Download Leggett-Garg Inequality and Quantum Delayed Choice Experiments

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?
Leggett-Garg (1985)
Sir Anthony James Leggett
Uni. of Illinois at UC
Prof. Anupam Garg
Northwestern University, Chicago
A. J. Leggett and A. Garg,
PRL 54, 857 (1985)
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.”
Leggett-Garg (1985)
Consider a dynamic system with a dichotomic quantity Q(t)
Dichotomic : Q(t) =  1 at any given time
Q1
Q2
Q3
...
t1
t2
t3
...
time
A. J. Leggett and A. Garg, PRL 54, 857 (1985)
PhD Thesis, Johannes Kofler, 2004
Two-Time Correlation Coefficient (TTCC)
Q1
Q2
t=0
t
Temporal correlation: Cij =  Qi Q j  =
...
Q3
time
2t . . .
1
N
N
Q
i
(r) Q (r)
j
= pij+(+1) + pij(1)
r=1
r  over an ensemble
Ensemble
Time ensemble (sequential)
Spatial ensemble (parallel)
1  Cij  1
Cij = 1  Perfectly correlated
Cij =1  Perfectly anti-correlated
Cij = 0  No correlation
LG string with 3 measurements
K3 = C12 + C23  C13
Q1
Q2
Q3
t=0
t
2t
K3 = Q1Q2 + Q2Q3  Q1Q3
Consider: Q1Q2 + (Q2  Q1)Q3
If Q1  Q2 :
1 + 0
Q1  Q2 : 1 + (2)
= 1
= 1 or 3
 Q1Q2 + Q2Q3  Q1Q3 = 1 or 3
 3 < Q1Q2 + Q2Q3  Q1Q3 < 1
K3
Macrorealism
(classical)
3  K3  1
Leggett-Garg Inequality (LGI)
time
time
TTCC of a spin ½ particle (a quantum coin)
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
Q1
Q2
Q3
t=0
t
2t
Time
C12 = x(0)x(t) =  x e-iHt x eiHt 
= x [xcos(t) + ysin(t)] 
 C12 = cos(t)
Similarly, C23 = cos(t)
and C13 = cos(2t)
Quantum States Violate LGI: K3 with Spin ½
Q1
Q2
Q3
t=0
t
2t
time
K3 = C12 + C23  C13 = 2cos(t)  cos(2t)
(/3,1.5)
Maxima (1.5) @
cos(t) =1/2
Quantum !!
K3
Macrorealism
(classical)
No violation !
0

2
t
3
4
LG string with 4 measurements
Q1
Q2
Q3
Q4
t=0
t
2t
3t
time
K4 = C12 + C23 + C34  C14
or,
K4 = Q1Q2 + Q2Q3 + Q3Q4  Q1Q4
Consider: Q1(Q2  Q4) + Q3(Q2 + Q4)
If Q2  Q4 :
0 + (2)
Q2  Q4 : (2) +
K4
Macrorealism
(classical)
0
 Q1 Q2 + Q 2 Q3 + Q 3 Q4  Q1 Q4
2  K4  2
Leggett-Garg Inequality (LGI)
time
= 2
= 2
= 2
Quantum States Violate LGI: K4 with Spin ½
Q1
Q2
Q3
Q4
t=0
t
2t
3t
time
K4 = C12 + C23 + C34  C14 = 3cos(t)  cos(3t)
(/4,22)
Quantum !!
Extrema (22) @
cos(2t) =0
K4
Macrorealism
(classical)
Quantum !!
0

(3/4,22)
2
t
3
4
LG string with M measurements
Q1
Q2
...
QM
t=0
t
...
Mt
Even,M=2L:
time
KM = C12 + C23 +    + CM-1,M  C1,M
or,
KM = Q1Q2 + Q2Q3 +    + QM-1QM  Q1QM
(Q1 + Q3)Q2 + (Q3+ Q5)Q4 +    + (Q2L-3 + Q2L-1)Q2L-2+ (Q2L-1  Q1)Q2L
Max: all +1  2(L1)+0.  M2
Min: odds +1, evens –1  2(L1)+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.  M2
Macrorealism
(classical)
Min: odds +1, evens –1  2L1.  M
KM
M+2  KM  (M2)
M  KM  (M2)
if M is even,
if M is odd.
M
Quantum States Violate LGI: KM with Spin ½
Q1
Q2
...
QM
t=0
t
...
Mt
time
KM = C12 + C23 +    + CM-1,M  C1,M = (M-1)cos(t)  cos{(M-1)t)}
Maximum:
Mcos(/M) @ t = /M
Macrorealism
(classical)
Note that for large M:
Quantum
Mcos(/M) M > M-2
KM
 Macrorealism
is always violated !!
M

