Transcript Quantum computing

Two Level Systems and Kondo-like
traps as possible sources of
decoherence in
superconducting qubits
Lara Faoro and Lev Ioffe
Rutgers University (USA)
Outline
• Decoherence in superconducting qubit [ experimental state of the art ]:
• low frequency noise (1/f noise)
• high frequency noise (f noise)
• We discuss two possible microscopic mechanisms for the fluctuators
• weakly interacting quantum Two Level Systems (TLSs)
• environment made by Kondo-like traps
• TLSs model:
• significant source of noise
• detailed characteristics of the noise power spectrum are in a
qualitative and quantitative disagreement with the data
• Kondo-like traps model:
• significant source of noise
• agreement with most features observed in the experiments
What are the sources of noise?
Electromagnetic fluctuations
of the circuit (gaussian)
Discrete noise due to
fluctuating background charges (BC)
trapped in the substrate or in the
junction
There are several experiments in different frequency regimes but
the dominant source of noise is yet to be identified!
Experimental picture of
the noise power spectrum
S  

1 
?
T

Origin of both types of noise are
Zimmerli et al. 1992
the same ?
Visscher et al. 1995
Zorin et al. 1996 Kenyon et al. 2000
Nakamura et al. 2001
Astafiev et al. 2004 Wellstood et al. 2004
Low frequency noise ( 1/f )

S   

• T 2- Temperature dependence of the noise
• 1/f spectrum up to frequency ~ 100-1000 Hz. [ where is the upper cut-off ??? ]
The intensity is in the range of   10 3  10  4 e Hz
at f=10Hz
• some samples clearly produce a telegraph noise but 1/f spectrum
points to numerous charges participating in generating the noise.
• This noise dominates T2 and it is greatly reduced by echo technique.
high frequency noise ( f )
Theoretical analysis
Upper level: use a proper model to study decoherence.
“fluctuators model” and not spin boson model
Paladino, Faoro, Falci and Fazio (2002)
Galperin, Altshuler, Shantsev (2003)
Lower level: understanding which is the microscopic mechanism
of decoherence that originate the fluctuators
Faoro, Bergli, Altshuler and Galperin (2004)
Faoro and Ioffe (2005)
Quantum TLSs model
H TLS  E z  t x
P E ,t  

t
e 2 d 2
with
 10 3  10  4

p2
Ei  E j  3
r
p  ed
2

p
P  dE r 3 

10 20
 3
cm eV
Relaxations for TLSs
• interaction with low energy phonons T>100 mk
• Many TLSs interacts via dipole-dipole interactions:
H int  
i
j i
ˆpi ˆp j  3rˆij ˆpi rˆij ˆpi 
4 rij3
ˆpi   iz pi
2

p
The effective strength of the interactions is controlled by

and it is always very weak.
Dipole and qubit interaction
Each dipole induces a change in the
island potential or in the gate charge
-
+
+
+
Q
pE
V 
Q
i.e. Qg  CV 
barrier
L  3nm
substrate
L  300nm
Charge Noise Power Spectrum:
S q   

i t




dt

Q
t

Q
0
e
g
g



G     dt  zi t  zi 0 e i t
i
ˆpi   iz pi
pi  edi

Rotated basis:
 zi  cos i zi  sin i xi
cos θi 
Ei
Ei
Ei  Ei2  ti2
sinθi 
ti
Ei
p
L
Dephasing rates for the dipoles
The weak interaction
• causes a width in each TLS
• at low frequency some of the TLSs become classical
Effective electric field
eff
H int
  hi t cos i zi  sin i xi 
i
2
i c
h
  k cos j
pure dephasing:
j i
2
ij
1
cosh E j
2
p 2 
 T
  
  
kij 
pi p j  3rˆij pi rˆij p j 
4 rij3
  10 3 T
N.B: density of thermally activated TLSs enough (Continuum)
Relaxation rates for the dipoles
Fermi Golden Rule
 
1
i
j
E

 E j    2
2
i
2
k ij sin2 i sin2 j
But in presence of large disorder, some of TLSs:
Ei  E j  0
2
p 
 T
  
 i1  sin2 i 
2
These dipoles become classical and will be responsible for 1/f noise
S q   at high frequency

G      dt  zi t  zi 0 e i t   sin2 i
i

i

  2Ei 2   2
p  2 V  2
S q      2 2  e
   e L 
white!
p 2
  
T

In the barrier...
 3
V  10 A
7
E
The density of TLSs ~ 0.1 / K too low!
Strongly coupled TLS
H  H Q  H TLS  H I

