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QI primitives for quantum-limited measurement

QI primitives for quantum-limited measurement David Wineland, Time & Frequency Division, NIST, Boulder. Summary: Quantum information experiments with trapped ions Ramsey interferometer, angular momentum picture entangled states for increased precision * spin squeezing

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QI primitives for quantum-limited measurement

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  1. QI primitives for quantum-limited measurement David Wineland, Time & Frequency Division, NIST, Boulder

  2. Summary: • Quantum information experiments with trapped ions • Ramsey interferometer, angular momentum picture • entangled states for increased precision * spin squeezing * spin “Schrödinger cat” states • efficient detection with ancilla qubits, * application to ion clocks QI primitives for quantum-limited measurement David Wineland, Time & Frequency Division, NIST, Boulder

  3. Some QI experiments with trapped ions ion trap electrodes  3-D harmonic well Use quantized motional mode as “data bus” (J. I. Cirac, P. Zoller, Phys. Rev. Lett. 74, 4091 (1995) Internal-states (qubits) Motional modes (e.g., center-of-mass mode) • • • • n=3 n=2 n=1 n=0 Motion quantum states

  4. 2P3/2 2P1/2 9Be+ qubits 2S1/2 electronic ground level hyperfine states F = 2, mF = 0 F = 1,mF = -1 B  119 G (coherence time ~ 10 s) two-photon stimulated-Raman transitions (vs. -waves) ~1.21 GHz

  5. 2P3/2 2P1/2 9Be+ qubits 2S1/2 electronic ground level hyperfine states F = 2, mF = 0 F = 1,mF = -1 B  119 G (coherence time ~ 10 s) two-photon stimulated-Raman transitions  spin/motion coupling • • • • ~1.25 GHz ~5 MHz

  6. 2P3/2 2P1/2 photon-count histograms | • • 0 10 20 30 | • • 0 10 20 30 Measurement: State-dependent laser scattering (“cycling” transition)

  7.  ei  enclosed area A Geometric phase gates: phase-space diagram for motional mode wavefunction: p  A x D(+) = D()D()exp{-i Im(*)} (concatenate small displacements) make force state-dependent to realize phase gate, e.g,: 12 12 12 i12 12 i12 12  12 • use optical dipole forces General formalism: Milburn, Schneider, James (1999) Sørensen & Mølmer (1999, 2000) Solano, de Matos Filho, Zagury (1999)

  8. Optical dipole forces  |e ion harmonic motion frequency m r b  >> m |↓, |↑ basic idea: use gradient of Stark shift to apply forces F, F to |↓ and |↑ b = r Polarization/intensity gradient standing wave b r

  9. Dipole forces to displace ions in phase space  |e ion harmonic motion frequency m r b  >> m |↓, |↑ basic idea: use gradient of Stark shift to apply forces to |↓ and |↑ b - r = m +  “walking” standing wave example: make F = - F

  10. 2-ion phase-space displacement (e.g., stretch mode): b - a = stretch +  apply forces for t = 2/ p assume F = - F “stretch” mode A ||: || i || || i || x in general: i  ei moving optical-dipole potential grating U(t= tgate) = exp[-i(/4)z1z2] HI = z1z2 (up to Z rotation) For N qubits H = JiJi, Ji = k ki i = {x,y,z} p ||: 2- qubit gate errors:  1  1 x 10-3 C. J. Ballance et al., arXiv:1512.04600  0.8  0.4 x 10-3 J. P. Gaebler et al., arXiv:1604.00032 || || || || x

  11. Mach-Zehnder interferometer: (e.g., for photons) incoming photon f 1 detect photon in path 1 or 2 50/50 beam splitters 2 Carl Caves; this AM review: V. Giovannetti, S. Lloyd, L. Maccone Science 306, 1330 (2004).