2
t
3
4
Evaluating K3
K3 = C12 + C23  C13
Hamiltonian : H = ½ z
ENSEMBLE
0
ENSEMBLE
0
ENSEMBLE
0
x
↗
x
↗
x
↗
x
↗
t=0
t
x(0)x(t)
= C12
x
↗
x(t)x(2t) = C23
x
↗
x(0)x(2t) = C13
2t
time
Evaluating K4
K4 = C12 + C23 + C34 C14
Joint Expectation Value
Hamiltonian : H = ½ z
ENSEMBLE
0
ENSEMBLE
0
ENSEMBLE
0
ENSEMBLE
0
x
↗
x
↗
x
↗
x(0)x(t)
x
↗
x
↗
x
↗
t=0
t
2t
= C12
x(t)x(2t) = C23
x
↗
x(2t)x(3t) = C34
x
↗
x(0)x(3t)
3t
time
= C14
Moussa Protocol
Joint Expectation Value
Dichotomic
observables
Target qubit (T)
A
↗

Probe qubit (P)
|+
Target qubit (T)

B
↗
A

A
x
↗
AB
x
↗
 AB
B
(1-  )I/2+|++|
Target qubit (T)
AB
B
O. Moussa et al, PRL,104, 160501 (2010)
Moussa Protocol
Probe qubit (P)
|+
Target qubit (T)

Dichotomic observable be, A = P  P
x
↗
A
(projectors)
Let| be eigenvectors and 1 be eigenvalues of X
Then, X=|++|||, and X1 = p(+1)  p(1).
Apply on the joint system: UA = |00|P1T + |11|P A
p(1) = ||1 = tr [ {UA {|++|} UA†} {||1}] = P
A = P+  P = p(+1)  p(1) = X1
Extension:
Probe qubit (P)
|+

Target qubit (T)

A
B
x
↗
AB   
A
Sample
13CHCl
3
(in DMSO)
Target:
Resonance Offset:
Ensemble of
~1018 molecules
13C
Probe: 1H
100 Hz
0 Hz
T1 (IR)
5.5 s
4.1 s
T2 (CPMG)
0.8 s
4.0 s
Experiment – pulse sequence
0
= Ax Aref
1H
Ax(t)+i Ay(t)
13C
=
V. Athalye, S. S. Roy, and T. S. Mahesh,
Phys. Rev. Lett. 107, 130402 (2011).
Ax(t) =  x(t)
Aref =  x(0)
Experiment – Evaluating K3
Q1
Q2
Q3
t=0
t
2t
time
K3 = C12 + C23  C13
= 2cos(t)  cos(2t)
Error estimate:  0.05
V. Athalye, S. S. Roy, and T. S. Mahesh,
Phys. Rev. Lett. 107, 130402 (2011).
t
( = 2100)
Experiment – Evaluating K3
50
100
t (ms)
150
200
LGI satisfied
(Macrorealistic)
LGI violated !!
(Quantum)
165 ms
Decay constant of K3 = 288 ms
V. Athalye, S. S. Roy, and T. S. Mahesh,
Phys. Rev. Lett. 107, 130402 (2011).
250
300
Experiment – Evaluating K4
Q1
Q2
Q3
Q4
t=0
t
2t
3t
time
K4 = C12 + C23 + C34  C14
= 3cos(t)  cos(3t)
Error estimate:  0.05
Decay constant of K4 = 324 ms
V. Athalye, S. S. Roy, and T. S. Mahesh,
Phys. Rev. Lett. 107, 130402 (2011).
( = 2100)
t
Quantum to Classical
13-C signal of chloroform
in liquid
Signal
 x
|1
|1
|0
time
| = c0|0 + c1|1
s =
|c0|2
c0c1*
|c0|2
c0*c1
|c1|2
eG(t) c0*c1
Quantum State
|0
eG(t) c0c1*
|c1|2
|c0|2
0
0
|c1|2
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
Single Particle at a time
Intensity so low that
only one electron at a time
• Not a wave of particles
4000 clicks
• Single particles interfere with themselves !!
C. Jönsson , Tübingen,
Germany, 1961
Single particle interference
• Two-slit wave packet collapsing
• Eventually builds up pattern
• Particle interferes with itself !!
Which path ?
• 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 ?
Movable wall;
measure recoil
Source
Source
No:
Movement of slit
washes out pattern
Albert Einstein
Crystal with
inelastic collision
No:
Change in wavelength
washes out pattern
• 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
Niels Bohr
Which path ?
• Short answer: no, we can’t tell
• Anything that blocks one slit washes out the
interference pattern
Bohr’s Complementarity principle
(1933)
 Wave and particle natures are complementary !!
 Depending on the experimental setup one
obtains either wave nature or particle nature
Niels Bohr
– not both at a time
Mach-Zehnder Interferometer
Open Setup
D0
1

