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NMR investigations of Leggett- Garg Inequality

NMR investigations of Leggett- Garg Inequality. V. Athalye 2 , H. Katiyar 1 , Soumya S. Roy 1 , Abhishek Shukla 1 , R. Koteswara Rao 3 T. S. Mahesh 1 1 IISER-Pune, 2 Cummins College, Pune , 3 IISc, Bangalore. Acknowledgements: .

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NMR investigations of Leggett- Garg Inequality

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  1. 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, Bangalore 4 University of Oxford

  2. Plan • NMR as a quantum testbed • Correlation Leggett-Garg Inequality • Entropic Leggett-Garg Inequality • Summary Athalye, Roy, TSM,PRL2011. Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]

  3. Nuclear Spins • Many nuclei have ‘spin angular momentum’ and ‘magnetic moment’ ħgB0 B0 |1 |0 • |0 + b|1 Coherent Superposition

  4. H0 H1cos(wt) Nuclear Magnetic Resonance (NMR) • Spectrometer Sample: 1015 spins RF coil Pulse/Detect ~ Superconducting coil

  5. Pseudopure State p1 |1 B0 p0 p1 p0 |0 = E 1    kT  = ~ 105 at 300 K, 12 T 1015 spins

  6. Pseudopure State p1 |1 B0 p0 p1 p0 |0 = E  =(1-  )1/2+|00| 1    kT  = pseudopure ~ 105 at 300 K, 12 T 1015 spins

  7. Pseudopure State p1 |1 RF B0 p0 p1 p0 |0 = E  =(1-  )1/2+|00|  =(1-  )1/2+|++| 1    kT  = pseudopure ~ 105 at 300 K, 12 T 1015 spins

  8. Resources • parahydrogens • (Jones &Anwar, • PRA 2004) • q-transducer • (Cory et al, • PRA 2007) Nonseparable State • ~ pure states Resource: Entanglement  > 1/3 2-qubit register  =(1-  )1/2+|00| UW UW   1/3 • Cory 1997 • Chuang 1997 • pseudopure • states Separable State Resource: Discord (in units of 2) Hemant, Roy, TSM, A. Patel, PRA2012

  9. NMR systems useful? Pseudopure |0000000 Preparation (scalability?) 7-qubit NMR register Shor’s algorithm No entanglement finite discord Chuang, Nature 2002 15 = 3 x 5 Open question: Is discord sufficient resource for quantum computation ?

  10. 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

  11. Correlation LGI (CLGI)

  12. Leggett-Garg (1985) Sir Anthony James Leggett Uni. of Illinois at UC  Prof. AnupamGarg Northwestern University, Chicago How to distinguish Quantum behavior From Classical ? 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.”

  13. LGI studies in various systems • N. Lambert et al, PRB 2001 • J.-S. Xu et al., Sci. Rep 2011 • Palacios-Laloyet al., Nature Phys. 2010 • M. E. Goggin et al., PNASUSA 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

  14. Leggett-Garg (1985) Consider a system with a dynamic dichotomic observable Q(t) Dichotomic : Q(t) =  1 at any given time Q1 Q2 Q3 . . . time t2 t3 . . . t1 A. J. Leggett and A. Garg, PRL 54, 857 (1985) PhD Thesis, Johannes Kofler, 2004

  15. Two-Time Correlation Coefficient (TTCC) = pij+(+1) + pij(1) Q1 Q2 . . . Q3 Time ensemble (sequential) time Ensemble r  over an ensemble t = 0 t 2t . . . Spatial ensemble (parallel) Cij= 1  Perfectly correlated Cij=1  Perfectly anti-correlated Cij= 0  No correlation 1 Cij 1 N 1  Temporal correlation: Cij =  QiQj = Qi(r)Qj(r) N r = 1

  16. LG string with 3 measurements K3 = C12 + C23 C13 K3 = Q1Q2 + Q2Q3 Q1Q3 Q1 Q2 Q3 Q1Q2+Q2Q3-Q1Q3 1 1 1 1 1 1 -1 1 1 -1 1 -3 1 -1 -1 1 -1 1 1 1 -1 1 -1 -3 -1 -1 1 1 -1 -1 -1 1 Q1 Q2 Q3 time t = 0 t 2t K3 Macrorealism (classical) 3  K3  1 Leggett-Garg Inequality (LGI) time

  17. TTCC of a spin ½ particle Consider : A spin ½ particle Hamiltonian : H = ½ z Maximally mixed initial State : 0 =½ 1 Dynamic observable: x eigenvalues 1 (Dichotomic ) Q1 Q2 Q3 Time t = 0 t 2t 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) PhD Thesis, Johannes Kofler, 2004

  18. Quantum States Violate LGI: K3 with Spin ½ K3 = C12 + C23  C13 = 2cos(t)  cos(2t) Q1 Q2 Q3 time (/3,1.5) t = 0 t 2t Quantum !! Maxima (1.5) @ cos(t) =1/2 K3 Macrorealism (classical) No violation ! 0 4  2 3 t

  19. Quantum States Violate LGI: K4 with Spin ½ Q4 K4 = C12 + C23 + C34  C14 = 3cos(t)  cos(3t) Q1 Q2 Q3 (/4,22) Quantum !! t = 0 t 2t 3t time Extrema (22) @ cos(2t) =0 K4 Macrorealism (classical) 0  2 3 4 (3/4,22) t

