Abstract
Topological phases exhibit some of the most striking phenomena in modern physics. Much of the rich behaviour of quantum Hall systems, topological insulators, and topological superconductors can be traced to the existence of robust bound states at interfaces between different topological phases. This robustness has applications in metrology and holds promise for future uses in quantum computing. Engineered quantum systems—notably in photonics, where wavefunctions can be observed directly—provide versatile platforms for creating and probing a variety of topological phases. Here we use photonic quantum walks to observe bound states between systems with different bulk topological properties and demonstrate their robustness to perturbations—a signature of topological protection. Although such bound states are usually discussed for static (timeindependent) systems, here we demonstrate their existence in an explicitly timedependent situation. Moreover, we discover a new phenomenon: a topologically protected pair of bound states unique to periodically driven systems.
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Introduction
Phases of matter have long been characterized by their symmetry properties, with each phase classified according to the symmetries that it possesses^{1}. The discovery of the integer and fractional quantum Hall effects in the 1980s has led to a new paradigm, where quantum phases of matter are characterized by the topology of their groundstate wavefunctions. Since then, topological phases have been identified in physical systems ranging from condensedmatter^{2,3,4,5,6,7,8,9} and highenergy physics^{10} to quantum optics^{11} and atomic physics^{12,13,14,15}.
Topological phases of matter are parametrized by integer topological invariants. As integers cannot change continuously, a consequence is exotic phenomena at the interface between systems with different values of topological invariants. For example, a topological insulator supports conducting states at the surface, precisely because its bulk topology is different to that of its surroundings^{8,9}. Creating and studying new topological phases remains a difficult task in a solidstate setting because the properties of electronic systems are often hard to control. Using controllable simulators may be advantageous in this respect.
Here we simulate onedimensional topological phases using a discrete time quantum walk^{16}, a protocol for controlling the motion of quantum particles on a lattice. We create regions with distinct values of topological invariants and directly image the wavefunction of bound states at the boundary between them. The controllability of our system allows us to make small changes to the Hamiltonian and demonstrate the robustness of these bound states. Finally, using the quantum walk, we can access the dynamics of strongly driven systems far from the static or adiabatic regimes^{17,18,19}, to which most previous work on topological phases has been restricted. In this regime, we discover a topologically protected pair of nondegenerate bound states, a phenomenon that is unique to periodically driven systems.
Results
Splitstep quantum walks
Discrete time quantum walks have been realized in several physical architectures^{20,21,22,23,24}. Here we use the photonic setup demonstrated in ref. 24 to implement a variation of these walks, the splitstep quantum walk^{25} of a single photon, with two internal states encoded in its horizontal, H〉, and vertical, V〉, polarization states. The quantum walk takes place on a one dimensional lattice (Fig. 1). One step of the splitstep quantum walk consists of four steps. First, a polarization rotation R(θ_{1}) of the single photon is achieved with a suitable wave plate (see Methods), then a polarizationdependent translation T_{1} of H〉 to the right by one lattice site using a calcite beam displacer. This is followed by a second rotation R(θ_{2}), and finally another translation T_{2} of V〉 to the left. The quantum walk is implemented by repeated applications of the onestep operator U(θ_{1}, θ_{2})=T_{2}R(θ_{2})T_{1}R(θ_{1}).
The propagation of the photon in the static experimental setup can be described by an effective timedependent Schrödinger equation with periodic driving. The dynamics of the quantum walk can be understood through the effective Hamiltonian H_{eff}(θ_{1}, θ_{2}), defined through where τ is the time required for one step of the quantum walk. Throughout this paper, we chose units such that Therefore, the quantum walk described by the evolution U(θ_{1}, θ_{2}) corresponds to a stroboscopic simulation of the effective Hamiltonian H_{eff}(θ_{1}, θ_{2}) viewed at unit time intervals. That is, after n steps of the quantum walk, the photon evolves according to meaning that the evolution under the quantum walk coincides with the evolution under H_{eff}(θ_{1}, θ_{2}) for integer multiples of τ.
