Non-equilibrium in cold atom systems

1. Non-equilibrium in cold atom systems K. Sengupta Indian Association for the Cultivation of Science, Kolkata Collaborators: Brian Clark, Michael Kobodurez, Steven Girvin, David Pekker Stephen Powell, Subir Sachdev, Christian Trezger

2. Outline • Introduction • Atoms as quantum emulators: the Bose Hubbard model • Emulating the Ising model: tilted lattices • Non-equilibrium Dynamics: A perspective • Dynamics of cold atom systems • Conclusion

3. Laser and Cold atoms in a trap Apply counter propagating laser: standing wave of light. The atoms feel a potential V = -a |E|2 Unit of energy

4. Energy Scales dEn = 5Er ~ 20 U U ~10-300 t The atoms can hop from one lattice site to the next: kinetic energy The neutral atoms interact by short-range potential: effectively on-site interaction. Ignore higher bands Model Hamiltonian For a deep enough potential, the atoms are localized : Mott insulator described by single band Bose-Hubbard model.

5. Coherence Pattern

8. Mott-Superfluid transition: preliminary analysis using Bose-Hubbard model Mott state with 1 boson per site Stable ground state for 0 < m < U Mott state is destabilized when the excitation energy touches 0. Adding a particle to the Mott state Removing a particle from the Mott state z Destabilization of the Mott state via addition of particles/hole: onset of superfluidity

9. Beyond this simple picture Higher order energy calculation by Freericks and Monien: Inclusion of up to O(t3/U3) virtual processes. Phase diagram for n=1 and d=3 Mean-field theory (Fisher 89) Quantum Monte Carlo studies for 2D systems: Trivedi and Krauth. O(t2/U2) theories MFT Superfluid Predicts a quantum phase transition with z=2 (except at the tip of the Mott lobe where z=1). Mott

10. Tilted optical lattice: Emulating the Ising model

11. Applying an electric field to the Mott state • Shift the center of the trap. This generates effective electric field for the atoms 2. If the atoms have a net spin, apply a linearly varying magnetic field. Generation of an effective tilt for the lattice: One can access regime where the effective electric field has the same magnitude as U which is impossible in standard condensed matter systems.

12. Construction of an effective model: 1D Parent Mott state

13. Neutral dipoles Resonantly coupled to the parent Mott state when U=E. Neutral dipole state with energy U-E. Two dipoles which are not nearest neighbors with energy 2(U-E).

14. Effective dipole Hamiltonian: 1D

15. Weak Electric Field For weak electric field, the ground state is dipole vaccum and the low-energy excitations are single dipole • The effective Hamiltonian for the dipoles for weak E: • Lowest energy excitations: Single band of dipole excitations. • These excitations soften as E approaches U. This is a precursor of the appearance of Ising density wave with period 2. • Higher excited states consists of multiparticle continuum.

16. Strong Electric field • The ground state is a state of maximum dipoles. • Because of the constraint of not having two dipoles on consecutive sites, we have two degenerate ground states • The ground state breaks Z2 symmetry. • The first excited state consists of band of domain walls between the two filled dipole states. • Similar to the behavior of Ising model in a transverse field.

17. Intermediateelectric field: QPT Quantum phase transition at E-U=1.853w. Ising universality.

18. Recent Experimental observation of Ising order (Bakr et al Nature 2010) First experimental realization of effective Ising model in ultracold atom system

19. A digression: Readout scheme for boson occupation • Let the system equilibrate and raise the optical lattice depth so that the boson number configuration is frozen. • Send in photons of “right” wavelength. • Even number of bosons in a lattice site undergo light assisted collisions and move out of the lattice. • Image the remaining atoms and thus obtain the parity of occupation of the original boson configuration.

20. Non-equilibrium Dynamics

21. Non-equilibrium dynamics of closed quantum system • Relation with cold atoms: These experimental systems (and ion traps ) provide • perfect realization closed zero temperature quantum systems • Conceptual issues with non-eq. dynamics: • Universality in out of equilibrium systems • Thermalization: Nature of the steady state at long times • Short time dynamics: universal signatures of quantum criticality • Understanding defect production in quantum systems • Study of dynamics in field theories in strongly coupled regime and its relation to gravity via ADS/CFT

