Learning about order from noise Quantum noise studies

Learning about order from noise Quantum noise studies of ultracold atoms Eugene Demler Harvard University Collaborators: Takuya Kitagawa, Susanne Pielawa, Adilet Imambekov, Ehud Altman, Vladimir Gritsev, Anatoli Polkovnikov, Mikhail Lukin Experiments: Bloch et al. , Dalibard et al. , Schmiedmayer et al.

Quantum noise Classical measurement: collapse of the wavefunction into eigenstates of x Histogram of measurements of x

Probabilistic nature of quantum mechanics Bohr-Einstein debate: EPR thought experiment (1935) “Spooky action at a distance” Aspect’s experiments with correlated photon pairs: tests of Bell’s inequalities (1982) + - 1 S + 2 - Analysis of correlation functions can be used to rule out hidden variables theories

Second order coherence: HBT experiments Classical theory Hanburry Brown and Twiss (1954) Quantum theory Glauber (1963) For bosons For fermions Used to measure the angular diameter of Sirius HBT experiments with matter

Shot noise in electron transport Variance of transmitted charge e- e- Shot noise Schottky (1918) Measurements of fractional charge Current noise for tunneling across a Hall bar on the 1/3 plateau of FQE Etien et al. PRL 79: 2526 (1997) see also Heiblum et al. Nature (1997)

Analysis of quantum noise: powerful experimental tool Can we use it for cold atoms?

Outline Quantum noise as a probe of equilibrium correlation functions in low dimensional systems. Interference experiments with independent condensates Quantum noise as a probe of dynamics. Interaction induced collapse of Ramsey fringes Goal: new methods of analyzing quantum many-body states of ultracold atoms

Interference experiments with cold atoms Analysis of thermal and quantum noise in low dimensional systems

Interference of independent condensates Experiments: Andrews et al. , Science 275: 637 (1997) Theory: Javanainen, Yoo, PRL 76: 161 (1996) Cirac, Zoller, et al. PRA 54: R 3714 (1996) Castin, Dalibard, PRA 55: 4330 (1997) and many more

Experiments with 2 D Bose gas Hadzibabic et al. , Nature 441: 1118 (2006) z Time of flight Experiments with 1 D Bose gas Hofferberth et al. , Nature Physics 4: 489 (2008) x

Interference of two independent condensates r’ r Assuming ballistic expansion 1 r+d d 2 Phase difference between clouds 1 and 2 is not well defined Individual measurements show interference patterns They disappear after averaging over many shots

Interference of fluctuating condensates d Polkovnikov, Altman, Demler, PNAS 103: 6125(2006) Amplitude of interference fringes, x 1 x 2 For independent condensates Afr is finite but Df is random For identical condensates Instantaneous correlation function

Fluctuations in 1 d BEC Thermal fluctuations Thermally energy of the superflow velocity Quantum fluctuations Weakly interacting atoms

Interference between Luttinger liquids Luttinger liquid at T=0 K – Luttinger parameter For non-interacting bosons For impenetrable bosons Finite temperature Experiments: Hofferberth, Schumm, Schmiedmayer and

Distribution function of fringe amplitudes for interference of fluctuating condensates Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006 Imambekov, Gritsev, Demler, Varenna lecture notes, c-m/0703766 is a quantum operator. The measured value of will fluctuate from shot to shot. L Higher moments reflect higher order correlation functions We need the full distribution function of

Distribution function of interference fringe contrast Hofferberth et al. , Nature Physics 4: 489 (2008) Quantum fluctuations dominate: asymetric Gumbel distribution (low temp. T or short length L) Thermal fluctuations dominate: broad Poissonian distribution (high temp. T or long length L) Intermediate regime: double peak structure Comparison of theory and experiments: no free parameters Higher order correlation functions can be obtained

