Nonlinear dynamics study in periodically driven quantum systems

Nonlinear Dynamics Study in Periodically Driven Quantum Systems HAILONG WANG (B.S.), LanZhou University A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE May 2014 DECLARATION I hereby declare that the thesis in it are my original work and it has been written by me in its entirety I have duly acknowledged all the sources of information which have been used in the thesis This thesis has also not been submitted for any degree in any university previously Hailong Wang 10 May 2014 ii Acknowledgements Many people have contributed to this dissertation in different ways, I wish to use this opportunity to extend my sincerest thanks and appreciations to them Without their help, this thesis could never been possible First and foremost, I would like to express my deepest gratitude to my supervisor A/Prof Jiangbin Gong for his invaluable supervision, guidance and patience His enthusiasm and insight have motivated me throughout the whole course of my PhD studies I enjoyed the great benefit of his inspiring discussions, and sharing of knowledge He always encouraged me to think independently and I am really grateful for the academic freedom he gave me I would also like to thank Derek Ho for his generous and enthusiastic help during the early parts of my research projects He is not only a good friend for sharing the joy of life but also a perfect research collaborator I learned a lot from the cooperation with him during my PhD studies My thesis would also be in no part possible without the help of Prof Yuanping Feng His warmhearted help in the early time of my graduate study is really important I am greatly encouraged by his continuing care and help I have benefited from much input in the final editing of the current manuscript Longwen read the entire thesis twice and offered invaluable critiques Wang Chen read the second chapter and caught several typos and inaccuracies I would also like to thank my fellow group members - Derek, Longwen, Dr Adam, Qifang, Yon Shin, Gaoyang, Dayang and Shencheng - for sharing the joy of learning physics Special thanks to my friends - Taolin, Hou Ruizheng, Tang Qinling, Wang Chen, Zhu Feng, Liusha, Feng Lin, Qin Chu, Zhu Guimei, Yang Lina, etc Thanks to their efforts, we have made a joyful life in the past few years Finally, my up-most appreciation goes to my beloved parents, who provide their unconditional love and support in every aspects of my life iii Contents DECLARATION ii Acknowledgements iii Contents iv List of Tables ix List of Figures xi Introduction 1.1 Overview 1.2 Quantum resonant kicked rotor 1.3 Extended kicked rotor with an extra phase parameter 1.4 Pumping in a class of kicked rotor systems 1.5 Thesis outline Kicked Rotor Systems 2.1 Overview 2.2 Periodically driven quantum systems 2.2.1 Floquet theorem 2.2.2 Stroboscopic observations and the Floquet operator 2.2.3 Periodically kicked systems 2.3 Kicked rotor model 2.3.1 Classical kicked rotor 2.3.2 Quantum