Loading in 2 Seconds...

Pulse Methods for Preserving Quantum Coherences T. S. Mahesh

Loading in 2 Seconds...

- By
**shina** - Follow User

- 60 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Pulse Methods for Preserving Quantum Coherences T. S. Mahesh' - shina

Download Now**An Image/Link below is provided (as is) to download presentation**

Download Now

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

Pulse Methods for Preserving Quantum Coherences

T. S. Mahesh

Indian Institute of Science Education and Research, Pune

Criteria for Physical Realization of QIP

- Scalable physical system with mapping of qubits
- A method to initialize the system
- Big decoherence time to gate time
- 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

- Coherence and decoherence
- Sources of signal decay
- Dynamical decoupling (DD)
- Performance of DD in practice
- Understanding DD
- DD on two-qubits and many qubits
- Noise spectroscopy
- Summary

- Coherence and decoherence
- Sources of signal decay
- Dynamical decoupling (DD)
- Performance of DD in practice
- Understanding DD
- DD on two-qubits and many qubits
- Noise spectroscopy
- Summary

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

Quantum System – Environment interaction Evolution U(t)

System Environment

System Environment

||E = (c0|0 + c1|1)|E

c0|0|E0 + c1|1|E1

Entangled

U(t)

|0|E |0|E0

|1|E |1|E1

U(t)

U(t)

System Environment

- = ||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

Coherence decays irreversibly

|E1(t)|E0(t)| = eG(t)

Decoherence

- Coherence and decoherence
- Sources of signal decay
- Dynamical decoupling (DD)
- Performance of DD in practice
- Understanding DD
- DD on two-qubits and many qubits
- Noise spectroscopy
- Summary

Decoherence

Incoherence

Amplitude

decay

Phase

decay

Depolarization

T1

process

T2

process

Relaxation

Decoherence

Incoherence

Amplitude

decay

Phase

decay

Depolarization

T1

process

T2

process

Relaxation

Echo

Signal

y

/2-x

+ d

d

y

Symmetric distribution of pulses removes incoherence

Decoherence

Incoherence

Amplitude

decay

Phase

decay

Depolarization

T1

process

T2

process

Relaxation

Time to reach equilibrium, (energy of spin-system is not conserved)

T1

Lifetime of coherences, (energy of spin-system is conserved)

T2

Bloch’s Phenomenological Equations (1940s)

Decoherence

Incoherence

Amplitude

decay

Phase

decay

Depolarization

T1

process

T2

process

Relaxation

Amplitude damping (T1 process, dissipative)

g(t) is net damping : eg., g(t) = 1 et/T1

1 0

0 (1g)1/2

p 0

0 1 p

E0 = p1/2

=

0g1/2

0 0

0 0

g1/20

E3 = (1 p)1/2

E1 = p1/2

(1g)1/20

0 1

E2 = (1 p)1/2

E()= ∑ Ek Ek†

k

Asymptotic state (t , g 1) :

In NMR,

p =

~ 0.5 + 104

1

1 + eE/kT

Amplitude damping (T1 process, dissipative)

Measurement of T1: Inversion Recovery

Equilibrium

Inversion

M() = 1 2exp( /T1)

Incoherence

Decoherence

Amplitude

decay

Phase

decay

Depolarization

T1

process

T2

process

Relaxation

Phase damping (T2 process, non-dissipative)

g(t) is net damping : eg., g(t) = 1 et/T2

1 0

0 (1g)1/2

0 0

0 g1/2

a 0

0 1-a

a b

b* 1-a

E0 =

E1 =

=

(t) =

E()= ∑ Ek Ek†

k

Stationary state (t , g 1) :

Phase damping (T2 process, non-dissipative)

Spin-Spin

Relaxation

dMx(t)Mx(t)

dt T2

=

Signal envelop: s(t) = exp( t/T2)

Transverse magnetization: Mx(t) Re{01(t)}

FWHH = /T2

Bloch’s equation :

Solution : Mx(t) = Mx(0) exp( t/T2)

- Coherence and decoherence
- Sources of signal decay
- Dynamical decoupling (DD)
- Performance of DD in practice
- Understanding DD
- DD on two-qubits and many qubits
- Noise spectroscopy
- Summary

Carr-Purcell (CP) sequence (1954)

Signal

y

/2y

y

y

Shorter 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)

Signal

x

/2y

x

x

Robust against errors in pulse !!!

S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958)

Dynamical effects are minimized Dynamical decoupling

Sampling points

1

2

3

4

time

j = T(2j-1) / (2N)

Linear in j

Time

Signal

CP

CPMG

No

pulse

Hahn

Echo

S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958)

Götz S. Uhrig

PRL 98, 100504 (2007)

CPMG (1958):

Uniformly distributed pulses

Uhrig 2007

(UDD):

Optimal distribution of p pulses for

a system with dephasing bath

j = T sin2 ( j /(2N+1) )

T = total time of the sequence

N = total number of pulses

Carr & Purcell, Phys. Rev (1954) .

