Quantum coherence and interactions in many body systems Collaborators: Ehud Altman, Anton Burkov, Derrick Chang, Adilet Imambekov, Vladimir Gritsev, Mikhail Lukin, Giovanna Morigi, Anatoli Polkonikov Eugene Demler Harvard University Funded by NSF, AFOSR, Harvard-MIT CUA

Condensed matter physics Atomic physics Quantum optics Quantum coherence Quantum information

Quantum Optics with atoms and Condensed Matter Physics with photons Interference of fluctuating condensates From reduced contrast of fringes to correlation functions Distribution function of fringe contrast Non-equilibrium dynamics probed in interference experiments Luttinger liquid of photons Can we get “fermionization” of photons? Non-equilibrium coherent dynamics of strongly interacting photons

Interference experiments with cold atoms

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

Interference of two independent condensates 1 2 r r+d d r’ Clouds 1 and 2 do not have a well defined phase difference. However each individual measurement shows an interference pattern

Nature 4877:255 (1963)

Interference of one dimensional condensates Experiments: Schmiedmayer et al., Nature Physics (2005,2006) Transverse imaging long. imaging trans. imaging Longitudial imaging Figures courtesy of J. Schmiedmayer

x1x1 d Amplitude of interference fringes, Interference of one dimensional 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)

Interference of two dimensional condensates LyLy LxLx LxLx Experiments: Hadzibabic et al. Nature (2006) Probe beam parallel to the plane of the condensates Gati et al., PRL (2006)

Interference of two dimensional condensates. Quasi long range order and the KT transition LyLy LxLx Below KT transitionAbove KT transition

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!? Experiments with 2D Bose gas Hadzibabic et al., Nature 441:1118 (2006)

Fundamental noise in interference experiments Amplitude of interference fringes is a quantum operator. The measured value of the amplitude will fluctuate from shot to shot. We want to characterize not only the average but the fluctuations as well.

Shot noise in interference experiments Interference with a finite number of atoms. How well can one measure the amplitude of interference fringes in a single shot? One atom: No Very many atoms: Exactly Finite number of atoms: ? Consider higher moments of the interference fringe amplitude,, and so on Obtain the entire distribution function of

Shot noise in interference experiments Interference of two condensates with 100 atoms in each cloud Coherent states Number states Polkovnikov, Europhys. Lett. 78:10006 (1997) Imambekov, Gritsev, Demler, 2006 Varenna lecture notes

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, cond-mat/ We need the full distribution function of

Interference of 1d condensates at T=0. Distribution function of the fringe contrast Narrow distribution for. Approaches Gumbel distribution. Width Wide Poissonian distribution for

Interference of 1d condensates at finite temperature. Distribution function of the fringe contrast Luttinger parameter K=5 Experiments: Schmiedmayer et al.

Interference of 2d condensates at finite temperature. Distribution function of the fringe contrast T=T KT T=2/3 T KT T=2/5 T KT

From visibility of interference fringes to other problems in physics

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 is a quantum operator. The measured value of will fluctuate from shot to shot. How to predict the distribution function of Yang-Lee singularity 2D quantum gravity, non-intersecting loops

Fringe visibility and statistics of random surfaces Proof of the Gumbel distribution of interfernece fringe amplitude for 1d weakly interacting bosons relied on the known relation between 1/f Noise and Extreme Value Statistics T.Antal et al. Phys.Rev.Lett. 87, (2001) Fringe visibility Roughness

Non-equilibrium coherent dynamics of low dimensional Bose gases probed in interference experiments

Studying dynamics using interference experiments. Thermal decoherence Prepare a system by splitting one condensate Take to the regime of zero tunneling Measure time evolution of fringe amplitudes

Relative phase dynamics Quantum regime 1D systems 2D systems Classical regime 1D systems 2D systems Burkov, Lukin, Demler, cond-mat/ Experiments: Schmiedmayer et al. Different from the earlier theoretical work based on a single mode approximation, e.g. Gardiner and Zoller, Leggett

Quantum dynamics of coupled condensates. Studying Sine-Gordon model in interference experiments J Prepare a system by splitting one condensate Take to the regime of finite tunneling. System described by the quantum Sine-Gordon model Measure time evolution of fringe amplitudes

Dynamics of quantum sine-Gordon model Power spectrum Gritsev, Demler, Lukin, Polkovnikov, cond-mat/ A combination of broad features and sharp peaks. Sharp peaks due to collective many-body excitations: breathers

Condensed matter physics with photons

Luttinger liquid of photons Tonks gas of photons: photon “fermionization” Chang, Demler, Gritsev, Lukin, Morigi, unpublished

Self-interaction effects for one-dimensional optical waves k k0k0 w w0w0 Nonlinear polarization for isotropic medium Envelope function

Self-interaction effects for one-dimensional optical waves Frame of reference moving with the group velocity Gross-Pitaevskii type equation for light propagation Competition of dispersion and non-linearity Nonlinear Optics, Mills

Self-interaction effects for one-dimensional optical waves BEFORE : two level systems and insufficient mode confinement Interaction corresponds to attraction. Physics of solitons (e.g. Drummond) Weak non-linearity due to insufficient mode confining Limit on non-linearity due to photon decay NOW: EIT and tight mode confinement Sign of the interaction can be tuned Tight confinement of the electromagnetic mode enhances nonlinearity Strong non-linearity without losses can be achieved using EIT

Controlling self-interaction effects for photons WcWc w w D w Imamoglu et al., PRL 79:1467 (1997) describes photons. We need to normalize to polaritons

Tonks gas of photons Photon fermionization Crystal of photons Is it realistic? Experimental signatures

Tonks gas of atoms Small g – weakly interacting Bose gas Large g – Tonks gas. Fermionized bosons Additional effects for for photons : Photons are moving with the group velocity Limit on the cross section of photon interacting with one atom

Tonks gas of photons Limit on strongly interacting 1d photon liquid due to finite group velocity Concrete example: atoms in a hollow fiber Experiments: Cornell et al. PRL 75:3253 (1995); Lukin, Vuletic, Zibrov et.al. Theory: photonic crystal and non-linear medium Deutsch et al., PRA 52:1394 (2005); Pritchard et al., cond-mat/

Atoms in a hollow fiber k w Without using the “slow light” points l=1mm A=10mm 2 L tot =1cm A – cross section of e-m mode Typical numbers

Experimental detection of the Luttinger liquid of photons Control beam off. Coherent pulse of non-interacting photons enters the fiber. Control beam switched on adiabatically. Converts the pulse into a Luttinger liquid of photons. “Fermionization” of photons detected by observing oscillations in g 2 c K – Luttinger parameter

Non-equilibrium dynamics of strongly correlated many-body systems g 2 for expanding Tonks gas with adiabatic switching of interactions 100 photons after expansion

Outlook Atomic physics and quantum optics traditionally study non-equilibrium coherent quantum dynamics of relatively simple systems. Condensed matter physics analyzes complicated electron system but focuses on the ground state and small excitations around it. We will need the expertise of both fields Next challenge in studying quantum coherence: understand non-equilibrium coherent quantum dynamics of strongly correlated many-body systems “…The primary objective of the JQI is to develop a world class research institute that will explore coherent quantum phenomena and thereby lay the foundation for engineering and controlling complex quantum systems…” From the JQI web page