Bandgap-Assisted Quantum Control of Topological Edge States in a Cavity

Quantum matter with exotic topological order has potential applications in quantum computation. However, in present experiments, the manipulations on topological states are still challenging. We here propose an architecture for quantum control of topological matter. We consider a topological superconducting qubit array with Su-Schrieffer-Heeger (SSH) Hamiltonian which couples to a microwave cavity. The light-matter interactions are analyzed by exploiting topological bandgap in the qubit array. With proper cavity-qubits couplings, edge states and topological phase transition can be spectroscopically probed by the cavity. And the reflection spectrum shows a signature of vacuum Rabi splitting for edge states. Moreover, with the protection of topological bandgap, cavity induces nonlocal interaction between edge states. Quantum interference of emissions from two edge states is discussed. Our work may pave a way for topological quantum state engineering.

Introduction-. Characterization of topological matter is a crucial issue in condensed matter physics [1]. A hallmark of topological phases is the existence of topological invariants, e.g., Chern number and Zak phase, defined on energy bands of the systems [2][3][4]. According to edge-bulk correspondence, topological states emerge in the bandgaps and give rise to many novel transport phenomena [5,6].
Due to their insensitivity to local decoherence, topological states have prospective applications in quantum information processing. In particular, zero-dimensional edge states, e.g., Majorana bound states are candidate to realize topological quantum computation [7][8][9], and have been observed experimentally in a range of materials, including semiconductor nanowires [10][11][12][13], ferromagnetic atomic chains [14] and iron-based superconductors [15]. However, the manipulations of edge states are rather challenging, for which reason topological materials with large bandgaps are explored [16][17][18].
Cavity quantum electrodynamics (QED), in which quantized electromagnetic fields are strongly coupled to an atomic system, was originally used for studying fundamentals of atomic physics and quantum optics [19]. With the superb control of quantum states, cavity QED is now applied to quantum information processing, in which the cavity field is proposed for manipulating, measuring, or transferring quantum states of atomic systems [20]. Circuit QED, in which a microwave transmission line resonator acting as a cavity is coupled to superconducting quantum circuit, is an extension of the cavity QED [21,22]. The on-chip circuit QED system is not only a good platform for studying fundamental physics in microwave regime [23], but also a very promising candidate for realizing quantum computation and simulations [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. The interacting qubits make it possible to explore many-body physics. For example, many-body localization [37,38], Mott insulator of photons [40] and correlated quantum walk [41] are observed in 1D qubit arrays. With these experimental achievements, superconducting qubit systems are hopeful to simulate topological matter [42][43][44][45][46].
In this work, we study the interaction between a microwave cavity and the topological matter of a superconducting qubit array, described by the Su-Schrieffer-Heeger (SSH) Hamiltonian [47] which has been experimentally realized in a periodic driving way [48]. Different from the electronic transport detections of Majorana fermions [13,[49][50][51], the cavity spectroscopy method we study here unveils the edge states and topological phase transition with proper cavity-qubit couplings. In the superconducting qubit array, strong qubit-qubit interactions can give rise to large bandgaps. We pinpoint the role of topological bandgap in quantum manipulation of edge states, especially for small qubit arrays.
Spectroscopic characterization of a topological qubit array by a cavity-. As schematically shown in Fig. 1(a), we study a theoretical model that a typical topological lattice in one-dimensional systems [52], with SSH interactions, is placed inside a cavity. Considering rapid progresses and flexible chip designs of superconducting quantum circuits, we here assume that the SSH array with N unit cells, formed by 2N superconducting qubits [43], is coupled to a microwave transmission line resonator, as schematically shown in Fig. 1(b). The Hamiltonian of the whole system is given as where ω c and ω 0 are the frequencies of the cavity and qubits, respectively. The parameter g iµ denotes the coupling strength of the cavity to the qubit µ in the ith unit cell. The operators of qubits A and B at the ith unit cell are σ + iA = |A i α i | and σ + iB = |B i β i | with the ground (excited) states |α i (|A i ) and |β i (|B i ), respectively. The second line in Eq. (1) represents the SSH interaction Hamiltonian with tunable coupling strengths t 1 and t 2 , which could be implemented in  Design of (a) with superconducting qubit circuits where the couplings t1 and t2 are tunable. The microwave transmission line resonator acting as a cavity is coupled to qubits. (c) Reflection spectrum of the qubit array with 8 qubits. The frequencies of qubits and driving field are respective ω0 and ω l ; ω± = ω0 ± 2t0. The reflection at ϕ = 0.25π is shown in the right panel. Here we consider cavity-qubits couplings g = g0(−1, 1, 1, 1, −1, 1, 1, 1) with g0/2π = 5 MHz. Other parameters are: ω0/2π = 6 GHz, t0/2π = 100 MHz, κ/2π = 20 MHz, γiA = γiB = 20 × 2π kHz. The white-dashed curves represent energy spectrum of the qubit array. (d) Cavity mediated couplings between qubits, denoted by the orange lines, in dispersive regime.