Meiboom & Gill, Rev. Sci. Instru. (1958).

Carr Purcell Sequence

time

0

T

Was believed to be optimal for N flips in duration T

1

2

3

4

5

6

7

j = T(2j-1) / (2N)

Linear in j

Uhrig Sequence

Uhrig, PRL (2007)

time

2

1

3

4

5

6

7

0

T

Proved to be optimal for N flips in duration T

j = T sin2 ( j /(2N+1) )

Hahn-echo (1950)

CPMG (1958)

PDD (XY-4)

(Viola et al, 1999)

UDD

(Uhrig, 2007)

CDDn = Cn = YCn−1XCn−1YCn−1XCn−1

C0 =

(Lidar et al, 2005)

- Coherence and decoherence
- Sources of signal decay
- Dynamical decoupling (DD)
- Performance of DD in practice
- Understanding DD
- DD on two-qubits and many qubits
- Noise spectroscopy
- Summary

Electron Spin Resonance

(-irradiated malonic acid

single crystal)

J. Du et al, Nature

461, 1265 (2009)

Time (s)

Time (s)

- Coherence and decoherence
- Sources of signal decay
- Dynamical decoupling (DD)
- Performance of DD in practice
- Understanding DD
- DD on two-qubits and many qubits
- Noise spectroscopy
- 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

Redfield Theory: semi-classical

System - > Quantum, Lattice - > Classical

System

Completely reversible

No decoherence

System+

Random field

(coarse grain)

Local field X(t)

time

Auto-correlation

function

G() = X(t) X*(t+) = dx1 dx2 x1 x2 p(x1,t) p(x1,t | x2, )

Fluctuations have finite memory: G() = G(0) exp(||/ c)

c Correlation Time

2c

1+ 2c2

Spectral density J() = G() exp(-i) d = G(0)

2c

1+ 2c2

J() =

G(0)

J()

c = 1

c0c0*eGtc0c1*

eGtc1c0* c1c1*

2 J()

2

= d

3

8

15

4

3

8

1

T2

1

T1

J(2) + J() + J(0)

J(2) + J()

0

Dipolar Relaxation in Liquids

L. Cywinski et al, PRB 77, 174509 (2008).

M. J. Biercuk et al, Nature (London) 458, 996 (2009)

0

Time-dependent Hamiltonian

exp(-iH(t) dt)

Magnus expansion

Cywiński, PRB 77, 174509 (2008)

M. J. Biercuk et al, Nature (London) 458, 996 (2009)

|x()|= e()

Fourier Transform of Pulse-train

F()

F()

2 J()

2

= F() d

0

J()

Modified Spectral density:

J’() = J() F()

2 J()

2

= F() d

0

Residual area contributes

to decoherence

Cywiński, PRB 77, 174509 (2008)

M. J. Biercuk et al, Nature (London) 458, 996 (2009)

- Coherence and decoherence
- Sources of signal decay
- Dynamical decoupling (DD)
- Performance of DD in practice
- Understanding DD
- DD on two-qubits and many qubits
- Noise spectroscopy
- Summary

Electron-nuclear entanglement

(Phosphorous donors in Silicon)

No DD

PDD

Wang et al,

PRL 106, 040501 (2011)

Levitt et al, PRL, 2004

Hamiltonian: H =h1Iz1 + h2Iz2+ hJ I1 I2

S. S. Roy & T. S. Mahesh,

JMR, 2010

Hz

HE

|01+|10

2

Eigenbasis of HE

Eigenbasis of Hz

|01−|10

|11

2

|11

90x , , , 90y ,

|10

|01

|00

Fidelity = 0.995

|00

1

2J

27s

j = N sin2 ( j /(2N+1) )

= 4.027 ms

2 ms

2 ms

5-chlorothiophene-2-carbonitrile

Singlet

Fidelity

S. S. Roy, T. S. Mahesh, and G. S. Agarwal,

Phys. Rev. A 83, 062326 (2011)

Product state

0110

01+10

Entanglement

0011

00+11

S. S. Roy, T. S. Mahesh, and G. S. Agarwal,

Phys. Rev. A 83, 062326 (2011)

- Coherence and decoherence
- Sources of signal decay
- Dynamical decoupling (DD)
- Performance of DD in practice
- Understanding DD
- DD on two-qubits and many qubits
- Noise spectroscopy
- Summary

Alvarez and D. Suter,

arXiv: 1106.3463 [quant-ph]

|x()|= e()

F()

2 J()

2

() = F() d

0

- Coherence and decoherence
- Sources of signal decay
- Dynamical decoupling (DD)
- Performance of DD in practice
- Understanding DD
- DD on two-qubits and many qubits
- Noise spectroscopy
- Summary

Dynamical decoupling can greatly enhance the coherence times,

some times by orders of magnitude

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.

Download Presentation

Connecting to Server..