Introduction

Compared to classical computation, quantum computation has unparallel advantages in solving problems like factoring a large number1. However, it is difficult to realize in practice due to decoherence caused by environments. In order to overcome this obstacle, topological quantum computation2,3,4,5,6 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 anyons7,8 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 system9, interface between topological insulator10,11, topological superconductors12,13,14,15,16,17, solid state system18, optical lattices19,20 and spin chains21. 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 controls22. 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 set23,24,25,26, quantum state steering or preparations27,28. 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 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 and denote the annihilation and creation operators for fermions or bosons at the spatial position j. “∗” stands for complex conjugate. The N × N matrix A0 (B0) with elements () should satisfy to guarantee the hermicity of H0, 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 fermions29 as an example. The Hamiltonian reads,

where J and Δ are hopping and pairing amplitude, respectively. is the fermionic annihilation (creation) operation at site j and μ represents the chemical potential. By the pioneering work29, one can find that there exist two different topological phases when parameters change. The quantum critical line separating those phases is given by and Δ = 0. To be specific, the parameter satisfying and Δ ≠ 0 is a nontrivial topological phase which can support a Majorana zero mode at the boundaries. In following, we set Δ = 1 and J =μ = 2 to ensure the existence of the Majorana zero mode in the Kitaev’s chain. The Majorana zero mode can be revealed by solving the secular equation of the BdG Hamiltonian,

where the elements of matrices A0 and B0 are

It can be found easily that Xi = Xi* and Yi = Yi* due to the time-reversal symmetry of the Hamiltonian.

Figure 1(a) demonstrates the eigenvalues of the BdG Hamiltonian, while Fig. 1(b–e) show the distribution of the left and right Majorana zero mode, respectively. As seen in this figure, the Majorana zero mode is located near the two boundary sites of the chain. Taking a chain of length N = 30 for concreteness, we show in the following that the Majorana zero mode can be achieved by controlling the chemical potential at the two ends of the Kitaev’s chain. Consider two control Hamiltonians and , the nonzero elements of matrices Ak given by Eq. (20) corresponding to the control Hamiltonian are and .

Figure 1
figure 1

The energy spectrum and spatial distributions of the BdG Hamiltonian describing the Kitaev’s chain with total number N = 30 of sites.

We have set the lattice spacing as units. There exists two Majorana modes in the band gap, i.e., the 30th and 31th eigenmodes. The 30th eigenmode is labeled by left mode and the 31th is labeled by right mode. (b,c) are the coefficients X30 and Y30 of the left mode, while (d,e) are the coefficients X31 and Y31 of the right mode.

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Suppose that the initial mode is an equally weighted superposition of all sites, namely the initial mode can be expressed as with . The form of Lyapunov function could be chosen as V = Q†PQ and the hermitian matrix P could be constructed in the following manner (see methods),

Here pi = 0, pT = − 1 and UT is the target eigenvector. Then the control field becomes and we choose Fk = 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 by for the left mode and 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 with θ ∈ [0, 2π].

Figure 2
figure 2

The dynamical evolution of system as a function of time with initial mode .

Ol and Or represent the occupations of the left and right mode, while Ol + Or approaching unit implies the other quasiparticle modes except the right and left modes are suppressed. (b,c) denote the dynamical evolution of the control fields f1(t) and f2(t), respectively.

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For a finite length N of the Kitaev’s chain, there exists a weak interaction between the left and right mode with the interaction strength λ ∝ e−N/ξ 29, where ξ is the coherence length. Obviously, the left and 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 with constraint that Dj(0) = 0 if Cj(0) ≠ 0 or Cj(0) = 0 if Dj(0) ≠ 0, 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 with . As expected, it converges to the right mode asymptotically.

Figure 3
figure 3

The dynamical evolution of system as a function of time with initial mode .

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Bose system

For the case of bosons, we take the noninteracting Su-Schrieffer-Heeger (SSH) model30 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 lattice31. The edge mode in the topological band has been shown in Ref. 31, which can be witnessed by the nontrivial Zak phase32 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 33 (see methods) in 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 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.

Figure 4
figure 4

The eigenvalue spectrum and spatial distributions of the Hamiltonian in the SSH model with N = 21 sites.

Two edge mode are found in the band gap, corresponding to the 11th and 32th eigenvectors. We label the 11th eigenvector as left mode while the 32th eigenvector is the right mode. (b) and (c) are the coefficients X11 and Y11 of the left mode while (d,e) are the coefficients X32 and Y32 of the right mode.

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Case 1

The initial mode is described by an arbitrary superposition of creation operators or annihilation 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 with . That is, the initial mode contains the creation operators of all sites in this control system. The Lyapunov function is taken as with , T = 32 and the control field is given by with F1 =2, 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 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 mode23, where the Lyapunov function is redefined as

Figure 5
figure 5

The dynamical evolution of system as a function of time with conventional Lyapunov technique and initial mode .

(c) denotes the dynamical behavior of the Lyapunov function V.

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Here, WT,η(Q) is a target eigenvector of the matrix with η(Q) ∈  (corresponding to the right mode when η = 0, i.e., WT, 0 = UT). The secular equation can be written as,

where λj, η represents the eigenvalues. It returns to the secular equation of the matrix when η(Q) = 0. The control field can be rewritten as , where η(Q) is implicitly defined as

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 F1 is an positive constant. We can choose the control field with F1 = 1 to guarantee . 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.

Figure 6
figure 6

The dynamical evolution of system as a function of time with unconventional Lyapunov technique.

The physical parameters are the same to the Fig. 5 except for the Lyapunov function.

