisticalMechanics and Its Applications vol 322 no 1ndash4 pp 345ndash3582003

[17] B Qiao H E Ruda and D Z Zhou ldquoDynamical equations ofquantum information and Gaussian channelrdquo Physica A vol363 no 2 pp 198ndash210 2006


[18] Q Bi L Guo andH E Ruda ldquoQuantumcomputing in decoher-ence-free subspace constructed by triangulationrdquo Advances inMathematical Physics vol 2010 Article ID 365653 10 pages2010

[19] B Qiao and J Fang Network Science and Statistical PhysicsMethod Beijing University Press Beijing China 2011 (Chi-nese)

[20] B Qiao and H E Ruda ldquoQuantum computing using entangle-ment states in a photonic band gaprdquo Journal of Applied Physicsvol 86 no 9 pp 5237ndash5244 1999

[21] G Mahler and V A Weberruszlig Quantum Networks SpringerBerlin Germany 1995

[22] S Trumm M Wesseli H J Krenner et al ldquoSpin-preservingultrafast carrier capture and relaxation in InGaAs quantumdotsrdquo Applied Physics Letters vol 87 no 15 Article ID 1531132005

[23] M A Nielsen and I L Chuang Quantum Computationand Quantum Information Cambridge University Press Cam-bridge UK 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


where 12059000

= |0⟩⟨0| is a projector [21] So we obtain a time-independent Hamiltonian as

119867 = 119880minus1

00(119905) (119905) 11988000 (

119905)

= 120596012059000

+ 120596112059011

+

1

2

119892 (12059001

+ 12059010) = 119867

0+ 1198671

(2)

where defining

1198670= 120596012059000

+ 120596112059011

1198671=

1

2

119892 (12059001

+ 12059010)

(3)

When the exciting optical field takes off 11988000(119905) = 1 we have

(119905) = 119867 So the corresponding Schrodinger equation forthe Hamiltonian in (2) is

119894

120597120595

120597119905

= 119867120595 (4)

which can reflect the same dynamics described by theHamiltonian in (1)

If considering the influence of environment the totalHamiltonian can be supposed as 119867 = 119867

0+ 1198671+ Δ119867

where Δ119867 comes from the interaction between the systemand environment which may introduce dissipation anddecoherence in the system Then a main frame to constructthe subdynamic systems is firstly to establish a subdynamicequation [13ndash19] as

119894

120597

120597119905

= Θ (5)

where is a projected state 119867 + Δ119867 = ΩΘΩminus1 and Ω is

a similarity operator These parameters and relations can beconstructed by the following several rules

(1) Propose projectors as

119875119899119896

=1003816100381610038161003816119899 otimes 120593119896⟩ ⟨119899 otimes 120593

119896

1003816100381610038161003816

for 119899 = 0 1 119896 = 0 1 2

(6)

and 119876119899119896

= 1 minus 119875119899119896 where 119899 corresponds to the index

of eigenvectors for Hamiltonian in system 119896 denotes theindex to the Hamiltonian in environment and 120593

119896is the 119896th

eigenvector of diagonal part of the interaction Hamiltonianbetween the system and environment

(2) Introduce the creation (destruction) operatorexpressed as

119862119899119896

= 119875119899119896119867119875119899119896

+ 119875119899119896Δ119867119876119899119896

1

119864119899119896

minus 119876119899119896 (

119867 + Δ119867)119876119899119896

119876119899119896Δ119867119875119899119896

119862dagger

119899119896= 119863119899119896

(7)

(3) Introduce the projected state in the subdynamicsystem as

= Ωminus1120595 (8)

where120595 is a solution of the original Schrodinger equation andthe similarity operatorΩ is defined by

Ω119899119896

= (119862119899119896

+ 119875119899119896)

Ω =

1

sum

119899=0

sum

119896

Ω119899119896

Ωminus1

=

1

sum

119899=0

sum

119896

Ωminus1

119899119896

=

1

sum

119899=0

sum

119896

(119875119899119896

+ 119863119899119896119862119899119896)minus1

(119875119899119896

+ 119863119899119896)

