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Out-of-equilibrium dynamics of inhomogeneous quantum systems
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Out-of-equilibrium dynamics of inhomogeneous quantum systems
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Out-of-equilibrium dynamics of inhomogeneous quantum systems b y Sunil Y esh w an th Ma y 13, 2016 A dissertation presen ted to the Univ ersit y of Southern California Graduate Sc ho ol in partial fulllmen t of the requiremen ts for the degree of Do ctor of Philosoph y , Ph ysics i ABSTRA CT The question of whether and ho w closed quan tum systems equilibrate lies at the foun- dations of statistical mec hanics. It in v olv es asking whether suc h systems can b e describ ed b y statistical equilibrium ensem bles and ho w suc h ensem bles can b e deriv ed from the equa- tions of motion[25]. Despite its imp ortance, this has remained a dicult question to tac kle. Exp erimen tal in v estigations require that the system b e w ell isolated from its en vironmen t and its dynamics b e con trolled and measured precisely . Ho w ev er, recen t exp erimen ts ha v e demonstrated the capabilit y to conduct precise studies of out-of-equilibrium dynamics in closed quan tum systems. F or example, in [39], a series of tub es con taining ultracold 87 Rb atoms w as created using a t w o dimensional optical lattice. The atoms conned to these one dimensional tub es exhibited a lac k of thermalization, after exp eriencing thousands of collisions with eac h other. Suc h exp erimen ts in v olving ultracold atoms trapp ed in optical lattices oer the abilit y to sim ulate condensed matter systems and test fundamen tal questions in statistical mec hanics. The parameters of optical lattices can b e tuned dynamically in order to sim ulate a v ariet y of mo dels. A toms are trapp ed in these lattices after b eing co oled to nanok elvin temp eratures and are w ell isolated. Their prop erties suc h as site o ccupations and momen tum distribution can b e measured precisely . In this dissertation, w e consider closed inhomogeneous systems that ha v e b een realized exp erimen tally using cold atoms. W e use analytical and n umerical tec hniques to explore the prop erties and applications of non-equilibrium dynamics in these systems. The rst problem w e consider is that of ho w long closed quan tum systems tak e to equi- librate. Despite its apparen t simplicit y , this is as y et an unresolv ed question. Starting with a rigorous denition of equilibration time, w e in v estigate the approac h to equilibrium in ho- mogeneous and disordered one dimensional systems. W e nd equilibration times that agree with ph ysical in tuition. The second problem that w e tac kle in v olv es probing the in terfaces b et w een co existing phases in trapp ed quan tum gases. W e prop ose a no v el approac h that relies on measuring the temp oral uctuations of observ ables. The capabilit y to measure the time ev olution of observ ables in cold atom systems has b een demonstrated in exp erimen ts. F or example, ii in [23], a series of one dimensional decoupled c hains con taining 10 to 18 atoms eac h w ere realized. The exp erimen talists w ere able to study the time dep endence of spatial correlations in eac h of these c hains follo wing a sudden c hange of the optical lattice depth. This w as made p ossible in part b y adv ances in single site imaging tec hniques, allo wing exp erimen talists to study the dynamics of individual trapp ed atoms [5]. This dissertation is organized as follo ws: In Section I, w e review some asp ects of exp erimen ts in v olving ultracold atoms. W e fo cus on the realization of the Hubbard mo del using optical lattices, and the Mott insulating and sup eruid phases. In Section I I, w e review some of the theoretical framew ork for studying non-equilibrium dynamics in closed quan tum systems. W e pro vide precise denitions of equi- libration and review some of the analytical approac hes used to study temp oral uctuations in free fermion systems. In Section I I I, w e in v estigate the approac h to equilibrium in clean and disordered quan tum systems. W e start b y dening equilibration time precisely , based on the long time a v erage v alue of an observ able. W e then consider clean systems, where, using analytical approac hes, w e nd constan t equilibration times for the case of clustering initial states with system parameters far from critical p oin ts. Ho w ev er, when w e consider small quenc hes around a critical p oin t, w e nd the equilibration time to exhibit a p o w er la w dep endence on the system size. W e then analyze noisy systems, where literature p oin ts at the existence of large equilibration time scales that div erge exp onen tially with system size. Sp ecically , w e consider the tigh t-binding mo del with diagonal impurities and giv e n umerical evidence for the existence of these exp onen tially large equilibration times. W e also nd that the equilibration time dep ends on our c hoice of observ able. Finally , w e consider y et another noisy system whose ev olution dynamics is randomly sampled from an ensem ble of unitary matrices. Here, w e are able to pro v e analytically that the equilibration time is constan t, th us sho wing that noise alone do es not guaran tee slo w equilibration. In Section IV, w e presen t a no v el tec hnique that uses temp oral uctuations to prob e the prop erties of cold quan tum gases conned in harmonic traps. The ground states of these systems exhibit a co existence of sup eruid and Mott insulating regimes, whic h are con v en- tionally detected using the lo cal compressibilit y [7, 59, 60]. Motiv ated b y a scaling analysis [20], w e prop ose an alternativ e approac h in v olving the out-of-equilibrium dynamics follo wing iii a w eak quenc h. In particular, w e sho w ho w the temp oral uctuations of the site o ccupations rev eals the lo cation of spatial b oundaries b et w een compressible and incompressible regions. W e demonstrate the feasibilit y of our approac h for sev eral mo dels using n umerical sim ula- tions. W e rst consider in tegrable systems, hard-core b osons (spinless fermions) conned b y a harmonic p oten tial, where space separated Mott and sup eruid phases co exist. W e then analyze a non-in tegrable system whic h has co existing c harge densit y w a v e and sup er- uid phases. The results of our sim ulations indicate that the temp oral v ariance of the site o ccupations is a more eectiv e detector of phase b oundaries than the lo cal compressibilit y . In Section V, w e study the exp erimen tal feasibilit y of the metho d prop osed in the previous section. Based on tec hniques applied in exp erimen tal literature [23] for measuring out-of- equilibrium b eha vior of site o ccupations, w e prop ose estimators of the temp oral v ariance. W e study the con v ergence prop erties of these estimators to examine their dep endence on the n um b er of measuremen ts carried out. Using n umerical exp erimen ts, w e demonstrate that the b oundary b et w een co existing regimes can b e sp otted correctly using as few as 30 measuremen ts. iv A CKNO WLEDGMENT My researc h and coursew ork at the Univ ersit y of Southern California w ere made p ossible b y the close supp ort of sev eral colleagues, friends, and family . I w as fortunate to ha v e Lorenzo Camp os V en uti as m y advisor. None of the researc h that I conducted at USC w ould ha v e b een p ossible without his men torship. His guidance and innite patience in discussing researc h and ph ysics concepts w ere in v aluable. I am also grateful for his thorough review and feedbac k on initial drafts of this dissertation. I w as similarly fortunate to ha v e Stephan Haas as a men tor and to collab orate with him on researc h pro jects. I greatly appreciate his constan t guidance and advice. I am grateful to Lorenzo Camp os V en uti, Stephan Haas, Christoph Haselw andter, Aiic hiro Nak ano, and Susum u T ak ahashi for taking the time to b e on m y qualifying examination committee. I w ould also lik e to thank the mem b ers on m y dissertation defense committee, Lorenzo Camp os V en uti, Stephan Haas, Chi Mak, Aiic hiro Nak ano, and Susum u T ak ahashi. I am grateful for their review of m y dissertation and for their insigh tful questions and discussion during m y defense. One of the b est things to emerge from m y sta y at USC has b een m y friendship with Stephen Pink erton and Scott MacDonald. I will not forget w orking with them on the late nigh t piles of homew ork and screening exam preparation material, accompanied b y coee and pancak es. As a studen t more than 9,000 miles from home, the constan t supp ort and encouragemen t of m y paren ts has mean t a lot to me. Starting from sc ho ol pro jects, they ha v e alw a ys motiv ated me to pursue m y in terests and in tro duced me to ph ysics, programming, and electronics. I am also grateful to m y brother Chandan for serious discussions on computer science and not-so-serious discussions ab out cats on the in ternet. A big thank y ou to m y in-la ws for the w arm w elcome to the family and for supp orting me through the nal y ears of m y Ph.D. And a sp ecial men tion of m y small nephew who brough t fun to ev ery o ccasion. I am forev er grateful to m y wife Sh w eta for her supp ort and encouragemen t. Marrying her w as the b est decision I ev er made. v LIST OF FIGURES 1 Fig. 1 (a) from [36], describing the parameters in the Hamiltonian. V 0 repre- sen ts the depth of the p oten tial w ells in whic h atoms are conned, forming a lattice-lik e structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Energy gap in the sp ectrum of the Mott phase. (Example for a system with n = 1 atom lo cated on eac h lattice site.. Figure from [34]. . . . . . . . . . . . . . . . . . . 4 3 Optical lattices depicting connemen t in 2 and 3 dimensions. Figure from [8]. . 5 4 Connemen t in an optical lattice and the sligh t oset of p oten tial from one site to another due to the Gaussian in tensit y prole of the laser. Figure from [57]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 Connemen t in an optical lattice and the sligh t oset of p oten tial from one site to another due to the Gaussian in tensit y prole of the laser.. Figure from [34]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6 Probabilit y distribution of an observ able, when time is considered as a uni- formly distributed random v ariable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7 An example where T eq 100. This sho ws the ev olution of an observ able n ` (t) = E [N ` (t)=L] with time. This observ able measures the n um b er of particles in the lo w est in sites 1 48 for a system of size L = 96 with 24 particles. The system is initialized at t = 0 with all 24 particles in the rst 24 sites. It is allo w ed to ev olv e in time under the inuence of a random onsite p oten tial. W e presen t futher details of suc h systems and their time ev olution as part of the discusson on noisy systems in subsection C. . . . . . . . . . . . . . . . . . 24 8 Equilibration times T L eq for the Losc hmidt ec ho as a function of the system size L for dieren t quenc h parameters h 1;2 . T L eq is computed b y solving n umerically for the rst solution of L (t) =L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 9 Equilibration times T L eq for the Losc hmidt ec ho close to the crosso v er region h'cL 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 10 Equilibration times T m z eq for the transv erse magnetization m z (t) =h z 1 (t)i as a function of the size L for dieren t quenc h parameters h 1;2 . T m z eq is computed exactly solving for the rst solution of m z (t) =m z . . . . . . . . . . . . . . . . . . . . . . . 30 vi 11 Equilibration timesT m z eq for the transv erse magnetization close to the crosso v er region h'c 0 L 1 . F or this v alues of parameters c 0 ' 13. . . . . . . . . . . . . . . . . . . 30 12 Relaxation of n ` (t) = E [N ` (t)=L] with time for lengths L = 16; 40; 64; 112, at lling = N=L = 0:25 and = `=L = 0:5. The random distribution has v ariance = 0:6 (W = 0:6 p 3 1:03). The ensem ble a v erage is computed using 1000 realizations at eac h time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 13 Scaling of equilibration time T eq (L) vs system size L, extracted from n ` (t). T eq is obtained from this rst solution of E[n ` (T eq )] = E[n ` (t)]. P arameters are as in Figure 12 . The ensem ble a v erages are obtained summing o v er 2000 realizations. The equilibration time is found to ob ey log(T eq ) = 0:021L 0:6266 + 2:53 with residual norm 0:17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 14 Relaxation of F(t) = E[log(L(t))]. P arameters are as in Figure 12 and an ensem ble a v erage w as p erformed o v er 1000 realizations at eac h instan t in time. 36 15 Scaling of the relaxation time T eq suc h that F(T eq ) = F , T eq = 1:39L 0:2771 0:847 with residual norm 0:05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 16 Plot of F (t) forL = 120, at = 1=2 (half lling). The v ariance F (t) is sho wn as error bar. Numerical results are obtained a v eraging o v er N samples = 7500 unitaries distributed according to the Haar measure and obtained with the algorithm outlined in [46]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 17 V erication of Eqs. (24) and (25), whic h predict AB / L and B A 2 / L 2 whenq = 1. A b est t to algebraic scaling giv es B A 2 /L 2:0 and AB /L 1:1 . W e consider a tigh t binding mo del at half lling with b oth the observ able and p erturbation set to ^ A = ^ B = P i (1) i ^ n i . B A 2 and AB are made dimensionless b y dividing b y their v alues at L 0 = 102, i.e., w e plot e AB = AB (L)= AB (L 0 ) and B e A 2 = B e A 2 (L)= B e A 2 (L 0 ). . . . . . . . . . . . . . . . . . . . . . 45 vii 18 (a) W edding cak e site o ccupation prole of hard-core b osons in a one- dimensional harmonic trap describ ed b y Eq. (26). The system consists of L = 500 sites and N = 250. The Hamiltonian parameters are = 10, = 0:2, = L 2 (J = 1 throughout). The phase b oundaries b et w een the Mott plateau lo cated at the trap cen ter and the adjacen t sup eruid regions can b e detected b y the con v en tional lo cal compressibilit y i (red) and b y the temp oral v ariance of the site o ccupations N 2 i (green) in tro duced in this w ork. (b) A closer lo ok at the sup eruid region for the system sho wn in (a) rev eals temp oral v ariance p eaks at the in terface b et w een the sup eruid and the Mott insulator. (c) Finite-size scaling of the maxim um temp oral v ariance of the site o ccupations and of the compressibilit y vs L for the Hamiltonian in Eq. (26). W e nd N 2 max / L 0:83 and max / L 0:05 . Both quan tities in this plot are made dimensionless b y dividing b y their v alues at L 0 = 50, i.e., e max = max (L)= max (L 0 ) and e N 2 max = N 2 max (L)=N 2 max (L 0 ). (d) Dep endence of the normalized temp oral v ariance (N 2 i = 2 ) on the quenc h amplitude for the Hamiltonian in Eq. (26) with all other parameters as in (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 19 (Color online) (a) Unit cell a v eraged site o ccupancy in the presence of a stag- gered p oten tial Eq. (30). This is a system with L = 500, N = 150, and parameters = 10, = 0:2, = 1=L 2 , V 0 = 1:5. (b) A closer lo ok at the sup eruid region for the system sho wn in (a) rev eals temp oral v ari- ance p eaks at the in terface b et w een the sup eruid and the Mott insulator. (c) Finite-size scaling of the maxim um temp oral v ariance of the site o ccupa- tions and of the compressibilit y vs L for the Hamiltonian Eq. (30). W e nd N 2 max /L 0:80 and max /L 0:14 . Both quan tities in this plot are made dimen- sionless b y dividing b y their v alues at L 0 = 50, i.e.,e max = max (L)= max (L 0 ) and e N 2 max = N 2 max (L)=N 2 max (L 0 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 viii 20 (a) Spatial prole of the temp oral v ariance of the site o ccupations N 2 i and of the lo cal compressibilit y i for the mo del in Eq. (33). W e initialize the system with 19 sites and 5 particles in the ground state with parameters J = 1, V = 8:0, V 0 = 0:5 and = 0:1225. The quenc h is p erformed b y c hanging the trap p oten tial from to + with = 0:0061. (b) Dep endence of the v ariance on the quenc h amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 21 (a) Distributions of the site o ccupations N i (t) at sites near the in terface b e- t w een the CD W-I and the sup eruid phase. (b) and (d) Distribution function of the site o ccupation at a site deep in the CD W-I regime (site i = 1) and at a site at the edge of the CD W-I domain (site i = 10), resp ectiv ely . (c) and (e) Time dep endence ofN i (t) corresp onding to (b) and (d), resp ectiv ely . These results are obtained from sim ulations with the same Hamiltonian and system parameters as in Fig. 20. Eac hN i (t) is sampled at N = 4 10 4 random times uniformly distributed in [0;T ] with T = 40~=J . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 22 Estimating the temp oral v ariance. (a) An example of the temp oral v ariance estimator s i for N T = 15. All parameters are the same as for Fig. 20. W e compute s i indep enden tly for eac h of the L = 19 sites of the system. (b) Scaling of the v ariance of the estimator s i at the site i = 10, with N d . The t sho ws var(s i ) N 0:98 T in accordance with our theoretical prediction that var(s i ) 1=N T : (c) Results of a n umerical exp erimen t to compute p M , the probabilit y that the maxim um of s i is lo cated at the maxim um of the exact v ariance i.e. site i2f9; 10g. 10 4 samples offs i g w ere used to compute this probabilit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ix 23 Estimating the temp oral v ariance. (a) An example of the temp oral v ariance estimator v at eac h site for N S = 3 and N T = 10. All parameters are the same as for Fig. 20. W e compute v indep enden tly for eac h of the L = 19 sites of the system. (b) Scaling of the v ariance of the estimator v at the site i = 10 with N T , for dieren t v alues of N S . The t sho ws var(s i ) N 0:97 T for the N S v alues considered, in accordance with our prediction that v is a consisten t estimator. (c) Results of a n umerical exp erimen t to compute p M , the probabilit y that fv i g and the exact v ariance ha v e maxima at the same sites (i = 9; 10). 10 4 samples offv i g w ere used to compute these probabilities. 