1. 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. NICE-BEC, June 4th - Session on “Non equilibrium dynamics”

2. 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? Q U A N T U M NOISE SYSTEM CLASSIC B A T H

3. 1 0.5 Specific realization in zero dimensions Shunted Josephson Junction Charge Noise with 1/f spectrum Bath: Zero temperature resistance ~ R J C V N (t)‏ I ext 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 arxiv/0908.0868 R/R Q superconductor noise insulator

4. Specific realizations in one dimension Dipolar atoms in a cigar shape potential Noise: fluctuations of the polarizing field Bath: immersion in a condensate Trapped ions Noise: Charge fluctuations on the electrodes Bath: Laser cooling Cigar shape potential: Bloch group (2004) - BEC immersion: Daley, Fedichev, Zoller (2004)

5. 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

6. Review of equilibrium physics in 1D: continuum limit a : average distance  (x) : displacement field Low-energy effective action: phonons (controls the quantum fluctuations) Luttinger parameter Haldane (1980)

7. Review : density correlations in 1D Crystalline correlation decay as a power law: Long-wavelength fluctuationsCrystal fluctuations Scale invariant, critical state Two types of low-lying density fluctuations

8. 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? lattice potentialphonons Effective action:

9. Review: Mott transition in 1D The quadratic term is scale invariant. How does the lattice change under rescaling ? Buchler, Blatter, Zwerger, PRL (2002) Quantum phase transition at K = 2 K > 2lattice decayscritical K < 2lattice growsMott insulator

10. Can we have non-equilibrium quantum critical states? Non-equilibrium quantum phase transitions? What are the effects of the external noise?

11. 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!

12. Effects of non-equilibrium noise Zero temperature bath induces both dissipation and fluctuations (satisfies FDT) External noise induces only fluctuations (breaks FDT) We can cast the quadratic action into a linear quantum Langevin equation:

13. Monroe group, PRL (06), Chuang group, PRL (08) Indications for short range spatial correlations Time correlations: 1/f spectrum The measured noise spectrum in ion traps

14. 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 , F 0  0 with F 0 /  = const.)

15. Non equilibrium critical state: Bragg spectroscopy 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) Add a periodic potential which modulates with time

16. Absorption spectrum in the non equilibrium critical state Equilibrium (F 0 =0) Non equilibrium (F 0 /η=2) Luther&Peschel(1973) Unaffected by noise Long wavelength limit: Near q 0 =2π/a: Strongly affected by the noise

17. Absorption spectrum in the non equilibrium critical state The energy loss can be negative  critical gain spectrum Near q 0 =2π/a:

18. Non equilibrium quantum phase transitions Add a static periodic potential (“lattice”) at integer filling Does the lattice induce a quantum phase transition? 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 What are the effects of the lattice on the correlation function? or

19. Non equilibrium Mott transition: scaling analysis Non equilibrium phase transition at How does the lattice change under rescaling ? 2x2 Keldysh action (non equilibrium quantum critical state) K F 0 /  pinned critical

20. 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

21. Summary: Quantum systems coupled to the environment show non equilibrium critical steady states and phase transitions F 0 /  2D superfluid 2D crystal critical K F 0 /  1.Critical steady state with power-law correlations (faster decay) 2.Negative response to external probes (“critical amplifier”) 4.High dimensions: novel phase transitions tuned by a competition of classical noise and quantum fluctuations E.G. Dalla Torre, E. Demler, T. Giamarchi, E. Altman - arxiv/0908.0868 (v2)‏ 3.Non equilibrium quantum phase transitions: a real-time RG approach

22. Non equilibrium phase transitions - coupled tubes Inter-tube tunneling: Phase transition at K 1D critical 2D superfluid F 0 / 

23. Non equilibrium phase transitions - coupled tubes F 0 /  Inter-tube repulsion K 1D critical 2D crystal Inter-tube tunneling K 1D critical 2D superfluid F 0 /  Both perturbations (actual situation) F 0 /  K 2D superfluid critical 2D crystal

24. 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

25. Time dependent variational approach Variational Hamiltonian The variational parameter f V (t) is determined self consistently by requiring a vanishing response of to any variation of f V (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 Original Hamiltonian