4 EC 1  2n g 
2
z 
E 
ng 
1
3
ng 
2
2
EJ
 x   z  t x  e z d E
2
  edE2  t 2
Eopt2  Eopt1
d 2 EC2
 2
L Eopt1
d  0.1nm EC  130eV Eopt1  30GHz
Eopt2  Eopt1  2 eV
Astafiev et al. 2004
ng
In the substrate...
p  2 V  2
S q      2 2  e
   e L 
S q    10
 3
V  10 A
9
17
 10
18
e2
Hz
Astafiev et al. 2004
• Comparison with experiments :
2
S q   

 2  10 6  108 e 2
S q   

2
TC
TC  120 mK
 10
15
 10
17
e2
Hz
S q   at low frequency

G     dt  zi t  zi 0 e i t   cos 2 i
i

i
2 1i
 2   1i 
2
p  2 V  T 2
S q      2 2  e
   e L  
• it has a 1/f dependence for   
  103 T
• it has only linear dependence on Temperature
• it has intensity in agreement with experimental data
What did we learn from the
dipole picture?
10 20
 3
cm eV
S q  

1
T  dependence
Number of thermally activated TLSs
f
nTLS 
T
 0V
W
W  1eV
T 2  dependence

N
TC
W
 0V  106  0V
T

2
nTLS
T 
  N
W 
Search for fluctuators of different nature ...
Andreev fluctuators model

Faoro, Bergli, Altshuler and Galperin (2004)
v
qubit
~


T
H I   v  c c  H T  z
T0
 

H E     c c H T  T0  c c   c c  

 

g    d                
0
g    2 2
T 2 dependence
• correlations are short range
• amplitude of oscillations increases with increasing 
  T
N
T
 0V  10  6  0V  1
W
T  20mK
Kondo-like traps model

H  H BCS  H d  H sd
U
H BCS    k ck ck   ck ck   h.c.
k
k
H d    d0 cdi  cdi  U  ndi ndi
i

i
H sd   Vki ck cdi  h.c.
i
k
ndi  cdi  cdi
i  2N 0 Vi
Kondo Temperature
   
U
1 
T  i
exp 
2i
 2i  U
i
K
0
d
0
d



 d0
2
Properties of the ground state
and the localized excited state
Weak coupling
Strong coupling
TK  
TK  
E s  Ed


doublet
TK*  0.3
d H s 0
TK


singlet
K  TK  TK* 
TK*  0.3
“Physics” of the Kondo-like traps
Density of states close
to the Fermi energy


dTK
 TK  TK* 
TK
0
 K    0  d K  K    0 
0

 0*
 TK
bare density
TK*  0.3  1010 Hz
W  1014 Hz
w
barrier
L
TK

0
d
weight of the Kondo resonance
Transition amplitude:
superconductor
Fast processes
A0  TK* 
2
Slow processes
A  A0 e
r


2
1
 Al 3 
A
t i  i    j t j
  

 TK* 
2


Superconductor coherence lenght
 10 4 TK* 
2
S q   at high frequency
• This noise is dominated by fast tunneling processes between traps
• effectively the motion of electrons between trap acts as resistor R
  
S    R coth

 2 
From the conductance G we calculate the resistance R
R 1  G  we 
1
2
2
  0V 
2


A
0
* 

T

K 
2
The noise power spectrum raises linearly with the frequency!
NB: Andreev fluctuators have the same but … 1 and 2 
S q   at low frequency
S q   

i t




dt

Q
t

Q
0
e
g
g


 i  Ai
d i
i
r

  we  
L   2   i2
i 
2
   min ,? but  max  A0  108 Hz
i
r
S q    we   
 L
2
• in the barrier :
 3
V  107 A
2
  0V  T 2


* 
  TK  
r  0 V T 

* 
L   TK 
  w 
 3
 0  10 A
3
2
estimates :   10 2  103 e
w  10  4
experimental value:   103  10 4 e
g  
Conclusions
• We have discussed a novel microscopic mechanism
(Kondo-like traps) that might be the dominant source of noise for dephasing
• But the “physics” of the device is complex : Kondo-like + TLSs
• TLSs are “killed” by the T-dependence!
• Our analysis cannot be done in greater details, due to the lack of
an analytical theory of kondo-like impurites with superconductor
• Try to measure 1/f noise after suppressing the superconductivity. We
expect reduction of 1/f noise
• Reasonable level of noise even only in the barrier.
• Different substrates
no changes in the intensity of the noise (NEC)
• relevant for phase qubit.