kicked rotor 2.3.3 Mapping onto the Anderson model 2.4 Experimental realizations and achievements 2.5 Kicked rotor variants 2.5.1 Double-kicked rotor model 2.5.2 Kicked Harper model 2.5.3 Extended kicked rotor model with an extra phase parameter Quantum Resonance and Spinor Representation v 1 7 8 10 11 11 15 16 18 20 20 21 24 25 Contents vi 3.1 Overview 3.2 Quantum resonance and translation symmetry 3.2.1 Translation symmetry 3.2.2 Quantum resonance in kicked rotor model 3.3 Extended Hilbert space and spinor representation 3.3.1 Bloch theorem 3.3.2 Spinor representation Analytical and Numerical Studies of Floquet Bands 4.1 Reduced Floquet matrix 4.2 Floquet band structure 4.2.1 Analytical solvable cases 4.2.2 Numerical study 4.3 Quasienergy spectrum 4.3.1 Quasienergy spectrum vs k 4.3.2 Quasienergy spectrum vs ⌧ 4.4 Floquet band eigenstates 4.4.1 A few remarks 4.4.2 Numerical study 25 26 27 28 29 30 34 39 39 42 43 44 48 48 49 52 53 54 Exponential Quantum Spreading in ORDKR under near Resonance Condition 5.1 Overview 5.2 ORDKR under high-order quantum resonance conditions 5.2.1 ORDKR 5.2.2 Band representation 5.2.3 Some spectral properties 5.3 EQS in ORDKR tuned near quantum resonance 5.3.1 Numerical results 5.3.2 Theoretical analysis 5.3.2.1 Single-band approximation 5.3.2.2 Pseudoclassical approximation 5.3.2.3 Quantitative investigation of EQS rates 5.4 Concluding remarks 57 57 59 60 60 64 67 67 69 69 72 75 80 Kicked-Harper model vs On-Resonance Double Kicked Rotor Model: From Spectral Difference to Topological Equivalence 6.1 Overview 6.2 KHM and ORDKR 6.3 Spectral differences and their dynamical implications 6.3.1 Summary of main numerical findings 6.3.2 Flat band and band symmetry in ORDKR 6.3.3 A theoretical bandwidth result and its dynamical consequence 6.4 Topological equivalence between ORDKR and KHM 6.4.1 Motivation and notation 81 81 82 85 85 89 91 94 95 vi Contents vii 6.4.2 Numerical findings 96 6.4.3 Proof of topological equivalence 98 6.5 Concluding remarks 105 Adiabatic Pumping and Non-adiabatic Pumping in a class of Kicked Rotor Models 107 7.1 Overview 107 7.2 Extended kicked rotor model and spinor representation 108 7.3 Adiabatic pumping 110 7.3.1 Numerical study 110 7.3.2 Theoretical analysis 112 7.3.3 Topological phase and quantized pumping 115 7.3.4 |0i-state and geometric pumping 117 7.4 Non-adiabatic pumping 121 7.5 Concluding remarks 125 Thesis Conclusions and Future Perspectives 127 8.1 Thesis conclusions 127 8.2 Future perspectives 128 Bibliography 129 Appendix A 137 Resonant Floquet operators in the spinor representation 137 A solvable case with ⌧ = 2⇡/3 138 Proof of the Existence of Flat band 139 Appendix B Expressions for reduced Floquet matrices 4.1 Reduced Floquet matrix for ORDKR 4.2 Reduced Floquet matrix for KHM Calculation of symmetric B matrix vii 141 141 141 143 145 List of Tables 4.1 4.2 4.3 4.4 4.5 4.6 case N = case N = critical k for KRM critical k for KHM critical k for ORDKR critical k for 