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NICE-BEC, June 4th - Session on “Non equilibrium dynamics” Quantum critical states and phase transitions in the presence of non equilibrium noise Emanuele G. Dalla Torre – Weizmann Institute of Science, Israel Collaborators: Ehud Altman – Weizmann Inst. Eugene Demler – Harvard Univ. Thierry Giamarchi – Geneve Univ. Quantum systems coupled to the environment External noise from the environment (classical) System Zero temperature thermal bath (quantum) The systems reaches a non-equilibrium steady state: Criticality? Phase transitions? Specific realization in zero dimensions Iext Shunted Josephson Junction Charge Noise with 1/f spectrum J R Bath: Zero temperature resistance C ~ VN(t) In the absence of noise this system undergoes a superconductor-insulator quantum phase transition at the universal value of the resistor The external noise shift the quantum phase transition away from its universal value R/RQ 1 insulator 0.5 arxiv/0908.0868 superconductor noise Specific realizations in one dimension Trapped ions Noise: Charge fluctuations on the electrodes Bath: Laser cooling Dipolar atoms in a cigar shape potential Noise: fluctuations of the polarizing field Bath: immersion in a condensate Cigar shape potential: Bloch group (2004) - BEC immersion: Daley, Fedichev, Zoller (2004) Outline 1. Review of the equilibrium physics in 1D (no noise) 2. Non equilibrium quantum critical states in one dim. A. Dynamical response B. Phase transitions 3. Extension to higher dimensions 4. Outlook and summary Review of equilibrium physics in 1D: continuum limit a : average distance (x) : displacement field Low-energy effective action: phonons Luttinger parameter (controls the quantum fluctuations) Haldane (1980) Review : density correlations in 1D Two types of low-lying density fluctuations Long-wavelength fluctuations Crystal fluctuations Crystalline correlation decay as a power law: Scale invariant, critical state Review: effects of a lattice in 1D Add a static periodic potential (“lattice”) at integer filling When does the lattice induce a quantum phase transition to a Mott insulator? Effective action: phonons lattice potential Review: Mott transition in 1D The quadratic term is scale invariant. How does the lattice change under rescaling ? K>2 K<2 lattice decays lattice grows critical Mott insulator Quantum phase transition at K = 2 Buchler, Blatter, Zwerger, PRL (2002) What are the effects of the external noise? Can we have non-equilibrium quantum critical states? Non-equilibrium quantum phase transitions? Effects of non-equilibrium noise Immersion in a BEC (or laser cooling) behaves as a zero temperature bath The external noise couples linearly to the density If we assume that the noise is smooth on an inter-particle scale, we can neglect the cosine term and retain a quadratic action! Effects of non-equilibrium noise We can cast the quadratic action into a linear quantum Langevin equation: Zero temperature bath induces both dissipation and fluctuations (satisfies FDT) External noise induces only fluctuations (breaks FDT) The measured noise spectrum in ion traps Monroe group, PRL (06), Chuang group, PRL (08) • Time correlations: 1/f spectrum • Indications for short range spatial correlations Crystalline correlations in the presence of 1/f noise Using the Langevin equation we can compute correlation functions: crystal correlations remain power-law, with a tunable power noise dissipation Non equilibrium quantum critical state! (Note: exact only in the scale invariant limit , F00 with F0 / = const.) Non equilibrium critical state: Bragg spectroscopy Add a periodic potential which modulates with time Goal: compute the energy transferred into the system In linear response, we have to compute density-density correlations in the absence of the potential (V=0) Absorption spectrum in the non equilibrium critical state Long wavelength limit: Equilibrium (F0=0) Luther&Peschel(1973) Unaffected by noise Near q0=2π/a: Non equilibrium (F0/η=2) Strongly affected by the noise Absorption spectrum in the non equilibrium critical state Near q0=2π/a: The energy loss can be negative critical gain spectrum Non equilibrium quantum phase transitions Add a static periodic potential (“lattice”) at integer filling Does the lattice induce a quantum phase transition? or What are the effects of the lattice on the correlation function? The Hamiltonian is not quadratic and we cannot cast into a Langevin equation Instead we use a double path integral formalism (Keldysh) and expand in small g Non equilibrium Mott transition: scaling analysis 2x2 Keldysh action (non equilibrium quantum critical state) How does the lattice change under rescaling ? K critical Non equilibrium phase transition at pinned F0 / Extension: General noise source We develop a real-time Renormalization Group procedure > -1 irrelevant: doesn’t affect the phase transition < -1 relevant: destroys the phase transition (thermal noise) = -1 marginal: non-equilibrium phase transition Summary: Quantum systems coupled to the environment show non equilibrium critical steady states and phase transitions E.G. Dalla Torre, E. Demler, T. Giamarchi, E. Altman - arxiv/0908.0868 (v2) 1. Critical steady state with power-law correlations (faster decay) 2. Negative response to external probes (“critical amplifier”) F0 / 3. Non equilibrium quantum phase transitions: a real-time RG approach 4. High dimensions: novel phase transitions tuned by a competition of classical noise and quantum fluctuations K 2D superfluid critical 2D crystal F0 / Non equilibrium phase transitions - coupled tubes Inter-tube tunneling: K Phase transition at 2D superfluid 1D critical F0 / Non equilibrium phase transitions - coupled tubes Inter-tube tunneling K Inter-tube repulsion 2D superfluid K 1D critical 1D critical 2D crystal F0 / F0 / Both perturbations (actual situation) K 2D superfluid critical 2D crystal F0 / Outlook : reintroduce backscattering In the presence of backscattering, the Hamiltonian is not quadratic Keldysh path integral enables to treat the cosine perturbatively (relevant/irrelevant) How to go beyond? We introduce a new variational approach for many body physics The idea: substitute the original Hamiltonian by a quadratic variational one Time dependent variational approach Variational Original Hamiltonian Hamiltonian The variational parameter fV(t) is determined self consistently by requiring a vanishing response of to any variation of fV (t). We show that this approach is equivalent to Dirac-Frenkel (using a variational Hamiltonian instead that a variational wavefunction) We successfully use it to compute the non linear I-V characteristic of a resistively shunted Josephson Junction