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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 with and . Subsequently, the control field can be straightforwardly taken as . We set , while the other coefficients vanish and F1 = 1 for numerical calculation. The occupations of the left and right mode are given in Fig. 7. As expected, the Lyapunov function reaches its minimum when the system arrives at the edge mode. The final mode could be approximately written as , showing that we have realized the edge mode. Note that the occupation difference could not guarantee that the final mode converges to the edge mode, which is distinct to the aforementioned cases . As the evolution of the coefficients of the operator is unitary (see equation (22)) when B = 0, the coefficients should satisfy , i.e., it is invariant during the evolution. From the numerical calculation, we can find that the final mode can be approximately written as , indicating that the coefficients of the other quasiparticle modes almost vanish.

Figure 7
figure 7

The dynamical evolution of system as a function of time with the Lyapunov function

. It can be found that and imply the other quasiparticle modes being suppressed.

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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 in the Fermi system as the initial mode without uncertainties, we can write the initial mode with uncertainties as 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 with fk(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.

Figure 8
figure 8

The fidelity versus (a) the uncertainties in the initial mode and (b) the perturbations in the control fields f1(t) and f2(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.

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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 . The results in Fig. 9 suggest that the fidelity keeps high even though the control fields have influences on the nearest-neighbor sites.

Figure 9
figure 9

The fidelity versus the chemical potential of nearest-neighbor sites of the boundaries affected by the control fields.

We describe this influence by . 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.

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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.

Figure 10
figure 10

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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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,

As an example, we focus on the Bose system whose parameters are the same as in Fig. 7 except that the control field f1(t) is replaced by the equation (11) with . 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.

Figure 11
figure 11

The dynamical evolution of system as a function of time with the square wave pulses.

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Finally, we would like to discuss on the experimental feasibility for the present control protocols. The SSH model can be experimentally realized by 87Rubidium atoms34 in 1D double-well optical lattice35. 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 f1(t) in Fig. 11(b)). In addition, we can realize the effective Kitaev’s chain in the quantum-dot-superconductor system37, 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 field38,39. Then the boundary chemical potential can be controlled by local gates40,41. Most recently, the observation of Majorana fermions in this system has also been observed in experiments41,42.

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,

For the case of fermions, we denote the Hamiltonian by , i.e., . The operators obey the anticommutation relations: and Define a time-dependent fermionic operator,

where the operators and are time-independent while the coefficients are time-dependent. It is easy to check that according to the anti-commutation relation . In the Heisenberg picture, the evolution of this operator satisfies (ħ = 1),

After a brief algebraic operation, the equation becomes

The evolution of coefficients Cj(t) and Dj(t) then can be written in a compact form of matrix,

where and . We use the Gothic letter 0 to denote the matrix in equation (16) corresponding to the Hamiltonian H0 in equation (12) for simplicity hereafter.

For the Fermi system, the quadratic Hamiltonian can be rewritten as up to a constant factor ( called Bogoliubov-de-Gennes(BdG) Hamiltonian), where . Clearly, . In fact, the equation (16) is actually the BdG-Schrödinger equation43, where Q is the quasi-particle wave function in the Nambu representation. One can claim that if εl is an eigenvalue of with corresponding eigenvector , l = 1, …, N:

is also an eigenvector with eigenvalue −εl, i.e.,

where Xl* = Y2N + 1 − l, Yl* = Y2N + 1 − l, and . Thus the eigenvalues come in pairs ±εl for the BdG Hamiltonian 33. Diagonalizing the BdG Hamiltonian, the quasi-particles can be represented by annihilation (creation) operators ,

where l = 1, …, N. In terms of the quasi-particle modes, the Hamiltonian can be written as , where are the energy of the quasi-particle .

Let one of the quasi-particle modes be the target mode which we want to prepare, e.g., where u and v are arbitrary constants. and are the annihilation and creation operators of the target mode, respectively. The goal is to design control fields that can drive any initial modes to the target one. It should be noticed that we cannot choose the target arbitrarily because it depends on the free Hamiltonian. In other words, we need a stationary target mode which does not evolve under the free Hamiltonian. As the edge mode is robust against perturbations, we focus on the preparation of it. The evolution described by the equation (16) is unitary since is hermitian. As a result the sum remains unchanged during the time evolution. To make the calculation clear, we write the target mode as , in which is an eigenvector of the BdG Hamiltonian , meanwhile it is also a solution of equation (16). Namely, , where and . Assume that there are K control Hamiltonians for the system in quadratic form: , k = 1, …, K. Together with the original Hamiltonian, the equation of motion for the coefficients in the operator becomes

where fk(t) is the control field.

There are many choices for the Lyapunov functions, for example, , and V3 = Q†PQ. Here, denotes the norm. Those Lyapunov functions are nonnegative and reach the minimum when the system arrives at the target. Apparently, different Lyapunov functions lead to different invariant set and different characteristics of convergence. In following, we choose V = Q†PQ as the Lyapunov function to show how our scheme works while the analysis for other Lyapunov functions are similar to it. To this end, it is instructive to deduce the first-order time derivative of the Lyapunov function,

where we have set by properly constructing the matrix P. In order to make the time derivative of V non-positive, one can design the control fields in the following style: with Fk > 0. Strictly speaking, the quantum system converges to the invariant set determined by the La Salle’s invariance principle, equivalent to the solution .

Note that the commutation relations for bosons: and are different from fermions. Keeping this difference in mind and by an analysis similar to the case of fermions, one can obtain a dynamical evolution of operator with

In this case, the matrix of BdG Hamiltonian is . Therefore, we can find that , where , σz is Pauli matrix and II is the N × N identity matrix. The dynamics of coefficients are not unitary in general except for B0 = 0. For this special situation, the control mechanism is analogous to the case of fermions.

Additional Information

How to cite this article: Shi, Z. C. et al. Preparation of topological modes by Lyapunov control. Sci. Rep. 5, 13777; doi: 10.1038/srep13777 (2015).