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Quantum Correlations in Nuclear Spin Ensembles

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

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Quantum Correlations in Nuclear Spin Ensembles

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

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

  3. Leggett-Garg (1985) Sir Anthony James Leggett Uni. of Illinois at UC  Prof. AnupamGarg 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.”

  4. Leggett-Garg (1985) Consider a dynamic system with a dichotomic quantity 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

  5. 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

  6. LG string with 3 measurements K3 = C12 + C23 C13 K3 = Q1Q2 + Q2Q3 Q1Q3 Consider: Q1Q2 + (Q2  Q1)Q3 If Q1 Q2 : 1 + 0 = 1 Q1 Q2 : 1 + (2) = 1 or 3  Q1Q2 + Q2Q3  Q1Q3 = 1 or 3 Q1 Q2 Q3 time t = 0 t 2t K3  3 < Q1Q2 + Q2Q3 Q1Q3 < 1 Macrorealism (classical) 3  K3  1 Leggett-Garg Inequality (LGI) time

  7. 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 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)

  8. 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

  9. LG string with 4 measurements Q4 K4 = C12 + C23 + C34  C14 or, K4 = Q1Q2 + Q2Q3+ Q3Q4 Q1Q4 Consider: Q1(Q2 Q4) + Q3(Q2 + Q4) If Q2 Q4 : 0 + (2) = 2 Q2 Q4 : (2) + 0 = 2  Q1Q2 + Q2Q3 + Q3Q4  Q1Q4 = 2 Q1 Q2 Q3 t = 0 t 2t 3t time Macrorealism (classical) K4 2  K4  2 Leggett-Garg Inequality (LGI) time

  10. 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) t = 0 t 2t 3t time Quantum !! Extrema (22) @ cos(2t) =0 K4 Macrorealism (classical) Quantum !! 0  2 3 4 (3/4,22) t

  11. LG string with M measurements . . . QM KM = C12 + C23 +    + CM-1,M C1,M or, KM = Q1Q2 + Q2Q3+    + QM-1QM Q1QM . . . Mt Even,M=2L: (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 Min: odds +1, evens –1  2L1.  M Q1 Q2 t = 0 t time Macrorealism (classical) (M2) KM M+2  KM  (M2) if M is even, M  KM  (M2) if M is odd. time M

  12. Quantum States Violate LGI: KM with Spin ½ . . . QM . . . Mt KM = C12 + C23 +    + CM-1,M C1,M = (M-1)cos(t)  cos{(M-1)t)} Q1 Q2 Macrorealism (classical) Maximum: Mcos(/M) @ t = /M t = 0 t time • Note that for large M: • Mcos(/M) M > M-2 • Macrorealism • is always violated !! Quantum (M2) KM 4 3 2  M t

  13. 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

  14. 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

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

  16. Moussa Protocol  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: 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 x↗ x↗ A A B Probe qubit (P) Probe qubit (P) |+ |+ A AB     Target qubit (T) Target qubit (T) Extension:   

  17. 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 Ensemble of ~1018 molecules

  18. Experiment – pulse sequence 0 = Ax Aref 1H Ax(t)+i Ay(t) Ax(t) =  x(t) Aref=  x(0) 13C  = V. Athalye, S. S. Roy, and T. S. Mahesh, Phys. Rev. Lett. 107, 130402 (2011). 

  19. 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 T. S. Mahesh, Phys. Rev. Lett. 107, 130402 (2011).  ( = 2100) t

  20. Experiment – Evaluating K3 50 100 150 200 250 300 t (ms) 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). 

  21. 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 T. S. Mahesh, Phys. Rev. Lett. 107, 130402 (2011).  ( = 2100) t

  22. Quantum to Classical 13-C signal of chloroform in liquid Signal  x |c0|2eG(t) c0c1* eG(t) c0*c1 |c1|2 time |1 |1 |0 |0 |c0|2c0c1* c0*c1 |c1|2 |c0|20 0 |c1|2 Quantum State Classical State rs = | = c0|0 + c1|1

  23. NMR implementation of a Quantum Delayed-Choice Experiment SoumyaSingha Roy, AbhishekShukla, and T. S. Mahesh Indian Institute of Science Education and Research, (IISER) Pune

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

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

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

  27. 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?

  28. Which path ? Crystal with inelastic collision Movable wall; measure recoil Source Source No: Change in wavelength washes out pattern No: Movement of slit 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 Albert Einstein

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

  30. Bohr’s Complementarity principle (1933) • Wave and particle natures are complementary !! • Depending on the experimental setup one obtains either wave nature or particle nature • – not both at a time Niels Bohr

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

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

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

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

  35. Mach-Zehnder Interferometer Open Setup S0 orS1 Intensities are independent of  i.e., no interference 

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

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

  38. Mach-Zehnder Interferometer Closed Setup Intensities are dependent of  Interference !! S0 orS1 

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

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

  41. Two schools of thought • Einstein, Bohm, …. • Apparent wave-particle duality • Reality is independent of observation • Hidden variable theory • Bohr, Pauli, Dirac, …. • Intrinsic wave-particle duality • Reality depends on observation • Complementarity principle

  42. Delayed Choice Experiment Wheeler’s Gedanken Experiment (1978) D0 1 BS2  D1 BS1 Delayed Choice BS2 0 Decision to place or not to place BS2 is made after photon has left BS1

  43. Delayed Choice Experiment Wheeler’s Gedanken Experiment (1978) D0 1 BS2  D1 BS1 Delayed Choice BS2 Complementarity principle : Results do not change with delayed choice Hidden-variable theory : Results should change with the delayed choice 0

  44. No longer Gedanken Experiment (2007)

  45. No longer Gedanken Experiment (2007) COMPLEMENTARITY SATISFIED

  46. Delayed Choice Experiment Wheeler’s Gedanken Experiment (1978) • 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 X  Complementarity principle : Results do not change with delayed choice Hidden-variable theory : Results should change with the delayed choice

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

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

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

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

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