Matter wave interferometry beyond classical limits

Transcription

1 Max-Planck-Institut für Quantenoptik Varenna school on Atom Interferometry,

2 The Plan Lecture 1 (Wednesday): Quantum noise in interferometry and Spin Squeezing Lecture 2 (Friday): Spin squeezed hyperfine states: Quantum interferometry Lecture 3 (Today): Spin squeezed motional states, Alternative strategies

3 Outline Spin Squeezing Of A Double Well BEC The Experimental System And Its Parameters Experimental Implementation Adiabatic Spin Squeezing Alternative Approaches To Quantum Metrology

4 Outline Spin Squeezing Of A Double Well BEC The Experimental System And Its Parameters Experimental Implementation Adiabatic Spin Squeezing Alternative Approaches To Quantum Metrology

5 Motivation Interferometers based on internal degrees of freedom Measurements of: time magnetic fields motional degrees of freedom Measurements of: gravity rotation acceleration...

6 Overview Squeezing motional degrees of freedom RF Atom chip Trap wire RF B x y z x Gravity B 0 Nonlinearity fixed, Rabi coupling Ω tuned by controlling the height of the potential barrier. J. Estève, Nature 455, 1216 (2008) C. F. Ockeloen, arxiv: (2013)

7 Experimental System Macroscopically occupied spatial two-mode system Double Well potential Single internal state Coupling via tunneling

8 Realizations Atom chips Heidelberg (Schmiedmayer) MIT (Ketterle) Paris (Reichel) Toronto (Thywissen) Heidelberg (Oberthaler) Haifa (Steinhauer) Optical traps Schumm et al. Nature Physics 1, 57 (2005) Jo et al. PRL 98, (2007) Maussang et al. PRL 105, (2010) LeBlanc et al. PRL 106, (2011) Albiez et al. PRL (2005), Estève et al. Nature 455, 1216 (2008) Levy et al. Nature 449, 579 (2007)

9 The Intuitive Picture Overlap between left and right mode tunneling

10 The Intuitive Picture Overlap between left and right mode tunneling

11 The Hamiltonian Milburn et al. PRA 55, 4318 (1997), Spekkens et al. PRA 59, 3868 (1999), Javanainen et al. PRA 60, 2351 (1999), Ananikian et al. PRA 73, (2006)

12 The Hamiltonian Milburn et al. PRA 55, 4318 (1997), Spekkens et al. PRA 59, 3868 (1999), Javanainen et al. PRA 60, 2351 (1999), Ananikian et al. PRA 73, (2006)

13 Excursion: Josephson Junctions JJ: Superconductors connected via a thin insulator. S L I S R BEC L BEC R Re(Ψ) Re(Ψ) x x

14 Excursion: Josephson Junctions JJ: Superconductors connected via a thin insulator. S L I S R BEC L BEC R Re(Ψ) Re(Ψ) x x

15 The Parameters: Coupling Calculation of the Josephson energy E J Solve the Gross-Pitaevskii equation Ananikian et al. PRA 73, (2006), Zapata et al. PRA 57, R28 (1998)

16 The Parameters: Coupling Calculation of the Josephson energy E J Solve the Gross-Pitaevskii equation E J follows from overlap integrals Φ l Φ r d 3 x or... Ananikian et al. PRA 73, (2006), Zapata et al. PRA 57, R28 (1998)

17 The Parameters: Coupling Calculation of the Josephson energy E J Solve the Gross-Pitaevskii equation E J follows from overlap integrals Φ l Φ r d 3 x or... (better) by analogy to a capacitor in electrodynamics Ananikian et al. PRA 73, (2006), Zapata et al. PRA 57, R28 (1998)

18 The Parameters: Nonlinearity Calculation of the Charging energy E C Solve the Gross-Pitaevskii equation Ananikian et al. PRA 73, (2006), Zapata et al. PRA 57, R28 (1998)

19 The Parameters: Nonlinearity Calculation of the Charging energy E C Solve the Gross-Pitaevskii equation Calculate from onsite integrals Φl,r 4 d 3 x or Ananikian et al. PRA 73, (2006), Zapata et al. PRA 57, R28 (1998)

20 The Parameters: Nonlinearity Calculation of the Charging energy E C Solve the Gross-Pitaevskii equation Calculate from onsite integrals Φl,r 4 d 3 x or (better) use E C = 2 µ N, where µ is the chemical potential Ananikian et al. PRA 73, (2006), Zapata et al. PRA 57, R28 (1998)