(9)

(4)The ruling equation is subdynamic Liouville equation

119894

120597

120597119905

120588 = [Θ 120588] (10)

where Θ operator is introduced by

Θ =

1

sum

119899=0

sum

119896

Θ119899119896

=

1

sum

119899=0

sum

119896

(119875119899119896119867119875119899119896

+ 119875119899119896Δ119867119876119899119896119862119899119896119875119899119896)

(11)

and the relationship of the aboveΘ operator with the original119867+Δ119867 can be proven by a kind of similarity transformation

Θ = Ωminus1

(119867 + Δ119867)Ω (12)

So it is obvious to use (12) and consider (8) one can obtainSKE (5) from the Schrodinger equation or the Liouvilleequation

3 Rabi Oscillation in Subdynamic Space

Using the Rabi oscillation between |1⟩ (or |1⟩⟨1|) and |0⟩

(or |0⟩⟨0|) as the working states of quantum computationis a reasonable approach of quantum computing based onquantum dot system [11 22] In the ideal situation Δ119867 = 0the Liouville equation for this two-level system with the Rabioscillation is expressed by

119894119880minus1

11(119905)

120597120588

120597119905

11988011 (

119905) = 119880minus1

11(119905) [119867 120588]119880

11 (119905)

= 119880minus1

11(119905)119867120588119880

11(119905) minus 119880

minus1

11(119905) 120588119867119880

11(119905)

= 119880minus1

11(119905) 11988011 (

119905) (119905) 119880minus1

11(119905) 12058811988011 (

119905)

minus 119880minus1

11(119905) 120588119880

11(119905) (119905) 119880

minus1

11(119905) 11988011(119905)

= (119905) 119880minus1

11(119905) 12058811988011 (

119905) minus 119880minus1

11(119905) 12058811988011 (

119905) (119905)

= [ (119905) 120588 (119905)] = 119894

120597119880minus1

11(119905) 120588119880

11(119905)

120597119905

= 119894

120597120588 (119905)

120597119905

(13)



120597119880minus1

11(119905)

120597119905

12058811988011(119905) + 119880

minus1

11(119905) 120588

12059711988011 (

119905)

120597119905

= 0

120588 = 11988011 (

119905) 120588 (119905) 119880minus1

11(119905)

= 12058811 |

1⟩ ⟨1| + 12058800 |

0⟩ ⟨0| + 12058810 |

1⟩ ⟨0|

+ 12058801 |

0⟩ ⟨1|

(14)


119899119898as

119894

119889

119889119905

12058811

=

1

2

119892 (12058801

minus 12058810)

119894

119889

119889119905

12058800

= minus119894

119889

119889119905

12058811

= minus

1

2

119892 (12058801

minus 12058810)

119894

119889

119889119905

12058810

=

1

2

119892 (12058800

minus 12058811)

(15)


= 1198801198901+ 1198811198902+1198821198903

(16)

with

119880 = 12058810

+ 12058801

119881 = 119894 (12058810

minus 12058801)

119882 = 12058811

minus 12058800

(17)


119894sdot 119890119895= 120575119894119895


119889

119889119905

(

119880

119881

119882

) = (

0 0 0

0 0 119892

0 minus119892 0

)(

119880

119881

119882

) (18)


Ωminus1(

(

(

119889119880

119889119905

119889119881

119889119905

119889119882

119889119905

))

)

= Ωminus1

(

0 0 0

0 minus0 119892

0 minus119892 0

)ΩΩminus1

(

119880

119881

119882

) (19)

119889

119889119905

(

) = (

Θ1

0 0

0 Θ2

0

0 0 Θ3

)(

) (20)


Ω(

(119905)

(119905)

(119905)

) = Ω(

minus119894119892 0 0

0 119894119892 0

0 0 0

)Ωminus1Ω(

(0)

