Quantum random walks Andre Kochanke Planck-Institute of Quantum Optics 7/27/2011
Quantum random walks
Jan 01, 2016
Quantum random walks. Andre Kochanke. 7/27/2011. Max-Planck-Institute of Quantum Optics. Motivation. Motivation. ?. ?. ?. ?. Overview. Density matrix formalism Randomness in quantum mechanics Transition from classical to quantum walks Experimental realisation. - PowerPoint PPT Presentation
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Quantum random walks
Andre Kochanke
Max-Planck-Institute of Quantum Optics 7/27/2011
2
Motivation
a.kochanke
what are we about to hear, so the following slides make sense at allmaybe: start with bean machine and then atom into the machine????gaussian at the bottom! BUT USE BINS
Motivation
3
?
??
?
a.kochanke
what are we about to hear, so the following slides make sense at allmaybe: start with bean machine and then atom into the machine????
4
Overview
• Density matrix formalismRandomness in quantum mechanics
• Transition from classical to quantum walks
• Experimental realisation
5
Density matrix approach
• Two state system 12
12
-1 10
a.kochanke
insert blochspheretrace?
6
Density matrix approach
• Two state system
• Density operator
12
12
0-1 1
a.kochanke
insert blochspheretrace?
7
Density matrix approach
• Density operator
Pure state Mixed state
12
12
0-1 1
a.kochanke
insert blochspheretrace?
8
Galton box
• Binomial distribution
a.kochanke
put in animation of the dots according to the density matrix.
9
Galton box
• Statistical mixture
• First four steps
a.kochanke
highlightingmehr matrizen bis lvl 5?
10
Quantum analogy
• Used Hilbert space
• Specify subspaces
0-1-2-3 1 2 3
11
Quantum analogy
• Evolution with shift and coin operators
0-1-2-3 1 2 3
12
Quantum analogy
• Evolution with shift and coin operators
0-1-2-3 1 2 3
13
Quantum analogy
• Evolution with shift and coin operators
0-1-2-3 1 2 3
14
Quantum analogy
• State transformation
• Density matrix transformation
15
Quantum analogy
Quantum analogy
16
Position
pc
pq
Variances
pc
pq
pc
pq
Position
Position
100 steps
17
• Phase shift• Transformed density matrix
• Average
• Decoherence effect
Decoherence
18
Different realisations
• C. A. Ryan et al., “Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor”, PRA 72, 062317 (2005)
• M. Karski et al., “Quantum Walk in Position Space with Single Optically Trapped Atoms”, Science 325, 174 (2009)
• A. Schreiber et al., “Photons Walking the Line: A Quantum Walk with Adjustable Coin Operations”, PRL 104, 050502 (2010)
• F. Zähringer et al., “Realization of a Quantum Walk with One and Two Trapped Ions”, PRL 104, 100503 (2010)
19
SetupCCD
Microwave
Dipole trap laser
Objective
Fluorescence picture
Cs
jF = 4;mF =4i
jF = 3;mF =3i
Mic
row
ave
M. Karski et al., Science 325, 174 (2009)
a.kochanke
new pic, focus on the standing wave trap!
20
Setup
Polarizations and
21
Setup
Polarizations and
22
Results
M. Karski et al., Science 325, 174 (2009)
23
Results
M. Karski et al., Science 325, 174 (2009)
Theoretical expectation
6 steps
24
Results
Theoretical expectationM. Karski et al., Science 325, 174 (2009)
6 steps
25
Results
Theoretical expectation
26
Results
Theoretical expectationM. Karski et al., Science 325, 174 (2009)
27
Results
Gaussian fitM. Karski et al., Science 325, 174 (2009)
28
Conclusion
• The density matrix formalism allows you to describe cassical and quantum behavior
• Karski et al. showed how to prepare a quantum walk with delocalized atoms
• The quantum random walk is not random at all
M. Karski et al., Science 325, 174 (2009)
29
30
References
• C. A. Ryan et al., “Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor”, PRA 72, 062317 (2005)
• M. Karski et al., “Quantum Walk in Position Space with Single Optically Trapped Atoms”, Science 325, 174 (2009)
• SOM for “Quantum Walk in Position Space with Single Optically Trapped Atoms”, Science 325, 174 (2009)
• A. Schreiber et al., “Photons Walking the Line: A Quantum Walk with Adjustable Coin Operations”, PRL 104, 050502 (2010)
• F. Zähringer et al., “Realization of a QuantumWalk with One and Two Trapped Ions”, PRL 104, 100503 (2010)
• M. Karksi, „State-selective transport of single neutral atoms”, Dissertation, Bonn (2010)• C. C. Gerry and P. L. Knight, „Introductory Quantum Optics“, Cambridge University Press,
Cambridge (2005)• M. A. Nielsen and I. A. Chuang, „Quantum Computation and Quantum Information“,
Cambridge University Press, Cambridge (2000)
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