To measure the topological qubit array, we assume that a probe field with the strength η and the frequency ω l = ω c is applied to the qubit array via the cavity. Thus, the dynamics of the reduced density matrix ρ of the whole system can be described by the master equatioṅ Here, κ is the decay rate of the cavity, γ iA and γ iB are the decay rates of the qubits A and B at the ith unit cell, respectively. The dissipation superoperator is defined as The energy spectrum corresponding to both bulk and edge states of the SSH array can be measured by the reflection of the probe field, as shown in Fig. 1(c). The reflection spectrum is obtained by solving the master equation in Eq. (2). Topological bandgap represents the energy separation between two bulk bands in topological phase. The cavityqubits couplings we choose here allow the observation of topological phase transition. In superconducting qubit circuits, topological phases have recently been demonstrated experimentally [48,[58][59][60][61][62][63][64][65]. However, the quantum operations on topological states have not been implemented. Below, we study the manipulation of topological states in superconducting qubit array via microwave fields.
Vacuum Rabi splitting for resonant coupling between the cavity and edge modes-. To show how to manipulate the qubit array by the quantized field in the cavity, we rewrite the states |A i and |B i of qubits A and B in the ith unit cell via eigenstates |Ψ j in the singleexcitation subspace of the qubit array [46], i.e., is the label of the jth eigenstate from the lowest to highest energies, |A i = σ + iA |G and |B i = σ + iB |G with |G being the ground state of the qubit array. Then, in the basis of these eigenstates, the Hamiltonian in Eq. (1) can be rewritten as with Ψ + j = |Ψ j G|, and ω j is the eigenenergy corresponding to the eigenstate |Ψ j . The parameter ξ j = ξ j g 0 is the effective coupling strength between the cavity and the jth eigenmode with ξ j = i (ξ 2i−1,j + ξ 2i,j ) under the assumption that qubits have the homogeneous couplings to the cavity with the strength g 0 , i.e., g iµ ≡ g 0 . Hereafter, we call Ψ + j bulk or edge modes when |Ψ j are bulk or edge eigenstates. In Fig. 2(a), we show |ξ j | for the qubit array size 2N = 36. The bulk modes have different couplings to the cavity because of their parities of wavefunctions. The oddparity bulk states have zero coupling. However, the evenparity bulk states are coupled to the cavity. Two edge states have equal coupling strength to the cavity, i.e., ξ 18 = ξ 19 .
In Fig. 2(b), we show energy splitting produced by the qubits-cavity couplings. We assume that the qubit frequency is ω 0 = 2π × 6 GHz. The anticrossing near the driving frequency ω l = 2π × 6 GHz represents the Rabi splitting due to the resonant interaction between the cavity and edge modes. Because of the degeneracy of two edge states, the anticrossing here represents the couplings between the cavity and two edge states. If the frequency of the cavity is at resonance for the transitions from the ground to bulk states with high energies, a large anticrossing, as shown in upper part of Fig. 2(b), is produced around ω l = 2π × 6.2 GHz. The large energy gap of the SSH Hamiltonian protects the Rabi splitting of edge states. In Fig. 2(c), the coupling strengths ξ N and ξ N +1 between the cavity and edge modes are plotted  versus the unit cell number N . When the qubit array is small, e.g., N ≤ 14, the edge states overlap with each other and form hybridized edge states with odd and even parities. The edge state with odd parity decouples from the cavity. With the increase of the unit cell number, two edge states are far separated from each other. The localized edge states lose parity, thus they have the same coupling strength to the cavity. We study the relation between the coupling strength ξ N (ξ N +1 ) and ϕ in Fig. 3. For example, when the qubit array has N = 6 unit cells, the coupling strengths are described by the black-solid and blue-dashed curves. When ϕ is small, the edge states have the same coupling to the cavity. However, the increase of ϕ leads to hybridized edge states with even and odd parities. We find that the hybridized regime becomes smaller with the increase of the system size, e.g., N = 18 (green-solid and blue-dash-dotted curves) and N = 78 (red-solid and orange-dotted curves) as we show here. We also find that in topological phase (i.e., ϕ < π/2), the hybridized edge state with even parity has the coupling strength ξ e = √ 2 cos ϕg 0 . The couplings for separated edge states ξ L =ξ R = √ cos ϕg 0 .
