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Non-linear Interferometry Beyond the Standard Quantum Limit

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Spin Squeezing and Non-linear Atom Interferometry with Bose-Einstein Condensates

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Abstract

In the experiments described in the previous chapter we detected coherent spin squeezed atomic quantum states. However the implementation of a full atom interferometer where the two modes are defined by two mean field wavefunctions in a double well potential is difficult. One of the problems is the limited range in which the system parameter \(\Uplambda\) can be tuned—especially the Rabi regime is not accessible for our setup.

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Notes

  1. 1.

    We use the excitation of these modes when changing the barrier height \(V_0\) abruptly for a calibration of \(V_0\)[3].

  2. 2.

    Technologically a coherent change of the frequency of one of the two electromagnetic fields is not possible in our experiment.

  3. 3.

    We use the labeling \(|{a}\rangle\) and \(|{b}\rangle\) for both the single particle states and the mean field modes which is not rigorously correct. However since we neglect external dynamics and assume perfect wavefunction overlap this labeling is justified.

  4. 4.

    Dipolar relaxation is a two body process meaning its rate \(L_2 \propto K_2N\) is proportional to the number of atoms in the trap. The loss coefficient for the \(|2,-1\rangle\) state was measured to \(K_2 = 8.8 \times 10^{-14}\,\text{cm}^3/\text{s}\) [26]

  5. 5.

    At the time this measurement was done the active magnetic field compensation was not yet installed. Therefore we measure a larger width as theoretically expected. The inelastic width extracted from the measurement shown in Fig. 4.5 agrees well with the theoretical prediction.

  6. 6.

    Experimentally we also found the best number squeezing at this magnetic field.

  7. 7.

    A lower Rabi frequency than for the usual coupling is chosen here in order to work in a higher \(\Uplambda\) situation.

  8. 8.

    We assume here without loss of generality \(\langle\hat J_y\rangle = 0\) such that the twist is symmetric to the \(J_y\) axis.

  9. 9.

    We assume small fluctuations \(\Updelta J^2_{\perp, {\rm max}}\) as compared to the total atom number \(\Updelta J^2_{\perp, {\rm max}} < {N}^{2}/4\) such that the Bloch sphere can be locally approximated by a plane.

  10. 10.

    The single-photon Rabi frequency for the coupling of the Zeeman sub-states is approximately \(2\pi \cdot 10 \) kHz.

  11. 11.

    Experimentally the phase of the coupling pulses at time t can be found by a measurement of the population imbalance versus pulse phase of a final \(\pi/2\) pulse. The zero crossings identify the two phases where the rotation axis hits the center of the spin state.

  12. 12.

    60 experimental repetitions define one dataset in all measurements done in context with the internal spin system.

  13. 13.

    In order to obtain better statistics we average the measurements for \(\alpha = 16^\circ\) and \(\alpha = 17^\circ\).

  14. 14.

    As explained in In Sect. 3.4.1 we always remove the photon shot noise in these experiments.

  15. 15.

    Depending on the relative position of dipole trap and optical lattice the central well can contain up to \(450\) atoms.

  16. 16.

    Experimentally the number of echo pulses should be kept minimal, since the coupling pulses introduce additional noise due to fluctuations of pulse phase and power.

  17. 17.

    The spacing between the wells is only \(5.7\mu\!\)m such that magnetic field fluctuations and the electromagnetic radiation fields for the coupling are homogeneous over the whole system.

  18. 18.

    Fluctuation measurements on a coherent spin state after a \(7\pi/2\) pulse in a Rabi cycle still show shot noise limited noise characteristics. This indicates negligible pulse power fluctuations in our experiment and suggests again that shot-to-shot magnetic field fluctuations are the main noise source.

  19. 19.

    As a reminder, \(p=\langle n_a/N\rangle\) is the probability for an atom to be found in mode \(|{a}\rangle\).

References

  1. Leggett A (2001) Bose–Einstein condensation in the alkali gases. Some fundamental concepts. Rev Mod Phys 73:307–356

    Article  ADS  Google Scholar 

  2. Pezzé L, Smerzi A, Berman GP, Bishop AR, Collins LA (2006) Nonlinear beam splitter in Bose–Einstein-condensate interferometers. Phys Rev A 74:033610

    Article  ADS  Google Scholar 

  3. Gati R (2007) Bose–Einstein condensates in a single double well potential. Ph.D. thesis, University of Heidelberg

    Google Scholar 

  4. Santarelli G et al (1999) Quantum projection noise in an atomic fountain. A high stability cesium frequency standard. Phys Rev Lett 82:4619–4622