21 The Parameters: Detuning

22 Control Knobs Tuning of χ = E C /2 and Ω = 2E J /N No Feshbach resonance available with 87 Rb no tuning of E C

23 Control Knobs Tuning of χ = E C /2 and Ω = 2E J /N No Feshbach resonance available with 87 Rb no tuning of E C Control of the barrier height V 0 tuning of E J Josephson energy (khz) barrier height 2 (khz) Charging energy (Hz)

24 Control Knobs Tuning of χ = E C /2 and Ω = 2E J /N No Feshbach resonance available with 87 Rb no tuning of E C Control of the barrier height V 0 tuning of E J Josephson energy (khz) barrier height 2 (khz) Charging energy (Hz) For N = 2000 atoms! Ω max 100 mhz to 4 Hz χ 1 Hz interactions never negligible

25 Outline Spin Squeezing Of A Double Well BEC The Experimental System And Its Parameters Experimental Implementation Adiabatic Spin Squeezing Alternative Approaches To Quantum Metrology

26 A Bosonic Josephson Junction (Array) Realization of an optical double well trap Control of the number of occupied wells by power in crossed dipole beam. Albiez et al. PRL (2005), Estève et al. Nature 455, 1216 (2008)

27 Realizable Parameter Regime Atom number in two neighboring wells: N = 1000 to N = 2000 Coupling Ω max 100 mhz to 4 Hz Nonlinearity χ 1 Hz Nonlinearity can not be switched off

28 Realizable Parameter Regime Atom number in two neighboring wells: N = 1000 to N = 2000 Coupling Ω max 100 mhz to 4 Hz Nonlinearity χ 1 Hz Nonlinearity can not be switched off Large amplitude Rabi oszillations not possible self trapping Coupling pulses (π/2, π, ) not easily realized

29 Mean Field Dynamics Of The Interacting Double Well Zero detuning: Ĥ = χ ˆ J z 2 Ω ˆ J x Nonlinearity M. Albiez, PRL 95, (2005) Zibold et al. PRL 105, (2010)

30 Mean Field Dynamics Of The Interacting Double Well Zero detuning: Ĥ = χ ˆ J z 2 Ω ˆ J x Nonlinearity Front of sphere M. Albiez, PRL 95, (2005) Zibold et al. PRL 105, (2010)

31 Mean Field Dynamics Of The Interacting Double Well Zero detuning: Ĥ = χ ˆ J z 2 Ω ˆ J x Nonlinearity Front of sphere Back of sphere M. Albiez, PRL 95, (2005) Zibold et al. PRL 105, (2010)

32 Mean Field Dynamics Of The Interacting Double Well Zero detuning: Ĥ = χ ˆ J z 2 Ω ˆ J x Nonlinearity Front of sphere Back of sphere Resulting trajectories on the Blochsphere M. Albiez, PRL 95, (2005) Zibold et al. PRL 105, (2010)

33 Self Trapping Double well: a Josephson oscillations b Self- trapping relative phase φ [π] population imbalance z relative phase φ [π] population imbalance z evolution time [ms] evolution time [ms] M. Albiez, PRL 95, (2005) Zibold et al. PRL 105, (2010)

34 Self Trapping Double well: Internal states: a Josephson oscillations b Self- trapping relative phase φ [π] population imbalance z relative phase φ [π] population imbalance z evolution time [ms] evolution time [ms] M. Albiez, PRL 95, (2005) Zibold et al. PRL 105, (2010)

35 Squeezing Strategies One-axis-twisting Requires non-adiabatic control over state rotations. difficult to realize.

36 Squeezing Strategies One-axis-twisting Requires non-adiabatic control over state rotations. difficult to realize. Adiabatic squeezing Nonlinearity relatively strong No atom number loss problems Relatively simple to implement. (At least at first sight) Prepare BEC in double well at low lattice Adiabatically ramp up the lattice

37 The Ground State Ramping the barrier height 0.1 Rabi 0.1 Josephson 0.1 Fock Jz fluctuations (db) Rabi Josephson Fock coherence 0.5 spin squeezing (db) Heisenberg limit regime parameter

38 Finite Temperature? Does temperature play a role? Condensation directly into the double well potential Initial state in thermal equilibrium

39 Finite Temperature? Does temperature play a role? Condensation directly into the double well potential Initial state in thermal equilibrium Energy scales: Minimal temperature: T 10 nk = 200 Hz h/k B Nonlinearity χ and coupling Ω in the few Hz range

40 Finite Temperature? Does temperature play a role? Condensation directly into the double well potential Initial state in thermal equilibrium Energy scales: Minimal temperature: T 10 nk = 200 Hz h/k B Nonlinearity χ and coupling Ω in the few Hz range Temperature is very important to consider. More careful treatment needed.