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˜ l) representation can be found from matrix elements of the latter operators These are explicitly calculated in this Appendix h#, ˜ M⌧ |#0 , ˜0 i = l| ˆ l +1 X l= h#, ˜ l|lihl|#0 , ˜0 ie l i⌧ l2 /2 (1) Using Eq (5.3), replacing ⌧ = 2⇡p/q , l = lq + ˜00 , and summing over l, ˜00 , one finds: l l h#, ˜ M⌧ |#0 , ˜0 i = (# l| ˆ l where the matrix M˜˜0 = ll ˜ ˜0 l,l #0 (2) ⇡pq)M˜˜0 ll exp( i⇡p˜2 /2) l In a completely similar manner one finds: h#, ˜ V |#0 , ˜0 i = (# l| ˆ l where: #0 )V˜˜0 (#) ll q V˜˜0 (#) = ll 1X e q ik cos(2⇡j/q+#/q) j=0 137 e i(˜ ˜0 )(2⇡j/q+#/q) l l (3) (4) For ⌧ = 2⇡p/q with both p and q being odd integer, we have (ordkr) U˜˜0 (#) ll =e i⌧ ˜2 /2 l q X e i⌧ ˜002 /2 i⇡(˜ ˜00 ) l l l e ˜00 =0 l q 1 X ik1 cos(2⇡j1 /q+#/q) i(2⇡j1 /q+#/q)(˜0 l e e q ˜00 ) l (5) j1 =0 N 1X e q ik2 cos(2⇡j2 /q+#/q) i(2⇡j2 /q+#/q)(˜00 ˜ l l) e j2 =0 A solvable case with ⌧ = 2⇡/3 For ORDKR under the resonance condition ⌧ = 2⇡/3, there are three Floquet bands and all of them can be analytically solved The eigenphases are found to be ⌦1 = ⌦, (6) ⌦0 = 0, ⌦ = ⌦, where ⌦ = cos (⌘), ✓ ◆ ⌘= cos + cos cos 3↵ , 3 k # ↵ = cos , p 3k # = sin (7) The eigenphase ⌦0 is independent of the Bloch phase # and so it gives a perfectly flat band The corresponding normalized eigenvector is (see sect.5.2.3 for notations): (0) u0 (#) = ⇥ ⇤ (0) u1 (#) = e i#/3 e 3i↵ cos( + ⇡/3) ⇥ ⇤ (0) u2 (#) = e 2i#/3 cos( ⇡/3) e 3i↵ = [6(1 ⌘)] 138 1/2 (8) .3 Proof of the Existence of Flat band Here we give a simple proof that whenever ⌧ = 2⇡p/q , with p and q odd and (ordkr) ˆ coprime integers, is an infinitely degenerate proper eigenvalue of U2⇡M/N , that genˆ (ordkr) erates a flat band in the band spectrum of U2⇡M/N From Eq (4) it is immediate that : V(# + ⇡) = RV † (#)R , R˜˜ := jl ˜ ( j ˜ ˜ 1)l , Hence, defining a matrix M1 := RM, and using that M is diagonal, whenever pq is an odd integer, the factorization may be rewritten as follows: U (ordkr) (#) = M† V † (#) M1 V(#) (9) Next we introduce the following unitary matrices S(#): S˜˜(#) = jl p q ˜ ˜˜ eij#/q e2⇡ipj l/q For any given #, S(#) defines a rotation of spin axes that carries V(#) to diagonal form ˇ For a generic q ⇥ q matrix-valued function A(#) of quasi-position let A(#) denote the matrix S(#)† A(#)S(#) Then (9) yields: ˇ ˇ ˇ ˇ ˇ U (ordkr) (#) = M† V † (#) M1 V(#) (10) ˇ ˇ Straightforward calculation shows that V(#) is diagonal, and M1 (#) is symmetric Therefore, the last equation can be rewritten in the form: ˇ ˇ ˇ ˇ ˇ U (ordkr) (#) = M⇤ V ⇤ (#) M1 V(#) (11) where ⇤ denotes complex conjugation It is then immediate that: ˇ ˇ (U (ordkr) )⇤ (#) = W(#) U (ordkr) (#) W † (#) , (12) ˇ where W(#) = V(#)M1 Eq (12) says that the matrix U (ordkr) (#) is unitarily equivalent to its own complex conjugate, hence its spectrum is invariant under complex conjugation The same is then true of the spectrum of U (ordkr) (#) , that is indeed ˇ unitarily equivalent to U (ordkr) (#) by construction This proves symmetry of the spec- trum with