Entanglement for two qubits interacting with a thermal field

Mikhail Mastyugin

The XXII International Workshop High Energy Physics and Quantum Field Theory

June 24 – July 1,  2015 Samara, Russia

• The magnetic flux through the hole superconductor takes discrete values

• Josephson tunneling contact - two superconductors S1 and S2, which separated by a thin dielectric layer

7 20 2 10

2

hcФ Гс см

e

Josephson predicted 2 effects: • The dependence of the superconducting current through the tunneling

barrier of the phase difference at the contact

Ic – The critical current • Feedback voltage at the contact with the derivative of the phase

difference time

Alternating current oscillates at a frequency

sincI I

2

h dV

e dt

2eV

Concepts of superconducting qubits

• Josephson coupling energy

• Josephson inductance

• The charge (Coulomb) energy

2c

j

IE

e

2 cosjc

LeI

2(2 )

2c

eE

C

• one-contact flux qubit and his basic

state

The potential energy U (φ) when β <1, the potential has a minimum;

when β> 1 - two minimums

2( )( ) [(1 cos ) ]

2e

jU E

0

2 cLI

Ф

Main characteristics of Josephson junctions:

Flux qubits:

• three-contact flux qubit

the potential energy U (φ) in the case of low inductance

β << 1 and f = 0, then the potential has two minimums at the points:

Standing in the local minima correspond to the two currents

1( , ) {2 2cos cos cos(2 ( ) 2 )}

2jU E f

0

1/ 2еФf

Ф

0; arccos1/ 2

2, 1 1/ (2 )L R cI I

Superconducting qubit stream with Josephson junctions :

• Two superconducting qubits interact with a superconducting

electric "resonator" (LC-circuit)

The scheme transitions in a three-level artificial atom Δ-type and effective

two-level atom with degenerate two-photon transition.

• A qubit interact with a superconducting electric

"resonator" (LC-circuit), the second qubit is outside the cavity

= 2eg

The Hamiltonian interaction

The evolution operator

The reduced density matrix

Initially, the resonator field is in single-mode thermal field

The atoms are in the form of coherent states

22 2

1 2 2 1=1

= ( ) ( ), (1)i ii

H g a R a R R R R R

Influence of atomic coherence and dipole-dipole interactionon the entanglement of two qubits induced by thermal noise

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

( ) = .(2)

U U U U

U U U U

U t U U U U

U U U U

( ) = ( ) (0) (0) ( ).A F F At Tr U t U t

(0) = ( ) | |,Fn

P n n n ( ) =(1 )

n

n

nP n

n

1 21 1 1 2 2 21) | (0) = cos | sin | , | (0) = cos | sin | .

i ie e

2) | (0) = cos | , sin | , ,ie

The matrix elements of the evolution operator

criterion Perez Horodetskih or "negative" = 2 ,i

i

2 2 2 2 2 211 14 41= 1 2 , = 2 , = 2 ,

A A AU a a U a a U a a

2 2 2 244 12 13 21 31= 1 2 , = = , = = ,

A B BU a a U U a U U a

2 224 34 42 43= = , = = ,

B BU U a U U a

22 33= =U Uexp ( )

2= [1 exp( )] 2 exp( (3 ) ] [1 exp( )] ,

4 2

gi t

gig t i t ig t

23 32= =U U

exp ( )2

= [1 exp( )] 2 exp( (3 ) ] [1 exp( )] ,4 2

gi t

gig t i t ig t

= exp cos sin 1,2 2 2

g g gA i t t i t

= exp ( ) 1 exp( )

2

gB i t ig t

2 2 2 2 2 2 2 2 2= , = 2( ), = 8( ) .a a a a a a a ag

1

11 12 13 14*12 22 2 24* *13 23 33 34* * *14 24 34 44

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) (4)

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

TA

t t t t

t t t tt

t t t t

t t t t

The initial coherent atomic not entangled statesThe initial coherent atomic not entangled states

Fig. 1. Time-dependent parameter entanglement for different initial coherent atomic states: , (dashed line) and

(solid line). The average number of photons in the mode . The constant dipole-dipole interaction of atoms .

1 21 1 1 2 2 21) | (0) = cos | sin | , | (0) = cos | sin | .

i ie e

1 2 / 4 1 2 0 1 2 1 2 / 4, / 4, 0

= 0,01n0,1

Fig.2. The time dependence of the parameter for the entanglement

of incoherent and coherent the initial states of atoms. The first case corresponds to the dashed, and the second - a solid line. The average number of photons in the mode . The constant dipole-dipole interaction of atoms . Phase states of atoms in both cases are the same .