The topological structure underlying splitstep quantum walks is revealed by studying the structure and symmetry of H_{eff}(θ_{1}, θ_{2}). H_{eff}(θ_{1}, θ_{2}) has a gapped spectrum, with two bands corresponding to opposite polarizations (Fig. 2a). Because the quantum walk is translationally invariant, each eigenstate is associated with a quasimomentum k and a superposition of H〉 and V〉. In addition, this class of quantum walks has a chiral symmetry described in ref. 25 (also detailed in the Methods), which requires that the polarization component of any eigenstate be confined to a particular great circle on the Bloch sphere. Therefore, as the quasimomentum k traverses the first Brillouin zone from −π to π, the polarization component of the eigenstate traces a closed path confined to that great circle, see Fig. 2b and Methods. The total number of times W that this closed path winds about the origin is the winding number and gives the topological invariant of H_{eff}(θ_{1}, θ_{2}).
Because W has to be an integer, it cannot be changed by small modifications of the effective Hamiltonian. That is, W can only change when the spectrum of H_{eff}(θ_{1}, θ_{2}) closes its gap while preserving chiral symmetry. For the splitstep quantum walk, two distinct topological phases with W=0 and W=1 exist as can be seen in the phase diagram shown in Fig. 2c. The two phases are separated by lines along which the gap closes.
As we mentioned above, nontrivial topological phases support localized states at their boundaries. Because our experimental setup allows access to individual lattice sites, we can probe this phenomenon by creating a boundary between regions where the dynamics are governed by two different gapped Hamiltonians H_{eff}(θ_{1−}, θ_{2}) and H_{eff}(θ_{1+}, θ_{2}), characterized by winding numbers W_{−} and W_{+} respectively. We create the boundary by making θ_{1} spatially inhomogeneous with θ_{1}(x)=θ_{1−} for lattice positions x<0 and θ_{1}(x)=θ_{1+} for x≥0. Although this breaks translational symmetry, the chiral symmetry remains (see Methods). When W_{−}≠W_{+}, it is expected that topologically robust localized states exist at the boundary near x=0. This can be understood in a heuristic fashion as follows. When W_{−}≠W_{+}, the winding number W_{−} of the bulk gapped Hamiltonian H_{eff}(θ_{1−}, θ_{2}) can only be changed to that of H_{eff}(θ_{1+}, θ_{2}) given by W_{+} by closing the gap of the system. Thus, near the boundary between these two regions, the energy gap closes, and it is expected that states exist within the gaps of the bulk spectra of H_{eff}(θ_{1−}, θ_{2}) and H_{eff}(θ_{1+}, θ_{2}). Because extended states do not exist inside the gap, such a state is necessarily localized at the boundary. That is, a change in topology at a boundary is accompanied by the presence of a localized state. Furthermore, we are able to show that the localized states are robust against perturbations^{26} such as small changes of quantum walk parameters or the presence of a static disordered potential caused by, for example, small spatial variations of rotation angles θ_{1} and θ_{2}.
Experimental confirmation of bound states
To probe the existence of the bound states, we initialize a photon next to the boundary between two topologically distinct quantum walks (Fig. 1). In the absence of bound states, the photon is expected to spread ballistically, with the detection probability at the origin quickly decreasing to zero. However, if there is a bound state, the bound state component of the initial state will remain near this boundary even after many steps.
We first implemented splitstep quantum walks with θ_{2}=π/2 and θ_{1−} and θ_{1+} such that W_{−}=W_{+}=1, as shown on the phase diagram in Fig. 3a. For initial photon polarizations of H〉 and V〉, shown in Fig. 3b,c respectively, the detection probability at the origin quickly decreases to zero. On the other hand, in Fig. 3d, with parameters chosen to create a boundary between topologically distinct phases W_{−}=0 and W_{+}=1, we observe the existence of at least one bound state as a peak in the probability distribution near the origin after four steps.
We note that our experiment is able to detect bound states with fewer steps than the theoretical study in ref. 25, because we use a sharper boundary between distinct topological phases. As a result, the bound states become narrower and are more easily detected.
We can test the robustness of these bound states against a variety of changes in microscopic parameters to confirm that they are topologically protected. In Fig. 3e, θ_{1−} and θ_{1+} are shifted from those of Fig. 3d while maintaining W_{−}=0 and W_{+}=1, and we confirmed the continued existence of a bound state. The quasienergy E of this localized state, that is, the eigenvalue of the effective Hamiltonian associated with this state, can be found by explicit calculation (Fig. 3f). We indeed find a single state at E=0.