D1
Single photon
BS1
0
Only one detector clicks at a time !!
Mach-Zehnder Interferometer
0  ei 1
2
Open Setup
 0
D0
1

D1
Single photon
BS1
0
Trajectory can be assigned
Mach-Zehnder Interferometer
Open Setup
D0
1

D1
0  ei 1
Single photon
2
BS1
0
Trajectory can be assigned
1
Mach-Zehnder Interferometer
Open Setup
D0
1

D1
Single photon
BS1
0
Trajectory can be assigned : Particle nature !!
Mach-Zehnder Interferometer
Open Setup
0  ei 1
2
0  ei 1
2
 0 ...... p(0)  1 / 2
 1 ...... p(1)  1 / 2
S0 or S1
Intensities are independent of 
i.e., no interference

Mach-Zehnder Interferometer
Closed Setup
D0
1

BS2
D1
Single photon
BS1
0
Again only one detector clicks at a time !!
Mach-Zehnder Interferometer
 1   ei  0  1 


1  ei 
1  ei 
 0
1
2
2
2
p (0)  cos2 ( / 2)
0
p (1)  sin 2 ( / 2)
1
D0
BS2

0  ei 1
D1
2
Single photon
BS1
0
Again only one detector clicks at a time !!
Mach-Zehnder Interferometer
Closed Setup
Intensities are dependent of 
Interference !!
S0 or S1

Mach-Zehnder Interferometer
Closed Setup
D0
1

BS2
D1
Single photon
BS1
0
BS2 removes ‘which path’ information
Trajectory can not be assigned : Wave nature !!
Photon knows the setup ?
D0
Open Setup

1
D1
Particle behavior
BS1
0
D0

1
Closed Setup
BS2
D1
Wave behavior
BS1
0
Two schools of thought
Bohr, Pauli, Dirac, ….
Einstein, Bohm, ….
• Intrinsic wave-particle duality
• Apparent wave-particle duality
• Reality depends on observation
• Reality is independent of observation
• Complementarity principle
• Hidden variable theory
Delayed Choice Experiment
Wheeler’s Gedanken Experiment (1978)
D0
1

BS2
D1
BS1
0
Delayed Choice BS2
Decision to place or not to place BS2
is made after photon has left BS1
Delayed Choice Experiment
Wheeler’s Gedanken Experiment (1978)
D0
1

BS2
D1
BS1
0
Delayed Choice BS2
Complementarity principle :
Hidden-variable theory :
Results do not change with
Results should change with
delayed choice
the delayed choice
No longer Gedanken Experiment (2007)
No longer Gedanken Experiment (2007)
COMPLEMENTARITY SATISFIED
Delayed Choice Experiment
Wheeler’s Gedanken Experiment (1978)
Bohr, Pauli, Dirac, ….
Einstein, Bohm, ….
• Intrinsic wave-particle duality
• Apparent wave-particle duality
• Reality depends on observation
• Reality is independent of observation
• Complementarity principle

• Hidden variable theory
Complementarity principle :
Hidden-variable theory :
Results do not change with
Results should change with
delayed choice
the delayed choice
X
Quantum Delayed Choice Experiment
D0
1

BS2
D1
BS1
0
Superposition of
present and absent !!
Quantum Delayed Choice Experiment
Open-setup
1
Closed setup
D0

1
BS2

D1
BS1
0
e-
BS2
D1
BS1
0
D0
e-
Quantum Delayed Choice Experiment
Open-setup
1
Quantum setup
Closed setup

BS1
BS2
D0
1

D1
BS1
0
BS1
0
0
e-
BS2
D0
1 
D1
e-
e-
D0
BS2
D1
Equivalent Quantum Circuits:
Open MZI
Closed MZI
Wheeler’s
delayed choice
Quantum
delayed choice
Continuous Morphing b/w wave & particle
|00
 = 0 : Particle nature
 = /4 : Complete superposition
 = /2 : Wave nature
Quantum Delayed Choice Experiment
Interference
No Interference
Visibility :
Open and Closed MZI
Open and Closed MZI
|p
 = 
|w
Phys. Rev. A, 2012
Open and Closed MZI
|p
 = 0.02
|w
 = 0.97
Phys. Rev. A, 2012
Quantum Delayed Choice Experiment
 = 
Phys. Rev. A, 2012
Quantum Delayed Choice Experiment
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