  12. Ramsey interferometry with qubits: /2 pulses  50/50 beam splitters applied radiation (“/2 pulses”) frequency   0 /2 pulse /2 pulse M T measure N 0 0 /2 “Signal” “shot noise” limit (independent of  - 0 projection noise only) Noise Õ  ((Õ - Õ)2)1/2

  13. Ramsey interferometer, angular momentum picture J = iSi (Si = ½, equivalent to ensemble of two-level systems) (R. Feynman et al. J. Appl. Phys. 28, 49 (1957)) e.g., Hi = ћ SzBz intial = …N = J = N/2, mJ = -N/2 (“coherent spin state”) (in rotating frame of applied field): z Bz = (o - )/ Bz = (o - )/ (Brf >> Bz) Brf/2 y J(0) x (o - )T = /2 Free precession Second Ramsey pulse First Ramsey pulse N Ñ(tf) Õ = Ñ(tf) = Ĵz + JÎ Ñ(tf) = N/2(1 + cos(o - )T) 0 0 /2  = (o - )T 

  14. Measurement uncertainty relations: Uncertainty relation for operators: For operators Õ1, Õ2, Schwartz inequality  (*) Õ1Õ2 ½|[Õ1,Õ2]| (measurement fluctuations on identically prepared systems) e.g. for Õ1 = x, Õ2 = p  position/momentum uncertainty relation Uncertainty relations for parameters and operators: For operator Õ() ( parameter) Õ  |dÕ/d| ,  Õ/ |dÕ/d| Example: In (*), let Õ1 = Õ, Õ2 = H (Hamiltonian)  ÕH ½|[Õ,H]| But, ћ dÕ/dt = i[H,Õ] + ћÕ/t ÕH ½ ћ | dÕ/dt| (for Õ/t = 0) Õ = |dÕ/dt| t, H t  ½ ћ

  15. Quantum limits to (angular momentum) rotation angle measurement For (0) = J, -J (“coherent” spin state) Uncertainty relation: y Jz = - N/2 x consider rotations about x axis: z y observed for coherent spin states: • Itano et al., PRA 47, 3554 (1993). • Santarelli et al., PRL 82, 4619 (1999). • … “projection noise” x x = Jy/Jz = J/|J|  = N-½ independent of o - 

  16. coherent spin state J(T) J(0) Angle sensitivity = WANT: “spin-squeezed” state J J(T) Jz J(0) J(T) B. Yurke et al., PRA33, 4033 (1986)

  17. Generate spin squeezing with HI= Jz2,  U = exp(-itJz2) • Sanders, Phys. Rev. A40, 2417 (1989) (nonlinear beam splitter for photons) • Kitagawa and Ueda, Phys. Rev. A47, 5138 (1993) (potentially realized by Coulomb interaction in electron interferometers) • Sørensen & Mølmer, PRL 82, 1971 (1999) (trapped ions) • Solano, de Matos Filho, Zagury PRA59, 2539 (1999) (trapped ions) • Milburn, Schneider and James, Fortschr. Physik 48, 801 (2000) (trapped ions) HI= (Jz) Jz U 1/2J |J|  =  = signal-to-noise improvement J2J J/ |J|

  18.  = /6 After /2 pulse about BRF Jz 0.9 anti-squeezing 0.8 0.7 squeezing 0.6 0.5   0.4 /2 Standard quantum limit  Simple experiment, N = 2 (9Be+): V. Meyer et al., PRL 86, 5870 (2001) apply HI Jx2. For N = 2,  cos + sin (entangled state!) BRF  J(0) |J|  = J2J But, J(0) shrinks with squeezing

  19. J J coherent state squeezed state Jz(tf) Jz(tf) -J -J    = (o - )T    0 0   1.07, N = 2 for  perfect V. Meyer et al., PRL 86, 5870 (2001)

  20. J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, • M. Rey, M. Foss-Feig, J. J. Bollinger • arXiv:1512.03756, accepted for Science HI= (Jz) Jz  = 2.5 (2), N = 85

  21. K. C. Cox, G. P. Greve, J. M. Weiner, and J. K. Thompson, PRL 116, 093602 (2016) 87Rb, | = |F=2, mF = 2, | = |3,3 measure Jz (via cavity frequency pulling) project to squeezed state  = 2.3 (3)  = 7.7 (3) N = 4 x 105

  22. O. Hosten, N. J. Engelsen, R. Krishnakumar, M. A. Kasevich Nature 529, 505 (2016) 87Rb, | = |F=2, mF = 0, | = |3,0 N = 5 x 105  = 10.1(2)

  23. Ramsey interferometry of J = 0 states: R1,2,…N(/2) entangling pulse T M measure |J| = 0 ! spin “Schrödinger cat” states After second Ramsey pulse, measure parity operator: (J. Bollinger et al., Phys. Rev. A54, R4649 (1996)) (two values possible: 1, -1)

