Preparation of topological modes by Lyapunov control

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.

Compared to classical computation, quantum computation has unparallel advantages in solving problems like factoring a large number 1 . However, it is difficult to realize in practice due to decoherence caused by environments. In order to overcome this obstacle, topological quantum computation [2][3][4][5][6] has been proposed, where the ground states are isolated from the rest energy spectrum by gaps, making it robust against perturbations. The topological quantum computation can be performed by braiding non-Abelian anyons 7,8 while the evolution of the system, protected by topology, is described by a nontrivial unitary transformation. The simplest example of the non-Abelian anyons is the Majorana fermions which are self-conjugate quasiparticles and have been extensively studied both theoretically and experimentally. Recently, the Majorana fermions are predicted to exist in fractional quantum Hall system 9 , interface between topological insulator 10,11 , topological superconductors 12-17 , solid state system 18 , optical lattices 19,20 and spin chains 21 . Although there are great progress in this field, how to prepare and manipulate Majorana fermions in quantum systems remains challenging.
Generally speaking, a quantum system cannot evolve into a desired state without any quantum controls 22 . While most readers are familiar with the feedback control, here we begin with introducing Lyapunov-based quantum control. The Lyapunov control refers to the use of Lyapunov function to design control fields for manipulating a dynamical system. In quantum mechanics, the evolution of system is governed by the Schrödinger equation and the system state can be described by a time-dependent vector. The Lyapunov function then can be defined as the distance between the time-dependent vector and the target vector. Until now, most studies of Lyapunov control focus on the analysis of largest invariant set [23][24][25][26] , quantum state steering or preparations 27,28 . In this work, we extend the application of Lyapunov control and apply it to manipulate many-body system, e.g., driving quasiparticles in a quantum many-body system.
To be specific, by the use of Lyapunov control technique, we present a method to manipulate the topological modes in both Fermi and Bose systems. For a Fermi system described by the Kitaev model, we show how to steer an arbitrary initial mode into the Majorana zero mode by manipulating the chemical potential of the boundary sites. The system can be driven into a special Majorana zero mode localized at one of the boundaries when the initial mode is represented only by creation or annihilation operators. For a Bose system described by the noninteracting Su-Schrieffer-Heeger (SSH) model, the control mechanism is similar to the Fermi system. Nevertheless, due to the vanishing off-diagonal block (pairing terms) in the Hamiltonian, it is impossible to drive an arbitrary superposition of operators with different Scientific RepoRts | 5:13777 | DOi: 10.1038/srep13777 sites into the target mode except for two special cases, namely, the modes can be solely described by creation (or annihilation) operators or by creation and annihilation operators at same site. An unconventional Lyapunov technique is also explored to achieve the target mode while the conventional Lyapunov control is not effective. The sensitivity of the fidelity to perturbations and uncertainties in the control fields and initial modes is also examined. Finally, we show that the control field can be replaced with square wave pulses, which might make the realization of the control much easier in experiments.

