Appendix A One-Dimensional Gross-Pitaevskii Simulations in the Transverse Potential

Appendix A One-Dimensional Gross-Pitaevskii Simulations in the Transverse Potential A.1 Effective Interaction Constant for the Transverse GPE Simulations Because of our elongated geometries (see Sect. 1.1.3.3) and also because the dynamics of the BEC in the double wells occurs essentially in the direction of splitting, it is often sufficient to resort to 1D simulations in the transverse horizontal direction. The 1D Gross-Pitaevskii equation in the (x)-direction reads i ψ t = 2 2 ψ 2m 2 x + V (x)ψ + g N ψ 2 ψ (A.1) where ψ(x) denotes the transverse wave function and g is the effective transverse interaction constant. We want to derive an analytical expression for g at short times right after splitting. We begin by assuming that the wave function is separable, and that the initial unsplit cloud is described by a 1D Thomas-Fermi profile ϕ 0 (z) for N atoms at ). The splitting process, which occurs within ms, can be seen as non-adiabatic with respect to the slow axial dynamics. We therefore that immediately after splitting, the axial profile remains unchanged. In the transverse direction, on the other hand, we assume that the wave function is close to the non-interacting transverse ground state. In particular, we assume that in the vertical transverse direction, the potential after splitting is harmonic with the frequency ω y and that the wave function φ(y) in (y)-direction is in the non-interacting Gaussian ground state. Under these assumptions, the effective transverse interaction constant reads g g 3D ϕ 0 (z) 4 dz φ(y) 4 dy (A.2) equilibrium in the initial (close-to) harmonic trap (frequencies ω (0) x,y,z Springer International Publishing Switzerland 2016 T. Berrada, Interferometry with Interacting Bose-Einstein Condensates in a Double-Well Potential, Springer Theses, DOI 10.1007/978-3-319-27233-7 219

220 Appendix A: One-Dimensional Gross-Pitaevskii Simulations ( = 2 a 2 5 32/3 2π s a (0) }{{} N 2.09 x ω z (0) ω y ωy ω (0) y ) 1/3 (A.3) For usual parameters, g /h = 0.30 Hz µm (to be compared for example to the axial 1D effective interaction constant g 1D = 2 ω = h 17 Hz µm. It means that in the transverse direction, where kinetic energy dominates, the effect of non-linearity is rather weak. A.2 Parameter Estimation of the BJJ The parameters of the BJJ, such as displayed in Fig. 3.14, are calculated by computing the left and right mode wave functions in the one-dimensional transverse potential obtained from the rf-dressing simulations beyond the RWA. More precisely: The ground and the first excited state wavefunctions φ g (x) and φ e (x) of the 1D GPE equation in the transverse potential are computed using the effective interaction constant introduced above. We resorted to an imaginary-time evolution using the standard split-operator method (with a symmetrized time step, ensuring that the local error at each time step is of order O( t 3 ) and the global error on the final result is of order O( t 2 ) [1]. The left and right mode wavefunctions are defined as linear combinations of φ g and φ e : φ L (x) = φ g + φ e 2 φ R (x) = φ g φ e 2 (A.4) (A.5) The parameters of the BJJ in the standard two-mode model (tunnel coupling and on-site interaction energies) are numerically computed using the integrals ( 2 ) J = 2m φ L φ R + φ L V φ R dx, (A.6) U L,R = g φ L,R 4 dx. (A.7) To compute the spatial derivatives, we make use of the fact that the operator is diagonal in momentum representation and compute the Fourier transforms of φ L,R. The parameters of the improved two-mode model (see Ref. [2]) can also be numerically computed using φ g and φ e.

Appendix A: One-Dimensional Gross-Pitaevskii Simulations 221 Great care has to be given to the choice of the space and time steps. Typical values are x = 5 nm (to be compared to the radial harmonic oscillator length 250 nm) and t = 5 µs (to be compared to the typical transverse oscillation period 0.5 ms). The same propagation method can be used to compute the GPE dynamics in the 1D potential. It can be adapted to the case of a time-dependent potential, for example to simulate the splitting and recombination ramps.

