Transcript Slide 1

Pulse Methods for Preserving
Quantum Coherences
T. S. Mahesh
Indian Institute of Science Education and Research, Pune
Criteria for Physical Realization of QIP
1. Scalable physical system with mapping of qubits
2. A method to initialize the system
3. Big decoherence time to gate time
4. Sufficient control of the system via time-dependent Hamiltonians
(availability of universal set of gates).
5. Efficient measurement of qubits
DiVincenzo, Phys. Rev. A 1998
Contents
1. Coherence and decoherence
2. Sources of signal decay
3. Dynamical decoupling (DD)
4. Performance of DD in practice
5. Understanding DD
6. DD on two-qubits and many qubits
7. Noise spectroscopy
8. Summary
Contents
1. Coherence and decoherence
2. Sources of signal decay
3. Dynamical decoupling (DD)
4. Performance of DD in practice
5. Understanding DD
6. DD on two-qubits and many qubits
7. Noise spectroscopy
8. Summary
Closed and Open Quantum System
Environment
Hypothetical
Environment
Coherent Superposition
An isolated 2-level quantum system
| = c0|0 + c1|1,
with |c0|2 + |c1|2 = 1
Density Matrix
rs = || = c0c0*|0 0| + c1c1*|1 1|+
c0c1*|0 1| + c1c0*|1 0|
=
c0c0*
c0c1*
c1c0*
c1c1*
Coherence
Population
Effect of environment
Quantum System – Environment interaction
Evolution U(t)
|0|E 
U(t)
|0|E0
System  Environment
|1|E 
U(t)
|1|E1
System  Environment
||E  = (c0|0 + c1|1)|E 
U(t)
c0|0|E0 + c1|1|E1
System  Environment
Entangled
Decoherence
r = ||E |E|
= c0c0*|0 0||E0 E0| + c1c1*|1 1||E1 E1| +
c0c1*|0 1||E0 E1| + c1c0*|1 0||E1 E0|
rs = TraceE[r] = c0c0*|0 0| + c1c1*|1 1|+
E1|E0 c0c1*|0 1| + E0|E1 c1c0*|1 0|
=
c0c0*
E0|E1 c1c0*
E1|E0 c0c1*
c1c1*
Coherence
Population
Coherence decays irreversibly
|E1(t)|E0(t)| = eG(t)
Decoherence
Contents
1. Coherence and decoherence
2. Sources of signal decay
3. Dynamical decoupling (DD)
4. Performance of DD in practice
5. Understanding DD
6. DD on two-qubits and many qubits
7. Noise spectroscopy
8. Summary
Signal Decay
13-C signal of chloroform
in liquid
Signal
 x
Time
Frequency
Signal Decay
Decoherence
Amplitude
decay
T1
process
Phase
decay
T2
process
Relaxation
Incoherence
Depolarization
Signal Decay
Decoherence
Amplitude
decay
T1
process
Phase
decay
T2
process
Relaxation
Incoherence
Depolarization
Incoherence
Individual (30 Hz, 31 Hz)
Net signal – faster decay
Time
Hahn-echo or Spin-echo (1950)
Echo
Signal
y
/2-x
t
t
+ d
 d
y
Symmetric distribution of  pulses removes incoherence
Signal Decay
Decoherence
Amplitude
decay
T1
process
Phase
decay
T2
process
Relaxation
Incoherence
Depolarization
Bloch’s Phenomenological Equations (1940s)
1
 Mx   0  0 0  M x   T2
d  

 
My   0 0 0  M y    0
dt   
 
0 0 
 Mz   0
 M z   0
0
1
T2
0
0  M x



0  M y 

eq 
1  M  M
z
z


T1 
M zeq
T1
T2
Time to reach equilibrium,
(energy of spin-system is not
conserved)
Lifetime of coherences,
(energy of spin-system is
conserved)

M
Bloch’s Phenomenological Equations (1940s)
1
 Mx   0  0 0  M x   T2
d  

 
My   0 0 0  M y    0
dt   
 
0 0 
 Mz   0
 M z   0
0
1
T2
0
Solutions in rotating frame:
M zeq
 t 
M x (t )  M x (0) exp  
 T2 
0
 t 
M y (t )  M y (0) exp  
 T2 
0

