Non classical correlations Non classical correlations ofof two interacting qubits two interacting qubits coupled to independent coupled to independent

reservoirsreservoirs

R. MiglioreR. MiglioreCNR-INFM, Research Unit CNISM of PalermoCNR-INFM, Research Unit CNISM of Palermo

Dipartimento di Scienze Fisiche ed Dipartimento di Scienze Fisiche ed AstronomicheAstronomiche

Università di Palermo, ItalyUniversità di Palermo, Italy

M. Scala, M.A. Jivulescu, M. Guccione, L.L. M. Scala, M.A. Jivulescu, M. Guccione, L.L. SánchezSánchez-Soto, A. Messina-Soto, A. Messina

dsfa

CNR-INFM

The system and its Hamiltonian Two coupled qubits interacting with two independent bosonic baths. Counter-rotating terms are present in the interaction Hamiltonian describing

the qubit-qubit coupling.

Microscopic derivation of the Markovian master equation in the weak damping limit

The system dynamics Stationary state at general temperatures Behavior of the entanglement at zero temperature presenting the phenomena

of sudden death and sudden birth as well as the presence of stationary entanglement for long times.

Effect of nonzero temperature on the entanglement dynamics.

Conclusive remarks

OUTLINEOUTLINE

M. Scala et al., J. Phys A: Math. and Theor. 41, 435304 (2008); M. Scala et al., in preparation.R. Migliore et al., Phys. Stat. Sol. B 246, 1013 (2009).

(2(1) (1) (1) (2) (2)2

)1 2S x xH

THE THE SYSTEM AND ITS HAMILTONIAN SYSTEM AND ITS HAMILTONIAN

† †E k k k j j j

k j

H a a b b

intS EH H H H

(1) † †i

)nt

(2( ) ( )k k k j j jk

xj

xH a a g b b

We consider two spin-1/2 like interacting systems (qubits)coupled with their own (uncorrelated) bosonic environments

Reservoir 2

T2

2

20

21

Reservoir 1

T1

110

11(1) (2)x x

Counter-rotating terms included in the interaction play a central role in the dynamics of the entanglement between the two systems.

(1)

The four-level energy spectrum The four-level energy spectrum of the bipartite systemof the bipartite system

IIII

I

Icos( 2) 00 sin( 2) 11I Ia

cos( 2) 10 sin( 2) 01II IIb

sin( 2) 10 cos( 2) 01II IIc

sin( 2) 00 cos( 2) 11I Id

2 2 2 22( 1 2) 1

1( ( ) ( ) )

2I II

By exploiting the fact that the Hamiltonian HS, in the uncoupled basis {|00>; |11>; |10>; |01>}, is block diagonal, it is straightforward to find its energy spectrum and the relative eigenstates

2 22 1( ) 2 1

1[ ( ) ]

2a bE

2 22 1( ) 2 1

1[ ( ) ]

2c dE 2 1 2 2

2 1

sin( )

I

2 22 1

sin( )

II

&

Here

The allowed transitions here sketched are characterized by the Bohr frequencies

Microscopic derivation of the Microscopic derivation of the Markovian master equation in the weak Markovian master equation in the weak

damping limitdamping limit From this Hamiltonian model, performing both the Born-Markov and the rotating wave

approximation, we find that the evolution of the two two-state systems is described by the equation:

†1 ( )k k k

k

B a a †

2 ( )k j jj

B g b b and the Kubo-Martin-Schwinger relation holds.

/,,

i B li ll li

Kl

Te

Assuming that the two reservoirs are independent and that both are

in a thermal state, with temperatures T1 and T2 respectively, one has: ,† ( ) (0)il mi lm

i Bd e B

with i.e. when † ( ) ( 00)l mB B , 0i lm l m

(2) (2)(2

1 11 )I x xJ a b da b c dc

(2) (2)12)

1(1II xII xJ a c da c b db

All the jump processes involve transitions between dressed states of the open system under study, described by the following operators relative to the coupling of the first (second) qubit with its own reservoir:

for the transitions b a and d c

for the transitions c a and d b

(2)

Rearranging the ME, we obtain a system of differential coupled equations, describing the time evolution of the populations of the dressed states |a>, |b>, |c> and |d> namely

Rearranging the ME, we obtain a system of differential coupled equations, describing the time evolution of the populations of the dressed states |a>, |b>, |c> and |d> namely

,

,

( 2 )( ) ( ) ( )

