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

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

Shot noise in electron transport Shot noise Schottky (1918) Variance of transmitted charge e-e- e-e- 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?

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

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:R3714 (1996) Castin, Dalibard, PRA 55:4330 (1997) and many more

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

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

x1x1 d Amplitude of interference fringes, Interference of fluctuating condensates For identical condensates Instantaneous correlation function For independent condensates A fr is finite but Df is random x2x2 Polkovnikov, Altman, Demler, PNAS 103:6125(2006)

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

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

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

Distribution function of interference fringe contrast Hofferberth et al., Nature Physics 4:489 (2008) Comparison of theory and experiments: no free parameters Higher order correlation functions can be obtained 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

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

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

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

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

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

Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic et al., Nature 441:1118 (2006) The onset of proliferation coincides with  shifting to 0.5! Fraction of images showing at least one dislocation 0 10% 20% 30% central contrast high T low T

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

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

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

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

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

Interaction induced collapse of Ramsey fringes. Multimode analysis 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 Low energy effective theory: Luttinger liquid approach

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

Interaction induced collapse of Ramsey fringes in one dimensional systems Fundamental limit on Ramsey interferometry 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 How to distinquish decoherence due to many-body dynamics?

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

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