M z (t )  M z  M z (0)  M z
eq
0  M x



0  M y 

eq 
1  M  M
z
z


T1 

M
eq
exp  Tt 

1

Mz
eq
Signal Decay
Decoherence
Amplitude
decay
T1
process
Phase
decay
T2
process
Relaxation
Incoherence
Depolarization
Effect of environment
r
r’ = E(r)
= ∑ Ek r Ek†
k
(operator-sum representation)
Amplitude damping (T1 process, dissipative)
g(t) is net damping : eg., g(t) = 1  et/T1
E0 =
p1/2
1
0
E1 = p1/2 0
0
0
(1g1/2
1/2
E2 = (1  p)1/2 (1g
0
0
1
E3 = (1  p)1/2 0
g1/2
E(r) = ∑ Ek r Ek†
k
Asymptotic state (t , g  1 :
p
r =
0
0
1p
In NMR,
p=
1
1 + eE/kT
~ 0.5 + 104
g1/2
0
0
0
Amplitude damping (T1 process, dissipative)
Measurement of T1: Inversion Recovery
Equilibrium
t
Inversion
M(t) = 1 2exp( t/T1)
Signal Decay
Decoherence
Amplitude
decay
T1
process
Phase
decay
T2
process
Relaxation
Incoherence
Depolarization
Phase damping (T2 process, non-dissipative)
g(t) is net damping : eg., g(t) = 1  et/T2
E0 = 1
0
0
(1g1/2
E(r) = ∑ Ek r Ek†
E1 = 0
0
r(t) =
k
Stationary state (t , g  1 :
r =
0
g1/2
a
b
b* 1-a
a
0
0
1-a
Phase damping (T2 process, non-dissipative)
Transverse magnetization: Mx(t)  Re{r01(t)}
Bloch’s equation :
dMx(t)
 Mx(t)
=
dt
T2
Spin-Spin
Relaxation
Solution : Mx(t) = Mx(0) exp( t/T2)
Signal envelop: s(t) = exp( t/T2)
FWHH = /T2
Contents
1. Coherence and decoherence
2. Sources of signal decay
3. Dynamical decoupling (DD)
4. Performance of DD in practice
5. Understanding DD
6. DD on two-qubits and many qubits
7. Noise spectroscopy
8. Summary
Carr-Purcell (CP) sequence (1954)
Signal
y
/2y
t
y
t
t
y
t
t
t
Shorter t is better (limited by duty-cycle of hardware)
H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954)
t
Meiboom-Gill (CPMG) sequence (1958)
Signal
x
/2y
t
x
t
t
x
t
t
t
Robust against errors in  pulse !!!
S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958)
t
CPMG
Dynamical effects are minimized

t

t
1
t
Dynamical decoupling

t
t
2

t
3
t
Sampling points
t
4
time
j = T(2j-1) / (2N)
Linear in j
Signal
No
pulse
Hahn
Echo
CP
CPMG
Time
S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958)
Dynamical Decoupling (DD)
CPMG (1958):
Uniformly distributed  pulses
Uhrig 2007
(UDD):
Optimal distribution of  pulses for
a system with dephasing bath
j = T sin2 ( j /(2N+1) )
T = total time of the sequence
N = total number of pulses
Götz S. Uhrig
PRL 98, 100504 (2007)
Carr & Purcell, Phys. Rev (1954) .
Meiboom & Gill, Rev. Sci. Instru. (1958).
Carr Purcell Sequence

0
1

2

3
j = T(2j-1) / (2N)
Uhrig Sequence



4
Linear in j

5

6

7
time
T
Was believed to be optimal for N  flips
in duration T
Uhrig, PRL (2007)





time
0
1
2
3
j = T sin2 ( j /(2N+1) )
4
5
6
7
T
Proved to be optimal for N  flips in
duration T
Dynamical Decoupling (DD)
Hahn-echo (1950)
CPMG (1958)
PDD (XY-4)
(Viola et al, 1999)
CDDn = Cn = YCn−1XCn−1YCn−1XCn−1
C0 = t
(Lidar et al, 2005)
UDD
(Uhrig, 2007)
Contents
1. Coherence and decoherence
2. Sources of signal decay
3. Dynamical decoupling (DD)
4. Performance of DD in practice
5. Understanding DD
6. DD on two-qubits and many qubits
7. Noise spectroscopy
8. Summary
DD performance
ION-TRAP qubits
M. J. Biercuk et al,
Nature 458, 996 (2009)
DD performance
Electron Spin Resonance
(g-irradiated malonic acid
single crystal)
J. Du et al, Nature
461, 1265 (2009)
Time (s)
Time (s)
DD performance
Solid State NMR
13C of Adamantane
Dieter et al,
PRA 82, 042306 (2010)
Dynamical Decoupling in Solids
13C of Adamantane
D. Suter et al,
PRL 106, 240501 (2011)
Contents
1. Coherence and decoherence
2. Sources of signal decay
3. Dynamical decoupling (DD)
4. Performance of DD in practice
5. Understanding DD
6. DD on two-qubits and many qubits
7. Noise spectroscopy
8. Summary
Sources of decoherence – dipole-dipole interaction
Randomly
fluctuating
local fields
Spin in a
coherent
state
Sources of decoherence – dipole-dipole interaction
Randomly
fluctuating
local fields
Spin looses
coherence
Source of Phase-damping – chemical shift anisotropy
B0
Redfield Theory: semi-classical
System - > Quantum, Lattice - > Classical
System
System+
Random field
(coarse grain)
dr
 i[ H , r ]
dt
Completely reversible
No decoherence
d r
 iH , r   R r  r eq
dt


