NMR investigations of Leggett- Garg Inequality

NMR investigations of Leggett-Garg Inequality

V. Athalye2, H. Katiyar1, Soumya S. Roy1, Abhishek Shukla1, R. Koteswara Rao3

T. S. Mahesh1

1IISER-Pune, 2Cummins College, Pune,

3IISc, Bangalore

Acknowledgements:

Usha Devi1, K. Rajagopal2, Anil Kumar3, and G. C. Knee4

1 Bangalore University,2 HRI & Inspire Inst., Virginia, USA,

3 IISc, Bangalore4 University of Oxford

Plan

• NMR as a quantum testbed

• Correlation Leggett-Garg Inequality

• Entropic Leggett-Garg Inequality

• Summary

Athalye, Roy, TSM, PRL 2011.

Hemant, Abhishek, Koteswar, TSM,arXiv: 1210.1970 [quant-ph]

ħgB0

Many nuclei have ‘spin angular momentum’ and ‘magnetic moment’

Coherent Superposition

a|0 + b |1 |0 |1

B0

Nuclear Spins

Spectrometer Sample:1015 spins

RF coilPulse/Detect

Superconductingcoil

H0

H1cos(wt)

~

Nuclear Magnetic Resonance (NMR)

1015 spins

Pseudopure State

p1

p0

= 1

~ 105 at 300 K, 12 T

E

kT =

B0 |0

|1 p1

p0

1015 spins

Pseudopure State

p1

p0

= 1

~ 105 at 300 K, 12 T

E

kT =

B0 |0

|1 p1

p0

pseudopure

=(1- )1/2+|00|

1015 spins

Pseudopure State

p1

p0

= 1

~ 105 at 300 K, 12 T

E

kT =

B0 |0

|1 p1

p0

RF

pseudopure

=(1- )1/2+|++| =(1- )1/2+|00|

2-qubit register

=(1- )1/2+|00|

> 1/3UW

NonseparableState

Resources

SeparableState

1/3UW

• parahydrogens (Jones &Anwar, PRA 2004)

• q-transducer (Cory et al, PRA 2007)

Resource:Entanglement

Resource:Discord(in units of 2)

Hemant, Roy, TSM, A. Patel, PRA2012

• pseudopure states

• Cory 1997• Chuang 1997

• ~ pure states

7-qubit NMR register

NMR systems useful?Pseudopure|0000000

Preparation

(scalability?)

Shor’salgorithm

15 = 3 x 5 Chuang, Nature 2002

No entanglementfinite discord

Open question:

Is discord sufficient resource for quantum computation ?

NMR system as a quantum testbed• Geometric Phases (Suter, 1988)

• Electromagnetically Induced Transparency (Murali, 2004)

• Contextuality (Laflamme, 2010)

• Delayed choice (Roy, 2012)

• Born’s rule (Laflamme, 2012)

•

•

•

Why NMR?

• Long life-times of quantum coherence

• Unmatched control on spin dynamics

Correlation LGI(CLGI)

Macrorealism“A macroscopic object, which has available to it two or more macroscopically distinct states, is at any given time in a definite one of those states.”

Non-invasive measurability“It is possible in principle to determine which of these states the system is in without any effect on the state itself or on the subsequent system dynamics.”

A. J. Leggett and A. Garg, PRL 54, 857 (1985)

Leggett-Garg (1985) Sir Anthony James LeggettUni. of Illinois at UC

Prof. Anupam GargNorthwestern University, Chicago

How to distinguishQuantum behaviorFrom Classical ?

• N. Lambert et al, PRB 2001

• J.-S. Xu et al., Sci. Rep 2011

• Palacios-Laloy et al., Nature Phys. 2010

• M. E. Goggin et al., PNAS USA 2011

• J. Dressel et al., PRL 2011

• M. Souza et al, NJP 2011

• Roy et al, PRL 2011

• G. C. Knee et al., Nat. Commun. 2012

• C. Emary et al, PRB 2012

• Y. Suzuki et al, NJP 2012

• Hemant et al, arXiv 2012

LGI studies in various systems

Consider a system with a dynamic dichotomic observable Q(t)

Dichotomic : Q(t) = 1 at any given time

timeQ1 Q2 Q3

t2 t3 . . .

