QUANTUM SIMULATORS JENS EISERT, FU BERLIN PROBING THE NON-EQUILIBRIUM

Transcription

1 QUANTUM SIMULATORS PROBING THE NON-EQUILIBRIUM JENS EISERT, FU BERLIN JOINT WORK WITH HENRIK WILMING, INGO ROTH, MARCEL GOIHL, MAREK GLUZA, CHRISTIAN KRUMNOW, TERRY FARRELLY, THOMAS SCHWEIGLER, JOERG SCHMIEDMAYER AND OTHERS

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3 WHAT THIS TALK IS ABOUT HOW DO SYSTEMS EQUILIBRATE AND THERMALIZE? OR NOT, IN CASE OF MANY-BODY LOCALIZATION?

4 WHAT THIS TALK IS ABOUT WHAT WINDOWS INTO PROBING THIS DO QUANTUM SIMULATORS PROVIDE?

5 WHAT THIS TALK IS ABOUT CAN QUANTUM SIMULATORS SHOW A QUANTUM ADVANTAGE, AND IF SO HOW COULD WE EVER FIND OUT?

6 REVISITING EQUILIBRATION arxiv:

7 EQUILIBRATION O H = X x2 h x HOW DO NON-INTEGRABLE MODELS EQUILIBRATE AND THERMALIZE? Gring, Kuhnert, Langen, Kitagawa, Rauer, Schreitl, Mazets, Smith, Demler, Schmiedmayer,, Science 337, 1318 (2012) Kaufman, Tai, Lukin, Rispoli, Schittko, Preiss, Greiner, Science 353, 794 (2016) Trotzky, Chen, Flesch, McCulloch, Schollwöck, Eisert, Bloch, Nature Physics 8, 325 (2012) Choi, Hild, Zeiher, Schauß,Rubio-Abadal, Yefsah, Khemani, Huse, Gross, Science 352, 1547 (2016)

8 EQUILIBRATION O H = X x2 h x Equilibration to time averages of local observables O := ho(t)i = lim T!1 Z T 0 tr( (t)o) =tr(!o) O Gring, Kuhnert, Langen, Kitagawa, Rauer, Schreitl, Mazets, Smith, Demler, Schmiedmayer,, Science 337, 1318 (2012) Kaufman, Tai, Lukin, Rispoli, Schittko, Preiss, Greiner, Science 353, 794 (2016) Trotzky, Chen, Flesch, McCulloch, Schollwöck, Eisert, Bloch, Nature Physics 8, 325 (2012) Choi, Hild, Zeiher, Schauß,Rubio-Abadal, Yefsah, Khemani, Huse, Gross, Science 352, 1547 (2016)

9 EQUILIBRATION O H = X x2 h x Equilibration to time averages of local observables O O := ho(t)i = lim T!1 Z T 0 tr( (t)o) =tr(!o) Invoke eigenstate therm hypothesis (ETH) tr \A ( eihe ) = eihe A tr \A e Deutsch, Phys Rev A 43, 2046 (1991) Srednicki, Phys Rev E 50, 888 (1994) Z H Thermalization: Form its own heat bath

10 EQUILIBRATION O Equilibration to time averages of local observables O O := ho(t)i = lim T!1 Z T Deviations from time average Var(O, H, ) :=(ho(t)i O) 2 applekok 2 e S 2(!) in terms of Renyi entropy S ( ) := 1 1 log (tr( )) 0 tr( (t)o) =tr(!o) Systems equilibrate if effective dimension S 2 (!) is large Reimann, Phys Rev Lett 101, (2008) Linden, Popescu, Short, Winter, Phys Rev E 79, (2009) Reimann, Kastner, New J Phys 14, (2012) Short, Farrelly, New J Phys 14, (2012). Gogolin, Eisert, Rep Prog Phys 79, (2016) H = X x2 h x

11 EQUILIBRATION O H = X h x x2 BUT WHEN IS THE EFFECTIVE DIMENSION LARGE? A DIFFERENT TAKE Wilming, Goihl, Roth, Eisert, arxiv:

12 VOLUME LAWS AND ENTANGLEMENT ERGODICITY A H = X x2 h x Generic eigenvectors { ei} of local Hamiltonians* are expected to satisfy a volume law for the entanglement entropy S ( A (e)) A Obviously, ETH implies sth much stronger FOR WHAT VALUES OF Overwhelming numerical (and analytical) evidence For > 1 only Rigol, Dunjko, Olshanii, Nature 452, 854 (2008) Fujita, Nakagawa, Sugiura, Watanabe, arxiv: Lu, Grover, arxiv: IS THIS MEANINGFUL? But then all Renyi entropies are equivalent Huang, arxiv: Vidmar, Hackl, Bianchi, Rigol, Phys Rev Lett 119, (2017) Lloyd, Pagels, Ann Phys 188, 186 (1988) Page, Phys Rev Lett 71, 1291 (1993) * lim!1 he H ei N = e Wilming, Goihl, Roth, Eisert, arxiv:

