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Quantum Correlations in Nuclear Spin Ensembles T. S. Mahesh Indian Institute of Science Education and Research, Pune

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Quantum or Classical ? How to distinguish quantum and classical behavior?

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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 Leggett Uni. of Illinois at UC Prof. Anupam Garg Northwestern University, Chicago

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Consider a dynamic system with a dichotomic quantity Q(t) Dichotomic : Q(t) = 1 at any given time time Q1Q1 Q2Q2 Q3Q3 t2t2 t3t3... Leggett-Garg (1985) A. J. Leggett and A. Garg, PRL 54, 857 (1985) PhD Thesis, Johannes Kofler, 2004 t1t1

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time Q1Q1 t = 0 Q2Q2 Q3Q3 tt... 2t2t Two-Time Correlation Coefficient (TTCC) Ensemble Time ensemble (sequential) Spatial ensemble (parallel) Temporal correlation: C ij = Q i Q j = Q i (r) Q j (r) N 1 r = 1 N 1 C ij 1 C ij = 1 Perfectly correlated C ij = 1 Perfectly anti-correlated C ij = 0 No correlation = p ij + (+1) + p ij ( 1) r over an ensemble

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LG string with 3 measurements K 3 = C 12 + C 23 C 13 K 3 = Q 1 Q 2 + Q 2 Q 3 Q 1 Q 3 3 K 3 1 Leggett-Garg Inequality (LGI) K3K3 time Macrorealism (classical) time Q1Q1 t = 0 Q2Q2 Q3Q3 tt2t2t Consider: Q 1 Q 2 + (Q 2 Q 1 )Q 3 If Q 1 Q 2 : 1 + 0 = 1 Q 1 Q 2 : 1 + ( 2) = 1 or 3 Q 1 Q 2 + Q 2 Q 3 Q 1 Q 3 = 1 or 3 3 < Q 1 Q 2 + Q 2 Q 3 Q 1 Q 3 < 1

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TTCC of a spin ½ particle (a quantum coin) Time Q1Q1 t = 0 Q2Q2 Q3Q3 tt2t2t 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 C 12 = x (0) x ( t) = x e -iH t x e iH t = x [ x cos( t) + y sin( t)] C 12 = cos( t) Similarly, C 23 = cos( t) and C 13 = cos(2 t)

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Quantum States Violate LGI: K 3 with Spin ½ time Q1Q1 t = 0 Q2Q2 Q3Q3 tt2t2t K 3 = C 12 + C 23 C 13 = 2cos( t) cos(2 t) K3K3 t 22 33 Macrorealism (classical) Quantum !! 44 0 No violation ! ( /3,1.5) Maxima (1.5) @ cos( t) =1/2

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Consider: Q 1 (Q 2 Q 4 ) + Q 3 (Q 2 + Q 4 ) If Q 2 Q 4 : 0 + ( 2) = 2 Q 2 Q 4 : ( 2) + 0 = 2 Q 1 Q 2 + Q 2 Q 3 + Q 3 Q 4 Q 1 Q 4 = 2 K 4 = C 12 + C 23 + C 34 C 14 or, K 4 = Q 1 Q 2 + Q 2 Q 3 + Q 3 Q 4 Q 1 Q 4 time Q1Q1 t = 0 Q2Q2 Q3Q3 tt2t2t3t3t Q4Q4 Macrorealism (classical) K4K4 time LG string with 4 measurements 2 K 4 2 Leggett-Garg Inequality (LGI)

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K 4 = C 12 + C 23 + C 34 C 14 = 3cos( t) cos(3 t) Quantum States Violate LGI: K 4 with Spin ½ Extrema ( 2 2) @ cos(2 t) =0 K4K4 Macrorealism (classical) Quantum !! t 22 33 44 0 ( /4,2 2) (3 /4, 2 2) time Q1Q1 t = 0 Q2Q2 Q3Q3 tt2t2t3t3t Q4Q4

