quantum correlations in nuclear spin ensembles t. s. mahesh indian institute of science education...

Quantum Correlations in

Nuclear Spin Ensembles

T. S. Mahesh

Indian Institute of Science Education and Research, Pune

Quantum or Classical ?

How to distinguish quantum and classical behavior?

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

Consider a dynamic system with a dichotomic quantity 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

Consider: Q1Q2 + (Q2 Q1)Q3

If Q1 Q2 : 1 + 0 = 1

Q1 Q2 : 1 + (2) = 1 or 3

Q1Q2 + Q2Q3 Q1Q3 = 1 or 3

3 < Q1Q2 + Q2Q3 Q1Q3 < 1

TTCC of a spin ½ particle (a quantum coin)

TimeQ1

t = 0

Q2 Q3

t 2t

Consider :

A spin ½ particle precessing about z

Hamiltonian : H = ½ z

Initial State : highly mixed state : 0 = ½ 1 + x ( ~ 10-5)

Dichotomic observable: x eigenvalues 1

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

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

C12 = cos(t)

Similarly, C23 = cos(t)

and C13 = cos(2t)

Quantum States Violate LGI: K3 with Spin ½

timeQ1

t = 0

Q2 Q3

t 2t

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

K3

t2 3


Quantum !!

40

No violation !

(/3,1.5)

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

Consider: Q1(Q2 Q4) + Q3(Q2 + Q4)

If Q2 Q4 : 0 + (2) = 2

Q2 Q4 : (2) + 0 = 2

Q1Q2 + Q2Q3 + Q3Q4 Q1Q4 = 2

K4 = C12 + C23 + C34 C14 or,

K4 = Q1Q2 + Q2Q3 + Q3Q4 Q1Q4

time

Q1

t = 0

Q2 Q3

t 2t 3t

Q4

Macrorealism(classical)K4

time

LG string with 4 measurements

2 K4 2

Leggett-Garg Inequality (LGI)

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

Quantum States Violate LGI: K4 with Spin ½

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

K4 Macrorealism(classical)

Quantum !!

Quantum !!

t2 3 40

(/4,22)

(3/4,22)

time

Q1

t = 0

Q2 Q3

t 2t 3t

Q4

Even,M=2L: (Q1 + Q3)Q2 + (Q3+ Q5)Q4 + + (Q2L-3 + Q2L-1)Q2L-2+ (Q2L-1 Q1)Q2L

Max: all +1 2(L1)+0. M2

Min: odds +1, evens –1 2(L1)+0. M+2

Odd,M=2L+1: (Q1 + Q3)Q2 + (Q3+ Q5)Q4 + + (Q2L-3 + Q2L-1)Q2L-2+ (Q2L-1 +Q2L+1)Q2L Q1Q2L+1

Max: all +1 2L–1. M2

Min: odds +1, evens –1 2L1. M

KM = C12 + C23 + + CM-1,M C1,M or,

KM = Q1Q2 + Q2Q3 + + QM-1QM Q1QM

time

Q1

t = 0

Q2

t

QM

Mt

. . .

. . .

LG string with M measurements

M+2 KM (M2) if M is even,

M KM (M2) if M is odd.


M

KM

time

(M2)

KM = C12 + C23 + + CM-1,M C1,M = (M-1)cos(t) cos{(M-1)t)}

Quantum States Violate LGI: KM with Spin ½

Maximum: Mcos(/M) @ t = /M

Note that for large M:

Mcos(/M) M > M-2

\ Macrorealism is always violated !!

2 3 4

tM

KM


Quantum(M2)

time

Q1

t = 0

Q2

t

QM

Mt

. . .

. . .

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

Target qubit (T) A B

x

↗(1- )I/2+|++|

AB

Moussa Protocol

Target qubit (T)

Probe qubit (P)

A

x

↗|+ A

Dichotomic observable be, A = P P (projectors)

Let| be eigenvectors and 1 be eigenvalues of X

Then, X=|++|||, and X1 = p(+1) p(1).

Apply on the joint system: UA = |00|P1T + |11|P A

p(1) = ||1 = tr [ {UA {|++|} UA†} {||1}] = P

A = P+ P = p(+1) p(1) = X1

Target qubit (T)

Probe qubit (P)

A B

x

↗|+

AB

Extension:

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

Ensemble of ~1018 molecules

Experiment – pulse sequence

1H

13C

= Ax Aref

Ax(t)+i Ay(t)

Ax(t) = x(t) Aref = x(0)

=

0

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

t

Experiment – Evaluating K3

timeQ1

t = 0

Q2 Q3

t 2t

K3 = C12 + C23 C13

= 2cos(t) cos(2t)

( = 2100)

Error estimate: 0.05



50 100 150 200 250 300 t (ms)

LGI violated !!(Quantum)

LGI satisfied(Macrorealistic)

Decay constant of K3 = 288 ms

165 ms


t


( = 2100)

Error estimate: 0.05

K4 = C12 + C23 + C34 C14

= 3cos(t) cos(3t)

time

Q1

t = 0

Q2 Q3

t 2t 3t

Q4

Decay constant of K4 = 324 ms


time

Signal x

Quantum to Classical

13-C signal of chloroformin liquid

| = c0|0 + c1|1

|0

|1

|0

|1

|c0|2 c0c1*

c0*c1 |c1|2

rs =|c0|2 0

0 |c1|2

|c0|2 eG(t) c0c1*

eG(t) c0*c1 |c1|2

Quantum State Classical State

NMR implementation of a

Quantum Delayed-Choice Experiment

Soumya Singha Roy, Abhishek Shukla, and T. S. Mahesh

Indian Institute of Science Education and Research, (IISER) Pune

Wave nature of particles !!