13 ENTANGLEMENT ERGODICITY A Entanglement ergodicity: A system is EG, if there ex a positive function g, such that for energy eigenvectors ei with energy density e the conditions hold For every sufficiently large lattice there exists a subsystem such that the reduced state A (e) =tr A c ( eihe ) fulfills S 2 ( A (e)) g(e)n The function g is sufficiently well behaved A 2-Renyi entropy taken for convenience Entanglement ergodicity is stable under short evolution Is what many-body localization is not about Wilming, Goihl, Roth, Eisert, arxiv:

14 EQUILIBRATION FROM ENTANGLEMENT ERGODICITY A Entanglement ergodicity: S 2 ( A (e)) g(e)n WHAT DOES THIS MEAN FOR EQUILIBRATION? Wilming, Goihl, Roth, Eisert, arxiv:

15 EQUILIBRATION FROM ENTANGLEMENT ERGODICITY A Entanglement ergodicity: S 2 ( A (e)) g(e)n ) Bound overlaps of products with energy eigenstates i Inside [e,e+ ] Energy density of i outside [e,e+ ] Anshu, New J Phys 18, (2016).,exploit ergodicity High diagonal entropy for all Renyi entropies for all 0 Equilibration: For any state in the same phase as a product, with energy density e, and a Hamiltonian with non-degenerate gaps in spatial dimension, there exists constants C and k(e) > 0 such that Var(O, H, ) applekok 2 Ce k(e)n/( +1) Wilming, Goihl, Roth, Eisert, arxiv:

16 LESSON A Lesson: From (very plausible) entanglement ergodicity, general strong equilibration follows ETH Wilming, Goihl, Roth, Eisert, arxiv:

17 GAUSSIFICATION AND A COLD ATOMIC EXPERIMENT In preparation

18 Gluza, Krumnow, Friesdorf, Gogolin, Eisert, PRL 117 (2016) Cramer, Dawson, Eisert, Osborne, PRL100, (2008) Calabrese, Cardy, Phys Rev Lett 96, (2006) GAUSSIFICATION A Reduced states become Gaussian in time, even if initial states are highly correlated Quenched non-interacting systems Gaussify in time

19 GAUSSIFICATION Gaussification: If A initial states have clustering correlations tr( AB) tr( A)tr( B) apple C A B e d(a,b)/ the Hamiltonian is quadratic and free-particle ergodic then, for any local observable A, any " > 0, there exists a relaxation time t rel independent of the system size such that for all t 2 [t rel,t rec ] tr(a(t) ) tr(a(t) G ) < " where G is a (possibly time-dependent) Gaussian state Proof techniques: Lieb-Robinson bounds, Bernstein-Spohn blocking, fermionic Lindeberg central limit theorem True for all planar lattices, non-gaussian initial state, bosons and fermions Gluza, Krumnow, Friesdorf, Gogolin, Eisert, PRL 117 (2016) Cramer, Dawson, Eisert, Osborne, PRL100, (2008)

20 GAUSSIFICATION A GREAT, BUT WHAT DO THE SECOND MOMENTS DO?

21 GAUSSIFICATION A H = X Free-particle ergodicity: A system is ergodic if there exists a time t h x x2 such that for all t 2 [t,cl], the propagator is suppressed as W j,k (t) <Ce t for some > 0 nx f j (t) = W j,k (t)f k WHEN IS THIS THE CASE? k=1

22 GAUSSIFICATION A H = X Free-particle ergodicity: A system is ergodic if there exists a time t h x x2 such that for all t 2 [t,cl], the propagator is suppressed as W j,k (t) <Ce t for some > 0 Kuzmin theorem: Suppose (a n ) are real numbers and the ) Always, for all translationally invariant local models* Gluza,Eisert, Farrelly, in preparation gaps n =(a n+1 a n ) are (i) increasing and (ii) satisfy n 2 [, 2 ] with > 0, then NX 1 n=0 e ia n apple cot( /4) apple 2 * Under very mild assumptions, basically the dispersion relation must not be flat and no points with E 00 (p) =E 000 (p) =0