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Even,M=2L: (Q 1 + Q 3 )Q 2 + (Q 3 + Q 5 )Q 4 + + (Q 2L-3 + Q 2L-1 )Q 2L-2 + (Q 2L-1 Q 1 )Q 2L Max: all +1 2(L 1)+0. M 2 Min: odds +1, evens –1 2(L 1)+0. M+2 Odd,M=2L+1: (Q 1 + Q 3 )Q 2 + (Q 3 + Q 5 )Q 4 + + (Q 2L-3 + Q 2L-1 )Q 2L-2 + (Q 2L-1 +Q 2L+1 )Q 2L Q 1 Q 2L+1 Max: all +1 2L–1. M 2 Min: odds +1, evens –1 2L 1. M K M = C 12 + C 23 + + C M-1,M C 1,M or, K M = Q 1 Q 2 + Q 2 Q 3 + + Q M-1 Q M Q 1 Q M timeQ1Q1 t = 0 Q2Q2 tt QMQM MtMt... LG string with M measurements M+2 K M (M 2) if M is even, M K M (M 2) if M is odd. Macrorealism (classical) M KMKM time (M 2)

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K M = C 12 + C 23 + + C M-1,M C 1,M = (M-1)cos( t) cos{(M-1) t)} Quantum States Violate LGI: K M with Spin ½ Maximum: Mcos( /M) @ t = /M Note that for large M: Mcos( /M) M > M-2 Macrorealism is always violated !! 22 33 44 t M KMKM Macrorealism (classical) Quantum (M 2) time Q1Q1 t = 0 Q2Q2 tt QMQM MtMt...

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Evaluating K 3 K 3 = C 12 + C 23 C 13 t = 0 tt2t2t x↗x↗ x↗x↗ x↗x↗ x↗x↗ x↗x↗ x↗x↗ time ENSEMBLE x (0) x ( t) = C 12 x ( t) x (2 t) = C 23 x (0) x (2 t) = C 13 ENSEMBLE 00 Hamiltonian : H = ½ z 00 00

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Evaluating K 4 K 4 = C 12 + C 23 + C 34 C 14 t = 0 tt2t2t x↗x↗ x↗x↗ x↗x↗ x↗x↗ x↗x↗ time x↗x↗ x↗x↗ x↗x↗ 3t3t ENSEMBLE x (0) x ( t) = C 12 x ( t) x (2 t) = C 23 x (0) x (3 t) = C 14 x (2 t) x (3 t) = C 34 Joint Expectation Value ENSEMBLE Hamiltonian : H = ½ z 00 00 00 00

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Moussa Protocol O. Moussa et al, PRL,104, 160501 (2010) Target qubit (T) Probe qubit (P) AB x↗x↗ |+ AB Joint Expectation Value A↗A↗ B↗B↗ AB Target qubit (T) Dichotomic observables Target qubit (T) AB x↗x↗ (1- ) I /2+ |+ +| AB

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Moussa Protocol Target qubit (T) Probe qubit (P) A x↗x↗ |+ AA Dichotomic observable be, A = P P (projectors) Let| be eigenvectors and 1 be eigenvalues of X Then, X =|+ +| | |, and X 1 = p(+1) p( 1). Apply on the joint system: U A = |0 0| P 1 T + |1 1| P A p( 1) = | | 1 = tr [ {U A {|+ +| } U A † } {| | 1 }] = P A = P + P = p(+1) p( 1) = X 1 Target qubit (T) Probe qubit (P) AB x↗x↗ |+ AB Extension:

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Sample 13 CHCl 3 (in DMSO) Target: 13 C Probe: 1 H Resonance Offset: 100 Hz 0 Hz T 1 (IR) 5.5 s 4.1 s T 2 (CPMG) 0.8 s 4.0 s Ensemble of ~10 18 molecules

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Experiment – pulse sequence 1H1H 13 C = A x A ref A x (t)+i A y (t) A x (t) = x (t) A ref = x (0) = 00 V. Athalye, S. S. Roy, and T. S. Mahesh, Phys. Rev. Lett. 107, 130402 (2011).

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t Experiment – Evaluating K 3 time Q1Q1 t = 0 Q2Q2 Q3Q3 tt2t2t K 3 = C 12 + C 23 C 13 = 2cos( t) cos(2 t) ( = 2 100) Error estimate: 0.05 V. Athalye, S. S. Roy, and T. S. Mahesh, Phys. Rev. Lett. 107, 130402 (2011).

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Experiment – Evaluating K 3 50100150200250300 t (ms) LGI violated !! (Quantum) LGI satisfied (Macrorealistic) Decay constant of K 3 = 288 ms 165 ms V. Athalye, S. S. Roy, and T. S. Mahesh, Phys. Rev. Lett. 107, 130402 (2011).

