ion trapping
TRANSCRIPT
-
7/30/2019 Ion trapping
1/27
Quantum information processing in ion traps IID. J. Wineland, NIST, Boulder
Part I,
Rainer Blatt
Lecture 1: Nuts and bolts
Ion trapology Qubits based on ground-state hyperfine levels Two-photon stimulated-Raman transitions
* Rabi rates, Stark shifts, spontaneous emission
Lecture 2: Quantum computation (QC) and quantum-limited measurement
Trapped-ion QC and DiVincenzos criteria
Gates Scaling Entanglement-enhanced quantum measurement
Lecture 3: Decoherence
Memory decoherence Decoherence during operations* technical fluctuations
* spontaneous emission
* scaling
Decoherence and the measurement problem
-
7/30/2019 Ion trapping
2/27
Ion trapping 101
Earnshaws theorem: In a charge free region, cannot confine a charged particle
with static electric fields.Proof: For confinement, must have (2(q)/2xi)trap location < 0 (xi {x,y,z})But from Laplaces equation: 2 = 0, cannot satisfy confinement condition for all xi.
B0
U0
Solution 1: Penning trap:
q U0 [2z2 x2y2]
Difficult to accomplish individual
ion addressing.(However, see: Ciaramicoli, Marzoli,
Tombesi, PRL 91, 017901 (2003))
Solution 2: RF-Paul trap:
= (x2 + y2 + z2)V0
cost + U0
(x2 + y2 + z2)
[ + + = + + = 0] (Laplace)
-
7/30/2019 Ion trapping
3/27
y
z
x
y x
axisztrap
end view
~2R
Special case:
linear RF trap(quadrupole mass
filter plugged on-axisWith
static fields)U
o
V0
cos T
t
Uc
(Get s and numerically)
-
7/30/2019 Ion trapping
4/27
Equations of motion
(classical treatment adequate)Mathieu equation
(2)
z-motion, qz = 0 (static harmonic well)
x,y motion, Mathieu equation:
plug into (2), find (recursion relation for) C2n
-
7/30/2019 Ion trapping
5/27
Solution in ith direction (i {x,y}):
|qi|
0 .2 .4 .6 .8 1.0
ai 0
-.2
.2
-.4
.4
.6
-.6
.8
1.0
stable
unstable
unstable
pseudo-potential strength
sta
tic
potential
strength
-
7/30/2019 Ion trapping
6/27|q |
0 .2 .4 .6 .8 1.0
ax 0
-.2
.2
-.4
.4
.6
-.6
.8
1.0
-.8-1.0
-1.2
ay0.2
-.2
.4
-.4
-.6
.6
-.8-1.0
.8
1.0
az
x = T/2
y = T/2
x 0
y 0
stable
ai Ui, ax + ay + az =0Simultaneous solution for x, y, z:
-
7/30/2019 Ion trapping
7/27
0.05
typically, we live here
|ai| , qi2
-
7/30/2019 Ion trapping
8/27
Heuristic approach: pseudo-potential approximation
assume mean ion position changes negligibly in duration 2/T
pseudo-potential from micromotion kinetic energy:
agrees with
Mathieu equation
limit |ai| , qi2
-
7/30/2019 Ion trapping
9/27
Digression: Optical dipole traps:
outer
electron
L Response of atomic core tolaser field is negligible
because of heavy mass.
ForL >> 0 (blue detuning)electron response out of phase with
electric force. Electron trapped in
ponderomotive (pseudo-potential)
laser potential (just like RF trap).
Trapping in field minima.
Core is attached to electron
Dispersion:ForL
-
7/30/2019 Ion trapping
10/27
(some) ion-trap realities
axisztrap
y x
EDC
Patch potentials:
Static potentials: pushes
ions away from trap axis
micromotion xsinTtcan cause X-tal heating
Fluctuating patch fields:
causes heating; COM
primarily affected
Source: unknown!(mobile electrons on
oxide layers,.. ??)
