Pulse Methods for Preserving Quantum Coherences T. S. Mahesh
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Transcript of Pulse Methods for Preserving Quantum Coherences T. S. Mahesh
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
Contents1. 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
Contents1. 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
EnvironmentEnvironment
Hypothetical
Coherent Superposition
| = c0|0 + c1|1, with |c0|2 + |c1|2 = 1
An isolated 2-level quantum system
rs = || = c0c0*|0 0| + c1c1
*|1 1|+
c0c1*|0 1| + c1c0
*|1 0|
c0c0* c0c1
*
c1c0* c1c1
*
Density Matrix
Coherence
Population
=
Effect of environmentQuantum System – Environment interaction Evolution U(t)
|0|E |0|E0
|1|E |1|E1
U(t)
U(t)
||E = (c0|0 + c1|1)|E U(t)
c0|0|E0 + c1|1|E1
System Environment
System Environment
System Environment
Entangled
Decoherencer = ||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* E1|E0 c0c1
*
E0|E1 c1c0* c1c1
*
=Coherence
Population
|E1(t)|E0(t)| = eG(t)
Coherence decays irreversibly
Decoherence
Contents1. 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
Time Frequency
13-C signal of chloroformin liquid
Signal x
Signal Decay
IncoherenceDecoherence
DepolarizationAmplitude decay
Phasedecay
T1 process
T2 process
Relaxation
Signal Decay
IncoherenceDecoherence
DepolarizationAmplitude decay
Phasedecay
T1 process
T2 process
Relaxation
Incoherence
Individual (30 Hz, 31 Hz)
Net signal – faster decay
Time
Hahn-echo or Spin-echo (1950)
y
t t
+ d
d
y
Symmetric distribution of pulses removes incoherence
Signal
Echo
/2-x
Signal Decay
IncoherenceDecoherence
DepolarizationAmplitude decay
Phasedecay
T1 process
T2 process
Relaxation
2
2
1
10
10
1
0 00 00 0 0 0
0 0 0 0 0
x Tx x
y y yTeq
z z z zT
M M Md M M Mdt
M M M M
M
eqzM
T1
Time to reach equilibrium, (energy of spin-system is not conserved)
T2Lifetime of coherences, (energy of spin-system is conserved)
Bloch’s Phenomenological Equations (1940s)
Bloch’s Phenomenological Equations (1940s)
2
2
1
10
10
1
0 00 00 0 0 0
0 0 0 0 0
x Tx x
y y yTeq
z z z zT
M M Md M M Mdt
M M M M
M
eqzM
1
2
2
exp)0()(
exp)0()(
exp)0()(
TtMMMtM
TtMtM
TtMtM
eqzz
eqzz
yy
xx
Solutions in rotating frame:
eqzM
0
0
Signal Decay
IncoherenceDecoherence
DepolarizationAmplitude decay
Phasedecay
T1 process
T2 process
Relaxation
Effect of environment
r r’ = E(r)
= ∑ Ek r Ek†
k(operator-sum representation)
Amplitude damping (T1 process, dissipative)
E0 = p1/2 1 00 (1g1/2
E1 = p1/2 0 g1/2
0 0
E2 = (1 p)1/2 (1g1/2 0 0 1
E3 = (1 p)1/2 0 0
g1/2 0
r = p 00 1 p
Asymptotic state (t , g 1 :
g(t) is net damping : eg., g(t) = 1 et/T1
In NMR, p =
~ 0.5 + 104
1 1 + eE/kT
E(r) = ∑ Ek r Ek†
k
M(t) = 1 2exp( t/T1)
Amplitude damping (T1 process, dissipative)
Measurement of T1: Inversion Recovery
Equilibrium
Inversion
t
Signal Decay
IncoherenceDecoherence
DepolarizationAmplitude decay
Phasedecay
T1 process
T2 process
Relaxation
Phase damping (T2 process, non-dissipative)
E0 = 1 00 (1g1/2
r = a 00 1-a
Stationary state (t , g 1 :
g(t) is net damping : eg., g(t) = 1 et/T2
E1 = 0 00 g1/2
r(t) = a bb* 1-a
E(r) = ∑ Ek r Ek†
k
Bloch’s equation : dMx(t) Mx(t) dt T2
=
Solution : Mx(t) = Mx(0) exp( t/T2)
Transverse magnetization: Mx(t) Re{r01(t)}
Phase damping (T2 process, non-dissipative)
Spin-SpinRelaxation
Signal envelop: s(t) = exp( t/T2)
FWHH = /T2
Contents1. 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)
y
t
Signal
t
/2y
tt
y
tt
y
t
Shorter t is better (limited by duty-cycle of hardware)
H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954)
Meiboom-Gill (CPMG) sequence (1958)
x
t
Signal
t
/2y
tt
x
tt
x
t
Robust against errors in pulse !!!
S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958)
CPMG
t t
t t
t t
t t
Sampling points
Dynamical effects are minimized Dynamical decoupling
time1 2 3 4
j = T(2j-1) / (2N) Linear in j
Time
Signal
CPNopulse
HahnEcho
CPMG
S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958)
Dynamical Decoupling (DD)
Optimal distribution of pulses for a system with dephasing bath
T = total time of the sequence
N = total number of pulses
j = T sin2 ( j /(2N+1) )
PRL 98, 100504 (2007)Götz S. Uhrig
Uniformly distributed pulsesCPMG (1958):
Uhrig 2007 (UDD):
Carr Purcell Sequence
j = T(2j-1) / (2N) Linear in jWas believed to be optimal for N flips in duration T
1 2 3 4
5 6 70time
Carr & Purcell, Phys. Rev (1954) .Meiboom & Gill, Rev. Sci. Instru. (1958).
1
3
4
5
6
70time
Uhrig Sequence
2
j = T sin2 ( j /(2N+1) )
Uhrig, PRL (2007)
T
T
Proved to be optimal for N flips in duration T
Hahn-echo (1950)
CPMG (1958)
PDD (XY-4) (Viola et al, 1999)
UDD (Uhrig, 2007)
Dynamical Decoupling (DD)
CDDn = Cn = YCn−1XCn−1YCn−1XCn−1
C0 = t(Lidar et al, 2005)
Contents1. 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
ION-TRAP qubits
M. J. Biercuk et al, Nature 458, 996 (2009)
DD performance
Time (s) Time (s)
DD performanceElectron Spin Resonance(g-irradiated malonic acidsingle crystal)
J. Du et al, Nature461, 1265 (2009)
13C of Adamantane
Dieter et al, PRA 82, 042306 (2010)
Solid State NMR
DD performance
Dynamical Decoupling in Solids
D. Suter et al,PRL 106, 240501 (2011)
13C of Adamantane
Contents1. 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
Spin in acoherent
state
Randomlyfluctuating local fields
Sources of decoherence – dipole-dipole interaction
Spin loosescoherence
Randomlyfluctuating local fields
Sources of decoherence – dipole-dipole interaction
Source of Phase-damping – chemical shift anisotropy
B0
Redfield Theory: semi-classical
System - > Quantum, Lattice - > Classical
],[ rr Hidtd
Completely reversibleNo decoherence
System
System+Random field(coarse grain)
eqRHidtd
rrrr
,
Local field X(t)
time
G(t) = X(t) X*(t+t) = dx1 dx2 x1 x2 p(x1,t) p(x1,t | x2, t)
Auto-correlation function
Fluctuations have finite memory: G(t) = G(0) exp(|t|/ tc)
tc Correlation Time
Auto-correlation
Spectral density J() = G(t) exp(-it) dt = G(0) 2tc
1+ 2tc2
Spectral density
J()
tc = 1
G(0) 2tc
1+ 2tc2
J() =
rr
,)( XXJdtd
(after secular approximation)
Spectral density
J()
tc = 1
G(0) 2tc
1+ 2tc2
J() =
1T1
J(2) + J()
J(2) + J() + J(0)1T2
3 8
15 4
3 8
Dipolar Relaxation in Liquids
G = d2 J() 2
0
c0c0
* eGt c0c1*
eGt c1c0* c1c1
*
Effect of decoupling pulses L. Cywinski et al, PRB 77, 174509 (2008).M. J. Biercuk et al, Nature (London) 458, 996 (2009)
0
exp(-i H(t) dt ) Magnus expansion
Time-dependent Hamiltonian
Filter Functions
|x(t)|= e(t)
Cywiński, PRB 77, 174509 (2008)M. J. Biercuk et al, Nature (London) 458, 996 (2009)
F()
t
= F() d2 J() 2
0
F(t)
Fourier Transform of Pulse-train
J(t)
Modified Spectral density: J’() = J() F()
Residual area contributes to decoherence
Filter Functions
Cywiński, PRB 77, 174509 (2008)M. J. Biercuk et al, Nature (London) 458, 996 (2009)
= F() d2 J() 2
0
Contents1. 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
Wang et al, PRL 106, 040501 (2011)
Two-qubit DDElectron-nuclear entanglement(Phosphorous donors in Silicon)
No DD PDD
S. S. Roy & T. S. Mahesh, JMR, 2010
Fidelity = 0.995
Two-qubit DD – in NMR Levitt et al, PRL, 2004
|00
|11
|01 |10
Eigenbasis of Hz
90x , , , 90y , 12J
Hamiltonian: H = h1Iz1 + h2Iz
2+ hJ I1 I2
Hz HE
Eigenbasis of HE
|01−|102
|01+|102
|00 |11
5-chlorothiophene-2-carbonitrile
Two-qubit DD – in NMR
2 ms 2 ms
27sj = Nt sin2 ( j /(2N+1) )t = 4.027 ms
UDD-7 on 2-qubits
SingletFidelity
S. S. Roy, T. S. Mahesh, and G. S. Agarwal,Phys. Rev. A 83, 062326 (2011)
Entanglement
Product state
0110
01+10
0011
00+11
UDD-7 on 2-qubits
S. S. Roy, T. S. Mahesh, and G. S. Agarwal,Phys. Rev. A 83, 062326 (2011)
Dynamical Decoupling in Solids
CPMG
UDD
RUDD
Abhishek et al
Uhrig, 2011
Dynamical Decoupling in Solids1H of Hexamethylbenzene
Abhishek et al
DD on single-quantum coherences
Dynamical Decoupling in Solids
1H of Hexamethyl Benzene
Abhishek et al
No DD RUDD
2q 4q 6q 8q
Abhishek et al
Dynamical Decoupling in Solids
Contents1. 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 Alvarez and D. Suter,arXiv: 1106.3463 [quant-ph]
|x(t)|= e(t)
F(t)
(t) = F(t) d2 J(t) 2
0
Contents1. 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.