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Goren Gordon , Gershon KurizkiWeizmann Institute of Science, Israel
Daniel LidarUniversity of Southern California, USA
QEC07 USC Los Angeles, USADec. 17-21, 2007
OutlineUniversal dynamical decoherence
control formalismBrief overview of
Calculus of VariationsAnalytical derivation of equation
for optimal modulationNumerical resultsConclusions
2 a tR t Fd G
' '
F F K K
y t y y t y
12
1 2 2 20 0
,''
,T t
E t tt
dt dt t t
Z
Z
Decoherence ScenariosIon trap Cold atom in (imperfect)
optical lattice
Ion in cavityKreuter et al. PRL 92 203002 (2004)
Keller et al. Nature 431, 1075 (2004)
Häffner et al. Nature 438 643 (2005)
Jaksch et al. PRL 82, 1975 (1999)Mandel et al. Nature 425, 937 (2003)
Universal dynamical decoherence control formalism
. .
a
j
e
j
j
a
j
H t e e
j j
j e
t
h c
system+modulation
bath
coupling g|
e||
j
j
a
ej
a t
2| R t tF t e t e
Fidelity of an initial excited state:
11
2
1 2*
2 110 0 2
2Re ai t tt t
t etR d t ttt
tt d
2ji t
ejj
t e 1 10
t
ai dt tt e
Average modified decoherence rate
Reservoir response(memory) function
Phasemodulation
Kofman & Kurizki, Nature 405, 546(2000); PRL 87, 270405 (2001); PRL 93, 130406(2004)Gordon, Erez and Kurizki, J. Phys. B, 40, S75 (2007) [review]
11
2
1 2*
2 110 0 2
2Re ai t tt t
t etR d t ttt
tt d
2 a tR t Fd G
Time-domain
Frequency-domain
2
ejG
1
2
1 10
/
1
2
t t
t i tt
F t
dt t e
System-bath coupling spectrum
Spectral modulation intensity
G()
Ft()
R t
Universal dynamical decoherence control formalism
Kofman & Kurizki, Nature 405, 546(2000); PRL 87, 270405 (2001); PRL 93, 130406(2004)Gordon, Erez and Kurizki, J. Phys. B, 40, S75 (2007) [review]
No modulation (Golden Rule)
0 1
2
a
t
a
t t
F
R G
1 10
t
ai dt tt e
Universal dynamical decoherence control formalismSingle-qubit decoherence control
Decay due to finite-temperature bath couplingProper dephasing
Multi-qudit entanglement preservationImposing DFS by dynamical modulationEntanglement death and resuscitation
Dephasing control during quantum computation
(Gordon & Kurizki, PRL 97, 110503 (2006))
(Gordon & Kurizki, PRA 76, 042310 (2007))
(Gordon et al. J. Phys. B, 40, S75 (2007))
R R
( )2 T taGt FR d
G()
A B
11
22
(Gordon, unpublished)
Brief overview of Calculus of Variations
Want to minimize the functional: 0
, ' , , 'T
y y F t y y dtF
0
, ' , , 'T
y y K t y y dt E KWith the constraint:
The procedure:
' '
F F K K
y t y y t y
1. Solve Euler-Lagrange equation
Get solution: ;y t 0 00 ; ' 0 'y y y y
;y t E
; , ' ;y t y t E K
E
2. Insert the solution to the constraint:
Get
3. Get solution as a function of the constraint:
Analytical derivation of optimal modulation
Want to minimize the average modified decoherence rate:
With the energy constraint (a given modulation energy):
11
2
1 2*
2 110 0 2
2Re ai t tT t
t etR d t ttT
tT d
2
0
T
adt Et
1 10
t
ai dt tt e
AC-Stark shift
g|
e|
a
a t
1 10
ti dt t
t e
Resonant field amplitude
g|
e|
t
r t
(Gordon et al. J. Phys. B, 40, S75 (2007))
Analytical derivation of optimal modulation
Want to minimize the average modified decoherence rate:
With the energy constraint (a given modulation energy):
11
2
1 2*
2 110 0 2
2Re ai t tT t
t etR d t ttT
tT d
2
0
T
adt Et
, 0'' tt t Z
1 110 1
1, sin
Tt tt dt t tt t
Tt Z
Euler-Lagrange equation for optimal modulation
0 ' 00
0i tt t e Use notation:
0a
0
t
at d
Analytical derivation of optimal modulation
'' , 0t t t Z
1 1 1 10
1, sin
Tt t dt t t t t t t
T Z
Euler-Lagrange equation for optimal modulation
0 ' 0 0
Using the energy constraint, one can obtain:
2
1 1 10 0
1,
T tE dt dt t t
E Z
12
1 2 2 20 0
,''
,T t
E t tt
dt dt t t
Z
Z
Equation for Optimal Modulation
Numerical resultsCompare optimal modulation to Bang-Bang (BB) control:
Viola & Lloyd PRA 58 2733 (1998)Shiokawa & Lidar PRA 69 030302(R) (2004)Vitali & Tombesi PRA 65 012305 (2001)Agarwal, Scully, Walther PRA 63, 044101 (2001)
a t
0
0
2| | /
t
a a
ti t it
t
t t d
t e d e
F t
Numerical resultsCompare optimal modulation to Bang-Bang (BB) control:
Viola & Lloyd PRA 58 2733 (1998)Shiokawa & Lidar PRA 69 030302(R) (2004)Vitali & Tombesi PRA 65 012305 (2001)Agarwal, Scully, Walther PRA 63, 044101 (2001)
a t
Numerical resultsCompare optimal modulation to Bang-Bang (BB) control:
Viola & Lloyd PRA 58 2733 (1998)Shiokawa & Lidar PRA 69 030302(R) (2004)Vitali & Tombesi PRA 65 012305 (2001)Agarwal, Scully, Walther PRA 63, 044101 (2001)
a t
/
/ 1/ ct
aG
DD condition
1/ ct 2 0aR Fd G
Numerical results1/ noisef
min max1/G
Optimal pulse shape
2 a tR t Fd G
Fidelity Rte
F. T.X
2 a tR t Fd G
Numerical results - multi-peakedG
Optimal pulse shape
• Dynamical decoupling and Bang-Bang modulations are environment-insensitive, i.e. ignore coupling
spectrum
• Optimal modulation “reshapes” (chirps) the pulse to minimize spectral overlap of the system-bath coupling and modulation spectra
• Current results using universal dynamical decoherence control are also applicable to decay and proper-dephasing, at finite- temperatures
• Extensions to multi-partite deocherence and entanglement optimal control underway…
“Know thy enemy” Thank you !!!