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t Experiment – Evaluating K 4 ( = 2 100) Error estimate: 0.05 K 4 = C 12 + C 23 + C 34 C 14 = 3cos( t) cos(3 t) time Q1Q1 t = 0 Q2Q2 Q3Q3 tt2t2t3t3t Q4Q4 Decay constant of K 4 = 324 ms V. Athalye, S. S. Roy, and T. S. Mahesh, Phys. Rev. Lett. 107, 130402 (2011).

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time Signal x Quantum to Classical 13-C signal of chloroform in liquid | = c 0 |0 + c 1 |1 |0 |1 |0 |1 |c 0 | 2 c 0 c 1 * c 0 * c 1 |c 1 | 2 s = |c 0 | 2 0 0 |c 1 | 2 |c 0 | 2 e (t) c 0 c 1 * e (t) c 0 * c 1 |c 1 | 2 Quantum StateClassical State

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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

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Wave nature of particles !! C. Jönsson, Tübingen, Germany, 1961

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Not a wave of particles Single particles interfere with themselves !! Intensity so low that only one electron at a time 4000 clicks C. Jönsson, Tübingen, Germany, 1961 Single Particle at a time

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Two-slit wave packet collapsing Eventually builds up pattern Particle interferes with itself !! Single particle interference

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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 ?

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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 ?

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Short answer: no, we can’t tell Anything that blocks one slit washes out the interference pattern Which path ?

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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

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Mach-Zehnder Interferometer Open Setup Single photon D0 D1 1 0 Only one detector clicks at a time !! BS1

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Mach-Zehnder Interferometer Open Setup Single photon D0 D1 1 0 Trajectory can be assigned BS1

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Mach-Zehnder Interferometer Open Setup Single photon D0 D1 1 0 Trajectory can be assigned BS1

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Mach-Zehnder Interferometer Open Setup Single photon D0 D1 1 0 Trajectory can be assigned : Particle nature !! BS1

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Mach-Zehnder Interferometer Open Setup S 0 or S 1 Intensities are independent of i.e., no interference

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Mach-Zehnder Interferometer Closed Setup Single photon D0 D1 1 0 Again only one detector clicks at a time !! BS1 BS2

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Mach-Zehnder Interferometer Single photon D0 D1 1 0 Again only one detector clicks at a time !! BS1 BS2

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Mach-Zehnder Interferometer Closed Setup S 0 or S 1 Intensities are dependent of Interference !!

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Mach-Zehnder Interferometer Closed Setup BS2 removes ‘which path’ information Trajectory can not be assigned : Wave nature !! Single photon D0 D1 1 0 BS1 BS2

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Photon knows the setup ? D0 D1 1 0 Open Setup Closed Setup D0 D1 1 0 BS2 BS1 Particle behavior Wave behavior

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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

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Delayed Choice Experiment Wheeler’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

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Delayed Choice Experiment Wheeler’s Gedanken Experiment (1978) 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

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No longer Gedanken Experiment (2007)

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COMPLEMENTARITY SATISFIED

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Bohr, Pauli, Dirac, …. Intrinsic wave-particle duality Reality depends on observation Complementarity principle Delayed Choice Experiment Wheeler’s Gedanken Experiment (1978) 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

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Quantum Delayed Choice Experiment Superposition of present and absent !! D0 D1 1 0 BS2 BS1

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Quantum Delayed Choice Experiment D0 D1 1 0 BS2 BS1 Open-setup e- D0 D1 1 0 BS2 BS1 Closed setup e-

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Quantum Delayed Choice Experiment D0 D1 1 0 BS2 BS1 Open-setup e- D0 D1 1 0 BS2 BS1 Closed setup e- D0 D1 1 0 BS2 BS1 Quantum setup

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Equivalent Quantum Circuits: Open MZI Closed MZI Wheeler’s delayed choice Quantum delayed choice

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Continuous Morphing b/w wave & particle |00 = 0 : Particle nature = /4 : Complete superposition = /2 : Wave nature

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Quantum Delayed Choice Experiment Interference No Interference Visibility :

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Open and Closed MZI

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|p|p |w|w = Phys. Rev. A, 2012

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Open and Closed MZI |p|p |w|w = 0.97 = 0.02 Phys. Rev. A, 2012

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Quantum Delayed Choice Experiment Phys. Rev. A, 2012 =

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Quantum Delayed Choice Experiment Phys. Rev. A, 2012

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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

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