Cavity induced coupling between two edge modes-. When the cavity is far detuned from qubits, i.e., g 0 ∆ 0 (let ∆ 0 = ω 0 −ω c ), virtual-photons-mediated interactions among qubits g 2 0 /∆ 0 can be obtained [38,56], as shown in Fig. 1(d). In terms of the eigenmodes of the qubit array, the effective coupling strengths between ith and jth eigenmodes are with ∆ j/k = ω j/k − ω c . The eigenmodes with j = 2N, 2N − 2, 2N − 4, · · · have collective coupling strengths ξ j = j √ 2N g 0 . The coefficients 2N and 2N −2 are shown in the inset of Fig. 3. Thus, these bulk modes have dominant termsξ jξk . Generally speaking, if the cavityqubit coupling g is given, effective couplings J jk are determined by the qubit-cavity detuning ∆ 0 , the number N of unit cells and qubit-qubit coupling strengths t 1 and t 2 .
As schematically shown in Fig. 4(a), when the detunings of the bulk modes to the cavity are much larger than those of the edge modes to the cavity, and coupling strengths of the bulk modes to the cavity are comparable to those of the edge modes to the cavity, then the cavity induced couplings between bulk modes or between the bulk modes and the edge modes are negligibly small. When the energy splitting induced by hybridization of edge states is negligible (i.e., ∆ N ∆ N +1 ), the cavity mediated effective interaction Hamiltonian only contains the coupling between two edge modes with the strength In Figs. 4(b) and 4(c), we show the excitation dynamics of the left-edge qubit (qubit A in the first unit cell is excited initially) in topological phase with ϕ = 0.1π and 0.3π, respectively. Figures 4(b) and 4(c) clearly show the population exchange between two edge states produced by the edge-mode coupling. In fact, finite topological bandgap makes the effective couplings between edge modes different from Eq. (5). In Figs. 4(d) and 4(e) with ϕ = 0.5π and 0.9π, the excitation propagates through the array and is bounded by the boundaries. In nontopological phase, excitation propagates along the qubit array with low velocity (see Fig. 4(e)), which is yielded by the smooth energy bands with large gap. Quantum interference induced by topological state coupling-. As schematically shown in Fig. 5(a), we further consider that the left edge qubit A 1 is coupled to a waveguide, in which a probe field passes through. The left-edge qubit mainly contributes to the left edge state. Near resonance driving for the edge mode, the topological bandgap makes the bulk states to be negligible. Then the left edge state can be driven by fields passing through the waveguide. The single photons transmission amplitude can be given as and the susceptibility χ = −i(t − 1)/t is where ∆ p is the detuning between the probe field and the left edge state. As schematically shown in Fig. 5(b), the parameters γ L and γ R are the decay rates for left and right edge states, Γ L comes from the coupling between the left-edge qubit and the waveguide. The transmission of the probe field as a function of the detuning ∆ p is shown in Fig. 5 (c) with J = 0 and 0.035Γ L , respectively. When there is no coupling between edge states, the transmission vanishes at the resonance. However, when there is the coupling between two edge states, a transparency windows for the probe field appears. This can be further confirmed by the susceptibility, which is plotted as a function of the detuning ∆ p in Fig. 5(d) in the parameter regime J Γ L . This transparency window, in which the distance between two peaks is less than 2J, is from the quantum interference as shown in Fig. 5(e), which is similar to electromagnetically induced transparency [66]. However, in the parameter regime J > Γ L , the transparency window, in which the distance between two peaks equals to 2J, is from the strong-coupling-induced energy splitting, which is similar to Autler-Townes splitting [67].

Conclusions and discussions-.
In summary, we study cavity control and manipulations on topological degrees of freedom in one-dimensional systems with the SSH Hamiltonian. We show that the coupling between the cavity and edge modes are protected by topological bandgap, and topological phase transitions can be measured via the reflection spectrum of the probe field through the cavity. Due to topologically protected bandgap, the Rabi splitting, resulted from the resonant coupling of edge modes to the cavity, can be observed. When the cavity is largely detuned from the edge modes, the long-range coupling between two edge states can be realized, this can further result in the quantum interference for emissions from two edge states when a qubit at the edge of the array is coupled to a waveguide. Meanwhile, we find that topological properties of systems can also be detected by the cavity even for a small size, but the edge states are hybridized in small systems, and the hybridized states possesses the parity properties. We show that the parity engineering can also yield the coupling between edge states, as long as the splitting between hybridized edge states is small comparing to the detuning between cavity and qubits.
We also propose an experimental setup for implementing our approach by coupling superconudcitng qubit arrays to a transmission line resonator. This is because the tunable coupling between superconducting qubits can be experimentally realized via cavity or other superconducting elements [37,38]. Moreover, the coupling strength between superconducting qubits can be sufficiently large such that the large topological bandgap of the system can be obtained, and thus the selective coupling of the edge states to the cavity is easier to be realized. We mention that our approach can also be applied to other systems. Our study on cavity QED for the topological matter might have potential applications in quantum information and quantum optics.