63 x CONTENTS Abstract ii A c kno wledgmen t v List of Figures vi I. Ultracold atoms trapp ed in optical lattices 1 A. Bac kground and Bose-Hubbard Hamiltonian 2 B. Exp erimen tal setup 4 C. Hard core Bosons and spinless F ermions 10 I I. Non-equilibrium dynamics of nite isolated quan tum systems 12 A. Non-equilibrium dynamics 12 B. F ree F ermions and co v ariance matrices 14 C. Losc hmidt ec ho 18 I I I. Equilibration times in clean and disordered quan tum systems 21 A. Setting and denitions 22 B. Clean systems 25 C. Quan tum Ising mo del 27 D. The Anderson mo del 31 E. A CUE (circular unitary ensem ble) mo del 36 IV. Small quenc h dynamics as an in v estigativ e to ol for trapp ed atom systems 41 A. Quenc hes and Observ ables 43 B. Hard-core b oson systems 46 C. Nonin tegrable systems 51 D. Conclusions 54 V. Ecien t measuremen t of temp oral uctuations 56 A. Motiv ation 56 B. Estimation with N s = 2 58 xi C. A general consisten t estimator 60 D. Conclusions 62 References 65 xii I. UL TRA COLD A TOMS TRAPPED IN OPTICAL LA TTICES Recen tly , exp erimen ts in v olving ultracold atoms trapp ed in optical lattices ha v e con- tributed signican tly to our understanding of quan tum systems b oth in and out of equilib- rium [9]. Optical lattices, also referred to as crystals of ligh t, allo w parameters of quan tum systems to b e tuned with great nesse. The atoms trapp ed in suc h lattices can also b e shielded from external noise after co oling to nanok elvin temp eratures, allo wing them to b e studied using simple theoretical mo dels. In fact, eorts are underw a y to study the man y- b o dy ph ysics of cold atoms in the absence of Earth’s gra vit y , b y establishing the Cold A tom Lab oratory in the In ternational Space Station[72]! The initial exp erimen ts on ultracold atoms w ere motiv ated b y the theoretical studies of b o- son lo calization and phase transitions, in v olving b osons in p erio dic and disordered p oten tials [26]. A t the time of [26], F ermi systems on a lattice had b een studied extensiv ely , fo cusing on phenomena suc h as Anderson lo calization and electron-electron in teraction leading to a metal-insulator transition [42]. The seminal theoretical w ork of Fisher et. al. examined the analogous problem for b osons. Their study fo cused on the Bose-Hubbard mo del, whic h represen ts in teracting b osons on a lattice. They found that this mo del exhibits a phase transition from a sup eruid to an insulating phase. Quan tum phase transitions are observ ed when the parameters of a Hamiltonian are v aried. Hence, the exp erimen tal demonstration of this phase transition w as a c hallenging task, as it required a system where the Bose-Hubbard Hamiltonian could b e realized and its parameters could b e tuned with precision. A feasible exp erimen t w as prop osed b y Jaksc h et al. [36]. This prop osal w as motiv ated b y adv ances in optical lattices in one and higher dimensions [55, 73], as w ell as the observ ation of a Bose-Einstein condensate in a gas of Rubidium atoms [3]. This mo del w as rst realized exp erimen tally b y Greiner et. al. [34], where they studied the transition from a sup eruid to an insulating phase in trapp ed 87 Rb atoms. In this section, w e review the Bose-Hubbard mo del and its exp erimen tal realization using ultracold atom trapp ed in optical lattices. The discussion in this section is based on the reviews presen ted in [9, 21, 25, 57]. While discussing details of the exp erimen tal setup, w e fo cus on the tec hniques used in [34]. Although there ha v e b een signican t adv ancemen ts in exp erimen ts since then (for example: mixtures of atomic sp ecies, fermionic condensates, 1 spatially v arying tunneling amplitudes), the conceptual basis remains the same. A. Bac kground and Bose-Hubbard Hamiltonian The Bose-Hubbard Hamiltonian is, H =J X <i;j> a y i a j +h:c + X i i n i + 1 2 U X i n i (n i 1) (1) It represen ts b osons on a lattice, with sites indexed b y i2f0; 1::::Lg. Here a y i and a i are creation and annihilation op erators that increase or decrease the n um b er of a b osons at a particular site. The rst term represen t the hopping of particles from one site to the other. In the sp ecic con text of optical lattices, it is related to the probabilit y that a particle conned in a p oten tial w ell lab eled i, tunnels to a dieren t p oten tial w ell j . U is a measure of in teraction b et w een b osons. It represen ts a repulsion that app ears when there are m ultiple b osons on the same lattice site. In the Hamiltonian of Eqn. (1), terms of the form n i n j with i6= j are absen t, indicating that the in teraction falls sharply as the distance b et w een particles increases. This corresp onds to the case where b oson w a v efunctions are sharply lo calized to individual sites. i represen ts an energy oset on the i-th lattice site. Most mo dern exp erimen ts are capable of of utilizing 3D lattices, formed b y the in tersec- tion of three orthogonal laser standing w a v es, fo cused to w aists at the p osition where the condensate is conned using a radio frequency (RF) trap. The RF trap has a frequency m uc h lo w er than that of the lasers used to create the optical lattice, and hence do esn’t ha v e to b e explicitly considered as a part of the system’s Hamiltonian. Ho w ev er, the laser b eams do not ha v e uniform in tensit y proles orthogonal to their direction of propagation as sho wn in Fig. (5). This prole is generally Gaussian (lo cally quadratic) and has to b e included in the i term in the Hamiltonian. Therefore, there is usually a term quadratic in i that con tributes to i . The study in [26] describ ed t w o distinct phases for a homogeneous system. A sup eruid phase o ccurs when the in teratomic in teractions quan tied b y U are small compared to the site-to-site hopping term, i.e. the ratio U J is small. The fact that U is small allo ws for 2 Figure 1. Fig. 1 (a) from [36], describing the parameters in the Hamiltonian. V 0 represen ts the depth of the p oten tial w ells in whic h atoms are conned, forming a lattice-lik e structure. signican t o v erlap b et w een w a v efunctions of particles lo calized at dieren t sites. In the U J limit, probabilities asso ciated with single particle w a v efunctions are spread ev enly o v er the en tire lattice, and the ground state of a system of N particles distributed o v er L sites can b e describ ed to a go o d appro ximation b y , j SF i = L X i=1 a y i ! N j0i = a y k=0 N j0i wherej0i is the v acuum. It is eviden t that suc h a phase, exhibits quan tum uctuations, i.e. n 2 i =hn 2 i ihn i i 2 6= 0 F or large, dilute systems, the o ccupation distribution at eac h site is P oisson distributed with n 2 i =hn i i. When w e consider the correlation matrix b y R ij =ha y i a j i w e can see that there are long range correlations presen t, indicated b y the presence of o-diagonal elemen ts. W e can also consider the co v ariance matrix in the momen tum basis, wherea y k anda k create and annihilate particles with momen tum k resp ectiv ely . In this basis the single particle densit y matrix has a m uc h simpler form, since all N particles are in a state with momen tum k = 0. ha y k a q i =N k;0 0;q 3 Figure 2. Energy gap in the sp ectrum of the Mott phase. (Example for a system with n = 1 atom lo cated on eac h lattice site.. Figure from [34]. On the other hand, a Mott phase is observ ed for larger v alues of U . This phase is c haracterized b y a commensurate n um b er of particles at all sites. F or a state with N L = n, i.e. n particles p er site, the ground state is giv en b y , j Mott i = L Y i=1 a y i n j0i When compared to the sup eruid state, the single particle densit y matrices for the Mott state are diagonal in the p osition basis and ha v e o diagonal elemen ts presen t in the momen- tum basis. The lo w v ariance in momen tum distribution in the sup eruid phase compared to the wide spread of momen ta in the Mott phase is the k ey to distinguishing these phases in exp erimen ts.The energy sp ectra for these phases diers signican tly . In the JU limit, a gap of magnitude U exists b et w een states with single particles and an y excited states as sho wn in the sc hematic Fig. (2). B. Exp erimen tal setup The theoretical in v estigations of Fisher et. al. in [26] suggested an exp erimen tal study of the transition from a sup eruid to a Mott insulating phase using liquid He 4 adsorb ed on p orous surfaces. Studying this quan tum phase transition requires a setup that allo ws ne tuning of the parameters of the Hamiltonian while sim ultaneously ensuring that the system is w ell shielded from external noise for a lifetime long enough to allo w accurate measure- men t. The rst observ ation of this phase transition of the Bose-Hubbard Hamiltonian used a condensate of Rubidium atoms trapp ed in an optical lattice. 4 Figure 3. Optical lattices depicting connemen t in 2 and 3 dimensions. Figure from [8]. A gas consisting of 87 Rb atoms w as co oled to form a Bose-Einstein condensate. The gas consisting of appro ximately 2 10 5 atoms w as then loaded in to an optical lattice, whic h w as formed b y the in tersection of three orthogonal standing w a v e laser b eams.The condensate w as conned b y a spherically symmetric radio frequency trap within a diameter of 26m. The optical lattice w as then gradually switc hed on, b y increasing the in tensit y of the lasers used to create the ligh t crystal. The gradual ramp w as p erformed in order to ensure that the ground state of the system after turning on the optical lattice retained the prop ert y of the BEC i.e. a macroscopic fraction of the atoms w as in the q = 0 momen tum state. W e no w describ e the realization of the Bose-Hubbard Hamiltonian using an optical lattice. When a t w o lev el atom is placed in an electric eld due to an mono c hromatic laser, a dip ole momen t is induced. This in teraction b et w een the laser and the atom pro duces an eectiv e p oten tial that scales with the in tensit y of the applied eld. When w e consider a standing w a v e pro duced b y a reected laser b eam, w e get an in tensit y prole that v aries along the direction of laser propagation as, V lattice (x) =V 0 cos 2 (kx) (2) wherek = 2= with the w a v elength of the laser. This creates a series of p oten tial w ells, whose depth dep ends on the strength of the laser eld. F or sucien tly high in tensit y of the laser eld, atoms are conned to the minima of this p oten tial with a small probabilit y of tunneling to adjacen t w ells. This connemen t creates a (1 dimensional) lattice lik e structure. T o get a b etter idea of the connemen t that o ccurs, consider the w a v efunction indexed b y q , q (x) of an particle placed in the p oten tial giv en b y Eqn. (2) [33, 57]. In general, there are an innite n um b er of solutions of the Sc hro dinger equation for a giv en momen tum q , n (1) q (x); (2) q (x); (3) q (x):::: (j) q (x)::: o whic h giv e rise to a band structure. Omitting the index 5 (j) for clarit y , w e get ~ 2 2m @ 2 @x 2 +V lattice (x) q (x) =E q q (x) Due to the p erio dic nature of the lattice, w e can use the Blo c h theorem to write q (x) =e ikx q (x) Substituting this form of the w a v efunction in the Sc hro dinger equation giv es us " 1 2m i~ @ @x +~k 2 +V 0 cos 2 (kx) # q (x) =E q q (x) T o solv e for the band structure of atoms in suc h an optical lattice, w e consider the equation in terms of q ~ 2 2m @ 2 @x 2 q (x) + V 0 2 (cos(2kx) 1) =E q q (x) Using the substitution x 0 =kx, w e get @ 2 @x 02 q (x 0 ) + V 0 4E R (2 2 cos(2x 0 )) = E q E R q (x 0 ) This simplies to the Mathieu equation [1], for whic h the standard form is @ 2 y @v 2 + (a 2q cos(2v))y = 0 An imp ortan t energy scale that emerges in this analysis is the c haracteristic energy of the lattice, also called the Recoil energy. E R = ~ 2 k 2 2m When the laser is switc hed o i.e. V 0 = 0, w e ha v e the trivial case where the eigenfunctions are just plane w a v es. As V 0 increases, w e see that spatial prole of probabilit y densities of eigenfunctions sho ws p erio dicit y . It has maxima at the minima of V (x), and these corresp ond to sites on the optical lattice as sho wn in Fig. (4) . W e can p erform a unitary transform on the Blo c h w a v efunctions, to obtain a basis where w a v e functions ha v e probabilit y densities that are sharply lo calized at eac h site. The w a v e- functions that constitute this basis are referred to as W annier orbitals. These are obtained b y taking a discrete F ourier transform as w n (xx i )/ X q e iqx i (n) q (x) 6 Figure 4. Connemen t in an optical lattice and the sligh t oset of p oten tial from one site to another due to the Gaussian in tensit y prole of the laser. Figure from [57]. Figure 5. Connemen t in an optical lattice and the sligh t oset of p oten tial from one site to another due to the Gaussian in tensit y prole of the laser.. Figure from [34]. where the constan t of prop ortionalit y is found b y normalization. This is the W annier orbital in the n-th energy band that is lo calized at a site with p osition x i . Th us far, w e ha v e ha v e restricted our discussion to the 1-dimensional case for simplicit y . In this case, a single standing w a v e laser connes atoms to planes and the eectiv e Bose- Hubbard Hamiltonian is one dimensional. Extending this to 3-dimensional case, the p oten tial that traps atoms in the lattice is V (x) =V 0 cos 2 (kx) +cos 2 (ky) +cos 2 (kz) 7 This results in a simple cubic three dimensional lattice, mo deled b y atoms conned to p oin ts. W e can no w use the result of [37] to understand ho w the Bose-Hubbard Hamiltonian of Eqn. (1) emerges from the optical lattice system studied so far. Their w ork assumes a lo w energy limit, where the energies in v olv ed are to o small to cause excitations from the n = 1 band to higher energy bands. Therefore atoms are assumed to exist in states giv en b y , (~ x) = X i a i w(~ x ~ x i ) The general Hamiltonian for trapp ed Bosons used in [37] is, H = d 3 x y (~ x) ~ 2 2m r 2 +V 0 (~ x) +V T (~ x) (~ x) + 2a s ~ 2 m d 3 x y (~ x) y (~ x) (~ x) (~ x) Here, V 0 is the p erio dic optical trapping p oten tial that giv es rise to Blo c h w a v efunctions. V T is an additional external p oten tial eg: harmonic trap, laser sp ec kle etc. y (~ x) is an op erator that creates a Boson at a p osition ~ x. Therefore the rst in tegral in the Hamiltonian represen ts single particle energies, and the second term in v olv es in teractions b et w een atoms. W e can no w mak e the substitution y (x) = P i a y i w(xx i ) to reco v er Eqn. (1) with the parameters of the Hamiltonian giv en b y , U = 4a s ~ 2 m d 3 xjw(~ x)j 4 J ij = d 3 xw (~ x~ x i ) ~ 2 2m r 2 +V 0 (~ x) w(~ x~ x j ) (3) i = d 3 xV T (~ x)jw(~ x~ x i )j 2 The en vironmen t of an atom trapp ed in this optical lattice and lo calized to a site can also b e appro ximated b y a harmonic p oten tial of frequency r ~ 2 k 2 2m p V 0 =E r W e can see that the strength of the harmonic p oten tial increases with V 0 . F or sucien tly high V 0 and for neutral atoms with short range in teratomic in teraction, w e can simply our description using a tigh t binding mo del. Generally , the tunneling probabilit y from site i to site j drops exp onen tially asjijj increases. In the tigh t binding mo del, w e assume that atoms are conned sucien tly to their sites so that tunneling is only p ossible to adjacen t sites, i.e. J ij =J i;j1 8 Suc h a mo del arises in the case where the in tegral for J ij in Eqn. (3) is v ery small for W annier orbitals corresp onding tojijj> 1. Ev aluating these in tegrals after substituting the W annier orbital functions, w e reco v er the dep endence of the Hamiltonian paramets U andJ on the strength and frequency of the laser eld as [43] J/E r V 0 E r 3=4 e 2 p V 0 =Er U/E r V 0 E r 3=4 The prediction of [27] is that the sup eruid-Mott-insulator phase transition can b e ob- serv ed as ratio U=J is increased. Observing the scaling of these t w o quan tities with V 0 , w e can see that the required phase transition ma y b e ac hiev ed b y increasing the p oten tial depthV 0 . The tunneling probabilit y from one lattice site to another falls exp onen tially with increasing V 0 and is th us more sensitiv e to c hanges in V 0 than U . This can b e illustrated b y noticing the pro cedure adopted in [34] to bring ab out a transition from the sup eruid to the Mott insulator phase b y increasing V 0 . The exp erimen t starts with V 0 = 0E r i.e. with the lattice turned o. It is then increased to 13E r whic h is the largest p oten tial depth for whic h a signature of the sup eruid phase is visible. It is then increased further to 20E r , at whic h p oin t sup eruid phase is no longer detectable. In order to study the state of the system at a giv en instan t of time, matter w a v e in terfer- ence with time-of-igh t absorption imaging is used. In this tec hnique, once the parameters U and J ha v e b een adjusted adiabatically to a desired v alue, the optical lattice p oten tial is suddenly turned o. This allo ws the atomic w a v efunctions, whic h w ere previously conned to sin usoidal w ells to in terfere with eac h other. As w e ha v e seen earlier, the w a v efunctions in the sup eruid phase ha v e a unique q = 0 momen tum, whereas the momen ta in the Mott phase ha v e a wider distribution. Absorption imaging allo ws exp erimen talists to study this distribution of momen ta. The long range phase coherence in the sup eruid phase pro duces an in terference pattern with a single sharp maxima, while the lac k of coherence in the Mott phase is indicated b y the absence of suc h maxima. In order to p erform measuremen ts of site o ccupation and particle momen ta, the lattice p oten tial w as turned o suddenly . This allo w ed in terference b et w een w a v efunctions that w ere 9 previously lo calized at lattice sites. The sup eruid phase has w a v efunctions that exhibit a high degree of coherence, and therefore yields a pattern with sharp maxima in in its momen tum distribution. On the other hand, the Mott insulator phase, is c haracterized b y Boson lo calization and displa ys no suc h sharp maxima in its momen tum distribution. C. Hard core Bosons and spinless F ermions In order to sim ulate the exact dynamics of a system of b osons whose ev olution is go v erned b y the Hamiltonian Eqn. (1), w e w ould require matrices with v ery high dimension. F or a system with at most N b osons o ccup ying eac h of L sites, this matrix has dimension N L , whic h sev erely limits the system sizes the particle n um b ers that w e can analyze. When the repulsion U b et w een b osons is v ery strong there is a large energy p enalt y for site o ccupancy exceeding 1 b oson. A ccordingly , in the limit where w e ha v e U J , w e can eectiv ely ignore the in teraction term in the Bose-Hubbard Hamiltonian prop ortional to ^ n i (^ n i 1). In suc h a scenario, w e restrict our analysis of the system to a 2 L dimensional subspace of the Hilb ert space where n i =f0; 1g. This is referred to as a hardcore b oson mo del. In the hard core b oson limit the Hamiltonian Eqn. (1) simplies to H =J X i a y i a i+1 +h:c + X i i n i (4) with the constrain t n i 1. A further simplication can b e p erformed using the Jordan-Wigner transformation [48, 64] from hard core b osons to spinless fermions. Using this, w e are able to study the system b y fo cusing only on the L-dimensional single particle sector of the Hilb ert space. This allo ws us to access systems with up toO(10 4 ) sites and p erform nite size scaling analyses. The Jordan-Wigner transformation allo ws us to construct the mapping, a y i =f y i i1 Y k=1 e if y k f k a i = i1 Y k=1 e if y k f k f i where n f y i ;f i o are op erators for spinless fermions. The terms in the hard core b oson Hamiltonian are of the form a y i a j =f y i i1 Y k=1 e if y k f k j1 Y k=1 e if y k f k f j 10 In the i =j case, this is trivial and w e get a y i a j =f y i f j F or ij 1, w e get a y i a j =f y i j1 Y i e if y k f k f j and for the sp ecic case i =j 1, w e ha v e a y i a j =f y i e if y i f i f j (5) W e can simplify this further b y noting that e if y i f i = 1 X k=0 (i) k k! (f y i f i ) k =f y i f i 1 X k=1 (i) k k! + 1 =f y i f i 1 X k=0 (i) k k! 1 ! + 1 = 1 +f y i f i (e i 1) = 1 2f y i f i Substituting this ab o v e giv es us, a y i a j =f y i 1 2f y i f i f j =f y i f j Therefore, the hard core b oson system is equiv alen t to a system of spinless fermions with Hamiltonian H =J X i f y i f i+1 +h:c + X i i f y i f i In the follo wing section, w e describ e ho w the state of suc h a system can b e describ ed in the single particle sector using a co v ariance matrix. W e also explore ho w this matrix can b e used to study its dynamics and compute exp ectation v alues of site o ccupations. 