1 2/ 2, 0 1 2/ 4, / 4

= 5n0,1 1 2  0

Figure 3. The time dependence of the parameter entanglement for coherent initial states of atoms

and different values of the constant dipole-dipole interaction: α = 0 (dashed line) and α = 0,1 (solid line). The average number of photons in the mode

Figure 4. The time dependence of the parameter entanglement for coherent initial states of atoms

and different values of the relative phase of the atomic states (solid line) and (dashed line). The average number of photons in the mode

The constant dipole-dipole interaction of atoms α = 0,1.

1 2 1 2 / 4, / 4, 0

= 0,1n

1 2/ 4, / 4

  0   / 2

= 0,05n

2) | (0) = cos | , sin | , ,ie

The initial coherent atomic entangled statesThe initial coherent atomic entangled states

Fig.5. Time-dependent parameter entanglement for initial entangled states (solid line), (dashed line), (dotted). The average number of photons in the mode , constant dipole-dipole interaction α=0,1 and

Fig.6. Time-dependent parameter entanglement for initial entangled states (solid line), (dashed line), (dotted). The average number of photons in the mode , constant dipole-dipole interaction α=0,1 and

/ 4 cos   0.4   / 2

= 0,1n3 / 3

/ 3

  0 / 2

= 0,1n

Hamiltonian

starting the atomic density matrix

Initially, the resonator field is in single-mode thermal field

Atoms are in coherent states

eigenfunctions

Entanglemen of two supercondacted qubets one of wich is a traped in a cavity

(0) = ( ) | |,Fn

P n n n ( ) =(1 )

n

n

nP n

n

1 21 1 1 2 2 21) | (0) = cos | sin | , | (0) = cos | sin | .

i ie e

1 2 3 4| = ( | , , 2 | , , 1 | , , 1 | , , ) ( = 1, 2,3,4).in in i n i n i n i nC X n X n X n X n i

1 2 1 1 1 2 1 2= (1/ 2) ( ) ( ) ( ),z zH a a g a a J

1 2 1 2(0) =| (0) | (0) (0) | (0) | .A

( ) ( )kiE t

k kk

t C t e Ф

Here

Where

energy eigenvalues

Where

2 2 2 21 2 3 4= 1/ | | | | | | | |in i n i n i n i nC X X X X

21

11

1= ;

2 2 1 2

n n

nXn n

2

12

1= ;

2 1n

nXn

2

13

3 2= ;

2 1n

n

nX

n

14 = 1;nX 2

1

21

1= ;

2 2 1 2

n n

nXn n

2

22

1= ;

2 1n

nXn

2

23

3 2= ;

2 1n

n

nX

n

24 = 1;nX 22

31

1= ;

2 2 1 2

n n

nXn n

2

32

1= ;

2 1n

nXn

2

33

3 2= ;

2 1n

n

nX

n

34 = 1;nX 22

41

1= ;

2 2 1 2

n n

nXn n

2

42

1= ;

2 1n

nXn

243 = ;

2 1n

nXn

44 1nX

2 21 2

2 2 2 4

3 2 , 3 2 ,

1 6 4 , 1 (6 4 )

n n n n

n n

n n

n n

1 2/ = ( 1) / 2, / = ( 1) / 2,n n n n n nE n A B E n A B

3 4/ = ( 1) / 2, / = ( 1) / 2.n n n n n nE n A B E n A B 2 2 2= 4 6 2 , = 2 4( 1) ( 1),n nA n B n = /J g

reduced atomic density matrix

criterion Perez Horodetskih or "negative"

21 1| = ( / )[| , ,1 (1/ ) | , ,0 ], = 0;E

2 2| = (1/ 2)[(1/ ) | , ,1 | , ,0 ( / ) | , ,0 ], / = ;E

3 2| = (1/ 2)[ (1/ ) | , ,1 | , ,0 ( / ) | , ,0 ], / = ,E 2= 1

0 =| , ,0 , 0 =E

11 12 13 14*12 22 23 24* *13 23 33 34* * *14 24 34 44

( ) = .A t

= 2 ,ii

* *11 12 13 23* * *12 22 14 241

13 14 33 34*

23 24 34 44

( ) = .T

A t

Fig. 1. The time dependence of the parameter ɛ(t) with and . The initial atomic states: (a) , (b) and (c)

= 0.01n = 0,1| , | ,

1 2| (0) = (1/ 2(| | ), | (0) = (1/ 2(| | )A A

Fig. 2. The time dependence of the parameter ɛ(t) for and α = 1. The initial atomic state .

Fig. 3. The time dependence of the parameter ɛ(t) for

and α = 1. The initial atomic state .

= 0.1n

= 10n

,

,