41 Collective Modes: Plasma Frequency Collective modes required to understand the system. Low energy excitations: Plasma oszillations High energy excitations: Self trapped modes

42 Collective Modes: Plasma Frequency Collective modes required to understand the system. Low energy excitations: Plasma oszillations High energy excitations: Self trapped modes

43 Collective Modes: Plasma Frequency Collective modes required to understand the system. Low energy excitations: Plasma oszillations High energy excitations: Self trapped modes Maximum plasma frequency: ω pl = 2π 60 Hz (Note ω pl,max ω trap ) (Reminder: temperature T = 200 Hz)

44 The Challenges: Finite Temperature (I) Many-body modes, resulting fluctuations and adiabatic transformations Hz 950Hz 1450Hz 20nK population energy Hz Josephson mode #

45 The Challenges: Finite Temperature (I) Many-body modes, resulting fluctuations and adiabatic transformations 5000 J z Distributions Hz 950Hz 1450Hz 20nK population energy Hz Josephson mode # Adiabatic Cooling

46 The Challenges: Finite Temperature (II) Phase diagram : Adiabats and isothermals in the number squeezing coherence plane (18.5) number squeezing (db) GS 5 1 (6.2) (1.5) (0.3) isothermal adiabatic coherence Adiabatic barrier ramping increases spin squeezing.

47 Challenges: Position Stability (I) For the double well trap position stability better then z = 100 nm required Solution 1: Active phase stabilization:

48 Challenges: Position Stability (I) For the double well trap position stability better then z = 100 nm required Solution 1: Active phase stabilization:

49 Solution 2: Passive stability The Block :

50 Challenges: Position Stability (II) Solution 3: Multi well system: Release the harmonic confinement in lattice direction Required position stability z ω 2 Analyze spin squeezing between two neighboring sites (local observables) Better statistics! Estève et al. Nature 455, 1216 (2008), Gross et al. PRA 84, (2011)

51 Outline Spin Squeezing Of A Double Well BEC The Experimental System And Its Parameters Experimental Implementation Adiabatic Spin Squeezing Alternative Approaches To Quantum Metrology

52 Squeezing Measurements Spin squeezing, necessary incredient: Number squeezing ξ 2 N = 4 2 Jˆ z N < 1

53 Squeezing Measurements Spin squeezing, necessary incredient: Number squeezing ξ 2 N = 4 2 Jˆ z N < 1

54 Squeezing Measurements Spin squeezing, necessary incredient: Number squeezing ξ 2 N = 4 2 Jˆ z N < 1 Low lattice depth Broad distribution of n = J z High lattice depth Narrow distribution of n = J z

55 How Slow Is Slow Enough? Goal: Optimize number squeezing Ramp speed number squeezing (db) barrier height time ramp up time (ms) ramp up time (ms) number squeezing (db) Best achieved number squeezing: Multi well: ξ 2 N 6 db Double well: ξ 2 N 2 db

56 Coherence Measurements (I) Number squeezing is not sufficient Spin squeezing: ξ 2 S = ξ2 N V 2 Coherence (Visibility in Ramsey experiment) crucial

57 Coherence Measurements (I) Number squeezing is not sufficient Spin squeezing: ξ 2 S = ξ2 N V 2 Coherence (Visibility in Ramsey experiment) crucial BUT: Spin rotations are very hard here

58 Coherence Measurements (I) Number squeezing is not sufficient Spin squeezing: ξ 2 S = ξ2 N V 2 Coherence (Visibility in Ramsey experiment) crucial BUT: Spin rotations are very hard here Solution: Measure coherence via time-of-flight interference

59 Coherence Measurements (II) Time-of-flight interference measurements single shots a b c d Single shot visibility very high Single spatial mode per well

60 Coherence Measurements (III) Coherence follows V from the ensemble averaged visibility a b c d 1d density longitudinal coordinate ( )