11 I I. NON-EQUILIBRIUM D YNAMICS OF FINITE ISOLA TED QUANTUM SYS- TEMS As discussed in the previous section, cold atoms trapp ed in optical lattices ha v e allo w ed exp erimen talists to mak e remark able progress in their abilit y to study the dynamics of closed quan tum systems. These exp erimen ts and asso ciated theoretical studies ha v e b een able to test imp ortan t questions regarding equilibration and thermalization, and their relation to in tegrabilit y , disorder and phase transitions for systems far from the thermo dynamic limit. In this section, w e review common theoretical metho ds for studying out-of-equilibration quan tum dynamics as w ell as some sp ecic imp ortan t results. The discussion in this section is based on the reviews presen ted in [25, 53, 78]. This section is organized as follo ws: in subsection A, w e presen t some imp ortan t results on out-of-equilibrium dynamics and thermalization in quan tum systems. W e describ e some basics of the theoretical framew ork generally used to analyze suc h problems. in subsection B, w e review the formalism generally used for analysis of free systems. This uses co v ariance matrices to simplify n umerical and analytical computation. in subsection C, w e review a few results for the Losc hmidt ec ho, has b een used to study temp oral uctuations and equilibration follo wing a quenc h. A. Non-equilibrium dynamics Consider a quan tum system initialized in the ground state of a Hamiltonian H 0 with densit y matrix giv en b y (0) =j (0)ih (0)j A t time t = 0, a quan tum quenc h is p erformed, where the Hamiltonian is suddenly c hanged to H6= H 0 . Suc h a quenc h can generally b e represen ted as H = H 0 +B where [B;H]6= 0. The system then pro ceeds to ev olv e in time as (t) =e iHt (0)e iHt (6) 12 In this and all future calculations, w e set ~ = 1 for simplicit y . H can b e sp ecied in terms of its sp ectral decomp osition H =E n jnihnj with energies E n and eigenfunctionsjni. Quan tities that can b e measured in exp erimen ts are observ ables, represen ted b y hermitian op erators. W e consider the time ev olution of the exp ectation v alue of a generic observ able A. W e can sp ecify A using the matrix A mn =hmjAjni and the state of the system at an y instan t of time can b e sp ecied using mn (t) =hmj(t)jni This allo ws us to compute the time ev olving exp ectation v alue for A, hA(t)i = tr((t)A) = tr( X p mp (t)A pn ) = X mn mn (t)A nm W e can write Eqn. (6) in terms of p erio dic functions oscillating with frequencies prop or- tional to the gaps of H as mn (t) = mn (0)e i(mn)t and as a result w e get the quasi p erio dic nature of the exp ectation v alue of the observ able hA(t)i = X mn (0)A mn e i(mn)t (7) Based on this, it is eviden t that for systems with a nite Hilb ert space, the temp oral b eha vior of the exp ectation v alue is a trigonometric p olynomial with a nite n um b er of terms. Eqn. (7) do es not con v erge in the limit t!1. The denition of equilibration for suc h closed systems in v olv es an equilibrium ensem ble with an equilibrium densit y matrix eq that satises [56] eq =(t) = lim T!1 1 T T 0 (t)dt F or an y observ able A, suc h a densit y matrix also satises, tr(A eq ) =hA(t)i =A(t) = lim T!1 1 T T 0 hA(t)idt =A 13 0 5 10 15 20 Time −10 −5 0 5 10 Observable 0.00 0.02 0.04 0.06 0.08 0.10 Probability density Figure 6. Probabilit y distribution of an observ able, when time is considered as a uniformly dis- tributed random v ariable. where w e useA(t) to denotehA(t)i for simplicit y . The ab o v e equation denes an equilib- rium v alueA forA. With this denition, w e can return to the original question of whether A(t) equilibrates, and if so, to what exten t. Considering time as a uniformly distributed random v ariable, it has b een sho wn in [56] that one can imp ose b ounds on the temp oral v ariance ofA(t), A 2 = [A eq (t)A(t)] 2 = [tr(A eq ) tr(A(t))] 2 is generally b ounded b y a quan tit y exp onen tially small in the system size. Suc h an approac h, whic h considers time as a uniformly distributed random v ariable can b e used construct a probabilit y distribution forA(t) as sho wn in Fig. (6). This distribution can b e studied to quan tify the exten t to whic h exp ectation v alues of observ ables equilibrate. The result ab o v e corresp onds to a b ound on the second momen t of this distribution. F or example, in [17], it w as found that qualitativ ely dieren t distributions can b e found dep ending on the magnitude of in the quenc h p erformed. B. F ree F ermions and co v ariance matrices As w e ha v e seen, a hard core b oson Hamiltonian that is quadratic in the op erators b y i and b i can b e mapp ed to an equiv alen t system of spinless F ermions via the Jordan-Wigner trans- formation. This fermionic Hamiltonian can b e written as H = X i;j a y i M ij a j =a y Ma 14 Discarding the in teractions b et w een particles in the system allo ws us to analyze the exact dynamics of the system using the co v ariance matrix R. The elemen ts of this matrix are giv en b y R ij (t) =ha y i a j i t = tr(a y i a j (t)) W e no w illustrate ho w the co v ariance matrix formalism is helpful for the problems w e are in terested in addressing, w e use it deriv e analytic expressions for a few imp ortan t quan tities. These the exact mean and v ariance for the exp ectation v alue of a general observ able. Firstly , since The matrix M can b e diagonalized to get the single particle states and energies for H . If the unitary matrix T diagonalizes M suc h that T y MT = , w e ha v e H =a y Ma =a y T y TMT y Ta =b y b where b =Ta andM = X k k jkihkj Here, the eigen v ectorsjki of M are the single particle energy eigenstates, that liv e in the L-dimensional 1-particle sector of the Hilb ert space. W e also ha v e b y k = X n T kn a y n whic h acts on the v acuum suc h that Hb y k j0i = k b y k j0i W e consider observ ables whic h can b e sp ecied in the the form A = X i;j a y i A ij a j =a y Aa where w e mak e no distinction in notation b et w een the matrix A and the op erator. Using the exp ectation v alue A(t) =hA(t)i = tr(A(t)) = tr( X i;j a y i A ij a j (t)) = X i;j A ij tr(a y i a j (t)) = X i;j A ij R ij (t) = X i;j A ij R T ji (t) = tr(AR T (t)) It is eviden t that the co v ariance matrix R T b eha v es similar to the densit y matrix (t) for exp ectation v alue calculations. It oers the adv an tage of b eing just L-dimensional as 15 describ ed earlier. The unitary transformation T whic h diagonalizes M can no w b e applied to R. W e consider the co v ariance matrix in the basis of single particle energy eigenstate op erators, S ij =hb y i b j i =h X m T im a y m X n T jn a n i = X mn T im ha y m a n iT y nj or more simply , S =TRT y and S T =TR T T y W e can no w compute the time ev olving exp ectation v alue A(t), A(t) = tr(AR(t) T ) = tr(AT y S T (t)T ) = tr AT y e it S T (0)e it T = tr AT y e it TT y S T (0)TT y e it T = tr Ae itM R T (0)e itM This sho ws that the co v ariance matrix ev olv es in time as R T (t) =e itM R T (0)e itM for a system with Hamiltonian H = a y Ma. As men tioned earlier, the temp oral v ariance A 2 is a quan tit y w e are in terested in. W e no w deriv e this temp oral v ariance in terms of the co v ariance matrix R. Essen tially , w e v erify the expression A 2 = X k;q F k;q F q;k X k (F k;k ) 2 as men tioned in [19, 77]. T o start with, w e can write the time dep endence A(t) = tr Ae itM R T (0)e itM in the basisjki of the single particle energy eigenstates of M as A(t) = tr e Ae it e R T (0)e it = X k;q hkjAjqie itq hqjR T (0)jkie it k = X k;q hkjAjqihqjR T (0)jkie it(q k ) 16 In terms of this, the innite time a v erage can b e written as A(t) = X k;q hkjAjqihqjR T (0)jki lim T!1 T 0 e it(q k ) dt = X k;q hkjAjqihqjR T (0)jki kq = X k hkjAjkihkjR T (0)jki = X k F kk using the notation F kq =hkjAjqihqjR T (0)jki. The in tegral o v er the time dep enden t term v anishes unless w e ha v e q = k . W e mak e the assumption that this condition is only satised when k = q . In other w ords, w e assume that the Hamiltonian is non-degenerate. In the researc h presen ted in this dissertation, w e v erify this non-degeneracy n umerically . W e can no w compute the temp oral v ariance A 2 =A(t) 2 A(t) 2 = lim T!1 T 0 tr Ae itM R T (0)e itM 2 X k F kk ! 2 = lim T!1 T 0 " X k;q hkjAjqihqjR T (0)jkie it(q k ) # 2 X k F kk ! 2 W e can ev aluate the term in paren theses, " X k;q hkjAjqihqjR T (0)jkie it(q k ) # 2 = X m;n X k;q hkjAjqihqjR T (0)jkihmjAjnihnjR T (0)jmie it(q k +nm) As in the case of the time a v erage, only certain terms surviv e when w e p erform the time a v eraging in tegral. The non-v anishing terms all ha v e q k + n m = 0 This is satised when q k = m n with q6= k , m6= n and q6= m. This w ould corresp ond to a gap degeneracy . W e also c hec k this condition n umerically for the Hamiltonians that w e consider in later sections. q =k and m =n q =m, k =n,q6=k and m6=n 17 Therefore w e ha v e lim T!1 T 0 " X k;q hkjAjqihqjR T (0)jkie it(q k ) # 2 = X k X q hkjAjkihkjR T (0)jkihqjAjqihqjR T (0)jqi + X k6=q hkjAjqihqjR T (0)jkihqjAjkihkjR T (0)jqi Using the denition F kq =hkjAjqihqjR T (0)jki, w e arriv e at a compact expression A(t) 2 = X kq F kk F qq + X k6=q F kq F qk = X k F kk ! 2 + X kq F kq F qk X k=q F kq F qk = X k F kk ! 2 + X kq F kq F qk X k F 2 kk Therefore, w e v erify that that temp oral v ariance can indeed b e computed as A 2 =A(t) 2 A(t) 2 = X k F kk ! 2 + X kq F kq F qk X k F 2 kk X k F kk ! 2 = X kq F kq F qk X k F 2 kk = tr(F 2 ) X k F 2 kk This expression greatly simplies n umerical computations of the temp oral v ariance for an y giv en observ able once w e kno w the elemen ts of the matrix F . C. Losc hmidt ec ho One of the imp ortan t observ ables in analyzing non-equilibrium dynamics is the Losc hmidt ec ho. It is giv en b y L(t) =jh (t)j (0)ij 2 and has found widespread applications in man y con texts suc h as quan tum c haos [31] and fundamen tal asp ects of quan tum statistical mec hanics [69, 75]. This quan tit y measures the delit y b et w een the state of the system at t and its state at t = 0. W e can rewrite it as L(t) =jh (0)je iHt j (0)ij 2 =h (0)je iHt j (0)ih (0)je iHt j (0)i =h (t)j (j (0)ih (0)j)j (t)i 18 in whic h case it can b e in terpreted as the exp ectation v alue of the pro jection op erator j (0)ih (0)j. A third alternate w a y to write the Losc hmidt ec ho is as j(t)j 2 where (t) =h (0)je iHt j (0)i = tr(e iHt (0)) In this subsection, w e deriv e a simple expression for the Losc hmidt ec ho that allo ws us to compute its time dep endence using co v ariance matrix elemen ts as w e did with other v ariables. This is non-trivial, b ecause unlik e observ ables considered th us far, the Losc hmidt ec ho is not the exp ectation v alue of an observ able is quadratic in op erators a y i and a i . W e rst use the result of [52]. This result giv es us a relation b et w een the densit y matrix of the system and its asso ciated co v ariance matrix. The densit y matrix of a system can b e written in the form =Ke H H = X ij X ij a y i a j where K is a normalization constan t to ensure tr() = 1 i.e. K = 1=tr(e a y Xa ). The result of [52] allo ws us to relate X to the co v ariance matrix as X T = ln 1 IR R e X = R T 1 IR T Based on this, w e can start to relate the delit y to the co v ariance matrix as (t) = tr(e iHt (0)) = 1 tr(e a y X 0 a ) tr e iHt e a y X 0 a These expressions in v olving traces can b e greatly simplied using a result from [40] whic h states that: tr e (A) e (B) = det 1 I +e A e B with the notation (A) =a y Aa. Using this result on the expression for delit y , w e get (t) = 1 det(1 I +e X 0 ) det 1 I +e iMt e X 0 = det(1 IR T 0 ) det 1 + e iMT R T 0 1 IR T 0 = det(1 IR T 0 +e iMt R T 0 ) W e reco v er the result of [66]. This allo ws us to compute the time ev olution of the Losc hmidt ec ho using only the Hamiltonian and the initial co v ariance matrix. W e can also 19 use this result to get some further insigh t ab out the b eha vior of the Losc hmidt ec ho for small times. In Eqn. (7) of [17], it is predicted that the Losc hmidt ec ho has a Gaussian b eha vior a small v alues of t , i.e. L(t) =j(t)j 2 e 2 t 2 1 2 t 2 for small t. Therefore, w e ha v e to obtain an expression for the delit y that is of the form (t)e i e 2 t 2 2 Hence, expanding our result for (t) to second order in t, w e ha v e, det((1 IR T 0 +R T 0 e itM ) = det(1 IR T 0 +R T 0 (1itM t 2 M 2 2 )) = det(1 IitR T 0 M t 2 R T 0 M 2 2 ) = det(Y ) W e are in terested in a p olynomial form for ln(). In order to simplify the ab o v e expression and con v ert the determinan t ev aluation to a trace, w e use the exp onen tial det(Y) = e tr[ln(Y)] Therefore, w e ha v e, ln[(t)] = tr[ln(1 IitR T 0 M t 2 R T 0 M 2 2 )] W e can p erform a T a ylor expansion of the logarithm around 1 I to get ln(IitR T 0 M t 2 R T 0 M 2 2 )itR T 0 M t 2 R T 0 M 2 2 1 2 (itR T 0 M + t 2 R T 0 M 2 2 ) 2 +O(t 3 ) W e no w expand the square term and drop terms of order t 3 and t 4 . This lea v es us with, ln(1 IitR T 0 M t 2 R T 0 M 2 2 )itR T 0 M t 2 R T 0 M 2 2 + t 2 (R T 0 M) 2 2 +::: Therefore, the logarithm of the delit y for short times is giv en b y ln[(t)] = tr itR T M t 2 R T M 2 2 + t 2 (R T M) 2 2 +::: This is in agreemen t with the result stated earlier regarding the Gaussian b eha vior of L(t) for small times. Comparing this to the exp ected expression for Losc himdt ec ho, w e get L(t) =j(t)j 2 e 2 t 2 with 2 tr R T M 2 + tr (R T M) 2 for small t. The co v ariance matrix approac h review ed th us far is v ery general and applicable to a wide v ariet y of systems. In sections 3 and 4 of this dissertation w e presen t studies of non- equilibrium dynamics under the inuence inhomogeneous Hamiltonians that utilize this ap- proac h. 20 I I I. EQUILIBRA TION TIMES IN CLEAN AND DISORDERED QUANTUM SYS- TEMS In this section, w e study the equilibration dynamics of closed nite quan tum systems, fo cusing on the time needed for the system to equilibrate. In particular, w e p erform nite size scaling analyses of the equilibration time T eq with the system size L for three classes of systems. The goal here is to ascertain whether the in tro duction of random inhomogeneit y (noise) to the system has an eect on the scaling of T eq with L. The con ten t of this section is based on w ork published in [14] and done b y the author in collab oration with Researc h Asst. Prof. Lorenzo Camp os V en uti and Prof. Stephan Haas. W e rst consider equilibration times in clean systems. Using the Losc hmidt ec ho as an observ able, w e are able to giv e general estimates for the scaling of the T eq as a function of length. Giv en a lo cal Hamiltonian H = P i h i and a sucien tly clustering initial state j 0 i, where the correlation g i;j =hh i h j ihh i ihh j i deca ys sucien tly fast as a function of ij at large separations, w e nd that T eq =O(L 0 ), i.e. the equilibration time is indep enden t of the system size. The scaling c hanges to T eq =O(L ) with a dynamical critical exp onen t in the case where b oth the initial state and the ev olution Hamiltonian are close to a critical p oin t. W e c hec k these general ndings explicitly for the Ising c hain in a transv erse eld. Ho w ev er, w e are cautious to note that these argumen ts ma y need mo dications for other observ ables e.g. suc h as those undergoing sp on taneous symmetry breaking. W e then turn our atten tion to systems with random impurities. Here, in tuition based on classical mo dels suggests that, in general, w e should exp ect a slo w er transien t approac h to equilibrium in when compared to clean systems. Indeed, it is kno wn that exp onen tially large time-scales are presen t in glassy systems [44]. T o c hec k and further understand this conjecture, w e a n umerical approac h to compute the equilibration time for the tigh t-binding mo del with diagonal impurities, sometimes called Anderson mo del [4]. A similar mo del (alb eit with pseudo-random diagonal elemen ts) w as recen tly studied [32], where it w as found that observ ables relax follo wing a p o w er-la w b eha vior. Suc h p o w er-la w equilibration pattern (observ ed also in [38]for another noisy system) is itself a signature of slo w equilibration. Ho w ev er, according to the denition of equilibration time prop osed earlier, T eq also dep ends on the long-time