Observation of pairs of bound states in quantum walks
Our experiment also reveals a new topological phenomenon unique to periodically driven systems, which can be probed by studying splitstep quantum walks with θ_{2}=0, and the θ_{1} parameters shown in Fig. 4a. With the appropriate choice of basis (see Methods), this quantum walk becomes equivalent to the one described by the onestep operator U=iTR(θ_{1}), where T=T_{1}T_{2} can be implemented with a single calcite beam displacer, extending the experiment to seven steps. This class of quantum walks can only realize a single topological phase characterized by the winding number W=0. Therefore, we do not expect bound states for spatially inhomogeneous θ_{1} based on winding numbers. However, the evolution of the probability distribution shown in Fig. 4b displays period2 oscillations in the vicinity of the origin. This observation strongly suggests the existence of at least two bound states whose quasienergies differ by π.
Again, we can demonstrate that these bound states are robust against small perturbations to the walk parameters. The parameters for Fig. 4c are chosen so that they are continuously connected with those for Fig. 4b. As expected, we observe the period2 oscillations in the evolution of probability distributions indicating the existence of a pair of bound states whose quasienergies differ by π.
We confirm the absence of bound states for splitstep quantum walks with θ_{2}=0 when θ_{1−} and θ_{1+} are continuously connected without crossing a phase boundary. The results in Fig. 4d,e show detection probabilities that quickly decrease to zero near the boundary for initial polarizations H〉 and V〉, respectively. As H〉 and V〉 span the space of internal states for the walker, this shows that indeed there is no bound state near x=0.
The existence of a pair of bound states with quasienergy difference π is a previously unknown topological phenomenon. It is a consequence of chiral symmetry, defined as the existence of an operator Γ that anticommutes with the Hamiltonian, Γ H Γ^{−1}=−H. An eigenstate ψ〉 with energy E therefore implies the existence of an eigenstate Γ^{−1}ψ〉 with energy −E. That is, states with energies E and −E generally come in pairs—the only exception is if E=−E. In a static system, a single state with energy zero can exist because its energy satisfies E=−E (ref. 26). It is topologically protected: it cannot be shifted in energy by weak, symmetrypreserving perturbations because a single state cannot be split into two. In a periodically driven system, because the effective Hamiltonian is defined through a onestep evolution operator by the quasienergies of H_{eff}(θ_{1}, θ_{2}) are defined only up to 2π. In particular, E=π and E=−π correspond to the same quasienergy, and, therefore, E=π represents another special value of quasienergy satisfying E=−E. Thus, like zeroenergy states of static systems, a single state with quasienergy π is topologically protected^{17,18}. The coexistence of E=0 and E=π states suggested by the period2 oscillations, observed in Fig. 4b, is verified through the explicit calculation of the quasienergy spectrum presented in Fig. 4f. In the Methods section, we give the characterisation of this structure in terms of topological invariants of periodically driven systems and prove their topological robustness.
Bound states under decoherence
One feature of our optical quantum walk setup is the ability to tune the level of decoherence^{24}. Each pair of beam displacers forms an interferometer, which can be intentionally misaligned to add temporal and spatial walkoff^{24}. This process, coupled with measurement of the photon, corresponds well to pure dephasing^{24}. If the system at step N is described by the density matrix ρ_{N}, it will evolve according to:
where p is the amount of dephasing and K_{i} are the associated Kraus operators. For p=0, equation 1 describes a pure quantum walk, while p=1 represents a system without any quantum coherence, that is, the evolution is described by a classical random walk. Although p=0 indicates that we do not introduce deliberate dephasing, some may arise from experimental imperfections.
Although the topological bound states observed in the paper are no longer eigenstates of the quantum walk under dephasing, signatures of bound states are still observable for a small number of steps. This result demonstrates that it is possible to study topological phenomena, for short times, in other systems that might be more prone to decoherence.
In Fig. 5, we show splitstep quantum walks with θ_{1} and θ_{2} parameters equivalent to those presented in Fig. 4b,d. However, here we have introduced additional decoherence at a level of p=0.2 according to equation 1. Fig. 5a shows the same period2 oscillation as before demonstrating that a pair of bound states can be seen with this small amount of decoherence. The absence of a bound state is confirmed when we chose W_{−}=W_{+}=0, and the results are shown in Fig. 5b.
In the presence of dephasing, the bound state observed in Fig. 4b gradually decays as the number of steps increases. However, for slow dephasing, this decay is slow and, for few steps, the probability distribution is still peaked near the boundary compared with the cases with no bound states (Fig. 5a). Fig. 5c summarizes the effect of decoherence on the probability distribution around the boundary region).