23 GAUSSIFICATION A Convergence to GGE: For generic free fermionic/bosonic TI and Hamiltonians and short range correlated states, one finds convergence to a Gaussian GGE, k A (t) G (t)k 1 = O(t 1 ) Largely generalizes beautiful work by Calabrese, Essler, Fagotti, PRL 106, (2011) No ambiguity in GGE (charges are const of motion) Lesson: Get here the full picture of equilibration and GGE convergence, including large independence of initial conditions

24 EQUILIBRATION FROM ERGODICITY A CAN ONE EXPERIMENTALLY FIND GAUSSIFICATION?* * Joerg Schmiedmayer, Thomas Schweigler et al

25 EXPERIMENTAL GAUSSIFICATION Yes, in the quantum field states of cold atoms on atom chips Density and phase quadratures (z) p n GP (z)+ (z)e i (z) [ (x), (y)] = i (x y)

26 EXPERIMENTAL GAUSSIFICATION Observe Gaussification from higher moments Connected correlation function P M = P z W (4) (z) z Z(4) (z) Full correlation function Lesson: Gaussification can be observed, but only phase differences can be directly measured Schweigler, Gluza, Rauer, Eisert, Schmiedmayer, in preparation HOW CAN ONE MEASURE THE MISSING QUADRATURES? Schweigler, Gluza, Eisert, Schmiedmayer, in preparation

27 TOMOGRAPHY OF INACCESSIBLE QUADRATURES Evolution in effective non-interacting (of sine-gordon) model H = Z L 0 dz ngp (z) 4m (@ z (z)) 2 + g (z) 2 = X k>0! k ( 2 k + 2 k)+g 2 0 Gluza, Schweigler, Krumnow, Schmiedmayer, Eisert, in preparation

28 TOMOGRAPHY OF INACCESSIBLE QUADRATURES Evolution in effective non-interacting (of sine-gordon) model H = Z L 0 dz ngp (z) 4m (@ z (z)) 2 + g (z) 2 = X k>0! k ( 2 k + 2 k)+g 2 0 Gluza, Schweigler, Krumnow, Schmiedmayer, Eisert, in preparation

29 RECOVERY OF ALL QUADRATURES Make it a convex (SDP) recovery problem Measure many phases in time slices Evolve under effective model Find most likely density profile in 2-norm, under Heisenberg constraint + i 0 Gluza, Schweigler, Krumnow, Schmiedmayer, Eisert, in preparation

30 RECOVERY OF ALL QUADRATURES Works very well: E.g., recurrences in quenched quantum systems Phase coherence as a function of time Rauer, Erne, Schweigler, Cataldini, Tajik, Schmiedmayer, Science 6 Apr (2018) Gluza, Schweigler, Krumnow, Schmiedmayer, Eisert, in preparation

31 QUENCHED COLD ATOMIC SYSTEMS ON ATOM CHIPS Lesson: Gaussification is observed; but equally interesting is a new window into cold atomic quantum simulators Quadratures in 1D bosons can be measured Consistent with interacting thermal state preparation Coming now: Entanglement Gluza, Schweigler, Krumnow, Schmiedmayer, Eisert, in preparation

32 ABSENCE OF THERMALIZATION arxiv: In preparation

33 MANY-BODY LOCALIZATION: INTERPLAY OF DISORDER AND INTERACTION Some systems stubbornly refuse to thermalize Many-body localization: Interplay of disorder and interactions

34 MANY-BODY LOCALIZATION: INTERPLAY OF DISORDER AND INTERACTION Anderson model: Particle hopping on a line under random potential H = X j ( jihj +1 + j +1ihj + f j jihj ) Static localization: Most eigenstates have clustering correlations Dynamic localization: E(sup t hn e ith mi ) apple c 1 e c 2dist(n,m) Many-body localization: Still a crime story

35 Log-growth of entanglement Matrix product, area-law eigenstates MANY-BODY LOCALIZATION: INTERPLAY OF DISORDER AND INTERACTION Some systems stubbornly refuse to thermalize H = LX x i x i+1 + y i Znidaric, Prosen, Prelovsek, PRB 77, (2008) Badarson, Pollmann, i=1moore, PRL 109, (2012) Kim, Chandran, Abanin, arxiv: Eisert, Osborne, Phys Rev Lett 97, (2006) y i+1 + i z z i+1 + h i z i Bauer, Nayak, J Stat Mech P09005 (2013) Luitz, Laflorencie, Alex, arxiv: Rich phenomenology, still a crime story l-bit Hamiltonian in terms of quasi-local com H (N eff ) e = X i! (1) i i z + X i,j Friesdorf, Werner, Goihl, Eisert, Brown, NJP 17, (2015) Chandran, Carresquilla, Kim, Abanin, Vidal, PRB 92, (2015)! (2) i,j z i z j + r Huse, Nandkishore, Oganesyan, Phys Rev B 90, (2014) Friesdorf, Werner, Goihl, Eisert, Brown, NJP 17, (2015) Brown, Goihl, Werner, Eisert, im preparation Slow information propagation Kim, Banuls, Cirac, Hastings, Huse, PRE 92, (2015) Slow equilibration Brown, Goihl, Werner, Eisert, im preparation