equilibrium v alue. Our ndings conrm a v ery slo w equilibration dynamics 21 c haracterized b y equilibration times div erging exp onen tially with the system size. F urther, w e nd that the exact nature of this exp onen tial gro wth in T eq dep ends on the c hoice of observ able. Disorder alone migh t not b e sucien t to guaran tee the presence of exp onen tially large relaxation timescales. T o illustrate this p oin t, w e analyze a second noisy system. In this case the ev olution op erator is deriv ed from a unitary matrix sampled from the circular unitary ensem ble (CUE). This mo del is similar to the ones previously considered in [10, 24], where, according to a dieren t denition, equilibration times decreasing algebraically with the size w ere predicted [24]. Using our denition w e pro v e analytically that, for the Losc hmidt ec ho, T eq =O(1). This section is organized as follo ws: in subsection A, w e set the stage for further analysis b y recapitulating imp ortan t denitions and outlining our metho dology for studying equilibration. w e b egin our analysis b y studying clean systems in subsection B and the quan tum Ising mo del in particular in subsection C. w e use n umerical metho ds to in v estigate equilibration timescales for a noisy tigh t- binding mo del in sub ection D. w e study y et another noisy system in subsection E, the ev olution dynamics here is based on the CUE (circular unitary ensem ble). w e presen t our conclusions in subsection F. A. Setting and denitions Before considering noisy systems, let us recall some elemen tary y et imp ortan t facts re- garding unitary equilibration. Consider a quan tum system initialized in a giv en state, and then allo w ed to ev olv e undisturb ed under the action of a time indep enden t Hamiltonian. The system is initialized in some state 0 whic h ev olv es unitarily in time as (t) = e itH 0 e itH . W e observ e non-trivial dynamics pro vided the initial state is not an eigenstate of the ev o- lution Hamiltonian. F urther, w e note that b ecause of the unitary nature of the ev olution, 22 (t) do es not con v erge in the strong sense as t!1. This is true irresp ectiv e of the Hilb ert space dimension, i.e. also in the thermo dynamic limit. One can then consider the p ossibilit y of a w eak er con v ergence b y lo oking at matrix elemen ts A (t) = tr [A (t)] where A is an observ able. The exp ectation v alue of suc h an observ able A corresp onds to some exp erimen tally accessible quan tit y . In general, these exp ectation v alues uctuate in time as hA(t)i. The timescale at whic h suc h an exp ectation v alue relaxes to equilibrium iden ties the e quilibr ation time of the particular dynamics. In the thermo dynamic limit (innite systems), equilibration times are t ypically extracted from the exp onen tial deca y of observ ables or correlation functions to w ards their equilibrium v alues. In this limit the sp ectrum b ecomes con tin uous and one c an ha v e limitA (1) = lim t!1 A (t) for some observ ables A (or appropriately rescaled observ ables) essen tially as a consequence of Riemann-Leb esgue lemma (see e.g. [17] and also [81] for a recen t discussion). F or nite systems, or more generally for systems with a discrete energy sp ectrum, the dynamics is quasi-p erio dic and suc h exp onen tial deca y cannot o ccur. A (t) is a trigonometric p olynomial, and hence, do es not admit an innite time limit. This motiv ates the need for rigorous alternativ e denitions of equilibration time and related quan tities. In these cases, a meaningful denition of equilibration time is: the rst time instan t at whic h the exp ectation v alue of an observ able equals its equilibrium v alue. The equilibrium v alue is giv en b y the time a v erageA := lim T!1 T 0 A (t)dt . In other w ords, giv en an observ able A and its equilibrium v alueA, T eq is the smallest t for whic hA(t) =A [11]. Since w e deal exclusiv ely with systems where A (t) is a trigonometric p olynomial, the time a v erageA := lim T!1 T 0 A (t)dt exists and is nite. Suc h a time a v erage coincides with the innite time limit when the latter exists, i.e. A =A (1). So the time a v erage can b e seen as a mathematical tric k, reminiscen t of Cesaro summation, to obtain the innite time limit when the function oscillates. Alternativ ely the time a v erage can mimic the actual measuremen t pro cess. In this case T is the observ ation time whic h one ma y argue to b e v ery large compared to the time scales of the unitary dynamics (see e.g. [35]). Clearly one ma y w an t to in v estigate the eect of a nite T , here for simplicit y w e will alw a ys tak e T!1. T o recapitulate, the standard denition of e quilibr ation time , extracted from the exp onen- tial deca y of some time dep enden t observ able or correlation function, do es not mak e sense for 23 0 100 200 300 400 500 19 20 21 22 23 24 t n l (t) n l (t) vs t Average of n l (t) = 19. Figure 7. An example where T eq 100. This sho ws the ev olution of an observ able n ` (t) = E [N ` (t)=L] with time. This observ able measures the n um b er of particles in the lo w est in sites 1 48 for a system of size L = 96 with 24 particles. The system is initialized at t = 0 with all 24 particles in the rst 24 sites. It is allo w ed to ev olv e in time under the inuence of a random onsite p oten tial. W e presen t futher details of suc h systems and their time ev olution as part of the discusson on noisy systems in subsection C. nite systems. As stated ab o v e the dynamics is almost p erio dic and no exp onen tial deca y is p ossible. Instead, in nite systems, exp ectation v alues A (t) start from a v alueA (0) whic h retains memory of the initial state 0 , and then, after an e quilibr ation time , approac h an a v erage v alueA and start uctuating around it with uctuations A = q A 2 A 2 due to the nite dimensionalit y of the system. Clearly the precise concept of equilibration time is to some exten t arbitrary , and dieren t denitions are p ossible. Ours is the follo wing: T eq is the rst time for whic h A (t) equals the a v erageA, i.e. is the rst solution of A (T eq ) =A. This denition is b oth simple to implemen t analytically and n umerically and ph ysically in tuitiv e. W e presen t an illustrativ e example in g. 7. T o commen t further on the v alidit y of the ab o v e denition of T eq Clearly the precise n umerical v alue of T eq is irrelev an t, whereas the imp ortan t infor- mation is con tained in the scaling dep endence of T eq on the system size, T eq (L). In principleA (t) could in tersectA at a rst time T 1 , deviate considerably from the a v erage and in tersect A again at T 2 , and p ossibly ha v e m ultiple in tersections up to T n b efore uctuations of order A start to set in. In this situation it seems that the 24 equilibration pro cess cannot b e captured b y a single T eq but rather consists of man y time scales. In all the situations encoun tered in our analysis, ho w ev er, w e found that equilibration could alw a ys b e captured b y a single time-scale i.e. for the classes of systems and observ ables w e consider, w e did not observ e considerable deviations of A(t) fromA for t>T eq . The ab o v e denition dep ends on the sp ecic observ able A and dieren t observ ables ma y ha v e, in principle, dieren t equilibration times. Ho w ev er, w e exp ect that for large classes of observ ables the scaling dep endence with L will b e of a similar form eg: p o w er la w, exp onen tial etc. Based on the exp erience with innite systems, where observ ables deca y to w ards their equilibrium v alue, one could dene T eq as that time for whic hA (t) is o by a smal l amoun t from its a v erage v alue A. This is, after all, the denition w e use in the thermo dynamic limit when the approac h to equilibrium is an exp onen tial deca y . A denition of this sort has b een used for instance in [24]. Suc h a denition, ho w ev er, in tro duces an additional parameter viz. our denition of a smal l amount. B. Clean systems T o p erform the scaling analysis outlined ab o v e, w e consider the Losc hmidt ec ho (LE). L (t) = h 0 je itH j 0 i 2 ; (8) wherej 0 i is the state in whic h w e initialize the system and H is the ev olution Hamiltonian. Equation (8) can b e in terpreted as the time ev olv ed exp ectation v alue of the observ able j 0 ih 0 j, and is also kno wn as survival pr ob ability . The Loshmidt ec ho can b e studied using a cum ulan t expansion. This appro ximates L (t) correctly for sucien tly large times of the order of T eq . This conclusion is based on n umerical exp erimen ts on the Ising mo del in a transv erse eld [17]. L(t) can b e written as a pro duct of L d terms (see b elo w), it should therefore b e clear that T a ylor expansion of lnL w orks b etter than the T a ylor expansion of L itself. The cum ulan t 25 expansion of Eq. (8) reads L (t) = exp " 2 1 X n=1 (t 2 ) n (2n)! hH 2n i c # ; (9) wherehi c stands for connected a v erage with resp ect to j 0 i. T runcating Eq. (9) up to the rst order w e obtain L (t)' exp [t 2 (H 2 )] (H 2 =hH 2 ihHi 2 ). Equating the short time expansion to the a v erage v alue for the Losc hmidt ec ho L w e get the follo wing expression for the equilibration time: T eq = s lnL H 2 : (10) W e mak e a few observ ations ab out the results deriv ed th us far: First of all, the equilibration time in Eq. (10) is in v ersely prop ortional to the square ro ot of an energy uctuation. This is not simply the in v erse of an energy gap as one migh t guess nav ely . Secondly , w e w ould lik e to compare Eq. (10), with another estimate of equilibration time whic h app eared recen tly in a single b o dy setting [79]. The estimate of [79] reads T eq 1=hE min i ave , where E min is the minim um energy gap a v eraged o v er an energy shell around the initial energy hHi [49]. So, apart from the order of a v erages, the equilibration time in [79] is in v ersely prop ortional to a standard deviation of an energy uctuation as m uc h as in Eq. (10). The denitions dier in the n umerator whic h tak es in to accoun t the man y-b o dy nature of the problem. Thirdly , the estimate Eq. (10) is v alid only for the equilibration time of the Losc hmidt ec ho, and in principle, dieren t observ ables migh t equilibrate with dieren t time scales. W e will no w pro vide argumen ts to estimate Eq. (10) whic h rst app eared in [17]. F or a lo cal Hamiltonian H = P i h i and a sucien tly clustering initial state j 0 i, (i.e. g i;j = hh i h j ihh i ihh j i deca ys sucien tly fast as a function of ij at large separations), all the cum ulan ts in Eq. (9) are extensiv e in the system size. (By extensiv e, w e mean that in a d-dimensional system of linear size L,hH 2n i c /L d ). This implies that at leading order in L, L (t)'e f(t)L d [50]. Therefore, b y Jensen’s inequalit y , e fL d L exp[L d max t f (t)], sho wing thatL is exp onen tially small in the system v olume (note that w e m ust ha v e f (t) 0). F or 26 this reason it is sometimes useful to consider the logarithm of the LE F (t) = lnL (t). The equilibration time for F w ould then b e giv en b y T F eq = q lnL=H 2 . No w, since lnL' f (t)L d , w e see that the equilibration times for L andF = lnL are exp ected to giv e the same scaling with resp ect to L. F rom equation (8) w e get then T eq = O (1): the equilibration time is indep enden t of the system size. This argumen t fails when one considers small quenc hes close to a quan tum critical p oin t as in this case the clustering prop erties of the initial state tends to break do wn. In this case j 0 i is the ground state of H 0 =H ( 0 ) where the external parameter 0 is close to a quan tum critical p oin t. One then suddenly c hanges the parameters b y a small amoun t 0 ! 0 + and the system is let ev olv e undisturb ed with the new Hamiltonian H =H ( 0 +). The small quenc h regime is sp ecied roughly b y minfL 1= ;L 2=d g with is the correlation length exp onen t. In this regime, p erturbation theory is applicable (see b elo w) and the a v erage LE reduces toLjh 0 j0ij 4 , wherej0i is the ground state of H [66]. The scaling prop erties of the delit y jh 0 j0ij close to a quan tum critical p oin t ha v e b een studied in [16] where it w as sho wn that 1jh 0 j0ij/ 2 L 2(d+) , where is the dynamical critical exp onen t, and the scaling dimension of the op erator driving the transition. Using similar scaling argumen ts one can sho w that the v ariance scales ashH 2 i c L 2(d) [17]. These results are v alid in the p erturbativ e regime wherejh 0 j0ij is not to o far from 1, i.e. minfL 1= ;L 2=d g where = (d + ) 1 [16]. F rom equation (8) w e no w get T eq = O L , i.e. the equilibration time div erges for large systems. Suc h a div ergence is reminiscen t of the critical slo wing do wn observ ed in quan tum Mon te Carlo algorithms. C. Quan tum Ising mo del As a concrete example w e no w consider equilibration times and in particular the prediction Eq. (10) for the quan tum Ising mo del undergoing a sudden eld quenc h. W e used the Hamiltonian H(h) = L X j=1 x j x j+1 +h z j ; (11) with p erio dic b oundary conditions. W e initialize the system in the ground state of Eq. (11) with parameter h = h 1 . A t time t = 0, w e c hange the parameter suddenly to h = h 2 , and allo w the state to ev olv e unitarily with the Hamiltonian H (h 2 ). 27 The mo del in Eq. (11) has critical p oin ts in the Ising univ ersalit y class at h c =1 with d = = = 1, separating an ordered phasejhj< 1 from a disordered paramagnetic region jhj > 1. F orjhj < 1 the order parameter h x i i b ecomes non-zero, th us breaking the Z 2 symmetry ( x i ! x i ) of the Hamiltonian. A ccording to Eq. (10) and the discussion of the previous section w e exp ect the follo wing b eha vior for the equilibration time as a function of the quenc h parameters h 2 ; h =h 2 h 1 and system size L: T eq / 8 > < > : L for h 2 =h c ; and hL 1 const: otherwise (12) As a rst test w e c hec k whether Eq. (12) is satised for the Losc hmidt ec ho itself. The LE for a sudden quenc h has b een computed in [54] (sup erscripts refer to to dieren t v alues of the coupling constan ts h (i) ) L (t) = Y k>0 h 1 sin 2 (# k ) sin 2 (2) k t=2 i Here: k = 2 p 1 +h 2 + 2h cos(k) is the single particle disp ersion # k =# (2) k # (1) k with # (i) k b eing the Bogoliub o v angles at parameters h i the quasimomen ta are quan tized according to k = (2n + 1)=L; n = 0; 1;:::L=2 1 (see [17] for further details). The ab o v e analytical expression for the time ev olution of the Losc hmidt ec ho allo ws to compute its equilibration time exactly . This is done b y solving n umerically for the rst solution ofL (t) =L for giv en quenc h parameters. In Fig. 8 w e plot the equilibration time T L eq of the Losc hmidt ec ho v ersus size for dieren t quenc h parameters. Indeed Eq. (12) is satised to a high accuracy . Moreo v er the transition b et w een the t w o b eha viors of Eq. (12) app ears to b e v ery sharp. F rom the n umerical analysis the follo wing b eha vior for T L eq emerges v alid outside the crosso v er region hcL 1 T L eq = 8 > < > : L 4 for h 2 =h c ; and hcL 1 c 4h otherwise ; 28 æ æ æ æ æ æ æ æ æ æ æ æ à à à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì ì ì 0 100 200 300 400 500 600 0 20 40 60 80 100 120 140 L T eq L æ h 1 =0.95 h 2 =1 à h 1 =0.97 h 2 =1 ì h 1 =0.99 h 2 =1 L� 4 Figure 8. Equilibration times T L eq for the Losc hmidt ec ho as a function of the system size L for dieren t quenc h parameters h 1;2 . T L eq is computed b y solving n umerically for the rst solution of L (t) =L. æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æææææææææææææææ 140 160 180 200 220 25 30 35 40 45 50 L T eq L æ T eq L L� 4 c 4 Δh Figure 9. Equilibration times T L eq for the Losc hmidt ec ho close to the crosso v er region h'cL 1 . where the constan t c in principle dep ends on h 1 ; h 2 , but for small quenc hes close to the critical p oin t tends to c' 6:1. The t ypical b eha vior in the crosso v er region h' cL 1 is depicted in Fig. 9. It is natural to ask whether the predictions of Eq. (12) are satised for observ ables other than the Losc hmidt ec ho. T o this end, w e consider the transv erse magnetization m z (t) = h z i (t)i whic h can b e computed as [6, 17] m z (t) = 1 L X k cos # (2) k cos (# k ) + sin # (2) k sin (# k ) cos t (2) k (13) F rom eq. (13) w e extract the equilibration time from the solution of m z (t) = m z = 29 æ æ æ æ æ æ æ æ æ æ æ æ à à à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì ì ì 0 100 200 300 400 500 600 0 20 40 60 80 100 120 140 L T eq m z æ h 1 =0.90 h 2 =1 à h 1 =0.95 h 2 =1 ì h 1 =0.99 h 2 =1 L� 4 Figure 10. Equilibration times T m z eq for the transv erse magnetization m z (t) =h z 1 (t)i as a function of the size L for dieren t quenc h parameters h 1;2 . T m z eq is computed exactly solving for the rst solution of m z (t) =m z . æ æ æ æ æ æ æ æ æ ææææ æ ææææææææ ææææææææ æææææææææææææææææææææææææææææææ 80 100 120 140 160 180 200 20 22 24 26 28 30 32 L T eq m z æ T eq m z L� 4 c’ 4 Δh Figure 11. Equilibration times T m z eq for the transv erse magnetization close to the crosso v er region h'c 0 L 1 . F or this v alues of parameters c 0 ' 13. L 1 P k cos # (2) k cos (# k ). The n umerical results, sho wn in Fig. 10, conrm that Eq. (10) is applicable to the transfer magnetization to o. The n umerical results can b e summarized as T m z eq = 8 > < > : L 4 for h 2 =h c ; and hc 0 L 1 c 0 4h otherwise with constan t giv en no w b y c 0 ' 15 for small quenc hes close to criticalit y . Finally , let us commen t on the approac h to equilibrium of the order parameter m x (t) = h x i (t)i. W e recall that, since a non-zero m x breaks the symmetry of the Hamilto- 30 nian, m x m ust b e computed as the clustering part of an equal time correlation function: h x i (t) x j (t)i jijj!1 ! [m x (t)] 2 . As usual in symmetry brok en phases this requires the thermo dynamic limit to b e tak en rst, but a nite size appro ximation for systems with p erio dic b oundary conditions can b e obtained b y considering the correlation at half c hain [m x L (t)] 2 :=h x i (t) x i+L=2 (t)i. W e do not exp ect form ula (10) to repro duce the equilibration time correctly for the order parameter, as it do es not distinguish whether w e are in the ordered phase or not. The b eha vior of m x (t) has recen tly b een obtained analytically in the thermo dynamic limit in the quenc h setting [12]. The results of [12] for m x (t) can b e summarized as follo ws. m x (t) = 0 iden tically for quenc hes starting in the disordered phase (jh 1 j > 1) as the