. . .

Leggett-Garg (1985)

A. J. Leggett and A. Garg, PRL 54, 857 (1985)

PhD Thesis, Johannes Kofler, 2004

t1

timeQ1

t = 0

Q2 Q3

t . . .

. . .

2t

Two-Time Correlation Coefficient (TTCC)

EnsembleTime ensemble (sequential)

Spatial ensemble (parallel)

Temporal correlation: Cij = Qi Qj = Qi(r)

Qj(r)N

1

r = 1

N

1 Cij 1 Cij = 1 Perfectly correlated

Cij =1 Perfectly anti-correlated

Cij = 0 No correlation

= pij+(+1) + pij

(1)

r over an ensemble

LG string with 3 measurements

K3 = C12 + C23 C13

K3 = Q1Q2 + Q2Q3 Q1Q3

3 K3 1

Leggett-Garg Inequality (LGI)

K3

time

Macrorealism(classical)

timeQ1

t = 0

Q2 Q3

t 2t

Q1 Q2 Q3 Q1Q2+Q2Q3-Q1Q3

1 1 1 11 1 -1 11 -1 1 -31 -1 -1 1-1 1 1 1-1 1 -1 -3-1 -1 1 1-1 -1 -1 1

TTCC of a spin ½ particle

TimeQ1

t = 0

Q2 Q3

t 2t

Consider :A spin ½ particle

Hamiltonian : H = ½ wz

Maximally mixed initial State : 0 = ½ 1 Dynamic observable: x eigenvalues 1 (Dichotomic )

C12 = x(0)x(t) = x e-iHt x eiHt

= x [xcos(wt) + ysin(wt)]

C12 = cos(wt)

Similarly, C23 = cos(wt)

and C13 = cos(2wt)PhD Thesis, Johannes Kofler, 2004

Quantum States Violate LGI: K3 with Spin ½

timeQ1

t = 0

Q2 Q3

t 2t

K3 = C12 + C23 C13 = 2cos(wt) cos(2wt)

K3

wt2 3

Macrorealism(classical)

Quantum !!

40

No violation !

(/3,1.5)

Maxima (1.5) @cos(wt) =1/2

K4 = C12 + C23 + C34 C14 = 3cos(wt) cos(3wt)

Quantum States Violate LGI: K4 with Spin ½

Extrema (22) @cos(2wt) =0

K4 Macrorealism(classical)

Quantum !!

wt2 3 40

(/4,22)

(3/4,22)

time

Q1

t = 0

Q2 Q3

t 2t 3t

Q4

Evaluating K3

K3 = C12 + C23 C13

t = 0 t 2t

x

↗

x

↗

x

↗

x

↗

x

↗

x

↗

time

ENSEMBLE x(0)x(t) = C12

x(t)x(2t) = C23

x(0)x(2t) = C13

ENSEMBLE

ENSEMBLE

0


0

0

Evaluating K4

K4 = C12 + C23 + C34 C14

t = 0 t 2t

x

↗

x

↗

x

↗

x

↗

x

↗

time

x

↗x

↗

x

↗

3t

ENSEMBLE x(0)x(t) = C12

x(t)x(2t) = C23

x(0)x(3t) = C14

x(2t)x(3t) = C34

Joint Expectation Value

ENSEMBLE

ENSEMBLE

ENSEMBLE


0

0

0

0

Moussa Protocol

O. Moussa et al, PRL,104, 160501 (2010)

Target qubit (T)

Probe qubit (P)

A B

x

↗|+

AB

Joint Expectation Value

A↗

B↗

ABTarget qubit (T)

Dichotomicobservables

Sample13CHCl3

(in DMSO)