(0)

(0)

) (21)

where

(

119880

119881

119882

) = Ω(

) = (

0 0 1

119894 minus119894 0

1 1 0

)(

) (22)


(

119880(119905)

119881 (119905)

119882 (119905)

) = (

1 0 0

0 cos 120579 minus sin 120579

0 sin 120579 cos 120579)(

119880(0)

119881 (0)

119882 (0)

) (23)



12058811 12058801 and 120588

10as zero) one gets

(

119880(0)

119881 (0)

119882 (0)

) = (

0

0

minus1

) (24)


(

119880(119905)

119881 (119905)

119882 (119905)

) = (

0

minus sin 120579 (119905)

minus cos 120579 (119905)) (25)




Δ119871 (120588) =

1

2

12057410(21205900112058812059010

minus 12059011120588 minus 120588120590

11) (26)

where 12059010

or 12059001



119889

119889119905

12058811

= minus119894

1

2

119892 (12058801

minus 12058810) minus 1205741012058811

119889

119889119905

12058800

= minus

119889

119889119905

12058811

= 119894

1

2

119892 (12058801

minus 12058810) + 1205741012058811

119889

119889119905

12058810

= minus11989412057512058810

minus 119894

1

2

119892 (12058800


1

2

1205741012058810

(27)



119894

119889

119889119905

Ωminus1

(

119880

119881

119882

)

= Ωminus1

(

minus

1

1198792

minus120575 0

120575 minus

1

1198792

119892

0 minus119892 minus

1

1198791

)ΩΩminus1

(

119880

119881

119882

)

+Ωminus1

(

0

0

minus

1

1198791

)

(28)


1+120574ph 1198791 =



(L + ΔL)(119880

119881

119882

) = (

minus

1

1198792

minus120575 0

120575 minus

1

1198792

119892

0 minus119892 minus

1

1198791

)(

119880

119881

119882

)

+(

0

0

minus

1

1198791

)

(29)


120588 = (12058800 12058811 12058822 12058801 12058810) or = (120588

00 12058811 12058822 12058801 12058810) by the

above formalism

119889

119889119905

Ωminus1120588 (119905) = Ω


minus1120588 (119905) = Θ120588 (119905)

=

119889

119889119905

120588 (119905)

(30)

with

Θ = Ωminus1

(L + ΔL) Ω

= 119875 (L + ΔL) 119875 + 119875 (L + ΔL) 119876119862119875

120588 (119905) = Ω120588 (119905) = (119875 + 119862) 120588 (119905) = sum

119899119898

(119875119899119898

+ 119862119899119898

) 120588 (119905)

(31)


119875 = sum

119899119898

119875119899119898

= sum

119899119898

1003816100381610038161003816120588119899119898

) (120588119899119898

1003816100381610038161003816

for 119899119898 = 00 11 01 10

(32)

with119876 = 1 minus 119875 (33)


formalism1003816100381610038161003816120588119899119898

) = (119875119899119898

+ 119862119899119898

)1003816100381610038161003816120588119899119898

) (34)

This 120588119899119898


119899119898 and then using 120588

119899119898 119897119899119898




) =1003816100381610038161003816120588119899119898

) + 119876119899119898

1

119872119899119899

minus 119876119899119898

(L + ΔL) 119876119899119898

sdot 119876119899119899 (

L + ΔL) 1003816100381610038161003816120588119899119899)

(35)




Ω = (

minus

1

120575

119892

120575

119892

120575

119892

0

1

119892


minus

1

119892


1 1 1

) (36)


(

)

=

((((

(

minus120575

119892

1205752+ 1198922

0

1205752

1205752+ 1198922

120575

119892

21205752+ 21198922

1

2

119892


1198922

21205752+ 21198922

1

2

120575

119892

1205752+ 1198922

minus

1

2

119892


1

2

1198922

1205752+ 1198922

))))

)

(

119880

119881

119882

)

(37)