    Article  ADS  Google Scholar 

  5. Vandersypen LMK, Chuang IL (2005) NMR techniques for quantum control and computation. Rev Mod Phys. 76:1037–1069

    Article  ADS  Google Scholar 

  6. Sørensen AS, Duan L, Cirac J, Zoller P (2001) Many-particle entanglement with Bose–Einstein condensates. Nature 409:63–6

    Article  ADS  Google Scholar 

  7. Kaufman AM et al (2009) Radio-frequency dressing of multiple Feshbach resonances. Phys Rev A 80:050701

    Article  ADS  Google Scholar 

  8. Widera A et al (2004) Entanglement interferometry for precision measurement of atomic scattering properties. Phys Rev Lett 92:160406

    Article  ADS  Google Scholar 

  9. Erhard M, Schmaljohann H, Kronjäger J, Bongs K, Sengstock K (2004) Measurement of a mixed-spin-channel Feshbach resonance in \(^{87}\)Rb. Phys Rev A 69:032705

    Google Scholar 

  10. Steel M, Collett M (1998) Quantum state of two trapped Bose–Einstein condensates with a Josephson coupling. Phys Rev A 57:2920–2930

    Article  ADS  Google Scholar 

  11. Sinatra A, Castin Y (2000) Binary mixtures of Bose–Einstein condensates. Phase dynamics and spatial dynamics. Eur Phys J D 8:319

    Article  ADS  Google Scholar 

  12. Li Y, Treutlein P, Reichel J, Sinatra A (2009) Spin squeezing in a bimodal condensate. spatial dynamics and particle losses. Eur Phys J B 68:365–381

    Article  ADS  Google Scholar 

  13. Poulsen UV, Mølmer K (2001) Positive-P simulations of spin squeezing in a two-component Bose condensate. Phys Rev A 64:013616

    Article  ADS  Google Scholar 

  14. Hall DS, Matthews MR, Wieman CE, Cornell EA (1998) Measurements of relative phase in two-component Bose–Einstein condensates. Phys Rev Lett 81:1543–1546

    Article  ADS  Google Scholar 

  15. Matthews MR et al (1998) Dynamical response of a Bose–Einstein condensate to a discontinuous change in internal state. Phys Rev Lett 81:243–247

    Article  ADS  Google Scholar 

  16. Treutlein P, Hommelhoff P, Steinmetz T, Hänsch TW, Reichel J (2004) Coherence in microchip traps. Phys Rev Lett 92:203005

    Article  ADS  Google Scholar 

  17. Cohen-Tannoudji C, Dupont-Roc J, Grynberg G (1992) Atom-photon interactions: basic processes and applications. Wiley, New York

    Google Scholar 

  18. Milburn G, Corney J, Wright E, Walls D (1997) Quantum dynamics of an atomic Bose–Einstein condensate in a double-well potential. Phys Rev A 55:4318–4324

    Article  ADS  Google Scholar 

  19. Javanainen J, Ivanov M (1999) Splitting a trap containing a Bose–Einstein condensate. Atom number fluctuations. Phys Rev A 60:2351–2359

    Article  ADS  Google Scholar 

  20. Bohi P et al (2009) Coherent manipulation of Bose–Einstein condensates with state-dependent microwave potentials on an atom chip. Nat Phys 5:592–597

    Article  Google Scholar 

  21. Chin C, Grimm R, Julienne P, Tiesinga E (2010) Feshbach resonances in ultracold gases. Rev Mod Phys 82:1225

    Article  ADS  Google Scholar 

  22. Hall DS, Matthews MR, Ensher JR, Wieman CE, Cornell EA (1998) Dynamics of component separation in a binary mixture of Bose–Einstein condensates. Phys Rev Lett 81:1539–1542

    Article  ADS  Google Scholar 

  23. Timmermans E (1998) Phase separation of Bose–Einstein condensates. Phys Rev Lett 81:5718–5721

    Article  ADS  Google Scholar 

  24. Li Y, Castin Y, Sinatra A (2008) Optimum spin squeezing in Bose–Einstein condensates with particle losses. Phys Rev Lett 100:210401

    Article  ADS  Google Scholar 

  25. Raghavan S, Smerzi A, Fantoni S, Shenoy SR (1999) Coherent oscillations between two weakly coupled Bose–Einstein condensates. Josephson effects, \(\pi\) oscillations, and macroscopic quantum self-trapping. Phys Rev A 59:620–633