2

(2 )( ) ( ) ( )

2

II I IIac II ac cr I bd

I II IIbd II I bd cr I ac

c c ct i t c t

c c ct i c t c t

( )

( ) ( )

( ( )( ) )

( ) )

(

( ) (

) dd

d

II II I I

II

acc cc

ccI I I ddId bb

a

t

c c c c

c c c ct

t

t

tt t

t

( )

( )

( ) ( )

(

( )

( ))

(

(

)

) ( )a I II I II

I

a aa

aa I II I dd

bb

b b

c

b b I

ct tc c c c

c c c c tt t

t t

t

,

,

( 2 )( ) ( ) ( )

2

( 2 )( ) ( ) ( )

2

I I IIab I ab cr II cd

I II Icd I I cd cr II ab

c c ct i t c t

c c ct i c t c t

( )( ) ( )

2

( )( ) ( )

2

I II I IIad da ad

I II I IIbc cb bc

c c c ct i t

c c c ct i t

and of the corresponding coherences:and of the corresponding coherences:

The decay rates (i I,II) and the cross terms are given by:The decay rates (i I,II) and the cross terms are given by:ic ,cr ic

When the temperatures of the two reservoirs T1 and T2 are both zero, the

rates and vanish.Physically this means that there is no possibility to create excitations in the bipartite system due to the interaction with the reservoirs.

ic ,cr ic

2 2

,11 ,22cos cos sin sin cos sin sin cos2 2 2 2 2 2 2 2

I II I II I II I III I Ic

2 2

,11 ,22cos sin sin cos cos cos sin sin2 2 2 2 2 2 2 2

I II I II I II I IIII II IIc

2 2

, ,11 ,22cos cos sin sin cos sin sin cos2 2 2 2 2 2 2 2

I II I II I II I IIcr I I Ic

2 2

, ,11 ,22cos sin sin cos cos cos sin sin2 2 2 2 2 2 2 2

I II I II I II I IIcr II II IIc

The excitation rates and the cross terms are obtained by substituting, with the corresponding quantities

The excitation rates and the cross terms are obtained by substituting, with the corresponding quantities

ic ,cr ic

,i ll ,i ll

Analytical solution I: the stationary Analytical solution I: the stationary statestate

, ( )( )I II

aa STI I II II

c c

c c c c

, ( )( )I II

bb STI I II II

c c

c c c c

, ( )( )I II

cc STI I II II

c c

c c c c

, ( )( )I II

dd STI I II II

c c

c c c c

We prove the existence of the following stationary solution, by imposing and the normalization condition

0 , , ,ii i a b c d

[ ( )] 1Tr t

there exists a stationary entanglement traceable back to the presence of counter-rotating terms in the interaction Hamiltonian describing the coupling between the two two-state systems.

there exists a stationary entanglement traceable back to the presence of counter-rotating terms in the interaction Hamiltonian describing the coupling between the two two-state systems.

when T1 = T2 = 0 K aa,ST= 1

We note, that the analysis of this Hamiltonian model allows to bring to the light the fact that, when T1T 2, it is wrong to apply the principle of

detailed balance in order to derive the stationary solution of the master equation of the system.This is due to the fact that the excitation rates are not related to the corresponding rates from the usual Boltzmann factor.

We note, that the analysis of this Hamiltonian model allows to bring to the light the fact that, when T1T 2, it is wrong to apply the principle of

detailed balance in order to derive the stationary solution of the master equation of the system.This is due to the fact that the excitation rates are not related to the corresponding rates from the usual Boltzmann factor.

Initial state: |11>Initial state: |11>

Linear entropy

SL() =1-Tr[2]

Linear entropy

SL() =1-Tr[2]

cos( 2) 00 sin( 2) 11I Ia cos( 2) 00 sin( 2) 11I Ia

Analytical solution II: Analytical solution II: dynamics of the dynamics of the entanglement entanglement at zero temperatureat zero temperature

( ) 1 ( ) ( ) ( )aa bb cc ddt t t t

( )( ) ( (0) (0)) (0)I I IIc t c c tbb bb dd ddt e e

( )( ) ( (0) (0)) (0)II I IIc t c c tcc cc dd ddt e e

( )( ) (0) I IIc c tdd ddt e

2[ ( )]

2( ) (0)I II I

Ic c c

i t

cd cdt e

[ ( )]2( ) (0)

I II I IIda

c c c ci t

ad adt e

[ ( )]2( ) (0)

