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

Ion oscillations in a Penny trap

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 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 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

Scheme of sideband cooling

Schrödinger’s cat

then what will you see when the chamber is open?

dead2

1alive

2

1

If

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

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)

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