    Google Scholar 

  26. Widera A et al (2005) Coherent collisional spin dynamics in optical lattices. Phys Rev Lett 95:190405

    Article  ADS  Google Scholar 

  27. Kitagawa M, Ueda M (1993) Squeezed spin states. Phys Rev A 47:5138–5143

    Article  ADS  Google Scholar 

  28. Bachor HA, Ralph TC (2004) A guide to experiments in quantum optics. Wiley, New York

    Book  Google Scholar 

  29. Sinatra A, Castin Y (1998) Phase dynamics of Bose–Einstein condensates. Losses versus revivals. Eur Phys J D 4:247–260

    Article  ADS  Google Scholar 

  30. Wright EM, Walls DF, Garrison JC (1996) Collapses and revivals of Bose–Einstein condensates formed in small atomic samples. Phys Rev Lett 77:2158–2161

    Article  ADS  Google Scholar 

  31. Yurke B, Stoler D (1986) Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion. Phys Rev Lett 57:13–16

    Article  ADS  Google Scholar 

  32. Greiner M, Mandel O, Hänsch TW, Bloch I (2002) Collapse and revival of the matter wave field of a Bose–Einstein condensate. Nature 419:51–54

    Article  ADS  Google Scholar 

  33. Ferrini G, Minguzzi A, Hekking FWJ (2008) Number squeezing quantum fluctuations and oscillations in mesoscopic Bose Josephson junctions. Phys Rev (A) 78:023606

    Article  ADS  Google Scholar 

  34. Pezzé L, Smerzi A (2009) Entanglement nonlinear dynamics and the Heisenberg limit. Phys Rev Lett 102:100401

    Article  MathSciNet  ADS  Google Scholar 

  35. Pezzé L, Collins LA, Smerzi A, Berman GP, Bishop AR (2005) Sub-shot-noise phase sensitivity with a Bose–Einstein condensate Mach-Zehnder interferometer. Phys Rev (A) 72:043612

    Article  ADS  Google Scholar 

  36. Appel J et al (2009) Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit. Proc Natl Acad Sci U S A. 106:10960–10965

    Article  ADS  Google Scholar 

  37. Fernholz T (2008) Spin squeezing of atomic ensembles via nuclear-electronic spin entanglement. Phys Rev Lett 101:073601

    Article  ADS  Google Scholar 

  38. Ramsey NF (1950) A molecular beam resonance method with separated oscillating fields. Phys Rev 78:695–699

    Article  ADS  Google Scholar 

  39. Ramsey NF (1949) A new molecular beam resonance method. Phys Rev 76:996

    Article  ADS  Google Scholar 

  40. Albiez M (2005) Observation of nonlinear tunneling of a Bose–Einstein condensate in a single Josephson junction. Ph.D. thesis, University of Heidelberg

    Google Scholar 

  41. Metcalf H, Vander Straten P (1999) Laser cooling and trapping. Springer, New York

    Book  Google Scholar 

  42. Gross C, Zibold T, Nicklas E, Estève J, Oberthaler MK (2010) Nonlinear atom interferometer surpasses classical precision limit. Nature 464:1165–1169

    Article  ADS  Google Scholar 

  43. Wineland D, Bollinger J, Itano W, Heinzen D (1994) Squeezed atomic states and projection noise in spectroscopy. Phys Rev A 50:67–88

    Article  ADS  Google Scholar 

  44. Sørensen AS, Mølmer K (2001) Entanglement and extreme spin squeezing. Phys Rev Lett 86:4431–4434

    Article  ADS  Google Scholar 

  45. Viola L, Lloyd S (1998) Dynamical suppression of decoherence in two-state quantum systems. Phys Rev (A) 58:2733–2744

    Article  MathSciNet  ADS  Google Scholar 

  46. Ferrini G, Spehner D, Minguzzi A, Hekking FWJ (2010) Anomalous decoherence rate of macroscopic superpositions in Bose Josephson junctions. Phys Rev A 82:033621

    Article  ADS  Google Scholar 

  47. Search CP, Meystre P (2003) Noise limits in matter-wave interferometry using degenerate quantum gases. Phys Rev (A) 67:061601

    Article  ADS  Google Scholar 

  48. Scully MO, Dowling JP (1993) Quantum-noise limits to matter-wave interferometry. Phys Rev (A) 48:3186–3190

    Article  ADS  Google Scholar 

  49. Cronin AD, Schmiedmayer J, Pritchard DE (2009) Optics and interferometry with atoms and molecules. Rev Mod Phys 81:1051

    Article  ADS  Google Scholar 

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  1. University of Heidelberg, Heidelberg, Germany

    Christian Groß

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  1. Christian Groß
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Groß, C. (2012). Non-linear Interferometry Beyond the Standard Quantum Limit. In: Spin Squeezing and Non-linear Atom Interferometry with Bose-Einstein Condensates. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25637-0_4

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  • DOI: https://doi.org/10.1007/978-3-642-25637-0_4

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