Results
In this part, we present the main results of this work by showing how well the topological modes can be prepared via the Lyapunov control. The details of calculation and simulation can be found in METHODS. Without loss of generality, we consider a quantum system described by quadratic Hamiltonian, where â j and ˆ † a j denote the annihilation and creation operators for fermions or bosons at the spatial position j. " * " stands for complex conjugate.
0 0 0 to guarantee the hermicity of H 0 , where "∼ " denotes transposition, and ε = −1 for fermions while ε = 1 for bosons. Since the commutation relations of fermions are different from bosons, we will study the control for the Fermi and Boson systems separately.
Fermi system. We take the 1D Kitaev's chain of spinless fermions 29 as an example. The Hamiltonian reads, where J and Δ are hopping and pairing amplitude, respectively. ( ) † a a j j is the fermionic annihilation (creation) operation at site j, and μ represents the chemical potential. By the pioneering work 29 , one can find that there exist two different topological phases when parameters change. The quantum critical line separating those phases is given by μ = J 2 and Δ = 0. To be specific, the parameter satisfying μ > J 2 and Δ ≠ 0 is a nontrivial topological phase which can support a Majorana zero mode at the boundaries. It can be found easily that X i = X i* and Y i = Y i* due to the time-reversal symmetry of the Hamiltonian. Here p i = 0, p T = − 1, and U T is the target eigenvector. Then the control field becomes and we choose F k = 10 for the numerical calculations. Figure 2 shows the occupations of the left and right mode as a function of evolution time, where the occupation is defined for the right mode. We observe that the initial mode asymptotically converges to the Majorana zero mode with time, and the control fields almost vanish when the system arrives at the target mode. Further simulations show that this proposal works for almost arbitrary initial modes. For example, it can also be driven to the Majorana zero mode when the initial modes are  right modes are degenerate when N/ξ  1. Therefore, it is impossible to drive an initial mode into one of the Majorana zero mode individually, if the initial mode includes both the creation and annihilation operators at the same site. However, when the initial mode can be represented by it might be possible to drive the initial mode into one of the Majorana zero mode. Figure 3 shows this possibility for driving the system into the right mode while the initial mode is As expected, it converges to the right mode asymptotically.
Bose system. For the case of bosons, we take the noninteracting Su-Schrieffer-Heeger (SSH) model 30 to show the control performance. The Hamiltonian reads where ε is a parameter to change the hoping amplitude J, 0 ≤ ε ≤ 1, and μ is the chemical potential. This model can be applied to describe bosons hopping in a double-well 1D optical lattice 31 . The edge mode in the topological band has been shown in Ref. 31, which can be witnessed by the nontrivial Zak phase 32 of the bulk bands. Thereby it can be taken as the target mode in this control system, and we choose the parameters J = 1, μ = 2, N = 21, and ε = 0.3 for the following numerical calculation. Firstly, we present the results of exact diagonalization of τ ⋅  Fig. 4(a) and give the coefficients of the edge mode in Fig. 4(b-e). It can be found that the edge mode is located near the first site of the chain, this suggests us to regulate the on-site chemical potential (energy) of site 1 to manipulate the system. Namely, the control Hamiltonian is suggested to be = . † H a a b 1 1 1 As the Hamiltonian is block diagonal, we could drive the system from an arbitrary initial mode to the target mode for two special cases listed below.
Case 1. The initial mode is described by an arbitrary superposition of creation operators or annihila tion operators only. Since the annihilation and creation operators that describe quasi-particle modes are decoupled each other, the control system can only converge to the annihilation or creation operators in the target mode, respectively. For the numerical calculations, we choose the initial mode described by a superposition of creation operators That is, the initial mode contains the creation operators of all sites in this control system. The Lyapunov function is taken as 32 , T = 32, and the control field is given by where Im(·) denotes the imaginary part of (·). Figure 5 shows the occupation of right mode as a function of evolution time t. It demonstrates that the operator ( ) a t does not completely converge to the right mode since the occupation of the right mode approaches 0.5814. On the other hand, when resolving the characteristic spectrum of the free and control Hamiltonian, one can find that the target mode is controllable for an arbitrary superposition of creation operators. Next, we adopt an implicit Lyapunov-based method to steer an arbitrary initial mode into the right mode 23 where λ j, η represents the eigenvalues. It returns to the secular equation of the matrix  b 0 when η(Q) = 0. The control field can be rewritten as η Here θ(t) is a slowly varying real function satisfying θ(0) = 0 and θ(t) > 0 for every t > 0. We set θ(t) = 0.5 t for simplicity. By taking the time derivative of V, one can find where F 1 is an positive constant. We can choose the control field Figure 6 demonstrates the dynamics of occupation of the right mode, we find that it can reach about 0.9887 when completing the control. Hence an arbitrary initial mode can be steered to the right mode by making use of the implicit Lyapunov function.
Case 2. The initial mode is an arbitrary superposition of creation and annihilation operators at the same site only, i.e., In this case, the Lyapunov function is chosen a bit different from before, which becomes 32 . Subsequently, the control field can be straightforwardly taken as . We set ( ) = .

