Bound States and Recurrence Properties of Quantum Walks

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)

Autrans

18.07.2013

What are Quantum Walks?

a single particle

A. H. Werner

Dynamics of

01

What are Quantum Walks?

a single particle

with internal degree

of freedom

A. H. Werner

Dynamics of

01

What are Quantum Walks?

a single particle

with internal degree

of freedom

on a lattice

A. H. Werner

Dynamics of

01

What are Quantum Walks?

a single particle

with internal degree

of freedom

on a lattice

in discrete timesteps

A. H. Werner

Dynamics of

01

What are Quantum Walks?

a single particle

with internal degree

of freedom

on a lattice

in discrete timesteps

strictly local

A. H. Werner

Dynamics of

01

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

02

source: ucm.es

source: wikipedia.org

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phase space of

trapped ions

Experimental Realisations

wave guide arrays

optical fibres

atom in

optical lattice

03

source: iap.uni-bonn.de/

source: Peruzzo et al. (2010) source: Matjeschk et al. (2012)

source: Schreiber et al. (2011)

A. H. Werner

Outline

Propagation properties

Bound states in interacting quantum walks

Recurrence properties

04 A. H. Werner

Outline

Propagation properties

Bound states in interacting quantum walks

Recurrence properties

04 A. H. Werner

1D Example: Coined Quantum Walk

05 A. H. Werner

1D Example: Coined Quantum Walk

05 A. H. Werner

1D Example: Coined Quantum Walk

05 A. H. Werner

1D Example: Coined Quantum Walk

Hilbert space: 05 A. H. Werner

1D Example: Coined Quantum Walk

Basis:  

Hilbert space: 05 A. H. Werner

1D Example: Coined Quantum Walk

Basis:  

Walk operator:

Hilbert space: 05 A. H. Werner

1D Example: Coined Quantum Walk

Basis:  

Walk operator:

Time evolution:

Hilbert space: 05 A. H. Werner

1D Example: Coined Quantum Walk

Basis:  

Walk operator:

Time evolution:

Hilbert space: 05 A. H. Werner

1D Example: Coined Quantum Walk

Basis:  

Walk operator:

Time evolution:

Coin operator:  

Hilbert space: 05 A. H. Werner

1D Example: Coined Quantum Walk

U

Basis:  

Walk operator:

Time evolution:

Coin operator:  

Hilbert space: 05 A. H. Werner

1D Example: Coined Quantum Walk

U

Basis:  

Walk operator:

Time evolution:

Coin operator:  

Shift operator:

Hilbert space: 05 A. H. Werner

1D Example: Coined Quantum Walk

U

S

Basis:  

Walk operator:

Time evolution:

Coin operator:  

Shift operator:

Hilbert space: 05 A. H. Werner

1D Example: Coined Quantum Walk

U

S

Basis:  

Walk operator:

Time evolution:

Coin operator:  

Shift operator:

Hilbert space: 05 A. H. Werner

1D Example: Coined Quantum Walk

U

S

Basis:  

Walk operator:

Time evolution:

Coin operator:  

Shift operator:

Hilbert space: 05 A. H. Werner

1D Example: Coined Quantum Walk

Basis:  

Walk operator:

Time evolution:

Coin operator:  

Shift operator:

Hilbert space: 05 A. H. Werner

1D Example: Coined Quantum Walk

Basis:  

Walk operator:

Time evolution:

Coin operator:  

Shift operator:

Hilbert space: 05 A. H. Werner

Example: Hadamard Walk

Walk operator

Hadamard Coin

Initial state

06

200 time steps

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Ballistic scaling Diffusive scaling

Asymptotic position distribution

Position observable  

Characteristic function of

Find minimal   for the existence of

07 A. H. Werner

1D Example: Hadamard Walk

08 A. H. Werner

Propagation properties

translation invariance

co

here

nce

09 A. H. Werner

Propagation properties

translation invariance

co

here

nce

Ballistic transport

Anderson localisation

Diffusive transport

Diffusive transport

09 A. H. Werner

Outline

Propagation properties

Bound states in interacting quantum walks

Recurrence properties

10 A. H. Werner

Outline

Propagation properties

Bound states in interacting quantum walks

Recurrence properties

10 A. H. Werner

1D Example: Coined Quantum Walk

Basis:  

Walk operator:

Time evolution:

Coin operator:  

Shift operator:

Hilbert space: 11 A. H. Werner

1D Example: Coined Quantum Walk

Basis:  

Walk operator:

Time evolution:

Coin operator:  

Shift operator:

Hilbert space: 11 A. H. Werner

Interacting Quantum Walks

Two particles on the line

Free evolution:  

Projection on collision space:  

Interaction on collision:

Coin on collision:  

12 A. H. Werner

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 A. H. Werner

Interacting Hadamard Walk

Free evolution Interaction

13 A. H. Werner

Fourier Description I

Walk operator

Free evolution

14 A. H. Werner

Fourier Description II

Write Walk operator in terms of   and  

15 A. H. Werner

Fourier Description II

Write Walk operator in terms of   and  

Conserved by

translation invariance.

External parameter

15 A. H. Werner

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

15

Family of 1D QWs with perturbation

at the origin indexed by

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Interacting Hadamard Walk

Compare with

free band structure

depth=p1-p2

Jointly diagonalize   and  

on a ring of length L

L=8 L=32

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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 A. H. Werner

Interacting Quantum Walks

17 A. H. Werner

Result: For all values   in the band gap, there is an

interaction   such that   is an eigenvalue of   .