00 12058811 12058801 12058810) and energy 119897

119899119898 for




Θ1003816100381610038161003816120588119899119898

) = (120588119899119898

1003816100381610038161003816119867

1003816100381610038161003816120588119899119898

) + (120588119899119898

1003816100381610038161003816119867119876119899119898

119862119899119898

1003816100381610038161003816120588119899119898

)

= 120579119899119898

1003816100381610038161003816120588119899119898

)

Θ1003816100381610038161003816120588119899119898

) = (120588119899119898

1003816100381610038161003816(119867 + Δ119867)

1003816100381610038161003816120588119899119898

)

+ (120588119899119898

1003816100381610038161003816(119867 + Δ119867)119876

119899119898119862119899119898

1003816100381610038161003816120588119899119898

)

= 1205791015840

119899119898

1003816100381610038161003816120588119899119898

)

(38)


rarr


to1205791015840



(

minus119894119892 0 0

0 119894119892 0

0 0 0

) 997888rarr (

(

minus

1

1198792


0 0

0 minus

1

1198792


0

0 0 minus

1

2

1198922

1198792(1198922+ 1205752)

(119890minus119894(1199051198792)

minus 1)

)

)

(39)


119894

119889

119889119905

= minus

1

1198792

minus

1

2

1198922

1198792(1198922+ 1205752)

(119905) = minus119890minusint(11198792)119889119905

[int

1

2

1198922

1198792(1198922+ 1205752)

119890int(11198792)119889119905

+ 119862]

= minus119890minus119894(1199051198792)

int

119905

0

119894

2

1198922

1198792(1198922+ 1205752)

119890119894(11198792)119905

119889119905

=

1

2

1198922

1198792(1198922+ 1205752)

(119890minus119894(1199051198792)

minus 1)

(40)



to 1205791015840

119899119898



(

(119905)

(119905)

(119905)

) = (

119890119894(11198792+radicminus120575

2minus1198922)

0 0


2minus1198922)

0

0 0 119890minus119894((12)(119892

21198792(119892

2+1205752))(119890minus119894(1199051198792)minus1))

)(

) (41)


2minus 1198922) by ∓119892

2




0

1015840 1

1015840 2

1015840

3

(42)




119880crot (| 00⟩ | 01⟩ | 10⟩ | 11⟩ )

= (|00⟩ | 01⟩ minus |11⟩ | 10⟩ )

(43)

with

119880crot = (

1 0 0 0

0 1 0 0

0 0 0 minus1

0 0 1 0

) (44)


119880crot (

119880119880

119881119881

119882119882

) = (

119880119880

119881119881

1198821198821015840

) (45)




119880crot (

)

= 119880crot (

0 minus

1

2

119894

1

2

0

1

2

119894

1

2

1 0 0

)(

119880

119881

119882

)(

0 minus

1

2

119894

1

2

0

1

2

119894

1

2

1 0 0

)(

119880

119881

1198821015840

)

= (

1015840

)

(46)




minus1




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597119880minus1

11(119905)

120597119905

12058811988011(119905) + 119880

minus1

11(119905) 120588

12059711988011 (

119905)

120597119905

= 0

120588 = 11988011 (

119905) 120588 (119905) 119880minus1

11(119905)

= 12058811 |

1⟩ ⟨1| + 12058800 |

0⟩ ⟨0| + 12058810 |

1⟩ ⟨0|

+ 12058801 |

0⟩ ⟨1|

(14)


119899119898as

119894

119889

119889119905

12058811

=

1

2

119892 (12058801

minus 12058810)

119894

119889

119889119905

12058800

= minus119894

119889

119889119905

12058811

= minus

1

2

119892 (12058801

minus 12058810)

119894

119889

119889119905

12058810

=

1

2

119892 (12058800

minus 12058811)

(15)


= 1198801198901+ 1198811198902+1198821198903

(16)

with

119880 = 12058810

+ 12058801

119881 = 119894 (12058810

minus 12058801)