symmetry remains un brok en. for quenc hes starting in the ordered phase (jh 1 j < 1) one has a dieren t b eha vior dep ending whether one ends up in the ordered or disordered phase. More sp ecically m x (t) = 8 > < > : Ae t= jh 2 j< 1 A 0 e t= p 1 + cos ( k 0 t +) +::: jh 2 j> 1 The constan ts A; A 0 ; ; k 0 ; all dep ends on h 1 ; h 2 and are giv en explicitly in [12]. F or quenc hes starting end ending in the ordered region, the equation m x (t) = m x = m x (1) = 0 has no real solution. Corresp ondingly at nite size, T m x eq m ust b e an increasing function of L and the simplest guess is T m x eq / L = L. Alternativ ely , for quenc hes ending in the disordered region, w e see that m x (t) = 0 has a nite solution ev en in the thermo dynamic limit. This leads us to exp ect that in this case T m x eq = O (L 0 ). (assuming that the scaling of T eq with L is in general, monotonically increasing, and constan t in the w orst case) D. The Anderson mo del W e no w turn to random systems, where, as men tioned earlier, v ery long relaxation times are kno wn to exist. The mo del w e consider here is the tigh t-binding mo del with random 31 diagonal disorder, sometimes referred to as the Anderson mo del [4]: H = L X j=1 h t c y j c j+1 +c y j+1 c j i c y i c i i : (14) Here, the diagonal elemen ts i are i.i.d (indep enden t and iden tically distributed) random v ariables. In all of our sim ulations w e will sample these diagonal elemen ts from a uniform distribution in the in terv al [W;W ]. Through the Jordan-Wigner mapping, Hamiltonian Eq. (14) equiv alen tly describ es an XX c hain ofL spins in a random magnetic eld. The Hamiltonian Eq. (14) can b e written in compact notation as H =c y M c , wherec = (c 1 ;:::;c L ) T ,M is the one particle Hamiltonian and the subscript refers to the random v ariables i . In the innite v olume limit, the sp ectrum ofM is giv en, with probabilit y one, b y (r 2 )+supp ( i ) = [2t; 2t]+supp ( i ) wherer 2 is the discrete Laplacian in 1D, (r 2 ) i;j = i;j+1 + i;j1 , and supp is the supp ort. In case of a uniform distribution w e ha v e (M ) = [2tW; 2t +W ]. Moreo v er, for an y nite amoun t of randomness, the sp ectrum is almost surely pure p oin t, i.e. consists only of eigen v alues, and the eigenfunctions are exp onen tially lo calized (see e.g. [28, 41]). Suc h a situation is referred to as a lo c alize d phase. Since in the absence of disorder the mo del Eq. (14) is a band conductor, in one spatial dimension there is a metal-insulator transition for an y ho w ev er small amoun t of disorder W > 0. T o study the equilibration prop erties of the Anderson mo del, w e pro ceed as follo ws. W e initialize the system in a state 0 , ev olv e it unitarily with one instance of Hamiltonian Eq. (14) in to (t), and consider the exp ectation v alue of some observ able A:A (t) = tr (A (t)). F or v ery large systems one exp ects that concen tration results will apply and that the single instanceA (t) will b e, with v ery large probabilit y , close to its ensem ble a v erage A (t) := E [A (t)] where w e denoted with E [] the a v erage o v er the random p oten tials i . When this happ ens, or more precisely when the relativ e v ariance 2 A=A 2 ! 0 as the system size increases, one sa ys thatA is self-a v eraging. W e will not b e concerned with this issue here, w e just notice that A (t) generally giv es the result of a h yp othetical measuremen t of A within the condence in terv al. The equilibration time is then obtained b y solving for the rst solution of A(T eq ) = A. T o b e concrete w e will consider a Gaussian initial state 0 with N particles uniquely sp ecied b y the co v ariance matrix R i;j = tr(c y i c j 0 ). Since Eq. (14) conserv es particle n um b er, 32 the ev olution is constrained to the sector with N particles. As observ ables, w e c ho ose a general quadratic op erator giv en b y A = P i;j a i;j c y i c j . In this case the exp ectation v alue A (t) can b e completely c haracterized in terms of one-particle matrices [19]: A (t) = tr [A (t)] = tr ae itM R T e itM : (15) T o b e sp ecic w e will study t w o particular quadratic observ ables: N ` := P ` j=1 c y i c i whic h coun ts the n um b er of particles presen t in the rst ` sites (sa y from left). In this case the thermo dynamic limit is giv en b y xing the particle densit y = N=L together with the observ able densit y =`=L and let L!1. As describ ed earlier in the con text of clean systems, a useful quan tit y to consider is the Losc hmidt ec ho. In this free F ermionic setting it can b e written as [40, 52] L (t) = det 1R T +R T e itM 2 : Because of the random nature of the problem w e do not exp ect the initial lo cations of the N particles to matter particularly . Hence w e will consider an initial state where all the N particles are pushed to the left, i.e. R i;i = 1 for i = 1; 2;:::;N and all other en tries zero. W e ha v e c hec k ed that initializing the particles at other sites do es not c hange our results. In an y case for pure initial states, R T is a (orthogonal) pro jector R T 2 = R T , meaning its eigen v alues are either 1 or 0. Therefore, in some basis, R T alw a ys has the aforemen tioned form, i.e. there exists a unitary matrix X suc h that X y R T X = R T N = diag (1; 1;:::; 1; 0;:::; 0) with N en tries 1 and (LN) zeros. The Losc hmidt ec ho then b ecomesL (t) = det 1R T N +R T N X y e itM X 2 . Our c hoice of initial state corresp onds to ha ving X = 1 I. Thanks to the simple form of R T N , one can ev aluate the determinan t using Laplace’s form ula and reduce it to a determinan t of an NN matrix, i.e. L (t) = det N X y e itM X 2 (16) where the op erator N [Y ] truncates the last LN ro ws and columns of Y . Since clearly k N [U]kkk for an y unitary matrix U and L-dimensional complex v ector , the eigen- v alues of N X y e itM X ,z i (t) ha v e mo dulus smaller than one. The LE can than b e written as L (t) =e F(t) =e Lf(t) 33 0 100 200 300 400 0.16 0.18 0.2 0.22 0.24 0.26 t n l (t) L=24 L=48 L=72 L=96 L=120 Figure 12. Relaxation of n ` (t) = E [N ` (t)=L] with time for lengths L = 16; 40; 64; 112, at lling =N=L = 0:25 and =`=L = 0:5. The random distribution has v ariance = 0:6 (W = 0:6 p 3 1:03). The ensem ble a v erage is computed using 1000 realizations at eac h time. ha ving dened f (t) = (1=N) P N i=1 lnjz i (t)j 2 . Pro videdf (t) has a limit asL!1, this sho ws thatL (t) is exp onen tially small in the system size. A ccordingly , since F (t) := lnL (t) is extensiv e, w e exp ect it to b e self-a v eraging and con v enien t to study . W e also consider the a v eraged Losc himdt ec ho L (t) = E [L (t)] and the ensem ble a v erage of the logarithm of the Losc himidt ec ho (LLE) F (t) = E [F (t)]. Note that b y Jensen’s inequalit y L (t)e F(t) . In Fig. 12 w e sho w the results of our n umerical sim ulations for the ensem ble a v eraged observ able n ` (t) = E [N ` (t)=L]. When computing the equilibration time for observ able A b y lo oking for the rst solution of A(T eq ) = A, the computationally most demanding part is the calculation of A, esp ecially for large system sizes. F or quadratic observ ables, this computation can b e simplied considerably . W e rst note that the time and ensem ble a v erages comm ute. This is essen tially a consequence of F ubini’s theorem and the fact that all our quan tities are b ounded. The time a v erage of A (t) for a particular realization can b e computed diagonalizing M . With the notation M =Q y Q, = diagf 1 ;:::; L g, and Q unitary , one hasA (t) = P n;m Q y AQ n;m Q y R T Q m;n e it(mn) . No w, simply note that, with probabilit y one, the sp ectrum n is non-degenerate. This implies that the time a v erage is giv en exactly b y A = X n Q y AQ n;n Q y R T Q n;n (17) The v alue A is then computed taking the ensem ble a v erage of A, A = E A , using 34 0 50 100 150 2 3 4 5 6 L log(T eq ) Best fit:ψ=0.6266 Fit with ψ=0.50 Simulation data Figure 13. Scaling of equilibration time T eq (L) vs system size L, extracted from n ` (t). T eq is obtained from this rst solution of E[n ` (T eq )] = E[n ` (t)]. P arameters are as in Figure 12 . The ensem ble a v erages are obtained summing o v er 2000 realizations. The equilibration time is found to ob ey log(T eq ) = 0:021L 0:6266 + 2:53 with residual norm 0:17. Eq. (17). In gure 13 w e plot the equilibration time obtained for n ` (t) as a function of system size L. Our n umerical results sho w that the equilibration time scales exp onen tially with the system size. F or the case of the LE and the LLE w e ha v e not b een able to compute the time a v erage analytically . It is kno wn that in case of non-degenerate man y b o dy energies, whic h is safe to assume in presence of randomness, one has L = P n jh 0 jfngij 4 [17] (wherejfngi = jn 1 ;:::;n L i’s are the man y-b o dy eigenfunctions satisfying P j n j = N ). F or our c hoice of initial state, the amplitudes are giv en b y h 0 jfngi = detV [1;2;:::;N] Q y V y fng where V [1;:::;N] is the NL matrix with ones on the diagonal and zero otherwise. V fng is formed in the same w a y but the ones on the diagonal are in corresp ondence of the ro w i for whic h n i = 1. The normalization of the w eigh ts is pro vided b y Cauc h y-Binet’s form ula X fng P i n i =N detV [1;2;:::;N] Q y V y fng detV fng TQV = detV [1;2;:::;N] Q y QV y [1;2;:::;N] = 1 The time a v erage Losc hmidt ec ho is then giv en b y L = X fng P i n i =N detV [1;2;:::;N] Q y V y fng 4 35 0 100 200 300 400 −60 −40 −20 0 t F(t) L=16 L=40 L=64 L=88 L=112 Figure 14. Relaxation of F(t) = E[log(L(t))]. P arameters are as in Figure 12 and an ensem ble a v erage w as p erformed o v er 1000 realizations at eac h instan t in time. The ab o v e sum, ho w ev er, con tains an exp onen tial n um b er of terms and is not practical for n umerical computations. T o ev aluate L and F, w e compute the -random a v erage L (t)' N 1 samples P N samples i=1 L i (t) (and similarly for F) using Eq. (16). T o p erform the a v eraging, w e use ensem bles with as man y as N samples = 2000 and N times random times uniformly distributed b et w een [T 0 ;T ]. An appro ximation to the innite time a v erage is then obtained via L' N 1 samples N 1 times (TT 0 ) 1 P N times j=1 P N samples i=1 L i (t j ). W e use a p ositiv e T 0 to get eliminate the p ossibilit y of considering times from the the initial transien t. This allo ws us to obtain more precise estimates with the same computational cost. T ypical, w e used are T' 1000,T 0 ' 750 for the system sizes sho wn in Fig.(14). W e had to use larger times of the order of T' 10000 with T 0 ' 1000 for systems of size exceeding L 128. In gure 14 w e plot the ensem ble a v eraged LLE time series F (t) at dieren t sizes L while in gure 15 w e sho w the equilibration times of the logarithmic Losc hmidt ec ho as a function of L . In this case as w ell, our n umerical results indicate equilibration times div erging exp onen tially in the system size. Based on the n umerical evidence, w e conjecture that in general, for an observ able O , the equilibration time in the Anderson mo del migh t satisfy lnT O eq =c O L +d O Here the co ecien ts c O ; d O ; are constan ts whic h dep end on the c hoice of observ able O . E. A CUE (circular unitary ensem ble) mo del Th us far, w e ha v e considered a clean system that displa y ed constan t or p o w er la w scaling of the equilibration time, and a random system that displa y ed exp onen tially large equilibration 36 0 50 100 150 200 2 3 4 5 6 L log(T eq ) Best fit:ψ=0.2771 Fit with ψ=0.5 Simulation data Figure 15. Scaling of the relaxation time T eq suc h that F(T eq ) = F , T eq = 1:39L 0:2771 0:847 with residual norm 0:05. times. This p oses the question of whether in tro ducing randomness to a system’s Hamiltonian is a guaran tee of suc h exp onen tial scaling of T eq withL. In this section w e consider a random matrix mo del for whic h w e are able to pro v e analytically that T eq 1. T o implemen t this mo del, w e x the ev olution op erator U at time t = 1 to b e tak en from the circular unitary ensem ble (CUE). This means that U is an LL unitary matrix sampled from the uniform Haar measure o v er the group U (L). A t other times, the ev olu- tion is dened via U t . The arbitrariness in the denition of U t for t2 R is xed in the follo wing w a y . An y unitary matrix U from the CUE can b e written as U =V y e i V whereV is again Haar distributed, e i := diag(e i 1 ;e i 2 ;:::;e i L ) and the phases i are distributed according to P () = C Q i<j e i i e i j 2 where C 1 = (2) L L! is the normalization con- stan t and i 2 [0; 2) (see e.g. [45]). Our mo del is describ ed b y taking a Hamiltonian H = := diag( 1 ; 2 ;:::; L ) and considering the a v erage o v er all isosp ectral Hamiltonians H 0 =V y HV withV Haar distributed. The dynamical ev olution is giv en b y e itH 0 . Giv en these considerations this mo del is equiv alen t to the ensem ble considered in [10, 24] after a v eraging all the energies E i with the CUE distribution P (). Let us no w turn to the computation of F (t) = E U ln det N e itH 0 . If t = n is an in teger, the ev olution op erator is giv en b y e itH 0 =U n where U n is unitary and Haar distributed, hence, at in teger times w e obtain F (n) F 0 = E U lnjdet N (U)j 2 ; indep enden t of n. W e observ ed, as it is natural to exp ect, that the function F (t) has a 37 limit as t!1 (see Fig. 16). In this case the limit m ust coincide with the time a v erage and with F(n) for n in teger, i.e. F (1) = F(n) = F. This sho ws that the relaxation time in this random systems is b ounded b y one. In fact in principle one could ha v e F (t) = F also for a time t smaller than one, ho w ev er our n umerics indicates that the rst o ccurrence of F (t) = F is indeed at T eq = 1 (see Fig. 16) indep enden t of the system size. W e w ould lik e to men tion at this p oin t the results of Ref. [11] where a dieren t sparse random ensem ble has b een constructed and an equilibration time gro wing as the system size has b een rep orted. These ndings sho w that random systems can giv e rise, in general, b oth to slo w and fast equilibration pro cesses and the correct equilibration time-scale can only b e obtained through an accurate in v estigation (although one exp ects faster equilibration to b e asso ciated to less sparse Hamiltonians). It turns out that the limiting v alue F (1) can b e obtained exactly . The distribution of eigen v alues of truncated matrices N (U) when U is Haar distributed has b een com- puted in [82]. The eigen v alues z i of N (U) are complex n um b ers in the unit disk jz i j 1. Calling r i = jz i j, and dening the probabilit y distribution of the mo duli P L;N (r) N 1 P N i=1 E U [ (rjz i j)], the CUE a v erage at in tegers time is then giv en b y F 0 =N 1 0 P L;N (r) ln r 2 dr: (18) Zyczk o wski and Sommers w ere able to compute the distribution of the mo duli P L;N (r) and obtained, with x =r 2 P (r) = 2r N (1x) LN1 (LN 1)! d dx LN 1x L 1x : The probabilit y densit y P (r) in the thermo dynamic limit at xed particle densit y N=L = ; L!1, w as also computed in [82] and is giv en b y P (r) = 1 1 2r (1r 2 ) 2 (19) for r< p and zero otherwise. Plugging Eq. (19) in to Eq. (18) one obtains, in the thermo- dynamic limit, the follo wing particularly simple result F (1) =L [ ln + (1) ln (1)] : (20) 38 0 1 2 3 4 5 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 t F(t) F(t) (μ ± σ) Asymptotic Figure 16. Plot of F (t) for L = 120, at = 1=2 (half lling). The v ariance F (t) is sho wn as error bar. Numerical results are obtained a v eraging o v er N samples = 7500 unitaries distributed according to the Haar measure and obtained with the algorithm outlined in [46]. Quite surprisingly Eq. (20) is the negativ e V on Neumann en trop y of the Gaussian state with co v ariance matrix R (n) = E [R (n)] obtained taking the ensem ble a v erage of the co v ari- ance matrix at in teger times t =n. The reasoning is the follo wing. First w e remind that the v on Neumann en trop y of a F ermionic Gaussian state W with co v ariance matrix W is giv en b y S vN ( W ) = tr W ln W = trW lnW + tr (1 IW ) ln (1 IW ) No w note that at in teger times the ev olution op erator is U n =U 0 with U 0 again unitary . A t suc h times the a v erage of the co v ariance is then R (n) = E U URU y and is prop ortional to the iden tit y b y Sc h ur’s lemma. The constan t is xed noting that trR (n) = trR =N , whic h implies R (n) = 1 I. The claim then follo ws trivially taking traces of diagonal op erators of the form ( 1 I) ln( 1 I). W e do not kno w the a v erage co v ariance at non-in teger times, ho w ev er if a limit t!1 exist it m ust coincide with R (n). In that case w e w ould ha v e F (1) =S R(1) [=S R(n) ]. Finally , w e also v eried that F (t) is indeed self a v eraging, i.e. the relativ e v ariance go es to zero as L increases. In particular, since F (t)/ L; the v ariance of the rescaled v ariable F (t)=L go es to zero. 39 F. Conclusions In this section w e considered equilibration in nite-dimensional isolated systems, and particularly concen trated on the time needed to reac h equilibrium and its scaling b eha vior with the system size. The standard denition of equilibration time extracted from the exp onen tial deca y of some observ able do es not w ork in nite system b ecause the dynamics is quasi-p erio dic and th us, no exp onen tial deca y can tak e place. Our denition of equilibration timeT eq is precisely the earliest instan t of time at whic h an observ able A (t) =h 0 jA (t)j 0 i reac hes its equilibrium v alue. It is giv en b y the earliest solution of A (t) =A. W e rst examined clean systems. Considering the Losc hmidt ec ho as a particular observ able, w e sho w ed that in the general situation of a gapp ed or clustering initial state, the equilibration time is indep enden t of system size, i.e. T eq =O (L 0 ). On the other hand, for small quenc hes close to a critical p oin t, w e found T eq =O L where w as the dynamical exp onen t. W e then turned to random systems and tac kled the tigh t-binding mo del with diagonal impurities as an example. Considering t w o dieren t observ ables A, w e found in b oth cases that lnT eq = c A L A +d A , where c A ;d A ; A are observ able-dep enden t constan ts. Ho w ev er, it remains to b e determined whether the exp onen tial div ergence of equilibration time is a general feature that is observ able indep enden t. Finally w e considered a no v el random matrix mo del, similar to the one considered in [10, 24]. F or this case, w e w ere able to pro v e that T eq = O (1). This sho ws that a higher degree of randomness in the time ev olution is not a guaran tee of slo w equlibration. This w ork on the CUE mo del also raises the question of whether the connection b et w een the a v erage of the logarithmic Losc hmidt ec ho and the v on Neumann en trop y has more general v alidit y . 