  20. Evaluating K3 K3 = C12 + C23 C13 Hamiltonian : H = ½ z x↗ x↗ ENSEMBLE x(0)x(t) = C12 0 x↗ x↗ x(t)x(2t) = C23 0 ENSEMBLE x(0)x(2t) = C13 x↗ x↗ 0 ENSEMBLE t = 0 t 2t time

  21. Evaluating K4 K4 = C12 + C23 + C34 C14 Joint Expectation Value Hamiltonian : H = ½ z x(0)x(t) = C12 x↗ x↗ 0 ENSEMBLE ENSEMBLE x(t)x(2t) = C23 x↗ x↗ 0 ENSEMBLE x↗ 0 x↗ x(2t)x(3t) = C34 ENSEMBLE x↗ x↗ x(0)x(3t) = C14 0 t = 0 t 2t 3t time

  22. Moussa Protocol Dichotomic observables Joint Expectation Value A↗ B↗ AB Target qubit (T)  x↗ A B Probe qubit (P) |+ AB  Target qubit (T) O. Moussa et al, PRL,104, 160501 (2010)

  23. Sample 13CHCl3 (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). 

  24. Experiment – pulse sequence 0 1/2 = Ax Aref 1H 90x PFG Ax(t)+i Ay(t) Ax(t) = cos(2tij) Ay(t) = sin(2tij) Ax(t)  x(t) 13C  = V. Athalye, S. S. Roy, and TSM,Phys. Rev. Lett. 107, 130402 (2011). 

  25. Experiment – Evaluating K3 K3 = C12 + C23  C13 = 2cos(t)  cos(2t) Q1 Q2 Q3 time t = 0 t 2t Error estimate:  0.05 V. Athalye, S. S. Roy, and TSM,Phys. Rev. Lett. 107, 130402 (2011).  ( = 2100) t

  26. Experiment – Evaluating K3 50 100 150 200 250 300 t (ms) LGI violated !! (Quantum) LGI satisfied 165 ms Decay constant of K3 = 288 ms V. Athalye, S. S. Roy, and TSM,Phys. Rev. Lett. 107, 130402 (2011). 

  27. Experiment – Evaluating K4 Q4 K4 = C12 + C23 + C34  C14 = 3cos(t)  cos(3t) Q1 Q2 Q3 t = 0 t 2t 3t time Error estimate:  0.05 Decay constant of K4 = 324 ms V. Athalye, S. S. Roy, and TSM,Phys. Rev. Lett. 107, 130402 (2011).  ( = 2100) t

  28. Entropic LGI (ELGI) A. R. Usha Devi, H. S. Karthik, Sudha, and A. K. Rajagopal, arXiv: 1208.4491 [quant-ph]

  29. System t1 System state: 1/2 Dynamical observable : Sz(t) = UtSzUt† Time Evolution: Ut = exp(iSxt) Q1 Q2 Q3 . . . time t2 t3 . . . A. R. Usha Devi et al, arXiv: 1208.4491 [quant-ph]

  30. ELGI bound t1 Information Deficit: Q1 Q2 Q3 . . .  time t2 t3 . . .  A. R. Usha Devi et al, arXiv: 1208.4491 [quant-ph]

  31. Extracting Probabilities tk Single-event: k For S = 1/2 P(0) = ½ P(1) = ½ Qk . . . time . . . Hemant, Abhishek, Koteswar, TSM,arXiv: 1210.1970 [quant-ph]

  32. Extracting Probabilities Qi Two-time joint: tj ti Qj . . . time Invasive . . . j

  33. Extracting Probabilities Qi Two-time joint: tj ti Qj . . . time . . .

  34. Extracting Probabilities Qi Two-time joint: tj ti Qj . . . time . . . Non-Invasive Measurement (NIM) P(0,qj) P(1,qj)

  35. System Two-time joint probability ancilla H C system Hemant, Abhishek, Koteswar, TSM,arXiv: 1210.1970 [quant-ph]

  36. Two-time joint probabilities t1 P(q1,q2) P(q1,q3) Q1 Q2 Q3 time t2 t3 Hemant, Abhishek, Koteswar, TSM,arXiv: 1210.1970 [quant-ph]

  37. Information Deficit Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph] CNOT

  38. Information Deficit Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph] CNOT Anti CNOT

  39. Information Deficit Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph] CNOT Anti CNOT NIM

  40. Legitimate Grand Probability A. R. Usha Devi et al, arXiv: 1208.4491 [quant-ph] t1       P(q2,q3) P(q1,q2) P(q1,q3) P’(q1,q2) = P(q1,q2,q3) P’(q1,q3) = P(q1,q2,q3) P’(q2,q3) = P(q1,q2,q3) Classical Probability Theory: q2 q3 q1 Q1 Q2 Q3 time t2 t3 Marginals Grand

  41. Extracting Grand Probability Three-time joint: Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]

  42. Illegitimate Joint Probability P(q1,q2,q3) is illegitimate !!  Violation of Entropic LGI Hemant, Abhishek, Koteswar, TSM, arXiv: 1210.1970 [quant-ph]

  43. 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 !!

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