119882 = 12058811

minus 12058800

(17)


119894sdot 119890119895= 120575119894119895


119889

119889119905

(

119880

119881

119882

) = (

0 0 0

0 0 119892

0 minus119892 0

)(

119880

119881

119882

) (18)


Ωminus1(

(

(

119889119880

119889119905

119889119881

119889119905

119889119882

119889119905

))

)

= Ωminus1

(

0 0 0

0 minus0 119892

0 minus119892 0

)ΩΩminus1

(

119880

119881

119882

) (19)

119889

119889119905

(

) = (

Θ1

0 0

0 Θ2

0

0 0 Θ3

)(

) (20)


Ω(

(119905)

(119905)

(119905)

) = Ω(

minus119894119892 0 0

0 119894119892 0

0 0 0

)Ωminus1Ω(

(0)

(0)

(0)

) (21)

where

(

119880

119881

119882

) = Ω(

) = (

0 0 1

119894 minus119894 0

1 1 0

)(

) (22)


(

119880(119905)

119881 (119905)

119882 (119905)

) = (

1 0 0

0 cos 120579 minus sin 120579

0 sin 120579 cos 120579)(

119880(0)

119881 (0)

119882 (0)

) (23)



12058811 12058801 and 120588

10as zero) one gets

(

119880(0)

119881 (0)

119882 (0)

) = (

0

0

minus1

) (24)


(

119880(119905)

119881 (119905)

119882 (119905)

) = (

0

minus sin 120579 (119905)

minus cos 120579 (119905)) (25)




Δ119871 (120588) =

1

2

12057410(21205900112058812059010

minus 12059011120588 minus 120588120590

11) (26)

where 12059010

or 12059001



119889

119889119905

12058811

= minus119894

1

2

119892 (12058801

minus 12058810) minus 1205741012058811

119889

119889119905

12058800

= minus

119889

119889119905

12058811

= 119894

1

2

119892 (12058801

minus 12058810) + 1205741012058811

119889

119889119905

12058810

= minus11989412057512058810

minus 119894

1

2

119892 (12058800


1

2

1205741012058810

(27)



119894

119889

119889119905

Ωminus1

(

119880

119881

119882

)

= Ωminus1

(

minus

1

1198792

minus120575 0

120575 minus

1

1198792

119892

0 minus119892 minus

1

1198791

)ΩΩminus1

(

119880

119881

119882

)

+Ωminus1

(

0

0

minus

1

1198791

)

(28)


1+120574ph 1198791 =



(L + ΔL)(119880

119881

119882

) = (

minus

1

1198792

minus120575 0

120575 minus

1

1198792

119892

0 minus119892 minus

1

1198791

)(

119880

119881

119882

)

+(

0

0

minus

1

1198791

)

(29)


120588 = (12058800 12058811 12058822 12058801 12058810) or = (120588

00 12058811 12058822 12058801 12058810) by the

above formalism

119889

119889119905

Ωminus1120588 (119905) = Ω


minus1120588 (119905) = Θ120588 (119905)

=

119889

119889119905

120588 (119905)

(30)

with

Θ = Ωminus1

(L + ΔL) Ω

= 119875 (L + ΔL) 119875 + 119875 (L + ΔL) 119876119862119875

120588 (119905) = Ω120588 (119905) = (119875 + 119862) 120588 (119905) = sum

119899119898

(119875119899119898

+ 119862119899119898

) 120588 (119905)

(31)


119875 = sum

119899119898

119875119899119898

= sum

119899119898

1003816100381610038161003816120588119899119898

) (120588119899119898

1003816100381610038161003816

for 119899119898 = 00 11 01 10

(32)

with119876 = 1 minus 119875 (33)


formalism1003816100381610038161003816120588119899119898

) = (119875119899119898

+ 119862119899119898

)1003816100381610038161003816120588119899119898

) (34)