Auto-correlation
Local field X(t)
time
Auto-correlation
function
G(t) =  X(t) X*(t+t)  =
 dx1 dx2 x1 x2 p(x1,t) p(x1,t | x2, t)
Fluctuations have finite memory: G(t) = G(0) exp(|t|/ tc)
tc  Correlation Time
Spectral density J() =  G(t) exp(-it) dt = G(0)
2tc
1+ 2tc2
Spectral density
J() = G(0)
2tc
1+ 2tc2
J()

d r
 J ( )X X , r
dt
tc = 1

(after secular approximation)
Spectral density
J() = G(0)
2tc
1+ 2tc2
J()

c0c0*
eGt c0c1*
eGt c1c0*
c1c1*
Dipolar Relaxation in Liquids
1  J(2) + J()
T1
1  3 J(2) + 15 J() + 3 J(0)
T2
8
4
8
tc = 1
G= 2



0
J() d
2
Effect of decoupling pulses
L. Cywinski et al, PRB 77, 174509 (2008).
M. J. Biercuk et al, Nature (London) 458, 996 (2009)
Time-dependent Hamiltonian

exp(-i  H(t) dt )
0
Magnus expansion
Filter Functions
|x(t)|= e(t)
= 2



0
Cywiński, PRB 77, 174509 (2008)
M. J. Biercuk et al, Nature (London) 458, 996 (2009)
J() F() d
2
Fourier Transform of Pulse-train
F()
F(t)
t
Filter Functions
= 2



0
J() F() d
2
Modified Spectral density:
J’() = J() F()
Residual area contributes
to decoherence
Cywiński, PRB 77, 174509 (2008)
M. J. Biercuk et al, Nature (London) 458, 996 (2009)
J(t)
Contents
1. Coherence and decoherence
2. Sources of signal decay
3. Dynamical decoupling (DD)
4. Performance of DD in practice
5. Understanding DD
6. DD on two-qubits and many qubits
7. Noise spectroscopy
8. Summary
Two-qubit DD
Two-qubit DD
Electron-nuclear entanglement
(Phosphorous donors in Silicon)
Wang et al,
PRL 106, 040501 (2011)
No DD
PDD
Two-qubit DD – in NMR
Levitt et al, PRL, 2004
Hamiltonian: H = h1Iz1 + h2Iz2+ hJ I1
S. S. Roy & T. S. Mahesh,
JMR, 2010
Hz
I2
HE
Eigenbasis of HE
Eigenbasis of Hz
90x , 1 ,  , 90y , 
2J 

|11
|00 |01+|10 |11
2
|10
|01
|00
|01−|10
2
Fidelity = 0.995
Two-qubit DD – in NMR
27s
2 ms
2 ms
j = Nt sin2 ( j /(2N+1) )
t = 4.027 ms
5-chlorothiophene-2-carbonitrile
UDD-7 on 2-qubits
Singlet
Fidelity
S. S. Roy, T. S. Mahesh, and G. S. Agarwal,
Phys. Rev. A 83, 062326 (2011)
UDD-7 on 2-qubits
Product state
0110
01+10
Entanglement
0011
00+11
S. S. Roy, T. S. Mahesh, and G. S. Agarwal,
Phys. Rev. A 83, 062326 (2011)
Dynamical Decoupling in Solids
CPMG
UDD
RUDD
Uhrig, 2011
Abhishek et al
Dynamical Decoupling in Solids
DD on single-quantum coherences
1H of Hexamethylbenzene
Abhishek et al
Dynamical Decoupling in Solids
RUDD
No DD
1H of Hexamethyl Benzene
Abhishek et al
Dynamical Decoupling in Solids
2q
4q
6q
8q
Abhishek et al
Contents
1. Coherence and decoherence
2. Sources of signal decay
3. Dynamical decoupling (DD)
4. Performance of DD in practice
5. Understanding DD
6. DD on two-qubits and many qubits
7. Noise spectroscopy
8. Summary
Noise Spectroscopy
|x(t)|= e(t)
F(t)
Alvarez and D. Suter,
arXiv: 1106.3463 [quant-ph]
(t) = 2



0
J(t) F(t) d
2
Contents
1. Coherence and decoherence
2. Sources of signal decay
3. Dynamical decoupling (DD)
4. Performance of DD in practice
5. Understanding DD
6. DD on two-qubits and many qubits
7. Noise spectroscopy
8. Summary
Summary
1. Dynamical decoupling can greatly enhance the coherence times,
some times by orders of magnitude
2. Various types of pulsed DD sequences are available. Best DD depends
on the spectral density of the bath, the state to be preserved, robustness
to pulse errors, etc.
3. Filter-functions are useful tools to understand the performance of DD.
4. DD on large number of interacting qubits also shows improved performance.