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a)ˆcos(0 txkEE
k
)ˆcos(ˆˆ 0 txkEE
)ˆcos()ˆˆ(2 txk
Rotating frame titi AA ee 2 )ˆcos( txk txkitxki ee ˆˆ titi AA ee
tixiktixik AA eeee ˆˆ Rotating wave approximation
kmotionmotion ,, kmotionmotion ,,
a
xikxik ee ˆˆ
tixiktixik AA eeeeH ˆˆint
xike xik ˆ1ˆ Another rotating frame titi tt eaeaikx ˆˆ1 †0
tititi
tititi
Att
Att
eeaeai
eeaeaiH
ˆˆ1
ˆˆ1
†
†
int
1 ,0 kx
tititi
tititi
tAtAA
tAtAA
eaieaie
eaieaieH
ˆˆ
ˆˆ
†
†
int
tixiktixik AA eeeeH ˆˆint
xike xik ˆ1ˆ Another rotating frame titi tt eaeaikx ˆˆ1 †0
1 ,0 kx
A
intH Carrier interaction
tititi
tititi
Att
Att
eeaeai
eeaeaiH
ˆˆ1
ˆˆ1
†
†
int
tititi
tititi
tAtAA
tAtAA
eaieaie
eaieaieH
ˆˆ
ˆˆ
†
†
int
tixiktixik AA eeeeH ˆˆint
xike xik ˆ1ˆ Another rotating frame titi tt eaeaikx ˆˆ1 †0
tititi
tititi
Att
Att
eeaeai
eeaeaiH
ˆˆ1
ˆˆ1
†
†
int
1 ,0 kx
tA
†int ˆˆ aaiH Red sideband interaction
1,,,1,int nnnnnniH 1,,,1,int nnnnniH Interaction strengthgiven by n
tititi
tititi
tAtAA
tAtAA
eaieaie
eaieaieH
ˆˆ
ˆˆ
†
†
int
tixiktixik AA eeeeH ˆˆint
xike xik ˆ1ˆ Another rotating frame titi tt eaeaikx ˆˆ1 †0
tititi
tititi
Att
Att
eeaeai
eeaeaiH
ˆˆ1
ˆˆ1
†
†
int
1 ,0 kx
tA
aaiH ˆˆ†int Blue sideband interaction
1,,,1,int nnnnnniH 1,,,1,1int nnnnniH Interaction strengthgiven by 1n
Resolved sideband coolingStep 1: Doppler cool
2Dopp
E
tDopp
t
E
2
MHz 52/
MHz 202/
Ion cooled to ground state: PRL 75, 4011 (1995)
Neutral atom cooled to ground state: PRX 2, 041014(2012)
Resolved sideband coolingStep 1: Doppler cool
Step 2: Pump to
2S1/2
2P1/2
370 nm
|
|
/2g p = 20 MHz
F=1
F=0
F=1
F=0
Ion cooled to ground state: PRL 75, 4011 (1995)
Neutral atom cooled to ground state: PRX 2, 041014(2012)
GHz 6.12A
Resolved sideband coolingStep 1: Doppler cool
Step 2: Pump to 2P1/2
/2g p = 20 MHzF=1
F=0
Ion cooled to ground state: PRL 75, 4011 (1995)
Neutral atom cooled to ground state: PRX 2, 041014(2012)
Step 3: Apply red sideband
tA
kk
1,, nn
Resolved sideband coolingStep 1: Doppler cool
Step 2: Pump to 2P1/2
/2g p = 20 MHzF=1
F=0
Ion cooled to ground state: PRL 75, 4011 (1995)
Neutral atom cooled to ground state: PRX 2, 041014(2012)
Step 3: Apply red sideband
tA
Step 4: Pump to
1,1,, nnn
Resolved sideband coolingStep 1: Doppler cool
Step 2: Pump to 2P1/2
/2g p = 20 MHzF=1
F=0
Ion cooled to ground state: PRL 75, 4011 (1995)
Neutral atom cooled to ground state: PRX 2, 041014(2012)
Step 3: Apply red sideband
tA
Step 4: Pump to
etc
Finish in !0, n
… How to check?
Measuring phonon number
Red sideband interaction strength given by n
Blue sideband interaction strength given by 1n
Ion cooled to ground state: PRL 75, 4011 (1995)
Neutral atom cooled to ground state: PRX 2, 041014(2012)
Assume thermal state with mean phonon number n
Probe red, blue sidebands for same duration
1n
n
P
Pblueex
redex
and spin state
Measuring phonon number
Ion cooled to ground state: PRL 75, 4011 (1995)
Neutral atom cooled to ground state: PRX 2, 041014(2012)
Assume thermal state with mean phonon number n
1n
n
P
Pblueex
redex
and spin state
Before:
5.0nasymmetry 1/3
After: 014.0nasymmetry 1/67
Quantum logic spectroscopyMotivation: Probe a “clock” transition when you don’t have a cycling transition
Spectroscopy ionLogic ion
Science 309, 749 (2005)
Quantum logic spectroscopyMotivation: Probe a “clock” transition when you don’t have a cycling transition
Spectroscopy ionLogic ion
Science 309, 749 (2005)
Quantum logic spectroscopy
Science 309, 749 (2005)
Step 1: Initialization n=1n=0
n=1n=0
Be+Al+
mAlBe0
Quantum logic spectroscopy
Science 309, 749 (2005)
Step 1: Initialization
mAlAlBemAlBe
00
n=1n=0
n=1n=0
Be+Al+
Step 2: Interrogate clock transition
mAlBe0
Quantum logic spectroscopy
Science 309, 749 (2005)
Step 1: Initialization
mAlAlBemAlBe
00
n=1n=0
n=1n=0
Be+Al+
Step 2: Interrogate clock transition
mAlBe0
Step 3: Drive red sideband on Al
mmAlBemAlAlBe
100
Quantum logic spectroscopy
Science 309, 749 (2005)
Step 1: Initialization
mAlAlBemAlBe
00
n=1n=0
n=1n=0
Be+Al+
Step 2: Interrogate clock transition
mAlBe0
Step 3: Drive red sideband on Al
mmAlBemAlAlBe
100
Step 4: Drive red sideband on Be
mAlBeBemmAlBe
010
Step 5: Read out Be
Quantum logic spectroscopy: Initialization sequence
n=1n=0
mF = 1/2
mF = 1/2
mF = 3/2
mF = 3/2
Al+
mF =5/2
mF = 5/2
…
- Carrier transition with 0 Fm
Quantum logic spectroscopy: Initialization sequence
n=1n=0
mF = 1/2
mF = 1/2
mF = 3/2
mF = 3/2
Al+
mF =5/2
mF = 5/2
…
- Carrier transition with 0 Fm
- Red sideband transition with 1 Fm
Quantum logic spectroscopy: Initialization sequence
n=1n=0
mF = 1/2
mF = 1/2
mF = 3/2
mF = 3/2
Al+
mF =5/2
mF = 5/2
…
- Carrier transition with 0 Fm
- Red sideband transition with 1 Fm
- Laser cool the motional mode (with Be)
Quantum logic spectroscopy: Initialization sequence
n=1n=0
mF = 1/2
mF = 1/2
mF = 3/2
mF = 3/2
Al+
mF =5/2
mF = 5/2
…
- Carrier transition with 0 Fm
- Red sideband transition with 1 Fm
- Laser cool the motional mode (with Be)