  24. e.g., N = 6 Õ = N - N (nonentangled) Õ = parity (entangled) 6 2 0 -6 Õ  0 2 ( - 0)T  Õ (single measurement)  = 1/N, independent of ( - 0)T (V. Meyer et al., PRL 86, 5870 (2001) (N = 2))

  25. Ramsey interferometry with two entangling pulses (D. Leibfried et al. Science ’04, Nature ‘05) entangling “/2” pulse entangling “/2” pulse M T t  measure all ions fluoresce no ions fluoresce

  26. S/N = C “win” if C > C Entangled state interferometry: (D. Leibfried et al. Science 2004, Nature 2005) “Signal” Noise N = 1 C = 0.84(1) > 3-1/2 = 0.58  = 1.45(2) N = 3 N = 4 T. Monz et al., PRL 106, 130506 (2011)  > 1 for N = 14 N = 5 C = 0.419(4) > 6-1/2 = 0..408  = 1.03(1) N = 6

  27. S/N = C “win” if C > C Entangled state interferometry: But! assumptions above: perfect probe oscillators, noise = projection noise Restrictions:  Huelga, Macchiavello, Pellizzari, Ekert, Plenio, Cirac PRL 79, 3865 (1997) (phase decoherence of atoms)  DJW et al. NIST J. Research 103, 259 (1998)  André, Sørensen, Lukin, PRL 92, 230801 (2004) (phase decoherence of source radiation)  C. W. Chou et al., PRL 106, 160801 (2011) (D. Leibfried et al. Science 2004, Nature 2005) “Signal” Noise N = 1 Future:  More and better  Apply entangled states in accurate clocks  …. C = 0.84(1) > 3-1/2 = 0.58  = 1.45(2) N = 3 N = 4 T. Monz et al., PRL 106, 130506 (2011)  > 1 for N = 14 N = 5 C = 0.419(4) > 6-1/2 = 0..408  = 1.03(1) N = 6

  28. Coulomb interaction 1P1 2P3/2 3P0 optical qubit  = 267 nm (F=1, mF = -1)  = 167 nm 2S1/2 Be+ hyperfine qubit (F=2, mF = -2) 1S0 Efficient detection with ancilla qubits (Al+ optical clock experiment, T. Rosenband et al.) transfer information to 9Be+ P. O. Schmidt et al., Science309, 749 (2005)

  29. Coulomb interaction 3P1 Efficient detection with ancilla qubits (Al+ optical clock experiment, T. Rosenband et al.) 2P3/2 3P0 2 1 0 n 1S0 Cool ions to ground state with Be+ |Be|n=0 P. O. Schmidt et al., Science309, 749 (2005)

  30. Coulomb interaction 3P1 Efficient detection with ancilla qubits (Al+ optical clock experiment, T. Rosenband et al.) 2P3/2 3P0 2 1 0 n 1S0 If  = |3P0|Be|n=0 P. O. Schmidt et al., Science309, 749 (2005)

  31. Coulomb interaction 3P1 Efficient detection with ancilla qubits (Al+ optical clock experiment, T. Rosenband et al.) 2P3/2 3P0 2 1 0 n 1S0 |n=0 Is “QND” measurement; can repeat to increase Fidelity F = 0.85  0.9994 D. Hume et al., Phys. Rev. Lett. 99, 120502 (2007) P. O. Schmidt et al., Science309, 749 (2005)

  32. (some things) to do:  reduce errors (2-qubit gate)  10-4  scale up  apply entangled states to accurate clocks  …….

  33. NIST IONS, February, 2016 Back row: Dave Leibrandt, John Bollinger, Christoph Kurz, Kevin Gilmore, Shon Cook, Raghu Srinivas, Dave Hume, David Allcock, Shaun Burd, Jwo-Sy Chen, Jim Bergquist, Sam Brewer, Dave Wineland Front row: Justin Bohnet, Kyle McKay, Yao Huang, James Chou, Susanna Todaro, Katie McCormick, Daniel Slichter, Didi Leibfried, Aaron Hankin, Stephen Erickson, Andrew Wilson, Yong Wan, Ting Rei Tan

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