Quantum-limited metrology (examples with trapped atomic ions) Dave - PowerPoint PPT Presentation

Quantum-limited metrology (examples with trapped atomic ions) Dave Wineland, Dept. of Physics, U Oregon, Eugene, OR & Research Associate, NIST, Boulder, CO

Summary: • Ramsey interferometer, angular momentum picture * application to spectroscopy/clocks • entangled states for increased precision * spin squeezing * spin “Schrödinger cat” states • efficient detection with ancilla qubits * application to spectroscopy/clocks • squeezed harmonic oscillator states Quantum-limited metrology (examples with trapped atomic ions) Dave Wineland, Dept. of Physics, U Oregon, Eugene, OR & Research Associate, NIST, Boulder, CO

Mach-Zehnder interferometer: Ligo uses large incoming photon coherent states & different geometry φ 1 detect photon 50/50 2 in path 1 or 2 beam splitters Carl Caves, … review: V. Giovannetti, Science 306 , 1330 (2004).

Ramsey interferometry with π /2 pulses like 50/50 beam splitters in Mach-Zehnder interferometer qubits (states | ↓〉 and | ↑〉 , E ↑ - E ↓ = ћ ω 0 ) applied radiation (“ π /2 pulses”) frequency ω near ω 0 π /2 π /2 M ↑ pulse pulse T measure N for each qubit: 0 0 π /2 “Signal” signal to noise independent of φ Noise ∆ Õ ≡ ( 〈 (Õ - 〈 Õ 〉 ) 2 〉 ) 1/2 (“projection noise”)

Ramsey interferometer, angular momentum picture J = Σ S i ( S i = ½, equivalent to ensemble of two-level systems) e.g., H i = ћ γ S z B 0 R. Feynman et al. J. Appl. Phys. 28 , 49 (1957)) J = N/2, m J = -N/2 (“coherent spin state”) (in rotating frame of applied field (frequency ω ): z B z = B 0 ( ω o - ω )/ ω o (B rf >> B z ) B rf /2 y 〈 J (0) 〉 ( ω o - ω )T = π /2 x Free precession Second Ramsey pulse First Ramsey pulse N 〈 Ñ ↑ (t f ) 〉 Õ = Ñ ↑ (t f ) = Ĵ z + JÎ 〈 Ñ ↑ (t f ) 〉 = N/2(1 + cos( ω o - ω )T) (t f = T) 0 0 π /2 ϕ = ( ω o - ω )T →

Measurement uncertainty relations : Uncertainty relation for operators: For operators Õ 1 , Õ 2 , Schwartz inequality (1) ∆ Õ 1 ∆ Õ 2 ≥ ½| 〈 [Õ 1 ,Õ 2 ] 〉 | (measurement fluctuations on identically prepared systems) for Õ 1 = x, Õ 2 = p, Eq. (1) gives position/momentum uncertainty relation Time/Energy uncertaintly relation: Uncertainty relations for parameters and operators For operator Õ( ξ ) ( ξ = parameter) ∆ Õ ≅ |d 〈 Õ 〉 /d ξ〉 | ∆ξ , ⇒ ∆ξ ≅ ∆ Õ /|d 〈 Õ 〉 /d ξ〉 | In (1), let Õ 1 = Õ, Õ 2 = 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

Quantum limits to (angular momentum) rotation angle measurement For J(0) =  J, -J 〉 (“coherent” spin state) Uncertainty relation:   y    ΔJ ΔJ 1 J , J 1 J x y x y z 2 2 J z = - N/2 x angle sensitivity = N -1/2 N   J ( 0 ) 0 , z      J ( 0 ) J ( 0 ) J/2 1 N x y 2 observed for coherent spin states: • Itano et al. , PRA 47, 3554 (1993). 0 • Santarelli et al. , PRL 82 , 4619 (1999). 0 π /2 • … “projection noise” phase sensitivity ∆φ = N -½ independent of φ

coherent spin state 〈 J (T) 〉 angle sensitivity = N -1/2 〈 J (0) 〉 ϕ = ( ω o - ω )T WANT: “spin-squeezed” state ∆ J z 〈 J (0) 〉 B. Yurke et al ., 〈 J (T) 〉 PRA 33 , 4033 (1986) ∆ J ⊥ Angle sensitivity = 〈 J (T) 〉

Generate spin squeezing with H I =  χ J z 2 , ⇒ U = exp(-i χ tJ z 2 ) test by first H I = (  χ J z ) J z rotating state to | 〈 J 〉 | Δθ(coheren t) 1/ 2J 1/ √ 2J κ =   κ = signal-to-noise improvement  ∆ J ⊥ √ 2J Δθ(squeeze d) ∆ J ⊥ / | 〈 J 〉 | ΔJ / J  Note: | 〈 J 〉 | shrinks with squeezing • Sanders, Phys. Rev. A 40 , 2417 (1989) (nonlinear beam splitter for photons) • Kitagawa and Ueda, Phys. Rev. A 47 , 5138 (1993) (potentially realized by Coulomb interaction in electron interferometers) • Sørensen & Mølmer, PRL 82 , 1971 (1999) (trapped ions) • Solano, de Matos Filho, Zagury PRA 59 , 2539 (1999) (trapped ions) • Milburn, Schneider and James, Fortschr. Physik 48, 801 (2000) (trapped ions)

apply H I ∝ J x 2 . For N = 2 , | ↓〉 | ↓〉 → cos( α )| ↓〉 | ↓〉 + i sin( α ) | ↑〉 | ↑〉 α = π /6 After π /2 pulse about B RF ∆ J z B RF 0.9 ξ anti-squeezing 0.8 〈 J (0) 〉 0.7 squeezing 0.6 0.5 0.4 ξ → π /2 Standard quantum limit θ | 〈 J 〉 | κ = ∆ J ⊥ √ 2J V. Meyer et al. , PRL 86 , 5870 (2001)