40 IV. SMALL QUENCH D YNAMICS AS AN INVESTIGA TIVE TOOL F OR TRAPPED A TOM SYSTEMS In this section, w e examine the feasibilit y of using out-of-equilibrium uctuations of ob- serv ables of a quan tum system to gain insigh t in to its equilibrium prop erties. W e consider sev eral classes of one-dimensional, nite, isolated quan tum systems that can b e realized exp erimen tally using ultracold atoms trapp ed in optical lattices. W e use analytical and n u- merical metho ds to study p ost small quenc h dynamics in these systems. The dynamics is c haracterized b y observ ables uctuating in time follo wing a transien t b eha vior. W e ev aluate whether the magnitude of these uctuations can b e used to prob e the presence of b oundaries b et w een dieren t regimes in the ground state of the system. The collectiv e b eha vior of ultracold atoms in optical lattices can b e tuned b y v arying the depth of lattice p oten tials, th us adjusting the ratio b et w een the strength of the on-site in ter- actionU and the hopping parameter J . In this manner, a quan tum phase transition b et w een a sup eruid (shallo w lattice) and a Mott insulator (deep lattice) can b e induced [34, 70, 71]. An imp ortan t feature in those exp erimen ts is the presence of a (to a go o d appro ximation) harmonic trap, whic h results in the co existence of sup eruid and Mott domains for a wide range of v alues of U=J [7, 58, 74]. Exp erimen ts with a few-site resolution [30], as w ell as single-site resolution [5, 68], ha v e b een able to resolv e the site-o ccupation proles and rev eal the c haracteristic w edding cak e structure in whic h Mott plateaus are ank ed b y sup eruid domains. This phenomenology , for sucien tly deep lattices, can b e describ ed within the Bose-Hubbard mo del [26, 37]. While adiabatically slo w v ariations of the lattice p oten tial can b e used as a tuning knob for quan tum phase transitions in systems of trapp ed atoms, quenc hing that p oten tial can b e utilized as a means of probing the dynamics. Using this approac h, in a recen t exp erimen t on quasi-one-dimensional quan tum gases in an optical lattice, it w as demonstrated that quasiparticle pairs transp ort correlations with a nite v elo cit y across the system, leading to an eectiv e ligh t cone for the quan tum dynamics [23]. Another p ossibilit y is to quenc h the harmonic trap [65, 67, 76]. It has b een recen tly sho wn theoretically that a statistical analysis of the temp oral uctuations in w eak quenc hes can b e used to study phase transitions [18, 20]. In this section, w e adapt this uctuation analysis to examine b oundaries b et w een spatially 41 co existing phases in trapp ed systems after a quenc h of the trapping p oten tial. In optical lattice exp erimen ts in whic h U=J is not to o large, but larger than the critical v alue for the formation of Mott insulating domains, it is c hallenging to accurately determine the b oundaries b et w een insulating and sup eruid regions. This b ecause the Mott insulator ma y exhibit sizable uctuations of the site o ccupancies so single shot measuremen ts of the latter in an insulating plateau ma y not lo ok all that dieren t from those in the sup eruid region close b y . F or the purp ose of accurately determining the b oundaries b et w een those domains, sev eral lo cal compressibilities ha v e b een prop osed in the literature [7, 59, 60, 74], including i :=@h^ n i i=@ i [7], as w ell as the site-o ccupation uctuations n 2 i :=h^ n 2 i ih^ n i i 2 [59, 60], wherehi stands for the quan tum exp ectation v alue and i is the lo cal c hemical p oten tial at site i. Here w e prop ose the use of an out-of-equilibrium quan tit y , the temp oral v ariance of the exp ectation v alues of site o ccupancies, as a precise indicator of b oundaries b et w een domains. N i (t) =h^ n i (t)i is the exp ectation v alue of ^ n i (t), the site-o ccupation op erator at site i and at time t (in the Heisen b erg picture). The temp oral v ariance of this exp ectation v alue is giv en b y N 2 i :=N 2 i N i 2 , where denotes the innite-time a v erage f = lim T!1 T 1 T 0 f(t)dt. Out-of-equilibrium dynamics can b e triggered b y making a small, sudden c hange in the conning p oten tial or the lattice depth. After suc h a c hange (referred to as a quenc h), the site o ccupation exp ectation v alues N i (t) oscillate in time. Our n umerical analysis of the temp oral v ariance ofN i (t), and of the compressibilit y i , sho ws that N 2 i has sev eral features that mak e it attractiv e as an indicator of spatial phase b oundaries. Sp ecically , when compared to i , (i) the temp oral v ariance sho ws a stronger div ergence with system size at the b oundary b et w een domains, i.e., N 2 i / L with an exp onen t whic h is larger than that for i (L is the linear system size); and (ii) N 2 i detects ner details in the o ccupation prole, whic h are not resolv ed b y i . The scaling of these quan tities with system size is motiv ated b y analytical results obtained for homogeneous systems. The scaling analysis also emphasizes the p oin t that b ey ond a certain system size, the temp oral v ariance is strictly larger than the lo cal compressibilit y . F urthermore, w e discuss an exp erimen tally feasible w a y to study temp oral uctuations, based on a small n um b er of temp oral sampling p oin ts. W e also sho w that a detailed analysis of the full temp oral distribution P N i ofN i rev eals that deep in the incompressible region, P N i 42 is a single p eak ed, appro ximately Gaussian, narro w distribution, whereas in the b oundaries with the sup eruid part P N i is a double-p eak ed function indicating bistabilit y and absence of equilibration. W e should stress that in the w ork presen ted in this section, a small quenc h is dened as one follo wing whic h the system is sucien tly close to the initial equilibrium state. The corresp onding p ost quenc h time uctuations of v arious observ ables are not exp onen tially small, as one w ould exp ect them to b e in the case of larger global quenc hes in generic systems [62]. This section is organized as follo ws: in subsection A, w e recapitulate results for homogeneous systems and presen t an o v erview of t w o imp ortan t quan tities: the temp oral v ariance N 2 i and the compress- ibilit y i . in subsection B, w e apply the prop osed tec hnique for iden tifying phase b oundaries to (in tegrable) hard-core b oson systems. W e also in v estigate nite size scaling prop erties of the v ariance, as in tegrable systems allo w us to obtain exact results for v ery large lattice sizes. w e extend this analysis to a (nonin tegrable) J -V -V 0 system in subsection C. nally , w e presen t our conclusions in subsection D. The con ten t of this section is based on w ork published in [77] and done b y the author in collab oration with Asso c. Prof. Marcos Rigol, Researc h Asst. Prof. Lorenzo Camp os V en uti and Prof. Stephan Haas. A. Quenc hes and Observ ables T emp oral uctuations follo wing a quan tum quenc h ha v e b een studied extensiv ely in the con text of homogeneous systems [53]. Since some of these results form the motiv ation for our analysis of inhomogeneous systems, w e briey review relev an t prior w ork. W e consider systems initialized in the ground state of a Hamiltonian ^ H 0 = P n E n jnihnj. The quan tum quenc h is then p erformed b y suddenly c hanging the Hamiltonian to ^ H = ^ H 0 + ^ B . F or deniteness, w e assume that the p erturbation ^ B is lo cal and extensiv e, i.e., ^ B = P i ^ B i 43 with ^ B i =O(L 0 ) in the system size L, where i denotes sites in a lattice. A t time t after the quenc h, the system’s state is giv en b y j (t)i = exp(it ^ H)j (0)i (setting ~ = 1). F or quenc hes with =O(L 0 ) and a generic observ able ^ A, the exp ectation v alueA(t) =h ^ A(t)i = h (t)j ^ Aj (t)i oscillates around an a v erage v alue with uctuations A 2 =h ^ A(t)i 2 h ^ A(t)i 2 that are exp onen tially small in the system v olume [51] (see, e.g., Ref. [19]). In other w ords, A 2 =O(e V ), where is a p ositiv e constan t. Ho w ev er, if the quenc h amplitude is comparativ ely small [i.e., O(L q ) for some exp onen t q> 0 to b e sp ecied], the original state is not completely destro y ed during the p ost quenc h time ev olution. As a result, suc h quenc h exp erimen ts can b e used to obtain information on the pre-quenc h state of the system. As sho wn in Ref. [20], the temp oral v ariance for suc h small quenc hes is of order 2 , and is giv en b y B A 2 = 2 2 X n>0 jZ n j 2 +O 3 ; (21) with Z n := A 0;n B n;0 = (E n E 0 ) and the notation A n;m =hnj ^ Ajmi. The subscript B in B A 2 indicates that the v ariance is computed for time ev olution follo wing a quenc h ^ B . A simple condition for neglecting the cubic term in Eq. (21) can b e written as 2 F 1, where F is the delit y susceptibilit y [80]. Using the scaling la w in Ref. [16], one obtains the condition minfL d=2 ;L 1= g, where is the correlation length critical exp onen t. Equation (21) sho ws an in triguing similarit y to the zero temp erature equilibrium isother- mal susceptibilit y AB dened b y h ()j ^ Aj ()i =h (0)j ^ Aj (0)i AB +O ( 2 ), wherej ()i is the ground state of ^ H = ^ H 0 + ^ B . Indeed, w e ha v e AB = 2 X n>0 ReZ n : (22) Moreo v er, using Eq. (21), w e see that, to second order in , w e ha v e B A 2 = A B 2 , where w e dene B(t) :=h ^ B(t)i =h (t)j ^ Bj (t)i. The same dualit y holds for the susceptibil- it y , i.e., AB = BA for Hermitian op erators ^ A; ^ B . F or systems with a non-zero sp ectral gap , one can further relate susceptibilities to quan tum uctuations. One can sho w that AA (2=)A 2 , where A 2 =h ^ A 2 ih ^ Ai 2 is the (zero-temp erature) quan tum uctuation of ^ A. Both, quan tum uctuations and generalized susceptibilities, are commonly used indicators of critical b eha vior in homogeneous systems. Here, w e adv o cate for temp oral uctuations 44 0 0.5 1 1.5 2 2.5 0 2 4 6 log(L/L 0 ) log(Δ B ˜ A 2 ) log(˜ χ AB ) Figure 17. V erication of Eqs. (24) and (25), whic h predict AB / L and B A 2 / L 2 when q = 1. A b est t to algebraic scaling giv es B A 2 / L 2:0 and AB / L 1:1 . W e consider a tigh t binding mo del at half lling with b oth the observ able and p erturbation set to ^ A = ^ B = P i (1) i ^ n i . B A 2 and AB are made dimensionless b y dividing b y their v alues at L 0 = 102, i.e., w e plot e AB = AB (L)= AB (L 0 ) and B e A 2 = B e A 2 (L)= B e A 2 (L 0 ). as a sup erior indicator. F or homogeneous systems (and extensiv e observ ables) one can sho w that all these quan tities are extensiv e in gapp ed, non-critical, systems [17, 20]. Instead, in the critical region (dened b y L) one can use scaling h yp othesis to predict that the b eha vior at criticalit y is [20]: A 2 L 2d2 A (23) AB L q ; q = 2d + A B (24) B A 2 L 2q : (25) Note that if the p erturbation ^ B is relev an t, one has d + B = 1= > 0 and the exp onen t q can b e written as q = d + 1= A . The ab o v e equations mak e it clear that the strongest div ergence is exhibited b y the temp oral uctuations. F or simplicit y , setting ^ A = ^ B , the exp onen ts satisfy 2d 2 A <q< 2q . W e v erify Eqs. (24) and (25) for a tigh t binding mo del of spinless fermions ^ H 0 = P L i=1 h J( ^ f y i ^ f i+1 + h:c:) i at half lling (and p erio dic b oundary conditions) and observ- able/p erturbation giv en b y ^ A = ^ B = P j (1) i ^ n i . W e nd that the scaling of AB and B A 2 is in accordance with the predictions of Eqs. (24) and (25) with q = 1 (see Fig. 17). Before w e discuss inhomogeneous systems in the con text of ultracold atom exp erimen ts, w e w ould lik e to clarify ho w the lo cal compressibilit y i relates to the observ ables in tro duced so far. Straigh tforw ardly , w e realize that i = AA with ^ A = ^ n i , whereasN i (t) :=h^ n i (t)i and its (temp oral) v ariance B N 2 i are the dynamical coun terparts. W e fo cus on a small p erturbation of the strength of a trapping p oten tial of the form ^ B = L 2 P i (ii 0 ) 2 ^ n i . 45 Here i 0 is the lo cation of the trap cen ter, and the p oten tial is normalized b y L 2 to ensure extensivit y of the Hamiltonian. (In principle, other quenc hes are clearly p ossible in whic h the system is p erturb ed b y v arying dieren t parameters, e.g., the lattice depth.) A dv o cating a lo cal densit y appro ximation, one can assume that a v ery large trapp ed system can b e divided in to extensiv e regions, where eac h region can b e considered appro ximately homogeneous. In this case, the scaling prediction for the temp oral v ariance of the site o ccupations, is B N 2 i () 2 8 > < > : O(1) i in the gapped region L i in the critical region ; with a new scaling exp onen t . A ccording to Eqs. (23)(25), w e exp ect this exp onen t to b e larger than the corresp onding ones for the compressibilit y and site-o ccupation uctuations. B. Hard-core b oson systems As a rst example for the prop osed analysis, let us explore ho w to detect spatial b ound- aries b et w een co existing phases in an in tegrable mo del, where w e can p erform n umerical sim ulations for v ery large systems. This also allo ws us to p erform a nite-size scaling analy- sis to compare the div ergence of the temp oral uctuations with that of the compressibilit y , demonstrating that the temp oral v ariance exhibits a stronger div ergence at the b oundary b et w een domains. W e examine a quan tum system of hard-core b osons in one dimension describ ed b y the Hamiltonian ^ H 0 =J L1 X i=1 ( ^ b y i ^ b i+1 + h:c:) + L X i=1 g i ^ n i ; (26) whic h can b e though t of as the limit U=J!1 of the Bose-Hubbard mo del [22]. In Eq. (26), ^ b y i ( ^ b i ) is the creation (annihilation) op erator of a hard-core b oson at site i, ^ n i = ^ b y i ^ b i , and g i describ es a harmonic conning p oten tial, with g i = L 2 (iL=2 +) 2 . The trap is shifted o-cen ter b y a small amoun t to remo v e degeneracies in the energy lev els and gaps of the Hamiltonian [see the discussion of Eq. (27)]. W e initialize the system in a ground state j (0)i of a lattice with L sites and N hard- core b osons. After p erforming a sudden quenc h on the trap p oten tial, ! + at time t = 0, the system ev olv es unitarily as j (t)i = exp(i ^ Ht)j (0)i. The p ost-quenc h 46 Hamiltonian is giv en b y ^ H = ^ H 0 + ^ B . The hard-core b oson Hamiltonian (26) can b e mapp ed on to a Hamiltonian quadratic in fermion op erators ^ f y i and ^ f i through the Jordan- Wigner transformation [22]. F rom that transformation, it follo ws that the site o ccupations of hard-core b osons and spinless fermions are iden tical. The F ermionic Hamiltonian can b e written as ^ H = P i;j ^ f y i M i;j ^ f j with M i;j = J( i;j+1 + i;j1 ) + ( + )g i i;j . The nonin teracting c haracter of the latter system allo ws one to write temp oral uctuations of site o ccupations (and in fact of an y quadratic observ able in the fermions) in terms of one- particle quan tities alone. Consider the general observ able ^ X = P i;j ^ f y i i;j ^ f j . One can sho w thath (t)j ^ Xj (t)i = X (t) = tr( ^ Xe it ^ H 0 ^ 0 e it ^ H 0 ) = tr(e itM Re itM ) where R is the co v ariance matrix of the initial state ^ 0 ; i.e., R i;j = tr(^ 0 ^ f y j ^ f i ) (note that the initial state do es not necessarily need to b e Gaussian). Let the one-particle Hamiltonian M ha v e the sp ectral represen tation M = P k k jkihkj (jki are one the particle eigenfunctions). Dening F k;q =hkjjqihqjRjki where ; R are one-particle op erators, the temp oral v ariance of X is then giv en b y X 2 = X k;q F k;q F q;k X k (F k;k ) 2 : (27) Note that Eq. (21) holds for sucien tly small and relies on the assumption of a non- degenerate man y-b o dy sp ectrum. Equation (27), on the other hand, relies on the assumption of non-resonan t conditions for the one-particle sp ectrum [19, 56], whic h has b een v eried in our n umerical calculations (for the non-degenerate case with a trap shift 6= 0 as men tioned earlier). T o compute the v ariance of the site o ccupations, w e tak e X =N i with (i) x;y = i;x i;y . Results of our n umerical sim ulations are sho wn in Fig. 18, where the site o ccupations are plotted along with the t w o measures of lo cal critical b eha vior w e wish to compare here. Clearly b oth quan tities are able to distinguish the sup eruid regions from the insulating plateau at the trap cen ter. The lo cal compressibilit y i v anishes in the plateau (band in- sulating) regions where the state is close to j1; 1;:::; 1i (trap cen ter) and near the trap b oundaries with state j0; 0;:::; 0i. Also, i is roughly constan t in the sup eruid region. In con trast, N 2 i uctuates strongly within the sup eruid regime, displa ying sharp p eaks delineating the insulating regime from its surroundings. A closer lo ok at the site o ccupation proles [Fig. 18(b)] rev eals that, due to the nite size of the system studied (whic h will also b e the case in exp erimen ts), the site o ccupations at the b oundary b et w een insulating 47 50 100 150 200 250 300 350 400 450 500 0 0.5 1 1.5 Site number (a) Initial site occupancy Temporal variance Local compressibility Trap potential 30 40 50 60 70 0 0.2 0.4 0.6 0.8 Site number (b) 0 0.5 1 1.5 2 −1 0 1 2 log e (L/L 0 ) (c) log e (e κ max ) log e (Δ e N 2 max ) 0 100 200 300 400 500 0 0.05 0.1 0 1 2 Site number δλ ΔN 2 i /δλ 2 (d) Figure 18. (a) W edding cak e site o ccupation prole of hard-core b osons in a one-dimensional harmonic trap describ ed b y Eq. (26). The system consists of L = 500 sites and N = 250. The Hamiltonian parameters are = 10, = 0:2, =L 2 (J = 1 throughout). The phase b oundaries b et w een the Mott plateau lo cated at the trap cen ter and the adjacen t sup eruid regions can b e detected b y the con v en tional lo cal compressibilit y i (red) and b y the temp oral v ariance of the site o ccupations N 2 i (green) in tro duced in this w ork. (b) A closer lo ok at the sup eruid region for the system sho wn in (a) rev eals temp oral v ariance p eaks at the in terface b et w een the sup eruid and the Mott insulator. (c) Finite-size scaling of the maxim um temp oral v ariance of the site o ccupations and of the compressibilit y vs L for the Hamiltonian in Eq. (26). W e nd N 2 max / L 0:83 and max / L 0:05 . Both quan tities in this plot are made dimensionless b y dividing b y their v alues at L 0 = 50, i.e.,e max = max (L)= max (L 0 ) and e N 2 max = N 2 max (L)=N 2 max (L 0 ). (d) Dep endence of the normalized temp oral v ariance (N 2 i = 2 ) on the quenc h amplitude for the Hamiltonian in Eq. (26) with all other parameters as in (a). 