Outline The goal The Hamiltonian The superfast cooling concept Results Technical issues (time...
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Transcript of Outline The goal The Hamiltonian The superfast cooling concept Results Technical issues (time...
Outline
• The goal
• The Hamiltonian
• The superfast cooling concept
• Results
• Technical issues (time allowing)
• All cooling techniques are based on a small coupling parameter, and therefore rate limited
• We propose a cooling method which is potentially faster than and with no limit on cooling rate
• Approach adaptable to other systems (micro-mechanical, segmented traps, etc).
Goal
ˆ†H/ = + . .
i KX
za a e h c
The Hamiltonian
0
ˆ†0H/ = + + . .
2
i KX t
z xa a e h c
0
Standing wave
• Assume we can implementboth and pulses
• We could implement the red-shift operator
impulsively using infinitely short pulses via the Suzuki-Trotter approx.
X P
Cooling at the impulsive limit
x yyxn i X P t niP tiX te e e
†2x yX P a a
,t n n
,T
with
and taking
Solution: use a pulse sequence to emulateo pulseo Wait (free evolution)o reverse-pulse
[Retzker, Cirac, Reznik, PRL 94, 050504 (2005)]
yP
IntuitionyX
yX
But, we have only have X
12
1!
, , ,exp
, ,A B A
k
B A B A A Be e e
A A B
†ai if free pB t H t a
†i ip pulse pA t H t a a
2 2 2exp if f f p f pt H t t P t t
But
The above argument isn’t realizable:
• We cannot do infinite number of infinitely short pulses
• Laser / coupling strength is finite Cannot ignore free evolution while pulsing
Quantum optimal control
How we cool
Apply the pulse and the pseudo-pulse
Repeat
Reinitialize the ion’s internal d.o.f.
Repeat
xXyP
Sequ
ence
Cycle
Optimal control
2 possible avenues:
• Search for an “optimal” target operator
• Search for an “optimal” cooling cycle
40
100 2 10 2
730 0.31laser
KHz MHz
Ca nm
Cycle A Cycle B Cycle C
Initial phonon count 3 5 7
Final phonon count 0.4 1.27 1.95
after 100 cycles 0.02 0.10 0.22
Cycle duration 4.4 2.7 0.8
No. of X,P pulses 6 3 3
No. of sequences 10 10 10
2
2
2
We can do even better• Cycles used were optimized for the impulsive limit
• Stronger coupling meansfaster cooling
Cycle A Cycle B Cycle C
Initial phonon count 3 5 7
Final phonon count 0.25 1.01 1.69
after 100 cycles 0.0004 0.007 0.025
Cycle duration 0.4 0.3 0.2
No. of X,P pulses 6 3 3
No. of sequences 10 10 10
2
2
2
We can do even better
1 2GHz
/ 2E t
Some additional points
• For linear ion traps, we can cool ions individually – not to the global ground state
• does not apply here, as we’re not measuring energy of an unknown Hamiltonian [Aharonov & Bohm, Phys. Rev. 122 5 (1961) ]
Technical issues• Implementation of with 3 evolutions
dependent on commutation relations
• Matrix exponentiation very problematic
• If calc. involves cut-off -s and -s doubly so
• Must do commutation relations analytically
• BCH series for 3 exponents contains thousands of elements in first 6 orders
• Computerized non-commuting algebra
yP
P X
Superfast cooling
• A novel way of cooling trapped particles
• No upper limit on speed
• Optimized control gives surprisingly good results, even when working with a single coupling
• Applicable to a wide variety of systems
• We will gladly help adapt to your system