This 120588119899119898


119899119898 and then using 120588

119899119898 119897119899119898




) =1003816100381610038161003816120588119899119898

) + 119876119899119898

1

119872119899119899

minus 119876119899119898

(L + ΔL) 119876119899119898

sdot 119876119899119899 (

L + ΔL) 1003816100381610038161003816120588119899119899)

(35)




Ω = (

minus

1

120575

119892

120575

119892

120575

119892

0

1

119892


minus

1

119892


1 1 1

) (36)


(

)

=

((((

(

minus120575

119892

1205752+ 1198922

0

1205752

1205752+ 1198922

120575

119892

21205752+ 21198922

1

2

119892


1198922

21205752+ 21198922

1

2

120575

119892

1205752+ 1198922

minus

1

2

119892


1

2

1198922

1205752+ 1198922

))))

)

(

119880

119881

119882

)

(37)


00 12058811 12058801 12058810) and energy 119897

119899119898 for




Θ1003816100381610038161003816120588119899119898

) = (120588119899119898

1003816100381610038161003816119867

1003816100381610038161003816120588119899119898

) + (120588119899119898

1003816100381610038161003816119867119876119899119898

119862119899119898

1003816100381610038161003816120588119899119898

)

= 120579119899119898

1003816100381610038161003816120588119899119898

)

Θ1003816100381610038161003816120588119899119898

) = (120588119899119898

1003816100381610038161003816(119867 + Δ119867)

1003816100381610038161003816120588119899119898

)

+ (120588119899119898

1003816100381610038161003816(119867 + Δ119867)119876

119899119898119862119899119898

1003816100381610038161003816120588119899119898

)

= 1205791015840

119899119898

1003816100381610038161003816120588119899119898

)

(38)


rarr


to1205791015840



(

minus119894119892 0 0

0 119894119892 0

0 0 0

) 997888rarr (

(

minus

1

1198792


0 0

0 minus

1

1198792


0

0 0 minus

1

2

1198922

1198792(1198922+ 1205752)

(119890minus119894(1199051198792)

minus 1)

)

)

(39)


119894

119889

119889119905

= minus

1

1198792

minus

1

2

1198922

1198792(1198922+ 1205752)

(119905) = minus119890minusint(11198792)119889119905

[int

1

2

1198922

1198792(1198922+ 1205752)

119890int(11198792)119889119905

+ 119862]

= minus119890minus119894(1199051198792)

int

119905

0

119894

2

1198922

1198792(1198922+ 1205752)

119890119894(11198792)119905

119889119905

=

1

2

1198922

1198792(1198922+ 1205752)

(119890minus119894(1199051198792)

minus 1)

(40)



to 1205791015840

119899119898



(

(119905)

(119905)

(119905)

) = (

119890119894(11198792+radicminus120575

2minus1198922)

0 0


2minus1198922)

0

0 0 119890minus119894((12)(119892

21198792(119892

2+1205752))(119890minus119894(1199051198792)minus1))

)(

) (41)


2minus 1198922) by ∓119892

2




0

1015840 1

1015840 2

1015840

3

(42)




119880crot (| 00⟩ | 01⟩ | 10⟩ | 11⟩ )

= (|00⟩ | 01⟩ minus |11⟩ | 10⟩ )

(43)

with

119880crot = (

1 0 0 0

0 1 0 0

0 0 0 minus1

0 0 1 0

) (44)


119880crot (

119880119880

119881119881

119882119882

) = (

119880119880

119881119881

1198821198821015840

) (45)




119880crot (

)

= 119880crot (

0 minus

1

2

119894

1

2

0

1

2

119894

1

2

1 0 0

)(

119880

119881

119882

)(

0 minus

1

2

119894

1

2

0

1

2

119894

1

2

1 0 0

)(

119880

119881

1198821015840

)

= (

1015840

)

(46)




minus1




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894

119889

119889119905

Ωminus1

(

119880

119881

119882

)

= Ωminus1

(

minus

1

1198792

minus120575 0

120575 minus

1

1198792

119892

0 minus119892 minus

1

1198791

)ΩΩminus1

(

119880

119881

119882

)

