Higher order connected correlation functions

Transcription

1 Higher order connected correlation functions AQuS, September 5-8, 2016 Thomas Schweigler Schmiedmayer Group, Atominstitut, TU Vienna Vienna Center for Quantum Science and Technology

2 Outline Introducing our system How to measure higher order correlation functions What are connected correlation function and what can we learn from them Thermal equilibrium Gaussification, build up of non-gaussian fluctuations

3 The system under study Two one-dimensional Bose gases of 87 Rb atoms: adjustable tunnelcoupling x z Theory: two coupled Lieb-Lininger type Hamiltonians Phase-density representation:

4 The system under study Two one-dimensional Bose gases of 87 Rb atoms: adjustable tunnelcoupling x z Variable transformation to relative and common degrees of freedom: Low energy effective theory: sine-grodon Hamiltonian quadradic Luttinger liquid non-quadradic

5 Experimental realization Atom chip: micro-fabricated wire structure on a silicon chip used to create a magnetic trap Atom chip: Double well trap: created through rf-dressed state potentials Experimental parameters: Highly elongated traps, ratio 1:100 T 40 nk N 5000 per well x z

6 Probing the system time-of-fligh expansion absorption imaging spatially resolved relative phase x z

7 Higher order correlation functions Observable: difference of the relative phase between two points in the gas Many experimental realizations Calculate joint momenta or N-point correlation functions Calculate joint cumulants or N-point connected correlation functions Note that this phase difference is not limited to [-π, π) Calculating connected correlation function 1. Calculate correlation functions up to order N 2. Calculate N-point connected correlation functions as a polynomial of correlation functions up to order N disconnected part

8 Connected correlation function If fluctuations Gaussian In pertubation theory: N-point connected correlation = sum of connected Feyman diagrams with N external legs for example, 4-point connected correlation function connected diagrams disconnected diagrams contribute do not contribute

9 Thermal equilibrium In thermal equilibrium: Quadratic Hamiltonian Gaussian fluctuations can be easily seen in classical field approximation Sine-Gordon in thermal equilibrium Phase fluctuations governed by two length scales (classical field approx): thermal choherence length healing length of the relative phase characterizes the importance of non-quadradic terms Grisins et al., PRA 87, (2013) Experimental preparation: slow evaporative cooling in coupled DW

10 Experimental correlation functions 4-point correlation function no coupling, J = 0 quadratic Hamiltonian Gaussian fluctuations no connected part intermediate coupling non-quadratic Hamiltonian non-gaussian fluctuations connected part strong coupling small relative phase quadratic approximation of cosine Gaussian fluctuations no connected part 2D cuts for z 3 = z 4 = 15 μm sine-gordon Hamiltonian

11 Integral measures theory: classical field approximation numerical shots produced by stochastic process homogeneous density finite imaging resolution considered 80% confidence intervals obtained from bootstrap similar temperatures, λ T = μm different tunnelcoupling strength

12 phase distributions uncoupled uncoupled cold slowly cooled fast cooled

13 Solitons sine-gordon model has solitonic solutions kink-soliton: fast cooled strongly coupled v = 0 l J = 7

14 Quench to zero tunnel coupling equilibrium state, intermediate J switch tunnel-coupling off low energy theory Experimental Results

15 Quench to zero tunnel coupling Fluctuations get bigger expected from dephasing in free theory Simplest theory, evolution with quadtratic Luttinger-liquid random process produces shots for initial non-gaussian disttribution evolve with

16 Quench to finite tunnel coupling equilibrium state, zero J switch tunnel-coupling on low energy theory Experimental Results not explained by sine-gordon alone Sine-Gordon theory: Dalla Torre et al., PRL 110(9), (2013)

17 Quench to finite tunnel coupling Experimental Results eq. theory for λ T = 16 μm

18 Conclusion Demonstrated a new way of analyzing cold atom experiments Observed non-gaussian states in a multimode system Found robust way to check whether the system is in thermal equilibrium Observed sine-gordon solitons Observed Gaussification and build up of non- Gaussian fluctuations

19 Thank you! Theory: Igor Mazets Sebastian Erne Valentin Kasper Thomas Gasenzer Jürgen Berges Experiment: Bernhard Rauer Federica Cataldini Tim Langen Jörg Schmiedmayer arxiv: