Bound States and Recurrence Properties of Quantum Walks

Transcription

1 Bound States and Recurrence Properties of Quantum Walks Autrans Albert H. Werner Joint work with: Andre Ahlbrecht, Christopher Cedzich, Volkher B. Scholz (now ETH), Reinhard F. Werner (Hannover) Andrea Alberti & Dieter Meschede (Bonn) Alberto F. Grünbaum (Berkeley) Luis Velázquez (Zaragoza)

2 What are Quantum Walks? Dynamics of a single particle 01

3 What are Quantum Walks? Dynamics of a single particle with internal degree of freedom 01

4 What are Quantum Walks? Dynamics of a single particle with internal degree of freedom on a lattice 01

5 What are Quantum Walks? Dynamics of a single particle with internal degree of freedom on a lattice in discrete timesteps 01

6 What are Quantum Walks? Dynamics of a single particle with internal degree of freedom on a lattice in discrete timesteps strictly local 01

7 Why? Step towards quantum-simulators Simulation of lattice systems in discrete time steps Simulation of one particle-effects Quantum Biology quantization of random walks Searching in graphs Quantum computer source: wikipedia.org source: ucm.es 02

8 Experimental Realisations atom in optical lattice optical fibres source: iap.uni-bonn.de/ phase space of trapped ions source: Schreiber et al. (2011) wave guide arrays source: Matjeschk et al. (2012) source: Peruzzo et al. (2010) 03

9 Outline Propagation properties Bound states in interacting quantum walks Recurrence properties 04

10 Outline Propagation properties Bound states in interacting quantum walks Recurrence properties 04

11 1D Example: Coined Quantum Walk 05

12 1D Example: Coined Quantum Walk 05

13 1D Example: Coined Quantum Walk 05

14 1D Example: Coined Quantum Walk Hilbert space: 05

15 1D Example: Coined Quantum Walk Basis: Hilbert space: 05

16 1D Example: Coined Quantum Walk Basis: Walk operator: Hilbert space: 05

17 1D Example: Coined Quantum Walk Basis: Walk operator: Time evolution: Hilbert space: 05

18 1D Example: Coined Quantum Walk Basis: Walk operator: Time evolution: Hilbert space: 05

19 1D Example: Coined Quantum Walk Basis: Coin operator: Walk operator: Time evolution: Hilbert space: 05

20 1D Example: Coined Quantum Walk Basis: Coin operator: Walk operator: Time evolution: U Hilbert space: 05

21 1D Example: Coined Quantum Walk Basis: Coin operator: Walk operator: Shift operator: Time evolution: U Hilbert space: 05

22 1D Example: Coined Quantum Walk Basis: Coin operator: Walk operator: Shift operator: Time evolution: S U Hilbert space: 05

23 1D Example: Coined Quantum Walk Basis: Coin operator: Walk operator: Shift operator: Time evolution: S U Hilbert space: 05

24 1D Example: Coined Quantum Walk Basis: Coin operator: Walk operator: Shift operator: Time evolution: S U Hilbert space: 05

25 1D Example: Coined Quantum Walk Basis: Coin operator: Walk operator: Shift operator: Time evolution: Hilbert space: 05

26 1D Example: Coined Quantum Walk Basis: Coin operator: Walk operator: Shift operator: Time evolution: Hilbert space: 05

27 Example: Hadamard Walk Walk operator Hadamard Coin Initial state 200 time steps 06

28 Asymptotic position distribution Position observable Characteristic function of Find minimal for the existence of Ballistic scaling Diffusive scaling 07

29 1D Example: Hadamard Walk 08

30 coherence Propagation properties translation invariance 09

31 coherence Propagation properties translation invariance Ballistic transport Anderson localisation Diffusive transport Diffusive transport 09

32 Outline Propagation properties Bound states in interacting quantum walks Recurrence properties 10

33 Outline Propagation properties Bound states in interacting quantum walks Recurrence properties 10

34 1D Example: Coined Quantum Walk Basis: Coin operator: Walk operator: Shift operator: Time evolution: Hilbert space: 11

35 1D Example: Coined Quantum Walk Basis: Coin operator: Walk operator: Shift operator: Time evolution: Hilbert space: 11

36 Interacting Quantum Walks Two particles on the line Free evolution: Projection on collision space: Interaction on collision: Coin on collision: 12

37 Example: Interacting Hadamard Walk Two particles on the line Free evolution: Projector on collision space: Interaction on collision: Interaction phase: Initial state: Walk preserves symmetric/antisymmetric subspaces 12