+Ωminus1

(

0

0

minus

1

1198791

)

(28)


1+120574ph 1198791 =



(L + ΔL)(119880

119881

119882

) = (

minus

1

1198792

minus120575 0

120575 minus

1

1198792

119892

0 minus119892 minus

1

1198791

)(

119880

119881

119882

)

+(

0

0

minus

1

1198791

)

(29)


120588 = (12058800 12058811 12058822 12058801 12058810) or = (120588

00 12058811 12058822 12058801 12058810) by the

above formalism

119889

119889119905

Ωminus1120588 (119905) = Ω


minus1120588 (119905) = Θ120588 (119905)

=

119889

119889119905

120588 (119905)

(30)

with

Θ = Ωminus1

(L + ΔL) Ω

= 119875 (L + ΔL) 119875 + 119875 (L + ΔL) 119876119862119875

120588 (119905) = Ω120588 (119905) = (119875 + 119862) 120588 (119905) = sum

119899119898

(119875119899119898

+ 119862119899119898

) 120588 (119905)

(31)


119875 = sum

119899119898

119875119899119898

= sum

119899119898

1003816100381610038161003816120588119899119898

) (120588119899119898

1003816100381610038161003816

for 119899119898 = 00 11 01 10

(32)

with119876 = 1 minus 119875 (33)


formalism1003816100381610038161003816120588119899119898

) = (119875119899119898

+ 119862119899119898

)1003816100381610038161003816120588119899119898

) (34)

This 120588119899119898


119899119898 and then using 120588

119899119898 119897119899119898




) =1003816100381610038161003816120588119899119898

) + 119876119899119898

1

119872119899119899

minus 119876119899119898

(L + ΔL) 119876119899119898

sdot 119876119899119899 (

L + ΔL) 1003816100381610038161003816120588119899119899)

(35)




Ω = (

minus

1

120575

119892

120575

119892

120575

119892

0

1

119892


minus

1

119892


1 1 1

) (36)


(

)

=

((((

(

minus120575

119892

1205752+ 1198922

0

1205752

1205752+ 1198922

120575

119892

21205752+ 21198922

1

2

119892


1198922

21205752+ 21198922

1

2

120575

119892

1205752+ 1198922

minus

1

2

119892


1

2

1198922

1205752+ 1198922

))))

)

(

119880

119881

119882

)

(37)


00 12058811 12058801 12058810) and energy 119897

119899119898 for




Θ1003816100381610038161003816120588119899119898

) = (120588119899119898

1003816100381610038161003816119867

1003816100381610038161003816120588119899119898

) + (120588119899119898

1003816100381610038161003816119867119876119899119898

119862119899119898

1003816100381610038161003816120588119899119898

)

= 120579119899119898

1003816100381610038161003816120588119899119898

)

Θ1003816100381610038161003816120588119899119898

) = (120588119899119898

1003816100381610038161003816(119867 + Δ119867)

1003816100381610038161003816120588119899119898

)

+ (120588119899119898

1003816100381610038161003816(119867 + Δ119867)119876

119899119898119862119899119898

1003816100381610038161003816120588119899119898

)

= 1205791015840

119899119898

1003816100381610038161003816120588119899119898

)

(38)


rarr


to1205791015840



(

minus119894119892 0 0

0 119894119892 0

0 0 0

) 997888rarr (

(

minus

1

1198792


0 0

0 minus

1

1198792


0

0 0 minus

1

2

1198922

1198792(1198922+ 1205752)

(119890minus119894(1199051198792)

minus 1)

)

)

(39)


119894

119889

119889119905

= minus

1

1198792

minus

1

2

1198922

1198792(1198922+ 1205752)

(119905) = minus119890minusint(11198792)119889119905

[int

1

2

1198922

1198792(1198922+ 1205752)

119890int(11198792)119889119905

+ 119862]