Discussions
Until now, we have achieved the goal of driving the initial mode of many-body system into a desired quasi-particle mode. The proposal needs to know exactly the system Hamiltonian and the initial mode, as well as to implement precisely the control fields. However, this may be difficult in practice. In experiments, we often encounter uncertainties in the initial modes, perturbations in the control fields, and uncertainties in the Hamiltonian. In previous section, the proposal has been implemented in the Fermi and Bose systems without any perturbations or uncertainties. In following, we discuss the effect of perturbations and uncertainties in the control fields, initial modes and Hamiltonian on the performance of the control.
We first examine the effect of uncertainties in the initial mode and perturbations in the control fields. Taking ( ) =â a 0 1 in the Fermi system as the initial mode without uncertainties, we can write the initial mode with uncertainties as ε ε ( ) = − ( ) + ′ˆâ a a 0 1 0 j with ε quantifying the uncertainties. The dependence of the fidelity on ε is plotted in Fig. 8(a). For the control field with perturbations, we write it as δ with f k (t) representing the perturbationless control field. The dependence of the fidelity on the perturbations is presented in Fig. 8(b). One can find from Fig. 8 that the fidelity is more sensitive to the uncertainties in the initial mode, while it is robust against the perturbations in the control fields. In fact, from the principle of the Lyapunov control, it is suggested that the fidelity of the control process is sensitive to the sign rather than the amplitude of the control fields. This observation can be used to understand the robustness against the perturbations in the control fields.
In a more realistic circumstance, individual controls on the boundary sites are difficult to implement, which means that the control on the boundary sites might affect the on-site chemical potential of their nearest neighbors. Suppose that the chemical potential of the nearest-neighbor sites, which is affected by the control fields, can be characterized by , i.e., the on-site chemical potential of 2nd and (N − 1)th site are replaced by ( )μ The results in Fig. 9 suggest that the fidelity keeps high even though the control fields have influences on the nearest-neighbor sites.
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, δH 0 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, 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 μ ′ j of the bulk sites is replaced with μ ε μ ′ = ( + ) 1 j j (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.
Since the form of control field generally takes , the amplitude of the control fields may change fast with time, which increases the difficulty in the realizations. It is believed that the square wave pulses can be readily achieved in experiments. Therefore we try to take the square wave pulses instead of for the control field. The principle to design the square wave pulses should satisfy, Figure 8. The fidelity versus (a) the uncertainties in the initial mode and (b) the perturbations in the control fields f 1 (t) and f 2 (t). Other parameters are the same as in Fig. 3. The control time is terminated when the fidelity reaches 99.15%. One can observe that the fidelity is still above 98% even though there are 10% perturbations in the control fields. . Other parameters are the same as in Fig. 8. δ = 0.5 means that the value of control fields on the nearest-neighbor site is the half of control fields on the boundary sites.
As an example, we focus on the Bose system whose parameters are the same as in Fig. 7 except that the control field f 1 (t) is replaced by the equation (11) with ′ = . F 0 04 1 . Figure 11 demonstrates the results for the square wave pulses of the control field and it can also achieve the edge mode eventually. On the other hand, we find that convergence time is shortened as well. Of course, the square wave pulses of the control fields can also be applied to the Fermi system.
Finally, we would like to discuss on the experimental feasibility for the present control protocols. The SSH model can be experimentally realized by 87 Rubidium atoms 34 in 1D double-well optical lattice 35 . The implementations of Lyapunov control require to perform operations defined by the control Hamiltonians with strengths defined by the control fields. In our case, the control Hamiltonians are the particle number operators of the boundary sites, and the control can be experimentally realized by manipulating the on-site chemical potentials of the boundary sites. The realization of Kitaev's chain requires spinless fermions, which can be prepared in an optical lattice by trapping the fermions and the BEC reservoir  with Feshbach molecules (the couplings between them can be induced by an rf-pulse) 36 . By driving the fermions with Raman laser to produce a strong effective coupling, the system in this situation is equivalent to the Kitaev's chain. In order to realize the control Hamiltonians, one can adopt additional lasers to control the chemical potentials of the boundary sites, where the intensity of lasers is simulated by square wave pulses (e.g., see f 1 (t) in Fig. 11(b)). In addition, we can realize the effective Kitaev's chain in the quantum-dot-superconductor system 37 , a linear array with quantum dots linked by s-wave superconductors with normal and anomalous hoppings. In this system, the chemical potential in each quantum dot can be controlled individually by gate voltages with a high degree of precision. Alternatively, the Kitaev's chain can also be achieved in the system which consists of a strong spin-orbit interaction semiconductor nanowire (in the low density limit) coupling to a superconductor in magnetic field 38,39 . Then the boundary chemical potential can be controlled by local gates 40,41 . Most recently, the observation of Majorana fermions in this system has also been observed in experiments 41,42 .
In summary, we present a scheme to prepare quasi-particle mode by Lyapunov control in the both Fermi and Bose systems. For the Fermi system, we choose the Kitaev's model as an illustration and specify the Majorana zero mode as the target mode. The results show that by controlling the chemical potential at the two boundary sites, the system can be driven asymptotically into one of the Majorana zero mode such as the right mode. In contrary, the situation for bosons is different due to the commutation relations. As an example, in the noninteracting SSH model, we show how to prepare the edge mode by the control fields. In particular, we apply the implicit Lyapunov-based technique to the boson system which provides us with a new way to steer the bosons. The robustness of the fidelity against perturbations and uncertainties is also examined. Finally, we try to replace the control fields with square wave pulses, which might help realize the control fields more easily in experiments since it is difficult to apply a fast time-varying control fields in practice.

Methods
In this part, we give the derivation of the control scheme, starting with the quadratic Hamiltonian,  . We use the Gothic letter  0 to denote the matrix in equation (16) corresponding to the Hamiltonian H 0 in equation (12) (16) is actually the BdG-Schrödinger equation 43 , where Q is the quasi-particle wave function in the Nambu representation. One can claim that if ε l is an eigenvalue of . Therefore, we can find that  τ , σ z is Pauli matrix and II is the N × N identity matrix. The dynamics of coefficients are not unitary in general except for B 0 = 0. For this special situation, the control mechanism is analogous to the case of fermions.