Download Generation of mesoscopic superpositions of two

Generation of Mesoscopic
Superpositions of Two Squeezed States
of Motion for A Trapped Ion
Shih-Chuan Gou (郭西川)
Department of Physics
National Changhua University of Education
國立彰化師範大學物理系
Schemes for possible realization of quantum
computer
•
Atom-cavity system
•
Ion trap
•
NMR
•
Quantum dots
•
Spintronics…
Reference:
“Generation of mesoscopic superpositions of two squeezed
states of motion for a trapped ion” , Phys. Rev. A 55, 3719
(1997).
S.-C Gou, J. Steinbach, and P.L. Knight,
Working principle of the ion trap
  x, y , z  

U0
2
2
2
2
z

x

y
r02  2Z 02

Penning trap:  +magnetic field
Paul trap:  +r.f.
Combined trap:  + magnetic field+r.f.
Linear and ring trap:…
Ion oscillations in a Penny trap
Realization of cavity QED in the ion trap
   0
homogeneous classical laser field
: annihilation and creation operators of the harmonic oscillator
z  e e  g g ,   e g ,   g e
Quantized CM motion
where the Lamb-Dicke parameter h is defined as
=
width of the ground-state wavepacket
of the trapped ion
wavelength of driving laser
Thus in the interaction picture, we have
where
Choose
   L  0  0, ,2 ,
 >0 blue sideband
0
 <0  red sideband
and
   L , 
h  1
(well-resolved sideband limit)
(Lamb-Dicke limit)
Thus to the leading order, we can engineer, for example, the l-photon-like
interaction if we have an l-th red sideband excitation
Quantum state engineering in ion trap
Theory:
Squeezed states [Cirac, et. al. (1993)]
Even and odd coherent states (Schrödinger cat states)
[de Matos Filho and Vogel (1996)]
Pair coherent states [Gou, Steinbach, Knight (1996)]

Experiment:
D. Wineland’s group (NIST)
Squeezed states
 ,  D S   0
where

D   exp a†   *a

displacement operator
  * 2  †2 
S    exp  a  a 
2
2

with
  rei
squeeze operator
squeezing factor
Thus for two quadrature phase operators
X 1  a †ei / 2  aei / 2 ,

X 2  i a †ei / 2  ae i / 2

the minimum uncertainty product X 1 2 X 2 2  1 is reserved
with
X 1 2  e 2 r X 2 2  e 2 r
Even and odd squeezed states
Now since

   ,    ,
even squeezed states
   ,    ,
odd squeezed states
a cosh r  a e
† i
a
2
i

sinh r     2  
2
† 2 2 i
 2a ae tanh r  a e
†

tanh 2 r  
 2

i


 e tanh r   
2
 cosh r

where
   cosh r   *ei sinh r
Hamiltonian for a 2-level ion in 2-D trap
y
= -2x
= 0
x
= 2x
Superposed electric fields
i  0t  k0 x 0 
 
E
x, y, t   E0e
=0
 E1ei 0 2 x t k1x 1 
 E2ei 0  2 x t k2 x 2   E3ei 0t k0 y 3 
The total Hamiltonian in the interaction picture
The evolution of the system can be described by a density matrix
obeying the master equation
accounts for the momentum transfer in the x-y plane due to spontaneous
emission described by the angular distribution
For a highly anisotropic trap (x<<y ), if hy << hx <<1 (Lamb-Dicke limit)
and << j, then the master equation is reduced to
Steady-state solution
of the master equation
d ss
0
dt
 ss  g g  vs
vibrational steady state
(dark state)



E0
E2
 tanh r ,
 tanh 2 r ,   1  0 , 2  1   2
E1
E1
Thus the eigenvalue  is determined by
The steady-state solutions depends on the parities of
the initial state
for initial state with even parity

c
n 0
2n
2n   vs     
for initial state with odd parity

c
n 0
2 n 1
2n  1   vs     
for initial state with mixed parity

c
n 0
n
n   vs  Pe      Po    
hx=0.02
hx=0.05
Number distribution P(n) of the vibrational steady state (grey bars) for
various Lamb-Dicke parameters. The ion is initially prepared in the
vacuum state. The number distribution of the even squeezed state, 1,
1 are shown in dark bars.
even squeezed state
  2,  1    2,  1
odd squeezed state
  2,  1    2,  1
Wigner distribution for even and odd squeezed states
Scheme of sideband cooling
Schrödinger’s cat
If
 
1
1
 alive 
 dead
2
2
then what will you see when the chamber is open?
Δ= 0
Δ=
Δ=

-
For example, one may use the following π-pulse sequence to generate
the number state n  of vibration:
g,0 
laser cooling
 e,1 
g,2 
e,2 
… e,n 
g,n 
laser off
Creation of entangled Schrödinger cat states
with ions [(Monre, Meekhof, King and Wineland (1996)]

1
  e ei  g e i
2

Various level schemes for the trapped ions
Measurement of quantum jump
Trapped ions as quantum computers
[Cirac, Zoller, (1995)]
Vibrational mode as a quantum data bus
(a) With the first laser pulse the state of ion 1 is mapped to the COM mode;
(b) the state of ion 2 is changed conditional on the state of the COM mode.
(NIST, Ion Storage Group)
10mm
The scheme of the linear trap used in the Innsbruck group: A radio-frequency field (16
MHz, about 1000 Volts) is applied to the elongated electrodes (red) to provide the trapping
in the radial direction. The ring-shaped electrodes at the two ends are responsible for the
trapping in the axial direction, on which a static electric field of the order of +2000 Volts is
applied. The ions (indicated by green dots ) oscillate in the radial and axial directions.
However, since the trapping frequency in the radial direction (4 MHz) is much larger than
that in the axial direction(700 kHz ), the ions arrange themselves in a linear string. The
distance between the ions is typically only a few µm.
breathing mode
center-of-mass motion
Experimental demonstration of the motion of a string of 7 ions.
(Figures by J.Eschner, F. Schmidt-Kaler, R. Blatt, Universität Innsbruck)
Perspectives of trapped ions
Merits:
•long decoherence times of the internal states of the ion
•high efficiency to prepare, coherently control and detection of the states
of the qubit using laser pulses
Challenges:
•fluctuations (intensity, frequency, phases…) of the driving lasers
•collisions with background gas in the vacuum chamber
•decoherence of the vibrational states that limits the number of operations
•deviation between the laser focus and the position of the ion
•difficulties to cool a string of ions to the ground state of motion