The corresponding eigenvectors   satisfy

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 A. H. Werner

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)

A. H. Werner

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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)

A. H. Werner

18

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)

A. H. Werner

18

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)

A. H. Werner

18

Outline

Propagation properties

Bound states in interacting quantum walks

Recurrence properties

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19

Outline

Propagation properties

Bound states in interacting quantum walks

Recurrence properties

A. H. Werner

19

Recurrence in Random Walks

„Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend

die Irrfahrt im Straßennetz“

George Pólya Does the walker return with certainty?

recurrent transient

20

Georg Pólya; Mathematische Annalen 84(2), (1921)

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Markov Process

21

Georg Pólya; Mathematische Annalen 84(2), (1921)

Countable state space  

Transition matrix  

Probability to move from   to    

Trajectory  

Fix initial state  

Probability to return to   in exactly   steps

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Return Probabilities

Return in exactly   steps:

First return after exactly

  steps (conditioned)

Generating functions

Recurrence:

22 A. H. Werner

Renewal Equation

Recurrence criterium:

Polya criterium:

23

Renewal equation:

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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 A. H. Werner

Return Amplitudes

Return after exactly   steps

Generating function

Conceptional problem: First return probabilities   .

25

Idea: Use renewal equation

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Return Amplitudes

Return after exactly   steps

Generating function

Conceptional problem: First return probabilities   .

25

Idea: Use renewal equation

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Simple Counter Example

26

time step

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Simple Counter Example

26

time step

Way out:

Directly use classical Polya criterium for  

M. Stefanak, I. Jex, T. Kiss Phys. Rev. Lett. 100, 020501 (2008)

A. H. Werner

Operational Approach

Test for return in each time step

projective measurement:

Modified dynamics:

1. Unitary time step  

2. Measurement: System in state   ?

Yes No

System returned.

End of experiment.

System in state

27 A. H. Werner

First Return Amplitudes

First return after   steps

Generating function

Total return probability

Definition:   recurrent iff   .

28 A. H. Werner

Renewal Equation

Generating function

  and   differ by rank   perturbation: Krein Formula

Identify scalar product on RHS with  

29

Renewal-equation

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Random Walk vs. Quantum Case

Random Walk:

probabilities

Quantum Case:

amplitudes

Return

First return

Return probability

Renewal equation

30

F.A. Grünbaum, L. Velázquez, A. H. Werner, R. F. Werner; Com. Math. Phys. 320(2) (2013)

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Return Probability of 1D Quantum Walks

31 A. H. Werner

Return Probability of 1D Quantum Walks

31

Critical dimension in quantum case:   .

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Recurrence Criteria

Characterization in terms of spectral measure  

For matrices

32 A. H. Werner

Recurrence Criteria

Characterization in terms of spectral measure  

For matrices

pure point singular continuous absolutely continuous

32 A. H. Werner

Recurrence Criteria

Characterization in terms of spectral measure  

Theorem:   is recurrent iff   has no absolutely

continuous component.

F.A. Grünbaum, L. Velázquez, A. H. Werner, R. F. Werner; Com. Math. Phys. 320(2) (2013)

32

pure point singular continuous absolutely continuous

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RAGE Theorem

  sequence of compact operators strongly convergent to

the identity,   unitary operator

33

pure point singular continuous absolutely continuous

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Comparison

  is recurrent iff

  contains no

absolutely continuous

component

Distinguishes between

singular and non-

singular spectrum

  is localized iff the

spectral measure   is

pure point

Distinguishes between

continuous and pure

point spectrum

Recurrence RAGE theorem

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34

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   .

A. H. Werner

35

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.

A. H. Werner

36

Expected Return Time

Given first return amplitudes

Consider expected return time

Proof idea: Identify   with winding number of the phase of the

Schur function on the unit circle

Result: If the pair   is recurrent, the expected

return time   is infinite or an integer!

  counts point masses in   .

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37

Expected return time

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38

Summary translation invariance

co

here

nce

Pure Point

spectrum

Singular continuous

spectrum

absolutely continuous

spectrum

A. H. Werner

Summary translation invariance

co

here

nce

Pure Point

spectrum

Singular continuous

spectrum

absolutely continuous

spectrum

A. H. Werner

Thank you for your

attention!

References I

A. Ahlbrecht, V.B. Scholz, A. H. Werner; J. Math. Phys. 52, 102201 (2011)

A. Ahlbrecht, H. Vogts, AHW, and R. F. Werner J. Math. Phys. 52, 042201 (2011)

G. Grimmett, S. Janson, P.F. Scudo; Phys. Rev. E, 69, 026119 (2004).

F.A. Grünbaum, L. Velázquez, A. H. Werner, 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, A. H. Werner, M. Keyl, T.

Schaetz, R. F. Werner; Phys. Rev. Lett. 109, 240503 (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)

A. H. Werner

References II

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, C. Silberhorn Phys.

Rev. Lett. 106, 180403 (2011)

H. Schmitz, R. Matjeschk, Ch. Schneider, J. Glueckert, M. Enderlein, T. Huber,

and T. Schaetz; Phys. Rev. Lett. 103, 090504 (2009)

ucm.es: http://pendientedemigracion.ucm.es/info/giccucm/index.php/Quantum_Computation.html

Zugriff: 03.07.13

wikipedia.org: http://en.wikipedia.org/wiki/D-Wave_Systems Zugriff. 03.07.13

iap.uni-bonn.de: http://quantum-technologies.iap.uni-bonn.de/ Zugriff. 03.07.13

F. Zähringer, G. Kirchmair,, R. Gerritsma, E. Solano, R. Blatt and C. F. Roos;

Phys. Rev. Lett. 104, 100503 (2010)

A. H. Werner