36 MANY-BODY LOCALIZATION: INTERPLAY OF DISORDER AND INTERACTION l-bit Hamiltonian in terms of quasi-local com H (N eff ) e = X i! (1) i i z + X i,j! (2) i,j z i z j + r BUT HOW TO FIND L-BIT HAMILTONIAN? Huse, Nandkishore, Oganesyan, Phys Rev B 90, (2014) Vosk, Altman, PRL 110, (2013) Rademaker, Ortuno, PRL 116, (2016) Pekker, Clark, Oganesyan, Refael,

37 MANY-BODY LOCALIZATION: INTERPLAY OF DISORDER AND INTERACTION Permute ki = (e)i, 2 S D s.t. Pauli algebra is exactly satisfied Start from Z (1) = z 1 2 L 1 and order 1 (e)i so that diagonal of (1) z (t) = X (1) he z ei eihe e (2) is ordered decreasingly, order 2 1 (e)i so that diag of 2 is ordered, repeat (Iterate to tensor network method) Goihl, Gluza, Krumnow, Eisert, arxiv: Kulshreshtha, Pal, Wahl, Simon, arxiv:

38 MANY-BODY LOCALIZATION Cold atomic quantum simulations of many-body localization Schreiber, Hodgman, Bordia, Lueschen, Fischer, Vask, Altman, Schneider, Bloch, Science 349, 842 (2015) Choi, Hild, Zeiher, Schauss, Rubio-Abadal, Yefzah, Khemani, Huse, Bloch, Gross, Science 352, 1547 (2016) HOW TO UNAMBIGUOUSLY MEASURE MBL? With Gross, Bloch, Goihl, Gluza, in preparation

39 MANY-BODY LOCALIZATION In-situ and parity-projected density-density correlations f Corr (k, t) = hn L/2 n L/2+k i hn L/2 ihn L/2 + ki y Corr (k, t) = X k Z T f Corr (k, t)k 2 f eq (T )= 1 d hp i i 0 (t) Lesson: Building Ton l-bit intuition, can devise feasible 0 witnesses of MBL discriminating from Anderson localization With Gross, Bloch, Goihl, Gluza, in preparation

40 TOWARDS QUANTUM ADVANTAGES Phys Rev X, today (2018) Quantum, in press (2018)

41 QUANTUM SIMULATORS SHOWING A QUANTUM ADVANTAGE Quest for quantum advantage of quantum devices Quantum simulators already outperform state-of-the art algorithms Trotzky, Chen, Flesch, McCulloch, Schollwöck, Eisert, Bloch, Nature Physics 8, 325 (2012) LACK OF IMAGINATION? Braun, Friesdorf, Hodgman, Schreiber, Ronzheimer, Riera, del Rey, Bloch, Eisert, Schneider, Proc Natl Acad Sci (2015) Choi, Hild, Zeiher, Schauss, Rubio-Abadal, Yefzah, Khemani, Huse, Bloch, Gross, Science 352, 1547 (2016)

42 QUANTUM SIMULATORS SHOWING A QUANTUM ADVANTAGE IBM/Google: Intermediate problems for superconducting qubits Boson sampling: Outperforms classical computers in terms of computational complexity Aaronson, Arkhipov, Th Comp 9, 143 (2013) Boixo, Isakov, Smelzanski, Babbush, Ding, Jiang, Bremner, Martinis, Neven, arxiv: (2016) HOW CAN ONE EVER BE SURE THAT THEY DO THE RIGHT THING? But: No efficient discrimination from classical devices

43 QUANTUM SIMULATORS SHOWING A QUANTUM ADVANTAGE Devise cold atomic simulators showing a quantum advantage Prepare product state Evolve for unit time under local Ham In-situ measure Gives computationally hard intermediate problem (sampling is hard up to an additive error in the total variation distance), but Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, Phys Rev X (2018), arxiv:

44 QUANTUM SIMULATORS SHOWING A QUANTUM ADVANTAGE Devise cold atomic simulators showing a quantum advantage it can be efficiently certified (with local measurements) Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, Phys Rev X (2018), arxiv:

45 SUMMARY ETH implies equilibration Quenched systems Gaussify New windows into quantum simulators Quantum simulators probing MBL Quantum simulators Quantum simulators can outperform classical ones This talk