I II I IIbc

c c c ci t

bc bct e

2[ ( )]

2( ) (0)II I II

IIc c c

i t

ac act e

2[ ( )]

2( ) (0)I II II

IIc c c

i t

bd bdt e

2[ ( )]

2( ) (0)I I II

Ic c c

i t

ab abt e

Exploiting the knowledge of the master equation solutions (when T1=T2=0 K)

we determine the amount of entanglement between the two-state systems and the entanglement dynamics, by analyzing the concurrence, a function introduced by Wootters and defined as

( * )y y y yR i : eigenvalues of the matrix

0

1

C

C

0 1C no entanglement

maximal entanglement

RESULTS I: INITIAL STATE |01>RESULTS I: INITIAL STATE |01>con

cu

rren

ce

1=2=10

flat spectrum ==0.1

T1=T2= 0 K

Damped Rabi-oscillations + stationary entanglement

due to the presence of counter-rotating terms in the x(1) x

(2) interaction Hamiltonian which are responsible for the presence of the component |11> in the ground state |a>

t

RESULTS II: INITIAL STATE |11> RESULTS II: INITIAL STATE |11> Birth and death of the entanglementBirth and death of the entanglement

1=2=10 ; flat spectrum: ==0.1 ; T1=T2= 0 K

Sudden death and birth of entanglement, phenomena which are well known in the recent literature on dissipative two-qubits dynamics. Note that also in this case the stationary entanglement is due to the structure of the qubit-qubit interaction Hamiltonian and not to the presence of a common environment.

Sudden death and birth of entanglement, phenomena which are well known in the recent literature on dissipative two-qubits dynamics. Note that also in this case the stationary entanglement is due to the structure of the qubit-qubit interaction Hamiltonian and not to the presence of a common environment.

con

cu

rren

ce

t

EFFECT OF NONZERO TEMPERATURES IEFFECT OF NONZERO TEMPERATURES IBy means of Laplace transforms, it is possible to obtain the solution of the master equation at generic T

1 and T

2

initial state |01> Effect on the oscillations

initial state |11> Effect on the short-time dynamics

T1=T

2=10 mK

T1=T

2=20 mK

T1=T

2=30 mK

The oscillatory phenomena, indicating coherence, are robust enough with respect to the temperature increasing

The oscillatory phenomena, indicating coherence, are robust enough with respect to the temperature increasing

con

cu

rren

ce

con

cu

rren

ce

t (sec) t (sec)

1=2=10GHz;

ohmic spectrum: 1=2=0.1

T1=T

2=10 mK

T1=T

2=20 mK

T1=T

2=30 mK

Initial state |11>: Effect on the long-time dynamicscon

cu

rren

ce T

1=T

2=5 mK

T1=T

2=10 mK

T1=T

2=15 mK

The stationary entanglement due to non-resonant interaction is less robust with respect to temperature: anyway it is experimentally detectable.

The stationary entanglement due to non-resonant interaction is less robust with respect to temperature: anyway it is experimentally detectable.

t (sec)

EFFECT OF NONZERO TEMPERATURES IIEFFECT OF NONZERO TEMPERATURES II

The larger the coupling constant λ the larger the amount of stationary entanglement.

SUMMING UPSUMMING UP

Derivation of the master equation for two coupled qubits interacting with two independent reservoirs.

Stationary solution: in general, for T1 T2, the detailed balance principle is not satisfied.

Dynamics at zero temperature: Damped Rabi oscillations and stationary entanglement due to the counter-rotating terms in the qubit-qubit interaction Hamiltonian.

Initial condition |11>: phenomena of sudden death and birth of entanglement.

Nonzero values for the reservoirs temperatures may destroy stationary entanglement, which is anyway visible for a reasonable temperature range.

WORK IN PROGRESS…WORK IN PROGRESS… Decoherence in superconducting Decoherence in superconducting

systems:systems: Structured environmentsStructured environments Non-markovian dynamicsNon-markovian dynamics

WORK IN PROGRESS…WORK IN PROGRESS… Decoherence in superconducting Decoherence in superconducting

systems:systems: Structured environmentsStructured environments Non-markovian dynamicsNon-markovian dynamics

M. Scala et al., J. Phys A: Math. and Theor. 41, 435304 (2008); M. Scala et al., in preparation.R. Migliore et al., Phys. Stat. Sol. B 246, 1013 (2009).