Increasing atomic clock precision with and without entanglement

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1 Lehman College Increasing atomic clock precision with and without entanglement Christopher C. Gerry Department of Physics and Astronomy Lehman College, The City University of New York Bronx, New York Collaborator: Richard Birrittella, Funding: NSF, US Army, US Air Force, PSC-CUNY

2 Outline Ramsey Spectroscopy and atomic clocks one atom at a time Ramsey spectroscopy with entangled atoms Ramsey spectroscopy with N atoms using atomic parity measurements Photonic interferometry using photon number parity Atomic clock based on atomic parity measurements How to do the measurements

3 Noise Reduction in Phase Shift Measurements: n SQL Standard quantum limit Shot-noise limit n HL Heisenberg limit Greatest sensitivity allowed by QM for linear phase shifts.

4 Ramsey method of separated fields

5 Ramsey spectroscopy with twolevel atoms or trapped ions: 0 E e E g

6 Bloch sphere picture of Ramsey spectroscopy Measure probablity of finding atom in the excited state: P e e e

7 Ramsey pulse of frequency Free evolution in frame rotating at after time T, 2 i T g e e 0 gives, 7

8 Second 2 Ramsey pulse: g g 2 e e e 2 g g e e e 2 it it 2 P cos e e 0 T 2

9 Ramsey Fringes P e Maximum a t. 0 0 Fringe width (resolution) T

10 0 T one atom 0 T N N atoms, Standard Quantum Limit (SQL) Detection is by Dehmelt s electron shelving technique:

11 Cesium beam atomic clock

12 Atomic clock definition of the second: periods of the radiation for the hyperfine transition: 33 Cs: F 3, M 0 F 4, M 0.

13 To further improve atomic clocks for greater precision and greater stability, need to make the Ramsey fringes narrower. One approach: Increase the free evolution time T.

14 Atomic Fountain clock: increases T

15 The Entanglement Advantage : Two entangled ions 0 2 g g e e Free evolution for time T : 2iT T g g e e e i T dis g e e g Implement a C-Not gate to disentangle: Pe cos(2 T) T 2T e e2 e g2 5

16 For maximally entangled state of N ions (J. Steinbach and C.C. Gerry, Phys. Rev. Lett. 8, 5528 (998)). g g g e e e 2 N 2 N 2 Pe cos( NT ) 2 0 NT Heisenberg-limited Improvement by a factor shot-noise limit. N over 6

17 Alternative Approach (Bollinger et al. PRA 54 (996)): Measure parity operator: N e ˆ N N ˆ cos N cos NT 0 N e is the number of atoms detected in their excited states T N total number of ions in the ensemble Heisenberg limited: 0 NT 7

18 J N x, y, z 2 i x, y, z i Dicke States J, J ij J, J g, g..., g, 2, 2, x y z N J, M, J, M J, J,..., J 2 J, J e, g2,..., gn g, e2,..., gn... g, g2,..., en N,... J, J e, e..., e, N N

19 Atomic Coherent States *, exp R zj z J 2 J J 2J JM, J R, J, J J, M, MJJ M i i z e 2, e tan 2. /2 N i/2 i /2, J, i e sin 2 e e cos 2 g J i i First pulse: 2 J J 2J JM, J exp i 2 J y J, J 2 J, M, MJJ M, J i g e 2. N i i i /2

20 Initial State: J, J,all atoms in their ground states. ˆ ˆ exp exp ˆ exp ˆ U i J y ij z i J y Jˆ 2 J, J Uˆ Jˆ Uˆ J, J 2J cos z Ncos 0 T, z

21 Suppose we prepare the state J, J J, J 2 Creation of this state replaces the first Ramsey pulse. Final state is then exp exp f i J y i J z 2 2 i Jz in 2 in 2 e e J, J e J, J J 0 f z f

22 Collective atomic, or SU(2), parity operator J exp i J J z No classical analog! 2J f J f cos 2 J N = cos N 0 0 T T

23 Definition of photon number parity Photon number operator: Parity Operator: nˆ aˆ aˆ, a ˆ, aˆ nˆ n n n, n 0,,2..., nˆ ˆ exp i nˆ ˆ n n n No classical analog!

24 Interferometry with a N00N state? ˆ measurement 0 0 N 2 N N

25 N /2 cos, even, ˆ N N N N /2 sin N N, N odd. N Heisenberg-limited Super-resolved interference fringes

26 Interferometry with coherent light ˆ ˆ ˆˆ S a a b b n cos SQL Standard quantum limit n S S n sin

27 Coherent light interferometry and parity measurements Dowling and collaborators

28 Photon number parity approach to interferometry with coherent light ˆ exp cos b n 0 Super-resolved: min n cos

29 Experiment: L. Cohen et al. Opt. Exp. 22, 0945 (204)

30 From L. Cohen et al. Opt. Exp. 22, 0945 (204)

31 For the initial state g g g 2 N Central Fringes ˆ cos N 0 T There is no entanglement involved!

32 For the maximally entangled state: g g g e e e J J J, J 2 N 2 N 2 2 Central Fringes

33 J 2J JM J z, J M J, M MJJ M /2 Entangled state. Jz, J

34 Alternative: Prob. of detecting J, J J J J, M J, M, P out out M M M J

35 nˆ nˆ nˆ ˆ J c z