respect to the zero eigenphase axis As the spectrum consists of q points on the unit circle (counting multiplicities), and q is odd, this symmetry entails that an 139 odd number of eigenvalues must be real, hence equal to ±1; on the other hand, from ˇ ˇ ˇ (11), det(U (ordkr) (#)) = |det(M1 (#)|2 |det(V(#))|2 = 1, so at most an even number of eigenvales may be equal to So is always an eigenvalue, independent of ✓, and this produces a flat band at zero eigenphase 140 Appendix B Expressions for reduced Floquet matrices For ~ = 2⇡M/N with M and N being coprime and odd integers, reduced N ⇥ N h ˜ Floquet matrix is given by U (') i = n,m P1 ˆ |m + l ⇥ N i eil' l= hn| U 4.1 Reduced Floquet matrix for ORDKR The Floquet operator of ORDKR is p2 UORDKR = ei 2~ e i K cos(q) ~ e ip 2~ e i L cos(q) ~ (13) The reduced N ⇥ N Floquet matrix is thus h ˜ UORDKR (') i n,m = X l= = X ˆ hn| UORDKR |m + l ⇥ N i eil' X N X l= l0 = m0 =0 p2 hn| ei 2~ e i K cos(q) ~ m0 + l ⇥ N ↵ ⌦ p2 L ⇥ m0 + l0 ⇥ N e i 2~ e i ~ cos(q) |m + l ⇥ N i eil' N 1 X ~ Z 2⇡ X K 0 = ei n d✓2 e i ~ cos(✓2 ) ei✓2 (m n) ei✓2 l N 2⇡ m0 =0 l0 = Z 2⇡ X L i ~ (m0 +l0 N )2 ⇥ e d✓1 e i ~ cos(✓1 ) ei✓1 (m m ) ei✓1 (l 2⇡ = N X m0 =0 ⇥ i ~ n2 e2 2⇡ e 2⇡ Z i ~ m02 l= 2⇡ d✓2 e i K cos(✓2 +⇡) i(✓2 +⇡)(m0 n) i⇡(m0 n) ~ e e Z 2⇡ d✓1 e i L cos(✓1 ) i✓1 (m m0 ) ~ e e i✓1 l0 N X l0 )N il' e ei(✓2 +⇡)l N l0 = l= 141 X (14) ei✓1 lN eil' To simplify, we make use of the Poisson summation formula X X e2⇡il' = (' (15) j), j= l= and obtain h ˜ UORDKR (') i n,m =e N X i ~ n2 i ~ m02 i⇡(m0 n) e e m0 =0 ⇥ N N 1 X i K cos( 2⇡ j2 N e ~ N ' N ) i( 2⇡ j2 N e ' N )(m0 n) j2 =0 N X ' N i L cos( 2⇡ j1 ~ N e ) i( 2⇡ j1 N e ' N )(m m0 ) j1 =0 N 1N 1N 1 X X X in ' = e Ne N (16) ' im N j2 =0 m =0 j1 =0 ⇥e ⇥e ~ i 2⇡2 ~ i 2⇡2 e i 2⇡ nj2 i K cos( 2⇡ j2 N ~ N e i 2⇡ m0 j1 N n2 m02 e e i L cos( 2⇡ j1 ~ N ' N ) i 2⇡ j2 m0 N e ' N ) i 2⇡ j1 m N e For the sake of illustration, we write the reduced Floquet matrix as a product of unitary matrices B B ˜ UORDKR (') = B @ B B B @ B B B @ e i 2⇡2 ~ 10 n2 2⇡ ~ 02 m CB CB C@ A 10 ei ' in N e CB CB C@ A B B B e @ C C C A 2⇡ e i N nj2 p N e i 2⇡ m0 j1 N p N ' im N CB K 2⇡ B CB ei ~ cos( N j2 A@ 10 CB CB AB @ C C C A ' N 10 ) e i L cos( 2⇡ j1 ~ N ' N CB CB C@ A 10 ) CB CB C@ A 2⇡ e i N j2 m p N 2⇡ N e i p j1 m N (17) 142 C C A C C A If we introduce an additional periodic phase parameter ↵ [0, 2⇡) to the ORDKR map, the Floquet operator becomes p2 UORDKR ↵ = ei 2~ e i K cos(q) ~ ip 2~ e e i L cos(q+↵) ~ (18) The corresponding reduced Floquet matrix is h ˜ UORDKR (', ↵) i n,m = N 1N 1N 1 X X X in ' e Ne N2 ' im N j2 =0 m =0 j1 =0 ⇥e i 2⇡2 ⇥ ei ~ e i 2⇡ nj2 i K cos( 2⇡ j2 N ~ N e i 2⇡ m0 j1 N n2 2⇡ ~ 02 m e e ' N (19) ) i 2⇡ j2 m0 N e ' N i L cos( 2⇡ j1 ~ N +↵) i 2⇡ j1 m N e Written again as a product of unitary matrices, B B ˜ UORDKR (', ↵) = B @ B B B @ B B B @ ' ein N 10 e i 2⇡2 ~ n2 2⇡ ~ 02 m CB CB C@ A 10 ei CB CB C@ A B B B e @ C C C A 2⇡ e i N nj2 p N e i 2⇡ m0 j1 N p N ' im N CB K 2⇡ B CB ei ~ cos( N j2 A@ 10 CB