38 Interacting Hadamard Walk Free evolution Interaction 13

39 Fourier Description I Walk operator Free evolution 14

40 Fourier Description II Write Walk operator in terms of and 15

41 Fourier Description II Write Walk operator in terms of and Conserved by translation invariance. External parameter 15

42 Fourier Description II Write Walk operator in terms of and Conserved by translation invariance. External parameter Walk in this variable is perturbed by on subspace of - constant functions Family of 1D QWs with perturbation at the origin indexed by 15

43 Interacting Hadamard Walk Jointly diagonalize on a ring of length L and L=8 L=32 Compare with free band structure depth=p 1 -p 2

44 Quantum Walk with Point Perturbation projection onto the subspace Finite rank perturbation essential spectrum unchanged Look for eigenvalues independent of Look for eigenvalues in band gap of Consistency condition: 16

45 Interacting Quantum Walks Result: For all values in the band gap, there is an interaction such that is an eigenvalue of. The corresponding eigenvectors satisfy 17

46 Example: Interacting Hadamard Walk Two particles on the line Free evolution: Projection on collision space: Interaction on collision: Interaction phase: Initial state: Walk preserves symmetric/antisymmetric subspaces 17

47 Interacting Hadamard Walk Result: Explicit formula for quasi-energy of the bound state. A.Ahlbrecht, A. Alberti, D.Meschede, V.B.Scholz, AHW, R.F. Werner New J. Phys. 14 (2012) Y.Lahini, M.Verbin, S.D.Huber, Y.Bromberg, R.Pugatch, Y.Silberberg; Phys. Rev. A 86, (2012) A.Schreiber, A.Gábris, P.Rohde, K.Laiho, M.Štefaňák,V.Potoček, C.Hamilton,I.Jex, C.Silberhorn Science (2012) 18

48 Interacting Hadamard Walk Result: Explicit formula for quasi-energy of the bound state. Effective theory of molecule as QW A.Ahlbrecht, A. Alberti, D.Meschede, V.B.Scholz, AHW, R.F. WernerNew J. Phys. 14 (2012) Y.Lahini, M.Verbin, S.D.Huber, Y.Bromberg, R.Pugatch, Y.Silberberg; Phys. Rev. A 86, (2012) A.Schreiber, A.Gábris, P.Rohde, K.Laiho, M.Štefaňák,V.Potoček, C.Hamilton,I.Jex, C.Silberhorn Science (2012) 18

49 Interacting Hadamard Walk Result: Explicit formula for quasi-energy of the bound state. Effective theory of molecule as QW A.Ahlbrecht, A. Alberti, D.Meschede, V.B.Scholz, AHW, R.F. WernerNew J. Phys. 14 (2012) Y.Lahini, M.Verbin, S.D.Huber, Y.Bromberg, R.Pugatch, Y.Silberberg; Phys. Rev. A 86, (2012) A.Schreiber, A.Gábris, P.Rohde, K.Laiho, M.Štefaňák,V.Potoček, C.Hamilton,I.Jex, C.Silberhorn Science (2012) 18

50 Interacting Hadamard Walk Result: Explicit formula for quasi-energy of the bound state. Effective theory of molecule as QW Molecule exponentially localized A.Ahlbrecht, A. Alberti, D.Meschede, V.B.Scholz, AHW, R.F. WernerNew J. Phys. 14 (2012) Y.Lahini, M.Verbin, S.D.Huber, Y.Bromberg, R.Pugatch, Y.Silberberg; Phys. Rev. A 86, (2012) A.Schreiber, A.Gábris, P.Rohde, K.Laiho, M.Štefaňák,V.Potoček, C.Hamilton,I.Jex, C.Silberhorn Science (2012) 18

51 Outline Propagation properties Bound states in interacting quantum walks Recurrence properties 19

52 Outline Propagation properties Bound states in interacting quantum walks Recurrence properties 19

53 Recurrence in Random Walks George Pólya Does the walker return with certainty? Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz recurrent transient Georg Pólya; Mathematische Annalen 84(2), (1921) 20

54 Markov Process Countable state space Transition matrix Probability to move from to Trajectory Fix initial state Probability to return to in exactly steps Georg Pólya; Mathematische Annalen 84(2), (1921) 21

55 Return Probabilities Return in exactly steps: First return after exactly steps (conditioned) Generating functions Recurrence: 22

56 Renewal Equation Renewal equation: Recurrence criterium: Polya criterium: 23

57 Recurrence in Time Discrete Quantum Systems Scenario: separable Hilbert space unitary operator evolution:, Question: Given, does the system return with certainty to this initial state? 24