Discussion
The bound states observed in Fig. 3d,e are direct analogues of the zeroenergy states in the SuSchriefferHeeger (SSH) model of polyacetylene^{27} and the JackiwRebbi model of a onedimensional spinless Fermi field coupled to a Bose field^{10}. Specifically, the SSH model describes conduction in conjugated organic polymers, of which polyacetylene is the simplest example. By studying this polymer from a topological perspective, they identified the formation of a 'topological soliton'^{27} that is responsible for the chargetransfer doping mechanism in this molecule. Despite these theoretical predictions, its topological nature has never directly been confirmed. With our system, we have simulated the same class of topological phases as the SSH model and demonstrated their robustness to system perturbations for the first time.
The experiment can be extended to other symmetry classes and higher dimensions. In particular, the beam displacer architecture could also implement a twodimensional walk^{28}, which would allow observation of topologically protected edge states. Furthermore, one could simulate many topological classes of static systems that have been theoretically predicted but have not yet been realized because of the lack of natural materials with suitable symmetries^{25}. In addition, novel topological phenomena, unique to periodically driven systems, are expected in other symmetry classes and dimensions. The versatility of photonic quantum walks makes them ideal tools for exploring these captivating phenomena^{17}.
Methods
Rotation operators implemented in the experiment
The implementation of splitstep quantum walks with a photon requires the rotations of polarization, written as R(θ). In this experiment, we used halfwave plates that implement where σ_{i} are the Pauli matrices such that σ_{z}H〉=H〉 and σ_{z}V〉=−V〉.
Winding numbers of splitstep quantum walk
In our theoretical proposal^{25}, we considered creating a boundary between regions with different topological numbers by varying the second rotation angle θ_{2}. We described the topological structure of the splitstep quantum walk in terms of the onestep evolution operator, or Floquet operator, U(θ_{1}, θ_{2})=T_{2}R(θ_{2})T_{1}R(θ_{1}) and associated chiral symmetry operator which depends only on θ_{1} and satisfies In this experiment, we implemented inhomogeneous splitstep quantum walks by varying the first rotation angle, θ_{1}. To maintain the chiral symmetry in the system, it is necessary to characterize the dynamics in terms of an alternative chiral symmetry operator that depends only on the second rotation angle, θ_{2}. In the following, we explain and define such a chiral symmetry operator. As a consequence of considering such a chiral symmetry operator, the present phase diagrams are slightly different from those in ref. 25.
Because the origin of time for a periodically driven system is arbitrary, we can characterize the topology of the splitstep quantum walk with a different initial time, namely in terms of the evolution operator U′(θ_{2}, θ_{1})=T_{1}R(θ_{1})T_{2}R(θ_{2}). This alternative choice corresponds to making a halfperiod shift of the origin of time. Using the momentumspace expressions and we see that U′(θ_{2}, θ_{1}) is different from U(θ_{1}, θ_{2}) only through the exchange of θ_{1} and θ_{2}, that is, U′(θ_{2}, θ_{1})=U(θ_{1}, θ_{2}). Therefore, it is clear that the chiral symmetry operator of U′(θ_{2}, θ_{1}) is given by and that the winding numbers of U′(θ_{2}, θ_{1}) are the same as those of U(θ_{1}, θ_{2}). The chiral symmetry of U′(θ_{2}, θ_{1}) only depends on the second rotation angle θ_{2}, and thus the symmetry is preserved even when θ_{1} is varied in space. Therefore, it is possible to construct inhomogeneous quantum walks with boundaries between topologically distinct phases, while preserving the required chiral symmetry across the entire system.
The splitstep quantum walk with θ_{2}=0
We studied the behaviour of the splitstep quantum walk U=T_{2}R(θ_{2})T_{1}R(θ_{1}) with θ_{2}=0, noting that R(θ=0)=σ_{z}. In the experiment, we implemented the quantum walk with Floquet operator U_{ex}=T_{2}T_{1}R(θ_{1}). In this section, we show that these two quantum walks are related through a unitary transformation, and therefore represent equivalent dynamics with equivalent topological properties.
The splitstep quantum walk with θ_{2}=0 is described by the Floquet operator It is simple to check that the unitary transformation acts on T=T_{2}T_{1} such that V^{−1}TV=iTσ_{z}, where ^ is the coordinate operator. Therefore, U(θ_{1}, 0) and U_{ex} are unitarily related through Apart from a global phase, the experimental implementation is equivalent to the splitstep quantum walk with θ_{2}=0.
