Squeezing generation and revivals

in a cavity-ion systemNicim Zagury

Instituto de Física, Universidade Federal

Rio de Janeiro, Brazil

colaboradores: R. Rangel. L. Carvalho

A ion inside a Paul trap in a cavity

…is being shined by two laser fields

The lasers and the cavity mode are quasi resonant to a electronic transition of the ion.

Level scheme

atomic transition frequency

cavity frequency

vibration frequency

laser frequencies

1 2ˆ ˆ( ( ) ( ) ( ( ) ( )1 2 .}

ˆ ˆ{ sin( ( ) )

}| | .

i tcav cav

i k x t t i k x t t

H g k x t ae

ig e ig e e g h c

A effective Hamiltonian

† † † †1 2ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ( ( )}{ )effV i ba ab a b ab

• =

1 21 2

cav cav cav cavg g g g

RWA + adiabatic elimination of the upper state:

/(2 )cav cavk m ( Lamb-Dicke parameter )

Master Equation

† † †1 ˆ[ , ] (2 )2eff

dV a a a a a a

dt i

Even though we have considered a bad cavity, we were able to obtain an analytical solution for the total density operator of the system

† † † †1 2ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ( ( )}{ )effV i ba ab a b ab

2 2 2 21 2 /16 0

A product of the vacuum of the cavity field and an ideal squeezed state of the motion of the ion

2 †2( ) exp ( ) / 2vibS b b

The system reaches the steady state:

†| 0| 0 00 | |( ) ( )vsteady ib vibcav S S

Very large squeezing:2

2 1 22

1 2 64e

For finite times the two subsystems are, in general, entangled, but periodically at times

and(arctan( 4 / ) ) /n n

' ( ) /n n they disentangle.

Remarkably, even though there is dissipation there is a complete “revival” of the state of the motion at t =

n and of the state of the field at t= 'n

The cavity field also returns periodically to the initial vacuum state

†| 0 0 |( ) ( )vib vibS S

Periodically, the state of motion returns to the same ideal squeezed state

1†( )

( ( ) 1)| |( ) ( ( )) ( ( ))

k

k

n t

n tk kt S t S t

The solution for the reduced density operators at any time are “squeezed thermal states”:

2 1

22 1

For small values of / , the maximum value of n(t)

is .25( / ) . This state is almost ideal squeezed sta an te.

1 2/ 0.6

1/ 0.4

Behaviour of n with time

Squeezing

0 4 8 12 16 200,0

0,1

0,2

0,3

0,4

0,5

/ 1 = 0.4

2/

1 = 0,9

(pC)2

(xV)2

t

2 2 2( ) ( ) ( ( ) 1)X t P t n t revivals

2 2 2 21 2 /16 0When the system

is always entangled and there is no revi

,

vals.

15 /t

11/t

110 /t

Remarks and conclusions

1. For 2 2 2 2

1 2 /16 0 , the two subsystems

disentangle periodically at given times n and n´

2. Although there is dissipation the the state of motion and the cavity field and ¨revive¨ completely at n and n´Respectively

3. At any time the reduced density matrices correspond to squeezed thermal states

4. These results can be easily generalized for a initial coherentstate

[1] H. Zeng and F. Lin, Phys. Rev. A 50, R3589 (1994).

[2] E. Massoni, M. Orszag, Opt. Comm. 190, 239 (2001)

[3] R. Rangel, E. Massoni, and N. Zagury, Phys. Rev. A 69, 023805 (2004).

References