Field-Driven Quantum Systems, From Transient to Steady State

Transcription

1 Field-Driven Quantum Systems, From Transient to Steady State Herbert F. Fotso The Ames Laboratory Collaborators: Lex Kemper Karlis Mikelsons Jim Freericks

2 Device Miniaturization Ultracold Atoms Optical Lattices Small potential difference Large electric fields Beyond linear response Ψ(t = t ) Prepared state? Ψ SS (t >> t ) Steady state Thermal State?

3 Models: Hubbard, Falicov-Kimball U(t) Hubbard, Falicov-Kimball J(t) H eq = <ij>,σ µ c iσ c iσ + U iσ i ) J ijσ (c iσ c jσ + h.c. J ijσ = J ij σ = J Hubbard n iσ n i σ J ij = and J ij = J F-K F-K: Heavy-light mixture. Electric field:e(t) = Eθ(t t ) Peierls substitution: J ijσ J ijσ e [ i ] R j R A(r,t)dr i

4 Methods Kadanoff Baym Keldysh Schwinger formalism Two-time contour ordered Green s function Contour ordered Greens function: t' t

5 NonEquilibrium DMFT Lattice to Impurity Need Inpurity Solver Exact solution for the Falicov-Kimball model U Λ(t, t ) G c imp(t, t ) = (1 w 1 ) [ (i c t + µ) δ c (t, t ) Λ(t, t ) ] 1 + w 1 [ (i c t + µ U) δ c (t, t ) Λ(t, t ) ] 1 Half-filling, w 1 =.5 J. K. freericks, Phys. Rev. B 77, 7519 (28)

6 Nonequilibrium Strong Coupling Expansion Atomic Green's function Hopping = o(n t4 ) operations K. Mikelsons, J. K. Freericks, and H. R. Krishnamurthy, Phys. Rev. L 19, 2642 (212)

7 Two-Time Green s Function G R,<,> (t, t ) extracted from contour ordered G c (t, t ) G R Density Of States G < Distribution Function Observables: J(t), E Kin (t), E Pot (t), E Tot (t)

8 Wigner Coordinates t = (t + t )/2 t max t rel = t t t max t t max t G R,< (, t rel ) t rel t t max t rel G R (, ω) Density Of States at time G < (, ω) Distribution Function at time

9 Thermalization Thermal State: FDT: G < (, ω) = 2iF T (ω)img R (, ω) ReG < (, ω) Joule heating: J E T E Kin =, E Pot = E Tot = U/4 H.F., K. Mikelsons and J. freericks, Scientific Reports 4, 4699 (213)

10 Density of states Falicov-Kimball E=1., U=3., T=.1. Hubbard U/J=24, U/T=4, E/U=(1,1,1) NESS F-K NESS Hubbard Im G R (, t rel ) Mixed Im G R (, t rel ) Mixed Dynamic Range Time Equilibrium NESS J t rel NESS Dynamic Range Time Equilibrium J t rel

11 Thermalization Thermal State: FDT: G < (, ω) = 2iF T (ω)img R (, ω) ReG < (, ω) Joule heating: J E T E Kin =, E Pot = E Tot = U/4

12 Relaxation Scenarios Using ReG < + Observables (J(t), E kin (t), E Pot (t), E Tot (t)) In both F-K and hubbard models: Monotonic Oscillatory Thermalization No Thermalization In the Hubbard model: A case of persistent oscillations H.F., K. Mikelsons and J. freericks, Scientific Reports 4, 4699 (213)

13 Monotonic thermalization F-K.4.3 E Tot (T = ) F-K Re G < (, t rel ) -.2 Max Energy, Current.2.1 E Pot current E Tot E kin -.4 t 2 rel t 1.2 Hubbard 6 E Tot (T = ) Hubbard Re G < (, t rel ) Max t 4 rel Energy, Current 4 2 E Pot E Tot current E kin t F-K, E=.5, U=1.5, T=.1. Hubbard, U/J=24, E/U=(1,,), U/T=4.

14 Monotonic approach to non-thermal state Re G < (, t rel ) Re G < (, t rel ) ( a ) F-K.4.8 ( b ) Max Energy, Current t rel ( c ) ( d ) Hubbard 6 Max t rel Energy, Current 4 2 E Pot current current E Tot (T = ) E Tot E kin -1 1 t E Tot (T = ) E Tot E kin Hubbard E Pot F-K t F-K, E=2., U=3., T=.1. Hubbard, U/J=24, E/U=1.25(1,,), U/T=4.