Topological invariants of 0 and π energy bound states
In this section, we show that the topological classification of periodically driven systems with chiral symmetry is given by Z×Z, and give explicit expressions for the topological invariants in terms of the wavefunctions of the bound states. These invariants provide another way to understand the topological protection of the 0 and πenergy bound states found in the experiment.
We consider the bound states at energy 0 because analogous arguments apply to the bound states at π. Suppose that there are N_{0} degenerate bound states with energy 0, which we label with α′=1,...,N_{0}. Because the chiral symmetry operator Γ anticommutes with the Hamiltonian, Γ^{2} commutes with H. When there is no conserved quantity^{29} associated with Γ^{2}, it is possible to choose the phase of Γ such that Γ^{2}=1. Because is an eigenstate of H with energy 0, we can choose the basis of zeroenergy states such that they are eigenstates of Γ. We denote the zeroenergy states in this basis as ψ_{α}^{0}〉 and their eigenvalues under Γ as Q_{α}^{0}. As Γ^{2}=1, Q_{α}^{0} is either +1 or −1.
We now show that the sum of eigenvalues, the integer represents the topological invariant associated with zeroenergy bound states. The invariant Q^{π} for πenergy bound states is constructed in an analogous fashion, where ψ_{α}^{π}〉 are the πenergy bound states. To show that these quantities are indeed topological invariants, we show that perturbations to the Hamiltonian that preserve the chiral symmetry cannot mix the zero and πenergy bound states with the same eigenvalues of Γ, and therefore cannot change the energies of these states away from 0 or π. Let H′ be a perturbation to the system such that {Γ, H′}=0. Now we evaluate the matrix element of {Γ, H′}=0 between the 0 (π) energy states. The result is
Thus, bound states with the same eigenvalues Q_{α} cannot mix, whereas those with different eigenvalues in general do mix and are not protected by chiral symmetry. Because one can break up any finite change of the Hamiltonian into successive changes of small perturbations, one can repeat this argument and show that the values Q^{0} and Q^{π} cannot change unless the bound states at 0 and π energies mix with the bulk states.
In the limiting case of the splitstep quantum with θ_{2}=0, θ_{1−}=−π, θ_{1}=π, we can analyse the bound states of the shifted evolution operator U′(θ_{2}, θ_{1})=T_{1}R(θ_{1})T_{2} with chiral operator The boundstate wavefunctions can be easily computed and one finds a single zeroenergy bound state with Q^{0}=1 and a single πenergy bound state with Q^{π}=−1. Because the pair of bound states found in the experiment arises in a situation that is continuously connected with this limiting splitstep quantum walk without closing the gaps, the observed pair is characterized by the same values of the topological invariants.
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How to cite this article: Kitagawa, T. et al. Observation of topologically protected bound states in photonic quantum walks. Nat. Commun. 3:882 doi: 10.1038/ncomms1872 (2012).
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Acknowledgements
We thank B.P. Lanyon, B.J. Powell and T.C. Stace for discussions. We acknowledge financial support from the ARC Centres of Excellence for Engineered Quantum Systems (Project number CE110001013) and Quantum Computation and Communication Technology (Project number CE110001027), Discovery and Federation Fellow programs and an IARPAfunded US Army Research Office contract. T.K., M.S.R., E.B. and E.D. thank DARPA OLE program, CUA, NSF under DMR0705472, AFOSR Quantum Simulation MURI, and the AROMURI on Atomtronics. T.K. thanks the Kodama foundation. I.K. and A.A.G. thank the Dreyfus and Sloan Foundations, ARO under W911NF070304 and DARPA's Young Faculty Award N660010912101DOD35CAP. A.A.G. acknowledges funding from the National Science Foundation under award number CHE1037992.
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M.A.B. and A.F. designed and performed experiments, collected and analyzed data, and wrote the paper. T.K., M.S.R., and E.B. developed the concepts, conceived the design of experiments, provided theoretical tools, analyzed the data and wrote the paper. I.K. provided theoretical support and wrote the paper. E.D., A.A.G, and A.G.W. supervised the project and edited the manuscript.
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Kitagawa, T., Broome, M., Fedrizzi, A. et al. Observation of topologically protected bound states in photonic quantum walks. Nat Commun 3, 882 (2012). https://doi.org/10.1038/ncomms1872
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DOI: https://doi.org/10.1038/ncomms1872
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