C. Jönsson , Tübingen, Germany, 1961

• Not a wave of particles• Single particles interfere with themselves !!

Intensity so low thatonly one electron at a time

4000 clicks

C. Jönsson , Tübingen, Germany, 1961

Single Particle at a time

• Two-slit wave packet collapsing• Eventually builds up pattern• Particle interferes with itself !!

Single particle interference

• A classical particle would follow some single path• Can we say a quantum particle does, too?• Can we measure it going through one slit or another?

Which path ?

• Einstein proposed a few ways to measure which slit the particle went through without blocking it

• Each time, Bohr showed how that measurement would wash out the wave function

Movable wall;measure recoil

Source

Crystal with inelastic collision

Source

No:Movement of slit washes out pattern

No:Change in wavelength washes out pattern

Niels BohrAlbert Einstein

Which path ?

• Short answer: no, we can’t tell• Anything that blocks one slit washes out the

interference pattern

Which path ?

Bohr’s Complementarity principle (1933)

Niels Bohr

Wave and particle natures are complementary !!

Depending on the experimental setup one

obtains either wave nature or particle nature

– not both at a time

Mach-Zehnder Interferometer

Open Setup

Single photon

D0

D1

1

0

Only one detector clicks at a time !!

BS1


Open Setup

Single photon

D0

D1

1

0

Trajectory can be assigned

BS1

02

10

ie


Open Setup

Single photon

D0

D1

1

0

Trajectory can be assigned

BS1

12

10

ie


Open Setup

Single photon

D0

D1

1

0

Trajectory can be assigned : Particle nature !!

BS1


Open Setup

S0 or S1

Intensities are independent of i.e., no interference

2/1)1(......12

10

2/1)0(......02

10

pe

pe

i

i


Closed Setup

Single photon

D0

D1

1

0

Again only one detector clicks at a time !!

BS1

BS2


Single photon

D0

D1

1

0

Again only one detector clicks at a time !!

BS1

BS2

2

10 ie

)2/(sin)1(

)2/(cos)0(

2

11

2

10

2

1010

2

2

p

p

eee iii


Closed Setup

S0 or S1

Intensities are dependent of

Interference !!


Closed Setup

BS2 removes ‘which path’ information

Trajectory can not be assigned : Wave nature !!

Single photon

D0

D1

1

0BS1

BS2

Photon knows the setup ?

D0

D11

0

Open Setup

Closed Setup

D0

D11

0

BS2

BS1

BS1

Particle behavior

Wave behavior

Two schools of thought

Bohr, Pauli, Dirac, ….

• Intrinsic wave-particle duality

• Reality depends on observation

• Complementarity principle

Einstein, Bohm, ….

• Apparent wave-particle duality

• Reality is independent of observation

• Hidden variable theory

Delayed Choice ExperimentWheeler’s Gedanken Experiment (1978)

Delayed Choice BS2

Decision to place or not to place BS2

is made after photon has left BS1

D0

D1

1

0

BS2

BS1


Delayed Choice BS2

Complementarity principle :

Results do not change with

delayed choice

D0

D1

1

0

BS2

BS1

Hidden-variable theory :

Results should change with

the delayed choice

No longer Gedanken Experiment (2007)

No longer Gedanken Experiment (2007)

COMPLEMENTARITY SATISFIED

Bohr, Pauli, Dirac, ….

• Intrinsic wave-particle duality

• Reality depends on observation

• Complementarity principle


Complementarity principle :

Results do not change with

delayed choice

Hidden-variable theory :

Results should change with

the delayed choice

Einstein, Bohm, ….

• Apparent wave-particle duality

• Reality is independent of observation

• Hidden variable theory X

Quantum Delayed Choice Experiment

Superposition of present and absent !!

D0

D11

0

BS2

BS1


D0

D11

0

BS2

BS1

Open-setup

e-

D0

D11

0

BS2

BS1

Closed setup

e-


D0

D11

0

BS2

BS1

Open-setup

e- e-

D0

D11

0

BS2

BS1

Closed setup

e-

D0

D11

0

BS2

BS1

Quantum setup

Equivalent Quantum Circuits:

Open MZI

Closed MZI

Wheeler’s delayed choice

Quantumdelayed choice

Continuous Morphing b/w wave & particle

|00

a = 0 : Particle nature

a = /4 : Complete superposition

a = /2 : Wave nature


Interference

No Interference

Visibility :

Open and Closed MZI

Open and Closed MZI |p

|w

=

Phys. Rev. A, 2012

Open and Closed MZI |p

|w = 0.97

= 0.02

Phys. Rev. A, 2012


Phys. Rev. A, 2012

=


Phys. Rev. A, 2012

Quantum Delayed Choice Experiment“Depending on the state of 13C spin, 1H spin can simultaneously exist in a superposition of particle-like to wave-like states !! Time to re-interpret Bohr’s complementarity principle?

Phys. Rev. A, 2012

quantum correlations in nuclear spin ensembles t. s. mahesh indian institute of science education...

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