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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 - PowerPoint PPT Presentation
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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
• Atom-cavity system
• Ion trap
• NMR
• Quantum dots
• Spintronics…
Schemes for possible realization of quantum computer
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,
Penning trap: +magnetic field
Paul trap: +r.f.
Combined trap: + magnetic field+r.f.
Linear and ring trap:…
Working principle of the ion trap
22220
20
0 22
,, yxzZr
Uzyx
Realization of cavity QED in the ion trap
homogeneous classical laser field
: annihilation and creation operators of the harmonic oscillator
eggeggeez , ,
0
where the Lamb-Dicke parameter is defined as
wavelength of driving laser=
width of the ground-state wavepacket of the trapped ion
Thus in the interaction picture, we have
where
Quantized CM motion
Choose
,2,,00 L
>0 blue sideband
<0 red sideband
Thus to the leading order, we can engineer, for example, the l-photon-like
interaction if we have an l-th red sideband excitation
and ,L
1(well-resolved sideband limit)
(Lamb-Dicke limit)
Quantum state engineering in ion trap
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)]
Theory:
Experiment:
D. Wineland’s group (NIST)
Squeezed states
0, SDwhere
aaD *†exp
2†2*
22exp aaS
displacement operator
squeeze operator
with ire squeezing factor
Thus for two quadrature phase operators
2/2/†2
2/2/†1 , iiii aeeaiXaeeaX
122
21 XXthe minimum uncertainty product is reserved
with reX 22
2 reX 221
Even and odd squeezed states
,, even squeezed states
,, odd squeezed states
22† sinhcosh reara i
rer i sinhcosh * where
tanhcosh
tanhtanh2
2
2
222††2
rer
rearaeaa
i
ii
Now since
x
y
= 0
=0
= -2x
= 2x
300220
110000
32
2
210,,
yktixkti
xktixkti
eEeE
eEeEtyxEx
x
Superposed electric fields
Hamiltonian for a 2-level ion in 2-D trap
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 y << x <<1 (Lamb-Dicke limit) and << j, then the master equation is reduced to
Steady-state solution of the master equation
vsss gg
0dt
d ss
vibrational steady state
21012
1
2
1
0 2 , ,tanh ,tanh rE
Er
E
E
(dark state)
Thus the eigenvalue is determined by
The steady-state solutions depends on the parities of the initial state
vsn
n nc 20
2
for initial state with even parity
for initial state with odd parity
vs
nn nc 12
012
for initial state with mixed parity
oevsn
n PPnc0
x=0.02 x=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, are shown in dark bars.
Wigner distribution for even and odd squeezed states
even squeezed state odd squeezed state1,21,2 1,21,2
Δ= -Δ= 0
Δ=
For example, one may use the following π-pulse sequence to generate the number state n of vibration:
g,0 e,1 g,2 e,2 … e,n g,n
laser coolinglaser off
Creation of entangled Schrödinger cat states with ions [(Monre, Meekhof, King and Wineland (1996)]
ii egee 2
1
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.
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.
10mm
center-of-mass motion
breathing mode
Experimental demonstration of the motion of a string of 7 ions.
(Figures by J.Eschner, F. Schmidt-Kaler, R. Blatt, Universität Innsbruck)
•high efficiency to prepare, coherently control and detection of the states of the qubit using laser pulses
Challenges:
Perspectives of trapped ions
Merits:
•difficulties to cool a string of ions to the ground state of motion
•long decoherence times of the internal states of the ion
•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