58 Return Amplitudes Return after exactly steps Generating function Conceptional problem: First return probabilities. Idea: Use renewal equation 25

59 Return Amplitudes Return after exactly steps Generating function Conceptional problem: First return probabilities. Idea: Use renewal equation 25

60 Simple Counter Example time step 26

61 Simple Counter Example Way out: Directly use classical Polya criterium for time step M. Stefanak, I. Jex, T. Kiss Phys. Rev. Lett. 100, (2008) 26

62 Operational Approach Test for return in each time step projective measurement: Modified dynamics: 1. Unitary time step 2. Measurement: System in state? Yes System returned. End of experiment. No System in state 27

63 First Return Amplitudes First return after steps Generating function Total return probability Definition: recurrent iff. 28

64 Renewal Equation Generating function and differ by rank perturbation: Krein Formula Identify scalar product on RHS with Renewal-equation 29

65 Random Walk vs. Quantum Case Random Walk: probabilities Quantum Case: amplitudes Return First return Return probability Renewal equation F.A. Grünbaum, L. Velázquez,, R. F. Werner; Com. Math. Phys. 320(2) (2013) 30

66 Return Probability of 1D Quantum Walks 31

67 Return Probability of 1D Quantum Walks Critical dimension in quantum case:. 31

68 Recurrence Criteria Characterization in terms of spectral measure For matrices 32

69 Recurrence Criteria Characterization in terms of spectral measure For matrices pure point singular continuous absolutely continuous 32

70 Recurrence Criteria Characterization in terms of spectral measure Theorem: is recurrent iff has no absolutely continuous component. pure point singular continuous absolutely continuous F.A. Grünbaum, L. Velázquez,, R. F. Werner; Com. Math. Phys. 320(2) (2013) 32

71 RAGE Theorem sequence of compact operators strongly convergent to the identity, unitary operator pure point singular continuous absolutely continuous 33

72 Comparison Recurrence is recurrent iff contains no absolutely continuous component RAGE theorem is localized iff the spectral measure pure point is Distinguishes between singular and nonsingular spectrum Distinguishes between continuous and pure point spectrum 34

73 Proof idea I: Measures on the unit circle Given a probability measure on, the unit circle, define for with two analytic functions Stieltjes function: Schur function : Boundary behaviour of and for characterizes. Theorem: The absolutely continuous part of is supported on the subset of, where. 35

74 Proof idea II Identify RHS of renewal equation with Schur function For to be recurrent we need Using this implies for the Schur function Since bounded by we need for almost all, which is equivalent to having no absolutely continuous part. 36

75 Expected Return Time Given first return amplitudes Consider expected return time Result: If the pair is recurrent, the expected return time is infinite or an integer! counts point masses in. Proof idea: Identify with winding number of the phase of the Schur function on the unit circle 37

76 Expected return time 38

77 coherence Summary translation invariance Pure Point spectrum Singular continuous spectrum absolutely continuous spectrum

78 coherence Summary translation invariance Thank you for your attention! Pure Point spectrum Singular continuous spectrum absolutely continuous spectrum

79 References I A. Ahlbrecht, V.B. Scholz, ; J. Math. Phys. 52, (2011) A. Ahlbrecht, H. Vogts, AHW, and R. F. Werner J. Math. Phys. 52, (2011) G. Grimmett, S. Janson, P.F. Scudo; Phys. Rev. E, 69, (2004). F.A. Grünbaum, L. Velázquez,, R. F. Werner; Com. Math. Phys. 320(2) (2013) A. Joye; CMP 307(1) (2011) A. Joye, M. Merkli; J. Stat. Phys. 140(6) (2010) M. Karski, L. Förster, JM. Choi, A. Steffen, W. Alt, D. Meschede, A. Widera; Science 325 (2009) N. Konno, J. Math. Soc. Japan Volume 57, Number 4 (2005) R. Matjeschk, A. Ahlbrecht, M. Enderlein, Ch. Cedzich,, M. Keyl, T. Schaetz, R. F. Werner; Phys. Rev. Lett. 109, (2012) A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O'Brien; Science, 329(5998) (2010) G. Pólya; Mathematische Annalen 84(2), (1921)

80 References II A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, C. Silberhorn Phys. Rev. Lett. 106, (2011) H. Schmitz, R. Matjeschk, Ch. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz; Phys. Rev. Lett. 103, (2009) ucm.es: Zugriff: wikipedia.org: Zugriff iap.uni-bonn.de: Zugriff F. Zähringer, G. Kirchmair,, R. Gerritsma, E. Solano, R. Blatt and C. F. Roos; Phys. Rev. Lett. 104, (2010)