Target: 13C Probe: 1H

Resonance Offset: 100 Hz 0 Hz

T1 (IR) 5.5 s 4.1 s

T2 (CPMG) 0.8 s 4.0 s

V. Athalye, S. S. Roy, and TSM, Phys. Rev. Lett. 107, 130402 (2011).

Experiment – pulse sequence

1H

13C

= Ax Aref

Ax(t)+i Ay(t)

Ax(t) = cos(2tij) Ay(t) = sin(2tij)

Ax(t) x(t)

=

0


1/2

90x

PFG

wt

Experiment – Evaluating K3

timeQ1

t = 0

Q2 Q3

t 2t

K3 = C12 + C23 C13

= 2cos(wt) cos(2wt)

(w = 2100)

Error estimate: 0.05



50 100 150 200 250 300 t (ms)

LGI violated !!(Quantum)

LGI satisfied

Decay constant of K3 = 288 ms

165 ms

V. Athalye, S. S. Roy, and TSM,Phys. Rev. Lett. 107, 130402 (2011).

wt


(w = 2100)

Error estimate: 0.05

K4 = C12 + C23 + C34 C14

= 3cos(wt) cos(3wt)

time

Q1

t = 0

Q2 Q3

t 2t 3t

Q4

Decay constant of K4 = 324 ms


Entropic LGI(ELGI)

A. R. Usha Devi, H. S. Karthik, Sudha, and A. K. Rajagopal,arXiv: 1208.4491 [quant-ph]

timeQ1 Q2 Q3

t2 t3 . . .

. . .

t1

System

A. R. Usha Devi et al,arXiv: 1208.4491 [quant-ph]

System state: 1/2

Dynamical observable : Sz(t) = Ut Sz Ut†

Time Evolution: Ut = exp(iwSxt)

Information Deficit:

timeQ1 Q2 Q3

t2 t3 . . .

. . .

t1

ELGI bound

A. R. Usha Devi et al,arXiv: 1208.4491 [quant-ph]

Extracting ProbabilitiesSingle-event:

timeQk

. . .

. . .

tk

For S = 1/2

P(0) = ½P(1) = ½

Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]

k

Extracting Probabilitiestime

Qj

. . .

. . .

tjti

QiTwo-time joint:

Invasivej


Qj

. . .

. . .

tjti

QiTwo-time joint:


Qj

. . .

. . .

tjti

QiTwo-time joint:

P(0,qj) P(1,qj)

Non-Invasive Measurement (NIM)

System

Two-time joint probability

CH

system

ancilla


Two-time joint probabilities

P(q1,q2) P(q1,q3)

time

Q1 Q2 Q3

t2 t3t1


Information Deficit

CNOT


Information Deficit

CNOT

AntiCNOT


Information Deficit

CNOT

AntiCNOT

NIM


Legitimate Grand Probability A. R. Usha Devi et al,arXiv: 1208.4491 [quant-ph]

Classical Probability Theory:

P’(q1,q2) = P(q1,q2,q3)q3

P’(q1,q3) = P(q1,q2,q3)q2

P’(q2,q3) = P(q1,q2,q3)q1

P(q1,q2)

P(q1,q3)

P(q2,q3)

Marginals Grand

time

Q1 Q2 Q3

t2 t3t1

Extracting Grand Probability

Three-time joint:


Illegitimate Joint Probability

P(q1,q2,q3)is illegitimate !!

Violation ofEntropic LGI


Summary• NMR spin-system violated correlation LGI for short time scales

indicating the quantumness of the system.

• The gradual decoherence lead to the ultimate satisfaction of

correlation LGI.

• NMR spins systems also violated entropic LGI in the expected

time interval

• The experimental grand probability P(q1,q2,q3) could not generate

the experimental marginal probability P(q1,q3) supporting the

theoretical prediction.

Thank You !!

NMR investigations of Leggett- Garg Inequality

Documents

Transcript of NMR investigations of Leggett- Garg Inequality