-
7/30/2019 Ion trapping
11/27
Idealized trap:RF electrodes
control electrodes
1'
2'
3'
4'
1
2
3
4
IONS
200 mApproximation:
gold-coated
alumina wafers
-
7/30/2019 Ion trapping
12/27
0.2 mm
Chris Myatt et al.
linear Paul (RF) trap
VRF
~ 500 V
RF ~ 50 250 MHz
-
7/30/2019 Ion trapping
13/27
~ 1 cm
-
7/30/2019 Ion trapping
14/27
-
7/30/2019 Ion trapping
15/27
0.2 mm
For9Be+, V0 = 500 V, T/2 = 200 MHz, R = 200 m
x,y/2 ~ 6 MHz
-
7/30/2019 Ion trapping
16/27
Ion Trap QC: Proposal: J. I. Cirac and P. Zoller,PRL 74, 4091 (1995)
Motion data bus
(e.g., center-of-mass mode)Laser beam
n=3
n=2
n=1
n=0
Stay in two lowest
motional states (motion qubit)
Internal-state qubit
-
7/30/2019 Ion trapping
17/27
n'
n
ladder states for selected motional mode
0
1
0
E - E =0
0
12
0 = optical frequency: single photon
need good laser frequency stability
memory and gate coherence limited by upper state lifetime (~ seconds)
0 = RF/microwave frequency memory coherence limited by upper-state lifetime (>> days)
sideband transitions weak at RF 2-photon optical stimulated-Raman transitions
frequency stability = RF modulator stability vary sideband coupling (Lamb-Dicke parameter) with k2 k1
gate decoherence: spontaneous-Raman scattering (fundamental limit)
-
7/30/2019 Ion trapping
18/27
Stimulated-Raman transitions:
Simple case: Motion: 1-D harmonic well (frequency M
),
Internal states: 3-level system
rrk ,G
bbk ,G
M
0
e
,0
,1
,0
,1
e,0
e,1
,2
b = blue
r= red
electric dipole
transitions
e >> >> 0 >> M
-
7/30/2019 Ion trapping
19/27
rrk ,G
bbk ,G
M
0
e
,0
,1
,0
,1
e,0
e,1
,2
b = blue
r= red
zero-point
wavefunction spread
-
7/30/2019 Ion trapping
20/27
(+ rotating wave approximation)
similarly:
-
7/30/2019 Ion trapping
21/27
Adiabatic elimination:
make ansatz:
-
7/30/2019 Ion trapping
22/27
rrk ,G
bbk ,G
M
0
e
,0,1
,0
,1
e,0
e,1
,2
b = blue
r= red
Stark shift of from blue laser
Add in other Stark shifts
-
7/30/2019 Ion trapping
23/27
Add in other Stark shifts
absorb Stark shifts into wave function amplitudes
near a resonance:
Rabi flopping
-
7/30/2019 Ion trapping
24/27
(Lamb-Dicke parameter)
Be+: P = 1 mW, w0 = 25 m, /2 = 100 GHz, /2 ~ 0.5 MHz
Carrier transitions:
Debye-Waller factor
Sideband transitions: n = n 1(n> = larger of n and n)
red sideband (n = n-1): can get from
Jaynes-Cummings Hamiltonian from cavity-QED(see, e.g., Raimond, Brune, Haroche, Rev. Mod. Phys. 73, 565 (01))
-
7/30/2019 Ion trapping
25/27
Complete picture:
Sum over excited states, typically:
2P3/2
2S1/2
2P1/2
can tune out differential Stark shifts
can tune out polarization sensitivity
For N ions, consider effects of 3N modes
Debye-Waller factors from spectator modes
sideband transitions: interference from two-mode transitions:
e.g. np -mr= M
-
7/30/2019 Ion trapping
26/27
Spontaneous emission:
rrk ,G
bbk ,G
0 ,1
,0
,1
,2
b = blue
r= red
e
e,1
e,0
e
,0
M
-
7/30/2019 Ion trapping
27/27
Quantum information processing in ion traps IID. J. Wineland, NIST, Boulder
Lecture 1: Nuts and bolts
Ion trapology Qubits based on ground-state hyperfine levels
Two-photon stimulated-Raman transitions* Rabi rates, Stark shifts, spontaneous emission
Lecture 2: Quantum computation (QC) and quantum-limited measurement
Trapped-ion QC and DiVincenzos criteria
Gates Scaling Entanglement-enhanced quantum measurement
Lecture 3: Decoherence
Memory decoherence Decoherence during operations
* technical fluctuations
* spontaneous emission
* scaling
Decoherence and the measurement problem