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Preparation of topological modes by Lyapunov control.

Shi ZC, Zhao XL, Yi XX - Sci Rep (2015)

Bottom Line: For Bose systems, taking the noninteracting Su-Schrieffer-Heeger (SSH) model as an example, we illustrate how to drive the system into the edge mode.The sensitivity of the fidelity to perturbations and uncertainties in the control fields and initial modes is also examined.The experimental feasibility of the proposal and the possibility to replace the continuous control field with square wave pulses is finally discussed.

View Article: PubMed Central - PubMed

Affiliation: School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China.

ABSTRACT
By Lyapunov control, we present a proposal to drive quasi-particles into a topological mode in quantum systems described by a quadratic Hamiltonian. The merit of this control is the individual manipulations on the boundary sites. We take the Kitaev's chain as an illustration for Fermi systems and show that an arbitrary excitation mode can be steered into the Majorana zero mode by manipulating the chemical potential of the boundary sites. For Bose systems, taking the noninteracting Su-Schrieffer-Heeger (SSH) model as an example, we illustrate how to drive the system into the edge mode. The sensitivity of the fidelity to perturbations and uncertainties in the control fields and initial modes is also examined. The experimental feasibility of the proposal and the possibility to replace the continuous control field with square wave pulses is finally discussed.

No MeSH data available.


The effect of uncertainties in the Hamiltonian on the fidelity.The influence of boundary Hamiltonian is depicted in (a). Each point is an average over 30 simulations in (b). The horizontal axis denotes the number of perturbations at each instance of time in the Kitaev’s chain. Other parameters are the same as in Fig. 8.
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f10: The effect of uncertainties in the Hamiltonian on the fidelity.The influence of boundary Hamiltonian is depicted in (a). Each point is an average over 30 simulations in (b). The horizontal axis denotes the number of perturbations at each instance of time in the Kitaev’s chain. Other parameters are the same as in Fig. 8.

Mentions: On the other hand, the Lyapunov control requires to know the system Hamiltonian exactly, which may be difficult in practice. One then may ask how does the control performance change if there exist uncertainties in the Hamiltonian. We now turn to study this problem. The Hamiltonian with uncertainties can be written as . Here, δH0 denotes the deviation (called uncertainties) of the Hamiltonian in the control system. This deviation might manifest in the hopping amplitude J, pairing Δ, or the chemical potential μ. As the control is exerted on the boundary sites only, we study the deviation in the boundary sites and the bulk sites, separately. Figure 10(a) shows the fidelity as a function of the deviations in the boundary Hamiltonian, (, where j = 1, N). It finds that the deviations caused by the boundary Hamiltonian do not have a serious impact on the fidelity. When the deviation happens in the bulk sites, for example, the on-site chemical potential of the bulk sites is replaced with (note that site j is randomly chosen from the bulk, and ε is an random number, ε ∈ [ − 0.02, 0.02]), we consider n (n = 1, …, 20) uncertainties appearing simultaneously at each instance of evolution time. In other words, we simulate n fluctuations for the on-site chemical potentials, where each fluctuation is generated for a randomly chosen site n, the value of fluctuations for chosen sites is randomly created and denoted by ε. By performing the extensive numerical simulations, we demonstrate the results in Fig. 10(b). It can be found that the quantum system is robust against small uncertainties since the fidelity is always larger than 97.9%. An interesting observation is that with the number of fluctuations increasing, the fidelity increases. This can be understood as follows. Firstly, the small deviations cannot close the gaps in the topological system, thus the fidelity would not deteriorate sharply. Secondly, although more uncertainties participate in the control procedure, the average of the uncertainties almost approaches zero as the average of the random number ε is zero.


Preparation of topological modes by Lyapunov control.

Shi ZC, Zhao XL, Yi XX - Sci Rep (2015)

The effect of uncertainties in the Hamiltonian on the fidelity.The influence of boundary Hamiltonian is depicted in (a). Each point is an average over 30 simulations in (b). The horizontal axis denotes the number of perturbations at each instance of time in the Kitaev’s chain. Other parameters are the same as in Fig. 8.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4561913&req=5

f10: The effect of uncertainties in the Hamiltonian on the fidelity.The influence of boundary Hamiltonian is depicted in (a). Each point is an average over 30 simulations in (b). The horizontal axis denotes the number of perturbations at each instance of time in the Kitaev’s chain. Other parameters are the same as in Fig. 8.
Mentions: On the other hand, the Lyapunov control requires to know the system Hamiltonian exactly, which may be difficult in practice. One then may ask how does the control performance change if there exist uncertainties in the Hamiltonian. We now turn to study this problem. The Hamiltonian with uncertainties can be written as . Here, δH0 denotes the deviation (called uncertainties) of the Hamiltonian in the control system. This deviation might manifest in the hopping amplitude J, pairing Δ, or the chemical potential μ. As the control is exerted on the boundary sites only, we study the deviation in the boundary sites and the bulk sites, separately. Figure 10(a) shows the fidelity as a function of the deviations in the boundary Hamiltonian, (, where j = 1, N). It finds that the deviations caused by the boundary Hamiltonian do not have a serious impact on the fidelity. When the deviation happens in the bulk sites, for example, the on-site chemical potential of the bulk sites is replaced with (note that site j is randomly chosen from the bulk, and ε is an random number, ε ∈ [ − 0.02, 0.02]), we consider n (n = 1, …, 20) uncertainties appearing simultaneously at each instance of evolution time. In other words, we simulate n fluctuations for the on-site chemical potentials, where each fluctuation is generated for a randomly chosen site n, the value of fluctuations for chosen sites is randomly created and denoted by ε. By performing the extensive numerical simulations, we demonstrate the results in Fig. 10(b). It can be found that the quantum system is robust against small uncertainties since the fidelity is always larger than 97.9%. An interesting observation is that with the number of fluctuations increasing, the fidelity increases. This can be understood as follows. Firstly, the small deviations cannot close the gaps in the topological system, thus the fidelity would not deteriorate sharply. Secondly, although more uncertainties participate in the control procedure, the average of the uncertainties almost approaches zero as the average of the random number ε is zero.

Bottom Line: For Bose systems, taking the noninteracting Su-Schrieffer-Heeger (SSH) model as an example, we illustrate how to drive the system into the edge mode.The sensitivity of the fidelity to perturbations and uncertainties in the control fields and initial modes is also examined.The experimental feasibility of the proposal and the possibility to replace the continuous control field with square wave pulses is finally discussed.

View Article: PubMed Central - PubMed

Affiliation: School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China.

ABSTRACT
By Lyapunov control, we present a proposal to drive quasi-particles into a topological mode in quantum systems described by a quadratic Hamiltonian. The merit of this control is the individual manipulations on the boundary sites. We take the Kitaev's chain as an illustration for Fermi systems and show that an arbitrary excitation mode can be steered into the Majorana zero mode by manipulating the chemical potential of the boundary sites. For Bose systems, taking the noninteracting Su-Schrieffer-Heeger (SSH) model as an example, we illustrate how to drive the system into the edge mode. The sensitivity of the fidelity to perturbations and uncertainties in the control fields and initial modes is also examined. The experimental feasibility of the proposal and the possibility to replace the continuous control field with square wave pulses is finally discussed.

No MeSH data available.