Goren Gordon , Gershon Kurizki
Weizmann Institute of Science, Israel
Daniel Lidar
University of Southern California, USA
QEC07 USC Los Angeles, USA
Dec. 17-21, 2007
Outline
 Universal dynamical decoherence

control formalism
 Brief overview of
 K   K 
F  F

  

 y t  y '

y
t  y ' 

Calculus of Variations
 Analytical derivation of equation  EZ t, t 
 ''  t  
for optimal modulation
 dt  dt Z t , t 
 Numerical results
 Conclusions
R  t   2  dG   a  Ft  

T
0
2
t1
1
0
2
2
2
Decoherence Scenarios
Keller et al.
Nature 431,
1075 (2004)
Ion trap
Cold atom in (imperfect)
optical lattice
Häffner et al.
Nature 438
643 (2005)
Jaksch et al. PRL 82, 1975 (1999)
Mandel et al. Nature 425, 937 (2003)
Ion in cavity
Kreuter et al.
PRL 92
203002 (2004)
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]
system+
H  t    a   a  t   e e 
modulation
bath
 j j j 
 t 
ej
j e  h.c. coupling
j
| j
|e
a
j

 ej
a
|g
j
Fidelity of an initial excited state:
2
 R  t t
F  t   e |  t 
Average modified
decoherence rate
Reservoir response

(memory) function
e
t
t1
2
R  t   Re  dt1  dt2   t1  t2   *  t1    t2  eia  t1 t2 
0
0
t
 t    ej e
2
j
 i j t
t
dt1 a  t1 

0
 t   e
i
Phase
modulation
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]
Time-domain
t
t1
2
R  t   Re  dt1  dt2   t1  t2   *  t1    t2  eia  t1 t2 
0
0
t

Frequency-domain
R  t   2  dG   a  Ft  

   ej     
2
Ft     t   / t
System-bath
coupling spectrum G
Spectral
modulation
intensity
2
1 t
it1
 t   
dt

t
e


1
1

0
2
t
 i  dt1 a  t1 
 t e 0

Ft()
G()
R t 

No modulation (Golden Rule)
 a t   0   t   1
 Ft      
 R  2 G a 
Universal dynamical decoherence
control formalism
 Single-qubit decoherence control (Gordon et al. J. Phys. B, 40, S75 (2007))
 Decay due to finite-temperature bath coupling
    R   R 
 Proper dephasing
1 2






R()  t   2  dGT   a     Ft  

1 2

 Multi-qudit entanglement preservation
 Imposing DFS by dynamical modulation
 Entanglement death and resuscitation
G()
(Gordon, unpublished)
 Dephasing control during
quantum computation
(Gordon & Kurizki, PRA 76, 042310 (2007))
A
(Gordon & Kurizki,
PRL 97, 110503 (2006))
B
Brief overview of Calculus of
Variations
Want to minimize the functional:
F  y, y '   F  t , y, y 'dt
T
0
K  y, y '   K  t , y, y 'dt  E
T
With the constraint:
0
The procedure:
1. Solve Euler-Lagrange equation
Get solution:
y  t;  
 K   K 
F  F

  


 y t  y '

y

t

y
'


y  0  y0 ; y '  0  y '0
2. Insert the solution to the constraint:
Get
   E
K  y  t;   , y '  t;     E
3. Get solution as a function of the constraint:
y  t; E 
Analytical derivation of optimal
modulation
Want to minimize the average modified decoherence rate:
T
t1
2
R T   Re  dt1  dt2   t1  t2   *  t1    t2  eia  t1 t2 
0
0
T
With the energy constraint (a given modulation energy):

T
dt   t   E
2
a
0
(Gordon et al. J. Phys. B, 40, S75 (2007))
AC-Stark shift
t
 i  dt1 a  t1 
0
 a t 
Resonant field amplitude
t
 i  dt1 t1 
0
 r t 
 t   e
 t   e
|e
a
|g
|e
 t 
|g
Analytical derivation of optimal
modulation
Want to minimize the average modified decoherence rate:
T
t1
2
R T   Re  dt1  dt2   t1  t2   *  t1    t2  eia  t1 t2 
0
0
T
With the energy constraint (a given modulation energy):

T
0
dt  a  t   E
2
Euler-Lagrange equation for optimal modulation
Use notation:
  t     t  ei0t
  t    d a  
t
0
  a  0
 ''  t   Z t ,   t    0
1 T
Z t ,   t     dt1  t  t1  sin   t     t1     t  t1  
T 0
  0   '  0  0
Analytical derivation of optimal
modulation
Euler-Lagrange equation for optimal modulation
  0   '  0  0
 ''  t   Z t ,   t    0
1 T
Z t ,   t     dt1  t  t1  sin   t     t1     t  t1  
T 0
 E 
Using the energy constraint,
one can obtain:
1 T
dt

0
E
t
 dt Z
0
1
t1 ,   t1 
Equation for Optimal Modulation
 EZ t ,   t 
 ''  t  

T
0
dt1

t1
0
dt2Z t2 ,   t2 
2
2
Numerical results
Compare 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 
 a  t     t    d a  
t
0
  t   e
i  t 
  t     d   ei
 Ft   |    |2 / t
t
0
Numerical results
Compare optimal modulation to
Bang-Bang (BB) control:
 a t 
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)
Numerical results
Compare 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)
G   a 
DD condition
 /
1/ tc
 a t 

1/ tc

R  2  dG   a  F    0

Numerical results
Fidelity  e Rt
1/ f noise
G   1/  min    max

R  t   2  dG   a  Ft  

Optimal pulse shape
X
F. T.
Numerical results
G   - multi-peaked

R  t   2  dG   a  Ft  

Optimal pulse shape
• Dynamical
decoupling and Bang-Bang modulations
environment-insensitive, i.e. ignore coupling spectrum
are
• 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 finitetemperatures
• Extensions
to multi-partite deocherence
optimal control underway…
“Know thy enemy”
and
entanglement
Thank you !!!