CB AB @ ' N 10 ) e i L cos( 2⇡ j1 ~ N ' N CB CB C@ A 10 +↵) C C C A 2⇡ e i N j2 m p N CB CB C@ A 2⇡ (20) The Floquet operator of KHM is i L cos(p) ~ e 143 i K cos(q) ~ , C C A N e i p j1 m N 4.2 Reduced Floquet matrix for KHM UKHM = e (21) C C A with reduced N ⇥ N Floquet matrix h ˜ UKHM (') i n,m X = l= 1 = e 2⇡ =e N = ˆ hn|UKHM |m + l ⇥ N ieil' i L cos(n~) ~ i L cos(n~) ~ N X N ' ein N e Z 2⇡ d✓e i K cos(✓) i✓(m n) ~ e N X X ei✓lN eil' l= e ' N i K cos( 2⇡ j ~ N ) i( 2⇡ j N e ' N (22) )(m n) j=0 i L cos(n~) ~ e i 2⇡ nj i K cos( 2⇡ j N ~ N e ' N ) i 2⇡ jm N e e ' im N j=0 For the sake of illustration, we write the reduced Floquet matrix as a product of unitary matrices B B ˜ UKHM (') = B @ B B B @ ' ein N i L cos(n~) ~ 10 e B B B e @ C C C A CB CB C@ A 2⇡ i e pN nj N ' im N CB B CB e A@ i K cos( 2⇡ j ~ N ' N 10 ) CB CB C@ A 2⇡ N eip jm N C C C A C C A (23) If we introduce an additional periodic phase parameter ↵ [0, 2⇡) to the KHM map, the Floquet operator becomes UKHM ↵ =e i L cos(p ↵) ~ e i K cos(q) ~ (24) The corresponding reduced Floquet matrix is h ˜ UKHM (', ↵) i n,m = N 1 X in ' e Ne N i L cos(n~ ↵) ~ j=0 144 e i 2⇡ nj i K cos( 2⇡ j N ~ N e ' N ) i 2⇡ jm N e e ' im N (25) Written as a product of unitary matrices, B B ˜ UKHM (', ↵) = B @ B B B @ e ' in N i L cos(n~ ↵) ~ B B B e @ 10 e C C C A CB CB C@ A 2⇡ i e pN nj N ' im N CB B CB e A@ i K cos( 2⇡ j ~ N ' N 10 ) C C C A CB CB C@ A 2⇡ N eip jm N (26) Calculation of symmetric B matrix D1 is a diagonal unitary matrix and F is a unitary matrix The corresponding matrix elements are [D1 ]n,n = ei 2⇡ ~ n indices m and n take values 0, 1, · · · , N , Fm,n = 2⇡ p ei N mn , N where ~ = 2⇡ M , and N We also assume k is an integer ranging ˜ from to N , and k is an integer ranging from to Q, with Q = (N 145 1)/2 From C C A B ⌘ F D1 F † and using the fact that M N is an odd number, we have Bm,n N X i 2⇡ [ M k2 +( N +m = e N N n)k] k=1 N 2⇡ M X = ( 1)k ei N [ k +(m N = N k=1 Q X n)k] ˜ +(m n)(2k)] ˜ 2⇡ M ei N [ (2k) ˜ k=1 Q X i 2⇡ [ M (N e N N ˜ ˜ 2k)2 +(m n)(N 2k)] ˜ k=1 i 2⇡ [ M N +(m n)N ] e N N Q 1 X i 2⇡ [2M k2 +2(m ˜ = + e N N N N = 1 + N N = + N N ˜ k=1 Q X 2⇡ (27) ˜ n)k] ˜2 + M N 2M N k+(m n)N 2(m n)k] ˜ ˜ ei N [2M k ˜ k=1 Q Xh ˜ k=1 Q X ˜ k=1 e ˜2 +2(m n)k) ˜ 2⇡ ei N (2M k i4⇡ M N ˜ k2 " 2⇡ ˜ k cos 4⇡ (m N # n) It is now seen that B is a symmetric matrix, i.e., Bm,n = Bn,m 146 ˜2 2(m n)k) ˜ + ei N (2M k i ... digesting quantum dynamics that is nevertheless in the deep quantum regime Therefore it provides us with a promising opportunity to study the quantum- classical correspondence in periodic driven quantum. .. limited to static systems, they can also appear in driven quantum systems In the last few years, the periodically driven quantum systems has been a fast-growing research field A variety of robust and... prominent model systems in the study of nonlinear dynamics The kicked rotor model was introduced by Casati, Chirikov, Izrailev and Ford [2], and later became one of the basic models to study quantum