22. Absence of thermalization in 1D Bose gas Blue detuned laser used to create tightly bound 1D tubes of Rb atoms. The depth of the lattice potential is kept large to ensure negligible tunneling between the tubes. The bosons are imparted a small kinetic energy so as to place them initially at a superposition state with momentum p0 and –p0. The evolution of the momentum distribution of the bosons are studied by typical time of flight experiments for several evolution times. The distribution of the bosons are never gaussian within experimental time scale; clear absence of thermalization in nearly-integrable 1D Bose gas. Kinoshita et al. Nature 2006

23. Dynamics in the Bose-Hubbard model

24. Projection operator formalism for Bose-Hubbard model Mott state Distinguishing between hopping processes Distinguish between two types of hopping processes using a projection operator technique Define a projection operator Divide the hopping to classes (b) and (c) Eliminate the “offending terms” by a cannonical transformation to get Heff

25. Building fluctuations over MFT Design a transformation which eliminate hopping processes of class (b) perturbatively in J/U. Obtain the effective Hamiltonian Use the effective Hamiltonian to compute the ground state energy and hence the phase diagram

26. Equilibrium phase diagram Reproduction of the phase diagram with remarkable accuracy in d=3: much better than standard mean-field or strong coupling expansion (of the same order) in d=2 and 3. Allows for straightforward generalization for treatment of dynamics

27. Non-equilibrium dynamics Consider a linear ramp of J(t)=Ji +(Jf - Ji) t/t. For dynamics, one needs to solve the Sch. Eq. Make a time dependent transformation to address the dynamics by projecting on the instantaneous low-energy sector. The method provides an accurate description of the ramp if J(t)/U <<1 and hence can treat slow and fast ramps at equal footing. Takes care of particle/hole production due to finite ramp rate Final equation to be solved Generic result: Defect density scales with ramp rate for slow ramps through critical point: Kibble-Zurek law. To be tested for realistic experimental systems.

28. Defect production and residual energy for linear dynamics One finds a plateau like behavior of both P and Q instead of the expected scaling behavior for large quench time. For typical values of h/Jc= 70ms while the ramp time is roughly around 20-30 ms. Absence of critical scaling: may be understood as the inability of the system to access the critical (k=0) modes. The dynamics is not adiabatic enough to display scaling. This is also the regime where the local theory works.

29. Order parameter dynamics 1- Fast quench from the Mott to the SF phase; study of superfluid dynamics. Single frequency pattern near the critical point; more complicated deeper in the SF phase. Strong quantum fluctuations near the QCP; justification of going beyond mft.

30. Study of non-linear ramp dynamics Protocol: with the exponent varying between 1 and 5 Non-monotonic initial slope signifying minimal defect production for slow to moderate ramps around

31. Experiments with ultracold bosons on a lattice: finite rate dynamics 2D BEC confined in a trap and in the presence of an optical lattice. Single site imaging done by light-assisted collision which can reliably detect even/odd occupation of a site. In the present experiment one detects sites with n=1. Ramp from the SF side near the QCP to deep inside the Mott phase in a linear ramp with different ramp rates. The no. of sites with odd n displays plateau like behavior and approaches the adiabatic limit when the ramp time is increased asymptotically. No signature of scaling behavior. Interesting spatial patterns. W. Bakr et al. arXiv:1006.0754

32. Dynamics in the dipole model

33. Quench dynamics across the quantum critical point Tune the electric field from Ei to Ef instantaneously Compute the dipole order Parameter as a function of time The time averaged value of the order parameter is maximal near the QCP

34. Dynamics with a finite rate: Kibble-Zureck scaling Change the electric field linearly in time with a finite rate v Theory Quantities of interest Experiment Scaling laws for finite –size systems Q ~ v2 (v) for slow(intermediate ) quench. These are termed as LZ(KZ) regimes for finite-size systems.

35. Kibble-Zureck scaling for finite-sized system Observation of Kibble-Zurek law for intermediate v with Ising exponents. Expected scaling laws for Q and F Dipole dynamics Correlation function

36. Conclusion • Ultracold atom systems emulates several spin and boson models and allow us to experimentally access the properties of these models in the strongly correlated regime. • They provide excellent examples of closed quantum systems. • Together, they provide us with the requisite test-bed for studying non-equilibrium dynamics of strongly correlated quantum systems. 4. In the Bose-Hubbard model, experimental regime do not (yet) correspond to the slow limit and thus do not exhibit KZ power law. • In the finite-size 1D dipole model, there is a range of ramp rate for which one expects to observe finite-size version of KZ scaling with universal Ising exponents.