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Single-point position and transition defects in continuous time quantum walks.

Li ZJ, Wang JB - Sci Rep (2015)

Bottom Line: The number of bound states is found to be critically dependent on the defect parameters, and the localized probability peaks can be readily obtained by projecting the state vector of CTQW on to these bound states.The interference between two bound states are also observed in the case of a transition defect.The spreading of CTQW probability over the line can be finely tuned by varying the position and transition defect parameters, offering the possibility of precision quantum control of the system.

View Article: PubMed Central - PubMed

Affiliation: Institute of Theoretical Physics, Shanxi University, Taiyuan, 030006, China.

ABSTRACT
We present a detailed analysis of continuous time quantum walks (CTQW) with both position and transition defects defined at a single point in the line. Analytical solutions of both traveling waves and bound states are obtained, which provide valuable insight into the dynamics of CTQW. The number of bound states is found to be critically dependent on the defect parameters, and the localized probability peaks can be readily obtained by projecting the state vector of CTQW on to these bound states. The interference between two bound states are also observed in the case of a transition defect. The spreading of CTQW probability over the line can be finely tuned by varying the position and transition defect parameters, offering the possibility of precision quantum control of the system.

No MeSH data available.


The probability distribution of CTQW with a single-point transition defect when t = 30, j0 = 0, jd = 1, 2, 5 and β = −0.5, 0.5.
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f8: The probability distribution of CTQW with a single-point transition defect when t = 30, j0 = 0, jd = 1, 2, 5 and β = −0.5, 0.5.

Mentions: When the CTQW does not start from the defect position, i.e., jd ≠ j0 = 0, Fig. 8 presents the probability distribution at time t = 30. The left panel, with β = −0.5 and thus no bound state existing, shows that the CTQW wave-packet is largely reflected with a smaller transmission peak observed at the same locations as the ballistic peaks of the free quantum walk. The right panel is the situation for β = 0.5, where two bound state exist. If the defect position is the nearest to the initial position of the CTQW, jd = j0 + 1, the eigen localization induced by two bound states accumulates the probability in the vicinity of the defect position and displays strong eigen-localisation (see Fig. 8(b)). Only considering the projections of the bound eigenvectors, we have and , which is nearly equal to the coordinate values in Fig. 8(b). If the defect position goes away from the initial position,  > 1, the factor in combinated projection makes the eigen-localization probability decay exponentially with increasing distance .


Single-point position and transition defects in continuous time quantum walks.

Li ZJ, Wang JB - Sci Rep (2015)

The probability distribution of CTQW with a single-point transition defect when t = 30, j0 = 0, jd = 1, 2, 5 and β = −0.5, 0.5.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4555049&req=5

f8: The probability distribution of CTQW with a single-point transition defect when t = 30, j0 = 0, jd = 1, 2, 5 and β = −0.5, 0.5.
Mentions: When the CTQW does not start from the defect position, i.e., jd ≠ j0 = 0, Fig. 8 presents the probability distribution at time t = 30. The left panel, with β = −0.5 and thus no bound state existing, shows that the CTQW wave-packet is largely reflected with a smaller transmission peak observed at the same locations as the ballistic peaks of the free quantum walk. The right panel is the situation for β = 0.5, where two bound state exist. If the defect position is the nearest to the initial position of the CTQW, jd = j0 + 1, the eigen localization induced by two bound states accumulates the probability in the vicinity of the defect position and displays strong eigen-localisation (see Fig. 8(b)). Only considering the projections of the bound eigenvectors, we have and , which is nearly equal to the coordinate values in Fig. 8(b). If the defect position goes away from the initial position,  > 1, the factor in combinated projection makes the eigen-localization probability decay exponentially with increasing distance .

Bottom Line: The number of bound states is found to be critically dependent on the defect parameters, and the localized probability peaks can be readily obtained by projecting the state vector of CTQW on to these bound states.The interference between two bound states are also observed in the case of a transition defect.The spreading of CTQW probability over the line can be finely tuned by varying the position and transition defect parameters, offering the possibility of precision quantum control of the system.

View Article: PubMed Central - PubMed

Affiliation: Institute of Theoretical Physics, Shanxi University, Taiyuan, 030006, China.

ABSTRACT
We present a detailed analysis of continuous time quantum walks (CTQW) with both position and transition defects defined at a single point in the line. Analytical solutions of both traveling waves and bound states are obtained, which provide valuable insight into the dynamics of CTQW. The number of bound states is found to be critically dependent on the defect parameters, and the localized probability peaks can be readily obtained by projecting the state vector of CTQW on to these bound states. The interference between two bound states are also observed in the case of a transition defect. The spreading of CTQW probability over the line can be finely tuned by varying the position and transition defect parameters, offering the possibility of precision quantum control of the system.

No MeSH data available.