Appendix B List of Symbols The numerical values specific to Rubidium 87 are given for the F = 1, m F = 1 hyperfine level of the ground state (5 2 S 1/2 ). Physical constants Reduced Planck constant: 1.05 10 34 m 2 kg s 1 k B Boltzmann constant: 1.38 10 23 m 2 kg s 2 K 1 a 0 Bohr radius: 5.29 10 11 m μ B Bohr magneton: 9.27 10 24 JT 1 μ 0 Vacuum permeability: 4π 10 7 VsA 1 m 1 The ideal Bose gas n 0 l ɛ l β T T c Ground state uniform density Quantum numbers k = 2π ( L lx ˆx + l y ŷ + l z ẑ ) Kinetic energy of plane (matter-)wave Inverse temperature in units of k B Thermal de Brolie wavelength Critical temperature for Bose-Einstein condensation The weakly interacting Bose gas m Mass of one boson: 1.44 10 25 kg a s s-wave scattering length: 5.32 10 9 m a l, a l Bosonic creation and annihilation operator in the mode l. ˆ, ˆ Bosonic field operators. V ( r) External potential g 3D = 4π 2 a s /m 3D interaction constant: 5.14 10 51 Jm 3 Springer International Publishing Switzerland 2016 T. Berrada, Interferometry with Interacting Bose-Einstein Condensates in a Double-Well Potential, Springer Theses, DOI 10.1007/978-3-319-27233-7 223

224 Appendix B: List of Symbols Elongated condensates a, a χ = Na s a /a 2 g 1D = 2 ω a s R TF1D μ TF1D Harmonic oscillator length in the transverse (longitudinal) direction χ-parameter (ratio of interaction energy over radial kinetic energy) Effective 1D interaction constant 1D Thomas-Fermi radius 1D Thomas-Fermi chemical potential Two-mode Bose-Hubbard model N Total atom number N L,R Number of atoms in the left (right) mode φ L ( r) Spatial wave function for the left (right) mode a L,R, a L,R Bosonic creation and annihilation operator Energies J Tunnel coupling energy U L,R On-site interaction energy constant in the left (right) mode U = (U L + U R ) /2 Averaged on-site interaction energy constant = EL 0 E0 R Difference of zero-point energies between the modes ɛ = (U L U R )(N 1)/2 + Full energy detuning Dimensionless parameters: ratio of tunneling and interaction energy = UN/2J MQST threshold: >2 γ = U/2J Rabi: γ 1/N, Josephson: 1/N γ N, Fock: γ N η = NJ/2U Phase diffusion threshold: η 1/4 Macroscopic observables n = (N L N R )/2 Half-number imbalance z = (N L N R )/N Normalized population imbalance φ = (φ L φ R )/2 Half-number imbalance φ = (φ L φ R )/2 Half-number imbalance cos(φ φ ) Coherence factor Squeezing factors ξ N = (N L N R )/ N Number-squeezing factor ξ φ = (φ) N Phase-squeezing factor ξ S = ξ N / cos φ (Coherent, or useful) spin-squeezing factor

Appendix B: List of Symbols 225 Magnetic trapping and rf-dressed potentials g F Landé factor: 2.00 κ = g F μ B Linear Zeeman shift: h 0.7 MHz/G V TB Trap bottom ω x,y,z Angular trap frequencies s Static field magnetic energy (over hbar) RF Radio-frequency field Rabi angular frequency ω rf frequency δ = ω s rf detuning α Tilt angle RF Amp rf dressing intensity (in each wire) in unit of I0 max = 79.5 mapp Critical splitting intensity (appearance of second minimum) RF c Amp Imaging Ɣ = 2π 6.07 MHz Natural line width σ 0 = 2.91 10 9 cm 2 Absorption cross section I sat = 1.67mW cm 2 Saturation intensity α = σ 0 /σ eff Absorption cross section correction factor p,σ p Number of detected photons per atom, (mean, std. dev) b,σ b Number of background photons per pixel (mean, std. dev.) Sensitivity limits φ SQL = 1/ N Standard quantum limit (or shot noise limit, quantum proj. noise) φ H = 1/N Heisenberg limit φ d Phase noise on tof phase estimation for coh. states Minimum detectable number squeezing ξ N,d Miscellaneous Phase diffusion R Phase diffusion rate τ coh = 1/R Phase coherence time Timings t φ Phase accumulation time Duration of the beam-splitter operation t BS