48 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −0.5 0 0.5 1 1.5 log e (L/L 0 ) (c) log e (e κ max ) log e (Δ e N 2 max ) 50 100 150 200 250 0 0.2 0.4 0.6 Unit cell (a) Trap potential Temporal variance Initial site occupancy Local compressibility 55 60 65 70 75 0 0.2 0.4 0.6 Unit cell (b) Figure 19. (Color online) (a) Unit cell a v eraged site o ccupancy in the presence of a staggered p oten tial Eq. (30). This is a system with L = 500, N = 150, and parameters = 10, = 0:2, = 1=L 2 , V 0 = 1:5. (b) A closer lo ok at the sup eruid region for the system sho wn in (a) rev eals temp oral v ariance p eaks at the in terface b et w een the sup eruid and the Mott insulator. (c) Finite- size scaling of the maxim um temp oral v ariance of the site o ccupations and of the compressibilit y vs L for the Hamiltonian Eq. (30). W e nd N 2 max /L 0:80 and max /L 0:14 . Both quan tities in this plot are made dimensionless b y dividing b y their v alues at L 0 = 50, i.e.,e max = max (L)= max (L 0 ) and e N 2 max = N 2 max (L)=N 2 max (L 0 ). and sup eruid domains c hange in a step wise fashion. N 2 i displa ys clear signatures of the presence of suc h steps in the site o ccupation proles, while they are barely reected in i . More imp ortan tly , a nite-size scaling analysis rev eals that the maxima of N 2 i div erge m uc h more rapidly with system size than the maxima of i . W e rst v eried that the size ` of the in termediate region b et w een the t w o band insulator states scales as ` L=c with c 4. A t to n umerical data [see Fig. 18(d)] rev eals p o w er-la w dep endencies on system size L (or equiv alen tly , on `) N 2 max /L 0:83 (28) max /L 0:05 (29) The scaling seen in Fig. 18(d) mak es apparen t that, b ey ond some system size (that will 49 dep end on the Hamiltonian parameters), the signal giv en b y N 2 i will exceed that of i . This means that the b oundaries b et w een domains can b e determined with higher condence using the temp oral measure, pro vided the systems are not to o small. In general, insulating states realized in exp erimen ts exhibit nonzero quan tum uctua- tions of the site o ccupancies. This is to b e con trasted to the quan tum uctuations of the site o ccupancies in the band insulating phases of Hamiltonian (26), whic h are alw a ys zero. In order to address what happ ens in the presence of nonzero quan tum uctuations of the site o ccupancies, while still retaining the adv an tages of dealing with mo dels mappable to nonin teracting ones, w e add a staggered p oten tial to Eq. (26) and consider ^ H 0 =J L1 X i=1 ( ^ b y i ^ b i+1 + h:c:) + L X i=1 [g i ^ n i +V 0 (1) i ^ n i ]: (30) The prop erties of systems whic h suc h a Hamiltonian ha v e b een previously studied for spin- less fermions [63] and hard-core b osons [61]. The ground state displa ys site-o ccupation uctuations within the insulating phase with a v erage site o ccupancy of 1/2. Those uctu- ations v anish as V 0 !1, in whic h case the insulator b ecomes a pro duct state of the form j0; 1; 0;:::; 1; 0i. A ccordingly , w e plot all quan tities in Fig. 19(a) a v eraged o v er (t w o site) unit cells. As seen in Fig. 19(a), for this mo del and the parameters c hosen, the insulat- ing plateau in the cen ter of the trap is larger relativ e to the size of the sup eruid domains than the one in the absence of the staggered p oten tial. Nonetheless, the sup eruid domains are clearly iden tiable using N 2 i and i (notice that i is nonzero also in the insulating domain). Studying the nite-size scaling of the maxim um of b oth quan tities, w e nd the temp oral v ariance and compressibilit y scaling to b e, N 2 max /L 0:80 (31) max /L 0:14 (32) [see Fig. 19(c)], resp ectiv ely . Therefore the same conclusions hold regarding b etter detectabil- it y of spatial phase b oundaries using N 2 i for sucien tly large system sizes. In terestingly , w e ha v e found that the systems sizes for whic h N 2 i starts to giv e a stronger signal than i are larger in the presence than in the absence of the staggered p oten tial. This results from ha ving nonzero c harge uctuations in the insulator in the cen ter of the trap. The dep endence of the normalized temp oral v ariance (N 2 i = 2 ) on the quenc h amplitude is depicted in Fig. 18(c) for the Hamiltonian in Eq. (26). As exp ected, the normalized 50 temp oral v ariance deca ys for increasing , follo wing a linear regime (N 2 i / 2 ) for small quenc hes (< 1=L 2 ). C. Nonin tegrable systems In order to sho w that the prop osed approac h w orks b ey ond in tegrable Hamiltonians suc h as the ones analyzed in the previous section, here w e consider a nonin tegrable mo del. W e should stress that the exp onen tial increase of the Hilb ert space with system size sev erely restricts the system sizes that can b e studied n umerically . W e fo cus on a system consisting of hard-core b osons with nearest and next-nearest in teractions (a J -V -V 0 mo del) in the presence of a harmonic trap, describ ed b y the Hamiltonian ^ H = L1 X i=1 J( ^ b y i ^ b i+1 + h:c:) +V ^ n i 1 2 ^ n i+1 1 2 +V 0 ^ n i 1 2 ^ n i+2 1 2 + L X i=1 i 2 ^ n i : (33) Note that, in order to maximize the size of insulating and sup eruid domains, in Eq. (33) w e only consider one half of what w ould b e the harmonic trap in an exp erimen t. In the absence of a trap, the phase diagram of Hamiltonian (33) has b een studied using the densit y matrix renormalization group tec hnique [47]. The comp etition b et w een nearest- neigh b or and next-nearest-neigh b or in teractions generates four phases: t w o c harge-densit y- w a v e insulator phases, a sup eruid (Luttinger-liquid) phase, and a b ond-ordered phase. In the presence of a trap, and for a suitable c hoice of the parameters, the same four phases can b e observ ed. W e fo cus our analysis on a parameter regime where the system exhibits a c harge densit y w a v e of t yp e one (CD W-I) in the cen ter of the trap, whic h is surrounded b y a sup eruid phase. The site o ccupations in the CD W-I phase are similar to those in the presence of the sup erlattice p oten tial analyzed in the previous section, when the a v erage site o ccupation p er unit cell is 1/2 (see Fig. 20). In con trast to the sup erlattice case, the CD W-I phase here is not due to the presence of a translationally symmetry breaking term but is stabilized b y the presence of in teractions. There are t w o other phases that ha v e larger unit cells, consisting of 4 sites for CD W-I I and 3 sites for b ond-order. The CD W-I phase is the b est suited for our purp oses b ecause w e are able to observ e sev eral unit cells that exhibit its 51 2 4 6 8 10 12 14 16 18 0 0.2 0.4 0.6 0.8 1 Site number Initial density Temporal variance/5 Local compressibility x 5 Trap potential (a) Interface Empty CDW−I regime 5 10 15 0.01 0.02 0.03 0 1 2 3 Site number δλ ΔN 2 i /δλ 2 (b) Figure 20. (a) Spatial prole of the temp oral v ariance of the site o ccupations N 2 i and of the lo cal compressibilit y i for the mo del in Eq. (33). W e initialize the system with 19 sites and 5 particles in the ground state with parameters J = 1, V = 8:0, V 0 = 0:5 and = 0:1225. The quenc h is p erformed b y c hanging the trap p oten tial from to + with = 0:0061. (b) Dep endence of the v ariance on the quenc h amplitude . exp ected prop erties. In Fig. 20(a), w e sho w results for a site-o ccupation prole exhibiting a CD W-I plateau surrounded b y a small sup eruid domain. In the same gure one can see that, at the edge of the CD W-I plateau, the lo cal compressibilit y i exhibits a m uc h w eak er signal than the temp oral uctuations N 2 i . (Note that w e used m ultiplicativ e factors to enhance i and reduce N 2 i so that b oth measures can app ear on the same scale). Also, notice that i do es not v anish in the CD W-I plateau, whic h exhibits nonzero site o ccupation uctuations. Since calculations for larger systems are prohibitiv ely large, a nite-size scaling analysis of the observ ables is not p ossible here. Nonetheless, from Fig. 20(a), it is eviden t that the temp oral v ariance is a b etter indicator of the in terface b et w een domains than the lo cal compressibilit y . In fact, compared to the in tegrable systems considered in the preceding section, the adv an tage of using N 2 i o v er i to iden tify in terfaces b et w een domains is enhanced, esp ecially taking in to accoun t the small system sizes considered here. In Fig. 20(b), w e plot N 2 i = 2 vs. Similarly to the results in the previous section, w e notice a decrease in the p eak heigh t with increasing , follo wing a linear regime (N 2 i / 52 6 7 8 9 10 11 −0.02 0 0.02 0 100 200 300 Site number <n i >−<n i > Probability density Interface (a) Empty CDW−I regime 0 20 40 60 80 100 0.938 0.94 0.942 tJ/¯ h 0 100 200 300 0 20 40 60 80 100 0.19 0.2 0.21 0.22 0.23 tJ/¯ h 0 100 200 300 (b) (c) (d) (e) Site 10 Interface Site 1 CDW−I regime Figure 21. (a) Distributions of the site o ccupations N i (t) at sites near the in terface b et w een the CD W-I and the sup eruid phase. (b) and (d) Distribution function of the site o ccupation at a site deep in the CD W-I regime (site i = 1) and at a site at the edge of the CD W-I domain (site i = 10), resp ectiv ely . (c) and (e) Time dep endence of N i (t) corresp onding to (b) and (d), resp ectiv ely . These results are obtained from sim ulations with the same Hamiltonian and system parameters as in Fig. 20. Eac hN i (t) is sampled at N = 4 10 4 random times uniformly distributed in [0;T ] with T = 40~=J . 2 ) at small . F or& 0:2, a qualitativ ely dieren t b eha vior sets in. This is b ecause the CD W-I domain is destro y ed b y the nal trap and an n = 1 Mott insulating domain app ears at the p oten tial minim um of the trap. The latter domain giv es rise to a large temp oral v ariance of the site o ccupations at its end, whic h is lo cated in sites that w ere formerly in the CD W-I regime. W e no w go b ey ond the second momen t analysis presen ted so far and examine the full probabilit y distribution P i (x) of the random v ariable N i (t) equipp ed with the time a v er- age measure. Based on the results for homogeneous systems [18, 20], w e exp ect P i (x) to b e a single p eak ed, appro ximately Gaussian, narro w distribution for sites i deep in the 53 (gapp ed) insulating regime. On the con trary , P i (x) is predicted to b e a double p eak ed distribution with a relativ ely large v ariance for (critical) in terface sites i. In a limiting, somewhat simplied case, P i (x) can b e appro ximated b y a t w o parameter distribution P i (x) = 1= q 2N 2 i (xN i ) 2 [18]. In Fig. 21(a), w e sho w the distribution P i (x) for sites near the in terface separating the insulating and sup eruid regions. F or sites i deep in the insulating region [Fig. 21(b)], the site o ccupations uctuate ab out one unique cen tral v alue, resulting in a singly-p eak ed distribution function. This signies measure concen tration, indicating lo cal equilibration in the nite system considered here [Fig. 21(c)]. In con trast, as one mo v es closer to the in terface [Fig. 21(d)], the exp ectation v alues of observ ables can b e appro ximated as [18] A(t)'A(t) +A 1 cos[(E 1 E 0 )t] +A 2 cos[(E 2 E 0 )t] +::: (34) with the remaining terms b eing negligible (the constan ts A 1;2 dep end on the initial state, ev olution Hamiltonian and rst excited states, see [18] for details). The probabilit y distri- bution then dev elops p eaks atA(t)jjA 1 jjA 2 jj and a relativ ely large v ariance [18]. This bistabilit y indicates a lac k of measure concen tration and a breakdo wn of lo cal equilibration [see Fig. 21(e)]. D. Conclusions W e ha v e studied v arious trapp ed systems whose ground states exhibit co existence of insulating and sup eruid domains, as relev an t to ultracold atom exp erimen ts in optical lattices. An analysis of the time ev olution of the site o ccupations N i , follo wing a small quenc h of the trapping p oten tial, allo w ed us to sho w that the temp oral v ariance of N i can b e used as an accurate to ol to lo cate b oundaries b et w een domains. W e found that the temp oral v ariance ofN i at those b oundaries exhibits a p o w er la w scaling with system size with an exp onen t that is greater than the one of a previously prop osed lo cal compressibilit y . F urthermore, w e p erformed a binning analysis to explicitly study the temp or al probabil- it y distribution of site o ccupancies. Suc h a temp oral distribution giv es the probabilit y of observing a giv en v alue of the site o ccupation in a large observ ation time-windo w [0;T ]. W e observ ed that the distributions are sharply p eak ed and appro ximately Gaussian for sites that 54 are deep in the insulating phase, while sites at the in terface displa y a bimo dal distribution, i.e., are c haracterized b y a lac k of measure concen tration. 55 V. EFFICIENT MEASUREMENT OF TEMPORAL FLUCTUA TIONS The discussion in this section expands on the analysis of small quenc hes presen ted in the previous section. As concluded earlier, the temp oral v ariance of the exp ectation v alue of site o ccupationsN i (t) is a v aluable to ol to prob e the b oundaries b et w een dieren t regimes in the ground state of a quan tum system. In this section w e consider the cost of measuring N 2 i in the lab. P art of this section is based on w ork published in [77] and done b y the author in collab oration with Asso c. Prof. Marcos Rigol, Researc h Asst. Prof. Lorenzo Camp os V en uti and Prof. Stephan Haas. This section is organized as follo ws: in subsection A, w e motiv ate the sub ject of this section i.e. the need for an ecien t measuremen t strategy . in subsection B, w e prop ose a simple estimator that in v olv es p erforming t w o indep en- den t measuremen ts at eac h instan t of time. W e use n umerical sim ulations to study the p erformance of this estimator. in subsection C, w e prop ose a more general consisten t estimator that is more accurate while requiring a greater n um b er of measuremen ts at eac h instan t of time . Once again, w e use n umerical sim ulation to study the p erformance of the estimator. in subsection D, w e summarize our conclusions. A. Motiv ation In the previous section, w e assumed that the exp ectation v alue A(t j ) =hA(t j )i could b e determined exactly for v arious times t j . In this section, w e tak e a deep er lo ok at the issue of estimating the temp oral v ariance A 2 using measuremen t data, k eeping in mind ultracold atom exp erimen ts. In these exp erimen ts, one t ypically obtains information ab out site o ccupations b y taking a snapshot of the system [5, 68] at a giv en time t j after the quenc h. A t this instan t of time, the depth of p oten tial w ells in the optical lattice is increased suddenly to a large v alue. This eectiv ely reduces tunneling probabilities b et w een adjacen t w ells sharply thereb y disabling in teractions b et w een adjacen t sites in the Bose-Hubbard. This 56 freezes the state of the system and allo ws subsequen t imaging to b e p erformed in order to determine site o ccupation n um b ers at an y giv en time. W e ha v e review ed exp erimen ts that implemen t suc h proto cols in section I I of this dissertation. In our sp ecic case, the observ able of in terest is the on-site o ccupation n um b er whic h can b e measured as outlined ab o v e. Ho w ev er, w e k eep our discussion general so that it can b e applied to an y observ able. If w e p erform a measuremen t of an observ able A at time t j , w e obtain one of the eigen v alues of A with probabilit y sp ecied b y the Born rule. If w e p erform N S measuremen ts of A at t j , let the result of the i-th measuremen t of A at t j b e A i (t j ). W e can obtain an estimate of the exp ectation v alue A(t j ) =hA(t j )i b y p erforming N S measuremen ts of A at the same instan t, a time t j after the quenc h. When A is compactly supp orted [for F ermions ^ n i (t j ) is actually Bernoulli distributed] the error in estimatingA(t j ) decreases exp onen tially with N S as a consequence of the Cherno b ound. One strategy to estimate the temp oral v ariance w ould b e to tak e N S sucien tly large suc h that A(t j ) can b e obtained with the desired precision. One then needs to rep eat the ab o v e pro cedure at N T dieren t times ft 1 ;t 2 ;:::;t N T g where t i 2 [0;T ] to estimate the temp oral v ariance. The question of ho w large should b e T to obtain a sucien tly go o d estimate A 2 T A 2 has b een addressed in [15]. T ypically taking T to b e one or t w o reviv al times giv es v ery accurate results. The reviv al time is the time required b y excitations to tra v el the length of the system, and is hence prop ortional to the linear size L [15]. The prop ortionalit y constan t is a time-scale whic h is of the order of the tunneling time ~=J . As a result, a total of N S N T measuremen t are required (in principle N s ma y dep end onj , but w e do not consider this generalization here). This form ulation can allo w us to ev aluate approac hes to appro ximate the temp oral v ariance A 2 . Essen tially , w e attempt to minimize the total n um b er of measuremen ts N S N T . The system is prepared N S N T times, in the same initial state 0 at time t = 0 and allo w ed to ev olv e unitarily thereafter with the same Hamiltonian parameters. Let us denote with A p (t j ) the result of the p-th measuremen t of A p erformed at time t j , p = 1;:::;N S , j = 1;:::;N T (i.e., one of the eigen v alues of A). The random v ariables A p (t j ) at dieren t times are indep enden t but not iden tically distributed (as opp osed to measuremen ts p erformed in equilibrium, in whic h case they are iden tically distributed). In the language of statistics, what w e w ould lik e to build is a c onsistent estimator of the 57 temp oral v ariance A 2 . A consisten t estimator is a metho d to obtain a giv en quan tit y with the prop ert y that, as the n um b er of data p oin t increases, the estimator con v erges to the actual parameter w e are trying to estimate (see e.g. Ref. [2]). In our case the data p oin ts are the random v ariables A p (t j ). These are Bernoulli distributed indep enden t random v ariables (i.e. tak e only t w o v alues, A p (t j )2f0; 1g) B. Estimation with N s = 2 The follo wing quan tit y can b e sho wn to b e a consisten t estimator of A 2 : s = 1 N T N T X j=1 A 1 (t j )A 2 (t j ) 1 N 2 T X i;j A 1 (t i )A 2 (t j ): (35) As stated earlier, w e use the notation A p (t j ) for the result of the p-th measuremen t of A p erformed at time t j . T aking the exp ectation v alue, one obtains E[s] = 1 N T N T X j=1 hA(t j )i 2 1 N T N T X j=1 hA(t j )i 2 ; (36) In this case, the sym b ol E denotes the exp ectation v alue after the 2N T measuremen ts. W e still ha v e to sp ecify ho w to c ho ose the N T times. If w e pic k the times uniformly distributed in [0;T ] and denote with T the exp ectation v alue op eration of suc h uniform times, then one has T[E[s]] = (A 2 ) T A T 2 =: A 2 T , where w e indicated f T = T 1 T 0 f(t)dt . The fact that the exp ectation v alue of our estimator is the exact temp oral v ariance means that s is an unbiase d estimator of the temp oral v ariance o v er a windo w [0;T ]. The question of ho w large should b e T to obtain a go o d estimate A 2 T A 2 has b een addressed in [15]. T ypically taking T to b e one or t w o reviv al times giv es v ery accurate results. The reviv al time is the time required b y excitations to tra v el the length of the system, and is hence prop ortional to the linear size L [15]. The prop ortionalit y constan t is a time-scale whic h is of the order of the tunneling time ~=J . Since s is an un biased estimator, the error in estimating the temp oral v ariance using s can b e quan tied using the v ariance of s, var(s) = T[E[s 2 ]]T[E[s]] 2 . W e sho w b elo w that var(s) =O (1=N T ); (37) 58 whic h sho ws the consistency of the estimator, i.e. when the n um b er of measuremen ts increases the estimator con v erges to the exact temp oral v ariance. The follo wing is a sk etc h of the deriv ation of Eqn.37. W e use F[] to represen t taking the exp ectation v alue o v er m ultiple measuremen ts at the same time, follo w ed b y an exp ectation v alue o v er dieren t uniformly distributed times i.e. F[] = T[E[]]. var[s] = F[s 2 ]F[s] 2 = F s 2 2 hAi 2 hAi 2 2 F[s 2 2 ] = F 2 4 1 N T X i A 1 i A 2 i 1 N 2 T X ij A 1 i A 2 j ! 