15 Relaxation Scenarios Using ReG < + Observables (J(t), E kin (t), E Pot (t), E Tot (t)) In both F-K and hubbard models: Monotonic Oscillatory Thermalization No Thermalization In the Hubbard model: A case of persistent oscillations

16 Relaxation Scenarios Using ReG < + Observables (J(t), E kin (t), E Pot (t), E Tot (t)) In both F-K and hubbard models: Monotonic Oscillatory Thermalization No Thermalization In the Hubbard model: A case of persistent oscillations What about integrability?

17 What about integrability? ETH, M. Rigol, V. Dunjko and M. Olshanii, Nature 452, 854 (28) M. Kollar, F. A. Wolf, and M. Eckstein, PRB 84, 5434 (211)

18 Distribution Function E-field along the diagonal E = E(1, 1, 1,...) Wigner distribution function n k (t) Band Energy, Band Velocity n k (t) n E(k),V (k) (t) n k (t) = ig < k+a(t) (t, t) E(k) = lim d J d d i=1 cos(k i) V (k) = lim d J d d i=1 sin(k i) Isolated system, Joule heating: T, n k (t).5 J (t) E

19 Time evolution of Wigner distribution function TOF measurements for heavy-light Fermi-Fermi mixtures Movies produced with Paraview/VTK H.F., J. Vicente and J. freericks, Phys. Rev. A (214)

20 Still Images, E=2., U=.25 t + 2π/U t + 2π/U +.5 2π/E t + 2π/U π/E t + 2π/U π/E

21 Still Images, E=2., U=.25 t + 2 2π/U t + 3 2π/U t + 4 2π/U t + 7 2π/U

22 Still Images, E=2., U=1. t + 1 2π/U t + 2 2π/U t + 3 2π/U t + 3 2π/U

23 Between the Transient and the steady state Desire solution from transient to steady state. The case of field-driven correlated systems Computational approaches: Sign problem in Quantum Monte Carlo Poor scaling Computational time and memory What about extrapolation from transient?

24 Beyond t max Want momentum dependent quantities at longer times Increase t max More memory, longer run times DMFT: Σ k (t, t ) = Σ(t, t ), G k (t, t ) = [ G k (t, t ) 1 Σ(t, t ) ] 1 Instead, extrapolate the self-energy Σ(t, t ) Σ(t, t ) also satisfies FDT: Σ < (, ω) = 2iF T (ω)imσ R (, ω) For FK model, we have an exact solution for the Steady State DMFT.

25 Monotonic Thermalization E=.5, U=1.5, T=.1. Monotonic thermalization System evolves through successive effective temperature states. 4 3 Transient G < Im[G < (ω)] 2 1 G < (T ) FDT G < ( = t max ) -2 2 ω H.F., K. Mikelsons and J. freericks, Scientific Reports 4, 4699 (213)

26 FDT for the Self-energy β = 1/T or T β=1/t fit to T Σ < (ω) T fit to β T ave FDT ω

27 ImΣ R,< (t rel ) completely defined by causality Steady State Steady State -.1 = t + 1. = t = t + 1. = t + 3. = t + 5. = t + 5. Im(Σ R (t rel )) = t + 7. = t + 9. = t Im(Σ R (t rel )) = t + 7. = t + 9. = t = t rel 2 4 t rel Steady State = t = t + 1. = t = t + 3. = t + 5. = t + 5. = t + 7. Im(Σ R (t rel )) = t + 7. = t + 9. = t Im(Σ R (t rel )) = t + 9. = t = t t rel 2 4 t rel

28 Monotonic Thermalization of Σ Relaxation completely captured by RealΣ < (t rel ). 2 Real(Σ < (t rel )) t rel

29 Extrapolate Σ to longer... Extrapolation to longer t max 2 = t = t Σ < (, t rel ) 1-1 Σ < (, t rel ) t 2 rel 2 = t t 2 Extrapolation rel Initial data = t Σ < (, t rel ) 1-1 Σ < (, t rel ) t 2 rel -2 t 2 rel Can calculate k-dependent quantities to longer t max

30 Conclusion and outlook Field-Driven Quantum System Different Relaxation scenarios What about integrability? Frustrated phase separation Extrapolation scheme to bridge the gap between transient and steady state Can calculate momentum dependent quantities at longer times How about oscilatory relaxation? Thank you!