= minus119890minus119894(1199051198792)

int

119905

0

119894

2

1198922

1198792(1198922+ 1205752)

119890119894(11198792)119905

119889119905

=

1

2

1198922

1198792(1198922+ 1205752)

(119890minus119894(1199051198792)

minus 1)

(40)



to 1205791015840

119899119898



(

(119905)

(119905)

(119905)

) = (

119890119894(11198792+radicminus120575

2minus1198922)

0 0


2minus1198922)

0

0 0 119890minus119894((12)(119892

21198792(119892

2+1205752))(119890minus119894(1199051198792)minus1))

)(

) (41)


2minus 1198922) by ∓119892

2




0

1015840 1

1015840 2

1015840

3

(42)




119880crot (| 00⟩ | 01⟩ | 10⟩ | 11⟩ )

= (|00⟩ | 01⟩ minus |11⟩ | 10⟩ )

(43)

with

119880crot = (

1 0 0 0

0 1 0 0

0 0 0 minus1

0 0 1 0

) (44)


119880crot (

119880119880

119881119881

119882119882

) = (

119880119880

119881119881

1198821198821015840

) (45)




119880crot (

)

= 119880crot (

0 minus

1

2

119894

1

2

0

1

2

119894

1

2

1 0 0

)(

119880

119881

119882

)(

0 minus

1

2

119894

1

2

0

1

2

119894

1

2

1 0 0

)(

119880

119881

1198821015840

)

= (

1015840

)

(46)




minus1




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(

minus119894119892 0 0

0 119894119892 0

0 0 0

) 997888rarr (

(

minus

1

1198792


0 0

0 minus

1

1198792


0

0 0 minus

1

2

1198922

1198792(1198922+ 1205752)

(119890minus119894(1199051198792)

minus 1)

)

)

(39)


119894

119889

119889119905

= minus

1

1198792

minus

1

2

1198922

1198792(1198922+ 1205752)

(119905) = minus119890minusint(11198792)119889119905

[int

1

2

1198922

1198792(1198922+ 1205752)

119890int(11198792)119889119905

+ 119862]

= minus119890minus119894(1199051198792)

int

119905

0

119894

2

1198922

1198792(1198922+ 1205752)

119890119894(11198792)119905

119889119905

=

1

2

1198922

1198792(1198922+ 1205752)

(119890minus119894(1199051198792)

minus 1)

(40)



to 1205791015840

119899119898



(

(119905)

(119905)

(119905)

) = (

119890119894(11198792+radicminus120575

2minus1198922)

0 0


2minus1198922)

0

0 0 119890minus119894((12)(119892

21198792(119892

2+1205752))(119890minus119894(1199051198792)minus1))

)(

) (41)


2minus 1198922) by ∓119892

2




0

1015840 1

1015840 2

1015840

3

(42)




119880crot (| 00⟩ | 01⟩ | 10⟩ | 11⟩ )

= (|00⟩ | 01⟩ minus |11⟩ | 10⟩ )

(43)

with

119880crot = (

1 0 0 0

0 1 0 0

0 0 0 minus1

0 0 1 0

) (44)


119880crot (

119880119880

119881119881

119882119882

) = (

119880119880

119881119881

1198821198821015840

) (45)




119880crot (

)

= 119880crot (

0 minus

1

2

119894

1

2

0

1

2

119894

1

2

1 0 0

)(

119880

119881

119882

)(

0 minus

1

2

119894

1

2

0

1

2

119894

1

2

1 0 0

)(

119880

119881

1198821015840

)

= (

1015840

)

(46)




minus1




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119880crot (

)

= 119880crot (

0 minus

1

2

119894

1

2

0

1

2

119894

1

2

1 0 0

)(

119880

119881

119882

)(

0 minus

1

2

119894

1

2

0

1

2

119894

1

2

1 0 0

)(

119880

119881

1198821015840

)

= (

1015840

)

(46)




minus1




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article the rabi oscillation in subdynamic system...

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