2 3 5 = F " 1 N 2 T X i X j A 1 i A 2 i A 1 j A 2 j + 1 N 4 T X ijkl A 1 i A 2 j A 1 k A 2 l 2 N 3 T X ijk A 1 i A 2 i A 1 j A 2 # W e can no w compute the exp ectation v alues of eac h of the terms individually to nd: F " 1 N 2 T X i X j A 1 i A 2 i A 1 j A 2 j # = (hA(t)i 2 ) T 2 + 1 N T (hA(t) 2 i 2 ) T 1 N T (hA(t)i 4 ) T F " 1 N 4 T X ijkl A 1 i A 2 j A 1 k A 2 l # =hA(t)i T 4 F " 2 N 3 T X ijk A 1 i A 2 i A 1 j A 2 # = 2(hA(t)i 2 ) T hA(t)i T 2 =) var[s] = F[s 2 ]F[s] 2 = 1 N T (hA(t) 2 i 2 ) T 1 N T (hA(t)i 4 ) T / 1 N T (38) In Eqn. (38) w e use that fact that the time a v erages of quan tum exp ectation v alues eac h scale asO(N 0 T ). In gure 22(a), w e sho w a t ypical realization of s i obtained b y taking N T = 15. One can compare Fig. 23 with Fig. 20(a), where the exact v ariance is plotted for the same Hamiltonian parameters. Clearly , the maxima at sites 9; 10 in Fig. 23(a) predicts a transition region in agreemen t with that in Fig. 20(a). Still, w e are primarily in terested in the ecacy of v i in lo cating the b oundary b et w een co existing phases. In other w ords, w e are in terested in kno wing whether the p osition of the maxima of v i coincides with that of the exact v ariance [sites 9 and 10 as seen in in Figs. 20 and 22(a)]. T o this end, w e compute the probabilit y 59 2 4 6 8 10 12 14 16 18 −0.02 0 0.02 0.04 Site si 1 2 3 4 5 6 7 8 9 10 −15 −10 −5 0 log e (NT) log e (var(s10)) Results of numerics −0.98log e (NT)−3.7 0 10 20 30 40 50 60 70 80 90 100 0.5 0.6 0.7 0.8 NT pM (a) (b) (c) Figure 22. Estimating the temp oral v ariance. (a) An example of the temp oral v ariance estimator s i for N T = 15. All parameters are the same as for Fig. 20. W e compute s i indep enden tly for eac h of the L = 19 sites of the system. (b) Scaling of the v ariance of the estimator s i at the site i = 10, with N d . The t sho ws var(s i )N 0:98 T in accordance with our theoretical prediction that var(s i ) 1=N T : (c) Results of a n umerical exp erimen t to compute p M , the probabilit y that the maxim um of s i is lo cated at the maxim um of the exact v ariance i.e. site i2f9; 10g. 10 4 samples of fs i g w ere used to compute this probabilit y . p M (N T ) = Prob(arg max i fv i g2f9; 10g) as a function of the n um b er of measuremen ts N T . W e plot p M (N T ) in Fig. 22(c) as a function of N T . W e observ e that this simple estimator that is restricted to t w o measuremen ts at eac h instan t of time as dened in Eq. (35) allo ws us to lo cate the b oundary with roughly 70% accuracy , using around 40 measuremen ts. In the follo wing section, w e presen t a more general estimator, that oers b etter accuracy . C. A general consisten t estimator W e no w consider a more general approac h to estimating the temp oral v ariance. The quan tum exp ectation v alue at a giv en instan t of time A(t j ) =hA(t j )i can b e estimated 60 using N S measuremen ts b y e j = 1 N S N S X p=1 A p (t j ); This estimate e j con v erges toA(t j ) in the large N S limit. With the quan tum v ariance at t j b eing 2 j = A 2 (t j ), Prob je j A(t j )j> j N S e 2 =4 or equiv alen tly Prob (je j A(t j )j>)e 2 N 2 S =(4 2 j ) Using the estimator e j for the exp ectation v alue at t j , w e can dene the follo wing estimator v for the the temp oral v ariance of A: v = 1 N T N T X j=1 (e j ) 2 with = 1 N T N T X j=1 e j : (39) Using E[] to denote exp ectation v alues tak en at eac h instan t of time for eac h of the inde- p enden t measuremen ts, w e nd, E[v] = 1 N T N S ( (N S 1) X j hA(t j )i 2 + X j hA 2 (t j )i ) 1 N 2 T X j6=k hA(t j )ihA(t k )i 1 N 2 T N S ( (N S 1) X j hA(t j )i 2 + X j hA 2 (t j )i ) : W e still ha v e to sp ecify ho w to c ho ose the N T times. If w e pic k the times randomly with uniform distribution in [0;T ] and denote withT[] the corresp onding time a v erage op eration (i.e., T a v erages uniformly o v er all N T indep enden t times t j ), w e obtain T[E[v]] = N S 1 N S (hA(t)i 2 ) T N T 1 N T hAi T 2 + 1 N S N T h (N T 1)hA(t) 2 i T (N S 1)(hA(t)i 2 ) T i ; where w e indicated f(t) T = T 1 T 0 f(t)dt. W e see that in the limit N S ;N T !1, the exp ectation v alue of this estimator tends to the exact v ariance (A 2 ) T A T 2 =: A 2 T . . The largest error term here is exp ected to b e 1 N S hA(t) 2 i T / 1 N S . This means that v is an asymptotically un biased estimator, i.e., when the n um b er of measuremen t increases 61 the estimator con v erges to the exact temp oral v ariance. F urthermore, w e ha v e n umerically c hec k ed that v is also a c onsistent estimator, meaning that the error on v , enco ded in var[v] = T[E[v 2 ]] T[E[v]] 2 , tends to zero as the n um b er of measuremen ts increases. In Fig. 23(b), w e sho w that var[v]N 1 T . W e no w test the feasibilit y of this approac h for distinguishing dieren t domains in trapp ed systems. W e p erform n umerical exp erimen ts on the Hamiltonian in Eq. (33). A ccording to our general recip e, w e p erform a small quenc h of the trapping p oten tial and measure the o ccupation n um b er at eac h site during the follo wing time ev olution. In this case, the observ able is the site o ccupation A = ^ b y i ^ b i and w e usev i to denote the corresp onding temp oral v ariance estimated according to Eq. (39) for i = 1;:::;L. As men tioned earlier, A p (t j ) is the result of the p-th measuremen t of A at time t j . In our n umerical exp erimen ts, this is obtained b y randomly generating one of the eigen v alues of A (0 or 1 for A = ^ b y i ^ b i ) with probabilities giv en b y the Born rule. In gure 23(a), w e sho w a t ypical realization of v i obtained taking N T = 10 and N S = 3, for a total of 30 measuremen ts. Once again, w e can compare these results to the exact v ariance as w ell as the simple estimator s i . W e obtain rst indications that this estimator is similarly capable of sp otting the b oundary b et w een co existing phases. F urthermore, w e v erify its consistency n umerically , nding that it con v erges to its exp ectation v alue with the v ariance deca ying as 1=N T for v arious v alues of N S . W e also w e compute the probabilit y p M (N T ) = Prob(arg max i fv i g2f9; 10g) as a function of the n um b er of measuremen ts N T for dieren t v alues of N S . p M (N T ) is plotted in Fig. 23(c) as a function of N T for dieren t N s . W e observ e that this estimator allo ws us to lo cate the b oundary with a 90% accuracy , using a total of around 40 measuremen ts. This is a signican t impro v emen t o v er the previously considered estimator, ev en while k eeping N S = 2. D. Conclusions In this section, w e analyzed the feasibilit y of the metho d prop osed in the previous section to use temp oral uctuations as an in v estigativ e to ol. Our analysis motiv ated b y the fact that ultracold atom exp erimen ts use shapshots to freeze the state of a system and measure observ ables at a giv en instan t of time. This requires that the system b e initialized p erfectly 62 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 −12 −10 −8 −6 log(NT) log(var(v)) 2 4 6 8 10 12 14 16 18 0 0.02 0.04 0.06 0.08 Site v(NS = 3,NT = 10) 5 10 15 20 25 30 35 40 45 50 0.7 0.8 0.9 1 NT pM NS = 2 NS = 3 NS = 4 NS = 5 NS = 2 NS = 3 NS = 4 NS = 5 (b) (c) (a) Figure 23. Estimating the temp oral v ariance. (a) An example of the temp oral v ariance estimator v at eac h site for N S = 3 and N T = 10. All parameters are the same as for Fig. 20. W e compute v indep enden tly for eac h of the L = 19 sites of the system. (b) Scaling of the v ariance of the estimator v at the site i = 10 with N T , for dieren t v alues of N S . The t sho ws var(s i )N 0:97 T for the N S v alues considered, in accordance with our prediction that v is a consisten t estimator. (c) Results of a n umerical exp erimen t to compute p M , the probabilit y thatfv i g and the exact v ariance ha v e maxima at the same sites (i = 9; 10). 10 4 samples offv i g w ere used to compute these probabilities. and the quenc h b e p erformed starting at t = 0 for eac h snapshot. Based on this, the exp erimen tal feasibilit y of our approac h dep ends strongly on b eing able to obtain signican t insigh ts using as few snapshots as p ossible. W e considered estimators of the exact temp oral v ariance that use a nite n um b er of snap- shots at eac h instan t, and require that snapshots b e tak en at a small, nite n um b er of time instan ts. W e v eried that they w ere indeed statistically un biased and consisten t. W e p er- formed n umerical sim ulations to analyze the con v ergence prop erties of these estimators.W e found that b oth estimators con v erge to their exact v alues as 1=N T with n um b er of snapshots b eing prop ortional to N T . Since the ob jectiv e is to analyze ground state prop erties of the system, iden tifying gen- eral features of the temp oral v ariance prole is more imp ortan t than quan tifying the exact 63 statistics of temp oral uctuations. Therefore, w e study the abilit y of our consisten t estima- tor to sho w prominen t p eaks at phase in terfaces that coincide with sites at whic h the exact v ariance sho ws large p eaks. W e use n umerical sim ulations to get an idea of the n um b er of measuremen ts required. W e use n umerical sim ulations with a JVV 0 mo del with 19 sites and 5 particles. W e found that sample v ariances obtained using a around 40 measuremen ts are sucien t for lo cating phase b oundaries with around 90 probabilit y . 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Abstract (if available)
Abstract
The question of whether and how closed quantum systems equilibrate lies at the foundations of statistical mechanics. It involves asking whether such systems can be described by statistical equilibrium ensembles and how such ensembles can be derived from the equations of motion [25]. Despite its importance, this has remained a difficult question to tackle. Experimental investigations require that the system be well isolated from its environment and its dynamics be controlled and measured precisely. However, recent experiments have demonstrated the capability to conduct precise studies of out-of-equilibrium dynamics in closed quantum systems. For example, in [39], a series of tubes containing ultracold ⁸⁷Rb atoms was created using a two dimensional optical lattice. The atoms confined to these one dimensional tubes exhibited a lack of thermalization, after experiencing thousands of collisions with each other. ❧ Such experiments involving ultracold atoms trapped in optical lattices offer the ability to simulate condensed matter systems and test fundamental questions in statistical mechanics. The parameters of optical lattices can be tuned dynamically in order to simulate a variety of models. Atoms are trapped in these lattices after being cooled to nanokelvin temperatures and are well isolated. Their properties such as site occupations and momentum distribution can be measured precisely. ❧ In this dissertation, we consider closed inhomogeneous systems that have been realized experimentally using cold atoms. We use analytical and numerical techniques to explore the properties and applications of non-equilibrium dynamics in these systems. ❧ The first problem we consider is that of how long closed quantum systems take to equilibrate. Despite its apparent simplicity, this is as yet an unresolved question. Starting with a rigorous definition of equilibration time, we investigate the approach to equilibrium in homogeneous and disordered one dimensional systems. We find equilibration times that agree with physical intuition. ❧ The second problem that we tackle involves probing the interfaces between coexisting phases in trapped quantum gases. We propose a novel approach that relies on measuring the temporal fluctuations of observables. The capability to measure the time evolution of observables in cold atom systems has been demonstrated in experiments. For example, in [23], a series of one dimensional decoupled chains containing 10 to 18 atoms each were realized. The experimentalists were able to study the time dependence of spatial correlations in each of these chains following a sudden change of the optical lattice depth. This was made possible in part by advances in single site imaging techniques, allowing experimentalists to study the dynamics of individual trapped atoms [5]. ❧ This dissertation is organized as follows: ❧ In Section I, we review some aspects of experiments involving ultracold atoms. We focus on the realization of the Hubbard model using optical lattices, and the Mott insulating and superfluid phases. In Section II, we review some of the theoretical framework for studying non-equilibrium dynamics in closed quantum systems. We provide precise definitions of equilibration and review some of the analytical approaches used to study temporal fluctuations in free fermion systems. ❧ In Section III, we investigate the approach to equilibrium in clean and disordered quantum systems. We start by defining “equilibration time” precisely, based on the long time average value of an observable. We then consider clean systems, where, using analytical approaches, we find constant equilibration times for the case of clustering initial states with system parameters far from critical points. However, when we consider small quenches around a critical point, we find the equilibration time to exhibit a power law dependence on the system size. We then analyze noisy systems, where literature points at the existence of large equilibration time scales that diverge exponentially with system size. Specifically, we consider the tight-binding model with diagonal impurities and give numerical evidence for the existence of these exponentially large equilibration times. We also find that the equilibration time depends on our choice of observable. Finally, we consider yet another noisy system whose evolution dynamics is randomly sampled from an ensemble of unitary matrices. Here, we are able to prove analytically that the equilibration time is constant, thus showing that noise alone does not guarantee slow equilibration. ❧ In Section IV, we present a novel technique that uses temporal fluctuations to probe the properties of cold quantum gases confined in harmonic traps. The ground states of these systems exhibit a coexistence of superfluid and Mott insulating regimes, which are conventionally detected using the local compressibility [7, 59, 60]. Motivated by a scaling analysis [20], we propose an alternative approach involving the out-of-equilibrium dynamics following a weak quench. In particular, we show how the temporal fluctuations of the site occupations reveals the location of spatial boundaries between compressible and incompressible regions. We demonstrate the feasibility of our approach for several models using numerical simulations. We first consider integrable systems, hard-core bosons (spinless fermions) confined by a harmonic potential, where space separated Mott and superfluid phases coexist. We then analyze a non-integrable system which has coexisting charge density wave and superfluid phases. The results of our simulations indicate that the temporal variance of the site occupations is a more effective detector of phase boundaries than the local compressibility. ❧ In Section V, we study the experimental feasibility of the method proposed in the previous section. Based on techniques applied in experimental literature [23] for measuring out-of-equilibrium behavior of site occupations, we propose estimators of the temporal variance. We study the convergence properties of these estimators to examine their dependence on the number of measurements carried out. Using numerical experiments, we demonstrate that the boundary between coexisting regimes can be spotted correctly using as few as 30 measurements.
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Yeshwanth, Sunil
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Core Title
Out-of-equilibrium dynamics of inhomogeneous quantum systems
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College of Letters, Arts and Sciences
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Doctor of Philosophy
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Physics
Publication Date
03/15/2016
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11/12/2015
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