Curriculum Vitae Dr. Tarik Berrada Theresiengasse 26/8, 1180 Vienna (Austria) born on 30/07/1985 in Brive-la-Gaillarde (France) French/Moroccan nationalities +43 (0)650 32 42 053 dr.tarik.berrada@gmail.com Researcher ID: H-3960-2014 Professional Experience 2014 2015 Analyst, Metalogic GmbH, Munich Development of statistical shortterm prediction methods for the energy market. 2014 2015 Postdoctoral researcher, Vienna University of Technology (J. Schmiedmayer s group) Experimental and theoretical research on the physics of ultracold atomic quantum gases. Studies 2009 2014 Ph.D. in physics, Vienna University of Technology (J. Schmiedmayer s group) Austrian-French research project in quantum physics between TU Vienna and Institut d Optique, Palaiseau, France. Development of new techniques for quantum noise reduction in high precision interferometric measurements with ultracold atoms. 12/05/2014 Ph.D. defence: Mach-Zehnder interferometry with interacting Bose- Einstein condensates in a double-well potential, passed with distinction (sehr gut). 2008 2009 Research assistant, University of Vienna (M. Arndt s group) Springer International Publishing Switzerland 2016 T. Berrada, Interferometry with Interacting Bose-Einstein Condensates in a Double-Well Potential, Springer Theses, DOI 10.1007/978-3-319-27233-7 227

228 Curriculum Vitae 2005 2008 Master of engineering, Télécom ParisTech, Paris Double degree with: Master of Science in quantum physics at Université Paris Sud XI, Orsay. 2003 2005 Preparatory class, Lycée Saint Louis, Paris Scholarships: 2011 2013 Fellowship of the Vienna Doctoral Program on Complex Quantum Systems (CoQuS) 2007 2008 French studies scholarship Other: 2012 Organizer of the annual summer school on Complex Quantum Systems, Vienna Languages French: Native speaker English: Fluent German: Bilingual, lived in Austria for more than 10 years Arabic: Fair oral skills List of Publications Publications Resulting from the Ph.D. thesis: 2014 Interferometry with non-classical motional states of a Bose-Einstein condensate, S. van Frank, A. Negreti, T. Berrada, R. Bücker, S. Montangero, J.-F. Schaff, T. Schumm, T. Callarco & J. Schmiedmayer, Nat. Commun. 5, 4009 (2014) 2013 Integrated Mach-Zehnder interferometer for Bose-Einstein condensates, T. Berrada, S. van Frank, R. Bücker, T. Schumm, J.-F. Schaff & J. Schmiedmayer, Nat. Commun. 4, 2077 (2013) Vibrational state inversion of a Bose-Einstein condensate: optimal control and state tomography, R. Bücker, T. Berrada, S. van Frank, J.-F. Schaff, T. Schumm, J. Schmiedmayer, G. Jäger, J. Grond & U. Hohenester, J. Phys. B 46, 104012 (2013) 2012 Momentum distribution of one-dimensional Bose gases at the quasicondensation crossover: Theoretical and experimental investigation, T. Jacqmin, B. Fang, T. Berrada, T. Roscilde & I. Bouchoule, Phys. Rev. A 86, 043626 (2012) Dynamics of parametric matter-wave amplification, R. Bücker, U. Hohenester, T. Berrada, S. van Frank, A. Perrin, S. Manz, T. Betz, J. Grond, T. Schumm, and J. Schmiedmayer, Phys. Rev. A 86, 013638 (2012) 2011 Sub-Poissonian Fluctuations in a 1D Bose Gas: From the Quantum Quasicondensate to the Strongly Interacting Regime, T. Jacqmin, J. Armijo, T. Berrada, K. Kheruntsyan & I. Bouchoule, Phys. Rev. L 106, 230405 (2011) Twin-atom beams, R. Bücker, J. Grond, S. Manz, T. Berrada, T. Betz, C. Koller, U. Hohenester, T. Schumm, A. Perrin & J. Schmiedmayer, Nature Physics 7, 608 611 (2011)

Curriculum Vitae 229 Two-Point Phase Correlations of a One-Dimensional Bosonic Josephson Junction, T. Betz, S. Manz, R. Bücker, T. Berrada, C. Koller, G. Kazakov, I. Mazets, H. Stimming, A. Perrin, T. Schumm & J. Schmiedmayer, Phys. Rev. L 106, 020407 (2011) Publications Resulting from the Master thesis: 2010 Influence of conformational molecular dynamics on matter wave interferometry, M. Gring, S. Gerlich, S. Eibenberger, S. Nimmrichter, T. Berrada, M. Arndt, H. Ulbricht, K. Hornberger, M. Müri, M. Mayor, M. Bröckmann & N. Doltsinis, Phys. Rev. A 81, 031604(R) (2010)