Sweep from Superfluid to Mottphase in the Bose-Hubbard model p.1/14

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1 Sweep from Superfluid to phase in the Bose-Hubbard model Ralf Schützhold Institute for Theoretical Physics Dresden University of Technology Sweep from Superfluid to phase in the Bose-Hubbard model p.1/14

2 Motivation quilibrium properties well understood for many systems, e.g., near a phase transition Dynamical (time-dependent) phase transitions response time typically diverges non-equilibrium properties Analogy to expanding universe effective horizon (universal behaviour) loss of causal contact Amplification of quantum fluctuations seeds for pattern formation etc. Physical example: Bose-Hubbard model Relation to real cosmic inflation? Sweep from Superfluid to phase in the Bose-Hubbard model p.2/14

3 Bose-Hubbard Model Ĥ = J(t) αβ M αβ â αâ β + U 2 (â α) 2 â 2 α α Interaction U, tunnelling rate J(t), lattice matrix M αβ Large integer filling n = ˆn α = â αâ α 1 Superfluid phase transition at J c = O(U/n) J U/n superfluid phase Ψ sf ( N α α) â 0 J U/n insulator Ψ (â α) n 0 α Sweep from Superfluid to phase in the Bose-Hubbard model p.3/14

4 Dynamical Phase Transition Zero temperature T = 0 xternal parameter superfluid J = J(t) Ψ(t) Two competing ground states Ψ < and Ψ > Actual quantum Ψ < Ψ > state Ψ(t) Sweeping through phase transition (critical point J c ) non-equilibrium dynamics J(t) Sweep from Superfluid to phase in the Bose-Hubbard model p.4/14

5 Number Fluctuations Normal-mode expansion with label κ, eigenvalues λ ( ) κ 1 t J(t) t + 8λ κ [Un + 2J(t)λ κ ] δˆn κ = 0 xponential sweep of tunnelling rate (experiments) J(t) = J 0 exp{ γt} Scaling solution with τ κ = 4λ κ J(t)/γ ( [ ]) Un δˆn τκ 2 κ = 0 τ κ γ superfluid Adiabaticity parameter ν = Un γ = µ Ψ(t) γ Ψ Fast ν 1 vs slow ν 1 sweep < Ψ > Sweep from Superfluid to phase in the Bose-Hubbard model p.5/14

6 Cosmic Horizon superfluid c(t) Ψ(t) Ψ < Ψ > t Analogue of cosmic horizon horizon (t) = dt J(t )Un t Sweep from Superfluid to phase in the Bose-Hubbard model p.6/14

7 Quantum Fluctuations c(t) Size of cosmic horizon always decreases d dt horizon(t) < 0 Oscillation λ r(t) horizon crossing freezing λ r(t) and squeezing PSfrag replacements δn κ (t) t t δn frozen κ Amplification of quantum fluctuations ω/2 Analogous to early universe (WMAP) Sweep from Superfluid to phase in the Bose-Hubbard model p.7/14

8 Frozen Number Fluctuations δˆn 2 κ = n 1 e 2πν + O(tλ κ e γt ) 2πν Off-site number correlations decay exponentially ˆn αˆn β ˆn α ˆn β = δˆn α δˆn β = O(γt e γt ) On-site number variations 2 (n α ) = ˆn 2 α ˆn α 2 = δˆn 2 α = n 1 e 2πν 2πν Fast ν 1 sweep: 2 superfluid (n α ) n Ψ(t) (Poissonian superfluid) Slow ν 1 sweep: 2 (n α ) 0 ( ) Ψ < Ψ > Sweep from Superfluid to phase in the Bose-Hubbard model p.8/14

9 Off-diagonal long-range order Phase fluctuations grow (instability) δ ˆφ 2 κ = ν 1 e 2πν γ 2 t 2 + O(γt ln λ κ ) 2πn Decay of off-diagonal long-range order â α(t)â β (t) n exp{ U 2 t 2 2 (n α )} Loss of phase coherence due to horizon peak at k = 0 decreases (Greiner et al) superfluid Ψ < Ψ > Note: revival for Ut 2πN (similar to spin echo) Ψ(t) Sweep from Superfluid to phase in the Bose-Hubbard model p.9/14

10 Decay of Superfluid superfluid Superfluid fraction defined via j = ϱ sf Φ is determined by â lâl+1 and thus also decreases n sf { U n exp 2 t 2 n 1 } e 2πν 2πν Rapid sweep ν 1 decay with exp{ nu 2 t 2 } (independent of γ) Ψ < Ψ > Adiabatic sweep ν 1 decay of superfluid fraction much slower exp{ nu 2 t 2 /(2πν)} R. S., M. Uhlmann, Y. Xu and U. R. Fischer, Sweeping from the superfluid to phase in the Bose-Hubbard model, cond-mat/ Ψ(t) Sweep from Superfluid to phase in the Bose-Hubbard model p.10/14

11 Similarities to Cosmic Inflation Release of energy (p)re-heating Robust against initial (small-scale) perturbations Universality (no fine-tuning) superfluid Ψ < Ψ > Amplification of quantum fluctuations But: different spectrum in general Preferred frame (rest frame of medium) No unique/constant propagation speed Neglect of (quantum) back-reaction Ψ(t) Sweep from Superfluid to phase in the Bose-Hubbard model p.11/14

12 Speculations... Postulate: No (locally) preferred frame Unique/constant propagation speed A = 1 dt d 3 r Φ 2 ( Φ) 2 2 t 2 scale-invariance A[λt, λr] = A[t, r] Dominated by (quantum) back-reaction? correct 1/k 3 -spectrum (conformal de Sitter metric) Was cosmic inflation just a phase transition? R. S., Phys. Rev. Lett. 95, (2005) Sweep from Superfluid to phase in the Bose-Hubbard model p.12/14

13 Summary Analogy between cosmic inflation and dynamical quantum phase transitions ffective cosmic horizons loss of causal contact non-adiabatic behaviour amplification of quantum fluctuations Relation to real cosmic inflation? superfluid Ψ(t) Ψ < Ψ > Sweep from Superfluid to phase in the Bose-Hubbard model p.13/14

14 Acknowledgements German Research Foundation (DFG): mmy-noether Programme Alexander von Humboldt foundation SF-Programme Cosmology in the Laboratory Pacific Institute of Theoretical Physics U-IHP ULTI, CIAR, NSRC many interesting discussions... Sweep from Superfluid to phase in the Bose-Hubbard model p.14/14