J J coherent state squeezed state 〈 J z (t f ) 〉 〈 J z (t f ) 〉 -J -J π π ϕ = ( ω o - ω )T → ϕ → 0 0 κ ≅ 1.07, N = 2 for Ψ perfect V. Meyer et al. , PRL 86 , 5870 (2001)

J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, A. M. Rey, M. Foss-Feig, J. J. Bollinger (NIST) Science 352 , 1297 (2016) H I = (  χ J z ) J z κ = 2.5 (2), N = 85

K. C. Cox, G. P. Greve, J. M. Weiner, and J. K. Thompson, PRL 116 , 093602 (2016) 87 Rb, | ↓〉 = |F=2, m F = 2 〉 , | ↑〉 = |3,3 〉 measure J z (via cavity frequency pulling) project to squeezed state κ exp = 2.3 (3) κ = 7.7 (3) N = 4 x 10 5 squeezing reduced by noise in feedback

O. Hosten, N. J. Engelsen, R. Krishnakumar, M. A. Kasevich Nature 529 , 505 (2016) 87 Rb, | ↓〉 = |F=2, m F = 0 〉 , | ↑〉 = |3,0 〉 Measure J z via frequency shift of coupled cavity z z N = 5 x 10 5 κ = 10.1(2)

Ramsey interferometry of 〈 J 〉 = 0 states: conventional Ramsey pulse R 1,2,…N ( π /2) entangling pulse T M measure | 〈 J 〉 | = 0 ! spin “Schrödinger cat” states ⇒ Heisenberg limit After second Ramsey pulse, measure parity operator: (J. Bollinger et al ., Phys. Rev. A 54 , R4649 (1996)) (two values possible: 1, -1)

e.g., N = 6 Õ = N ↑ - N ↓ (nonentangled) Õ = parity (entangled) 6 2 〈 Õ 〉 → 0 -6 0 2 π ( ω - ω 0 )T → 〈 Õ 〉 (single measurement) ∆ϕ = 1/N, independent of ( ω - ω 0 )T “Heisenberg limit”

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

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

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

Efficient detection with ancilla qubits (Al + optical clock experiment, NIST, Boulder) Coulomb interaction 1 P 1 2 P 3/2 3 P 0 optical (F=1, m F = -1) qubit λ = 167 nm λ = 267 nm 2 S 1/2 Be + hyperfine (F=2, m F = -2) 1 S 0 qubit transfer information to 9 Be + P. O. Schmidt et al., Science 309 , 749 (2005)

Efficient detection with ancilla qubits (Al + optical clock experiment, NIST, Boulder) Coulomb interaction 2 P 3/2 3 P 1 3 P 0 2 0 n 1 1 S 0 Cool ions to ground state with Be + : | ↓〉 Be |n=0 〉 P. O. Schmidt et al., Science 309 , 749 (2005)

Efficient detection with ancilla qubits (Al + optical clock experiment, NIST, Boulder) Coulomb interaction 2 P 3/2 3 P 1 3 P 0 2 0 n 1 1 S 0 If Ψ = | 3 P 0 〉 | ↓〉 Be |n=0 〉 P. O. Schmidt et al., Science 309 , 749 (2005)

Efficient detection with ancilla qubits (Al + optical clock experiment, NIST, Boulder) Coulomb interaction 2 P 3/2 3 P 1 3 P 0 2 0 n 1 1 S 0 Is “QND” measurement; can repeat to increase Fidelity F = 0.85 → 0.9994 D. Hume et al. , Phys. Rev. Lett. 99 , 120502 (2007) Now, extensions to molecules: C.–W. Chou et al., Nature 545, 203 (2017). P. O. Schmidt et al., Science 309 , 749 (2005)

Sensitive mechanical motion detection e.g. single harmonic motion α i D( α i ) ≡ exp[ α i a  - α i *a] with α = ( ∆ x + i ∆ p/m ω x )/(2x 0 )) here, κ ∝ α I / ground-state wave packet spread ground state (n =0) wavefunction

Sensitive mechanical motion detection e.g. single harmonic motion (D( α ) ≡ exp[ α a  - α *a] with α = ( ∆ x + i ∆ p/m ω x )/(2x 0 ))

generated with parametric drive trap potential modulated at 2 ω x κ ≅ 7.3 S. C. Burd et al., Science 364 , 1163 (2019).

Photon metrology: • UV fibers: Y . Colombe et al., Optics Express, 22 , 19783 (2014) recipe: http://www.nist.gov/pml/div688/grp10/index.cfm • Detection without optics; D. Slichter, V. Verma MoSi