Interference between interacting 1 d Bose liquids. Distribution function of the interference amplitude Distribution function of Quantum impurity problem: interacting one dimensional electrons scattered on an impurity Conformal field theories with negative central charges: 2 D quantum gravity, non-intersecting loop model, growth of random fractal stochastic interface, high energy limit of multicolor QCD, … 2 D quantum gravity, non-intersecting loops Yang-Lee singularity

Fringe visibility and statistics of random surfaces Mapping between fringe visibility and the problem of surface roughness for fluctuating random surfaces. Relation to 1/f Noise and Extreme Value Statistics Fringe visibility Roughness Analysis of sine-Gordon models of the type

Experiments with 2 D Bose gas Hadzibabic, Dalibard et al. , Nature 441: 1118 (2006) z Time of flight x Typical interference patterns low temperature higher temperature

Experiments with 2 D Bose gas Hadzibabic et al. , Nature 441: 1118 (2006) x integration over x axis z z Contrast after integration 0. 4 integration over x axis 0. 2 z high T integration over x axis Dx middle T low T 0 z 0 10 20 30 integration distance Dx (pixels)

Experiments with 2 D Bose gas Integrated contrast Hadzibabic et al. , Nature 441: 1118 (2006) 0. 4 fit by: middle T 0. 2 0 low T Exponent a high T 0 10 20 30 integration distance Dx if g 1(r) decays exponentially with : 0. 5 0. 4 0. 3 high T 0 if g 1(r) decays algebraically or exponentially with a large : 0. 1 low T 0. 2 0. 3 central contrast “Sudden” jump: BKT transition

Experiments with 2 D Bose gas. Proliferation of thermal vortices Hadzibabic et al. , Nature 441: 1118 (2006) 30% Fraction of images showing at least one dislocation 20% 10% low T high T 0 0 0. 1 0. 2 0. 3 central contrast 0. 4 The onset of proliferation coincides with a shifting to 0. 5!

Fringe contrast in two dimensions. Evolution of distribution function Experiments: Kruger, Hadzibabic, Dalibard

Quantum noise as a probe of non-equilibrium dynamics Ramsey interferometry and many-body decoherence

Ramsey interference 1 0 Atomic clocks and Ramsey interference: Working with N atoms improves the precision by. t

Interaction induced collapse of Ramsey fringes Two component BEC. Single mode approximation Ramsey fringe visibility time Experiments in 1 d tubes: A. Widera et al. PRL 100: 140401 (2008)

Spin echo. Time reversal experiments Single mode approximation The Hamiltonian can be reversed by changing a 12 Predicts perfect spin echo

Spin echo. Time reversal experiments Expts: A. Widera, I. Bloch et al. No revival? Experiments done in array of tubes. Strong fluctuations in 1 d systems. Single mode approximation does not apply. Need to analyze the full model

Interaction induced collapse of Ramsey fringes. Multimode analysis Low energy effective theory: Luttinger liquid approach Luttinger model Changing the sign of the interaction reverses the interaction part of the Hamiltonian but not the kinetic energy Time dependent harmonic oscillators can be analyzed exactly

Time-dependent harmonic oscillator See e. g. Lewis, Riesengeld (1969) Malkin, Man’ko (1970) Explicit quantum mechanical wavefunction can be found From the solution of classical problem We solve this problem for each momentum component

Interaction induced collapse of Ramsey fringes in one dimensional systems Only q=0 mode shows complete spin echo Finite q modes continue decay The net visibility is a result of competition between q=0 and other modes Decoherence due to many-body dynamics of low dimensional systems Fundamental limit on Ramsey interferometry How to distinquish decoherence due to many-body dynamics?

Interaction induced collapse of Ramsey fringes Single mode analysis Kitagawa, Ueda, PRA 47: 5138 (1993) Multimode analysis evolution of spin distribution functions T. Kitagawa, S. Pielawa, A. Imambekov, et al.

Summary Experiments with ultracold atoms provide a new perspective on the physics of strongly correlated many-body systems. Quantum noise is a powerful tool for analyzing many body states of ultracold atoms Thanks to: Harvard-MIT