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





5. Understanding DD



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





5. Understanding DD



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



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



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



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



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





5. Understanding DD



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)





5. Understanding DD



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





5. Understanding DD



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





5. Understanding DD



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)


CPMG

UDD

RUDD

Abhishek et al

Uhrig, 2011

Dynamical Decoupling in Solids1H of Hexamethylbenzene

Abhishek et al

DD on single-quantum coherences


1H of Hexamethyl Benzene

Abhishek et al

No DD RUDD

2q 4q 6q 8q

Abhishek et al






5. Understanding DD



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





5. Understanding DD



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.

Documents

Pulse Methods for Preserving Quantum Coherences T. S. Mahesh