61 Optimizing Number Squeezing And Coherence Knowing the ramp speed (a few Hz/ms), to which height should we ramp to optimize spin squeezing? 1 1 < cos( φ ) > Barrier height Time < cos( φ ) > Barrier height Time (db) 0 (db) 0 N umber squeezing ξ 2 N N umber squeezing ξ 2 N Barrier height (Hz) Barrier height (Hz)

62 A Spin Squeezed Josephson Junction The experimental Phase diagram 0 2 ξ N (db) squeezing Number db - 3 db - 6 db Phase coherence <cos(φ)> 2

63 A Spin Squeezed Josephson Junction The experimental Phase diagram 0 2 ξ N (db) squeezing Number db - 3 db - 6 db Phase coherence <cos(φ)> 2 Double well (green): ξ 2 S Multi well (red/blue): ξ 2 S = 2.3 db = 3.8 db

64 Room For Improvement 3.8 db squeezing is nice, but can it be done better? T. Berrada, Nat Commun doi: /ncomms3077, (2013)

65 Room For Improvement 3.8 db squeezing is nice, but can it be done better? In principle: YES T. Berrada, Nat Commun doi: /ncomms3077, (2013)

66 Room For Improvement 3.8 db squeezing is nice, but can it be done better? In principle: YES In practice: very hard in optical traps T. Berrada, Nat Commun doi: /ncomms3077, (2013)

67 Room For Improvement 3.8 db squeezing is nice, but can it be done better? In principle: YES In practice: very hard in optical traps Why? Maximum plasma frequency is tiny temperature in the pk regime reqired T. Berrada, Nat Commun doi: /ncomms3077, (2013)

68 Room For Improvement 3.8 db squeezing is nice, but can it be done better? In principle: YES In practice: very hard in optical traps Why? Maximum plasma frequency is tiny temperature in the pk regime reqired Better: Split 1D condensate along tight direction Atom Chips = 7.8 db demonstrated. ξ 2 S T. Berrada, Nat Commun doi: /ncomms3077, (2013)

69 Outline Spin Squeezing Of A Double Well BEC Alternative Approaches To Quantum Metrology

70 Alternative Approaches To Quantum Metrology Encode phase information in the variance (Twin Fock states) R. Bücker, Nature Physics 7, 608 (2011), B. Lucke, Science 334, 773 (2011), C. Gross, Nature 480, 219 (2011)

71 Alternative Approaches To Quantum Metrology Encode phase information in the variance (Twin Fock states) R. Bücker, Nature Physics 7, 608 (2011), B. Lucke, Science 334, 773 (2011), C. Gross, Nature 480, 219 (2011) NOON states: N-fold enhanced fringe period (Ions) D. Leibfried, Nature 438, 639 (2005)

72 Spin Squeezing For Optical Lattice Clocks Alkaline earth atoms (Mg, Sr, Yb,... ) Use Singlet Triplet intercombination line (mhz) A priory: No interactions, atoms isolated Use long range interactions for squeezing: Rydberg atoms A. Derevianko, RMP 83, 331 (2011), L. I. R. Gil, arxiv: v2 (2013)

73 Spin Squeezing With Rydberg Dressing (I) Rydberg atoms Very strong van-der-waals interactions V = C 6 /d 6 V 100 d = 1 µm Blockade radius R b = 6 C 6 / Ω 10 µm L. I. R. Gil, arxiv: v2 (2013)

74 Spin Squeezing With Rydberg Dressing (II) Rydberg dressing Admix small fraction of Rydberg state d = (1 ɛ) g + ɛ r Resulting soft core potential V = V 0 = const. in blockade Hamiltonian (fully blockaded) One axis-twisting! Ĥ = Ω ˆ J x + V 0 ˆ J z 2 /2 + δ ˆ J z L. I. R. Gil, arxiv: v2 (2013)

75 Spin Squeezing With Rydberg Dressing (III) Soft core interaction V Hz Very fast squeezing O(1 ms) Interaction switchable via Laser dressing L. I. R. Gil, arxiv: v2 (2013)

76 GS (18.5) (6.2) (1.5) (0.3) isothermal adiabatic coherence 2 ξ N 250Hz Hz 1450Hz Josephson mode # db - 3 db 20nK - 6 db Phase coherence <cos(φ)> Summary Dynamics of a interacting two level system Adiabatic spin squeezing number squeezing (db) Finite temperature effects population energy Hz Spin squeezing spatial DOF (db) Number squeezing Alternative strategies for Quantum Metrology

77 THE END